Magnetic field effects on spin relaxation in heterostructures

Effect of a magnetic field on electron spin relaxation in quantum wells is studied theoretically. We have shown that Larmor effect and the cyclotron motion of the carriers can either jointly suppress D’yakonov-Perel’ spin relaxation or compensate each other. The spin relaxation rates tensor is derived for any given direction of the external field and arbitrary ratio of bulk and structural contributions to the spin splitting. Our results are applied to the experiments on electron spin resonance in SiGe heterostructures, and enable us to extract spin splitting value for such quantum wells.

Figure 1

Frames of axes. $z\Vert \left[001\right]$ is a growth axis. Axes $x_1\Vert \left[1\overline{1}0\right]$ and $y_1\Vert \left[110\right]$. The frame of axis $x^{\prime }$, $y^{\prime }$, and $z^{\prime }$ is connected with the external magnetic field: $z^{\prime }\Vert {\mathbf{\omega }}_L$, $y^{\prime }$ is a perpendicular to ${\mathbf{\omega }}_L$ lying in a quantum well plane, and $x^{\prime }$ is chosen so that $x^{\prime }$, $y^{\prime }$, and $z^{\prime }$ form a right-hand triple.

Figure 2

Longitudinal spin relaxation rate vs magnetic field tilt angle $\theta$ for an SiGe-based quantum well. Spin relaxation rate is measured in the units $2{\tau }_1{k}^2{\delta }^2∕{\hslash }^2$ where $\delta$ is a coefficient at the first harmonics of spin splitting. Solid curve corresponds to Rashba term ($\alpha =\delta$, $\beta =0$), dashed curve to Dresselhaus term ($\alpha =0$, $\beta =\delta$). The contribution of the third harmonics is neglected. At $\theta =0$ the parameter ${\omega }_C{\tau }_1=1$. The parameters of calculation are as follows: $g=2$, $m=0.196m_0$. The inset shows the nonmonotonous behavior of $1∕T_1\left(\theta \right)$ when Dresslhaus term dominates.

Figure 3

Longitudinal spin relaxation rate vs magnetic field tilt angle $\theta$ in the case of equal Dresselhaus and Rashba terms $(\alpha =\beta )$ for different orientations of magnetic field. The spin relaxation rate is presented in units $2{\tau }_1{k}^2{\alpha }^2∕{\hslash }^2$. The values of parameters used in the calculation are the same as for Fig. 2.

Figure 4

Transverse relaxation time vs magnetic field tilt angle $\theta$ for SiGe structure. (a) Open squares show the ESR linewidth measured in Ref. 12. Solid curve presents the results of calculation when only one (Rashba or Dresselhaus) linear in $\mathit{k}$ term dominates. Parameters ${\omega }_C=2.96\times {10}^{11}{\mathrm{s}}^{-1}$ at normal magnetic field orientation, ${\omega }_L=5.93\times {10}^{10}{\mathrm{s}}^{-1}$, and momentum relaxation time ${\tau }_1={10}^{-11}\mathrm{s}$ were taken from experiment. Spin precession frequency is ${\Omega }_{k_F}=4\times {10}^8{\mathrm{s}}^{-1}$. (b) Dresselhaus and Rashba terms are equal $(\alpha =\beta )$. Spin relaxation rate is measured in the units of ${\tau }_1{k}^2{\alpha }^2∕{\hslash }^2$. Parameters of calculation are the same as for Fig. 2(a).