Longitudinal conductance of mesoscopic Hall samples with arbitrary disorder and periodic modulations

We use the Kubo-Landauer formalism to compute the longitudinal (two-terminal) conductance of a two-dimensional electron system placed in a strong perpendicular magnetic field and subjected to periodic modulations and/or disorder potentials. The scattering problem is recast as a set of inhomogeneous, coupled linear equations, allowing us to find the transmission probabilities from a finite-size system computation. The results we present are exact for noninteracting electrons within a spin-polarized lowest Landau level: the effects of the disorder and the periodic modulation are fully accounted for. When necessary, Landau level mixing can also be incorporated straightforwardly into the same formalism. In particular, we focus on the interplay between the effects of the periodic modulation and those of the disorder, when the later is dominant. This appears to be the relevant regime to understand recent experiments [S. Melinte et al., Phys. Rev. Lett. 92, 036802 (2004)], and our numerical results are in qualitative agreement with these experimental results. The numerical techniques we develop can be generalized straightforwardly to many-terminal geometries, as well as other multichannel scattering problems.

Figure 1

A sketch of the model geometry of the Hall sample with periodic boundary conditions in the $y$-direction, and its contact to leads on both $x$-axis ends. The lower diagram shows a detailed view of the left edge. The first $N_c$ eigenstates $\mid X_1⟩,\mid X_2⟩\dots ,\mid X_{N_c}⟩$ near the edge belong to different conduction channels. We assume that each such conduction channel is attached to external leads at both edges of the sample.

Figure 2

A sketch of the toy model. The “sample” with $n=5t_2$ bonds is connected to semi-infinite leads with $t_1$ bonds.

Figure 3

Transmission rate for the toy model corresponding to $n=20$, $t_1=1$ and $t_2=0.5$ and 0.3, respectively. $T\left(E\right)$ is a symmetric function, so only the interval $E>0$ is shown.

Figure 4

Longitudinal conductance of a small sample with square periodic potential but no disorder potential. $\varphi ∕{\varphi }_0=q∕p=2$, $B=5.44\mathrm{T}$, $L_x=1.248\mu \mathrm{m}$, $L_y=1.17\mu \mathrm{m}$, corresponding to a total of 1860 states in the lowest Landau level.

Figure 5

Longitudinal conductance for pure triangular periodic potential with $A=0.01\mathrm{meV}$ (no disorder) and $q∕p=3\left(B9.42\mathrm{T}\right)$. $L_x=2.004\mu \mathrm{m}$, $L_y=2.028\mu \mathrm{m}$. This sample has 9256 states in the LLL, divided into 52 channels. The inset shows a semilogarithmic plot of the original data set.

Figure 6

(Color online) Longitudinal conductance for triangular periodic potential plus weak disorder, for $q∕p=7∕3$ $(B=7.33\mathrm{T})$. $L_x=1.998\mu \mathrm{m}$, $L_y=2.028\mu \mathrm{m}$. This sample has 7176 states divided into 52 channels. The inset shows a semilogarthmic plot of the data set. The amplitude of disorder is about $1\mathrm{meV}$, which is only a fraction of the size of the largest gap. However, some of the expected smaller gaps are already filled in by disorder.

Figure 7

(Color online) Longitudinal conductance for a large disorder potential. $L_x=2.432\mu \mathrm{m}$, $L_y=2.418\mu \mathrm{m}$, $B=4.71\mathrm{T}$. The thick line corresponds to disorder only, while the thin line corresponds to disorder plus a small periodic potential ($A=0.01\mathrm{meV}$, $p∕q=2∕3$). In both cases $T=1\mathrm{mK}$ and $V=1\mu \mathrm{eV}$. The inset is a semilogarithmic plot of the original data sets showing the increase of conductance in the off-resonance tunneling regime, induced by the small periodic potential.

Figure 8

Contour plot of the disorder potential used to calculate the conductance of Fig. 7. Solid lines are equipotentials for $0\mathrm{meV}$, and dashed lines for $0.025\mathrm{meV}$. These energies are located within the central peak of conductance. Parts of the contour go along the edge at $y=\pm L_y∕2$.

Figure 9

Another example of a weak triangular potential imposed on a large disorder potential. Here, ${\sigma }_{xx}$ is plotted as a function of filling factor $\nu$. Panels (a) and (c) show the results with both periodic and disorder potentials, panels (b) and (d) are for disorder only. Panels (a) and (b) are $T=V=0$ data, panels (c) and (d) are measured at $T=1\mathrm{mK}$ and $V=1\mu \mathrm{V}$. The parameters are $B=7.85\mathrm{T}$, $p∕q=2∕5$, $L_x=2.432\mu \mathrm{m}$, $L_y=2.418\mu \mathrm{m}$, 11036 states divided in 62 channels.

Figure 10

(Color online) Longitudinal conductance of a sample with a fixed disorder potential but varying strength of triangular periodic potential. Parameters for this sample are: $L_x=5.134\mu \mathrm{m}$, $L_y=4.992\mu \mathrm{m}$, $p∕q=2∕5$, $B=5.233\mathrm{T}$, leading to 31616 states divided amongst 128 channels. The data correspond to $T=5\mathrm{mK}$ and $V=1\mu \mathrm{eV}$. The inset shows the filling factors as a function of energy for the same energy range as the main plot.

Figure 11

The disorder potential used in the calculation for Fig. 10, generated by summing random Coulomb scatterers. Solid and dashed contours are at energies $-0.02\mathrm{meV}$ and $0.09\mathrm{meV}$, respectively, which are at the center and right edge of the broadest conductance peak $(A=0.1\mathrm{meV})$ in Fig. 10.