Temperature dependence of structural, dynamical, and electronic properties of amorphous Bi₂Te₃: an ab initio study

Pair correlation function (PCF), defined as the normalized possibility (relative to the uniform probability) of finding an atom at distance r from the central atom, has been widely used to character the structure of liquids and glass. The total PCFs for Bi₂Te₃ at different temperatures obtained from our ab initio MD simulations are plotted in figure 1(a). The first peaks in total PCF become stronger with decreasing temperature, demonstrating that the system is more ordered at the glassy state. In order to compare the fraction of homopolar bonds of amorphous Bi₂Te₃ and Sb₂Te₃, we calculated the total and partial pair-correlation functions (PCFs) $g\left(r\right)$ of amorphous Bi₂Te₃ and Sb₂Te₃ in figure S1 is available online at stacks.iop.org/NJP/21/093062/mmedia. Similar to total PCF (in figure S1(a)), ${g}_{{\rm{Bi}}-{\rm{Te}}}\left(r\right)$ and ${g}_{{\rm{Sb}}-{\rm{Te}}}\left(r\right)$ in figure S1(b) resemble each other except for the small offset of the peak position. However, the partial PCFs of homopolar bonds (figures S1(c) and (d)) for Bi₂Te₃ and Sb₂Te₃ are different. There is an obvious primary peak at 3 Å and 2.9 Å in ${g}_{{\rm{Sb}}-{\rm{Sb}}}\left(r\right)$ and ${g}_{{\rm{Te}}-{\rm{Te}}}\left(r\right)$ for Sb₂Te₃, respectively. In contrast, the peaks around 3 Å in ${g}_{{\rm{Bi}}-{\rm{Bi}}}\left(r\right)$ and ${g}_{{\rm{Te}}-{\rm{Te}}}\left(r\right)$ for Bi₂Te₃ are very small, demonstrating the fraction of homopolar bonds in amorphous Bi₂Te₃ in the first nearest shell is much lower than that in Sb₂Te₃. Therefore, amorphous Bi₂Te₃ exhibits a more ordered amorphous network. Previous research has indicated that homopolar bonds are linked to the stability of tetrahedral fragments in GeTe and GST, but they do not significantly strengthen octahedral motifs [27]. Coordination number (CN) is the number of atoms in the first-neighbor shell of a given atom. The fraction of various CN from 1 to 6 for Bi₂Te₃ at different temperatures calculated using a uniform cutoff value of 3.3 Å are shown in figure 1(b). It can be seen that Bi-centered atoms are mostly 3 and 4-coordination while Te-centered atoms have predominantly 2 or 3-coordination. As the temperature decreases from 1173 to 300 K, the percentage of 4-coordination Bi-centered atoms gradually increases, the same are for the 3-coordination Te-centered atoms. It is interesting to note that the fractions of the 4- and 5 coordinated Bi atoms increases noticeably as the temperature is lowered. Some 6-coordinated Bi atom also emerged at low temperature. These results indicate that high-coordination Bi configurations are gradually formed in the amorphous state of Bi₂Te₃. Previous studies have shown that when a cutoff value of 3.3 Å is used for Sb₂Te₃ at 300 K, the total CNs are 3.77 and 2.58 for Sb and Te, respectively. In this work, the total CNs of amorphous Bi₂Te₃ for Bi and Te at 300 K are 3.81 and 2.57, respectively, similar to those observed in Sb₂Te₃.

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To further characterize the structures of the liquid and amorphous, we also calculated bond-angle distribution functions ${g}_{3}\left(\theta \right)$ in Bi₂Te₃ at different temperatures as shown in figure 2. Partial bond-angle distribution functions for three bonded atoms are divided into two groups: Bi-centered (the left half of figure 2) and Te-centered (the right half). One can see that among these partial bond-angle distributions, the most prominent peaks are located in Te-Bi-Te and Bi-Te-Bi, namely at 90° and 167°, which are the characteristic peaks of octahedral geometry. The peaks at the bond angle of 90° and 167° becomes sharper upon cooling, revealing that the octahedral configuration may exist in Bi₂Te₃ and becomes more order with the decrease in temperature. In comparison, the main peaks in Bi-Bi-Te and Te-Te-Bi are around 55°, and there are no obvious peaks in Bi-Bi-Bi and Te-Te-Te. These results indicate that the interaction between Bi and Te atoms are very strong, while there are not many homopolar bonds (such as Bi-Bi and Te-Te bonds) in Bi₂Te₃.

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To further explore the local structure in Bi₂Te₃, we use the ACA method proposed recently [26], which is a full three-dimensional atomistic analysis method that can identify and characterize the local structure more clearly. Collective alignment is the first part of ACA method, which is used to determine whether there is a local structure order in the system. In this part of analysis, 2000 clusters for Bi and Te types of central atoms respectively are randomly selected from the MD simulation trajectories at each temperature of 1173 K, 673 K and 300 K. Each cluster consists of one central atom and six nearest neighboring atoms. The center atoms of these clusters are overlapped to each other, and then a modified Lenard-Jones-type attractive potential and a strong harmonic potential are used to perform a MD-simulated annealing to align the orientations of these clusters. Rigid rotation is applied to the clusters as well as translational operations until their overall mean-square distance is minimized. After the collective alignment, three-dimensional atomic density distributions smoothed with Gaussian smearing are obtained. The collective alignments of Bi-centered and Te-centered results at 1173, 673 and 300 K are plotted in figure 3. At 1173 K, Bi- and Te-centered clusters exhibit only a few high-density spots, showing that liquid Bi₂Te₃ is relatively disordered. As the temperature drops to 673 K, defective octahedral configuration is gradually formed in the Bi₂Te₃ system. In the glassy state, the higher atomic-density contour plots with clearer octahedral configurations are shown in the right of figure 3, revealing the short-range order (SRO) of the system is further enhanced at 300 K. On the whole, distorted octahedral is the main SRO of the Bi₂Te₃ glass around the Bi atoms. Although Te-centered clusters also exhibit octahedral configuration, the volume of the center sphere is significantly larger than that of other six spheres around it, indicating that the existence of vacancies may lead to defective octahedral-like SRO. In comparison, the volume of each sphere for Bi-centered clusters is almost the same, demonstrating that the Bi-centered clusters consist mainly of relatively perfect octahedrons.

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One-on-one cluster-template alignment, another part of the ACA method, is applied to further quantify the octahedral SRO in amorphous Bi₂Te₃. In this scheme, a more direct definition of structure fitting score f is used to describe the structural similarity between given a template and selected cluster:

$$\begin{eqnarray}&&{\rm{\Delta }}{r}_{T}^{2}=\begin{array}{c}{\rm{\min }}\\ C=1,\,\ldots ,\,{n}_{c}\end{array}{\rm{\Delta }}{r}_{C,T}^{2},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&f=\,{\rm{\min }}\left(\displaystyle \frac{1}{{n}_{T}}\displaystyle \sum _{T=1}^{{n}_{T}}{\rm{\Delta }}{r}_{T}^{2}\right),\end{eqnarray} \tag{ 2 }$$

where ${\rm{\Delta }}{r}_{C,T}^{2}$ is the distance between an atom C in the cluster and an atom T in the template. The term ${\rm{\Delta }}{r}_{T}^{2}$ is the minimal square distance between the atom T in the template and all ${n}_{c}$ atoms in the cluster. When the value of f is zero, the structural similarity between selected cluster and template is perfect; on the contrary, a larger f indicates the selected cluster has large deviation from the template. The role that defective octahedrons play in the amorphous Ge-Sb-Te alloy have been studied using the same scheme in our previous work [28], and thus it is interesting to compare the ACA results of the one-on-one cluster-template alignment of amorphous Bi₂Te₃ with those in Sb₂Te₃ at 300 K. The Bi (Sb)-centered and Te-centered clusters in amorphous Bi₂Te₃ with Sb₂Te₃ with different templates are plotted in figures (a) and (b), respectively. Five Bi-centered or Te-centered atomic clusters of 3-, 4-, 5-fold defective octahedrons and a 6-fold perfect octahedron as well as a 4-fold tetrahedron as shown in figure 4(c) are used as the templates in the alignment. Each Bi(Sb)-centered or Te-centered cluster extracted from the Bi₂Te₃ and Sb₂Te₃ system at 300 K is aligned with these five templates to determine their structural similarity to the templates. If the alignment score is less than 0.2, the cluster is identified as corresponding to its template cluster. From figure 4(a), we can see that in the Sb₂Te₃ system, the highest proportion of configuration are 5-fold clusters, while 6- and 4-fold clusters also occupy a higher proportion. Remarkably, for Bi₂Te₃, the 6-fold cluster occupies a very prominent proportion, notably higher than 5-fold and 4-fold clusters. On the other hand, as shown in figure 4(b), Te-centered clusters in Bi₂Te₃ and Sb₂Te₃ show similar fractions in glassy state with respect to the five templates, except that the population of the 4-fold Te-centered clusters in Bi₂Te₃ is slightly higher than that in Sb₂Te₃. There are almost no tetrahedrons in the Bi₂Te₃ or Sb₂Te₃ systems at 300 K. For Te-centered clusters of amorphous Bi₂Te₃, the proportion of 6-fold perfect octahedron is very small, there are mainly defective octahedrons with 3-, 4-, 5-fold octahedrons, resulting in the non-uniform volume of central and surrounding spheres appears in collective alignments result of Te-centered clusters at 300 K (figure 3). By comparison, we find that most of the distorted octahedral clusters in the Bi₂Te₃ glass sample exist in the Bi-centered configuration, and the SRO in Bi₂Te₃ is stronger than that in the Sb₂Te₃ system at 300 K. Introducing a small amount of Sc alloy into Sb₂Te₃ could significantly enhance the crystallization speed at elevated temperatures [13], and Qiao et al [29] found that Sc forms much more robust octahedral motifs than Sb and Te, which significantly and clearly reduces the stochasticity of nucleation. Therefore, the stable octahedral structure in amorphous state could be regarded as a crystal seed to enhance crystallization speed.

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Peierls-like distortion (PLD) is a significant feature in the chalcogenide PCMs [30–32]. On one hand, Peierls distortion is reinforced during the spontaneous structural relaxation of amorphous PCMs, resulting in the structural and electronic properties of the glass deviating from those of resonantly-bonded crystals, which lead to the resistance drift phenomenon [30]. On the other hand, Peierls distortion has been found to be the structural origin during the quench, which is associated with fragile-to-strong liquid crossover and metal-to-semiconductor transition in supercooled liquid PCMs [33, 34]. As can be seen from the results above, most configurations in Bi₂Te₃ systems are defective/distorted octahedral clusters. PLD can result in the instability of the perfect octahedral arrangement, leading to 3-fold, 4-fold, 5-fold, and 6-fold octahedrons as seen from the one-on-one cluster-template alignment. In order to quantitatively understand the PLD in amorphous Bi₂Te₃ and Sb₂Te₃, we calculate the PLD factor of Bi-centered and Sb-centered 6-fold octahedrons plot the results in figures 5(a) and (b), respectively. Here the PLD factor is obtained by averaging the differences of long- and short-bonds in the diagonal lines of each 6-fold octahedron [35]. Bi-centered or Sb-centered clusters with the alignment score with respect to the octahedron template in the range from f ≤ 0.15 to ≤ 0.40 are used to calculate the PLD factor. As one can see from figure 5 that there are not many Bi-centered clusters, in the Bi₂Te₃ amorphous with alignment score f ≥ 0.3, while there is still substantial amount of Sb-centered clusters in Sb₂Te₃ glass with the score higher than 0.3. Nevertheless, for f ≤ 0.4, 97.1% of the Bi-centered clusters and 94.8% of the Sb-centered clusters can be counted for. The PLD factors in Bi₂Te₃ system are peaked around 0.25–0.35 Å, while that of the Sb₂Te₃ system are around 0.35–0.45 Å, which demonstrates that the degree of distortion for octahedrons in Bi₂Te₃ system is smaller than that in the Sb₂Te₃ system.

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Insights on the connectivity of a topological network of the amorphous materials can be obtained from analysis of relevant rings distribution by using RINGS code [36]. The comparison of the primitive rings for Bi₂Te₃ and Sb₂Te₃ at 300 K are computed in figure 6. Similar to glassy state Sb₂Te₃ [18], 4-fold rings in Bi₂Te₃ are the dominant structural motif among these n-fold rings, the fraction of which is even larger than Sb₂Te₃. Previous studies have pointed out that the 4-fold rings ABAB are the basic structural elements for fast phase transition in PCMs [37–40], which could reduce the stochasticity of nucleation [41]. In our previous work for amorphous Sb₂Te₃, we found that these defective octahedrons are connected with each other via 4-fold rings [18]. Highly coordinated (such as 6-fold and 5-fold) octahedrons can provide more structural sites for forming 4-fold ring; therefore, less distorted octahedral motifs in amorphous Bi₂Te₃ are more conducive to the formation of these 4-fold rings.

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The liquid-to-glass transition temperature is usually determined using the Wendt–Abraham parameter defined as ${g}_{\min }/{g}_{\max }$ [42]. Here, ${g}_{\min }$ and ${g}_{\max }$ correspond to the values of PCF at its first minimum and first maximum, respectively. According to the results of PCF above (figure 1(a)), the Wendt–Abraham parameter of Bi₂Te₃ from 1173 to 300 K are computed and plotted in figure 7(a). During the quenching process, Wendt–Abraham parameters are divided into two segments for linear fitting: the line of 1173–723 K is labeled in green and the line of 673–300 K is labeled in red. It is worth noting that the slopes of the two straight lines are quite different, revealing that the liquid-to-glass transition temperature of Bi₂Te₃ system is between 673 and 723 K. Moreover, the self-diffusion coefficient D of Bi₂Te₃ upon cooling are also simulated, which is usually described by the Arrhenius equation:

$$\begin{eqnarray}&&D={D}_{0}\exp \left(-\displaystyle \frac{{E}_{\alpha }}{{{k}}_{{\rm{B}}}T}\right)\,,\end{eqnarray} \tag{ 3 }$$

where ${E}_{\alpha }$ is denoted as the activation energy and D₀ as the pre-exponential factor. The relationship between ln(D) and 1/T is shown in figure 7(b). With the calculated results made linear fitting, the calculated activation energy is 0.340 eV between 1173 and 723 K (blue line). As the temperature decreases continuously, the calculated activation energy becomes 0.073 eV from 673 to 300 K (orange line). A sharp change in activation energy may imply a liquid-to-glass transition, which is consistent with the analysis of Wendt–Abraham parameter.

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To further understand the dynamical properties of the Bi₂Te₃ system during transition from liquid to glassy state, we simulated the self-part of the time-dependent van Hove correlation function, defined by [43]

$$\begin{eqnarray}&&{G}_{s}\left(\mathop{r}\limits^{ \rightharpoonup },t\right)=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}\delta \left[\mathop{r}\limits^{ \rightharpoonup }-{\mathop{r}\limits^{ \rightharpoonup }}_{i}\left(t\right)+{\mathop{r}\limits^{ \rightharpoonup }}_{i}\left(0\right)\right],\end{eqnarray} \tag{ 4 }$$

where N is the number of atoms in the system, and ${\mathop{r}\limits^{ \rightharpoonup }}_{i}\left(t\right)$ is the time-dependent position of the ith atom ( i = 1, ..., N). The function $4\pi {r}^{2}{{G}}_{s}\left(\mathop{r}\limits^{ \rightharpoonup },t\right){\rm{d}}r$ is the probability to find an atom within time t in the vicinity Dr of points at the distance r given that initially the atom was located at the origin at t = 0. The self-part of the van Hove correlation function ${{G}}_{s}\left(r,t\right)$ of Bi₂Te₃ at different time intervals (0.3, 0.6, 0.9, 1.5, and 3.0 ps) upon quenching is shown in figure 8. As the temperature decreases from 1173 to 723 K, ${{G}}_{s}\left(r,t\right)$ exhibits an obvious change with time, indicating that Bi₂Te₃ is still in the liquid state, and the diffusivity of Bi and Te atoms is similar. Remarkably, when the temperature decreases to 623 K, the peak of ${{G}}_{s}\left(r,t\right)$ changes very little with time, indicating that Bi₂Te₃ changes to a solid. Although the cooling interval is only 50 K (723 to 623 K), the change of the peak for ${{G}}_{s}\left(r,t\right)$ is relatively large, which is consistent with the analysis of the Wendt–Abraham parameter such that liquid-to-glass transition temperature of Bi₂Te₃ system is in the range of 673 and 723 K. As temperature continuously decreases to 300 K, ${{G}}_{s}\left(r,t\right)$ again changes very little with time, showing that Bi₂Te₃ has been completely transformed into a glassy state.

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It should be noted that a liquid–liquid transition (LLT) in PCM liquids in the rapid cooling process has been detected experimentally by femtosecond pulse x-ray diffraction, but it is difficult to capture in our AIMD simulations due to the excessive cooling rate and the inadequate equilibration time at each temperature step [33, 34]. In future research, we will try to simulate the transition in supercooled PCM liquids with thousands of atoms and sufficient equilibration time during superfast cooling by using new simulations such as the deep potential for molecular dynamics (DeePMD), which is based on machine learning [44].

The electronic density of states (DOS) of Bi₂Te₃ and Sb₂Te₃ at 300 K are presented in figure 9. The results of DOS are calculated by the HSE06 functional, which can produce accurate band gaps than can the PBE functional. For Bi₂Te₃ and Sb₂Te₃, the lower valence band (−14∼−8 eV) originates from Te-5s, Bi/Sb-s states; however, Te-s states of Bi₂Te₃ dominate in −12∼−10 eV, and that of Sb₂Te₃ play a dominant role in −15∼−11 eV. The upper valence band (−6∼ 0 eV) is contributed from Bi/Sb-p and Te-p states. The contribution to the low conduction band (0–2 eV) is similar to the upper valence band, which is dominated by Te-p and Bi/Sb-p states. It is noteworthy that the contribution of Bi-p states in Bi₂Te₃ make major contributions to the low conduction band; in contrast, Sb-p and Te-p states of Sb₂Te₃ contribute equally to the low conduction band. There is a strong hybridization between Te-p states and Bi-/Sb-p states for the two amorphous compounds in the energy of −6 to 2 eV. Furthermore, both of Bi₂Te₃ and Sb₂Te₃ in glass state are semiconductor, and the energy gap of Bi₂Te₃ is 0.310 eV, which is slightly larger than the gap of 0.244 eV for Sb₂Te₃. The energy gap of crystalline Bi₂Te₃ is also larger than that of crystalline Sb₂Te₃, the corresponding energy gaps are 0.202 and 0.157 eV respectively by HSE06 calculations [45]. In addition, the Bader charge [46, 47] can give more electronic information based on the electronic charge density. The results of the Bader analysis for Bi₂Te₃ and Sb₂Te₃ at 300 K are presented in figure 10. A positive indicates the loss of electrons and a negative indicates the gain of electrons. On average, each Bi atom loses about 0.64 electrons and each Sb atom donates about 0.44 electrons. The scatter of the Bader charge for the same type of atom in amorphous Bi₂Te₃ is smaller than that in amorphous Sb₂Te₃, indicating that scatter of the short-range configurations in Bi₂Te₃ is relatively limited, which is consistent with the conclusion that the distortion of octahedrons is smaller in amorphous Bi₂Te₃ than in Sb₂Te₃. Similar to what is shown by previous research of amorphous Sb₂Te₃ and Sc₂Te₃ [41], Bader charge analysis shows larger charge transfer in amorphous Bi₂Te₃ than that in Sb₂Te₃, leading to larger energy penalty for forming Bi-Bi and Te-Te homopolar bonds due to stronger Coulomb interactions, a fact which explains why amorphous Bi₂Te₃ has fewer homopolar bonds than does Sb₂Te₃.

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