Anisotropy reduction and tunability of hole-spin qubit g-factor in strained parabolic Ge/SiGe quantum wells

Abstract

Hole-spin qubits in planar Ge/SiGe heterostructures have attracted significant attention in recent years owing to their favorable electrical characteristics and prolonged coherence times. However, the strong spin-orbit interaction also makes them susceptible to charge noise and inhomogeneous strain. This is further exacerbated by the highly anisotropic g-factor of the planar design. Although there are some known strategies to suppress charge noise, one approach is to engineer an isotropic g-factor. In this work we analyze how qubit confinement profile affects the g-factor of hole-spin qubits. We show that decreasing the characteristic in-plane qubit confinement length reduces the g-factor anisotropy. We perform analytical and numerical analysis to compare two types of quantum wells: square wells and parabolic wells. We show that square wells have limited tunability, while parabolic wells offer broader tunability, making them more promising for qubit engineering.

Figures (10)

• Figure 1: A plot of the ground state hole wavefunction probability (charge density) along the x-z plane for a $40$ lattice constant ($a=a_{\ce{Si_{0.2}Ge_{0.8}}}=0.5613~\text{nm}$) thick quantum well with (a) a $20\,\text{nm}$ and (b) a $5\,\text{nm}$ full-width-half-maximum lateral confinement. The anisotropic nature of the hole effective mass skews the wavefunction along one direction. This asymmetry can be made symmetric by tuning the confinement strength which, intuitively, should mitigate the anisotropy in certain geometry-sensitive properties such as the effective hole g-factor.
• Figure 2: A schematic diagram of the heterostructure. A Ge-rich layer, which can be either a square or a parabolic well, is embedded between $\ce{Si_{0.2}Ge_{0.8}}$ barriers and grown atop a relaxed $\ce{Si_{1-c_0}Ge_{c_0}}$ buffer layer. The structure is capped by an insulating oxide layer, above which voltage gates are placed to provide additional electrostatic confinement to produce the dot.
• Figure 3: A plot of the strain $\varepsilon_\parallel(z)$ for both the square well (solid) and the parabolic well (dashed). The heterostructure is strain-balanced using Eq. \ref{['eq:strain-balancing']} as constraint. This prevents the formation of dislocations by matching the average lattice constant of the heterostructure to that of the buffer. Since the well has a higher Ge concentration relative to the relaxed buffer, it is compressive strained while the barrier region is tensile strained. The strain transition point is abrupt in a square well and occurs at the well-barrier interface. In contrast, the Ge concentration gradient in a parabolic well is more gradual which allows it to distribute the strain more evenly. This moves the strain transition point inside the well region instead of being at the interface.
• Figure 4: A plot of the effective qubit width as a function of the quantum well width $d_w$. Here we consider the idealized quantum wells provided by Eqs. \ref{['eq:ideal-square']} and \ref{['eq:ideal-para']}. In the case of a square well, the qubit width initially increases linearly with $d_w$ as indicated in Eq. \ref{['eq:square-scaling']}. When $d_w \in \left[19.5\,\text{nm},38.3\,\text{nm}\right]$, the system enters a hybridized regime where the effect of $V_\text{bias}$ can no longer be treated perturbatively. Beyond the transition region, the system is purely dominated by $V_\text{bias}$. In this triangular well limit, the qubit width saturates to a fixed value determined by the electric field strength $F$. In contrast, the qubit width in the parabolic well is accurately estimated by Eq. \ref{['eq:parabolic-scaling']} for both values of $F$. This is because $V_\text{bias}$ only shifts the energy and position of the qubit while keeping its width the same, as we can see from Eq. \ref{['eq:parabolic-potential-rewrite']}. It is important to note, however, that the behavior of a realistic finite well potential is significantly different from this idealized case for a sufficiently large $d_w$.
• Figure 5: Contour plots of effective g-factor and anisotropies for the square and parabolic quantum well. The square well g-factor anisotropy plateaus for large $d_w$ because the system becomes dominated by the triangular potential $V=-e F z$. In contrast, the parabolic well continuously changes with $d_w$, demonstrating the potential for qubit engineering. Importantly, we verify our intuition in both well types that increasing the lateral confinement strength to make $l_z \approx l_\parallel$ reduces the g-factor anisotropy.