Enhanced valley splitting in Si layers with oscillatory Ge concentration

The valley degeneracy in Si qubit devices presents problems for their use in quantum information processing. It is possible to lift this degeneracy by using the wiggle well architecture, in which an oscillatory Ge concentration couples the valleys. This paper presents the basic theory of this phenomenon together with model calculations using the empirical pseudopotential theory to obtain the overall magnitude of this effect and its dependence on the wavelength of the concentration oscillations. We derive an important selection rule which can limit the effectiveness of the wiggle well in certain circumstances.

Figure 1

The smooth potential $V\left(z\right)=V_{\text{str}}\left(z\right)+V_{\text{osc}}\left(z\right)$ felt by an electron in the proposed SiGe heterostructure, shown for an average Ge concentration in the well of 0%, 10%, and 20%, and an electric field $F/\epsilon =8.5$ mV/nm. The atomistic disorder potential $V_d\left(\mathbf{r}\right)$ is not visible on this relatively coarse scale. The Si layer with Ge added in a modulated fashion occupies the half space $z<0$. The barrier occupies the $z>0$ region. The width of the barrier is 1 nm and its height is 1 eV.

Figure 2

Envelope functions of the ground state ${\left|\psi \left(z\right)\right|}^2=|F^{+}{\left(z\right)|}^2+{\left|F^{-}\left(z\right)\right|}^2$, plotted for ${\overline{n}}_{\text{Ge}}=0$ (a), 0.1 (b), and 0.2 (c) at $q=3.7{\text{nm}}^{-1}$. These are the solutions of Eq. (36).

Figure 3

The WW contribution to the valley splitting ${\mathrm{\Delta }}_w$, plotted as a function of the wave vector of the Ge concentration $q=\frac{2\pi }{\lambda }$. ${\overline{n}}_{\text{Ge}}=0.05$ (blue curve), 0.10 (yellow curve), 0.15 (green curve), and 0.20 (red curve). The first ($3.7{\text{nm}}^{-1}$) and third ($19.6{\text{nm}}^{-1}$) peaks correspond to the long-wavelength WW and short-wavelength WW, respectively. The intermediate peak near $q=9.8{\text{nm}}^{-1}$ is due to a second order effect in ${\overline{n}}_{\text{Ge}}$. The electric field applied is $F=0.1$ V/nm.

Figure 4

The logarithms of the height for each peak plotted as as a function of ${\overline{n}}_{\text{Ge}}$. The heights for $q=3.7{\text{nm}}^{-1}, q=9.8{\text{nm}}^{-1}$, and $q=19.6{\text{nm}}^{-1}$ are shown by blue circles, blue squares, and red circles, respectively. The data points are connected by a guide to the eye. Note that the points for $q=19.6{\text{nm}}^{-1}$ are referred to the vertical scale on the right. Linear fits to the data give slopes of 1.42, 1.98, and 0.99, which agrees very well with the expected values 1.5, 2, and 1.