Criteria for the appearance of a periodical component in the polaron distribution function

In this paper we establish criteria of instability of the homogeneous large polaron gas versus the periodic state. We study, in the framework of the Bogolyubov approach of the equilibrium distribution functions, how this transition depends on the parameters of the polar medium (dielectric parameters, electron-phonon coupling constant, polaron density). The temperature dependence of the transition is analyzed, indicating that the stability of the periodic phase can be achieved in a given range of temperatures, depending on the material and the polaron density. The comparison of the total energies of the periodic and homogeneous phases does confirm that the stability of the periodic phase corresponds to the lowest-energy configuration.

Figure 1

(a) The interpolaron pair potential as a function of the relative distance and (b) the Fourier transform $\sigma \left(k\right)$ of the collective potential $K\left(r\right)$ shown for different values of ${\epsilon }^{*}∕{\epsilon }_{\infty }$ (we fix $\alpha =15$ and $n={10}^{18}{\mathrm{cm}}^{-3}$).

Figure 2

The dependence of the critical temperature $T_{\mathrm{cr}}^{\left(1\right)}$ on (a) ${\epsilon }^{*}∕{\epsilon }_{\infty }$ for different values of the electron-phonon coupling constant $\alpha$ and (b) the electron-phonon coupling constant $\alpha$ for different values of ${\epsilon }^{*}∕{\epsilon }_{\infty }$ (we fix $n={10}^{18}{\mathrm{cm}}^{-3}$).

Figure 3

The dependence of the wave vector $k_{\mathrm{opt}}$ on ${\epsilon }^{*}∕{\epsilon }_{\infty }$ for different values of the electron-phonon coupling constant $\alpha$ (we fix $n={10}^{18}{\mathrm{cm}}^{-3}$ and $T=T_{\mathrm{cr}}^{\left(1\right)}$).

Figure 4

The dependence of the critical value of the electron-phonon coupling constant ${\alpha }_{\mathrm{cr}}$ on the density $n$ for different values of ${\epsilon }^{*}∕{\epsilon }_{\infty }$.

Figure 5

The dependence of the critical temperature $T_{\mathrm{cr}}^{\left(1\right)}$ on the density $n$ for different values of the electron-phonon coupling constant $\alpha$ (we fix ${\epsilon }^{*}∕{\epsilon }_{\infty }=1.05$). The inset shows the dependence on the polaron density of the temperature interval $\Delta T=T_{\mathrm{cr}}^{\left(1\right)}-T_{\mathrm{cr}}^{\left(2\right)}$ (see Sec. 3) for the same values of $\alpha$.

Figure 6

The dependence of $\mid \gamma \mid (\gamma =i\mid \gamma \mid )$ on the temperature (a) for different values of ${\epsilon }^{*}∕{\epsilon }_{\infty }$ (we fix $n={10}^{18}{\mathrm{cm}}^{-3}$ and $\alpha =20$) and (b) for different electron-phonon coupling constants $\alpha$ (we fix $n={10}^{18}{\mathrm{cm}}^{-3}$ and ${\epsilon }^{*}∕{\epsilon }_{\infty }=1.05$). The two crossing points of each line with the horizontal axis define the temperature range in which the periodic phase can exist. For a given ${\epsilon }^{*}∕{\epsilon }_{\infty }$ in (a) or $\alpha$ in (b) the periodic phase of the polaron distribution can occur only in the region delimited by the respective curve.

Figure 7

The dependence of the wave number $k_{\mathrm{opt}}$ (solid lines) of the periodic phase on the temperature $T$ and for different values of coupling constant $\alpha$ (we fix ${\epsilon }^{*}∕{\epsilon }_{\infty }=1.05$). The dashed, dot, and dashed-dot lines show the wave number $k_{\mathrm{opt}}$ at the critical temperature $T_{\mathrm{cr}}^{\left(1\right)}$ for different values of $n$. In the region on the right of these lines the periodic polaron distribution cannot exist $(T>T_{\mathrm{cr}}^{\left(1\right)})$.

Figure 8

The ratio $\mid U_{\text{periodic}}\mid ∕\mid U_{\mathrm{hom}}\mid (U<0)$ of the potential energies of the periodic and homogeneous phases as a function of $\alpha$ for different values of ${\epsilon }^{*}∕{\epsilon }_{\infty }$ (we fix $n={10}^{19}{\mathrm{cm}}^{-3}$). The curves are referred to a temperature 5% below the critical one. All the curves but the solid one are limited on the left by the critical value of $\alpha$.