Using molecular dynamics, the effect of an atomic mass difference on a localized phonon mode in SiGe alloys was investigated. Phonon dispersion relations revealed that a change in atomic mass causes the optical and acoustic modes to shift frequency. The results indicate that the local mode is sensitive to both Si and Ge atomic mass changes; reducing the Si atomic mass shifts the local mode to higher frequencies, and increasing the Ge atomic mass shifts the local mode to lower frequencies. Furthermore, the results suggest that the local mode originates from the Si–Ge bond vibration. Although the Si–Si, Si–Ge, and Ge–Ge optical mode frequencies are well approximated by the two-body harmonic oscillator model, a much heavier effective mass than that of the Si–Ge pair must be assumed to reproduce the local mode frequency. A plausible interpretation of the local mode is a collective vibration of Ge clusters embedded within the Si lattice.

The increasing global demand for renewable sources of energy has been the driving force for thermoelectric (TE) research over the past few decades.1–7 The ideal thermoelectric will maintain efficient interconversion between heat energy and electricity, providing a means for harnessing waste heat. The most difficult aspect concerning this theme of research maximizes the figure of merit, defined as zT = S2σT/k, where S2 is the Seebeck coefficient, σ the electrical conductivity, T the absolute temperature, and k the thermal conductivity.8 A majority of TE materials have zT values around 1,9,10 but an extreme case of 40011 has also been reported. Recent advances have increased the benchmark of zT to 2, which has been realized for materials including SnSe12 and Cu2Se.13 A zT value of 2 is too low to be economically competitive with large-scale electronics and has limited TE materials to niche applications, such as deep space exploration and wearable14 and portable devices. According to the zT expression, lowering the thermal conductivity would improve the TE efficiency, but the intertwining relationship between thermal and electrical conductivities proves challenging.

A variety of group IV semiconductors, including silicon nanowires,15–19 superlattices,20–22 and SiGe alloys,23–25 have been established as prospective TE materials26–28 due to low thermal conductivities,29–34 low toxicity, compatibility with metal–oxide–semiconductors (MOS), structural integrity at high temperatures,35,36 and ability to realize both n-type37,38 and p-type39,40 properties. SiGe alloys have thermal conductivities 1–2 orders of magnitude lower41–44 than those of their parent elements, Si45,46 and Ge,47,48 and the alloy limit has been surpassed using SiGe superlattices.34 Depending on the atomic configuration,49 thermal conductivity values are able to be manipulated. Despite thermal conductivity having been studied extensively, the present state of technology restricts bulk SiGe alloys to high temperature (900–1200 K) applications. Parameters governing the thermal conductivity, such as phonon lifetimes and group velocities, can be characterized by phonon dispersion relations, an area lacking for SiGe alloy research.

Usuda et al.50 obtained phonon spectra for bulk Si1−xGex alloys by conducting an inelastic x-ray scattering experiment. The subsequent work conducted by Yokogawa et al.51 reported the production of an “anomalous phonon mode” occurring around 13 meV. This mode is almost independent of the wave number, suggesting that it is a localized vibration mode. Pagés et al.52 reported that the frequency of the Ge–Ge bond will shift lower when the Ge–Ge bond is surrounded by Si atoms. The anomalous mode occurs at frequencies lower than that of the Ge–Ge optical mode, and for these reasons, Yokogawa et al. speculated that the anomalous mode is a local vibrational mode comprising Ge–Ge pairs or Ge clusters, surrounded by Si atoms. In this study, the term “local mode” will be used in place of “anomalous mode” hereinafter.

A follow-up investigation on the nature of this local mode was conducted using molecular dynamic (MD) simulations.53 This study revealed that a SiGe alloy, which consists of Si–Si, Si–Ge, and Ge–Ge bonds, will always produce this mode. Contrastingly, the SiGe compound comprising a configurational periodicity of Si–Ge–Si–Ge⋯ bonds, spanning long distances, failed to produce this mode.53 Despite the inability of the SiGe compound to produce the local mode, investigating the role that Si–Ge bonds play in thermal transport should not be neglected. Using MD, a further study of the local mode dependency on compositional cluster size was recently carried out.54 These simulation results demonstrated the strong dependency of the local mode on Si systems with small Ge clusters. Conversely, Ge rich SiGe alloys comprising Si clusters, small and large, produced the lowest local mode intensities. In addition, Si–Ge optical intensities increased with the local mode intensity.

The speculation by Yokogawa et al. is open to interpretation regarding which bonds are the dominant contributors to the local mode. To elaborate, Ge clusters comprising Ge–Ge bonds, surrounded by Si atoms (Si–Si bonds), are expected to produce the local mode. The combination of the Ge clusters and Si atoms would also lead to the formation of Si–Ge bonds within the periphery of the clusters. The local mode occurs at a low frequency, suggesting that the main constituent species is the Ge–Ge bond (Ge clusters). Whether Si–Si or Si–Ge imposes a stronger influence on the local mode production still needs further scrutiny.

In an alloy, the difference in atomic mass between constituent species causes mass disorder to occur within the lattice. The lattice positions are maintained, but the distribution of the constituent species is randomized. Constituent elements with large mass differences, such as that of SiGe, enhance grain boundary scattering, phonon-impurity scatter,55 and mass disorder scatter.56 This is observed in SiGe related structures, wherein bulk Si and bulk Ge have thermal conductivities at least 1 order of magnitude higher than those of SiGe alloys. It has also been reported that Anderson localization may occur in binary isotopic alloys.57 This possibility may extend to SiGe alloys if atomic masses were altered. The local mode is speculated to play a role in the lowering of thermal conductivity,51 and insight into how this mode reacts to structural changes could assist with phonon engineering for optimal devices.

To observe the production of the local mode, SiGe alloys were simulated using MD to produce phonon dispersion relations. By reducing the Si atomic mass or by increasing the Ge atomic mass, the effects on the local mode were revealed.

A bulk Si50Ge50 alloy model of the cuboidal shape was generated with dimensions 16.6 × 2.22 × 2.22 nm3 in the x, y, and z crystallographic directions (Fig. 1). All calculations shown in this paper were performed using the same structure, but the Si or the Ge atomic mass was changed each time. The same bulk Si50Ge50 alloy model was utilized in the study by Yokogawa et al.51 and also the follow-up simulation studies.53,54 This model produced a phonon dispersion relation with optical mode frequencies and the local mode, in good agreement with their experimental data. The default Si and Ge atomic masses were set to 28.09 and 72.63 amu, respectively. To implement an atomic mass difference, either the Si atomic mass was reduced or the Ge atomic mass was increased (Table I). Increasing the Si atomic mass would bring the Si atomic mass closer to that of Ge, and likewise, reducing the Ge atomic mass would bring the Ge atomic mass closer to that of Si. By enacting only a Si mass reduction or a Ge mass increase, any possible outstanding effects caused by the Si or Ge mass changes would be pronounced, and which species has a stronger influence on the production of the local mode would be made clear.

FIG. 1.

Atomistic model of the bulk SiGe alloy displayed using the molecular visualization program (VMD).58 Si atoms are depicted in yellow, and Ge atoms are depicted in pink.

FIG. 1.

Atomistic model of the bulk SiGe alloy displayed using the molecular visualization program (VMD).58 Si atoms are depicted in yellow, and Ge atoms are depicted in pink.

Close modal
TABLE I.

Si and Ge atomic masses per SiGe alloy case. Case A is the SiGe alloy with no change in atomic mass for both elements, Case B is an example of a reduction in the Si atomic mass, and Case C shows how the Ge atomic mass increase was implemented.

Si atomic mass (amu)Ge atomic mass (amu)
Case A (unaltered SiGe alloy) 28 73 
Case B (reduction of Si mass) 16 73 
Case C (increased Ge mass) 28 118 
Si atomic mass (amu)Ge atomic mass (amu)
Case A (unaltered SiGe alloy) 28 73 
Case B (reduction of Si mass) 16 73 
Case C (increased Ge mass) 28 118 

We employed a simulation package (LAMMPS)59 to perform geometry optimization and MD simulation in the NVE ensemble (constant number of atoms, volume, and energy) at 300 K. Periodic boundary conditions were imposed in the x, y, and z crystallographic directions to emulate a three-dimensional bulk system.

The potential parameters governing the interatomic forces between interacting atoms were taken from a set of parameterized values published by Tomita et al.60 and of the original Stillinger–Weber61 potential, which are given as

PE=ij>iϕ2rij+iijk>jϕ3(rij,rik,θijk),
(1)
ϕ2rij=εijAijBijrijσijpijrijσijqijexpσijrijaijσij,
(2)
ϕ3rij,rik,θijk=εijkλijkexpγijσijrijaijσij+γikσikrikaikσik×cosθijk+cosθ0ijk2,
(3)

where ϕ2rij is the two-body term describing the pair interaction, with rij defining the interatomic distance between atoms i and j. ϕ3rij,rik,θijk is the three-body term describing the triplet interaction between atoms i, j, and k and sets the bond angle, θijk, between the lines ij and ik. ɛij, Aij, Bij, σij, pij, qij, and aij are the adjustable parameters for the two-body component (ϕ2rij), and ɛijk, λijk, γij, σij, aij, γik, σik, aik, and θ0ijk are the adjustable parameters for the three-body component (ϕ3rij,rik,θijk). We employed a single interatomic potential function (potential parameters of Si) to observe only the effect of a mass difference and to ensure that no effects from the potential difference occurred.

The phonon dispersion relations were predicted by using the spectral energy density (SED),62 which is calculated using the space–time Fourier transform of the atomic trajectories obtained in the MD simulation. For a given phonon mode, Φκ,ω, where κ is the wavevector and ω is the angular frequency, the SED can be calculated as

Φκ,ω=14πτ0NTabBmb0τ0nx,y,zNTu̇anx,y,zb;t×expiκrnx,y,z0iωtdt2,
(4)

where τ0 is the phonon lifetime, NT is the total unit cells in the system, mb is the atomic mass of atom b, B is the total number of atoms, displaced, u̇a, in direction a at time t, and r represents the equilibrium positions for each unit cell (nx,y,z). In this work, phonon dispersions along the Γ − X (100) direction were extracted. The displayed phonon dispersion relations are a merge of the first and second Brillouin zones, comprising the acoustic [k = (0, 0, 0) ∼ (1, 0, 0)] and optical [k = (2, 0, 0) ∼ (3, 0, 0)] phonon modes. This enables the individual phonon modes to be displayed on one plot.

As the SED method is sensitive to background noise, two types of non-linear digital filtering were applied. Often used in image and signal processing, median filtering involves selecting a data point and treating this point by taking the median value of its surrounding neighbors. By repeating this procedure for all the data points, background noise can be reduced while preserving a significant portion of the raw data. Mean filtering is a similar process, wherein the mean value is outputted instead of the median. In this study, two sets of median filtering followed by one set of mean filtering were applied. These procedures ensured that the resulting data were not excessively affected by the filtering, preventing significant alteration of the data.

This SED method has been tested on carbon nanotubes62 and argon and silicon systems,63,64 with Feng et al.65 confirming the usability of this method for bulk systems. Tomita et al.60 successfully reproduced thermal conductivities and phonon dispersions of Ge(1−x−y)SixSny alloys by calculating the dynamical structure factor, a type of spatiotemporal Fourier transform. These results were found to be in reasonable agreement with the literature. The SED calculation method in this study was also utilized in the report by Yokogawa et al.,51 producing a phonon dispersion relation, which is in good agreement with their experimental data. It is therefore assumed that the phonon dispersions produced in this current study will also be viable.

In Fig. 2, the alloy with standard Si and Ge masses [Fig. 2(a)] has been simulated with Si and Ge potential parameters, but the altered Si mass alloys [Figs. 2(b) and 2(c)] have been simulated with the Si potential only. This results in an overall shift of all the phonon modes to higher frequencies. A reduction in Si atomic mass caused the Si–Si and Si–Ge optical bands to shift to higher frequencies [Figs. 2(b) and 2(c)], with the Si–Si band as the prominently shifted mode. The same observation can be seen in the corresponding SED intensities [Figs. 3(c) and 3(d)]. The Ge–Ge optical mode remains unshifted. The local mode spectrum reveals a slight shift to higher frequencies. The phonon dispersion relations are accompanied by a color bar on the side to gauge the magnitude of the SED of each phonon mode. According to the color shading, the acoustic modes and the Ge–Ge optical mode have the strongest intensities. Applying a Si only potential [Figs. 2(b) and 2(c)] caused Si–Si to occur at high intensities for k wavevectors 0–0.4 pi/a, and Si–Ge occurs at high intensities between k = 0.4 pi/a and k = 1 pi/a. The phonon dispersions in this study are a merge of Brillouin zone indices, k = (0, 0, 0) ∼ (1, 0, 0) and k = (2, 0, 0) ∼ (3, 0, 0) for the acoustic and optical modes, respectively. For this reason, the local mode appears to merge with the acoustic modes. However, the previous study we conducted on the cluster size dependency of the local mode54 revealed that this mode only appears in the same Brillouin zone as the optical modes and is almost independent of the wave number.

FIG. 2.

Phonon dispersion relations of SiGe alloys with unaltered atomic masses (a) and Si atomic mass reduced to 20 (b) and 16 amu (c).

FIG. 2.

Phonon dispersion relations of SiGe alloys with unaltered atomic masses (a) and Si atomic mass reduced to 20 (b) and 16 amu (c).

Close modal
FIG. 3.

(a)–(d) Spectral energy densities of the SiGe alloys with Si masses reduced to 20, 18, and 16 amu in the local mode (a) and the optical modes of Ge–Ge (b), Si–Ge (c), and Si–Si (d).

FIG. 3.

(a)–(d) Spectral energy densities of the SiGe alloys with Si masses reduced to 20, 18, and 16 amu in the local mode (a) and the optical modes of Ge–Ge (b), Si–Ge (c), and Si–Si (d).

Close modal

Phonon dispersion relations in Fig. 2 show that the local mode overlaps with the acoustic modes. To better distinguish the local mode, the SED intensities in Fig. 3 were extracted from k = 0 pi/a where no overlap occurs. The effect of reducing the Si atomic mass entails the shifting of Si related optical modes (Si–Si and Si–Ge) to higher frequencies but only in minute amounts for the local mode. Regarding the acoustic modes, there was no distinguishable sign of a change in the group velocity. These results demonstrate that a reduction in Si atomic mass has a stronger effect on the Si–Si mode.

To observe the effect of an increasing Ge atomic mass on the local mode intensity, Ge masses were increased to 81 and 85 amu [Figs. 4(b) and 4(c)]. The increase in the Ge atomic mass caused a near negligible effect on the optical modes. Furthermore, in accordance with the color shading, the acoustic modes and the Ge–Ge optical mode produced the highest intensities. The acoustic branches did not exhibit clear signs of a shift in the group velocity. To further test the effect of an increasing Ge atomic mass, SiGe alloys with Ge masses increased within the range of 118–293 amu were simulated.

FIG. 4.

Phonon dispersion relations and the corresponding density of states of the SiGe alloys with unaltered atomic masses (a) and Ge atomic mass increased to 81 (b) and 85 amu (c).

FIG. 4.

Phonon dispersion relations and the corresponding density of states of the SiGe alloys with unaltered atomic masses (a) and Ge atomic mass increased to 81 (b) and 85 amu (c).

Close modal

Figure 5 shows the phonon dispersion relations and density of states data for two of the SiGe alloys with an increased Ge atomic mass. When the mass was increased from 118 to 293 amu, the phonon branches for the Ge–Ge optical mode, and the local mode, shifted to lower frequencies and also reduced in intensity. The frequency of the Si–Si optical mode remains unchanged, whereas the Si–Ge mode has shifted minimally to lower frequencies. The pattern observed from all the phonon dispersion relations in this study is that specific optical modes will only shift frequency when their constituent element has been adjusted in atomic mass. However, the local mode and the Si–Ge optical mode have shifted frequency for every atomic mass change we simulated.

FIG. 5.

Phonon dispersion relations and corresponding density of states representing the SiGe alloy with Ge atomic masses of 118 (a) and 293 amu (b).

FIG. 5.

Phonon dispersion relations and corresponding density of states representing the SiGe alloy with Ge atomic masses of 118 (a) and 293 amu (b).

Close modal

The phonon bands depicting the acoustic modes of Ge 118 amu [Fig. 5(a)] have steeper slopes when compared to those of Ge 293 amu [Fig. 5(b)]. The slope of the acoustic phonon bands in Fig. 5(b) indicates that a larger Ge mass has caused a reduction in the group velocities. The low frequency modes have a propagating character, and Fig. 5 indicates that an increasing Ge mass has a stronger influence on these modes.

The SED intensities corresponding to the data shown in Fig. 5 are given in Fig. 6, along with the addition of another SiGe alloy with Ge increased to 207 amu for additional comparison. In accordance with the results shown in the phonon dispersions in Fig. 5, Figs. 6(a)6(c) reveal that an increase in Ge atomic mass will incur a shift to lower frequencies for the local mode, Ge–Ge, and Si–Ge. The local mode frequencies [Fig. 6(a)] are also lower in intensity, and it is expected that further increments to the Ge atomic mass would ensue further reductions in frequency and intensity. The Si–Si band remains unshifted [Fig. 6(d)].

FIG. 6.

(a)–(d) Spectral energy densities of the SiGe alloys with Ge masses increased to 118, 207, and 293 amu in the local mode (a) and the optical modes for Ge–Ge (b), Si–Ge (c), and Si–Si (d).

FIG. 6.

(a)–(d) Spectral energy densities of the SiGe alloys with Ge masses increased to 118, 207, and 293 amu in the local mode (a) and the optical modes for Ge–Ge (b), Si–Ge (c), and Si–Si (d).

Close modal

The local mode shifts frequency when the atomic mass of any constituent species is changed. To discuss this frequency shift, optical mode and local mode frequencies were calculated using a simple harmonic oscillator approximation.

Figure 7 shows the peak frequencies of each optical mode and local mode as a function of Si and Ge masses. The dashed lines are the predicted frequencies using the simple harmonic oscillator model. The harmonic oscillator frequencies, ω, are given by ω=k/mr, where k is the force constant and mr is the reduced mass given by mr = (m1m2)/(m1 + m2), with m1 and m2 the masses of Si and Ge, respectively. The force constants were set to 9.5, 5, and 9 eV/Å for Si–Si, Si–Ge, and Ge–Ge modes, respectively. To deduce the k force constants, the force curve of the corresponding two-body term was fitted by a quadratic function at the minimum. Regarding the local mode, we utilize the Si–Ge force constant (5 eV/Å) because the local mode is dependent on both Si and Ge masses and also appears in the same Brillouin zone as the optical modes. To calculate the local mode harmonic oscillator frequencies, the Si–Ge reduced mass was multiplied by a fitting parameter, N, which was set to a value of 15. The reduced masses for the SiGe alloys with altered Si and Ge masses are given in Tables II and III, respectively.

FIG. 7.

(a) Optical mode frequencies, calculated harmonic oscillator frequencies, and local mode frequencies for the SiGe alloys with Si atomic masses ranging from 16 to 28 amu. The Ge atomic mass was set to 73 amu. (b) Optical mode frequencies, calculated harmonic oscillator frequencies, and local mode frequencies for the SiGe alloys with Ge atomic masses ranging from 73 to 293 amu. The Si atomic mass was set to 28 amu.

FIG. 7.

(a) Optical mode frequencies, calculated harmonic oscillator frequencies, and local mode frequencies for the SiGe alloys with Si atomic masses ranging from 16 to 28 amu. The Ge atomic mass was set to 73 amu. (b) Optical mode frequencies, calculated harmonic oscillator frequencies, and local mode frequencies for the SiGe alloys with Ge atomic masses ranging from 73 to 293 amu. The Si atomic mass was set to 28 amu.

Close modal
TABLE II.

Reduced masses applied to find the harmonic oscillator frequencies of the local mode plotted in Fig. 7(a). Reduced masses were calculated using adjusted Si atomic masses and a Ge atomic mass of 73 amu.

Si mass (amu)Reduced mass (10−26 kg)
16 2.19 
18 2.41 
20 2.62 
28 3.38 
Si mass (amu)Reduced mass (10−26 kg)
16 2.19 
18 2.41 
20 2.62 
28 3.38 
TABLE III.

Reduced masses applied to find the harmonic oscillator frequencies of the local mode plotted in Fig. 7(b). Reduced masses were calculated using adjusted Ge atomic masses and a Si atomic mass of 28 amu.

Ge mass (amu)Reduced mass (10−26 kg)
73 3.38 
118 3.78 
163 3.99 
207 4.12 
252 4.21 
293 4.27 
Ge mass (amu)Reduced mass (10−26 kg)
73 3.38 
118 3.78 
163 3.99 
207 4.12 
252 4.21 
293 4.27 

The optical mode frequencies in Fig. 7(a), showing the Si mass dependency, follow the same pattern as that of the calculated harmonic oscillator frequencies. Similarly, the local mode also shifts frequencies in a manner that follows the behavior of the calculated harmonic oscillator frequencies. Figure 7(b) shows a plot of the optical modes, harmonic oscillators, and local mode frequencies of the SiGe alloys with increased Ge atomic masses. To obtain the harmonic oscillator frequencies, the same force constants applied in Fig. 7(a) for the Si–Si, Si–Ge, Ge–Ge, and local modes are also applied in Fig. 7(b). The optical modes in Fig. 7(b) shift frequencies in a similar manner to that of the calculated harmonic oscillator frequencies.

Our recently published work54 reported that the local mode changes intensity with respect to the cluster size. Furthermore, the local mode never appeared in the SiGe compound that comprises only Si–Ge bonds. The configuration of the local mode is speculated to comprise small Ge clusters surrounded by Si atoms. In addition, the results shown in this current study reveal that the local mode is sensitive to both Si and Ge atoms. Depicted in Fig. 8 is the proposed model for the local mode. This figure is a simplified schematic but is useful for providing an image to visualize the local mode with regard to the phrasing “small Ge clusters surrounded by Si atoms.” It is speculated that the overall motion of the local mode is a collective vibration comprising movement from the small Ge clusters surrounded by the Si–Ge bonds.

FIG. 8.

Proposed vibrational model of the local mode. Small Ge clusters embedded within the Si lattice altogether vibrate as a whole, with an emphasis on the Si–Ge vibration at the cluster surface.

FIG. 8.

Proposed vibrational model of the local mode. Small Ge clusters embedded within the Si lattice altogether vibrate as a whole, with an emphasis on the Si–Ge vibration at the cluster surface.

Close modal

Figure 9 shows the local mode frequencies as a function of N that emulates an increasing effective mass of the Si–Ge harmonic oscillator. The local mode spectrum is reproduced with the values of N between 9 and 30. It should be noted that N does not represent the cluster size directly. As the Ge cluster size increases, the number of Si–Ge bonds connecting the Ge cluster and the surrounding Si lattice will also increase. Furthermore, an increase in the number of Si–Ge bonds will also increase the force constant. The theory is as follows: the effective mass increases with the Ge cluster size, and the number of Si–Ge bonds connected to the Ge cluster is proportional to the Ge cluster surface area. An increase in Ge cluster size shifts the local mode frequency, and an increase in the Si–Ge bonds at the cluster surface will also shift the local mode frequency. The effect of a larger volume (increasing Ge cluster size) is stronger than the effect of an increasing surface area (increasing number of Si–Ge bonds). Therefore, an increasing Ge cluster size will affect the local mode frequency shift, more so than the effect of an increasing number of Si–Ge bonds. Resultantly, the harmonic oscillators will shift to lower frequencies when the Ge cluster size increases. Although Fig. 9 is a rough estimate, the plot is a useful guide to visualize the predicted frequencies of the local mode as a function of varying effective mass.

FIG. 9.

Calculated harmonic oscillator frequencies of the local mode using the Si–Ge force constant (5 ev/Å) and a reduced mass of 3.38 × 10−26 kg (Si 28 amu and Ge 73 amu), plotted as a function of N effective mass. Indicated on the plot is the range of N values needed to multiply with the Si–Ge reduced mass in order to produce the expected local mode frequencies. To the left of the predicted local mode frequencies, a density profile of the SiGe alloy (Ge 50%) is attached to serve as a reference, indicating the local mode frequency range.

FIG. 9.

Calculated harmonic oscillator frequencies of the local mode using the Si–Ge force constant (5 ev/Å) and a reduced mass of 3.38 × 10−26 kg (Si 28 amu and Ge 73 amu), plotted as a function of N effective mass. Indicated on the plot is the range of N values needed to multiply with the Si–Ge reduced mass in order to produce the expected local mode frequencies. To the left of the predicted local mode frequencies, a density profile of the SiGe alloy (Ge 50%) is attached to serve as a reference, indicating the local mode frequency range.

Close modal

A challenge concerning thermoelectric research is the ability to restrict the movement of phonons while preserving the mobility of electrons. One possible solution is to localize specific carriers. The localization of wave functions, arising from the destructive interference of coherent scattering, namely, Anderson localization,66 has a tendency to occur where strong disorder is present. Anderson localization of electrons hinders the power factor and was therefore considered as a non-desired phenomenon in thermoelectrics.67 Contrastingly, the localization of phonons has proven beneficial to such devices by suppressing thermal transport and resultantly lowering the thermal conductivity.68 The combination of delocalizing electrons, which helps enhance electrical conductivity, and the localization of phonons, which obstructs thermal transport, is ideal for thermoelectric devices if successfully implemented.67 

Qiu et al.69 simulated periodic and aperiodic SiGe superlattices to observe the effect of aperiodicity and rough interfaces on thermal transport. The authors reported that the aperiodic superlattice with a perfect interface produced the lowest phonon transmittance within the region of 60–100 cm−1 (1.8–3 THz).69 In addition, this superlattice in particular produced the lowest thermal conductivity. Similarly, Juntunen et al.68 also simulated a range of periodic and aperiodic SiGe superlattices to observe the effects of Anderson localization on thermal transport. The phonon transmission spectrum for an aperiodic superlattice with regions of atomic mixing at some of the interfaces revealed that Anderson localization occurs approximately between 1 and 3 THz. However, aperiodic superlattices with no interfacial mixing demonstrated localization spanning 0.4–9 THz. Although the SiGe alloy models in this current paper are not superlattices, there is atomic disorder within the lattice, and the local mode occurs around 2–4 THz, which is similar to the frequency range observed by Qiu et al. and Juntunen et al.

The similarities between the local mode in this current paper and the phonon transmission data reported by Juntunen et al. and Qiu et al. suggest that there is a possibility that the local mode is caused by Anderson localization. Despite this speculation, it is difficult to characterize and visualize this type of localization in three-dimensional systems.67 Although the local mode appears in bulk SiGe alloys, whether the mode originates from Anderson localization has yet to be confirmed. Investigating this query goes beyond the scope of this paper but would form a natural progression to the next phase of this research topic. The possible effect on thermal conductivity also gives promise for the future of thermoelectrics.

The local mode occurs at the lowest frequency of all the optical modes, which suggests that the Ge cluster is the main component of the local mode configuration. However, the results in this study show that the Si–Ge bond and the local mode exhibit a similar pattern of behavior, in which both modes shift frequency regardless of whether Si has been reduced or whether Ge has been increased. This suggests that a change in the density of Si–Ge bonds will also influence the intensity of the local mode. The initial query proposed whether Si or Ge has the stronger influence on the production of the local mode can be reinterpreted into a concept where neither element has a stronger influence; instead, both species are necessary. It has, however, been established that Si–Ge bonds alone cannot produce the local mode.53 Si–Si bonds surrounding the Ge–Ge bonds form Si–Ge bonds within the periphery of the Ge clusters, causing the production of the local mode. Further work entailing thermal conductivity calculations of these alloys would provide concrete conclusions on the effect this local mode has on thermal transport.

In order to observe the effect of the changing atomic mass on the local mode, bulk SiGe alloys varying in Si or Ge atomic mass were simulated to extract phonon dispersion data. In accordance with the calculated harmonic oscillator frequencies, the optical modes and the local mode shift to higher frequency when the Si atomic mass is reduced and to lower frequencies when the Ge atomic mass is increased. The Ge–Ge optical mode did not shift frequency when the Si atomic mass was reduced. The Si–Si optical mode did not shift when the Ge atomic mass was increased. The Si–Ge optical mode and the local mode, however, shifted frequency regardless of whether the atomic mass was increased or decreased, suggesting that the Si–Ge bond is a dominant constituent to the local mode configuration. Harmonic oscillator frequencies of the local mode, deduced using the Si–Ge force constant, were calculated as a function of increasing effective mass. These results indicated the range of effective masses predicted to produce the local mode frequencies of the SiGe alloy, comprising Si 28 and Ge 73 amu. Subsequent work investigating the thermal conductivities of these alloys could provide useful insight for device engineering, where the implementation of mass disorder within the configuration, can be considered.

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

The authors have no conflicts to disclose.

The data that support the findings of this study are available within the article.

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