Anomalous Resistance Ridges along Filling Factor $\nu =4i$

We report anomalous structure in the magnetoresistance of ${\mathrm{SiO}}_2/\mathrm{Si}(100)/{\mathrm{SiO}}_2$ quantum wells. When Landau levels of opposite valleys are driven through coincidence at the Fermi level, the longitudinal resistance displays elevations at filling factors that are integer multiples of 4 ($\nu =4i$) accompanied by suppression on either side of $\nu =4i$. This persists when either the magnetic field or the valley splitting is swept, leading to resistance ridges running along $\nu =4i$. The range of field over which they are observed points to the role of spin degeneracy, which is directly confirmed by their disappearance with the addition of an in-plane magnetic field. The data suggest a new type of many-body effect arising from the combined degeneracy due to the valley and the spin degrees of freedom.

Figure 1

(a) Layer structure. (b) Quantum well potential. The electric fields in the oxide layers are independently controlled by $V_{\mathrm{FG}}$ and $V_{\mathrm{BG}}$. (c) Only out-of-plane valleys are occupied. (d) The resulting DOS, with and without valley splitting ${\Delta }_{\mathrm{V}}$. (e) Gray-scale plot of $d^2{G}_{\mathrm{SD}}/d{V}_{\mathrm{SD}}^2$ at $B=5.5\mathrm{T}$, $T=4.2\mathrm{K}$. Solid lines labeled $n^{+}=0$ and $n^{-}=0$ mark lines of zero electron density in the lower and the upper valley subbands as extracted using $\alpha =0.46\mathrm{meV}/{10}^{15}{\mathrm{m}}^{-2}$. (f) Data at 1.5 K. Ridges along $\nu =16$, 20, 24, and 28 are marked by arrows. Solid lines are the same as those in (e). (g) Sample configuration. (h) Diagram of the spin-degenerate LLs as a function of potential bias $\delta n$. (${\Delta }_{\mathrm{V}}\propto \delta n$). Thicker lines represent the lowest LLs for the two valley subbands $+$ and $-$. Dotted lines schematically mark where the ridges are observed.

Figure 2

(a) $R_{xx}$ as a function of filling factor $\nu$ at various values of $V_{\mathrm{BG}}$ (fixed intervals, data offset) at $B=5.5\mathrm{T}$, 350 mK. (b) Gray-scale plot of the same data as (a). A monotonic background has been subtracted for clarity. White numbers indicate filling factors. (c) and (d) $R_{xx}$ and $R_{xy}(B)$ taken at different $V_{\mathrm{BG}}$ keeping $n$ constant. Ridges are seen at $\nu =12$, 16, and 20 but not 8 for $V_{\mathrm{BG}}=25\mathrm{V}$, $V_{\mathrm{FG}}=0.83\mathrm{V}$.

Figure 3

(a) $d^2{G}_{\mathrm{SD}}/d{V}_{\mathrm{SD}}^2$ at 1.5 K, $\theta =55.6°$, $\mathbf{B}=9.735\mathrm{T}$, $B_{\perp }=5.5\mathrm{T}$. (b) Direction of the sample with respect to the field. (c) Schematic diagram of spin-split LLs.

Figure 4

(a) $R_{xx}$ of the bonded-SOI sample (8 T, 350 mK). (b) Evolution of the lowest six orbital LLs from the front layer to those of the back. (${\Delta }_{\mathrm{SAS}}$ has been adjusted for each orbital to reflect its dependence on concentration.)