Trion formation in a two-dimensional hole-doped electron gas

The interaction between a single hole and a two-dimensional, paramagnetic, homogeneous electron gas is studied using diffusion quantum Monte Carlo simulations. Electron-hole relaxation energies, pair-correlation functions, and electron-hole center-of-mass momentum densities are reported for a range of electron-hole mass ratios and electron densities. We find numerical evidence of a crossover from a collective excitonic state to a trion-dominated state in a density range in agreement with that found in recent experiments on quantum-well heterostructures.

Figure 1

Electron-hole relaxation energies for mass ratios $m_h/m_e=0.5$, 1, 2, 4, and 8, and HEG densities $r_s=1$, 2, 4, 6, and 10 a.u. The brown dashed line indicates the isolated exciton energy at $-4{\mathrm{Ry}}^{*}$, and the magenta dotted, black short-dashed, red long-dashed, green dotted-dashed, and blue dotted-dotted-dashed lines show isolated trion energies at mass ratios $m_h/m_e=0.5$, 1, 2, 4, and 8, respectively. Fits of the electron-hole relaxation energies at each mass ratio have been constrained to tend to the respective trion energies at $r_s/a_0^{*}\to \infty$. Error bars are smaller than the symbols.

Figure 2

Total energy of an isolated trion as a function of $\mu /m_e$ (circles). The energy of the neutral exciton is also shown (dashed line). Error bars are smaller than the symbols.

Figure 3

Cumulative (integrated) electron-hole PCFs normalized to show the total electron weight within a circle of radius $r$ centered on the hole. Data are plotted for constant mass ratio $m_h/m_e=1$ and HEG densities $r_s=1$, 2, 4, 6, and 10 a.u. (left panel), and for constant HEG density $r_s=6$ a.u. and mass ratios $m_h/m_e=0.5$, 1, 2, 4, and 8 (right panel).

Figure 4

Values of the electron-hole PCF at $r=0$ (on-top PCF) and at its first minimum. The solid line is a least-squares fit of the on-top PCF data to Eq. (4). The PCF values 0.25, 0.5, and 0.75 are represented with dotted gray lines. Error bars are smaller than the symbols. Note the logarithmic scales.

Figure 5

Crossover of hole-in-HEG system from the high-density collective excitonic state to a localized trion immersed in a low-density HEG as a function of mass ratio $m_h/m_e$ and density parameter $r_s/a_0^{*}$. Experimental data showing the evolution of absorption and photoluminescence spectra from the Fermi-edge singularity to discrete trion and exciton peaks are shown as colored areas, and are consistent with our results.

Figure 6

Momentum density for four representative points of our “phase diagram”: left column, $m_h/m_e=1$; right column, $m_h/m_e=8$; top row, $r_s=1$ a.u.; bottom row, $r_s=6$ a.u. Error bars are shown but are sometimes smaller than the symbols. Our results are normalized such that ${\int }_0^{\infty }2\pi \overline{k}\rho \left(\overline{k}\right)d\overline{k}=\pi k_F^2$.