Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an $n$-GaAs microcavity

Faraday rotation up to ${19}^{\circ }$ in the absence of an external magnetic field is demonstrated in an $n$-type bulk GaAs microcavity under circularly polarized optical excitation. This strong effect is achieved because (i) the spin-polarized electron gas is an efficient Faraday rotator and (ii) the light wave makes multiple round trips in the cavity. We introduce a concept of Faraday rotation cross section as a proportionality coefficient between the rotation angle, electron spin density and optical path and calculate this cross section for our system. From independent measurements of photoinduced Faraday rotation and electron spin polarization we obtain quantitatively the cross section of the Faraday rotation induced by free electron spin polarization ${\sigma }_F^{\exp }=-(2.5\pm 0.6)\times {10}^{-15}$ rad$\times$cm${}^2$ for photon energy 18 meV below the band gap of GaAs, and electron concentration $2\times {10}^{16}$ cm${}^{-3}$. It appears to exceed the theoretical value ${\sigma }_F^{\mathrm{th}}=-0.7\times {10}^{-15}$ rad$\times$cm${}^2$, calculated without fitting parameters. We also demonstrate the proof-of-principle of a fast optically controlled Faraday rotator.

Figure 1

(a) The $n$-doped GaAs microcavity sample with substrate used in this study (a) and the calculated transmission spectrum of the cavity. (b) We pumped at an energy of 1.59 eV and probed at the cavity resonance, about 18 meV below the band gap.

Figure 2

The three variants of the setup used in photoinduced Faraday rotation experiments (PEM: photoelastic modulator; BS: Babinet-Soleil compensator; AOM: acousto-optic modulator). Upper panel: (a) cw-Faraday rotation and Hanle effect; (b) nuclear spin cooling for measurement of the Knight field; and (c) time-resolved Faraday rotation induced by microsecond long pulses. Lower panel: cryostat, Helmholtz coils used to apply longitudinal and/or transverse magnetic field, and detection setup used is common in all three variants.

Figure 3

(a) Left scale: cw-Faraday rotation as function of the pump power (closed circles). Right scale: electron polarization calculated with Eq. (5) plotted as a function of $G{\tau }_s/n_e$ (solid line). (b) The full width at half maximum of the Hanle curves versus pump power density. Extrapolation to zero pump power density gives $B_{\mathrm{FWHM}}=3.3$ G, corresponding to spin relaxation time of 160 ns. Inset shows the Hanle curve acquired at 0.3 W/cm${}^2$.

Figure 4

(a) Scheme for optical cooling of nuclear spins at relatively large pump power in a longitudinal field (left) and detection in a small transverse field at lower pump power (right). (b) Typical Hanle curve versus total field $B_T=B_N+B_x$. Depending on the initial value of $B_T$, Faraday rotation will exhibit different time evolutions as $B_T$ relaxes to $B_x$. (c)–(e) Time evolution of Faraday rotation during cooling in a longitudinal field (yellow region), and detection in a weak transverse field $B_x=$1.5 G, for the three cases shown in panel (b) (red curves are fit to the data using Eq. (8)). In the shaded area (during cooling) the signal was not recorded to allow for a change in the lock-in sensitivity. (f) The extracted nuclear field as a function of longitudinal field. The observed Knight field of about 0.4 G is independent of sample temperature.

Figure 5

The spectrum of PL intensity (open square) and the PL circular polarization from the GaAs microcavity (open circle). The solid lines are fit by the calculated PL intensity and electron spin polarization assuming Fermi distributions of electrons.

Figure 6

(a) Ratio of Hartree-Fock and state-filling contributions to the Faraday rotation cross section as function of electron Fermi energy, calculated as functions of detuning, $\Delta E$, after Eqs. (18) and (16) respectively. The parameters of GaAs were used. (b) Schematic diagram in the detuning. Fermi energy axes showing dominant contributions to spin Faraday rotation: filled area corresponds to $|{\sigma }^{\mathrm{HF}}|>|{\sigma }^{\mathrm{sf}}|$.

Figure 7

(a) Experimentally measured temperature-dependent Faraday rotations as function of magnetic field at different temperatures: ${\theta }_F^{2K}$ (solid squares), ${\theta }_F^{5K}$ (solid circles), ${\theta }_F^{10K}$ (solid triangles), ${\theta }_F^{20K}$ (solid stars). (b) Comparison between measured Faraday rotation $\Delta {\theta }_F$ (left scale, symbols) and calculated temperature dependent spin polarization $\Delta {\rho }_e^{2K}$, $\Delta {\rho }_e^{5K}$, $\Delta {\rho }_e^{10K}$ (right scale, lines).

Figure 8

TRFR for circularly polarized 600-ns pulses (open squares). The solid black lines superimposed on experimental data are exponential fits of the rise and decay of the Faraday rotation, and the solid green curve in the figure bottom shows the train of pulses.

Figure 9

Right scale: Faraday rotation angle (crosses) and its fit (blue line) using Eq. (5) with measured spin relaxation time. Left scale: Temperature dependence of decay time (squares) and its linear fit.