Intervalley coupling for interface-bound electrons in silicon: An effective mass study

Orbital degeneracy of the electronic conduction band edge in silicon is a potential roadblock to the storage and manipulation of quantum information involving the electronic spin degree of freedom in this host material. This difficulty may be mitigated near an interface between Si and a barrier material, where intervalley scattering may couple states in the conduction ground state, leading to nondegenerate orbital ground and first excited states. The level splitting is experimentally found to have a strong sample dependence, varying by orders of magnitude for different interfaces and samples. The basic physical mechanisms leading to such coupling in different systems are addressed. We expand our recent study based on an effective mass approach, incorporating the full plane-wave expansions of the Bloch functions at the conduction band minima. Physical insights emerge naturally from a simple Si/barrier model. In particular, we present a clear comparison between ours and different approximations and formalisms adopted in the literature and establish the applicability of these approximations in different physical scenarios.

Figure 1

Energies and envelope functions for a conduction band offset $U_0=$ 3 eV. (a) Comparison between the expectation value for the ground-state energies obtained variationally by solving the finite differences equation through the steepest descent method (squares) and using the trial wave function in Eq. (12) (diamonds). Exact results for $U_0\to \infty$ from Ref. 57 are also shown (solid line). [(b)–(d)] Envelope functions squared (probability densities) for the indicated electric fields. The solid line is obtained solving the finite differences equation variationally (through the steepest descent method). The dashed line is obtained from the trial function in Eq. (12). In (b), $F=1.00$ V/nm and the two approaches are in good agreement, while at lower electric fields [(c) and (d)], the functional form in Eq. (12) leads to somewhat more localized states near the interface.

Figure 2

Intervalley coupling as a function of the conduction band offset $U_0$. All data correspond to an external field of 10${}^{-2}$ V/nm. The absolute value of the $\delta$ function contribution $|V_{\delta }|$ and the evanescent tail contribution $|V_E|$ are depicted separately, as well as the real and imaginary parts of the total coupling $V_{VO}=V_{\delta }+V_E$. The offset for Si/SiGe interfaces is 0.1–0.2 eV and for Si/SiO${}_2$ interfaces is 3 eV.

Figure 3

Data points give the calculated intervalley coupling as a function of applied electric field, both in a log scale, for barrier height $U_0=3$ eV. A linear fit for the data points with $V_{VO}$ given in meV and $F$ in V/nm is included (dotted line through the points), leading to an estimated $\lambda \sim 0.27$Å. The vertical dashed lines correspond to the experimental values of the breakdown field for SiGe and SiO${}_2$. The dotted horizontal line represents the measured valley coupling (or half the splitting) reported in Ref. 21. The inset shows the same results presented in Ref. 19, covering a much narrower range of field values, for $U_0=3$ eV (solid line) and 125 meV (dotted line). The solid straight line is a possible prediction from the model in Ref. 26, calculated from Eq. (17), taking $\alpha =0.43$ Å and $\langle \frac{\partial u}{\partial z}\rangle =F$.