Strong spin-orbit interaction and helical hole states in Ge/Si nanowires

We study theoretically the low-energy hole states of Ge/Si core/shell nanowires. The low-energy valence band is quasidegenerate, formed by two doublets of different orbital angular momenta, and can be controlled via the relative shell thickness and via external fields. We find that direct (dipolar) coupling to a moderate electric field leads to an unusually large spin-orbit interaction of Rashba type on the order of meV which gives rise to pronounced helical states enabling electrical spin control. The system allows for quantum dots and spin qubits with energy levels that can vary from nearly zero to several meV, depending on the relative shell thickness.

Figure 1

Schematic drawing of the systems studied in this paper. Top: Excerpt of a Ge/Si nanowire with core radius $R$ and shell thickness $R_s-R$, where the $z$ axis corresponds to the axis along the wire. The nanowires are typically several micrometers in length and can therefore be considered infinitely extended, hosting a 1D hole gas inside their cores. The surrounding Si shell influences the hole spectrum through static strain. Bottom: Quantum dots of (effective) length $L$ form when the holes are subject to additional confinement in the $z$ direction. This can be realized via gates (Refs. 4, 27, and 28) or, in principle, by surrounding the Ge with layers of barrier material during growth (Ref. 44).

Figure 2

Low-energy hole spectrum of a Ge nanowire as a function of the longitudinal wave number $k_z$. In the unstrained case, $\gamma =0$, the plot is independent of $R$, with ${\hslash }^2/\left(m{R}^2\right)\simeq 0.76\text{meV}$ for $R=10\text{nm}$. Due to time-reversal invariance and cylindrical symmetry, each line is a twofold degeneracy, where red (blue) indicates quantum numbers $F_z=\pm 1/2$ ($F_z=\pm 3/2$). At $k_z=0$ the spectrum is quasidegenerate, with the lowest states having $L_z\approx 0$ (ground states) and $|L_z|=1$ (excited states) character. Dashed red lines result from the effective 1D model for the lowest subspace, where $k_z$ is treated perturbatively. The top figure is a plot of the low-energy sector of a strained system, $\gamma =40\%$, illustrating strong dependence on the Si shell thickness.

Figure 3

Top: Hole energy spectrum in a nanowire-based QD (Ge/Si core/shell, $R=5\text{nm}$), for both a thin and a thick shell, as a function of confinement length $L$. Each line corresponds to a Kramers pair, and dashed lines represent $\Delta$ for comparison. Bottom: Level splitting of the two lowest Kramers doublets as a function of relative shell thickness $\gamma$ and for different lengths $L$. Static strain, induced via the shell, allows continuous tuning of the energy gap over several meV, an attractive feature for spin-qubit applications. For details, see Appendix .

Figure 4

Dispersion relation for holes in a Ge nanowire of $R=10\text{nm}$, negligible shell, and an applied electric field $E_x$ along $x$, calculated from $H_{\mathrm{LK}}^{\mathrm{eff}}+H_{\mathrm{DR}}$, Eqs. (3) and (6), with $H_{\mathrm{DR}}$ as the DRSOI Hamiltonian. Hole bands of lower (higher) energy are plotted blue (red). The RSOI is about 100 times smaller than the DRSOI and thus negligible. Note that the DRSOI shows qualitatively similar features to the standard Rashba SOI with dispersion curves shifted along $k_z$ against each other.

Figure 5

Top: Splitting of the lowest valence band when an electric field $E_x=6\text{V}/\mu \text{m}$ is applied to a Ge/Si nanowire of $R=5\text{nm}$ and $R_s=6.5\text{nm}$. Ground (excited) hole states are plotted blue (red). $E_{\mathrm{SO}}>1.0\text{meV}$, a large value compared to that for InAs (Refs. 6 and 40), and the degeneracy at $k_z=0$ may be lifted via a magnetic field (see Fig. 6). The conventional RSOI for holes is negligible. Bottom: Plot of $\langle J_y\rangle$ for the above system, where $\langle J_x\rangle$ and $\langle J_z\rangle$ are zero throughout. In the ground state, the nanowire carries opposite spins in opposite directions with $|\langle J_y\rangle |\ge 1/2$.

Figure 6

Top: Hole spectrum of Fig. 5 (top) in the presence of $B_x=1\text{T}$. The magnetic field opens a gap of $0.30\text{meV}$ at $k_z=0$, corresponding to a $g$ factor above 5. Bottom: Plot of the ground state spin, $\langle J_x\rangle$ and $\langle J_y\rangle$, where $\langle J_z\rangle =0$ throughout. At energies within the gap, the Ge/Si nanowire features helical hole states with $E_{\mathrm{SO}}>1.0\text{meV}$, $\left|k_z\right|\simeq 90\mu {\text{m}}^{-1}$, and $|\langle J_y\rangle |\ge 1/2$.