Theory of single electron spin relaxation in Si/SiGe lateral coupled quantum dots

We investigate the spin relaxation induced by acoustic phonons in the presence of spin-orbit interactions in single electron Si/SiGe lateral coupled quantum dots. The relaxation rates are computed numerically in single and double quantum dots, in in-plane and perpendicular magnetic fields. The deformation potential of acoustic phonons is taken into account for both transverse and longitudinal polarizations, and their contributions to the total relaxation rate are discussed with respect to the dilatation and shear potential constants. We find that in single dots the spin relaxation rate scales approximately with the seventh power of the magnetic field, in line with a recent experiment. In double dots the relaxation rate is much more sensitive to the dot spectrum structure, as it is often dominated by a spin hot spot. The anisotropy of the spin-orbit interactions gives rise to easy passages, special directions of the magnetic field for which the relaxation is strongly suppressed. Quantitatively, the spin relaxation rates in Si are typically two orders of magnitude smaller than in GaAs due to the absence of the piezoelectric phonon potential and generally weaker spin-orbit interactions.

Figure 1

Orientation of the double dot in the coordinate system $\mathrm{x̂}=[100]$, $\mathrm{ŷ}=[010]$. The potential minima, sketched by two circles, are parametrized by the position vectors $\pm \mathbf{d}$ or by the distance $2d$ and the angle $\delta$. The in-plane magnetic field orientation is given by the angle $\gamma$.

Figure 2

Calculated energy spectrum of the Si double quantum dot with respect to the interdot distance at zero magnetic field. The states are labeled (colored) according to the irreducible representations ${\Gamma }_i$ of ${\text{C}}_{\text{2v}}$. On the right-hand side we give the highest orbital momentum of associated single dot states (Fock-Darwin states). The tunneling energy $T$ is also shown.

Figure 3

Calculated tunneling energy vs interdot distance for zero magnetic field calculated by exact numerical diagonalization (dotted line), exact LCSDO formulas (thin solid line), and leading order approximation [Eq. (9), thick solid line]. The tunneling energy for a finite perpendicular magnetic field ($B_z=2\hspace{0.5em}\text{T}$, dashed line) is given for comparison.

Figure 4

Calculated energy spectrum of the DQD with interdot distance $2d/l_0=2.5$ plotted against the perpendicular magnetic field. The thick line indicates ${\Gamma }_{\text{S}}^{\uparrow }$, the lowest state with opposite spin polarization to that of the ground state.

Figure 5

Spin relaxation rates of a single electron in a single quantum dot vs in-plane field for $\gamma ={135}^{°}$. The total rate (solid black line) and its contributions of the TA phonons (solid red line) and the LA phonons (solid blue line) are shown. The spin hot spot at $B_{\parallel }\approx 8.3\hspace{0.5em}\text{T}$ causes the spin relaxation rate to increase up to the orbital relaxation rate (dash-dotted line). The three dots give the experimental data of Ref. 71 fitted by a $B^7$ curve (dotted line).

Figure 6

Ratio of the relaxation rate of the approximations and the exact numerics vs in-plane magnetic field. The low-$B$-field limit, represented by Eqs. (14) and (15), is shown by the solid lines, and the numerically evaluated isotropic average approximation, Eq. (16), by the dashed lines. The contributions of the TA and LA phonons are given in the top and bottom panels, respectively. The magnetic field is in plane and $\gamma ={135}^{°}$. The constant line at 1 (dotted) is a guide to the eye.

Figure 7

Calculated spin relaxation rate in a DQD with tunable interdot distance in a perpendicular magnetic field. The rate is given in inverse seconds by the color with the scale on the right. The $y$ axis is calibrated in interdot distance (left) and tunneling energy at $B=0$ (right).

Figure 8

Calculated spin relaxation rate in a DQD as a function of the interdot distance and the orientation of the in-plane magnetic field ($B_{\parallel }=4\hspace{0.5em}\text{T}$). The rate is given in inverse seconds by the color with the scale on the right. The $y$ axis is calibrated in interdot distance (left) and tunneling energy at $B=0$ (right). The dot’s main axis is along $[100]$.

Figure 9

Calculated spin relaxation rate in a DQD as a function of the interdot distance and the orientation of the in-plane magnetic field ($B_{\parallel }=4\hspace{0.5em}\text{T}$). The rate is given in inverse seconds by the color with the scale on the right. The $y$ axis is calibrated in interdot distance (left) and tunneling energy at $B=0$ (right). The dot’s main axis is along $[110]$.