Theory of Valley Splitting in Si/SiGe Spin-Qubits: Interplay of Strain, Resonances and Random Alloy Disorder

TL;DR

This study addresses the critical challenge of valley splitting in Si/SiGe spin-qubits by developing a comprehensive envelope-function theory that unifies strain effects, random alloy disorder, and nontrivial inter-valley resonances across neighboring Brillouin zones. The authors derive a multi-valley coupled envelope equation, decompose the intervalley coupling into deterministic and random parts, and show that the valley splitting follows a Rice distribution with parameters set by the deterministic and random contributions. A key finding is that shear strain can unlock a long-period wiggle-well resonance (q = 2k1), yielding a predominantly deterministic enhancement of $E_{VS}$, in contrast to the conventional $2k0$ picture. The framework extends existing theories and provides a computationally efficient tool for optimizing epitaxial profiles to suppress spin-valley hotspots, with implications for robust, scalable Si/SiGe spin qubits.

Abstract

Electron spin-qubits in silicon-germanium (SiGe) heterostructures are a major candidate for the realization of scalable quantum computers. A critical challenge in strained Si/SiGe quantum wells (QWs) is the existence of two nearly degenerate valley states at the conduction band minimum that can lead to leakage of quantum information. To address this issue, various strategies have been explored to enhance the valley splitting (i.e., the energy gap between the two low-energy conduction band minima), such as sharp interfaces, oscillating germanium concentrations in the QW (known as wiggle wells) and shear strain engineering. In this work, we develop a comprehensive envelope-function theory augmented by an empirical nonlocal pseudopotential model to incorporate the effects of alloy disorder, strain, and non-trivial resonances arising from interactions between valley states across neighboring Brillouin zones. We apply our model to analyze common epitaxial profiles studied in the literature with a focus on wiggle well type structures and compare our results with previous work. Our framework provides an efficient tool for quantifying the interplay of these effects on the valley splitting, enabling complex epitaxial profile optimization in future work.

Figures (6)

• Figure 1: (a) First Brillouin zone of the face-centered cubic (fcc) lattice. The degeneracy between the six equivalent conduction band minima in Si near the $X$-points is lifted by strain. (b) Energy diagram for the conduction band ground states in Si/SiGe quantum dots. Biaxial strain due to lattice mismatch between Si and SiGe leads to a separation of the two valleys oriented along $[001]$ and $[00\overline{1}]$ from the other four conduction band ground state valleys. The heterostructure potential and alloy disorder finally lift the remaining degeneracies. (c) Cubic unit cell of relaxed bulk Si (diamond crystal) composed of two interpenetrating fcc sub-lattices separated by a non-primitive translation along a quarter of the face diagonal. Atoms of the first and second fcc sub-lattice are shown in red and blue, respectively. (d) Biaxial strain along $[100]$ and $[010]$ yields a tetragonal crystal with reduced symmetry. Similar to the relaxed crystal, the two sub-lattices are interchangeable by a nonsymmorphic screw symmetry Feng2022Woods2024. (e) Additional shear strain along $[110]$ further reduces the symmetry to an orthorhombic system. The displacement between the two sub-lattices is controlled by Kleinman's internal ionic displacement parameter $\zeta$ (blue arrows). The nonsymmorphic screw symmetry, which maps the two sub-lattices onto each other, is broken by the shear strain.
• Figure 2: (a) Plot of the coefficients $C_{n}^{\left(2\right)}$ governing the magnitude of the deterministic contribution to the valley splitting as a function of shear strain $\varepsilon_{x,y}$. The coefficients have been computed from Eq. \ref{['eq: bandstructure coefficient C2']} using the plane wave expansion coefficients of the Bloch factors for strained silicon at the conduction band minimum. In addition to shear strain, biaxial tensile strain arising from the $\mathrm{Si}/\mathrm{Si}_{0.7}\mathrm{Ge}_{0.3}$ heterostructure was assumed, see Eq. \ref{['eq: biaxial strain QW']}. Coefficients with odd $n$ show a linear dependence on shear strain, while coefficients with even $n$ are practically constant. (b) Same as (a) for the coefficients $C_{n}^{\left(4\right)}$ given by Eq. \ref{['eq: coefficients C4']} governing the magnitude of the disorder-induced contribution to the valley splitting.
• Figure 3: Impact of interface width on valley splitting. (a) Mean valley splitting $\langle E_{\mathrm{VS}}\rangle$ and shape parameters $\nu=2\vert\Delta_{\mathrm{det}}\vert$ and $\sigma=\sqrt{2\Gamma}$ of the Rice distribution in the absence of shear strain $\varepsilon_{x,y}=0$ as a function of the QW interface width. The interface width is given in units of monolayers $\mathrm{ML}=a_{0}/4\approx0.1357\,\mathrm{nm}$ of relaxed Si. The gray shaded region indicates the deterministically enhanced regime according to Eq. \ref{['eq: separatrix condition']} and the red shaded region shows the [25%, 75%] quantile of the Rice distribution. Deterministic enhancements are observed for sharp interfaces with up to one ML width. (b) Same as (a), but for small shear strain $\varepsilon_{x,y}=0.05\%$. The deterministically enhanced regime is extended to about five MLs. (c) Absolute values of the individual components $\Delta_{\mathrm{det},n}$ contributing to the deterministic valley splitting, see Eq. \ref{['eq: Delta det compact']}, as a function of the interface width. In the absence of shear strain, the resulting valley slitting is dominated by the $n=0$ resonance with a small correction induced by the $n=-2$ resonance. (d) In the case of small shear strain $\varepsilon_{x,y}=0.05\%$, the $n=-1$ resonance provides the dominant contribution over almost the entire parameter domain. In the simulation, an electric field of $F=10\,\mathrm{mV}/\mathrm{nm}$ was assumed.
• Figure 4: Mean valley splitting in the wiggle well heterostructure as a function of wave number. (a) In the absence of shear strain $\varepsilon_{x,y}=0\%$ deterministic enhancements are observed only at $q=2k_{0}$ (short-period wiggle-well), which increases with increasing amplitude $X_{\mathrm{ww}}$. Away from the resonance, the non-zero valley splittings are disorder-dominated. (b) A small amount of shear strain $\varepsilon_{x,y}=0.1\,\%$ unlocks a new resonance at $q=2k_{1}$ (long-period wiggle-well). (c) Mean valley splitting for fixed amplitude $X_{\mathrm{ww}}=5\%$ and different values of shear strain $\varepsilon_{x,y}$. A constant electric field $F=5\,\mathrm{mV}/\mathrm{nm}$ and biaxial tensile strain, see Eq. \ref{['eq: biaxial strain QW']}, was assumed in the simulations.
• Figure 5: (a) Mean valley splitting of the long-period wiggle-well with fixed wave number $q=0.32\times2\pi/a_{0}\approx2k_{1}$ as a function of shear strain $\varepsilon_{x,y}$ and Ge amplitude $X_{\mathrm{ww}}$. The dashed line is the separatrix defined in Eq. \ref{['eq: separatrix condition']} that separates disorder-dominated and deterministically enhanced regimes. For large shear strains, a deterministic enhancement is achieved already for very small Ge amplitudes. The red line indicates the constant shear strain value considered in panel (b). (b) Mean valley splitting as a function of wave number $q$ and amplitude $X_{\mathrm{ww}}$ for fixed shear strain $\varepsilon_{x,y}=0.025\,\%$. Three domains with valley splitting enhancements can be observed, namely, at very low $q\approx0$ (neary constant uniform Ge concentration in the QW), near $q\approx k_{1}$ (lower harmonic/ Ge spike, only for sufficiently high $X_{\mathrm{ww}}$) and near $q\approx2k_{1}$ (long-period wiggle-well). The dashed line is again the separatrix. (c) Left: Epitaxial profile and ground state electron density distribution for the uniform Ge concentration in the well ($q=0$, $X_{\mathrm{ww}}=5\%$) at shear strain $\varepsilon_{x,y}=0.025\,\%$. The low value of the ratio $\nu/\left(2\sigma\right)$ indicates that this configuration falls into the disorder-dominated regime. Right: Power spectral density (PSD) of the product $\left(U_{\mathrm{QW}}\left(z\right)+U_{F}\left(z\right)\right)\left|\psi_{0}\left(z\right)\right|^{2}$ as a function of the wave number $k$. The contribution to the valley splitting is dominated by the $2k_{1}$-resonance and is thus larger than predicted by the $2k_{0}$-theory. The inset shows the statistical distribution of the intervalley coupling parameter in the complex plane in the range of $-200\,\text{$\mathrm{\text{\textmu}eV}$}\leq\mathrm{Re}\left(\Delta\right),\mathrm{Im}\left(\Delta\right)\leq200\,\mathrm{\text{\textmu}eV}$. (d) Same as (c), but at ($q=k_{1}$, $X_{\mathrm{ww}}=20\%$). The product of the potential and the wave function yields a strong Fourier component at $2k_{1}$, although the potential has only half of the frequency $q=k_{1}$. The configuration has a moderate deterministic enhancement $\nu/\left(2\sigma\right)\approx0.62$. (e) Same as (c), but for a long-period wiggle-well at ($q=2k_{1}$, $X_{\mathrm{ww}}=15\%$). The $q=2k_{1}$-periodic modulation of the potential provides a significant deterministic enhancement of the valley splitting $\nu/\left(2\sigma\right)\approx1.41$. A constant electric field $F=5\,\mathrm{mV}/\mathrm{nm}$ and biaxial (tensile) strain was assumed in all simulations.