Optimization of Si/SiGe Heterostructures for Large and Robust Valley Splitting in Silicon Qubits

TL;DR

The paper tackles the problem of small and device-to-device fluctuating valley splitting in Si/SiGe quantum wells used for silicon spin qubits. It introduces a variational optimization framework for designing Ge concentration profiles within the epitaxial stack, incorporating strain and random alloy disorder through a multi-valley envelope-function theory. By tuning objectives and enforcing feasibility via spectral and budget constraints, the method recovers known designs and proposes a novel modulated wiggle well that achieves large, reproducible valley splitting with wide tunability via a vertical electric field. This advances scalable silicon quantum architectures by enabling switchable qubits with on-demand valley splitting while controlling disorder-induced variability.

Abstract

The notoriously low and fluctuating valley splitting is one of the key challenges for electron spin qubits in silicon (Si), limiting the scalability of Si-based quantum processors. In silicon-germanium (SiGe) heterostructures, the problem can be addressed by the design of the epitaxial layer stack. Several heuristic strategies have been proposed to enhance the energy gap between the two nearly degenerate valley states in strained Si/SiGe quantum wells (QWs), e.g., sharp Si/SiGe interfaces, Ge spikes or oscillating Ge concentrations within the QW. In this work, we develop a systematic variational optimization approach to compute optimal Ge concentration profiles that boost selected properties of the intervalley coupling matrix element. Our free-shape optimization approach is augmented by a number of technological constraints to ensure feasibility of the resulting epitaxial profiles. The method is based on an effective-mass-type envelope-function theory accounting for the effects of strain and compositional alloy disorder. Various previously proposed heterostructure designs are recovered as special cases of the constrained optimization problem. Our main result is a novel heterostructure design we refer to as the "modulated wiggle well," which provides a large deterministic enhancement of the valley splitting along with a reliable suppression of the disorder-induced volatility. In addition, our new design offers a wide-range tunability of the valley splitting ranging from about 200 $μ$eV to above 1 meV controlled by the vertical electric field, which offers new perspectives to engineer switchable qubits with on-demand adjustable valley splitting.

Figures (6)

• Figure 1: (a) Brillouin zone of the face centered cubic crystal with six valley states along the edges between the $\Gamma$ point and the six $X$ points. In biaxially strained Si/SiGe QWs, the four valley states along the $k_{x}$ and $k_{y}$ directions (blue) are energetically separated from the two low-energy valleys along the $k_{z}$ directions (red). (b) Schematic illustration of the ground and excited valley state wave function (including the rapidly oscillating Bloch factors) in the effective confinement potential of a $\mathrm{Si}/\mathrm{Si}_{0.7}\mathrm{Ge}_{0.3}$ QW. The energy splitting $E_{\mathrm{VS}}$ between the two valley states is usually small in conventional QWs with smooth interfaces. Disorder in the SiGe alloy leads to a significant volatility of the valley splitting. The inset shows a typical realization of alloy disorder in a conventional Si/SiGe QW.
• Figure 2: Optimization results for a fixed Ge budget $x_{\mathrm{Ge}}=5\%$ distributed within the QW. (a) Optimization with the cost functional $J_{0}^{\left(A\right)}$ (maximum deterministic component) at a cutoff wave number $k_{c}=0.5\times2\pi/a_{0}$ yield a modulated wiggle well. (b) A similar structure is found when optimizing with the cost functional $J_{0}^{\left(B\right)}$ (reliable enhancement) at the same cutoff wave number. (c) Optimization with the cost functional $J_{0}^{\left(C\right)}$ (minimum random component) at the same $k_{c}$ yields a narrow well. (d) Epitaxial profile with a Ge spike obtained by maximizing the deterministic component (with cost functional $J_{0}^{\left(A\right)}$ ) at a reduced cutoff wave number $k_{c}=0.07\times2\pi/a_{0}$. (e) Conventional wiggle well with wave number $2k_{1}$ for comparison. In all plots, the first column shows the epitaxial Ge profile and the second column is the potential energy and ground state envelope wave function at $F=5\,\mathrm{mV/nm}$. The third column shows the power spectral density (PSD) of the function $S\left(z\right)$, see Eq. \ref{['eq: function S']}, and the fourth column illustrates the statistical distribution of the intervalley coupling parameter $\Delta$ in the complex plane.
• Figure 3: Modulated wiggle well optimizing the reliable enhancement (B) for different total Ge concentrations $x_{\mathrm{Ge}}$ in the QW. (a) Optimized epitaxial profiles for different Ge budgets $x_{\mathrm{Ge}}=\left\{ 2\%,4\%,6\%,8\%,10\%\right\}$. (b) Deterministic $\nu$ and disorder-induced contributions $\sqrt{2\Gamma}$ to the intervalley coupling parameter for the modulated wiggle wells optimized at different Ge concentrations. The labels indicate the respective Ge concentration in percent.
• Figure 4: (a) Mean valley splitting $\langle E_{\mathrm{VS}}\rangle$ of the modulated wiggle well (red), the Ge spike (green) and the conventional long-period wiggle well (blue) as a function of the applied electric field $F$. All structures contain in total $5\%$ Ge in the QW domain. Due to the asymmetry in their epitaxial profiles, the modulated wiggle well and the Ge spike exhibit a strong, non-symmetric field dependency, which allows for tuning into a high and low valley splitting regime. For comparison, the field dependency of the conventional wiggle well is only weak and fully symmetric under inversion of the field strength $F\leftrightarrow-F$. The shaded regions indicate the [25% ,75%] percentile of the Rice distribution (\ref{['eq: Rice dist']}). (b) Sensitivity of the mean valley splitting on the electric field $\partial\langle E_{\mathrm{VS}}\rangle/\partial F$ as a function of the applied electric field. The field sensitivity of two non-symmetric structures is peaked around the design field strength $F_{\mathrm{opt}}$. In contrast, the conventional wiggle well shows only a weak field dependency. (c) Cross section of the total potential energy landscape and the electronic ground state envelope wave function for the modulated wiggle well at different electric fields. (d) Same cross sections for the Ge spike. The localization of the envelope wave function is strongly dependent on the electric field such that different parts of the non-symmetric epitaxial profiles are probed in the low and high valley splitting regimes.
• Figure 5: Optimization under inverted objectives. (a) Minimization of the deterministic component (D) leads to a (nearly) uniform Ge distribution in the QW. The PSD (third column) shows a sharp dropout around $2k_{1}$, such that the resulting valley splitting is fully disorder-induced. (b) Maximization of the disorder-induced component (E) of the intervalley coupling parameter yields a profile similar to the uniform Ge distribution, but with an increased Ge content in the region that overlaps with the envelope wave function. (c) Maximization of the valley splitting variance (F) leads to a wiggle-well type structure with non-zero baseline Ge concentration.