Valley Polarization in Si(100) at Zero Magnetic Field

The valley splitting, which lifts the degeneracy of the lowest two valley states in a ${\mathrm{SiO}}_2/\mathrm{Si}(100)/{\mathrm{SiO}}_2$ quantum well, is examined through transport measurements. We demonstrate that the valley splitting can be observed directly as a step in the conductance defining a boundary between valley-unpolarized and -polarized regions. This persists to well above liquid helium temperature and shows no dependence on magnetic field, indicating that single-particle valley splitting and valley polarization exist in (100) silicon even at zero magnetic field.

Figure 1

The source-drain conductance $G_{\mathrm{SD}}$ at 4.2 K as a function of $V_{\mathrm{FG}}$ at different values of $V_{\mathrm{BG}}$. A and B mark features due to second spatial-subband and upper valley subband occupation, respectively. The inset shows the experimental setup with a TEM image of a SIMOX structure.

Figure 2

Left: The conductance at $V_{\mathrm{BG}}=60\mathrm{V}$, 4.2 K. The lower two graphs show first and second derivatives. The feature marked A is due to the second spatial subband. B marks the feature due to the upper valley while C marks the onset of conduction. Right: Gray-scale plot of the second derivative showing the evolution of A, B, and C with $(V_{\mathrm{BG}},V_{\mathrm{FG}})$. White squares show calculated onsets of second spatial-subband occupation [17]. The abrupt horizontal line at $V_{\mathrm{FG}}=2\mathrm{V}$ is an experimental artifact.

Figure 3

(a) $d^2{G}_{\mathrm{SD}}/d{V}_{\mathrm{FG}}^2$ at 5.5 T, 4.2 K. (b) Same data as (a) but with different contrast. Solid squares joined by curves are estimated onsets of second spatial-subband occupation [17, 24]. Solid straight lines show calculated lines of even ${\nu }^{+}$ and ${\nu }^{-}$, while dotted lines represent ${\nu }^{\mathrm{tot}}=4i$, where $i$ is an integer [25]. The region beneath ${\nu }^{-}=0$ is completely valley polarized. (c) Density of states at zero magnetic field with and without valley splitting $\Delta$. (d) Resultant Landau levels. Thicker lines also correspond to valley subband edges.

Figure 4

Temperature dependence at $B=0\mathrm{T}$, $V_{\mathrm{BG}}=60\mathrm{V}$. Left panel: $G_{\mathrm{SD}}$. Right panel: $d^2{G}_{\mathrm{SD}}/d{V}_{\mathrm{FG}}^2$. Arrows mark valley subband edges extracted using $\alpha =0.46\mathrm{meV}/{10}^{15}{\mathrm{m}}^{-2}$.