Spin polarization recovery and Hanle effect for charge carriers interacting with nuclear spins in semiconductors

We report on theoretical and experimental study of the spin polarization recovery and Hanle effect for the charge carriers interacting with the fluctuating nuclear spins in the semiconductor structures. We start the theoretical description from the simplest model of static and isotropic nuclear spin fluctuations. Then we describe the modification of the polarization recovery and Hanle curves due to the anisotropy of the hyperfine interaction, finite nuclear spin correlation time, and the strong pulsed spin excitation. For the latter case, we predict the appearance of the resonant spin amplification in the Faraday geometry and of the quantum Zeno effect. The set of the experimental results for various structures and experimental conditions is chosen to highlight the specific effects predicted theoretically. We show that the joint analysis of the spin polarization recovery and the Hanle effect is a very valuable tool for addressing carrier spin dynamics in semiconductors and their nanostructures.

Figure 1

Spin polarization calculated after Eqs. (7a) (solid blue curve) and (7b) (solid red curve) for transverse and longitudinal magnetic field, respectively. The dashed curves show the approximations (11). The left inset shows the QD with randomly oriented nuclear spins and a single electron spin. The right top and bottom insets illustrate the spin precession with the frequency ${\mathit{\Omega }}_{\mathrm{tot}}$ (blue arrow) in the Faraday and Voigt geometries, respectively.

Figure 2

The spin polarization in zero magnetic field, ${\mathrm{\Omega }}_L=0$, calculated after Eq. (13) as a function of the anisotropy parameter $\lambda$. The schematics show the distributions of the random nuclear field for the corresponding values of $\lambda$ in the coordinate frame shown in the inset, see Eq. (12).

Figure 3

(a) Hanle and PRC (red) calculated after Eq. (6) for the parameters ${\delta }_x={\delta }_y=\delta$ and ${\delta }_z=0.1\delta$ ($\lambda =0.1$). The dashed curves show Eqs. (14). (b) The same curves for ${\delta }_x={\delta }_y=\delta$ and ${\delta }_z=10\delta$ ($\lambda =10$) and approximations (15).

Figure 4

Hanle (blue) and PRC (red) for finite nuclear spin correlation time calculated after Eq. (18) for the parameters (a) ${\tau }_s\delta =100$, ${\tau }_c\delta =10$; (b) ${\tau }_s\delta =3$, ${\tau }_c\delta =0.3$. The dashed curves show Eqs. (25) and (24) in panel (a) and (27) in panel (b).

Figure 5

Energy levels in the QD and transitions between them. The red and blue arrows denote the electron and heavy hole spins, respectively. The excitation with ${\sigma }^{+}$ polarized light is shown by the magenta arrow, black arrows denote the trion recombination, and the blue arrow at the top shows the trion spin relaxation, which leads to the spin generation in the ground state.

Figure 6

The dependence of the spin polarization on the repetition period of the pump pulses for weak pulses, $Q\to 1$ (blue curve) [Eq. (34)] and strong pulses, $Q=0$ (red curve). The black dotted and dashed curves show the approximate Eqs. (37) and (38), respectively. The calculations are performed with ${\tau }_s\delta =100$.

Figure 7

PRC (red) and Hanle curve (blue) for the parameters ${\tau }_s\delta =1000$, $T_R\delta =10$ for weak, $Q\to 1$, (dotted curves) and $\pi$ pump pulses, $Q=0$, (solid curves).

Figure 8

PRC (red) and Hanle curve (blue) for the short repetition period, $T_R\delta =0.1$, for (a) weak, $Q\to 1$, and (b) strong, $Q=0$, pump pulses calculated for ${\tau }_s\delta =10$. The black dotted curve in (b) shows Eq. (37) with account for Eq. (40).

Figure 9

The same as in Fig. 8 but for $T_R\delta =1$ (and ${\tau }_s\delta =100$).

Figure 10

The visibility of RSA in the Faraday geometry, Eq. (41), calculated for ${\tau }_s\delta =100$.

Figure 11

PRC (solid red curve) calculated after Eq. (45) for the parameters ${\delta }_{\mathrm{T}}=\delta$, ${\mathrm{\Omega }}_L^{\mathrm{T}}={\mathrm{\Omega }}_L$, ${\tau }_0\delta =0.2$ in the limit ${\delta }_{\mathrm{T}}^2{\tau }_s^{\mathrm{T}}{\tau }_0\gg 1$. The red dashed and black solid curves show $P\left({\mathrm{\Omega }}_L\right)$ [Eq. (7b)] and $G\left({\mathrm{\Omega }}_L^{\mathrm{T}}\right)/G\left(0\right)$ [Eq. (44)], respectively.

Figure 12

$n$-doped bulk GaAs (sample 1). PRC (red) and RSA (blue) signals in dependence of magnetic fields $B_{\parallel }$ and $B_{\perp }$, respectively, at the same detection conditions. They are fitted with Eq. (11b) for PRC and Eq. (11a) for zero RSA peak, as shown by the black dashed lines. Gray dashed line represents the basic 2 to 1 ratio of the amplitudes. $E=1.493$ eV, $P_{\text{pump}}=1$ W/${\mathrm{cm}}^2$, $f_{\text{m}}=50$ kHz, and $T=1.8$ K.

Figure 13

$n$-type ZnSe/${\mathrm{Zn}}_{0.85}{\mathrm{Mg}}_{0.15}\text{Se}$ single QW (sample 2). PRC (red) and RSA (blue) signals detected at the same experimental conditions. Black dashed lines are the fits using Eq. (27). $E=2.8037$ eV, $P_{\text{pump}}=0.35$ W/${\mathrm{cm}}^2$, $f_{\text{m}}=50$ kHz, and $T=1.8$ K.

Figure 14

ZnSe/${\mathrm{Zn}}_{0.89}{\mathrm{Mg}}_{0.11}{\mathrm{S}}_{0.18}{\mathrm{Se}}_{0.82}$ single QW with resident holes (sample 3). (a) KR signal demonstrating two oscillating components at $B_{\perp }=1$ T and corresponding $g$ factors. (b) PRC (red) and RSA (blue) signals with fittings (black dashed lines) using Eq. (27). The green dashed line shows the unmodified fit for the RSA peak. The gray dashed line at 1/3 is the reference of the basic case. $E=2.812$ eV, $P_{\mathrm{pump}}=0.35$ W/${\mathrm{cm}}^2$, $f_{\mathrm{m}}=100$ kHz, and $T=1.8$ K.

Figure 15

$p$-doped (In,Ga)As/GaAs QDs (sample 4). PRC (blue) and RSA (red) signals at the same experimental conditions. Black dashed line is a fit using Eq. (45) [11]. $E=1.392$ eV, $P_{\text{pump}}=0.35$ W/${\mathrm{cm}}^2$, $f_{\mathrm{m}}=25$ kHz, and $T=1.8$ K.