On the two-phonon relaxation of excited states of boron acceptors in diamond

The relaxation of holes from excited states of boron acceptors in diamond with the emission of two optical phonons is studied theoretically. To describe the wave function of acceptor states, an electron-like Hamiltonian with an isotropic effective mass is used. The wave function of the ground state is determined by the quantum-defect method. The probability of the transition is calculated in the adiabatic approximation. It is assumed that the phonon dispersion law is isotropic and the phonon frequency is quadratically dependent on the wave-vector modulus, with the maximum and minimum frequencies $\omega_{\operatorname{max}}$ and $\omega_{\operatorname{min}}$ at the center and boundary of the Brillouin zone, respectively. A high sensitivity of the probability of the transition to the characteristic of phonon dispersion $\omega_{\operatorname{max}}$-$\omega_{\operatorname{min}}$ is revealed, especially for the transition with the energy $E_T$ in the range $2\hbar\omega_{\operatorname{min}}\le E_{T}<\hbar\omega_{\operatorname{min}}+\hbar\omega_{\operatorname{max}}$. Depending on the energy of the transition and on the phonon dispersion, the two-phonon relaxation rate at the low-temperature limit varies from extremely small values ($<$ 10$^{8}$ с$^{-1}$) near the threshold $E_{T} = 2\hbar\omega_{\operatorname{min}}$ to extremely large values ($>$ 10$^{12}$ с$^{-1}$) in the “resonance” range $\hbar\omega_{\operatorname{min}}+\hbar\omega_{\operatorname{max}}\le E_{T}\le 2\hbar\omega_{\operatorname{max}}$.