Anomalous temperature-induced volume contraction in GeTe

The recent surge of interest in phase-change materials GeTe, ${\mathrm{Ge}}_2{\mathrm{Sb}}_2{\mathrm{Te}}_5$, and related compounds motivated us to revisit the structural phase transition in GeTe in more detail than was done before. The rhombohedral-to-cubic ferroelectric phase transition in GeTe has been studied using high-resolution neutron powder diffraction on a spallation neutron source. We determined the temperature dependence of the structural parameters in a wide temperature range extending from 309 to 973 K. The results of our studies clearly show an anomalous volume contraction of 0.6% at the phase transition from the rhombohedral-to-cubic phase. In order to better understand the phase transition and the associated anomalous volume decrease in GeTe, we have performed phonon calculations based on the density functional theory. Results of the present investigations are also discussed with respect to the experimental data obtained for single crystals of GeTe.

Figure 1

Structure of the low-temperature phase of GeTe (space group $R3m$) shown in the pseudocubic (thick black lines), hexagonal (dotted lines), and rhombohedral (thick blue/gray lines) representations.

Figure 2

Temperature dependence of diffraction diagrams for GeTe measured at (a) wide $d$ range, and $d$ ranges corresponding, respectively, to the cubic (b) ${\left(222\right)}_c$, (c) ${\left(220\right)}_c$, and (d) ${\left(200\right)}_c$ reflections. Note the phase transition at $T_c=600$ K, development of the rhombohedral distortions below $T_c$, and vanishing splitting of the diffraction peaks at the onset of the rhombohedral-to-cubic phase transition. (e), (f) and (g) show the intensity vs $d$ plots corresponding to (b), (c) and (d), respectively.

Figure 3

Results of the Rietveld profile refinement for the (a) low-temperature rhombohedral phase of GeTe and (b) high-temperature cubic phase of GeTe. The refinements were performed by taking into account the impurity phase ${\mathrm{GeO}}_2$. The orange and green small vertical lines indicate the positions of the Bragg peaks corresponding to the GeTe and ${\mathrm{GeO}}_2$ phases, respectively. The blue curves at the bottom denote differences between the measured and calculated intensities. The conventional Rietveld discrepancy parameters for refinement (a) were $R_p=20.9\%,R_{wp}=21.3\%,R_e=8.48\%$, and ${\chi }^2=6.323$, and those for refinement (b) were $R_p=26.6\%,R_{wp}=18.9\%,R_e=11.9\%$, and ${\chi }^2=2.509$.

Figure 4

Temperature variations of (a) distortion parameter $\Delta x$ and (b) deviation $\Delta \alpha$ of the distortion angle from ${90}^{\circ }$. The present NPD data (circles) are compared to the neutron-diffraction experiments performed on single crystals (squares) by Chattopadhyay et al. [8]. The continuous lines are just guides to the eye.

Figure 5

Temperature dependences of short $\left(s\right)$ and long $\left(l\right)$ Ge-Te bond lengths determined in the present NPD measurements (circles) and those provided by the single-crystal neutron diffraction [8] (squares). The continuous lines are just guides to the eye.

Figure 6

Pair-distribution function $G\left(r\right)$ calculated for the cubic GeTe phase with the static distortions $\Delta x_0=0,\Delta x_1=7.94\times {10}^{-3}$, and $\Delta x_2=14.56\times {10}^{-3}$. Simulation are performed at the experimental $r$ range and $Q_{max}=15 {\AA }^{-1}$ [13].

Figure 7

Temperature dependence of the GeTe pseudocubic unit-cell volume $\left(V\right)$. The present neutron powder diffraction results (circles) are compared to the single-crystal neutron-diffraction data (squares) obtained by Chattopadhyay et al. [8]. Solid lines denote results of the DFT-QHA calculations for the low- and high-temperature phases of GeTe.

Figure 8

Experimental (symbols) and theoretical (lines) isotropic temperature factors $B$ determined for Ge (circles) and Te (squares) atoms in the rhombohedral and cubic phases of GeTe. The solid lines denote calculated $B$ factors for the rhombohedral and ideal cubic GeTe structures. Dashed lines correspond to calculated $B$ factors for the cubic GeTe structure with static distortion $\Delta x=14.56\times {10}^{-3}$ ($s=2.84$ Å, $l=3.19$ Å). Calculations are performed within the harmonic approximation. Theoretical isotropic temperature factors are evaluated as $B=8{\pi }^2\langle U\rangle$, where $\langle U\rangle$ is the trace of the $U_{ij}$ tensor [20].