Valley-degenerate two-dimensional electrons in the lowest Landau level

We report low temperature magnetotransport measurements on a high mobility ($\mu =325000{\mathrm{cm}}^2/\mathrm{V}\mathrm{s}$) two-dimensional electron system on a H-terminated Si(111) surface. We observe the integral quantum Hall effect at all filling factors $\nu \le 6$ and find that $\nu =2$ develops in an unusually narrow temperature range. An extended, exclusively even numerator, fractional quantum Hall hierarchy occurs surrounding $\nu =3/2$, consistent with twofold valley-degenerate composite fermions (CFs). We determine activation energies and estimate the CF mass.

Figure 1

(a) $R_{xx}$, $R_{yy}$ (left), and ${\rho }_{xy}$ (right) vs $B$ at $T=90\mathrm{mK}$ and $n_s=3.75\times {10}^{11}{\mathrm{cm}}^{-2}$ with filling factors listed on the top axis; the fractional states at $\nu =\frac{8}{5},\frac{10}{7},$ and $\frac{4}{3}$ have plateaus in ${\rho }_{xy}$. (b) $R_{xx}$ and $R_{yy}$ vs $B$ for $n_s=3.75\times {10}^{11}{\mathrm{cm}}^{-2}$. Seven FQH states are visible at $T=110\mathrm{mK}$; the filling factors are labeled on the top axis. Top panel: ${\rho }_{xy}$ vs $B$ for $T=110\mathrm{mK}$. The derivative is also shown to accentuate the fine structure, matching the $R_{xx}$ minima.

Figure 2

(a) Resistance data (with measurement errors) versus temperature used in Shubnikov–de Haas analysis. Open (closed) symbols are from $R_{xx}$ ($R_{yy}$). Note that only three data points are available for all $R_{yy}$ data. (b) Mass of composite fermions assuming that the gaps probed by Shubnikov–de Haas (SdH) oscillations have the form ${\Delta }_{\text{CF}}=\hslash {\omega }_c^{\text{CF}}$. Open symbols indicate values extracted from maxima. See text for a discussion of the statistical and systematic errors involved in this measurement. (c) Equivalent CF gap energy calculated from the masses. The gap value at $\nu =4/3$ is calculated from activation energy data and is approximately the same for both $R_{xx}$ and $R_{yy}$.

Figure 3

(a) Activation energy measurements as a function of $B$ in units of Coulomb energy. Vertical bars show the $95\%$ confidence errors from fits to the linear portions of Arrhenius plots. Closed (open) symbols indicate data from $R_{yy}$ ($R_{xx}$) measurements. The $\nu =2$ data shows an enhanced energy gap greater than the Landau level spacing ($E_{\text{LL}}$) and indicates the scatter between $R_{xx}$ and $R_{yy}$ data. (b) Arrhenius plot of $R$ for $\nu =2$ and 6 at $n_s=4.74\times {10}^{11}{\mathrm{cm}}^{-2}$. While the $\nu =6$ dependence is linear over a wide range of temperatures, the $\nu =2$ data show a very narrow linear region. Both $R_{xx}$ (open symbols) and $R_{yy}$ (closed) show this effect for $\nu =2$. (c) Plot of the linear portion of ${\sigma }_{yy}$ versus ${\Delta }_{\nu }/T$, showing the difference between the critical conductivities ${\sigma }_{yy}^C={\sigma }_{yy}\left(1/T\to 0\right)$ at different filling factors and densities.