Dependency of a localized phonon mode intensity on compositional cluster size in SiGe alloys

To keep up with increasing energy demands, researchers have been driven to develop alternative, cost-effective, and sufficient renewable energy resources, paving the way for thermoelectric (TE) applications.^1–3 The majority of semiconductors, however, do not have high enough TE efficiencies to be able to compete with existing electricity producing methods. Consequently, the development of a device with a high TE efficiency is the most challenging aspect of this research theme.⁴ Since the late 1950s,⁵ the SiGe alloy has shown promise as an ideal TE material^6,7 owing to its low thermal conductivity,⁸ good stability at high temperatures,^9,10 ability to consist of both n-type^11,12 and p-type¹³ properties, and ease of implementation with other components.¹⁴ Currently, bulk SiGe is limited to high temperature TE applications, so the prediction of phonon behavior^15–17 and its role in thermal management¹⁸ are essential for the advancements of semiconductor technology.

As the reduction in thermal conductivity is one method whereby the TE efficiency can increase,^12,19,20 the study of changes in phonon dispersion relations is needed. Obtaining such phonon spectra is difficult due to the limitations of instrumental capabilities and inelastic neutron scattering, for example requiring large sample sizes. In addition, Raman spectroscopy can only access the optical phonon modes from the Brillouin zone center. In order to extract the acoustic phonon mode data, other spectroscopic techniques are required. Reports of phonon dispersion relations of SiGe alloy crystals are lacking. Usuda et al.²¹ managed to produce the phonon spectra of optical and acoustic modes of bulk SiGe alloys using inelastic x-ray scattering (IXS). Using Molecular Dynamics (MD), Tomita et al.²² succeeded in simulating the phonon dispersion relations of bulk Ge(_1−x−y)Si_xSn_y alloys. However, for the purpose of developing an extended formulation of a potential, an emphasis was placed on investigating thermal conductivity rather than focusing on the differences in phonon dispersion data. There is a clear need for more insight into the changes occurring within bulk SiGe alloy phonon dispersion relations.

Yokogawa et al.²³ conducted an IXS experiment on bulk SiGe alloys and observed the occurrence of an unidentified phonon mode, termed by the authors as the “anomalous phonon mode.” This mode occurred in the low frequency region at ∼13 meV and was nearly independent of the wave number, suggesting localized behavior. Pagès et al.²⁴ reported that the Ge–Ge optical mode frequency will shift to a lower energy when the Ge–Ge atoms are surrounded by Si atoms. The unknown mode occurs at a frequency lower than that of Ge–Ge, leading to the speculation by Yokogawa et al., where the origin of the unknown mode is a local vibrational mode, formed by Ge pairs, surrounded by Si atoms.

Our previous MD simulation study²⁵ reported that a bulk SiGe alloy, comprising Si–Si, Si–Ge, and Ge–Ge bonds, will produce this anomalous, local mode. Contrastingly, a SiGe compound comprising a Si–Ge–Si–Ge configuration will never produce this mode, indicating that Si–Ge bonds alone are insufficient in describing this unknown mode. Si–Si or Ge–Ge bonds are also needed. Atomistic disorder in bulk SiGe alloys leads to the reduction in thermal conductivity. Interestingly, the addition of atomistic disorder to bulk SiGe alloys with pores already implemented into the material does not affect the thermal conductivity.²⁶ The dependency of the local mode on such structural changes, in particular the effect of Si or Ge cluster size on the production of the local mode, has yet to be studied. To provide further insight into the relation between cluster size and the local mode, SiGe models with diverse spatial distributions of Si and Ge were generated. These alloy models with clusters of varying percentage Si or Ge composition were simulated to investigate the dependency of the local mode intensity on compositional cluster size.

Using MD, phonon dispersion relations were simulated to observe the production of the local mode and the effect of changing the cluster sizes within a bulk SiGe alloy. A group of bulk alloys covering a composition range of Si₈₀Ge₂₀ (Si rich), Si₅₀Ge₅₀ (Ge 50%), and Si₂₀Ge₈₀ (Ge rich) was simulated with varying cluster-size distributions per model. Within the alloy, these bonds link together to form small, intermediate, and large sized clusters of the Si or Ge crystal domain, giving a cluster size distribution that varies per model. This brings forth the possibility of measuring the effect of clusters of varying Si or Ge composition on the production of the local mode. To avoid confusion, the term “cluster” used in this study refers to the clusters of the Si or Ge crystal domain, formed within the SiGe alloy, as opposed to the definition of a cluster of SiGe crystal grains.

The MD simulation code LAMMPS²⁷ was utilized throughout this study. Bulk Si_xGe_1−x models were generated with an initial lattice constant (a_i), given as a_i = 5.658 − 0.227x. These models of cuboidal shape were defined by 30a_i × 4a_i × 4a_i in the x, y, and z crystallographic directions, with units of Å. Three groups of SiGe alloys were tested with compositions, Si₈₀Ge₂₀ (20% Ge), Si₅₀Ge₅₀ (50% Ge), and Si₂₀Ge₈₀ (80% Ge), with each alloy comprising 3840 atoms. Of each % Ge group, the mixing of Si and Ge atoms within the structures differed, allowing variation in cluster size distribution to be created. In this way, the dependency of the local mode on small, intermediate, or large cluster sizes could be tested. The Si and Ge atoms in the atomistic pictures presented in this paper are denoted by the colors yellow and pink, respectively. According to Yokogawa et al.,²³ both Si and Ge atoms are related to the local mode but the Ge atom provides a dominant contribution. For the purpose of analyzing the size of Ge clusters in each alloy, the Ge cluster-size distribution was also recorded.

A geometry optimization preceded the calculation of the phonon properties, after which the MD simulation of the NVE ensemble was conducted, whereby the velocities and positions of the atoms are integrated under conditions of a constant number of atoms and a constant amount of volume and energy. Periodic boundary conditions were imposed in the x, y, and z directions in order to emulate a three-dimensional bulk system.

The potential governing the forces between the atoms used in this study is a modified version²² of the Stillinger–Weber potential,²⁸ given as

where $ϕ2rij$ is the two-body term describing the pair interaction and r_ij defines the interatomic distance between atoms i and j. $ϕ3rij,rik,θijk$ is the three-body term that describes the triplet interaction between atoms, i, j, and k, and sets the bond angle, θ_ijk, between the lines i–j and i–k. ɛ_ij, A_ij, B_ij, σ_ij, p_ij, q_ij, and a_ij are the adjustable parameters for the two-body component, $ϕ2rij$⁠, and ɛ_ijk, λ_ijk, γ_ij, σ_ij, a_ij, γ_ik, σ_ik, a_ik, and θ_0ijk are the adjustable parameters for the three-body component, $ϕ3rij,rik,θijk$⁠.

To obtain the phonon dispersion relations along the Γ − X (1, 0, 0) direction, the spectral energy density (SED)²⁹ $Φκ,ω$ with a wavevector κ and angular frequency ω was calculated as

where τ₀ is the phonon lifetime, N_T is the total number of unit cells in the system, m_b is the atomic mass of atom b, displaced, $u̇a$⁠, in the direction a at a time t, B is the total number of atoms per n_x,y,z unit cell, and r represents the equilibrium positions for each n_x,y,z unit cell. In this paper, phonon dispersions showing the individual phonon mode branches as well as dispersion relations given as a combination of the acoustic and optical modes, are shown.

The spatio-temporal Fourier transform formula utilized in this study is the same as the method reported in the study by Yokogawa et al.²³ The simulated phonon dispersion relations were in good agreement with those produced by their experimental data, giving a reason to believe that the use of this calculation method is viable for the alloys simulated in this current study.

In Fig. 1, the local mode appears in the same Brillouin zones as the optical modes, suggesting that this local mode is influenced by or is of optical origin. It was previously established that a SiGe compound, comprising solely Si–Ge bonds, did not produce the local mode.²⁵ Figure 1 is that of an alloy, indicating that the local mode will appear when Si–Ge bonds are in the presence of Si–Si or Ge–Ge bonds whereas Si–Ge bonds alone are insufficient.

The models presented in Figs. 2(a)–2(c) and the Ge cluster size distribution in Fig. 2(e) depict the difference in cluster sizes between the alloys termed alloy 1, alloy 2, and alloy 3. Alloy 1 is formed of mostly smaller sized Ge clusters, in comparison to alloy 3, containing two large Ge clusters, one of which comprises over 1000 Ge atoms. Alloy 1 comprises a configuration close to that of a SiGe model wherein the atom substitution is ideally random. Alloy 1 can therefore be regarded as a reference model with respect to the alloys varying in cluster sizes in this paper.

As shown in Figs. 2(a)–2(c), the phonon dispersion relations are accompanied on the side by a color map, indicating the intensity of each phonon mode density of states. The spectral energy density (SED) governs the color map. The color shading begins with the shade black, representing zero to minimal phonon intensity, and is denoted by low SED values. The gradual transition from red to yellow indicates an increasing density of states intensity, denoted by higher SED values. Figure 2(d) is a depiction of the local mode intensities for all three alloys, given by the spectral energy density (SED) values plotted as a function of frequency, within the range of 0–6 THz. The local mode intensities occurring around 3–4 THz [Fig. 2(d)] are in good agreement with the experimental value.²³ There is some difficulty differentiating between the local mode, longitudinal acoustic (LA) mode, and transverse acoustic (TA) mode, as the modes overlap. To distinguish the local mode in a clear manner, the intensities shown in this study were retrieved from the point at k = 0 π/a where no overlap occurs.

The LA mode produces the steepest branch and therefore has the highest group velocity. These phonons are also assumed to be the main carriers of heat in these alloys. The local mode, which has no clear phonon propagation, overlaps with the acoustic modes, which cause the greatest amount of propagation, indicating a possible effect on thermal conductivity. The local mode and LA mode produced by alloy 1 [Fig. 2(a)] have a stronger intensity than the local mode and LA mode of alloy 3 [Fig. 2(c)]. Although the difference is not outstanding, the LA mode for alloy 1 is slightly steeper than that of alloy 3 and produces a higher intensity of the local mode. Yokogawa et al. observed a reduction in thermal conductivity of a Ge rich SiGe alloy, which was attributed to the local mode merging with the TA mode, causing a broadening of the TA mode.²³ Thermal conductivity calculations for the alloys in this study would provide concrete conclusions about the effect of the local mode on thermal management.

The highest intensity of the local mode was produced by the alloy with smaller sized clusters, and the lowest intensity, by the alloy with the larger clusters. There is a reduction in the Si–Ge density of states as the cluster sizes increase [Figs. 2(a)–2(c)], also corresponding to a reduction in the local mode intensities [Fig. 2(d)]. For all of the alloys in this study, the local mode intensity increasing with the Si–Ge mode intensity was observed. Alloys with small Ge clusters can form more Si–Ge bonds, and therefore, the Si–Ge optical intensity increases. Alloys with larger clusters, however, will instead lead to a higher amount of Si–Si and Ge–Ge bond formation, giving a higher density of states for those particular optical modes. Torres et al.³⁰ theoretically produced Raman intensities of Si_1−yGe_y mixed crystals in order to investigate the nature of atom substitution (whether it is random or caused by clustering/anticlustering). The authors utilized their own formulated percolation model and an ab initio model, forming a comparison of the results between the two methods. A composition domain of y = 0.16, 0.71, and 0.84 was tested, and the results showed a dependency of the Raman intensities on clustering, and anticlustering, by incorporating an order parameter, k, which quantified the clustering. For compositions y = 0.16 and y = 0.84, Si–Si and Ge–Ge Raman intensities, respectively, seemed to be almost invariant to the level of clustering/anticlustering. The Raman intensities for the minority species (Ge–Ge for y = 0.16 and Si–Si for y = 0.84), however, were favored when clustering occurred. On the other hand, a high degree of anticlustering (negative k values) favored the Si–Ge Raman intensities for all three compositions. Interestingly, the Si–Ge phonon density of states produced for the models in the current study exhibits a similar behavior to these reported results. Alloy 1, comprising small clusters and therefore tending toward the nature of anticlustering, produces the highest Si–Ge density of states intensity. Alloy 3, however, comprising large Si and large Ge clusters and therefore tending toward higher order parameter k values, produces the highest density of states intensity for Si–Si and Ge–Ge.

There is a clear relation between the local mode intensity and Si–Ge optical intensity observed in Fig. 2. Furthermore, the local mode appears in the same Brillouin zone as the optical modes, as depicted in Fig. 1. Despite the relation between the local mode intensity and Si–Ge optical intensity, it was already established that a structure comprising solely Si–Ge bonds will not produce the local mode. The origin of the local mode can, therefore, only partly be described as localized vibration caused partially by Si–Ge bonds.

Figure 3 shows the phonon intensities produced by alloy 4, alloy 5, and alloy 6. In a Si rich alloy, as the Ge cluster-size distribution changes, the intensity of the local mode will change. Alloy 4, comprising small Ge clusters, produced the highest local mode intensity, whereas alloy 6, comprising large Ge clusters, produced the smallest local mode intensity [Fig. 3(a)]. Similar to Si₅₀Ge₅₀ alloys, the local mode intensities are highest [Fig. 3(a)] when the Si–Ge intensities are also high [Fig. 3(d)]. Alloy 6 has a high intensity of Ge–Ge [Fig. 3(c)] and Si–Si [Fig. 3(e)] due to the large Ge and Si clusters, respectively.

These Si rich alloys could be viewed as Si rich models comprising Ge clusters surrounded by Si pairs, with each model differing in the density of Ge–Ge pairs. Alloy 6 has a high density of Ge–Ge pairs, due to the large Ge clusters, but a low intensity of the local mode. Contrastingly, alloy 4 comprises small Ge clusters and therefore incurs a low density of Ge–Ge pairs, producing the highest local mode intensity. This observation reveals that a high density of Ge–Ge pairs does not necessarily mean a high local mode intensity will occur. Similarly, a high density of Si–Si pairs will also not produce a high local mode intensity. Alloy 4 has the highest local mode intensity, as well as the highest Si–Ge optical mode intensity, indicating the importance of Si–Ge bonds when attempting to observe the local mode.

In Ge rich alloys, the local mode intensity changes with the Si cluster-size distribution. Figures 4(a) and 4(b) show that alloy 7, with small Si clusters, gives the highest local mode intensity whereas alloy 9, with large Si clusters, gives the smallest intensity. Smaller Si clusters will form more Si–Ge bonds [Fig. 4(d)], which corresponds to a higher local mode intensity [Fig. 4(a)]. Contrastingly, larger Si clusters will form more Si–Si [Fig. 4(e)] and less Si–Ge bonds, leading to the production of a smaller local mode intensity.

The structure of the Ge rich alloys can be viewed as Si clusters surrounded by Ge atoms, so it is the Si cluster-size distribution that changes per model. Figure 4 shows that small Si clusters will promote the production of the local mode. Similarly, the Si rich alloys can be viewed as Ge clusters surrounded by Si atoms, whereby the Ge cluster-size distribution changes per model. Figure 3 shows that small Ge clusters will produce high local mode intensities. In comparison to the results for Ge 50% alloys and the Si rich alloys, the Ge rich local mode intensities are much less. The Si rich models produce higher local mode intensities, suggesting that the Ge cluster-size distribution is more influential on the local mode. Overall, the observations taken from this paper reveal that the local mode is produced by small Ge clusters surrounded by Si pairs, producing Si–Ge bonds from which the local mode vibration occurs.

It was previously observed²⁵ that the local mode will not be produced by a structure comprising only Si–Ge bonds. A standalone Si–Ge bond will not be able to produce the local mode, but when this bond is surrounded by Ge–Ge bonds (Ge cluster) and Si–Si (the surrounding Si), the local mode is produced. The origin of the local mode is considered to be small Ge clusters surrounded by Si atoms, with Si–Ge bonds being formed in the periphery of the Ge cluster. The vibrational movement is predicted to be a collective vibration combining the stretching and compression of the Si–Ge bonds, vibration from the encircling Si atoms, and vibration from the Ge cluster. The local mode occurs in the low frequency region, below that of the Ge–Ge optical bond frequency, indicating that the Ge cluster provides a dominant contribution to the production of the local mode. The collective vibration is, therefore, predicted to include an emphasis on the movement of the Ge–Ge bonds (Ge cluster). However, large Ge clusters do not produce high local mode intensities, indicating that the surrounding bonds (Si–Si, Si–Ge) contribute more to the local mode production than originally anticipated. The models with small Ge clusters produce the highest local mode intensities, indicating that the Ge cluster plays an important role (the local mode occurs in the low frequency region), but the surrounding bonds (Si–Ge and Si–Si) also influence the local mode intensity. Larger Ge clusters would incur more Ge–Ge and Si–Si bond formation and less Si–Ge bond formation and therefore diminishes the effect of the surrounding Si and Si–Ge bonds on the production of the local mode.

To summarize, SiGe alloys comprising 20%–80% Ge, varying in cluster size distribution, were simulated in order to observe the effect of cluster size on the production of the local mode. Si rich alloys with small Ge clusters, and Ge 50% alloys with a mix of small and intermediate sized clusters, produced the highest local mode intensities. Of the Ge rich alloys, the structures with small Si clusters produced the highest local mode intensities but were lacking when compared to Si rich and Ge 50% alloys. An increase in Si–Ge optical activity was observed when an increase in the local mode intensity occurred, indicating this mode is possibly of optical origin. The local mode, however, does not appear in a SiGe compound structure, suggesting that Si–Ge is only partially responsible for the production of this mode. In conclusion, the local mode is speculated to be a product of localized vibration caused by Ge clusters surrounded by Si pairs.

This work was supported by CREST Project No. JPMJCR19Q5 of the Japan Science and Technology Corporation (JST).