The Integer Quantum Hall Effect
Aly Abouzaid, Jun Dai, Simon Feiler, Philipp Kessler, and Chris Waddell
The integer quantum Hall effect concerns the transport properties of a 2-dimensional electron system in the presence of a magnetic field. We analyze the theoretical underpinnings of this effect, emphasizing in particular the role of disorder and topological considerations. We proceed to review the classic experiment of von Klitzing et al. [1], which provided the earliest experimental confirma- tion of exact quantization of Hall conductivity. Finally, we discuss metrological applications of the quantum Hall phenomenon.
I. INTRODUCTION (a) (b)
In a 2-dimensional electron system (2DES), Ohm’s Law J = σE gives the relationship between the applied electric field and the induced current density at the level of linear response theory, where the conductivity σ is a rotationally-invariant 2 2 matrix of the form × σ σ σ = xx xy . (1) σ σ − xy xx The components σxx, σxy are referred to as the longitu- dinal and Hall conductivities, respectively. Remarkably, it is observed that, when a strong magnetic field is ap- plied orthogonal to the plane of the 2DES (see schematic FIG. 1. (a) Schematic of a Hall bar apparatus, with orthog- in Fig. 1(a)), the Hall conductivity varies stepwise with onal B-field; a current is applied in the y-direction, and the the strength of the magnetic field (see Fig. 1(b)), with longitudinal (y-direction) and transverse (x-direction) voltage plateaux at the universal quantized values are measured. (b) Hall resistance Rxy and longitudinal resis- eν tance Rxx as a function of magnetic field strength; for the σxy = , ν N . (2) quantum Hall system, σxx = 0 if and only if Rxx vanishes, Φ0 ∈ −1 and σxy = Rxy . Figure taken from [3]. where we have defined the fundamental constant Φ0 = 2π~/e, known as the flux quantum. Moreover, the longi- tudinal conductivities vanish precisely on these plateaux where one has pˆ = i~ ∂x, ∂y for a system constrained − (also seen in Fig. 1(b)), indicating the absence of lon- to the (x, y)-plane, and me and e are the electron mass gitudinal transport. This is the (integer) quantum Hall and charge. It will be advantageous to use Landau gauge effect (QHE). The observation that the Hall conductivity is exactly A(x, y) = xByˆ , (4) quantized as in equation (2) for any magnetic field strength was first realized experimentally by von Klitzing whereupon the Hamiltonian can be written as et al. [1]. Since then, a rich body of literature (see e.g. 1 [2]) has developed to provide a theoretical foundation for Hˆ = pˆ2 + (ˆp + eBxˆ)2 . (5) 2m x y this phenomenon, involving the confluence of a number e of important concepts, such as the role of disorder in lo- Evidently, the momentum py is a good quantum number, calizing quantum states, and the relevance of topological so we may investigate product eigenstates of the form considerations. We elaborate upon these developments iky y below, beginning with an elementary description of a sin- ψky (x, y) = e fky (x) , (6) 10 gle electron in the presence of a magnetic field. 11 upon which the Hamiltonian acts as 1 II. THEORY ˆ 2 2 Hψky = pˆx + (~ky + eBxˆ) ψky 2me 2 2 (7) A. A First Approach: Landau Levels pˆx meωB 2 2 = + xˆ + ky`B ψky , 2me 2 The quantum Hamiltonian for a single particle in a where we have recalled the expression ωB = eB/me for magnetic field B = A = B zˆ is given by ∇ × the cyclotron frequency of an electron in a uniform mag- 1 2 netic field, and we have implicitly defined the magnetic Hˆ = pˆ + eA(ˆx, yˆ) , (3) p length `B ~/eB. This is manifestly the Hamiltonian 2me ≡ 2 for a 1-dimensional quantum harmonic oscillator (QHO) (a) centred at x = k `2 ; if we define operators − y B
1 † 1 aˆ = pˆx ieBxˆ , aˆ = pˆx + ieBxˆ , √2~eB − √2~eB (8) then we recover the typical QHO commutation relations, and we can implement the standard QHO analysis. The 1 energy levels are En = ~ωB n + , where n N is 2 ∈ the eigenvalue of the number operator ˆ =a ˆ†aˆ. Conse- N quently, if we neglect spin, then the operatorsp ˆy and ˆ form a complete set of commuting observables, and weN (b) can index all energy eigenstates ψn,ky (x, y) by the corre- sponding quantum numbers (n, ky). The eigenenergies of the system thus arrange them- selves into highly degenerate Landau levels (LLs), in- dexed by the quantum number n, with the degeneracy corresponding to a continuum of possible momenta ~ky. In reality, the Hall geometry has finite dimensions Lx,Ly, so the momenta must be quantized, and we expect a cut- off at large momenta, due to the non-zero lattice spac- ing within a given material; the degeneracy is thus large but finite. If we model the edges of the Hall sample with FIG. 2. (a) U-shaped confining potential; dots represent an ‘infinite well’ confining potential, then the y-momenta states localized around a given x-position. (b) Deformation of 2π the U-shaped potential, due to applied voltage and impurities. are quantized as ky N. Furthermore, since the eigen- ∈ Ly Figures taken from [4]. states in a given LL are harmonic oscillator eigenstates 2 centred at x = ky`B, the restriction x < Lx/2 can − | | 2 be interpreted as a restriction ky < Lx/2` on allowed 2 | | B ky, or equivalently, by the equilibrium position ky`B of momenta. The degeneracy in a single LL is then roughly the state in the x-direction; the eigenstates continue− to
2 be localized in the x-direction in the presence of a uni- Z Lx/2`B Ly LxLy BLxLy form electric field. We may therefore assume that the NLL = dk = 2 = . (9) 2π 2 2π` Φ −Lx/2`B B 0 effect of the confining potential is to increase the energy associated with the states localized near the edge, which are referred to as edge states or edge modes, as depicted 1. From Landau Levels to the QHE in Fig. 2(a). (The unperturbed energy eigenstates local- ized in the bulk are strongly suppressed near the edges if ` L , so the confining potential only slightly deforms In Section II C, a fully quantum mechanical treatment B x of the multi-particle QHE is given; however, our discus- these states.) sion of LLs already allows for an intuitive semi-classical If the potential can be approximated as linear in the derivation, following [4], in the case of integer filling frac- vicinity of the edges, so that tion ν N/NLL, interpreted as the number of filled LLs. ≡ ∂V One such derivation is found in Appendix B, where we V (x) (x x±) , (11) ∂x x=x± assume a large sample, so that edge effects may be ne- ≈ − 11 glected; it is seen that an applied electric field lifts the then one might classically anticipate that an electron lo- degeneracy in each LL, and one recovers the correct ex- calized near the edges should have velocity pression for the conductivities. In reality, one cannot ne- glect edge effects; in fact, the physics of the near-edge re- 1 ∂V v = yˆ , (12) gion is essential to an understanding of the QHE. We will −eB ∂x x=x± therefore introduce an approximately U-shaped confining potential V (x), as depicted in Fig. 2(a), with a steep in- since this satisfies the Lorentz force law; for a single cline near the edges of the sample in the x-direction; one electron, this should also hold quantum mechanically by has Ehrenfest’s theorem. That is, the edge states result in an effectively 1-dimensional current flowing at the edge of the material, as though a wire were placed along the eVH = V (x+) V (x−) , (10) 10 − periphery of the sample; since the current flows in one where x± are the positions of the edges, and VH is the direction, the edge states are said to be chiral. applied Hall voltage. In the previous section, the degen- If precisely an integer number ν N of LLs is filled eracy in a single Landau level was indexed by momentum with non-interacting electrons, then∈ the induced current Physics 305 Notes Physics 3053 Notes (a) Chris Waddell E E (a) Chris Waddell is seen to be E E November 23, 2018 5~ÊB/2 Z Z November 23, 2018 dky ν dx ∂V eνVH 5~ÊB/2 I νe vy(ky) = yˆ = yˆ , 2 3~ÊB/2 ≈ − 2π 2πB `B ∂x Φ0 Disorder (13) 3~ÊB/2 Disorder from which we may immediately deduce the desired con- ~ÊB/2 ductivity in equation (2). We note that, as seen from the ~ÊB/2 above computation, the conductivity is actually insensi- DOS DOS tive to the precise form of the potential in the interior, DOS DOS provided that the potential is sufficiently weak in the interior that the only states near the Fermi energy are (b) Localized E ‡xy edge states (see Fig. 2(b)). This illustrates the crucial 3e/ 0 role played by the edge states in charge transport; more- 7~ÊB/2 over, this invariance to small variations in the potential Extended will be of particular interest in the next section, where 5~ÊB/2 2e/ 0 we consider the role of disorder. EF
3~ÊB/2
‹ DOS 2 3 B. Hall Plateaux and the Role of Disorder FIG. 3. (a) Broadening of Landau levels due to microscopic So far, our analysis has only been applicable to the case disorder. (b) States at the edges of the Landau bands are of integer filling fractions, and is thus unable to account localized, whereas states in the centre are extended; if the for the plateaux in the values of the Hall conductivity Fermi energy lies between bands, then the conductivity lies that are observed as the magnetic field is varied. The on a plateaux. ingredient missing from the above discussion is the vital role played by the impurities present in a realistic system. 7 1 7
Disorder in a system can be modelled by adding a to represent the correlation length7 of such a state, giving1 7 a characteristic length scale4 for the distribution of the small, random electronic potential to the Hamiltonian; in 3 a quantum Hall system, provided that this random po- wavefunction. In the limite4 ξ , the state becomes 3 → ∞ tential is much smaller than the energy associated with an extended state. 2 the electronic cyclotron motion, this can be treated in In the quantum Hall system, there is another way to analyze the localization effect.2 Recall that a classical perturbation theory. The disorder potential then leads to 6 6 splitting in the highly degenerate Landau levels, so that electron in a magnetic field undergoes cyclotron motion 6 the density of states, previously δ-function localized to 6 y˙(t) the LL energies, becomes broader and turns into bands, x(t) = x0 R sin(ωBt + φ) =7 x0 + − 8 ωB as shown in Fig. 3(a). Between these bands reside only 5 (16) 11 x˙(t) the edge states, whose energies deviate significantly from 5 y(t) = y + R cos(ω t + φ) = y5 9; 0 B 0 7 the LL energies. − ωB The way in which disorder induces Hall plateaux, how- ever, is through the mechanism of localization, wherein one may therefore argue that the quantized versions of the ‘centre position’ variables4 x0, y0 are given by 6 quantum states are restricted to small regions of the to- 3 10 tal system, rather than diffusing throughout the system. 1 d 1 d The genericity of localization in the presence of disorder xˆ0 =x ˆ yˆ , yˆ0 =y ˆ + xˆ ,6 (17) 912 − ω dt ω dt 8 is a well-studied phenomenon (see e.g. [5–9]), including B B in the case of the QHE [10]; most notably, in the cele- whence one finds from the Schr¨odingerequation, and the brated tight-binding model canonical commutation relations forx, ˆ yˆ and their deriva- 10 X X tives, that H = t c† c + U c† c (14) − m n n n n hm,ni n ˆ 2 ∂V (ˆx, yˆ) ˆ 2 ∂V (ˆx, yˆ) i~x˙ 0 = i`B , i~y˙0 = i`B . (18) ∂y0 − ∂x0 of Anderson [5], with constant hopping t and on-site Thus, in expectation value, a single electron moves along terms Un which are i.i.d. random variables in some fixed equipotential contours. Generically, the equipotential range Un [ W, W ], the energy eigenstates are found to be localized∈ states− of the form contours of a random potential will be closed orbits; con- sequently, an electron restricted to move along such a ψ(r) 2 e−|r−r0|/ξ (15) contour will be bound to a small region, whose size is | | ∼ comparable to the distance between impurities. In par- in the thermodynamic limit n , provided that W is ticular, the states that are most localized are those re- → ∞ larger than some critical value Wc. Here, ξ is understood stricted to equipotentials contours near the peaks and 4 troughs of the disorder potential, which are also at the that the latter condition should actually refer to a mo- upper and lower edges of the Landau band (see Fig. bility gap. In Appendix E, the Kubo formula 3(b)). X m Jy n n Jx m m Jx n n Jy m The relevance of the localization phenomenon comes σxy = i~ h | | ih | | i − h |2 | ih | | i . (En Em) from the fact that, to a first approximation, a localized n=6 m − state does not transport charge, and therefore does not (21) contribute to the current. If the Fermi level lies within for the conductivity in the energy eigenstate m of a the region of extended states, then the conducting elec- quantum mechanical system was derived, where| ni are trons can move freely, as in a metal. However, if we eigenstates of the Hamiltonian with energy eigenvalues| i increase the electron density in the system or reduce the En. This formula must be somewhat modified to de- magnetic field, so that the Fermi level lies within the scribe the conductivity in the many-body ground state; region of localized states, we do not gain any current- using the current defined in equation (20), the appropri- carrying states (and the material is said to be an Ander- ate reformulation for our purposes is son insulator); therefore, the conductivity remains the 2 Z 2 2 0 ie X d k d k α β same, a fact which is responsible for the plateaux ob- σxy = (uk , uk0 ) , (22) ~ (2π)2 (2π)2 S served over a broad range of filling fractions ν (see Fig. Eα
Properties of Semiconductor Surface Inversion Layers in the Electric Quantum Limit*
FRANK STERN AND W. E. HowARD I&M S'watson Research Center, Yorktomn Heights, Rem Fork (Received 10 July 196'I)
The strong surface electric Geld associated with a semiconductor inversion layer quantizes the motion normal to the surface. The bulk energy bands split into electric sub-bands near the surface, each of which is a two-dimensional continuum associated with one of the quantized levels. We treat the electric quantum limit, in which only the lowest electric sub-band is occupied. Within the effective-mass approximation, we have generalized the energy-level calculation to include arbitrary orientations of (1) the constant-energy ellipsoids in the bulk, (2) the surface or interface, and (3) an external magnetic Geld. The potential asso- ciated with a charged center located an arbitrary distance from the surface is calculated, taking into account screening by carriers in the inversion layer. The bound states in the inversion layer due to attractive Cou- lomb centers are calculated for a model potential which assumes the inversion layer to have zero thickness. The Born approximation is compared with a phase-shift calculation of the scattering cross section, and is found to be reasonably good for the range of carrier concentrations encountered in InAs surfaces. The low- temperature mobility associated with screened Coulomb scattering by known charges at the surface and in the semiconductor depletion layer is calculated for InAs and for Si (100) surfaces in the Born approxima- tion, using a potential that takes the inversion-layer charge distribution into account. The InAs results are in good agreement with experiment. In Si, but not in InAs, freeze-out of carriers into inversion-layer bound states is expected at low temperatures and low inversion-layer charge densities, and the predicted behavior is in qualitative agreement with experiment. An Appendix gives the phase-shift method for two- dimensional scattering and the exact cross section for scattering by an unscreened Coulomb potential.
I. INTRODUCTION in the s direction, with a continuum for motion in the plane parallel to the surface. N e-type inversion layer is produced at the surface The quantization of energy levels in inversion layers A of a p-type semiconductor when the energy bands has been anticipated for many years, ' but the two- near the surface are bent down enough that the bottom dimensional nature of the electron gas when only one of the conduction band lies near or below the Fermi electric sub-band is occupied was only recently con- level. This band bending can be introduced by applying Grmed by experiments on e-type inversion layers on a an electric Geld to the surface, in a conGguration like (100) surface of silicon in the presence of a magnetic that shown in Fig. 1, or by the presence of positive Geld perpendicular to the surface. ' charges at or near the surface associated with impurity The principal of this is to ions or other Coulomb centers. purpose paper study the e8ect of charged centers near the surface on the The electric field associated with an inversion layer properties of electrons in the inversion layer. To do is strong enough to produce a potential well whose this we Grst Gnd the average potential due to such a width in the s direction, the direction perpendicular charge seen by the inversion layer electrons, including to the surface, is small compared to the wavelengths for the first time the effect of screening in this quasi- of the carriers. Thus the energy levels of the electrons two-dimensional system. From this potential we Gnd are grouped in what we call electric sub-bands, each 5 both the scattering cross section of the electrons and of which corresponds to a quantized level for motion the inversion-layer bound states that result when the so we see instantly that potential is attractive. METALLIC GATE ELECTRODE The calculated scattering rate is compared with e2 X Z dkxdky σ = (α) . (29) GATE " experimental results on inversion-layer mobility at xy (2π)2 Fxy VOLTAGE INSULATOR ~ α low temperatures in InAs' and in Si.4 These experi- ments measure the conductance between the e-type By an extension of the Gauss-Bonnet theorem, the in- n- TYPE n-TYPE INVERSION contacts of Fig. 1, and determine the inversion-layer tegral of the Berry curvature xy(α) over the Brillouin DIFFUSED LAYER F CONTACT electron mobility by using the Hall e6ect4 or magneto- zone is 2πCα, with Cα an exact integer, referred to as ' the Chern number. We thus recover Hall quantization resistance. Thus an experimental curve of inversion- p-TYPE SEMICONDUCTOR 2 2 J. R. Schrieffer, in Semiconductor Surface Physics, edited by e X e FIt. 1. Schematic metal-insulator —semiconductor structure R. H. Kingston (University of Pennsylvania Press, Philadelphia, σxy = Cα Z . (30) used for inversion-layer experiments. The inversion-layer 2π~ ∈ 2π~ FIG. 4. Schematics of a metal-insulator-semiconductorelectron struc- Pennsylvania, 1957),p. 55. α concentrationture, with an n-typeis changed inversionby changing layer as 2DEG.the gate Thevoltage. traditionalThe 2 A. B. Fowler, F. F. Fang, W. E. Howard, and P. J. Stiles, inversion-layer conductance is measured a small MOSFET uses silicon as substrate and SiOby applyingas insulator [19]. Phys. Rev. Letters 16, 901 (1966); in Proceedings of the Inter- This argument reveals something deep about the impor- potential diGerence between the n-type contacts2 and measuring national Conference on the Physics of Semiconductors, kyoto, 1966, tance of topological order in determining the properties the resulting current. Q'. Phys. Soc. Japan Suppl. 21, 331 (1966)g. 8S. of a material. * Kawaji and Y. Kawaguchi, in Proceedings of the Inter- Some of the results of this work were presented at the Chicago national Conference on the Physics of Semicondlctors, Eyoto, 1W6, meeting of the American Physical Society, March, 1967; Bull. PJ. Phys. Soc. Japan SuppL 2l., 336 (1966)). Am.Fig.Phys. 5 showsSoc. 12, the2/5 MOSFET(1967). band diagram near the sur- 4 F. F. Fang and A. B. Fowler (to be published). III. EXPERIMENTAL REALIZATIONS face of the substrate. The z-axis is chosen to be perpen-168 816 dicular to the device, starting at the substrate-insulator interface (z = 0 A)˚ and going deeper into the substrate Experimental probing of the QHE was instigated in for increasing z-values. Band-bending is observed at the [15], where the Hall conductivity was measured for the junction; the lower edge of the conduction band E is at 2-dimensional electron gas (2DEG) formed in a MOS in- c a minimum near the substrate-insulator interface, with version layer (c.f. below). Similar techniques were subse- precise shape determined by the gate voltage. The curve quently employed in the seminal work of von Klitzing [1], g(z) represents the charge carrier density for electrons in where the QHE plateaux of Fig. 1 were first observed; the lowest energy subband E , which becomes localized this work was honoured with the Nobel Prize in physics 0 at the interface, leading to the 2DEG [19]. The energies in 1985. Even though some aspects of the QHE, such E and E of higher subbands are also shown. as the vanishing of the longitudinal resistance, had been 1 2 163 knownSEMICONDUCTOR prior to the work of von KlitzingSURFACE [15, 16], the INVERSION LAYERS 817 quantized Hall resistance had not been theoretically an- ticipated. As discussed above, this discovery heraldedcon- the layer mobility pervasiveversus roleinversion-layer of topology in condensedfree-carrier matter physics, RBITRARY UNITS centration canwhichbe constructed. remains and active area of research (e.g. in studies (4- FOLD) . 20— Because the ofinversion the fractionallayer Hall effectis thin and— theof quantumthe order spinof Hall ef- Ei (2-FOLD) 100 4 or less—fect).the Inmobility the following,is sensitive we provide someto scattering experimental de- tails regarding the von Klitzing experiment; more recent associated with the surface, and thus provides a power- experimental investigations involving graphene systems — FERMI .LEVEL ful tool for studyinginclude [17,surface-scattering 18] (see Appendix F). mechanisms. If the Coulomb centers near the surface are positively 0 I 120A charged, they attractA. 2DEGelectrons, in the inversionand in layergeneral of a MOSFETlead E to bound states. The properties of these bound states LLI I 00) SURFACE depend strongly Theon mainthe experimentalamount of challengescreening in observingby in- the LLI = Iol'cm-' version-layer QHE is the manufacture of a 2DEG, whose transport -20 electrons, and are among the most = ~ properties in the presence of a magnetic field may then L Sx IOIOcm interesting results of our work. In particular, we 6nd 'cm be probed. This can be realized using a quantum well K=75Q that in Si inversionwith widthlayers comparablethe tobound the destates Broglie wavelengthin the of absence of screeningthe electron.are deep This splitsenough the energyto trap spectrumthe 6rst into dis- electrons whichtinctenter subbands;the inversion at sufficientlylayer low temperatures,at low tem- only the peratures. As lowestthe gate energyvoltage subbandis is occupied,increased andand the electronsmore are effectively confined to 2 dimensions. In [1], the n-type electrons are added,inversionsome layer ofwill a Metal-Oxide-Semiconductor-Field-enter the electric sub- FIG. 2. Surface potential well and surface charge distribution band and will Effect-Transistorcontribute to (MOSFET)screening, wasthus usedweakening (see schematic in for a FIG.representative 5. PotentialSi wellsurface. (Ec) nearThe the substrate-insulatorcurve labeled E, inter-gives the the attractive potential.Fig. 4). AAt metallica sufficiently gate electrodehigh is locatedinversion- above theenergyfaceof (zthe= 0bottomA).˚ Theof chargethe carrierconduction density bandg(z) ofnear the lowestthe semi- p-type silicon substrate, which is electrically insulated by layer electron concentration, the screening will reach conductor-insulatorsubband E0 hasinterface a maximumat near@=0. theAlso surface,shown resultingare the in theposition an SiO2 layer. This setup allows the charge carrier den-of the 2DEGFermi [19].level for 10" electrons/cm' in the inversion its full value, and the levels will so shallow layer, sity of theenergy inversion layer to be adjustedbe by varying theand the positions of the 6rst two excited states, as calculated by that the orbitsgateof voltage.inversion-layer electrons bound to Howard {Ref. 10). The zero of energy is taken to be the bottom adjacent Coulomb centers overlap. Vnder these condi- of the lowest electric sub-band. The upper curve is the charge distribution of carriers in the lowest electric sub-band, taken from tions the bound states merge with the bottom of the Ref. 10. Also shown is a vertical line at the average inversion-layer lowest electric sub-band and effectively disappear. In thickness as computed from the approximate relations given in InAs, on the other hand, the bound states merge with Eqs. {22) and {42). the bottom of the lowest electric sub-band even in the absence of screening, and effects associated with bound dinger equation for motion perpendicular to the surface states are not expected. for an arbitrary surface orientation and for arbitrary Except for the calculation of Landau levels in bulk. ellipsoidal constant-energy surfaces. Appendix A, all of our results apply to zero magnetic The electron potential well near the semiconductor- field. In addition we restrict ourselves to low tempera- insulator interface is shown for a typical case in Fig. 2. tures, for which only the lowest electric sub-band is We wish to 6nd the energy levels E and envelope occupied by electrons. We call this limiting case the functions f belonging to self-consistent solutions of the electric quantum limit. effective-mass equation The organization of the remainder of the paper is summarized by the following Table of Contents: Section 2. Electric Sub-bands where T is the kinetic-energy operator and p is the 3. Coulomb Potentials and Screening electrostatic potential, which in turn is the solution of 4 Bound States Poisson's equation A. General Considerations B. Bound States for a Model Potential Py = —4n.p/~. 5. Impurity Scattering A. Born Approximation Here p is the charge density given by the fixed charge B. Validity of the Born Approximation in the depletion layer plus the charge in the inversion- Comparison with Experiment A. InAs layer states, and ~ is the dielectric constant. The B. Si boundary conditions on P are Discussion and Conclusions as 3'~ Appendix A. Landau Levels $~0 ~, (3a) Appendix B. Screened Coulomb Potential Appendix C. Two-Dimensional Scattering where ~,. and ff:;, are the static dielectric constants of 2. ELECTRIC SUB-BANDS the semiconductor and the insulator, respectively. The effective-mass equation (1) is perhaps a poorer In this section we consider the energy levels of approximation in the inversion-layer problem than in inversion-layer electrons moving in a potential well some other applications, since the width (in the s which depends only on s, the distance from the surface. direction) of the inversion-layer envelope function f We 6nd the effective masses for motion parallel to the is only of the order of 20 A in some cases, and therefore ggrf@ce g,nd the corresponding one-dimension@1 Scb.ro- not much larger than atomic dimensions, The boundary 6
1. The Hall Bar and Four-Terminal Measurement IV. APPLICATIONS
In order to accurately determine the longitudinal and transverse resistances, one needs to perform a precise The robustness of the Hall quantization to microscopic measurement of both the current and voltage in the Hall disorder and changes in the macroscopic properties of a bar system. Starting with the longitudinal resistance, one material makes the measurement of the Hall conductiv- could imagine applying the simple setup of Fig. 6(a). ity an appealing metrological standard. The relation be- The 2DEG is connected with two ohmic contacts to a tween the fine structure constant α and the von Klitzing voltage supply U and the current meter I. The equiva- constant RK is given by lent circuit in Fig. 6(b) shows the additional measured resistances of the current meter R , the powersupply iC 2π µ c R and the two contacts R , which would lead to ~ 0 iV contacts RK 2 = . (31) inaccurate measurement results. A much improved re- ≡ e 2α sult can be achieved by using the so called four-terminal measurement seen in Fig. 6(c). Here, the voltage supply In SI, the vacuum permeability µ0 and the speed of light is changed to a current source I, which drives a constant are fixed constants; thus, the fine structure constant can current through the 2DEG. A high impedance-voltmeter be inferred directly from a measurement of RK . The op- V is used to measure the longitudinal voltage drop of the erative impediments to this method are then limitations 2DEG. Its contacts (Fig. 6(d)) have ideally no voltage in how well the precision of a reference resistor can be 152 Diffusive classical transport in two-dimensionaldrop due to electron the high gases impedance measurement. maintained. At present, two electrical units need to be realized in terms of the metre and the kilogram in order (a) U (b) RiV U R to make electrical units measurable in the SI. Generally, I iC E I the ohm (Ω) and the watt (W ) are chosen, as these units can be most accurately determined experimentally [21]. Due to a theorem of Thompson and Lampard [22], the ca- j Rcontact R2DEG Rcontact pacitance of a particular class of capacitor can be related Fig. 10.8 (a) Two-terminal measure- (c) I (d) I R to a single length measurement. With this calculable ca- ment with a bar geometry. (b) Equiv- iC alent circuit with contact resistances pacitance, the resistance of a resistor can be determined E I Rcontact and the internal resistances through bridge techniques [23], with a precision of up to of the voltage source R (typically iV W 8 < 50 Ω)andoftheammeterR iC R R 1 part in 10 , about two orders of magnitude less precise (typically 10 Ω). (c) Four-terminal j Rcontact 2DEG contact measurement≤ with a bar geometry. than the reproducibility of the von Klitzing constant. Fi- L Rcontact (d) Equivalent circuit with contact re- Rcontact nally, a DC current comparator bridge may be used to sistances Rcontact,theinternalresis- U no tance of the voltmeter R (typically determine RK relative to the reference resistance. This iV current U > 10 MΩ)andtheinternalresistanceof R procedure constitutes the second most accurate method the current source (typically > 10 MΩ). iV for the determination of α; the most accurate involves comparing the experimentally observed anomalous mag- FIG.In general, 6. the(a) setup Two-terminal shown schematically, measurement which is called for a a two-terminal 2DEG. (b) measurement,willnotleadtothemeasurementofthetwo-dimensional netic moment ae of the electron to the famous QED pre- Equivalentelectron gas resistance, circuit showing because the the resistances additional of the electrical measured contacts, resis- diction, evaluated up to tenth order in [24]. tances,Rcontact,theinternalresistanceofthevoltagesource, which lead to inaccurate results. (c)R Four-terminaliV, and that of measurement.the ammeter, RiC (d), are Equivalent connected in circuit series. According which allows to the equivalentthe resis- Historically, the QHE has also been used to define the tancecircuit measurement shown in Fig. 10.8(b) of only we obtain the 2DEG for the [20]. measured resistance R = SI unit of resistance, from which all other electric units U/I = R2DEG +2Rcontact + RiV + RiC.Asignificantimprovementcan be achieved using the four-terminal arrangement depicted schematically must be derived. However, this definition is likely to be inFor Fig. the10.8(c). measurement Two narrow sideof the contacts transverse have been Hall attached resistance, to the abrogated during the revision of the International System onebar whichadds leave two the additional current distribution contacts and on the the electric transverse field essentially edges of Units, to take place in early 2019, wherein fixed values ofundisturbed, the 2DEG but (as allow seen the voltage in Fig. to be1), picked leading up along to the the electron typical will be assigned to seven physical constants, including gas. In the measurement setup the voltage source has been replaced by Hallacurrentsourcewhichdeliversawell-definedcurrent bar geometry. Using the same principle,I independent one achieves of the Planck constant and the elementary charge, thereby reasonablythe size of the good load resistance. values for The the resistances Hall voltage of the voltage drop contacts and the do making the von Klitzing constant a defined quantity (see Hallnot play resistance. a role, because no current will flow through the voltmeter due Appendix G for further discussion). to its very large internal resistance. As a consequence, the electric field in the two-dimensional electron gas is given by E = U/L,thecurrent density is j = I/W and therefore the specific resistivity| | takes the value | | U W ρxx = . [1] K. v. Klitzing, G. Dorda,I andL M. Pepper. New method [3] J. Weis and K. von Klitzing. Metrology and microscopic If afor magnetic high-accuracy field is applied determination normal to the of plane the of fine-structure the two-dimensional con- picture of the integer quantum Hall effect. Phil. Trans. R. electronstant gas, based the pattern on quantized of field lines Hall will resistance. be changed asPhys. depicted Rev. in Soc., 369:3954–3974, 2011. doi:10.1098/rsta.2011.0198. Fig. 10.9.Lett., Near 45(6):494–497, the current contacts 1980. which are equipotential lines of the [4] David Tong. The Quantum Hall Effect. http://www. electric field, the equipotentials are forced to run parallel to the edge of the[2] contact, R. E. Prange roughly as and long S. as M. the Girvin.distance fromThe the Quantum contact is lessHall than Ef- damtp.cam.ac.uk/user/tong/qhe.html, January 2016. the widthfect.W Graduateof the sample. texts The in contemporaryfield lines of the current physics. density, Springer- how- [5] P. W. Anderson. Absence of diffusion in certain random ever,Verlag, must be 1987. at theISBN Hall angle 9783540962861. relative to the direction of the electric lattices. Phys. Rev., 109(5):1492–1505, 1958. field. Far away from the current contacts (much further than W )the edges of the sample force the field lines of j into a direction parallel to 7
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Applied Physics Letters, 95(17):172105, org/utils/common/pdf/CGPM/CGPM24.pdf. October 2009. doi:10.1063/1.3254329. [19] Frank Stern and W. E. Howard. Properties of semi- conductor surface inversion layers in the electric quan- Appendix A: Classical Hall Effect tum limit. Phys. Rev., 163:816–835, Nov 1967. doi: 10.1103/PhysRev.163.816. URL https://link.aps. org/doi/10.1103/PhysRev.163.816. At the classical level, one can analyze the Hall effect [20] Thomas Ihn. Semiconductor nanostructures: by considering the average motion of a single electron quantum states and electronic transport. Ox- in a Hall bar geometry with orthogonal magnetic field ford Univ. Press, 2010. ISBN 9780199534425. B = Bzˆ and in-plane electric field E = Exˆ. The Lorentz doi:10.1093/acprof:oso/9780199534425.001.0001. force law, modified by a Drude model-type friction term, URL http://dx.doi.org/10.1093/acprof: gives oso/9780199534425.001.0001. [21] B. Jeckelmann and B. Jeanneret. The Quantum Hall d m me v = e E + v B v , (A1) Effect as an Electrical Resistance Standard. S´eminaire dth i − h i × − τ h i Poincar´e, 2:39–51, 2004. where v is an ensemble average for the electron velocity, [22] A.M. Thompson and D.G. Lampard. A New Theorem m is theh i electron mass, and τ is the mean free time. At in Electrostatics and its Application to Calculable Stan- e equilibrium, d v = 0, so the above yields dards of Capacitance. Nature (London), 177:888, 1956. dt h i [23] A Jeffery, R E Elmquist, J Q Shields, L H Lee, M E m 0 = eE eB v v Cage, S H Shields, and R F Dziuba. Determination of − x − h yi − τ h xi the von Klitzing constant and the fine-structure constant m (A2) 0 = eE + eB v v , through a comparison of the quantized Hall resistance − y h xi − τ h yi 8 that is, impurities and edge effects; see Fig. 7.) Hamilton’s equa- tions yield mex˙ = p+eA, and the sensible single-electron E m/τ eB v ˆ ˆ e x = h xi . (A3) current operator is I = ex˙ ; in second quantization, this Ey − eB m/τ vy − − h i becomes e X Given that the current is J = ne v , we have immedi- Iˆ ∇ † = ψn,ky i~ + eA ψn,ky cn,ky cn,ky , − h i −me h | − | i ately that n,ky (B4) 1 m/τ eB E = J ρJ . (A4) where we have obviously continued to neglect spins. In ne2 eB m/τ ≡ − Landau gauge, we must therefore evaluate expectation values We have therefore found longitudinal and Hall resistivi- ties ieBxˆ i~ ψn,ky ∂x ψn,ky , i~ ψn,ky ∂y + ψn,ky . m B mω − h | | i − h | ~ | i B (B5) ρxx = 2 , ρxy = = 2 , (A5) ne τ ne ne The first quantity vanishes, since the momentum expec- where ωB = eB/m denotes the typical cyclotron fre- tation value in an eigenstate of a QHO is zero. The latter quency for an electron in a uniform magnetic field. Note yields that the Hall resistivity varies linearly with the magnetic 2 meE meE field B in the classical case, rather than stepwise. ~ky eB ky` + = , (B6) − B eB2 B
2 meE given that the QHO states are centred at k ` + 2 . Appendix B: Landau Levels with Constant Electric − y B eB Field and the QHE
The analysis of a single electron in the presence of a magnetic field, discussed in Section II A, may be repeated with the addition of an in-plane electric field E = Exˆ, implemented through electrostatic potential energy term Φ(x, y) = Ex. The Hamiltonian is thus modified to − ˆ 1 2 2 H = pˆx + (ˆpy + eBxˆ) eExˆ . (B1) 2me −
The momentum py is still a good quantum number, so we can again consider product eigenstates as in equation (6), upon which the Hamiltonian acts as h i ˆ 1 2 2 Hψky = pˆx + (~ky + eBxˆ) eExˆ ψky 2me − 2 2 h pˆ m ω m E 2i x e B 2 e (B2) = + xˆ + ky`B 2 ψky 2me 2 − eB FIG. 7. Approximate bending of Landau levels in a 2DES. 2 In each diagram, the pink curve represents the Hall poten- h~kyE mE i + 2 ψky . tial. The upper and lower insets illustrate the differentiation B − 2B between compressible and incompressible regions. Diagrams This is now a QHO Hamiltonian with a trivial offset in (a) and (d) illustrate the case of a 2DES in thermal equilib- the energy; the energy eigenvalues are then rium, diagrams (b) and (e) the case of a small applied Hall voltage, and diagrams (c) and (f) the case of a large applied 2 Hall voltage. Figure taken from [3]. 2 1 ~E mE En,ky = ~ω n + + ky , (B3) B 2 B − 2B2 If we neglect electron-electron interactions, then the which breaks the degeneracy in each LL. many-body ground state is simply a Slater determinant We can use the above to derive the QHE for the case of the eigenstates ψ such that n 1, 2, . . . , ν . The of integer filling fraction ν, so that the Fermi energy E n,ky F expectation value for the current operator∈ { is then} lies between Landau levels. We assume that the sample is sufficiently large that edge effects can be neglected; the ν e X X meE applied Hall voltage then manifests as a uniform electric GS Iˆ GS = yˆ h | | i −me B field E = Exˆ throughout the sample, the scenario that n=1 ky (B7) we have just analyzed. (In reality, the electrostatic po- eν = ELxLyyˆ tential within the bulk will not be exactly linear, due to −Φ0 9
P k where we have used the result of equation (9) (neglecting Assuming a terminating series solution g(r) = k ckr , spin degeneracy). It follows that the expectation for the one finds recurrence relation current density is ˆx = 0, ˆy = eEν/Φ0, from which J J − 2 2 2me ~ωB we may deduce the desired conductivities (k+2) m ck+2 = (m+k+1) E ck. (C4) − ~2 2 − eν σxx = 0 , σxy = . (B8) Φ0 The series must terminate at both sufficiently large and small (possibly negative) k; if k0 is the first index such
In addition to eliding the role of edge states in the that ck0 = 0, then we must have k0 = 2 m. We then 6 − ± QHE, this derivation is also unsatisfactory in that it ne- have ck0 , ck0+2, . . . , ck0+2j non-zero for some j 0, and ≥ glects the two-fold degeneracy that should be associated ck0+2j+2 = 0; for this to be true, we must have with spin-1/2 particles. The reason that this is generally a reasonable approximation is that the electron spins will ~ωB E = (m + k + 2j + 1) also lead to a Zeeman splitting [20] 0 2 (C5) ~ωB 1 = (m m + 2j + 1) , E E gµ B. (B9) 2 ± → ± 2 B which are indeed the previously determined Landau level Since we are interested in the strong field and low tem- energies. In particular, the lowest Landau level (LLL) perature regime, the energy cost associated with flipping corresponds to taking the negative sign and j = 0; in a given spin is typically very large, so we may neglect the this case, c−m−2 is the only non-zero coefficient in our spin degeneracy. series solution, and we have eigenfunction
imφ m −r2/4`2 ψm,LLL e r e B . (C6) Appendix C: Landau Level Wavefunctions in ∼ Symmetric Gauge
Appendix D: QHE from Gauge Invariance Although it has been utilitarian to use Landau gauge when approaching the problem of a single particle in a magnetic field, changing to a rotationally symmetric Although it is intuitively clear from the discussion in gauge may also provide some insight; in particular, it Section II B that the existence of Hall plateaux can be ex- will be valuable for analyzing the rotationally symmet- plained through localization whilst the Fermi energy lies ric Corbino disk geometry of Appendix D. We therefore within a ‘mobility gap’ in the spectrum, it remains un- choose to work in symmetric gauge clear why the Hall resistivity at these plateaux takes the 2 observed values ρxy = 2π~/νe , since these values were yB xB 1 naively derived with the assumption that all states in the A = xˆ + yˆ = r B . (C1) − 2 2 −2 × filled Landau levels should be involved in charge trans- port. The apparent robustness of the quantum Hall state We may anticipate that angular momentum Lz = i~∂φ to the macroscopic characteristics of the system and the − should be a good quantum number; on states of the form effects of disorder has motivated a formulation of this imφ ψm(r, φ) e f(r), the Hamiltonian acts as effect on the general grounds of gauge invariance. We ∼ present such an argument, initially due to Laughlin [11] ˆ 1 2 ~m eB 2 and revised by Halperin [25], which integrates the pre- Hψm = pˆr + ( + rˆ) ψm 2me rˆ 2 vious discussion regarding the role of localized states in ~2 preserving exact Hall quantization, following the discus- = ∂r(r∂r)ψm (C2) sion in [4]. −2mer 2 2 2 Given the observed insensitivity of the QHE to the meω ~mωB ~ m + B rˆ2 + + ψ . global geometric features of the sample, we may choose 8 2 2m rˆ2 m e for the purposes of the following argument a ‘Corbino Anticipating a wavefunction damped at large radii for disk’ geometry, consisting of a two-dimensional conduc- the sake of normalizability, we may define without loss tive annulus encircling a solenoid with magnetic flux Φ, − 2 2 as depicted in Fig. 8. The disk is also subject to a uni- of generality f(r) = g(r)e r /4`B . The time-independent form magnetic field B. The vector potential throughout Schr¨odingerequation then yields the disk can be taken to be 1 r2 m2 g00(r) + 1 g0(r) g(r) ˆ Br Φ ˆ r − `2 − r2 A(r) = Aφ(r)φ = + φ . (D1) B (C3) 2 2πr (m + 1)meωB 2meE = g(r) . If impurities are sufficiently small to be neglected for the ~ − ~2 10
choosing a more convenient gauge in which to approach the quantum mechanical problem, wherein an explicit contribution of the flux to the energy spectrum is now absent. For an extended state, whose wavefunction ψ is non-vanishing throughout the disk, one sees that for such a gauge transformation to leave the wavefunction single-valued, one requires eΦ Z; that is, Φ must be 2π~ ∈ an integer multiple of the flux quantum Φ0. On the other hand, for a localized state, we can choose the position of the branch cut for the phase such that it coincides with a region where the wavefunction vanishes; thus, we have no such quantization criterion in this case. As a result, we see that the energies of localized states are insensitive to the flux Φ, whereas those of the extended states are FIG. 8. Corbino disk geometry with magnetic field B, sensitive to the non-integer part of Φ/Φ0. threaded by flux Φ. Figure taken from [4]. One might expect, therefore, that if the system be- gins in an energy eigenstate ψm, and Φ is increased from zero to Φ0, then the system will end in the state ψm, time being, the Hamiltonian is then given that the flux has changed by an integer multiple of Φ0. However, if the change occurs as an adiabatic 1 h 2 2i (quasistatic) time-dependence in the Hamiltonian, over H = p + eA + p + eA − Φ 2m r r φ φ a time period T ω 1, then the states will in fact un- B 1 h ~2 ~ eBr eΦ 2i dergo spectral flow; although the spectrum will return = ∂ r∂ + i ∂ + + 2m − r r r − r φ 2 2πr to its initial form, the indiviual states ψm will be shifted (D2) ψm ψm+1. We can argue heuristically that this should occur→ on the basis of the above observation that increas- If the flux Φ is taken to vanish, then we simply have the ing Φ results in outward radial transport. If we have typical QHE Hamiltonian; in this case, the LLL wave- ν Landau levels, the result is that charge νe is trans- functions derived in symmetric gauge (see Appendix C) ported outward through any concentric ring− within the are of the form disk in a time T ; this is interpreted as a radial current
2 2 Ir = e/T . The state nearest the outermost edge of the imφ m −r /4`B − ψm(r, φ) e r e , m Z . (D3) disk can only flow to the edge; we therefore transport ∼ ∈ one ‘edge mode’ to the outer edge of the disk for each In particular, these wavefunctions are peaked near filled Landau level. The introduction of the flux is also r √2m` . (In fact, our analysis has neglected the max ≈ B accompanied by an induced EMF in the disk as a result fact that the Corbino disk has inner and outer bound- of Faraday’s law, with φ = ∂tΦ Φ0/T . We have aries, so the Hamiltonian should actually have confin- thus recovered that anE EMF− in the≈ azimuthal − direction ing potentials at these edges; however, the wavefunctions results in a current in the radial direction, from which we which are peaked within the available radial range of the may infer the typical Hall resistivity disk are suppressed at these boundaries, so we may as- sume that the Corbino disk supports energy eigenstates φ 2π~ which are only slight deformations of the typical LLL ρrφ = E = 2 . (D6) Ir νe states.) If Φ is non-zero, one can verify that the LLL eigenstates are generalized to The key realization is that, since the flux term in the
2 2 Hamiltonian can always be gauged away for localized imφ m+eΦ/2π~ −r /4`B ψm e r e . (D4) states, these states do not undergo spectral flow; in par- ∼ ticular, they do not transport charge as the flux Φ is The naive interpretation is that an increase in the flux increased. Nonetheless, we have found that, regardless Φ should result in outward radial motion of the electron, of how many states in each Landau level actually par- or charge transport. ticipate in transport, we recover the same values for the We can attempt to ‘gauge away’ the non-zero flux term Hall plateaux. from the Hamiltonian, meaning that we can perform a gauge transformation of the form
ψ(r, φ) ψ˜(r, φ) = e−ieΦφ/2π~ψ(r, φ) (D5) Appendix E: Linear Response and the Kubo → Formula on our wavefunctions, whereupon acting with the Hamil- tonian HΦ on ψ˜ is tantamount to acting with the Hamil- One motivation for considering two-point functions tonian HΦ=0 on the state ψ. This simply amounts to in quantum mechanics is that these objects encode 11 the linear response behaviour of a quantum mechani- where explicitly, one has cal system. Consider some unperturbed, multi-particle, 1 Heisenberg-picture Hamiltonian H0, to which we add a A = Ee−iωt (E8) time-dependent source term iω X for our AC electric field. Applying the above generalized H (t) = φ (t) (t) , (E1) source i Oi analysis to the present situation, we find i Z t 1 X 0 0 −iωt0 where i(t) are Heisenberg picture observables; we will δ J = dt [J (t ),J (t)] E e , (E9) O h ii ω h j i i j assume that the sources φi are small, so that we can use ~ j −∞ perturbation theory. By definition, states in the interac- tion picture evolve according to where we have manually restored factors of ~. If we henceforth fix the state Ω to be an eigenstate m of | i | i i∂t ψ(t) I = Hsource ψ(t) I ; (E2) the unperturbed Hamiltonian, with H0 m = Em m , we | i | i have at this level in perturbation theory| i | i the solution to this first order ODE is the state 0 −iEmt/~ 0 iEmt/~ [Jj(t ),Ji(t)] = e Jj(t )Ji(t)e ψ(t) I = U(t, t0) ψ(t0) I , (E3) h i h i −iEmt/~ 0 iEmt/~ | i | i e Ji(t)Jj(t )e − h 0 0 i where we have implicitly defined the unitary operator = e−iH0t /~J (t0)J (t)eiH0t /~ h j i i −iH t0/ 0 iH t0/ iH0(t−t0) −iH(t−t0) 0 ~ 0 ~ U(t, t0) = e e e Ji(t)Jj(t )e − h 0 i Z t (E4) = Jj(0)Ji(t t ) 0 0 h − i = T exp i Hsource(t )dt , 0 − Ji(t t )Jj(0) t0 − h − i = [J (0),J (t t0)] ; h j i − i with T denoting time-ordering within the Taylor expan- (E10) sion of the exponential (see e.g. [26] for details). In particular, if the system is initialized in some state Ω in that is, the two-point function depends only on the time | i the distant past t , then the expectation value for difference s = t t0. We may therefore rewrite the above → −∞ − the operator i at later times (in the presence of sources as O φj) is Z ∞ 1 X −iω(t−s) δ Ji = ds [Jj(0),Ji(s)] Eje . i φ = Ω U( , t) i(t)U(t, ) Ω h i ~ω 0 h i hO i h | −∞ O −∞ | i j Z t 0 0 (E11) = Ω i(t) + i dt [Hsource(t ), i(t)] + ... Ω Specifically, we have h | O −∞ O | i Z t 0 0 X −iωt + i dt [H (t ), (t)] , δ Ji = σij(ω)Eje (E12) i φ=0 source i h i ≈ hO i −∞ h O i j (E5) implicitly defining the frequency-space response function where we have truncated to first order in the sources. We Z ∞ can express this result as 1 iωs σij(ω) = ds [Jj(0),Ji(s)] e . (E13) ~ω 0 h i δ hOii ≡ hOiiφ − hOiiφ=0 t Now, if we insert a complete set of eigenstates for the X Z (E6) = i dt0 [ (t0), (t)] φ (t0) , unperturbed Hamiltonian, then we find h Oj Oi i j j −∞ 1 Z ∞ X σ (ω) = ds eiωs m J (0) n n J (s) m which is referred to as the Kubo formula. This expres- ij ω h | j | ih | i | i ~ 0 n sion encapsulates the system’s response to turning on the various sources at the linearized level. m J (s) n n J (0) m − h | i | ih | j | i In the quantum Hall case, the field to be turned on is an 1 X Z ∞ electric field E(t) = Ee−iωt, the source is some current = ds eiωs ω density J, and the response function of interest is the ~ n=6 m 0 conductivity σij. Working in the Weyl gauge At = 0, the m J n n J m e−i(Em−En)s/~ source term is simply the coupling of the current to the h | j| ih | i| i gauge potential m J n n J m e−i(En−Em)s/~ − h | i| ih | j| i H = J A , (E7) (E14) source − · 12 where we have defined Jk Jk(0), and noted that the effect term with n = m cancels. If≡ we integrate along a slightly −1 gse 1 deformed contour, or equivalently if we shift the fre- Rxy = n + (F1) quency space pole slightly via ω ω + i, then we can Φ0 2 perform the convergent integrals → emerges, with gs a factor accounting for both spin and sub-lattice degeneracy. This phenomenon can be under- Z ∞ h eiωs−se−i(Em−En)s/~ i∞ dseiωs−se−i(Em−En)s/~ = stood from a simple tight-binding analysis of the hexago- 0 iω s i(Em En)/~ 0 nal graphene structure [28–31], which reveals a relativis- − − − i~ tic dispersion relation near the two inequivalent corners − . of the Brillouin zone, known as Dirac points; the struc- → ~ω + En Em − (E15) ture near the Dirac points is expected to dominate the dynamics in the continuum limit. The half-integer effect We would now like to take a limit ω 0, to find the Hall is then seen to be a result of the electron-hole degener- conductivity for a DC electric field.→ Given that acy in this model, in the limit that the effective Semenoff mass of the charge carriers vanishes.
1 1 ~ω 2 2 + (ω ) , ~ω + En Em ≈ En Em − (En Em) O Appendix G: Resistance Standard − − − (E16) we see that to lowest finite order in ω, As noted in Section IV, two electrical standard units are required in order to represent or measure all other X m Jj n n Ji m m Ji n n Jj m σij(0) = i~ h | | ih | | i − h |2 | ih | | i . electric units in SI. It is known that the resistance value of (En Em) n=6 m − a wired resistor drifts over time [32]. On the other hand, (E17) the Josephson effect and the quantum Hall effect together In particular, σxy σxy(0) is what we refer to as the provide an accurate and reproducible means of realiz- ≡ Hall conductivity in the state m . ing two such electrical units in terms of non-electrical SI | i units. It was therefore realized that the world-wide con- sistency of electrical measurements could be improved by Appendix F: QHE in Graphene assigning constant values to the von Klitzing constant RK and the Josephson constant KJ = 2e/h. In 1990, aggregating data from all relevant experiments prior to As an inherently 2-dimensional structure, graphene June 1988, RK was assigned the value also provides an experimentally realizable quantum Hall system. In [17], high-mobility graphene samples were ob- RK−90 = 25812.807 Ω . tained, using a mechanical exfoliation technique similar to that applied in [27]. Graphene layers were deposited This was essentially a weighted average of measurements on a degenerately doped Si substrate, with a 300 nm performed as in Section IV and values based on the cal- insulating SiO2 layer. The number of layers in each pre- culation of the fine structure constant in terms of the pared sample was inferred from both atomic force mi- anomalous magnetic moment of the electron [21]. To croscopy (AFM) profiles and chromatic shifts in optical ensure reproducibility, Technical Guidelines for Reliable microscopy images, allowing single-layer samples to be Measurements of the Quantized Hall Resistance [33] have isolated. To fabricate the Hall-bar type apparatus, elec- been established, ensuring that the used QHE device is tron beam lithography was used to pattern samples, fol- working in the correct realm. lowed by a Cr/Au thin-film deposition. While the assigned RK−90 remains valid at the time of While the operating principle behind experimental writing, it will most likely be abrogated at the revision of studies of the QHE in graphene is similar to the case of the International System of Units (SI) [34]. As mentioned semiconductor devices, the theoretical description must in Section IV, this revision will result in redundancy in be modified; in particular, a half-integer quantum Hall the definition of the von Klitzing constant.