Hole spin relaxation in semiconductor quantum dots

Hole spin relaxation time due to the hole–acoustic-phonon scattering in $\mathrm{GaAs}$ quantum dots confined in quantum wells along (001) and (111) directions is studied after the exact diagonalization of Luttinger Hamiltonian. Different effects such as strain, magnetic field, quantum dot diameter, quantum well width, and the temperature on the spin relaxation time are investigated thoroughly. Many features that are quite different from the electron spin relaxation in quantum dots and quantum wells are presented with the underlying physics elaborated.

Figure 1

SRT vs the diameter $d$. Curve with $•$: exact diagonalization result with the energy sufficiently converged. Curve with $\times$: perturbation result. Curve with $엯$: exact diagonalization result but with only the lowest two heavy-hole and the lowest two light-hole levels used as basis functions. Curve with $∎$: exact diagonalization result but with only the 16 energy levels of $H_0$ as basis functions. (a) $T=0.2\mathrm{K}$ and (b) $T=4\mathrm{K}$.

Figure 2

SRT vs temperature $T$ for QD with $a=5\mathrm{nm}$ and $d=20\mathrm{nm}$. (a) Results under different magnetic fields. Curve with $∎$: $B=0.2\mathrm{T}$. Curve with $▴$: $B=0.6\mathrm{T}$. Curve with $•$: $B=1\mathrm{T}$. (b) Results at $B=1\mathrm{T}$ with contributions from different branches of phonons specified. Curve with $∎$: contribution from longitudinal phonons. Curve with $▴$: contribution from the transverse phonons of first branch. Curve with $◆$: contributions from the transverse phonons of the second branch. Curve with $•$: The total SRT.

Figure 3

SRT and the energy spectrum of the lowest two heavy-hole (dashed curve) and light-hole (solid curve) states vs strain for a QD in (001) quantum well at $a=5\mathrm{nm}$, $d=20\mathrm{nm}$, and $B=1\mathrm{T}$ at $T=4\mathrm{K}$. Note the scale of the energy spectrum is on the right-hand side of the figure. Inset: solid curve, total SRT; chain curve, SRT induced by the hole-phonon scattering due to piezoelectric coupling; dashed curve, SRT induced by the hole-phonon scattering due to deformation potential.

Figure 4

SRT vs strain for a QD in (001) quantum well at $a=20\mathrm{nm}$, $d=20\mathrm{nm}$, and $B=1\mathrm{T}$ at $T=4\mathrm{K}$. Solid curve: total SRT. Chain curve: SRT induced by the hole-phonon scattering due to piezoelectric coupling. Dashed curve: SRT induced by the hole-phonon scattering due to deformation potential.

Figure 5

SRT vs the diameter $d$ for QD’s at $a=20\mathrm{nm}$ and $B=1\mathrm{T}$ under different strains. Curve with $•$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0$, $\times$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0.07$; $◻$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=-0.07$; $▵$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=-0.09$; $◆$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=-0.3$. (a) $T=0.2\mathrm{K}$; (b) $T=2\mathrm{K}$.

Figure 6

SRT vs the magnetic field $B$ for a QD in (001) quantum well at $a=20\mathrm{nm}$, $d=20\mathrm{T}$, and $T=4\mathrm{K}$ under different strains. Curve with $•$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0$; $∎$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0.01$; $◆$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0.03$; $엯$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=-0.01$; $◻$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=-0.02$; $\times$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=-0.03$; $+$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=-0.2$.

Figure 7

SRT vs strain at $a=5\mathrm{nm}$, $d=20\mathrm{nm}$, $B=1\mathrm{T}$, and $T=4\mathrm{K}$. Solid curve: total SRT. Chain curve: SRT induced by the hole-phonon scattering due to piezoelectric coupling. Dashed curve: SRT induced by the hole-phonon scattering due to deformation potential.

Figure 8

SRT vs the diameter $d$ for a QD in (111) quantum well at $a=5\mathrm{nm}$ and $B=1\mathrm{T}$ under different strains. Curve with $•$: ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0$; $∎$: ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0.12$; $▴$: ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0.28$. (a): $T=0.2\mathrm{K}$, (b) $T=4\mathrm{K}$.

Figure 9

SRT vs the magnetic field $B$ at $a=5\mathrm{nm}$, $d=20\mathrm{T}$, and $T=4\mathrm{K}$ under different strains. Curve with $•$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0$; $\times$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0.01$; $+$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0.02$; $∎$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0.12$; $▴$, ${ϵ}_{\Vert }∕{ϵ}_{\Vert }^0=0.28$.

Figure 10

SRT vs the well width $a$ at $d=20\mathrm{nm}$, $B=1\mathrm{T}$, and $T=4\mathrm{K}$. Solid curve, (001) quantum well; dashed curve, (111) quantum well.