aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Quantum Hall Effects
Mark O. GoerbigLaboratoire de Physique des Solides, CNRS UMR 8502Universit´e Paris-Sud, FranceOctober 21, 2009
Preface
The present notes cover a series of three lectures on the quantum Hall effectgiven at the Singapore session “Ultracold Gases and Quantum Information” at
Les Houches Summer School
Les HouchesSummer School have covered in several aspects quantum Hall physics, and S.M. Girvin’s series of lectures in 1998 [4] have certainly become a reference inthe field. Girvin’s lecture notes were indeed extremely useful for myself whenI started to study the quantum Hall effect at the beginning of my Master andPhD studies.The present lecture notes are complementary to the existing literature inseveral aspects. One should first mention its introductory character to the field,which is in no way exhaustive. As a consequence, the presentation of one-particlephysics and a detailed discussion of the integer quantum Hall effect occupy themajor part of these lecture notes, whereas the – certainly more interesting –fractional quantum Hall effect, with its relation to strongly-correlated electrons,its fractionally charged quasi-particles and fractional statistics, is only brieflyintroduced.Furthermore, we have tried to avoid as much as possible the formal aspectsof the fractional quantum Hall effect, which is discussed only in the framework oftrial wave functions `a la Laughlin . We have thus omitted, e.g., a presentation ofChern-Simons theories and related quantum-field theoretical approaches, suchas the Hamiltonian theory of the fractional quantum Hall effect [5], as muchas the relation between the quantum Hall effect and conformal field theories.Although these theories are extremely fruitful and still promising for a deeperunderstanding of quantum Hall physics, a detailed discussion of them wouldrequire more space than these lecture notes with their introductory charactercan provide.Another complementary aspect of the present lecture notes as comparedto existing textbooks consists of an introduction to Landau-level quantisationthat treats in a parallel manner the usual non-relativistic electrons in semicon-ductor heterostructures and relativistic electrons in graphene (two-dimensionalgraphite). Indeed, the 2005 discovery of a quantum Hall effect in this amazingmaterial [6, 7] has given a novel and unexpected boost to research in quantumHall physics.As compared to the (oral) lectures, the present notes contain slightly moreinformation. An example is Laughlin’s plasma analogy, which is described inSec. 4.2.5, although it was not discussed in the oral lectures. Furthermore,I have decided to add a chapter on multi-component quantum Hall systems,which, for completeness, needed to be at least briefly discussed. These lectures are also available on the preprint server,http://arxiv.org/abs/cond-mat/9907002
Before the Singapore session of
Les Houches Summer School , this series oflectures had been presented in a similar format at the (French) Summer School ofthe Research Grouping “Physique M´esoscopique” at the Institute of ScientificResearch, Carg`ese, Corsica, in 2008. Furthermore, a longer series of lectureson the quantum Hall effect was prepared in collaboration with my colleagueand former PhD advisor Pascal Lederer (Orsay, 2006). Its aim was somewhatdifferent, with an introduction to the Hamiltonian theories of the fractionalquantum Hall effect and correlation effects in multi-component systems. Asalready mentioned above, the latter aspect is only briefly introduced within thepresent lecture notes and a discussion of Hamiltonian theories is completelyabsent. The Orsay series of lectures was repeated by Pascal Lederer at the
Ecole Polytechnique F´ed´erale in Lausanne Switzerland, in 2006, and at theUniversity of Recife, Brazil, in 2007. The finalisation of these longer and moredetailed lecture notes (in French) is currently in progress. The graphene-relatedaspects of the quantum Hall effect have furthermore been presented in a seriesof lectures on graphene (Orsay, 2008) prepared in collaboration with Jean-No¨elFuchs, whom I would like to thank for a careful reading of the present notes. ontents B = 0 . . . . . . . . . . . . 212.1.1 Hamiltonian of a free particle . . . . . . . . . . . . . . . . 212.1.2 Dirac Hamiltonian in graphene . . . . . . . . . . . . . . . 232.2 Hamiltonians for Non-Zero B Fields . . . . . . . . . . . . . . . . 262.2.1 Minimal coupling and Peierls substitution . . . . . . . . . 262.2.2 Quantum mechanical treatment . . . . . . . . . . . . . . . 272.3 Landau Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 Non-relativistic Landau levels . . . . . . . . . . . . . . . . 292.3.2 Relativistic Landau levels . . . . . . . . . . . . . . . . . . 302.3.3 Level degeneracy . . . . . . . . . . . . . . . . . . . . . . . 342.3.4 Semi-classical interpretation of the level degeneracy . . . 362.4 Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.1 Wave functions in the symmetric gauge . . . . . . . . . . 392.4.2 Wave functions in the Landau gauge . . . . . . . . . . . . 41 x -direction . . . . . . . . . . . . . . . . . . . . . . . . . . 475 3.2 Conductance of a Single Landau Level . . . . . . . . . . . . . . . 483.2.1 Edge states . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3 Two-terminal versus Six-Terminal Measurement . . . . . . . . . . 513.3.1 Two-terminal measurement . . . . . . . . . . . . . . . . . 513.3.2 Six-terminal measurement . . . . . . . . . . . . . . . . . . 523.4 The Integer Quantum Hall Effect and Percolation . . . . . . . . . 543.4.1 Extended and localised bulk states in an optical measure-ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.2 Plateau transitions and scaling laws . . . . . . . . . . . . 583.5 Relativistic Quantum Hall Effect in Graphene . . . . . . . . . . . 60 ν = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 925.2.1 Quantum Hall ferromagnetism . . . . . . . . . . . . . . . 935.2.2 Exciton condensate in bilayer systems . . . . . . . . . . . 955.2.3 SU(4) ferromagnetism in graphene . . . . . . . . . . . . . 985.3 Multi-Component Wave Functions . . . . . . . . . . . . . . . . . 995.3.1 Halperin’s wave function . . . . . . . . . . . . . . . . . . . 995.3.2 Generalised Halperin wave functions . . . . . . . . . . . . 102 A Electronic Band Structure of Graphene 105B Landau Levels of Massive Dirac Particles 111 hapter 1
Introduction
Quantum Hall physics – the study of two-dimensional (2D) electrons in a strongperpendicular magnetic field [see Fig. 1.1(a)] – has become an extremely im-portant research subject during the last two and a half decades. The interestfor quantum Hall physics stems from its position at the borderline between low-dimensional quantum systems and systems with strong electronic correlations,probably the major issues of modern condensed-matter physics. From a theoret-ical point of view, the study of quantum Hall systems required the elaborationof novel concepts some of which were better known in quantum-field theoriesused in high-energy rather than in condensed-matter physics, such e.g. chargefractionalisation, non-commutative geometries and topological field theories.The motivation of the present lecture notes is to provide in an accessible man-ner the basic knowledge of quantum Hall physics and to enable thus interestedgraduate students to pursue on her or his own further studies in this subject. Wehave therefore tried, whereever we feel that a more detailed discussion of someaspects in this large field of physics would go beyond the introductory characterof these notes, to provide references to detailed and pedagogical references orcomplementary textbooks.
Our main knowledge of quantum Hall systems, i.e. a system of 2D electrons ina perpendicular magnetic field, stems from electronic transport measurements,where one drives a current I through the sample and where one measures boththe longitudinal and the transverse resistance (also called Hall resistance). Thedifference between these two resistances is essential and may be defined topo-logically: consider a current that is driven through the sample via two arbitrarycontacts [C1 and C4 in Fig. 1.1(a)] and draw (in your mind) a line betweenthese two contacts. A longitudinal resistance is a resistance measured betweentwo (other) contacts that may be connected by a line that does not cross the7 Introduction
Hallresistance longitudinalresistance
C1 C2 C3 C4C5C6(a)
I 2D electron gas I H a ll r e s i s t an c e magnetic field BR H (b) Figure 1.1: (a) 2D electrons in a perpendicular magnetic field (quantum Hallsystem). In a typical transport measurement, a current I is driven throughthe system via the contacts C1 and C4. The longitudinal resistance may bemeasured between the contacts C5 and C6 (or alternatively between C2 andC3). The transverse (or Hall) resistance is measured, e.g., between the contactsC3 and C5. (b) Classical Hall resistance as a function of the magnetic field.line connecting C1 and C4. In Fig. 1.1(a), we have chosen the contacts C5and C6 for a possible longitudinal resistance measurement. The transverse re-sistance is measured between two contacts that are connected by an imaginaryline that necessarily crosses the line connecting C1 and C4 [e.g. C3 and C5 inFig. 1.1(b)]. Evidently, if there is a quantum
Hall effect, it is most natural to expect thatthere exists also a classical
Hall effect. This is indeed the case, and its historygoes back to 1879 when Hall showed that the transverse resistance R H of a thinmetallic plate varies linearly with the strength B of the perpendicular magneticfield [Fig. 1.1(b)], R H = Bqn el , (1.1)where q is the carrier charge ( q = − e for electrons in terms of the elementarycharge e that we define positive in the remainder of these lectures) and n el isthe 2D carrier density. Intuitively, one may understand the effect as due tothe Lorentz force, which bends the trajectory of a charged particle such thata density gradient is built up between the two opposite sample sides that areseparated by the contacts C1 and C4. Notice that the classical Hall resistance isstill used today to determine, in material science, the carrier charge and densityof a conducting material.More quantitatively, the classical Hall effect may be understood within theDrude model for diffusive transport in a metal. Within this model, one considers istory of the (Quantum) Hall Effect p described by the equation of motion d p dt = − e (cid:18) E + p m b × B (cid:19) − p τ , where E and B are the electric and magnetic fields, respectively. Here, we con-sider transport of negatively charged particles (i.e. electrons with q = − e ) withband mass m b . The last term takes into account relaxation processes due to thediffusion of electrons by generic impurities, with a characteristic relaxation time τ . The macroscopic transport characteristics, i.e. the resistivity or conductivityof the system, are obtained from the static solution of the equation of motion, d p /dt = 0, and one finds for 2D electrons with p = ( p x , p y ) eE x = − eBm b p y − p x τ ,eE y = eBm b p x − p y τ , where we have chosen the magnetic field in the z -direction. In the above ex-pressions, one notices the appearence of a characteristic frequency, ω C = eBm b , (1.2)which is called cyclotron frequency because it characterises the cyclotron mo-tion of a charged particle in a magnetic field. With the help of the Drudeconductivity, σ = n el e τm b , (1.3)one may rewrite the above equations as σ E x = − en el p x m b − en el p y m b ( ω C τ ) ,σ E y = en el p x m b ( ω C τ ) − en el p y m b , or, in terms of the current density j = − en el p m b , (1.4)in matrix form as E = ρ j , with the resistivity tensor ρ = σ − = 1 σ (cid:18) ω C τ − ω C τ (cid:19) = 1 σ (cid:18) µB − µB (cid:19) , (1.5)where we have introduced, in the last step, the mobility µ = eτm b . (1.6)0 Introduction
From the above expression, one may immediately read off the Hall resistivity(the off-diagonal terms of the resistivity tensor ρ ) ρ H = ω C τσ = eBm b τ × m b n el e τ = Ben el . (1.7)Furthermore, the conductivity tensor is obtained from the resistivity (1.5), bymatrix inversion, σ = ρ − = (cid:18) σ L − σ H σ H σ L (cid:19) , (1.8)with σ L = σ / (1 + ω C τ ) and σ H = σ ω C τ / (1 + ω C τ ). It is instructive todiscuss, based on these expressions, the theoretical limit of vanishing impurities,i.e. the limit ω C τ → ∞ of very long scattering times. In this case the resistivityand conductivity tensors read ρ = (cid:18) Ben el − Ben el (cid:19) and σ = (cid:18) − en el Ben el B (cid:19) , (1.9)respectively. Notice that if we had put under the carpet the matrix character ofthe conductivity and resistivity and if we had only considered the longitudinal components, we would have come to the counter-intuitive conclusion that the(longitudinal) resistivity would vanish at the same time as the (longitudinal)conductivity. The transport properties in the clean limit ω C τ → ∞ are thereforeentirely governed, in the presence of a magnetic field, by the off-diagonal, i.e.transverse, components of the conductivity/resistivity. We will come back tothis particular feature of quantum Hall systems when discussing the integerquantum Hall effect below. Resistivity and resistance
The above treatment of electronic transport in the framework of the Drudemodel allowed us to calculate the conductivity or resistivity of classical diffusive2D electrons in a magnetic field. However, an experimentalist does not measurea conductivity or resistivity, i.e. quantities that are easier to calculate for atheoretician, but a conductance or a resistance . Usually, these quantities arerelated to one another but depend on the geometry of the conductor – theresistance R is thus related to the resistivity ρ by R = ( L/A ) ρ , where L is thelength of the conductor and A its cross section. From the scaling point of viewof a d -dimensional conductor, the cross section scales as L d − , such that thescaling relation between the resistance and the resistivity is R ∼ ρL − d , (1.10)and one immediately notices that a 2D conductor is a special case. From thedimensional point of view, resistance and resistivity are the same in 2D, and theresistance is scale-invariant. Naturally, this scaling argument neglects the factthat the length L and the width W (the 2D cross section) do not necessarily istory of the (Quantum) Hall Effect H a ll r e s i s t an c e magnetic field B l ong i t ud i na l r e s i s t an c e B c (a) D en s i t y o f s t a t e s EnergyE F h ω C (b) Figure 1.2: (a) Sketch of the Shubnikov-de Haas effect. Above a critical field B c , the longitudinal resistance (grey) starts to oscillate as a function of themagnetic field. The Hall resistance remains linear in B . (b) Density of states(DOS). In a clean system, the DOS consists of equidistant delta peaks (grey) atthe energies ǫ n = ¯ hω C ( n + 1 / E F denotes the Fermi energy.coincide: indeed, the resistance of a 2D conductor depends in general on theso-called aspect ratio L/W via some factor f ( L/W ) [8]. However, in the caseof the transverse Hall resistance it is the length of the conductor itself thatplays the role of the cross section, such that the Hall resistivity and the Hallresistance truely coincide, i.e. f = 1. We will see in Chap. 3 that this conclusionalso holds in the case of the quantum Hall effect and not only in the classicalregime. Moreover, the quantum Hall effect is highly insensitive to the particulargeometric properties of the sample used in the transport measurement, such thatthe quantisation of the Hall resistance is surprisingly precise (on the order of10 − ) and the quantum Hall effect is used nowadays in the definition of theresistance standard. A first indication for the relevance of quantum phenomena in transport mea-surements of 2D electrons in a strong magnetic field was found in 1930 withthe discovery of the Shubnikov-de Haas effect [9]. Whereas the classical re-sult (1.5) for the resistivity tensor stipulates that the longitudinal resistivity ρ L = 1 /σ (and thus the longitudinal resistance) is independent of the magneticfield, Shubnikov and de Haas found that above some characteristic magneticfield the longitudinal resistance oscillates as a function of the magnetic field.This is schematically depicted in Fig. 1.2(a). In contrast to this oscillation inthe longitudinal resistance, the Hall resistance remains linear in the B field, inagreement with the classical result from the Drude model (1.7).The Shubnikov-de Haas effect is a consequence of the energy quantisation of2 Introduction the 2D electron in a strong magnetic field, as it has been shown by Landau atroughly the same moment. This so-called
Landau quantisation will be presentedin great detail in Sec. 2. In a nutshell, Landau quantisation consists of thequantisation of the cyclotron radius, i.e. the radius of the circular trajectory ofan electron in a magnetic field. As a consequence this leads to the quantisationof its kinetic energy into so-called Landau levels (LLs), ǫ n = ¯ hω C ( n +1 / n is an integer. In order for this quantisation to be relevant, the magnetic fieldmust be so strong that the electron performs at least one complete circularperiod without any collision, i.e. ω C τ >
1. This condition defines the criticalmagnetic field B c ≃ m b /eτ = µ − above which the longitudinal resistance startsto oscillate, in terms of the mobility (1.6). Notice that today’s samples of highestmobility are characterised by µ ∼ cm /Vs = 10 m /Vs such that one mayobtain Shubnikov-de Haas oscillations at magnetic fields as low as B c ∼ σ L = e Dρ ( E F ) (1.11)turns out to be proportional to the density of states (DOS) ρ ( E F ) at the Fermienergy E F rather than the electronic density, Due to Landau quantisation, theDOS of a clean system consists of a sequence of delta peaks at the energies ǫ n = ¯ hω C ( n + 1 / ρ ( ǫ ) = X n g n δ ( ǫ − ǫ n ) , where g n is takes into account the degeneracy of the energy levels. These peaksare eventually impurity-broadened in real samples and may even overlap [seeFig. 1.2(b)], such that the DOS oscillates in energy with maxima at the positionsof the energy levels ǫ n . Consider a fixed number of electrons in the sample thatfixes the zero-field Fermi energy the B -field dependence of which we omit in theargument. When sweeping the magnetic field, one varies the energy distancebetween the LLs, and the DOS thus becomes maximal when E F coincides withthe energy of a LL and minimal if E F lies between two adjacent LLs. Theresulting oscillation in the DOS as a function of the magnetic field translatesvia the relation (1.11) into an oscillation of the longitudinal conductivity (orresistivity), which is the essence of the Shubnikov-de Haas effect. Notice, however, that the Fermi energy and thus the DOS is a function of the electronicdensity. Furthermore we mention that in a fully consistent treatment also the diffusion con-stant D depends on the density of states and eventually the magnetic field. This affects theprecise form of the oscillation but not its periodicity. Naturally, this is a crude assumption because if the density of states ρ ( ǫ, B ) depends onthe magnetic field, so does the Fermi energy via the relation Z E F dǫ ρ ( ǫ, B ) = n el . However, the basic features of the Shubnikov-de Haas oscillation may be understood whenkeeping the Fermi energy constant. istory of the (Quantum) Hall Effect ρ xy ( h / e ) ρ Ω xx ( k ) / / /V x V y I x magnetic field B[T] Figure 1.3: Typical signature of the quantum Hall effect (measured by J. Smet,MPI-Stuttgart). Each plateau in the Hall resistance is accompanied by a van-ishing longitudinal resistance. The classical Hall resistance is indicated by thedashed-dotted line. The numbers label the plateaus: integral n denote the IQHEand n = p/q , with integral p and q , indicate the FQHE. An even more striking manifestation of quantum mechanics in the transportproperties of 2D electrons in a strong magnetic field was revealed 50 years laterwith the discovery of the integer quantum Hall effect (IQHE) by v. Klitzing,Dorda, and Pepper in 1980 [10]. The Nobel Prize was attributed in 1985 to v.Klitzing for this extremely important discovery.Indeed, the discovery of the IQHE was intimitely related to technologicaladvances in material science, namely in the fabrication of high-quality field-effect transistors for the realisation of 2D electron gases. These technologicalaspects will be briefly reviewed in separate a section (Sec. 1.2).The IQHE occurs at low temperatures, when the energy scale set by thetemperature k B T is significantly smaller than the LL spacing ¯ hω C . It consistsof a quantisation of the Hall resistance, which is no longer linear in B , as onewould expect from the classical treatment presented above, but reveals plateausat particular values of the magnetic field (see Fig. 1.3). In the plateaus, the4 Introduction
Hall resistance is given in terms of universal constants – it is indeed a fractionof the inverse quantum of conductance e /h , and one observes R H = (cid:18) he (cid:19) n , (1.12)in terms of an integer n . The plateau in the Hall resistance is accompaniedby a vanishing longitudinal resistance. This is at first sight reminiscent of theShubnikov-de Haas effect, where the longitudinal resistance also reveals minimaalthough it never vanished. The vanishing of the longitudinal resistance at theShubnikov-de Haas minima may indeed be used to determine the crossover fromthe Shubnikov-de Haas regime to the IQHE.It is noteworth to mention that the quantisation of the Hall resistance (1.12)is a universal phenomenon, i.e. independent of the particular properties of thesample, such as its geometry, the host materials used to fabricate the 2D elec-tron gas and, even more importantly, its impurity concentration or distribution.This universality is the reason for the enormous precision of the Hall-resistancequantisation (typically ∼ − ), which is nowadays – since 1990 – used as theresistance standard, R K − = h/e = 25 812 .
807 Ω , (1.13)which is also called the Klitzing constant [11, 12]. Furthermore, as alreadymentioned in Sec. 1.1.2, the vanishing of the longitudinal resistance indicatesthat the scattering time tends to infinity [see Eq. (1.9)] in the IQHE. This isanother indication of the above-mentioned universality of the effect, i.e. thatIQHE does not depend on a particular impurity (or scatterer) arrangement.A detailed presentation of the IQHE, namely the role of impurities, may befound in Chap. 3. Three years after the discovery of the IQHE, an even more unexpected effectwas observed in a 2D electron system of higher quality, i.e. higher mobility:the fractional quantum Hall effect (FQHE). The effect ows its name to the factthat contrary to the IQHE, where the number n in Eq. (1.12) is an integer, aHall-resistance quantisation was discovered by Tsui, St¨ormer and Gossard with n = 1 / The subscript K honours v. Klitzing and 90 stands for the date since which the unit ofresistance is defined by the IQHE. istory of the (Quantum) Hall Effect n = 1 /
3, as well as any n = 1 /q with q being an oddinteger, is due to the formation of a correlated incompressible electron liquid with extremely exotic properties [14], which will be reviewed in Chap. 4. Asfor the IQHE, the discovery and the theory of the FQHE was awarded a NobelPrize (1998 for Tsui, St¨ormer and Laughlin).After the discovery of the FQHE with n = 1 / a plethora of other types ofFQHE has been dicovered and theoretically described. One should first mentionthe 2 / / n = 2 / n = 3 / p/ (2 sp ± s and p . This series has found a compellinginterpretation within the so-called composite-fermion (CF) theory according towhich the F QHE may be viewed as an I QHE of a novel quasi-particle thatconsists of an electron that “captures” an even number of flux quanta [15, 16].The basis of this theory is presented in Sec. 4.4. Another intriguing FQHE wasdiscovered in 1987 by Willet et al. , with n = 5 / / n = p/q with odd denominatorshad been observed in monolayer systems. From a theoretical point of view, itwas shown in 1991 by Moore and Read [18] and by Greiter, Wilczek and Wen[19] that this FQHE may be described in terms of a very particular, so-called Pfaffian , wave function, which involves particle pairing and the excitations ofwhich are anyons with non-Abelian statistics. These particles are intensivelystudied in today’s research because they may play a relevant role in quantumcomputation. The physics of anyons will be introduced briefly in Sec. 4.3.Finally, we would mention in this brief (and naturally incomplete) historicaloverview a FQHE with n = 4 /
11 discovered in 2003 by Pan et al. [20]: it doesnot fit into the above-mentioned CF series, but it would correspond to a FQHEof CFs rather than an IQHE of CFs.
Recently, quantum Hall physics experienced another unexpected boost with thediscovery of a “relativistic” quantum Hall effect in graphene, a one-atom-thicklayer of graphite [6, 7]. Electrons in graphene behave as if they were relativisticmassless particles. Formally, their quantum-mechanical behaviour is no longerdescribed in terms of a (non-relativistic) Schr¨odinger equation, but rather bya relativistic 2D Dirac equation [21]. As a consequence, Landau quantisationof the electrons’ kinetic energy turns out to be different in graphene than inconventional (non-relativistic) 2D electron systems, as we will discuss in Sec. 2.This yields a “relativistic” quantum Hall effect with an unusual series for theHall plateaus. Indeed rather than having plateaus with a quantised resistanceaccording to R H = h/e n , with integer values of n , one finds plateaus with n = ± n ′ + 1), in terms of an integer n ′ , i.e. with n = ± , ± , ± , ... . Thedifferent signs in the series ( ± ) indicate that there are two different carriers, The quantity n determines the filling of the LLs, usually described by the Greek letter ν ,as we will discuss in Sec. 2. Introduction electrons in the conduction band and holes in the valence band, involved in thequantum Hall effect in graphene. As we will briefly discuss in Sec. 1.2, onemay easily change the character of the carriers in graphene with the help of theelectric field effect.Interaction effects may be relevant in the formation of other integer Hallplateaus, such as n = 0 and n = ± n = ± n ′ + 1) characteristic of the relativistic quantum Hall effect.Furthermore, a FQHE with n = 1 / As already mentioned above, the history of the quantum Hall effect is intimitelyrelated to technological advances in the fabrication of 2D electron systems withhigh electronic mobilities. The increasing mobility allows one to probe the finestructure of the Hall curve and thus to observe those quantum Hall states whichare more fragile, such as some exotic FQHE states (e.g. the 5/2, 7/2 or the 4/11states). This may be compared to the quest for high resolutions in optics: thehigher the optical resolution, the better the chance of observing tinier objects.In this sense, electronic mobility means resolution and the tiny object is thequantum Hall state. As an order of magnitude, today’s best 2D electron gases(in GaAs/AlGaAs heterostructures) are characterised by mobilities µ ∼ cm /Vs. The samples used in the discovery and in the first studies of the IQHE were so-calles metal-oxide-semiconductor field-effect transistors (MOSFET). A metalliclayer is seperated from a semiconductor (typically doped silicon) by an insulatingoxide (e.g. SiO ) layer (see inset I in Fig. 1.4). The chemical potential in themetallic layer may be varied with the help of a gate voltage V G . At V G = 0, theFermi energy in the semiconductor lies in the band gap below the acceptor levelsof the dopants [Fig. 1.4(a)]. When lowering the chemical potential in the metalwith the help of a positive gate voltage V G >
0, one introduces holes in the metalthat attract, via the electric field effect, electrons from the semiconductor to thesemiconductor-insulator interface. These electrons populate the acceptor levels,and as a consequence, the semiconductor bands are bent downwards when theyapproach the interface, such that the filled acceptor levels lie now below theFermi energy [Fig. 1.4(b)].Above a certain threshold of the gate voltage, the bending of the semicon-ductor bands becomes so strong that not only the acceptor levels are below theFermi energy, but also the conduction band in the vicinity of the interface whichconsequently gets filled with electrons [Fig. 1.4(c)]. One thus obtains a con-finement potential of triangular shape for the electrons in the conduction band, wo-Dimensional Electron Systems conductionbandacceptorlevelsvalenceband conductionbandacceptorlevelsvalenceband conductionbandacceptorlevelsvalenceband E z F E z F E z F (a)(b) (c) V V GG metal oxide(insulator) semiconductormetal oxide(insulator) semiconductor metal oxide(insulator) III V G zz EEE
2D electrons
Figure 1.4: MOSFET. The inset I shows a sketch of a MOSFET. (a) Levelstructure at V G = 0. In the metallic part, the band is filled up to the Fermienergy E F whereas the oxide is insulating. In the semiconductor, the Fermienergy lies in the band gap (energy gap between the valence and the conductionbands). Close to the valence band, albeit above E F , are the acceptor levels. (b) The chemical potential in the metallic part may be controled by the gatevoltage V G via the electric field effect. As a consequence of the introductionof holes the semiconductor bands are bent downwards, and above a thresholdvoltage (c) , the conduction band is filled in the vicinity of the interface withthe insulator. One thus obtains a 2D electron gas. Its confinement potentialof which is of triangular shape, the levels (electronic subbands) of which arerepresented in the inset II .8 Introduction dopants
AlGaAs z E F GaAs dopants
AlGaAs z E F GaAs (a) (b)
2D electrons
Figure 1.5: Semiconductor heterostructure (GaAs/AlGaAs). (a)
Dopants areintroduced in the AlGaAs layer at a certain distance from the interface. TheFermi energy lies below the in the band gap and is pinned by the dopant levels.The GaAs conduction band has an energy that is lower than that of the dopantlevels, such that it is energetically favourable for the electrons in the dopantlayer to populate the GaAs conduction band in the vicinity of the interface. (b)
This polarisation bends the bands in the vicinity of the interface between thetwo semiconductors, and thus a 2D electron gas is formed there on the GaAsside.the dynamics of which is quantised into discrete electronic subbands in the per-pendicular z -direction (see inset II in Fig. 1.4). Naturally, the electronic wavefunctions are then extended in the z -direction, but in typical MOSFETs onlythe lowest electronic subband E is filled, such that the electrons are purely2D from a dynamical point of view, i.e. there is no electronic motion in the z -direction.The typical 2D electronic densities in these systems are on the order of n el ∼ cm − , i.e. much lower than in usual metals. This turns out to beimportant in the study of the IQHE and FQHE, because the effects occur, as wewill show below, when the 2D electronic density is on the order of the density ofmagnetic flux n B = B/ ( h/e ) threading the system, in units of the flux quantum h/e . This needs to be compared to metals where the surface density is on theorder of 10 cm − , which would require inaccessibly high magnetic fields (onthe order of 1000 T) in order to probe the regime n el ∼ n B . The mobility in MOSFETs, which is typically on the order of µ ∼ cm /Vs,is limited by the quality of the oxide-semiconductor interface (surface rough-ness). This technical difficulty is circumvented in semiconductor heterostruc-tures – most popular are GaAs/AlGaAs heterostructures – which are grownby molecular-beam epitaxy (MBE), where high-quality interfaces with almostatomic precision may be achieved, with mobilities on the order of µ ∼ cm /Vs. These mobilities were necessary to observe the FQHE, which wasindeed first observed in a GaAs/AlGaAs sample [13]. wo-Dimensional Electron Systems (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(c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300 nm
SiOgraphene (2D metal)(insulator)doped Si (metal) G V Figure 1.6: Schematic view of graphene on a SiO substrate with a doped Si(metallic) backgate. The system graphene-SiO -backgate may be viewed as acapacitor the charge density of which is controled by a gate voltage V G .In the (generic) case of GaAs/AlGaAs, the two semiconductors do not pos-sess the same band gap – indeed that of GaAs is smaller than that of AlGaAs,which is chemically doped by donor ions at a certain distance from the interfacebetween GaAs and AlGaAs [Fig 1.5(a)]. The Fermi energy is pinned by thesedonor levels in AlGaAs, which may have a higher energy than the originally un-occupied conduction band in the GaAs part, such that it becomes energeticallyfavourable for the electrons in the donor levels to occupy the GaAs conductionband in the vicinity of the interface. As a consequence, the energy bands of Al-GaAs are bent upwards, whereas those of GaAs are bent downwards. Similarlyto the above-mentioned MOSFET, one thus obtains a 2D electron gas at theinterface on the GaAs side, with a triangular confinement potential. Graphene, a one-atom thick layer of graphite, presents a novel 2D electronsystem, which, from the electronic point of view, is either a zero-overlap semi-metal or a zero-gap semiconductor, where the conduction and the valence bandsare no longer separated by an energy gap. Indeed, in the absence of doping,the Fermi energy lies exactly at the points where the valence band touches theconduction band and where the density of states vanishes linearly.In order to vary the Fermi energy in graphene, one usually places a grapheneflake on a 300 nm thick insulating SiO layer which is itself placed on top ofa positively doped metallic silicon substrate (see Fig. 1.6). This sandwichstructure, with the metallic silicon layer that serves as a backgate, may thus beviewed as a capacitor (Fig. 1.6) the capacitance of which is C = QV G = ǫ ǫAd , (1.14)where Q = en D A is the capacitor charge, in terms of the total surface A , V G is the gate voltage, and d = 300 nm is the thickness of the SiO layer with the0 Introduction dielectric constant ǫ = 3 .
7. The field-effect induced 2D carrier density is thusgiven by n D = αV G with α ≡ ǫ ǫed ≃ . × cm − V . (1.15)The gate voltage may vary roughly between −
100 and 100 V, such that onemay induce maximal carrier densities on the order of 10 cm − , on top ofthe intrinsic carrier density which turns out to be zero in graphene, as will bediscussed in the next chapter. At gate voltages above ±
100 V, the capacitorbreaks down (electrical breakdown).In contrast to 2D electron gases in semiconductor heterostructures, the mo-bilities achieved in graphene are rather low: they are typically on the order of µ ∼ − cm /Vs. Notice, however, that these graphene samples are fabri-cated in the so-called exfoliation technique, where one “peals” thin graphite crys-tals, under ambiant condictions, whereas the highest-mobility GaAs/AlGaAslaboratory samples are fabricated with a very high technological effort. Themobilities of graphene samples are comparable to those of commercial silicon-based electronic elements. hapter 2 Landau Quantisation
The basic ingredient for the understanding of both the IQHE and the FQHE isLandau quantisation, i.e. the kinetic-energy quantisation of a (free) charged 2Dparticle in a perpendicular magnetic field. In this chapter, we give a detailedintroduction to the different aspects of Landau quantisation. We have chosen avery general presentation of this quantisation in order to account for both a non-relativistic and a relativistic 2D particle some properties of which, such as thelevel degeneracy, are identical. In Sec. 2.1, we introduce the basic Hamiltoniansfor 2D particles in the absence of a magnetic field and discuss both Schr¨odinger-and Dirac-type particles, and discuss the case of a non-zero B -field in Sec. 2.2.Sec. 2.3 is devoted to the discussion of the LL structure of non-relativistic andrelativistic particles. B = 0 In this section, we introduce the basic Hamiltonians which we treat in a quantum-mechanical manner in the following parts. Quite generally, we consider a Hamil-tonian for a 2D particle that is translation invariant, i.e. the momentum p = ( p x , p y ) is a constant of motion, in the absence of a magnetic field. Inquantum mechanics, this means that the momentum operator commutes withthe Hamiltonian, [ p , H ] = 0, and that the eigenvalue of the momentum operatoris a good quantum number. In the case of a free particle, this is a very natural assumption, and one has forthe non-relativistic case, H = p m , (2.1) All vector quantities (also in the quantum-mechanical case of operators) v = ( v x , v y ) arehence 2D, unless stated explicitly. Landau Quantisation in terms of the particle mass m . However, we are interested, here, in the motionof electrons in some material (in a metal or at the interface of to semiconductors).It seems, at first sight, to be a very crude assumption to describe the motionof an electron in a crystalline environment in the same manner as a particle infree space. Indeed, a particle in a lattice in not described by the Hamiltonian(2.1) but rather by the Hamiltonian H = p m + N X i V ( r − r i ) , (2.2)where the last term represents the electrostatic potential caused by the ionssituated at the lattice sites r i . Evidently, the Hamiltonian now depends on theposition r of the particle with respect to that of the ions, and the momentum p is therefore no longer a constant of motion or a good quantum number.This problem is solved with the help of Bloch’s theorem: although an ar-bitrary spatial translation is not an allowed symmetry operation as it is thecase for a free particle (2.1), the system is invariant under a translation by anarbitrary lattice vector if the lattice is of infinite extension – an assumptionwe make here. In the same manner as for the free particle, where one definesthe momentum as the generator of a spatial translation, one may then define agenerator of a lattice translation. This generator is called the lattice momen-tum or also the quasi-momentum . As a consequence of the discreteness of thelattice translations, not all values of this lattice momentum are physical, butonly those within the first Brillouin zone (BZ) – any vibrational mode, be it alattice vibration or an electronic wave, with a wave vector outside the first BZcan be described by a mode with a wave vector within the first BZ. Since theselecture notes cannot include a full course on basic solid-state physics, we referthe reader to standard textbooks on solid-state physics [25, 26].The bottom line is that also in a (perfect) crystal, the electrons may bedescribed in terms of a Hamiltonian H ( p x , p y ) if one keeps in mind that themomentum p in this expression is a lattice momentum restricted to the firstBZ. Notice, however, that although the resulting Hamiltonian may often bewritten in the form (2.1), the mass is generally not the free electron mass buta band mass m b that takes into account the particular features of the energybands – indeed, the mass may even depend on the direction of propagation, The statement that p is a constant of motion remains valid also in the case of a relativisticparticle. However, the Hamiltonian description depends on the frame of reference because theenergy is not Lorentz-invariant, i.e. invariant under a transformation into another frame ofreference that moves at constant velocity with respect to the first one. For this reason aLagrangian rather than a Hamiltonian formalism is often prefered in relativistic quantummechanics. Although this may seem to be a typical “theoretician’s assumption”, it is a very goodapproximation when the lattice size is much larger than all other relevant length scales, suchas the lattice spacing or the Fermi wave length. In GaAs, e.g., the band mass is m b = 0 . m , in terms of the free electron mass m . asic One-Particle Hamiltonians for B = 0 23 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(c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: A sublattice : B sublattice aa aa δ δδ (a) (b) KK’ K K’K’K M’’ M’MM’’M’M * * Γ y x Figure 2.1: (a)
Honeycomb lattice. The vectors δ , δ , and δ connect nn carbon atoms, separated by a distance a = 0 .
142 nm. The vectors a and a are basis vectors of the triangular Bravais lattice. (b) Reciprocal lattice ofthe triangular lattice. Its primitive lattice vectors are a ∗ and a ∗ . The shadedregion represents the first Brillouin zone (BZ), with its centre Γ and the twoinequivalent corners K (black squares) and K ′ (white squares). The thick partof the border of the first BZ represents those points which are counted in thedefinition such that no points are doubly counted. The first BZ, defined in astrict manner, is, thus, the shaded region plus the thick part of the border. Forcompleteness, we have also shown the three inequivalent cristallographic points M , M ′ , and M ′′ (white triangles).such that one should write the Hamiltonian more generally as H = p x m x + p y m y . The above considerations for electrons in a 2D lattice are only valid in the caseof a
Bravais lattice, i.e. a lattice in which all lattice sites are equivalent from acrystallographic point of view. However, some lattices, such as the honeycomblattice that describes the arrangement of carbon atoms in graphene due to thesp hybridisation of the valence electrons, are not Bravais lattices. In this case,one may describe the lattice as a Bravais lattice plus a particular pattern of N s sites, called the basis . This is illustrated in Fig. 2.1(a) for the case ofthe honeycomb lattice. When one compares a site A (full circle) with a site B(empty circle), one notices that the environment of these two sites is different:whereas a site A has nearest neighbours in the directions north-east, north-westand south, a site B has nearest neighbours in the directions north, south-westand south-east. This precisely means that the two sites are not equivalentfrom a crystallographic point of view – although they may be equivalent froma chemical point of view, i.e. occupied by the same atom or ion type (carbon4 Landau Quantisation
Figure 2.2: Energy bands of graphene. The valence band touches the conduc-tion band in the two inequivalent BZ corners K and K ′ . For undoped graphene,the Fermi energy lies precisely in the contact points, and the band dispersionin the vicinity of these points is of conical shape.in the case of graphene). However, all sites A form a triangular Bravais latticeas well as all sites B. Both subsets of lattice sites form the two sublattices , andthe honeycomb lattice may thus be viewed as a triangular Bravais lattice witha two-atom basis, e.g. the pattern of two A and B sites connected by the vector δ . In order to calculate the electronic bands in a lattice with N s Bravais sublat-tices, i.e. a basis with N s sites, one needs to describe the general electronic wavefunction as a superposition of N s different wave functions, which satisfy eachBloch’s theorem for all sublattices [25, 26]. Formally, this may be described interms of a N s × N s matrix, the eigenvalues of which yield N s different energybands. In a lattice with N s different sublattices, one therefore obtains one en-ergy band per sublattice, and for graphene, one obtains two different bands forthe conducting electrons, the valence band and the conduction band.The Hamiltonian for low-energy electrons in reciprocal space reads H ( k ) = t (cid:18) γ ∗ k γ k (cid:19) , (2.3)which is obtained within a tight-binding model, where one considers electronichopping between nearest-neighbouring sites with a hopping amplitude t . Be-cause the nearest neighbour of a site A is a site B and vice versa [see Fig.2.1(a)], the Hamiltonian is off-diagonal, and the off-diagonal elements are re-lated by complex conjugation due to time-reversal symmetry [ H ( − k ) ∗ = H ( k )].As already mentioned above, the lattice momentum k is restricted to the firstBZ, which is of hexagonal shape and which we have depicted in Fig. 2.1(b) for asic One-Particle Hamiltonians for B = 0 25completeness. The precise form of the functions γ k is derived in Appendix A[Eq. (A.9)]. The band structure is obtained by diagonalising the Hamiltonian,and one finds the two bands, labelled by λ = ± , ǫ λ ( k ) = λt | γ k | , which areplotted in Fig. 2.2. The valence band ( λ = − ) touches the conduction band( λ = +) in the two inequivalent corners K and K ′ of the first BZ. Because thereare as many electrons in the π -orbitals, that determine the low-energy conduc-tion properties of graphene, as lattice sites, the overall energy band structureis half-filled. This is due to the two spin orientations of the electrons, whichallow for a quantum-mechanical double occupancy of each π -orbital. As a con-sequence, the Fermi energy lies exactly in the contact points K and K ′ of thetwo bands unless the graphene sheet is doped, e.g. with the help of the electricfield effect, as described in Sec. 1.2 of the previous chapter.The inset in Fig. 2.2 shows the band dispersion in the vicinity of the con-tact points K and K ′ , the linearity of which is sufficient to describe the low-energy electronic properties in graphene, i.e. when all relevant energy scales aremuch smaller than the full band width. The conical form of the two bandsis reminiscent of that of relativistic particles, the general dispersion of whichis E = ± p m c + p c , in terms of the light velocity c and the particle mass m . If the latter is zero, one obtains precisely E = ± c | p | , as in the case of low-energy electrons in graphene (inset of Fig. 2.2), which may thus be treated as massless Dirac fermions . Notice that in the continuum description of electronsin graphene, we have two electron types – one for the K point and another onefor the K ′ point. This doubling is called valley degeneracy which is two-foldhere.The analogy between electrons in graphene and massless relativistic particlesis corroborated by a low-energy expansion of the Hamiltonian (2.3) around thecontact points K and K ′ , at the momenta K and K ′ = − K [see Fig. 2.1(a)], k = ± K + p / ¯ h , where | p / ¯ h | ≪ | K | . One may then expand the function γ ± K + p / ¯ h to first order, and one obtains formally H = t (cid:18) ∇ γ ∗ K · p ∇ γ K · p (cid:19) = v (cid:18) p x − ip y p x + ip y (cid:19) = v p · σ where σ = ( σ x , σ y ) in terms of the Pauli matrices σ x = (cid:18) (cid:19) , σ y = (cid:18) − ii (cid:19) and σ z = (cid:18) − (cid:19) and where we have chosen to expand the Hamiltonian (2.3) around the K point. Here, the Fermi velocity v plays the role of the velocity of light c , which is thoughroughly 300 times larger, c ≃ v . The details of the above derivation maybe found in Appendix A. The above Hamiltonian is indeed formally that ofmassless 2D particles, and it is sometimes called Weyl or Dirac Hamiltonian. Indeed, in graphene, the relevant low-energy scales are in the 10 −
100 meV regime,whereas non-linear corrections of the band dispersion become relevant in the eV regime. Notice that γ ± K = 0 by symmetry. One obtains a similar result at the K ′ point, see Eq. (A.15) in Appendix A. Landau Quantisation
We will discuss, in the remainder of this chapter, how the two Hamiltonians H S = p m b and H D = v p · σ , (2.4)for non-relativistic and relativistic particles, respectively, need to be modifiedin order to account for a non-zero magnetic field. B Fields
In order to describe free electrons in a magnetic field, one needs to replace themomentum by its gauge-invariant form [27] p → Π = p + e A ( r ) , (2.5)where A ( r ) is the vector potential that generates the magnetic field B = ∇ × A ( r ). This gauge-invariant momentum is proportional the electron velocity v , which must naturally be gauge-invariant because it is a physical quantity.Notice that because A ( r ) is not gauge invariant, neither is the momentum p .Remember that adding the gradiant of an arbitrary derivable function λ ( r ), A ( r ) → A ( r )+ ∇ λ ( r ), does not change the magnetic field because the rotationalof a gradient is zero. Indeed, the momentum transforms as p → p − e ∇ λ ( r )under a gauge transformation in order to compensate the transformed vectorpotential, such that Π is gauge-invariant. The substitution (2.5) is also called minimal substitution .In the case of electrons on a lattice, this substitution is more tricky becauseof the presence of several bands. Furthermore, the vector potential is unbound,even for a finite magnetic field; this becomes clear if one chooses a particulargauge, such as e.g. the Landau gauge A L ( r ) = B ( − y, , B × L y , where L y is themacroscopic extension of the system in the y -direction. However, it may beshown that the substitution (2.5), which is called Peierls substitution in thecontext of electrons on a lattice, remains correct as long as the lattice spacing a is much smaller than the magnetic length l B = r ¯ heB , (2.6)which is the fundamental length scale in the presence of a magnetic field. Be-cause a is typically an atomic scale ( ∼ . l B ≃
26 nm / p B [T],this condition is fulfilled in all atomic lattices for the magnetic fields, which maybe achieved in today’s high-field laboratories ( ∼
45 T in the continuous regimeand ∼
80 T in the pulsed regime). Higher magnetic fields may be obtained only in semi-destructive experiments , in whichthe sample survives the experiment but not the coil that is used to produce the magnetic field. amiltonians for Non-Zero B Fields H ( p ) → H ( Π ) = H ( p + e A ) = H B ( p , r ) . Notice that because of the spatial dependence of the vector potential, the result-ing Hamiltonian is no longer translation invariant, and the (gauge-dependent)momentum p is no longer a conserved quantity. We will limit the discussion tothe B -field Hamiltonians corresponding to the Hamiltonians (2.4) H BS = [ p + e A ( r )] m b (2.7)for non-relativistic and H BD = v [ p + e A ( r )] · σ (2.8)for relativistic 2D charged particles, respectively. In order to analyse the one-particle Hamiltonians (2.7) and (2.8) in a quantum-mechanical treatment, we use the standard method, the canonical quantisa-tion [28], where one interprets the physical quantities as operators that act onstate vectors in a Hilbert space. These operators do in general not commutewith each other, i.e. the order matters in which they act on the state vectorthat describe the physical system. Formally one introduces the commutator [ O , O ] ≡ O O − O O between the two operators O and O , which aresaid to commute when [ O , O ] = 0 or else not to commute. The basic physicalquantities in the argument of the Hamiltonian are the 2D position r = ( x, y ) andits canonical momenta p = ( p x , p y ), which satisfy the commutation relations[ x, p x ] = i ¯ h, [ y, p y ] = i ¯ h and [ x, y ] = [ p x , p y ] = [ x, p y ] = [ y, p x ] = 0 , (2.9)i.e. each component of the position operator does not commute with the mo-mentum in the corresponding direction. This non-commutativity between theposition and its associated momentum is the origin of the Heisenberg inequalityaccording to which one cannot know precisely both the position of a quantum-mechanical particle and, at the same moment, its momentum, ∆ x ∆ p x > ∼ h and∆ y ∆ p y > ∼ h .As a consequence of the commutation relations (2.9), the components of thegauge-invariant momentum no longer commute themselves,[Π x , Π y ] = [ p x + eA x ( r ) , p y + eA y ( r )] = e ([ p x , A y ] − [ p y , A x ])= e (cid:18) ∂A y ∂x [ p x , x ] + ∂A y ∂y [ p x , y ] − ∂A x ∂x [ p y , x ] − ∂A x ∂y [ p y , y ] (cid:19) , Landau Quantisation where we have used the relation [ O , f ( O )] = dfd O [ O , O ] (2.10)between two arbitrary operators, the commutator of which is a c-number or anoperator that commutes itself with both O and O [28]. With the help of thecommutation relations (2.9), one finds that[Π x , Π y ] = − ie ¯ h (cid:18) ∂A y ∂x − ∂A x ∂y (cid:19) = − ie ¯ h ( ∇ × A ) z = − ie ¯ hB, and, in terms of the magnetic length (2.6),[Π x , Π y ] = − i ¯ h l B . (2.11)This equation is the basic result of this section and merits some further discus-sion. ˆ As one would have expected for gauge-invariant quantities (the two com-ponents of Π ), their commutator is itself gauge-invariant. Indeed, it onlydepends on universal constants and the (gauge-invariant) magnetic field B , and not on the vector potential A . ˆ The components of the gauge-invariant momentum Π are mutually conju-gate in the same manner as x and p x or y and p y . Remember that p x gen-erates the translations in the x -direction (and p y those in the y -direction).This is similar here: Π x generates a “boost” of the gauge-invariant mo-mentum in the y -direction, and similarly Π y one in the x -direction. ˆ As a consequence, one may not diagonalise at the same time Π x and Π y ,in contrast to the zero-field case, where the arguments of the Hamiltonian, p x and p y , commute.For solving the Hamiltonians (2.7) and (2.8), it is convenient to use the pairof conjugate operators Π x and Π y to introduce ladder operators in the samemanner as in the quantum-mechanical treatment of the one-dimensional har-monic oscillator. Remember from your basic quantum-mechanics class that theladder operators may be viewed as the complex position of the one-dimensionaloscillator in the phase space, which is spanned by the position ( x -axis) and themomentum ( y -axis),˜ a = 1 √ (cid:18) xx − i pp (cid:19) and ˜ a † = 1 √ (cid:18) xx + i pp (cid:19) , More precisely we have used a gradient generalisation of this relation to operator functionsthat depend on several different operators,[ O , f ( O , ..., O J )] = J X j =1 ∂f∂ O j [ O , O j ]which is valid if [[ O , O j ] , O ] = [[ O , O j ] , O j ] = 0 for all j = 1 , ..., N . andau Levels x = p ¯ h/m b ω and p = √ ¯ hm b ω are normalisation constants in terms ofthe oscillator frequency ω [28]. The fact that the position x and the momentum p are conjugate variables and the particular choice of the normalisation constantsyields the commutation relation [˜ a, ˜ a † ] = 1 for the ladder operators.In the case of the 2D electron in a magnetic field, the ladder operators playthe role of a complex gauge-invariant momentum (or velocity), and they read a = l B √ h (Π x − i Π y ) and a † = l B √ h (Π x + i Π y ) , (2.12)where we have chosen the appropriate normalisation such as to obtain the usualcommutation relation [ a, a † ] = 1 . (2.13)It turns out to be helpful for future calculations to invert the expression for theladder operators (2.12),Π x = ¯ h √ l B (cid:0) a † + a (cid:1) and Π y = ¯ hi √ l B (cid:0) a † − a (cid:1) . (2.14) The considerations of the preceding section are extremely useful in the calcu-lation of the level spectrum associated with the Hamiltonians (2.7) and (2.8)of both the non-relativistic and the relativistic particles, respectively. The un-derstanding of this level spectrum is the issue of the present section. Becauseelectrons do not only possess a charge but also a spin, each level is split intotwo spin branches separated by the energy difference ∆ Z ǫ = gµ B B , where g isthe g -factor of the host material and µ B = e ¯ h/ m the Bohr magneton. In or-der to simplify the following presentation of the quantum-mechanical treatmentand the level structure, we neglect this effect associated with the spin degree offreedom. Formally, this amounts to considering spinless fermions . Notice, how-ever, that there exist interesting physical properties related to the spin degreeof freedom, which will be treated separately in Chap. 5. In terms of the gauge-invariant momentum, the Hamiltonian (2.7) for non-relativistic electrons reads H BS = 12 m b (cid:0) Π x + Π y (cid:1) . The analogy with the one-dimensional harmonic oscillator is apparent if onenotices that both conjugate operators Π x and Π y occur in this expression in aquadratic form. If one replaces these operators with the ladder operators (2.14),0 Landau Quantisation one obtains, with the help of the commutation relation (2.13), H BS = ¯ h ml B (cid:2) a † + a † a + aa † + a − (cid:0) a † − a † a − aa † + a (cid:1)(cid:3) = ¯ h ml B (cid:0) a † a + aa † (cid:1) = ¯ h ml B (cid:18) a † a + 12 (cid:19) = ¯ hω C (cid:18) a † a + 12 (cid:19) , (2.15)where we have used the relation ω c = ¯ h/m b l B between the cyclotron frequency(1.2) and the magnetic length (2.6) in the last step.As in the case of the one-dimensional harmonic oscillator, the eigenvaluesand eigenstates of the Hamiltonian (2.15) are therefore those of the numberoperator a † a , with a † a | n i = n | n i . The ladder operators act on these states inthe usual manner [28] a † | n i = √ n + 1 | n + 1 i and a | n i = √ n | n − i , (2.16)where the last equation is valid only for n > a on the groundstate | i gives zero, a | i = 0 . (2.17)This last equation turns out to be helpful in the calculation of the eigenstatesassociated with the level of lowest energy, as well as the construction of statesin higher levels n (see Sec. 2.4.1) | n i = (cid:0) a † (cid:1) n √ n ! | i . (2.18)The energy levels of the 2D charged non-relativistic particle are thereforediscrete and labelled by the integer n , ǫ n = ¯ hω C (cid:18) n + 12 (cid:19) . (2.19)These levels, which are also called Landau levels (LL), are depicted in Fig. 2.3(a)as a function of the magnetic field. Because of the linear field-dependence ofthe cyclotron frequency, the LLs disperse linearly themselves with the magneticfield.
The relativistic case (2.8) for electrons in graphene may be treated exactly inthe same manner as the non-relativistic one. In terms of the ladder operators(2.12), the Hamiltonian reads H BD = v (cid:18) x − i Π y Π x + i Π y (cid:19) = √ hvl B (cid:18) aa † (cid:19) . (2.20) andau Levels (a) (b) n=00 B ene r g y magnetic field ene r g y magnetic field B n=0n=1n=2n=3n=4 +,n=4+,n=3+,n=2+,n=1−,n=1−,n=2−,n=3−,n=4
Figure 2.3: Landau levels as a function of the magnetic field. (a)
Non-relativisticcase with ǫ n = ¯ hω C ( n + 1 / ∝ B ( n + 1 / (b) Relativistic case with ǫ λ,n = λ (¯ hv/l B ) √ n ∝ λ √ Bn .One notices the occurence of a characteristic frequency ω ′ = √ v/l B , whichplays the role of the cyclotron frequency in the relativistic case. Notice, however,that this frequency may not be written in the form eB/m b because the bandmass is strictly zero in graphene, such that the frequency would diverge. In order to obtain the eigenvalues and the eigenstates of the Hamiltonian(2.20), one needs to solve the eigenvalue equation H BD ψ n = ǫ n ψ n . Because theHamiltonian is a 2 × ψ n = (cid:18) u n v n (cid:19) , and we thus need to solve the system of equations¯ hω ′ a v n = ǫ n u n and ¯ hω ′ a † u n = ǫ n v n , (2.21)which yields the equation a † a v n = (cid:16) ǫ n ¯ hω ′ (cid:17) v n (2.22)for the second spinor component. One notices that this component is an eigen-state of the number operator n = a † a , which we have already encountered inthe preceding subsection. We may therefore identify, up to a numerical factor,the second spinor component v n with the eigenstate | n i of the non-relativistic Hamiltonian (2.15), v n ∼ | n i . Furthermore, one observes that the square of the Sometimes, a cyclotron mass m C is formally introduced via the equality ω ′ ≡ eB/m C .However, this mass is a somewhat artificial quantity, which turns out to depend on the carrierdensity. We will therefore not use this quantity in the present lecture notes. Landau Quantisation energy is proportional to this quantum number, ǫ n = (¯ hω ′ ) n . This equationhas two solutions, a positive and a negative one, and one needs to introduceanother quantum number λ = ± , which labels the states of positive and neg-ative energy, respectively. This quantum number plays the same role as theband index ( λ = + for the conduction and λ = − for the valence band) in thezero- B -field case discussed in Sec. 2.1. One thus obtains the level spectrum [29] ǫ λ,n = λ ¯ hvl B √ n (2.23)the energy levels of which are depicted in Fig. 2.3(b). These relativistic Landaulevels disperse as λ √ Bn as a function of the magnetic field.Once we know the second spinor component, the first spinor component isobtained from Eq. (2.21), which reads u n ∝ a v n ∼ a | n i ∼ | n − i . One thenneeds to distinguish the zero-energy LL ( n = 0) from all other levels. Indeed,for n = 0, the first component is zero as one may see from Eq. (2.17). In thiscase one obtains the spinor ψ n =0 = (cid:18) | n = 0 i (cid:19) . (2.24)In all other cases ( n = 0), one has positive and negative energy solutions,which differ among each other by a relative sign in one of the components. Aconvenient representation of the associated spinors is given by ψ λ,n =0 = 1 √ (cid:18) | n − i λ | n i (cid:19) . (2.25) Experimental observation of relativistic Landau levels
Relativistic LLs have been observed experimentally in transmission spectroscopy,where one shines light on the sample and measures the intensity of the trans-mitted light. Such experiments have been performed on so-called epitaxialgraphene [31] and later on exfoliated graphene [32]. When the monochro-matic light is in resonance with a dipole-allowed transition from the (partially)filled LL ( λ, n ) to the (partially) unoccupied LL ( λ ′ , n ± n to the first unoccupied one n + 1.The transition energy is ¯ hω C , independently of n , and one therefore observesa single absorbtion line (cyclotron resonance). In graphene, however, there aremany more allowed transitions due to the presence of two electronic bands, theconduction and the valence band, and the transitions have the energies∆ n,ξ = ¯ hvl B hp n + 1) − ξ √ n i , Epitaxial graphene is obtained from a thermal graphitisation process of an epitaxiallygrown SiC crystal [30] andau Levels )( DLL )( DLL )( CLL )( CLL )( BLL )( BLL )( ALL T r an s i t i on ene r g y ( m e V ) sqrt(B)
10 20 30 40 50 60 70 80 900.860.880.900.920.940.960.981.00 R e l a t i v e t r an s m i ss i on Energy (meV)
10 20 30 40 50 60 70 800.960.981.00 BE L L L L L L Be2cE ~ L E E AB CD BE L L L L L L Be2cE ~ L E E AB CD (D)(C)(B) R e l a t i v e t r an s m i ss i on Energy (meV) (A)0.4 T1.9 K transition Ctransition B
Bénergie [meV] energy [meV] r e l a t i v e t r an s m i ss i on t r an s m i ss i on ene r g y [ m e V ] r e l a t i v e t r an s m i ss i on (a) (b)(c) Figure 2.4: LL spectroscopy in graphene (from Sadowski et al. , 2006). (a)
For a fixed magnetic field (0.4 T), one observes resonances in the transmissionspectrum as a function of the irradiation energy. The resonances are associatedwith allowed dipolar transitions between relativistic LLs. (b)
These resonancesare shifted as a function of the magnetic field. (c)
If one plots the resonanceenergies as a function of the square root of the magnetic field, √ B , a lineardependence is observed as one would expect for relativistic LLs.4 Landau Quantisation where ξ = + denotes an intraband and ξ = − an interband transition. Onetherefore obtains families of resonances the energy of which disperses as ∆ n,ξ ∝√ B , as it has been observed in the experiments [see Fig. 2.4(c), where we showthe results from Sadowski et al. [31]]. Notice that the dashed lines in Fig. 2.4(c)are fits with a single fitting parameter (the Fermi velocity v ), which matcheswell all experimental points for different values of n . In the preceding subsection, we have learnt that the energy of 2D (non-)relativisticcharged particles is characterised by a quantum number n , which denotes theLLs (in addition to the band index λ in for relativistic particles). However,the quantum system is yet underdetermined, as may be seen from the followingdimensional argument. The original Hamiltonians (2.7) and (2.8) are functionsthat depend on two pairs of conjugate operators, x and p x , and y and p y ,respectively, whereas when they are expressed in terms of the gauge-invariantmomentum Π or else the ladder operators a and a † the Hamiltonians (2.15) and(2.20) depend only on a single pair of conjugate operators. From the originalmodels, one would therefore expect the quantum states to be described by two quantum numbers (one for each spatial dimension). This is indeed the case inthe zero-field models (2.4), where the quantum states are characterised by thetwo quantum numbers p x and p y , i.e. the components of the 2D momentum.For a complete description of the quantum states, we must therefore searchfor a second pair of conjugate operators, which necessarily commutes with theHamiltonian and which therefore gives rise to the level degeneracy of the LLs– in addition to the degeneracy due to internal degrees of freedom such as thespin or, in the case of graphene, the two-fold valley degeneracy.In analogy with the gauge-invariant momentum, Π = p + e A ( r ), we considerthe same combination with the opposite relative sign, ˜Π = p − e A ( r ) , (2.26)which we call pseudo-momentum to give a name to this operator. One maythen express the momentum operator p and the vector potential A ( r ) in termsof Π and ˜Π , p = 12 ( Π + ˜Π ) and A ( r ) = 12 e ( Π − ˜Π ) . (2.27)Notice that, in contrast to the gauge-invariant momentum, the pseudo-momentum depends on the gauge and, therefore, does not represent a physical quantity. However, the commutator between the two components of the pseudo-momentumturn out to be gauge-invariant, h ˜Π x , ˜Π y i = i ¯ h l B . (2.28) The quantum states are naturally only degenerate if one neglects the Zeeman effect. We will nevertheless try to give a physical interpretation to this operator below, within asemi-classical picture. andau Levels x and Π y , as well as the mixed commutators between the gauge-invariant momentum and the pseudo-momentum, h Π x , ˜Π x i = 2 ie ¯ h∂A x ∂x , h Π y , ˜Π y i = 2 ie ¯ h∂A y ∂y , (2.29) h Π x , ˜Π y i = ie ¯ h (cid:18) ∂A x ∂y + ∂A y ∂x (cid:19) = − h ˜Π x , Π y i . These mixed commutators are unwanted quantities because they would in-duce unphysical dynamics due to the fact that the components of the pseudo-momentum would not commute with the Hamiltonian, [ ˜Π x/y , H ] = 0. However,this embarrassing situation may be avoided by choosing the appropriate gauge,which turns out to be the symmetric gauge A S ( r ) = B − y, x, , (2.30)with the help of which all mixed commutators (2.29) vanish such that the com-ponents of the pseudo-momentum also commute with the Hamiltonian.Notice that there exists a second popular choice for the vector potential,namely the Landau gauge, which we have already mentioned above, A L ( r ) = B ( − y, , , (2.31)for which the last of the mixed commutators (2.29) would not vanish. Thisgauge choice may even occur simpler: because the vector potential only de-pends on the y -component of the position, the system remains then translationinvariant in the x -direction. Therefore, the associated momentum p x is a goodquantum number, which may be used to label the quantum states in additionto the LL quantum number n . For the Landau gauge, which is useful in the de-scription of geometries with translation invariance in the y -direction, the wavefunctions are calculated in Sec. (2.4.2). However, the symmetric gauge, thewave functions of which are presented in Sec. (2.4.1), plays an important rolein two different aspects; first, it allows for a semi-classical interpretation moreeasily than the Landau gauge, and second, the wave functions obtained fromthe symmetric gauge happen to be the basic ingredient in the construction oftrial wave functions `a la Laughlin for the description of the FQHE, as we willsee in Chap. 4.The pseudo-momentum, with its mutually conjugate components ˜Π x and ˜Π y ,allows us to introduce, in the same manner as for the gauge-invariant momentum Π , ladder operators, b = l B √ h (cid:16) ˜Π x + i ˜Π (cid:17) and b † = l B √ h (cid:16) ˜Π x − i ˜Π (cid:17) , (2.32)6 Landau Quantisation η B rR
Figure 2.5: Cyclotron motion of an electron in a magnetic field around theguiding centre R . The grey region indicates the quantum-mechanical uncer-tainty of the guiding-centre position due to the non-commutativity (2.39) of itscomponents.which again satisfy the usual commutation relations [ b, b † ] = 1 and which, in thesymmetric gauge, commute with the ladder operators a and a † , [ b, a ( † ) ] = 0, andthus with the Hamiltonian, [ b ( † ) , H B ] = 0. One may then introduce a numberoperator b † b associated with these ladder operators, the eigenstates of whichsatisfy the eigenvalue equation b † b | m i = m | m i . One thus obtains a second quantum number, an integer m ≥
0, which is nec-essary to describe, as expected from the above dimensional argument, the fullquantum states in addition to the LL quantum number n . The quantum statestherefore become tensor products of the two Hilbert vectors | n, m i = | n i ⊗ | m i (2.33)for non-relativistic particles. In the relativistic case, one has ψ λn,m = ψ λn,m ⊗ | m i = 1 √ (cid:18) | n − , m i λ | n, m i (cid:19) (2.34)for n = 0 and ψ n =0 ,m = ψ n =0 ⊗ | m i = (cid:18) | n = 0 , m i (cid:19) (2.35)for the zero-energy LL. How can we illustrate this somewhat mysterious pseudo-momentum introducedformally above? Remember that, because the pseudo-momentum is a gauge-dependent quantity, any physical interpretation needs to be handled with care. andau Levels m b ¨ r = − e (˙ r × B ) ⇔ ( ¨ x = − ω C ˙ y ¨ y = ω C ˙ x (2.36)which is nothing other than the electron’s acceleration due to the Lorentz force.These equations may be integrated, and one then finds˙ x = Π x m b = − ω C ( y − Y )˙ y = Π y m b = ω C ( x − X ) ⇔ y = Y − Π x eB x = X + Π y eB (2.37)where R = ( X, Y ) is an integration constant, which physically describes a con-stant of motion. This quantity may easily be interpreted: it represents the centreof the electronic cyclotron motion (see Fig. 2.5). Indeed, further integration ofthe equations (2.37) yields the classical cyclotron motion x ( t ) = X − r sin( ω C t + φ ) and y ( t ) = Y + r cos( ω C t + φ ) , where the phase φ is another constant of motion. The cyclotron motion itselfis described by the velocities (or else the gauge-invariant momenta) Π x /m andΠ y /m . Whereas the energy depends on these velocities that determine theradius r of the cyclotron motion, it is completely independent of the positionof its centre R , which we call guiding centre from now on, as one would expectfrom the translational invariance of the equations of motion (2.36).In order to relate the guiding centre R to the pseudo-momentum ˜Π , we useEq. (2.27) for the vector potential in the symmetric gauge, e A ( r ) = eB (cid:18) − yx (cid:19) = 12 ( Π − ˜Π ) . The postions x and y may then be expressed in terms of the momenta Π and ˜Π , y = ˜Π x eB − Π x eBx = − ˜Π y eB + Π y eB . A comparison of these expresssions with Eq. (2.37) allows us to identify X = − ˜Π y eB and Y = ˜Π x eB . (2.38)This means that, in the symmetric gauge, the components of the pseudo-momentum are nothing other, apart from a factor to translate a momentum8 Landau Quantisation into a position, than the the components of the guiding centre, which are natu-rally constants of motion. In a quantum-mechanical treatment, these operatorstherefore necessarily commute with the Hamiltonian, as we have already seenabove. Furthermore, the commutation relation (2.28) between the componentsof the pseudo-momentum, [ ˜Π x , ˜Π y ] = i ¯ h /l B induces the commutation relation[ X, Y ] = il B (2.39)between the components of the guiding-centre operator. This means that thereis a Heisenberg uncertainty associated with the guiding-centre position of aquantum-mechanical state – one cannot know its x - and y -components simulta-neously, and the guiding centre is, therefore, smeared out over a surface∆ X ∆ Y = 2 πl B (2.40)(see grey region in Fig. 2.5). This minimal surface plays the same role as thesurface (action) h in phase space and therefore allows us to count the numberof possible quantum states of a given (macroscopic) surface A , N B = A ∆ X ∆ Y = A πl B = n B × A , where we have introduced the flux density n B = 12 πl B = Bh/e, (2.41)which is nothing other than the magnetic field measured in units of the fluxquantum h/e . Therefore, the number of quantum states in a LL equals thenumber of flux quanta threading the sample surface A , and each LL is macro-scopically degenerate . We will show in a more quantitative manner than in theabove argument based on the Heisenberg inequality that the number of statesper LL is indeed given by N B when discussing, in the next section, the electronicwave functions in the symmetric and the Landau gauges.Similarly to the guiding-centre operator, we may introduce the cyclotronvariable η = ( η x , η y ), which determines the cyclotron motion and which fullydescribes the dynamic properties. The cyclotron variable is perpendicular tothe electron’s velocity and may be expressed in terms of the gauge-invariantmomentum Π , η x = Π y eB and η y = − Π x eB , (2.42)as one sees from Eq. (2.37). The position of the electron is thus decomposed intoits guiding centre and its cyclotron variable, r = R + η . Also the componentsof the cyclotron variable do not commute, and one finds with the help of Eq.(2.11) [ η x , η y ] = [Π x , Π y ]( eB ) = − il B = − [ X, Y ] . (2.43) Mathematicians speak of a non-commutative geometry in this context, and the charged2D particle may be viewed as a pardigm of this concept. igenstates N independent quantum-mechanical electrons at zero-temperature. In the absence of a magnetic field, electrons in a metal, due totheir fermionic nature and the Pauli principle which prohibits double occupancyof a single quantum state, fill all quantum states up to the Fermi energy, whichdepends thus on the number of electrons itself. The situation is similar in thepresence of a magnetic field: the electrons preferentially occupy the lowest LLs,i.e. those of lowest energy. But once a LL is filled, the remaining electrons areforced to populate higher LLs. In order to describe the LL filling it is thereforeuseful to introduce the dimensionless ratio between the number of electrons N el = n el × A and that of the flux quanta, ν = N el N B = n el n B = hn el eB , (2.44)which is called filling factor . Indeed the integer part, [ ν ], of the filling factorcounts the number of completely filled LLs. Notice that one may vary the fillingfactor either by changing the particle number or by changing the magnetic field.At fixed particle number, lowering the magnetic field corresponds to an increaseof the filling factor. The algebraic tools established above may be used calculate the electronic wavefunctions, which are the space representations of the quantum states | n, m i , φ n,m ( x, y ) = h x, y | n, m i . Notice first that one may obtain all quantum state | n, m i from a single state | n = 0 , m = 0 i , with the help of | n, m i = (cid:0) a † (cid:1) n √ n ! (cid:0) b † (cid:1) m √ m ! | n = 0 , m = 0 i , (2.45)which is a generalisation of Eq. (2.18). Naturally, this equation translates intoa differential equation for the wave functions φ n,m ( x, y ).A state in the lowest LL ( n = 0) is characterised by the condition (2.17) a | n = 0 , m i = 0 , (2.46)which needs to be translated into a differential equation. Remember from Eq.(2.12) that a = ( l B / √ h )(Π x − i Π y ) and, by definition, Π = − i ¯ h ∇ + e A ( r )where we have already represented the momentum as a differential operator inposition representation, p = − i ¯ h ∇ . One then finds a = − i √ (cid:20) l B ∂ x − i∂ y ) + x − iy l B (cid:21) , We limit the discussion to the non-relativistic case. The spinor wave functions for rela-tivistic electrons are then easily obtained with the help of Eqs. (2.34) and (2.35). Landau Quantisation where ∂ x and ∂ y are the components of the gradient ∇ = ( ∂ x , ∂ y ), and one seesfrom this expression that it is convenient to introduce complex coordinates todescribe the 2D plane. We define z = x − iy , z ∗ = x + iy , ∂ = ( ∂ x + i∂ y ) / ∂ = ( ∂ x − i∂ y ) /
2. The lowest LL condition (2.46) then becomes a differentialequation, (cid:18) z l B + l B ¯ ∂ (cid:19) φ n =0 ( z, z ∗ ) = 0 , (2.47)which may easily be solved by the complex function φ n =0 ( z, z ∗ ) = f ( z ) e −| z | / l B , (2.48)where f ( z ) is an analytic function, i.e. ¯ ∂f ( z ) = 0, and | z | = zz ∗ . This meansthat there is an additional degree of freedom because f ( z ) may be any analyticfunction. It is not unexpected that this degree of freedom is associated with thesecond quantum number m , as we will now discuss.The ladder operators b and b † may be expressed in position representation ina similar manner as a , and one obtains the space representation of the differentladder operators, a = − i √ (cid:18) z l B + l B ¯ ∂ (cid:19) , a † = i √ (cid:18) z ∗ l B − l B ∂ (cid:19) b = − i √ (cid:18) z ∗ l B + l B ∂ (cid:19) , b † = i √ (cid:18) z l B − l B ¯ ∂ (cid:19) . (2.49)In the same manner as for a state in the lowest LL, the condition for the referencestate with m = 0 is b | n, m = 0 i = 0, which yields the differential equation (cid:0) z ∗ + 4 l B ∂ (cid:1) φ ′ m =0 ( z, z ∗ ) = 0with the solution φ ′ m =0 ( z, z ∗ ) = g ( z ∗ ) e −| z | / l B , in terms of an anti-analytic function g ( z ∗ ) with ∂g ( z ∗ ) = 0. The wave function φ n =0 ,m =0 ( z, z ∗ ) must therefore be the Gaussian with a prefactor that is bothanalytic and anti-analytic, i.e. a constant that is fixed by the normalisation.One finds φ n =0 ,m =0 ( z, z ∗ ) = h z, z ∗ | n = 0 , m = 0 i = 1 p πl B e −| z | / l B , (2.50)and a lowest-LL state with arbitrary m may then be obtained with the help ofEq. (2.45), φ n =0 ,m ( z, z ∗ ) = i m √ m p πl B m ! (cid:18) z l B − l B ¯ ∂ (cid:19) m e −| z | / l B = i m p πl B m ! (cid:18) z √ l B (cid:19) m e −| z | / l B . (2.51) igenstates z m of analytic functions. In an arbitraryLL, the states may be obtained in a similar manner, but they happen to bemore complicated because the differential operators (2.49) no longer act on theGaussian only but also on the polynomial functions. They may be expressed interms on Laguerre polynomials.To conclude the discussion about the wave functions in the symmetric gauge,we calculate the average value of the guiding-centre operator in the state | n =0 , m i . With the help of Eqs. (2.32) and (2.38), one may express the componentsof the guiding-centre operator in terms of the ladder operators b and b † , X = l B i √ b † − b ) and Y = l B √ b † + b ) , (2.52)and the ladder operators act, in analogy with Eq. (2.16), on the states | n, m i as b † | n, m i = √ m + 1 | n, m + 1 i and b | n, m i = √ m | n, m − i . The average value of the guiding-centre operator is therefore zero in the states | n, m i , h R i ≡ h n = 0 , m | R | n = 0 , m i = 0 , but we have h| R |i = Dp X + Y E = l B Dp b † b + 1 E = l B √ m + 1 . (2.53)This means that the guiding centre is situated, in a quantum state | n, m i , some-where on a circle of radius l B √ m + 1 whereas its angle (or phase) is completelyundetermined.The symmetric gauge is the natural gauge to describe a sample in the formof a disc. Consider the disc to have a radius R max (and a surface A = πR max ).How many quantum states may be accomodated within the circle? The quan-tum state with maximal m quantum number, which we call M , has a radius l B √ M + 1, which must naturally coincide with the radius R max of the disc.One therefore obtains A = πl B (2 M + 1), and the number of states within thedisc is then, in the thermodynamic limit M ≫ M = A πl B = n B × A = N B , (2.54)in agreement with the result (2.41) obtained from the argument based on theHeisenberg uncertainty relation. If the sample geometry is rectangular, the Landau gauge (2.31), A L ( r ) = B ( − y, , p x = ¯ hk is a good2 Landau Quantisation quantum number due to translational invariance in the x -direction. One maytherefore use a plane-wave ansatz ψ n,k ( x, y ) = e ikx √ L χ n,k ( y ) , for the wave functions. In this case, the Hamiltonian (2.7) becomes H BS = ( p x − eBy ) m + p y m = p y m + 12 mω C ( y − y ) , (2.55)where we have defined y = kl B . (2.56)The Hamiltonian (2.55) is nothing other than the Hamiltonian of a one-dimensionaloscillator centred around the position y , and the eigenstates are χ n,k ( y ) = H n (cid:18) y − y l B (cid:19) e − ( y − y ) / l B , in terms of Hermite polynomials H n ( x ) [28]. The coordinate y plays the roleof the guiding centre component Y , the component X being smeared over thewhole sample length L , as it is dictated by the Heisenberg uncertainty relationresulting from the commutation relation (2.39) [ X, Y ] = il B .Using periodic boundary conditions k = m × π/L for the wave vector inthe x -direction, one may count the number of states in a rectangular surface oflength L and width W (in the y -direction), similarly to the above arguments inthe symmetric gauge. Consider the sample to range from y min = 0 to y max = W , the first corresponding via the above-mentioned condition (2.56) to thewave vector k = 0 and the latter to a wave vector k max = M × π/L . Twoneighbouring quantum states are separated by the distance ∆ y = ∆ kl B =∆ m (2 π/L ) l B = 2 πl B /L , and each state therefore occupies a surface σ = ∆ y × L = 2 πl B , which agrees with the result (2.40) obtained above with the help of theconsideration based on the Heisenberg uncertainty relation. The total numberof states is, as in the symmetric gauge and the general argument leading to Eq.(2.41), M = N B = n B × LW = n B × A , i.e. the number of flux quanta threading the (rectangular) surface A = LW . hapter 3 Integer Quantum HallEffect
The quantum-mechanical treatment of the 2D electron in a perpendicular mag-netic field is the backbone for the understanding of the basic properties of thequantum Hall effect. However, we need to relate the kinetic-energy quantisa-tion to the resistance quantisation, which is the essential feature of the IQHE.In the present chapter, we discuss the transport properties of electrons in theIQHE, namely the somewhat mysterious role that disorder plays in this type oftransport. Remember from the introduction that the Hall resistance is quan-tised with an astonishingly high precision (10 − ), such that it is now used as thestandard of resistance [see Eq. (1.13)]. The resistance quantisation in the IQHEtherefore does reflect neither a particular disorder distribution nor a particularsample geometry. Nevertheless, disorder turns out to play an essential role inthe occurence of the IQHE, as we will see in this chapter.We will first consider, in Sec. 3.1, the motion of a 2D electron in a perpen-dicular magnetic field when also an external electrostatic potential is present,such as the one generated by disorder or the confinement potential that definesthe sample boundaries. In Sec. 3.2, we then calculate the conductance of asingle LL within a mesoscopic picture and discuss the difference between a two-terminal and a six-terminal transport measurement in Sec. 3.3. Furthermore,we discuss, in Sec. 3.4, the IQHE within a percolation picture and present somescaling properties that characterise the plateau transitions. We terminate thischapter with a short discussion of the pecularities of the relativistic quantumHall effect in graphene the understanding of which requires essentially the sameingredients as the IQHE in non-relativistic quantum Hall systems.434 Integer Quantum Hall Effect L W + ++ _ _ y x µ L µ Rmax y y min Figure 3.1: Potential landscape of an electrostatic potential in a sample. Themetallic contacts are described by the chemical potentials µ L and µ R for theleft and right contacts, respectively. We consider L ≫ W ≫ ξ ≫ l B , where ξ is the typical length scale for the variation of the electrostatic potential. Thesample is confined in the y -direction between y max and y min . The thin linesindicate the equipotential lines. When approaching one of the sample edges,they become parallel to the edge. The grey lines indicate the electronic motionwith the guiding centre moving along the equipotential lines. The electron turnsclockwise around a summit of the potential landscape, which is caused e.g. bya negatively charged impurity ( − ), and counter-clockwise around a valley (+).At the sample edges, the equipotential lines due to the confinement potentialconnect the two contacts on the left and on the right hand side. lectronic Motion in an External Electrostatic Potential We consider a system the length L of which is much larger than the width W (see Fig. 3.1). This may be modeled by a confinement potential V conf ( y ) thatonly depends on the y -direction, i.e. the system remains translation-invariant inthe x -direction with respect to this potential. In addition to the confinement,we consider a smoothly varying electrostatic potential V imp ( x, y ) that is causedby the impurities in the sample. This impurity potential breaks the translationinvariance in the x -direction as well as that in the y -direction, which is alreadybroken by the confinement potential. The Hamiltonian of a 2D particle in aperpendicular magnetic field then needs to be completed by a potential term V ( r ) = V conf ( y ) + V imp ( x, y ) , (3.1)which creates a potential landscape that is schematically depicted in Fig. 3.1. In a first step, we consider a potential V ( r ) that varies smoothly on the scaleset by the magnetic length, i.e. ξ ≫ l B , where ξ describes the characteristiclength scale for the variation of V ( r ). Notice first that the external electrostaticpotential lifts the LL degeneracy because the Hamiltonian H = H B + V ( r = R + η ) no longer commutes with the guiding-centre operator R , in contrast to the“free” Hamiltonian H B , [ H, R ] = [ V, R ] = 0. Physically, this is not unexpected:the guiding centre is a constant of motion due to translation invariance, i.e. itdoes not matter whether the electron performs its cyclotron motion around apoint R or R in the 2D plane as long as the cyclotron radius is the same.However, the electrostatic potential V ( r ) breaks this translation invariance andthus lifts the degeneracy associated with the guiding centre.In the case where the electrostatic potential varies smoothly on a lengthscale set by the magnetic length and does not generate LL mixing, i.e. when |∇ V | ≪ ¯ hω C /l B , we may approximate the argument r in the potential (3.1) bythe guiding-centre variable R , V ( r ) ≃ V ( R ) . (3.2)Notice that this approximation may seem unappropriate when we consider theconfinement potential in the y -direction which may vary abruptly when ap- Naturally, the system is also confined in the x -direction, but since we consider a samplewith L ≫ W , the system appears as translation-invariant in the x -direction when one considersintermediate length scales. The latter may be taken into account with the help of periodicboundary conditions that discretise the wave vector in the x -direction, as we have seen in thepreceding chapter within the quantum-mechanical treatment of the 2D electron in the Landaugauge (see Sec. 2.4.2). This approximation may be viewed as the first term of an expansion of the electrostaticpotential in the coherent (or vortex) state basis, where the states are maximally localisedaround the guiding-centre position R [33]. Integer Quantum Hall Effect proaching the sample edges. The confinement potential with translation invari-ance in the x -direction will be discussed separately in the following subsection.As a consequence of the non-commutativity of the potential term V ( R ) withthe guiding-centre operator, the latter quantity acquires dynamics, as may beseen from the Heisenberg equations of motion i ¯ h ˙ X = [ X, H ] = [
X, V ( R )] = ∂V∂Y [ X, Y ] = il B ∂V∂Yi ¯ h ˙ Y = [ Y, V ( R )] = − il B ∂V∂X , (3.3)i.e. ˙ R ⊥ ∇ V . Here, we have used the commutation relation (2.39) for theguiding-centre components and Eq. (2.10). The Heisenberg equations of motionare particularly useful in the discussion of the semi-classical limit because theaveraged equations satisfy the classical equations of motion, h ˙ R i ⊥ ∇ V, (3.4)which means that, within the semi-classical picture, the guiding centres movealong the equipotential lines of the smoothly varying external electrostatic po-tential . This feature, which is also called the Hall drift , v D = E × B B = h ˙ R i = −∇ V × B eB , (3.5)in terms of the (local) electric field E = −∇ V /e , is depicted in Fig. 3.1 by thegrey lines.In the bulk, the potential landscape is created by the charged impurities inthe sample, and the electrons turn clockwise on an equipotential line around asummit that is caused by a negatively charged impurity and counter-clockwisearound a valley created by a positively charged impurity. If the equipotentiallines are closed, as it is the case for most of the equipotential lines in a poten-tial landscape, an electron cannot move from one point to another one over amacroscopic distance, e.g. from one contact to the other one. An electron mov-ing on a closed equipotential line can therefore not contribute to the electronictransport, and the electron is thus localised . Notice that this type of localisationit different from other popular types. Anderson localisation in 2D, e.g., is dueto quantum interferences of the electronic wave functions [34]. Here, however,the localisation is a purely classical effect. The high-field localisation is also In order to illustrate this point, consider a hiking tour in the mountains, e.g. around LesHouches in the French Alps. To go from one point to another one at the same height, oneusual needs go downhill as well as uphill. It is very rare to be able to stay on the same heightunless one wants to turn in circles that are just the closed countour lines which correspond toclosed equipotential lines in our potential landscape. For those who participated at the LesHouches session which was outsourced to Singapore, where there are no mountains and wherethe whether is anyway too hot for hiking, just look at a hiking map of some mountainousregion. Then search for countour lines that connect one border of the map to the oppositeborder. It turns out to be very hard to find such lines as compared to a large number of closedcountour lines. lectronic Motion in an External Electrostatic Potential y min and y max (see Fig. 3.1). In this case, the equipotential lines are open andtherefore connect the two different electronic contacts. The electrons occupyingquantum states at these equipotential lines then contribute to the electronictransport, in contrast to those on closed equipotential lines in the bulk. Thesequantum states are called extended states , as opposed to the localised states discussed above. The difference between localised and extended states turnsout to be essential in the understanding of the IQHE, as we will see below (Sec.3.4). x -direction Although the above semi-classical considerations yield the correct physical pic-ture of localised and extended states, it is based on the assumption that theelectrostatic potential varies smoothly on the scale set by the magnetic length,such that we may replace the electron’s position by that of its guiding centre [Eq.(3.2)]. This assumption is, however, problematic in view of the confinement po-tential which varies strongly at the sample edges, i.e. in the vicinity of y min and y max . We will therefore treat the y -dependent confinement potential in a quan-tum treatment. Naturally, the appropriate gauge for the quantum-mechanicaltreatment is the Landau gauge (2.31), which respects the translation invariancein the x -direction, and the Hamiltonian (2.55) becomes H = p y m + 12 mω C ( y − y ) + V conf ( y ) . (3.6)Remember that for a fixed wave vector k in the x -direction, the position aroundwhich the one-dimensional harmonic oscillator is centred is fixed by Eq. (2.56), y = kl B . We may therefore expand the confinement potential, even in the caseof a strong variation, around this position, V ( y ) ≃ V ( y = kl B ) − eE ( y )( y − y ) + O (cid:18) ∂ V∂y (cid:19) , where the local electric field is given in terms of the first derivative of thepotential at y , eE ( y ) = − ∂V conf ∂y (cid:12)(cid:12)(cid:12)(cid:12) y . This expansion yields the Hamiltonian H = p y m + 12 mω C ( y − y ′ ) + V conf ( y ) − mv D ( y ) , In the semi-classical picture the extended states are also called skipping orbits . Integer Quantum Hall Effect where the local drift velocity reads v D = E ( y ) /B and the position of theharmonic oscillator is shifted, y → y ′ = y + eE ( y ) /mω C . Notice that thelast term is quadratic in the electric field E ( y ) and therefore a second-orderterm in the expansion of the confinement potential. We neglect this term in thefollowing calculations. The final Hamiltonian then reads H = p y m + 12 mω C ( y − y ′ ) + V conf ( y ′ ) , (3.7)where we have replaced the argument y by the shifted harmonic-oscillator posi-tion y ′ , which is valid at first order in the expansion of the confinement potential.One therefore obtains the energy spectrum ǫ n,y = ¯ hω C (cid:18) n + 12 (cid:19) + V ( y ) , (3.8)where we have omitted the prime at the shifted harmonic-oscillator position tosimplify the notation. One therefore obtains the same LL spectrum as in theabsence of a confinement potential, apart from an energy shift that is determinedby the value of the confinement potential at the harmonic-oscillator position,which may indeed vary strongly. This position y plays the role of the guiding-centre position, as we have already mentioned in the last chapter, where we havecalculated the electronic wave functions in the Landau gauge (2.4.2). One thusobtains a result that is consistent with the semi-classical treatment presentedabove. We now calculate the conductance of a completely filled LL for the geometrydepicted in Fig. 3.1, i.e. when all quantum states (described within the Landaugauge) of the n -th LL are occupied. In a first step, we calculate the current ofthe n -th LL, which flows from the left to the right contact, with the help of theformula [35] I xn = − eL X k h n, k | v x | n, k i , (3.9)i.e. as the sum over all N B quantum channels labelled by the wave vector k = 2 πm/L , with the velocity h n, k | v x | n, k i = 1¯ h ∂ǫ n,k ∂k = L π ¯ h ∆ ǫ n,m ∆ m , in terms of the dispersion relation (3.8). Notice that the velocity in the y -direction is zero because the energy does not disperse as a function of the y - This relation may be obtained from the Heisenberg equations of motion, i ¯ h ˙ x = [ x, H ] =( ∂H/∂p x )[ x, p x ] = i∂H/∂k , where we have used Eq. (2.10) and p x = ¯ hk . One thereforeobtains the operator equation ˙ x = 1¯ h ∂H∂k , which we evaluate in the state | n, k i . In the last step we have used the periodic boundaryconditions. onductance of a Single Landau Level (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(c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(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) y maxn+1 ν = n ν = n − (b) yx ν = n+1 y y max y maxn n−1 n+1nn−1 (a) µ max Figure 3.2: Edge states. (a)
The LLs are bent upwards when approaching thesample edge, which may be modeled by an increasing confinement potential.One may associate with each LL n a maximal value y nmax of the y -componentwhere the LL crosses the chemical potential µ max . (b) At each position y nmax ,the filling factor decreases by a jump of 1. The n -th edge state is associatedwith the jump at y nmax and the gradient of the confinement potential imposes adirection to the Hall drift of this state (chirality) . This chirality is the same forall edge states at the same edge.component of the wave vector. The above expression is readily evaluated with∆ m = 1, and one obtains h n, k | v x | n, k i = Lh ( ǫ n,m +1 − ǫ n,m ) . With the help of this expression, the current (3.9) of the n -th LL becomes I n = − eL X m Lh ( ǫ n,m +1 − ǫ n,m ) , and one notices that all terms in the sum cancel apart from the boundary terms ǫ n,m min and ǫ n,m min , which correspond to the chemical potentials µ min and µ max , respectively. The difference between these two chemical potentials maybe described in terms of the (Hall) voltage V between the upper and the loweredge, µ max − µ min = − eV . One thus obtains the final result I n = − eh ( µ max − µ min ) = e h V. (3.10)This means that each LL contributes one quantum of conductance G n = e /h to the electronic transport and n completely filled LLs contribute a con-ductance G = n − X n ′ =0 G n ′ = n e h . (3.11) Notice that, because the lowest LL is labelled by n ′ = 0, the last one has the index n − n completely filled levels. Integer Quantum Hall Effect
Notice furthermore that this is a particular example of the Landauer-B¨uttikerformula of quantum transport G n = e h T n through a conduction channel n , where T n is the transmission coefficient of thechannel [36, 35, 37]. Because T n + R n = 1, in terms of the reflexion coefficient,the above result (3.10) indicates that each filled LL may be viewed as a conduc-tion channel with perfect transmission T n = 1, i.e. where an injected electronis not reflected or backscattered. The astonishing feature of perfect transmission, which is independent of thelength L (or more precisely of the aspect ratio L/W , see the discussion in Sec.1.1.2 of the introduction) or the particular geometry of the sample, may beunderstood from the edge-state picture which we have introduced above (seeFig. 3.2). Consider the upper edge, without loss of generality. The current-transporting edge state of the n -th LL is the one situated at y nmax , where the n -thLL crosses the Fermi energy and where the filling factor jumps from ν = n +1 to n . Due to the upward bent of the confinement potential a particular direction isimposed on the electronic motion, which is nothing other than the Hall drift (seeFig. 3.1). This uni-directional motion is also called chirality of the edge state.Notice that this is the same chirality which one expects from the semi-classicalexpression (3.5) for the drift velocity. The chirality is the same for all edge states n at the same sample edge where the gradient of the confinement potential doesnot change its direction. Therefore, even if an electron is scattered from onechannel n to another one n ′ at the same edge it does not change its directionof motion, and the electron cannot be backscattered unless it is scattered tothe opposite edge with inverse chirality. However, in a usual quantum Hallsystem, the opposite edges are separated by a macroscopic distance ∼ W , andbackscattering processes are therefore strongly (exponentially) suppressed in theratio l B /W between the magnetic length, which determines the spatial extensionof quantum-mechanical state, and the macroscopic sample width W . Noticethat the quantum Hall system at integer filling factors ν = n is therefore a veryunusual electron liquid: it is indeed a bulk insulator with perfectly conducting(non-dissipative) edges. Strictly speaking the filling factor does not jump not abruptly when one takes interactionsbetween the electrons into account. In this case, two incompressible strips, of ν = n + 1 and ν = n are separated by a compressible strip of finite width. The picture of chiral electrontransport remains, however, essentially the same when considering such compressible regions. wo-terminal versus Six-Terminal Measurement L W + ++ __ y x µ L µ Rmax yy min µ µ maxmin = µ= µ LR Figure 3.3: Two-terminal measurement. The current is driven through thesample via the left and the right contacts, where one also measures the voltagedrop and thus a resistance. The upper edge is in thermodynamic equilibriumwith the left contact (blue), whereas the lower one is in equilibrium with theright contact (red). The chemical potential drops abruptly when the upper edgereaches the right contact, and when the lower edge reaches the left contact. Dis-sipation occurs in these hot spots (red dots). The measured resistance betweenthe two contacts thus equals the Hall resistance.
In the preceding section Sec. (3.2), we have calculated the conductance ofa single LL (and n filled LLs) within a so-called two-terminal measurement,where we inject a current in the left contact with chemical potential µ L andcollect the outcoming current at the right contact with µ R . As a consequence ofEq. (3.10), this current builds up a voltage V between the upper and the lowersample edge. This voltage drop is therefore associated with a Hall resistance,which is the inverse of the conductance G = ne /h , R H = G − = he n , (3.12)and which coincides with the contact (or interface) resistance of a mesoscopicsystem [37]. However, the voltage drop V L between the left and the right contactis given by the difference of the chemical potentials in the contacts, µ R − µ L = − eV L , and the associated longitudinal resistance V L /I is non-zero, in contrastto what we have seen in the introduction. This is due to the fact that the2 Integer Quantum Hall Effect difference between the longitudinal and the Hall resistance is not clearly definedin such a two-terminal measurement.This feature may be understood from Fig. 3.3. Indeed, due to the above-mentioned absence of backscattering, the chemical potential is constant alonga sample edge, but there is a potential difference between the two edges. Thismeans that the chemical potential must change somewhere along the edge. Con-sider the upper edge that is fed with electrons by the left contact, i.e. the upperedge is in thermodynamic equilibrium with the left contact and the chemicalpotentials therefore coincide, µ L = µ max (see Fig.3.3). Now, when the upperedge touches the right contact which is at a different chemical potential µ R , thechemical potential of the upper edge must rapidly relax to be in equilibriumwith the right contact. In the same manner, the lower edge is in equilibriumwith the right contact, µ min = µ R , and abruptly changes when touching the leftcontact. The rapid change in the chemical potential is associated with a dissi-pation of energy (at so-called hot spots ) that has been observed experimentally[38]. In this experiment, the sample was put in liquid helium and the heatingat the hot spots caused a local vaporisation of the helium observable in form ofa fountain of gas bubbles.Due to the equivalence of the chemical potentials µ L = µ max and µ min = µ R ,the voltage drops V , between the upper and the lower edge, and V L betweenthe current contacts are equal, V = V L . An unexpected consequence of thisequation is that in a resistance measurement between the two contacts, in thetwo-terminal configuration, the two-terminal resistance equals the Hall resis-tance, R R − L = R H = he n , (3.13)and not the (vanishing) longitudinal resistance, when the bulk is insulating (at ν = n ). A more sophisticated geometry that allows for the simulaneous measurementof a well-defined longitudinal and Hall resistance is the six-terminal geometry,with two additional contacts at the upper and two at the lower edge [see Fig.3.4(a)]. These additional contacts (2 and 3 at the upper and 5 and 6 at the loweredge, the left and the right contacts being labelled by 1 and 4, respectively) areused to measure a voltage, i.e. they have ideally an infinitely high internalresistance to prevent electrons to leak out of or into the sample. The chemicalpotential therefore remains constant at the upper edge µ L = µ = µ , as wellas that at the lower edge µ R = µ = µ , and one measures a zero-resistance, R L = ( µ − µ ) /eI = ( µ − µ ) /eI = 0, as one expects from the calculationof the conductance through n LLs (see Sec. 3.11), which is entirely transverse.The conductance matrix is thus off-diagonal, as well as the resistance matrix, G = n e h − n e h ! and R = (cid:18) − he nhe n (cid:19) , (3.14) wo-terminal versus Six-Terminal Measurement I IR ~ 56 2 3 41 µ − µ = µ − µ µ − µ = 0
5L HL 3 µ = µ (a) (b) µ = µ R ~ µ = µ = µ
56 R
Figure 3.4: (a)
Six-terminal measurement. The current I is driven throughthe sample via the contacts 1 and 4. Between these two contacts the chemicalpotential on the upper edge µ L (blue) does not vary because the electrons donot leak out or in at the contacts 2 and 3, where one measures the longitudi-nal resistance. In the same manner, the chemical potential µ R (red) remainsconstant between the contacts 5 and 6 on the lower edge. The longitudinalresistance measured between 2 and 3 as well as between 5 and 6 is therefore R L = ( µ − µ ) /eI = ( µ − µ ) /eI = 0. The Hall resistance is determined bythe potential difference between the two edges and thus measured, e.g. betweenthe contacts 5 and 3, where µ − µ = µ R − µ L , and thus R H = ( µ − µ ) /eI . (b) Four-terminal measurement in the van-der-Pauw geometry. In a Hall-resistancemeasurement, one drives a current through the sample via the contacts 1 and3 (connected by the continuous blue line) and measures the Hall resistance viathe contacts 2 and 4 (dashed blue line). In a measurement of the longitudinalresistance, the current is driven through the sample via the contacts 1 and 4(continuous red line) and one measures a resistance between the contacts 2 and3 (connected by the dashed red line).4
Integer Quantum Hall Effect and one precisely measures the diagonal elements of the resistance matrix, thelongitudinal resistance, between the contacs 3 and 2 (or 6 and 5). The off-diagonal elements, i.e. the Hall resistance, may e.g. be measured between thecontacts 5 and 3 [as shown in Fig. 3.4(a)], and one measures then the result R H = G − n = h/e n obtained from the calculation presented in Sec. 3.11because of the voltage drop V = ( µ L − µ R ) /e = ( µ − µ ) /e between the upperand the lower edge.Finally we mention that there exists an intermediate geometry that consistsof four terminals (van-der-Pauw geometry), where the resistance measurementsare equally well defined [Fig. 3.4(b)]. If one labels the contacts from 1 to 4 ina clockwise manner, one may measure a Hall resistance between the contacts2 and 3 while driving a current through the sample by the contacts 1 and 3[blue lines in Fig. 3.4(b)]. In this case, one may use the clear topologicaldefinition mentioned at the beginning of the introduction. If one connects thecontacts 2 and 3 by an imaginary line through the sample (dashed blue line)it necessarily crosses the imaginary line (continuous blue) which connects thecurrent contacts 1 and 3 through the sample. This is precisely the topologicaldefinition of a Hall-resistance measurement.Similarly, one may measure the longitudinal resistance between the contacts2 and 3 if one drives a current through the sample via the contacts 1 and 4.In this case, the imaginary line (dashed red) which connects the contacts 2 and3 where one measures a resistance does not need to cross the line (continuousred) between the contacts 1 and 4 at which one injects and collects the current,respectively. As we have already mentioned at the beginning of the introduction,this defines topologically a measurement of the longitudinal resistance.These considerations show that a resistance measurement, although it doesnot depend on the microscopic details of the sample, depends nevertheless onthe geometry in which the contacts are placed at the sample [39, 35]. Thisaspect is often not sufficiently appreciated in the literature, namely the factthat one measures, in a two-terminal geometry, a Hall resistance between thecontacts that are used to inject and collect the current and not a longitudinalresistance, as one may have expected naively, when the system is in the IQHEcondition. Until now we have shown that the Hall resistance is quantised [Eq. (3.12)] when n LLs are completely filled, i.e. when the filling factor is exactly ν = n . However,we have not yet explained the occurence of plateaus in the Hall resistance, i.e.a Hall resistance that remains constant even if one varies the filling factor,e.g. by sweeping the magnetic field, around ν = n . In order to explain the Strictly speaking, we have not gained anything because the quantum treatment allows usonly to determine the Hall resistance at certain points of the Hall curve, those at the magnetic he Integer Quantum Hall Effect and Percolation νε ε ε n−1 n−1 n−1 (a) (b) (c) density of states density of states density of statesextended stateslocalised states RR B HL h/e n h/e (n+1) RR B=n h/e n HL R BR
L HF FF
EEE
Figure 3.5: Quantum Hall effect. The (impurity-broadened) density of statesis shown in the first line for increasing fillings (a) - (c) described by the Fermienergy E F . The second line represents the impurity-potential landscape thevalleys of which become successively filled with electrons when increasing thefilling factor, i.e. when lowering the magnetic field at fixed particle number.The third line shows the corresponding Hall (blue) and the longitudinal (red)resistance measured in a six-terminal geometry, as a function of the magneticfield. The first figure in column (c) indicates that the bulk extended states arein the centre of the DOS peaks, whereas the localised states are in the tails.6 Integer Quantum Hall Effect constance of the Hall resistance over a rather large interval of magnetic fieldaround ν = n , we need to take into account the semi-classical localisation ofadditional electrons (or holes) described in Sec. 3.1. This is shown in Fig. 3.5,where we represent the filling of the LLs (first line), the potential landscape ofthe last partially-filled level (second line) and the resistances as a function ofthe magnetic field, measured in a six-terminal geometry (third line). We startwith the situation of n completely filled LLs [column (a) of Fig. 3.5], whichwe have extensively discussed above: the LL n (and its potential landscape) isunoccupied. In a six-terminal measurement, one therfore measures the Hallresistance R H = h/e n and a zero longitudinal resistance, as we have seen inEq. (3.14).In column (b) of Fig. 3.5, we represent the situation where the LL n getsmoderately filled by electrons when the magnetic field B is decreased. Theseelectrons in n populate preferentially the valleys of the potential landscape, ormore precisely the closed equipotential lines that enclose these valleys. Theelectrons in the LL n are thus (classically) localised somewhere in the bulk anddo not affect the global transport characteristics, measured by the resistances,because they are not probed by the sample contacts. Therefore, the Hall re-sistance remains unaltered and the longitudinal resistance remains zero despitethe change of the magnetic field. This is the origin of the plateau in the Hallresistance.If one continues to lower the magnetic field, the regions of the potential land-scape in the LL n occupied by electrons become larger, and they are eventuallyenclosed by equipotential lines that pass through the bulk and that connect theopposite edges. In this case, an electron injected at the left contact and trav-elling a certain distance at the upper edge may jump into the state associatedwith this equipotential line and thus reach the lower edge. Due to its chirality,the electron is then backscattered to the left contact, which causes an increasein the longitudinal resistance. Indeed, if one measures the resistance betweenthe two contacts at the lower edge, a potential drop is caused by the electronthat leaks in from this equipotential connecting the upper and the lower edge.It is this potential drop that causes a non-zero longitudinal resistance. At thesame moment the Hall resistance is no longer quantised and jumps to the next(lower) plateau, a situation that is called plateau transition . This situation ofelectron-filled equipotential lines connecting opposite edges, which are thus ex-tended states [see first line of Fig. 3.5(c)] as opposed to the bulk localised states ,arises when the LL n is approximately half-filled. Notice that these extendedstates, which are found in the centre of the DOS peaks [see upper part of Fig.3.5(c)], are bulk states in contrast to the above-mentioned edge states, whichare naturally also extendedThe clean jump in the Hall resistance at the plateau transition accompanied fields corresponding to ν = hn el /eB = n . If we substitute the filling factor in Eq. (3.12), wesee immediately that R H = h/e ν = B/en el , i.e. one retrieves the classical result for the Hallresistance. Remember that due to the label 0 for the lowest LL, all LLs with n ′ = 0 , ..., n − n is then the lowest unoccupied level. he Integer Quantum Hall Effect and Percolation et al. , 2008, on a 2D electronsystem on a n -InSb surface. The figures (a) - (g) show the local DOS at varioussample voltages, around the peak obtained from a dI/dV measurement (h).Figure (i) shows a calculated characteristic LDOS, and figure (j) an STS resulton a larger scale.by a peak in the longitudinal one is only visible in the six- (or four-)terminalmeasurement. As we have argued in Sec. 3.3.2, there is no clear cut between thelongitudinal and the Hall resistivity in the two-terminal configuration, wherethe resistance measured between the current contacts is indeed quantised inthe IQHE. At the plateau transition, however, the chemical potential at theedges is no longer constant because of backscattered electrons and the resistanceis no longer quantised. One observes indeed the resistance peak associatedwith the longitudinal resistance in the six- or four-terminal configuration. As aconsequence, one measures, at the plateau transition, the superposition of theHall and the longitudinal resistances.If one increases even more the filling of the LL n , the same arguments applybut now in terms of hole states . The Hall resistance is quantised as R H = h/e ( n + 1), and the holes (i.e. the lacking electrons with respect to n + 1completely filled LLs) get localised in states at closed equipotential lines aroundthe potential summits. As a consequence, the longitudinal resistance drops tozero again. The physical picture presented above, in terms of localised and extended bulkstates, has recently been confirmed in scanning-tunneling spectroscopy (STS)of a 2D electron system that was prepared on an n -InSb surface instead of themore common GaAs/AlGaAs heterostructure [40]. Its advantage consists of itsaccessibility by an “optical” (surface) measurement that cannot be performedif the 2D electron gas is buried deep in a semiconductor heterostructure. In an8 Integer Quantum Hall Effect
STS measurement one scans the sample and thus measures the local density ofstates at a certain energy that can be tuned via the voltage between the tipof the electron microscope and the sample. When measuring the differentialconductance dI/dV , which is proportional to the DOS, one observes a peakthat corresponds to the centre of a LL [Fig. 3.6(h)] where the extended statesare capable of transporting a current between the different electric contacts, asmentioned above. Whereas the quantum states at energies corresponding toclosed equipotential lines of the impurity landscape are clearly visible as closedorbits in Fig. 3.6(a),(b) and (f),(g), the states in the vicinity of the peak aremore and more extended, as shown by the spaghetti-like lines in Figs. 3.6(c),(d)and (e), as one would expect from the arguments presented above.
The physical picture presented above suggests that the plateau transition in theHall resistance is related to a percolation transition, where initially separatedelectron-filled valleys start to percolate between the opposite sample edges be-yond a certain threshold of the filling. Because of the second-order characterof a percolation transition, this scenario suggests that the plateau transition isa second-order quantum phase transition described by universal scaling laws,where the control parameter is just the magnetic field B [41, 42]. We finish thischapter on the IQHE with a brief overview over these scaling laws, and refer theinterested reader to the literature [41, 42] and the class given by G. Batrouni atthe same Singapore session of Les Houches Summer School 2009. The phase transition occurs at the critical magnetic field B c and is charac-terised by an algebraically diverging correlation length ξ ∼ | B − B c | − ν , (3.15)where ν is called the critical exponent . In the same manner, the temporalfluctuations are described by a correlation “length” ξ τ that is related to thespatial correlation length ξ , ξ τ ∼ ξ z ∼ | B − B c | − zν , (3.16)where z is called dynamical critical exponent . It is roughly a measure of theanisotropy between the spatial and temporal fluctuations, which is often en-countered in non-relativistic condensed-matter systems. At the phase transition B c , the longitudinal and transverse resistivities ρ L/H are described in terms of universal functions that are functions of the ratio τ /ξ τ Although we use the same Greek letter ν for the critical exponent, it must not be con-funded with the filling factor, which plays no role in this subsection. Notice that in relativity, time is considered as the “fourth” dimension, and Lorentz in-variance would require that spatial and temporal fluctuations be equivalent, i.e. z = 1. he Integer Quantum Hall Effect and Percolation T(K) ( ∆ B) −1 ( ∆ B ) − N = 1N = 1 N = 0N = 1N = 1 (cid:18) d(cid:26)xydB (cid:19)max (cid:18)d(cid:26)xydB (cid:19)max
Figure 3.7: Experiment by Wei et al. , 1988. The width of the transition ∆ B and of the derivative of the Hall resisitivity ∂ρ xy /∂B , measured as a functionof temperature, reveals a scaling law with an exponent 1 /zν = 0 . ± .
04, forthe transition between the filling factors 1 → N = 0 ↓ ), 2 → N = 1 ↑ ) and3 → N = 1 ↓ ).between the (imaginary) time τ , which is proportional to the inverse tempera-ture, ¯ h/τ = k B T [41, 42] and the temporal correlation length ξ τ , ρ L/H = f L/H (cid:18) τξ τ (cid:19) = f L/H (cid:18) ∆ B zν T (cid:19) , (3.17)where we have defined ∆ B ≡ | B − B c | . In the case of an AC (alternatingcurrent) measurement at frequency ω , another dimensionless quantity, namelythe ratio between the frequency and the temperature, ¯ hω/k B T , needs to betaken into account such that the universal function reads ρ ACL/H = f L/H (cid:18) τξ τ , ¯ hωk B T (cid:19) . However, we do not consider an alternating current here. Equation (3.17) thenyields the scaling of the width of the peak in the longitudinal resistance (or elsethe plateau transition) ∆ B ∼ T /zν . (3.18)A measurement of this width by Wei et al. [43] has confirmed such criticalbehaviour with an exponent 1 /zν = 0 . ± .
04 (see Fig. 3.7).Furthermore, one may distinguish between the two exponents ν and z withina measurement of the plateau-transition width as a function of the electric field0 Integer Quantum Hall Effect E via current fluctuations. One may identify the energy fluctuation eEξ atthe correlation length ξ with the energy scale ¯ h/ξ τ ∼ ¯ h/ξ z set by the temporalfluctuation ξ τ , which yields E ∼ ξ − (1+ z ) ∼ ∆ B ν (1+ z ) , and thus∆ B ∼ E /ν (1+ z ) . (3.19)Other measurements by Wei et al. [44] have shown that these types of fluctu-ations yield z ≃
1, i.e. ν ≃ .
3. The precision of the measured critical expo-nent has since been improved – more recent experiments [45, 46] have revealed ν = 2 . ± . ν class = 4 / ∼ l B of the wave functions associated with the equipotential lines enhance percola-tion, i.e. the electron puddles in the potential valleys may percolate before theytouch each other in the classical sense. A model that takes into account thiseffect has been proposed by Chalker and Coddington [47], though with simplify-ing assumptions for the puddle geometry, and one obtains a critical exponent ν = 2 . ± . z = 2 for non-interactingelectrons, whereas the measured value z ≃ ν ≃ .
59) than the measured one when interactions are not taken intoaccount [50].
We finish this chapter on the IQHE with a short presentation of the relativis-tic quantum Hall effect (RQHE) in graphene, which is understandable in thesame framework of LL quantisation and (semi-classical) one-particle localisa-tion as the IQHE in a non-relativistic 2D electron system. Indeed, the abovearguments also apply to relativistic electrons in graphene, but we need to takeinto account the two different carrier types, electrons and holes, which carry adifferent charge. This is not so much a problem in the case of the impurity po-tential with its valleys and summits: in a particle-hole transformation, a valleybecomes a summit and vice versa . Furthermore, the direction of the Hall drift Notice, however, that due to the universality of the scaling laws and the fluctuations at alllength scales, the results are expected to be independent on these microscopic assumptions. The particle-hole transformed landscape corresponds to an impurity distribution in whichone interchanges negatively and positively charged impurities. elativistic Quantum Hall Effect in Graphene case ν=0 B n=0n=1n=2n=3n=4n=−1n=−2n=−3n=−4 ene r g y magnetic field (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) completelyfilled levelshalf−filled central level max y ene r g y n=0+, n=1+, n=2+, n=3n=0+, n=1+, n=2+, n=3 (K’)(K) (b)(a) Figure 3.8: (a)
Mass confinement for relativistic Landau levels. Whereas theelectron-like LLs ( λ = +) are bent upwards when approaching the sample edge( y max ), the hole-like LLs ( λ = − ) are bent downwards. The fate of the n = 0LL depends on the valley (parity anomaly) – in one valley ( K ), the level energydecreases, whereas it increases in the other valley ( K ′ ). (b) Filling of the bulkLandau levels at ν = 0. All electron-like LLs ( λ = +) are unoccupied whereasall hole-like LLs ( λ = − ) are completely filled. The n = 0 LL is altogetherhalf-filled.changes in this transformation. Because of the universality of the quantum Halleffect, both types of impurity distributions related by particle-hole symmetryyield the same quantisation of the Hall resistance. The picture of semi-classicallocalisation therefore applies also in the case of relativistic electrons in graphene.The situation is different for the confinement potential. An ansatz of theform V ( y ) – remember that the Hamiltonian of electrons in relativistic grapheneis a 2 × V ( y − y max/min ) → ∞ at the sample edge con-fines electrons but not the holes of the valence band for which we would need V ( y − y max/min ) → −∞ for an efficient confinement. A possible confinementpotential may be formed with the Pauli matrix σ z , V conf ( y ) = V ( y ) σ z = (cid:18) V ( y ) 00 − V ( y ) (cid:19) , (3.20)which, together with the Hamiltonian (2.8), yields the Hamiltonian which cor-responds to the non-relativistic model (3.6). For a constant term M = V ( y )the contribution (3.20) plays the role of a mass of a relativistic particle (see alsoAppendix B). Therefore, the confinement (3.20) is sometimes also called massconfinement . The corresponding energy spectrum, which one obtains within thesame approximation as in Sec. 3.1 via the replacement y → y = kl B in the2 Integer Quantum Hall Effect
Landau gauge, reads [c.f. Eq. (B.8) in Appendix B] ǫ λn,y = λ s M ( y ) + 2 ¯ h v l B n, (3.21)and is schematically represented in Fig. 3.8(a). Notice that Eq. (3.21) is onlyvalid for n = 0 – indeed, the n = 0 acquires a non-zero energy M ( y ), which isnegative for our particular choice (see Appendix B). This feature is sometimescalled parity anomaly in high-energy physics. Remember that in the case ofgraphene, one has two inequivalent low-energy points in the first BZ which giverise to a relativistic energy spectrum. The Dirac Hamiltonians (2.4) and (2.8)for the zero- B and magnetic-field case, respectively, applies principally only toone of the two valleys (say K ), whereas that for the other valley is given by − H D (or − H BD ) if one interchanges the A and B components [c.f. Eq. (A.16)in Appendix A]. The confinement term (3.20) therefore reads − V conf ( y ) in theother valley, i.e. with a negative mass. The n = 0 LL thus shifts to positiveenergies in the second valley, and the two-fold valley degeneracy is lifted in thislevel. A more detailed discussion of the mass confinement (3.20) in graphenemay be found in the Appendix B.This type of confinement may seem to be somewhat artificial, whereas theconfinement in the non-relativistic case is easier to accept. Notice, however, thatthe whole model of massless Dirac fermions [second Hamiltonian in Eq. (2.4)]only describes the physical properties at length scales that are large comparedto the lattice spacing (in graphene). In the true lattice model, the electrons arenaturally confined because one does not allow for hopping from a lattice siteat the edge into free space. The expression (3.20) is therefore only an effective model to describe confinement. We notice that, although the effective modelyields a qualitatively correct picture, the fine structure of the dispersion at theedge depends on the edge geometry [51]. For further reading, we refer theinterested reader to the literature [21].With the help of these preliminary considerations, we are now prepared tounderstand the RQHE in graphene – the semi-classical localisation is the sameas in the non-relativistic case, and the confinement, which needed to be adoptedto account for the simultaneous presence of electron- and hole-like LLs, yieldsthe edge states which are responsible for the quantum transport and, thus, theresistance quantisation. The RQHE was indeed discovered in 2005 by two differ-ent groups [6, 7], and the results are shown in Fig. 3.9 [7]. The phenomenologyof the RQHE is the same as that of the IQHE in non-relativistic LLs: one ob-serves plateaus in the Hall resistance while the longitudinal resistance vanishes.Notice that one may vary the filling factor either by changing the B -field atfixed carrier density [Fig. 3.9(a)] or one keeps the B -field fixed while changingthe carrier density with the help of a gate voltage [Fig. 3.9(c)]. The lattermeasurement is much easier to perform in graphene than in non-relativistic 2Delectron gases in semiconductor heterostructures.In spite of the similarity with the non-relativistic IQHE, one notices, in Fig.3.9, an essential difference: the quantum Hall effect is observed at the filling elativistic Quantum Hall Effect in Graphene V =15V Density of statesB=9TT=30mK T=1.6K ∼ ν∼ 1/ν
Figure 3.9: Measurement of the relativistic quantum Hall effect (Zhang et al. ,2005). (a)
RQHE at fixed carrier density ( V G = 15 V) at T = 30 mK. Thefilling factor is varied by sweeping the magnetic field. (b) Sketch of the DOSwith the Fermi energy between the LLs n = 0 and + , n = 1. (c) RQHE at fixedmagnetic field ( B = 9 T) at higher temperatures, T = 1 . Integer Quantum Hall Effect factors ν = ± n + 1) , (3.22)in terms of the LL quantum number n , whereas the IQHE is observed at ν = n (or ν = 2 n if the LLs are spin-degenerate). The step in units of 4 is easy tounderstand: each relativistic LL in graphene is four-fold degenerate (in additionto the guiding-centre degeneracy), due to the two-fold spin and the additionaltwo-fold valley degeneracy. However, there is an “offset” of 2. This is due to thefact that the filling factor ν = 0 corresponds to no carriers in the system, i.e.to a situation where the Fermi energy is exactly at the Dirac point (undopedgraphene). In this case, one has a perfect electron-hole symmetry, and the n = 0 LL must therefore be half-filled [see Fig. 3.8(b)], or else: there are asmany electrons as holes in n = 0. According to the considerations presented inSec. 3.4, this does not correspond to a situation where one observes a quantumHall effect due to percolating extended states. Indeed, the system turns outto be metallic at ν = 0 with a finite non-zero longitudinal resistance [6, 7]. Asituation, where one would expect a quantum Hall effect, arises when the centralLL n = 0 is completely filled (or completely empty). As a consequence of thefour-fold level degeneracy, one obtains the quantum Hall effect at ν = 2 (or ν = −
2) observed in the experiments (see Fig. 3.9). This is the origin of theparticular filling-factor sequence (3.22) of the RQHE in graphene. hapter 4
Strong Correlations and theFractional Quantum HallEffect
In the preceding chapter, we have seen that one may understand the essentialfeatues of the IQHE within a one-particle picture, i.e. in terms of Landauquantisation; at integer filling factors ν = n , which correspond to n completelyfilled LLs, an additional electron is forced, as a result of the Pauli principle,to populate the next higher (unoccupied) LL [see Fig. 4.1(a)]. It therefore,needs to “pay” a finite amount of energy ¯ hω C [or √ hv/l B )( √ n − √ n −
1) inthe case of the RQHE in graphene] and is localised by the impurities in thesample, due to the classical Hall drift which forces the electron to move onclosed equipotential lines. The system is said to be incompressible because onemay not vary the filling factor and pay only an infinitesimal amount of energy –indeed in the case of a fixed particle number, consider an infinitesimal decreaseof the magnetic field which amounts to an infinitesimal change of the surface2 πl B occupied by each quantum state. Since the total surface of the systemremains constant, the infinitesimal increase of 2 πl B may not be accomodatedby an infinitesimal change in energy, due to the gap between the LL n − n where at least one electron must be promoted to. This gives rise to a zerocompressibility.In view of this picture of the quantum Hall effect, it was therefore a bigsurprise to observe a FQHE at a filling factor ν = 1 /
3, with the correspondingHall quantisation R H = h/e ν = 3 h/e [13], and, later, at a large set of otherfractional filling factors. Indeed, if only the kinetic energy is taken into account,the ground state at ν = 1 / As before, we neglect the electron spin to render the discussion as simple as possible. Therole of spin will be discussed briefly in the last chapter on multi-component systems.
Strong Correlations and the Fractional Quantum Hall Effect h ω C x x x x (a) (b) X X
Figure 4.1: (a)
Sketch of a completely occupied LL. An additional electron (greycircle) is forced to populate the next higher LL because of the Pauli principle. (b)
Sketch of a partially filled LL. Because of the presence of unoccupied states in theLL (crosses), the Pauli principle does not prevent an additional electron (greycircle) to populate the next higher LL. The low-energy dynamical properties ofthe electrons are described by excitations within the same LL (no cost in kineticenergy), and inter-LL excitations are now part of the high-energy degrees offreedom.LL which is only one-third filled.Notice that we have neglected so far the mutual Coulomb repulsion betweenthe electrons, which happens to be responsible for the occurence of the FHQE.The relevance of electronic interactions is discussed in the next section (Sec.4.1). In Sec. 4.2, we present the basic results of Laughlin’s theory of theFQHE, such as the ground-state wave functions, fractionally charged quasi-particles and the interpretation of Laughlin’s wave function in terms of a 2Done-component plasma. The related issue of fractional statistics is introducedin a section apart (Sec. 4.3), and we close this chapter with a short discussionof different generalisations of Laughlin’s wave function, such as CF theory orthe Moore-Read wave function in half-filled LLs.
As already mentioned above, the situation of a partially filled LL is somewhatopposite to that of n completely occupied levels, where one observes the IQHE.This difference is summarised in Fig. 4.1 and it is also the origin of the differentrole played by the Coulomb repulsion between the electrons. In the case of n completely filled LLs, one has a non-degenerate (Fermi-liquid-like) ground state,where the interactions may be treated within a perturbative approach. Indeed,any type of excitation involves a transition between two adjacent LLs that areseparated by an energy gap of ¯ hω C [see Fig. 4.1(a)], and we need to comparethe Coulomb energy at the characteristic length scale R C = l B √ n + 1 to thisgap, V C ¯ hω C ∼ me / ǫ ¯ h / ( Bn ) − / , In order to simplify the discussion, we consider only the IQHE in non-relativistic quantumHall systems, but the arguments apply also to the RQHE in graphene. he Role of Coulomb Interactions r s = me ǫ ¯ h n − el for the 2D Coulomb gas in Fermi liquid theory [52, 53]. The last expression isobtained by identifying the Fermi energy E F = ¯ h k F / m , in terms of the Fermiwave vector k F , with the energy of the last occupied LL ¯ hω C n . The perturba-tive approach allows one, e.g., to describe collective electronic excitations in theIQHE, such as magneto-plasmon modes (the 2D plasmon in the presence of amagnetic field) or magneto-excitons (inter-LL excitations that acquire a disper-sion due to the Coulomb interaction) [54], or else the corresponding modes ofthe RQHE in graphene [55, 56].In the case of a partially filled LL n the situation is inverted: for an electronicexcitation, there are enough unoccupied states in the LL n for an electron of thesame level to hop to. From the point of view of the kinetic energy, there is noenergy cost associated with such an excitation ( low-energy degrees of freedom)whereas an excitation to the next higher (unoccupied) LL costs an energy ¯ hω C .Inter-LL excitations may then be neglected as belonging to high-energy degreesof freedom [Fig. 4.1(b)]. Notice that all possible distributions of N electronswithin the same partially filled LL n therefore have the same kinetic energy,which effectively drops out of the problem. The macroscopic degeneracy maybe lifted by phenomena due to other energy scales, such as those associated withthe impurities in the sample or else the electron-electron interactions. The firsthypothesis (impurities) may be immediately discarded as the driving mechanismof the FQHE because, in contrast to the IQHE, the FQHE only occurs in high-quality samples with low impurity concentrations. Indeed, the hierarchy ofenergy scales in the FQHE may be characterised by the succession¯ hω C > ∼ V C ≫ V imp , (4.1)and we therefore need to consider seriously the Coulomb repulsion, which governthe low-energy electronic properties in a partially filled LL. Notice that we thusobtain a system of strongly-correlated electrons for the description of which allperturbative approaches starting from the Fermi liquid are doomed to fail. Theonly hope one may have to describe the FQHE is then a well-educated guess ofthe ground state.The most natural guess would be that the electrons in a partially filled LLbehave as classical charged particles that form a crystalline state in order tominimise their mutual Coulomb repulsion. Such a state is also called Wignercrystal (WC) because it was first proposed by Wigner in 1934 [57]. A WC hasindeed been thought – before the discovery of the FQHE – to be the ground stateof electrons in a partially filled LL [58]. Even if the WC is the ground state atvery low filling factors, as it has been shown experimentally [59], this state may As for the IQHE, impurities play nevertheless an important role in the localisation ofquasi-particles, which we need to invoke later in this chapter in order to explain the transportproperties of the FQHE. Strong Correlations and the Fractional Quantum Hall Effect not allow for an explanation of the FQHE. Indeed, the WC is a state that breaksa continuous spatial symmetry (translation invariance) and any such state hasgapless long-wave-length excitations (
Goldstone modes ). The Goldstone modeof the WC (as of any other crystal) is the acoustic phonon the energy of whichtends to zero at zero wave vector. One may thus compress the WC by changingthe occupied surface in an infinitesimal manner or else by adding an electronwithout changing the macroscopic surface and pay only an infinitesimal amountof energy. The ground state is therefore compressible, i.e. it is not separatedby an energy gap from its single-particle excitations, a situation that is at oddswith the FQHE.
As a consequence of the above-mentioned considerations on the WC, one thusneeds to search for a candidate for the ground state that does not break anycontinuous spatial symmetry and that has an energy gap. Such a state is the incompressible quantum liquid which was proposed by Laughlin in 1983 [14]the basic features of which we present in the present section. We consider,here, only the FQHE in the lowest LL (LLL), for simplicity. There are differentprescriptions to generalise the associated wave functions to higher LLs, e.g. withthe help of Eq. (2.18) (see MacDonald, 1984). Experimentally, several FQHEstates have been observed in the next higher LL n = 1 although the majorityof FQHE states is found in the LLL. In order to illustrate – one cannot speak of a derivation – Laughlin’s wave func-tion, we first need to remember the one-particle wave function of the LLL andthen consider the corresponding two-particle wave function. We have alreadyseen in Sec. 2.4.1 that a one-particle wave function in the LLL is described interms of an analytic function times a Gaussian, ψ ∼ z m ′ e −| z | / , in terms of the integer m ′ = 0 , ..., N B −
1, where we have absorbed now (andin the remainder of these lecture notes) the magnetic length in the definition ofthe complex position, z = ( x − iy ) /l B .Consider, in a second step, an arbitrary two-particle wave function. Thiswave function must also be an analytic function of both postions z and z of the first and second particle, respectively, and may be a superposition ofpolynomials, such as e.g. of the basis states ψ (2) ( z, Z ) ∼ Z M z m e − ( | z | + | z | ) / , (4.2) There is even some slight indication for a 1/5 FQHE state in the next excited LL n = 2[60]. We neglect the numerical prefactors here that account for the normalisation of the wavefunctions. aughlin’s Theory Z = ( z + z ) / z = ( z − z ). The quantum number m plays the role ofthe relative angular momentum between the two particles, and M is associatedwith the total angular momentum of the pair. Because of the analyticity of theLLL wave functions, m must be an integer, and the exchange of the positions z and z imposes on m to be odd because of the electrons’ fermionic nature.Laughlin’s wave function [14] is a straight-forward N -particle generalisationof the two-particle wave function (4.2), ψ Lm (cid:0)(cid:8) z j , z ∗ j (cid:9)(cid:1) = Y k The variational parameter in Laughlin’s wave function (4.3) is nothing otherthan the exponent m , with respect to which we would, in principle, need tooptimised the wave function in order to approximate the true ground state ofthe system. Notice, however, that due to the LLL analyticity condition andfermionic statistics, the exponent is restricted to odd integers, m = 2 s + 1, interms of the integer s . Furthermore, this variational parameter turns out tobe fully determined by the filling factor ν , as we will show with the followingargument. Consider Laughlin’s wave function as a function of the position z k of somearbitrary but fixed electron k . There are N − z k − z l ) m ,one for each of the remaining N − l , occuring in the ansatz (4.3),such that the highest power of z k is m ( N − Y k Figure 4.2: Haldane’s pseudopotentials for the Coulomb interaction in the LLs n = 0 and n = 1. Notice that we have plotted the pseudopotentials for bothodd and even values of the relative angular momentum m even though only oddvalues matter in the case of fermions.the real-space form of the interaction potential. Indeed, if a pair of electrons isin a quantum state with relative angular momentum m , the average distance be-tween the electrons is | z | ∼ l B √ m . Haldane’s pseudopotential v m is thereforeroughly the value of the original interaction potential at the relative distance l B √ m , v m ≃ V (cid:16) | z | = l B √ m (cid:17) , (4.9)and the small- m components of Haldane’s pseudopotentials correspond to theshort-range components of the underlying interaction potential. Figure 4.2shows the pseudopotential expansion for the Coulomb interaction in the lowest( n = 0) and the first excited ( n = 1) LL.Haldane’s pseudopotentials are extremely useful in the description of the N -particle state as well. Indeed, the N -particle interaction Hamiltonian V maybe rewritten in terms of pseudopotentials as V = X i 0. Therefore, Laughlin’s wavefunction is even the exact ground state of the model (4.11). Furthermore, itis the only zero-energy state because if one keeps the total number of particlesand flux fixed, any other state different from that described by Laughlin’s wavefunction involves a particle pair in a state with an angular momentum quantumnumber different from m . If it is smaller than m , this particle pair is affectedby the associated non-zero pseudopotential m ′ and thus costs an energy on theorder of v m ′ > 0. If the particle pair is in a momentum state with m ′ > m ,there is at least another pair with m ′′ < m in order to keep the filling factorfixed, and this pair raises the energy. These general arguments show that anyexcited state involves a finite (positive) energy given by a pseudopotential v m ′ ,with m ′ < m , which plays the role of an energy gap . In this sense, the liquidstate described by Laughlin’s wave function is indeed an incompressible statethat already hints at the possibility of a quantum Hall effect if we can identifythe correct quasi-particle of this N -particle state that becomes localised by thesample impurities.Notice that the above considerations are based on an extremely artificialmodel interaction (4.11) that has, at first sight, very little to do with the physicalCoulomb repulsion. However, the model is often used to generate numerically(in exact-diagonali-sation calculations) the Laughlin state, which may then becompared to the Coulomb potential decomposed in Haldane’s pseudopotentials.This procedure has shown that theL aughlin state generated in this mannerhas an overlap of more than 99% with the state obtained from the Coulombpotential [63, 64], which is amazingly high for a wave function obtained froma well-educated guess. This high accuracy of Laughlin’s wave function maybe understood in the following manner: when one decomposes the Coulombinteraction potential in Haldane’s pseudopotentials, one obtains a monotonicallydecreasing function when plotted as a function of m (see Fig. 4.2). Furthermore,the component v is much larger than v and all other pseudopotentials v m with higher values of m . These higher terms may be treated in a perturbativemanner and do not change the ground state which is protected by the above-mentioned gap on the order of v > v m , with m > One has v /v ≃ . m do not play any physical role because of the fermionic nature of theelectrons. Strong Correlations and the Fractional Quantum Hall Effect function (4.3) has indeed a lower energy than the previously proposed WC.Again the reason for this unexpected feature is the capacity of Laughlin’s wavefunction, which varies as r m when two particles i and j approach each otherwith r = | z i − z j | , to screen the short-range components of the interactionpotential. Notice that for a WC of fermions, the corresponding N -particle wavefunction decreases as r , as dictated by the Pauli principle. Until now, we have discussed some ground-state properties of Laughlin’s wavefunction. We have seen that the Laughlin state at ν = 1 /m is insensitive tothe short-range components of the interaction potential described by Haldane’spseudopotentials v m ′ with m ′ < m , whereas excited states must be separatedfrom the ground state by a gap characterised by these short-range pseudopoten-tials. However, we have not characterised so far the nature of the excitations.There are two different sorts of excitations: (i) elementary excitations (quasi-particles or quasi-holes) that one obtains by adding or removing charge from thesystem, and (ii) collective excitations at fixed charge. The latter are simply acharge-density-wave excitation which consist of a superposition of particle-holeexcitations at a fixed wave vector q (the momentum of the pair) and which maybe shown to be gapped at all values of q . Its dispersion reveals a minimum(called magneto-roton minimum ) at a non-zero value of the wave vector thatindicates a certain tendency to form a ground state with modulated chargedensity, such as a WC. The characteristic dispersion relation of these collectiveexcitations is shown in Fig. 4.3(a). However, we do not discuss collectiveexcitations here and refer the interested reader to the literature for a moredetailed discussion [65, 1, 4], and concentrate here on a presentation of theelementary excitations. Quasi-holes Elementary excitations are obtained when sweeping the filling factor slightlyaway from ν = 1 /m . Remember that there are two possibilities for varying thefilling factor: adding charge to the system by changing the electronic density oradding (or removing) flux by varying the magnetic field. Remember further [seeEq. (4.4)] that the number of flux is intimitely related to the number of zerosin Laughlin’s wave function. We therefore consider the ansatz ψ qh (cid:0) z , (cid:8) z j , z ∗ j (cid:9)(cid:1) = N Y j =1 ( z j − z ) ψ Lm (cid:0)(cid:8) z j , z ∗ j (cid:9)(cid:1) (4.12)for an excited state. Each electron at the positions z j thus “sees” an additionalzero at z . In order to verify that this wave function adds indeed another fluxquantum to the system, we may expand Laughlin’s wave function (4.3) formally aughlin’s Theory mm+1 (a) (b) Figure 4.3: (a) Dispersion relation for collective charge-density-wave excita-tions (Girvin, MacDonald and Platzman, 1986; Girvin, 1999). The continuouslines have been obtained in the so-called single-mode approximation (Girvin,MacDonald and Platzman, 1986) for the Laughlin states at ν = 1 / 3, 1/5 and1/7, whereas the points are exact-diagonalisation results (Haldane and Rezayi,1985; Fano, Ortolani and Colombo, 1986). The arrows indicated the character-istic wave vector of the WC state at the corresponding densities. (b) Quasi-holeexcitation. Each electron jumps from the state m to the next-higher angularmomentum state m + 1.in a polynomial, ψ Lm ( { z j , z ∗ j } ) = X { m i } α m ,...,m N z m ... z m N N e − P j | z j | / , where the α m ,...,m N describe the expansion coefficients. We now choose theposition z at the centre of the disc, in which case the wave function of theexcited state (4.12) simply reads ψ qh ( { z j , z ∗ j } ) = X { m i } α m ,...,m N z m +11 ... z m N +1 N e − P j | z j | / , i.e. each exponent is increased by one, m i → m i + 1. This may be illustratedin the following manner: each electronjumps from the angular momentum state m to a state in which the angular momentum is increased by one (see Fig. 4.3),leaving behind an empty state at m = 0. The excitation is therefore called a quasi-hole as we have already suggested by the subscript in Eq. (4.12). Thisalso affects the quantum state with highest angular momentum M , i.e. wehave increased the sample size by the surface occupied by one flux quantum,while keeping the number of electrons fixed. Furthermore, this quasi-hole is Naturally, the total surface of the quantum Hall system remains constant, but physicallywe have slightly increased the B -field. Each quantum state occupies then an infinitesimally Strong Correlations and the Fractional Quantum Hall Effect associated with a vorticity if one considers the phase of the additional factor inEq. (4.12), ψ qh ( z = 0 , { z j , z ∗ j } ) ∝ N Y j e − iθ j × ψ Lm (cid:0)(cid:8) z j , z ∗ j (cid:9)(cid:1) , i.e. each particle that circles around the origin z = 0 experiences an additionalphase shift of 2 π as compared to the original situation described in terms ofLaughlin’s wave function (4.3). This is reminiscent of the vortex excitation ina type-II superconductor [66].We have seen above that one can create a quasi-hole excitation at the postion z by introducing one additional flux quantum, N B → N B + 1, which lowers thefilling factor by a tiny amount. However, we have not yet determined the chargeassociated with this elementary excitation. This charge may be calculated byconsidering the filling factor fixed, i.e. we need to add some (negative) chargeto compensate the extra flux quantum in the system. From Eq. (4.4), we noticethat the relation between the extra flux ∆ N B and the compensating extra charge∆ N is simply given by m ∆ N = ∆ N B ⇔ ∆ N = ∆ N B m . (4.13)This very important result is somewhat unexpected: in order to compensate oneadditional flux quantum (∆ N B = 1), one would need to add the m -th fraction ofan electron . The charge deficit caused by the quasi-hole excitation is therefore e ∗ = em , (4.14)i.e. the quasi-hole carries fractional charge . Quasi-particles In the preceding paragraph, we have considered a quasi-hole excitation that isobtained by introducing an additional flux quantum in the system [or, math-ematically, an additional zero in the Laughlin wave function, see Eq. (4.12)].Naturally, one may also lower the number of flux quanta by one in which caseone obtains a quasi-particle excitation with opposite vorticity as compared tothat of the quasi-hole excitation. This opposite vorticity suggests that we use aprefactor Q Nj =1 ( z ∗ j − z ∗ ), instead of Q Nj =1 ( z j − z ) as in the expression (4.12),in order to create a quasi-particle excitation at the position z . Remember,however, that the resulting wave function would have unwanted components inhigher LLs because the analyticity condition of the LLL is no longer satisfied.In order to heal the quasi-particle expression, one formally projects it into theLLL, ψ qp (cid:0) z , (cid:8) z j , z ∗ j (cid:9)(cid:1) = P LLL N Y j =1 ( z ∗ j − z ∗ ) ψ Lm (cid:0)(cid:8) z j , z ∗ j (cid:9)(cid:1) . (4.15) smaller surface 2 πl B , such that the system may accomodate for one more quantum state, M = N B → N B + 1. aughlin’s Theory tunnelingevent µ µ L R (a) (b) µ µ L R ν=1/3ν=1/3 tunnelingevent ν=0 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) sg V sg V sg V sg V Figure 4.4: Experimental setup for the observation of fractionally chargedquasi-particles. In addition to the usual geometry, one adds, at the upper andthe lower edges, side gates that are used to deplete the region around the gatesby the application of a voltage V sg . The filling factor is chosen to be ν = 1 / (a) Weak-backscattering limit. The incompressible liquid has a bottleneck atthe side gates, i.e. the edges are so close to each other that a tunneling eventbetween them has a finite probability. A particle injected at the left contactmay thus be backscattered (grey arrow) in a region filled by the incompressibleLaughlin liquid, although the majority of the particles reaches the right contact(black arrows). (b) Strong-backscattering limit. If one increases the side-gatevoltage V sg , the incompressible ν = 1 / ν = 0). In this case, backscattering is themajority process (black arrow), and a tunneling may occur over the depletedregion such that a particle injected at the left contact may still reach the rightone (grey arrows).There are several manners of taking into account this projection P LLL . A com-mon one consists of replacing each occurence of the non-analytic variables z ∗ j (and powers of them) in the polynomial part of the wave function by a deriva-tive with respect to z j in the same polynomial [67]. By partial integration, thisamount to deriving the Gaussian factor by ( ∂ z j ) m which, up to a numericalprefactor, yields exactly the non-analytic polynomial factor z ∗ j m . We will en-counter this projection scheme again in the discussion of the CF generalisationof Laughlin’s wave function (Sec. 4.4.1). That the fractional charge of Laughlin quasi-particles is not only a mathe-matical concept but a physical reality has been proven in a spectacular manner From now on, we use the term “(Laughlin) quasi-particles” generically in order to denotequasi-particles and quasi-holes. Strong Correlations and the Fractional Quantum Hall Effect in so-called shot-noise experiments on the ν = 1 / Inthese experiments, one constrains the quantum Hall system with the help ofside gates (see Fig. 4.4) that are used to deplete the region in their vicinity viathe application of a gate voltage V sg . As a consequence of this depletion thequantum Hall system has a bottleneck where the corresponding edge states arebrought into spatial vicinity [Fig. 4.4(a)] or where the incompressible quantumliquid may even be cut into two parts separated by a completely depleted bar-rier [Fig. 4.4(b)]. In the first case, an injected charge may be backscattered ina tunneling event at the bottleneck over a region filled by the ν = 1 / I (over a certain time interval) but simultaneously the (square of the) currentfluctuation ¯(∆ I ) which is proportional to the carrier charge. If the elementarycharged excitations are e ∗ = e/ e ∗ = e/ e in the case of atunneling process over a depleted region [Fig. 4.4(b)]. A compelling physical picture of Laughlin’s wave function (4.3) and the prop-erties of its elementary excitations (4.12) and (4.15) with fractional charge hasbeen provided by Laughlin himself [14], in terms of an analogy with a classical2D one-component plasma . In the present subsection, we present the basic ideasand results of this plasma analogy, for completeness and pedagogical reasons.However, no new results will come out of this analogy here, as compared tothose derived above.Remember from basic quantum mechanics that the modulus square of aquantum-mechanical wave function may be interpreted as a statistical probabil-ity distribution. For Laughlin’s wave function (4.3), one obtains the probabilitydistribution (cid:12)(cid:12) ψ Lm ( { z j } ) (cid:12)(cid:12) = Y i Hamiltonian U cl . Themock Hamiltonian may be obtained exactly from this identification, − βU cl = ln (cid:12)(cid:12) ψ Lm ( { z j } ) (cid:12)(cid:12) , (4.16)and one obtains, by choosing somewhat artificially β = 2 /q , U cl = − q X i Process in which a particle A moves on a path C around anotherparticle B. In three space dimensions, one may profit from the third direction( z -direction) to lift the path over particle B and thus to shrink the path intoa single point. (b) Process equivalent to moving A on a closed path around Bwhich consists, apart from a topologically irrelevant translation, of two succes-sive exchanges of A and B.of unit charge at the position z . In order to maintain charge neutrality, theimpurity needs to be screened by the plasma particles. Since the charge of eachplasma particle is q = m = 2 s + 1 and thus greater than unity, one needs 1 /q plasma particles to screen the impurity of charge one. Remember that eachplasma particle represents one electron of unit charge in the original Laughlinliquid. One therefore obtains the same charge fractionalisation of the Laughlinquasi-particle (4.14), e ∗ = e/m , as in the original quantum model. One of the most exotic consequences of charge fractionalisation in 2D quantummechanics, exemplified by Laughlin quasi-particles, is fractional statistics . Re-member that, in three space dimensions, the quantum-mechanical treatment oftwo and more particles yields a superselection rule according to which quantumparticles are, from a statistical point of view, either bosons or fermions . Thissuperselection rule is no longer valid in 2D (two space dimensions), and onemay find intermediate statistics between bosons and fermions. The correspond-ing particles are called anyons , because the statistics may be any . The presentsection is meant to illustrate these amazing aspects of 2D quantum mechanics,and we try to avoid a too formal or mathematical treatment. We refer, again,the interested reader to the more detailed literature [70].In order to illustrate the different statistical (i.e. exchange) properties oftwo quantum particles in three and two space dimensions, consider a particle Athat moves adiabatically on a closed path C in the xy -plane around another oneB of the same species (see Fig. 4.5). We choose the path to be sufficiently faraway from particle B and the two particles to be sufficiently localised such thatwe can neglect corrections due to the overlap between the two correspondingwave functions. Notice first that such a process T is equivalent, apart from atopologically unimportant translation, to two successive exchange processes E ,in which one exchanges the positions of A and B. Algebraically, this may be ractional Statistics E = T or E = ±√T , (4.21)modulo a translation.Let us discuss first the three-dimensional case. Because of the presence of thethird direction ( z -direction), one may elevate the closed path in this directionwhile keeping the position of particle A fixed in the xy plane. We call theelevated path C ′ . Furthermore, one may now shrink the closed loop C ′ into asingle point at the position A without passing by the position of particle B whichremains in the xy -plane. This final (point-like) path is called C ′′ . Although thisprocedure may seem somewhat formal, a quantum-mechanical exchange processdoes principally not specify the exchange path in order to define whether aparticle is a boson or a fermion, but only its topological properties. From atopological point of view, all paths that may be continuously deformed into eachother define a homotopy class [71]. Equation (4.21) must therefore be viewed asan equation for homotopy classes in which a simple translation and an alloweddeformation are irrelavant. As a consequence of these considerations, the simplepoint-like path C ′′ at the position of particle A, which may be formally describedby C ′′ = 1, is in the same homotopy class as the original path C . Therefore, theassociated processes are the same, and one has T = T ( C ) = T ( C ′′ ) = and thus E = √ , (4.22)where the last equation is symbolic in terms of the one operator. It indicatesthat the quantum-mechanical operator E , corresponding to particle exchange,has two eigenvalues that are the two square roots of unity, e B = exp(2 iπ ) = 1and e F = exp( iπ ) = − 1. This is precisely the above-mentioned superselectionrule, according to which all quantum particles in three space dimensions areeither bosons ( e B = 1) or fermions ( e F = − C enclosing the second particle Binto a single point at the position of A, without passing by B itself. This meansthat the position of B must be an element of the path at a certain moment ofthe shrinking process, which cannot profit from a third dimension in order toelevate the loop on which it moves above the xy -plane. The single point stillrepresents a homotopy class of paths, but these paths do not enclose anotherparticle, and C is therefore an element of another homotopy class, i.e. the oneof all paths starting from A and enclosing only the particle B. If there are morethan two particles present, the homotopy classes are described by the integernumber of particles enclosed by the paths in this class. From an algebraic pointof view, the exchange processes are no longer described by the two roots of unity,1 and − 1, but by the so-called braiding group , and the classification in bosonsand fermions is no longer valid. In the simplest case of Abelian statistics, one There are more complicated cases of non-Abelian statistics, in which the exchange pro-cesses of more than two different particles no longer commute, but we do not discuss this casehere and refer the reader to the review by Nayak et al. [70]. Strong Correlations and the Fractional Quantum Hall Effect needs to generalise the commutation relation ψ ( r ) ψ ( r ) = ± ψ ( r ) ψ ( r ) , (4.23)for bosons and fermions, respectively, to ψ ( r ) ψ ( r ) = e iαπ ψ ( r ) ψ ( r ) , (4.24)where α is also called the statistical angle . One has α = 0 for bosons and α = 1 for fermions, and all other values of α in the interval between 0 and 2 for anyons . Sometimes anyonic statistics is also called fractional statistics – indeedall physical quasi-particles, such as those relevant for the FQHE, have an anglethat is a fractional (or rational) number, but there is no fundamental objectionthat irrational values of the statistical angle should be excluded.Before discussing the anyonic nature of Laughlin quasi-particles, we need tomention an important issue in these statistical considerations. We know thatfermions are forced to satisfy Pauli’s principle which excludes double occupancyof a single quantum state, whereas the number of bosons per quantum state isunrestricted. What about anyons then? In the context of quantum fields thePauli principle yields, via Eq. (4.23) for r = r = r , ψ ( r ) ψ ( r ) = 0 . For an arbitrary statistical angle, one obtains in the same manner, from Eq.(4.24), (cid:0) − e iαπ (cid:1) ψ ( r ) ψ ( r ) = 0 , (4.25)which may be viewed as a generalised Pauli principle for 2D anyons [72]. Onlyif α = 0 modulo 2, we may have ψ ( r ) ψ ( r ) = 0 in order to satisfy Eq. (4.25).Otherwise, when α = 0 modulo 2, we necessarily have ψ ( r ) ψ ( r ) = 0. Anyonsare, thus, from an exclusion-principle point of view more similar to fermionsthan to bosons. We may now apply the above general statistical considerations to the case ofLaughlin quasi-particles. The basic idea is to describe the statistical angle asan Aharonov-Bohm phase due to some gauge field that is generated by the fluxbound to the charges included in a closed loop ∂ Σ. This closed loop, aroundwhich a quasi-particle moves adiabatically, encloses a surface Σ. The gauge fieldis not to be confunded with the one which generates the true magnetic field B – it is rather a mock (or fake) field A M (with B M = ∇ × A M ) that generatesthe flux bound, e.g., by the electrons in the Laughlin liquid via the relation(4.4). We consider the case where the area Σ is filled with N el (Σ) electronscondensed in an incompressible quantum liquid described by Laughlin’s wavefunction (4.3) and N qh (Σ) quasi-hole excitations (4.12), such that there are twocontributions to B M = | B M | , B M Σ = N flux he = [ mN el (Σ) + N qh (Σ)] he . (4.26) eneralisations of Laughlin’s Wave Function ∂ Σ, is given byΓ A − B = 2 π e ∗ h I ∂ Σ d r · A M ( r ) = 2 π e ∗ h Z Σ d r B M ( r ) , where e ∗ = e/m is the charge of the quasi-particle and where we have usedStoke’s theorem to convert the line integral of A M on ∂ Σ into a surface integralof B M over the area Σ. The Aharonov-Bohm phase has therefore two contri-butions, one Γ el that stems from the electrons condensed in the Laughlin liquidand the other one Γ qh that is due to the enclosed quasi-holes. One obtains fromEq. (4.26) Γ el = 2 π e ∗ e mN el = 2 πN el , (4.27)for the enclosed electrons, i.e. an integer times 2 π . Notice that this contributionto the Aharonov-Bohm phase may not be interpreted in terms of a statisticalangle because it does not describe a true exchange process: the involved particlesare not of the same type – we have chosen a quasi-particle to move on a pathenclosing condensed electrons. However, had we chosen an electron rather thana quasi-hole to move along the path ∂ Σ, the Aharonov-Bohm phase,Γ el − el = 2 π ee mN el (Σ) , would give rise to a statistical angle α = mN el (Σ). If we have only one electronenclosed by the path, N el (Σ) = 1, the statistical angle is simply the odd integer m , which is equal to 1 (modulo 2), as it should be for fermions.A more interesting situation arises when the path encloses Laughlin quasi-holes, in which case the Aharonov-Bohm phase readsΓ el = 2 π e ∗ e N qh = 2 π N qh m . (4.28)Consider a single quasi-hole in the area Σ, N qh = 1: one encounters the ratherunusual situation in which the Aharonov-Bohm phase is a fraction of 2 π , andthe associated statistical angle is α = 1 /m . This illustrates that Laughlin quasi-holes are indeed anyons with fractional statistics, as we have argued above. Although Laughlin’s wave function (4.3) has been extremely successful in the de-scription of the FQHE at ν = 1 / / 5, it is not capable of describing all ob-served FQHE states. Indeed, there are e.g. FQHE states at ν = 2 / , / , / , ... corresponding to the series p/ (2 p + 1), or more generally to p/ (2 sp + 1), in Remember that the statistical angle is defined with respect to an exchange process E which is the square root of the process T considered here [Eq. (4.21)]. The relation betweenthe statistical angle and the Aharononv-Bohm phase is therefore Γ = 2 πα and not πα . Strong Correlations and the Fractional Quantum Hall Effect vortex ν∗ = 1ν∗ = 2 electronfree flux quantumwith 2s (bound) flux quantacomposite fermion ν = 2/5 ν = 1/3 theoryCF Figure 4.6: Schematic view of composite fermions. The electronic state at ν = 1 / ν ∗ = 1,where each vortex bound to an electron carries 2 s (here s = 1) flux quanta. Inthe same manner a CF filling factor ν ∗ = 2 gives rise to an (electronic) FQHEstate at ν = 2 / s and p , which may be accounted for within composite-fermion (CF) theory, which we present below. Furthermore, even-denominatorFQHE states have been observed at ν = 5 / / n = 1), and, in wide quantum wells or bilayer quantum Hall systems, at ν = 1 / ν = 1 / ν = 5 / / Pfaffian wave function. Both the CF and the Pfaffian wave functions are sophisticatedgeneralisations of Laughlin’s original idea. Soon after the discovery of the most prominent FQHE state at ν = 1 / 3, alot of other states have been observed at the filling factors ν = p/ (2 sp + 1).In a first theoretical approach, these states were interpreted in the frameworkof a hierarchy scheme [62, 75] according to which the quasi-particles of theLaughlin (parent) state, such as ν = 1 / 3, condense themselves into a Laughlin-type (daughter) state, due to their residual Coulomb repulsion – remember thatthe Laughlin quasi-particles are charged with charge e ∗ = e/m . In this picture,the 2 / / − P Nj | z j | / 4) being an ubiquitous factor which finally eneralisations of Laughlin’s Wave Function ψ Lm ( { z j } ) = Y k 2, which corresponds to the limit p → ∞ in Eq.(4.34). In this limit the effective magnetic field (4.31) vanishes, B ∗ = 0, andone may then expect the corresponding phase to be described in terms of ametallic state, such as a Fermi liquid that one would obtain for electrons whenthe magnetic field vanishes. A natural ansatz for the N -particle wave functionof such a Fermi-liquid state is given by the Slater determinant ψ F L = det (cid:0) e i k i · r j (cid:1) , where the N electrons occupy the states described by the wave vectors k , i = 1 , ..., N , the modulus of which is delimited by the Fermi wave vector | k i | ≤ k F , and r j is the position of the j -th particle. Notice that this stateis nevertheless unappropriate in the description of a state in the LLL. Indeed,if the scalar product in the exponent is rewritten in terms of complex variables, k i · r j = ( k i z ∗ j + k ∗ i z j ) / 2, one realises that the Fermi-liquid state violates theLLL condition of analyticity. Formally, one may again avoid this problem byprojecting the Fermi-liquid state into the LLL, and one obtains indeed a state, ψ ν =1 / F L = P LLL Y k Coulomb interaction in a strong magnetic field. As we have alreadymentioned in Sec. 4.2.2, such an interaction may yield a discrete two-particlespectrum, in contrast to a repulsive interaction in the absence of a magneticfield. As a consequence, pairing may occur at certain relative angular momentafor particular pseudopotential sequences and for sufficiently high filling factors. In the present case, one may exclude s -wave pairing, i.e. in the relative angularmomentum state with m = 0 due to the Pauli principle, and the most naturalcandidate would therefore be p -wave pairing in the relative angular momentumstate m = 1 [19].A wave function that accounts for p -wave pairing was proposed by Mooreand Read in 1991 [18], ψ MR ( { z j } ) = Pf (cid:18) z i − z j (cid:19) Y k 2. However, it does not have the correct statisticalproperties. This problem is healed by the first factor Pf[1 / ( z i − z j )] whichrepresents the Pfaffian of the N × N matrix M ij = 1 / ( z i − z j ). The Pfaffianmay be viewed as the square root of the more familiar determinant, Pf( M ) = p det( M ), and has the same anti-symmetric properties as the determinant inan exchange of two particles i and j , such that it generates a fermionic wavefunction. Notice, furthermore, that this Pfaffian seems, at first sight, to takeaway some of the zeros such that one could expect the filling factor to increase.However, the function Q k Brief Overview ofMulticomponentQuantum-Hall Systems In the preceding chapters, we have completely neglected the physical conse-quences of possible internal degrees of freedom, apart from an occasional degen-eracy factor that has been smuggled in to account for experimental data. Thischoice has been made simply for pedagogical reasons, but it is clear that oneprominent internal degree of freedom – the electronic spin – may not be putunder the carpet so easily. Naively, one may expect that each LL is split intotwo distinct spin-branches separated by the energy gap ∆ Z due to the Zeemaneffect. If this gap is large, one may use the same one-particle arguments as inthe case of the IQHE, but now for each spin branch separately: once the low-est spin branch of a paticular LL is completely filled, additional electrons mustovercome an energy gap that is no longer given by the LL separation but by∆ Z . This would indeed not change the presented explanation of the IQHE –instead of a localised electron in the next higher LL, one simply needs to invokelocalisation in the upper spin branch.Also in the case of the FQHE, the explanation would need to be modifiedonly in the fine structure if the Zeeman gap is sufficiently large. If the electronsfill partially the lower spin branch of the lowest (or any) LL, one may omit alltransitions to the upper spin branch and argue that they constitute the high-energy degrees of freedom, in the same manner as inter-LL excitations in thecase of the “spinless” fermions which we have discussed in Sec. 4.1.However, the situation is not so easy as the above picture might suggest.Indeed, already in 1983 Halperin pointed out [84] that the Zeeman gap in GaAs,890 Brief Overview of Multicomponent Quantum-Hall Systems with a g -factor of g = − . 4, is ∆ Z = gµ B B = g (¯ he/ m ) B ≃ . B [T] K andtherefore much smaller than both the LL separation ¯ hω C = (¯ he/m ) B ≃ B [T]K, due to the rather small band mass ( m = 0 . m , in terms of the bareelectron mass m , in GaAs), and the Coulomb energy scale V C = e /ǫl B ≃ p B [T] K with a dielectric constant of ǫ ≃ 13. For a characteristic field of 6T, for which one typically reaches the LLL condition ν = 1, one therefore hasthe energy scales∆ Z ≃ ≪ e ǫl B ≃ 120 K < ∼ ¯ hω C ≃ 140 K , (5.1)in GaAs. The situation is qualitatively the same in graphene, where one findsfor a field of 6 T∆ Z ≃ ≪ e ǫl B ≃ 620 K < ∼ √ hvl B ≃ , (5.2)for g ≃ ǫ ≃ . 5, which are the appropriate values for graphene on a SiO substrate. The inevitable consequence of these considerations is that, even if one mayneglect the kinetic energy scale in a low-energy description of a partially filledLL, one cannot do so with the Zeeman energy scale. One must therefore takeinto account the electron spin within a two-component picture in which eachquantum state | n, m i is doubled, | n, m ; σ i with σ = ↑ and ↓ . Another multi-component system that we have already discussed is preciselygraphene, not only because of the tiny Zeeman gap which requires to take intoaccount the electronic spin, but also because of its double valley degeneracy dueto the two inequivalent Dirac points situated at the corners K and K ′ in thefirst BZ. Each quantum state | n, m i therefore occurs in four copies, | n, m ; σ i with σ = ( K, ↑ ), ( K, ↓ ), ( K ′ , ↑ ) and ( K ′ , ↓ ). Formally this four-fold degeneracymay be described with the help of an SU(4) spin, whereas the two-fold spindegeneracy in GaAs, e.g., is represented by the usual SU(2) spin. Notice thatit is very difficult in graphene to lift the valley degeneracy, and the associatedenergy scale is expected to be on the same order of magnitude as the Zeemangap, i.e. it is tiny with respect to the one set by the Coulomb interactions. A third multi-component system that we would like to mention consists of a dou-ble quantum well [see Fig. 5.1(a)]. These bilayer systems, which are fabricated Remember that this field is somewhat arbitrary because the situation ν = 1 may also beobtained easily for other fields by varying the gate voltage V G . Naturally, the dielectric constant depends on the dielectric environment around thegraphene sheet and thus also on the substrate. he Different Multi-Component Systems Wd2t (a) (b) W W ∆ sb Figure 5.1: (a) Profile of a double quantum well. The two wells are separatedby a distance d that is typically on the same order of magnitude as the wellwidth W , d ∼ W ∼ 10 nm. In the presence of a tunneling term t betweenthe two wells, the electronic subband is split into a symmetric and an anti-symmetric combination, separated by the energy scale ∆ SAS = 2 t . (b) Widequantum well. In a wide quantum well the energy gap between the occupiedlowest electronic subband and the unoccupied first excited subband, ∆ sb , isdecreaased as compared to a narrow quantum well.by molecular-beam epitaxy, consist of two quantum wells spatially separated byan insulating barrier that is on the same order of magnitude as the width of eachof the wells. Formally, each of the wells (layers) may be described in terms of anSU(2) pseudo-spin , σ = ↑ for an electron in the left well and σ = ↓ for one in theright well. In contrast to the true electron spin, the Coulomb interaction doesnot respect this SU(2) symmetry – indeed, the repulsion is stronger betweenparticles within the same layer (i.e. with the same pseudo-spin orientation)than between particles in different layers (with opposite pseudo-spin orienta-tion) because, in the second case, electrons may not be brought together closerthan the distance d between the layers. In order to minimise the interactionenergy, it is therefore favourable to charge both layers equally. Alternatively,this may be viewed as some capacitive energy, if one interprets the two-layer sys-tem in terms of a capacitor, that favours an equal charge distribution betweenthe two layers as compared to a charging of only one layer. Notice, further-more, that tunneling, with the tunneling energy t , between the two quantumwells lifts the pseudo-spin degeneracy: whereas the symmetric superposition | + i = ( | ↑i + | ↓i ) / √ |−i = ( | ↑i − | ↓i ) / √ SAS = 2 t [seeFig. 5.1(a)], but it may be strongly reduced experimentally with the help of ahigh potential barrier separating the two wells. The term ∆ SAS , which playsthe role of a Zeeman gap (though in the x -quantisation axis), may become thelowest energy scale in the system, such that the SU(2) pseudo-spin symmetry2 Brief Overview of Multicomponent Quantum-Hall Systems breaking only stems from the difference in the Coulomb interaction betweenparticles in the same and in different layers. Another quantum Hall system that may be characterised as a multi-componentsystem is a wide quantum well [Fig. 5.1(b)]. Indeed, the samples which revealthe highest mobilities are those fabricated in wide quantum wells, where thewell width w is often much larger than the magnetic length l B . As comparedto a narrow quantum well, the energy difference between the lowest and thefirst excited electronic subbands, which are the energy levels of the confinementpotential in the z -direction, is strongly decreased. Although the Fermi level stillresides in the lowest electronic subband (pseudo-spin σ = ↑ ), the energy gap tothe next unoccupied one (pseudo-spin σ = ↓ ) may then become smaller thanthe relevant Coulomb energy scale. In the same manner as for the electronicspin, one must therefore no longer discard higher electronic subbands. In a firstapproximation one may restrict the calculations to these two lowest subbands[85, 81] although the next higher subbands also shift to lower energies and needeventually be taken into account. Similarly to the quantum-Hall bilayer, which issometimes also used in the description of the large quantum well, the Coulombinteraction decomposed in these electronic subband states is not pseudo-spinSU(2)-symmetric.In the remainder of this chapter, we discuss some aspects of correlated statesthat one encounters in multi-component quantum Hall systems in general, start-ing (Sec. 5.2) with the completely spin-polarised state at ν = 1 (quantum Hallferromagnet) and its various manifestations in the different quantum Hall sys-tems described above. We will not discuss, for reasons of space limitation, theamazing physical properties of the elementary excitations of the quantum Hallferromagnet, which is a topological spin-texture state (skyrmion) , and refer theinterested reader to the literature [86, 87, 4, 3]. In the line of the precedingchapter, we have chosen to discuss a generalisation of Laughlin’s wave function,which we owe to Halperin [84], in order to account for the electronic spin (Sec.5.3). These wave functions are further generalised to even more componentsthan two, and we close this section with a discussion of their possible use in thedescription of multi-component FQHE states. ν = 1 If one takes into account internal degrees of freedom, the state at ν = 1 is nolonger simply a Slater determinant of all occupied quantum states in the lowestLL, but one must take into account the macroscopic degeneracy due to the factthat each state | n, m i may now be occupied by 0, 1 or 2 particles. In thissense the situation at ν = 1 is much more similar to the FQHE in a partiallyfilled LL than to the IQHE which one obtains for completely filled LL [86], and he State at ν = 1 93the macroscopic degeneracy is again lifted by the mutual Coulomb interactionsbetween the electrons. We first consider the generic case of electrons at ν = 1 in the conventionalmonolayer quantum Hall system while taking into account their physical spin.In view of the above-mentioned energy arguments, we completely neglect theZeeman effect, which would otherwise trivially lift the macroscopic degeneracyat ν = 1 by polarising all electron spins. Because of the fact that two electrons,with opposite spin, may now occupy the same quantum state | n, m i , the electronpair may in principle be in a relative angular momentum state with m = 0 –the Pauli principle, which only applies to fermions of the same species, does nolonger prevent this quantum number to be odd. Indeed, such an electron pair isdescribed by a two-particle wave function with the rather unspectacular poly-nomial factor ( z i, ↑ − z j, ↓ ) = 1, where z i, ↑ is the position of an arbitrarily chosenspin- ↑ electron and z j, ↓ that of a spin- ↓ electron. Such an electron pair thereforeinteracts via the Haldane pseudopotential v , which is the largest pseudopoten-tial in the case of a repulsive Coulomb interaction because it characterises theinteraction at the shortest possible length scale (see Fig. 4.2). Since v ≃ v ,the system thus tends to avoid double occupancy, and the ground state is de-scribed by the fully anti-symmetric (orbital) wave function (4.7) regardless ofwhether the electron at the position z j is spin- ↑ or spin- ↓ .Notice that, although both spinless and spin-1/2 electrons are described bythe same wave function, the physical origin of these ground states is different:in the case of spinless fermions, it is simply the non-degenerate wave functiondescribed by a Slater determinant, whereas in the case of electrons with spin,the state is formed in order to minimise the mutual Coulomb repulsion.Because the orbital wave function (4.7) for electrons with spin at ν = 1 isfully anti-symmetric, the spin wave function describing the internal degrees offreedom must be fully symmetric, e.g. χ F M = | ↑ , ↑ , ..., ↑ N i , (5.3)in order to form an overall wave function that is anti-symmetric. The subscriptindicates the index of the particle that the spin is associated with. The globalwave function, therefore, reads ψ ν =1 ,F M = Y k Because the spontaneous spin polarisation in the quantum Hall ferromagnetchooses, in the absence of a Zeeman effect, an arbitrary direction in the three-dimensional spin space, one is confronted with a spontaneous SU(2) symmetrybreaking. As a consequence of this broken continuous symmetry, there exists agapless collective excitation (Goldstone mode) the energy of which tends to zeroin the long wave-length limit. Indeed, even if we have chosen the ferromagnetin Eq. (5.3) to be oriented in the z -direction, any other orientation, such as theone described by the wave function | ↓ , ↓ , ..., ↓ N i or N O j =1 | + j i = | + , + , ..., + N i , where the + j sign indicates the symmetric superposition | + j i = ( | ↑ j i + | ↓ j i ) / √ j -th electron, would also describe a groundstate. The Goldstone mode in the large wave-length limit may then be viewedas a global rotation of all spins into another ground-state configuration, whichnaturally does not imply an energy cost.In the case of a ferromagnet, the Goldstone mode is nothing other than thespin-density wave that disperses as ω ∝ q in the small wave vector limit, ql B ≪ 1. At first sight, this mode seems in contradiction with the observationof a quantum Hall effect at ν = 1, even in the absence of a Zeeman effect, whichrequires a gap as we have seen above. Notice, however, that this gap needs tobe a transport gap in which a quasi-particle moves independently from a quasi-hole in order to transport a current. This is not the case in a spin wave with ql B ≪ 1, but one obtains freely moving quasi-particles and quasi-holes in thelimit ql B ≫ 1. In this limit, the spin-wave dispersion tends to a finite value thatis given by the exchange energy between particles of different spin orientationand that is proportional to the interaction energy scale e /ǫl B , as in the case ofthe FQHE [87].There are more exotic spin-texture excitations (skyrmions), which are de-scribed by a topological quantum number associated with the winding of thespin-texture. These are gapped excitation which carry an electric charge related Remember that for a crystaline ground state (WC), the Goldstone mode is the acousticphonon, as we have briefly discussed in the previous chapter in Sec. 4.1. he State at ν = 1 95to this topological quantum number. As mentioned above, a detailed discussionof these amazing excitations is beyond the scope of the present lecture notes. The ν = 1 in a bilayer system is remarkably different from the quantum Hallferromagnet described in the preceding subsection. Although the electronicinteractions still favour a fully anti-symmetric orbital wave function (4.7) andthus a symmetric, i.e. ferromagnetic, pseudo-spin wave function, the interactionpotential is no longer SU(2) symmetric in the pseudo-spin degree of freedom. As we have already mentioned above, a charge imbalance Q between the twoquantum wells (layers) is penalised by a charging energy, E C = Q / C , in termsof the capacitance C = ǫ A /d , where A is the area of the 2D system. Because Q = − eνn el A = − eνn B A = − eν A / πl B when all electrons reside in a singlelayer and Q = 0 if they are equally distributed between the two layers, oneobtains an energy cost E C N el ∼ ν e ǫl B dl B , per particle in the charge-imbalenced state, in agreement with a more sophisti-cated microscopic calculation [87]. In terms of the pseudo-spin magnetisation,this means that in the ground-state configuration, with a homogeneous chargedistribution over both layers, all pseudo-spins are oriented in the xy -plane. Re-member that a pseudo-spin ↑ corresponds to an electron in the upper layer and ↓ to one in the lower layer, and a configuration as the one described in Eq. (5.3)is therefore excluded, whereas the symmetric and anti-symmetric combinations χ + = N O j =1 | + j i and χ − = N O j =1 |− j i , with |± j i = ( | ↑ j i ± | ↓ j i ) / √ x - and the y -direction, respectively, may be generalised bychoosing any other direction described by the angle φ in the xy -plane, χ φ = N O j =1 | φ j i , (5.5)where | φ j i ≡ [ | ↑i + exp( iφ ) | ↓i ] / √ 2. The states χ + and χ − are obtained for φ = 0 and φ = π (modulo 2 π ), respectively.Contrary to the case of the spin ferromagnet with full SU(2) symmetry,where a general state would be described in terms of two angles θ and φ , thedifferent possible easy-plane pseudo-spin ferromagnetic are characterised by theangle φ which may vary between 0 and 2 π . The low-energy degrees of freedom Naturally, such an anti-symmetric orbital wave function is only physical if the layerseparation d is not too large (as compared to the magnetic length) – otherwise one wouldsimply have completely decoupled layers. Brief Overview of Multicomponent Quantum-Hall Systems ν = 1 − − −− − − −+ + + ++ + + − − − −− − −+ + ++++ + (b)(a) H a ll v o l t age magnetic field (T)−1010 I = II = −I holeelectron Figure 5.2: Hall resistance measurement used to detect excitonic condensation,adopted from (Eisenstein and MacDonald, 2004). (a) Counterflow configuration,in which one drives a current I ↑ = I through the upper layer that is flowing inthe opposite direction as that, I ↓ = − I in the lower layer. The hole componentof the excitonic quantum state in one layer thus moves in the same direction asthe electron component in the other one. (b) The two curves schematically rep-resent, when taking into account only excitonic superfluidity, the Hall resistancein both layers within the counterflow configuration. Because of the relative signbetween the currents in the two layers, the measured Hall resistances are ofopposite sign. Electrons with no interlayer correlations yield the usual linear B -field dependence of the Hall resistance in order to compensated the Lorentzforce acting on them individually. In the case of exciton condensation (around B = 5 T), charge tranport is due to a uniform current of charge-neutral excitons,which are not affected by the Lorentz force, and the Hall resistance vanishes,as it has been observed in the experiments (Kellogg et al. , 2004; Tutuc et al. ,2004). he State at ν = 1 97are therefore described by a different universality class that turns out to be thesame as the one that describes superfluidity or superconductivity. The relationbetween superfluidity and the easy-plane pseudo-spin ferromagnet in bilayersystems at ν = 1 may indeed be understood in the following manner: on theaverage, the average filling factor per layer is ν ↑ = ν ↓ = 1 / interlayer exciton that satisfiesbosonic statistics [Fig. 5.2(a)]. Below a certain temperature, these bosonscondense into a collective state that is nothing other than the exciton superfluid [88, 89, 90, 87]. The phase coherence between the different excitons is preciselydescribed by the angle φ .The first experimental indication of excitonic superfluidity in bilayer quan-tum Hall systems was a zero-bias anomaly in tunneling experiments [91]. Indeed,if one injects a charge in a tunneling experiment into one of the layers and collectsit in a contact at the other layer, the tunneling conductance dI z /dV is expectedto be weak in the case of uncorrelated electrons because of the Coulomb repul-sion between electrons in the opposite layers. However, below a critical valueof d/l B , where one expects the interlayer correlations to be sufficiently strongto form a phase-coherent excitonic condensate, the injected electron systemat-ically finds a hole in the other layer, such that tunneling between the layersis strongly enhanced. This strong enhancement, which due to its reminiscencewith the Josephson effect in superconductors [66] is also called quasi-Josephsoneffect , has indeed been observed experimentally [91].Another strong indication for excitons in bilayer quantum Hall systems stemsfrom transport measurements in the counterflow configuration, where the cur-rent in the upper layer I ↑ = I flows in the opposite direction as compared tothat in the lower layer I ↓ = − I [see Fig. 5.2(a)]. From a technical point ofview, it is indeed possible to contact the two layers separately such that onemay measure the Hall resistance (and also the longitudinal resistance) in bothlayers independently. In the case of exciton condensation, the charges involvedin transport are zero because the excitons are charge-neutral objects, which arenot coupled to the magnetic field and thus not affected by the Lorentz force.In addition to a vanishing longitudinal resistance, one would therefore expect avanishing Hall resistance because no density gradient between opposite edges isbuilt up to compensate the Lorentz force [89, 90]. This is schematically shown inFig. 5.2(b). The simultaneous vanishing of the Hall and longitudinal resistanceswas indeed observed in 2004 by two different experimental groups [92, 93]. Contrary to the Josephson effect, only the tunneling conductance dI z /dV is stronglyenhanced whereas the tunneling current remains zero in the quasi-Josephson effect in bilayersystems. Brief Overview of Multicomponent Quantum-Hall Systems The arguments in favour of a quantum Hall ferromagnetism may easily be gener-alised to the case of graphene, where the Coulomb interaction respects to greataccuracy the four-fold spin-valley degeneracy, as we have described above. In or-der to avoid confusion about the filling factor, one first needs to remember thatthe filling factor ν G in graphene is defined with respect to the charge-neutralpoint, which happens to be in the centre of the central n = 0 LL (see Sec. 3.5).Two of the four (degenerate) spin-valley branches are therefore completely filledat ν G = 0, which in non-relativistic quantum Hall systems would correspondrather to a filling factor ν = 2. Similarly the filling factor ν = 1 would cor-respond to a graphene filling factor ν G = − 1, whereas ν G = 1 implies threecompletely filled spin-valley branches ( ν = 3).Let us first consider the filling factor ν G = − In the same manner asfor the spin quantum Hall ferromagnet at ν = 1, the short-range component v of the Coulomb potential is screened in the completely anti-symmetric orbitalwave function (4.7), and the spin part of the wave function must therefore becompletely symmetric. Notice, however, that one may now distribute the elec-tron over the four internal states | m ; K, ↑i , | m ; K, ↓i , | m ; K ′ , ↑i and | m ; K ′ , ↓i .The general spin wave function is therefore a superposition of all these states χ SU(4) = N O m =1 ( u m, | m ; K, ↑i + u m, | m ; K, ↓i + u m, | m ; K ′ , ↑i + u m, | m ; K ′ , ↓i ) , (5.6)where the complex coefficients u m,i satisfy the normalisation condition P i =1 | u m,i | =1. In the case of global coherence, all coefficients are independent of the guiding-centre quantum number m , u m,i = u i , and one thus obtains the spin wave func-tion of an SU(4) ferromagnetism [94, 95, 96, 97]. These arguments may also begeneralised to the case of ν G = 0, where two branches are completely filled [97],but the ground state does not reveal the same degeneracy as the SU(4) ferro-magnet at ν G = ± 1. Indeed, a general argument on K -component quantumHall system shows that one has generalised ferromagnetic states at all integervalues of the filling factor ν = 1 , ..., K − ν G = 0 , ± ν G = ± , ± , ... of the RQHE which may be explained by LL quantisation within the pictureof non-interacting relativistic particles. In the same manner as for the spinquantum Hall ferromagnet, the gapless spin-density-wave modes, which reveala higher degeneracy due to the larger SU(4) symmetry, do not imply that thecharged modes are also gapless. Indeed, the elementary charged excitations ofthe SU(4) quantum Hall ferromagnet are generalised skyrmions [97, 99] whichare separated by a gap from the ground state, which therefore describes an The filling factor ν G = 1 is related to ν G = − ulti-Component Wave Functions Until now, we have considered a multi-component quantum Hall effect at theinteger filling factor ν = 1 (or other integer fillings in the case of graphene)that is described in terms of the Vandermonde determinant (4.7) Q k 1) and (1 , , ν = 1 / Brief Overview of Multicomponent Quantum-Hall Systems SU(2) symmetric, such as for the true electron spin. In this case, one mayshow that ( m, m, n ) states are only eigenstates of the total-spin operator, whichcommutes with the interaction Hamiltonian, if n = m (i.e. in the ferromag-netic state) or if n = m − Physical relevance of Halperin states A physically relevant Halperin state is e.g. the unpolarised (3 , , 2) state whichwould occur at a filling factor ν = 2 / 5. Remember from the discussion of CFtheory in Sec. 4.4 that there is also a (naturally polarised) CF candidate, with p = 2 completely filled CF LLs, to describe the ground state at this filling fac-tor. Which of them is now the better one? This question could be answeredwithin exact-diagonalisation calculations, which showed that, in the absence ofa Zeeman effect, the true ground state is described in terms of the unpolarisedHalperin wave function (3 , , 2) [106]. Notice, however, that the energy differ-ence between the two states is extremely small, as may be seen from variationalcalculations [76], such that the polarised CF state becomes the ground stateabove a critical value of the energy ∆ Z associated with the Zeeman effect. Thiscritical value would therefore describe a phase transition between an unpolarisedand a fully polarised FQHE state. Such transitions have indeed been observedin polarisation experiments, where the strength of the Zeeman effect was variedby a simultaneous change in the magnetic field and in the electronic density[107, 108]. We would finally mention that Halperin’s wave function may easily be gener-alised to describe possible FQHE states in systems with a larger number ofcomponents, such as the four spin-valley components in graphene. This gener-alised wave function for K -component quantum Hall systems may be writtenas a product ψ SU ( K ) m ,...,m K ; n ij (cid:16)n z (1) j , z (2) j , ..., z ( K ) j K o(cid:17) = ψ Lm ,...,m K × ψ intern ij (5.16)of a product of Laughlin wave functions ψ Lm ,...,m K = K Y j =1 N j Y k j 2. One may indeed have different “degrees” ofinvertibility that are described by the rank of the matrix. Consider, e.g., thefully anti-symmetric wave function with m i = n ij = m . In this case, Eq. (5.17)actually consists only of one single equation relating the component filling fac-tors, i.e. 1 = m ( ν + ... + ν K ) = mν , and all other lines of the matrix equationare simply copies of the first one. The rank of this matrix is 1, i.e. only thetotal filling factor is fixed, ν = 1 /m [SU( K ) ferromagnet] whereas in the caseof an invertible matrix the rank is K and the K lines in the matrix equation(5.17) represent (linearly) independent equations. If the rank of an exponentmatrix is smaller than K but larger than 1, the resulting state is neither a fullSU( K ) ferromagnet nor a state with completely fixed component filling factors(or polarisations) – it is rather a state with some intermediate ferromagneticproperties.04 Brief Overview of Multicomponent Quantum-Hall Systems As for two-component Halperin wave functions (5.8), a generalisation ofLaughlin’s plasma analogy allows one to distinguish between physical (i.e. ho-mogeneous) and unphysical states (which show a phase separation of at leastsome of the components). Indeed, the exponent matrix M must have onlypositive eigenvalues in order to describe a homogeneous state [104]. We finallymention that M encodes not only information concerning the filling factors(5.18), but fully describes the quantum Hall state (5.16), such as its topologicaldegeneracy, the charges of its quasi-particle excitations as well as the statisticalproperties of the latter [110]. ppendix A Electronic Band Structureof Graphene In this appendix, we calculate the band structure of graphene in the tight-binding model [111], the results of which we have summarised in Sec. 1.2.3.Because graphene’s honeycomb lattice consists of two distinct sublattices A andB, the electronic wave function ψ k ( r ) = a k ψ ( A ) k ( r ) + b k ψ ( B ) k ( r ) , (A.1)is a superposition of two wave functions, for the A and B sublattice, respectively,where a k and b k are complex functions of the quasi-momentum k . Both ψ ( A ) k ( r )and ψ ( B ) k ( r ) are Bloch functions with ψ ( j ) k ( r ) = X R l e i k · R l φ ( j ) ( r + δ j − R l ) , (A.2)in terms of the atomic wave functions phi ( j ) ( r + δ j − R l ) centred around theposition R l − δ j , where δ j is the vector which connects the sites R l of the under-lying Bravais lattice with the site of the j atom within the unit cell. Typicallyone chooses the sites of one of the sublattices, e.g. the A sublattice, to coincidewith the sites of the Bravais lattice such that δ A = 0.With the help of these wavefunctions, we may now search the solutions ofthe Schr¨odinger equation Hψ k = ǫ k ψ k , where H is the full Hamiltonian for electrons on a lattice, which is of the type(2.2) mentioned in Sec. 2.1. Here, we have chosen an arbitrary representation,which is not necessarily that in real space. Multiplication of the Schr¨odingerequation by ψ ∗ k from the left yields the equation ψ ∗ k Hψ k = ǫ k ψ ∗ k ψ k , which may The wavefunction ψ k ( r ) is, thus, the real space representation of the Hilbert vector ψ k . Appendix a a δ a AB BB 12 3 Figure A.1: Tight-binding model for the honeycomb lattice.be rewritten in matrix form with the help of Eqs. (A.1) and (A.2)( a ∗ k , b ∗ k ) H k (cid:18) a k b k (cid:19) = ǫ k ( a ∗ k , b ∗ k ) S k (cid:18) a k b k (cid:19) . (A.3)Here, the Hamiltonian matrix is defined as H k ≡ ψ ( A ) ∗ k Hψ ( A ) k ψ ( A ) ∗ k Hψ ( B ) k ψ ( B ) ∗ k Hψ ( A ) k ψ ( B ) ∗ k Hψ ( B ) k ! = H † k , (A.4)and the overlap matrix S k ≡ ψ ( A ) ∗ k ψ ( A ) k ψ ( A ) ∗ k ψ ( B ) k ψ ( B ) ∗ k ψ ( A ) k ψ ( B ) ∗ k ψ ( B ) k ! = S † k (A.5)accounts for the non-orthogonality of the trial wavefunctions. The eigenvalues ǫ k of the Schr¨odinger equation yield the energy bands, and they may be obtainedfrom the secular equation det (cid:2) H k − ǫ λ k S k (cid:3) = 0 , (A.6)which needs to be satisfied for a non-zero solution of the wavefunctions, i.e. for a k = 0 and b k = 0. The label λ denotes the energy bands, and it is clear thatthere are as many energy bands as solutions of the secular equation (A.6), i.e.two bands for the case of two atoms per unit cell.From now on, we neglect the overlap of wave functions on neighbouring sites,such that the overlap matrix (A.5) simply becomes the one matrix times thenumber of particles N due to the normalisation of the wave functions. Thesecular equation then tells us that the energy bands are just the eigenvalues ofthe Hamiltonian matrix (A.4). Furthermore, one notices that because the twosublattices are equivalent from a chemical point of view, we have ψ ( A ) ∗ k Hψ ( A ) k = ψ ( B ) ∗ k Hψ ( B ) k , and the diagonal terms therefore contribute just a constant shift lectronic Band Structure of Graphene H AB k ≡ ψ ( A ) ∗ k Hψ ( B ) k = N t AB k , with the hopping term t AB k ≡ X R l e i k · R l Z d r φ ( A ) ∗ ( r − R k ) Hφ B ) ( r + δ AB − R m ) , (A.7)where δ AB is a vector that connects an A site to a B site.In order to obtain the basic band structure of graphene, it is sufficient toconsider a hopping only between nearest-neighbouring sites described by the hopping amplitude t ≡ Z d r φ A ∗ ( r ) Hφ B ( r + δ ) , (A.8)where we have chosen δ AB = δ (see Fig. A.1). Notice that one may also takeinto account hopping to sites that are further away such as the next-nearestneighbours which turn out to be on the same sublattice and which would thusyield diagonal terms to the Hamiltonian matrix. However, whereas we have t ∼ A on the A sublattice (Fig. A.1), wemay see that the hopping term (A.7) consist of three terms corresponding tothe nearest neighbours B , B , and B , all of which have the same hoppingamplitude t . However, only the site B is described by the same lattice vector(shifted by δ ) as the site A and thus yields a zero phase to the hopping matrix.The sites B and B correspond to lattice vectors shifted by a = √ a e x + √ e y ) and a ≡ a − a = √ a − e x + √ e y ) , respectively, where a = | δ | = 0 . 142 nm is the distance between nearest-neighbour carbon atoms. Therefore, they contribute a phase factor exp( i k · a )and exp( i k · a ), respectively. The hopping term (A.7) may therefore be writtenas t AB k = tγ ∗ k = (cid:0) t BA k (cid:1) ∗ , where we have defined the sum of the nearest-neighbour phase factors γ k ≡ e i k · a + e i k · a . (A.9)The band dispersion may now easily be obtained by solving the secularequation (A.6), ǫ λ ( k ) = λ (cid:12)(cid:12) t AB k (cid:12)(cid:12) = λt | γ k | , (A.10)and is plotted in Fig. 2.2. The band dispersion is obviously particle-hole sym-metric, and the valence band ( λ = − ) touches the conduction band ( λ ) in theinequivalent points ± K = ± π √ a e x , Appendix which one determines by setting γ ± K = 0 and which coincide with the twoinequivalent BZ corners K and K ′ . Because the whole band structure is half-filled in undoped graphene, as we have mentioned in Sec. 2.1, the Fermi energylies exactly in these points K and K ′ . Continuum Limit The low-energy electronic properties may be obtained by expanding the bandstructure in the vicinity of these points, and the low-energy Hamiltonian isobtained simply by expanding the sum of the phase factors (A.9) around K and K ′ , γ ± p ≡ γ k = ± K + p = 1 + e ± i K · a e i p · a + e ± i K · a e i p · a ≃ e ± i π/ [1 + i p · a ] + e ∓ i π/ [1 + i p · a ]= γ ± (0) p + γ ± (1) p By definition of the Dirac points and their position at the BZ corners K and K ′ , we have γ ± (0) p = γ ± K = 0. We limit the expansion to first order in | p | a .Notice that, in order to simplify the notations, we have used a system of unitswith ¯ h = 1, i.e. where the momentum has the same units as the wave vector.The first order term is given by γ ± (1) p = i √ a h ( p x + √ p y ) e ± i π/ + ( − p x + √ p y ) e ∓ i π/ i = ∓ a p x ± ip y ) , (A.11)which is obtained with the help of sin( ± π/ 3) = ±√ / ± π/ 3) = − / 2. This yields the effective low-energy Hamiltonian H ξ p = ξv ( p x σ x + ξp y σ y ) , (A.12)in terms of the Fermi velocity v ≡ ta h . (A.13)The index ξ = ± denotes the valleys K and K ′ , and one obtains at the K pointthe Dirac Hamiltonian mentioned in (2.4) H D = v p · σ , (A.14)whereas the low-energy Hamiltonian at the K ′ point reads H ′ D = − v p · σ ∗ , (A.15)with σ ∗ = ( σ x , − σ y ). Both Hamiltonians yield the same energy spectrum whichis therefore two-fold valley-degenerate . lectronic Band Structure of Graphene K ( ξ = +) and K ′ ( ξ = − ) in a compact form, H ξD = ξH D = ξv p · σ . (A.16)10 Appendix ppendix B Landau Levels of MassiveDirac Particles Mass Confinement of Dirac Fermions at B = 0 Even in the absence of a magnetic field, electronic confinement in graphene turnsout to be quite tricky because a simple-minded approach in terms of a potential V conf = V ( y ) cannot confine Dirac electrons. This fact is due to an intrinsi-cally relativistic effect that is called the Klein paradox , according to which a(massless) relativistic particle may transverse a potential barrier without beingbackscattered [112]. This effect may be understood in the following manner:consider an incident electron in the region with V = 0 the energy of which isslightly above the Fermi energy. In the potential barrier, the Dirac point isshifted to a higher energy that corresponds to the barrier height and the Fermienergy lies now in the valence band, where the electron may still find a quantumstate (with the same wave-vector direction and the same velocity v ) – insteadof moving as an electron in the conduction band, it thus simply moves in thesame direction as an electron in the valence band [Fig. B.1(a)]. This is in starkcontrast with quantum mechanical tunneling of a non-relativistic particle, forwhich the transmission probability through a potential barrier is exponentiallysuppressed because of a lacking quantum state at the same energy as that ofthe incident electron.The problem is circumvented by a so-called mass confinement V conf = V ( y ) σ z = (cid:18) V ( y ) 00 − V ( y ) (cid:19) , (B.1)and we discuss first the simpler case of a constant mass term M σ z that needs tobe added to the Dirac Hamiltonian. That this term yields indeed a mass maybe seen from the Dirac Hamiltonian at B = 0 H mD = v p · σ + M σ z = (cid:18) M v ( p x − ip y ) v ( p x + ip y ) − M (cid:19) , (B.2)11112 Appendix E F electron in CB electronin VB electron in CB (a) barrier M(y)(b) Figure B.1: (a) Klein tunneling through a barrier. An incident electron inthe conduction band (CB) above the Fermi energy, which is at the Dirac pointbefore the barrier, transverses the barrier as en electron above the Fermi energyin the valence band (VB). The valence band is partially emptied because theDirac point has shifted to a higher energy corresponding to the barrier height. (b) Mass confinement. A gap opens when the particle approaches the edge,which becomes a forbidden region where no quantum state can be found at theenergy corresponding to that of the incident electron.the diagonalisation of which yields the energy spectrum ǫ λ ( p ) = λ p v | p | + M , which is gapped at zero momentum. This is nothing other than the dispersionrelation of a relativistic particle with mass m such that M = mv . Qualita-tively one may see from Fig. B.1(b) why a mass confinement is more efficientthan a potential barrier. Indeed, when the particle approaches the edge with M ( y ) = 0 a gap opens. An electron slightly above the Dirac point may thenonly propagate in the region with M = 0, whereas at the edge its energy lies inthe gap which is a forbidden region, and the electron is thus confined.Similarly to the B = 0 case, one may find the energy spectrum of the massiveDirac Hamiltonian (B.2) in a perpendicular magnetic field, which reads, in termsof the ladder operators a and a † , H BD = (cid:18) M v (Π x − i Π y ) v (Π x + i Π y ) − M (cid:19) = M √ ¯ hvl B a √ ¯ hvl B a † − M ! . (B.3)Its eigenvalues may be obtained in the same manner as in the M = 0 case (c.f.Sec. 2.3.2), and one obtains ǫ λn = λ s M + 2 ¯ h v l B n (B.4) The sign λ = − corresponds to the anti-particle. andau Levels of Massive Dirac Particles n = 0.Special care needs to be taken in the discussion of the central LL n = 0,which necessarily shifts away from zero energy. The associated quantum state(2.24) is zero in the first component u , whereas the second component is givenby v = | i . In order to satisfy the second line in the eigenvalue equation H BD ψ = ǫ ψ ⇔ M √ ¯ hvl B a √ ¯ hvl B a † − M ! (cid:18) | i (cid:19) = ǫ (cid:18) | i (cid:19) , one needs to fulfil √ hvl B a † u = ( ǫ + M ) v ⇔ ǫ + M ) | i , (B.5)such that the only solution is ǫ = − M . The relativistic n = 0 LL is thereforeshifted to negative energies and does no longer satisfy particle-hole symmetry.This effect is called parity anomaly and depends on the sign of the mass .In the case of graphene, we need to remember that there are two copiesof the energy spectrum, one at the K point and one at the K ′ point. Aswe have disussed in Appendix A, the Hamiltonian (B.3) describes the low-energy properties at the K point whereas we need to interchange the A and Bsublattices at the K ′ point and add a global sign in front of the off-diagonalterms [see Eq. (A.16)], H B ′ D = − M −√ ¯ hvl B a −√ ¯ hvl B a † M ! = − H B ′ D . (B.6)Naturally, the eigenstates of this Hamiltonian are the same as those of theHamiltonian (B.3) at the K point, but the eigenvalues change their sign. Dueto the particle-hole symmetry of the levels (B.4), the global sign does not affectthe energy spectrum for n = 0. However, the n = 0 LL, which does not respectparticle-hole symmetry, must again be treated apart, and one finds in the samemanner as for the K point the condition corresponding to Eq. (B.5), − √ hvl B a † u = ( ǫ − M ) v ⇔ ǫ − M ) | i . 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IR ~ 56 2 3 41 R ~ µ − µ = µ − µ µ − µ = 0 µ = µ L 3 µ = µ L µ = µ = µ ensity of statesB=9TT=1.6K ∼ ν∼ 1/ν V =15VT=30mKg -224 case ν=0 B n=0n=1n=2n=3n=4n=−1n=−2n=−3n=−4 ene r g y magnetic field (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) completelyfilled levelshalf−filled central level BE L L L L L L Be2cE ~ L E E AB CD BE L L L L L L Be2cE ~ L E E AB CD (D)(C)(B) R e l a t i v e t r an s m i ss i on Energy (meV) (A)0.4 T1.9 K lectron in CB electron in VB electron in CB -224 case ν=0 B n=0n=1n=2n=3n=4n=−1n=−2n=−3n=−4 ene r g y magnetic field (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) completelyfilled levelshalf−filled central level R e l a t i v e t r an s m i ss i on Energy (meV) .0 0.5 1.0 1.5 2.001020304050607080 )( DLL )( DLL )( CLL )( CLL )( BLL )( BLL )( ALL T r an s i t i on ene r g y ( m e V ) sqrt(B) 120 4 Magnetic Field B (T)00.51.01.52.0 ρ Ω xx ( k ) / / /V x V y I x4/75/3 4/38/57/5 123456