Effects of large disorder on the Hofstadter butterfly

Motivated by the recent experiments on periodically modulated, two-dimensional electron systems placed in large transversal magnetic fields [S. Melinte et al., Phys. Rev. Lett. 92, 036802 (2004)], we investigate the interplay between the effects of disorder and periodic potentials in the integer quantum Hall regime. In particular, we study the case where disorder is larger than the periodic modulation, but both are small enough that Landau level mixing is negligible. We carry extensive numerical calculations to understand the relevant physics in the lowest Landau level, such as the spectrum and nature (localized or extended) of the wave functions. Based on our results, we propose a qualitative explanation of the new features uncovered recently in these transport measurements.

Figure 1

The two-lead geometry considered: the finite-size 2DES has periodic boundary conditions in the $y$-direction, and is attached to metallic leads at the $x=\pm L_x∕2$ ends.

Figure 2

Averaged Fourier amplitudes of two types of disorder potential as a function of wave vector $q=\mid \mathbf{q}\mid$. For both Coulomb and Gaussian model, $V{\left(q\right)}^2$ is averaged over 116 disorder realizations. The relation between $V\left(q\right)$ and $V\left(\mathbf{r}\right)$ and relevant parameters are discussed in the text. The standard deviation, ${\left(L_x{L}_y\right)}^{-1}⟨\int d\mathbf{r}{V}^2\left(\mathbf{r}\right)⟩$ for the Coulomb model is $3.2\times {10}^{-7}{\mathrm{eV}}^2$, and $2.1\times {10}^{-7}{\mathrm{eV}}^2$ for the Gaussian model.

Figure 3

Profile of the disorder potential obtained from our Coulomb model on a $3.11\mu \mathrm{m}\times 2.96\mu \mathrm{m}$ sample with the $V^{\prime }\left(\mathbf{r}\right)$ correction at the $x=\pm L_x∕2$ edges. The disorder potential varies between $-3\mathrm{meV}$ and 3 meV, on a spatial length-scale much larger than $l=12.03\mathrm{nm}$. The critical region containing extended states is in the vicinity of $E=0.06\mathrm{meV}$. The contours are shown for $E=0.0575\mathrm{meV}$ (dashed), 0.17 meV (thick solid), and 0.31 meV (thin solid). These energy values correspond to classical filling factors $\nu =0.45$, 0.56, 0.66.

Figure 4

Semiclassical (dashed line) and quantum (solid line) filling factors for the disorder potential shown in Fig. 3, but different amplitudes of the triangular periodic potential (a) $A=0$, (b) $A=0.05\mathrm{meV}$, (c) $A=0.5\mathrm{meV}$, and (d) $A=5\mathrm{meV}$. As expected, agreement exists only in the limit $A\to 0$.

Figure 5

Semiclassical (dashed line) and quantum (full line) density of states calculated from corresponding filling factors in Fig. 4. We show only the center of the disorder-broadened lowest Landau level, where the density of states is large.

Figure 6

Semilog plot of the amplitude of Green’s function matrix element between the two edge states near $x=\pm L_x∕2$, as a function of energy, for $\delta ={10}^{-7}\mathrm{eV}$ (black line) and $\delta ={10}^{-8}\mathrm{eV}$ (grey line). $\delta$ is the small imaginary part in the Green’s function, see the Appendix. Only the disorder potential of Fig. 3 is applied in this calculation. Inset, the same quantity for $\delta ={10}^{-8}\mathrm{eV}$, over a larger energy range.

Figure 7

The effect of a triangular periodic potential on the critical energy regime. The disorder potential used here (not shown) supports a narrow interval of extended states centered at about $-0.6\mathrm{meV}$. As the amplitude $A$ of the periodic potential increases, the range of extended states increases dramatically. The left panel shows results for disorder-only and two relatively weak periodic potentials, while the right panel shows two larger periodic potentials where the three-subband structure expected for $\varphi ∕{\varphi }_0=3∕2$ is clearly seen.

Figure 8

A disorder potential of Gaussian type on a roughly $3\mu \mathrm{m}\times 3\mu \mathrm{m}$ square. The three lines are equipotential contours close to the critical regime, with energies of $-0.1\mathrm{meV}$ (dashed), 0 meV (thick solid), and 0.1 meV (thin solid). Cyclic boundary condition are applied in the $y$ direction.

Figure 9

Green’s functions for a sample with Gaussian disorder and various periodic potentials. The calculation included 20216 states with $\varphi ∕{\varphi }_0=3$. Similar to results shown in Fig. 9, we see that the periodic potentials widen the critical region.

Figure 10

(Color online) Projected local density of states ${\rho }_P(\mathbf{r};E)$ for $E=-0.504\mathrm{meV}$. Panel (a) shows the profile of the disorder potential, and the equipotential contour (black line) corresponding to $E=-0.504\mathrm{eV}$. The other three panels show ${\rho }_P(\mathbf{r};E)$ for (b) disorder only; (c) disorder plus triangular periodic potential with $A=0.1\mathrm{meV}$; (d) disorder plus square periodic potential with $A=0.1\mathrm{meV}$. The width and length of the sample are both 1.6 μm, and $\varphi ∕{\varphi }_0=3∕2$. Increased brightness corresponds to larger values.

Figure 11

(Color online) The same as in Fig. 10, but for an energy $E=-0.124\mathrm{meV}$ close to the band center.

Figure 12

(Color online) Number of grid points for which ${\rho }_P(\mathbf{r};E)>100$ as a function of $E$. This quantity characterizes the “spatial extent” of the wave function. The upper panel compares results for disorder only $(A=0)$ and disorder plus triangular potentials with $A\pm 0.01\mathrm{meV}$. The difference observed for the two signs is a consequence of particle-hole asymmetry of the triangular potential. The lower panel shows results for disorder only and disorder plus a square potential.

Figure 13

Equipotential contours at filling factor $\nu =0.3$ (upper panel) and $\nu =0.7$ (lower panel) for a $3\mu \mathrm{m}\times 3\mu \mathrm{m}$ sample with disorder plus a small triangular periodic potential with $A>0$. The shaded regions correspond to energies below the respective equipotential. In the semiclassical approximation, the shaded areas are filled with electrons, with the maximum density of $1∕2\pi l^2$, whereas the white areas are completely devoid of electrons. Quantum-mechanically, one expects interference in the regions with small periodic “bubbles,” induced by the periodic potential (see Figs. 10, 11).