Spin-flip Raman scattering of the $\Gamma \text{−}X$ mixed exciton in indirect band gap (In,Al)As/AlAs quantum dots

The band structure of type-I (In,Al)As/AlAs quantum dots with band gap energy exceeding $1.63$ eV is indirect in momentum space, leading to long-lived exciton states with potential applications in quantum information. Optical access to these excitons is provided by mixing of the $\Gamma$- and $X$-conduction-band valleys, for which their spins may be oriented by resonant spin-flip Raman scattering. This access is used to study the exciton spin-level structure by accurately measuring the anisotropic hole and isotropic electron $g$ factors. The spin-flip mechanisms for the indirect exciton and its constituents as well as the underlying optical selection rules are determined. The spin-flip intensity is a reliable measure of the strength of $\Gamma \text{−}X$-valley mixing, as evidenced by both experiment and theory.

Figure 1

(a) Band alignment in (In,Al)As/AlAs QDs as a function of dot diameter for the valence (VB) and conduction (CB) bands. The energy difference between $\Gamma$- and $X$-electron states is denoted by $\Delta E_{\Gamma X}$. For a particular diameter, $\Gamma \text{−}X$ crossover occurs, corresponding to a gap energy $E_{\Gamma X}$. (b) The PL spectrum of an (In,Al)As/AlAs QD ensemble at $T=1.8$ K; excitation photon energy $E_{\mathrm{exc}}=2.33$ eV. The exciton recombination times $\tau$ across the ensemble are shown by open diamonds (right scale). (c) Resonantly excited PL of direct and indirect excitons around the $\Gamma \text{−}X$ crossover. The laser photon energies are marked by arrows. (d) Difference between laser energy and the peak position of the direct (circles) and indirect (triangles) exciton PL.

Figure 2

(a) Stokes and anti-Stokes SFRS spectra for magnetic fields of 4 and 5 T in tilted geometry with crossed linear polarization. Resonant excitation close to the $\Gamma \text{−}X$ crossing at 1.636 eV. (b) Dependences of Raman shifts on the magnetic field; lines are linear fits.

Figure 3

(a) $g$-factor angle dependence for heavy hole, $X$-valley electron and indirect exciton. Dashed line is the fit for $g_{\mathrm{hh}}\left(\theta \right)$, solid line is the calculation for $g_{\mathrm{Ex}}\left(\theta \right)$; see text. Note that the points $g_{\mathrm{Ex}}\left({15}^{\circ }\right)$ and $g_{\mathrm{Ex}}\left({45}^{\circ }\right)$ are evaluated from the measured $\mathrm{e}$ and hh $g$ factors. (b) Magnetic-field-induced circular polarization degree of the QD photoluminescence measured in Faraday geometry at different magnetic fields and a laser power of 1 mW/${\mathrm{cm}}^2$; $E_{\mathrm{exc}}=2.33$ eV, $T=1.8$ K. (c) Calculated energies of angle-dependent bright and dark exciton states at the $\Gamma \text{−}X$-crossing point, $B=5$ T. The center of gravity is taken as zero.

Figure 4

Cross-circularly and cocircularly polarized SFRS spectra measured at $B=4$ T and $E_{\mathrm{exc}}=1.644$ eV.

Figure 5

Schemes of the SFRS Stokes processes. The curved lines represent the incident and outgoing light, the tilted arrows indicate the mixed electron spin states, and the carriers participating in the spin-flip processes are shown in blue. Acoustic phonon energy $\hslash {\omega }_{\mathrm{ph}}^{\prime }$ is equal to $g_{\mathrm{hh}}{\mu }_{\mathrm{B}}B$.

Figure 6

The $\mathrm{e}$- and hh-SFRS resonance profiles as measure of the spin-flip scattering efficiencies; $B=5$ T, $T=1.8$ K, $\theta ={75}^{\circ }$. The red curve represents the theoretically modeled $\mathrm{e}$-SFRS efficiency based on Eq. (9).

Figure 7

(a) Simulated Raman spectrum for an $X$-valley state, after Eq. (10), and scheme of the spectral bandpass of the monochromator slits. Inset: The smooth tail of the $X$-valley line for $\Omega <1$. (b) Simulated Raman spectrum for $\Gamma$- and $X$-valley states in the case of very weak $\Gamma \text{−}X$ coupling ($\delta =0.008$ meV).

Figure 8

Simulated SFRS resonance profiles based on Eqs. (A1) and (A2) for different $p$ values.

Figure 9

(a) Simulated SFRS spectra around $\zeta =0$ for different $u$, after Eq. (A4). Inset: The simulations of Eq. (A4) (solid line) and Eq. (A5) (dashed line) for $u=0$. (b) Raman line including spectral width of monochromator slits; Eq. (A4) convoluted with Gaussian function.