Resonance energies and linewidths of Rydberg excitons in ${\mathrm{Cu}}_2\mathrm{O}$ quantum wells

Rydberg excitons are the solid-state analogs of Rydberg atoms and can, e.g., for cuprous oxide, easily reach a large size in the region of $\mu \mathrm{m}$ for principal quantum numbers up to $n=25$. The fabrication of quantum welllike structures in the crystal leads to quantum confinement effects and opens the possibility to study a crossover from three-dimensional to two-dimensional excitons. For small widths of the quantum well (QW), there are several well-separated Rydberg series between various scattering thresholds, leading to the occurrence of electron-hole resonances with finite lifetimes above the lowest threshold. By application of the stabilization method to the parametric dependencies of the real-valued eigenvalues of the original three-dimensional Schrödinger equation, we calculate the resonance energies and linewidths for Rydberg excitons in QWs in regimes where a perturbative treatment is impossible. The positions and finite linewidths of resonances at energies above the third threshold are compared with the complex resonance energies obtained within the framework of the complex-coordinate-rotation technique. The excellent agreement between the results demonstrates the validity of both methods for intermediate sizes of the QW-like structures, and thus for arbitrary widths.

Figure 1

B-spline functions with (a) equidistant and (b) nonequidistant spacing of knots and additional ghost knots at the boundaries. When the first and last B splines are removed, the functions satisfy vanishing boundary conditions.

Figure 2

Spectrum of the Hamiltonian (8) with respect to (a) the threshold $E_{1,1}\left(L\right)$ and (b) the threshold $E_{2,2}\left(L\right)$ as a function of the QW width. Note that the energy scales differ from the energy scale in Fig. 4, where absolute values are given. The angular quantum number is $m=1$, and the box size parameter is ${\rho }_{max}=500\mathrm{nm}$. The five resonant states at $L=8\mathrm{nm}$, which are investigated with the stabilization method and the complex-coordinate-rotation method are highlighted in (b) by red markers.

Figure 3

Application of the stabilization method for the QW width $L=8\mathrm{nm}$ and $m=1$ for the resonances highlighted in Fig. 2. Note that here we show absolute resonance energies, which differ from the values in Fig. 2, where the threshold energy $E_{2,2}\left(L\right)$ has been subtracted. (a) Stabilization diagram in the region from ${\rho }_{max}=300\mathrm{nm}$ to ${\rho }_{max}=700\mathrm{nm}$. Segments belonging to the studied resonant states are highlighted by the colored solid and dashed lines. They refer to states 1–5 in Fig. 2. (b) Logarithm of the density of states as a function of energy $E$. Colored peaks marked as a and b are obtained from the solid and dashed segments in (a), respectively. For a better visualization, $\varrho \left(E\right)$ is shifted upwards by a small value ${\varrho }_0$, so $log\left(\varrho \right)$ is bounded from below by a constant value. The crosses and dots represent $\varrho \left(E\right)$ calculated by Eq. (11). The dashed lines represent the fits to the Lorentzian shape given in Eq. (12).

Figure 4

Eigenvalues of the complex-coordinate-rotated Hamiltonian (13) for the QW width $L=8\mathrm{nm}$ and the angular momentum quantum number $m=1$. Note that the resonance positions are shifted compared to Fig. 2, since no threshold energy has been subtracted. The first five threshold energies $E_{i,j}$ are shown as colored lines. The five selected resonances, which have been analyzed by application of the stabilization method, are marked by red markers, while all other resonances are represented by blue dots. The even parity resonance below the threshold $E_{2,1}$ is marked by an arrow.