Resonant impurity and exciton states in a narrow quantum well

An analytical investigation of resonant impurity and exciton states in a narrow quantum well (QW) is performed. We employ the adiabatic multisubband approximation assuming that the motions parallel and perpendicular to the heteroplanes separate adiabatically. The coupling between the Coulomb states associated with the different size-quantized subbands ($N=1$, 2, …) is taken into account. In the two- and three-subband approximation the spectrum of the complex energies of the impurity electron and the exciton optical absorption coefficient are derived in an explicit form. The spectrum comprises a sequence of series of quasi-Coulomb levels $\left(n\right)$ where only the series belonging to the ground subband $N=1$ is truly discrete while the excited series $N⩾2$ consist of quasi-discrete energy levels possessing non-zero widths ${\Gamma }_{Nn}$. Narrowing the QW leads to an increase of the binding energy and to a decrease of the resonant energy width ${\Gamma }_{Nn}$ and the resonant energy shift $\Delta E_{Nn}$ of the impurity electron. Displacing the impurity center from the midpoint of the QW causes the binding energy to decrease while the width ${\Gamma }_{Nn}$ and the corresponding shift $\Delta E_{Nn}$ both increase. A Lorentzian form is recovered for the exciton absorption profile. The absorption peak is narrowed and blue shifted for a narrowing of the quantum well. A successful comparison with existing numerical data is performed. For GaAs QW’s it is shown that the resonant states analyzed here are sufficiently stable to be observed experimentally.

Figure 1

(a) A schematic form of the potentials $V_{NN}\left(\rho \right)$ (6) and quasi-Coulomb discrete $\left(n\right)$ and continuous $\left(q\right)$ spectra (10) adjacent to the ground $(N=1)$ and first excited $(N=2)$ size-quantized levels ${\epsilon }_N={\hslash }^2{\pi }^2{N}^2∕2m_e{d}^2$ in the the QW of width $d$. The density of states ${\rho }_{\perp }\left(11\right)$ is plotted as a function of the energy $E$ (7, 10) in the (b) single-subband (12) and (c) multisubband approximations providing the widths ${\Gamma }_{Nn}$ (31, 57).

Figure 2

The dimensionless resonant width ${\Gamma }_2\left(1\right)∕\mathcal{R}$ (31) of the ground impurity state $(n_0=1)$ adjacent to the $N=2$ subband plotted as a function of the relative impurity position $\mid b\mid ∕(d∕2)$ and the width of the QW $d$ ($\mathcal{R}$ and $a_0$ are the impurity Rydberg constant and the Bohr radius, respectively).

Figure 3

The dependence of the resonant shift $\Delta E_{\perp 2}^{\left(r\right)}\left(1\right)$ (36) of the ground impurity state $(n_0=1)$ adjacent to the $N=2$ subband given in terms of the impurity Rydberg constant $\mathcal{R}$ on the relative impurity position $\mid b\mid ∕(d∕2)$ and the width of the QW $d$ scaled to the impurity Bohr radius $a_0$.

Figure 4

The relative coefficient of the exciton absorption $\Lambda (\omega ,{\nu }_0)$ (45) calculated from Eq. (54) for the different widths $d$ in the vicinity of the ground peak $({\nu }_0\gtrsim 1)$ (56). The parameter $s=(\hslash \omega -E_g-{\epsilon }_2)∕4{\mathcal{R}}^{\left(\mathrm{ex}\right)}$ is the shift of the photon energy $\hslash \omega$ related to the edge of absorption $E_g+{\epsilon }_2$ given in terms of the exciton Rydberg constant ${\mathcal{R}}^{\left(\mathrm{ex}\right)}$.