Band structure of a two-dimensional electron gas in the presence of two-dimensional electric and magnetic modulations and a perpendicular magnetic field

Two-dimensional (2D) periodic electric modulations of a 2D electron gas split each Landau level into the well-known butterfly-type spectrum described by a Harper-type equation multiplied by an envelope function. This equation is slightly modified for 2D magnetic modulations but the spectrum remains qualitatively the same. The same holds if both types of modulations are present. The modulation strengths do not affect the structure of the butterfly-type spectrum, they only change its scale or its envelope. The latter is described by the ratio $\alpha$ of the flux quantum $h∕e$ to the flux per unit cell. Exact numerical and approximate analytical results are presented for the energy spectrum as a function of the magnetic field. For integer $\alpha$ the internal structure collapses into a band for all cases. The bandwidth at the Fermi energy depends on the modulation strength, the electron density, and, when both modulations are present, on the phase difference between them. In the latter case if the modulations have a $\pi ∕2$ phase difference, the bandwidth at the Fermi energy is nearly independent of the magnetic field and the commensurability oscillations of the diffusive contribution to the resistivity disappear.

Figure 1

Approximate band structure, for $n=0,\dots ,10$, as a function of $\hslash {\omega }_c$ for square electric [(a) and (c)] or magnetic [(b) and (d)] modulations of long period $(a=200\mathrm{nm})$, in (a) and (b), and short period $(a=80\mathrm{nm})$ in (c) and (d). The crossings of the dotted curves with the Landau levels give the asymptotic flat-band positions and the dashed line, in panels (a) and (b), is the $\delta =1$ curve.

Figure 2

The same as in Fig. 1 for both electric and magnetic in-phase modulations of long (a) and short (b) period. The thin solid curves show the Fermi energy for temperature $T=1\mathrm{K}$ and electron density $n_e={10}^{11}{\mathrm{cm}}^{-2}$.

Figure 3

Left panels: The approximate bandwidth $\Delta$ of the 10th Landau level as a function of the modulation period $a$ at magnetic field $B=0.5\mathrm{T}$ for modulation strengths (a) $V=0.1\mathrm{meV}$ and $\hslash \omega =0$, (b) $V=0$ and $\hslash \omega =0.1\mathrm{meV}$, and (c) $V=\hslash \omega =0.1\mathrm{meV}$. Right panels: The constant term of the square of the velocity in arbitrary units, given by Eqs. (17) or (18), versus the modulation period $a$ at magnetic field $B=0.5$. Panels (d), (e), and (g) correspond to panels (a), (b), and (c), respectively. The two periods used in Figs. 1, 2 are $80\mathrm{nm}$, indicated by the triangle, and $200\mathrm{nm}$.

Figure 4

The approximate bandwidth of the Landau level closest to the Fermi level as a function of the magnetic field for different electron densities $n_e=0.5$ (a), 1.0 (b), 2.0 (c), and $5.0\times {10}^{11}{\mathrm{cm}}^{-2}$ (d). The solid curves denote broadening with both modulations present and of the same strength. The dotted (dashed) curves show the broadening when only the electric (magnetic) modulation is present.

Figure 5

Exact and approximate ($\alpha$ integer) spectrum, in units of $\hslash {\omega }_c$, versus the cyclotron energy $\hslash {\omega }_c$ for the $n=2$ Landau level and a period $a=80\mathrm{nm}$. The upper panel is for an electric modulation, the middle one for a magnetic one, and the bottom one for both modulations present. Notice the continuous bands for $\alpha$ integer: In this case the exact, numerically obtained bandwidth coincides with the one obtained analytically.

Figure 6

Broadened DOS for the exact (thin solid curves: $\Gamma =0.01\mathrm{K}$, thick curves: $\Gamma =0.5\mathrm{K}$) and the $\alpha$ integer spectrum approximation (dotted curves: $\Gamma =0.01\mathrm{K}$, dashed curves: $\Gamma =0.5\mathrm{K}$). The magnetic field is such that $\alpha =2\mathrm{in.}$ (a), $\alpha =7∕3\mathrm{in.}$ (b), and $\alpha =5∕2\mathrm{in.}$ (c). $V=\hslash {\omega }_0=0.1\mathrm{meV}$.

Figure 7

Band structure with both modulations present and phase shifted by $\varphi =\pi ∕2$. The Fermi level at temperature $T=5\mathrm{K}$ is also indicated by the thin solid curve. The $\delta \sim 1$ curve shows a constant bandwidth leading to a washout of the commensurability oscillations of the diffusive contributions to the resistivity.

Figure 8

Bandwidth at the Fermi energy as a function of B for different phases $\varphi =\pi ∕4$ (a), $\varphi =\pi ∕2$ (b), and $\varphi =\pi$ (c) between the two modulations.

Figure 9

The same as in Fig. 7 for different electric modulation strengths $V$, as indicated, and fixed magnetic one $\hslash {\omega }_0=0.1\mathrm{meV}$ at phase shift of $\pi ∕2$.

Figure 10

Exact and approximate (dotted curves) spectrum for various phase differences between the two modulations $\varphi =\pi ∕4$ (a), $\varphi =\pi ∕2$ (b), $\varphi =3\pi ∕4$ (c), and $\varphi =\pi$ (d).