Long-term dynamics of the electron-nuclear spin system of a semiconductor quantum dot

A quasiclassical theoretical description of polarization and relaxation of nuclear spins in a quantum dot with one resident electron is developed for arbitrary mechanisms of electron-spin polarization. The dependence of the electron-nuclear spin dynamics on the correlation time ${\tau }_c$ of electron-spin precession, with frequency $\Omega$, in the nuclear hyperfine field is analyzed. It is demonstrated that the highest nuclear polarization is achieved for a correlation time close to the period of electron-spin precession in the nuclear field. For these and larger correlation times, the indirect hyperfine field, which acts on nuclear spins, also reaches a maximum. This maximum is of the order of the dipole-dipole magnetic field that nuclei create on each other. This value is nonzero even if the average electron polarization vanishes. It is shown that the transition from short correlation time to $\Omega {\tau }_c\gtrsim 1$ does not affect the general structure of the equation for nuclear-spin temperature and nuclear polarization in the Knight field but changes the values of parameters, which now become functions of $\Omega {\tau }_c$. For correlation times larger than the precession time of nuclei in the electron hyperfine field, it is found that three thermodynamic potentials $\left(\chi ,\mathit{\xi },\varsigma \right)$ characterize the polarized electron-nuclear spin system. The values of these potentials are calculated assuming a sharp transition from short to long correlation times, and the relaxation mechanisms of these potentials are discussed. The relaxation of the nuclear-spin potential is simulated numerically showing that high nuclear polarization decreases relaxation rate.

Figure 1

Time scale for different regimes of hyperfine interaction in quantum dot. (1) Short correlation time regime $\left({\tau }_c<{\Omega }^{-1}\right)$. During ${\tau }_c$ the electron and nuclear spins rotate around a small angle. (2) Intermediate correlation time regime $\left({\Omega }^{-1}<{\tau }_c<{⟨\omega ⟩}^{-1}\right)$. During ${\tau }_c$ the electron spin rotates around a large angle with constant angular velocity. (3) Long correlation time regime $\left({\tau }_c>{⟨\omega ⟩}^{-1}\right)$. During ${\tau }_c$ the electron and nuclear spins change their direction. For ${⟨{\omega }_{\text{dd}}⟩}^{-1}>{\tau }_c>{⟨\omega ⟩}^{-1}$ the total nuclear spin, $I_{\Sigma }$ is conserved, but $\Omega$ changes direction as a result of the nuclear-spin precession in the nonuniform electron hyperfine field. $⟨{\omega }_{\text{dd}}⟩$ is the frequency of nuclear-spin precession in the local magnetic field of the neighboring nuclei. For ${⟨{\omega }_{\text{dd}}⟩}^{-1}<{\tau }_c$, the dipole-dipole interaction between spins of neighboring nuclei changes $I_{\Sigma }$.

Figure 2

(Color online) Nuclear-spin-relaxation time on the resident electron vs correlation time ${\tau }_c$ (curves 1 and 2) and vs. frequency $\Omega$ (curves 3 and 4) of the electron-spin precession in the nuclear hyperfine field. Solid curves 1 and 3 show the relaxation time for the nuclear polarization component along $\mathbf{\Omega }$. Dashed curves 2 and 3 show the relaxation time of the nuclear polarization component transversal to $\mathbf{\Omega }$. Dependences of relaxation time on ${\tau }_c$ (1 and 2) were calculated for a constant value of $\Omega$. They are normalized to the value of $T_{1,e}$ at $\Omega {\tau }_c=1$. Dependences of relaxation time on $\Omega$ (3.4) were calculated for a constant value of ${\tau }_c$, and are normalized to the value of $T_{1,e}$ at $\Omega =0$ The nuclear dynamic polarization is ineffective for $\Omega {\tau }_c⪢1$ because in this region the value of the relaxation time for the nuclear-spin longitudinal component increases fast, and the leakage factor in the Eq. (20) decreases. The difference between times $T_{1,e}$ for longitudinal and transverse component of nuclear polarization in the region $\Omega {\tau }_c⪢1$ increases $\epsilon$ in Eq. (20) but decreases nuclear polarization.

Figure 3

(Color online) Different mechanisms for the hyperfine interaction between the resident electron and the nuclei. Nuclear spins precess in the mean Knight field and are polarized by the oriented resident electron. The rates of these effects are proportional to the average electron spin. Polarized nuclei create a mean Overhauser field on the electron, and a mean indirect Weiss field on each other. Their polarization also relaxes via interaction with the resident electron. The rates of these three mechanisms are proportional to the value of nuclear polarization.

Figure 4

Dependence of nuclear polarization on the mean spin of the resident electron. Here $T_{1e}\left(0\right)=T_{1l}$. Curve 1 was calculated using the box model with ${\Omega }_{\text{max}}{\tau }_c⪡1$, curve 2 using the SCIB model with ${\Omega }_{\text{max}}{\tau }_c⪡1$, and curve 3 using the SCIB model with ${\Omega }_{\text{max}}{\tau }_c=10$.

Figure 5

(Color online) Relations between the ENSS’s description under illumination and in the darkness. Under illumination the ENSS state is described by the average electron spin, $s_0$, and the inverse nuclear-spin temperature, $\beta$. The Knight field of the electron creates the mean polarization of the cooled nuclei. After a sharp transition to darkness, the ensemble of quantum dots splits in two subensembles with two different electron-spin projections, $s_{\Omega }=\pm 1/2$, on the nuclear hyperfine field. The ENSS is characterized by an electron potential, $\varsigma$, by a nuclear potential, $\mathit{\xi }$, and by a nuclear inverse spin temperature, ${\chi }_{s_{\Omega }}$. The relaxation of these thermodynamic potentials is due to the dipole-dipole interaction between nuclear spins and electron phonon interaction. This relaxation transfers nuclear spin to the crystalline lattice (on a time $T_{\mathit{\xi }}$), energy diffusion from the quantum dot to the environment (on a time $T_{\chi }$), and energy from the scattering of phonons (on a time $T_{\varsigma }$).

Figure 6

(Color online) Nuclear-spin polarization as a function of the inverse nuclear-spin temperature for a spherical quantum dot with infinite wall. The solid curve is the averaged Langevin distribution [Eq. (31)] for infinite numbers of nuclei, the dashed curve, and the calculation for the SQDIB model [Eq. (30)] with $N=489$. Circles are the Monte Carlo simulation, also with $N=489$.

Figure 7

(Color online) Conservation of the quantum dot’s parameters. (a) With the dipole-dipole interaction switched off, the spin components $I_{\Sigma ,x}$, $I_{\Sigma ,y}$, and $I_{\Sigma ,z}$, given by curves 1, 2, and 3, respectively, the spin modulus, $I_{\Sigma }$, given by curve 4, and the energy of the system, $E$, given by curve 5, are all conserved. (b) When the dipole-dipole interaction is switched on, the spin components are no longer conserved, and change their values chaotically instead (curves 1, 2, $z$ component is now shown). But the spin modulus, curve 3, is still conserved within numerical error.

Figure 8

(Color online) Spin correlator, $G$, vs time for different nuclear temperatures, as indicated, on a quantum dot with $N=1365$ nuclei. Solid lines are for the numerical simulation, whereas dashed lines are a Gauss approximation with fitting parameter $T_{\mathit{\xi }}$ following Eq. (32). (1) $\beta \hslash ⟨\omega ⟩\approx 3.5$, ${\rho }_{\Omega }=0.410$, $T_{\mathit{\xi }}⟨\omega ⟩=820\pm 15$, (2) $\beta \hslash ⟨\omega ⟩\approx 0.18$ (${\rho }_{\Omega }\approx 0.051$, $T_{\mathit{\xi }}⟨\omega ⟩=160\pm 15$, (3) $\beta \hslash ⟨\omega ⟩\approx 0.078$, ${\rho }_{\Omega }=0.036$, $T_{\mathit{\xi }}⟨\omega ⟩=90\pm 15$, (4) $\beta \hslash ⟨\omega ⟩\approx 0.05$ ${\rho }_{\Omega }=0.027$, $T_{\mathit{\xi }}⟨\omega ⟩=60\pm 15$.

Figure 9

(Color online) Dependence of nuclear-spin-relaxation time on nuclear-spin polarization. Circles are for $N=251$, squares for $N=485$, diamonds for $N=895$, and triangles for $N=1365$. All points coincide well with the straight line calculated from Eq. (28), with slope $T_2\gamma \approx 3$.