Role of Nuclear Quadrupole Coupling on Decoherence and Relaxation of Central Spins in Quantum Dots

Strain-induced gradients of local electric fields in semiconductor quantum dots can couple to the quadrupole moments of nuclear spins. We develop a theory describing the influence of this quadrupolar coupling on the spin correlators of electron and hole “central” spins localized in such dots. We show that when the quadrupolar coupling strength is comparable to or larger than the hyperfine coupling strength between nuclei and the central spin, the relaxation rate of the central spin is strongly enhanced and can be exponential. We demonstrate a good agreement with recent experiments on spin relaxation in hole-doped (In,Ga)As self-assembled quantum dots.

Figure 1

Spin correlator $C_2(t)=⟨{\widehat{\sigma }}_z(t){\widehat{\sigma }}_z(0)⟩$ at ${\gamma }_c=0$, ${\gamma }_{\parallel }\equiv ⟨{\gamma }_{\parallel }⟩=1$, $N=700$ nuclear spins, and $\mathit{B}=0$, shown up to times (a) $t=5/{\gamma }_{\parallel }$ and (b) $t=0.5/{\gamma }_{\parallel }$ (time $t$ is in units of $1/{\gamma }_{\parallel }$). Here $\beta ={\gamma }_{\perp }/{\gamma }_{||}$ is the hyperfine coupling anisotropy.

Figure 2

(a) The real time central spin-spin correlator for different magnitudes of the static field ${\gamma }_c$ in spin-$1/2$ bath. (b) Exponential fit (dashed red) of the spin correlator (blue) for spin-$1/2$ bath at ${\gamma }_c=12$ in units of ${\gamma }_{\parallel }$. (c) Central spin correlator in spin bath with $I=1$ and quadrupole coupling ${\gamma }_Q^i=4{\gamma }_c^i$. (d) Exponential fit (dashed red) of spin correlator in spin-1 bath at ${\gamma }_c=12$. In all cases: $\beta =0.2$ and $\mathit{B}=0$.

Figure 3

For $\beta =0.2$: (a) Typical Overhauser field dynamics for ${\gamma }_c=0$ (horizontal line) and ${\gamma }_c=8$ (fluctuating curve). (b) Real time Overhauser field correlator $C_2^{\mathrm{bath}}(t)=⟨B_{nz}(t)B_{nz}(0)⟩$.

Figure 4

(a) Real time spin correlator and (b) frequency power spectrum $P(\omega )=\int dt{e}^{i\omega t}⟨{\widehat{\sigma }}_z(t){\widehat{\sigma }}_z(0)⟩$ for different values of external out-of-plane magnetic field $B_z$; ${\gamma }_c=8$, and $\beta =0.2$.

Figure 5

Central spin correlator at $\beta =1$ in zero external field: (a) Real time correlator and (b) frequency power spectrum.