Influence of well-width fluctuations on the binding energy of excitons, charged excitons, and biexcitons in $\mathrm{GaAs}$-based quantum wells

We present a first-principle path integral Monte Carlo (PIMC) study of the binding energy of excitons, trions (positively and negatively charged excitons) and biexcitons bound to single-island interface defects in quasi-two-dimensional ${\mathrm{GaAs}∕\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}$ As quantum wells. We discuss in detail the dependence of the binding energy on the size of the well-width fluctuations and on the quantum-well width. The numerical results for the well-width dependence of the exciton, trions and biexciton binding energy are in good quantitative agreement with the available experimental data.

Figure 1

Dependence of the height of the localization potential $V_{e\left(h\right)}^{\mathrm{loc}}$, Eq. (5), for electrons (open dots) and holes (full dots) on the well width $L$ for a well-width fluctuation of $1\mathrm{ML}$. Inset: the lowest energy level in the potential, $V_{e\left(h\right)}$, vs the well width.

Figure 2

Left: electron ${\rho }_e\left(z\right)$ and hole ${\rho }_h\left(z\right)$ density matrix in the QW (dotted lines indicate QW walls). Right: QW width dependence of the effective electron–electron (ee) and electron–hole (eh) potentials, see Eq. (4).

Figure 3

(a) Binding energies of various excitonic complexes vs the diameter of the $1\mathrm{ML}$ quantum well width fluctuation for a QW width of $L=60\mathrm{\AA }$ and temperature, $T=1.5\mathrm{K}$; (b) the same as (a) but now for the relative increase of the binding energy.

Figure 4

Average distance between the constituents of the different excitonic complexes as a function of $D$ for (a) equally charged and (b) oppositely charged particles.

Figure 5

Localization energy of an electron (a) and hole (b) in the different excitonic complexes as a function of diameter of the localization island for a one monolayer fluctuation and QW width of $L=60\mathrm{\AA }$. Dotted lines are the corresponding localization energies of a single electron and hole in the same localization potential.

Figure 6

(a) Localization energy gain, $\delta E_{\mathrm{loc}}$ [Eq. (15)], and energy gain, $\delta E$, of the exciton and biexciton vs diameter of the localization island. (b) Comparison of the same energies for positive and negative trions. The same parameters are used as in Fig. 5.

Figure 7

(a), (b) Radial distribution function of electrons (a) and holes (b) for exciton, positive and negative trions, and the single electron (hole) as a function of distance, $r_{e\left(h\right)}$, from the center of the localization potential. The same parameters are used as in Fig. 5.

Figure 8

The same as Fig. 3, but now as a function of the number of monolayers, $N$, for a fixed QW width of $L=40\mathrm{\AA }$, temperature $T=1.5\mathrm{K}$ and diameter of the localization potential, $D=400\mathrm{\AA }$.

Figure 9

The same as Fig. 4, but now as a function of the number of monolayers, $N$. Same parameters are used as in Fig. 8.

Figure 10

Exciton binding energies for isotropic (a) and anisotropic (b) hole mass vs quantum well width. Solid square symbols are experimental data of Ref. 10 and circular symbols with a plus sign are obtained from the stochastic variational calculations (Refs. 13, 14). Full (dashed) lines with small symbols are PIMC results when the localization is included (not included).

Figure 11

Trion binding energies for isotropic (a) and anisotropic (b) hole mass vs quantum well width. Symbols are experimental data of Refs. 1, 2 and theoretical calculations of Ref. 22. Full (dashed) lines with symbols are PIMC results when localization is included (not included).

Figure 12

The same as in Fig. 11 but now for the biexciton binding energy. Symbols are experimental data of Refs. 4, 5, 6, 7, 8, 9, 10.

Figure 13

The biexciton to exciton binding energy ratio (Haynes factor), (a) vs QW width and (b) vs normalized localization potential. Symbols: experimental data for ${\mathrm{GaAs}∕\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}\mathrm{QWs}$ (Ref. 10). Curves with symbols: PIMC results. Solid (dashed) curve is the exciton binding energy with (without) localization included; (b) $E_B\left(X_2\right)∕E_B\left(X\right)$ vs localization strength $E_{\mathrm{loc}}∕E_B\left(X_2\right)$. Squares are the experimental data; Rhombii are PIMC results.