Modeling of the thermalization of trapped paraexcitons in ${\mathrm{Cu}}_2\mathrm{O}$ at ultralow temperatures

We theoretically analyze the relaxation of paraexcitons in cuprous oxide due to exciton-phonon and exciton-exciton scattering. Particular attention is paid to the evolution of the distribution function as well as to the cooling process of the exciton gas. The results underline the importance of interexcitonic collisions at moderate and higher densities which prevent the formation of the typical bottleneck and accelerate the thermalization at ultralow temperatures significantly. Furthermore, we discuss the impact of strain on the exciton-phonon coupling and show that the overall cooling process of the excitons benefits from strain induced effects. However, for very low lattice temperatures ($T\ll 1.0\mathrm{K}$), the process of thermalization is slow, and the excitons might not reach the lattice temperature within their finite lifetime.

Figure 1

Distribution function at $t=0\mathrm{n}\mathrm{s}$ (solid blue line), $0.05\mathrm{n}\mathrm{s}$ (dash dotted red line), $0.5\mathrm{n}\mathrm{s}$ (dashed black line), and $1.0\mathrm{n}\mathrm{s}$ (solid black line) with $n={10}^{12}\mathrm{c}{\mathrm{m}}^{-3}$ (left panel), $n={10}^{14}\mathrm{c}{\mathrm{m}}^{-3}$ (middle panel), and $n={10}^{16}\mathrm{c}{\mathrm{m}}^{-3}$ (right panel). Excitons undergo X-X, X-${\mathrm{Ph}}^{\mathrm{LA}}$, and X-${\mathrm{Ph}}^{\mathrm{TA}}$ scattering at a lattice temperature of $T_{\mathrm{Bath}}=0.1\mathrm{K}$.

Figure 2

Distribution function after $t=10\mathrm{n}\mathrm{s}$ (left column), $50\mathrm{n}\mathrm{s}$ (middle column), and $100\mathrm{n}\mathrm{s}$ (right column) at a bath temperature of $T_{\mathrm{Bath}}=0.1\mathrm{K}$ and an exciton density of $n={10}^{12}\mathrm{c}{\mathrm{m}}^{-3}$ (top row), $n={10}^{13}\mathrm{c}{\mathrm{m}}^{-3}$ (middle row), and $n={10}^{14}\mathrm{c}{\mathrm{m}}^{-3}$ (bottom row). The different curves include the following scattering processes: X-${\mathrm{Ph}}^{\mathrm{LA}}$ (dash-dotted red curve); X-${\mathrm{Ph}}^{\mathrm{LA}}$ and X-${\mathrm{Ph}}^{\mathrm{TA}}$ (dashed blue curve); X-X, X-${\mathrm{Ph}}^{\mathrm{LA}}$, and X-${\mathrm{Ph}}^{\mathrm{TA}}$ (solid black curve).

Figure 3

Development of the exciton's “effective temperature” $T_{\mathrm{eff}}$ for $n={10}^{14}\mathrm{c}{\mathrm{m}}^{-3}$ at a lattice temperature of $1\mathrm{K}$ (top panel), $0.5\mathrm{K}$ (middle panel), and $0.1\mathrm{K}$. The different curves include the following scattering processes: X-${\mathrm{Ph}}^{\mathrm{LA}}$ (dash-dotted red curve); X-${\mathrm{Ph}}^{\mathrm{LA}}$ and X-${\mathrm{Ph}}^{\mathrm{TA}}$ (dashed blue curve); X-X, X-${\mathrm{Ph}}^{\mathrm{LA}}$, and X-${\mathrm{Ph}}^{\mathrm{TA}}$ (solid black curve). The dashed black curve shows the respective bath temperatures.

Figure 4

Deformation potential $D^{\mathrm{TA}}$ for the interaction between the strain shifted $1s$ yellow paraexcitons and TA phonons (dashed blue line) as a function of the trap depths $V_0$. The dashed black line indicates the maximum value for $D^{\mathrm{TA}}$.