Nonequilibrium Phonon Dynamics and Its Impact on the Thermal Conductivity of the Benchmark Thermoelectric Material SnSe

Thermoelectric materials play a vital role in the pursuit of a sustainable energy system by allowing the conversion of waste heat to electric energy. Low thermal conductivity is essential to achieving high-efficiency conversion. The conductivity depends on an interplay between the phononic and electronic properties of the nonequilibrium state. Therefore, obtaining a comprehensive understanding of nonequilibrium dynamics of the electronic and phononic subsystems as well as their interactions is key for unlocking the microscopic mechanisms that ultimately govern thermal conductivity. A benchmark material that exhibits ultralow thermal conductivity is SnSe. We study the nonequilibrium phonon dynamics induced by an excited electron population using a framework combining ultrafast electron diffuse scattering and nonequilibrium kinetic theory. This in-depth approach provides a fundamental understanding of energy transfer in the spatiotemporal domain. Our analysis explains the dynamics leading to the observed low thermal conductivity, which we attribute to a mode-dependent tendency to nonconservative phonon scattering. The results offer a penetrating perspective on energy transport in condensed matter with far-reaching implications for rational design of advanced materials with tailored thermal properties.


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Comparison of experimental PDS with theoretical models with varying e-ph coupling parameter (Ge-ph).9

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Phonon band character as a function of q-position for without and with the non-analytical correction (NAC).

Phonon spectrum of low temperature Pnma phase of SnSe
We calculated the phonon band structure and phonon density of states of bulk SnSe, and the results are plotted in Figure S2A.The lowest frequency branches are mostly contributed by Sn atoms and are separated by a gap from the high-frequency ones, dominated by Se atoms.Having calculated the TDS, we compared the result given by phonon spectra with and without non-analytical correction (NAC) describing a correction arising from macroscopic field caused by atomic displacement induced polarization of non-metallic materials. 2The NAC brings modification to the phonon spectra (Figure S2A), particularly the presence of polar phonons, 3,4 and related phonon band shifting.Despite the change in phonon spectra, the TDS shown in Figure 1(b) of the main text is almost independent of the application of NAC as the TDS patterns with and without NAC are hardly distinguishable (Figure S2B).Similarly, the implication of NAC should be considered for the lineprofile in    Figure 3(a) and Figure S5(a) focus on comparing the experimental and simulated PDS data at q´ = (0, -1/8, 1/8).We have expanded our analysis to include PDS data for q´ = (0, -1/7, 1/7) and (0, -1/6, 1/6) in Figure S6(a) and (b).These intermediate points within the experimental pattern exhibit a similar trend in the 10 to 30 ps range to q´ = (0, -1/8, 1/8).Figure S9 contains a direct comparison between the dEph/dt and dEe/dt flows for the selected branches, offering insights into the dominant energy flow at specific times.Furthermore, the resulting mode contribution to the PDS are compared to the total simulated PDS.To analyze the contribution of particular phonon modes to the thermal transport, we divided the phonon modes into three sets according to their character and the importance of the Umklapp scattering (Figure 5(a)).We evaluated the mean change of the phonon population (Figure S12) induced by the laser pulse for each of the selected phonon sets as follows:

Comparison of experimental PDS with theoretical models with varying e-ph coupling parameter (Ge
where Nq denotes total number of the q-points q, Nm is number of phonon modes m in the mode set S (acoustic, opt-A, opt-B), nqm stands for the time dependent phonon population of the mode m at the q-point q and nqm 0 is initial phonon population.Then, the energy accumulated in those phonon-mode sets (Figure 5(b)) is obtained from the phonon spectral dependence: where ℏ  stands for the energy of phonon mode m at the q-point q.

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Figure Description Page S1Pnma crystal structures of layered SnSe in different crystallographic directions.2

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Figure S1.Crystal structure of SnSe in the Pnma phase projected in (a) three dimensions, (b) along the c axis, (b) along the b axis, and (d) along a axis.Color codes: gray (Sn) and green-yellow (Se).The structures were created using the Visualization for Electronic and Structural Analysis (VESTA) software. 1

Figure S2 A
Figure S2 A. Left -Phonon spectrum of SnSe along its high-symmetry points.Phonon bands without and with the non-analytical correction (NAC) are depicted.Right -total phonon density of states shown in shaded grey, and projection to the Sn and Se atoms in red and green lines, respectively.NAC is considered.

Figure 1 (
f) (main text) that analyze the TDS between the Γ(020)-Γ(011) points.FigureS2C show that the total TDS curves with or without NAC are identical and differences in the mode contributions are marginal.This indicates that the modifications of the phonon spectrum, occurring especially for higher optical phonons, are irrelevant for the observed TDS patterns.The mode resolved contributions prove that the dominant contributions are provided by the lowest lying (FigureS2 C) acoustic contribution (FigureS3), where only near the Brillouin zone edge the optical mode contribution can play role as line profiles indicate.The analysis in the main article considers data including the NAC.

Figure
Figure S2 B. TDS patterns comparison for calculation (a) without and (b) with the nonanalytical correction (NAC).

Figure
Figure S2 C. TDS line profile comparison (a) without and (b) with the non-analytical correction (NAC).The regions extracted for line of profiles are indicated by white rectangles in the Figure S2B).

Figure S8 .
Figure S8.Phonon band character as a function of q-position along a path in the [100] zone for (a) without and (b) with the non-analytical correction (NAC).The color of the solid lines denotes the mean value of the phonon eigenvectors' projection to the direction of the q-vector, assuming eigenvectors related to all atoms in the unit cell.

Figure S9 .
Figure S9.Relation of the e-ph and ph-ph energy flows to the phonon modes contributing the total PDS intensity.(a-b) The relative change of the PDS with respect to the PDS at pump probe time delay < 0 ps is depicted.Total PDS intensity and dominant phonon mode contributions are depicted.(e-h) Comparison of the absolute values of the e -ph (colored dashed lines) and ph-ph (colored solid lines) energy flows related to the selected modes in (a-b).

Figure S10 .
Figure S10.Phonon branch dependent lifetime of imaginary part of the Umklapp ph-ph scattering process self-energy along the [100] zone axis.The color bar denotes the phonon lifetime, (a) to (c) acoustic phonons, (d) to (h) Optical phonons.

Figure S11 .
Figure S11.Lattice thermal conductivity tensor.The diagonal elements are depicted.

Figure S12 .
Figure S12.Mode-resolved mean population difference in the [100] zone as a function of time.Acoustic and optical modes are resolved.OPT-1 includes optical branches (O1-O5, O7, O16-19), and OPT-2 includes the remaining optical modes with short phonon lifetimes.