Magnetic refrigeration with paramagnetic semiconductors at cryogenic temperatures
MMagnetic refrigeration with paramagnetic semiconductors at cryogenictemperatures
Alexander Vlasov, Jonathan Guillemette,
2, 3
Guillaume Gervais, and Thomas Szkopek McGill University, Department of Electrical and Computer Engineering, Montreal, QC, Canada,H3A 0E9 McGill University, Department of Physics, Montreal, QC, Canada, H3A 2T8 John Abbott College, Department of Physics, Montreal, QC, Canada, H9X 3L9 (Dated: 24 September 2018)
We propose paramagnetic semiconductors as active media for refrigeration at cryogenic temperatures byadiabatic demagnetization. The paramagnetism of impurity dopants or structural defects can provide theentropy necessary for refrigeration at cryogenic temperatures. We present a simple model for the theoreticallimitations to specific entropy and cooling power achievable by demagnetization of various semiconductorsystems. Performance comparable to that of the commonly used paramagnetic salt cerous magnesium nitratehydrate (CMN) is predicted.The adiabatic cooling of paramagnetic spins achievesrefrigeration on the basis of exchange of entropy in athermodynamic cycle. The exchange of heat by magneti-zation of spins was first observed in iron , and the mag-netocaloric phenomenon was first applied to magnetic re-frigeration by Giauque in 1933 . The methods and prac-tice of magnetic refrigeration have since been developedover many years , with the state-of-the-art involving theadiabatic demagnetization of electron spins in paramag-netic salt powders at the milliKelvin temperature scale,and nuclear spins in metals at the sub-milliKelvin tem-perature scale. There is renewed interest in developingnew methods and materials for cryogenic refrigeration,driven by the development of new technologies such asquantum information processing that rely upon thegeneration of cryogenic temperatures.New materials for adiabatic demagnetization are beingdeveloped, such as the intermetallic compound YbPt Sn,which combines high specific entropy at
T < . Fundamental extensions to re-frigeration by adiabatic demagnetization are also beingdeveloped. For example, continuous cycle cooling can beachieved by the exchange of spin entropy via a spin cur-rent, leading to the spin Peltier effect. Cooling has beenexperimentally demonstrated in a mesoscopic device ,spurring interest in the field of spin caloritronics . Spinentropy is believed to play a dominant role in the largethermopower of the transition metal oxide Na x Co O at cryogenic temperatures . The enhancement of mag-netocaloric effects at quantum critical points, wherechanges in entropy induced by magnetic field are large,is an active area of research . Another extension to theprinciple of refrigeration by manipulation of entropy withmagnetic field is refrigeration by adiabatic magnetization of superconductors .We propose paramagnetic semiconductors, such as Si:Pshown schematically in Fig. 1a), as the active mediumfor refrigeration by adiabatic demagnetization. Paramag-netism can originate from un-ionized impurity dopants orstructural defects, both of which lead to localized mag-netic moments. The thermodynamic cycle is that of a Si:P P SiSi SiSi SiSiSiSi B ln2 Temperature, T E n t r op y , S / N s B i = m T B f = m T B a) b) g μ B B D demagnetization m agne t i z a t i on w a r m i ng FIG. 1. a) A doped semiconductor, such as Si:P, exhibitsparamagnetism at low temperature. Un-ionized donor states D exhibit Pauli paramagnetism, with a Zeeman splitting of gµ B B in the case of a spin-1/2 donor atom. b) Paramag-netic entropy enables a refrigeration cycle, illustrated herewith B i = 300 mT, B f = 10 mT and a spin-1/2 system.From an initial state (1) at a temperature T i , a large field B i leads to (isothermal) magnetization to a low entropy state(2). Adiabatic demagnetization to a field B f leads to a low en-tropy state (3) at a low temperature T f = T i · B f /B i , coolingthe paramagnetic refrigerant. Warming of the paramagneticrefrigerant returns the system to its initial state (1). typical adiabatic demagnetization refrigeration cycle, asshown in Fig.1b. The spins of impurity dopants are po-larized at a high field B i at an initial temperature T i toachieve a state of low entropy S ( B i /T i ). An adiabaticsweep of magnetic field to a final B f < B i maintainsconstant entropy S ( B f /T f ) = S ( B i /T i ), thus bringingspins to a final temperature T f = T i · B f /B i . The reser-voir of low temperature spins can thus be used as a re-frigerant. Ever present diamagnetism does not make a a r X i v : . [ c ond - m a t . m e s - h a ll ] J un contribution to field dependent entropy because diamag-netic magnetization M is temperature independent, andMaxwell’s relation implies ( ∂ S /∂B ) T = ( ∂M/∂T ) B ∼ S ( B/T ) is proportional to the concentra-tion of impurity dopant spins. There is a wide variety ofsemiconductor hosts and impurity dopants, giving a widerange of experimentally accessible impurity dopant prop-erties and concentrations. Lastly, semiconductors are themost common substrate used for electronic and micro-electromechanical (MEMS) devices, and thus offers thepossibility of monolithic integration of solid state refrig-erant with active devices. Silicon is particularly appeal-ing because of its wide availability and well understoodproperties.We develop a simple theoretical model for refrigerationby semiconductor paramagnetism, focussing our discus-sion on impurity doped semiconductors although simi-lar principles apply to semiconductors with structuraldefects. An essential requirement for effective coolingis that the paramagnetic specific heat of donor boundelectron spins (or acceptor bound hole spins) dominatesover phonon specific heat and itinerant electron specificheat. This condition has been experimentally shown tobe satisfied at low temperatures and moderate magneticfields in heavily doped silicon, for example . Thephonon specific heat is C phonon = (12 / · ( π k B N A /V m ) · ( T / Θ D ) , where N A is Avogadro’s number, Θ D is the De-bye temperature of the semiconductor, V m is the molarvolume and k B the Boltzmann constant. The paramag-netic specific heat is taken to be that of an ideal ensembleof non-interacting spin-1/2 impurity dopants, leading tothe Schottky form , C spin = N s k B · (cid:18) gµ B B k B T (cid:19) · sech (cid:18) gµ B B k B T (cid:19) , (1)where N s is the spin density, µ B is the Bohr magnetonand g the Land´e g-factor. The generalization to spin- J impurity dopants is straightforward . Itinerant electronsin semiconductors with impurity dopant concentrationsexceeding the critical density of the metal-insulator tran-sition (MIT) leads to a third contribution to specific heat, C electron = π / · k B T · DOS ( E F ), where DOS ( E F ) is thedensity of states (DOS) at the Fermi energy.There are several heavily doped semiconductors inwhich paramagnetism has been observed. As will be dis-cussed further below, the density N s of impurity boundspins depends upon the impurity dopant concentration N D in a non-trivial manner. Theoretical specific heatcurves at a modest magnetic field of B = 100 mTare shown in Fig. 2 based on experimentally mea-sured spin densities N s for a variety of semiconductorsystems. Fig. 2a) shows the theoretical specific heatof Si:P with the maximum spin-1/2 density reportedto date, N s = 3 × cm − at a P doping density N D = 1 × cm − , as inferred from specific heatmeasurements . Fig. 2b) shows the theoretical specific a) 1 KTemperature, T
100 mK10 mK 10 K1100.10.010.001 e l e c t r o n phonon dono r s p i n S pe c i f i c H ea t, C [ µ J / K cc ] Si:P1100.10.010.001 S pe c i f i c H ea t, C [ µ J / K cc ]
100 1 KTemperature, T
100 mK10 mK 10 K e l e c t r o n phonon dono r s p i n
4H SiC:Nb) 1 KTemperature, T
100 mK10 mK 10 K1100.10.01 S pe c i f i c H ea t, C [ µ J / K cc ] e l e c t r o n phonon dono r s p i n c) FIG. 2. The theoretical specific heat C of several heavilydoped semiconductor systems, including contributions fromparamagnetism (red line), itinerant electrons (blue line) andphonons (green line). The applied magnetic field is B =100 mT and the spin density N s is taken from experimentalmeasurements. a) Si:P at a P density N D = 1 . × cm − and corresponding spin-1/2 density N s = 3 × cm − asmeasured by specific heat . The free electron density wasestimated as n = N D − N s . b) 4H-SiC:N at an N den-sity N D = 7 . × cm − and corresponding spin-1/2 den-sity N s = 3 . × cm − as measured by electron spinresonance . The free electron density was estimated as n = N D − N s . c) GaN:Mn at an Mn density N D = 4 . × cm − and corresponding spin-5/2 density N s = N D as measured bymagnetization . The free electron density was measured tobe n (cid:54) . × cm − . Ge:As J = 1/26.5×10 /cc S p i n E n t r op y , ∆ S [ J / K cc ] –1 –2 –3 –4 –5 –6 –7 Si:P J = 1/23.0×10 /cc a-Si J = 1/26.3×10 /cc J = 1/23.0×10 /cc J = 15.3×10 /cc GaAs:Mn J = 5/21.1×10 /cc GaN:Si J = 1/26.0×10 /cc GaN:Mn J = 5/24.5×10 /cc Ga Mn N J = 5/22.7×10 /cc YbPt Sn J = 1/21.3×10 /cc CMN J = 1/21.6×10 /cc semiconductors FIG. 3. The theoretical spin entropy ∆ S = k B N s ln(2 J + 1) of several semiconductor systems is compared along with theintermetallic compound YbPt Sn and the commonly used paramagnetic salt CMN . The experimentally measured spin- J and spin density N s are indicated for each material system. The semiconductor systems shown include Ge:As , Si:P , a-Si ,4H-SiC:N , neutron irradiated 4H-SiC , GaAs:Mn , GaN:Si , GaN:Mn and Ga − x Mn x N . heat of 4H-SiC:N with the maximum spin-1/2 densityreported to date, N s = 3 × cm − at an N dopingdensity N D = 7 . × cm − , as measured by electronspin resonance . Fig. 2c) shows the theoretical specificheat of GaN:Mn at an Mn density N D = 4 . × cm − ,where the spin-5/2 density N s = N D follows ideal para-magnetic behaviour as seen in direct measurement ofmagnetization . The paramagnetic specific heat dom-inates over both phonon and electron contributions attemperatures of T ∼ T (cid:54) , S = N s k B [ln { x ) } − x tanh ( x )] (2)where the dimensionless ratio x = gµ B B/ k B T deter-mines the paramagnetic entropy per spin S /N s . Thegeneral case of a spin- J impurity is more complex, where J and x determine S /N s . The most commonly usedimpurity dopants give spin-1/2 paramagnetism. There isa wide range of electron g-factors accessible in semicon-ductors, from g = +2 in silicon through to g < − . Because the cooling cycle occurs under idealadiabatic conditions where ∆ S = 0, the practical im-plication of g-factor selection is that it determines themagnetic field B i that is required to polarize spins intoa low entropy state at the initial temperature T i .The spin density N s is the most critical parameter thatdetermines the cooling characteristics of the refrigerator.The maximum paramagnetic spin- J entropy that is avail-able for refrigeration is ∆ S = N s k B ln(2 J +1). The max-imum heat ∆ Q that can be absorbed by the refrigerant at the final field B f is,∆ Q = (cid:90) ∞ T f C spin dT = JN s | g | µ B B f (3)in the limit of a highly polarized state of the refrigerant, k B T f (cid:28) | g | µ B B f , prior to warming. In other words, themaximum heat that can be absorbed by N s spins is theenergy required to fully depolarize the N s spins at theterminal field B f .Importantly, electron spin relaxation times T in semi-conductors can vary by many orders of magnitude, de-creasing to short times in the case of heavily doped semi-conductors. For example, T < N s > cm − in silicon . Over the time scale of a demagnetiza-tion, typically of the order of 10 s to 10 s, the im-purity spins and lattice phonons are effectively in ther-mal equilibrium, and a single temperature T describesthe semiconductor lattice and spins. The rapid thermalequilibration time admits the possibility of isothermal heat extraction by demagnetization of the semiconduc-tor. In the isothermal limit, the rate of heat extraction˙ Q from the environment by the semiconductor refriger-ant is ˙ Q = T ˙ S = T ( ∂S/∂B ) T ˙ B . In the case of theideal spin-1/2 paramagnetic system, the heat extractionrate reaches a peak value of ˙ Q peak = 0 . · N s µ ˙ B at amagnetic field µ B B peak = 0 . · k B T .The density N s of impurity bound spins, which is crit-ical to refrigerator performance, itself depends upon theimpurity dopant concentration N D . At impurity dopantconcentrations well below the critical density N c of theMIT, single electrons are bound to impurity dopants and N s = N D . As the impurity concentration approachesthe critical density N c , the spin density N s exhibits amaximum before decreasing with increasing N D . Exper-imental measurements of specific heat in Si:P , electronspin resonance in Ge:As and 4H-SiC:N , and electronspin resonance in GaN:Si all show a rapid drop in spindensity N s as the dopant density approaches the crit-ical density N c . The microscopic mechanisms respon-sible for the loss of paramagnetism as N D approaches N c varies from one material system to another. Anti-ferromagnetic ordering has been postulated in the formof singlet formation at doubly occupied donor sites andsinglet formation of spatially proximate donor sites ,per the theory of Bhatt and Lee . From a thermody-namic perspective, the onset of spin ordering depletes thespin entropy that would otherwise be available for use in amagnetic refrigeration scheme. At doping densities above N c , electron delocalization reduces the density of local-ized electrons that contribute directly to paramagnetism.The Pauli paramagnetism of the delocalized electron gasabove the MIT gives a temperature independent magne-tization, and thus provides no entropy for magnetic re-frigeration per the relation ( ∂ S /∂B ) T = ( ∂M/∂T ) B ∼ S = N s k B ln(2 J + 1) available for magnetic refrigeration fora variety of material systems is shown in Fig. 3,on the basis of the experimentally measured spin den-sity N s and angular momentum J . Several commondoped semiconductor systems are compared, includingGe:As , Si:P , 4H-SiC:N , GaAs:Mn , GaN:Si andGaN:Mn . Among these doped semiconductors, thegeneral trend is that a greater spin density is achievablein larger gap semiconductors. The critical dopant den-sity N c at which electron delocalization occurs is verywell approximated by N c = (0 . /a ∗ B ) for a wide rangeof semiconductors , where a ∗ B is an effective Bohr radiusthat tends to smaller values for larger gap semiconductorsand larger impurity ionization energies. In short, tighterconfinement of localized states enables a greater densityof localized states without the onset of delocalization.Paramagnetism from localized electronic states canbe introduced into semiconductors with structural de-fects. Amorphous silicon is typically prepared in a hydro-genated state, a-Si:H, to passivate dangling bonds. Theresidual spin density is N s ∼ cm − , but can be sig-nificantly enhanced to a density N s = 6 . × cm − by thermal annealing to remove passivating hydrogenatoms and produce a-Si . Neutron irradiation offersanother alternative method of imparting paramagnetismto semiconductors , with a spin-1 density of N s =5 . × cm − achieved in 4H-SiC. The paramagnetismof neutron irradiated 4H-SiC is attributed to divacan-cies V Si V C with a magnetic moment of 2 µ B . Lastly,we note that significant paramagnetism can persist indilute magnetic semiconductors such as Ga − x Mn x N ,where spin-5/2 paramagnetism of Mn at an effectiveconcentration x eff = 0 .
061 was observed in a crystalwith Mn fraction x = 0 . N s = 2 . × cm − to be achieved. For compari-son, a commonly used paramagnetic salt for magnetic cryo-refrigeration is cerous magnesium nitrate hydrate(CMN) , where the Ce ions give a spin-1/2 densityof N s = 1 . × cm − . The magnetic semiconductorsystem Ga − x Mn x N gives a higher density of spins withhigher momentum J than that of CMN.We finally consider the minimum temperature T f thatis achievable by adiabatic demagnetization. There isa critical temperature T c below which the spin systemundergoes a transition to an ordered state, depletingentropy available for refrigeration and setting a limitto the achievable base temperature . In the absenceof the itinerant electrons of a metallic state, it is ex-pected that magnetic dipole-dipole interactions set thetemperature scale at which ordering occurs, k B T c ∼ ( µ / π ) · µ B (cid:10) /r (cid:11) , with (cid:10) /r (cid:11) ∼ N s . There is thusa trade-off between specific spin entropy ∆ S ∝ N s andthe minimum temperature T c ∝ N s , while the strengthof spin-spin interaction varies from material to material.Ordering temperatures are typically determined exper-imentally. For example, the ordering temperature ofCMN is T c = 2 mK at N s = 1 . × cm − , andthe ordering temperature of the recently discovered in-termetallic compound YbPt Sn is T c = 250 mK at N s = 1 . × cm − . There is a paucity of experi-mental work reported on the spin ordering temperatureof magnetic semiconductors. With spin densities com-parable to or lower than that of CMN, semiconductorparamagnetism is expected to persist down to the ∼ mKtemperature scale.In conclusion, the measured spin- J and spin density N s of paramagnetic semiconductors provides spin en-tropy comparable to that of paramagnetic salts used inadiabatic demagnetization refrigerators. Paramagneticsemiconductors are thus appealing as solid state cryo-refrigerants that can be monolithically integrated withactive electronic components. Integration at this scaleoptimizes cooling by minimizing the volume of materialto be cooled, and minimizing the number of interfacesthat can contribute to parasitic thermal resistance. Thechemical stability of semiconductors is also favourable incomparison with hydrated paramagnetic salts. Nonethe-less, further work is required to identify the most promis-ing semiconductor system, as their is a wide variety of im-purity dopants and structural defects available to choosefrom. Significant experimental work is required to quan-tify the paramagnetic contribution to specific heat ofsemiconductors at cryogenic temperatures, and to deter-mine the electron spin ordering temperatures that will setthe temperature limit to magnetic refrigeration in thesematerials.We thank Pascal Pochet and Shengqiang Zhou forbringing to our attention recent developments in neu-tron irradiated semiconductors. The authors acknowl-edge financial support from the Canada Research ChairsProgram, the Canadian Institute for Advanced Research,and the Canadian Natural Sciences and Engineering Re-search Council. E. Warburg and L. H¨onig, Ueber die w¨arme, welche durch peri-odisch wechselnde magnetisirende kr¨afte im eisen erzeugt wird,Annal. der Phys. 256, 814835 (1883). W. F. Giauque and D. P. MacDougall, Attainment of tempera-tures below 1 ◦ absolute by demagnetization of Gd (SO ) · , 768 (1933). F. Pobell, Matter and Methods at Low Temperatures (Springer,New York, 1992). J. M. Hornibrook, J. I. Colless, I. D. Conway Lamb, S. J. Pauka,H. Lu, A. C. Gossard, J. D. Watson, G. C. Gardner, S. Fallahi, M.J. Manfra, and D. J. Reilly, Cryogenic Control Architecture forLarge-Scale Quantum Computing, Phys. Rev. Applied , 024010(2015). M. Mohseni, P. Read, H. Neven, S. Boixo, V. Denchev, R. Bab-bush, A. Fowler, V. Smelyanskiy and J. Martinis, Commercializequantum technologies in five years, Nature , 171 (2017). D. Jang, T. Gruner, A. Steppke, K Mistumoto, C. Geibel and M.Bando, Large magnetocaloric effect and adiabatic demagnetiza-tion refrigeration with YbPt Sn, Nature Comms. , 8680 (2015). J. Flipse, F. L. Bakker, A. Slachter, F. K. Dejene and B. J.van Wees, Direct observation of the spin-dependent Peltier effect,Nature Nanotech. , 166 (2012). G. E. W. Bauer, E. Saitoh and B. J. van Wees, Spin Caloritronics,Nature Mat. , 391 (2012). Y. Wang, N. S. Rogado, R. J. Cava and N. P. Ong, Spin entropyas the likely source of enhanced thermopower in Na x Co O , Na-ture , 425 (2003). B. Wolf et al. , Magnetocaloric effect and magnetic cooling near afield-induced quantum-critical point, Proc. Nat. Acad. Sci. ,6862 (2011). F. Dolcini and F. Giazotto, Adiabatic magnetization of supercon-ductors as a high-performance cooling mechanism, Phys. Rev. B , 024503 (2009). N. Kobayashi, S. Ikehata, S. Kobayashi, and W. Sasaki, MagneticField Dependence of Specific Heat of Heavily Phosphorus DopedSilicon, Solid State Comm. , 1147 (1979). M. Lakner, H. v. L¨ohneysen, A. Langenfeld and P. W¨olfe, Local-ized magnetic moments in Si:P near the metal-insulator transi-tion, Phys. Rev. B , 17064 (1994). S. Wagner, M. Lakner, and H. v. L¨ohneysen, Specifc heat ofSi:(P,B) at low temperatures, Phys. Rev. B , 4219 (1997). R.K. Pathria, Statistical Mechanics (Butterworth-Heinemann,New York, 1996). A. G. Zabrodskii, Magnetic ordering in doped semiconductorsnear the metalinsulator transition, phys. stat. sol. (b) , 33(2004). M. Zaj¸ac, J. Gosk, M. Kami´nska, A. Twardowski, T. Szyszko andS. Podsiadlo, Paramagnetism and antiferromagnetic dd couplingin GaMnN magnetic semiconductor, Appl. Phys. Lett. , 2432(2001). G. Feher, Electron Spin Resonance Experiments on Donors inSilicon. I. Electronic Structure of Donors by the Electron NuclearDouble Resonance Technique, Phys. Rev. , 1219 (1959). H. Kosaka, A. A. Kiselev, F. A. Baron, K. W. Kim and E.Yablonovitch, Electron g-factor Engineering in III-V Semicon-ductors for Quantum Communications, Elect. Lett. , 464(2001). G. Feher and E. A. Gere, Electron Spin Resonance Experimentson Donors in Silicon. II. Electron Spin Relaxation Effects, Phys.Rev. , 1245 (1959) A. Wolos, Z. Wilamowski, M. Piersa, W. Strupinski, B. Lucznik,I. Grzegory, and S. Porowski, Properties of metal-insulator tran-sition and electron spin relaxation in GaN:Si, Phys. Rev. B ,165206 (2011). R. N. Bhatt and P. A. Lee, Scaling Studies of Highly DisorderedSpin-1/2 Antiferromagnetic Systems, Phys, Rev. Lett. , 344(1984). P. P. Edwards and M. J. Sienko, Universality aspects of themetal-nonmetal transition in condensed media, Phys. Rev. B ,2575 (1978). M. Pawlowski, M. Piersa, A. Wolos, M. Palczewska, G. Strz-elecka, A. Hruban, J. Gosk, M. Kami´nska, and A. Twardowski,Mn Impurity in Bulk GaAs Crystals, Acta. Phys. Pol. , 825(2005). M. Stutzmann and D. K. Biegelsen, Electron-spin-lattice relax-ation in amorphous silicon and germanium, Phys. Rev. B ,6256 (1983). Y. Wang, Y. Liu, E. Wendler, R. H¨ubner, W. Anwand, G. Wang,X. Chen, W. Tong, Z. Yang, F. Munnik, G. Bukalis, X. Chen,S. Gemming, M. Helm, and S. Zhou, Defect-induced magnetismin SiC: Interplay between ferromagnetism and paramagnetism,Phys. Rev. B , 174409 (2015). W. F. Giauque, R. A. Fisher, E. W. Hornung, and G. E. Brodale,Magnetothermodynamics of Ce Mg (NO ) · O. I. Heat ca-pacity, entropy, magnetic moment from 0.5 to 4.2 ◦ K with fieldsto 90 kG along the a crystal axis. Heat capacity of Pyrex 7740glass in fields to 90 kG, J. Chem. Phys.58