QQuantum Shape Effects
A theoretical treatise on the quantum-mechanical influence of geometry in the thermodynamics of strongly confined nanostructures
PhD Thesis
September 2020
Energy InstituteIstanbul Technical University
ALHUN AYDIN a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b lhun AYDIN, a Ph.D. student of ITU Energy Institute 301142001 successfully defended thethesis entitled "QUANTUM SHAPE EFFECTS", which he prepared after fulfilling the require-ments specified in the associated legislations, before the jury whose signatures are below. Thesis Advisor: Prof. Dr. Altu ˘g ¸S˙I ¸SMAN ..............................Istanbul Technical University
Jury Members: Prof. Dr. Özgür E. MÜSTECAPLIO ˘GLU ..............................Koç University
Assoc. Prof. Dr. Z. Fatih ÖZTÜRK ..............................Istanbul Technical University
Prof. Dr. Jonas FRANSSON ..............................Uppsala University
Assoc. Prof. Dr. Ahmet Levent SUBA ¸SI ..............................Istanbul Technical University
Date of Submission : 26 June 2020Date of Defense : 15 September 2020 © 2020 Alhun Aydın3 o the love of wisdom, oreword Isn’t it ironic that forewords are actually written at the very last? I don’t know where tostart, so let me start from the very beginning. It was all gas and dust cloud... Well, okay maybenot from that much beginning. . . I had the passion for science, curiosity for nearly everythingduring my entire life. Still, I think I can name a few milestones on my path leading to science.I was greatly fascinated by physics, when I had watched the movie "Back to the Future" withtremendous excitement during my preschool childhood years. My encounter with Goldbachconjecture in the first year of high school was another defining point for my enthusiasm toexplore and think upon the unsolved problems in mathematics and physics. Contrary to themounting evidences on that direction, initially I was not thinking of choosing math or physicsas a major. The chance was on my side and thanks to my "lower than expected score" onthe national exam, I chose physics to study and I got enrolled to Koç University with a fullmerit-scholarship, which was a great decision as things stand. But there were two importantturning points in my ever-changing career plan between music and science. The first one wasthe "Quantum teleportation" topic given to me by my then supervisor Tekin DEREL˙I (I owe hima great deal of gratitude for his guidance in science) for the investigation during the independentstudy course when I was a junior undergraduate. I remember that I’d worked day and night withan immense excitement and joy. I had the same feeling when I joined to my advisor Altu ˘g¸S˙I ¸SMAN’s Nano Energy Research Group and read their papers to work it out. Thanks to theirapproaches coupled with my own attitude, I have never considered research as a job or as a duty,rather I always felt that I am just doing what I enjoy. I’ll do my best to keep this amateur spirit.The story of my thesis research started around August 2015, when Altug suddenly draw asquare within a square (image below) and said "This thing must rotate!" by pointing the innerone. At the moment I saw it (and we discussed on it), I was literally amazed and in a senseI felt that it was destined to be my Ph.D. topic. Then its unofficial name became "dönen ¸sey"(meaning "the rotating thing" in Turkish)."Did you look at dönen ¸sey?", he was constantly keep asking. We were discussing at the timewhether it should rotate or not. Fatih was asking to me with his never-ending cheerful mood "Isit rotating or not huh?" (with a big smile on his face). From my initial calculations in September2015, I noticed a free energy difference between two different angular configurations of theinner square, suggesting that it should somehow rotate. From September 2015 to the summer of2016, I had to deal with courses and the infamous qualification exam. Finally, in August 2016,5fter I came back from California out of left field, I started to focus on "the rotating thing" andat that time we were sure that there is a finite amount of torque and so, under quasistatic processit must rotate!The time I spent during Ph.D. helped me a lot to shape my thoughts about life, nature, humannature, universe, reality and everything. These years of doing Ph.D. was definitely highly crucialfor my development both in terms of knowledge and experience not only in academia, but alsoin life. Due to the interdisciplinary nature of our Institute, I tried my best to write the thesisunderstandable also by the non-experts of the field. Okay, now comes the thanks part.Firstly, I’d like to express my great and warm gratitude to my advisor Altu ˘g ¸S˙I ¸SMAN. Tome, he has been much more than a scientific advisor, he is a very good friend. I’ve always felthis true care and sincere guidance. His dialogue with his students was exemplary. I’ve learnt alot of things from him and not just knowledge but also principles and virtues. His wit, patience,kindness and work-ethic were invaluable for me. He once and for all revived my passion forscience by his exceptional scientific approach. I know that whichever path I choose for myfuture, I’ll be doing scientific research at least in some part of the 24 hours. The unbreakableintellectual bond that we established between us constitutes a large part of the things that Igained during my Ph.D. time. Certainly, I’d like to thank also to his wife Ay¸se KA ¸SLILAR¸S˙I ¸SMAN for her warm friendship, realistic advices and mutual sincerity. I am waiting our nextdinner-table-conversations/discussions impatiently.Z. Fatih ÖZTÜRK was definitely the person who brings joy to my time during Ph.D. with hisgreat humor that usually makes me rolling on the floor laughing. I’ll remember his conversationswith my advisor on work agenda and memories, with a big smile on my face. I thank to himalso for his tireless support, kindness and cheerfulness all the time. Likewise, I’d like to thank tothe other members of Nano Energy Research Group; Sevan KARABETO ˘GLU, Gülru BABAÇSCHÜBLER, Co¸skun FIRAT and Türker ¸SAH˙IN for their valuable supports and friendships. Iappreciate all academic, administrative and employee staff of ITU Energy Institute for providingme a comfortable and beautiful workplace. For nearly 8 years, Energy Institute was like my truehome, considering my usual late-evening/night, weekend and even holiday workings.I’d like to thank to my thesis steering committee member Özgür E. MÜSTECAPLIO ˘GLU(as well as his research group members) for his viewpoints, comments, advices and kindness.I’d like to express my gratitude to Jonas FRANSSON for inviting me to Uppsala UniversityPhysics & Astronomy Department and for his sincere supports along with valuable advicesand a fruitful collaboration. I am very grateful to Ronnie KOSLOFF for accepting me into hisresearch group and transferring me a portion of his vast and valuable knowledge and experiencealong with our joyful conversations and moments in our picnics and gatherings. I’m thankful toA. Levent SUBA ¸SI for his detailed feedback on editing of the thesis. I thank to all jury membersfor their comments and suggestions on the final form of the thesis. I’d like to thank also to NickTREFETHEN, ˙Inanç ADAG˙IDEL˙I, Gökhan Barı¸s BA ˘GCI, Mauro PATERNOSTRO, ObinnaABAH, Peter SALAMON and Raam UZDIN for our fruitful discussions.I appreciate the hospitalities of Uppsala University, Department of Physics & Astronomyand The Hebrew University of Jerusalem, Fritz Haber Research Center for Molecular Dynam-ics. I thank to "Uppsala Multidisciplinary Center for Advanced Computational Science" and"Computer Cluster Service of Fritz Haber Center for Molecular Dynamics" for providing meaccess to their computational resources. I thank to Israel Ministry of Foreign Affairs, Scientificand Technological Research Council of Turkey and AIM Energy Technologies for their partialsupports during my Ph.D. work. 6pecial thanks go to my former home (though I still feel at home when I’m there), Koç Uni-versity, for preparing me to the academia, with its world-class education, academic and socialenvironment. Most parts of this thesis are written in its lounges and terraces with spectacu-lar forest and Black Sea views. I thank to Istanbul Technical University for providing a verynice academic environment, residential resources and financial supports on several conferencesabroad. I really enjoyed my time in its beautiful campus at Ayaza˘ga.Life does not pass without good friends; a Ph.D. is never done. Luckily, I have many goodones. My close friend Serhat TET˙IKOL was the first person who listen to, comment on and crit-icize the core ideas of this thesis work. He helped me to look from different angles as he alwaysblended the careful listening with his candid opinions. Our highly enjoyable intellectual dis-cussions, accompanied by coffee and fresh air in the daytime and whisky and nuts in the night,helped a lot to polish my mind. Our deep talks are infamous for ending up with the most unex-pected conclusions. I acknowledge all members of Papa John’s Pizza Group (shortly Papa Johnsor PJPG) for being such a coherent group, I’ll really miss our extremely funny and geeky partiesat the New Energy Technologies Lab. Work hours (in fact after work hours as well) could notbe without my coworker friends Murat Ferhat DO ˘GDU, Can AKSAKAL, Ahmet GÜLTEK˙IN,Osman ÜRPER, Ertu ˘grul DEM˙IR, Zeynep CAMTAKAN, Berker YURTSEVEN, Utku HAR-MANKAYA, Tu ˘gçin KIRANT, Gökçen GÖKÇEL˙I, U ˘gur KAHVEC˙I, Neslihan KOYUNCU,EDAG friends as well as many other good friends who have worked with me in Energy In-stitute. Together we shared loads of nice memories, activities and joy. I’m grateful for all.My dear friends Erelcan YANIK and Burak ÖZAYDIN deserve also big credit, as members ofour intriguing philosophical discussion group "Bitli Sinir", for our collective thinking withoutboundaries. I am still having a small hope for someday being able to regularly play basket-ball with my dear friend Erelcan. I especially thank to my childhood friend Ça ˘gın Ç˙IÇEK forour coherently absurd chit-chats, heart-to-heart talks, game-days and horror-nights with cooldrinks, and to Yelda Ç˙IÇEK for enriching our meetings even more. I thank to Bü¸sra TAYFURand Berkay SAK˙IN for our laugh-filled lion milk table evenings, to my like-minded friend LucaTONDELLI for our enjoyable chats, activities and holidays, as well as to many other fantasticfriends I’ve met during my business trip at Hawaii; a heaven on earth was the most entertain-ing with you all. Meetings with my undergrad circle, Samet ÖZCAN, Güven YILMAZ, Çı ˘gılYILMAZ, Elif ÖZPA ˘GDA, Fatih ONUR, ˙Irem LAÇ˙IN, Gökçehan AKO ˘GUZ, Melek ARI andAtakan ARASAN, were occasional but enjoyably memorable. I love you all.I spent one year of my Ph.D. at Uppsala, Sweden and at Jerusalem, Israel. My time at Up-psala was instructive and pleasant with Jonas, Henning, Juan David, Andreas, Johann, Emel,Paramita, Seif, Johannes, Oladunjoye, Francesco, Tomas, Anna and other nice people at theVilla and at the Angstrom Buildings. My Jerusalem times were far beyond my initial expecta-tions, I never wanted my time there to end, thanks to good friendships of Ronnie, Marcel, Bar,Aviv, Efrat, Florian, Suleiman, Sayak, Roie, Laura, Estefania, Itay, Oren, both Ksenia’s, Han,Atılgan, Müslüm, Igor, Raam and other nice people at the Fritz Haber Center, also thanks to myroommates Sheng Fu, Ellen and dear Fransiscus Ismael with whom I feel we’ve developed alifelong friendship bond. I’ll always miss our picnics, gatherings, trips and discussions. Beauti-ful people that I’ve met during conferences, seminars, meetings and workshops, I thank to luckfor living those nice moments with you all which I won’t forget.My cute, lovely sister Aslı, my sweet mother Hanife and my supportive father Öztürk, aswell as my beloved aunts Nurten, Nur¸sen and Emine and my dear cousin Yasemin, I owe mygreatest gratitude to all of you for the love and support that you have given to me.7astly, I thank to my friends who I couldn’t mention here and are not necessarily relatedwith the course of my Ph.D. thesis but surely important to me. I am glad to have/had you in mylife.Wow, wordy! Seems like I have been waiting for this... Here we go! Alhun AYDINPhysicistSeptember 2020
Note: In this arXiv version of the thesis, several stylistic adaptations have been made fromthe original ITU format, for the sake of online reading compatibility. ummary QUANTUM SHAPE EFFECTS
How does geometry affect the physical properties of matter? Geometry has been considered asan important mathematical concept for understanding physical reality since the time of ancientGreek philosophers. The geometry of a physical object is associated with its sizes and overallshape. Sizes are characterized by volume, surface area, peripheral length and number of verticesof the object. With the development of quantum mechanics in the beginning of the last century, itis seen that nature has different appearances depending on the scale of physical systems. Physicsof nanoscale (a billionth of a meter) is governed by quantum mechanics and nanomaterials havesome superior properties in comparison with their macroscale counterparts.Quantum mechanics taught us that matter exhibits both particle-like and wave-like charac-teristics, the so called wave-particle duality. Wave behavior is associated with the de Brogliewavelength, which is usually quite small for our macro world. Wave nature of particles becomeprominent when they are confined in domains with sizes that are comparable to their de Brogliewavelengths. In such a case, the physical properties of particles are affected by the confine-ment domain and so quantum size effects appear. Utilization of quantum size effects lead totailoring and enhancing various properties of materials, constituting the backbone of modernnanoscience and nanotechnology.Unlike size, shape is not so straightforward to define. In almost all systems size and shapeeffects coexist and interfere with each other. Is there any way to separate them? Can wechange the shape of a domain without altering its sizes and focus on pure shape effects (that iscompletely overlooked)?This thesis shows the separation of quantum size and shape effects from each other. Wepropose the existence and explore the consequences of a new type of physical effect whichwe call quantum shape effect. We introduce a size-invariant shape transformation on nesteddomains which can be realized in core-shell nanostructures. Performing a rotation on the corestructure causes a variation of the shape of the shell structure where the particles are confined.During this rotation all size parameters of the confined domain stay constant. By this waywe perfectly separate quantum size and shape effects from each other and investigate quantumshape effects alone. Shape not only becomes a control parameter on the material properties, butalso leads to novel physical behaviors which have never been seen before.In the thesis, after the introduction to the topic, literature overview and a review of quan-tum size effects, we introduce quantum shape effects in the third chapter in detail. We solvetime-independent Schrödinger equation numerically for the confinement domains that are con-stituted by the nested structures with various geometries. Eigenvalue spectrum for each angularconfiguration is obtained and used to calculate partition function and all other thermodynamic9uantities. It is demonstrated that the thermodynamic properties of non-interacting particlesstrongly confined in nested nanostructures significantly change with shape. Next, we developan analytical method to predict the quantum shape dependence of thermodynamic state func-tions as well as to develop a physical insight to the quantum shape effect phenomenon. Otherknown methods such as Weyl density of states and first two terms of Poisson summation formulacannot predict any shape-dependence in the thermodynamic properties. Our analytical method-ology is based on the quantum boundary layer approach. Considering the overlaps of quantumboundary layers forming in nested domains reveals the information about the shape dependenceof the properties of confined particles. We call the analytical model as overlapped quantumboundary layer method and its accuracy is quite good at estimating the functional behaviorsof shape-dependent thermodynamic properties. Influence of various boundary conditions andquantum size effects on quantum shape effects are also investigated.Thermodynamic properties such as internal energy, free energy, entropy and specific heat ofparticles are examined under quantum shape effects for particles obeying Maxwell-Boltzmannand Fermi-Dirac statistics. Their behavior shows exotic characteristics that are previously un-seen in the thermodynamics of confined non-interacting gases. Shape dependence of the chem-ical potential of electrons produces a novel kind of quantum oscillations which are intrinsicallydifferent than density- or size-dependent quantum oscillations.Due to quantum shape effects, free energies of various angular configurations are differentfrom each other. This suggests a spontaneous rotation of the core structure as a result of thetorque generated by the particles confined within the shell structure to minimize their free en-ergy. Formation of non-uniform and asymmetric pressure distribution even at thermodynamicequilibrium is the principal cause of this torque of quantum origin.From the application point of view, quantum shape effects lead to some novel heat engineand refrigeration cycles, opening up new possibilities in nanoscale thermodynamics. We pro-pose the existence of a new thermodynamic process under constant shape, we call isoformalprocess. The thermodynamic cycles featuring isoformal process are examined and they showvarious novel properties that are not encountered in conventional thermodynamic cycles. It isalso possible to design new nanoscale energy conversion devices based on quantum shape ef-fects. A number of possible applications are presented in the fifth chapter. As a whole, thisthesis constitutes a comprehensive investigation of the theory, methodology and applications ofquantum shape effects in thermodynamics, which hopefully have a great potential to bring newideas and advances to the field of nanoscale physics and energy.10 zet
KUANTUM ¸SEK˙IL ETK˙ILER˙I
Geometri maddenin fiziksel özelliklerini ne ¸sekilde etkiler? Antik Yunan filozoflarının za-manından beri geometri fiziksel gerçekli ˘gi anlamada önemli bir matematiksel kavram olarakgörülmü¸stür. Bir fiziksel nesnenin geometrisi genelde o nesnenin ölçeksel büyüklükleri (ebat-ları) ve bütünsel ¸sekli ile ili¸skilendirilir. Bir nesnenin ebatları Lebesgue ölçüsü altında o nes-nenin hacmi, yüzey alanı, çevresel uzunlu˘gu ve kö¸se sayıları ile belirlenir. Örne ˘gin üç boyutlubir nesne için esas ölçeksel büyüklük hacim iken, yüzey alanı, çevresel uzunluk ve barındırdı ˘gıkö¸se sayıları ise üç boyutlu nesnenin dü¸sük boyutlu ölçeksel büyüklükleri olarak tanımlanır.Geçti ˘gimiz yüzyılın ba¸sında kuantum mekani ˘ginin ke¸sfedilmesi ile birlikte do ˘ganın fiziksel sis-temlerin boyutuna ve ölçe ˘gine göre farklı davranı¸sları oldu˘gu gözlemlendi. Metrenin milyardabirini ifade eden nano ölçek fizi ˘gi günümüzde kuantum mekani ˘gi yasaları tarafından anla¸sıla-biliyor. Nano ölçekteki fizik bilim insanları açısından oldukça cazip bir konu zira hem teorikhem deneysel olarak gösterildi˘gi üzere nano ölçek malzemeler birçok açıdan makro ölçektekimalzemelere göre üstün özelliklere sahip. Enerji bilim ve teknoloji alanında da nanoyapılarınuygulaması gün geçtikçe artmakta ve bu malzemelerin fiziksel davranı¸slarının do ˘gru ve de-taylı bir biçimde anla¸sılması önem arz etmektedir. Geçmi¸ste teoride sınırlı kalan birçok fizik-sel olgu, günümüzde laboratuvar olanaklarının hızlı geli¸simi sayesinde deneysel olarak göster-ilebilmekte, hatta bir kısmı ticari olarak uygulanabilmektedir.Kuantum fizi ˘gi maddenin hem dalga hem de parçacık davranı¸sları gösterdi ˘gini ortaya koy-mu¸stur. Buna dalga-parçacık ikili ˘gi diyoruz. Parçacıkların dalga karakteri de Broglie dalga boy-ları ile ölçülür ki bu genelde oldukça küçüktür. Parçacıkların içinde sınırlandı ˘gı domenin ebat-ları de Broglie dalga boyları ile kar¸sıla¸stırılabilir bir mertebede ise parçacıkların dalga do ˘gasıönem kazanır. Böyle bir durumda, parçacıkların fiziksel özellikleri tutuklanma domenindenetkilenir ve kuantum ölçek etkileri ortaya çıkar. Kuantum ölçek etkileri malzemelerin çe¸sitliözelliklerini belirlenen amaca uygun ve daha iyi hale getirmeye yol açarak ça ˘gımız nanobilimve nanoteknolojisinin temel ta¸sını olu¸sturur.Bir nesnenin ebatlarının veya ölçe ˘ginin aksine ¸seklini tanımlamak ve sayısalla¸stırmak çokdaha zordur. Neredeyse tüm fiziksel sistemlerde ölçek ve ¸sekil bir arada birbiri içine geçmi¸s bir¸sekilde bulunur. Bir nesnenin büyülü ˘günü ve ¸seklini ayırıp, ayrı ayrı inceleme altına almak olasımıdır? Sınırlandırılmı¸s bir domenin ebatlarını de ˘gi¸stirmeden ¸seklini de ˘gi¸stirmek ve bu sayedeyalnızca ¸sekil etkilerini incelemek mümkün müdür? Sınırlandırılmı¸s domenlerdeki parçacıklarüzerinde ¸sekil etkilerini ölçek etkilerinden tamamen ayrı inceleme olasılı ˘gı literatürde ¸simdiyekadar göz ardı edilmi¸stir.Bu tezde kuantum ölçek etkileri ve kuantum ¸sekil etkileri birbirinden tamamen ayrılmı¸stır.Tezde kuantum ¸sekil etkileri olarak adlandırdı ˘gımız yeni bir fiziksel etkinin varlı ˘gını ortaya11oyuyor ve sonuçlarını tetkik ediyoruz. Birbirinin içine geçmi¸s domenlerde ölçekten ba ˘gımsız¸sekil de ˘gi¸stirimi tekni ˘gini gösteriyoruz. Literatürde oldu ˘gu gibi bu tarz iç içe geçmi¸s domenlerdeneysel olarak çekirdek-kabuk nanoyapılarında gösterilebilir. Çekirdek nanoyapıda gerçek-le¸stirilecek döndürme hareketi, kabuk ile çekirdek yapı arasında tutuklanmı¸s parçacıkların sınır-landı ˘gı domenin ¸seklini de ˘gi¸stirir. Bu dönme hareketi esnasında parçacıkların bulundu ˘gu domeninbütün ölçek de ˘gi¸skenleri aynı kalır. Bu sayede kuantum ölçek ve ¸sekil etkileri birbirindentamamen ayrılarak yalnızca kuantum ¸sekil etkilerini inceleme olana ˘gı olu¸sur. ¸Sekil, malzemeözellikleri üzerinde bir kontrol de ˘gi¸skeni haline gelmekle kalmaz, aynı zamanda daha öncegörülmemi¸s yepyeni fiziksel davranı¸sların ortaya çıkmasına sebebiyet verir.Tezin ilk bölümünde tezin motivasyonu ve tez konusu tanıtılarak, literatür incelemesi ileberaber tez çalı¸smasının ana çıktıları verilmi¸stir. ˙Ikinci bölümde kuantum ölçek etkilerininistatistiksel termodinamik özelinde bir derlemesi yapılmı¸stır. Kuantum ¸sekil etkilerinin nasılortaya çıktı˘gı ve temelleri tezin üçüncü bölümünde detaylı bir biçimde incelenmi¸stir. Çe¸sitli ge-ometrilerdeki iç içe geçmi¸s nanoyapılardan olu¸san tutuklama domenleri için zamandan ba ˘gım-sız Schrödinger denklemi sayısal olarak çözülmü¸stür. Çekirdek nanoyapı dedi ˘gimiz içteki ob-jenin her bir açısal durumu için özde ˘ger görüngesi elde edilmi¸s ve bu özde ˘gerler kullanılarakbölü¸süm fonksiyonu ile beraber di˘ger termodinamik büyüklükler hesaplanmı¸stır. ˙Iç içe geçmi¸snanoyapılarda tutuklanmı¸s etkile¸smeyen parçacıkların termodinamik özelliklerinin ¸sekil ba ˘gım-lılı ˘gı ortaya konmu¸stur. Ardından bu ¸sekil ba˘gımlılı ˘gını öngörmek için analitik bir yöntemgeli¸stirilmi¸stir. Geli¸stirilen yöntem tutuklanmı¸s parçacıkların termodinamik özelliklerinin ¸sekilba ˘gımlılıklarının fonksiyonel davranı¸sını do ˘gru öngörmekle kalmayıp kuantum ¸sekil etkisi ol-gusuna fiziksel bir kavrayı¸s getirmeyi ba¸sarmı¸stır. Analitik yöntemimiz kuantum sınır tabakayakla¸sımı üzerine geli¸stirilmi¸stir. Kuantum sınır tabakaların üst üste bindi ˘gi (örtü¸stü˘gü) böl-gelerin büyüklü ˘gü parçacıkların bulundu ˘gu domenin ¸sekil bilgisini ta¸sır. Bu örtü¸sen bölgelerinmiktarları içteki nanoyapının dönü¸sü sırasında açıyla beraber de ˘gi¸sir. Bu sayede domenin ter-modinamik özelliklerini bu örtü¸sme bölgelerini de göz önüne alan bir efektif hacim ile ili¸sk-ilendirmek mümkün olur. Örtü¸sen kuantum sınır tabaka modelimiz iç içe domenlerde güçlü bir¸sekilde tutuklanmı¸s parçacıkların termodinamik özelliklerini oldukça iyi bir do ˘grulukla anal-itik olarak öngörmektedir. Kuantum ¸sekil etkilerinin çe¸sitli sınır ko¸sullarındaki davranı¸sı vekuantum ölçek etkileri sebebiyle de ˘gi¸simi de tezin bu bölümünde incelenmi¸stir.Tezin dördüncü bölümünde parçacıkların iç enerji, Helmholtz serbest enerji, entropi veözgül ısı gibi termodinamik özellikleri kuantum ¸sekil etkileri altında Maxwell-Boltzmann veFermi-Dirac istatistikleri çerçevesinde ayrı ayrı incelenmi¸stir. Kuantum ¸sekil etkileri sebebiylebu termodinamik büyüklüklerin daha önce etkile¸smeyen gazların termodinami˘ginde görülmemi¸silginç fiziksel davranı¸slar gösterdi ˘gi görülmü¸s ve bu davranı¸sların kökenleri ve mekanizmalarıkurulan analitik modelin de yardımıyla açıklı ˘ga kavu¸sturulmu¸stur. Ayrıca elektronların kimyasalpotansiyelinin ¸sekil ba ˘gımlılı ˘gının yo ˘gunluk veya ölçe ˘ge ba ˘glı kuantum salınımlardan temeldefarklı olan ba¸ska bir tür kuantum salınımı gösterdi ˘gi ortaya konmu¸stur. Bu özgün kuantumsalınımının özellikle özgül ısıda güncel deneysel olanaklarla gösterilebilecek büyüklükte de˘gi¸sim-lere yol açtı ˘gı gözlenmi¸stir.Kuantum ¸sekil etkilerinden ötürü farklı açısal durumların serbest enerjileri birbirinden farklıolmaktadır. Bu da tutuklanmı¸s parçacıkların serbest enerjilerini minimize etme amacıyla dönebilmeserbestli ˘gi bulunan iç nanoyapı üzerinde tork uygulayaca ˘gını ve iç nanoyapının kendili ˘gindendönerek serbest enerjinin minimum oldu ˘gu açıda duraca˘gını i¸saret eder. Nano ölçekte tutuk-lanmı¸s yapılarda termodinamik denge durumunda dahi geometrik simetrinin bozuldu ˘gu herdurumda asimetrik ve düzensiz bir basınç da˘gılımı olu¸sur. ˙Iç içe geçmi¸s domenlerde bu tez12alı¸sması kapsamında ortaya konan kuantum-mekaniksel torkun ortaya çıkmasının temel ne-deni budur.Kuantum ¸sekil etkileri nano ölçek termodinami˘ginde yeni uygulama olanakları da açar.Bu tezde sabit ¸sekil durumunda izoformal proses olarak adlandırdı˘gımız yeni bir termodi-namik prosesin varlı ˘gını ortaya koyduk. ˙Izoformal prosese dayalı özgün ısıtma ve so ˘gutmaçevrimleri önerdik. ˙Iki izotermal, iki izoformal prosesten olu¸san bir termodinamik çevirimiile iki izentropik, iki izoformal prosesten olu¸san bir termodinamik çevrimin analizlerini yaptıkve alı¸sılagelmi¸s termodinamik çevrimlerde kar¸sıla¸sılmayan bazı özellikler içerdi˘gini gördük.Termodinamik çevrimlerin yanı sıra kuantum ¸sekil etkilerine dayalı yeni nano ölçek enerjidönü¸süm cihazlarının tasarlanması da mümkündür. Bu ba ˘glamda muhtemel uygulamalarınbirkaçı tezin be¸sinci bölümünde sunulmu¸stur. Tek malzemeli tek kutuplu termoelektrik ciha-zlar ve kuantum Szilard ısı makineleri kuantum ¸sekil etkilerinin uygulanabilece˘gi birçok farklıalandan sadece birkaçıdır. Kuantum ¸sekil etkilerinin termodinamikte ortaya çıkı¸sı, teorisi, yön-temleri ve uygulamaları kapsamlı bir ¸sekilde bu tezde incelenmi¸stir. Tezle ili¸skili yapılan bazıçalı¸smaların da gösterdi ˘gi üzere, bu tezde ortaya konan yeni fiziksel etki ve uygulamaları nanoenerji bilimi ve teknolojisinde yepyeni fikirlere ve geli¸smelere yol açma potansiyeline sahiptir.13 ontents
A.1 Details of Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . 104A.2 Global and Local Boundary Perturbations . . . . . . . . . . . . . . . . . . . . 106A.3 Local Momentum Flux Approach . . . . . . . . . . . . . . . . . . . . . . . . 107A.4 Energy Moments and Density Distributions . . . . . . . . . . . . . . . . . . . 109A.5 The First Order Quantum Boundary Layer Approach . . . . . . . . . . . . . . 111A.6 Pressure - Momentum Flux Equivalence . . . . . . . . . . . . . . . . . . . . . 114A.6.1 Equivalence for Maxwell-Boltzmann statistics . . . . . . . . . . . . . 114A.6.2 Equivalence for Fermi-Dirac and Bose-Einstein statistics . . . . . . . . 11615
Introduction
We, humans, are curious animals. Science, and even technology, are still first and foremostdriven by this unceasing curiosity, despite the gradual changes in the priorities of people dur-ing recent decades towards more economic and vanity-driven concerns. It’s maybe not clearwhether technology has made our lives easier or science has made us wiser, yet, one thing isclear that we are not just wondering things, but also understanding the nature of some thingsby using our intelligence, senses and the tools that we have created so far. The more we knowthings, the more we realize how large the lower part of the iceberg of knowledge might possiblybe, and even more our intellectual curiosity grows.Quantum theory, the physics of the tiny scales, is one of the biggest products in this questof the roads leading to the true nature of things. Nature behaves surprisingly different at smallscales. Many phenomena discovered at quantum realm are exceedingly counter-intuitive to usliving in and experiencing a macroscopic world. But still we know by now that some thingsare not what they appear to be. For instance, it’s nearly impossible to see the roundness of theEarth by the naked eye, when you look to the horizon. Similarly, we don’t notice in our macroworld the weird behaviors appearing at quantum scales. Even so, quantum effects actually playa significant role in our modern life, since many devices that we use today such as transistors,lasers, navigation devices and magnetic resonance imagers, directly rely on the principles ofquantum mechanics.So, how is the physics of the small scales? First of all, by small we mostly mean nanoscale,which is sometimes also called quantum scale. Nanometer is one billionth of meter, and ap-proximately one hundred thousandth of a human hair. We encounter with quantum phenomenaat not only small scale, but also low temperature and low mass conditions. At least one ofthese conditions need to be satisfied in order for quantum effects to appear. Under these terms,nature exhibits some phenomena that cannot be explained by classical, pre-quantum, physics.Quantum mechanics has shown that matter has a probabilistic wave nature, see Fig. 1.1 where16he famous double-slit experiment is illustrated. This concept underlies the roots of many dif-ferent quantum phenomena such as Heisenberg uncertainty principle, coherence/decoherence,entanglement, superposition, tunneling, wavefunction collapse during a quantum measurement,zero-point energy, Casimir effect, discreteness of certain physical quantities, indistinguishabil-ity of identical particles and so on. Some weird consequences of these phenomena are: inherentuncertainty in position and velocity of particles (Heisenberg uncertainty principle), intercon-nectedness of particles that have a shared past (quantum entanglement) and inexistence of anobjective reality before interacting with the particles (quantum probabilistic nature of wave-function) et cetera . All these quantum phenomena have surprised the world a lot and still havebeen continuing to surprise even scientists. We’ll discuss more deeply on the wave nature ofparticles in the next chapter.What we will focus on in this thesis is the thermodynamics at quantum scales. Beforethat, let’s briefly mention what thermodynamics deals with. Thermodynamics is the branch ofphysics that deals with the relationships between heat, work and other forms of energy. Let’sconcretize this over a simple example: Consider a gas confined in a macroscopic cylinder bya piston as shown in Fig. 1.2. For simplicity, we assume a weightless piston, depicted by thered bar that can move up or down without any friction. On top of the piston, there is a weight.The gas has a temperature and a pressure which are in equilibrium with the environment. Inother words, both the temperature and the pressure of the gas are constant in the beginning.Now, let’s give some heat to the system by bringing it in contact with a heat reservoir having ahigher temperature than the gas. The temperature of the gas will rise and because of that the gaspressure will start to increase. However, remember the piston was free to move and the outsidepressure is constant, it is just the atmospheric pressure plus the pressure exerted by the weight.Therefore, the system will try to keep the pressure in equilibrium with the outside pressure.To do that it will expand its volume and push the weight upwards, thereby doing work on theweight. This is one of the simplest heat engines. It converts heat into work or potential energy.Refrigerators, air conditioners, power plants, internal combustion transportation vehicles are allexamples of thermodynamic machines driven by thermodynamic principles.Going hand in hand with the above example, thermodynamics has started as a phenomeno-logical theory during the beginning of the industrial revolution (nowadays it’s called the firstindustrial revolution by the trendy perspective). In the late 19th century, mainly with the effortsof James Clerk Maxwell, Ludwig Boltzmann and Josiah Willard Gibbs, it has developed intoan analytical theory that is explainable by the statistical properties of a macroscopic physical Figure 1.1:
Double-slit experiment with (a) photons, (b) very large particles, (c) electrons.Photons and electrons passing the double-slit form interference patterns on the screen, showingthe wave nature of particles. Particles on (b) are too large/massive to exhibit their wave nature.17 igure 1.2:
A simple thermodynamic machine that converts heat into work. A gas at constantpressure held in a cylinder by a movable, frictionless piston, carrying a weight with mass m .Moving piston does work on the weight.system composed of many particles or states. Thus, statistical mechanics, one of the pillarsof modern physics, was born. Statistical mechanics uses statistical methods and microscopicphysical laws to explain the thermodynamic behaviors of macroscopic systems. It is also calledstatistical thermodynamics when it is applied to explain, in particular, thermodynamic proper-ties of systems.Thermodynamics is usually considered as one of the most well-established disciplines of allscience. It has very few (simple) premises, yet has a broad applicability with great success. Butstill, the statistical nature of its laws is both its strength and also its weakness. Despite its powerof applicability, the quantum mechanical origins (which is different than classical microscopicorigins) of thermodynamic laws are still a mystery. What is the meaning of temperature of aquantum object? What is entropy at quantum scale? How does heat behave in the quantumrealm? Quantum thermodynamics has emerged as a stand-alone research field quite recently toanswer all these questions and more. Before even coming to all these questions, we argue thateven in equilibrium quantum statistical thermodynamics, there are interesting phenomena thatyet to be discovered. How small can we reduce the sizes of our devices? What is the influenceof system’s geometry, on its physical properties? Nanoscale manipulation has reached a levelso extreme that facing these questions are now inevitable. To this end, in this thesis, we willtheoretically and computationally explore the quantum-mechanical influence of geometry in thethermodynamics of confined systems. A more detailed description of the thesis topic is givenin next section.From a broader perspective, utilization of quantum phenomena has a potential to revolu-tionize the current state of computing, communication, and energy technologies, to make themfaster, safer and more efficient than the conventional ones. In more detail, quantum computingcan perform certain type of operations significantly faster by making use of quantum super-position of states. Quantum cryptography in principle provides a secure informationtransfer. Design and fabrication of new nanodevices and nanomaterials improves the efficiency(e.g in thermoelectrics),
7, 8 stiffness (e.g graphene),
9, 10 sensitivity (e.g quantum sensors) and18apacity (e.g super/ultracapacitors) in energy generation, conversion and storage technolo-gies. Nanotechnology offers to design materials with the desired mechanical, electrical, opticaland thermal properties. Nanostructured materials can make the solar cells cheaper and moreefficient,
14, 15 which may increase the usability of solar energy drastically. Quantum thermody-namics stands on the ground of all these seemingly distinct research fields. Substantial develop-ment can only be obtained when a solid quantum thermodynamic framework is established. Asof 2020, today’s conventional computers/smartphones use transistors with 7 nm semiconductormanufacturing processes, the scale of which was 32 nm ten years ago, 180 nm twenty years agoand 600 nm thirty years ago and 10µm in 1971. Until recent years, we were still on the edge ofthe validity of classical thermodynamics. However, with the leap forwards in nanofabrication,we are now more aware of the major obstacles in the way of outstanding breakthroughs. Under-standing and controlling the heat dissipation and enabling efficient cooling at quantum scalesare still ongoing challenges.Due to the substantial influence on vastly different fields of modern life, countries spendhuge resources on nanoscale/quantum research and development. For nanoscience and nan-otechnology researches, the United States has invested nearly $27 billion since 2001, and an-nounced $1 . billion for the 2019 budget of the National Nanotechnology Initiative, whilethe European Union spends around e Very recently, both the European Commission and the United States Congressannounced their massive research programmes for quantum science and technologies, whichare e $1 . billion for the National QuantumInitiative respectively.
18, 19
Numerous projects on quantum and nanoscale thermodynamics fieldhave been supported under these umbrella projects during recent years. Considering the trendand the direction of technology, it is almost certain that funding of these areas will continueincreasingly. But regardless of the ongoing trend, nanoscale thermodynamics field providesme enough personal, scientific motivation to work on because it seeks for answers to deep andfundamental questions of physics.
In statistical physics, physical properties of systems are calculated through a probability dis-tribution function by summing over all possible values of quantum state variables (which areinfinitely many in general) in each degree of freedom. A degree of freedom is any parameterthat specifies a state. Quantum state variables determine the energy levels of a quantum system.Although energy levels are essentially discrete, it is customary to use continuum approxima-tion and thereby replacing the infinite summations with integrals. Continuum approximationassumes the energy spectrum to be continuous rather than discrete. This assumption providessimpler, analytical expressions for physical quantities and make it easy to see and interpret thefunctional relations and the main physics.Even though continuum approximation works well at macroscale, it fails to capture the char-acteristics of the systems confined at nanoscale, due to the reasons that we will explore in detailin the next chapter. Instead of replacing the summations directly with integrals, approachingthem using better mathematical tools like Poisson summation formula (PSF) has been studiedin the literature. First implementations of PSF have been done on Casimir effect and on latticesums during 1960’s and 70’s. Later, this methodology has been extended to study the finite-19ize effects or quantum size effects in thermodynamic properties of confined systems.
20, 23–40
Ithas been realized that there is a connection between PSF and Weyl conjecture (which describesthe asymptotic behavior of the eigenvalues of the Laplacian).
20, 41
Weyl conjecture has alsobeen used to obtain the quantum size effect corrections to thermodynamic expressions.
20, 42, 43
Even a much more powerful concept called quantum boundary layer, allowing to get quantumsize effects without solving the Schrödinger equation explicitly, has been developed in 2006 bySisman and his Nano Energy Research Group.
As a result of quantum size effects, considering the discreteness of the energy spectra byusing the infinite summations leads to many interesting results in the thermodynamic and trans-port properties of confined systems. For example, extensivity rule breaks down at nanoscaleand thermodynamic quantities become non-additive.
20, 29
Pressure of a confined gas becomesa tensorial quantity even at thermodynamic equilibrium. Although thermodynamics has al-ways been shown as a theory of continuous variables, in 2014, intrinsic discrete nature in thethermodynamic properties of confined and degenerate Fermi gases has been shown.
47, 48
Asmanifestations of quantum size effects, dimensional transition points in thermodynamic prop-erties of Maxwell-Boltzmann gases have been explored. Discrete and Weyl density of statesconcepts are introduced to represent the thermodynamic properties of confined systems moreaccurately.
43, 50
The phase diagram of quantum oscillations in confined and degenerate Fermigases has been constructed and the oscillations are successfully predicted by the half-vicinitymodel.
In all these studies quantum size effects have been explored.So far in literature, shape effects have never been investigated solely, because size and shapeeffects were inherently linked to each other. In this thesis, by building on the previously acquiredknowledge on quantum confinement effects, we separate them from each other and can focusonly on shape. Shape of an object is described as the geometric information which is invariantunder Euclidean similarity transformations such as translation, rotation, reflection and uniformscaling. We propose and explore a new type of quantum effect, which we call the quantumshape effect . The topic of this thesis is to introduce and examine quantum shape effects onthe thermodynamic properties of nanoscale systems. We establish the theoretical backgroundof this new effect and design novel nanoscale energy devices based on them. Fundamentalquestions we are going to answer (Spoiler: The answer is YES to all) in this thesis are listedbelow:• Is there a way to change the shape of an object without changing its sizes?• Does shape enter as a separate control variable to the thermodynamic state space?• How do thermodynamic potentials and functions change solely with shape?• Is it possible to analytically predict the shape-dependence of physical quantities?• Can we design novel nanoscale heat engines based on quantum shape effects?We’ve introduced our first results on quantum shape effects in March 2017 at the 5th Quan-tum Thermodynamics Conference, held in Oxford, United Kingdom. Subsequently within thenext years, we’ve presented our works on developing quantum shape effects in several other in-ternational conferences, workshops and seminars. We’ve published the fundamental results ofthis thesis in Ref. The thesis constitutes an introduction to and a comprehensive examinationof quantum shape effects in the thermodynamics of confined systems along with a thoroughreview of its now established background. 20 .3 Literature Overview
In this section, we will summarize the research that has been done in literature related to thesubject of this thesis. Some concepts might appear to be mentioned without explicit definitions,however, they would hopefully be clear during the second chapter of the thesis.Development of new techniques and technologies makes it possible to create and manip-ulate nanoscale systems much easier than before.
Many nanoscale systems exploitingquantum effects and having great application possibilities have been realized in recent years,such as nanomotors, single-atom heat engines etc.
Study of confined systems and size-dependent phenomena are very active and promising research areas since they can revolutionizeour understanding of thermodynamics at nanoscale as well as contributing to the developmentof nanoscale energy conversion and storage devices with excess properties.
Quantum size effects have shown to be of great importance in nanoscale thermodynamics,in fact it has been shown that they put some fundamental limitations on work extraction fromnon-equilibrium states. Quantum size effects are also very fundamental in nanoscale trans-port phenomena.
89, 93
Indeed, they result to even some conceptual changes in physics such asquestioning the meaning of "conductivity" at nanoscale and preference to use the word "con-ductance". Conductance quantization, De Haas-van Alphen effect, Shubnikov-De Haas effectare some of the important manifestations of quantum size effects which pretty much shaped themodern nanoscience and nanotechnology. Quantum size effects are especially important fornanoscale energy conversion and storage technologies, in particular for thermoelectricity.
8, 95, 96
Charge and heat transport even in a single molecular junction have attracted recent interest.
As an energy application of quantum size effects, thermosize effects, which can be consideredas a sub-branch of thermoelectric effects, proposed by Sisman and Müller in 2004 and havebeen studied extensively during the last decade.
29, 100–114
Quantum size effects is a broad topicon its own, but to fully characterize the geometry of confined systems, we need more than justsize.The role of geometry in physics is quite deep and diverse. Gravity, one of the fundamentalforces in physics, is explained by the geometry of spacetime in general relativity. Along withtopologically protected properties of matter, geometric properties like geometric phases andforces attract considerable attention nowadays. Berry phase induced forces and usingthose to drive tiny nanomotors have been studied for that matter.
A more related problem to the topic of this thesis comes from the spectral geometry. In1966, Polish mathematician Mark Kac popularized a fascinating problem in spectral geome-try by posing an elegant question: "Can one hear the shape of a drum?".
Of course at firstthought, it’s reasonable that differently shaped drums will sound different which is supportedby our daily-life experiences. For example, sound difference between high tom and low tomis due to their diameter difference. (Asking this question mathematically is something else ofcourse.) What has been asked actually is "Are there differently shaped drums that sound ex-actly the same ?" Kac didn’t know the answer, however studies related to this problem actuallygo back to the end of the 17th century, when an English polymath Robert Hooke observednodal patterns in vibrating plates. About a century later, a German physicist and musician ErnstChladni made first systematic experiments on the phenomenon. When a plate fixed at the mid-dle is set into vibration, excited modes of vibration can be visualized by sand grains poured onthe plate as shown in Fig. 1.3. Sand grains accumulate towards the non-oscillating nodal linesof the plate and form patterns unique to each mode, which is called Chladni patterns. In 1821,21 French mathematician Sophie Germain made a partial mathematical description. Membraneoscillations (also called cymatics) or in general the wave phenomenon, are mathematically for-mulated by Helmholtz partial differential equation. Solution of the Helmholtz equation for adomain gives eigenvalues and corresponding eigenfunctions, which define the vibrational andoscillational characteristics of the domain.
While a result answering Kac’s question has been published in 1964 by J. Milnor for 16-dimensional tori, it was not until 1992 that the first generalized mathematical proof answeringKac’s question has been announced by Gordon, Webb and Wolpert.
They created two(pair of) domains having different shape but exactly the same eigenvalue spectrum (see Fig.1.4). These kinds of domains are called isospectral. Their method of proof is based on Sunada’stheory, though there is more than one way to show the existence of differently shaped yetisospectral domains.
Besides showing it mathematically, there has been attempts to showisospectrality also experimentally, but of course it requires very precise equipments as well asideal conditions.
Hearing the shape is technically called an inverse problem. Shape determines the sound,but the catch is figuring out the shape from the sound. This inverse problem actually has beensolving in everyday life of several living organisms, Fig. 1.5. Mammals like bats and dolphinsconstantly use echolocation techniques to navigate and communicate. They send sound wavesinto environment and from their reflections they determine the object’s distance, shape andtype. Bats actually "see" things by hearing them. It is remarkable that evolution results to suchingenious solutions for complicated problems.In recent years, following the advancements in pattern recognition and numerical tech-niques, shape recognition based on eigenvalues of the Laplacian has become an attractivefield.
For non-rigid shape analysis a method called "Shape-DNA" based on Laplace-Beltrami spectra is introduced. Shape-DNA is basically a numerical fingerprint of any two- orthree-dimensional manifold based on its Laplacian eigenvalue spectrum. Recognition of objectsbased on Shape-DNA can be done in digital environment. Uniform scaling of different objectsis possible by the same method.
Although isospectral objects exist, their occurrence is ex-tremely improbable in daily life. This makes methods of shape recognition based on eigenvaluespectrum attractive even for commercial applications.Kac’s question can also be generalized into the quantum mechanical applications. Since
Figure 1.3:
Chladni patterns produced by scattered sand on an elastic, rectangular plate. Whenthe plate is forced to vibrate (say through a loud speaker), grains end up in the places with zerovibration, thereby visualizing the modes of vibration on the plate surface.22 igure 1.4:
An example of 2D isospectral domains. Left and right shapes have identical eigen-value spectrum, therefore two drums made by these shapes sound exactly the same.types of boundary conditions determine the solutions of differential equations, boundary con-ditions dramatically affect the energy spectra of the particles confined in a domain.
Cana gas feel the size and shape of its confinement domain? This question has also been takeninto consideration in several papers.
The important thing to realize is that Weyl terms(surface/volume, periphery/volume and edge/volume) are not enough to represent full charac-teristics of the domain. Weyl conjecture is just another approximation (but a really good one)that only holds true in the asymptotic limit. Although there may be some special absolutelyisospectral domains, there are infinitely many domains whose Weyl terms are completely iden-tical, yet properties of gas confined in these domains can still be different. In an absolutemathematical sense, there are differently shaped domains that gas cannot feel on which oneit’s filled. However, these kinds of isospectral domains are extremely rare and very speciallyarranged domains.
All these works done in spectral geometry are profoundly related with the main theme ofthis thesis. Laplacian eigenvalue spectrum is equivalent to the solutions of time-independentSchrödinger equation. While Weyl conjecture is giving some good information about the spec-
Figure 1.5:
An inverse problem of hearing the shapes in nature. Bats and dolphins use echolo-cation techniques to navigate, which relies on the solution of an inverse problem of "If an objectsounds like that, where is it and what is its shape?".23rum, it has limitations in strongly confined systems. The field is very rich both mathematicallyand physically, and research on open questions is still ongoing.
A very nice comprehen-sive review of the subject with its connections to physics can be found in Ref.
Research on quantum forces and torques is also very much related with our thesis as we alsointroduce a new kind of quantum force and torque due to quantum shape effects. Some aspectsof matter wave related forces such as quantum statistical forces due to boundary effects and quantum-classical hybrid systems with matter wave pressure are investigated inliterature. One of the remarkable consequences of the wave-like properties of quantum par-ticles are quantum mirages at quantum corrals.
They represent excellent examples offluctuation-induced forces like Casimir forces which are also subject to size and shape relatedgeometric effects.
Electromagnetic waves exert pressure. Several type of actuators and motors
67, 188, 189 have been proposed based on this radiation pressure. From the experience of these kinds ofsmall pressures, measurement of as tiny as piconewton forces are possible.
Like theradiation pressure produced by light, matter waves can also exert pressure. In literature, semi-classical approaches are used for the calculation of fluid-like properties of matter waves.
Since semi-classical or quantum hydrodynamic approach to calculate properties of matter wavesis quite adopted in literature, it may be proper to use similar kind of hydrodynamictransport approach in our calculations as well. Quantum kinetic theory and Wigner functionmethods are also used occasionally in literature.
Another closely related intriguing phenomenon is quantum backflow effect in whichflow of negative probability, or in other words, negative current of particles with entirely positivemomenta occurs. Although it is really a striking quantum phenomenon, the subject is largelyoverlooked and pretty much unexplained. One-sided momentum flux calculations for localpressures in this thesis will be done similar to the ones done in the calculation of the backfloweffect.Coherence of matter waves may also enhance matter wave related effects.
Quantumforces that may appear even in superconductors are also proposed in literature.
New methodsfor quantifying macroscopicity degrees of quantum phenomena may widen our view andfurther the enthusiasm beyond. Quantum gases confined under time dependent boundaries are also interesting to examine and may shed some light on the time-dependent studies.
We’ll start with an overview of quantum size effects in statistical thermodynamics. In the fol-lowing chapter, we first explain the quantum-mechanical origins of the size effects in confinednanostructures. We discuss how size effects can change the thermodynamic behaviors of sys-tems. We cover some necessary mathematical tools to calculate quantum size effect correctionsto the thermodynamic expressions. We review the quantum boundary layer method which isone of the most powerful methods for obtaining quantum size effect expressions as well asfor understanding their underlying physical mechanisms. We are particularly interested in thismethodology because we’ll extend it to explain also quantum shape effects.In the third chapter, we separate size and shape effects from each other completely andintroduce the quantum shape effects. Signatures of shape effects in eigenspectrum will be dis-cussed. We examine the shape dependence of partition function and extend quantum boundary24ayer methodology to analytically predict quantum shape effects as well. Different confinementdomains and boundary conditions will be considered. Furthermore, the influence of quantumsize effects on quantum shape effects will be discussed.Investigation of change in thermodynamic properties due to shape effects is done in thefourth chapter. Shape enters as a new control variable on thermodynamic state functions andstate space, opening up a whole new, before unexplored dimension in thermodynamics. Quan-tum shape effects on internal energy, free energy and entropy of confined systems are investi-gated. Non-uniform density distribution of particles causes a non-uniform pressure distributionin nested confinement domains. Due to quantum shape effects, an asymmetric non-uniformdistribution occurs which induce a quantum torque. Examination of the pressure and torque isdone from both local and global approaches. We further investigate quantum shape effects inelectron gases using Fermi-Dirac statistics.Several energy applications of quantum shape effects are proposed and explored during thefifth chapter. A new thermodynamic process called isoformal process is introduced and two newheat engine cycles are designed based on quantum shape effects. In this chapter we mentionalso a quantum Szilard engine variant and a single-material unipolar thermoelectric effect calledthermoshape effect, which are not directly parts of the thesis but mentioned anyway as they aredone by groups including the author of this thesis. Also, they constitute significant examples ofthe applications of quantum shape effects.The main outputs and highlights of this thesis are itemized below:• Quantum size and shape effects are completely separated from each other through a size-invariant shape transformation, which allows one to focus on purely shape-dependentphysical properties of confined systems.• Shape variation alone is able to change the thermodynamic state functions while all otherphysical parameters and geometric size variables are constant.• Equilibrium statistical properties of the particles confined in an arbitrary domain, alongwith their size and shape dependence, are analytically estimated with a reasonable accu-racy by extending the quantum boundary layer method.• The analytical methodology not only gives physical insights about the existence and un-derlying mechanisms of quantum shape effects, but also provides opportunity to predictthem without doing cumbersome numerical calculations.• Existence of size effects increase the quantum confinement whereas the appearance ofshape effects leads to a decrease in the effective confinement.• Quantum shape effects cause decrements in internal energy and Helmholtz free energy,whereas they have a more complicated effect on the entropy of the system.• Entropy and free energy of the confined system can decrease simultaneously and sponta-neously due to quantum shape effects, which is a unique phenomenon in the thermody-namics of non-interacting gases.• Quantum shape effects cause a breakdown of extensivity of the thermodynamic quantities,just like size effects. 25 Quantum shape effects give rise to quasistatic spontaneous rotation and/or Casimir-liketranslational motion of the objects that are freely movable inside the confinement domain.• Thermodynamically stable configurations in nested confinement domains are determinedby the symmetric periodicity points, whereas configurations other than that are unstable.• A quantum torque is induced due to the non-uniform pressure exerted by matter waves.• In the thermodynamics of electrons obeying Fermi-Dirac statistics, chemical potentialoscillates with the varying shape parameter for fixed number of particles, temperature andsizes. This causes oscillatory behaviors in all thermodynamic properties of confined anddegenerate Fermi gases regardless whether they intrinsically exhibit quantum oscillationsor not.• A new type of thermodynamic process, called isoformal process, based on keeping thedomain shape constant, is proposed and new thermodynamic cycles featuring the isofor-mal process are introduced and examined.• Quantum shape effects provide a novel way to design new type of nanoscale heat enginesand energy harvesting devices.The results that are found and the topics that are explored in this thesis are related to andcan shed light into many diverse areas of physics and mathematics, such as spectral geometry,quantum thermodynamics, quantum billiards, quantum corrals, bound states in waveguides,localization, topological properties, dimensional transitions, local properties, quantum hydro-dynamics, quantum transport, cross effects, quantum backflow, shape recognition, nanoscaleenergy technologies and so on. 26
A Review of Quantum Size Effects inStatistical Thermodynamics
In this chapter, we’ll introduce and review some primary concepts and methodologies that we’veused in this thesis.
Our main purpose in this thesis is to develop a comprehensive understanding on the role of con-finement geometry in the thermodynamic properties of physical systems at nanoscale. There-fore, in this review chapter we first mention what do we mean by confinement and how physicalsystems change behavior when confinement geometry is changed. Later on in this chapter, we’lldiscuss on how to quantify these changes and explore the influence of quantum size effects inthe thermodynamics of confined systems.
The foundations of quantum theory laid in 1900, when German physicist Max Planck (acciden-tally) discovered that the radiation is quantized. Yet the theory found a meaningful ground todevelop, after French physicist Louis de Broglie had suggested the hypothesis of matter waves, id est matter exhibit wave-like behavior, in 1924 in his PhD thesis. This behavior has beendemonstrated many times experimentally and now sits on the central part of our current under-standing of the universe. Before quantum mechanics, behaviors of particles are modeled as ifthey are points or hard spheres. Consider you have a box filled with point particles randomlymoving around inside the box. Particles can only bounce from the walls of the box when theycome infinitely close (touching basically) to the boundaries. On the other hand, if they are notpoint particles but waves, it would be more accurate to think of them occupying a finite amountof space without having a precise pointwise coordinate location. In that case, particles can feel27he presence of boundaries without strictly touching them. In other words, they can reflect backfrom the walls without necessarily coming really close to them.Fundamentally, a quantum system is described by an abstract mathematical object calledwavefunction to which all the quantum weirdness essentially boils down. The wavefunction isa square integrable complex-valued quantity that carries the possible outcomes of the measure-ments made on the system. For example, a position wavefunction of a single particle carries theinformation about the possible locations that particle can be found at a given momentum andtime. The motion of the particle depends on the time evolution of its wavefunction, which isdescribed by the Schrödinger equation, i (cid:126) ∂∂t Ψ( r , t ) = − (cid:126) m ∇ Ψ( r , t ) + V ( r , t )Ψ( r , t ) , (2.1)where (cid:126) = h/ (2 π ) , ∇ is the Laplace operator, r is the position vector, m is particle’s mass, V is the confinement potential, t denotes time and Ψ is the wavefunction in position basis. Inthermodynamics, we deal with the equilibrium properties of systems and so we are interestedin stationary solutions. To obtain the stationary states of a quantum system, we need to solvethe time-independent Schrödinger equation. By using the method of separation of variables, wereach an eigenvalue equation called the Helmholtz equation. Under zero potential, V( r )=0, it iswritten as ∇ ψ ( r ) + k ψ ( r ) = 0 (2.2)This equation is a wave equation and it is the general form of time-independent Schrödingerequation, where the wavenumber k corresponds to k = √ mE/ (cid:126) and E denotes the energy ofthe particle.Now we have an equation describing the wave nature of quantum particles and we’ll use it inthe next subsection. Before that, let’s mention a bit more about the wave nature of particles andhow do we quantify it. Essentially, what we are interested in is the position space representation,because we want to understand the geometry effects. In position space, the wave character ofparticles is quantified by their de Broglie wavelengths, which is defined as λ dB = h/p , where h is Planck’s constant and p is the momentum of the particle. For massive particles, momentumis the multiplication of particle’s mass and velocity, p = mv . This means, larger the particle’smass, smaller its de Broglie wavelength. This is the reason why wave-like behavior is moreapparent in subatomic particles but not in our macroscopic world.In condensed matter physics, we mostly deal with a collection of particles rather than asingle particle. Both the statistical behaviors and the wave nature of a collection of particles canbe captured by the so called thermal de Broglie wavelength, which is given by the followingexpression, λ th = h √ πmk B T , (2.3)where k B is Boltzmann’s constant and T is temperature. In Eq. (2.3), individual velocities ofparticles are replaced by the mean velocity corresponding to their average kinetic energies. Byuse of statistical mechanics, we don’t have to deal with the individual behaviors of astronomicnumber of particles, instead we can capture their statistical behavior which provides a quitegood representation even for dozens of particles.In addition to mass of the particles, thermal de Broglie wavelength says that the temperatureof the system is also important for the appearance of wave nature. Like mass, it is also inverselyproportional with the wavelength. Despite all, λ th by itself is not enough to see the difference28 igure 2.1: Differentiation of macro and nano scales are coming from comparisons of thethermal de Broglie wavelength of particles and the sizes of the domain that the particles confinedwithin. Depending on the nature of the system and particle statistics, one can replace the thermalde Broglie wavelength with the appropriate characteristic de Broglie wavelength.between macro world and nano world. What is important is its comparison with the systemsizes. Wave nature does not disappear magically at macroscale. What happens is our macroworld is too big in comparison with the thermal de Broglie wavelength. Recall the point particlevs wave-like particle example. Our macro world is on the order of meters, whereas the thermalde Broglie wavelength of a free electron gas at room temperature, for example, is around 4.3nanometers. There is orders of magnitude difference. Compared to our macro world,electrons are like point particles, although their actual behavior is wave-like. Note that thisanalysis was for a subatomic particle electron, one of the lightest massive particles. For atomsand molecules, the order of magnitude difference is even larger. So the essential thing separatingthe nano world from macro one is the comparison of the thermal de Broglie wavelength ofparticles with the sizes of the domain where those particles are confined. In Fig. 2.1, such acomparison is given. When L is much larger than λ th , we can assume the particles to behave aspoint-like. When L is comparable with λ th , wave-like behavior of particles become apparent.Hence, electrons confined in domains with nanoscale dimensions exhibit its wave nature evenat room temperature.It is necessary also to mention here that the statistical properties are not the same for allkinds of particles. In terms of statistical behavior, particles split up into two groups; Fermionsand Bosons, satisfying the Fermi-Dirac statistics and Bose-Einstein statistics respectively. Inquantum mechanics, particles are indistinguishable. Further to that, it is not just we cannotdistinguish this electron from that electron, it is meaningless even to talk about such a thing. (So, it is like asking to draw a triangle having two sides. It’s wrong by definition.) Degeneracyof particles comes into play when Fermions or Bosons are considered as confined particles.Quantum degeneracy brings a separate correction to the characteristic de Broglie wavelength(it can be chosen as thermal, mean or most probable de Broglie wavelengths). In such a case,rather than thermal de Broglie wavelength, it is more useful to consider Fermi wavelength forFermions for example. Therefore, λ c , the characteristic de Broglie wavelength, is a matter ofchoice which should be done according to the statistical nature of the particular system. Wewill mention more on this in section 4.9. When the indistinguishability of particles is ignored,we use the Maxwell-Boltzmann statistics, which gives accurate results for low density and hightemperature systems. To keep the discussion simple, we keep using the thermal de Broglie29avelength and demonstrate our formalism considering the Maxwell-Boltzmann statistics. Quantum confinement occurs when the motion of particles is restricted in at least one direction.For example, graphene is a 2D material (it consists of a single-layer of carbon atoms) in whichelectrons are free to move in two directions but restricted or confined in one direction. Accord-ingly, carbon nanotubes and quantum wires are considered as 1D structures that are confinedin two directions. When the particles are confined in all directions, the structure is called aquantum dot, hence 0D. In Fig. 2.2, different structures of carbon-based materials confined invarious directions can be seen as examples. The strength of confinement in a particular directionis determined by the confinement parameter of that direction, α = h √ mk B T L , (2.4)where L is the length in the confined direction. α is the ratio of the most probable de Brogliewavelength in Maxwell-Boltzmann statistics to the domain’s length. In terms of thermal deBroglie wavelength, it is stated as α = ( √ π/ ∗ ( λ th /L ) . Therefore, magnitude of the quantumconfinement of a domain depends on the particles’ mass, system’s temperature and characteris-tic domain size.Depending on the value of the confinement parameter, we can refer the domain as uncon-fined or free ( α ≈ ), weakly confined ( α < . ), confined ( . < α < ) and strongly confined( α > ). Note that α values give just a rough scale, since the transitions between these confine-ment regimes are not sharp but smooth. Mathematically, α enters as a proportionality constantof quantum states to statistical expressions of thermodynamic quantities and determines the es-sential discreteness in energy spectrum. Although the difference between each quantum state is Figure 2.2:
Allotropes of carbon at various dimensions. Numbers describing the dimensionsdenote the number of free directions for charge carriers. Note that the particles are appeared tobe confined on the surface of the fullerene, but its overall size is less than 1nm, which makes it0D as it is isotropically and strongly confined from all directions.30lways one, the number of quantum states within an energy interval is determined by α . Thus,it scales the energy spectrum.Quantum confinement plays a significant role in solid-state physics and material science.It paves a way not only to the discovery of materials with better properties in comparison totheir conventional counterparts, but also to the development of nanoscale devices that are moreefficient and better at certain tasks. Some well-known and well-studied examples of confinedsystems are the charge carriers (electrons and holes) and phonons in nanoscale metals and semi-conductors in addition to ultracold atoms in optical traps. Theoretical study of these confinedsystems is usually done using the particle in a box model. Consider a non-relativistic singlequantum particle confined in a 1D domain having length L , Fig. 2.3. The potential inside thewell is zero and infinite at the outside, meaning the walls are impenetrable. Even a quantumparticle cannot tunnel (leak) through the walls of an infinite well. This is a quite accurate modelfor the behaviors of conduction band electrons in metals for example. We will stick with theimpenetrable boundaries throughout the thesis in order to maximize the effect of confinementgeometry on the particles as quantum tunneling leads to leakages and reduces the geometryinfluence.To solve the particle in a box model, we first solve the time-independent Schrödinger equa-tion, which is basically the Helmholtz equation in Eq. (2.2). Since the confinement potential isinfinite outside the box and zero inside, the boundary conditions are Dirichlet so that ψ (0) = 0 and ψ ( L ) = 0 , which means the wavefunction goes to zero at the fixed ends of the box. Underthese boundary conditions, energy eigenvalues (corresponding to the energy levels of the sys-tem) of the particle confined in 1D domain with length L can be obtained from the solution ofEq. (2.2) as E n = (cid:126) π m (cid:16) nL (cid:17) , (2.5)where n denotes the modes of the wave (also corresponds to the quantum states in this particularexample), integers running from 1 to ∞ . The important thing to notice here is that energy levelsare not continuous but discrete. This is one of the properties of matter which was unexplainable Figure 2.3:
Stationary states correspond to standing waves in systems exhibiting wave-likebehavior, if the system has a fixed geometry. Leftmost subfigure shows the energy levels andthe corresponding wavefunctions of a single particle confined in an infinite potential well. Thefigure on the center shows a picture of a guitar’s vibrating strings. Rightmost subfigure showsthe fixed nodes of the guitar strings leading to the formation of wave modes.31y the classical physics. Discreteness of energies of confined particles is a direct result of theirwave characteristic. Eigenfunctions (corresponding to the wavefunctions) describing the spatialbehavior of the wave modes are also obtained as ψ n = (cid:114) L sin (cid:16) nπxL (cid:17) , (2.6)where x is the position defined between the edges of the domain. The reason for the appear-ance of these modes is because the domain is fixed at both ends. The length of the domain isassociated with the confinement of the particles. Smaller the length, higher the confinementand higher the energy of the particle by Eq. (2.5). In this regard, spatial confinement givesrise to the discreteness in momentum and energy space. Visualization of energy eigenfunctionscan be seen in Fig. 2.3. In 1D systems, each mode corresponds to a different energy and awavefunction. Modulus square of the wavefunction gives the probability of finding the particlefor a given state at a certain location inside the box.The resulting particle in a box eigenfunctions are extremely similar to the vibrating guitarstrings. This is because, they are the results of the same phenomenon that is described by thesame mathematical equation (differing only by constants); so they have the same physics. Justlike in the particle in a box example, guitar strings are fixed at both ends and they can vibrateonly in discrete set of modes. There is a fundamental mode, which has the lowest energy, lowestfrequency and highest wavelength. Higher harmonics of guitar strings correspond to the excitedstates of a confined quantum particle.Another important consequence of the wave nature of particles is the existence of the non-zero ground state (or zero point) energy. By their very nature, wave modes and quantum statevariables start from their ground state value n = 1 , corresponding to their fundamental mode.In Fig. 2.4, approximations on the representation of energy spectrum of particles can be seen.Classically, energy spectrum is considered to be continuous and starts from zero. As we haveseen from the particle in a box example, the true nature of the energy spectrum is discrete and itstarts from a non-zero value called the ground state. At macroscale, continuum approximationworks very well because the separation between the levels is inversely proportional to the sizeof the system. The larger the size, the higher the accuracy of the continuum approximation. Forweakly confined systems, we use so called bounded continuum approximation, which considersthe non-zero value of the ground state while still assuming a continuous spectrum for the rest.Despite taking a continuous spectrum, the bounded continuum approximation is very powerful.In addition to giving quite accurate results, it also properly captures the boundedness of thedomain and generates all quantum size effect correction terms. For strongly confined systems,however, even this approximation fails and one needs to consider the discreteness of the spec-trum. We’ll discuss more on the use of bounded continuum approximation and quantum sizeeffects in the next section. We talked about quantum confinement in a general way. But in fact, we only dealt with a 1Dmodel so far. Size of a 1D domain is characterized by its length. Once you know the length,you have the energy levels. But what about the higher dimensions? How do we quantify thesize in general? When we say the sizes of an object at macroscale, what does that even mean?To understand these, let’s continue our discussion with the 2D model. Think of a simple 2D32omain, for example a square with side length L . It has an area of L , a periphery of L and4 vertices (cusps or corners). We can quantify the sizes of a square with three numbers, whichmeans if we know the values of these size parameters, we can construct that particular squarewithout having any additional knowledge. For 2D domains, area is defined as the bulk geometricsize parameter, whereas the peripheral lengths and number of vertices are the lower dimensionalgeometric size parameters. Now you may have noticed that we actually missed something in our1D domain’s size analysis. What about the number of vertices in 1D domains? A 1D domainwith length L has 2 vertices, beginning and the end points of the domain. But isn’t it trivial?The answer is no! Can we think of a 1D domain with different number of vertices? The answeris of course yes! Just add some new vertices to the domain or remove the ones on the edges.Consider for instance the keyboard of a guitar. Depending on the location that you press on thekeyboard, it generates different sound. The reason is you are adding another fixed node whenyou press anywhere on the keyboard. Although the actual length of the string doesn’t change,adding another node creates two 1D domains with different lengths and they sound accordingto their new lengths. An example of this can also be seen in Fig. 2.5 comparing columns II andIII in the 1D domain row.In Fig. 2.5 size characterization of domains with different dimensions is illustrated. For 3Dobjects, the sizes are volume, surface area, peripheral lengths and number of vertices. Theseare altogether named as geometric size variables. In measure theory, this definition coincideswith the standard Lebesgue measure (or more generally the Hausdorff measure if one considersnon-integer continuous dimensions). Depending on the dimension of the object, the bulk termbecomes different (volume for 3D, area for 2D, length for 1D and vertices for 0D) while theremaining ones are named as lower dimensional geometric size variables. Sizes of a 3D objectare characterized by these four variables. If all four of these variables are the same for twoobjects, they are considered to have the same sizes, but they don’t necessarily have to haveexactly the same shape, as we shall see later. This point is very crucial and actually constitutesthe central point of the thesis. However, we’ll continue to our review in this chapter and turnback to this point in Chapter 3.As an explicit example, geometric size variables of a cube are shown in Fig. 2.6. Any 3D Figure 2.4:
Classical and quantum representations of the energy spectrum. Energy spectra areconsidered to be continuous in classical physics by the continuum approximation. For boundedsystems, a better approximation called bounded continuum approximation considers the non-zero value of the ground state E . Without any approximation, actual energy spectra of confinedsystems are discrete. 33 igure 2.5: A 3D domain is characterized by four different geometric size parameters: volume V , surface area A , peripheral lengths P and the number of vertices N V . Similarly, 2D domainis by A , P , N V and 1D by P , N V . This is the standard Lebesgue measure. Sizes of an objectcan be changed while keeping its general shape constant by the process called uniform scaling(compare columns I and II). Between columns II and III, both size and shape of the objects arechanged (except the last row where the size is unchanged). Figure 2.6:
Geometric size variables of a cube. It consists of 1 volume, 6 square surfaces,12 lines and 8 points. Powers of the length L generates the size variables of correspondingdimensions.object has a single volume. A cube has a surface area consists of 6 squares with the side length L of the cube. Surface area is a lower-dimensional (2D) property in this case. Periphery of thecube is the total lengths of the line segments that are present on the object which are the sidesof squares. There are 12 of them. Note that it is not × because all line segments are sharedby 2 different squares. The lowest dimensional elements are the number of vertices of whichthe cube has 8. This analysis may look too simple, however, it is actually indispensable for theunderstanding of the quantification of sizes of a domain and they play a significant role on thephysical properties of confined systems.By definition, thermodynamics deals with the average values of physical quantities, as theprinciples of thermodynamics are statistical in origin. This fact is expressed by the concept ofthermodynamic limit, which is defined as the limit of infinite number of particles ( N → ∞ ) andinfinite volume ( V → ∞ ) , yet still finite density ( N/ V = constant). Consequently, in classicalthermodynamics, there are no finite-size effects, because there is no finite size. In other words,34hermodynamic properties classically independent of the scale. Nanoscale thermodynamics, onthe other hand, challenges the thermodynamic limit by restricting finite number of particles intoa finite volume. For a single or very few particles, both quantum and thermal fluctuations hinderthe ability to determine the thermodynamic properties. Even if there are finite, but enoughnumber of particles to be able to do statistics, then the methods of statistical thermodynamicscan be extended into the nanoscale systems by proper methods which will be described in thefollowing section. In statistical physics, physical quantities are represented by infinite summations over quantumstate variables of the microscopic quantities that are weighted over the relevant statistical distri-bution function: X = (cid:88) i x i f i , (2.7)where X is a physical quantity, i is the quantum state variable, x is the microscopic quantityof X, f is the relevant distribution function. Classically, the summations are replaced by theintegrals under continuum approximation, which assumes a continuous spectrum of states. Thisapproximation fails at nanoscale, because the separation between the energy levels are inverselyproportional with the domain sizes. Namely, the higher the confinement, the more apparent thediscrete nature of energy levels. Calculation of summations directly can be cumbersome insome cases. After all, they are infinite sums needing truncation operations during numericalcalculations and they may depend on many degrees of freedom. Besides, most of the time itis very hard to predict the functional relations and dependencies of X on its control variablesinto the calculated quantities. To tackle with these problems, two distinct but interconnectedmathematical tools are employed in the literature.Before explaining the tools, in order to quantify our discussion and make it more physicallyintuitive, we’ll first mention an important concept called partition function. The partition func-tion is a powerful statistical concept that relates microscopic properties to macroscopic ones viathe probability theory, Fig. 2.7. For a monatomic gas the partition function is written as ζ = (cid:88) ε exp (cid:18) − εk B T (cid:19) . (2.8)Derivation of the partition function depends on the maximum entropy principle along withseveral constraints like fixed number of particles in a volume as well as their temperature. ε denotes the energy values obtained from the Schrödinger equation, which is microsopicin origin. k B T contains the temperature which is an average, macroscopic quantity. They arebrought together in a probabilistic weight factor, called Boltzmann factor, through the maxi-mization of entropy subject to fixed number of particles and total energy. Summation over allstates gives the statistical property called the single-particle partition function. Derivation ofthe partition function is done under thermodynamic equilibrium, that is the maximum entropycondition.To show the application of the methods we’ll use the partition function, since it captures theessence of the methods without loss of generality, as any other thermodynamic state function35 igure 2.7: Single-particle partition function with the Boltzmann statistical weight, providinga probabilistic connection between microscopic and macroscopic properties of a system.can be easily calculated from the partition function. However, it should be noted that partitionfunction is not necessary at all to obtain thermodynamic properties (which can be obtainedwithout using it but rather using the free energy), rather it is a convenient tool to derive them.Nevertheless, it also simplifies our investigations due to the fact it gives a self-contained quantityto explore and demonstrate the size and shape effects explicitly.
Let’s return to back to the mathematical tools that we refer. To calculate infinite sums, there areseveral approaches like Poisson summation formula and Abel-Plana formula as well as Euler-Maclaurin formula, which is for finite sums. The most convenient one for the evaluation of thesums encountered in statistical physics is the Poisson summation formula (PSF). PSF relatesthe original summation to the summation of its Fourier transform. The steps to get the reducedversion of PSF for even functions has been given in the Appendix of my Master of Sciencethesis. Here, we’ll focus more on the understanding of PSF. For even functions (which appliesto all thermodynamic state functions) the PSF can be written as ∞ (cid:88) i =1 f ( i ) = (cid:90) ∞ f ( i ) di − f (0)2 + 2 ∞ (cid:88) s =1 (cid:90) ∞ f ( i ) cos(2 πsi ) di. (2.9)PSF consists of three terms. You may have noticed that the first term actually corresponds tothe continuum approximation (i.e. replacement of the sum with integral). As it is shown inFig. 2.8, it is a reasonable approximation for systems having less significance on near groundstates which corresponds to the vanishing values of the confinement parameter α . Look atthe accuracy of the black curve representing the colored columns of the summations. It hasbeen shown clearly in our other study that in order for continuum approximation to be used,occupation probabilities of the excited states need to be substantial. If the system is so confined that its energy is close to ground state, the first term of PSFfails to represent the summation, see Fig. 2.9a. Discreteness of the sum and the smoothnessof the integral can be easily noticed in Fig. 2.9a. It should be noted that the summation startsnot from zero but from unity, unlike the integral. Therefore, integral representation contains thehalf of the zeroth value incorrectly. Bounded continuum approximation corrects this impropercalculation by removing the false contribution from the half of the function’s zeroth value bythe integral, Fig. 2.9b. The correction term is called the zero correction and corresponds to thesecond term of the PSF. Physically, zero correction excludes the false contribution of the zeroth36 igure 2.8:
The usual (unbounded) continuum approximation, which corresponds to the re-placement of infinite sums by integrals. It is a good approximation for systems with a lesserdegree of discreteness. Subfigure in the rightmost figure shows the ratio of the summation andintegral representations. It fails for not very small α values.quantum state, since there is no such state since all quantum states start from unity. All usualquantum size effect corrections to the thermodynamic properties come from this second term.For a system to be considered as confined, either the temperature is very low ( λ th is large),or the sizes are very small so that the confinement parameter is near to or larger than unity. Eventhough bounded continuum approximation is accurate up to confinement values near unity, itconsiderably fails for α > . The third term of PSF becomes important for such stronglyconfined systems. Note that integral curve passes from the middle points of the summationcolumns. For { i − . } integral representation mistakenly calculates over the top whereas for { i + 0 . } it falls short, Fig.2.9c. In total, this surplus (green triangles) and deficiency (orangetriangles) roughly compensate each other, except for strong confinements. The third term ofPSF is called the discreteness correction and corrects this total oscillatory difference betweenintegral and the summation.Despite the power of PSF, it has some weaknesses as well. You can only use it if you canobtain the energy eigenvalues analytically, which can be done only for some limited number ofgeometries like rectangular, cylindrical and spherical. Even for cylindrical and spherical ones, Figure 2.9:
Representations of (a) unbounded and (b) bounded continuum approximations viathe Poisson summation formula (PSF). Consideration of the second term of the PSF enhancesthe usual continuum approximation into the bounded domains. (c) The third term of PSF cor-responds to the discreteness correction and responsible from recovering the intrinsic discretenature of the actual summation. 37 igure 2.10:
An inverse problem of hearing the shapes. Surface area determines the pitch ofthe toms in a drum, physically showing the relationship between the size and the eigenvaluespectrum. Light waves are shown for analogy.we have to use some approximations to obtain analytical expressions for energy eigenvalues.If you want to explore quantum size effects for arbitrary geometries, PSF cannot do the job.Fortunately, there is another mathematical tool for that: the Weyl conjecture.
In 1911, German mathematician Hermann Weyl conjectured a law describing the asymptoticbehavior of the Laplace-Beltrami operator. It is known as one of the earliest works on spec-tral geometry, the field of mathematics concerning with the relationships between spectrum ofbounded differential operators and geometric structures of manifolds. Spectral geometry andWeyl’s result had considerable influence on the mathematical formulation of quantum mechan-ics and the understanding of geometry effects in confined systems.
The question of "Can onehear the shape of a drum?" that we mentioned in the introduction chapter is related with theWeyl conjecture. This question is about an inverse problem which is determining the shape of adrum from its sound, see Fig. 2.10. On the other side, the Weyl’s conjecture "solves" the directproblem, which is determining the behavior of the spectrum by using its geometric properties.Of course, Weyl’s conjecture cannot be used to answer Kac’s question, because it is valid onlyat asymptotics. But this result definitely has intrigued Kac’s question. In particular, the Weyl’sconjecture gives the asymptotic behavior of the number of eigenvalues less than k for Helmholtzequation, which can be written in its general, d-dimensional, closed form as, Ω d ( k ) = d (cid:88) n =0 (cid:18) − (cid:19) d − n (cid:18) k √ π (cid:19) n V n Γ (cid:2) n +22 (cid:3) , (2.10)where V n is the n -dimensional volume according to Lebesgue measure ( V → V , V → A , V → P , V → N V ). Note that Eq. (2.10) is analytical, as the summation over n serves forgenerating the lower-dimensional correction terms. This result is known as the Weyl conjecture,but actually the term conjecture is a misnomer here, as it is proved and generalized by VictorIvrii in 1980. It can readily be seen that Eq. (2.10) contains all geometric size variables inseparate terms, just like the first two terms of PSF gives. From the relation p = (cid:126) k wavenumberequals to the momentum with the proportionality constant (cid:126) . Equations depending only on k are said to be written in momentum space.In statistical physics, by using a useful concept called density of states, integrals over quan-tum states can be replaced by integral over energy. Thereby, one can reduce multiple integrals38nto a single one. In other words, density of states represents the Jacobian when we change thecoordinate systems from Cartesian to spherical one. Density of states describe the number ofstates within an infinitesimal momentum or energy interval. Note that Weyl conjecture givenin Eq. (2.10), gives the number of states up to a wavenumber. Hence, the derivative of Eq.(2.10) with respect to wavenumber gives the density of states in momentum space. Likewise,density of states in energy space can be obtained by writing the Eq. (2.10) in energy space anddifferentiating it with respect to energy.Weyl density of states (WDOS) in k -space can easily be obtained by differentiating Eq.(2.10) with respect to k . For this reason, we won’t write it explicitly, but instead below wewrite WDOS in energy space, which can also be used directly to calculate the thermodynamicproperties. W DOS d ( ε ) = d (cid:88) n =0 (cid:18) − (cid:19) d − n n V n (cid:2) n +22 (cid:3) λ nth (cid:18) εk B T (cid:19) n − , (2.11)Weyl conjecture gives identical results with the first two terms of PSF. Quantum size effectsare studied in literature by using Weyl conjecture or first two terms of PSF in the calculation ofinfinite summations.
24, 26–30, 42, 106, 225, 227
A comprehensive investigation of Weyl density of stateshas been done in Refs.
43, 50
Weyl conjecture and shape dependence is also significant when calculating forces that de-pend on the geometric structure of systems such as Casimir force.
Geometric Weyl cor-rections shed light on the understanding of the ultraviolet divergences and cutoffs encounteredin the calculation of Casimir force.
Weyl conjecture may be used also to obtain analyticalexpressions for such geometric dependencies.
Now let’s see our tools in action. A visual summary of the formalism of quantum statisticalthermodynamics is presented in Fig. 2.11. Consider large number of non-interacting parti-cles confined in a domain. First, the Schrödinger equation is solved considering the boundaryconditions exposed by the confinement domain, then the obtained eigenvalues are used in theappropriate partition function. In the figure, Maxwell-Boltzmann weight is used but the formal-ism can directly be applied to other statistics as well. There are two options. If one knows theeigenvalues analytically (which is only possible for limited number of geometries), the first twoterms of PSF can be used to obtain the partition function under quantum size effects. Other-wise, WDOS concept can be used for any geometry. After obtaining the partition function, anythermodynamic state function can be easily calculated by standard thermodynamic relations.For example, partition function by using either PSF or WDOS in three different dimensionsalong with quantum size effect corrections is written as ζ = V λ th − A λ th + P λ th − N V , (2.12a) ζ = A λ th − P λ th + N V , (2.12b) ζ = P λ th − N V . (2.12c)Here λ dth (dimensional powers of the thermal de Broglie wavelength) can be thought of theamount of d -dimensional volume roughly one particle occupies in an d -dimensional space.39 igure 2.11: Quantum statistical thermodynamics formalism for bounded continuum systems.PSF or WDOS concepts can be used to obtain quantum size effect corrections to the thermody-namic quantities.Then, we can define the de Broglie density n dB , which is the reciprocal of λ dth . In d -dimensionsit is defined as n dB,d = λ − dth . This quantity is also called as quantum density or phase spacedensity in literature, but to make it distinguishable from other different quantities with similarnames, we call it de Broglie density, which is an appropriate terminology since it defines thedensity in the de Broglie volume. Additive nature of the lower dimensional terms is not due toan assumption, but directly comes from the exclusion of zero-energy states from the bulk term.The partition function tells us very important things. If we pay attention to the functionaldependencies, we see that the partition function is predominantly proportional with the bulksizes of the particular dimension ( ζ ∝ V in 3D, ζ ∝ A in 2D, ζ ∝ P in 1D). Also, thetemperature dependency changes according to dimension as ζ ∝ T / in 3D, ζ ∝ T in 2D and ζ ∝ √ T in 1D, so ζ ∝ T n/ in n D generally. In this regard, the partition function can alsobe interpreted as the number of small D-dimensional cubes with the size of thermal de Brogliewavelength that can fit into a confinement domain.
Both PSF and WDOS are useful mathematical tools to obtain quantum size effect correctionsto the thermodynamic properties of particles at nanoscale. However, they won’t provide deeperinsights on how and why these correction terms appear. For example, why they add up with al-ternating sign? What is the meaning of the constants that multiply each individual term? Thesequestions are answered in 2006, when the concept of quantum boundary layer has been devel-oped and used as a new method for the calculation and understanding of the underlying mech-anisms of quantum size effects. This method is crucial in the context of quantum statisticalthermodynamics in the sense that it makes it possible to obtain the thermodynamical quanti-ties with quantum confinement corrections without needing to explicitly solve the Schrödingerequation or use Weyl conjecture, practically leaving out the occasionally burdensome statisticalcalculations. 40 .3.1 Quantum thermal probability density distribution
Quantum thermal probability density distribution, describing the number density distributionof quantum particles at thermal equilibrium, lies at the heart of the quantum boundary layerconcept. Consider particles confined in a 1D domain. Due to their wave nature, particleshave tendency to stay away from the boundaries of the domain, generating a non-uniform den-sity distribution even in thermodynamic equilibrium. Quantum boundary layer (QBL) methodapproximates this non-uniform density distribution with a uniform one by introducing emptylayers on boundaries, thereby constituting an effective size. How this is done can be seen inFig. 2.12. Black curve represents the exact ensemble-averaged quantum-mechanical particlenumber density distribution, in short we call quantum thermal density , given by the followingexpression: n ( x ) = (cid:10) | Ψ( x ) | (cid:11) ens = (cid:80) i exp( − ε i β ) | Ψ i ( x ) | (cid:80) i exp( − ε i β ) (2.13)where | Ψ( x ) | is the quantum-mechanical probability density and β = 1 / ( k B T ) . True densitydistribution of particles is non-uniform; forms a plateau at the center and gradually goes tozero to the boundaries. After around 2 δ , the density becomes uniform, but it is non-uniformwithin around 2 δ thickness near to boundaries. This is a local information about the occupationprobabilities of the confined particles. On the other hand, in thermodynamics, we deal withthe global properties of matter. At this point, we can make an approximation by replacing thenon-uniform distribution with a uniform one around the center and completely empty layer nearto the boundaries, red-dashed curve in Fig. 2.12. This assumption gives us the possibility tostill define global properties but also consider the wave behavior of particles. Two variablesdetermine the thickness of the QBL, the height of the plateau (maximum density value) andthe total area under the curves which has to equal to unity due to the law of conservation ofprobability. By these two constraints, the thickness of QBL is obtained as δ = h √ πmk B T = λ th , (2.14)which is not so surprisingly related to the thermal de Broglie wavelengths of particles, and itturns out to be conveniently quarter of it.QBL contains the statistical and quantum nature of particles in an approximate and unifiedway, just like the effective mass concept, capturing the approximate influence of potential fieldson particles, in solid state physics. After the thickness is determined (which turns out to begeometrically universal ), QBL method is basically based on the replacement of apparent ge-ometric size variables with the effective ones. In the calculation of thermodynamic properties,this replacement allows to some extent eliminate the difference between the conventional inte-gral approach and the exact summations over the eigenvalues of the Schrödinger equation. Inother words, QBL constitutes a bridge between classical and quantum pictures of thermody-namic and transport expressions. Even the Weyl conjecture is not needed.Methodology of QBL is summarized in Fig. 2.12. The partition function consists of aninfinite sum of which calculation might be numerically cumbersome. By using the standard for-malism of statistical mechanics, sums can be replaced by integrals, with a minor trick where theactual volume is also replaced by the effective volume which can be found by QBL formalism.Rather than partition function, free energy could also very well be used for this transformation.The methodology is pictured more visually in Fig. 2.13. Classically, particles are uni-formly distributed inside the domain, whereas in confined domains at nanoscale, they form a41 igure 2.12: Quantum boundary layer (QBL) methodology in a nutshell. Ensemble-averagedquantum-mechanical density distribution of confined particles is non-uniform (black curve) andit can be approximated by a uniform one (red-dashed curve) via the introduction of QBL δ . QBLmethod can be applied to the calculation of any quantity X by a summation over a distributionfunction f . Quantum size effect corrections on thermodynamic and transport properties canexactly and analytically be obtained from their conventional (bulk) expressions just by replacingthe apparent sizes with the effective ones.non-uniform distribution due to the wave nature. Existence of a nearly empty layer near toboundaries can be explicitly seen also in the contour plot of the 2D domain’s density. The ap-proximation of the actual density with a uniform middle region and empty near-boundary regioncan be seen on the rightmost subfigure of Fig. 2.13. Although QBL is constructed by consid-ering the 1D domain, it is straightforwardly applicable to the higher dimensional domains likethe one shown in Fig. 2.13.In Fig. 2.14, calculation of the effective area in a 2D domain is shown. The 2D squaredomain has length L and QBL dictates a reduction in length by δ from every outside boundary.This makes the effective square domain’s length L − δ . If one calculates the effective area,it can be readily found that it is A eff = L − Lδ + 4 δ . Note that L is the periphery of thedomain and is the number of vertices, so A eff = A − δ P + δ N V . This is the origin of theconstants appearing in quantum size effect terms. To subtract the QBL area from the actualdomain, we first subtract the peripheral QBL areas ( δ times four side lengths of the square) butthen the square areas at the corners are subtracted twice which needs to be added to correct the Figure 2.13:
Dressing QBL to a 2D square domain and constructing the effective area. Approx-imating the non-uniform quantum density by an effectively uniform one via the introduction ofQBL. 42 igure 2.14:
An example showing explicitly how geometric size correction terms appear.calculation. This is the origin of the alternating sign in the corrections.By the QBL procedure, effective volume in d -dimensions is written as V eff ,d = V d − V qbl,d .More explicitly, effective volumes in various discrete dimensions are written as V eff = V − δ A + δ P − δ N V , (2.15a) A eff = A − δ P + δ N V , (2.15b) L eff = L − δ N V . (2.15c)which can be generalized by the expression V eff ,d = (cid:80) dn =0 ( − δ ) d − n V n for the d -dimensionaleffective volume. We can now apply the QBL procedure to find the partition function, which can be approximatedin d -dimension by the QBL method as ζ d = (cid:88) ε f ( ε ) ≈ ζ cl,d V eff ,d V d , (2.16)where ζ cl,d = V d /λ dth is the classical partition function expression in d -dimension, that can befound in textbooks. The classical partition function is found by direct conversion of summationinto integral.In terms of QBL thickness δ , partition function using QBL method can be written analyti-cally as ζ d = d (cid:88) n =0 ( − δ ) d − n (4 δ ) d V n = n dB,d V eff ,d (2.17)where n dB,d = (4 δ ) − d is d -dimensional de Broglie density and V eff ,d is d -dimensional effectivevolume. As is seen, partition function actually becomes the product of de Broglie density andthe effective volume. In this sense, partition function can be interpreted also as the number ofparticles occupying the effective volume with the de Broglie density.43q. (2.17) can be written in its open form for different dimensions: ζ = n dB, V (cid:18) − δ AV + δ PV − δ N V V (cid:19) , (2.18a) ζ = n dB, A (cid:18) − δ PA + δ N V A (cid:19) , (2.18b) ζ = n dB, P (cid:18) − δ N V P (cid:19) . (2.18c)where the de Broglie density becomes n dB,d = (4 δ ) − d . These are mathematically equivalentto the ones found by the WDOS method in subsection 2.2.3. Note that the expressions aren’tgeometry specific, i.e. they are valid for any geometry. If one can properly dress the QBL to adomain, one does not need to worry about the lower-dimensional geometric size elements. Theyemerge as a result of this dressing. Therefore, QBL method not only generates the same termsbut also underlies the existence of these terms in the equations. Note that all quantum size effectterms depend on the thickness of QBL which is of quantum nature. When Planck constant ishypothetically set to zero, they all disappear. In this regard, they are genuinely quantum. QBL is primarily defined by considering regular domains. Still, it is directly applicable toarbitrary domains and accurately predicts the thermodynamic properties with high precision.One can think of many arbitrary shapes. To examine, we choose three characteristic shapeswhich are significant for our discussion due to their unique properties. The first shape weconsider is a square domain having an infinitesimally small point/dot at its center. Classically,infinitesimally small points shouldn’t change any property of the system, because they don’teven have any size. However, due to the quantum nature of particles, they can feel the existenceof even an infinitesimal boundary where they form a quantum boundary layer. This can be seenfrom the density distributions given in Fig. 2.15. For comparison, we’ll execute the analysisby comparing our shape (II) with a plain square (I). Temperature is taken to be
K for all thecalculations in this subsection. Errors due to numerical calculations are ensured to be negligibleby the procedures explained in Appendix A.1.We calculate the partition functions for both shapes numerically (exact) first. We find morethan . difference between their partition functions. The difference might sound tiny atfirst, but when you think that this is caused just by a single point with zero thickness, thatdifference is not trivial at all. Then we calculate partition functions using QBL method. Thecalculation for plain square is easy as we have already shown before, ζ I = A I eff / (4 δ ) where A eff = 14 × − − × − δ + 4 δ . This expression gives less than − relative error,which is remarkably accurate (100 times smaller than the difference itself) for such a smalldomain size. A plain square was easy, but what about the square with a dot inside? Can QBLmethod provide accurate expressions for that as well? The answer is yes! The difference ofthis shape from a plain square is the creation of the infinitesimally small dot or a single vertex.But the tricky point is we shouldn’t add this vertex point to the number of vertices! We shouldsubtract. The reason is creation of a vertex inside the domain is quite different than creatingit on the boundary (such as corners). When you create a point inside of the domain (whichis considered as an internal boundary), you are reducing the effective sizes of the domain by44 igure 2.15: The first arbitrary domain example. Comparison of a plain square domain anda square domain with an infinitesimally small point at the center. Even an infinitesimal pointcreates considerable amount of change in the partition function and QBL methodology canpredict this with quite good accuracy. Blue and red colors represent the lower and higher densityregions.the amount of the space evacuated by its QBL. By keeping the self-similarity of the domain,we can approximate the evacuated area near the single point as δ . Removing one δ exactlymeans removing a single vertex point from the number of vertices. Then, the effective area in thepartition function of the square with dot shape becomes A II eff = 14 × − − × − δ +3 δ .This expression, not only provides physical explanations to the changes in partition function,but also relatively accurately represents it with a relative error around . .The second domain that we consider is the one shown in Fig. 2.16 where ITU (stands forthe abbreviation of Istanbul Technical University) letters are formed by infinitesimally thin lineboundaries inside a rectangle (constituting internal boundaries as opposed to the external oneswhich are the rectangle’s boundaries). We again numerically solve the Schrödinger equationfor this domain and find the partition function as ζ IT U = 15 . . Now let’s calculate whatQBL method gives. As we always do, we calculate the effective area to find the partitionfunction. So we need the surface area, periphery and number of vertices for this domain. Thesurface area is just the area of the rectangle, A = 27 . × . nm , since infinitesimal linesdon’t contribute to the surface area. Periphery calculation in this domain needs to be handledcarefully. As it is seen from the density distributions of particles in this domain, QBL of externalboundaries are calculated by periphery times δ , whereas QBLs of internal boundaries are twiceof the periphery times δ . This is because internal boundaries lead to the evacuation of particleson both sides of the boundary, as QBL is formed in both sides of internal boundaries, unlikethe external ones, Fig. 2.16. Therefore, QBL’s of internal boundaries have δ thickness (exceptwhen they are closer to boundaries than δ ). Periphery calculation in this domain then becomes: P = 2 × (27 . .
0) + 2 × (4 × . . . nm (vertical lines have 7.0nm, horizontallines have 5.0 and 4.5 nm lengths).We are almost done with the effective area calculation. But the trickiest part comes at thelast. How do we calculate number of vertices in this domain with both external and internalboundaries? Should we just sum them up? The answer is no! It is clear from the densitydistribution in Fig. 2.16 that QBL does not form evenly among the different vertices. For45xample around outer corners, QBL evacuates lots of space, whereas around inner corners (e.g.the bottom of the letter U) instead of an evacuation QBL is even invited to enter more closerinto the cusps. This is the case also for the open cusps (e.g. the ends of the letter I). Is it possibleto account these effects at once? Or do we have to deal with each and every case separately?Fortunately, we found an approximate but reasonable formula for the calculation of the effectivenumber of vertices in a confined system. The effective number of vertices in its general form iswritten as N V = (cid:88) n cot( a n / , (2.19)where a is an interior angle of the domain and the sum is over all interior angles of the domain.When we calculate the area of QBL covering a boundary, it is simply the subtraction ofvertex contribution ( N V δ ) from the periphery (that is exposed by the particles) times QBLthickness ( P δ ). Note that the vertices contribution can be positive or negative depending on theangle of the vertex. In case of an acute angle, vertices contribution is positive since it causesthe overlaps of QBLs at the vertex. On the other hand, it is negative in case of a reflex anglebecause of the reverse effect (filling rather than evacuation). The amount of overlap area at ◦ angle is exactly δ .The number of vertices formula is approximate because it assumes that the angle a n isbetween infinitely long boundaries. Due to that, it diverges at ◦ and ◦ angles. Nevertheless,we can calculate the effective number of vertices with quite good precision. There are eight of ◦ angles (4 corners of the rectangle, 2 inner angles of the bottom of U letter and 2 faced downangles of the top of T letter), two of ◦ angles (2 outer angles of the bottom of U letter) andthere are seven angles which are open (the ends of I and T letters and upper tips of U letter). Wecan treat these open cusps as ◦ , since the formula goes to infinity for ◦ . Then, the numberof vertices becomes N V = 8 cot(90 /
2) + 2 cot(270 /
2) + 7 cot(270 /
2) = − . This is value andother negative values for number of vertices are perfectly fine as they should be interpreted ashow the confined particles feel them. By using A , P , N V , we can find the effective area andthen the partition function as ζ QBLIT U = 15 . . The relative error of this result is around . ,which is very low, considering the approximations we did.The last domain we’d like to consider is even more complex, which is a tractricoid, seeFig. 2.17. Numerical calculations reveal that the partition function for the particles confinedin this domain is ζ trac = 8 . . It is easy to calculate the surface area and periphery forthis domain. For the number of vertices, it is clear that there are four of ◦ angles (cornersof the rectangle). Also, the bottom tip of the tractricoid is very close to the outer boundaryand evacuates all the space from particles around it. Therefore it can be thought as if it closesthe bottom by creating additional two ◦ angles. On the top and middle of the tractricoid,there are three very small capes. These are so small that, they can be treated as ◦ cusps. Figure 2.16:
The second arbitrary domain example. Application of the QBL methodology toan arbitrary domain. Unlike the previous domains that are considered, internal and externalboundaries affect the system differently. 46 igure 2.17:
The third arbitrary domain example. Application of the QBL methodology toan arbitrary domain. Unlike the previous domains that are considered, internal (defining thetractricoid) and external (defining the outer rectangle) boundaries affect the system differently.This is justified by also the density distribution profiles. Then, number of vertices become N V = 6 cot(90 /
2) + 3 cot(270 /
2) = 3 . With that the QBL approach gives ζ QBLtrac = 8 . .This value has a relative error of less than . .We’ve investigated three very distinct arbitrary domains and demonstrated the accuracy ofQBL method in predicting the partition function, from which all thermodynamic quantities canbe derived. Note that the partition function is a basic statistical property, which suggests QBLmethod can be used to predict any other statistical property such as transport properties fromthe kinetic theory. In this sense, QBL method is a quite general and accurate tool to predictquantum size effects on the statistical properties of confined particles.This analysis completes this chapter and our preparation for the subject. In the next chapter,we’ll enter to the main topic and the results of this thesis.47 The Nature of Quantum Shape Effects
After the necessary introduction and the review, we are now in position to start the thesis subject.In this chapter, we will introduce a new effect that we’ve recently discovered in nanoscaleconfined systems. As the name of the thesis suggests, we call it quantum shape effect . In thischapter, we’ll explore the essence of quantum shape effects by separating them from quantumsize effects. We’ll look at the eigenspectra of different confinement shapes and compare themwith the difference in eigenspectra due to size effects. After that, the shape dependence ofthe single-particle partition function will be examined. We’ll then advance the usual quantumboundary layer method to be able to predict the shape dependencies in an analytical way. Thedevelopment and implementation of the analytical method is one of the most essential partsof this thesis, because it gives also a physical understanding to the effect that we’ve proposed.We conclude this introductory chapter for quantum shape effects by presenting them in variousadditional confinement domains and examining their occurrence conditions via playing with thesizes.Influence of geometry manifests itself on the discrete energy spectrum of the confined sys-tem of particles, Fig. 3.1. In fact, it is somewhat natural to expect this, because the boundaryconditions are defined directly by the geometry of the system. On the other hand, as we haveseen before, different geometries can have the same spectrum: isospectral domains. Moreover,different energy spectra may possibly lead to the same statistical property, because of the aver-aging process that is inherent in the quantum statistical thermodynamics formalism. Therefore,the analysis of the influence of geometry in confined systems is by no means trivial and thephysics behind this phenomenon is actually very rich and interesting.
We have explored size effects in the previous chapter. We’ve identified and quantified themvery precisely by using geometric size variables, V , A , P , N V . Each size variable has a distinctrole on the thermodynamic properties of a confined system, for instance their sign alternates48 igure 3.1: A simplified picture of how geometry effects appear in general. When particlesconfined in very small domains, the differences in their discrete energy spectrum can lead tosubstantial variations in their quantum statistical properties.and their magnitudes are very different in contributions to the partition function. Let’s turn ourattention to the shape. We’ll question; How we can define the shape of an object? How doesshape differ from size? Are they inherently linked to each other? Is there any way to separatethem and focus on their individual effects? Do size and shape influences the thermodynamicsin a similar way or not?Although shape might be seen as a somewhat vague concept at first thought, there are rigor-ous mathematical definitions of it. Here we consider the shape as an object’s geometric infor-mation that is invariant under Euclidean similarity transformations such as translation, rotation,reflection and uniform scaling.
Left and right subfigures in Fig. 3.2 show the same anddistinct shapes respectively. On the left table, a triangle undergoes Euclidean similarity trans-formations which preserves its shape. Note that they do not necessarily preserve the size, e.g.in uniform scaling. On the right table, all shapes are different than each other, although someor all of their geometric size variables might be the same, e.g. in isospectral domains.Changes in geometric size variables like V , A , P and N V changes the size, but also theychange the shape of a particular object. Generally in literature, aspect ratios of sizes or plaingeometry of structures, curvature values, etc. are considered as indicators of the shape charac-teristics of a domain. However, when aspect ratios of a domain changes, size variablesalso change along with shape simultaneously. Similarly, a change in plain geometry (e.g. fromcube to sphere) cannot be done without changing geometric size variables of a domain, i.e. V , A , P and N V . In other words, in those type of processes size and shape effects are inherentlylinked and cannot easily be separated from each other. Even if there exist domains with distinctshapes having the same sizes (like isospectral domains), it is not quite possible to transformthem from one form to another by continuous boundary deformations while still preserving thesize-invariance. But what we would like to focus on is the pure shape effect. To do that weneed to be able to keep all the geometric size variables constant but still being able to change49 igure 3.2: Understanding the shape of an object. Left table shows the objects under Euclideansimilarity transformations that preserve the shape. Right table shows the objects having distinctshapes, regardless of their topological or spectral properties.the shape in a continuous fashion. We see that it is possible and rather easy.
It is possible to continuously change the shapes of confinement domains without changing thevalues of geometric size variables. The technique that we propose is called the size-invariantshape transformation and the procedure is as follows: Take a domain, for instance a 2D squaredomain as in Fig. 3.3. Then, remove a smaller region from inside and fill the remaining domainwith particles. Performing a rotation or translation on the inner region changes the shape ofthe confinement domain perceived by particles. This simple process keeps all geometric sizevariables constant during the rotation, while still changing the shape of the confinement domain.Therefore, this is a pure shape transformation. But, what is quantum about it? As we shall seelater on, in order to observe any difference on the physical properties of the particles due to thischange, we need quite strong confinements and the effect vanishes in the limit of λ th << L where L is a characteristic domain size (not to mention this notion can be directly generalizedinto any dimension so that λ dth << V d ). Just like quantum size effects, the quantum shape effectis a direct consequence of the wave nature of particles. Things will be more concrete when wecorroborate our discussion with the analytical methodology.We choose to focus on the rotational transformation in this thesis, as it preserves symmetrywith respect to the origin and thus simplifies the discussions. In rotational case, shape transfor-mation is characterized by the rotation angle of the inner square region, which we denote by θ .We choose the configuration II in Fig. 3.3 as θ = 0 ◦ and clockwise direction is chosen as thepositive angle. 50 igure 3.3: Procedure of the size-invariant shape transformation, creating differently shapedconfinement domains, without changing their sizes, out of nested objects. (I) Take a simpledomain filled with particles. (II) Remove a region from the domain, where the removed partbecomes a region that particles cannot penetrate. (III) Perform rotation or translation to theinner region. (IV) Resulting shape is different than the one before the transformation, whiletheir sizes ( V , A , P , N V ) remain unchanged. Domains’ shape can be changed size-invariantly, but what about their quantum statistical prop-erties? Does their energy spectrum even change or have we just discovered some new isospec-tral domains? In order to explore this, we need to solve the time-independent Schrödingerequation for different θ values corresponding to the different angular configurations shown inFig. 3.4. After some initial tests and considerations for the determination of system sizes, sidelengths of outer and inner squares of the confinement domain are chosen as L o = 21 . and L i = 13 . so that confinement parameter α of the domain is fixed to unity. For our 2Dnested square domain, the confinement parameter is calculated as α ∗ = ( √ π/ ∗ ( λ th /L ∗ ) where L ∗ = 2 A / P is the harmonic mean size of the nested domain in transverse direction. Theclosest distance between the outer and inner boundaries of the domain is called the apex lengthand its maximum and minimum values are . and . respectively occurring at θ = 0 ◦ and θ = 45 ◦ , for the chosen parameters. The variation of apex length (see Fig. 3.8) with therotation angle can be analytically found for the nested square domain as a = L o − L i θ ) + cos( θ )] . (3.1)For this particular shape the angular configuration ranges only from θ = 0 ◦ to θ = 45 ◦ , sincefrom θ = 45 ◦ to θ = 90 ◦ it is the mirror symmetry of the former and θ = 0 = nπ/ for anyinteger n value. Therefore, solving the 2D Schrödinger equation for the configurations between θ = 0 ◦ and θ = 45 ◦ is sufficient. Here, θ = 45 ◦ is called the symmetric periodicity angle of thisparticular domain. Analytical solution for these kinds of irregular domains is not possible, sowe solve the Schrödinger equation numerically using the finite element method. The rotationdiscretization steps are chosen to be small. The solutions are obtained with ∆ θ = 1 ◦ and51 igure 3.4: Solving the 2D Schrödinger equation for blue regions in the nested square confine-ment domain and finding the eigenvalues for each angular configuration.also with a higher resolution ∆ θ = 0 . ◦ . It is seen that results do not change and perfectlymatch with each other. The details of how we implement this numerical calculation are givenin Appendix A.1.Solving the 2D Schrödinger equation for the blue regions in Fig. 3.4 gives the energyeigenvalues of that particular angular configuration. We compare the energy spectra of θ =0 ◦ (red curve) and θ = 45 ◦ (blue curve) configurations in Fig. 3.5a and 3.5b respectively.Although the magnitudes of their energy spectra are quite similar, the values are different thaneach other, which can be more explicitly seen when we zoom in the first fifty eigenvalues,Fig. 3.5b. A relatively large difference in their ground states can also be seen. During theshape transformation, eigenvalues smoothly transforms from red curve to the blue one. Forcomparison we plot the eigenspectra of square confinement domains with different sizes; sidelengths L = 10 nm (orange curve) and L = 15 nm (purple curve), Fig. 3.5c and 3.5d respectively.This analysis allows us to compare respective influences of quantum size and shape effects inthe energy eigenspectrum. Quantum size effects more apparently affect the energy spectrum Figure 3.5:
Looking at the energy eigenspectra. (a), (b) Purely due to shape difference, differentangular configurations lead to difference in eigenspectra, which is more apparent near to groundstate. (c), (d) Difference in eigenspectra due to quantum size effects. The gap between spectrais much clear since energy is inversely proportional to length.52 |||||||||||||||| || || | | ˜ = ° |||| |||| ||||||||| || || | | ˜ = ° |||| |||| |||| |||| ||||| | | ˜ = ° |||| |||| |||| ||||||||| | | ˜ = ° |||| |||| |||| |||| ||||| | | ˜ = ° |||| |||| |||| |||| ||||| | | ˜ = ° |||| |||| |||| |||| ||||| | | ˜ = ° |||| |||| |||| |||| ||||| | | ˜ = ° |||| |||| |||| |||| ||||| | | ˜ = ° |||| |||| |||| |||| ||||| | | ˜ = ° Figure 3.6:
Emergence of spectral gaps. The first twenty eigenvalues in their dimensionlessform (normalized to k B T ) in a number line plot. The results are shown for the range between θ = 0 ◦ and θ = 45 ◦ with ∆ θ = 5 ◦ steps.because of the inversely proportional dependence of energy to the length (here in 2D shapesthe area; length squared). Conversely, shape difference has a much less apparent but definitelydistinct influence on the energy spectrum.Examination of the eigenspectra in number line plot gives more information about whatis going on in energy space. In Fig. 3.6, first twenty dimensionless energy eigenvalues areshown in a number line plot from θ = 0 ◦ to θ = 45 ◦ with ∆ θ = 5 ◦ steps. Note that energy levelspacing of eigenvalues are much less than k B T . As it can be seen, at θ = 0 ◦ the energy spectrumlooks roughly uniform. At θ = 5 ◦ , separation of ground state from the remaining spectrum isoccurring. At θ = 10 ◦ , the first and even the second excited states also separate themselves fromthe rest of the spectrum. Emergence of spectral gaps between eigenvalues become very muchapparent after θ = 20 ◦ . Note that in all subfigures of Fig. 3.6 the same number of eigenvalues(which is 20) are shown. The reason why they look like fewer is because there are degeneracieswhich are much more explicit especially after θ = 15 ◦ . Moreover, there is a slight decrement onthe magnitudes of energy eigenvalues on average, as spectral gaps emerge. We’ll see the reasonof this explicitly during the next section.To get even more intuition about how shape affects the spectrum we can plot the eigen-functions. In Fig. 3.7, first six eigenfunctions for θ = 0 ◦ (left subfigure) and θ = 45 ◦ (rightsubfigure) can be seen. Higher the amplitude the redder and lower the amplitude the bluer in Figure 3.7:
Behavior of the wavefunctions. First six eigenvalues and eigenfunctions for θ = 0 ◦ (left) and θ = 45 ◦ (right). Red color shows the positive amplitudes, blue color denotes thenegative amplitudes in the rainbow color spectrum and the green parts are zero.53he rainbow color spectrum. The zero value is denoted by the green color in the middle of thespectrum. The ground state eigenfunction is always positive everywhere, while excited stateeigenfunction can take positive and negative values. Energy eigenvalues in their dimension-less forms ( ˜ ε = ε/ ( k B T ) where T = 300 K) are given within the corresponding eigenfunctiongraphs. For θ = 0 ◦ configuration, energy eigenvalues are very close to each other, while thereis a huge gap between the fourth and fifth eigenvalues of θ = 45 ◦ configuration compared to the θ = 0 ◦ configuration. The difference between their ground state values is nearly double. Theonly degeneracy in the first six eigenvalues of θ = 0 ◦ occurs in second and third eigenvalues,whereas θ = 45 ◦ has a four-fold degeneracy in its first four eigenvalues with a two-fold degen-eracy following right after. These differences are really substantial ones and play significantroles in the variation of quantum statistical properties as we shall see next. We see that energy spectrum changes with the size-invariant shape transformation. The nextstep is to see how quantum statistical properties change. We do this analysis over the single-particle partition function having Boltzmann factor at room temperature, T = 300 K. We usebare electron mass for the particles and we can think of them as an ideal low density electrongas. This chapter constitutes more of a mathematical examination of quantum shape effects tounderstand their origins better.We obtained the eigenvalues for a 2D domain because the perpendicular direction to thetransverse directions does not affect quantum shape effects. Shape dependence originates fromthe particular 2D shape of the domain and that’s why in this chapter we focus only on 2Ddomain. As long as one ensures the ergodicity, one can use Maxwell-Boltzmann statistics evenfor a single-particle system, by considering many copies of the considered system. We willopen the third direction to contain large enough number of particles inside the system whilekeeping the density at low to satisfy the Maxwell-Boltzmann statistics, during the calculationof thermodynamic properties for the 3D structures in fourth chapter.After solving the Schrödinger equation repeatedly for each angular configuration and ob-taining eigenvalues, we use them to calculate partition function. Indeed, partition functionchanges with changing rotation angle of the inner region θ in Fig. 3.8. This is the first cleardemonstration of pure quantum shape effects: changing a quantum statistical quantity due tojust shape difference. Variation of partition function with respect to angle has a tanh func-tion (sigmoid) behavior. It has the lowest value at θ = 0 ◦ , gradually rises from there up until θ = 45 ◦ where makes its maximum. There is . difference between the minimum and max-imum of the partition function, which is a considerable change. Since this is a quantum effect,its magnitude also depends sensitively on temperature. The smaller the temperature, the largerthe quantum shape effect.The change in partition function suggests the changes in other thermodynamic quantities aswell, but their examination will be done in the next chapter. Now we will try to understand moreabout quantum shape effects by exploring their underlying mechanisms as well as to attemptanalytically predicting them. 54 igure 3.8: Variation of the partition function with respect to the rotation angle of the innersquare region θ . Temperature is taken to be room temperature, T = 300 K. Subfigure shows thepartition function normalized to its value at the initial angular configuration ( θ = 0 ◦ ). Domainparameters are shown in the right figure. L o , L i and a are the outer square length, inner squarelength and apex length at the symmetric periodicity angle respectively. How quantum shape effects work? Why does the partition function change in such a way andhow do we interpret its behavior? The best way to understand the mechanisms of quantumshape effects is trying to predict them analytically. Recall that quantum boundary layer methodwas giving very precise analytical results for quantum size effects. To see if it can help also inshape effects, let’s look at first the quantum thermal density distributions of particles confinedin our confinement domains. In Fig. 3.9a, quantum thermal densities of particles for fourdifferent angular configurations are plotted. It shows where the particles are most probablylocated and from where they stay away. Expectedly, particles stay away from the regions nearto all boundaries. It is clear by looking the reddish regions that the rotation of inner object from ◦ to ◦ causes particles to confine into four local regions than their initial closed-pipe-likeformation. During this transition, position of the boundaries of the inner object varies inside thedomain and it comes really close to the boundaries of the outer one. Remember that quantumboundary layer was leading to an evacuation of the regions from particles near to boundaries.If, during the rotation, quantum boundary layers of outer and inner objects overlap, this wouldlead to a double evacuation of those regions, which is not possible of course and needs to becorrected. After this initial logic, we tested our argument by dressing the quantum boundarylayer to our domain. As you can see in Fig. 3.9b, quantum boundary layers of outer and innerboundaries indeed overlap with each other for some degrees, e.g. for θ = 25 ◦ and θ = 45 ◦ inthe figure. In addition, existence and the amount of overlap changes with the rotation angle θ .We are looking for a parameter or a quantity to reveal the hidden information about the shape ofthe confinement domain analytically. We have overlap areas in our hands now. Let’s see if it cangive some idea about what’s going on in our confinement domain during the shape variation.Formation of overlapping areas can be seen in more detail in Fig. 3.10a. Gray regionsdenote the quantum boundary layers of outer and inner boundaries. After dressing the quantumboundary layer, effective boundaries arise which determine the effective area, cyan region. This55 igure 3.9: For four different angular configurations, (a) quantum thermal density distributionsof particles confined in our specially designed domain, (b) formation of the overlaps of outerand inner objects’ quantum boundary layers carrying information about the intrinsic shape ofthe confinement domain.is the region where the non-uniform density distribution of particles is replaced by a uniformone and particles effectively occupy that region only. As is seen, quantum boundary layersof outer and inner boundaries overlap with each other at this angular configuration, which aredenoted by the orange color. Normally, as we have seen in Chapter 2, the effective area of adomain is calculated by subtracting the area evacuated by the quantum boundary layer from theactual area, e.g. in 2D case: A eff = A − A qbl = A − δ P + δ N V . On the other hand, inthis particular case, where the quantum boundary layers of outer and inner boundaries overlap,the usual QBL procedure improperly subtracts these overlap areas twice from the actual area.Therefore, we need to add these overlap regions to the usual effective domain in order to find theproper effective domain. From now on, we denote the usual non-overlap versions of effectivesizes with superscripts as V eff , A eff , P eff or N V eff depending on the intrinsic dimension of thedomain. Then, the real effective area can be found as A eff = A eff + A ovr where A eff = A−A qbl .This we call the overlapped quantum boundary layer method which may be seen as an extensionof the usual quantum boundary layer methodology.More explicitly, for the selected nested square confinement domain, the effective area isexpressed in overlapped quantum boundary layer method as A eff = [( L o − δ ) − ( L i + 2 δ ) ] + A ovr = ( L o, eff − L i, eff ) + A ovr = A eff + A ovr , where the overlap area can analytically beobtained as a function of θ by using simple geometric relations as follows, A ovr = tan θ (cid:20) L i, eff (cid:18) θ (cid:19) − L o, eff sin θ (cid:21) . (3.2)We can calculate the accuracy of the analytically found A eff by comparing it with the onethat can be numerically found from A eff = ζλ th , where ζ is the exact partition function inwhich the numerically solved eigenvalues are used. This equation gives the effective area ifwe could have had the chance of calculating it exactly. The comparison of the accuracy of ouranalytical method in estimating the effective area is given in Fig. 3.10b where the variation of56 igure 3.10: (a) Quantum boundary layer is dressed to the confinement domain for a particularrotation angle. Cyan, orange and gray colors denote effective, overlap and excluded regions.(b) The amount of normalized effective area varies with the angular configuration. Purple andorange curves show the exact and analytical results respectively. θ ∗ denotes the angle afterwhich overlaps start. (c) Change in partition function with respect to the rotation angle. Exactand analytical results are represented by red and blue curves respectively.the normalized effective area with respect to θ is plotted. Purple and orange curves representthe results of exact (numerical) and analytical (overlapped quantum boundary layer method)calculations. As is seen, the overlap areas start to be formed after the critical value of therotation angle, θ ∗ which is found as θ ∗ = Round π arctan L o, eff (cid:113) L i, eff − L o, eff − L i, eff L i, eff − L o, eff . (3.3)Note that Eq. (3.2) is valid for θ ∗ ≤ θ ≤ ◦ and A ovr = 0 for ◦ ≤ θ < θ ∗ , as there is nooverlap in that range of angular configurations. For the sizes ( L o and L i ) and the temperature( T = 300 K) considered in this analysis, the critical angle corresponds to θ ∗ = 14 ◦ .Values in Fig. 3.10b suggest that effective area actually takes up less space than half of theactual area. In other words, excluded regions due to quantum boundary layers are larger thanthe effective areas occupied by particles. This shows us that the confinement is extremely strongin this domain with the chosen parameters. In fact, we need such strongly confined domains tomake quantum shape effects appear. Because on the contrary case, there won’t be substantialoverlap or any overlap at all to generate the quantum shape effects. This point will becomeclearer when we examine the quantum size effects on quantum shape effects at the end of thischapter.In Fig. 3.10c, accuracy of our analytical method is tested over the partition function. Themaximum and mean relative errors of analytical effective area expression are around and respectively, so it correctly predicts the functional behavior of the partition function withrespect to changes in θ . Our analytical method explains also why the partition function has asigmoid variation and increases with increasing shape effects. The partition function is directlyproportional with the effective area of the domain. As a result of this, it mimics the behaviorof the effective area. Since the angular transformation from ◦ to ◦ creates overlaps andincreases the effective area by a sigmoid-like fashion, the variation of the partition function hasalso the same behavior. This is a direct consequence of the chosen confinement geometry. Insome other geometries, one can expect different behaviors, but partition function will alwaysmimic the behavior of the effective areas (or volumes or whatever the effective bulk term isproportional). 57ote that both Weyl density of states and the first two terms of PSF cannot predict anyshape-dependent change in partition function. These methods fail to predict quantum shapeeffects. (Third term of PSF would include them, that term cannot be obtained analytically forarbitrary shapes.) On the other hand, our overlapped quantum boundary layer method pre-dicts and represents the quantitative and qualitative nature of the shape dependence of partitionfunction reasonably well. The discrepancy between the numerical and analytical methods mostprobably comes from the all-or-none nature of the quantum boundary layer method. A finerapproximation to the non-uniform density distribution might give more accurate results, whichis discussed in Appendix A.5. So far we’ve only tested our method in the nested square confinement domain. Many other typesof domains can be designed using the size-invariant shape transformation technique. Let’s ex-plore for instance nested triangle and nested rectangle domains. For a fair comparison wewant to choose our new confinement domains as close as possible to the nested square domainin terms of their confinement strengths. In this regard, we need to determine a parameter tocompare the confinements of arbitrary domains. The usual confinement parameter defined inChapter 2 in Eq. (2.4) cannot do the job, because it is defined only for regular rectangular do-mains and it is specific to a chosen direction. Conversely, in nested domains the geometry is socomplex that a single parameter cannot define the confinement of even a particular direction,let alone the whole domain. Nevertheless, we won’t need to be such precise in the determi-nation of the comparison parameter, after all we just try to make a fair comparison betweenthe thermodynamic properties of the domains. As long as we find a sound parameter and usethe same parameter in the comparison of different domains, it should be fine. To this end, thecharacteristic length is defined as the harmonic mean size of the nested domain in transversedirection, which is found by L ∗ = 2 A / P . Then, the characteristic confinement parameter fornested domains becomes α ∗ = ( √ π/ ∗ ( λ th /L ∗ ) . Characteristic confinement parameter of thenested square domain with the considered parameters is unity. For nested triangular domain,lengths of outer and inner triangles are chosen as L o = 23 . and L i = 9 . respectively.For nested rectangular domain, long and short sides of outer rectangle are L o,l = 16 . and L o,s = 8 . , of inner rectangle are L i,l = 6 . and L i,s = 2 . . By choosing theseparameters, we managed to fix the apex length at the symmetric periodicity angles to andconfinement parameters to unity, α ∗ = 1 for all domain types.Quantum thermal densities of nested triangular and rectangular domains are shown in Fig.3.11. Symmetric periodicity angles of nested triangle and rectangle domains are θ = 60 ◦ and θ = 90 ◦ respectively. During the rotation in nested square domain there was a formation offour-fold degeneracy in the ground state. In nested triangle and rectangle domains respectivelythree-fold and two-fold degeneracy are formed as expected, which can be inferred also from therightmost subfigures of Fig. 3.11.The formation of distinct degeneracies of energy levels for our considered confinement do-mains can be seen more explicitly in Fig. 3.12. Variations of the first twenty eigenvalues withrespect to rotation angles are plotted where y -axis shows the number and the color bar denotesthe magnitude of the dimensionless energy eigenvalues. Clear formation of lines in Fig. 3.12occurs at the multiples of 4, 3 and 2 for nested square (Fig. 3.12a), triangle (Fig. 3.12b) and58 igure 3.11: Dimensionless quantum thermal density distributions of nested triangular andnested rectangular domains.rectangle (Fig. 3.12c) respectively, providing a direct evidence of four-fold, three-fold andtwo-fold degeneracies from the eigenvalue spectra. While the signatures of degeneracies in theeigenspectra are nearly perfect for lower eigenvalues, deformations of the spectral lines occur athigher eigenvalues especially near the critical values of rotation angles. This analysis shows thatsome properties of the arbitrarily confined domains can be inferred directly from their eigen-spectra, as long as the domain has distinct geometric features such as the occurrence of localconfinement regions as in our domains.In Fig. 3.13, comparisons of the overlapped quantum boundary layer method with the ex-act numerical results for the estimation of effective area and partition function are shown fornested triangle and rectangle domains. The accuracy of our analytical method shows similarcharacteristic to the one in nested square domain, though the mean relative errors are doubledin this case. Amount of effective areas compared to whole domain is similar (around half) in allthree (nested square, triangle, rectangle) cases. It is seen that in all cases, estimation accuracyof the analytical method increases with the formation of degeneracies in energy levels, whereaserrors are large during the transition regions from non-overlap to overlap cases. Errors decreasewith the increments in overlaps. Because the increments of overlaps lead to the formation ofdisconnected subregions inside the domain, which makes the geometry simpler to tackle. These
Figure 3.12:
Formation of n -fold degeneracies in the eigenspectra. First 20 dimensionlessenergy eigenvalues ( ˜ ε = ε/k B T ) for (a) nested square ( n = 4 ), (b) nested triangle ( n = 3 ) and(c) nested rectangle ( n = 2 ) domains. 59 igure 3.13: Comparisons of numerical and analytical results for different domains. Variationof (a), (c) effective area and (b), (d) partition function due to changes in angular configurationfor the nested triangle and rectangle domains respectively.are noticeable from the apparent formation of the splits of high density regions for the configu-rations near the critical value of θ ∗ , see Fig. 3.11. This suggests that finer approximation to thequantum thermal density may increase the accuracy indeed. In our considered cases, quantum confinement conditions were extremely strong in terms oftwo ways; the domain sizes were really small and the boundaries were perfectly impenetrableso that the wave function was zero at the boundaries, i.e. the Dirichlet boundary conditions. Wecan also examine the opposite and intermediate cases to understand more about the nature ofquantum shape effects. The opposite case is imposing the Neumann boundary condition wherethe wavefunction has a finite value on boundaries but its derivative is zero.Before examining these conditions in our confinement domain, let’s compare the quantumthermal densities that are occurring in Dirichlet and Neumann boundary conditions in simple 1Ddomains, Fig. 3.14. This comparison allows us to see how quantum boundary layer transformsin Neumann case in comparison with the Dirichlet case. Unlike in the Dirichlet boundaryconditions, particles tend to stay close to boundaries rather than staying away from them. Infact, their behavior is the exact opposite of the Dirichlet conditions.In Fig. 3.15a and Fig. 3.15b partition function changing with the rotation angle is shownfor Dirichlet and Neumann boundary conditions respectively. The functional behavior remainsunchanged when the boundary conditions are changed from Dirichlet to Neumann, but themagnitude of the partition function increased because the domain is unconfined in the Neumanncase where the leakage of wavefunction from boundaries is at its maximum. In Fig. 3.15c(3.15d), mixed boundary conditions, outer boundary Dirichlet (Neumann) and inner boundaryNeumann (Dirichlet), are examined. Interestingly, for mixed boundary conditions the functionalbehavior of the partition function is reversed. The reason of this can be understood again byconsidering the overlaps of quantum boundary layers.60 igure 3.14:
Quantum thermal density distributions of particles confined in 1D domains withDirichlet (blue curve) and Neumann (red curve) boundary conditions at the both ends. Uniformclassical distribution is shown by the orange line. Particles evacuate (accumulate) near the (to)boundaries for Dirichlet (Neumann) boundary conditions.
Advent of quantum shape effects depends on the strength of the confinement and the strongconfinement is a prerequisite for the appearance of quantum shape effects. This fact becamemuch more clear with the investigation of quantum boundary layer overlaps. If outer and innerboundaries do not close enough to each other, overlaps won’t emerge and quantum shape effectsdiminish. At this point, it is reasonable to ask how quantum size effects influence the magnitudeand behavior of quantum shape effects.In this section, we choose two different methods to change the characteristic confinement ofthe domain and then compare the variation of two different angular configurations. Remember,the characteristic confinement was defined as α ∗ = ( √ π/ ∗ ( λ th /L ∗ ) which was giving arough estimate (the roughness comes from the fact that a 2D information is reduced into a singleparameter) of the information about the size-confinement of the domain. The first method ofchanging the characteristic confinement is keeping the outer square length fixed and changingonly the inner square length, see top-left subfigure in Fig. 3.16. It can be inferred even from theimage that size-invariant shape transformation’s influence on the shape of the domain decreaseswhen the confinement is reduced (from right to left).Variation of partition function by increasing the characteristic confinement this way can beseen in Fig. 3.16a. Red and blue curves represent the θ = 0 ◦ and θ = 45 ◦ configurationsrespectively. When the confinement increases two configurations start to differ. Note that theprevious examination of partition function in this chapter is done considering α ∗ = 1 value,which corresponds to the end points of the curves in Fig. 3.16a. When the confinement isdecreased, the difference between two most distinct configurations vanishes, see the subfigureshowing the percentage difference in Fig. 3.16a. This is because the domain becomes lessand less sensitive to the changes in inner object, when inner object size is so small. Imagineyourself as a particle traveling inside the domains of θ = 0 ◦ and θ = 45 ◦ configurations withconfinement α ∗ = 0 . and then compare your experience with the α ∗ = 1 one. In which oneyou will feel more difference between the θ = 0 ◦ and θ = 45 ◦ configurations? For instance,imagine you started from the bottom and you want to reach the top, there is a very narrowbottleneck at θ = 45 ◦ for α ∗ = 1 , whereas you will experience a more or less similar path in61 igure 3.15: Rows from top to bottom show the domains with different boundary conditions,variation of partition function with θ , quantum thermal density distributions and different typeof overlaps of quantum boundary layers. Columns from left to right show the results for thenested square domains with (a) Dirichlet, (b) Neumann, (c) outer boundary Dirichlet, innerboundary Neumann and (d) outer boundary Neumann, inner boundary Dirichlet boundary con-ditions. Partition function has opposite behavior in mixed boundary conditions, compared tothe Dirichlet and Neumann boundary conditions.62 igure 3.16: Influence of quantum size effect on quantum shape effect. Changing confinementvia changing inner square size by keeping the outer square size fixed (top-left) and changingouter and inner square sizes by keeping the apex length constant (top-right). Partition functionvs characteristic confinement for (a) top-left and (b) top-right confinement methods. The legendapplies to both figures.between θ = 0 ◦ and θ = 45 ◦ for α ∗ = 0 . .The other type of confinement variation is changing both the outer and inner square lengthstogether so that the apex length is kept constant, which is seen in the top-right subfigure of Fig.3.16. In this type of change, the self-similarity of the domain is preserved much better than theprevious one. As is seen in Fig. 3.16b, even for α ∗ = 0 . , the difference between ◦ and ◦ configurations is around (see the percentage difference subfigure), which was near to zeroin the previous confinement type for the same alpha value of 0.8. The percentage differencebetween the partition function values of ◦ and ◦ configurations gradually increases with theconfinement, but in a slow pace. By the way, it is also seen that the characteristic confinementparameter that we choose did a good job in comparing different type of confinements. Becausethe values of partition function are very close to each other despite the confinement type isdifferent in both pictures. This shows that the harmonic mean for calculating the characteristicsizes is a reasonable choice in these type of comparisons. We conclude this chapter here withthis analysis, and we’ll go into the thermodynamics of quantum shape effects in the next chapter.63 Quantum Shape Effects in NanoscaleThermodynamics
Last chapter we have seen that the partition function has distinct values per rotation angle ofthe inner object in nested confinement domains. In this chapter we’ll go further to examine thethermodynamic properties of the particles confined in domains that undergo size-invariant shapetransformation. We assume quasi-static processes all the time so that any change is happeningslowly enough to keep the system always in equilibrium with the environment. This allowsus to focus on the thermodynamic properties and their shape dependence in strongly confinedsystems. Throughout our analysis in this thesis, we use this assumption. Non-equilibrium andfinite-time processes are out of scope of this thesis. Even for equilibrium processes, as we shallsee, many undiscovered and fascinating novel thermodynamic behaviors are hidden to be comeout by quantum shape effects.
Thermodynamic state space of simple systems (consisting of unconfined particles in the absenceof external force fields) has two macroscopic degrees of freedom or two dimensions becauseclassically thermodynamic state functions have only temperature and volume dependencies (orany other set of two state variables). Quantum size effects add three new variables to this statespace. In addition to the volume, we have surface area, periphery and vertices dependencies,making the thermodynamic state space five dimensional. One can go beyond these dimensionsby keeping them constant and playing with another variable, and so dimension. Remember wehave changed the partition function of a system by keeping all these five degrees of freedomconstant, but changing only the shape which is characterized by the rotation angle θ . Thus,shape of a system becomes the new thermodynamic control variable. It can characterize ther-modynamic properties of a system independently of any other thermodynamic state variables.An example of this can be seen in Fig. 4.1 where θ plays the role of opening a new dimension64 igure 4.1: An example of a higher dimensional thermodynamic diagram. Shape variationadds a new dimension to the conventional temperature-entropy ( T - S ) diagram, making a three-dimensional thermodynamic temperature-entropy-shape ( T - S - θ ) diagram.in the conventional T - S diagram, thereby allowing the access of taking advantage from a three-dimensional space of thermodynamic states even if all size parameters are kept constant. Theresulting diagram is a sketch of a novel three-dimensional thermodynamic temperature-entropy-shape ( T - S - θ ) diagram. Similar diagrams can also be drawn for pressure-volume relationship,allowing to exploit the opening of a new dimension for the amount of possible extractable workfrom a system with the help of quantum shape effects.For a given number of particles in a system, thermodynamic state space now becomes six-dimensional, X ( T, V ) QSE −−→ X ( . . . , A , P , N V ) QShE −−−→ X ( . . . , θ ) , (4.1)where X is any thermodynamic state function with particle number N . Note that shape pa-rameter does not necessarily have to be θ . As long as a shape transformation is size invariant,the relevant parameter can be considered as the shape variable, e.g. coordinate positions inthe translational change of inner object instead of rotational. In translational changes it canbe even more complicated by adding many additional control variables to the thermodynamicstate space. Extension of thermodynamic state space paves the way for designing new ther-modynamic processes and cycles with interesting and useful properties, as we shall see in nextchapter where we discuss the applications. Thermodynamic analysis of confined systems relies on the methods of statistical physics. There-fore, we would like to construct a reliable model that we can characterize by the statisticalmethods. Our first and foremost condition is to satisfy the basic statistical conditions. To beable to perform a statistical analysis, we need to have a large number of particles inside theconfinement domain, so that
N >> . In fact, even for a single-particle system, one can usethe statistical methods because the ensemble averages would be equal to the time averages, as65ong as the system behaves ergodic. However, we will ensure the sufficiently large number ofparticles to safely use the methods of statistical mechanics and to cope with the fluctuations.Throughout our analysis, we choose the particle density as n = 5 × m − which gives theparticle number in our systems as N = 1012 , unless it is stated otherwise. Beside physicalconstraints, the reason why we don’t want to choose a much larger particle number is that forour initial analysis, we would like to use the Maxwell-Boltzmann particle statistics, due to thereasons of its simplicity in expressions and possibility of getting analytical results easier. Be-sides, understanding of quantum shape effects in Maxwell-Boltzmann statistics will shed lightinto its understanding under other statistics.Maxwell-Boltzmann statistics is used to describe the average distribution of particles on en-ergy states in thermal equilibrium for high temperature and low density conditions. These con-ditions can be quantified by comparing the classical density of the particles inside the domainwith their de Broglie densities. Classical density ( n cl = N/ V ) should be much smaller than thede Broglie density, n cl << n dB , for Maxwell-Boltzmann conditions to be satisfied. Increas-ing the temperature decreases the de Broglie wavelengths of particles which causes a higherde Broglie density. Similarly, decreasing the density basically means smaller n cl , which againenlarges the gap between the classical and de Broglie densities, therefore contributing to the n cl << n dB condition. Since volume is constant throughout our analyses, decreasing the den-sity corresponds to decreasing the number of particles. If n cl << n dB condition is not satisfied,indistinguishability property of particles becomes prominent, so Bose-Einstein or Fermi-Diracstatistics needs to be used depending on the symmetric or antisymmetric nature of the particles’wavefunctions respectively under a particle exchange. In this regard, Bose-Einstein or Fermi-Dirac statistics are inherently quantum-mechanical and called together as quantum statistics.This terminology sometimes is extended to call the Maxwell-Boltzmann statistics as classicalstatistics. However, we think this is not quite appropriate terminology as it may cause reader notto expect any quantum behaviors in the systems obeying Maxwell-Boltzmann statistics, whichis definitely not true. Although the statistics itself is derived using the classical mechanics, it’sjust an asymptotic case of the quantum statistics and quantum nature of particles can still be in-corporated to the system by their discrete eigenspectrum that is determined by the Schrödingerequation.We can only satisfy n cl << n dB condition, at room temperature with bare electron mass,by increasing the sizes of at least one direction. We extend the domain in the third directionlongitudinally (see Fig. 4.2) without making any changes in the transverse directions whichwill be responsible from quantum shape effects. By this way, we fill the domain with manyparticles. Adding the third dimension not only gives us possibility to satisfy the conditions ofMaxwell-Boltzmann statistics, but also makes our discussions more realistic, since all materialsare intrinsically 3D. Moreover, there are some nanoarchitectures, e.g. core-shell nanostructures,which are suitable candidates for the demonstration of quantum shape effects and very close toour models. The length of the longitudinal direction is chosen as L l = 762 . . If we calculate theconfinement parameter of the longitudinal direction at room temperature, it gives α l = 0 . ,which means the third direction is nearly free (unconfined). Having at least one direction un-confined is in fact necessary to be able to satisfy the conditions of Maxwell-Boltzmann statisticsas it is impossible to do it with all directions confined. Nevertheless, the shape of the domain ispreserved in transverse direction, so quantum shape effects imposed by the transverse directioneigenvalues are still there. 66emember we have solved the Schrödinger equation for 2D domain in the previous chapter.We changed our domain into a 3D one now, so we need to solve the Schrödinger equation forour new 3D confinement domain. However, this is actually an unnecessarily hard task in termsof computational point of view. The first difficulty is we need a lot more mesh points in a 3Ddomain, which will increase the numerical solution time substantially compared to 2D case.The second difficulty is the truncation point of the summation over longitudinal momentumvalues would be much higher than the 2D transverse case, which means we have to calculatemuch more eigenvalues.Fortunately, we don’t need to worry about any of these numerical difficulties, because choos-ing a relatively long length in longitudinal direction gives us possibility to obtain analyticalexpressions without solving the Schrödinger equation explicitly for that direction. Using theorthogonality of eigenstates and the quadratic energy-momentum dispersion relation as well astaking the advantage from the product rule of exponents in Maxwell-Boltzmann distributionfunction give us opportunity to separate the partition function into two parts as numerical andanalytical and calculate each part individually. By this way, the partition function for our 3Ddomain can be captured as ζ = (cid:88) k exp ( − ˜ ε k ) ≈ (cid:18)(cid:90) exp ( − ˜ ε l ) di l − f l (0)2 (cid:19) (cid:88) t exp ( − ˜ ε t )= (cid:18) √ π α l − (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) analytical (cid:88) t exp ( − ˜ ε t ) (cid:124) (cid:123)(cid:122) (cid:125) numerical (4.2)where energy eigenvalues are represented in their dimensionless form as ˜ ε k = ε k β = ε k / ( k B T ) with k representing the 3D domain eigenvalues (which are hard to obtain with acceptable preci-sion in a reasonable time, but as we shall see there is no need for that). ˜ ε l and ˜ ε t are the dimen-sionless energy eigenvalues for 1D longitudinal direction and 2D transverse directions respec-tively. Since the structure in longitudinal direction is generated by a straightforward extension ofthe structure in transverse direction, the translational energy eigenvalues are analytically knownfor the longitudinal direction, which are in their dimensionless form ε l / ( k B T ) = ˜ ε l = ( α l i l ) where i l is the quantum state variable for momentum component in longitudinal direction. Then,the summation of longitudinal direction can be analytically approximated by the first two terms Figure 4.2:
Addition of the third dimension and turning the domain into a core-shell nanowire.The outer (shell) structure is fixed and the inner (core) structure is rotatable.67f PSF, which results a very simple expression shown within parentheses in Eq. (4.2). Althoughthe usual continuum approximation (the first term in the parenthesis) would also work for thelongitudinal direction, we will use the safer one and consider also the quantum size effect cor-rections coming from the nearly free direction. The numerical part has already been solved inthe previous chapter, so the partition function is successfully constructed.Note that our approach to finite-size thermodynamics is different than Hill’s approach. InHill’s nanothermodynamics, the thermodynamic limit is taken and the system is partitioned intoan ensemble of identical subsystems.
In that case, extensivity is broken due to the inter-actions of subsystems via the subdivision potential.
In non-interacting case, the subdivisionpotential is associated with the addition or removal energy cost of the subsystems.
It is notpossible to predict the existence of quantum shape effects by using Hill’s nanothermodynamicapproaches, because it is a limited approach where the thermodynamic limit is used and finite-size corrections are manually added into the expressions of continuum approximation. On theother hand, in our approach we are not using thermodynamic limit at all, since the volumeand number of particles are both finite in our system. We use the fundamental thermodynamicexpressions in their "summation-over-all-states" form, which are coming directly from the sta-tistical mechanics framework via the entropy maximization under constant number of particlesand energy. The information about the finiteness of the system is embedded within the eigen-states that we considered in our distribution functions. The thermodynamic expressions thatwe used are true independently from the fact that whether the system is infinite or finite. Ex-tensivity in our case is naturally (i.e. without manually adding any terms) broken due to thequantum-mechanical corrections of non-existing zero-energy states.We won’t examine the partition function again in this chapter, because adding the longi-tudinal direction does not change the analysis of the shape dependence of partition function.Now it’s time to explore the thermodynamic properties of the particles confined in our nesteddomains.
When a system is in thermodynamic equilibrium, we can define the thermodynamic state func-tions such as internal energy, free energy and entropy. Thermodynamics is concerned with thechanges in state functions, rather than their absolute values. Nevertheless, we give the normal-ized values of each thermodynamic state function rather than their relative values, to providean insight about their magnitudes in different systems. Their relative values can also be easilyseen from the figures that are given. Each thermodynamic state function is normalized by either
N k B T or N k B depending on the appropriate normalization factor. We used per particle nor-malization to prevent unnecessary amplification of magnitudes by the particle number which ischosen to be not so small.The primary thermodynamic quantity we would like to investigate is the internal energy,which is the energy of a system associated with the average random motion of its microscopicconstituents. For an ideal monoparticular gas, the internal energy consists of only the transla-tional kinetic energies of particles. The internal energy of a system with N identical particles isdefined in statistical mechanics as U = N (cid:88) k p k ε k (4.3)68here p k is the probability of the system occupying the momentum state k , which is given bythe probability density function p k = exp ( − ˜ ε k ) (cid:80) k exp ( − ˜ ε k ) = f k (cid:80) k f k = f k ζ (4.4)where f k = exp ( − ˜ ε k ) is the Maxwell-Boltzmann distribution function. Again, this becomesseparable because the product rule of exponents can be exploited in Maxwell-Boltzmann statis-tics. The partition function here serves as a normalization constant of the probabilities, ensur-ing that they add up to one. Putting Eq. 4.4 into Eq. 4.3 shows that the internal energy isprobabilistically the ensemble averaged expectation value of single particle energy eigenvalues, U = N (cid:104) ε k (cid:105) ens . Thermodynamically, there are two types of energy exchanges between a sys-tem and its environment: the work exchange (organized form of energy) and the heat exchange(unorganized form of energy). The change in internal energy corresponds to the sum of workand heat exchanges by the first law of thermodynamics ( dU = d ¯ W + d ¯ Q ).Internal energy is related also with the confinement of the domain. When the confinementincreases, energy eigenvalues shift up on the spectrum (due to their inverse proportionalitywith size) and since the internal energy is basically their ensemble average, the internal en-ergy increases as well. Increase in internal energy under an isothermal process is also calledconfinement energy, since it is solely due to the confinement.Similar to the partition function, internal energy can also be decomposed (thanks to MBstatistics) into the numerical and analytical parts as follows, UN k B T = (cid:88) k p k ˜ ε k = (cid:80) k f k ˜ ε k (cid:80) k f k = (cid:80) t (cid:80) l f t f l (˜ ε t + ˜ ε l ) (cid:80) t (cid:80) l f t f l (4.5a) ≈ (cid:80) t f t ˜ ε t (cid:0)(cid:82) f l di l − f l (0) / (cid:1) + (cid:80) t f t (cid:0)(cid:82) f l ˜ ε l di l (cid:1)(cid:80) t f t (cid:0)(cid:82) f l di l − f l (0) / (cid:1) (4.5b) = (cid:80) t f t ˜ ε t (cid:80) t f t (cid:124) (cid:123)(cid:122) (cid:125) numerical + 12 − α l / √ π (cid:124) (cid:123)(cid:122) (cid:125) analytical (4.5c)where f l = exp ( − ˜ ε l ) and f t = exp ( − ˜ ε t ) are the distribution functions for longitudinal andtransverse directions respectively. For a better representation than the continuum approxima-tion, we always use bounded continuum approximation (using the first two terms of PSF) forthe analytical parts. Note that sometimes the second term can give exactly zero (e.g. in theenergy integrals in Eq. 4.5b), in those cases the bounded continuum approximation does notlead to any improvement.Change in internal energy with rotation angle θ is shown in Fig. 4.3 by green curve forthree different nested confinement domains, calculated from Eq. (4.5c). Dashed-black curvesshow the results of analytical calculation by the overlapped quantum boundary layer method.Internal energy of the system smoothly decreases from θ = 0 ◦ to θ = 45 ◦ in a sigmoid-likefashion. During this variation, all physical parameters like number of particles, temperatureof the system as well as all geometric sizes are constant, except the rotation angle of the corestructure.The shape-dependent behavior of internal energy is very similar in all three examined nestedconfinement domain cases. As an indicator of the reasonable selection of characteristic confine-ment parameter, the internal energy values for all cases are quite close to each other. Internal69 igure 4.3: Variation of internal energy (normalized to
N k B T ) with respect to the shape changevia θ , for (a) nested square, (b) nested triangle and (c) nested rectangle domains. Green anddashed-black curves represent the exact (Eq. 4.5) and analytical (Eq. 4.6) calculations respec-tively.energy decreases with the rotation angle of the core structure. Higher internal energy means thatthe system is effectively more confined in higher internal energy states. The reason of this be-havior can be very well understood under the overlapped quantum boundary layer framework.The amount of overlap increases with the rotation angle and increase in overlap volume (sincethe domain is 3D, now we have overlap volumes all along the longitudinal direction instead ofoverlap areas which was in the 2D case). More overlap volume means there are more availableeffective volume inside the domain which can be occupied by particles and so the domain is ef-fectively less confined. Note that now we are talking about an effective confinement instead ofan apparent confinement, because the confinement is initially defined to be depend only on thesizes of the domain but not its intrinsic shape. Keeping the sizes constant means the confinementdoes not change. On the other hand, effectively the space occupied by particles increases due tothe overlap regions and so contributing to the decrease in confinement and internal energy. Thisdecrease of internal energy explicitly shows deconfinement just by change of shape.We realize that the dependency of quantum size and shape effects on the confinement ofthe system is quite opposite of each other. Both effects appear under confinements of particles.However, unlike quantum size effects, appearance of quantum shape effects leads to a decrementin the effective confinement of the domain. This is because quantum shape effects are causedby the overlap volumes which are contributing to the relaxation of the effective confinementby increasing the effective volume inside the domain. This behavior is evident also from thedecrease in internal energy due to quantum shape effects. Overlaps of quantum boundary layersprovide a nice physical explanation for this relaxation of confinement due to quantum shapeeffects.The analytical expression that is obtained for internal energy to plot dashed-black curves isgiven by the overlapped quantum boundary layer method as U A N k B T = (cid:18) (cid:19) bl + (cid:32) − T V ∂ V qbl ∂T (cid:33) QSE + (cid:32) V T V eff ∂∂T V ovr V (cid:33) QShE . (4.6)where A subscript denotes that the expression is completely analytical. The first term on theright-hand side represents the bulk term, as denoted by the subscript bl , which is a result fromcontinuum approximation. The second term is the quantum size effect term that is characterizedby the excluded quantum boundary layer volume ( V qbl ) and the conventional effective volume70 . It gives the usual quantum size effect corrections to the internal energy of Maxwell-Boltzmann gases. The third term is the quantum shape effect term, the only term that dependson the overlap volume and the shape characterization parameter θ . Thanks to the mathematicalstructure of the Maxwell-Boltzmann distribution, it is possible to express classical, quantumsize effect and quantum shape effect terms explicitly and separately from each other. Note thatthis additivity of quantum size and shape effect terms are not an assumption but directly comesfrom the nature of MB statistics. Bulk term coming from the continuum approximation incor-rectly calculates surface modes as well. The quantum size effect corrections actually fix thismiscalculation by excluding the zero energy modes. Surface modes are quantum mechanicallyforbidden because they correspond to i = 0 states. On the other hand, in bounded systemsstates start from a non-zero value, the ground state value, which is i = 1 . Additive quantumsize effect term naturally corrects this by use of PSF or Weyl formulas. This is very convenientfor the examination of distinct features of quantum size and shape effects.In addition, since we consider a single-particle picture (not the many-body picture), weassume no quantum correlations between the wave functions of the particles. All states arecompletely independent from each other. Hence, all the sums are diagonal. This also justifiesthe additive nature of the quantum size and shape effect corrections which are semi-classical,not genuinely quantum thermodynamic.It is also worth to mention that averaged internal energy of particles in our confinementdomains is much higher than its classical value based on continuum approximation which is3/2. Quantum size effects are responsible from this confinement energy, quantum shape effectsjust cause a reduction of it. In fact, if we wouldn’t have considered the overlap volumes, thedashed black curves in Fig. 4.3 would have stayed constant on their ◦ values (with no overlap)which corresponding to the first two terms of Eq. (4.6).Lastly, the maximum and mean relative errors of the analytical internal energy expressionare respectively . and . for nested square, . and . for triangle and . and . for rectangle domains. We give both the maximum and the mean relative errors in orderto provide a more detailed information about the nature of errors which speak for the accuracyof the analytical expressions that we derived. Investigation of internal energies of particles inside our nested confinement domains allowsus to learn interesting things about the nature of quantum shape effects. Now we move onto another important thermodynamic potential which is the Helmholtz free energy (we simplycall free energy). Under constant temperature condition, change in free energy equals to thetotal reversible work exchange (means the maximum possible total work exchange). Underconstant temperature as well as pressure conditions, on the other hand, the related quantity isthe Gibbs free energy in which the changes represent the maximum non-mechanical work (thework exchange except expansion/contraction process).For N number of identical particles obeying the Maxwell-Boltzmann statistics, free energyis written as F = − k B T ln Z (4.7)where Z = ζ N /N ! is the N -particle partition function. Here it is assumed that single-particlepartitions are independent and the indistinguishability of particles are ensured by N ! . Although71his is an approximation, as we shall see later during the comparison of results with the onesusing quantum statistics (FD in particular), it is a good approximation for the systems that areconsidered here. Eq. (4.7) then can be rewritten as F = − N k B T ln ζ + k B T ln N ! . Usuallythe term ln N ! is approximated by Stirling’s approximation as ln N ! ≈ N ln N − N , but wewon’t use this approximation since the calculation is already easy in its exact form. Like wehave done in internal energy expression, we can write the free energy as the sum of numericaland analytical parts, FN k B T = − ln (cid:34)(cid:88) t f t ( f l di l − f l (0) / (cid:35) + ln N ! N = − ln (cid:32)(cid:88) t f t (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) numerical − ln (cid:18) √ π α l − (cid:19) + ln N ! N (cid:124) (cid:123)(cid:122) (cid:125) analytical (4.8)where the numerical part consists of the logarithm of transverse direction partition functionwith negative sign and the analytical part contains both the longitudinal counterpart and thecontribution due to the indistinguishability of particles.Nature tends to minimize the free energy. Therefore, free energy specifies also the directionof a spontaneous thermodynamic transition from one state to the other. Examining the free en-ergy of our system under the shape variation allows us to determine which angular configurationminimizes the free energy of the system and so to which configuration the system "wants" totransform to become stable. The variation of free energy with respect to θ is shown in Fig. 4.4.Free energy of all systems decreases with increasing rotation angle or in other words increasingoverlap volumes (minimizing the effective confinement). This means nature favors the exis-tence of quantum shape effects. The reason is quite clear from the perspective of the overlappedquantum boundary layer theory. Minimization of free energy is equivalent to the maximizationof the effective volume (minimization of the effective confinement) under constant temperature.Increment in overlap volumes means there are effectively more available space to be occupiedby particles. Nature prefers to increase the overlap volumes to minimize the free energy.The minimum free energy configurations correspond to the symmetric periodicity angles ofthe domains ( θ = 45 ◦ , θ = 60 ◦ and θ = 90 ◦ for nested square, triangle and rectangle domainsrespectively). Consider the outer shell is fixed and the inner core is free to rotate. Assuming zero Figure 4.4:
Variation of free energy (normalized to
N k B T ) with respect to the shape change via θ , for (a) nested square, (b) nested triangle and (c) nested rectangle domains. Blue and dashed-black curves represent the exact (Eq. 4.8) and analytical (Eq. 4.9) calculations respectively.72riction and neglecting the interactions, if the initial state of a system is prepared in a config-uration other than the minimum free energy one, the inner structure would spontaneously startto rotate until it reaches to the symmetric periodicity angle. Even an external torque needs tobe applied to keep the inner structure stable under unstable angular configurations. In short, therotating thing will rotate! Though it seems counter-intuitive at first, this spontaneous movementof course does not violate any physical law. Configurations except the minimum free energy oneare actually unstable, despite the existence of thermodynamic equilibrium. A classical analogof this situation can be seen in the example of squeezing a spring. When you squeeze a spring,you store potential energy which can turn into kinetic energy when you release it. Similarly,preparing the confinement domain in a configuration other than the symmetric periodicity angleputs system into an unstable state, provided that the inner structure is free to rotate without anyfriction.By using the overlapped quantum boundary layer method, fully analytical expression forfree energy can be obtained as F A N k B T = (cid:18) − ln V N (4 δ ) − (cid:19) bl + (cid:20) − ln (cid:18) − V qbl V (cid:19)(cid:21) QSE + (cid:34) − ln (cid:32) V ovr V (cid:33)(cid:35) QShE , (4.9)where the first term is the classical free energy expression, while second and third terms corre-spond to the quantum size and shape effects corrections respectively. Just like size effect terms,shape effect terms also cause a breakdown of extensive nature of the thermodynamic state func-tions. Dashed black curves in Fig. 4.4 are the results of this expression in Eq. (4.9). It capturesthe true behavior of free energy with respect to a shape variation very well. The maximum andmean relative errors of the analytical free energy expression are respectively . and . fornested square, . and . for triangle and . and . for rectangle domains.Existence of free energy variation suggests that there should be some mechanical forcesacting and a torque that must be generated inside the confinement domain. We will explore thistorque along with its quantum-mechanical origins in Sections 4.7 and 4.8. Before going intothat, we will examine another very important thermodynamic property of our systems, calledentropy. Entropy is one of the primary concepts of thermodynamics and in fact of all physics, and itmay be difficult to grasp as there are several definitions, explanations and even derivations of it.Entropy is an extensive measure of the number of microscopic configurations corresponding tothe macroscopic state of a thermodynamic system.In Maxwell-Boltzmann statistics, entropy reads S = − N k B (cid:88) k p k ln p k (4.10)Putting Eq. (4.4) into the above equation generates the terms in internal energy and free energyin a way: S = U/T − F/T , indeed it is true that entropy is written by taking the differenceof Eq. (4.5) and Eq. (4.8). We avoid rewriting all these terms again for brevity. In Fig. 4.5,73ntropy of the confined system with respect to the rotation angle is plotted for nested square,triangle and rectangle domains. In all cases, entropy has a decreasing behavior with respect tothe starting position of the inner structures, which is θ = 0 ◦ for all cases. We have seen thedecreasing behavior of free energy as well during the previous free energy section. The sponta-neous decrease of both free energy and entropy is very interesting and a unique behavior in thethermodynamics of ideal gases. It even almost somewhat counter-intuitive at first thought. Thedirection of thermodynamic processes, which is determined by the minimum of free energy,is usually associated with the higher entropy state. Moreover, quantum shape effects cause aneffective expansion of the domain via overlap volumes and entropy should, in general, increasein the direction of expansion. The larger the occupiable space, the larger the number of micro-scopic possibilities and larger the entropy. On the other hand, this logic seems to be not workingin systems exhibiting quantum shape effects.In order to understand what’s going on, we need to look closer to the thermodynamic statefunctions of the system. From the initial configuration angle to the symmetric periodicity an-gles, not only free energy (Fig. 4.4) and entropy (Fig. 4.5) of the system decreases, but also theinternal energy (Fig. 4.3) decreases. In fact, the decrease in internal energy is larger than thedecrease in entropy and this is why free energy also decreases. Since the decrement in inter-nal energy of the system is larger than the decrement in its entropy, free energy must decreaseduring a shape transformation.Recall that we are dealing with a system that is in contact with a heat bath so that theprocesses are isothermal. This means, in order to keep the temperature constant during theprescribed shape transformation the system has to exchange heat with the environment. Indeed,the decrease in internal energy is due to the heat loss of the system to the environment. Thesystem decreases its entropy, by transferring it to the environment in a heat transfer process.Total entropy of system plus environment is unchanged as we assume the processes are quasi-static and so reversible. Therefore, everything is perfectly consistent also with the second lawof thermodynamics.In fact, there are examples of this phenomenon, spontaneous decrease of entropy, in nature.A perfect example is the formation of a snowflake. A snowflake has much lower entropy thana liquid drop of water, yet it forms spontaneously. At temperatures below the freezing point ofwater, the rate of energy reduction overcomes that of entropy, resulting a lower free energy statefor a snowflake than the liquid water. Thus, nature favors the formation of a snowflake. Con- Figure 4.5:
Variation of entropy (normalized to
N k B ) with respect to the shape change via θ , for(a) nested square, (b) nested triangle and (c) nested rectangle domains. Red and dashed-blackcurves represent the exact (Eq. 4.10) and analytical (Eq. 4.11) calculations respectively.74ersely at higher temperatures, entropy term overcomes again and frozen water turns into theliquid water. The crucial point here is that nature does not follow the direction of increasingentropy but rather decreasing free energy.The reason why this is happening in our systems under quantum shape effects can be under-stood through the analytical analysis of entropy. Analytical expression of entropy is obtainedby overlapped quantum boundary layer method as S A N k B = (cid:18) ln V N (4 δ ) + 52 (cid:19) bl + (cid:34) ln (cid:18) − V qbl V (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) S IQSE − T V ∂ V qbl ∂T (cid:124) (cid:123)(cid:122) (cid:125) S IIQSE (cid:35)
QSE + (cid:34) ln (cid:32) V ovr V (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) S IQShE + V T V eff ∂∂T V ovr V (cid:124) (cid:123)(cid:122) (cid:125) S IIQShE (cid:35)
QShE (4.11)In Fig. 4.5, comparison of the numerical and analytical results, which are in very good agree-ment, can be seen. The maximum and mean relative errors of the analytical entropy expressionare respectively . and . for nested square, . and . for triangle and . and . for rectangle domains.Unlike in free energy and internal energy, both quantum size and shape effect correctionsconsist of two distinct terms in entropy expression. Let’s examine these terms individually tounderstand their contributions to entropy. This will allow us also to understand the behaviorsof internal and free energies better, as they contain the same terms, compare Eqs. (4.6), (4.9)and (4.11). The first term in the analytical entropy expression in Eq. (4.11) is the classical termwhich is also known as Sackur–Tetrode equation. The second term (denoted by QSE subscriptand within the square brackets) is the QSE correction to entropy and it has two distinct terms.The first term of QSE correction to entropy is exactly the same term that free energy has onlywith the opposite sign as expected. It has a negative contribution to entropy because V qbl / V is always smaller than one. The other QSE term in entropy expression is exactly the sameterm that internal energy has as a QSE correction. This term has a positive contribution to theentropy because the derivative of V qbl with respect to temperature is negative. All in all, bothterms contribute and the end result is the decrement of entropy due to QSE terms for the giventemperature and sizes.There are also two shape-dependent terms in entropy expression, which are competing witheach other on the determination of the overall behavior. Again, the first shape-dependent term isthe free energy correction with the opposite sign. Its contribution to entropy is always positivebecause of V > V ovr by definition. Hence, this term does not cause any unconventionalbehaviors in free energy and entropy. In other words, when effective volume increases, entropyalso increases and free energy decreases. The second shape-dependent term, on the contrary,contribute to entropy negatively. Since increasing temperature reduces the thickness of QBLsand cause overlap volumes to diminish, V ovr is inversely proportional with temperature and sothe contribution of S IIQShE term is negative. At this point it should be stressed that these terms arenot entropy but size/shape corrections inside the genuine entropy expression which is alwayspositive. The S notation is for convenience.The unusual behavior of entropy directly comes from this second QShE term. Of coursein overall, S IQShE and S IIQShE terms compete with each other (as one has positive, the other has75egative sign). However, the course of competition is determined by the second QShE term. S IIQShE is always larger than S IQShE within the spanned range of θ , so its functional behavior hasa larger effect on the sum of two. For the nested rectangle domain, however, S IIQShE decreasesits decreasing trend (so the derivative) with θ after around θ ≈ ◦ . S IIQShE varies much less withshape after that point and the functional behavior of S IQShE starts to dominate which is causingthe entropy to increase rather than decrease with θ , see Fig. 4.5c. But this functional change issolely due to the behavior of S IIQShE term that depends on the temperature sensitivity of overlapvolumes. Let’s recall how we define the overlap volumes. We shall give the analytical expres-sions of overlap volumes and effective volumes only for the nested square domain, becausethey are much simpler and easier to grasp. The effective volume of the nested square domain isfound as V eff = [( L o − δ ) − ( L i + 2 δ ) ]( L l − δ ) + V ovr = ( L o, eff − L i, eff ) L l, eff + V ovr . Theoverlap volume can be obtained as, V ovr = L l, eff tan θ (cid:20) L i, eff (cid:18) θ (cid:19) − L o, eff sin θ (cid:21) , (4.12)for θ ∗ ≤ θ ≤ ◦ , where θ ∗ is given by Eq. (3.3). For ◦ ≤ θ < θ ∗ , V ovr = 0 and there is nooverlap. After this recall, the temperature sensitivity of the overlap volume (see in Eq. (4.11))can then easily be found by its temperature derivative, ∂ V ovr ∂T = −√ θ (cid:18) θ (cid:19) L l, eff δT (cid:115) V ovr L l, eff + δT V ovr L l, eff . (4.13)The first term dominates as long as L ∗ (the characteristic length of transverse sizes, definedin Section 3.6) is much smaller than the longitudinal size of the domain, which is the casehere. The dominant first term in Eq. (4.13) is negatively proportional with the square root ofthe overlap volumes. That’s why entropy decreases while overlap volume increases. To put itdifferently, increase in overlap volume affects entropy in two different competing ways: one isthe increase in entropy due to increment of effective volume, the other is decrease in entropydue to negative temperature sensitivity of effective volume. The overall behavior depends onthe result of the competition of these two terms. This is what happens in the nested rectanglecase. Depending on the shape of the domain and the angular configuration, different QShEterms determine the behavior of entropy.Above, we explained the peculiar behavior of entropy by examining the analytical expres-sions that we derived. The physical intuition for such a behavior can be conceived by a thermo-dynamic temperature analysis. Temperature is fundamentally associated with the distributionof the population of particles on the energy states of the system. When a system’s tempera-ture drops, the particles accumulate into the ground state and lower excited states, whereas iftemperature rises, they accumulate into the higher excited states. In size variations (such as avolumetric expansion), energy levels mainly shift (see Fig. 3.5). On the other hand, when thereis an effective shape expansion (the change of effective volume by the formation of overlap vol-umes), energy levels behave in a much more complicated way. When we rotate the system from θ = 45 ◦ to θ = 0 ◦ configuration, we cause ground state energy to rise and lower excited statesbecome near ground state in θ = 0 ◦ configuration. Higher excited states do not change muchduring shape transformation. If we look at the picture from the distribution point of view, nowthere are more particles occupying the ground state and lower excited states than before, so thetemperature drops. Therefore, distribution of occupied states changes in favor of a temperature76rop from θ = 45 ◦ to θ = 0 ◦ configuration. As an analogy for the process, one can consider thepopulation inversion in a laser in which case the most of the constituents in the system occupieshigher excited states than the lower ones. In our case, this is not the case, but the physical reasonof this behavior is again the change in population distribution over the states. With this analogy,we may call the process happening in our system as "population mixing".At this point, we would like to explain a small misprediction of our proposed overlappedQBL approach. Depending on the type of nested confinement geometry, there might be somesharp changes (e.g. the cusp in Fig. 4.5c black dashed curve), in the analytical calculations.There are no sharp changes in actual behaviors of the functions (Fig. 4.5c red curve). This typeof cusps may be apparent only for certain geometry types and there may be additional ones insome other more exotic ones. The misprediction of the analytical method is also very clear forthe regions where the overlaps does not start ( θ ≤ θ ∗ ). This is of course due to the simplisticnature of the approximation to define the QBL. In Appendix A.5, we will mention the possibleimprovements that can be done on the developed methodology, which can resolve these mis-predictions, though the geometrical calculations become extremely complicated. Nevertheless,even the simplest QBL approach correctly captures the functional behaviors of the thermody-namic state functions. Examination of heat capacity is essential for a better understanding of the thermal properties ofa particular system. Heat capacity is defined as the amount of heat that is required to changethe temperature of a system by a certain amount. Under constant volume, it is described mathe-matically as the derivative of internal energy with respect to temperature. Specific heat capacitygives the heat capacity per particle, providing a quantity that is independent of how large thesystem is.An important distinction we would like to mention here is we actually calculate the specificheat capacity at constant Weyl parameters. Note that in quantum shape effect calculations,not only volume, but also surface area, peripheral lengths and number of vertices are constant.These geometric size parameters are called Weyl parameters and thus we calculate the heatcapacity at constant Weyl parameters.In Fig. 4.6, we plot the specific heat changing with the rotation angle for our three differentdomain configurations. Specific heat does not monotonically increase in all nested domainconfigurations, rather it makes a small peak after its initial increase and then slightly decreasesforming a plateau at the higher angles. Despite the slight decrease at the higher angles, thefigures show that the specific heat grows when quantum shape effects increase. This meansa confined system’s capacity to store heat can be increased by help of quantum shape effects.Specific heat is also a convenient property for the experimental demonstrations of quantumshape effects.Dashed-black curves in Fig. 4.6 are the results of the analytical overlapped QBL approach.Our analytical model correctly predicts the saturation values of the specific heat at the higherangles. It deviates from the exact calculations for the lower angles, a behavior that is similarto the previously examined thermodynamic state functions. The maximum and mean relativeerrors of the analytical specific heat capacity result are respectively . and . for nestedsquare, . and . for triangle and . and . for rectangle domains. Since internal77 igure 4.6: Variation of specific heat capacity at constant Weyl parameters (normalized to
N k B ) with respect to the shape change via θ , for (a) nested square, (b) nested triangle and (c)nested rectangle domains. Solid-gold and dashed-black curves represent the exact and analyticalcalculations respectively.energy is written in a piecewise form under the overlapped QBL approach, a discrete jumpoccurs at the critical angle in heat capacity, which is the main reason of larger errors. Thesecond jump in Fig. 4.6c is again due to the temperature sensitivity of the overlap volume.Improvements on overlapped QBL approach may resolve these mispredictions, please refer toAppendix A.5. Let’s recall the change in free energy in our systems. We argue that under zero friction, innerstructure should rotate spontaneously to the equilibrium angle which minimizes the free energy,if the system is prepared at any other angular configuration in the first place. But in order forthe inner structure to be able to rotate, there needs to be a torque exerted on the inner structureby the confined particles. Mathematically, negative derivative of free energy with respect tothe rotation angle should give us the amount of torque that is exerted by particles to the innerstructure. Consequently, the torque at a given configuration angle θ reads τ = − ∂F∂θ ≈ − F θ +∆ θ − F θ ∆ θ = − N k B T ∆ θ ln (cid:18) ζ θ ζ θ +∆ θ (cid:19) , (4.14)where θ and θ + ∆ θ subscripts denote initial and perturbed rotational states respectively. Forthe numerical solution continuous derivative is not possible. To tackle with this, we perturbthe system from its initial configuration angle a tiny amount and then solve the Schrödingerequation again for the resulting perturbed system. ∆ θ denotes the amount of the rotationalperturbation, which has to be very small for a good approximation of the derivative. Here it ischosen as ∆ θ = 0 . . We have tested smaller and larger ∆ θ values and up to ∆ θ = 1 , the errordue to the discrete differentiation is negligible.The analytical expression of torque can be found by taking the derivative of Eq. (4.9) withrespect to θ , which quite simply gives τ A = N k B T ( ∂ V ovr /∂θ ) / V eff . For nested square domain,it is derived in its open form as τ A N k B T = sec θ V ovr L l, eff tan θ + tan θ csc θ ( L o, eff cos θ − L i, eff ) (cid:113) V ovr L l, eff tan θ tan θ V ovr L l, eff tan θ + L o, eff − L i, eff . (4.15)78 igure 4.7: Variation of torque (normalized to maximums of the exact ones in each case) with θ , for (a) nested square, (b) nested triangle and (c) nested rectangle domains. Purple and dashed-black curves represent the exact (Eq. 4.14) and analytical (Eq. 4.15) calculations respectively.In Fig. 4.7, we plot the torque versus rotation angle graphs by comparing the numerical andanalytical results for nested square, triangle and rectangle domains. The torque is normalized byconsidering maximum value of the exact solutions in each case. The maximum values predictedby Eq. (4.15) are not the same with exact ones, but the functional behaviors are captured quitewell. The amount of maximum torque exerted by the particles in nested square domain isaround 1 nNnm (nano-Newton nanometer). We explore the reasons why the maximum torqueoccurs at those particular θ values will be clear during the next section where we examine thenon-uniformity of the pressure distributions along the walls of inner structure.The torque equals to zero for ◦ and ◦ configurations. This means in order for innerstructure to rotate from ◦ to ◦ , one needs to generate an infinitesimally small perturbationfrom the unstable ◦ configuration to initiate the symmetry breaking. With this, one can alsochoose the direction of the rotation which is significant for engine/rotor systems.In Fig. 4.8, we investigate the quantum torque generated by the system for various confine-ment strengths in nested square domain. Since this torque is a result of quantum shape effects,its value should strictly depend on the magnitude of the confinement. The confinement param-eter of the transverse direction for all of our cases we examined so far was unity α t = 1 . . Weexplore two additional confinements of α t = 0 . and α t = 1 . by scaling up and down thedomain sizes. Variation of torque with the rotation angle for these various confinement casesis plotted in Fig. 4.8. As we expect, higher the confinement strength, larger the torque. An-other interesting thing in this analysis is the maximum torque angle is not fixed for a particulardomain geometry but it also changes with the amount of confinement.It was already clear from the variation of free energy with θ that the torque can be calculated.However, this doesn’t explain how such a torque can appear. This torque is of quantum origin.There is no mechanical or any other classical kind of force take place here. Confined particlesshould exert pressure to the walls of the inner square structure at equilibrium. The only possibleexplanation for the existence of a finite amount of torque is that the pressure exerted to the wallsof the inner square structure should be non-uniform so that there would be a moment imbalancewhich would generate a torque. Next step is to check how is the pressure profile on the walls ofinner square domain and whether it is non-uniform or not.79 igure 4.8: Dependence of the torque on the magnitude of confinement. Torque varies withrotation angle for various confinement strengths for the nested square domain. The values arenormalized to the maximum value of the torque for α t = 1 . How do we find the pressure distribution along a surface? Classically, a gas pressure at equi-librium is uniform and the pressure is equal at any point on the boundaries. On the other hand,in our case the pressure is caused by the confined particles at nanoscale which are behaving ac-cording to the laws of quantum mechanics. We have already seen that the density distributionsof these particles are non-uniform as opposed to the classical picture. We could not directlyuse the density distribution to obtain the pressure distribution without incorporating some addi-tional tools like QBL method perhaps. Classical thermodynamic equation of state is not valid atnanoscale where the pressure becomes a tensorial quantity. Moreover, the equation of state be-comes strongly depend on the sizes and the types of boundary deformation in strongly confinedsystems.Thermodynamically, pressure is the free energy response of a system against a boundarydeformation. Therefore, the most direct and fundamental way to investigate a pressure distribu-tion is making small deformations (perturbations) on the boundaries along which the pressuredistribution we want and looking to the system’s free energy response to these deformations.To this end, we can create either local or global perturbations as it’s shown in Fig. 4.9. First,we create tiny local perturbations on the walls of the inner square region. For convenience, weconduct the analysis on pure 2D system, as the third direction does not contribute to the creationof the non-uniform pressure profile. We already have the solutions for the unperturbed case. Toprobe the free energy response of the system, we need to separately solve the Schrödinger equa-tion again and again for each and every single perturbation we apply to the system. We havedone a quite extensive analysis of the proper nature of the perturbations for the calculation ofthe accurate pressure distribution. The details of these analyses are given in Appendix sectionsA.1, A.2 and A.3. We repeat this procedure of creating local perturbations along the length ofthe wall. To generate its profile, we calculate the pressure exerted by particles onto a singleperturbation by P L = 1∆ w (cid:18) − ∆ F → p ∆ h (cid:19) = − F p − F ∆ w ∆ h (4.16)where P L is the local pressure exerted on a single perturbation, ∆ w is the width and ∆ h is theheight of the perturbation. and p subscripts denote the unperturbed and perturbed versions ofthe free energy respectively. 80 igure 4.9: Probing the pressure profile along the inner domain walls by generating tiny localperturbations and looking to the free energy response of the system. By a similar procedure onecan create global perturbations as well to verify the integral of local pressure distribution.In Fig. 4.9, we plot the pressure profile along the inner domain wall of nested square casefor ◦ (maximum torque angle). ˜ L i ( r ) is the normalized position along the inner square wallwith respect to the middle point. As we expected, the pressure distribution along the boundaryis indeed non-uniform. In fact, it resembles to the density distribution near to the boundary.(At this point, one might ask how can particles transfer momentum and exert pressure withoutbeing able to touch to the walls. It is clear that particle density is zero at the boundaries andvery close to zero within δ (QBL thickness) away from the boundaries. Yet, particles cantransfer momentum because they are actually not point particles but matter waves. They areremotely interacting with the boundaries through their wavefunctions. The same pressure hasbeen examined in literature also under the name of matter-wave pressure. We use the notionof confined particles, it is a matter of choice to call them matter-waves or confined particles asthey are equivalent by the de Broglie relation.) The non-uniform pressure distribution that isshown in Fig. 4.10 is consistent with both the density distributions and also the torque valuesthat we calculated earlier. Moreover, this non-uniform pressure distribution explains the veryexistence of the torque. From the Fig. 4.10, we expect the inner square to rotate clockwise andthis is what the torque values say in Fig. 4.7.In addition to this approach, we analytically showed the equivalence of pressure and mo-mentum flux for all types of particle statistics, see the derivation in Appendix A.6. This showsthe consistency of our methodology from two separate physical approaches; thermodynamicsand transport theory. Therefore, the non-uniform pressure distribution is exactly equivalent tothe momentum flux distribution across the boundary.81e can analyze further and look for the total amount of pressure exerted on the inner bound-aries. To do that, we can either sum the local pressures generated by the local perturbations, orquite simply we can generate a perturbation with the width of the domain wall itself and lookfor the free energy response of the system. Inner square has 4 sides. We can either choose aside to apply this perturbation and multiply by 4 later, or we can even completely increase theouter length size at once, see the last row of Fig. 4.9. All the methods that we mention here givethe same result under a negligible error margin (summing local perturbations give a larger errorsince the errors of individual calculations add up). Global pressure then can be calculated by P G = − F p − F L i ∆ h , (4.17)where here F p is the free energy under global perturbation, different than the one in Eq. (4.16)which was local. Since we are interested in total pressure exerted on the inner square, we dividethe expression with L i . By using torque and pressure, we can also find the application pointsof the forces. Dividing Eq. (4.14) by Eq. (4.17) gives the point of application of the pressure.Variation of pressure and application points with respect to the rotation angle θ is given fornested square domain in Figs. 4.11a and 4.11b along with the comparison of numerical andanalytical results. The analytical expression for pressure is simply given by P AG = N k B T V eff . (4.18)It should be noted that Eq. (4.18) contains the effective volume instead of the apparent one.Ratio of Eq. (4.15) to Eq. (4.18) gives the application points of forces analytically. Analyticalpressure represents the true behavior extremely well and the accuracy of the analytical expres-sion for application points is similar to the accuracy of the torque expression.The point of application for nested square domain has already shown as an example in Fig.4.10. Application points of forces are shifted from center due to non-uniform pressure and socreate a moment on the body causing the rotation of the inner structure, Fig. 4.11c. As expected,the variation of application point with respect to rotation angle has the same character of thevariation of the torque with respect to angle, Fig. 4.11b. The reason why the torque is zero for ◦ and ◦ degree configurations can be clearly inferred from Fig. 4.11d, where it is shown thatpressures distributions are symmetric in those configurations, unlike in between angles.It should also be noted that even though pressure forces decrease with increasing angle, theincrease in moment arm length is much larger. Therefore, the main mechanism determining the Figure 4.10:
Normalized pressure distribution along the inner boundary of nested square do-main prepared at ◦ angular configuration. Blue dashed line shows where the point of applica-tion is. 82 igure 4.11: (a) Normalized pressure, (b) normalized application points changing with θ . Or-ange and cyan colors represent exact and black dashed curves represent the analytical results.(c) Analysis of the torque. (d) Pressure forces on the inner boundaries for various angularconfigurations. Figure 4.12: (a) Isothermal expansion (size variation) (b) Isothermal effective expansion (shapevariation at constant sizes) Changes in thermodynamic state functions with respect to (c) size(macroscopic), (d) size (nanoscale), (e) rotation angle. Blue, red and green curves representfree energy, entropy and internal energy respectively.83orque is the change of moment arm length or the application points. The farther the momentarm, the higher the torque, vice versa.Before passing on to the next section, we would like to mention another novelty of the quan-tum shape effect which is related with the spontaneous decrease of entropy and free energy. Inthe classical isothermal expansion of the gases, internal energy stays constant, free energy de-creases since entropy of the system increases, Figs. 4.12a and 4.12c. We know that due toquantum size effects, internal energy of a system does not stay constant during an isothermalexpansion, Fig. 4.12d. Conversely, in an isothermal shape expansion (which corresponds to aneffective expansion due to decrement of overlap volumes), internal energy decreases so muchthat not only free energy, but also entropy decreases, Figs. 4.12b and 4.12e. Fig. 4.12 summa-rizes how quantum shape effects cause novel behaviors at nanoscale which haven’t seen beforein classical systems or confined systems with only size effects.We complete our analysis here for the thermodynamic properties of Maxwell-Boltzmanngases confined in nested nanodomains. In the next section, we explore the thermodynamicproperties of systems obeying Fermi-Dirac statistics. Particularly, we focus on the free electrongas. Most of the candidate systems for quantum shape effects such as semiconductors and met-als can be modelled by free electron approximation and it is easier to focus on how QShE affectsthese systems as the model itself does not complicated with the details which are unnecessaryfor our investigation here.
Statistical behaviors of particles in nature are captured by two very distinct types of statis-tics, namely Fermi-Dirac and Bose-Einstein. Charge carriers in materials behave accordingto the Fermi-Dirac statistics. Electrons in various semiconductor and metallic systems havebeen studied extensively due to their paramount importance on nanoscale electronics and nano-engineered devices in general. Based on the research in the literature, we think that our nesteddomain architectures could possibly be realized in such systems. Besides, thermodynamic andtransport behaviors of charge carriers in semiconductors and metals carry valuable informationabout the material’s mechanical, electronic and energetic properties. Due to these reasons itis crucial to investigate quantum shape effects on specific and more realistic systems such aselectrons in low-dimensional semiconductors/metals. To accomplish that, we cannot rely onMaxwell-Boltzmann statistics anymore of course and we will use the Fermi-Dirac statistics.Even though Maxwell-Boltzmann statistics is still valid on those systems, we would like to ex-tend our investigation and be not restricted by low density and/or high temperature conditions.As a concrete example, we consider the electrons in a Gallium Arsenide (GaAs) semiconductornanowires in this section.Fermi-Dirac distribution function reads f = 1 / [exp(˜ ε − Λ)+1] where
Λ = µ/ ( k B T ) and µ ischemical potential. Spin degree of freedom is just a factor of two and ignored in our calculationsand in the expressions for brevity. Note that we don’t have the restriction of longitudinal size tobe relatively long anymore. Nevertheless, we keep our domain as it is in Maxwell-Boltzmanncase, to provide a better comparison with the previous cases. We analyze only the nested squaredomain in this section.Unlike in Maxwell-Boltzmann case, we cannot separate the distribution function and parti-tion function into the products of numerical and analytical parts unfortunately. This is because84n Fermi-Dirac (also in Bose-Einstein) statistics, one cannot separate the chemical potentialfrom the energy term on the exponential due the structure of their distribution functions. Ei-ther we have to solve the Schrödinger equation for the whole 3D confined system, which iscomputationally very demanding, or we have to find a way around.Choosing one direction relatively long again favors on the simplicity of this problem. En-ergy eigenvalues of this kind of system can be separated as ˜ ε = ˜ ε t + ˜ ε l . Although we cannotcompletely separate the analytical part from the numerical, since we know the eigenvalues ofthe longitudinal part analytically, we don’t have to sum, instead we can integrate them. Thelongitudinal part is relatively long so that the first two terms of PSF work very well on its rep-resentation. Therefore, we use the first two terms of PSF to the longitudinal direction, withoutseparating them from the other parts. The expression for the Fermi-Dirac partition functionbecomes Z = (cid:88) ε ln [1 + exp(Λ − ˜ ε )] (4.19a) = (cid:88) ε t (cid:90) d ˜ ε l ln [1 + exp(Λ − ˜ ε t − ˜ ε l )] (4.19b) = (cid:88) ε t − L l λ th Li (cid:0) − e Λ − ˜ ε t (cid:1) + 12 Li (cid:0) − e Λ − ˜ ε t (cid:1) , (4.19c)where λ th = h/ (cid:112) πm eff k B T and m eff = 0 . m e is the effective mass of conduction bandelectrons of GaAs, where m e being the bare electron mass. The summation is over the eigen-values of the transverse part whereas the kernel terms are the results of the PSF. In this sense,expressions that are derived this way can be considered as semi-analytical.Until this section, we always used the bare electron mass in our calculations. Having thespecified effective mass in GaAs means thermal de Broglie wavelength of electrons in thesestructures are roughly four times larger than the case we considered before. That is to say,we can enlarge our domain sizes a bit more than the previous cases so that the manufacturingthese kinds of materials in labs might be more realizable. With ease we can scale the transversedomain sizes in all directions by a scale parameter s . So, instead of L o ( L i ), we have L o s ( L i s ), s being the scale parameter we would like to choose. For s = 1 , the previously examined domainsizes are recovered. We choose s = 3 . for the calculations in this section. This value givesouter and inner square lengths as 64nm and 41nm respectively. Since we apply this to bothdirections in transverse plane, we don’t even have to calculate new eigenvalues for our newdomains, we can scale the numerically solved transverse eigenvalues easily by ˜ ε/s . Naturally,we tested these scaled eigenvalues for selected numerical results and no errors have been found.Temperature is again 300K in all calculations.The number of particles ( N ) is defined by the summation of the distribution function overall eigenvalues, N = (cid:80) ε f . The semi-analytical expression for number of particles is N = (cid:88) ε t − L l λ th Li (cid:16) − e Λ (cid:48) (cid:17) + 12 Li (cid:16) − e Λ (cid:48) (cid:17) , (4.20)where Λ (cid:48) = Λ − ˜ ε t is shortened for brevity. From the expression above (Eq. 4.20), chemi-cal potential can be numerically calculated as an inverse problem. To ensure large number ofparticles inside our confinement domain as well as to comply with doped GaAs semiconductor85onduction band electron concentrations, we choose three different electron densities to exam-ine, ranging from . n dB to n dB where n dB is the de Broglie density which is the reciprocalof the thermal de Broglie wavelength λ th . Note that these n dB and λ th contain now the effectivemass for GaAs electrons, instead of the bare electron mass which was used until this subsection.We’ve examined partition function in Maxwell-Boltzmann case. For comparison, in Fig.4.13 we plot the variation in partition function with θ . The functional behavior is relativelysimilar to the Maxwell-Boltzmann case for low densities. In degenerate regime however, wesee an oscillatory functional behavior, which is not expected at first sight. We explore more onthis oscillatory behavior during the following subsections.To compare the relative changes in the partition function with respect to shape betweencases, we calculate the relative differences of the minimum and maximum values of the functionin each case. From lowest to highest density cases, the relative differences are . , . and . respectively. The trend of the relative differences with density reflects the oscillatorydependence of the quantities in degenerate cases.In order to prevent any confusion, it should be stressed here that the degeneracy conceptused in this section of 4.9 is different than the degeneracy of energy levels which we’ve dis-cussed in the previous chapter. Degeneracy in the energy level context means the energy levelscorresponding to the same energy. On the other hand, degenerate regime in the context of elec-trons in solids (e.g. in semiconductors or metals) means the density of the electrons is verylarge so that the effect of Pauli exclusion principle and the exchange interaction is prominent.In statistical mechanics, the Pauli exclusion principle is taken into account by the Fermi-Diracdistribution function. Chemical potential is defined as the rate of change in free energy with respect to the changein particle number. Under constant temperature and volume, it is the derivative of Helmholtzfree energy with respect to particle number. The physical meaning of chemical potential issubject to various definitions. To grasp it as a thermodynamical concept, it may be better ifwe establish some analogies with the other concepts like temperature and pressure. The heattransfers from higher temperature to lower temperature to equalize the temperatures. By thesame token, volume expands from higher pressure to lower pressure to equalize the pressures.Similarly, particles move from higher chemical potential to lower chemical potential to equalize
Figure 4.13:
Change in partition function with rotation angle for various electron densities (a) . n dB ≈ . × (non-degenerate regime), (b) n dB ≈ . × (weakly degenerate regime)and (c) n dB ≈ . × (moderately degenerate regime).86he chemical potentials. Therefore, the chemical equilibrium is represented by the equality ofchemical potentials. It has a major role in solid state physics, especially in the physics ofsemiconductors, among all other fields.From Eq. (4.20), we can calculate the chemical potential µ as an inverse solution for a fixednumber of particles. Since there are no analytical expressions for inverses of the polylogarithmfunctions, we solve this inverse problem numerically. As a result, the shape-dependence of thechemical potential for three different electron densities is given in Fig. 4.14. At low densities,Fig. 4.14a, chemical potential does not show any oscillation. The negativity of chemical poten-tial suggests that electrons’ Fermionic behavior is not present and they almost obey Maxwell-Boltzmann statistics. In Fig. 4.14b, the density is increased by a factor 10 and chemical po-tential becomes positive. This means the electrons are in weakly degenerate conditions whichloosely corresponds to a transition regime from Maxwell-Boltzmann to Fermi-Dirac statistics.The peak that is shown in Fig. 4.14b is the first and the single peak and it monotonically de-creases after the peak. When we increase the density further in Fig. 4.14c, chemical potentialstarts to oscillate with the shape variation. This is a quite interesting result, because normallychemical potential is not an intrinsically oscillatory quantity. What we mean by intrinsicallyis, some physical quantities exhibit quantum oscillations with respect to changes in sizes andthe reason of these oscillations are due to the proximities and weights of the states near toideal Fermi surface (or Fermi line in 2D, Fermi point in 1D nanostructures). The contributionsof states near to Fermi surface are weighted by the occupancy variance function. Therefore,the physical quantities containing this occupancy variance function as a factor exhibit size- ordensity-dependent quantum oscillations. During my PhD, we have introduced a semi-analyticalmodel called half-vicinity model to understand and predict the oscillatory behaviors of quan-tum confined systems. Differently from size-dependent oscillations, this chemical potentialoscillations shown in Fig. 4.14c is not coming from the behavior of occupancy variance, butthey are a result of the influence of a complex shape variation on the system’s thermodynamicbehavior. In other words, when we decrease the size of the system while keeping the numberof particles constant, chemical potential increases just monotonically with confinement and nooscillatory behavior is observed. On the other hand, in order to fix the number of particles to acertain value, while changing the domain shape smoothly with θ , requires chemical potential tobehave oscillatory. This is another novel type of behavior that is seen in the thermodynamics ofconfined nanostructures, due to quantum shape effects.Relative differences of min-max values in each case are found as . , . and . re-spectively for Fig. 4.14a, 4.14b and 4.14c. This suggests quantum shape effects have much moredominant behavior for low density conditions where Maxwell-Boltzmann statistics is obeyed.When the degeneracy increases, the influence of shape reduces. This is understandable, becausethe degeneracy is inversely proportional to the Fermi wavelengths of particles. Like thermal deBroglie wavelength, there is another concept called Fermi wavelength which includes the in-fluence of degeneracy in terms of chemical potential into the de Broglie wavelength. Althoughde Broglie wavelength is still an important quantity for Fermionic particles, Fermi wavelengthprovides more detailed and more accurate information about the quantum statistical nature ofparticles. Fermi wavelength can be given in terms of the de thermal de Broglie wavelengthas λ F = λ th (cid:112) π/ Λ for 1D Fermi systems. The larger the chemical potential, the smaller theFermi wavelength and lesser the quantum wave behaviors of particles. In fact, it is possible todefine another confinement parameter for strongly degenerate ( Λ >> ) Fermionic systems, α F = α/ √ Λ F for 1D systems for example, where Λ F is the dimensionless chemical poten-87 igure 4.14: Variation of dimensionless chemical potential ( Λ ) with rotation angle for electrondensities (a) . n dB ≈ . × (non-degenerate regime), (b) n dB ≈ . × (weaklydegenerate regime) and (c) n dB ≈ . × (moderately degenerate regime). Figure showsthe change in behavior of chemical potential as the density goes from Maxwell-Boltzmannstatistics conditions to Fermi-Dirac statistics ones.tial at 0K. However, we would like to continue our discussion of the conditions of MB andFD statistics based on the comparison of the de Broglie densities. n << n dB corresponds toMB, n ≈ n dB and n >> n dB corresponds to FD statistics. These conditions correspond toa comparison between the mean distance between particles and their de Broglie wavelengths.This kind of comparison gives a measure of the degeneracy in solid state systems. Our analysisshows that low density conditions are favorable for the maximization of quantum shape effects.When Fermi level rises (i.e. degeneracy increases), average de Broglie wavelength of particlesdecreases and quantum shape effect decreases. In degenerate conditions, however, appearanceof a new type of oscillatory behavior is also worthy of note. We’ve seen that examination of quantum shape effects in Fermi-Dirac statistics has revealednew behaviors. We wonder how thermodynamic state functions of electrons behave under quan-tum shape effects. Semi-analytical expressions of internal energy, free energy and entropy arederived as follows respectively U = k B T (cid:88) ε t − L l λ th Li (cid:16) − e Λ (cid:48) (cid:17) − ˜ ε t L l λ th Li (cid:16) − e Λ (cid:48) (cid:17) + ˜ ε t Li (cid:16) − e Λ (cid:48) (cid:17) , (4.21) F = k B T ( N Λ − Z ) , (4.22) S = k B (cid:88) ε t − L l λ th Li (cid:16) − e Λ (cid:48) (cid:17) + 12 Li (cid:16) − e Λ (cid:48) (cid:17) + Λ (cid:48) L l λ th Li (cid:16) − e Λ (cid:48) (cid:17) − Λ (cid:48) Li (cid:16) − e Λ (cid:48) (cid:17) . (4.23)In Figs. 4.15, 4.16 and 4.17, variations of internal energy, free energy and entropy with theshape parameter θ is presented respectively. The analysis shows that for non-degenerate or very88 igure 4.15: Change in internal energy with shape for densities (a) n = 0 . n dB , (b) n = n dB and (c) n = 10 n dB . Figure 4.16:
Change in free energy with shape for densities (a) n = 0 . n dB , (b) n = n dB and(c) n = 10 n dB .weakly degenerate conditions (subfigures (a) and (b)), the functional behaviors of thermody-namic state functions are similar to the ones in Maxwell-Boltzmann statistics, as expected. Onthe other hand, all thermodynamic state functions exhibit oscillatory behaviors in the degener-ate regime. Internal energy and free energy are the quantities that do not contain occupancyvariance function as a factor in their expressions. Hence, these oscillations are not the alreadyknown density or size oscillations. Origin of these oscillations is the chemical potential oscil-lations due to quantum shape effects. The fact that chemical potential oscillates with shape forfixed number of particles, causes oscillations in all other quantities that depend on chemicalpotential.It is seen in Figs. 4.15c and 4.16c that the functional behaviors of internal energy and freeenergy are very similar to each other. On the other hand, oscillatory behavior of entropy, Fig.4.16c, is different than those of internal energy and free energy. Combined effect of chemicalpotential oscillations and shape-dependent occupancy variance oscillations might be the resultof this different oscillatory behavior.Differences of extremum points of internal energy function changing with shape are givenfor n = 0 . n dB , n = n dB and n = 10 n dB respectively as . , . and . . Similarly,differences of extremum values of free energy are , and . Finally, the differencesin extremum values of entropy are . , . and . . Influence of quantum shape effectsare larger to internal energy for low density conditions consistently, whereas changes in entropydue to quantum shape effects are more or less the same in all cases. Interestingly, variationin free energy due to quantum shape effects has somewhat unusual behavior. There is a hugeinfluence of quantum shape effects for weakly degenerate conditions. The reason of this is89 igure 4.17: Change in entropy with shape for densities (a) n = 0 . n dB , (b) n = n dB and (c) n = 10 n dB . Figure 4.18:
Change in specific heat capacity at constant Weyl parameters with shape for den-sities (a) n = 0 . n dB , (b) n = n dB and (c) n = 10 n dB .because free energy changes sign when chemical potential passes from zero. That’s why, it hasa higher sensitivity to the changes in shape around those values of chemical potential. Almostnegligible response of entropy to the changes in electron density might be the result of the factthat the configurational entropy doesn’t change much during density variations. The sizes andthe shape of the domain that is considered are the same in all cases, which doesn’t change theconfigurational entropy much.Another important property to examine is the electronic heat capacity at constant Weyl pa-rameters (includes the constant volume condition). A semi-analytical expression for electronicheat capacity at constant Weyl parameters is obtained as follows, C W = k B (cid:88) ε t − L l λ th Li (cid:16) − e Λ (cid:48) (cid:17) − ε t L l λ th Li (cid:16) − e Λ (cid:48) (cid:17) − ˜ ε t L l λ th Li − (cid:16) − e Λ (cid:48) (cid:17) + 12 Li − (cid:16) − e Λ (cid:48) (cid:17) − (cid:104)(cid:80) ε t − L l λ th Li (cid:16) − e Λ (cid:48) (cid:17) − ˜ ε t L l λ th Li − (cid:16) − e Λ (cid:48) (cid:17) + Li − (cid:16) − e Λ (cid:48) (cid:17)(cid:105) (cid:80) ε t − L l λ th Li − (cid:0) − e Λ (cid:48) (cid:1) + Li − (cid:0) − e Λ (cid:48) (cid:1) . (4.24)In Fig. 4.18, specific heat capacity varies with shape for various densities. The increase inheat capacity in low degrees of angle is preserved in all cases, whereas for high degrees of angleoscillations start to appear in degenerate case.Differences between the extremum values of specific heat function are given respectively as , and for n = 0 . n dB , n = n dB and n = 10 n dB .90 Applications For Nano Energy Science andTechnology
Up to now, we’ve introduced a novel effect, which we call the quantum shape effect, appearingin the thermodynamics of nanoscale systems that are confined in a particular way. During Chap-ter 3 we explore the fundamentals of the quantum shape effects and constructed its theory alongwith an analytical approach. During the fourth chapter we investigate quantum shape effects inthe thermodynamics of nanoscale confined systems. In this chapter, we examine the possibleapplications of quantum shape effects, specifically focusing on their energy applications.Nano energy science and technology is developing as a research field on its own and it dealswith understanding the workings of energy devices and thermodynamic machines at nanoscale.Exploration of quantum heat engines and nanoscale energy conversion devices constitutes an-other part of the thesis. Both theoretically and from the application point of view, investigationof the role of quantum shape effects in such systems is crucial for the development of new nanoenergy technologies. In this chapter we focus on how the newly introduced quantum shapeeffects can play a role on the nano energy applications.
With the addition of shape as a control parameter in thermodynamics, it is possible to definenew thermodynamic processes. Recall Fig. 4.1 where new parameters open up new dimensionsin thermodynamic state space. Each variable gives rise to distinct thermodynamic processes. Togive an example, keeping volume constant in a thermodynamic cycle, leads to isochoric process.Similarly, the constancy of pressure, temperature or entropy as well as zero heat transfer giverise to isobaric, isothermal, isentropic and adiabatic processes respectively. In a similar manner,we shall call the isoformal process when the shape of the working substance is constant in athermodynamic process. In this section, we construct quantum heat engines driven by quantumshape effects. The working substance of our engine is the particles confined in our nested square91omain shape. In our analyses, all processes are assumed to be infinitely slow (quasistatic),hence reversible, and no coherence exists among energy levels. For convenience, we executeour analyses in this chapter for the conditions where Maxwell-Boltzmann and Fermi-Diracstatistics give the same results, hence the density, temperature and the sizes are n = 5 × m − , T = 300 K and s = 1 (the scale parameter defined in Sec. 4.9). The first cycle we would like to introduce is a Stirling-like heat engine cycle. In the usual Stir-ling cycle, there are two isothermal (constant temperature) and two isochoric (constant volume)processes. In our engine, rather than two isochoric processes, we have two isoformal processes,where all geometric size variables and most importantly the shape of the confined system arekept constant.The cycle consists of four consecutive steps as it’s seen in Fig. 5.1: ( ) Isoformal heataddition → : The temperature of the system increases from K to
K. ( ) Isothermalshape-confinement (isothermal effective compression) → : Inner structure is rotated from ◦ to ◦ position by performing work. During this process heat is also given to the system tokeep the temperature constant at K. This is contrary to the classical Stirling cycle, in whichthe corresponding process is the isothermal compression of the gas that causes heat to be re-leased to the environment to keep the temperature constant. Contrarily, here system absorbsheat from the environment, even though it is an isothermal effective compression process. ( ) Isoformal heat rejection → : The temperature of the system decreases from K back to
K, while keeping the inner structure at the ◦ position. ( ) Isothermal shape-deconfinement(isothermal effective decompression) → : Inner structure spontaneously rotates from ◦ to ◦ degree position by doing work on its surroundings and rejects heat by keeping the temper-ature constant. This is again a quite uncommon process since both work generation and heatrejection take place during the same isothermal process.In Fig. 5.2, T - S and τ - θ diagrams of the cycle are shown. Unlike in classical thermody-namic cycles, the work exchange during high temperature shape transformation (from steps to ) is less than that of low temperature one (from steps to ), Fig. 5.2b. Additionally, thedirections of work and heat exchanges under isothermal shape transformation processes are thesame, unlike the ones in isothermal volume variation, Fig. 5.2a and 5.2b. In other words, thedirections of work are exactly opposite of the Stirling cycle. In the classical Stirling cycle, theisothermal expansion is carried out by means of volume increase. Work output occurs due toexpansion, at the higher temperature (higher pressure) side whereas work input is given duringthe contraction, at the lower temperature (lower pressure) side. Since the work output is largerthan the work input, the net work is positive. On the other hand, in this modified version ofStirling cycle, there is no expansion but an increment of effective confinement (or decrement ofeffective volume) as a result of quantum shape effects. That’s why, instead of work extraction,work has to be done on the system during → . Since quantum shape effects are larger atlower temperature, work extraction occurs on the lower temperature side of the cycle.It should also be noted that, one does not necessarily should mechanically rotate the innerstructure in a closed system to be able to realize → and → processes. Instead, a particleflow can also be considered in a nano channel made by nested structures where the inner one istwisted from ◦ to ◦ and from ◦ to ◦ along the channel. So, the particles would be exposedto varying confinement shapes during their flow through the channel.92 igure 5.1: Stirling-like thermodynamic cycle based on quantum shape effects. The cyclecontains four processes: ( ) isoformal (shape preserving) heat addition, ( ) isothermal shape-confinement, ( ) isoformal heat rejection, ( ) isothermal shape-deconfinement. Work genera-tion occurs at cold side, during the isothermal shape-deconfinement process. Figure 5.2:
Temperature-entropy ( T - S ) and torque-shape ( τ - θ ) diagrams. (a) T - ˜ S diagram,where ˜ S = S/ ( N k B ) . (b) ˜ τ - θ diagram, where ˜ τ = τ / ( N k B T ) . From state to and to ,no change occurs in τ - θ diagram since θ is kept constant during those changes. In addition, thetorque is zero at θ = 0 ◦ and θ = 45 ◦ configurations, as we’ve seen before. When the amount ofoverlap volumes increases with decreasing temperature, higher amount of torque occurs and thishappens at lower temperatures (low temperature side of the cycle in Fig. 5.1) on the contrary toclassical expectations. 93ote that heat and work exchanges are possible without changing the volume or other sizeparameters. All of them are kept as constant during the cycle, so the only work exchangemechanism is the shape transformation through the rotation process. Hence, by using the firstand second laws of thermodynamics, we can write dU = δQ + δW = T dS − τ dθ for thederivations of heat and work exchanges during the processes. For this Stirling-like cycle, heatand work exchanges at each process are then determined as follows Q = U ( T H , ◦ ) − U ( T C , ◦ ) Q = T H [ S ( T H , ◦ ) − S ( T H , ◦ )] W = − (cid:90) τ ( T H , θ ) dθ = U ( T H , ◦ ) − U ( T H , ◦ ) − Q Q = U ( T C , ◦ ) − U ( T H , ◦ ) Q = T C [ S ( T C , ◦ ) − S ( T C , ◦ )] W = − (cid:90) τ ( T C , θ ) dθ = U ( T C , ◦ ) − U ( T C , ◦ ) − Q . (5.1)Using Eq. (5.1), heat input and net work output can be determined respectively as Q in = Q + ( Q − Q ) and W net = W + W . We obtain the cycle efficiency ( W net /Q in ) as for T H = 300 K and T C = 200 K. A refrigeration cycle can readily be obtained just by reversingthe power cycle presented in Fig. 5.1.
The second cycle we would like to introduce is an Otto-like heat engine cycle. Otto cycleconstitutes the working principle of Otto engines which are used in automobiles with gasoline.In the usual Otto cycle, there are two isentropic (adiabatic reversible) and two isochoric pro-cesses. But now like we did in the previous subsection, rather than isochoric processes we haveisoformal processes (constant shape).The cycle consists of four consecutive steps as it’s seen in Fig. 5.3: ( ) Adiabatic shape-confinement → : Inner structure is rotated from ◦ to ◦ position by performing work onthe system without any heat addition. On the contrary to our classical expectations, the tem-perature of the system drops from 300K to 282K, in spite of the fact that the system undergoesan effective compression. The peculiarity of the process should be expounded more here. Aswe saw in Section 4.5 the entropy of a Maxwell-Boltzmann gas increases when we change theconfiguration from 45 to 0 degree isothermally. Accordingly, under an isentropic process, tokeep the entropy constant, the system has to decrease its temperature. However, since the pro-cess is isentropic, there is no heat exchange possibility. In that case, without any heat rejection,decrement of temperature during work input is completely an unexpected behavior. What hap-pens is, by changing the configuration from 45 to 0 degree, we are actually forcing the particlesto redistribute themselves over energy states and to fill more heavily the states near to groundstate. This is why temperature decreases during this process. ( ) Isoformal heat removal → :On the contrary to the usual Otto cycle instead of heat addition, we have heat removal after thefirst adiabatic process in this cycle. Hence, the temperature of the system drops further from282K to 250K, for constant shape. ( ) Adiabatic shape-deconfinement → : Inner structurespontaneously rotates from ◦ to ◦ degree position and does work on its surroundings. Sinceno heat added during the process, the temperature of the system raises from 250K and 270K. ( )94 igure 5.3: Otto-like thermodynamic cycle based on quantum shape effects. The cycle con-sists of four processes and four temperatures denoted by different colors: ( ) adiabatic shape-confinement, ( ) isoformal (shape preserving) heat removal, ( ) adiabatic shape-deconfinement,( ) isoformal heat addition. The cycle generates work during adiabatic shape-deconfinementprocess. Isoformal heat addition → : In order to complete the cycle we raise the temperature from270K to 300K by adding heat to the system at a constant size and shape. The sign of the heatexchange is the opposite of the usual Otto cycle.Transition from state 1 to 2 corresponds to the compression stage in the usual Otto cycle,since there is an effective shape confinement when the structure rotates from ◦ to ◦ . In clas-sical heat engines, when you compress the system, its temperature rises by the thermodynamicequation of state. In cycles featuring isoformal process, however, the temperature of the systemactually drops during an effective "compression". The origin of this peculiar behavior alongwith the sign changes in work and heat exchanges of Stirling-like and Otto-like heat cyclesrespectively lies in the forenamed behavior of entropy.For this Otto-like cycle, heat and work exchanges at each process are then determined bydifferences of internal energies of each state W = U ( T , ◦ ) − U ( T , ◦ ) Q = U ( T , ◦ ) − U ( T , ◦ ) W = U ( T , ◦ ) − U ( T , ◦ ) Q = U ( T , ◦ ) − U ( T , ◦ ) . (5.2)Using Eq. (5.2), heat input and net work output can be determined respectively as Q in = Q and W net = W + W . We obtain the cycle efficiency ( W net /Q in ) as . Again, the coolingcycle can easily be obtained just by reversing the heat pump cycle presented in Fig. 5.2.95 igure 5.4: Temperature-entropy ( T - S ) and torque-shape ( τ - θ ) diagrams. (a) T - ˜ S diagram,where ˜ S = S/ ( N k B ) . (b) ˜ τ - θ diagram, where ˜ τ = τ / ( N k B T ) . In this section, we would like to mention another heat engine called the quantum Szilard enginewhich constitutes a backbone for the reconciliation of thermodynamics, information theoryand quantum mechanics. It has become particularly popular topic during the recent decade,especially because of its interdisciplinary nature and fundamental importance to the many openquestions in physics today. Here we construct a quantum Szilard engine without featuring anexplicit Maxwell’s demon and we explore the role of quantum size and shape effects in thisquantum Szilard heat engine variant. Our purpose in this section is to mention the relevance ofthis important problem to confinement effects. Introduction and examination of the problem infull details can be found in our published article in Ref.
First, let’s start with a brief background of the problem, the Szilard’s paradox. Considera container with a single molecule confined inside. Then divide the container into two equalparts by a partition having zero thickness. Classically, insertion of the partition can be donewithout any work consumption due to its zero thickness. Depending on where particle is (whichis determined by a measurement), the partition moves left or right like a piston, and by theisothermal expansion one can extract work from the engine. In order to complete the cycle,one needs to remove the partition from the system, which also can be done without any workconsumption. One can repeat this cycle in a cyclic manner. As is seen, one can extract workby only taking heat from the heat reservoir. This clearly violates Kelvin–Planck statement ofthe second law of thermodynamics, as there is no heat rejected to the environment and all theheat absorbed is converted to work with efficiency. For the resolution of the paradox,many attempts have been done over the course of the years. The widely accepted resolution ofthe problem is this: in order to operate the cycle, there has to be a "demon" (an informationprocessing being historically called as demon) who records the which-side information of theparticle and in order to complete the cycle, the demon needs to erase it. According to Landauer’sprinciple, it is this erasure process that resulting a heat dissipation and the corresponding workexchange in an isothermal process.In the quantum versions of the Szilard engine, the same logic (erasure resolution) has beenfollowed in the literature. However, we see that there are other additional mechanisms in a quan-tum Szilard engine cycle that saves the second law of thermodynamics. By placing solenoids96 igure 5.5:
A schematic of the quantum Szilard engine from our study in Ref. [248]. Aquantum Szilard engine setup composed of three components, system (the box with a singleparticle inside), measuring device (whose sole purpose is to localize the particle via quantummeasurement) and heat bath (keeps all processes isothermal). (I → II) Partition is inserted intothe container. Insertion divides the container into two equal parts and generates an entangledstate of the particle’s position. (II → III) Quantum measurement is performed to localize theparticle into one side of the box. (III → IV) Particle expands the partition and work can beextracted from the system. (IV → I) The partition is removed from the container at the boundary,which completes the cycle.and attaching magnetic rods to the piston at both sides, regardless of which side the pistonmoves, one can convert the motion of the piston into electrical power by the solenoids, and eveninto direct current power by a passive diode bridge. This technique allows one to extract workfrom a Szilard engine, without knowing the which side the particle is and therefore withoutusing any information processing device like Maxwell’s demon.Let’s consider our quantum Szilard engine variant which has this type of setup with solenoids.There are four stages in the cycle: insertion, measurement, expansion and removal of the parti-tion, Fig. 5.5. Because of quantum size and shape effects, there has to be work done to insertthe partition into the system even if the partition has zero thickness. This is because even azero-thickness partition has a finite effective size due to quantum boundary layer in confinedsystems.
42, 44
After the insertion of the partition, particle finds itself in superposition of being two sidesat the same time. In that stage, the cycle won’t operate, because the pressure at both sideswill be exactly equal to each other. In order to break the quantum superposition and localizethe particle, one needs to perform a quantum measurement on the particle (which destroys theentanglement that is formed with the separation), only then particle is localized and one canextract work by expansion. This fact is independent of whether we gather, process or evenuse the which-side information or not. After the expansion, removal of the partition is a trivialprocess in a single-particle Szilard engine. Since the partition will be exactly at boundary of the97ontainer where the wavefunction goes to zero, removal of the partition does not change anythermodynamic property of the system.In our article, we quantify work, heat and energy exchanges during the thermodynamiccycle of quantum Szilard engine and obtained them analytically using the quantum boundarylayer method. In this section, we won’t examine the cycle fully, but we will give a brief exampleof how quantum size and shape effects play a role in the quantum version of the Szilard engine,as well as how quantum boundary layer helps us to understand the physical mechanisms of theprocesses. What we will focus on is the insertion process, where size and shape effects maketheir explicit difference. Numerical simulation of the insertion process is given in Fig. 5.6 alongwith the changes in free energy, entropy and internal energy during the insertion.
Figure 5.6:
Numerical simulation of the quasistatic isothermal insertion process for a Szilardbox with sizes L x = 20 nm, L y = 10 nm and at temperature T = 300 K. y is the depth of thepiston inserted into the domain. (a) Quantum-mechanical thermal probability density distribu-tion of the particle is non-uniform inside the container due to confinement effects. Magnitudesof the probability density distributions are represented by the rainbow color scale, where redand blue colors denote higher and lower density regions respectively. Partition with zero thick-ness enters the container in y -direction, y = 1 nm. (b) Partition enters almost halfway of thecontainer, y = 4 nm. Although it has no thickness, confined particle perceives a finite effectivethickness of δ . (c) Partition is at y = 7 nm depth. (d) Partition separates the container into twoequally sized parts, y = 10 nm. Particle has equal probability to be at both sides. Variation of(e) free energy, (f) entropy and (g) internal energy with respect to partition’s penetration depth y in nm’s.As is seen from the thermal probability density distribution of the particle inside the do-main, in Fig. 5.6(a,b,c and d), although the partition has zero thickness, the particle perceivesan effective thickness of δ as quantum boundary layer methodology suggests. Insertion ofthe zero-thickness wall reduces the effective domain inside the box, even though the apparentvolume (here it is area) stays the same. This is a size effect. Thermodynamic state functionsact accordingly by the influence of quantum size effects. However, when the partition comesvery close to the bottom boundary, we see overlaps of quantum boundary layers of the partitionand the bottom boundary. These overlaps suggest there is a quantum shape effect as well in this98omain. Accordingly, after around y = 8 nm depth, we start to see deviations from the usual be-haviors of thermodynamic state functions, Fig. 5.6(e,f and g). Work and heat exchanges duringinsertion is respectively calculated by the differences of free energy and entropy in the initialand final positions of the partition. Therefore, both quantum size and shape effects play rolein the determination of the insertion work which is finite, contrary to the classical case whereinsertion work is said to be zero. In the future, we are planning to investigate more on the quan-tum shape effects in 1D confined systems to understand their behaviors more fundamentally.We won’t go into further detail in this subject as we said before and will continue to this chapterwith a novel energy conversion approaches that exploit quantum shape effects. Thermoelectricity is the direct conversion of thermal energy to electric energy and vice versa.When two materials with different charge transport properties like different density of chargecarriers combined in a junction under a temperature difference, an electrical potential differenceis induced in response to the temperature gradient. Building up of the electrical potential froma temperature gradient is called the Seebeck effect, whereas generation of cooling and heatingfrom an electrical current is called the Peltier effect. Usually thermoelectric junctions consist ofn-type and p-type thermocouples where electrons and holes are responsible for charge transportrespectively. Provided that one end is at fixed chemical potential, difference in the electronictransport properties of electrons and holes causes a difference in the electrochemical potentialat the other end of the junction. This potential difference can be utilized as an electric volt-age at the separate end of the thermocouple and it is called the thermoelectric voltage. In asimilar fashion, when operated reversely, the electric voltage can be converted into a tempera-ture difference at both ends. Due to their durability, silent operation, scalability, precision andeco-friendliness, thermoelectric generators and coolers are of great interest both academicallyand commercially. Downsides of thermoelectrics are their high costs and relatively low energyconversion efficiency rate, although the expectation for their theoretical efficiency was muchhigher due to the nature of direct conversion process (being free of energy losses that happen inother multi-stage conversions of energy from one form to another).During last two decades, numerous studies have been done in thermoelectric research. It hasbeen shown that performance of thermoelectric devices can be improved by taking advantageof materials at nanoscale. Rather than focusing the conventional route, some studies have beenfocused on designing different kind of junctions both to understand the thermoelectric effectbetter at small scales and to contribute to the development of novel energy devices. One ap-proach is utilizing quantum size effects not only for improving the thermoelectric performanceof the materials individually, but also to directly design thermoelectric devices based on quan-tum size effects. Instead of creating p-n junctions, it has been proposed to design junctionswith the same material but having different sizes. When the temperature gradient is applied,an electrochemical potential difference is induced in those materials even if all their materialproperties are the same except their sizes. The electrochemical potential difference occurs dueto difference in quantum size effects on Seebeck coefficients of the pillars of the junction. Thisis called thermosize effect and it has been proposed by Sisman and Müller. Thermosize effecthas been studied during last decades.
29, 100–114, 254
We can aggregate these type of thermoelec-tric mechanisms under the umbrella term that we call single-material unipolar thermoelectrics.99 igure 5.7:
A schematic of a thermosize junction with narrow and wide graphene nanoribbons.Narrow and wide ribbons are separated by insulator on the middle, thermally and electricallyconnected on the cold side and only thermally connected on the hot side. Although the junctionis made of the same material, their size difference leads to an electric voltage on the hot side.An example of such setup is shown in Fig. 5.7 where we examined the thermosize effect ingraphene nanoribbon junctions.Apart from the topics about quantum shape effects on thermodynamics considered in thisthesis, recently we investigated whether quantum shape effects can also affect transport proper-ties of charge carriers and result a new type of single-material unipolar thermoelectric device.
We designed a thermoelectric-like junction shown in Fig. 5.8. We used Landauer formalismalong with Datta’s number of modes approach and tight-binding model to calculate transportproperties of core-shell nanostructures (basically the same nested-square domain we consideredin this thesis).
To maximize the influence of shape difference on the system, we consid-ered pure ballistic weakly degenerate transport regime with positive but very low chemicalpotential. We used GaAs as the shell structure which has a low effective mass. We conductedour analysis at low temperatures relative to room temperature.We showed that due to quantum shape effects on Seebeck coefficient under a tempera-ture difference, thermoshape potential is induced. At maximum thermopower region, the ther-moshape voltage is in the order of several millivolts per Kelvin for cold side temperature of 20Kwhen temperature difference of 2K. Thermoshape junction constitutes not only a viable setup
Figure 5.8:
A schematic of a thermoshape junction with core-shell nanostructures having ◦ and ◦ core structure angular configurations on the left and right pillar respectively. Eventhough each pillar of the junction is made of the same material having exactly the same sizes,only the shape difference leads to an electric voltage on the hot side.100or the macroscopic manifestation and demonstration of quantum shape effects, but also theirfirst possible device application. 101 Conclusion
In this thesis, we introduced a new aspect of finite-size effects. We proposed the quantum shapeeffect which appears in strongly confined structures. We provided a thorough examination ofthe origins and the nature of quantum shape effects in the thermodynamics of non-interactingparticles confined in impenetrable nanodomains. The main motivation was to explore this pre-viously unnoticed effect and understand the fundamental physics behind it. We extended thequantum boundary layer methodology to capture the quantum shape effects in addition to thequantum size effects. We supported our findings both with numerical simulations and analyticalcalculations. Finally, we constructed quantum heat engines driven by quantum shape effects, inorder to demonstrate possible thermodynamic applications of the effect.Quantum shape effect is essentially a result of the wave nature of particles and the discreteenergy spectrum caused by the confinement potential. To observe the effect, the conditionsshould be in favor of a comparable de Broglie wavelength with the domain sizes. Required con-ditions are not sufficient of course unless one does not create the conditions where boundariesof the system get really close to each other so that quantum boundary layers start to overlap.Core-shell nanoarchitectures are suitable candidates for the realization of nested domains. Inrecent years, research on core–shell semiconductor nanocrystals is also developed giving theprecise control of size and shape of these structures. Other candidate nanoarchitectures forquantum shape effects are easily shapeable nanostructures like graphene sheets or nanoribbons.Controlling material properties by shape can open up a whole new direction in material science.Waveguides are the systems where the optical analogue of quantum shape effects can be ob-served. Optical versions of this effect can be exploited for instance in double-clad fibers wherethe pumping energy’s absorption efficiency may be tuned by smooth shape variation.
Onecan also artificially create confinement conditions using optical traps where ultracold atoms orthe system of interest is being confined.
There are some advantages that make quantum shape effects realizable in labs and someapplications. For example, while most quantum systems can operate at cryogenic temperatures,quantum shape effects can appear even at room temperatures. It is not just a single and isolated102ffect, but can be demonstrated indirectly via many different physical mechanisms some ofwhich are presented in the previous chapter. However, the effect is strong only in very small,extremely confined systems which may be challenging to manufacture smoothly with ease. Theeffect depends strongly on the disorders and imperfections on the boundaries of the materialor inside the confinement domain. If the imperfections are small and random, which is mostlythe case, we expect quantum shape effect to survive because the influence of disorders will beaveraged out for all shape configurations and they will behave as fluctuations over the overallquantum shape effect behavior.Since the quantum shape effect is completely a new discovery, there are abundant things tostudy both from theoretical and application sides. It is a general, material-independent effectthat can possibly be seen in many other exotic nanoscale systems like perhaps topological orsuperconductor materials.
As a near future work, we are planning to investigate quantumshape effects in Bose-Einstein Condensates, where it may be possible for instance to controlthe condensation temperature by shape. Thermodynamic and transport properties of variousFermionic or Bosonic systems can be also explored under quantum shape effects. Topologicalthermodynamics in systems having more than one hole (note that nested domains investigatedhere have a single hole which is the core structure) can also be investigated.Exploration of pure 1D systems is also of great theoretical importance as they constitute thesimplest but, in many ways, exact systems that can be understood both physically and mathe-matically. We’ve already had some interesting results on such systems which may shed lightonto various nanoscale thermodynamic and transport phenomena. A more clear and analyticalunderstanding of energy moments would also help to clarify the notions of pressure, flux anddirectional quantities at quantum scale.We restricted ourselves to quasistatic and time-independent processes in this thesis. Ourfuture plan is to explore the quantum shape effects in interacting, time-dependent systems aswell as with finite (constant, harmonic, quartic, etc.) confinement potentials. Opening electricalor magnetic fields, would possibly yield even more interesting direct or cross effects and maylead to further novel nanoscale devices. Study of the shape effects in quantum coherent sys-tems is also a nice direction of research that would also help to the understanding of quantumthermodynamic systems. 103
Appendix: Supplementary Information
Here we give some additional details about the calculations done in the thesis. Theoreticalworks and analytical calculations are conducted by use of pen and lots of paper. A mathemati-cal computation and symbolic manipulation software
WOLFRAM MATHEMATICA is extensivelyused for calculating algebraic equations, generating figures and post-processing data. A numer-ical finite element analysis software
COMSOL MULTIPHYSICS is used for calculating eigenval-ues, eigenfunctions and other kind of numerical quantities, as well as for graphics and visualrepresentations.Also, in this Appendix A, we mention some attempts that we’ve done and approaches we’vetried to implement for the improvement of the results and understanding of some parts of thethesis work. They don’t necessarily contribute to the presented final results in the main parts ofthis thesis, but definitely increased our understanding and knowledge of the effect that we aredealing with.
A.1 Details of Numerical Calculations
We first created our confinement geometry with Dirichlet boundary conditions in
COMSOL soft-ware and used
Coefficient Form PDE in the
Mathematics module for the solution of Schrödingerequation for our arbitrary domain. The software uses finite element methods to numericallysolve partial differential equations. There are three different finite element solution methodsembedded which are MUMPS, PARDISO and SPOOLES. In terms of accuracy, we haven’tnoticed any detectable difference in our tests for the stationary Schrödinger equation under theboundary conditions considered in this study. However, flexibilities and memory allocationproperties of these methods are different. From our trials, we see that MUMPS is the fastestand the most flexible one in terms of allowing optional memory allocation. Hence, we choseMUMPS method.We wanted to have a high numerical precision in our eigenvalues. Using the correct meshingis therefore a crucial issue in finite element method. The software has a bunch of mesh types104 igure A.1:
Dense triangular meshing of our confinement domain in
COMSOL environment.Local refinements are possible when necessary.and a wide range of mesh control parameters, including adaptive mesh capabilities (which isan important feature because of the local tiny perturbations that we consider in some cases).After our tests, we realized that free triangular mesh gives better accuracy than other types ofmeshes for our domains. Investigation of mesh errors is done by considering the number ofmesh elements inside the minimum wavelength corresponding to the highest energy eigenvalueconsidered during the calculation of sums, so that increasing the number of mesh points won’tcause any difference in the eigenvalue solutions. To this aim, we divided the wavelength cor-responding to the largest eigenvalue that we truncate after, to the maximum size of our meshelement and found the number of mesh points inside the minimum wavelength. Then by in-creasing the number of mesh elements, we made a convergence analysis and determined thenumber of mesh elements accordingly. Around 70000 mesh elements are more than enoughfor the considered domains in the thesis. (Considered element size parameters of the meshingare: maximum element size: 0.1 nm, minimum element size: 0.000424 nm, maximum elementgrowth rate: 1.1, curvature factor: 0.2, resolution of narrow regions: 1) An example of themeshing that we used is given in Fig. A.1.In addition to
COMSOL software, for comparison we’ve made similar finite element calcu-lations also in
MATHEMATICA software using its built-in
NDEigensystem function. The resultswere the same, so we choose
COMSOL for our eigenvalue calculations and
MATHEMATICA forall other calculations and data processing.Summations in statistical quantities are over all possible states which are infinitely many. Nowonder, sums that we use are naturally convergent and one can truncate after some finite amountof terms. To determine the truncation point of eigenvalues, we made a truncation analysis. Theproper number of eigenvalues depends on the particle’s mass, temperature and the sizes of thedomain. By choosing bare electron mass, room temperature and nanoscale domain sizes (takingtransverse confinement parameter as unity), we determined the optimum number of eigenvaluesfor our arbitrary domain that we target to solve. We make sure that truncation errors are alwaysnegligible in all of our calculations. 105 igure A.2:
Comparison of global and local boundary perturbation approaches. Keeping theaxial symmetry is important.We repeated the same solution procedure for each and every angular configuration that wewanted to consider. To span the angular ranges from ◦ to the symmetric periodicity angles, weused sampling. For the correct sensitivity of the sampling, we compared our results for ∆ θ = 1 ◦ and ∆ θ = 0 . ◦ which we see no difference in the results.Numerical calculations are tested also with exactly solvable analytical models and it is en-sured that all steps of numerical calculations are consistent and produce negligible total errors. A.2 Global and Local Boundary Perturbations
Based on the physical quantity that is interested in, one can probe system’s free energy responseto the global or local perturbations on the boundaries, which are summarized in Fig. A.2. Tocalculate the torque (a global effect) exerted on inner square structure, we create small angularperturbations. We look to the difference of the unperturbed and perturbed free energies withrespect to the angular perturbation, which gives the torque. Similarly, a linear perturbation onall boundaries of inner square is done to calculate the exerted pressure.As a justification of the existence of torque, we argue that pressure distribution shouldchange non-uniformly along the walls of inner square. In order to find this pressure distri-bution, we implemented a local perturbation approach. We created tiny (having very smallwidth and height) perturbations along the walls of inner square. There are many subtle pointsto consider when creating such local perturbations. How large can they be both in terms ofwidth and height? Can and should they be infinitesimally thin? Should we create them inwards(removing a tiny part from the domain) or outwards (adding a tiny part to the domain)? Wemade extensive trials conceiving all of these along with many other considerations. There is noway to obtain results precisely and without affecting the system of course. By creating eventiny perturbations, we unavoidably change the actual area, periphery and number of vertices ofour domain. This leads to some variation in eigenvalue spectrum which we wouldn’t want tohappen. In order to reduce this influence, we could create zero width perturbations (creatinga perpendicular line boundary) instead of finite width ones. Hence, actual area stays the same106 igure A.3:
Comparison of outward and inward local perturbations.whether it is perturbed or unperturbed, but just the periphery and vertices change. However, inspite of keeping the actual area constant during the zero width perturbations, the effective areawhich is felt by the system changes due to the quantum boundary layer that is generated by theperturbation even if it has zero thickness.Another thing to notice is while global perturbations preserve the Euler characteristic (atopological invariant defined as the sum of the number of vertices and faces minus the edges ofa shape) of the domain, local perturbations do not. Thus, not only shape but also size variablesof the domain changes, when the Euler characteristic is not preserved. Tiny changes in area,periphery and number of vertices occur. Since the changes are tiny as the perturbation, lowesteigenvalues are almost not affected from these variations, but only higher eigenvalues of whichthe contributions to the global physical quantities are less. All in all, based on our tests, fi-nite width perturbations give more accurate results than other local perturbations, whereas zerowidth perturbations give much more smoother graphs.One other distinction of the perturbations is whether they are inwards or outwards, see Fig.A.3 for their comparison. Both of them affects the domain in a distinct way such as the sufficientmesh density differs. In the outward one, domain barely feels the existence of the perturbation,therefore much finer mesh is required than the inward perturbation. We have found an analyticalformula representing the influence of finite inward (+) and outward (-) perturbations to theexerted pressure, which results to the following modified expression of pressure: P ± = n cl k B T (cid:20) ±
12 ∆ hL + 2 δ (cid:18) L ∓ ∆ h (cid:19) ± (2 δ ) (cid:18) L ∓ ∆ h ) ∓ L (cid:19)(cid:21) (A.1)which is derived originally from Eq. (4.18). δ is the thickness of quantum boundary layer. Thesecond term in the square bracket is the second order term in Taylor expansion, the third termis the quantum size effect correction and the last term represents the second order quantum sizecorrection. The second and third term are more or less comparable to each other and their be-havior is determined by the ratio of ∆ h/L as well as temperature. As long as the perturbationdepth ∆ h is much smaller than L , errors coming from this finite difference operation are rea-sonably small. Both perturbation methods give similar results, and we used the inward one dueto its mesh efficiency. A.3 Local Momentum Flux Approach
There is one other approach we’ve tried for the calculation of torque, pressure and applicationpoints. It is a local approach again, but rather than making perturbations along the boundariesof the domain, we find the local quantum-mechanical momentum flux distributions inside the107omain. We sought for a consistency between the pressure determined from free energy byglobal boundary perturbations and the local momentum flux of the particles. We exploited theequivalence of pressure to the momentum flux, which we showed it in Appendix A.6. Ourpurpose was to find the local positions near to boundaries of the inner square walls where localmomentum flux equals to the pressure exerted to the wall. Thereby, we wanted to show theformation and existence of a pressure boundary layer. If we could generalize it to any domainlike it has been done in quantum boundary layer, then it might be possible to obtain the localpressure at any position along the domain boundaries without doing extensive calculations.Unlike the other approaches, eigenvalues are not sufficient to calculate the exact local prop-erties, we need eigenfunctions as well. We numerically calculated eigenfunctions correspondingto the already obtained eigenvalues. Just like in the standard derivation of probability flux inquantum mechanics, one can obtain the momentum flux by using the Schrödinger equation inconjunction with continuity equation as follows: ∂ρ p ∂t + (cid:126) ∇ · j p = 0 , ˆ H Ψ = i (cid:126) ∂ Ψ ∂t ⇒ j p = m (cid:0) Ψ ∗ ˆv Ψ − Ψ ˆv Ψ ∗ (cid:1) = ˆv ρ p (A.2)where ρ p = m Ψ ∗ ˆv Ψ is momentum density, ˆv = − ( i (cid:126) /m ) (cid:126) ∇ is the velocity operator and j p ismomentum flux operator. Note that j p contains the summation of both left and right fluxes de-scribed by the Wronskian W (Ψ , Ψ ∗ ) and gives zero at equilibrium. However, we are interestedin the flux towards the inner domain walls which is a one-sided flux. Total momentum flux in Figure A.4:
Finding the local momentum flux - pressure equivalence line (pressure boundarylayer) near to the boundary where pressure exerts. Point of applications can be found by inte-grating the local momentum flux along the pressure boundary layer that is parallel to the actualboundary. 108ne direction can simply be found by using statistical mechanics, J p = (cid:88) ε j p f = − N (cid:126) m (cid:80) ε exp( − ˜ ε )Ψ ∗ (cid:126) ∇ Ψ (cid:80) ε exp( − ˜ ε ) = − N (cid:126) m (cid:68) Ψ ∗ (cid:126) ∇ Ψ (cid:69) ens = N m (cid:68) Ψ ∗ (cid:126) ∇ Ψ (cid:69) ens (A.3)where f is the relevant distribution function (here the Maxwell-Boltzmann is shown as an ex-ample) and ens subscript with brackets denotes the ensemble average. This approach is knownalso as quantum hydrodynamics formalism. Calculation of point of applications now canbe done by integrating momentum flux times their application points along the wall and divid-ing it to the boundary layer integral of momentum flux. All of which can be obtained usingeigenvalues and eigenfunctions.In Fig. A.4, we show the line where local momentum flux equals to the non-uniform pres-sure exerted on a tiny partition of the wall. The local momentum flux along this equivalenceline is plotted for the particular part of the domain that is shown in the Figure. A small peakcan be seen near to the region where two boundaries approach themselves. This is an expectedbehavior of such system, as we will see the similar type of functional behaviors in our pure 1Danalyses in Section A.4. Calculation of application points and torque are also given in Fig. A.4.We have tried various numerical attempts to make the approach work, however in the end,we abandoned this approach as it gives very low numerical precision even though we obtainedcorrect functional behaviors most of the time. Besides, there is no need to force in this direction,since the local pressure distribution that is calculated from the free energy by local perturbationsprovides quite reasonable and clean results. Nevertheless, we acquired some physical intuitionabout the results which strengthens our interpretations.
A.4 Energy Moments and Density Distributions
For the sake of understanding more of the local properties of our system, we investigated en-ergy moments of the ensemble-averaged densities. Here as an example we used the Maxwell-Boltzmann distribution, but any other distribution can also be used. The general expression forvarious energy moment densities is written in its dimensionless (normalized to classical values)form ˜ n ( m ) ε = n ( m ) ε n ( m ) cl,ε = (cid:80) ε ˜ ε m/ exp( − ˜ ε ) | Ψ( r ) | V (cid:80) ε ˜ ε m/ exp( − ˜ ε ) , (A.4)where m superscript denotes the order of the energy moment. For m = 0 , it gives the usualparticle density whereas for m = 1 and m = 2 , it gives the momentum and energy densitieswith position dependence. Therefore ˜ n ( m ) ε ( r ) represents the particle, momentum and energydensity distributions for a particular coordinate inside the domain so that n (0) ε = n , n (1) ε = p and n (2) ε = u . Numerical simulation results of the energy densities for nested square domain isgiven in Fig. A.5.Examination of different moments of energy may provide valuable information about thelocal properties of the system. Particle density distribution, as we have shown before, givescrucial information about the behavior of particles confined in such a domain shape, onto whichthe quantum boundary layer methodology is built. It also provides a tentative visual explanationabout the existence of torque. Similarly, momentum distribution inside the domain, which isobtained when m = 1 gives information about how the momentum of particles is distributed109 igure A.5: Energy moment densities together with the corresponding quantities. (a) Particle(b) isotropic momentum and (c) energy density distributions inside the nested square domainfor θ = 25 ◦ configuration.along the confinement domain. This justifies the non-uniform distribution of momentum andenergy inside the domain. Note that despite intuitively correct, it does not always mean thatthe less particle localized in a region the less momentum flux. Proper examination can be doneusing momentum flux density distribution. Although momentum flux density is a tensorialquantity, in this section for simplicity we only plot isotropic momentum density, which is avectorial quantity just like the energy flux density. In other words, by taking the square root of itssquare we turned it into a scalar quantity first, but since it is a flux density, it becomes vectorial.Energy density distribution ( m = 2 ) also gives useful information about why confinementenergy is higher for some angular configurations and lower for some others. We comparedparticle density, isotropic momentum and energy densities for the nested square domain in Fig.A.5.Quantum boundary layer theory was constructed considering 1D domain and it has beenproven to be successful on many different confined systems with higher dimensions. To thisend, here we would like to investigate again 1D system and plot the distributions of energymoment densities in Fig. A.6 for confinement values of α = 0 . and α = 1 correspondingto the moderate and strong confinements. Blue, red and purple curves represent the particledensity, momentum density and the energy density respectively. The results are in coherencewith the Fig. A.4 where we’ve found the momentum flux distribution from a completely dif-ferent approach. Characteristic peaks of momentum and energy densities are present. One canfind the corresponding fluxes by multiplying the expressions with the velocity of the particulardirection. Momentum flux equals to the internal energy density in 1D case, since there is nooff-diagonal terms for the possibility of velocity combinations in 1D.Using Poisson summation formula, for 1D case, it is possible to obtain analytical expres-sions for density distributions for all moments. The expression we found for Maxwell-Boltzmannstatistics is written as ˜ n ( m ) ε = (cid:80) i ( αi ) m exp ( − ( αi ) ) | Ψ( r ) | V (cid:80) i ( αi ) m exp ( − ( αi ) ) ≈ (cid:20) − F (cid:18) m + 12 ; 12 ; − π ˜ x α (cid:19)(cid:21) (cid:20) − F (cid:18) m + 12 ; 12 ; − π (1 − ˜ x ) α (cid:19)(cid:21) (A.5)where F ( a ; b ; c ) is Kummer confluent hypergeometric function. In the Fig. A.6, we compared110 igure A.6: Comparisons of exact and analytical expressions for two different confinementstrengths (Left figure α = 0 . and right figure α = 1 ) of density distributions of differentmoments. Solid curves represent the exact (numerical) calculations whereas the dashed curvesrepresent the analytical calculations based on Eq. (A.5). Blue (cyan), red (orange) and purple(pink) curves represent the exact (analytical) results for m = 0 , m = 1 and m = 2 momentsrespectively.the accuracies of the analytical results with the exact ones. As is seen from Fig. A.6, analyticaland numerical results are in a very good agreement for α = 0 . , especially for m = 0 and m = 1 cases. Our analytical representation gives accurate results for all moments as long asthe confinement is not very strong so that the conditions of bounded continuum is satisfied. Itis also seen that for large confinement values like α = 1 , the results of numerical and analyticalcalculations start to deviate. The larger the confinement, the higher the deviation. The analysisin Fig. A.6 explains also the energy moment densities that are compared in Fig. A.5. Higherthe moment, closer the density near to the boundaries. A.5 The First Order Quantum Boundary Layer Approach
As we said before, in order for quantum shape effects to appear in confined systems, overlaps ofquantum boundary layers are essential. The method suggested in this thesis is quite successfulas it is shown by many examples. Nevertheless, a room for improvement to the methodologyis also possible which we would like to mention here. The usual quantum boundary layermethodology is an approximate one as is known. It approximates the gradually changing densitydistribution with a step function which can only have values 1 or 0. Calculating overlaps withthis method was possible and indeed proved to be accurate enough to represent quantum shapeeffects in most of the confined systems in this thesis. On the other hand, in reality overlaps donot start abruptly when the distance between boundaries is shorter than δ as it can be clearlyseen from Fig. 3.10 where there is a variation in effective area even before the critical valuewhere the overlaps start according to the quantum boundary layer method. Evacuation of theregion near to boundaries start not from δ or even δ , but visibly around . δ , see Fig. A.7 blackcurve. We can make a better approximation to the exact density distribution of particles nearto boundaries. Instead of the usual stepwise approximation, we can approximate the densitywith a ramp function, Fig. A.7. Let’s call the stepwise one as th order and name the ramp oneas the st order quantum boundary layer approaches. This should definitely give much more111ccurate results than the usual th order quantum boundary layer approach. The comparison ofboth approaches is given in Fig. A.7 where they are compared with the exact distribution. Figure A.7:
Approximations of quantum boundary layer for density distributions. Ensemble-averaged quantum density distribution of particles confined in a 1D domain with length L .Solid black and dot-dashed green curves represent the exact and classical density distributionsrespectively. th order quantum boundary layer is presented by dotted red lines where the exactdensity distribution is approximated by empty regions with thickness ˜ δ near to boundaries anduniform density region in the remaining parts. st order quantum boundary layer denoted bydashed blue lines is the linear approximation to the non-uniform density region with thickness δ . Here ˜ n = n/n cl , ˜ δ = δ/L and ˜ x = x/L where x is the position.The shape dependence of partition function has a sigmoid type functional behavior whichcan be mimicked by tanh function. We have found hybrid ways to approximate the effectivearea by using the st order approach for less complicated geometries and th order for others.However, we need to establish an elegant methodology with a physical meaning rather than justdoing some mathematical tricks.We’ve done several attempts to construct an analytical or at least a numerical approachfor the development of st order quantum boundary layer methodology. In this st order QBLmethod, the overlaps of the opposing boundaries start from δ distance instead of δ . Thereforeit is possible to capture even tiny details about the density distribution. Similar to the th ordermethod, middle regions are equal to / (1 − δ/L ) , but there is a gradual decrease near tothe boundaries, rather than a sharp one. The main difficulty is unlike the th order method,in this one, overlaps would have a weight factor. Near boundary overlaps would have verydifferent contribution to the effective density (or effective area) than the far boundary overlaps.This brings additional variables to the problem of overlap volume calculations and makes theanalytical geometric calculations highly complicated due to sensitive positional dependence ofthe QBL. Nevertheless, we defined a new parameter called the area amplitude renormalizationfactor, as its name suggests, it renormalizes the contributions of the overlaps to the effectivearea. A rough picture of the methodology is presented in Fig. A.8. For the simplest cases of112 igure A.8: Comparison of th and st order quantum boundary layer approaches. In the domainboundary analysis, real boundaries are shown by black color. The th order quantum boundarylayers are represented by solid orange and cyan colors respectively, whereas corresponding st order ones are shown by dashed lines. Details of area amplitude renormalization is not givenhere.nested square domain (which are ◦ and ◦ degree configurations) we predicted the correcteffective area with relative errors much less than . , which is remarkably accurate. However,implementing the same procedure is very difficult analytically for the degrees in between, letalone the arbitrary geometries.In order to test whether our analytical approach could be reliable at all for any kind ofdomain, we decided to invoke numerical methods. We applied a Monte Carlo method to testthe accuracy of the proposed methodology for arbitrary domains. The algorithm (see Fig. A.9)goes like this: (1) First, we draw our domain’s boundaries and assign random points insideof our confinement domain. (2) Then for each point, we take the distance from the point tothe normal (perpendicular) of the closest boundary. (3) We calculate the weight factor of thatpoint according to the area amplitude renormalization factor of st order approach and assignthat value for that particular point. (4) We repeat this procedure for each and every point,thereby generating the domain’s density distribution as a whole. The approach gives the correctfunctional behavior with some fluctuations and with moderate errors overall. Of course, thereare many non-trivial things to be considered, like how we calculate perpendicular distancesfrom and to the angled boundaries. Numerical errors are also concern here.Instead of the st order approach, another possible improvement of QBL could be to gointo the heart of the problem of determining the QBL thickness. Instead of using the first twoterms of PSF, we may use elliptic theta functions as full solutions for the partition functionunder infinite summation form and then calculate QBL thickness from the full solutions, whilekeep staying within th order approach. How does QBL thickness modify in strongly confinedregions? Should we stick with the overlap approach or can we come up with a better, even moreaccurate one? We don’t know yet. The improvements on QBL methodology is still ongoing.113 igure A.9: Implementation of Monte Carlo method for the calculation of density distributionsby analytical st order quantum boundary layer approach. A.6 Pressure - Momentum Flux Equivalence
In this section, we derive and show the mathematical equivalence of pressure and momen-tum flux in rectangular confinement domains for Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein statistics. We start with the derivations in Maxwell-Boltzmann statistics.
A.6.1 Equivalence for Maxwell-Boltzmann statistics
Thermodynamic pressure, Helmholtz free energy (for large N ) and partition function in Maxwell-Boltzmann statistics are given respectively as P = − (cid:18) ∂F∂V (cid:19) , (A.6) F = − N k B T (ln ζ − ln N + 1) , (A.7) ζ = (cid:88) { i ,i ,i } exp( − ˜ ε ) , (A.8)where ˜ ε = ε/k B T are dimensionless energy eigenvalues of a simple 3D rectangular domain.When we plug partition function into the free energy and then the resulting expression into the114ressure, we get the pressure by following steps: P = − L L ∂∂L ( − N k B T ln ζ ) = N k B TL L ∂α ∂L ∂∂α ln ζ = N k B TL L (cid:18) − α L (cid:19) ζ ∂ζ∂α = − n cl k B T α ζ (cid:88) (cid:0) − α i exp( − ˜ ε ) (cid:1) = 2 n cl k B T (cid:80) ( α i ) exp( − ˜ ε ) (cid:80) exp( − ˜ ε ) = 2 n cl k B T (cid:80) ( α i ) f (cid:80) f . (A.9)Pressure exerted to the L × L side (the area) of a rectangular box is then obtained as P = 2 n cl k B T (cid:80) ( α i ) f (cid:80) f . (A.10)To derive the pressure, we relied on thermodynamics equations. Now let’s derive the mo-mentum flux by considering the transport equations. From the transport theory, scalar momen-tum flux, translational kinetic energy and distribution function are given respectively as J p = 1 V m (cid:88) v f, (A.11) mv = k B T ( α i ) , (A.12) f = exp(Λ) exp( − ˜ ε ) . (A.13)Combining these equations gives the momentum flux J p = 1 V k B T exp(Λ) (cid:88) ( α i ) exp( − ˜ ε ) . (A.14)From the number of particles expression N = (cid:88) exp(Λ) exp( − ˜ ε ) , (A.15)fugacity is written as exp(Λ) = n cl V (cid:80) exp(˜ ε ) . (A.16)When we plug the fugacity into Eq. (A.14), we get momentum flux J p = 1 V k B T n cl V (cid:80) exp(˜ ε ) (cid:88) ( α i ) exp( − ˜ ε ) . (A.17)By simple reordering, it becomes J p = 2 n cl k B T (cid:80) ( α i ) exp(˜ ε ) (cid:80) exp(˜ ε ) = 2 n cl k B T (cid:80) ( α i ) f (cid:80) f . (A.18)So it is shown that pressure, Eq. (A.10), is equal to momentum flux, Eq. (A.18), in Maxwell-Boltzmann statistics. 115 .6.2 Equivalence for Fermi-Dirac and Bose-Einstein statistics The derivation in Fermi-Dirac and Bose-Einstein statistics are done together in this section.Thermodynamic pressure, Helmholtz free energy and partition function are given respectivelyas P = − (cid:18) ∂F∂V (cid:19) , (A.19) F = N µ − k B T Z, (A.20) Z FDBE = (cid:88) { i ,i ,i } ln [1 ± exp(Λ − ˜ ε )] , (A.21)where the upper and lower signs correspond to Fermi-Dirac and Bose-Einstein statistics respec-tively in all equations that they appear. Plugging the free energy into pressure gives P = − k B TL L ∂∂L ( N Λ − Z ) = − N k B TL L (cid:18) ∂ Λ ∂L − N ∂Z∂L (cid:19) . (A.22)There are derivatives in Eq. (A.22) which need to be found analytically. Derivative of dimen-sionless chemical potential with respect to L can be found by taking the advantage of the factthat number of particles does not depend on the changes in L . From the derivative of N withrespect to L we get ∂N∂L = 0 → ∂ Λ ∂L = − L (cid:80) ( α i ) f (1 ∓ f ) (cid:80) f (1 ∓ f ) . (A.23)Also, the derivative of partition function with respect to L can be found directly as ∂Z∂L = − L (cid:88) f (cid:80) ( α i ) f (1 ∓ f ) (cid:80) f (1 ∓ f ) + 2 L (cid:88) ( α i ) f. (A.24)By plugging these derivatives into Eq. (A.22), pressure becomes P = 2 n cl k B T (cid:80) ( α i ) f (cid:80) f . (A.25)This expression has exactly the same form of the one derived in Maxwell-Boltzmann statisticsin the previous section. The only difference comes from the distribution functions.The distribution function in Fermi-Dirac and Bose-Einstein statistics is given as f FDBE = 11 ± exp( − Λ + ˜ ε ) . (A.26)Combining Eq. (A.11) and Eq. (A.12) with Eq. (A.26) gives the momentum flux as J p = 1 V k B T (cid:88) ( α i ) ± exp( − Λ + ˜ ε ) . (A.27)From the number of particles N = (cid:80) f , and the expression for density V = n cl N , (A.28)momentum flux then becomes J p = 2 n cl k B T (cid:80) ( α i ) f (cid:80) f . (A.29)which is the same as the expression Eq. (A.18). Thus, it is seen here that pressure and momen-tum flux are mathematically equivalent independent of statistics.116 ibliography D. Kondepudi and I. Prigogine.
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