Formation of a large polaron crystal from a homogeneous, dilute polaron gas

We introduce a very simple method to study the phase transition from the homogeneous dilute large polaron gas toward the formation of a cubic crystal within the framework of the Vlasov [Many-Particle Theory and its Applications to Plasma (Gordon and Breach, New York, 1962)] nonlinear kinetic equation and the equations for the Bogolyubov [Problems of Dynamic Theory in Statistical Physics. Selected Works on Statistical Physics (Moscow State University, Moscow, 1979)] distribution functions. Two different critical temperatures $T_1^{\mathrm{cr}}$ and $T_2^{\mathrm{cr}}$ are introduced, defining the range of stability of the crystal phase $(T_2^{\mathrm{cr}}<T<T_1^{\mathrm{cr}})$. The existence and properties of the crystal phase are discussed as a function of the ionicity parameters of the medium, the electron-phonon Fröhlich coupling constant and the density of the dilute polaron gas. In particular, the lattice parameter is found to increase when the ionicity of the medium decreases whereas it is independent of the Fröhlich constant $\alpha$ and the polaron density. Furthermore, it increases upon decreasing the temperature from $T_1^{\mathrm{cr}}$ toward $T_2^{\mathrm{cr}}$, showing a behavior different from that observed for common solids. We find also that a drift velocity of the polaron gas tends to reduce the stability of the crystal. To illustrate the simplicity of the calculation scheme, we apply the same technique to study the formation of the argon (Ar) crystal starting from the atomic gas interacting through a reliable Lennard-Jones potential. We calculate the crystallization temperature and the Ar atom distribution. The lattice parameter is an increasing function of the temperature and the second critical temperature is not defined. We try to relate the different dependence on the temperature of the lattice parameter to the features of the polaron-polaron and atom-atom interactions.

Figure 1

The coefficients $a_1$, $a_2$, and $a_3$ [Eq. (16), $k=k_{\mathrm{opt}}$] are shown as function of $T$ for a polaron gas with ${\epsilon }^{*}∕{\epsilon }_{\infty }=1.00$, $n={10}^{18}{\mathrm{cm}}^{-3}$, and $\alpha =15$. The critical temperature $T_1^{\mathrm{cr}}$ is found when $a_1$, $a_2$, and $a_3$ become zero.

Figure 2

The position dependent density $\rho \left(x\right)=F_1(x,0,0)$ for different parameters of the polaron gas: (a) $\alpha =20$, $n={10}^{18}{\mathrm{cm}}^{-3}$, $T=13\mathrm{K}<T_1^{\mathrm{cr}}$, and different values of ${\epsilon }^{*}∕{\epsilon }_{\infty }$; (b) ${\epsilon }^{*}∕{\epsilon }_{\infty }=1.05$, $n={10}^{18}{\mathrm{cm}}^{-3}$, $T=10\mathrm{K}<T_1^{\mathrm{cr}}$, and different values of $\alpha$; (c) $\alpha =25$, ${\epsilon }^{*}∕{\epsilon }_{\infty }=1.02$, $T=50\mathrm{K}<T_1^{\mathrm{cr}}$, and different values of $n$. The horizontal lines $\mathbf{(}\rho \left(x\right)=1\mathbf{)}$ correspond to the homogeneous gas. $a_0^{*}$ is the effective Bohr radius (see the Appendix).

Figure 3

(a) Dependence of the dimensionless parameter $\zeta ={⟨x^2⟩}^{0.5}∕a$ on ${\epsilon }^{*}∕{\epsilon }_{\infty }$ (we fix $T=13\mathrm{K}$ and $n={10}^{18}{\mathrm{cm}}^{-3}$), for different values of $\alpha$ (let us note that the periodic phase is absent if $\alpha =17.5$ and ${\epsilon }^{*}∕{\epsilon }_{\infty }>1.05$ or $\alpha =20$ and ${\epsilon }^{*}∕{\epsilon }_{\infty }>1.08$). The region $\zeta ⩾1$ (limited by the thin solid line) corresponds to the absolute instability of the periodic phase. (b) Dependence of $\zeta$ on the temperature, for different values of ${\epsilon }^{*}∕{\epsilon }_{\infty }$ (we fix $\alpha =25$ and $n=5\times {10}^{18}{\mathrm{cm}}^{-3}$).

Figure 4

${\alpha }_{\mathrm{cr}}$ as function of $n$, for different values of ${\epsilon }^{*}∕{\epsilon }_{\infty }$. We require the Lindemann criterion $\zeta ⩽0.1$ be satisfied. The ending point of each line defines $T_2^{\mathrm{cr}}$ (see text).

Figure 5

$k_{\mathrm{opt},x}$ as function of $T$ [the temperature dependence of $\sigma (k_x,k_{\varrho })$ is taken into account] for different values of ${\Omega }_x$. We fix ${\epsilon }^{*}∕{\epsilon }_{\infty }=1.05$, $n={10}^{19}{\mathrm{cm}}^{-3}$, and $\alpha =20$. $a_0^{*}$ is the effective Bohr radius (see the Appendix).

Figure 6

$T_1^{\mathrm{cr}}$ as function of ${\Omega }_x$ for $n={10}^{19}{\mathrm{cm}}^{-3}$, ${\epsilon }^{*}∕{\epsilon }_{\infty }=1.05$, and several values of $\alpha$.

Figure 7

The phase diagram of the polaron gas in the plane $({\Omega }_x,T)$. The two curves enclose the region of existence of the periodic phase. We fix $\alpha =20$, ${\epsilon }^{*}∕{\epsilon }_{\infty }=1.05$, and $n={10}^{19}{\mathrm{cm}}^{-3}$.

Figure 8

Polaron distribution function for (a) $T=80\mathrm{K}$, $\alpha =20$, ${\epsilon }^{*}∕{\epsilon }_{\infty }=1.05$, $n={10}^{19}{\mathrm{cm}}^{-3}$, and several values of ${\Omega }_x$ and (b) $T=105\mathrm{K}$, $\alpha =20$, ${\Omega }_x=0.25$, $n={10}^{19}{\mathrm{cm}}^{-3}$, and several values of ${\epsilon }^{*}∕{\epsilon }_{\infty }$. $a_0^{*}$ is the effective Bohr radius (see the Appendix).

Figure 9

The coefficients $a_1$, $a_2$, and $a_3$ [see Eq. (16)] as function of $T$ calculated for the Lennard-Jones potential [Eq. (18)] with $r_0=3.408\mathrm{\AA }$ and $\epsilon =0.0103\mathrm{eV}$. The critical temperature $T_{\mathrm{cr}}$ is found at the point where $a_1$, $a_2$, and $a_3$ become zero.

Figure 10

The dependence of the wave vector $k_{\mathrm{opt}}$ on the temperature for the argon crystal.

Figure 11

The distribution function of the Ar atoms in the crystal for several values of $T$. We fix $n=2.1\times {10}^{22}{\mathrm{cm}}^{-3}$. The horizontal line $\mathbf{(}\rho \left(x\right)=1\mathbf{)}$ corresponds to the homogeneous gas. $a_0^{*}$ is the effective Bohr radius (see the Appendix).