Hole spin manipulation in inhomogeneous and nonseparable electric fields

The usual models for electrical spin manipulation in semiconductor quantum dots assume that the confinement potential is separable in the three spatial dimensions and that the ac drive field is homogeneous. However, the electric field induced by the gates in quantum dot devices is not fully separable and displays significant inhomogeneities. Here we address the electrical manipulation of hole spins in semiconductor heterostructures subject to inhomogeneous vertical electric fields and/or in-plane ac electric fields. We consider Ge quantum dots electrically confined in a Ge/GeSi quantum well as an illustration. We show that the lack of separability between the vertical and in-plane motions gives rise to an additional spin-orbit coupling mechanism (beyond the usual linear and cubic in momentum Rashba terms) that modulates the principal axes of the hole gyromagnetic $g$ matrix. This nonseparability mechanism can be of the same order of magnitude as Rashba-type interactions, and enables spin manipulation when the magnetic field is applied in the plane of the heterostructure even if the dot is symmetric (disk shaped). More generally, we show that Rabi oscillations in strongly patterned electric fields harness a variety of $g$-factor modulations. We discuss the implications for the design, modeling, and understanding of hole spin qubit devices.

Figure 1

The hole spin qubit device. The Ge quantum well (red) is embedded between ${\mathrm{Ge}}_{0.8}{\mathrm{Si}}_{0.2}$ barriers (light blue). The difference of potential between the central front gate (C) and the grounded side gates (L/R/T/B) shapes a hole quantum dot in this well. Unless otherwise specified, the Ge well is $L_{\mathrm{W}}=16$ nm thick, the upper barrier $L_{\mathrm{B}}=50$ nm thick, the diameter of the C gate is $d=100$ nm, and the gap with the side gates is $s=20$ nm. All gates are embedded in ${\mathrm{Al}}_2{\mathrm{O}}_3$, and are insulated from the heterostructure by 5 nm of this material. The substrate below the 150-nm-thick lower barrier acts as an effective back gate, used to independently tune the depth of the quantum dot and the vertical electric field. We assume, as in Ref. [37], that the ${\mathrm{Ge}}_{0.8}{\mathrm{Si}}_{0.2}$ barriers are not fully relaxed, and experience residual in-plane strain ${\varepsilon }_{xx}={\varepsilon }_{yy}={\varepsilon }_{\parallel }=0.26\%$ and out-of-plane strain ${\varepsilon }_{zz}={\varepsilon }_{\perp }=-0.19\%$. Consequently, the strains in the Ge well are ${\varepsilon }_{\parallel }=-0.63\%$ and ${\varepsilon }_{\perp }=+0.47\%$. The yellow contour is the isodensity surface that encloses 90% of the ground-state hole charge at $V_{\mathrm{C}}=-40$ mV and $V_{\mathrm{bg}}=0$ V.

Figure 2

(a) Map of the Rabi frequency $f_{\mathrm{R}}$ as a function of the magnetic field angles $\theta$ and $\phi$ defined in Fig. 1, for opposite drives $\delta V_{\mathrm{L}}=-\delta V_{\mathrm{R}}=(V_{\mathrm{ac}}/2)cos{\omega }_{\mathrm{L}}t$ ($V_{\mathrm{C}}=-40$ mV, $V_{\mathrm{bg}}=0$ V, $V_{\mathrm{ac}}=1$ mV and $B=1$ T). (b) Cut along the dashed gray line in (a) at constant magnetic field $B=1$ T (green) and at constant Larmor frequency $f_{\mathrm{L}}={\omega }_{\mathrm{L}}/\left(2\pi \right)=5$ GHz (orange).

Figure 3

(a) Map of the electrical confinement potential $V(x,z)$ in the $xz$ symmetry plane at $y=0$, computed at $V_{\mathrm{C}}=-40$ mV and $V_{\mathrm{bg}}=0$ V. (b), (c) Components of the electric field $E_x=-\partial V(x,z)/\partial x$ and $E_z=-\partial V(x,z)/\partial z$ (close up on the dot area). (d) Map of the ac potential $V_{\mathrm{ac}}(x,z)$ in the $xz$ plane at $y=0$, computed for a $V_{\mathrm{L}}=-1$ mV pulse on the L gate. (e), (f) Components of the ac electric field $E_{\mathrm{ac},x}=-\partial V_{\mathrm{ac}}(x,z)/\partial x$ and $E_{\mathrm{ac},z}=-\partial V_{\mathrm{ac}}(x,z)/\partial z$ (close-up on the dot area). The blue lines delineate the different materials. The origin of coordinates is at the middle of the well along the C gate symmetry axis.

Figure 4

(a) Map of the Rabi frequency $f_{\mathrm{R}}(\mathbf{B}\parallel \mathbf{x})$ at constant magnetic field $B=1$ T as a function of $V_{\mathrm{C}}$ and $V_{\mathrm{bg}}$ (opposite drives on the L and R gates). (b) Same for $f_{\mathrm{R}}(\mathbf{B}\parallel \mathbf{z})$. Note the different scales in (a) and (b). The hole is pulled by the electric field from the well to the top SiGe/${\mathrm{Al}}_2{\mathrm{O}}_3$ interface in the white areas.

Figure 5

(a) Correlations between the Rabi frequency $f_{\mathrm{R}}(\mathbf{B}\parallel \mathbf{x})$ at constant magnetic field and the dot size ${\ell }_z$, collected from the map of Fig. 4 (opposite drives on the L and R gates, NS mechanism). Dotted black lines connect points at the same $V_{\mathrm{C}}$, while symbol colors label points at the same $V_{\mathrm{bg}}$. The dashed orange line is a guide to the eye with slope $s=4$. (b) Same for $f_{\mathrm{R}}(\mathbf{B}\parallel \mathbf{x}+\mathbf{y})$ versus dot size $r_{\parallel }$ and a drive on the L gate only, collected from Fig. 7 (conventional $g$-TMR plus NS mechanism). The dashed orange line is a guide to the eye with slope $s=2$.

Figure 6

Rabi frequency for an in-plane magnetic field ($\theta ={90}^{\circ })$ as a function of $\phi$ for constant $f_{\mathrm{L}}=5$ GHz and different $V_{\mathrm{C}}$'s ($V_{\mathrm{bg}}=0$ V). The hole is driven with opposite modulations $\delta V_{\mathrm{L}}=-\delta V_{\mathrm{R}}=(V_{\mathrm{ac}}/2)cos{\omega }_{\mathrm{L}}t$ on the L and R gates. (b) Same for a hole driven by a modulation $\delta V_{\mathrm{L}}=V_{\mathrm{ac}}cos{\omega }_{\mathrm{L}}t$ on the L gate only.

Figure 7

(a) Map of the Rabi frequency $f_{\mathrm{R}}$ as a function of the magnetic field angles $\theta$ and $\phi$ defined in Fig. 1, for a drive $\delta V_{\mathrm{L}}=V_{\mathrm{ac}}cos{\omega }_{\mathrm{L}}t$ on the L gate only ($V_{\mathrm{C}}=-40$ mV, $V_{\mathrm{bg}}=0$ V, $V_{\mathrm{ac}}=1$ mV and $B=1$ T). (b) Cut along the dashed gray line in (a), at constant magnetic field $B=1$ T (green), and at constant Larmor frequency $f_{\mathrm{L}}={\omega }_{\mathrm{L}}/\left(2\pi \right)=5$ GHz (orange). (c) Map of the Rabi frequency $f_{\mathrm{R}}(\mathbf{B}\parallel \mathbf{x}+\mathbf{y})$ at constant magnetic field $B=1$ T as a function of $V_{\mathrm{C}}$ and $V_{\mathrm{bg}}$ (drive on the L gate only). The hole is pulled by the electric field from the well to the top SiGe/${\mathrm{Al}}_2{\mathrm{O}}_3$ interface in the white areas.

Figure 8

(a) Rabi frequency $f_{\mathrm{R}}(\mathbf{B}\parallel \mathbf{x})$ at constant $f_{\mathrm{L}}=5$ GHz as a function of the upper barrier thickness $L_{\mathrm{B}}$ for different $V_{\mathrm{C}}$'s and $V_{\mathrm{bg}}=0$ V (opposite drives on L and R gates, NS mechanism). (b) Same for $f_{\mathrm{R}}(\mathbf{B}\parallel \mathbf{x}+\mathbf{y})$ (drive on the L gate only, conventional $g$-TMR plus NS mechanism).

Figure 9

(a) Map of the Rabi frequency $f_{\mathrm{R}}$ as a function of the magnetic field angles $\theta$ and $\phi$ defined in Fig. 1 for a drive $\delta V_{\mathrm{C}}=V_{\mathrm{ac}}cos{\omega }_{\mathrm{L}}t$ on the C gate ($V_{\mathrm{C}}=-40$ mV, $V_{\mathrm{bg}}=0$ V, $V_{\mathrm{ac}}=1$ mV, and $B=1$ T). (b) Cut along the dashed gray line in (a) at constant magnetic field $B=1$ T (green) and at constant Larmor frequency $f_{\mathrm{L}}={\omega }_{\mathrm{L}}/\left(2\pi \right)=5$ GHz (orange).