Effect of disorder and strain on the operation of planar Ge hole spin qubits

TL;DR

This work addresses how disorder and gate-induced nonuniform strain affect planar Ge hole spin qubits in Ge/Si$_{1-x}$Ge$_x$ heterostructures. It introduces a multiscale framework combining atomistic valence-force-field strain, COMSOL-based gate-induced thermal strain, a $oldsymbol{k}oldsymbol{ullet}oldsymbol{p}$ model with nonuniform strain, and NEMO3D tight-binding validation to quantify EDSR in realistic devices. The key finding is that strain gradients generate a $k$-linear Dresselhaus SOC that dominates EDSR, yielding a broad range of $f_\pi$ (roughly $30{-}300$ MHz) with strong dependence on the magnetic-field and drive-field orientation, and achieving substantial improvements over uniform-strain models. This approach provides device-specific insights for optimizing hole spin qubits in Ge/SiGe heterostructures and offers a path toward more accurate predictions and scalable design rules for qubit operation.

Abstract

Germanium quantum dots in strained $\text{Ge}/\text{Si}_{1-x}\text{Ge}_{x}$ heterostructures exhibit fast and coherent hole qubit control in experiments. In this work, we theoretically and numerically address the effects of random alloy disorder and gate-induced strain on the operation of planar Ge hole spin qubits. Electrical operation of hole quantum dot spin qubits is enabled by the strong Rashba spin-orbit coupling (SOC) originating from the intrinsic SOC in the Ge valence band as well as from the structural inversion asymmetry inherent in the underlying 2D hole gas. We use the atomistic valence force field (VFF) method to compute the strain due to random alloy disorder, and thermal expansion models in COMSOL Multiphysics to obtain the strain from a realistic gate-stack of planar hole quantum dot confinement. Recently, spin-orbit coupling terms $\propto k$ have been shown to be induced by strain inhomogeneity. Our hybrid approach to realistic device modeling suggests that strain inhomogeneity due to both random alloy disorder and gate-induced strain make a strong contribution to the linear-$k$ Dresselhaus spin-orbit coupling, which eventually dominates hole spin EDSR; and there exist specific in-plane orientations of the global magnetic field $\mathbf{B}$ and the microwave drive $\mathbf{\tilde{E}}_{\text{ac}}$ for maximum EDSR Rabi frequency of the hole spin qubit. The current model including strain inhomogeneity accurately predicts the EDSR Rabi frequency to be $\!\sim\!100$ MHz for typical electric and magnetic fields in experiments, which represents at least an order of magnitude improvement in accuracy over phenomenological models assuming uniform uniaxial strain. State-of-the-art atomistic tight binding calculations via nano-electronic modeling (NEMO3D) are in agreement with the $\mathbf{k}{\cdot}\mathbf{p}$ description.

Figures (5)

• Figure 1: Design for a Ge hole quantum dot a) with realistic gatestack, b) cross-section of the design showing the 10 nm thick Ge layer between the lower and upper Ge/Si$_{0.2}$Ge$_{0.8}$ layers of thickness 60 nm and 10 nm, respectively. The first layer of Al source and drain gates (20 nm thick) sit atop the heterostructure, followed by a dielectric Al$_2$O$_3$ layer (17 nm), and a second layer of Pd/Ti electrodes (5/35 nm). The hole quantum dot forms directly under the Pd/Ti top gate(TG), as indicated by the oval shape in the Ge layer in b). The angular orientations of the applied in-plane magnetic field $\mathbf{B}$ and the ac electric field $\tilde{E}_{\text{ac}}$ with respect to the $[100]$ crystallographic axis is denoted by $\theta$ and $\phi$, respectively.
• Figure 2: Hole qubit EDSR Rabi frequency $f_\pi$ calculated from the k.p model under uniform uniaxial strain. The angles of $\mathbf{B}$ and $\Tilde{\mathbf{E}}_{\text{ac}}$ w.r.t. the [100] axis are denoted by $\theta$ and $\phi$, respectively. At $F_z{=}1.5$ MV/m (a-d) and $F_z{=}15$ MV/m (e-h)top-gate fields, with uniaxial strain assumption, $f_\pi$ is evaluated for a smaller dot first, a),e) $a_d{=}20$ nm, $\lvert B\rvert{=}200$ mT, b),f) $a_d{=}20$ nm, $\lvert B\rvert{=}670$ mT; and then for a larger dot, c),g) $a_d{=}45$ nm, $\lvert B\rvert{=}200$ mT, d),h) $a_d{=}45$ nm, $\lvert B\rvert{=}670$ mT. Here $a_d$ signifies the dot radius, and $\lvert B\rvert$ denotes the magnitude of the applied magnetic field. As suggested by the color bars, the EDSR Rabi frequency improves with increasing the dot radius as well as the applied $\mathbf{B}$ strength. $f_\pi(\theta,\phi)$ largely follows the trend of the dot product $\mathbf{B}\cdot\Tilde{\mathbf{E}}_{\text{ac}}$. As shown in d),h), for large dot at higher magnetic field the strong orbital vector potential terms cause the $f_\pi(\theta,\phi)$ trend to deviate from the dot product $\mathbf{B}\cdot\Tilde{\mathbf{E}}_{\text{ac}}$ relation. The microwave drive amplitude is given by $E_{\text{ac}}{=}10$ kV/m.
• Figure 3: tight binding simulation of the heterostructure profile in NEMO3D a) with random alloy disorder induced displacement vector field data of Ge atoms, calculated via the valence force field method. On the top panel, displacement amplitudes over a $150\times150\times10$$\text{nm}^3$ box around the quantum dot with in-plane radius $a_d{=}45$ nm and quantum well width $L_z{=}10$ nm is shown. A small section at the center of the dot is magnified to elaborate the randomness close to the buffer interfaces, as well as the variation of the displacement magnitudes across the quantum well. b) Variation of the uni-axial strain tensor components $\varepsilon_{xx},\,\varepsilon_{yy},\,\varepsilon_{zz}$ and shear strain tensor components $\varepsilon_{xy},\,\varepsilon_{xz},\,\varepsilon_{yz}$ with $z$ (across the QW). c) NEMO3D simulation of hole EDSR Rabi frequency at the top-gate field $F_z{=}1.5$ MV/m (top) and $F_z{=}15$ MV/m (bottom). The external magnetic field is $\lvert B\rvert{=}670$ mT. The $f_\pi(\theta,\phi)$ trend shows stark difference compared to the uniaxial case. The microwave drive amplitude is $|E_{\text{ac}}|{=}10$ kV/m.
• Figure 4: Cumulative strain profile in realistic Ge hole quantum dot. The spatial profile of the strain tensor components $\varepsilon_{ij}(x,y)$ in the Ge layer (z=60 nm cut) of the device in Fig. \ref{['fig1:sketch']} due to random alloy disorder as well as gate contraction. The features along the $y$-axis here are due to the Al source and drain electrodes as shown in the Fig. \ref{['fig1:sketch']} schematic. The atomic granularity comes from the random alloy disorder; while the long-range feature due to the gate contractions is signified by the COMSOL mesh data superimposed atop the atomistic VFF data.
• Figure 5: Hole qubit properties with full strain profile. For a circular dot of radius $a_d{=}45$ nm, a) The qubit $g$-factor is anisotropic and exhibits large $\sim \pi/4$ rotation of the principal axes. For all subfigures a-e), the top and bottom panels signify results for $F_z{=}1.5$ MV/m and $F_z{=}15$ MV/m, respectively. b) At $\lvert B\rvert{=}670$ mT, fast EDSR is predicted with gate time $\sim 10$ ns, and a similar $f_\pi(\theta,\phi)$ trend to the tight binding calculation is observed. Here, $|E_{\text{ac}}|{=}10$ kV/m. c) The $\propto \!\lvert B\rvert^2$ variation of the hole qubit dephasing rate ($(T_{2,RTN}^*)^{-1}$) due to the random telegraph noise from a single charge defect, with uniaxial strain (red) and inhomogeneous strain (green). The single charge defect is assumed to be situated at (50$\sqrt{2}$,50$\sqrt{2}$,0). d) The linear variation of the EDSR Rabi frequency $f_\pi$ w.r.t. the magnetic field strength $\lvert B\rvert$. In the presence of strain inhomogeneity (denoted by 'ih' on the green axes), $f_\pi$ is $\sim100\times$ higher than the uniaxial strain scenario (red). e) Relaxation rate ($\Gamma_1$) as a function of magnetic field strength.