Spin relaxation in diluted magnetic semiconductor quantum dots

Electron spin relaxation induced by phonon-mediated $s\text{−}d$ exchange interaction in a II-VI diluted magnetic semiconductor quantum dot is investigated theoretically. The electron–acoustic phonon interaction due to piezoelectric coupling and deformation potential is included. The resulting spin lifetime is typically on the order of microseconds. The effectiveness of the phonon-mediated spin-flip mechanism increases with increasing Mn concentration, electron spin splitting, vertical confining strength, and lateral diameter, while it shows nonmonotonic dependence on the magnetic field and temperature. An interesting finding is that the spin relaxation in a small quantum dot is suppressed for strong magnetic field and low Mn concentration at low temperature.

Figure 1

(a) Magnetic field dependence of the Mn correlation functions $G^{-+}$ (solid curve) and $G^{+-}$ (dashed curve) at $T=1\mathrm{K}$. (b) Phonon absorption (solid line) and emission (dashed line) factor. (c) and (d) Piezoelectric coupling (solid curves) and deformation potential (dashed curves) contributions to the product $S\Gamma$ as functions of $z_0$ and $\mid {\Delta }_0\mid$, respectively.

Figure 2

The renormalized spin-dependent electron energy spectra as a function of magnetic field at $T=\left(\mathrm{a}\right)$ 1 and (b) $20\mathrm{K}$, with fixed $z_0=2\mathrm{nm}$, $d=16\mathrm{nm}$, $x=0.002$. The spin-up (spin-down) levels are denoted by solid (dashed) lines.

Figure 3

Spin-flip scattering rate of the lowest spin-up level and spin-down level as a function of magnetic field at $T=1$ and $20\mathrm{K}$, with the same structure as in Fig. 2. (a) Spin-up level, $T=1\mathrm{K}$; (b) spin-down level, $T=1\mathrm{K}$; (c) spin-up level, $T=20\mathrm{K}$; (d) spin-down level, $T=20\mathrm{K}$. The total spin-flip scattering rate and the contributions from PZ and DP are denoted by solid lines, dashed lines, and short-dashed lines, respectively.

Figure 4

Renormalized spin-dependent electron energy spectra as a function of magnetic field, at $T=\left(\mathrm{a}\right)$ 1 and (b) $50\mathrm{K}$, with fixed $z_0=2\mathrm{nm}$, $d=16\mathrm{nm}$, $x=0.01$. The solid and dashed lines denote the spin-up and spin-down levels, respectively.

Figure 5

Spin-flip scattering rate of the lowest Zeeman doublet versus the magnetic field at $T=1$ and $50\mathrm{K}$, with the same structure as in Fig. 4. (a) Spin-up level, $T=1\mathrm{K}$; (b) spin-down level, $T=1\mathrm{K}$; (c) spin-up level, $T=50\mathrm{K}$; (d) spin-down level, $T=50\mathrm{K}$. The total spin-flip scattering rate and the contributions from PZ and DP are denoted by solid, dashed, and short-dashed lines, respectively.

Figure 6

Renormalized spin-dependent electron energy spectra vs the magnetic field at $T=\left(\mathrm{a}\right)$ 1 and (b) $50\mathrm{K}$, with fixed $z_0=2\mathrm{nm}$, $d=16\mathrm{nm}$, $x=0.05$. Spin-up and spin-down levels are denoted by solid and dashed lines, respectively.

Figure 7

Spin-flip scattering rate of the lowest Zeeman doublet as a function of magnetic field at $T=1$ and $50\mathrm{K}$, with the same structure as in Fig. 6. (a) Spin-up level, $T=1\mathrm{K}$; (b) spin-down level, $T=1\mathrm{K}$; (c) spin-up level, $T=50\mathrm{K}$; (d) spin-down level, $T=50\mathrm{K}$. The total spin-flip scattering rate and the contributions from PZ and DP are denoted by solid lines, dashed lines, and short-dashed lines, respectively.

Figure 8

Renormalized spin-dependent electron energy spectra vs the magnetic field at $T=\left(\mathrm{a}\right)$ 1 and (b) $10\mathrm{K}$, with fixed $z_0=2\mathrm{nm}$, $d=40\mathrm{nm}$, $x=0.001$. Spin-up and spin-down levels are denoted by solid and dashed lines, respectively.

Figure 9

Spin-flip scattering rate of the lowest Zeeman doublet as a function of magnetic field at $T=1$ and $10\mathrm{K}$, with the same structure as in Fig. 8. (a) Spin-up level, $T=1\mathrm{K}$; (b) spin-down level, $T=1\mathrm{K}$; (c) spin-up level, $T=10\mathrm{K}$; (d) spin-down level, $T=10\mathrm{K}$. The total spin-flip scattering rate and the contributions from PZ and DP are denoted by solid lines, dashed lines, and short-dashed lines, respectively.

Figure 10

(a) Renormalized energy of the lowest spin-up (solid line) and spin-down (dashed line) levels vs the temperature. (b) and (c) show the the spin-flip scattering rate of the spin-up and spin-down levels vs the temperature, respectively. The total spin-flip scattering rate and the PZ and DP contributions are denoted by solid, dashed, and short-dashed lines, respectively. Here we take $B=4\mathrm{T}$, $z_0=2\mathrm{nm}$, $d=16\mathrm{nm}$, and $x=0.01$ in the numerical calculation.

Figure 11

(a) and (b) show $1∕{\tau }_{\pm }$ vs the vertical confining length $z_0$ at fixed $B=1\mathrm{T}$, $T=1\mathrm{K}$, $d=16\mathrm{nm}$, $x=0.001$. (c) and (d) show $1∕{\tau }_{\pm }$ vs the lateral diameter $d$ at fixed $B=4\mathrm{T}$, $T=1\mathrm{K}$, $x=0.002$. The total spin-flip scattering rate and the PZ and DP contributions are denoted by solid, dashed, and short-dashed lines, respectively.