Protected valley splitting against interface disorder toward scalable silicon electron spin qubits

TL;DR

The paper tackles the problem of small and highly variable valley splitting in Si/SiGe quantum wells caused by atomic-scale interface disorder. It shows that CMOS-compatible uniaxial shear strain can robustly enhance valley splitting and suppress disorder-induced variability by activating a new inter-band coupling channel ($2k_1$) between bulk valleys, in addition to the conventional intra-band channel ($2k_0$). An atomistic pseudopotential framework plus an envelope-function model demonstrates that the inter-BZ coupling becomes dominant under strain and is remarkably resilient to interface steps and alloy disorder. This provides a practical path toward gate uniformity and scalability for silicon-based spin qubits and quantum processors.

Abstract

Regardless of various material design strategies, experimentally achieving substantial and controllable valley splitting in Si/SiGe quantum wells remains a central challenge for ensuring high gate uniformity. This difficulty arises from unavoidable atomic-scale disorder at the interface, caused by alloy randomness, which suppresses valley splitting and, more critically, induces large variations. Here, we demonstrate that CMOS-compatible uniaxial strain can substantially enhance valley splitting, rendering it immune to interface disorder. Atomistic pseudopotential calculations show that uniaxial strain linearly restores the valley splitting suppressed by interfacial disorder, with a large enhancement rate, while keeping disorder-induced variations within a narrow distribution. We reveal that uniaxial strain introduces a new coupling channel between bulk valleys in adjacent Brillouin zones through a small momentum transfer, which markedly reduces the susceptibility of valley splitting to interfacial disorder. These findings establish a viable route to improve gate uniformity in silicon-based spin qubits, paving the way for scalable quantum processors.

Figures (7)

• Figure 1: (a) Valley splitting energy in (001)-oriented $53~\mathrm{ML}$-$\mathrm{Si}$ QWs calculated using the atomistic SEPM. Black solid square denotes the valley splitting in the ideal $\mathrm{Si} / \mathrm{Ge}$ QW, while black solid triangles represent those in $\mathrm{Si} / \mathrm{SiGe}$ QWs with interfacial alloy disorder. The different lines correspond to three SiGe alloy configurations. Red crosses indicate experimentally reported valley splittings in $\mathrm{Si}/\mathrm{SiGe}$ QWs with interfacial alloy disorder PhysRevApplied.13.034068PhysRevLett.128.146802paqueletwuetzAtomicFluctuationsLifting2022. An electric field of $10~\mathrm{MV} / \mathrm{m}$ is applied to drive the electron to the upper interface. (b) Valley splitting energy in (001)-oriented $\mathrm{Si}(53 \mathrm{ML}) /\left(\mathrm{Ge}_4 \mathrm{Si}_4\right)_4$ QW calculated using the atomistic SEPM. An electric field of $10~\mathrm{MV} / \mathrm{m}$ is applied in the [001] direction. The square represents the ideal QW with an atomically flat interface. Scattered triangles mark the presence of interface alloy disorder, where every single $\mathrm{Si} / \mathrm{Ge}$ interface introduces a broadening width, set at 2 ML.
• Figure 2: Schematic illustration of the valley coupling mechanism. The right panel shows the conduction band edges of bulk Si along $k_z \parallel [001]$ over two adjacent Brillouin zones, including the $\Delta_1$ and $\Delta_{2^{\prime}}$ bands. Solid and dashed curves correspond to the cases without ($\varepsilon_{xy}=0$) and with ($\varepsilon_{xy}\neq 0$) shear strain, respectively. $V_{+-}^{\Gamma}$ and $V_{+-}^X$ denote the intra-BZ and inter-BZ valley couplings. On the left, three boxes highlight successive steps that link the bulk band structure to valley splitting in Si QWs. Box-I (“band folding”) illustrates the folding of the bulk $\Delta_1$ and $\Delta_{2^{\prime}}$ bands at $\pm k_0$ onto the $\Gamma$ point, resulting in pairs of degenerate parabolic bands along $k_x \parallel [100]$. Box-II (“confinement”) shows that quantum confinement quantizes these folded bands, yielding discrete orbital states. Here, the abrupt interface potential provides Fourier components at $2k_0$ and $2k_1$, which lift the degeneracy of the lowest $\pm k_0$ states and introduce additional inter-band couplings (red); while the $2k_0$ process (solid) leads directly to valley splitting, the $2k_1$ process (dashed) leaves the lowest two levels unaffected. Finally, Box-III (“shear strain”) illustrates how shear strain activates the $2k_1$ channel, allowing it to contribute indirectly to the interaction between the lowest $\pm k_0$ states (solid blue line) and thereby enhancing the valley splitting.
• Figure 3: Dependence of inter-BZ valley coupling $V_{+-}^X$ (blue) and intra-BZ valley couplings $V_{+-}^{\Gamma}$ (red) on shear strain in a 53 ML-thick Si QW. Solid squares represent the scenario of an ideal Si QW, and solid triangles correspond to Si QWs incorporating various interfacial alloy configurations, corresponding to the representative cases in Fig. 1(a). $V_{+-}^X$ is extracted using Eq. \ref{['eq.vs']}, with atomistically calculated $E_{\mathrm{VS}}$ as input. $V_{+-}^{\Gamma}$ remains constant with strain, corresponding to $E_{\mathrm{VS}} / 2$ in the absence of shear strain.
• Figure 4: (a) Valley splitting energy in (001)-oriented $53~\mathrm{ML}$-$\mathrm{Si}/\mathrm{Ge}$ QWs calculated using the atomistic SEPM. Black solid squares represent results for $\mathrm{Si} / \mathrm{Ge}$ QWs with atomically flat interfaces, while black solid circles depict the case with interface steps. Red crosses indicate experimentally reported valley splittings in $\mathrm{Si}/\mathrm{SiGe}$ QWs with interfacial alloy disorder PhysRevApplied.13.034068PhysRevLett.128.146802paqueletwuetzAtomicFluctuationsLifting2022. The solid/dashed lines represent fits based on Eq. \ref{['eq.vs']}. The inset illustrates the configuration of a single-step interface. (b) Dependence of inter-BZ valley coupling $V_{+-}^X$ (blue) and intra-BZ valley couplings $V_{+-}^{\Gamma}$ (red) on shear strain in a 53 ML-thick Si QW. Solid squares represent the scenario of an ideal Si QW, and solid circles correspond to Si QWs with the interface step. A uniform electric field of $10~\mathrm{MV} / \mathrm{m}$ is applied during the atomistic SEPM simulation.
• Figure S1: Wave functions of six lowest electronic bound states in a Si(53 ML)/Ge QW subjected to an electric field of $10~\mathrm{MV} / \mathrm{m}$ along the [001] direction. Panels (a-c) depict Si/Ge QW structures at different shear strains. The red and blue lines at specific energy levels represent valley pairs of different orbital states $|n, \pm\rangle$. The zero energy reference is set to the CBM of bulk Si.