Classical dynamics and semiclassical analysis of excitons in cuprous oxide

Excitons, as bound states of electrons and holes, embody the solid state analog of the hydrogen atom, whose quantum spectrum is explained within a classical framework by the Bohr-Sommerfeld atomic model. In a first hydrogenlike approximation the spectra of excitons are also well described by a Rydberg series, however, due to the surrounding crystal environment deviations from this series can be observed. A theoretical treatment of excitons in cuprous oxide needs to include the band structure of the crystal, leading to a prominent fine-structure splitting in the quantum spectra. This is achieved by introducing additional spin degrees of freedom into the system, making the existence and meaningfulness of classical exciton orbits in the physical system a nontrivial question. Recently, we have uncovered the contributions of periodic exciton orbits directly in the quantum mechanical recurrence spectra of cuprous oxide [J. Ertl et al., Phys. Rev. Lett. 129, 067401 (2022)] by application of a scaling technique and fixing the energy of the classical dynamics to a value corresponding to a principle quantum number $n=5$ in the hydrogenlike case. Here, we present a comprehensive derivation of the classical and semiclassical theory of excitons in cuprous oxide. In particular, we investigate the energy dependence of the exciton dynamics. Both the semiclassical and quantum mechanical recurrence spectra exhibit stronger deviations from the hydrogenlike behavior with decreasing energy, which is related to a growing influence of the spin-orbit coupling and thus a higher velocity of the secular motion of the exciton orbits. The excellent agreement between semiclassical and quantum mechanical exciton recurrence spectra demonstrates the validity of the classical and semiclassical approach to excitons in cuprous oxide.

Figure 1

Schematic view of the band structure. The conduction band belongs to irreducible representation ${\mathrm{\Gamma }}_6^{+}$ while at the $\mathrm{\Gamma }$-point a ${\mathrm{\Gamma }}_7^{+}$ and ${\mathrm{\Gamma }}_8^{+}$ valence band exists. These bands are separated by the spin-orbit coupling $\mathrm{\Delta }$. Transitions from the upper and lower valence band to the conduction band result in the yellow and green exciton series, respectively.

Figure 2

PSOS for the symmetry planes normal to $\left[100\right]$ (top row) and normal to $\left[1\overline{1}0\right]$ (bottom row) at $n_0=3$ (left), $n_0=5$ (middle), and $n_0=10$ (right). A selection of orbits is shown as insets, labeled by their winding numbers $M_1=1$ for the nearly circular orbits and $M_1:M_2$ for the periodic orbits on the two-dimensional tori. Their positions in the PSOS are marked by corresponding symbols. Coordinates and momenta are given in the scaled units (9) and thus approximately cover the same range for all values of $n_0$. The figure extends previous results for $n_0=5$ in the plane $\perp \left[100\right]$ presented in Ref. [19].

Figure 3

Top: Enlarged part of the PSOS for the plane normal to $\left[100\right]$ at $n_0=5$, where two fixed points are surrounded by elliptic and hyperbolic structures, respectively. Bottom: Stable and unstable periodic orbit with winding numbers $M_1:M_2=5:1$ corresponding to the two fixed points.

Figure 4

Three-dimensional orbits with winding numbers $M_1:M_2:M_3=16:1:2$. Four distinct orbits with different orientation and position of their maxima exist. The projection onto the $yz$ plane is shown below each orbit.

Figure 5

Sum ${\lambda }_i+1/{\lambda }_i$ for the nearly circular orbits in the different symmetry planes, where ${\lambda }_{\parallel }$ and ${\lambda }_{\perp }$ describe the stability of the orbits in the plane and out of the plane, respectively. The directions normal to the symmetry plane are stable for the plane normal to $\left[100\right]$ (blue curve) and unstable for the plane normal to $\left[1\overline{1}0\right]$ (orange curve). Both orbits are stable with respect to perturbations in the plane with almost identical stability eigenvalues ${\lambda }_{\parallel }$ (red and green curves).

Figure 6

Sum ${\lambda }_i+1/{\lambda }_i$ for the two-dimensional orbits in the plane normal to $\left[100\right]$ for $n_0=3$ (top), $n_0=5$ (middle), and $n_0=10$ (bottom). ${\lambda }_{\parallel }$ describes the stability of the orbits in the plane, ${\lambda }_{\perp }$ describes the stability of the orbits out of the plane. The stability eigenvalues for $n_0=3$ and 10 extend results presented for $n_0=5$ in the Supplemental Material of Ref. [20].

Figure 7

Difference of the action ${\tilde{S}}_{\mathrm{po}}$ of the periodic orbits (po) at $n_0=5$ and the corresponding two-dimensional orbits ${\tilde{S}}_{\perp \left[100\right]}$ in the plane normal to $\left[100\right]$ normalized by their winding number $M_2$ over the ratio of winding numbers $M_1/M_2$. Six different series of three-dimensional orbits with winding numbers $M_2:M_3$ are shown as solid points, connected by lines to guide the eye. The action of two-dimensional orbits in the plane normal to $\left[100\right]$ provides a lower border for the families of three-dimensional orbits. For increasing ratio $M_1/M_2$ the nearly circular orbit (indicated by a blue circle) is reached.

Figure 8

Second derivative of the function $J_2=g_E\left(J_1\right)$ with respect to $J_1$ for the two-dimensional orbits in the plane normal to $\left[100\right]$ at $n_0=3$ (top), 5 (middle), and 10 (bottom). The function of $g_E^{\prime \prime }$ for $n_0=5$ has already been presented in the Supplemental Material of Ref. [20].

Figure 9

Eigenvalue spectra for $n_0=3$ (top), $n_0=5$ (middle), and $n_0=10$ (bottom). Each eigenvalue contributes a delta peak to the quantum mechanical density of states (48). The lowest axis label for the effective quantum number $n_{\mathrm{eff}}$ is valid for all three parts of the figure. The three upper labels for the ratio $\tilde{\mathrm{\Delta }}/\mathrm{\Delta }$ of the scaled and real physical spin-orbit coupling are valid for the corresponding individual plots. Part of the spectrum for $n_0=5$ has already been presented in Ref. [20].

Figure 10

Recurrence spectra for $n_0=3$ (top), $n_0=5$ (middle), and $n_0=10$ (bottom). The semiclassical amplitudes are shown as colored bars. Amplitudes for orbits in the plane normal to $\left[100\right]$ are shown in blue, amplitudes for orbits in the plane normal to $\left[1\overline{1}0\right]$ in green, and amplitudes for three-dimensional orbits are shown in red. The amplitudes are labeled with the winding numbers of the corresponding orbits. The quantum recurrence spectra (black line) are shifted upwards for better visibility. The recurrence spectrum for $n_0=5$ has already been shown in Ref. [20].