Exciton spectroscopy of optical reflection from wide quantum wells

Optical spectroscopy of resonant reflection has been used for both experimental and theoretical studies of the exciton-light interaction in wide, high-quality quantum-well structures with the widths exceeding the exciton Bohr radius by an order of magnitude or more. The light mixes the low-lying confined exciton states which are captured by the generalized model developed by M. M. Voronov et al. [Phys. Solid State 49, 1792 (2007)]. We demonstrate that the high-energy confined states in the wide QWs can still be described by the standard model in which the amplitude reflection coefficient from the QW is a sum of individual size-quantized exciton resonances. The excitonic parameters extracted from fitting the experimental spectrum to the standard model agree with those obtained by the numerical solution of the two-particle Schrödinger equation in a rectangular quantum well. The measured and microscopically calculated spectra are compared with those found with the widely used model of the center-of-mass exciton quantization and the polaritonic model. The comparison shows that the two approximate models considerably underestimate the interaction of confined excitons with light because they ignore the strong modification of the exciton wave function near the QW interfaces.

Figure 1

Normal-incidence reflection spectrum of the 240-nm $\mathrm{GaAs}/{\mathrm{Al}}_{0.28}{\mathrm{Ga}}_{0.72}\mathrm{As}$ QW (blue curve) and the fit by Eqs. (1) and (2) (dashed red curve). Numbers indicate the exciton resonances corresponding to the quantization of exciton center-of-mass motion in this wide QW. The inset shows the radiative broadenings of different excitonic resonances, $\hslash {\mathrm{\Gamma }}_{0j}$, extracted from the reflection spectrum (blue dots) and obtained in the microscopic modeling (red squares). Sample temperature $T=1.5$ K.

Figure 2

QW-width dependencies of (a) the exciton radiative decay rates $\hslash {\mathrm{\Gamma }}_{0j}$ and (b) the energy shifts $\hslash \delta {\omega }_{0,j}$ for several of the lowest exciton transitions in $\mathrm{GaAs}/{\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ QWs computed in the microscopic model.

Figure 3

Theoretically modeled reflection spectra of heterostructures with rectangular GaAs QWs with width $L=100$, 150, 200, and 250 nm. The spectra are calculated using the standard model, Eqs. (1, 2, 3, 4, 5, 6) (blue curves), and the generalized model, Eq. (7) (red curves). In (a) and (b) the blue and red curves coincide. Dashed curves calculated as $R_j={\left|r_j\right|}^2$ show the artificially separated exciton resonances $j=1$ and $j=2$ in the low-energy spectral region. Vertical lines indicate the energy positions of the bare exciton states (blue lines) and their radiative shift (red lines).

Figure 4

Normalized integral difference of the spectra calculated using the total matrix $M$ in Eq. (8) and the modified matrix with a pair of off-diagonal matrix elements ${\mathrm{\Omega }}_{j{j}^{\prime }}={\mathrm{\Omega }}_{j^{\prime }j}$ set to zero. The horizontal axes indicate $j$ and $j^{\prime }$; the vertical axis is the normalized difference in percent. The bottom right panel shows the error estimate as a function of the number of coupled states and QW width.

Figure 5

(a) Cross section of the numerically obtained wave function for the ninth exciton state, ${\mathrm{\Phi }}_9\left(z\right)={\phi }_9(z,z,0)$, in the 240-nm GaAs/AlGaAs QW (dots). The cosine function (red line) is the best-fit wave function ${\mathrm{\Phi }}_9^{\mathrm{c}.\mathrm{m}.}\left(z\right)$ of the c.m. model. Arrows marked by $L_d$ show the transition layer. (b) The in-plane effective exciton Bohr radius introduced in Eq. (12) as a function of coordinate $z=z_e=z_h$ for the ninth state (dots) and the first state (black dashed line) of the exciton in the QW. (c) The local in-plane characteristic length $a_0$ defined by Eq. (13) as a function of $z$. Red lines are calculated according to Eq. (16).

Figure 6

(a) Product ${\mathrm{\Phi }}_9\left(z\right)cos\left(qz\right)$ for the ninth exciton state in the 240-nm GaAs/AlGaAs QW. Gray shading indicates the area which gives no integral contribution to the decay rate ${\mathrm{\Gamma }}_{0,9}$. Light blue areas are the contribution in the c.m. model. Red areas are the additions from the microscopic model. (b) Dependence of $\hslash {\mathrm{\Gamma }}_{0j}$ on the exciton level index $j$ calculated in the microscopic (blue dots) and c.m. (red dots) models. The inset shows the same dependence for small $j$. (c) Ratio of the radiative damping rates calculated in the microscopic and c.m. models vs the exciton state number $j$. The errors indicate the inaccuracy of the microscopic calculation, in particular, for $j\ge 15$, where the confined exciton states overlap with the unbound electron-hole continuum.

Figure 7

Comparison of the measured reflection spectrum of the 240-nm QW (black curve) with those calculated in the polaritonic model accounting for only the heavy-hole exciton polariton contribution (red curve) and for both the heavy- and light-hole exciton polaritons (green curve). The model parameters are $\hslash {\omega }_0=1.5158$ eV, $\hslash {\omega }_{LT}^{hh}=0.075$ meV, $\hslash {\mathrm{\Gamma }}_{hh}=0.07$ meV, $\hslash {\mathrm{\Gamma }}_{lh}=0.12$ meV, $L^{★}=196$ nm.