Dynamics of a Cr spin in a semiconductor quantum dot: Hole-Cr flip-flops and spin-phonon coupling

A detailed analysis of the photoluminescence (PL) intensity distribution in singly Cr-doped CdTe/ZnTe quantum dots (QDs) is performed. First of all, we demonstrate that hole-Cr flip-flops induced by an interplay of the hole-Cr exchange interaction and the coupling with acoustic phonons are the main source of spin relaxation within the exciton-Cr complex. This spin flip mechanism appears in the excitation power dependence of the PL of the exciton as well as in the intensity distribution of the resonant PL. The resonant optical pumping of the Cr spin which was recently demonstrated can also be explained by these hole-Cr flip-flops. Despite the fast exciton-Cr spin dynamics, an analysis of the PL intensity under magnetic field shows that the hole-Cr exchange interaction in CdTe/ZnTe QDs is antiferromagnetic. In addition to the Cr spin dynamics induced by the interaction with carriers' spin, we finally demonstrate using time resolved optical pumping measurements that a Cr spin interacts with nonequilibrium acoustic phonons generated during the optical excitation inside or near the QD.

Figure 1

(a) PL of the exciton in a Cr-doped QD (QD1) for three different excitation powers at $T=5$ K. (b) Excitation power dependence of the PL intensities of the different exciton-Cr lines. The solid line is a linear fit. (c) Excitation power dependence of the PL intensity ratios of the central and low energy lines ($I_{\left(\mathrm{C}\right)}/I_{\left(\mathrm{LE}\right)}$) and central and high energy lines ($I_{\left(\mathrm{C}\right)}/I_{\left(\mathrm{HE}\right)}$).

Figure 2

(a) Linearly polarized PL spectra of the exciton in a Cr-doped QD (QD2) and configuration of the resonant excitation and detection in the optical pumping experiments. (b) and (c) resonant optical pumping experiment on QD2 for cross-circularly polarized excitation detection (b) and cross-linearly polarized excitation detection (c). (d) Energy detuning dependence of the resonant PL intensity (${\mathrm{I}}_1$ at the beginning and ${\mathrm{I}}_2$ at the end of the pump pulse) detected in (C) for cross-linearly polarized excitation detection. (e) Excitation power dependence of the resonant PL transient detected in (C) for cross-linear excitation detection.

Figure 3

(a) Diagram of the energy levels in a Cr-doped QD (Cr alone and exciton-Cr). Only the three low energy Cr spin states $S_z=0$ and $S_z=\pm 1$ are presented. Both the bright excitons (${|\Uparrow }_h{\downarrow }_e\rangle$ in blue and ${|\Downarrow }_h{\uparrow }_e\rangle$ in red) and the dark excitons (${|\Uparrow }_h{\uparrow }_e\rangle$ or ${|\Downarrow }_h{\downarrow }_e\rangle$ in green) are presented. Res. Exc. stands for the energy of the laser used in the resonant optical pumping experiments. (b) PL intensity map of the magnetic field dependence of the exciton in QD1 in the Faraday configuration. (c) Circularly polarized PL spectra at ${\mathrm{B}}_z=0\mathrm{T}$ and ${\mathrm{B}}_z=7\mathrm{T}$ in $\sigma +$ and $\sigma -$ polarization.

Figure 4

(a) Zero magnetic field PL of the exciton-Cr complex in QD3 and (b) intensity map of the magnetic field dependence of the PL of QD3. (c) Excitation power dependence of the PL of QD3 at ${\mathrm{B}}_z=10\text{T}$ in $\sigma -$ (left) and $\sigma +$ (right) polarization. (d) Corresponding excitation power dependence of the PL intensity ratios $I_{\left(\mathrm{C}\right)}/I_{\left(\mathrm{LE}\right)}$ and $I_{\left(\mathrm{C}\right)}/I_{\left(\mathrm{HE}\right)}$.

Figure 5

Proposed spin relaxation paths within the exciton-Cr complex ($X$-Cr) for an excitation of the bright exciton ${|\Downarrow }_h{\uparrow }_e\rangle$ in $\sigma -$ polarization (top) and of the bright exciton ${|\Uparrow }_h{\downarrow }_e\rangle$ in $\sigma +$ polarization (bottom). ${\tau }_h$ is the spin flip time of the hole, ${\tau }_{h\mathrm{Cri}}$ is a hole-Cr flip-flop times, and ${\tau }_{nr}$ is the nonradiative recombination time of the dark excitons.

Figure 6

(a) Relaxation time ${\tau }_{h\mathrm{Cr}}$ between the states ${|+1\rangle |\Downarrow }_h;{\downarrow }_e\rangle$ and ${|0\rangle |\Uparrow }_h;{\downarrow }_e\rangle$ (or ${|-1\rangle |\Uparrow }_h;{\uparrow }_e\rangle$ and ${|0\rangle |\Downarrow }_h;{\uparrow }_e\rangle$) as a function of the energy splitting between those states, calculated at a temperature $T=10$ K with a Gaussian hole wave function (parameters $l_z=1,25$ nm and $l_{\perp }=3$ nm) and ${\mathrm{\Delta }}_{lh}=25$ meV (red), ${\mathrm{\Delta }}_{lh}=50$ meV (black). The other QD parameters are listed in Table 1 of the Appendix. The CdTe parameters used in the calculation can be found in Table 2 of the Appendix. (b) Temperature variation of the relaxation time for an energy splitting $E=1$ meV and ${\mathrm{\Delta }}_{lh}=25$ meV (red), ${\mathrm{\Delta }}_{lh}=50$ meV (black).

Figure 7

Calculated excitation power dependence of the population ratios $I_{\left(\mathrm{C}\right)}/I_{\left(\mathrm{HE}\right)}$ and $I_{\left(\mathrm{C}\right)}/I_{\left(\mathrm{LE}\right)}$ at (a) ${\mathrm{B}}_z=10\mathrm{T}$ in $\sigma -$ polarization (${\rho }_{{|0\rangle |\Downarrow }_h,{\uparrow }_e\rangle }/{\rho }_{{|-1\rangle |\Downarrow }_h,{\uparrow }_e\rangle }$ and ${\rho }_{{|0\rangle |\Downarrow }_h,{\uparrow }_e\rangle }/{\rho }_{{|+1\rangle ,|\Downarrow }_h,{\uparrow }_e\rangle }$, respectively), at (b) ${\mathrm{B}}_z=0\mathrm{T}$ and at (c) ${\mathrm{B}}_z=10$ T in $\sigma +$ polarization (${\rho }_{{|0\rangle |\Uparrow }_h,{\downarrow }_e\rangle }/{\rho }_{{|+1\rangle |\Uparrow }_h,{\downarrow }_e\rangle }$ and ${\rho }_{{|0\rangle |\Uparrow }_h,{\downarrow }_e\rangle }/{\rho }_{{|-1\rangle ,|\Uparrow }_h,{\downarrow }_e\rangle }$, respectively). The parameters used in the model are ${\tau }_b=0.25\mathrm{ns}, {\tau }_d=20\mathrm{ns}, {\tau }_h=10\mathrm{ns}, {\tau }_{\mathrm{Cr}}=50\mathrm{ns}, {\tau }_{h\mathrm{Cr}1}=2\mathrm{ns}$, and ${\tau }_{h\mathrm{Cr}2}=20\mathrm{ns}$. The QD parameters are listed in Table 1 of the Appendix.

Figure 8

Two wavelength pump-probe experiments performed on QD2. (a) Configuration 1: PL transients recorded for a probe laser tuned on an excited state of the dot and at $d=4\mu \mathrm{m}$ from the dot. Red: probe on; black: probe off. (b) Configuration 2: PL transient recorded for overlapped pump and probe and a for probe laser at $E_{\mathrm{probe}}=2014$ meV, close to the resonant laser ($E_{\mathrm{pump}}=2001$, 1 meV); red: probe on; black: probe off. (c) Evolution of the pumping transient intensity as a function of the distance between the probe laser and the dot. (d) Evolution of the pumping transient intensity as a function of the probe power, for a probe laser at $d=4\mu \mathrm{m}$ from the dot. (e) Evolution of the pumping transient intensity as a function of the probe power for a probe laser at $E_{\mathrm{probe}}=2014$ meV.

Figure 9

(a) Intensity map of the magnetic field dependence of the $\sigma -$ PL spectra of a Cr-doped QD calculated with the parameters listed in Table 1. A thermalization on the exciton-Cr levels with ${\mathrm{T}}_{\mathrm{eff}}=40$ K is used for a simplified description of the PL intensity distribution.