Tuning network topology and vibrational mode localization to achieve ultralow thermal conductivity in amorphous chalcogenides

Amorphous chalcogenide alloys are key materials for data storage and energy scavenging applications due to their large non-linearities in optical and electrical properties as well as low vibrational thermal conductivities. Here, we report on a mechanism to suppress the thermal transport in a representative amorphous chalcogenide system, silicon telluride (SiTe), by nearly an order of magnitude via systematically tailoring the cross-linking network among the atoms. As such, we experimentally demonstrate that in fully dense amorphous SiTe the thermal conductivity can be reduced to as low as 0.10 ± 0.01 W m−1 K−1 for high tellurium content with a density nearly twice that of amorphous silicon. Using ab-initio simulations integrated with lattice dynamics, we attribute the ultralow thermal conductivity of SiTe to the suppressed contribution of extended modes of vibration, namely propagons and diffusons. This leads to a large shift in the mobility edge - a factor of five - towards lower frequency and localization of nearly 42% of the modes. This localization is the result of reductions in coordination number and a transition from over-constrained to under-constrained atomic network.


Supplementary
Te 75 sample, we do not observe any sharp peak that is indicative of crystalline regions. We repeated this measurement for another set of samples that were prepared for our thermal conductivity measurement with 80 nm of ruthenium coating and the results are presented in Supplementary Fig. 1(b). In these samples, due to the existence of an 80 nm Ru coating and Si substrate, the Te peaks are not completely captured in the XRD data. Here, the only peak that appears in the XRD measurement is at ∼23 degrees. The other peaks in the XRD spectra (labeled accordingly) belongs to Ru transducer and Si substrate [4][5][6]. Supplementary Figure 2 shows the diffraction pattern for three SiTe compositions with thermal conductivity of 0.1 W m −1 K −1 . All the samples except for the a-Si 30 Te 70 are deposited in a single run. From these results, we conservatively conclude that the as-deposited SiTe films remain amorphous after the deposition. However, to ensure the accuracy of our results we perform transmission electron microscopy. Transmission electron microscopy. For this purpose, cross section view Focused Ion Beam (FIB) chips were prepared from the center of each wafer piece. In order to protect the area of interest during sample preparation, the site was coated in the FIB with e-beam SiOx and then tungsten. As part of the FIB sample preparation process, the FIB chip was micro-manipulated onto an Omniprobe stub grid. A JEOL ARM transmission electron microscope operated at 200 keV was used to collect all images and spectra.
Supplementary Figure 4 shows high resolution TEM results for a-Si 11 Te 89 which has the highest concentration of Te amongst our SiTe alloy composition series. For these measurement two different FIB chip were prepared from different parts of the sample and both of the TEM results suggest that the a-Si 11 Te 89 film is uniform, homogeneous with amorphous structure. The fact that the silicon substrate maintained its crystal structure is an indicative that the film has not been damaged during sample preparation process. Supplementary Figure 4. TEM micrographs for the sample with nominal composition of a-Si 10 Te 90 . According to these results the film is uniformly amorphous. Little oxygen is in a-Si 10 Te 90 , however, the oxygen level in the a-Si 10 Te 90 is similar that in Si substrate, so the oxygen could be from air exposure after FIB-cut. Nitrogen in CNx is enriched to the interface of CNx/Si-substrate.
Supplementary Figure 6. TEM micrographs for a-Si 20 Te 80 , measured for a second time from a different region of the wafer. According to these results the film is compositionally non-uniform. Regions with bright contrast are 1.6nm to 4nm large, appear mostly amorphous. Darker regions are polycrystalline. The prevalent lattice spacing is 3.4A (see diffractogram from selected SiTe region). Figure 7. TEM micrographs and its corresponding EELS mapping for a-Si 20 Te 80 .

Supplementary
Supplementary Figure 12. TEM micrographs for Si 50 Te 50 and Si 70 Te 30 before and after mapping. The region specified in red, is where the mapping was performed. As can be seen, SiTe has a high sensitivity to electron beam, and as a result, TEM imaging/mapping causes further segregation in the film. A JEOL ARM transmission electron microscope operated at 200 keV was used to collect all images and spectra.
Supplementary Figure 13. TEM micrographs for Se 20 Te 80 with nominal thicknesses of 5, 10, 20, and 40 nm. The micrographs for 10 and 20 nm Se 20 Te 80 indicate ordered regions in the film. We hypothesise that this crystallinity is not intrinsic to the film and is the result of sample preparation during FIB process.
Raman Spectroscopy. The Raman spectera presented in the main manuscript is taken from the same samples that the thermal conductivity were measured after etching off the 80 nm ruthenium transducer. Due to sensitivity of these films to any external stimuli, the films are prone to damage during the etch off process.
Therefore, another batch of samples were prepared without any metal coating for Raman measurement. Another important parameter that needs to be taken into account for estimating the thermal conductivity of the second layer, is thermal boundary conductance (TBC) at the interfaces. As can be seen in Supplementary   Fig. 17(a), the sensitivity of our analysis to the thermal conductivity of Si 20 Te 80 , even for a 5 nm thick film is much higher than the thermal boundary conductance. Therefore, in our analysis, we assume infinite TBC for the G 1/2 and G 2/3 interfaces and only fit for the thermal conductivity of the Si 20 Te 80 layer which is the main source of thermal resistance in the stack. Using a separate set of samples, we measure the thermal conductance across Ru/ 10 nm CN/Si and subtract the resistance from the layer thermal conductivity to account for the boundary resistances. However, since the resistance of the Si 20 Te 80 is large, subtracting the boundary conductance from the layer has negligible effect on the measured value. A representative experimental data and its corresponding fit is demonstrated in Supplementary Fig. 17(c) for a 5 and 40 nm thick Si 20 Te 80 film.
As can be seen, although we assume unrealistic values for the TBC, due to negligible sensitivity, the model perfectly fits to the experimental data. To ensure the accuracy of our measurements for thermal conductivity, we calculate the total thermal resistance between Ru and Si layer for different film thicknesses. Then, by applying a linear fit to the thermal resistance data, we obtain the thermal conductivity that is independent of the TBCs. We apply this to 2 different sample configurations with different interlayers to ensure the accuracy of our measurements and negligible impact of TBC on our thermal conductivity measurements. Based on these two sample configurations, we measure the thermal conductivity of the Si 20 Te 80 film to be 0.1 ± 0.01 W m −1   [11] [(C n H 2n−1 NH 3 ) 2 PbI 4 single-crystalline 0.099-1.25 Giri et al. [12] isoBA 2 PbI 4 layered crystalline 0.10 ± 0.02 Zhao et al. [13] SnSe crystalline 0.23 ± 0.03 Lee et al. [14] CsSnI 3 single-crystalline 0.38 ± 0.04 Supplementary Figure 20. Thermal conductivity of 20 nm thick Si 20 Te 80 from room temperature up to 300°C . The film started to delaminate for temperatures above 300°C. The uncertainty is calculated by assuming 10% variations in the Si 20 Te 80 film thickness.
Sound speed measurements. In order to measure the speed of sound in the SiTe films across different compositions, we use picosecond acoustics. In this technique, the absorption of laser pulse on the Ru surface, launches mechanical strain waves from the surface to the underlying layers. The strain waves travel at the speed of sound in the corresponding layer. At the interface between the layers, however, depending on the acoustic impedance defined as (Z = ρ × E) where ρ is density and E is Young's modulus, the waves are partially reflected and the remainder is transmitted. The reflected waves from each interface travels all the way back to the surface and influence the thermoreflectivity of the transducer. For the case where acoustic impedance at the interface is significant and a large portion of the strain waves are reflected, using a picosecond time resolution, the echoes can be detected with the probe beam in the TDTR signal as troughs and peaks. These echoes are only detectable when there are a few number of layers and the acoustic impedance between the layers of interest is large. For instance, for a-Si and Te sample studied here, due to lack of sufficient acoustic impedance at the interfaces, we have not been able to detect any echoes. Therefore, we report the sound speed for these two samples from the literature [15]. On the other hand, as can be seen from the solid line in Supplementary Fig.   21, the acoustic impedance at the Ru/Si interface is sufficiently large to influence our TDTR signal. In this case, the troughs corresponds to the reflected waves from the Ru/Si interface. Addition of CN x /Si 20 Te 80 /CN x layers between Ru and Si, depicted as solid circles, as well as changing the decay rate in the TDTR signal, add additional peaks. These peaks correspond to the interface between CN x /Si. By measuring the time between the first trough corresponding to Ru/CN x interface and the first peak corresponding to the CN x /Si, we estimate the time it takes for the strain waves to travel across the CN x /Si 20 Te 80 /CN x stack. In this case, since the sound speed in CN x is unknown, it is difficult to deconvolve the sound speed of SiTe from from that of the CN x . In order to accurately pinpoint the sound speed in Si 20 Te 80 layer, we deposited another batch of samples without the CN x interlayer: 80 nm Ru/5-40 nm Si 20 Te 80 /5 nm W/Si (see Supplementary Fig. 22). The W layer between the film and the Si substrate is to ensure sufficient reflection from the backside interface. The measured sound speed from 40 nm thick Si 20 Te 80 is 2150 ± 100 m s −1 . Using the obtained sounds speed for Si 20 Te 80 we estimate the sound speed in CN x to be 7500 ± 900 m s −1 which is well within the range for amorphous diamond like carbon. The nitrogen content in CN x is nearly 20%. Supplementary Note 3 Molecular dynamics simulations. In order to exclusively investigate the effect of mass scattering on a-SiTe alloy, molecular dynamics (MD) simulations were performed by randomly substituting the mass of Si with that of Te (127 u). We use non-equilibrium molecular dynamics (NEMD) and equilibrium molecular dynamics (NMD) methods to calculate the thermal conductivity of a-Si 20 -heavySi 80 . For both techniques, we use Stillinger-Weber interatomic potential that has been widely used to characterize the thermal properties of Si.
The simulation procedure and details of these techniques are given elsewhere [16][17][18]. Supplementary Fig.   23 (a) demonstrates the temperature profile across the simulation box. For these simulations two heat bath are assigned at the beginning and in the middle of the simulation box. The temperature of the hot and cold regions are set to 550 and 450 K, respectively. By measuring the heat flux transferred between these two heat baths, the thermal conductivity of the a-Si 20 -heavySi 80 is determined to be 0.50 W m −1 K −1 . This calculation is in good agreement with our EMD green-kubo calculations as shown in Supplementary Fig. 23 (b). This thermal conductivity is a factor 5 higher than the measured thermal conductivity for a-Si 20 Te 80 . Therefore, we conclude that although there is a large atomic mass mismatch between Si and Te, the mass scattering alone in a-Si 20 -heavySi 80 cannot explain the ultralow thermal conductivity of this material.
Supplementary Figure 23. Thermal conductivity for a-Si 20 -heavySi 80 from (a) non-equilibrium and (b) equilibrium molecular dynamic calculations. Both method result in thermal conductivity of ∼0.50 W m −1 K −1 . The heavySi mass in these simulations is similar to that of the Te.

Supplementary Note 4
Minimum limit thermal conductivity. A lower limit to the thermal conductivity of materials is estimated using the minimum limit model derived from the kinetic theory of gases that works on the basis of propagating modes or phonons [19]: where k B is the Boltzmann constant, v g is the average sound velocity in the material, and n is the number density. The average sound velocity can be written in terms of the longitudinal (v L ) and transverse (v T ) sound velocities as: In this study, for consistency, we assume that transverse sound velocity is 60% of that of the longitudinal mode, ν T = 0.6ν L . For Si 19 Te 81 , using our sound velocities derived from the picosecond acoustics measurements, we measure a longitudinal sound velocity of ν L = 2150 ± 100 m s −1 , which is in good agreement with previously reported values (ν L = 2030 m s −1 ) for the sound speed of a thicker SiTe film [20]. Other parameters that are used as an input to calculate the minimum limit such as number density and sound velocities across different compositions are given in Supplementary Note 3. Using these parameters, we determine a minimum thermal conductivity of 0.24 W m −1 K −1 for Si 19 Te 81 at room temperature. This estimate for the minimum thermal conductivity, however, is still more than a factor of two higher that the measured thermal conductivity of Si 19 Te 81 (0.10 W m −1 K −1 ). In fact, for nearly all compositions of SiTe studied in this work, Eq. 1 overpredicts the measured thermal conductivities, as shown in Supplementary Fig. 2(c). This discrepancy increases as the SiTe coordination number decreases and the alloy transitions into an under-constrained network.
Although phonon-mediated minimum limit to thermal conductivity described above and presented in Eq.
1 has served as a successful approach to predict the thermal conductivity of a variety of disordered crystals and amorphous materials, several recent works have experimentally measured values well below this limit.
This has motivated others to model the thermal conductivity in amorphous solids as a form of energy hopping between localized vibrational eigenstates. According to Allen and Feldman (AF) [21], a large portion of heat in disordered solids is transferred by quantized vibrations that are neither localized nor propagating. These delocalized non-propagating vibrational modes, diffusons, carry heat by diffusion with wavelength on the order of the inter-atomic spacing. Based on the AF formalism, Agne et al. [22] suggested a modified minimum limit model for heat transport in disordered solids that relies on the concept of diffusons rather than propagating modes. They argued that in a disordered solid, the lower bound to thermal conductivity occurs when the thermal transport is entirely driven by diffusons. This approach, albeit with the heat transfer carrier length scale being fundamentally different from those modeled in Eq. 1, leads to a similar functional form for the thermal conductivity of disordered materials: According to this diffuson-mediated minimum model, in one period of oscillation, each vibrating carrier will make two attempts to transfer energy where P is the probability of a successful energy transfer. In the high temperature limit and maximum diffusivity where P = 1, the calculated thermal conductivity is ∼37% lower than the phonon minimum limit model. By applying this diffuson-mediated minimum limit model to Si 19 Te 81 , we find a thermal conductivity of 0.14 W m −1 K −1 , which is in a better agreement with the measured value.
However, considering the fact that this model is supposed to set the lower bound to thermal conductivity, it still predicts 40% higher thermal conductivity than the measured value. This implies that the thermal transport in Si 19 Te 81 is driven by other atomistic properties that impede the transfer of energy beyond those accounted for in the diffuson limit.
To resolve this, we revisit an assumption that was made in the diffuson-mediated thermal conductivity (Eq. 3), which assumes 100% of attempts to transfer energy are successful between diffusons. As discussed earlier, since the coordination number in SiTe decreases by increasing the Te concentration, the alloy transitions from an over-constrained to an under-constrained network. This reduction in the number of bonds per atom eliminates the number of pathways through which diffusons can interact, and leads to a reduction in the probability of their successful energy transfer. To account for this probability, we assume that there is linear a relationship between the coordination number and the probability of successful energy transfer. We take the element with the highest possible coordination number, in this case Si, as the maximum probability of successful energy transfer P = 1.
Then, we calculate P for each alloy's configuration by normalizing their coordination number with respect to pure Si, i.e., P = (<r m >/<r max >) 1/3 = (<r Si x Te 1−x >/ <r Si >) 1/3 . Using this assumption, P changes from 1 to 0.8 for Si with < r > = 4 and Te with < r > = 2. By applying this condition, we calculate the diffuson-mediated thermal conductivity for Si 20 Te 80 as 0.12 W m −1 K −1 , in better agreement with the measured values of our SiTe alloys across the compositional phase space, shown in Fig. 2(c) in the main manuscript.
Supplementary Table 3. The longitudinal ν long , total sound speed ν g , number density n, coordination number < r >, and probability of successful transfer P for different amorphous chalcogenide compositions used in thermal conductivity estimation.
Composition ν long ν g n × 10 28 < r > P κ min,P κ min,D κ exp  [23]. b Due to low T g in tellurium film, it turned to crystalline during the deposition process.

Supplementary Note 5
Diffusivity and thermal conductivity calculations. In order to determine the contribution of non-propagating delocalized diffusons in thermal transport, it is necessary to calculate the thermal conductivity due to diffuson contribution. For this, we use formalism proposed by Allen-Feldman as follows: where V is the system volume, i is the mode number, N is the total number of modes, ω i is frequency of ith mode, C(ω i ) and D(ω i ) are the frequency dependant specific heat and mode diffusivity, respectively. These parameters can be obtained by: Where S i j is the heat current operator with unit of J/m 2 /s which can be obtained with the knowledge of eigenvectors, dynamical matrix, and the minimum distance between the pairs: To obtain the AF thermal conductivity we developed our own script using MATLAB to perform the above calculations. For this, we imported the dynamical matrix, eigenvector, position, and frequency of the modes from the force constant calculations into our model and estimated the diffusivity and thermal conductivity. It is worthwhile mentioning that there is a program, General Utility Lattice Program (GULP), which automatically performs the AF thermal conductivity calculations for a limited number of potentials such as harmonic, Lennard-Jones, and Stillinger-Weber (SW) to name a few. However, since we did not use any of the existing potentials for calculating the thermal conductivity of a-Si 20 Te 80 , we had to write our own script. Nonetheless, we used GULP to validate our script by calculating the diffusivity and AF thermal conductivity of a-Si and a-SiO 2 and comparing the results with those obtained by GULP. For this, we used SW for a-Si and Beest-Kramer-van Santen (BKS) for a-SiO 2 . The BKS potential is similar to those used for a-SiO 2 in previous studies [24][25][26]. In order to create a structure that is closer to a-SiTe 2 , we replace the mass of oxygen to that of tellurium in a-SiO 2 system (a-Si 127 O 2 ).
To calculate the diffusivity in a-Si, we use an amorphous system proposed by Barkema and Mousseau [27] with 1000 atoms at 300 K and under zero pressure. For a-Si 127 O 2 , we used melt-quench process with 720 atoms to obtain a uniform amorphous structure. The calculation parameters such as melt temperature, broadening factor, and cut-off frequency are similar to those of Ref [24][25][26]. To calculate the diffusivities in a-Si 20 Te 80 a Lorentzian broadening of 1δ ω ave = 0.4360 cm −1 and a cutoff frequency if 1 cm −1 was used for thermal conductivity calculation. Allen and Feldman recommended a broadening factor larger than the average frequency spacing δ ω ave . Considering this, we observe changing the broadening factor from 1δ ω ave = 0.4360 cm −1 to 5δ ω ave = 2.1801 cm −1 leads to small increase in the estimated thermal conductivity (∼3%). Amorphous a-Ge 20 Te 80 vs. a-78 Si 20 Te 80 . GeTe alloy is a well-known phase-change/thermoelectric material which has been extensively studied in terms of its thermal properties. Similar to silicon, germanium is a 4-coordinated element and forms a short range ordered tetrahedron upon mixing with Te. Atomic structure of SiTe studied here in many cases such as coordination number and radial distribution function resembles that of the GeTe. Supplementary Figure 26 shows the radial distribution function for our a-78 Si 20 Te 80 where the mass of silicon atoms are replaced by that of germanium and a-Ge 20 Te 80 from Ref. [28]. Due to similarities between GeTe and SiTe, it is interesting to investigate how much atomic masses of the constituent elements in a-Si 20 Te 80 would change the thermal properties. For this, we repeat our simulations for a-Si 20 Te 80 and change the silicon mass to that of germanium, a-78 Si 20 Te 80 . The result for this modified alloy system is presented in Supplementary Fig. 27. As can be seen, due to higher average atomic mass, the frequency of the modes have dropped from ∼400 cm −1 to ∼300 cm −1 . However, this has negligible impact on the thermal conductivity of the a-78 Si 20 Te 80 . As discussed in the manuscript, this is because all the modes above ∼100 cm −1 are localized and do not contribute to heat transfer. Although the force constants in a-78 Si 20 Te 80 has not been developed for a-Ge 20 Te 80 alloy system, our estimated thermal conductivity is in good agreement with experimentally reported values that spans from 0.1 to 0.23 W m −1 K −1 [1,10,29,30].
Supplementary Figure 27. (a) Vibrational density of states (vDOS) for a-Si 20 Te 80 and a-78 Si 20 Te 80 (similar mass to a-GeTe) obtained from force constant calculation on 300 atom supercell (b) diffusivity of modes calculated from AF formalism, (c) inverse participation ratio calculated from vibrational mode eigenvectors (d) Accumulative thermal conductivity as a function of modes frequency.
An important factor in the thermal transport mechanism is the contribution from the electrons in the total thermal conductivity. A common approach to estimate the thermal conductivity due to electron contribution is the widely used empirical equation proposed by Wiedemann-Franz (WF): k e = LT /ρ where k p and k e are thermal conductivities due to phonon and electron contribution, respectively, L is the Lorenz number 2.44 × 10 −8 W Ω K −2 , T is temperature, and ρ is the electrical resistivity. According to Bailey [31] and Petersen et al. [32], the electrical resistivity of the a-Si x Te 1−x for 0.02 < x< 0.4 is in the range of 0.1-5000 Ω m. We confirmed that for the a-Si 20 Te 80 composition the electrical resistivity in our samples is at least 100 Ω m, consistent with this range. This corresponds to electronic contribution, k e , of 7×10 −5 -1 ×10 −9 W m −1 K −1 which are orders of magnitude lower than the measured thermal conductivity. This indicates that SiTe is electrically insulating in its room temperature unperturbed state. This is also confirmed by a more recent study, Koo et al. [33], which showed threshold switching behavior of SiTe with resistance before switching as high as (> 1 GΩ at 0.1 V). The high resistivity of a-SiTe is attributed to the existence of deep trap states.