Exact Multi-Valley Envelope Function Theory of Valley Splitting in Si/SiGe Nanostructures

Abstract

Valley splitting in strained Si/SiGe quantum wells is a central parameter for silicon spin qubits and is commonly described with envelope-function and effective-mass theories. These models provide a computationally efficient continuum description and have been shown to agree well with atomistic approaches when the confinement potential is slowly varying on the lattice scale. In modern Si/SiGe heterostructures with atomically sharp interfaces and engineered Ge concentration profiles, however, the slowly varying potential approximation underlying conventional (local) envelope-function theory is challenged. We formulate an exact multi-valley envelope-function model by combining Burt-Foreman-type envelope-function theory, which does not rely on the assumption of a slowly varying potential, with a valley-sector decomposition of the Brillouin zone. This construction enforces band-limited envelopes, which satisfy a set of coupled integro-differential equations with a non-local potential energy operator. Using degenerate perturbation theory, we derive the intervalley coupling matrix element within this non-local model and prove that it is strictly invariant under global shifts of the confinement potential (choice of reference energy). We then show that the conventional local envelope model generically violates this invariance due to spectral leakage between valley sectors, leading to an unphysical energy-reference dependence of the intervalley coupling. The resulting ambiguity is quantified by numerical simulations of various engineered Si/SiGe heterostructures. Finally, we propose a simple spectrally filtered local approximation that restores the energy-reference invariance exactly and provides a good approximation to the exact non-local theory.

Figures (5)

• Figure 1: (a) First Brillouin zone of the face-centered cubic (fcc) lattice. In biaxially strained SiGe/Si/SiGe QWs grown in [001] direction, the degeneracy between the six equivalent conduction band minima near the $X$-points is lifted. The two valley states at $\mathbf{k}_{0}^{\pm}=\left(0,0,\pm k_{0}\right)^{T}$ (shown in red) are energetically far below the other four higher-energy valley states (shown in blue). (b) Decomposition of the first Brillouin zone of the fcc lattice into non-overlapping valley-specific sectors $S\left(\mathbf{k}_{0}\right)$. The sectors of the two low-energy valleys highlighted in red define the two-valley model (\ref{['eq: two-valley model-2']}).
• Figure 2: (a) Comparison of the eigenstates of a wiggle well heterostructure (\ref{['eq: wiggle well']}) with wave number $q=2k_{1}$ and Ge concentration $X_{\mathrm{ww}}=0.1$ for a QW with thickness $h=72\,\mathrm{ML}$. Absolute squares of slowly varying envelopes are shown as solutions of the the non-local eigenvalue problem (\ref{['eq: effective Schroedinger equation Fourier space']}) and its local approximation (\ref{['eq: effective Schroedinger equation (local)']}). In addition, the projected local envelope according to Eq. (\ref{['eq: filtered envelope']}) is shown as a dashed line for the ground state. The inset shows a zoom on the vertical electron density distribution, which reveals a spurious modulation with frequency around $2k_{1}$ on the local envelope, which is suppressed in the exact non-local model. (b) Power spectral densities (PSD) of the ground state envelopes. The local envelope equation (\ref{['eq: effective Schroedinger equation (local)']}) does not impose any band-limitation on the envelope, such that it in general contains Fourier components outside of is valley-sector. In contrast, the valley sector restriction is handled exactly in the non-local model (\ref{['eq: effective Schroedinger equation Fourier space']}). Note that while $F_{\mathbf{k}_{0}^{+}}^{\left(0\right)}$ is restricted to $k\in\left(0,2\pi\left(1-\varepsilon_{z,z}\right)/a_{0}\right)$, the domain is shifted by $-k_{0}$ to $k\in\left(-k_{0},+k_{1}\right)$ for the slowly varying envelope $f_{\mathbf{k}_{0}^{+}}^{\left(0\right)}$. The PSDs are scaled to unity at $k=0$.
• Figure 3: Valley splitting in a conventional QW with smoothed interfaces. (a) Dependence of the valley splitting on the QW interface width $\sigma$. The local model shows a strong (unphysical) dependence on the reference energy, which is illustrated by global offsets ranging from $U_{0}=0\,\mathrm{eV}$ to $U_{0}=1\,\mathrm{eV}$ (color-coded). The projected local envelope approach (dashed, orange) closely approximates the exact result (red, solid). (b) The magnitude of the ambiguity measure $2\left|R\right|$ increases with decreasing interface width. The inset shows a parametric plot of $R$ in the complex plane. (c) Valley splitting as a function of the QW width $h$. The projected-local model remains in very good agreement with the non-local result, while the local model deviates strongly for thin wells where the slowly varying potential approximation breaks down.
• Figure 4: Valley splitting as a function of the vertical electric field $F$ for a mirror-symmetric QW. The exact non-local and projected-local models satisfy $E_{\mathrm{VS}}\left(F\right)=E_{\mathrm{VS}}\left(-F\right)$ as required by symmetry. The conventional local model exhibits an unphysical asymmetry and a strong dependence on the reference energy (illustrated by offsets $U_{0}$ with the same color-coding as in Fig. \ref{['fig: conventional QW']}).
• Figure 5: Valley splitting in unconventional heterostructures. (a) Valley splitting in a wiggle well with Ge amplitude $X_{\mathrm{ww}}=0.05$ and varying wave number $q$. The dominant resonances at $q=2k_{0}$ and $q=2k_{1}$ are captured by both the local and the non-local model, but the quantitative results near $q=2k_{1}$ show a strong unphysical reference energy dependence when using the local model. (b) Numerical results for a Ge spike with amplitude $x_{s}=0.6$ and width $\sigma_{s}=2\,\mathrm{ML}$. The predictions of local and non-local theory differ significantly due to the sharp feature in the mesoscopic potential, which leads to unphysical contributions from short-wavelength components in the local EFT. The projected-local envelope approach is in good qualitative agreement with the exact result from the non-local model, but it tends to overestimate the actual valley splitting.