Spin and valley polarization dependence of resistivity in two dimensions

We demonstrate that spin polarization and valley polarization have quantitatively similar effects on the resistivity of a two-dimensional electron gas in a silicon-on-insulator quantum well. In so doing, we also examine the dependence on disorder, leading to a coarse but global phenomenology of how the resistivity depends on its key parameters: spin and valley polarization, density, disorder, and temperature.

Figure 1

(a) Schematic diagram of the sample. (b) Dependence of valley splitting ${\Delta }_{\mathrm{V}}$ on $\delta n$. (c) Spin- and valley-split subbands. (d) Single-particle phase diagram at constant density. (e) A three-dimensional visualization of (d) with $n$ as a further parameter. (f) A phase diagram in terms of polarization.

Figure 2

Magnetotransport at $T=1.4$ K as a function of $\delta n$. The left columns (a), (c), and (e) show data at negative $\delta n$ (${\Delta }_{\mathrm{V}}=0$, $P_{\mathrm{V}}=0$) while the right columns (b), (d), and (f) show data at positive $\delta n$. Each row corresponds to different density where (a) and (b), $n=2.5\times {10}^{15}$ m${}^{-2}$; (c) and (d), $n=3.5\times {10}^{15}$ m${}^{-2}$; and (e) and (f): $n=4.5\times {10}^{15}$ m${}^{-2}$. For each density (row), identical colors are used for identical values of $\left|\delta n\right|$. Small arrows are guides to the eye, marking the feature described in the body text. $P_{\mathrm{V}}$ in parentheses (Ref. 11) show values expected at zero magnetic field. Roman numerals in (c) and (d) show corresponding regions in Fig. 1d where the dotted line boundaries are straight line guides to the eye joining points A, B, and C.

Figure 3

Comparison of the resistivity between spin-degenerate–valley-polarized and spin-polarized–valley-degenerate electrons at equal out-of-plane bias. (a) $\left|\delta n\right|=1\times {10}^{16}$ m${}^{-2}$ and (b) $\left|\delta n\right|=2\times {10}^{16}$ m${}^{-2}$. Spin polarized data are taken at $B=28$ T.

Figure 4

Temperature dependence of resistivity (squares: $n=2.5\times {10}^{15}$ m${}^{-2}$, circles: $n=4.5\times {10}^{15}$ m${}^{-2}$). Data at $B=0$ T are shown in the top rows [(a)–(e)] while the lower rows [(f)–(j)] show data at $B$ $=$ 28 T. Each column corresponds to a value of $\delta n$. Polarization ($P_{\mathrm{S}},P_{\mathrm{V}}$) for $n=2.5\times {10}^{15}$ m${}^{-2}$ is indicated in parentheses for each graph. Polarization at $n=4.5\times {10}^{15}$ m${}^{-2}$ is the same except in (d) where $P_{\mathrm{V}}=0.87$ and (i) and (j) where the degree of spin-valley polarization is substantial but not full.