Integer Quantum Hall Effect on a Six-Valley Hydrogen-Passivated Silicon (111) Surface

We report magnetotransport studies of a two-dimensional electron system formed in an inversion layer at the interface between a hydrogen-passivated Si(111) surface and vacuum. Measurements in the integer quantum Hall regime demonstrate that the expected sixfold valley degeneracy for these surfaces is broken, resulting in an unequal occupation of the six valleys and anisotropy in the resistance. We hypothesize the misorientation of Si surface breaks the valley states into three unequally spaced pairs, but the observation of odd filling factors is difficult to reconcile with noninteracting electron theory.

Figure 1

(a) Schematic cross-section of a H-Si(111) substrate contact bonded to a SOI substrate. A $p^{+}$ layer in the SOI defines the gate, where blue arrows depict the electric field. A 2DES is formed at the H-Si(111) surface within an encapsulated cavity. (b) The H-Si(111) substrate has four $n^{+}$ contacts numbered accordingly. Tilted magnetic fields are applied in the $x\mathrm{\text{−}}z$ plane. (c) A $1\mu \mathrm{m}\times 1\mu \mathrm{m}$ AFM image of atomic steps on a H-Si(111) surface in relation to the crystal directions and the contacts of the device. (d) The projection of the six valleys for the Si(111) surface with pairs of valleys labeled $A$, $B$, and $C$.

Figure 2

(a) $R_{xy}$ (green), $R_{(12,34)}$ (blue), and $R_{(13,24)}$ (red) versus $B_{\perp }$ for the device at $T=300\mathrm{mK}$ and $n=6.5\times {10}^{11}{\mathrm{cm}}^{-2}$. Plateaus in $R_{xy}$ are labeled by horizontal bars, and the numbers indicate their corresponding filling factor $\nu$. The inset is a schematic energy level diagram depicting the three energy scales affecting $\mathcal{D}(E)$ at $B_{\perp }=1\mathrm{T}$. (b) The same data from (a) are plotted within $0\le B_{\perp }\le 2\mathrm{T}$. Hall resistance is plotted as $R_{xy}/B_{\perp }$, and the divergence is removed for $B_{\perp }\le 0.1\mathrm{T}$. Open symbols ($R_{xy}/B_{\perp }$) and closed symbols ($R_{xx}$) correspond to calculations having all six valleys equally occupied. The inset depicts $\mathcal{D}(E)$ for the 7 K model at $B=0$.

Figure 3

Data and calculations represented at $n=6.5\times {10}^{11}{\mathrm{cm}}^{-2}$ and $T=300\mathrm{mK}$. For the color scales represented in (a), (c), and, (d), a maximum in $d^2{R}_{xx}/d{B}_{\perp }^2$ or $d{E}_F/dn$ corresponds to a minimum in $R_{xx}$. (a) A surface plot of $d^2{R}_{(12,34)}/d{\nu }^2$ versus $1/\cos \theta$ for filling factors within $5\le \nu \le 11$. At ${\theta }_c$, the $R_{xx}$ minimum at $\nu =8$ disappears while a minimum at $\nu =10$ is at a peak. (b) Schematic energy level diagram following the 7 K model predicting the twofold energy levels from the A valleys (spin up and spin down) crossing one another at ${\theta }_c$. (c) A plot of $d^2{R}_{(12,34)}/d{B}_{\perp }^2$ versus $\nu$ and $1/\cos \theta$. The overlaid schematic fan diagram (dashed lines) represents expectations from the 7 K model with $\Delta \nu \simeq 8$. (d) A plot of a calculated $d{E}_F/dn$ versus $\nu$ and $1/\cos \theta$ using the 7 K model. The same fan diagram used in (c) is overlaid on top of calculations.

Figure 4

$R_{xx}$ minima at $\nu =3$, 4, and 6 are divided by their respective $R_{xx}$ values at $\theta =0°$ and plotted within $-15°\le \theta \le 25°$. Data represent measurements at $T=300\mathrm{mK}$ and $n=6.5\times {10}^{11}{\mathrm{cm}}^{-2}$.