Tunable $g$ factor and phonon-mediated hole spin relaxation in Ge/Si nanowire quantum dots

We theoretically consider $g$ factor and spin lifetimes of holes in a longitudinal Ge/Si core/shell nanowire quantum dot that is exposed to external magnetic and electric fields. For the ground states, we find a large anisotropy of the $g$ factor which is highly tunable by applying electric fields. This tunability depends strongly on the direction of the electric field with respect to the magnetic field. We calculate the single-phonon hole spin relaxation times $T_1$ for zero and small electric fields and propose an optimal setup in which very large $T_1$ of the order of tens of milliseconds can be reached. Increasing the relative shell thickness or the longitudinal confinement length further prolongs $T_1$. In the absence of electric fields, the dephasing vanishes and the decoherence time $T_2$ is determined by $T_2=2T_1$.

Figure 1

Sketch of a Ge/Si core/shell NW aligned with the $z$ axis of the coordinate system. Electric gates (blue) induce confinement along the $z$ axis and define a QD. The electric field $\mathit{E}$ lies perpendicular to the wire in the $xy$ plane and the magnetic field $\mathit{B}$ lies in the $xz$ plane.

Figure 2

Effective $g$ factor $g_{\text{eff}}$ as a function of the angle $\theta$ defined by $\mathit{B}=\left|\mathit{B}\right|(\sin \theta ,0,\cos \theta )$ for $\mathit{E}\parallel \widehat{\mathit{x}}$ (a) and $\mathit{E}\parallel \widehat{\mathit{y}}$ (b). We vary $\left|\mathit{E}\right|$ from 0 to $10\mathrm{V}/\mu \mathrm{m}$. For $\left|\mathit{E}\right|=0$, we find $g_{\text{eff}}\left(0\right)\approx 0.14$ and $g_{\text{eff}}(\pi /2)\approx 5.7$. It is clearly visible that $g_{\text{eff}}$ is affected much stronger by changes in $\left|\mathit{E}\right|$ for $\mathit{E}\parallel \widehat{\mathit{x}}$ than for $\mathit{E}\parallel \widehat{\mathit{y}}$. Even though the curves in (a) seem to overlap for $\left|\mathit{E}\right|\ge 6\mathrm{V}/\mu \mathrm{m}$, $g_{\text{eff}}(\pi /2)$ still decreases for growing fields and $g_{\text{eff}}$ remains anisotropic. We choose $R=10\text{nm}$ and $R_s=13\text{nm}$ for the NW and a QD confinement length of $z_g\approx 80\text{nm}$.

Figure 3

Relaxation rate $T_1^{-1}$ at $\left|\mathit{E}\right|=0$ for $\mathit{B}\parallel \widehat{\mathit{x}}$ (red, solid) and $\mathit{B}\parallel \widehat{\mathit{z}}$ (blue, solid). For $\mathit{B}\parallel \widehat{\mathit{x}}$, we plot the contributing phonon branches $F_{\pm 1}$ and $T$ (dashed). We find maximal values $T_{1,\text{max}}^{-1}(\mathit{B}\parallel \widehat{\mathit{z}})\approx 11{\text{ms}}^{-1}$ and $T_{1,\text{max}}^{-1}(\mathit{B}\parallel \widehat{\mathit{x}})\approx 60{\text{s}}^{-1}$. Note the nonmonotonic behavior of $T_1^{-1}$ as a function of ${\omega }_{Z,\text{eff}}$. The NW and QD parameters are chosen as in Fig. 2.

Figure 4

Relaxation rates $T_1^{-1}$ for $\mathit{E}\parallel \widehat{\mathit{x}}$ with $\left|\mathit{E}\right|=0.1\mathrm{V}/\mu \mathrm{m}$ for $\mathit{B}\parallel \widehat{\mathit{x}}$ (red, solid) and $\mathit{B}\parallel \widehat{\mathit{z}}$ (blue, solid). For comparison we plot $T_1^{-1}$ at $\left|\mathit{E}\right|=0$ (dotted). We find maximal values $T_{1,\text{max}}^{-1}(\mathit{B}\parallel \widehat{\mathit{z}})\approx 3.2{\mu \mathrm{s}}^{-1}$, $T_{1,\text{max}}^{-1}(\mathit{B}\parallel \widehat{\mathit{x}})\approx 5.8{\mu \mathrm{s}}^{-1}$. Rotating the electric field so that $\mathit{E}\parallel \widehat{\mathit{y}}$ yields the same curve for $\mathit{B}\parallel \widehat{\mathit{z}}$. Remarkably, for $\mathit{B}\parallel \widehat{\mathit{x}}$ almost no difference between the curves at $\mathit{E}\parallel \widehat{\mathit{y}}$ (dashed) and $\left|\mathit{E}\right|=0$ (dotted) is observed. We use the NW and QD parameters given below Fig. 2.