Quantitative modeling of spin relaxation in quantum dots

We use numerically exact diagonalization to calculate the spin-orbit- and phonon-induced triplet-singlet relaxation rate in a two-electron quantum dot exposed to a tilted magnetic field. Our scheme includes a three-dimensional description of the quantum dot, the Rashba and the linear and cubic Dresselhaus spin-orbit coupling, the ellipticity of the quantum dot, and a full angular description of the magnetic field. We are able to find reasonable agreement with the experimental results of  Meunier et al. [Phys. Rev. Lett. 98, 126601 (2007)] in terms of the singlet-triplet energy splitting and the spin relaxation rate, respectively. We analyze in detail the effects of the spin-orbit factors, magnetic-field angles, and dimensionality and discuss the origins of the remaining deviations from the experimental data.

Figure 1

(a) Energy of singlet ground state (thick lines) and three lowest triplet states (thin lines) in a two-electron harmonic ($\hslash {\omega }_0=2.5$ meV, $\hslash {\omega }_z=11.9$ meV), elliptic ($\delta =1.3$) quantum dot as a function of the magnetic-field strength for three different tilting angles of the magnetic field, $\theta ={65}^{\circ }$, ${50}^{\circ }$, and ${35}^{\circ }$. In the triplet states the dashed, solid, and dash-dotted lines indicate $S_z=-1,0,$ and $+1$, respectively. (b) Singlet-triplet energy splitting for the three cases of (a). Open circles correspond to the experimental data in Ref. 15. (c) Variation of $\Delta E\left(B\right)$ as a function of $\phi$ for $B=0.5$ T (solid line), 1.5 T (dash-dotted line), and 2.5 T (dashed line), respectively. The results for $B=1.5$ T and 2.5 T have been multiplied by factors of 1.4 and 2.4, respectively, to set the scale.

Figure 2

Calculated total triplet-singlet relaxation rate when the magnetic-field tilting angle is $\theta ={50}^{\circ }$ (solid line) in comparison with the experimental data in Ref. 15 (circles with error bars). The other parameters are the same as in Fig. 1. The solid line shows the total rate. The dashed and dash-dotted lines show the contributions from the piezoelectric and deformation potential couplings, respectively. The spin-orbit coupling parameters are $\gamma =11.15\mathit{eV}{\AA }^3$ and $\alpha =0$ for the Dresselhaus and Rashba terms, respectively.

Figure 3

Scaled relaxation rates according to Eq. (19) for four different sets of spin-orbit coupling strengths. $\alpha$ and $\gamma$ are in units of $\mathrm{meV}\AA$ and $\mathit{eV}{\AA }^3$, respectively.

Figure 4

Dependence of the relaxation rate on the magnetic-field angles $\theta$ and $\phi$. The field strength is fixed to $B=2$ T. The other parameters are the same as in Fig. 1.

Figure 5

Comparison between two-dimensional and three-dimensional calculations, in both cases with a tilted magnetic field of $\theta ={50}^{\circ }$. (a) Energy levels of the lowest singlet and triplet states. (b) Singlet-triplet energy splitting. (c) Relaxation rate. Parameters are the same as in Fig. 2.