Hole–spin dephasing time associated with hyperfine interaction in quantum dots

The spin interaction of a hole confined in a quantum dot with the surrounding nuclei is described in terms of an effective magnetic field. We show that, in contrast to the Fermi contact hyperfine interaction for conduction electrons, the dipole-dipole hyperfine interaction is anisotropic for a hole, for both pure or mixed hole states. We evaluate the coupling constants of the hole-nuclear interaction and demonstrate that they are only 1 order of magnitude smaller than the coupling constants of the electron-nuclear interaction. We also study, theoretically, the hole–spin dephasing of an ensemble of quantum dots via the hyperfine interaction in the framework of frozen fluctuations of the nuclear field, in the absence or in the presence of an applied magnetic field. We also discuss experiments which could evidence the dipole-dipole hyperfine interaction and give information on hole mixing.

Figure 1

Time dependence of the out-of-plane, $R_z\left(\tau \right)$, and in-plane, $R_x\left(\tau \right)$, components of the ensemble-averaged spin polarization, calculated for different anisotropy factors: pure hh $\left(\alpha =0\right)$, mixed heavy-light holes $\left(\alpha =0.5\right)$, conduction electrons $\left(\alpha =1\right)$, and pure lh $\left(\alpha =2\right)$.

Figure 2

Anisotropy dependence of the steady-state values $R_x\left(\infty \right)$ and $R_z\left(\infty \right)$. For $\alpha ⪢1$ (not shown in this figure), $R_z\left(\infty \right)\to 0$ and $R_x\left(\infty \right)\to \frac{1}{2}$.

Figure 3

Time dependence of the transverse $R_x\left(\tau \right)$, $R_y\left(\tau \right)$, and the longitudinal $R_z\left(\tau \right)$ components of the ensemble hole averaged spin. The magnetic field is applied along the $z$ direction. Results for pure hh and lh are, respectively, shown on the upper (hh, $\alpha =0$) and bottom part (lh, $\alpha =2$) of this figure. The curves are calculated for different values of the applied magnetic field given by the reduced $\delta$ value $\left(\delta =B/{\Delta }_0\right)$.

Figure 4

The upper part: time dependence of the transverse $R_x\left(\tau \right)$, $R_y\left(\tau \right)$, and the longitudinal $R_z\left(\tau \right)$ components of the ensemble hole averaged spin for a magnetic field applied along the $z$ direction. The bottom part: time dependence of the longitudinal $R_x^1$ and transverse $R_y^1$ and $R_z^1$ components of the ensemble-averaged spin for a magnetic field applied along the $x$ direction. The curves are calculated for $\delta =0,0.5,1,2,4$. The anisotropy factor is $\alpha =0.5$.

Figure 5

Time dependence of the longitudinal $R_x^1\left(\tau \right)$ and transverse $R_y^1\left(\tau \right)$ and $R_z^1\left(\tau \right)$ components of the ensemble hole averaged spin (hh case, $\alpha =0$). The magnetic field is applied along the $x$ direction. The curves are calculated for different magnetic fields and represented as a function of $\delta$ (some $\delta =4$ curves have been omitted for clarity).

Figure 6

Time dependence of the longitudinal $R_x^1\left(\tau \right)$ and transverse $R_y^1\left(\tau \right)$ and $R_z^1\left(\tau \right)$ components of the ensemble hole averaged spin (lh case, $\alpha =2$). The magnetic field is applied along the $x$ direction. The curves are calculated for different magnetic fields and represented as a function of $\delta$.

Figure 7

Magnetic field dependence of (a) the steady-state value of the spin $z$-component $R_z\left(\infty \right)$ for an external magnetic field applied along $z$ direction and several anisotropy factor values $\alpha =0,0.25,0.5,1,2$, $\left(\delta =B_z/{\Delta }_0\right)$; (b) the steady-state value of the spin $z$-component $R_z^1\left(\infty \right)$ for an external magnetic field applied along $x$ direction and several anisotropy factor values $\alpha =0,0.25,0.5,1,2$ $\left(\delta =B_x/{\Delta }_0\right)$.