Electrical Spin Driving by $g$-Matrix Modulation in Spin-Orbit Qubits

In a semiconductor spin qubit with sizable spin-orbit coupling, coherent spin rotations can be driven by a resonant gate-voltage modulation. Recently, we have exploited this opportunity in the experimental demonstration of a hole spin qubit in a silicon device. Here we investigate the underlying physical mechanisms by measuring the full angular dependence of the Rabi frequency, as well as the gate-voltage dependence and anisotropy of the hole $g$ factor. We show that a $g$-matrix formalism can simultaneously capture and discriminate the contributions of two mechanisms so far independently discussed in the literature: one associated with the modulation of the $g$ factor, and measurable by Zeeman energy spectroscopy, the other not. Our approach has a general validity and can be applied to the analysis of other types of spin-orbit qubits.

Figure 1

(a) Scanning electron micrograph of a device similar to the one measured, with false colors. The white dashed lines outline the Si nanowire, 25 nm large and 8 nm thick, extending between $S$ and $D$ (yellow). Gates $G1$ and $G2$ (brown) are 35 nm long and separated by ${\mathrm{Si}}_3{\mathrm{N}}_4$ spacers (cyan); microwave pulses are applied to $G2$. (b) Bias triangles of the double dot under study at $V_d=5\mathrm{mV}$. The quoted numbers denote the equivalent excess charges inferred from Pauli blockade; $V_0$ and $V_1$ mark points where spins are manipulated during pulse sequences. (c) Spectroscopy of the spin-orbit qubit under Pauli spin blockade [yellow star in (b)]; $I_d$ is plotted as a function of microwave frequency $f$ and $|\mathit{B}=(B,0,0)|$. The slope of the resonance lines gives $g_L^{*}$ and $g_R^{*}$. (d) Rabi oscillations as a function of microwave burst time $\tau$ for $\mathit{B}=(0,0,-0.36)\mathrm{T}$. A current offset of 200 fA is subtracted for clarity.

Figure 2

(a) Inset: Current trace as a function of microwave frequency $f$ for burst duration $\tau =20\mathrm{ns}$ at $\mathit{B}=(0,216,216)\mathrm{mT}$. We fit with $I_d\propto (f_r^2/a){\sin }^2(\pi \tau \sqrt{a})$, where $a=(f-f_0{)}^2+f_r^2$, with $f_r$, $f_0=|g^{*}|{\mu }_{B}B/h$ the Rabi and resonance frequency, respectively [28]. Blue and orange histograms display the dispersion of $f_0$ when $V_{G2}$ is pulsed to point $V_0$ and $V_1$, respectively. The Gaussian fits yield $g^{*}(V_0)=2.013\pm 0.001$ and $g^{*}(V_1)=2.010\pm 0.001$ (including the field uncertainty). (b) Typical isosurface of $\mathrm{\Delta }{E}^2=g^{*2}{\mu }_B^2{B}^2$ in the measurement frame $\{\mathit{x},\mathit{y},\mathit{z}\}$. The blue lines are the principal magnetic axes $\mathit{X}$, $\mathit{Y}$, and $\mathit{Z}$ of the ellipsoids at $V_{G2}=V_0$. (c) $g^{*}$ as a function of the field angles $\theta$ and $\phi$, reconstructed from six values of $g^{*}$ uniquely defining the ellipsoids. (d) Experimental cartography of $g^{*}$. $\theta$ and $\phi$ are stepped by 10° between 0° and 180°. The whole plot is obtained by symmetry ($\mathit{B}\to -\mathit{B}$).

Figure 3

(a) Angular dependence of the Rabi frequency. The two strong minima [($\phi \approx 0°$, $\theta \approx 90°$) and ($\phi \approx 180°$, $\theta \approx 90°$)] suggest that the spin-orbit field ${\mathit{B}}_{\mathrm{SO}}$ is in plane, perpendicular to nanowire axis (along $\mathit{x}$). The map is recorded at constant Zeeman splitting; see the main text. (b) Analogous map obtained from the fit of (a) with Eq. (3).

Figure 4

(a) IZ-EDSR and (b) $g$-TMR contributions to $f_r$. IZ-EDSR is the most relevant contribution, though $g$-TMR cannot be neglected, as it can reach $\sim 30\%$ of the experimental Rabi frequencies. Since $|{\mathit{f}}_r|\le |{\mathit{f}}_{\mathrm{TMR}}|+|{\mathit{f}}_{\mathrm{IZR}}|$, the measured values of Fig. 3 are not exactly the sum of plots (a) and (b) (see [30]).