Theory of photoexcited carrier relaxation across the energy gap of phase-ordered materials

We develop a theory to describe the energy relaxation of photoexcited carriers in low-temperature ordered states with a band-gap opening. Carrier relaxation time $\tau$ near and below transition temperature $T_{\mathrm{c}}$ is formulated by examining the contributions from different carrier-phonon scatterings to the relaxation rate. Transverse acoustic phonon modes are found to play a crucial role in carrier relaxation; their heat capacity determines the $\tau$ divergence near $T_{\mathrm{c}}$. Remarkable agreements with the theory and experimental data on two different materials which exhibit contrasting $\tau \left(T\right)$ behaviors are also demonstrated.

Figure 1

(a) Emission-reabsorption process of HEPs across the energy gap with width $2\Delta$. (b) Diagram of three-phonon scattering processes through which HEPs dissipate. Only the scatterings that involve TA modes are relevant to the lack of divergence in $\tau$. (c) Phonon density of states of TA modes with cutoff frequency ${\Omega }_{\mathrm{TA}}$, and those of LA modes with ${\Omega }_h$. The maximum frequency of the LEP is ${\Omega }_l$ (see text for its definition).

Figure 2

Domains on the $\omega$-${\omega }^{\prime }$ plane appearing in the double integral $K_{\mathrm{TA}}^{\left(1\right)}$ given by Eq. (8) for (a) $2{\Omega }_l<{\Omega }_{\mathrm{TA}}$ and (b) ${\Omega }_l<{\Omega }_{\mathrm{TA}}<2{\Omega }_l$.

Figure 3

(a) Numerical result of $\tau \left(T\right)$ based on Eq. (22). See text for detailed numerical conditions. (b) Inverse square law of $\tau$ at $T=T_{\mathrm{c}}$ with respect to $p$. (c) $T$ dependence of $I_{\mathrm{LA}}$ and $I_{\mathrm{TA}}$ defined by Eqs. (24) and (25). Because $I_{\mathrm{TA}}$ differs from zero at $T=T_{\mathrm{c}}$, the $\tau$ divergence at $T_{\mathrm{c}}$ is strongly suppressed.

Figure 4

Experimental data of ${\tau }_{\exp }$'s in (a) the C${}_{60}$ polymer (after Ref. 18) and (b) Bi2212 (after Ref. 12) and their numerical reproductions based on Eq. (22). The data for (a) $T\le T_{\mathrm{c}}=60$ K and (b) 75 K collapse onto the theoretical curves with $p=0.3$ and $p=0.07$, respectively.