# Spin Transport in a Quantum Hall Insulator
## 3 Results and Discussion  
Figure 1a shows the resonant reflection spectra for an optical transition between the zero Landau level states of heavy holes in the valence band and zero Landau level states of electrons in the conduction band, measured in a quantum Hall insulator at filling factors \(\
u=2\) and \(\
u=1.4\). Clearly, when \(\
u=2\), there is no PRR signal without photoexcitation. This trivial result indicates that the absorption of resonance radiation with its subsequent re-emission becomes possible only after the appearance of Fermi holes (vacancies) at the zero Landau level. Due to optical transitions associated with the upper spin sublevel of the zero electron Landau level, the PRR signal emerges as the filling factor is dropped to 1.4. Therefore, the PRR signal is unaffected by turning on the laser diode. By contrast, when it comes to the Hall insulator at the filling factor \(\
u=2\), switching the laser diode on results in a PRR signal, implying the generation of a macroscopic number of non-equilibrium magnetoexcitons. Using a rectangular pulse generator to modulate the laser diode current, we measured the decay time \(\tau\) of the resonant reflection signal from the photoexcitation point after the end of the pumping pulse. Thus, we found that an increase in the pumping pulse duration \(\tau_{\text{p}}\) or the peak power \(P_{\text{p}}\) leads to a rise in the number of magnetoexcitons excited in the Hall insulator per pulse. In this case, the pulse repetition time was set to tens of milliseconds, greatly beyond all possible characteristic transient and relaxation processes in the system under investigation. Figure 2 shows the dependence of the PRR signal decay time \(\tau\) on the pumping pulse duration \(\tau_{\text{p}}\) at a constant repetition time \(T_{\text{p}}\). On this curve, we can distinguish three characteristic regions. Region 1 is associated with the low density of non-equilibrium spin excitations, where the PRR signal is weakly dependent on the width of the pumping pulse. In region 2, the decay time of the PRR signal increases by approximately 300 \(\mu\)s. Finally, in region 3, we observe a sharp drop in the value of \(\tau\), which is related to spin excitations escaping the pumping area [14]. At the end of the last region, the relaxation time \(\tau\) is measured to be 30 \(\mu\)s. The velocity of spin excitations in the magnetofermionic condensate escaping the excitation spot can be estimated as \(\upsilon\approx\text{d}/2\tau\), where d is the diameter of the pumping spot. Considering d = 1.5 mm, the calculated velocity comes to approximately 25 m/s. This result demonstrates that spin excitations have excellent prospects for signal transmission in Hall insulators. However, it should be noted that the resonance reflection  
Figure 1: (**a**) PRR spectra of a 2 DES (sample 1) in the state of a Hall conductor, \(\
u=1.4\), and a Hall insulator, \(\
u=2.0\), with (green curves) and without (black curves) non-equilibrium spin excitations. (**b**) The diagram of the experimental setup with a double-fiber technique. (**c**) The optical transition scheme for PRR. The zero and the first Landau levels for electrons in the conduction band of an GaAs/ AlGaAs quantum well are designated by \(0_{\text{e}}\) and \(1_{\text{e}}\), respectively. The zero Landau level for heavy holes is denoted by \(0_{\text{hh}}\). technique does not discriminate between the relaxation of a non-equilibrium magnetoexciton to the ground state and the escape (diffusion) of a magnetoexciton from the pumping spot. A drop in the number of Fermi holes included in magnetoexcitons is observed in both cases. To understand the dependence of the spin excitation dynamics in the excitation area on the number of the pumped excitons, we measured additional PL spectra of a two-dimensional electron gas when non-equilibrium magnetoexcitons were present. These experiments allowed us to determine the generalized momentum distribution function of spin excitations.