Soft-phonon mediated structural phase transition in GeTe

Inelastic neutron scattering experiments on powder samples of GeTe together with density functional theory investigations of the phonon dynamics in the low- and high-temperature phases of GeTe crystal are reported. The dispersion of phonons in the high-temperature rocksalt phase show soft branches with the lowest one at the $\Gamma$ high-symmetry point. The structural phase transition in GeTe is reconsidered and shown to be driven by the con-densation of exactly three components of the triply degenerate optical transverse soft-phonon mode at the Brillouin zone center. The mechanism proposed allows us to explain the formation of structural distortions in the low-temperature ferroelectric phase of GeTe revealed by various experiments. A displacive nature of the phase change in crystalline GeTe is supported by the results of the present theoretical studies.

Figure 1

Phonon-dispersion relations for the rocksalt GeTe at the optimized lattice constant of 6.0116 Å. Transverse and longitudinal modes are marked by solid and dashed lines, respectively. Imaginary frequencies are denoted by negative values [${\omega }^2(\mathbf{k},j)<0$]. The high symmetry points are labeled according to the Brillouin zone of the $Fm\overline{3}m$ space group. Note the degeneracy of transverse acoustic branches along the $\Gamma \text{−}X\text{−}K$ and $\Gamma \text{−}L$ paths.

Figure 2

Change in relative energy ($\Delta E$) as a function of amplitude of the frozen ${\Gamma }_4^{-}$ mode ($Q$). The energy of the ideal rocksalt GeTe ($Fm\overline{3}m$ space group) is taken as the reference. Relative displacement of the Ge and Te sublattices ($u$) are denoted by the upper axis. The dashed curve corresponds to the polynomial fit $\Delta E=\frac{1}{2}a{Q}^2+\frac{1}{4}b{Q}^4+\frac{1}{6}c{Q}^6$.

Figure 3

Phonon-dispersion relations for the rhombohedral GeTe at the lattice parameters satisfying the configuration [$Q_{\text{min}}$,$\Delta E_{\text{min}}\left(Q_{\text{min}}\right)$]. Transverse and longitudinal modes are marked by solid and dashed lines, respectively. The high symmetry points $x(-\frac{1}{3},\frac{1}{6},\frac{1}{6}$), $T\equiv Z(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, $F(0,\frac{1}{2},\frac{1}{2})$, and $L(\frac{1}{2},0,0)$ are labeled according to the Brillouin zone of the $R3m$ space group. The TO and LO modes are denoted by superscripts $T$ and $L$, respectively.

Figure 4

(a) Comparison of the experimental [2] (solid lines) and calculated (dashed lines) Raman spectra for rhombohedral GeTe crystal at scattering geometries $Z\left(XX\right)\overline{Z}$ and $Z\left(XY\right)\overline{Z}$. Theoretical peaks are represented by a Lorentzian with experimental HWHMs and intensities calculated at the following experimental conditions: $T=82$ K and a He-Ne laser beam of 632.8 nm wavelength. For clarity, experimental HWHMs of peaks ($\sim 20$ cm${}^{-1}$) were artificially reduced to 2.5 cm${}^{-1}$. (b) Simulated Raman spectrum of a polycrystalline sample in backscattering geometry.

Figure 5

Generalized phonon density of states for rhombohedral GeTe measured at 150 K (symbols) for two incident neutron energies $E_i$ and calculated at the ground state (solid line). The dashed line denotes the Gaussian convolution of theoretical $G\left(E\right)$ with the experimental resolution of 1.7 meV. Experimental and calculated $G\left(E\right)$ are normalized to unity.