Valley splitting in low-density quantum-confined heterostructures studied using tight-binding models

A detailed study of reduced-basis tight-binding models of electrons in semiconducting quantum wells is presented. The focus is on systems with degenerate valleys, such as silicon in silicon germanium heterostructures, in the low-density limit, relevant to proposed quantum computing architectures. Analytic results for the bound states of systems with hard-wall boundaries are presented and used to characterize the valley splitting in silicon quantum wells. The analytic solution in a no-spin-orbit model agrees well with larger tight-binding calculations that do include spin-orbit coupling. Numerical investigations of the valley splitting for finite band offsets are presented that indicate that the hard-wall results are a good guide to the behavior in real quantum wells.

Figure 1

Sketch of different versions of a simple tight-binding model, all related by basis transformations. White regions are negative, striped regions positive. (a) Single-band model with one $p$-like orbital per atom and one atom per unit cell (length $a∕2$). Parameters are $\epsilon$ (on-site), $v$ (nearest-neighbor), and $u$ (second-nearest-neighbor). (b) Grouping two atoms together in a single cell yields a two-band model with two atoms per unit cell (length $a$), and each atom with one $p$-like orbital. (c) A basis transformation on the middle model using symmetric and antisymmetric combinations yields a two-band model with one atom of two orbitals per unit cell (length $a$). The parameters are: ${\epsilon }_s=\epsilon -v,{\epsilon }_p=\epsilon +v$, $V_{ss}=u-v∕2,V_{pp}=u+v∕2,V_{sp}=v∕2$.

Figure 2

Bulk dispersions for the one- and two-band models with the same parameters. The one-band phase is $\phi =k^{\left(1\right)}a∕2$ and the two-band phase is $\varphi =k^{\left(2\right)}a$ (the one-band cell is half as long as the two-band cell). The parameters are: $\epsilon =3.0\mathrm{eV}$, $v=\mathrm{0.682\hspace{0.17em}640}\mathrm{eV}$, $u=\mathrm{0.611\hspace{0.17em}705}\mathrm{eV}$. Also shown are two phases having the same energy as discussed in Sec. 2B. For clarity only one energy for the one-band model is shown.

Figure 3

Valley splitting (magnitude of the energy difference between the states of the lowest doublet) in a strained $\mathrm{Si}$ quantum well at zero applied field with hard-wall boundary conditions. Numerical results calculated with NEMO’s $s{p}^3{d}^{5}s*$ model, NEMO $spds$, and the two-band model presented here, two-band (exact), are shown as symbols with no lines. The approximate two-band splitting from Eq. (26), two-band (Approx), along with a fit of the NEMO results to Eq. (26), $spds$-fit, are shown as lines with no symbols, although they are calculated at the same points as the numerical results.

Figure 4

Ground state $S$- and $Z$-coefficients for a quantum well of 30 cells as calculated in the two-band model.

Figure 5

First excited state $S$- and $Z$-coefficients for a quantum well of 30 cells as calculated in the two-band model.

Figure 6

Valley splitting (magnitude of the energy difference between the states of the lowest doublet) versus well width for quantum wells with band offsets ranging from 0.1 to $10\mathrm{eV}$. The behavior is qualitatively similar for the entire range of band offsets, indicating that the hard-wall limit is a good starting point for real quantum wells. For typical experimental well widths, roughly 20 unit cells, the hard-wall limit is a reasonably good estimate of the splitting in wells of depth $0.1\mathrm{eV}$ or greater.

Figure 7

Fractional change in the valley splitting (see Sec. 3) versus well width for well depths ranging from 0.01 to $10\mathrm{eV}$.

Figure 8

Unit cells of the four-band model showing on-site and nearest-neighbor parameters.

Figure 9

Computation of anion and cation centers in the four-band model for a quantum well of four unit cells.