Lecture 7: the Integral Quantum Hall Effect References Introduction

Phys 769: Selected Topics in Condensed Matter Physics Summer 2010

Lecture 7: The integral quantum Hall eﬀect

Lecturer: Anthony J. Leggett TA: Bill Coish

References

R. E. Prange, S. M. Girvin, The Quantum Hall Eﬀect (2nd edition), Springer 1990 D. Yoshioka, The Quantum Hall Eﬀect, Springer 2002 J. Jain, Composite Fermions, Cambridge University Press 2007

Introduction

In a magnetic field, a system of charged particles (e.g. electrons) essentially has its dimen- sionality reduced by one, since the paths are bent into circles. At the classical level we see the effects of this even in 3D systems, e.g. in cyclotron resonance in metals. At the quantum level, the effects are more spectacular because the closed orbits now become quantized. However, in 3D metals the effects of this are somewhat blurred by the third dimension, and the only reason we can see anything interesting is that (e.g.) “extremal” regions of the Fermi surface tend to contribute an anomalously large amount; this is what leads to the well-known dHvA, Shubnikov-de Haas and other effects in bulk 3D metals (cf. lecture 4). The effect of quantization of closed orbits comes out much more strikingly for systems which start off genuinely “2D”; however, even given this condition the requirements to see something interesting are quite stringent. We need

(a) a scattering time which is much larger that the inverse of the cyclotron resonance

frequency ωc ≡ eB/m

(b) a mean free path much larger than the “magnetic length” (see below) deﬁned by 1/2 lM ∼ (~/eB)

−2 (d) a coverage (number of electrons per unit area) n such that n ∼ lM ∼ eB/~

1 inversion layer

Vg + + + + p-Si Al AlGaAs GaAs + + + + (p-type) depletion layer SiO2

Since (at least until recently) the maximum attained value of B was around 10 T, the maximum value of n which can be tolerated is of order 1012 cm−2.

The main types of system used for QHE experiments are

(a) Si MOSFET’s, and

(b) GaAs-AlGaAs heterostructures. (refs.: Prange et al. section I.8, II, VI, Yoshioka section I.1)

Si MOSFET’s: the width of the inversion layer is typically ∼25-50 A(width˚ of the depletion layer is much greater). The eﬀective mass of an electron at the bottom of the conduction band of Si is

0.2me (with six different “valleys” in bulk, often reduced to 2 near a surface) (note lattice has cubic symmetry) so the energy of the first “transverse” excited state is ∼100 K, and at the temperatures of interest the system may be regarded as 2D. Typical areal densities (controlled by the gate voltage) are in the range 1011-1012 cm−2, so that (taking the spin and “valley” degeneracy into account) the Fermi energy is in the range 4-40 K. This is very small compared to the width of the conduction band, so that the effective-mass approximation for the 2D motion should be very good. The mobility of electrons in the inversion layer seems to be limited mostly by scattering by the ionized acceptors in the depletion layer, and is usually not more than ∼ 40000 cm2/V sec. The main advantage of the Si MOSFET system is that the areal density may be easily controlled by changing the gate voltage ( ≈ 12 for Si, m∗/m ∼ 0.2).

2 GaAs-GaAlAs heterostructures: width of inversion layer ∼ 100 A(again,˚ width of “depletion” layer is much greater). Ef- fective mass of electron in GaAs conduction band ∼ 0.07me (with only one “valley”). Generally, parameters similar in order of magnitude to Si MOSFET case. But mobility can be considerably greater (since the donors are far separated from the inversion region), up to ∼ 3 × 107 cm2/V sec. ( ≈ 11).

4 8 −2 [Also: electrons on surface of liquid He (but n . 10 cm ⇒ TF . 2 mK)]

Conductances etc. in 2D:

Fundamental deﬁnitions: Consider a rectangular 2D block of material, length L, width W , then can deﬁne resistance R and y conductance Σ tensors by W x X X Vi ≡ RijIj,Ii ≡ ΣijVj (1) j j L

Hence Σ is matrix inverse of R. Note carefully that (e.g.) R11 means the ratio of V1 to I1 with I2 = 0, while Σ11 means the ratio of I1 to V1 with V2 = 0; these two quantities are not necessary inverses! In particular, note that since e.g. Rxx = Σyy/det Σ, it is perfectly possible to have Rxx and

Σxx simultaneously zero (provided that Σxy and Rxy are nonzero).

Resistivity ≡ resistance (tensor) × cross-sectional “area” (length in 2D) ⊥ to current flow / length over which voltage drops. In 3D dim ρ ∼ dim (RL), but in 2D resistance and resistivity have same dimension, and in fact R (resistance of a square) = ρ (as a tensor relation). For the “Hall resistance” things are even simpler: if magnetic field is out of page, RH is defined to be the voltage drop across y-dimension (W ) / current flowing in x-direction (with y-direction open-circuited so Iy = 0). Evidently in this case RH ≡ ρH, with no dependence on L or W . (By contrast, Rxx = (L/W )ρxx). Generally, in the QHE it is much more convenient to discuss “ances” rather than “ivities”.

We will see, below, that the quantum Hall states (integer or fractional) are characterized by the property σxx = 0, σxy = const. States of this kind have a remarkable property: under very wide conditions the Hall conductance ΣH ≡ I/V measured in any four-terminal setup with the topology shown on the ﬁgure is independent of the geometry and equal to σxy! R R R This follows simply from div j = 0 and V = − E · ds, since I = j⊥ds = σxy E · ds

3 Voltage leads

y I I V x

Current leads

−1 = σxyV ⇒ Σ⊥ ≡ I/V = σxy = RH . So the “Hall” resistance is actually a special case of a more general four-terminal resistance.

The quantum Hall eﬀect: phenomenology

The initial observations (von Klitzing et al., 1980) were that when the areal density of carriers ns is held ﬁxed and the magnetic ﬁeld B varied (or vice versa) there are a number of plateaux in the Hall resistance, corresponding to value h/ne2 where n is an integer. Over the length of each plateau the longitudinal resistance is zero within experimental error. Thus

ΣH ≡ Σxy ≡ 1/RH and Σxx = 0, and we get the graph plotted in terms of conductances, 1 as shown. The plateaux are centered on values of nsh/eB which correspond to integers. This is the integral quantum Hall eﬀect (IQHE).

Subsequently (Tsui et al., 1982) it was discovered that there exist also some rational fractional values of nsh/eB (≡ ν), around which Hall plateaux can be centered; the corresponding value of the Hall conductance is ν(e2/h). Almost without exception, the value of ν and which this “fractional quantum Hall effect” (FQHE) occurs are fractions with odd denomi- nators (originally 1/3, then 1/5, 2/5, 4/7 . . . ); the one definitively known counterexample2 (as of June 2010) is ν = 5/2, with some evidence also for ν = 7/2 and possibly ν = 19/8. The general behavior of the FQHE is similar to that of the IQHE, but it seems to be less robust against the effects of temperature and impurity scattering.

1 For historical reasons experimentalists conventionally plot RH versus B, so that the graphs, while qualitatively similar to the one shown, do not possess the periodicity shown along either axis. 2The phenomena which occur at ν = 1/2, though very interesting, are not examples of the FQHE.

4 3

1 2 3

As far as is currently known, the integral (or rational-fractional) value of ΣH in units of e2/h on the plateaux is exact, and in fact it now forms a basis for metrology (see Prange et al., ch. II). This is at ﬁrst sight very surprising, since not only variations in material properties such as the eﬀective mass but even deviations from perfection in the geometry 8 ought to be very large compared to 1 part in 10 ; nevertheless, the measured values of RH seem to be independent of all such variations to about this accuracy!

As we shall see in a moment, there is absolutely no mystery about the fact that at exactly 2 integral values of the “ﬁlling factor” ν ≡ nsh/eB the Hall resistance in units of h/e is exactly 1/ν; the mystery is why this result is maintained over a ﬁnite range of ν. Also, it appears that no “naive” theory can explain the appearances of plateaux at non-integral values of ν (the FQHE).

Classical considerations3

Let’s start by considering the simple problem of the motion of a free electron in an external electric ﬁeld E and magnetic ﬁeld B, in general not mutually parallel. Newton’s equation is d2r m = eE + ev × B (2) dt2 with the general solution

r(t) = v0t + r0(t) (3) where 2 v0 ≡ E × B/B , r0(t) = r0(cos ωct, sin ωct, 0), ωc ≡ eB/m (4) 3Yoshioka, section 2.1.2.

5 In words, the motion of the electron is a superposition of a steady drift in the direction perpendicular to both E and B, with velocity E/B, and a “cyclotron” (circular) motion of arbitrary amplitude around the magnetic ﬁeld.

It is clear that this solution can be generalized to a situation when the “field” E ≡ −∇V acting on the electron is varying in space (but the magnetic field is constant). In fact, let us E make the ansatz r(t) ≡ r1(t) + r2(t), where r1 and r2 are constrained to satisfy B dr (t) d2r (t) e dr 1 = E(r ) × B/B2, 2 = 2 × B (5) dt 1 dt2 m dt that is, the “guiding center” r1(t) moves along equipotentials in the plane normal to B, while r2(t) performs cyclotron motion around the guiding center. It is clear that this ansatz satisfies Newton’s equation up to terms of order of the gradients of E, so in the limit of sufficiently slow variation of the macroscopic potential it should be a good approximation.

Quantum mechanics of a single electron in a magnetic ﬁeld4

Consider a single electron moving freely in a plane (the xy-plane) under the influence of a magnetic field B normal to the plane. Classically, this is the E = 0 limit of the problem studied above, so the drift velocity is zero and the electron simply executes a periodic circular motion in the plane with the cyclotron frequency ωc ≡ eB/m (or in the more general case eB/m∗). If we invoke the correspondence principle, we would infer that at least in the large-amplitude (semiclassical) limit quantum-mechanical effects would give rise to energy levels spaced by h/τ ≡ ~ωc. Let’s now see in detail how this comes about: The free-electron Hamiltonian in the field Bzˆ has the simple form: 1 Hˆ = m(ˆv2 +v ˆ2) (6) 2 x y where v is the (kinematic) velocity operator: this is related to the canonical momentum operator pˆ by vˆ ≡ m−1(pˆ − eA(r)) (7) where the electromagnetic vector potential A(r) is related to the magnetic field B by B = curl A. The components of the velocity fail to commute for finite A: ie ie [ˆv , vˆ ] = ~ (∂ A − ∂ A ) ≡ ~ B (8) x y m2 x y y x m2 4Yoshioka, sections 2.2-3.

6 −1 Let us introduce, as well as the characteristic unit of time ωc , a characteristic length 1/2 lM ≡ (~/eB) whose signiﬁcance will shortly become clear, and introduce a dimensionless velocity V by 2 1/2 v ≡ (lM/τc)V ≡ (e~B/m ) V (9) Then the commutation relations of the components of Vˆ are

[Vˆx, Vˆy] = i (10) and the Hamiltonian has the form 1 Hˆ = ω (Vˆ 2 + Vˆ 2) (11) 2~ c x y The problem deﬁned by eqns. (10) and (11) is of course nothing but that of the simple harmonic oscillator, and we can immediately write down the energy levels: 1 E = (n + ) ω (12) n 2 ~ c which of course perfectly satisﬁes the correspondence principle.

However, since the original density of states (before the magnetic field was turned on) was proportional to the area of the sample, and since the application of the magnetic field cannot change the average DOS if taken over a sufficiently large energy range, it follows that the energy levels we have found must be massively degenerate. Quantitatively, the original DOS 2 (per spin state per valley) was m/2π~ per unit area of surface. The new DOS is simply 1/~ωc, so it follows that the degeneracy per unit area must be

2 N = ~ωc(m/2π~ ) ≡ eB/h (13)

In other words, for any given level n there is exactly one state per flux quantum h/e in the plane. The different values of n are said to correspond to different Landau levels: for our purposes the most important is the lowest Landau level (LLL) corresponding to n = 0.

We now define a quantity which is central to the QHE, namely the filling factor ν. For simplicity I first consider the case of a single “valley” and a single spin population. Then of ν is ν ≡ filling factor ≡ no. of electrons / flux quantum (14)

(this is actually more general)

7 Representations of Landau-level wave functions

For the simple case considered so far (electrons moving freely in plane in constant magnetic field) the massive degeneracy of the wave functions corresponding to a given level implies that many different representations are possible for the wave functions. However, if the degeneracy is split (e.g. by a spatially varying electrostatic potential) then much of this freedom is lost. Irrespective of this, and whatever the form of the potential, we always have some freedom in choosing the gauge of the magnetic vector potential (recall that given any vector potential A(r) which satisfies B = curl A, we can always make the substitution A → A + ∇χ, where χ is an arbitrary single-valued scalar function, without changing the magnetic field B). Generally speaking, for any given choice of scalar potential there exists a “natural” choice of gauge which makes the calculations of the eigenfunctions easiest (though it is easy to find only in a few simple cases such as those to be discussed).

Common to all representations is a particular characteristic length, which we have already met, namely the magnetic length 1/2 lM ≡ (~/eB) (15) Note that (a) this quantity is completely independent of all materials parameters (b) the area 2 corresponding to a single ﬂux quantum is 2πlM (c) the numerical value of lM corresponding to a ﬁeld of 1 T is approximately 250 A.˚

One possible representation is naturally associated with the so-called Landau gauge

Ax = −yB, Ay = Az = 0 (16)

The states corresponding to the n-th Landau level are given by

2 ψn(x, y) = exp(ikx) · φn(y − klM ) (17) where φn(x) is the n-th linear oscillator state. Since for a ﬁnite slab the wave function must satisfy periodic boundary conditions at the edges, k can take only the values 2πp/Lx where p is integral. Also, if the states are conﬁned in the y-direction with length Ly, k can (from 2 2 (17)) take values only between 0 and Ly/lM . This gives a total of LxLy/2πl states per Landau level which is right.

This representation is very convenient for considering the effect of a constant dc electric field E. Suppose the field is in the y-direction. Then the Hamiltonian is still invariant against translation in the x-direction, so we can take out a factor exp ikx as before and write

ψn(x, y) = exp(ikx) · f(y) (18)

8 where f(y) obeys the Schr¨odingerequation 2 ∂2 1 − ~ + ( k − eBy)2 − eEy f(y) = Ef(y) (19) 2m ∂y2 2m ~ (E is the total eigenvalue, not that associated only with the y-motion).

We deﬁne a “guiding center” coordinate y0 by 2 4 2 y0(k) = −klM + eEmlM/~ (20) and thus obtain ( ) 2 ∂2 m E 2 − ~ + mω2(y − y )2 f(y) = E + eEy − f(y) (21) 2m ∂y2 c 0 0 2 B

−y2/4l2 The solution is evidently a SHO eigenfunction Hn(y)e M with energy 1 m E 2 E(n, y ) = (n + ) ω − eEy + (22) 0 2 ~ c 0 2 B The interpretation of the three terms is that the first terms is the KE of cyclotron motion, the second is the PE of the guiding center in the electric field potential −eEy and the third is the KE associated with the classical drift velocity |v0| = E/B, which is in the x-direction: to see that the QM expectation value of v0 is the same as the classical value, write 1 1 v = hp + eyBi = ( k + eyB) (23) 0 m x m ~ 2 Using the relation (above) between y0 and k and the definition of lM, this becomes 2 2 v0 = ωchy − y0i + elME/~ = E/B (24) so j = (eE/B)/L (since the expectation value of hxi vanishes for any SHO eigenstate ψn(x)). Note that the above solution is exact5, independently of the relative magnitude of E and

B. Also note that the spacing of y0 and hence the DOS is independent of E.

It is intuitively plausible that the states we have constructed can be used to find an approximate solution to the problem of quantum motion in a magnetic field and an arbitrary strong, but slowly varying potential. In fact, it is fairly clear that provided that the electrostatic potential (or more generally any kind of external potential, which should play the same role) is slowly varying on the scale of lM, we can obtain a solution which looks locally like that above one, with E replaced by the local value of −∇V . The locus of the “guiding center” of such a state should be a contour of constant V , and the current carried by an electron in it should be given by e j = ev = − (zˆ × ∇V ) (25) B 5In the nonrelativistic limit. Clearly if we were to take the limit B → 0 at finite E, the drift velocity predicted by either the classical or quantum calculation → ∞ and eventually we have to worry about relativistic corrections. 9 Notice that this current is divergence-free, as must be the case for an energy eigenstate:

div j ∼ div (zˆ × ∇V ) ≡ 0 (26)

We should expect intuitively (and it can be conﬁrmed from a more detailed analysis) that the average area corresponding to a single state is the area of a single ﬂux quantum. Note however that depending on the detailed geometry of the potential, some states may be localized in the neighborhood of a potential minimum or maximum (circulating clockwise around a minimum and anticlockwise around a maximum), while others may be extended over the whole system. This feature is crucial for teh explanation of the nonzero plateau widths in the QHE, see below.

We need still to consider one more representation, the so-called circular or symmetric one. This is most naturally associated with the radial gauge, which in plane polar coordinates has 1 the form Ar = 0, Aθ = 2 rB. It is possible to write down the exact form of (a possible set of) degenerate eigenfunctions (cf. above): in fact, the n = 0 (LLL) have the form, when expressed in terms of z = x + iy,

l 2 2 ψ0,l(r, θ) ≡ ψ0,l(z) = const. z exp −|z| /4lM (27) However, it is actually more instructive to examine the approximate form of the wave functions for large l. In terms of polar coordinates r, θ we have ( ) 2 1 ∂ ∂ 1 ∂ ie r2 2 − ~ r + − B ψ(r, θ) = Eψ(r, θ) (28) 2m r ∂r ∂r r2 ∂θ 2 ~

It is clear that it is possible to choose ψ(r, θ) to be of the form exp ilθ Rl(r) where to preserve single-valuedness l must be integral: for a reason which will become clear we also choose it to be positive. Then the radial wave function has the form 2 1 d dR 1 − ~ r l + (l − r2/2l2 )2 R (r) = ER (r) (29) 2m r dr dr r2 M l l

It is intuitively clear that for large l the wave function Rl will be conﬁned to a region close to the value rl of r given by 2 1/2 rl ≡ (2l lM) (30) 2 Notice that a circular path with this radius encloses an area of 2πl lM, i.e. exactly l quanta 2 2 of ﬂux. For large l we can approximate the derivative term by d Rl/dr , and expand the

“potential” term in powers of r − rl up to second order: in this way we obtain 2 d2R 1 − ~ l + mω2(r − r )2 R = ER (31) 2m dr2 2 c l l l

10 This equation, when expressed in terms of the variable x ≡ r − rl, is independent of l and is exactly of the form of the TISE of a SHO with frequency ωc. Hence the solutions are the usual Hermite polynomials Hn(r − rl) with energies En = (n + 1/2)~ωc. For the p LLL, (n = 0), the extent of the wave function in the radial direction is ∼ ~/mωc ∼ lM. −1/2 1/2 Since the spacing between rings is ∼ l lM, each radial wave function overlaps ∼ l of its neighbors (orthogonality is automatically guaranteed, within a given LL, by the angular function). Note that the approximation is not valid for very small values of l.

With a view to a thought-experiment to be considered below, we need a slight generalization of the above argument. Consider a “Corbino-disk” geometry, that is, the above disk with a circular hole in its center through which we can apply an AB ﬂux Φ. The generalization of eqn. (28) to this geometry is

2 ( 2 2) ~ 1 ∂ ∂ 1 ∂ iΦ ie r − r + 2 − − B ψ(r, θ) = Eψ(r, θ) (32) 2m r ∂r ∂r r ∂θ Φ0 2 ~

The l-th state still encloses exactly l quanta of ﬂux; the only diﬀerence is that now we have

2 1/2 rl = (2(l − Φ/Φ0)lM) (33)

With this choice of rl, eqn. (31) is unchanged.

Topological considerations

Generally speaking, the occurrence of integer or rational-fraction quantum numbers in QM is a result either of some symmetry of the problem, or of topological considerations, or of both. However, as emphasized by Thouless,6 those numbers that are the consequence of topology are usually much more stable against small perturbations than the symmetry-derived kind.

6J. Math. Phys. 35, 5362 (1994).

11 To illustrate this point, consider the (meta) stability of a circulating-current state of 4He in an annulus. If the annulus were exactly cylindrical in shape (i.e. possessed exact invariance under rotation around its central axis), then angular momentum would be a good quantum number and independently of any He-He interactions we could attribute the metastability of the rotating current to its conservation. However, in real life there will certainly be small departures from cylindrical symmetry (both static and dynamic) and those will spoil the conservation of angular momentum. On the other hand, provided only that the amplitude of the order parameter is everywhere nonzero on some path around the ring, then we can deﬁne the “winding number” n ≡ H ∇ϕ·dl; this will be conserved irrespective of the detailed geometry of the ring, provided only that we can neglect exponentially rare ﬂuctuations of the Langer-Fisher type, and it is this feature, not the symmetry, which is generally believed to play the crucial role in stabilizing the circulating-current state.

+ In the case of the QHE (whether integral or fractional), the very high degree of robustness of the Hall resistance + against small changes (and in some cases even large ones) of materials parameters, geometry, etc., suggest very strongly that the origin of the aﬀect is topological, B and essentially all explanations in the literature rely on this feature, at least by implication. I start with what is probably the simplest approach, due to Laughlin and Halperin. For the moment I assume a single “valley” and spin index, so that the ﬁlling factor ν is just the number of electrons/Landau level.

Consider a simple Corbino-disk geometry, with current leads attached to the inner and outer edges, to which is applied a uniform field B plus an “Aharonov-Bohm” flux, which may vary in time. The total flux through the hole, or equally through any circular orbit within the disk, is thus an arbitrary function of time.

As above, we may choose a radial gauge and write the energy eigenfunctions of a given LL n for fixed flux in the form ψnl(r, ϕ) = exp ilϕ Hn(r − rl) where rl is the radius of an orbit enclosing exactly l quanta of flux. A point to notice is that the quantity rl will in general depend on the AB flux Φ. These eigenstates carry no current, in either the angular or the radial direction (in the angular case, the “canonical” angular momentum l~ is just canceled, when we average over the radial wave functions, by the “gauge” term in the expression for the current).

12 Now imagine that we slowly increase the AB ﬂux through the hole, thereby generating an emf V = −∂ΦAB(t)/∂t around the disk. Assume that the single-electron wave functions evolve adiabatically, i.e. so that they are solutions to the TISE for the “instantaneous” value of ΦAB; then it is clear that the guiding center of each state, rl, will move outwards. In fact, when we have increased the ﬂux by one unit h/e ≡ ϕ0, each state will have exactly replaced its outer neighbor (and one state will have been added at the inside edge and disappeared at the outer edge). Now, if (and only if!) each state contains an integral number n of electrons, then the net result will have been the transport of n electrons from the inner to −1 the outer edge of the disk. This gives a current ne/τ, where τ ≡ ϕ0(dΦAB/dt) ≡ ϕ0V , 2 and thus a Hall conductance ne/ϕ0 ≡ ne /h.

So far, so good, but all we have done is to recover the “naive” result that if there are exactly n electrons per flux quantum, i.e. exactly n LL’s are filled, then we get a conductance ne2/h. We still have to explain the existence of finite plateaux. The explanation, surprisingly, lies in the existence of disorder and hence of localized states. As usual in the theory of the QHE, we argue that since the experimentally observed effect is essentially independent of geometrical details we may choose any convenient geometry, and following Halperin we imagine that the disorder is confined to a section of the disk of intermediate radius, with two ideal “guard rings” inside and outside it (see Fig. 2). Then, as regards the guard rings, the states are just as previously, and in particular a change in the AB flux results in the motion of exactly one state across each of them. Consider now the situation in the disordered region. Here, according to our earlier arguments, we expect to find within a given LL both localized states (in which the electrons circulate around “hills’ or “valleys” of the potential) and “extended” states, which extend right around the disk; in the latter the behavior is qualitatively similar to that in the guard rings (and in particular there is no angular current in an energy eigenstate). A crucial consideration is that, barring some rather pathological cases7, the eigenstates at a given energy either wind right around the disk or enclose a finite number of isolated hills or valleys; the latter case corresponds to the edges of the band and the former to the middle. Thus, it follows that (excluding pathologies) the “band” corresponding to a given LL separates into three distinct regions: a region at the upper end where all orbits are localized and any electrons in them circulate (clockwise) around “peaks,” a middle region where all the states are extended around the

7For example involving “inland seas.”

13 whole disk, and a lower region where the states are all localized and any electrons in them circulate (anticlockwise) around “valleys” (troughs) in the local potential. It is intuitively clear that the (unique) energy of the extended states in the guard rings lies somewhere in the “extended” region of the disordered-region spectrum.

It is now obvious that the crucial question (within a single-electron picture) is: Where does the Fermi energy (chemical potential) lie? n=1 If it lies in the extended region, then the states of the guard rings will not be filled with an integral number of electrons, and the transported current will not in general correspond to an integral localized number of electrons transported per unit change of the AB flux. If on the other hand the chemical potential lies anywhere in the 8 localized region , then the extended states of a given LL will be extended n=0 either all full or all empty and our argument regarding the guard rings goes through: for each integral change of the AB flux, n electrons are transported across both the inner and the outer guard rings. A final, vital step in the argument is that since the localized states are not af- fected by the AB flux, their energies and thus their occupation factors cannot change; thus, for any given electron transported across the inner guard ring into the disordered region, one must leave this region and cross the outer guard ring. Thus the total current between the disks is exactly (ne2/h)V , where n is the number of LL’s whose extended regions are occupied. Note that the fraction of states in the disordered region (or in a more realistic model, in the whole system) affects only the length of the plateau, not the quantized conductance itself. Also note that the above argument gives no particular reason to believe that all the plateaux have the same length: the localized fraction could well itself be a function of n.

Edge states

While the Corbino-disk geometry provides (in my opinion) the simplest argument, it has the drawback that the Hall resistance cannot be easily measured in it. An alternative approach to the IQHE relies on the concept of edge states. For this we need to consider a diﬀerent geometry. In this case we may imagine that near the walls of the physical sample the potential rises drastically, so that viewed in cross-section the proﬁle V (y) is roughly as

8Or of course in the “gap” between LL’s, though this only happens for discrete (integral) values of the ﬁlling factor. 14 I y I

indicated. (There must also be some irregularities, not shown, in the potential in the bulk of the system in order to get the ﬁnite density of localized states, which is necessary to produce a ﬁnite width for the plateaux.)

Consider initially the case ∆V = 0, (so that V (y) is entirely due to electrostatic aﬀects within the sample itself). We consider e.g. the n = 0 LL and assume that F is well above the “bulk” energy (∼ ~ωc), so that all the extended states in the bulk are ﬁlled. As we approach m 2 the edge, the energy rises sharply (recall from above that En = ~ωc + V (yn) + 2 (E/B) ), so that in this region there are extended states (which circle the sample) at the Fermi level.

Let’s consider the current carried by a given state at the point x, y, for the moment ignoring 1 2 2 any spatial variation of the “drift” term 2 m(E /B ) in the energy. From the results above, the velocity of the state is given by v = E(r) × B/B2. Since the transverse extent of the wave function is just the magnetic length and independent of position on the contour, the probability density is constant and equal to L−1, when L is the length of the contour. Hence

(cf. above) the current In carried is (eE/BL) (along the contour).

Next we ask: what is the distance ∆y to the next state (in the next allowed volume of the guiding-center locus)? We know that the total area “occupied” by the state (that is, the inverse of the DOS/unit area) is just ϕ0/B, and this condition is satisﬁed if we assume that the distance between allowed orbits is independent of position on the orbit and equal to

ϕ0/BL. (If we were to assume that the inter-orbit distance varies along the orbit, we would get a value for the current In which is itself position-dependent, which cannot be the case for an energy eigenstate). Consequently, we get a relation between In and ∆y,

In = eE∆y/ϕ0 (34)

But eE∆y is simply the diﬀerence in energy between the levels En and En+1, so we ﬁnally get e I = (E − E ) (35) n h n+1 n

15 (This formula is actually valid for the localized states as well as the extended ones, but in the former case the resultant current does not contribute to the measured Hall eﬀect.)

It is possible to derive eqn. (2) by an alternative method, which makes it clear that the neglect of the drift term in the energy is not essential (Prange et al., p. 79): we imagine changing the “single-valuedness” boundary condition around the orbit so that ψ(x + L) = exp 2πiα ψ(x). (This can be achieved, physically, by imposing a ﬂux αϕ0 and redeﬁning ψ so as to get rid of the gauge term in the KE.) Then it is easy to show that e ∂Hˆ Iˆ = (36) n h ∂α Taking the expectation value and using the Feynman-Hellman theorem gives In = −(e/h)×

(∂En/∂α). When we vary α from 0 to 1, the eﬀect is to shift each allowed orbit to the neighboring value (cf. the earlier discussion of the Laughlin-Halperin argument). Thus, integrating from 0 to 1, we arrive again at eqn. (2).

It is clear that if we sum eqn. (2) over the orbits of a given LL which lie between y and y + ∆y we obtain for the total contribution in ∆I to the current e e ∆I = (µ(y + ∆y) − µ(y) ≡ ∆µ (37) h h 1 2 where µ(y) ≡ −eEy + 2 m(E/B) is the electrochemical potential. In the bulk of the system (though not necessarily close to the boundaries) this can usually be safely equated9 to the usual electrostatic potential, so we get ∆I = (e2/h)∆V for ﬁlled LL, or for the conductivity

2 σxy = ne /h (38) where n is the number of ﬁlled LL’s. This is of course just the IQHE.

A What happens near the walls? Let’s continue to assume that the electrochemical potential µ is constant around the edges (e.g. detach the voltage and current leads). Then, if µ is measured relative to the ﬂoor of the potential, it is easy to see that there will be a current of magnitude eµ/h running B around the edge of the sample. Since from (4) we can obtain this result separately for the top and bottom edges, it is clear that attaching current leads (but no voltage) will not change this result (since current conservation must be maintained).

9If one objects that this is not necessarily true to one part in 108, the answer is that the conductivity should strictly speaking be defined as the ratio of current to electrochemical potential rather than simply to voltage. 16 Now suppose we apply an electrochemical potential difference ∆µ (or, what is equivalent under most circumstances, a voltage ∆V ≡ e∆µ) between the voltage leads A and B. It is now clear that the quantity ∆µA,b (the electrochemical drop between lead A and the bulk) will be increased and ∆µB,b decreased. Consequently, more current will flow along the top edge (say to the left) and less along the bottom edge (to the right). The total current that flows in and out of the system through the current leads is, for each filled LL, e I = (∆µ − ∆µ ) = (e2/h)V (39) h A,b B,b again giving the IQHE. It therefore looks as if the Hall current flows entirely through the “edge” states, while the bulk of the sample remains inert.10

The above considerations give a convincing physical picture of the origin of IQHE, but one might be a little “current” worried that they rest on an independent-electron-type loop picture. Is it possible to obtain the eﬀect more generally from the properties of the many-body wave function, without making the independent-electron approx- “voltage” imation? The following elegant, if not 100% rigorous, loop argument is due to Thouless and co-workers11. Suppose that we take our Hall bar and join up both the sides and the ends by a loop of the same material (thus making as it were a fraction of a torus). We apply a time-dependent AB ﬂux

Φv(t) through the loop attached to the voltage leads, and measure the current IJ (t) around the loop connected to the current leads. The Hall conductance is the ratio IJ /(dΦv/dt).

Since the perturbation caused by Φv(t) is of the form δHˆ = −IˆvΦv(t), we have 1 Σ = Re I : I (ω) (40) H iω v J where A : B is the standard linear response function. Writing out the latter in terms of matrix elements, we get (ϕJ ≡ 2πΦJ /Φ0, etc.) P P ΣH(ϕj, ϕv) = i h0|Iv IJ − IJ Iv|0i (41) ~ 2 2 (E0 − Hˆ ) (E0 − Hˆ ) ∂Hˆ P ∂Hˆ ∂Hˆ P ∂Hˆ ≡ i h0| − |0iΦ−2 ~ 2 2 0 ∂ϕv (E0 − Hˆ ) ∂ϕJ ∂ϕJ (E0 − Hˆ ) ∂ϕv where |0i is the groundstate (with energy E0) and P projects this oﬀ (so that the energy denominator is never zero). Now, it is easy to demonstrate that the change of the GSWF

Ψ0 with (any) ϕv is given by

∂Ψ0 P ∂Hˆ = |Ψ0 > (42) ∂ϕv E − Hˆ ∂ϕv 10This is actually a rather delicate point, which appears not to be deﬁnitively resolved at present, either theoretically or experimentally. 17 11see e.g. J. Math. Phys. 35, 5362 (1994). and so the above expression for ΣH can be rewritten Z −2 N ∗ ∗ ΣH(ϕJ , ϕv) = i~ Φ0 d r (∂Ψ0 /∂ϕv ∂Ψ0 /∂ϕJ − ∂Ψ0 /∂ϕJ ∂Φ0/ ∂ϕv) (43)

Let us now integrate this relation over both ϕJ and ϕv from 0 to 2π. The average conductance ΣHis then given by 2 Z 2π Z 2π Z e i N ∗ ∗ ΣH = · dϕJ dϕv d r(∂Ψ0 /∂ϕv · ∂Ψ0/ ∂ϕj − ∂Ψ0 /∂ϕj ∂Ψ0/ ∂ϕv) h 2π 0 0 (44) An essential assumption, now, is that the groundstate returns to its original value (of course modulo 2nπ in phase) when ϕJ → ϕJ + 2π and ϕv → ϕv + 2π. Now with that constraint there is a general theorem that the quantity in is 2π times an integer n which defines the “first Chern class” of the mapping ϕJ , ϕv → Ψ0, so we finally obtain

2 ΣH = ne /h (45)

(Thouless argues, in eﬀect, that ΣH can be identiﬁed with ΣH, but in any case the above result guarantees the IQHE in an “average” sense.) The number n is essentially the number of times the phase of Ψ0 “wraps” around the torus.

Before leaving the IQHE, one point should be emphasized: While we have concentrated 2 on establishing that ΣH = ne /h, it is equally vital to the explanation of the experimental results that Σxx = 0 (if this were not so, it would be difficult to understand the independence of Σxy from the details of the geometry). In the “single-electron” picture given above, this effect is rather trivial since provided the extended and localized states occupy separate energy regions, it follows that whenever the Fermi energy lies in the “localized” region Σxx must be zero; since we have shown that this position of EF is also a necessary and sufficient condition for quantization of Σxy, the two phenomena must always go together, as they of course do experimentally. It is an interesting question whether one could prove this more generally, perhaps by a slight extension of the Thouless argument.

What kinds of affect would we expect to destroy the IQHE? The most obvious is temperature, since when kBT becomes an appreciable fraction of ~ωc we expect that electrons would be excited (e.g.) out of the LLL into higher Landau levels and the condition of perfect filling of the extended states would no longer be realized (in fact this happens at the weaker 12 condition kBT ∼ impurity bandwidth ). It turns out that the effect is also destroyed when we attempt to pass too large a current through the system: this is believed to be due to thermal “avalanching,” cf. Yoshioka section 3.4.2.

12In fact, once a nonzero number of extended states get partially populated, the longitudinal conductance −1 Rxx becomes nonzero and hence we can no longer set Rxy = Σxy .