Low-frequency dynamics in cooperative Jahn-Teller systems

Abstract

We present a detailed study of the low-frequency dynamics of systems undergoing strain-induced cooperative Jahn-Teller transitions. Exact expressions relating the phonon susceptibilities to the electronic susceptibilities are given. The phonon and the electron response functions are calculated explicitly when the mean-field static correlation functions are used. In the limit that the decay rate of electronic fluctuations is much slower than the phonon frequency, we find that $\frac{{{\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}}_{\mathrm{QQ}}(\ensuremath{\omega})}{\ensuremath{\omega}}$ and $\frac{{{\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}}_{\mathrm{OO}}(\ensuremath{\omega})}{\ensuremath{\omega}}$ have a central peak. As the temperature is lowered toward the structural-phase-transition temperature the width of the central peak approaches zero and the central-peak intensity diverges for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}=0$. In the opposite limit we obtained the mean-field expressions for the phonon dispersion relation, the sound velocity, the elastic constant, and the ultrasonic attenuation coefficient. We also found a relation connecting the width of the central peak to the attenuation coefficient. In the fast-relaxation regime an explicit calculation of the response functions is possible using the exact static correlation functions. From this calculation we obtain the phonon dispersion relation, the sound velocity, the elastic constant, and the attenuation coefficient in terms of the exact static electronic susceptibilities. Our expression for the elastic constant reduces in the mean-field approximation to the result derived before by several authors. A calculation of the linewidths of the crystal-field manifolds is also given. In addition we comment on the asymptotic behavior near the critical point and the connection between our findings and the results of recent studies of elastic phase transitions which were obtained using renormalization-group techniques.