Spin-exchange kinetics of excitons in ${\mathrm{Cu}}_2\mathrm{O}$: Transverse acoustic phonon mechanism

The interconversion between orthoexcitons and lower-lying paraexcitons in ${\mathrm{Cu}}_2\mathrm{O}$ is studied by time resolving the orthoexciton and paraexciton luminescence following picosecond photoexcitation. The buildup of paraexciton numbers and the orthoexciton decay are observed over the temperature range $2–20\mathrm{K}$. The low-temperature limit of the conversion rate is accurately measured. The temperature dependence of the interconversion rate suggests that the dominant mechanism for ortho-para interconversion is the emission or absorption of a single transverse acoustic (TA) phonon. We propose a microscopic model in which rotation of the lattice induces the spin flip through spin-orbit coupling. The energy splitting between orthoexcitons and paraexcitons is modified by applying stress to the crystal, and the the interconversion rate as a function of this splitting is found to be consistent with the single TA phonon mechanism.

Figure 1

(a) Schematic diagram of the exciton relaxation kinetics at low densities. Wavy arrows denote phonon emission; straight arrows photon emission. $\hslash \Omega$ is the optical phonon energy. The time-integrated photoluminescence is given for (b) $T=1.66\mathrm{K}$ and (c) $T=18\mathrm{K}$. For $T>15\mathrm{K}$, the ${\mathrm{X}}_p-{\Gamma }_{25}^{-}$ line starts to overlap with the ${\mathrm{X}}_o-{\Gamma }_{15}^{-}$.

Figure 2

Trace (a) is the buildup of spectrally integrated ${\mathrm{X}}_p-{\Gamma }_{25}^{-}$ paraexciton spectrum following a $5\mathrm{ps}$ laser pulse. This raw data includes a base line from previously produced paraexcitons plus a background due to scattered light from the laser and strong orthoexciton luminescence. Trace (c), obtained at an energy just below the paraexcitons spectrum, shows the transient scattered light. Trace (b) equals trace (a) minus trace (c) and corresponds to the buildup of paraexcitons at $T=1.66\mathrm{K}$ plus the residual decaying signal from paraexcitons created in previous pulses.

Figure 3

Normalized temporal behaviors of excitons after subtracting the residual paraexciton signal. Circles and dots are fits of Eqs. (3, 4) to the data. The conservation of total exciton number is confirmed by adding the two curves at each temperature (not shown). The sum decays via the nonradiative recombination process.

Figure 4

Temperature dependence of the down-conversion rates, $D=(D+{\gamma }_o)-{\gamma }_p$, determined from the orthoexciton decay rates ($D+{\gamma }_o$, solid triangles) and paraexciton buildup rates ($D+{\gamma }_o$, open triangles), assuming ${\gamma }_o={\gamma }_p$ and a value of ${\gamma }_p$ that is determined from the paraexciton decay. Our data are within the error bars of orthoexciton decay rates reported by Weiner et al.[6] The solid line corresponds to the TA-phonon process discussed in the text, scaled by a single fitting parameter. The temperature dependence, approximately given by Eq. (6), depends strongly on the phonon velocity involved. The dashed line is the LA-phonon process,[10] involving 3.5 times larger phonon velocity.

Figure 5

Normalized orthoexciton decay curves at $T=2\mathrm{K}$ for $\Delta =3.4$, 4.5, 5.7, 6.9, 9.3, and $12\mathrm{meV}$. The exchange splitting $\Delta$ is determined from the orthoexciton shift with stress, as detailed in Ref. 21. Our measured values of orthoexciton lifetime are approximately 30% larger than those reported by Denev and Snoke.[10] Their measurement may have been affected by the significantly shorter non-radiative decay of excitons in their sample.

Figure 6

Stress dependence of the orthoexciton lifetime at $T=2\mathrm{K}$. The dots are measured lifetimes as a function of the energy splitting. The solid line corresponds to the TA-phonon process with the same scaling parameter determined as in the temperature dependence of Fig. 4. The dashed line corresponds to the LA-phonon process.