Intricate short-range order in GeSn alloys revealed by atomistic simulations with highly accurate and efficient machine-learning potentials

GeSn alloys hold promise for silicon-compatible integrated applications in electronics, photonics, and topological quantum devices. However, understanding their intricate structures using density functional theory (DFT) calculations is hindered by spatiotemporal constraints. To overcome this limitation, we develop highly accurate and efficient machine-learning interatomic potentials based on a neuroevolution potential approach with farthest point sampling on a comprehensive DFT data set. The application of the developed machine-learning potential in large-scale atomistic simulations bridges the spatiotemporal gap between modeling and advanced characterizations, and facilitates the discovery of structural intricacies in GeSn alloys. Through extensive statistical sampling, we identify a type of short-range order (SRO) that is distinguished by both its structural signature and electronic band gap from the SRO structure previously predicted. Modeling based on a large simulation cell reveals the coexistence of nano SRO domains with various degrees of ordering, demonstrating a complex spatial heterogeneity of SRO structure. Our study not only reinforces the significance of fine-level structural information in alloys, but it also constitutes an effective framework for exploring SRO in a broad range of complex alloys based on highly accurate and effective machine-learning potentials.

Figure 1

Comparison of training processes using a full data set (a)–(d) and a farthest point sampling (FPS) reduced data set (e)–(h). The root-mean-square error (RMSE) for energy (a) and force (b) is illustrated during NEP training with the full data set [276009 structures, denoted as “Full” in the corresponding subplots (a)–(d) for brevity] and the testing data set (30668 structures) as a function of the number of generations. Panels (c) and (d) depict the comparisons between NEP predictions and DFT reference values for energy and force, respectively, on the testing data set. In (e)–(h), the same analyses as (a)–(d) are presented, but the model is trained with an FPS data set [denoted as “FPS” in the corresponding subplots (e)–(h) for brevity], which consists of a significantly reduced number of 137 structures sampled from the full data set, as illustrated in panel (i) in the two-dimensional principal component descriptor space. Training with the full data set achieves excellent accuracy, with RMSE of 0.3 meV/atom for energy and 15.9 meV/Å for force. The FPS approach, using the same training hyperparameters, further enhances predictive accuracy on the same testing data set, achieving an RMSE of 0.2 meV/atom for energy and 15.0 meV/Å for force.

Figure 2

Energy and structural variations arising from different types of SROs in ${\mathrm{Ge}}_{0.75}{\mathrm{Sn}}_{0.25}$ alloys. (a) The superposition of eight independent MC/MLP trajectories and one MC/DFT trajectory from our prior work [15] reveals that the total energy fluctuates around the two distinct energy levels: the high-energy basin (blue dashed line) and the low-energy basin (red dashed line), separated by $\sim 0.4\mathrm{eV}$. Both basins are well below the average energy (magenta dashed line) from random sampling (magenta dots). For clarity, the energy variation of MC/MLP-7 is separately shown in (c). Correspondingly, (b) illustrates the variation in the first-nearest-neighbors Sn-Sn SRO parameters (1NN, ${\alpha }_{\text{SnSn}}^1$, red), 2NN (${\alpha }_{\text{SnSn}}^2$, blue), and 3NN (${\alpha }_{\text{SnSn}}^3$, green). Regular SRO (R-SRO) exhibits a strong 1NN Sn-Sn depletion, a mild 2NN Sn-Sn repulsion, and a moderate 3NN Sn-Sn enhancement. Enhanced SRO (E-SRO) features a complete 1NN Sn-Sn depletion, a substantial 2NN Sn-Sn repulsion, and a significant 3NN Sn-Sn enhancement. Panel (d) depicts atomic configurations of random (special quasirandom structure), R-SRO, and E-SRO structures obtained from MC/MLP-7. Ge and Sn atoms are represented by gold and gray, respectively.

Figure 3

Variation in the calculated electronic band structure with SRO in ${\mathrm{Ge}}_{0.75}{\mathrm{Sn}}_{0.25}$. (a) A random (SQS) distribution results in a direct band gap at $\mathrm{\Gamma }$ of 0.07 eV. (b) R-SRO increases the direct band gap to 0.24 eV. (c) E-SRO leads to a further increase of a direct band gap to 0.40 eV. The corresponding Bloch spectral weight is color-coded in the bar on the right.

Figure 4

(a) Energy variation in MC sampling using the developed NEP MLP for ${\mathrm{Ge}}_{0.75}{\mathrm{Sn}}_{0.25}$ alloys based on a 64 000-atom supercell with the dimension ($11.7\times 11.7\times 11.7{\mathrm{nm}}^3$) comparable to the volume probed by APT [19]. (b) The large simulation cell is evenly divided into $n\times n\times n$ domains. (c) The probability density function of ${\alpha }_{\text{SnSn}}^1$ for a domain size of $\sim$ 12 nm (division number $n=1$), 6 nm ($n=2$), 2.4 nm ($n=5$), and 1.2 nm ($n=10$), respectively. While the average $\overline{{\alpha }_{\text{SnSn}}^1}=0.78$ is independent of domain size, its standard deviation $\sigma$ increases with $n$. Particularly, for the domain size $\sim 1.2\mathrm{nm}$ the peak at 1.0, indicating the complete depletion of Sn-Sn bonds, is highlighted by a red circle.

Figure 5

(a) The cumulative distribution of ${\alpha }_{\text{SnSn}}^1$ for domain size $\sim 1.2\mathrm{nm}$ ($n=10$) shows 10% of the domains yield an ${\alpha }_{\text{SnSn}}^1<0.6$, and these domains are called less-ordered domains (LODs). (b) Top: A conceptual and schematic diagram shows 26 neighboring domains (ND), colored in green, surrounding a blue-colored LOD. Bottom: The probability density functions of ${\alpha }_{\text{SnSn}}^1$ for LOD and ND are nearly separated, yielding an average $\overline{{\alpha }_{\text{SnSn}}^1}$ of $0.52\pm 0.08$ and $0.81\pm 0.10$ for LOD and ND, respectively.

Figure 6

Distribution of local atomic SRO parameters of 1NN Sn-Sn in ${\mathrm{Ge}}_{0.75}{\mathrm{Sn}}_{0.25}$ alloys, obtained from MC/DFT using a 64-atom supercell and MC/MLP using a 4096-atom supercell, showing that MC/MLP using a larger supercell with our developed MLP can reproduce the same average 1NN Sn-Sn SRO parameters and their distribution as those from MC/DFT using a smaller supercell.

Figure 7

MC/DFT confirmation of the coexistence of two energy basins. The MC/DFT sampling trajectory (in red) initiates from the structure within the low-energy basin, and remains in the low-energy basin for a considerable number of MC steps, consistent with the MC/MLP samplings (in gray, blue, green). Subsequently, it transitions to a high-energy basin, aligning with previous MC/DFT sampling [15] (in orange) and MC/MLP samplings (in gray, blue, green).

Figure 8

(a) Variation in SRO parameters of 1NN Sn-Sn (${\alpha }_{\text{SnSn}}^1$, red), 1NN Ge-Ge (${\alpha }_{\text{GeGe}}^1$, orange), and 1NN Ge-Sn (${\alpha }_{\text{GeSn}}^1$, blue) as a function of MC steps in MC/MLP-7 trajectory. (b) The corresponding energy variation as a function of MC steps in MC/MLP-7 trajectory. A slight jump in the low-energy basin is observed. Interestingly, this small energy variation is not captured by the 1NN Sn-Sn SRO parameters, nor the 1NN Ge-Ge or 1NN Ge-Sn SRO parameters. Instead, the subtle total energy change is found well captured when including the 2NN and 3NN Sn-Sn SRO parameters, as shown in Fig. 2.