Towards accurate models for amorphous GeTe: Crucial effect of dispersive van der Waals corrections on the structural properties involved in the phase-change mechanism

The effect of van der Waals dispersion correction in combination with density functional theory is investigated on a canonical amorphous phase-change material. Density functional theory (DFT), using the generalized gradient approximation, usually fails to reproduce the structure of amorphous tellurides, which manifests by an overestimation of the interatomic bond distances, and particularly the Ge-Te one involved in local geometries (tetrahedral or defect octahedral). Here, we take into account dispersion forces in a semiempirical way and apply such DFT simulations to amorphous GeTe. We obtain a substantial improvement of the simulated structure factor and pair-correlation function, which now reproduce the experimental counterparts with an unprecedented accuracy, including on a recent partial contribution from anomalous x-ray scattering and from x-ray absorption. A detailed analysis of the corresponding structures indicates that the dispersion correction reduces the Ge-Te bond length, increases the fraction of tetrahedral germanium, and reduces the presence of heteropolar so-called fourfold ABAB rings. Given that these structural features have been stressed to be central for the understanding of the phase-change mechanism, the present results challenge our current understanding of the crystal to amorphous transformation at play.

Figure 1

A comparison of the total structure factor $S_{\mathrm{expt}}\left(k\right)$ measured by Kohara et al. [29], Ghezzi et al. [48], Piarristeguy et al. [49], and Stellhorn et al. [50]. The top inset shows the low-wave-number region and highlights the three principal peaks, whereas the bottom inset represents the long-wave-number domain (structure functions have been shifted).

Figure 2

Calculated (vdW, black) XRD structure factor $S_X\left(k\right)$ of amorphous GeTe compared to experimental results from Piarristeguy et al. [49] (red curves, duplicated). The green and blue curves represent results from a simulation without dispersion correction and correspond to PBEsol and PBE schemes, respectively. The black lines correspond to the decomposition into partials $x_i{x}_j{f}_i{f}_j{S}_{ij}/{\langle f\rangle }^2$ for the vdW case.

Figure 3

Calculated XRD interference function for the vdW (black), PBEsol (green), and PBE (blue) simulations, compared to experimental data (red, duplicated) from Ref. [49].

Figure 4

Computed EXAFS spectra of amorphous GeTe (vdW and PBEsol without vdW correction), compared to absorption results from Noé et al. (black circles [57]) and Van Eijk (red squares [58]).

Figure 5

(a) Calculated difference functions ${\mathrm{\Delta }}_{\text{Ge}}S\left(k\right)$ and ${\mathrm{\Delta }}_{\text{Te}}S\left(k\right)$ for amorphous GeTe for the vdW (black curves) and the PBE (red curves) simulations, compared to experimental data from AXS [50]. (b) Same quantities but for a comparison between vdW (black) and PBEsol (green).

Figure 6

Calculated total pair-correlation function $g\left(r\right)$ of amorphous GeTe (black) compared to results from XRD (red [48], duplicated) and a simulation without the dispersion correction: PBEsol (green, shifted) and PBE (blue, shifted).

Figure 7

Computed partial pair-correlation functions $g_{ij}\left(r\right)$ $(i,j=$ Ge,Te) in amorphous GeTe (vdW, black curves; PBEsol, green curves), compared to RMC results (red curves) from Piarristeguy et al. (left [49]) and Stellhorn et al. (right [50]). The insets on the left show the corresponding quantities for the three independent quenches for the vdW simulation. The insets on the right show a comparison between the vdW (black) and the PBE (red) simulations.

Figure 8

Computed bond angle distribution Te-Ge-Te and Ge-Te-Ge in amorphous GeTe.

Figure 9

(a) Standard deviation $\langle {\sigma }_{\theta }\rangle$ for arbitrary angle numbers $n$ around Ge atoms in DFT-D2 (vdW) simulated amorphous GeTe split into two categories: Ge atoms having six low $\langle {\sigma }_{\theta }\rangle$ (blue), and Ge atoms not having such six $\langle {\sigma }_{\theta }\rangle$ (red). Brackets indicate system averages. The arrows indicate the relevant angles ${}_k{\mathrm{Ge}}_m\left(n\right)$ serving for the discussion. Here, $m$ and $k<m$ are the Ge neighbors, labeled according to their distance with respect to the central Ge atom. (b) Corresponding Ge-centered angles $\langle \overline{\theta }\rangle$. Colored arrows indicate angles which can be considered as rigid [because of (a)].

Figure 10

Resulting bond angle distribution of identified tetrahedral (blue) and pyramidal (red) geometries in DFT-D2 (vdW, thick lines) and PBEsol (thin lines) simulated amorphous GeTe. The PBE simulation (not shown) is close to PBEsol. The broken black lines correspond to reference compounds having a perfect pyramidal $\left({\mathrm{As}}_2{\mathrm{Se}}_3\right)$ [70] or tetrahedral geometry $\left({\mathrm{Ge}}_{20}{\mathrm{Se}}_{80}\right)$ [63].

Figure 11

Calculated ring statistics (red) in amorphous GeTe and corresponding ABAB statistics (black) for PBE (a), PBEsol (b), and DFT-D2 (vdW) (c) simulations.

Figure 12

Snapshot of an amorphous GeTe configuration with distorted five-membered rings represented.