Analysis of the exciton-exciton interaction in semiconductor quantum wells

The exciton-exciton interaction is investigated for quasi-two-dimensional quantum structures. A bosonization scheme is applied including the full spin structure. For generating the effective interaction potentials, the Hartree-Fock and Heitler-London approaches are improved by a full two-exciton calculation which includes the van der Waals effect. With these potentials the biexciton formation in bilayer systems is investigated. For coupled quantum wells the two-body scattering matrix is calculated and employed to give a modified relation between exciton density and blueshift. Such a relation is of central importance for gauging exciton densities in experiments which pave the way toward Bose-Einstein condensation of excitons.

Figure 1

Effective interaction potentials for the strictly two-dimensional system: full calculation results (solid lines), Heitler-London approximation (dashed lines), and Hartree-Fock approximation (dot-dashed lines) plotted vs distance in units of the three-dimensional (3D) (bulk) exciton Bohr radius $a_B$. The vertical axis is given in units of the bulk exciton Rydberg energy $R{y}^{\ast }$. Antisymmetric channels are displayed in gray, while symmetric channels have black lines.

Figure 2

Asymptotic behavior of the potentials on a double-logarithmic plot for the strictly two-dimensional system: full calculation (solid), Heitler-London (dashed), and multipole expansion (dot-dashed); symmetric states lie on top of antisymmetric ones.

Figure 3

The van der Waals effect in the strictly two-dimensional system. Deviation of the full calculation from the direct potential for asymptotic large distances (solid line) and fitted van der Waals like potential $\propto 1/R^6$ (dashed line).

Figure 4

Haynes factor $f_H=B_{XX}/B_X$ for the different potentials in the strictly two-dimensional system: full calculation (solid line), Heitler-London potential (dashed line), and Hartree-Fock potential (dot-dashed line) plotted vs mass ratio $\sigma =m_e/m_h$. The dots are numerically exact values from Ref. 32.

Figure 5

Biexciton binding energy in the bilayer system plotted vs charge separation at $\sigma =0.5$ (solid line) and $\sigma =0.3$ (dashed line). The arrows denote the corresponding critical charge separation $d_{\text{crit}}$. Curves in the inset show the symmetric potential $U^s\left(R\right)$ for $d=0.8$, 0.9, 1.0, and 1.1 in units of the 3D exciton Bohr radius $a_B$.

Figure 6

Critical charge separation in units of the bulk exciton Bohr radius $a_B$ plotted vs mass ratio $\sigma =m_e/m_h$.

Figure 7

Interaction potentials for sample A: full calculation results (solid lines), Heitler-London potentials (dashed lines), and Hartree-Fock potentials (dot-dashed lines) plotted vs distance. Antisymmetric channels have gray lines, while symmetric channels have black ones.

Figure 8

Comparison of the full calculation results for sample A (solid lines) and sample B (dashed lines). Antisymmetric channels have gray lines, while symmetric channels have black ones.

Figure 9

Real (solid line) and imaginary (dashed line) parts of the scattering amplitude as a function of momentum for sample A. Included is the Maxwell distribution at $T=100\text{K}$ to show the relevant values for $q$. The ordinate has been normalized to the capacitor value [Eq. (16)].

Figure 10

Correction factors to the capacitor formula in dependence on temperature for sample A (solid lines) and sample B (dashed lines). The upper panel shows $f_1$ (blueshift), while the lower one shows $f_2$ (broadening).

Figure 11

Exciton pair-correlation function vs distance at a temperature of $T=6\text{K}$. The solid line refers to the calculated exciton-exciton potential for sample A. The dashed line represents ideal bosons, while the dot-dashed line holds for ideal fermions.