Longitudinal and transverse electric field manipulation of hole spin-orbit qubits in one-dimensional channels

Holes confined in semiconductor nanostructures realize qubits where the quantum-mechanical spin is strongly mixed with the quantum orbital angular momentum. The remarkable spin-orbit coupling allows for fast all electrical manipulation of such qubits. We study an idealization of a cmos device where the hole is strongly confined in one direction (thin film geometry), while it is allowed to move more extensively along a one-dimensional channel. Static electric bias and ac electrical driving are applied by metallic gates arranged along the channel. In quantum devices based on materials with a bulk inversion symmetry, such as silicon or germanium, there exist different possible spin-orbit coupling based mechanisms for qubit manipulation. One of them, the $g$-tensor magnetic resonance, relies on the dependence of the effective $g$-factors on the electrical confinement. In this configuration, the hole is driven by an ac field parallel to the static electric field and perpendicular to the channel (transverse driving). Another mechanism, which we refer to here as iso-Zeeman electric dipole spin resonance, is due to the Rashba spin-orbit coupling that leads to an effective time-dependent magnetic field experienced by the pseudospin oscillating along the quantum channel (longitudinal driving). We compare these two modes of operation, and we describe the conditions in which the magnitudes of the Rabi frequencies are the largest. Different regimes can be attained by electrical tuning where the coupling to the ac electric field is made either weak or strong. Spin-orbit coupling can also be tuned by strains, with, in particular, a transition from a mostly heavy- to a mostly light-hole ground state for in-plane tensile strains. Although large strains always reduce the Rabi frequency, they may increase the qubit lifetimes even faster, which calls for a careful optimization of strains and electric fields in the devices. We also discuss the choice of channel material and orientation. The study is relevant to the interpretation of the current experiments on the manipulation of hole qubits and as a guide to the development of quantum devices based on silicon and germanium.

Figure 1

(a) Three-dimensional perspective with a system of coordinates and alignment with respect to the crystallographic axes. The structure has the smallest length $L_z$ along the direction of strong confinement $z=\left[001\right]$. (b) Energy potential profiles along the $x$, $y$, and $z$ directions (see the main text). A static electric field ${\mathcal{E}}_y$ is applied along $y$. ac electric fields are applied along $y$ and $x$ and lead to the $g$-TMR and IZ-EDSR effects, respectively. (c) Color maps of the probability densities in the $\left(xy\right)$ and the $\left(yz\right)$ planes that follow from the fundamental envelope functions numerically computed with the four-band LK model [39]. In this calculation, $L_z=10\mathrm{nm}$, $L_y=30\mathrm{nm}$, the electric field is ${\mathcal{E}}_y=0.5\mathrm{mV}/\mathrm{nm}$, and $x_0\equiv {\left(\pi \hslash \right)}^{1/2}/{\left(m_{0}K\right)}^{1/4}=10\mathrm{nm}$ [note that $x_0$ is introduced here for convenience as a mass-independent characteristic length; the actual extent of the ground-state wave function of heavy and light holes along $x$ is given by Eq. (12)].

Figure 2

(a) The function $\mathit{F}\left({\alpha }_{\parallel }\right)=\langle k_y^2\rangle L_y^2/{\pi }^2$ and (b) its derivative ${\mathit{F}}^{\prime }\left({\alpha }_{\parallel }\right)$, where ${\alpha }_{\parallel }$ is the dimensionless parameter defined by Eq. (10). The blue curve corresponds to the numerical calculation; the orange and green curves correspond to the weak electric field and the strong electric field asymptotics [Eq. (11)], respectively.

Figure 3

Inverse spin-orbit length as a function of the static electric field along $y$. We compare the semianalytical formula Eq. (C5) of Appendix pp3 (solid line) with the linear approximation Eq. (23) (dashed line) for a silicon channel with dimensions $L_z=4\mathrm{nm}$ and $L_y=30\mathrm{nm}$.

Figure 4

Maps of the Rabi frequency for the $g$-TMR effect [maps (a) and (b)] and the IZ-EDSR effect [maps (c) and (d)] as a function of the lateral electric field ${\mathcal{E}}_y$ and biaxial strain ${\varepsilon }_{\parallel }$. The maps are obtained from a numerical solution of the four-band $k·p$ model [39] in silicon. The lengths that characterize the lateral confinement are $x_0\equiv {\left(\pi \hslash \right)}^{1/2}/{\left(m_{0}K\right)}^{1/4}=10\mathrm{nm}$ and $L_y=30\mathrm{nm}$. The height of the semiconductor channel is $L_z=4\mathrm{nm}$ for maps (a) and (c), and it is $L_z=10\mathrm{nm}$ for maps (b) and (d). The dashed black lines outline the constant strains and electric field cuts shown in Figs. 5 and 6. The red dashed lines mark the critical strain ${\varepsilon }_{\parallel }^{*}$ that separates the mostly heavy-hole from the mostly light-hole ground state.

Figure 5

Comparison of the maximal $g$-TMR and the IZ-EDSR Rabi frequencies as given by Eq. (14), the $g$-factors Eqs. (B9), (B13), and (25) (dashed lines), with numerical calculations based on the four-band $k·p$ model [39] (full lines). The material is silicon and the parameters are ${\varepsilon }_{\parallel }=0\%$ for (a) and (b), ${\varepsilon }_{\parallel }=0.7\%$ for (c) and (d), $L_z=4\mathrm{nm}$ for (a) and (c), $L_z=10\mathrm{nm}$ for (b) and (d), $B=1\mathrm{T}$, ${\mathcal{E}}_{x/y}^{\text{ac}}=(1/30)\mathrm{mV}/\mathrm{nm}$, $L_y=30\mathrm{nm}$, and $x_0=10\mathrm{nm}$.

Figure 6

$g$-TMR [(a) and (b)] and IZ-EDSR [(c) and (d)] Rabi frequency dependence on biaxial strain. We compare the numerics [39] (full line with symbols) with the semianalytical formulas (dashed lines). The dashed vertical lines in magenta mark the critical strain ${\varepsilon }_{\parallel }^{*}$ that separates the heavy- and light-hole ground states. The material is silicon and the parameters are $L_z=4\mathrm{nm}$ for (a) and (c), $L_z=10\mathrm{nm}$ for (b) and (d), $B=1\mathrm{T}$, ${\mathcal{E}}_{x/y}^{\text{ac}}=(1/30)\mathrm{mV}/\mathrm{nm}$, $L_y=30\mathrm{nm}$, and $x_0=10\mathrm{nm}$.

Figure 7

(a) Double logarithmic plot of the IZ-EDSR Rabi frequency as a function of the static electric field ${\mathcal{E}}_y$ along $y$. (b) Dependence of the $g$-factors on the logarithm of the electric field ${\mathcal{E}}_y$. The data are obtained from the numerical solution of the four-band $k·p$ model [39]. The material is silicon and the relevant lengths are $x_0=10\mathrm{nm}$, $L_y=30\mathrm{nm}$, and $L_z=10\mathrm{nm}$.

Figure 8

Comparison between the maximal $g$-TMR and IZ-EDSR Rabi frequencies of a silicon quantum dot as a function of the static electric field along $y$. The parameters are $L_z=4\mathrm{nm}$, ${\varepsilon }_{\parallel }=0\%$ for (a), $L_z=10\mathrm{nm}$, ${\varepsilon }_{\parallel }=0\%$ for (b), $L_z=4\mathrm{nm}$, ${\varepsilon }_{\parallel }=0.7\%$ for (c), $L_z=10\mathrm{nm}$, ${\varepsilon }_{\parallel }=0.7\%$ for (d), $B=1\mathrm{T}$, ${\mathcal{E}}_{x/y}^{\text{ac}}=(1/30)\mathrm{mV}/\mathrm{nm}$, $L_y=30\mathrm{nm}$, and $x_0=15\mathrm{nm}$.

Figure 9

Comparison between the maximal $g$-TMR and IZ-EDSR Rabi frequencies of a germanium quantum dot as a function of the static electric field along $y$. The parameters are $L_z=4\mathrm{nm}$, ${\varepsilon }_{\parallel }=0\%$ for (a), $L_z=10\mathrm{nm}$, ${\varepsilon }_{\parallel }=0\%$ for (b), $L_z=10\mathrm{nm}$, ${\varepsilon }_{\parallel }=1\%$ for (c), $B=1\mathrm{T}$, ${\mathcal{E}}_{x/y}^{\text{ac}}=(1/30)\mathrm{mV}/\mathrm{nm}$, $L_y=30\mathrm{nm}$, and $x_0=10\mathrm{nm}$.

Figure 10

Maximal IZ-EDSR Rabi frequency as a function of the static electric field along $y$ and $z$. The parameters are ${\varepsilon }_{\parallel }=0\%$, $B=1\mathrm{T}$, ${\mathcal{E}}_x^{\text{ac}}=(1/30)\mathrm{mV}/\mathrm{nm}$, $x_0=10\mathrm{nm}$, $L_y=30\mathrm{nm}$, $L_z=4\mathrm{nm}$ for (a) and (c), and $L_z=10\mathrm{nm}$ for (b) and (d). Parts (a) and (b) are for silicon, and (c) and (d) are for germanium.