Exciton-exciton interaction and biexciton formation in bilayer systems

We report quantum Monte Carlo calculations of biexciton binding energies in ideal two-dimensional bilayer systems with isotropic electron and hole masses. We have also calculated exciton-exciton interaction potentials and pair-distribution functions for electrons and holes in bound biexcitons. Comparing our data with results obtained in a recent study using a model exciton-exciton potential [Schindler and Zimmermann, Phys. Rev. B 78, 045313 (2008)], we find a somewhat larger range of layer separations at which biexcitons are stable. We find that individual excitons retain their identity in bound biexcitons for large layer separations.

Figure 1

(Color online) Biexciton binding energy $E_b$ as a function of layer separation $d$ for electron/hole mass ratio $\sigma =0.3$. Panel (a) shows the binding energy for layer separations close to the critical separation; panel (b) shows the binding energy for small layer separations.

Figure 2

(Color online) Biexciton binding energy $E_b$ as a function of layer separation $d$ for electron/hole mass ratio $\sigma =0.5$. Panel (a) shows the binding energy for layer separations close to the critical separation; panel (b) shows the binding energy for small layer separations.

Figure 3

(Color online) Biexciton binding energy $E_b$ as a function of layer separation $d$ for equal electron and hole masses $\left(\sigma =1\right)$. The square shows Schindler and Zimmermann’s estimate of the critical point at which the biexciton ceases to be bound (Ref. 8).

Figure 4

(Color online) The region of biexciton stability from DMC calculations compared with that found by Schindler and Zimmermann (Ref. 8) and Meyertholen and Fogler (Ref. 21). The critical points were found by extrapolating the biexciton binding energies to zero using the fitting form set out in Ref. 21. The statistical errors are comparable to the size of the symbols.

Figure 5

(Color online) Exciton-exciton interaction potential $E_I\left(R\right)$ as a function of center-of-mass separation $R$ with $\sigma =1$. The solid lines show the fit to the DMC data. Dashed lines show the dipole-dipole interaction energy [Eq. (4)].

Figure 6

(Color online) Exciton-exciton interaction potential $E_I\left(R\right)$ as a function of hole-hole separation $R$ for the heavy-hole case $\left(\sigma =0\right)$ with $d=0.9$, 1.0, and $1.1a_B^{\ast }$. The solid lines show the interaction potential from Ref. 8.

Figure 7

(Color online) Exciton-exciton interaction potential $E_I\left(R\right)$ as a function of (constrained) center-of-mass separation $R$ for $d=0.9a_B^{\ast }$ and $\sigma =0$ and 1.

Figure 8

(Color online) PDF $g_{\mathit{\text{eh}}}^{\text{single}}\left(r\right)$ for an isolated electron-hole pair from the exact solution of Eq. (2) in Ref. 7, shown at several layer separations for $\sigma =1$.

Figure 9

(Color online) Extrapolated electron-electron PDF $g_{\mathit{\text{ee}}}^{\text{ext}}\left(r\right)$ for bound biexcitons with $\sigma =1$. The hole-hole and electron-electron PDFs are identical for equal electron and hole masses. The inset shows the maximum of the PDF in greater detail.

Figure 10

(Color online) Extrapolated electron-hole PDF $g_{\mathit{\text{eh}}}^{\text{ext}}\left(r\right)$ for the biexciton system with $\sigma =1$ and several bilayer separations $d$.

Figure 11

(Color online) Biexciton electron-hole PDF relative to the single-exciton PDF, $2g_{\mathit{\text{eh}}}^{\text{ext}}\left(r\right)-g_{\mathit{\text{eh}}}^{\text{single}}\left(r\right)$, at $\sigma =1$ and several bilayer separations $d$.