Impact of the valley degree of freedom on the control of donor electrons near a Si/SiO${}_2$ interface

We analyze the valley composition of one electron bound to a shallow donor close to a Si/barrier interface as a function of an applied electric field. A full six-valley effective mass model Hamiltonian is adopted. For low fields, the electron ground state is essentially confined at the donor. At high fields the ground state is such that the electron is drawn to the interface, leaving the donor practically ionized. Valley splitting at the interface occurs due to the valley-orbit coupling, $V_{\mathrm{vo}}^I=\left|V_{\mathrm{vo}}^I\right|e^{i\theta }$. At intermediate electric fields, close to a characteristic shuttling field, the electron states may constitute hybridized states with valley compositions differing from the donor and the interface ground states. The full spectrum of energy levels shows crossings and anticrossings as the field varies. The degree of level repulsion, thus, the width of the anticrossing gap, depends on the relative valley compositions, which vary with $|V_{\mathrm{vo}}^I|$, $\theta$ and the interface-donor distance. We focus on the valley configurations of the states involved in the donor-interface tunneling process, given by the anticrossing of the three lowest eigenstates. A sequence of two anticrossings takes place and the complex phase $\theta$ affects the symmetries of the eigenstates and level anticrossing gaps. We discuss the implications of our results on the practical manipulation of donor electrons in Si nanostructures.

Figure 1

(a) Double-well potential in the $z$ direction formed by the Coulombic donor potential plus the triangular interface/electric field potential. $d$ is the distance between the donor and interface. (Main panel) Symmetry of levels at the donor and at the interface. Every level is described by six coefficients corresponding to the six valleys of Si conduction band: $(x,-x,y,-y,z,-z)$. This defines the valley composition of each state. At the interface, the mass anisotropy breaks the valley degeneracy in a doublet ($z,-z$) and a quadruplet ($x,-x,y,-y$). The doublet degeneracy is lifted due to the valley orbit coupling $(V_{\mathrm{vo}}^I=|V_{\mathrm{vo}}^I\left|e^{i\theta }\right)$ arising in a sharp $\left(001\right)$ interface,[9, 12, 14] as shown in (b). $C_I^z$ and $C_I^{-z}$ are defined in Eq. (6), and $\overline{C_I^{-z}}=-C_I^{-z}$. At an isolated donor, the valley orbit coupling leaves a nondegenerate ground state with $A_1$ symmetry, well separated (splitting $\sim$12 meV) from the other five levels.[20, 21, 22] The binding energies given on the right of each level are experimental values for bulk P donors.[21] The lines join the valley compositions at donor and interface that are connected by symmetry.

Figure 2

Full spectrum of eigenvalues for $d=4a^{*}$ in a wide range of electric fields. For small values of the electric field, the lowest six eigenvalues correspond to the donor states and the highest six to interface states. We use here $V_{\mathrm{vo}}^I=\left|V_{\mathrm{vo}}^I\right|e^{i\pi /3}$ with $|V_{\mathrm{vo}}^I|=\lambda F$ and $\lambda =1.36$ Å.

Figure 3

Evolution of the valley population for the three lowest eigenvalues around the characteristic field (see Fig. 2) for two different distances, (a) $d=4a^{*}$ and (b) $d=5a^{*}$. We use $V_{\mathrm{vo}}^I=\left|V_{\mathrm{vo}}^I\right|e^{i\pi /3}$ with $|V_{\mathrm{vo}}^I|=\lambda F$ and $\lambda =1.36$ Å. The top panels reproduce the eigenvalues involved in the lowest-energy anticrossing, which is, in fact, a sequence of two anticrossings. All other panels show the valley population in the second excited state, the first excited state, and the ground state (GS) in different lines. The red (solid) curves correspond to the weight of the $\pm z$ valleys or longitudinal weight (at donor and interface), and the green (dashed) curves are the weight of the $\pm x$ and $\pm y$ valleys or transversal weight (at donor and interface). Labels D or I refer to donorlike or interface-like states in terms of real space location. Here D is a combination of the six valleys and I involves the $z$ and $-z$ valleys.

Figure 4

Three lowest eigenvalues around the anticrossings region, corresponding to $d=4a^{*}$ for different values of the phase $\theta$ of the valley-orbit coupling at the interface ($V_{\mathrm{vo}}^I=\left|V_{\mathrm{vo}}^I\right|e^{i\theta }$). $|V_{\mathrm{vo}}^I|=\lambda F$ and $\lambda =1.36$ Å. The extra panel on the right top corner shows the values of the two anticrossing gaps versus $\theta$. The dashed line represents the gap for the first anticrossing between the GS and the first excited state while the solid line is the gap between the first and second excited states. Note that for $\theta =0$ and $\pi$, $V_{\mathrm{vo}}^I$ is a real quantity and one of the anticrossings has zero gap (namely it is actually a two-level crossing). The two gaps are equal for $\theta =\pi /2$ which corresponds to a pure imaginary $V_{\mathrm{vo}}^I$. Our results are obviously invariant for $\theta \leftrightarrow -\theta$.

Figure 5

Three lowest eigenvalues for $d=4a^{*}$ close to the characteristic field with the valley orbit coupling at the interface $|V_{\mathrm{vo}}^I|=\lambda F$ with $\lambda =0.215$ Å as calculated by Sham and Nakayama.[9] We use $\theta =\pi /3$ as the phase of $V_{\mathrm{vo}}^I$ [compare with Fig. 4c].