Coupling between acoustic and soft transverse optical phonons leads to negative thermal expansion of GeTe near the ferroelectric phase transition

GeTe is a well-known ferroelectric and thermoelectric material that undergoes a structural phase transition from a rhombohedral to the rocksalt structure at $\sim 600–700$ K. We present a first-principles approach to calculate the thermal expansion of GeTe in the rhombohedral phase up to the Curie temperature. We find the minimum of the Helmholtz free energy with respect to the structural parameters, including the internal atomic displacement, in a manner similar to the traditional Grüneisen theory, explicitly accounting for the variation of the static elastic energy with respect to all structural parameters. We obtain the temperature dependence of the structural parameters of rhombohedral GeTe in very good agreement with experiments. In particular, we correctly reproduce a negative volumetric thermal expansion of GeTe near the phase transition. We show that the negative thermal expansion is induced by the coupling between acoustic and soft transverse optical phonons, which is also responsible for the low lattice thermal conductivity of GeTe.

Figure 1

Primitive unit cell of GeTe at (a) 0 K and (b) above the Curie temperature, generated using vesta software [24]. The low-temperature rhombohedral structure becomes more similar to the rocksalt structure as temperature increases: the angle between the primitive lattice vectors $\theta$ becomes closer to ${60}^{\circ }$ and the Te internal atomic position $(\tau ,\tau ,\tau )$ approaches (0.5,0.5,0.5) in reduced coordinates.

Figure 2

(a) Phonon dispersion of GeTe calculated using GGA-PBE exchange-correlation functional neglecting and accounting for screening (solid black lines and dashed red lines, respectively). The frequencies of the zone center Raman-active modes were taken from the measurements of Refs. [22] (red circles) and [56] (green squares). (b) Phonon density of states of GeTe calculated neglecting and including screening (solid black line and dashed red line, respectively) and measured by Refs. [23] (blue circles and red squares) and [57] (green triangles). The integral of the density of states over frequency is normalized to unity.

Figure 3

Structural parameters of GeTe as a function of temperature: (a) lattice constant $a$, (b) angle $\theta$, and (c) internal atomic coordinate $\tau$. Solid black lines represent our calculations. Red circles and blue squares correspond to the measurements of Refs. [17, 18], respectively. Dashed black lines represent our calculations shifted by the difference between our calculated values and the experimental values of Ref. [17] at 300 K.

Figure 4

Structural parameters of GeTe as a function of temperature: (a) lattice constant, (b) angle, and (c) internal atomic coordinate. Solid black lines represent the results obtained using our approach, while dashed red lines correspond to the standard method (see text for full explanation).

Figure 5

(a) Volumetric thermal expansion of GeTe: our calculation (solid black line), experiment [17] (red circles), and our calculation shifted by the difference between our and the experimental value at 300 K (dashed black line). (b) Computed volumetric thermal expansion including and neglecting acoustic soft optical mode coupling, shown in solid black and dashed red lines, respectively.

Figure 6

Temperature dependence of (a) average generalized Grüneisen parameters defined for each structural parameter ($a$, lattice constant; $\theta$, angle; $\tau$, internal atomic coordinate), and (b) normalized compliance matrix elements (see text for full explanation).

Figure 7

TO mode frequency versus temperature: our calculation (solid black line) and experiment [22] (red circles).

Figure 8

Elastic constants of GeTe as functions of temperature: (a) $C_{11}+C_{12}$, (b) $C_{13}$ (solid black line), and $C_{33}$ (dashed red line).