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            Frequency Dependent Electrical Transport in the Integer Quantum Hall Effect

            arXiv:cond-mat/0303198v1 [cond-mat.mes-hall] 11 Mar 2003

            Frequency Dependent Electrical Transport in the Integer Quantum Hall E?ect
            Ludwig Schweitzer

            1

            Introduction

            It is well established to view the integer quantum Hall e?ect (QHE) as a sequence of quantum phase transitions associated with critical points that separate energy regions of localised states where the Hall-conductivity σxy is quantised in integer units of e2 /h (see, e.g., [1,2]). Simultaneously, the longitudinal conductivity σxx gets unmeasurable small in the limit of vanishing temperature and zero frequency. To check the inherent consequences of this theoretical picture, various experiments have been devised to investigate those properties that should occur near the critical energies En assigned to the critical points. For example, due to the divergence of the localisation length ξ(E) ∝ |E ? En |?? , the width ? of the longitudinal conductivity peaks emerging at the transitions is expected to exhibit power-law scaling with respect to temperature, system size, or an externally applied frequency. High frequency Hall-conductivity experiments, initially aimed at resolving the problem of the so-called low frequency breakdown of the QHE apparently observed at ? 1 MHz, were successfully carried out at microwave frequencies (? 33 MHz) [3]. The longitudinal ac conductivity was also studied to obtain some information about localisation and the formation of Hall plateaus in the frequency ranges 100 Hz to 20 kHz [4] and 50–600 MHz [5]. Later, frequency dependent transport has been investigated also in the Gigahertz frequency range below 15 GHz [6,7] and above 30 GHz [8,9,10]. Dynamical scaling has been studied in several experiments, some of which show indeed power law scaling of the σxx (ω) peak width as expected, ? ? ω κ [6,11,12], whereas others do not [7]. The exponent κ = (?z)?1 contains both the critical exponent ? of the localisation length and the dynamical exponent z which relates energy and length scales, E ? L?z . The value of ? = 2.35 ± 0.03 is well known from numerical calculations [13,14], and it also coincides with the outcome of a ?nite size scaling experiment [15]. However, it is presently only accepted as true that z = 2 for non-interacting particles, and z = 1 if Coulomb electron-electron interactions are present [16,17,18,19,20]. Therefore, a theoretical description of the ac conductivities would clearly contribute to a better understanding of dynamical scaling at quantum critical points.

            2

            Ludwig Schweitzer

            Legal metrology represents a second area where a better knowledge of frequency dependent transport is highly desirable because the ac quantum Hall e?ect is applied for the realization and dissemination of the impedance standard and the unit of capacitance, the farad. At the moment the achieved relative uncertainty at a frequency of 1 kHz is of the order 10?7 which still is at least one order of magnitude to large [21,22,23,24]. It is unclear whether the observed deviations from the quantised dc value are due to external in?uences like capacitive and inductive couplings caused by the leads and contacts. Alternatively, the measured frequency e?ects that make exact quantisation impossible could be already inherent in an ideal non-interacting two-dimensional electron gas in the presence of disorder and a perpendicular magnetic ?eld. Of course, applying a ?nite frequency ω will lead to a ?nite σxx (ω) and in turn will in?uence the quantisation of σxy (ω), but it remains to be investigated how large the deviation will be. Theoretical studies of the ac conductivity in quantum Hall systems started in 1985 [25] when it was shown within a semiclassical percolation theory that for ?nite frequencies the longitudinal conductivity is not zero, thus in?uencing the quantisation of the Hall-conductivity. The quantum mechanical problem of non-interacting electrons in a 2d disordered system in the presence of a strong perpendicular magnetic ?eld B was tackled by Apel [26] using a variational method. Applying an instanton approximation and con?ning to the high ?eld limit, i.e., restricting to the lowest Landau level (LLL), an analytical solution for the real part of the frequency dependent longitudinal conductivity could be presented, σxx (ω) ∝ ω 2 ln(1/ω 2 ), a result that should hold if the Fermi energy lies deep down in the lower tail of the LLL. Generalising the above result, both real and imaginary parts of the frequency dependent conductivities were obtained in a sequence of papers by Viehweger and Efetov [27,28,29]. The Kubo conductivities were determined by calculating the functional integrals in super-symmetric representation near non-trivial saddle points. Still, for the ?nal results the Fermi energy was restricted to lie within the energy range of localised states in the lowest tail of the lowest Landau band and, therefore, no proposition for the critical regions at half ?llings could be given. The longitudinal conductivity was found to be
            2 σxx (ω) = c ω 2 lnb (1/ω 2 ) ? ie2 ω2lB ?(EF ) ,

            (1)

            with the density of states ?(EF ), and two unspeci?ed constants b and c [27]. The real part of the Hall-conductivity in the same limit was proposed as
            2 σyx (ω) = e2 /h (? ω/Γ )2 8πlB ne , h

            (2)

            2 h where lB = ? /(eB) is the magnetic length, Γ 2 = λ/2πlB is the second moment of the white noise disorder potential distribution with disorder strength λ, and ne denotes the electron density. The deviation due to frequency of the Hall-conductivity from its quantised dc plateau value can be perceived from

            Frequency dependent QHE

            3

            an approximate expression proposed by Viehweger and Efetov [29] for the 2nd plateau, e.g., for ?lling factor ν ≈ 2, σyx (ω) = e2 /h 2? ω h 2 ? 2 1 ? ω 2 /ωc Γ
            2

            (2 ? ν) .

            (3)

            Again, Γ is a measure of the disorder strength describing the width of the disorder broadened Landau band, and ωc = eB/me is the cyclotron frequency with electron mass me . According to (3), due to frequency a deviation from the quantised value becomes apparent even for integer ?lling. Before reviewing the attempts which applied numerical methods to overcome the limitations that had to be conceded in connexion with the position of EF in the analytical work, and to check the permissiveness of the approximations made, it is appropriate here to mention a result for the hopping regime. Polyakov and Shklovskii [30] obtained for the dissipative part of the ac conductivity a relation which, in contrast to (1), is linear in frequency ω, ? σxx (ω) = K ?ξω . (4)

            This expression has recently been used to successfully describe experimental data [31]. Here, ξ is the localisation length, ? the dielectric constant, and the pre-factor is K = 1/6 in the limit hω ? kB T . Also, the frequency scaling of ? the peak width ? ? ω κ was proposed within the same hopping model [30]. Turning now to the numerical approaches which were started by Gammel and Brenig who considered the low frequency anomalies and the ?nite size p scaling of the real part of the conductivity peak σxx (ω, Ly ) at the critical point of the lowest Landau band [32]. For these purposes the authors utilised the random Landau model in the high ?eld limit (lowest Landau band only) [1,13] and generalised MacKinnon’s recursive Green function method [33] for the evaluation of the real part of the dynamical conductivity. In contrast to the conventional quadratic Drude-like behaviour the peak value decayed linp 0 early with frequency, σxx (ω) = σxx ? const. |ω|, which was attributed to the long time tails in the velocity correlations which were observed also in a semiclassical model [34,35]. The range of this unusual linear frequency dependence varied with the spatial correlation length of the disorder potentials. A second p result concerns the scaling of σxx (ω, Ly ) at low frequencies as a function of the 2 p system width Ly , σxx (ωLy ) ∝ (ωL2 )2?η/2 , where η = 2 ? D(2) = 0.36 ± 0.06 y [32] is related to the multi-fractal wave functions [36,37,38,39], and to the anomalous di?usion at the critical point with η ? 0.38 [40] and D(2) ? 1.62 [41]. The frequency scaling of the σxx (ω) peak width was considered numerically for the ?rst time in a paper by Avishai and Luck [42]. Using a continuum model with spatially correlated Gaussian disorder potentials placed on a square lattice, which then was diagonalised within the subspace of functions pertaining to the lowest Landau level, the real part of the dissipative conductivity was evaluated from the Kubo formula involving matrix elements

            4

            Ludwig Schweitzer

            of the velocity of the guiding centres [43]. This is always necessary in single band approximations because of the vanishing of the current matrix elements between states belonging to the same Landau level. As a result, a broadening of the conductivity peak was observed and from a ?nite size scaling analysis a dynamical exponent z = 1.19 ± 0.13 could be extracted using ν = 2.33 from [13]. This is rather startling because it is ?rmly believed that for noninteracting systems z equals the Euclidean dimension of space which gives z = d = 2 in the QHE case. A di?erent theoretical approach for the low frequency behaviour of the σxx (ω) peak has been pursued by Jug and Ziegler [44] who studied a Dirac fermion model with an inhomogeneous mass [45] applying a non-perturbative calculation. This model leads to a non-zero density of states and to a ?nite bandwidth of extended states near the centre of the Landau band [46]. Therefore, the ac conductivity as well as its peak width do neither show power-law behaviour nor do they vanish in the limit ω → 0. This latter feature of the model has been asserted to explain the linear frequency dependence and the ?nite intercept at ω = 0 observed experimentally for the width of the conductivity peak by Balaban et al. [7], but, up to now, there is no other experiment showing such a peculiar behaviour.

            2

            Preliminary considerations – basic relations

            In the usual experiments on two-dimensional systems a current Ix (ω) is driven through the sample of length Lx and width Ly . The voltage drop along the current direction, Ux (ω), and that across the sample, Uy (ω), are measured from which the Hall-resistance RH (ω) = Uy (ω)/Ix (ω) = ρxy (ω) and the longitudinal resistance Rx (ω) = Ux (ω)/Ix (ω) = ρxx (ω)Lx /Ly are obtained, where ρxy and ρxx denote the respective resistivities. To compare with the theoretically calculated conductivities one has to use the relations in the following, only in Corbino samples σxx (ω) can be experimentally detected directly. L The total current through a cross-section, Ix (ω) = 0 y jx (ω, r)dy, is determined by the local current density jx (ω, r) which constitutes the response to the applied electric ?eld jx (ω, r) =
            u∈{x,y}

            σxu (ω, r, r′ )Eu (ω, r ′ ) d2 r ′ .

            (5)

            The nonlocal conductivity tensor is particularly important in phase-coherent mesoscopic samples. Usually, for the investigation of the measured macroscopic conductivity tensor one is not interested in its spatial dependence. Therefore, one relies on a local approximation and considers the electric ?eld to be e?ectively constant. This leads to Ohm’s law j = σE from which the resistance components are simply given by inverting the conductivity tensor

            Frequency dependent QHE

            5

            σ

            ρxx ρxy ρyx ρyy

            =

            σxx σyy

            1 ? σxy σyx

            σyy σyx σxy σxx

            .

            (6)

            For an isotropic system we have σxx = σyy and σyx = ?σxy which in case of zero frequency gives the well known relations ρxx = σxx , 2 2 σxx + σxy ρxy = ?σxy . 2 2 σxx + σxy (7)

            From experiment one knows that whenever ρxy is quantised ρxx gets unmeasurable small which in turn means that σxx → 0 and ρxy = 1/σyx . Therefore, to make this happen one normally concludes that the corresponding electronic states have to be localised. In the presence of frequency this argument no longer holds because electrons in localised states do respond to an applied time dependent electric ?eld giving rise to an alternating current. Also, both real and imaginary parts have to be considered now
            R I σxx (ω) = σxx (ω) + i σxx (ω), R I σxy (ω) = σxy (ω) + i σxy (ω) .

            (8)

            Assuming an isotropic system, the respective tensor components of the ac resistivity ρuv (ω) = ρR (ω) + i ρI (ω) with u, v ∈ {x, y} can be written as uv uv ρR (ω) = uv ρI (ω) = uv
            R 2 2 I R I R I σvu (δσxx + δσxy ) + 2σvu (σxx σxx + σxy σxy ) 2 2 R I R I (δσxx + δσxy )2 + 4(σxx σxx + σxy σxy )2 I 2 2 R R I R I σvu (δσxx + δσxy ) + 2σuv (σxx σxx + σxy σxy ) 2 2 R I R I (δσxx + δσxy )2 + 4(σxx σxx + σxy σxy )2

            (9)

            (10)

            2 R I 2 R I with the abbreviations δσxx ≡ (σxx )2 ? (σxx )2 and δσxy ≡ (σxy )2 ? (σxy )2 .

            3

            Model and Transport theory

            We describe the dynamics of non-interacting particles moving within a twodimensional plane in the presence of a perpendicular magnetic ?eld and random electrostatic disorder potentials by a lattice model with Hamiltonian H=
            r

            wr |r r| ?
            <r=r >


            Vrr′ |r r ′ | .

            (11)

            The random disorder potentials associated with the lattice sites are denoted by wr with probability density distribution P (wr ) = 1/W within the interval [?W/2, W/2], where W is the disorder strength, and the |r are the lattice base vectors. The transfer terms
            r′

            Vrr′ = V exp ? i e2 /? h
            r

            A(l) dl ,

            (12)

            6

            Ludwig Schweitzer
            ???? ???? ????? ? ?
            ? ? ?? ?

            ?????? ????? ?????
            ? ? ??

            ?

            ?? ?? ??×? ? ? ? ? ??

            ?

            ?

            ? ??

            ??

            ? Fig. 1. Disorder averaged imaginary part of σxx (E, ω, ε) at energy E/V = ?3.35, frequency ? ω/V = 0.001, and ε/V = 0.0008 as a function of sample length Lx = h N a. The system width is Ly /a = 32 and the number of realisations amounts to 29 for Lx /a ≤ 5 · 106 and to 8 for larger lengths

            which connect only nearest neighbours on the lattice, contain the in?uence of the applied magnetic ?eld via the vector potential A(r) = (0, Bx, 0) in their phase factors. V and the lattice constant a de?ne the units of energy and length, respectively. The electrical transport is calculated within linear response theory using the Kubo formula which allows to determine the time dependent linear conductivity from the current matrix elements of the unperturbed system σuv (EF , T, ω) = πe2 ?


            dE
            ?∞

            f (E) ? f (E + hω) ? ω

            (13)

            × Tr vu δ(EF ? H)?v δ(EF + hω ? H) , ? v ? where f (E) = (exp[(E ? EF )/(kB T )] ? 1)?1 is the Fermi function. The area of the system is ? = Lx Ly , vu = i/? [H, u] signi?es the u-component of the ? h velocity operator, and δ(E + hω ? H) = i/(2π)(Gω,+ (E) ? Gω,? (E)), where ? Gω,± (E) = ((E + hω ± iε)I ? H)?1 is the resolvent with imaginary frequency ? iε and unit matrix I. For ?nite systems at temperature T = 0 K, ensuring the correct order of limits for size ? and imaginary frequency iε, one gets with γ = ω + 2iε/? h σuv (EF , ω) = lim lim e2 1 ε→0 ?→∞ h ?? ω h
            EF EF ?? ω h

            dE Tr (? γ)2 [uGω,+ vG? ] h

            ? (? ω)2 [uGω,+ vG+ ] + 2iε [uv(Gω,+ ? G? )] h
            EF ?? ω h

            ?
            ?∞

            dE Tr (? ω)2 [uGω,+ vG+ ? uGω,? vG? ] h

            (14)

            Frequency dependent QHE
            ???? ?

            7

            ? ?

            ???
            ?

            ??

            ?

            ????

            ? ? ???

            ?

            ??

            ?? ??×? ? ? ? ?

            ?

            ??
            ?

            ? Fig. 2. The variance of ? σxx (E, ω, ε) versus system length Lx calculated for the averages shown in Fig. 1 exhibiting an empirical power-law ? L0.5 x

            = lim lim +
            ?∞

            e2 1 ε→0 ?→∞ h ?? ω h
            EF ?? ω h

            EF

            σuv (E, ω, ε, Lx , Ly ) dE
            EF ?? ω h

            ? σuv (E, ω, ε, Lx , Ly ) dE .

            (15)

            One can show that the second integral with the limits (?∞, EF ?? ω) does h not contribute to the real part of σuv (EF , ω) because the kernel is identical zero, but we were not able to proof the same also for the imaginary part. Therefore, using the recursive Green function method explained in the next ? section, we numerically studied ? σuv (E, ω, ε, Lx , Ly ) and found it to become very small only after disorder averaging. As an example we show in Fig. 1 ? the dependence of ? σxx , averaged over up to 29 realisations, on the length of the system for a particular energy E/V = ?3.35 and width Ly /a = 32. ? Also the variance of σxx (E, ω, ε) gets smaller with system length following an empirical power law ? (Lx /a)?0.5 (see Fig. 2). In what follows we neglect the second integral for the calculation of the imaginary parts of the conductivities and assume that only the contribution of the ?rst one with limits (EF ? ? ω, EF ) matters. Of course, on has to be particularly careful h even if the kernel is very small because with increasing disorder strength the energy range that contributes to the integral (?nite density of states) tends to in?nity. Therefore, a rigorous proof for the vanishing of this integral kernel is highly desirable.

            4

            Recursive Green function method

            A very e?cient method for the numerical investigation of large disordered chains, strips and bars that are assembled by successively adding on slice at a time has been pioneered by MacKinnon [47]. This iterative technique relies on

            8

            Ludwig Schweitzer
            0.6 0.5 0.4 0.3 0.2 0.1 0

            ?? ?

            ? ? ?

            ?

            ?? ?? ?

            ??

            ?? ?
            ?

            ?? ?

            ?? ?

            Fig. 3. The real and imaginary parts of σxx (E, ω) as a function of energy and frequency ? ω/V = 2·10?4 ( ), 5·10?4 ( ), 1·10?3 ( ), 2·10?3 ( ). For comparison, h the cyclotron frequency is ? ωc ≈ 1.57 V for αB = 1/8 which was chosen for the h magnetic ?ux density. The disorder strength is W/V = 1 and the maximal system width amounts to Ly /a = 96 with periodic boundary conditions applied

            the property that the Hamiltonian H (N +1) of a lattice system containing N +1 slices, each a lattice constant a apart, can be decomposed into parts that describe the system containing N slices, H (N ) , the next slice added, HN +1,N +1 , ′ and a term that connects the last slice to the rest, HN = HN,N +1 + HN +1,N . Then the corresponding resolvent is formally equivalent to the Dyson equation G = G0 + G0 V G where the ‘unperturbed’ G0 represents the direct sum ′ of H (N ) and HN +1,N +1 , and V corresponds to the ‘interaction’ HN . The essential advantage of this method is the fact that, for a ?xed width Ly , the system size is increased in length adding slice by slice, whereas the size of the matrices to be dealt with numerically remains the same [47,48]. A number of physical quantities like localisation length [49,50,51,52], density of states [51,53], and some dc transport coe?cients [33,54,55] have been calculated by this technique over the years. Also, this method was implemented for the evaluation of the real part of the ac conductivity in 1d [56,57] and 2d systems [32,58,59]. Further e?orts to included also the real and imaginary parts of the Hall- and the imaginary part of the longitudinal conductivity in quantum Hall systems were also successfully accomplished [60,61,62]. The iteration equations of the resolvent matrix acting on the subspace of such slices with indices i, j ≤ N in the N -th iteration step can be written as Gi,j
            ω,±,(N +1)

            = Gi,j

            ω,±,(N )

            + Gi,N

            ω,±,(N )

            HN,N +1 GN +1,N +1 HN +1,N GN,j

            ω,±,(N +1)

            ω,±,(N )

            Frequency dependent QHE

            9

            0
            ? ? ? ?

            ?? ?

            ?0.1 0

            0.1

            0.2
            ?? ?

            0.3

            ? ? ?

            0.4
            ?

            0.5

            0.6

            Fig. 4. The imaginary part of σxx (E, ω) as a function of the real part. Data are taken from Fig. 3

            ? GN +1,N +1 = [(E + hω ± iε)I ? HN +1,N +1 ? HN +1,N GN,N Gi,N +1
            ω,±,(N +1) ω,±,(N +1)

            ω,±,(N +1)

            ω,±,(N )

            HN,N +1 ]?1

            = Gi,N

            ω,±,(N )

            HN,N +1 GN +1,N +1

            ω,±,(N +1) ω,±,(N )

            GN +1,j

            = GN +1,N +1 HN +1,N GN,j

            ω,±,(N +1)

            .

            (16)

            The calculation of the ac conductivities starts with the Kubo formula (15) by setting up a recursion equation for ?xed energy E, width Ly = M a, and imaginary frequency ε, which, e.g., for the longitudinal component reads σxx (E, ω, ε, N ) = e2 e2 S xx = Tr 2 N hM N a hM N a2
            N

            (? γ)2 xi Gω,+ xj G? h ij ji
            i,j

            ? (? ω)2 xi Gω,+ xj G+ + 2iεδij x2 (Gω,+ ? G? ) h i ij ji ij ji The iteration equation for adding a new slice is given by
            ω,+ ? ? xx xx 1 2 SN +1 = SN + Tr AN RN +1 + DN RN +1 DN RN +1 ω,+ ω,+ + + 4 3 + BN RN +1 ? DN RN +1 DN RN +1 ? CN RN +1 . ω,±,(N +1)

            . (17)

            (18)

            ω,± with RN +1 ≡ GN +1,N +1 , and a set of auxiliary quantities as de?ned in the appendix. The coupled iteration equations and the auxiliary quantities are evaluated numerically, the starting values are set to be zero. In addition, coordinate translations are required in each iteration step to keep the origin xN +1 = 0 which then guarantees the numerical stability.

            5

            Longitudinal conductivity σxx(E, ω)

            In this section we present our numerical results of the longitudinal conductivity as a function of frequency for various positions of the Fermi energy

            10

            Ludwig Schweitzer
            ??? ?

            ??? ???? ???? ???? ???
            ?

            ? ?? ? ? ? ?? ????

            ?? ? ? ? ??

            ?????

            ????

            ?

            ???

            Fig. 5. The real (?) and imaginary ( ) part of σxx (EF , ω) in units of e2 /h at EF /V = ?3.5 as a function of frequency. Further parameters are the disorder strength W/V = 1.0, system width Ly /a = 32 and ε/V = 0.0004

            within the lowest Landau band. The real and imaginary parts of σxx (EF , ω) were calculated for several frequencies, but, for the sake of legibility, only four of them are plotted in Fig. 3 versus energy. While the real part exhibits a positive single Gaussian-like peak with maximum ≈ 0.5 e2 /h at the critical energy, the imaginary part, which is negative almost everywhere, has a double structure and vanishes near the critical point. ? σxx (EF , ω) almost looks like the modulus of the derivative of the real part with respect to energy. Plotting ? σxx (EF , ω) as a function of ? σxx (EF , ω) (see Fig. 4) we obtain for frequencies ω < ω ? a single, approximately semi-circular curve that, up to a minus sign, closely resembles the experimental results of Hohls et al. [31]. However, for larger ω our data points deviate from a single curve. 5.1 Frequency dependence of real and imaginary parts

            The behaviour of the real and imaginary part of the longitudinal ac conductivity in the lower tail of the lowest Landau band (E/V = ?3.5) is shown in Fig. 5. We ?nd for the imaginary part a linear frequency dependence for small ω which is, apart from a minus sign, in accordance with (1) [27]. The real part can nicely be ?tted to ∝ ω 2 log2 [V /(? ω)]2 in conformance with (1) h and b = 2, but disagrees with the ?ndings in [26] where b = 1 was proposed. 5.2 Behaviour of the maximum of σxx (ω)

            The frequency dependence of the real part of the longitudinal conductivity peak value was already investigated in [32] where for long-range correlated disorder potentials a non-Drude-like decay was observed. We obtained a similar behaviour also for spatially uncorrelated disorder potentials in a lattice p p model [61]. In Fig. 6 the di?erence |? σxx (ω) ? σxx (0)| is plotted versus frequency in a double logarithmic plot from which a linear relation can be

            Frequency dependent QHE
            ? ??? ???? ???? ???? ? ??

            11

            ? ?? ?

            ??

            ? ?? ???

            ?

            ?

            ????

            ?

            ????

            ????

            Fig. 6. The non-Drude decrease of the peak value of the longitudinal conductivity p p as a function of frequency. A linear behaviour of |? σxx (ω) ? σxx (0)| is clearly p 2 observed with σxx (0)/(e /h) = 0.512 using the following parameters E/V = ?3.29, W/V = 0.1, Ly /a = 32, ε/V = 0.0004

            discerned. A linear increase with frequency was found for the imaginary part of the longitudinal conductivity at the critical point as well [61]. The standard explanation for the non-Drude behaviour in terms of long time tails in the velocity correlations, which were shown to exist in a QHE system [32,35], seems not to be adequate in our case. For the uncorrelated disorder potentials considered here, it is not clear whether the picture of electron motion along equipotential lines, a basic ingredient for the arguments in [32], is appropriate. 5.3 Scaling of the σxx (ω) peak width

            The scaling of the width of the conductivity peaks with frequency is shown in Fig. 7 where both the σxx (ω) peak width expressed in energy and, due to the knowledge of the density of states, in ?lling factor are shown to follow
            1

            ? ,?

            ? ??

            0.1 10? 10?? 10?? ? 10??

            Fig. 7. Frequency scaling of the σxx (ω) peak width. The width in energy ?E ( ) and the width in ?lling factor ?ν (?) show a power-law ? ? ω κ with an exponent κ = (?z)?1 = 0.21

            12
            1.2 1.1
            ? ?
            ?

            Ludwig Schweitzer
            0.01
            ??? ? ?
            ?

            ?

            1
            ??

            ??

            ?

            ? ?? ?

            ? ? ??

            ??

            0.9

            ?

            ?

            0.005



            Filling factor 0 0 0.005 Frequency
            ?

            ?

            0.8 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Filling factor

            0.01

            Fig. 8. Frequency dependent Hall-conductivity: The ?gure on the l.h.s shows σxy (ω) versus ?lling factor for two system sizes Ly /a = 64 (?) and Ly /a = 128 ( ). On the r.h.s., the relative deviation of the Hall-conductivity from the dc value, σxy (ω)/σxy (0) ? 1, is plotted as a function of frequency for ν = 0.88

            a power-law ? ? ω κ with κ = (?z)?1 = 0.21 [60]. Taking ? = 2.35 from [14] we get z = 2.026 close to what is expected for non-interacting electrons. Therefore, the result reported in [42] seems to be doubtful. However, spatial correlations in the disorder potentials as considered in [42] may in?uence the outcome. Alternatively, one could ?x z = 2 and obtain ? = 2.38 in close agreement with the results from numerical calculations of the scaling of the static conductivity [63] or the localisation length [14]. The experimentally observed values κ ? 0.42 [6] and κ = 0.5 ± 0.1 [12] are larger by a factor of about 2. This is usually attributed to the in?uence of electron-electron interactions (z = 1) which were neglected in the numerical investigations.

            6

            Frequency dependent Hall-conductivity

            The Hall-conductivity due to an external time dependent electric ?eld as a function of ?lling factor ν is shown in Fig. 8 for system widths Ly /a = 64 and Ly /a = 128, respectively. While σxy (ω) has already converged for EF lying in the upper tail of the lowest Landau band a pronounced shift can be seen in the lower tail of the next Landau band. For ν = 1.3 a system width of at least Ly /a = 192 was necessary for σxy (ω) to converge. This behaviour originates in the exponential increase of the localisation length with increasing Landau band index. Due to the applied frequency hω/V = 0.008 the σxy ? plateau is not ?at, but rather has a parabola shape near the minimum at ν = 1.0, similar to what has been observed in experiment [24]. An example of the deviation of σxy (ω) from its quantised dc value is shown on the right hand side of Fig. 8 where σxy (ω)/σxy (0) ? 1 is plotted versus frequency for ?lling factor ν = 0.88. A power-law curve ? ω 0.5 can be ?tted to the data points. Using this empirical relation, we ?nd a relative deviation of the order

            Frequency dependent QHE

            13

            of 5 · 10?6 when extrapolated down to to 1 kHz, the frequency usually applied in metrological experiments [21,22,23,24]. Therefore, there is no quantisation in the neighbourhood of integer ?lling even in an ideal 2d electron gas without contacts, external leads, and other experimental imperfections. Recent calculations, however, show that this deviation can be considerably reduced, even below 1 · 10?8 , if spatially correlated disorder potentials are considered in the model [64].

            7

            Conclusions

            The frequency dependent electrical transport in integer quantum Hall systems has been reviewed and the various theoretical developments have been presented. Starting from a linear response expression a method has been demonstrated which is well suited for the numerical evaluation of the real and imaginary parts of both the time dependent longitudinal and the Hallconductivity. In contrast to the analytical approaches, no further approximations or restrictions such as the position of the Fermi energy have to be considered. We discussed recent numerical results in some detail with particular emphasis placed on the frequency scaling of the peak width of the longitudinal conductivity emerging at the quantum critical points, and on the quantisation of the ac Hall-conductivity at the plateau. As expected, the latter was found to depend on the applied frequency. The extrapolation of our calculations down to low frequencies resulted in a relative deviation of 5 · 10?6 at ? 1 kHz when spatially uncorrelated disorder potentials are considered. Disorder potentials with spatial correlations, likely to exist in real samples, will probably reduce this pronounced frequency e?ect. Our result for the frequency dependence of the σxx peak width showed power-law scaling, ? ? ω κ , where κ = (?z)?1 = 0.21 as expected for noninteracting electrons. Therefore, electron-electron interactions have presumably to be taken into account to explain the experimentally observed κ ≈ 0.5 Also, the in?uence of the spatial correlation of the disorder potentials may in?uence the value of κ. The frequency dependences of the real and imaginary parts of the longitudinal conductivities, previously obtained analytically for Fermi energies lying deep down in the lowest tail of the lowest Landau band, have been con?rmed by our numerical investigation. However, the quadratic behaviour found at low frequencies for the real part of σxx (ω) has to be contrasted with the linear frequency dependence that has been proposed for hopping conduction. p Finally, a non-Drude decay of the σxx peak value with frequency, σxx (0)? p σxx (ω) ∝ ω, as reported earlier for correlated disorder potentials, has been observed also in the presence of uncorrelated disorder potentials. A convincing explanation for the latter behaviour is still missing.

            14

            Ludwig Schweitzer

            Appendix
            The iteration equations of the auxiliary quantities (required in (18)) can be ? written as [62] with VN ≡ HN,N +1 = HN +1,N
            ? ω,+ ? ? 1 2 AN +1 = VN +1 RN +1 AN + DN RN +1 DN RN +1 VN +1

            (19) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39)

            ? ω,+ ω,+ ? + 3 4 2 1 BN +1 = VN +1 RN +1 BN ? DN RN +1 DN + DN RN +1 DN RN +1 VN +1(20) ? ω,+ + + 4 3 CN +1 = VN +1 RN +1 CN + DN RN +1 DN RN +1 VN +1

            FN +1 = GN +1 = HN +1 =
            1 DN +1 2 DN +1 3 DN +1

            = = = = = = = =

            ? ω,+ ω,+ + 3 4 VN +1 RN +1 FN + DN RN +1 DN RN +1 VN +1 ? ? ? 11 10 ω,? VN +1 RN +1 GN + DN RN +1 DN RN +1 VN +1 ? ω,? ω,? 11 ? 10 VN +1 RN +1 HN + DN RN +1 DN RN +1 VN +1 ? ω,+ ? 1 VN +1 RN +1 DN RN +1 VN +1 ? ω,+ ? 2 VN +1 RN +1 DN RN +1 VN +1 ? ω,+ + 3 VN +1 RN +1 DN RN +1 VN +1 ω,+ ? + 4 VN +1 RN +1 DN RN +1 VN +1 ? ? ? 5 VN +1 RN +1 DN RN +1 VN +1 ? + + 6 VN +1 RN +1 DN RN +1 VN +1 ? ? 10 ω,? VN +1 RN +1 DN RN +1 VN +1 ? ω,? 11 ? VN +1 RN +1 DN RN +1 VN +1 ? ω,? 14 ω,? VN +1 RN +1 DN RN +1 VN +1 ? ω,+ 15 ω,+ VN +1 RN +1 DN RN +1 VN +1 ? ? 16 ? VN +1 RN +1 DN RN +1 VN +1

            4 DN +1 = 5 DN +1 6 DN +1

            10 DN +1 = 11 DN +1 14 DN +1

            15 DN +1 = 16 DN +1

            ? + + 1 1 h EN +1 = VN +1 RN +1 EN + (? ω)I RN +1 VN +1 ? ? ? 2 2 h EN +1 = VN +1 RN +1 EN + (? ω)I RN +1 VN +1 ω,+ ω,+ ? 3 3 h EN +1 = VN +1 RN +1 EN + (? ω)I RN +1 VN +1 ? ω,? ω,? 4 4 EN +1 = VN +1 RN +1 EN + (? ω)I RN +1 VN +1 , h

            where the auxiliary quantities are de?ned by
            N ? AN = VN i,j N ? BN = VN i,j

            G? i xi (? γ)2 Gω,+ ? 2iεδij I xj G? VN h ij jN N

            (40)

            h h Gω,+ xi (? γ)2 G? + 2iεδij I + (? ω)2 G+ xj Gω,+ VN (41) ij ij jN Ni

            Frequency dependent QHE
            N ? CN = VN i,j N ? FN = VN i,j N ? GN = VN i,j N ? HN = VN i,j N ? 1 DN = VN i N ? 2 DN = VN i N ? 3 DN = VN i N 4 DN

            15

            (? ω)2 G+ i xi Gω,+ xj G+ VN h ij N jN (? ω)2 Gω,+ xi G+ xj Gω,+ VN h ij Ni jN (? ω)2 G? i xi Gω,? xj G? VN h ij N jN (? ω)2 Gω,? xi G? xj Gω,? VN h ij Ni jN (? γ)G? i xi Gω,+ VN h N iN (? γ)Gω,+ xi G? VN h Ni iN (? ω)Gω,+ xi G+ VN h Ni iN (? ω)G+ i xi Gω,+ VN h N iN
            i N

            (42)

            (43)

            (44)

            (45)

            (46)

            (47)

            (48)

            =

            ? VN

            (49)

            ? 5 DN = VN i N ? 6 DN = VN i N ? 10 DN = VN i N ? 11 DN = VN i N ? 14 DN = VN i N ? 15 DN = VN i N ? 16 DN = VN i

            (? ω)G? i xi G? VN h N iN (? ω)G+ i xi G+ VN h N iN (? ω)G? i xi Gω,? VN h iN N (? ω)Gω,? xi G? VN h iN Ni (? ω)Gω,? xi Gω,? VN h Ni iN (? ω)Gω,+ xi Gω,+ VN h Ni iN (? ω)G? i xi G? VN h N iN

            (50)

            (51)

            (52)

            (53)

            (54)

            (55)

            (56)

            16

            Ludwig Schweitzer
            N ? 1 EN = VN i N ? 2 EN = VN i N ? 3 EN = VN i N ? 4 EN = VN i

            (? ω)G+ i G+ VN h N iN (? ω)G? i G? VN h N iN (? ω)Gω,+ Gω,+ VN h Ni iN (? ω)Gω,? Gω,? VN . h Ni iN

            (57)

            (58)

            (59)

            (60)

            For the translation of the coordinates xi → xi ? a one gets AN BN CN FN GN HN
            1/2′ DN 3/4′ DN 5′ DN 6′ DN 10/11′ DN 14′ DN 15′ DN 16′ DN
            ′ ′ ′ ′ ′ ′

            1 5 2 3 = AN + HN ? 2aDN + a2 EN + aHN

            (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73) (74)

            = = = = = = = = = = = = =

            2 1 4 3 BN + HN ? HN + a(HN ? HN ) 6 1 4 2 CN + HN ? 2aDN + a2 EN + aHN 4 15 3 4 3 FN ? aDN + 2aDN ? aDN ? aHN ? a2 EN 16 2 5 11 10 GN + a(DN + DN ) ? 2aDN + a2 EN + aHN 10 14 11 5 4 HN ? aDN + 2aDN ? aDN ? aHN ? a2 EN 1/2 3 D N + HN 3/4 4 D N + HN 5 2 DN ? aEN 6 1 DN ? aEN 10/11 5 + HN DN 14 4 DN ? aEN 15 3 DN ? aEN 16 2 DN ? aEN ,

            with the following abbreviations
            1 1 2 HN = a(DN + DN ) 2 3 4 HN = a(DN + DN ) 3 HN 4 HN 5 HN ? ω,+ aVN (RN ω,+ ? aVN (RN ? ω,? aVN (RN ? RN )VN + RN )VN ? RN )VN

            (75) (76) (77) (78) . (79)

            = = =

            ? ? ?

            Frequency dependent QHE

            17

            References
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            18

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