Coherence-decoherence transition in a spin-magnetoexcitonic ensemble in a quantum Hall system

The recent experimental studies of extremely long-lived macroscopic ensembles of spin-cyclotron excitons (magnetoexcitons) which have to obey the Bose-Einstein statistics signal the emergence of an excitonic coherent phase. In the present paper the theory of a weakly interacting Bose gas of spin-cyclotron excitations is developed in terms of a virial correction to the single-magnetoexciton energy. The condition for coherent-incoherent phase transition is discussed. It is expected to be strongly related to the studied long-distance interexcitonic correlations. The results obtained theoretically are discussed in terms of their agreement with specific experimental data.

Figure 1

Illustration of the $S_z=1$ cyclotron spin-flip exciton at $\nu =2$ filling. The thick purple arrow indicates the effective promotion of a spin-flip electron from the zero to the first Landau level at nonzero exciton momentum $\mathbf{q}$. The vertical axis corresponds to the energy hierarchy, and the horizontal conventionally describes the $K$ space for $\mathbf{q}$ vectors or the real 2D space because the average distance between the positions of a promoted electron and an effective “hole” in the real space is equal to $l_B^2\mathbf{q}\times \widehat{z}$ (see the text). Shown are the cyclotron and Zeeman gaps, $\hslash {\omega }_c$ and ${\epsilon }_{\mathrm{Z}}$, as well as the Coulomb shift $\mathrm{\Delta }{E}_{\mathrm{C}}$.

Figure 2

Dispersion of a CSFE magnetoexciton calculated for a GaAs symmetrically doped QW 30 nm wide with an electron density of $2\times {10}^{11}{\mathrm{cm}}^{-2}$.

Figure 3

Functions $U(q,\mathrm{\Phi })$ and $V\left(p\right)$ measured in units of $e^2/\kappa l_B$ (see text). Both are calculated with the same form factor that was used to calculate the CSFE dispersion (see Fig. 2) and are appropriate for the quantum well used in the experiment (see Fig. 5 below). (a) The dependence $U(q,\mathrm{\Phi })$ on angle $\mathrm{\Phi }$ between ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ is shown for specific value $q=|{\mathbf{q}}_1|=|{\mathbf{q}}_2|=0.9$. (b) The function $V\left(p\right)=W_1+W_2$, where $\mathrm{\Phi }=0$, is calculated for $q_1=0.9+p$ and $q_2=0.9-p$. The coherent case ${\mathbf{q}}_1\equiv {\mathbf{q}}_2$ is marked by solid dots.

Figure 4

Energies are measured in units of $e^2/\kappa l_B$ and calculated with the previously used form factor (see Figs. 2 and 3) (a) Virial correction in the incoherent magnetoexcitonic gas: ${\mathcal{E}}_{\mathrm{vir}}\left(q\right)/\overline{n}=w_1\left(q\right)+w_2\left(q\right)$ (see text). (b) Virial correction in the coherent state: ${\tilde{\mathcal{E}}}_{\mathrm{vir}}\left(q\right)/\overline{n}=v\left(q\right)$.

Figure 5

The experimentally observed transition from the incoherent magnetoexcitonic phase to the coherent state at magnetic field $B=4.12\mathrm{T}$, which is accompanied by the increase in the oscillator strength of the photoinduced resonant reflection (PRR) normalized to one magnetoexciton. The temperature range where the phase transition occurs is in close agreement with the estimate presented in the text.