Theory of low-temperature thermal expansion of glasses.

Abstract

We have developed a theory of low-temperature thermal expansion of glasses explaining a number of existing experimental data. We assume that thermal expansion, like many other low-temperature properties of glasses, is determined by associated two-level systems (TLS's); this concept has been introduced to explain these properties by Anderson, Halperin, and Varma and by Phillips. Our theory is based on the Karpov-Klinger-Ignat'ev model of two-level systems in glasses. The deformation potential of the TLS's is calculated. We have shown that it consists of two parts: The larger part (of the order of 0.3 eV) is responsible for the observed transport properties of glasses; however, it does not contribute to the thermal expansion of glasses. The latter is caused by a relatively small second part of the deformation potential which is, within logarithmic accuracy, proportional to the TLS's interlevel spacing E.This is why at low temperatures the coefficient of thermal expansion of glasses is approximately a linear function of the temperature. Its sign is determined by a microscopic structure of the TLS. We have calculated the Gr\"uneisen parameter \ensuremath{\Gamma}. It appears to be of the order of (${\mathrm{scrE}}_{a}$/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{D}$${)}^{2/3}$\ensuremath{\simeq}100, where ${\mathrm{scrE}}_{a}$ is an energy of the order of 30 eV and ${\ensuremath{\omega}}_{D}$ is the Debye frequency. Such large values of \ensuremath{\Gamma} are connected with the softness of local anharmonic potentials that produce the TLS's in glasses. Our principal result is the dependence of the coefficient of thermal expansion \ensuremath{\alpha} on the time of experiment, ${\ensuremath{\tau}}_{\mathrm{expt}}$. It is shown that if \ensuremath{\alpha}<0, then after heating glass it is at first contracted and afterwards, after the time about ${10}^{\mathrm{\ensuremath{-}}8}$ sec (at T=0.3 K), a slow expansion begins. At ${\ensuremath{\tau}}_{\mathrm{expt}\mathrm{\ensuremath{\simeq}}1}$ sec the parameter \ensuremath{\Gamma} can have the absolute value of about (1/3) of that at ${\ensuremath{\tau}}_{\mathrm{expt}\mathrm{\ensuremath{\simeq}}{10}^{\mathrm{\ensuremath{-}}8}}$ sec. Such behavior of the thermal expansion coefficient is due to the fact that the contribution of the TLS's with large relative tunnel splitting (${\ensuremath{\Delta}}_{0}$/E\ensuremath{\simeq}1) is negative while that of the TLS's with small relative tunnel splitting (${\ensuremath{\Delta}}_{0}$/E\ensuremath{\ll}1) is positive. The latter, however, have large times of relaxation which can be comparable with the time of experiment. Finally, we discuss the relative role of the TLS's and free-electron contributions to the thermal expansion of metallic glasses.