Detection of a Landau Band-Coupling-Induced Rearrangement of the Hofstadter Butterfly

The spectrum of 2D electrons subjected to a weak 2D potential and a perpendicular magnetic field is composed of Landau bands with a fractal internal pattern of subbands and minigaps referred to as Hofstadter’s butterfly. The Hall conductance may serve as a spectroscopic tool as each filled subband contributes a specific quantized value. Advances in sample fabrication now finally offer access to the regime away from the limiting case of a very weak potential. Complex behavior of the Hall conductance is observed and assigned to Landau band-coupling-induced rearrangements within the butterfly.

Figure 1

(a) Repeated Hofstadter butterfly. Hall conductance contributions in units of $2e^2/h$ of partially filled Landau bands are indicated in dominant gaps. The green line tracks the Fermi energy as the Landau band $n=13$ is filled with decreasing flux. The resulting Hall sequence 0-1-0-1 can be identified. For other bands the ${\Phi }_0/\Phi$ range is shown at the top. (b) Landau fan chart for modulation $V_0=2.4\mathrm{meV}$ and period $a=100\mathrm{nm}$ in single band approximation. Weak 2D modulation subdivides blue Landau bands into repeated butterflies.

Figure 2

Longitudinal and Hall resistivities for $n_{\mathrm{s}}=4.32\times {10}^{11}{\mathrm{cm}}^{-2}$. The modulation period is $102.7\pm 0.5\mathrm{nm}$. Dotted lines mark flatband positions. The inset shows the sample geometry. Landau bands are denoted by their orbital index.

Figure 3

(a) Expected Hall conductance for Landau bands $n=13$ and 14 in red, obtained when taking the biggest gap sizes of Fig. 1a as plateau widths. Black lines are their convolution with a fitted Gaussian of width $0.013\hslash {\omega }_{\mathrm{c}}$. (b) Schematic representation of the density of states. (c) Measured Hall conductance (black) and longitudinal resistivity (blue) for bands $n=13$ and 14. (d),(e) are like (a) and (c) but for bands $n=17$ and 18.

Figure 4

Consequences of Landau band coupling. Top panel: calculation of bands $n=37$ and $K=2$ ignoring coupling with other bands. Hall conductances for major gaps are marked. Middle panel: deformations due to coupling. As in the top panel, $E_{\mathrm{F}}$ traces are plotted for three densities. Jumps of $E_{\mathrm{F}}$ across gaps produce Hall plateaus. The expected Hall conductance behavior is shown in the red and blue insets. Bottom panel: measured Hall conductance for $n_{\mathrm{s}}$ from $4.32\times {10}^{11}{\mathrm{cm}}^{-2}$ for (a) to $4.17\times {10}^{11}{\mathrm{cm}}^{-2}$ for (c). Here, ${\rho }_{xx}$ shows the boundary of the 37th band for each Hall trace.