Experimental evidence for the significance of optical phonons in thermal transport of tin monosulfide

The understanding of the lattice dynamics is essential for engineering the thermal transport properties in quantum materials. Based on the canonical point of view, acoustic phonons are believed to be the principal thermal carriers in heat flow. Here, in this work, optical phonons are elucidated to play a pivotal role in determining the lattice thermal conductivity in thermoelectric material SnS by using the state-of-the-art inelastic neutron scattering technique combined with first-principles calculations. Additionally, in contrast to acoustic phonons, optical phonons are observed to exhibit pronounced softening and broadening with temperature. Our observations not only shed light on the significance of the optical phonons in thermal transport but also provide a vital clue to suppress the propagation of optical phonons to optimize the thermoelectric performance of SnS.


Introduction
The search for semiconducting material with extreme thermal transport properties is critical in the realm of modern technological applications. Materials with high thermal conductivity are imperative for the heat dissipation in microelectronic devices. By contrast, low thermal conductivity is relevant to achieving high performance in thermal barrier coatings [1], phase change memories [2] as well as thermoelectric energy conversion [3,4]. In general, heat is involved by electron or lattice vibration (phonon). Nevertheless, in a semiconducting system, heat is mostly transferred via the lattice vibration [5], in which phonon is a key factor determining the thermodynamic properties. For crystalline solids, it is deeply convinced that acoustic phonons are the major heat carriers owing to the low group velocity and short lifetime of optical phonons. In recent years, optical phonons are experimentally and theoretically unraveled to make significant contributions to thermal transport, but such intriguing phenomenon has been only reported in a handful of quantum materials. For instance, Pang et al reported that the longitudinal optical branch contributes nearly one-third to thermal transport in nuclear fuel UO 2 using the inelastic neutron scattering (INS) technique [6]. In thermoelectric materials PbSe [7] and Mg 2 Si [8], the high-energy optical phonons were theoretically estimated to account for 25% and 30% of the lattice thermal conductivity, more surprisingly, the optical modes were even predicted to contribute 75% to heat conduction in the phase change material Ge 2 Sb 2 Te 5 [9]. Thereby, we believe that the exploration of the lattice dynamics in systems where thermal conduction is dominated by optical phonons is of fundamental interest.
The IV-VI semiconductor SnSe with an extraordinary figure of merit of 2.6 has attracted intense attention over last several years because of its intrinsically ultralow lattice thermal conductivity driven by the giant lattice anharmonicity and high thermopower facilitated by the multi-band degeneracy [10][11][12][13][14][15]. SnS as an analog compound of SnSe, however, has long been recognized as a poor thermoelectric material for its low carrier concentration. Lately, a great improvement of the power factor through alkaline metal doping has ignited the research enthusiasm on SnS, while a higher lattice thermal conductivity impedes its thermoelectric performance as appealing as that of SnSe [16][17][18]. Interestingly, optical modes have been recently invoked to play a central role in the thermal transport of SnS based on first-principles calculations [19,20], which might open an avenue to improve its thermoelectric performance. Benefiting from the advances in the neutron scattering technique, time of flight INS can get access to an intuitive recognition about phonon excitations by measuring four-dimensional scattering function. Therefore, we can have a better understanding of the role of optical phonons in heat transport of SnS. Unfortunately, to date, the detailed experimental examination of the contributions from acoustic phonons to the lattice thermal conductivity and the excitation behaviors of intermediate and high-energy optical phonons throughout the Brillouin zone are not yet reported, although Lanigan-Atkins et al have performed INS experiments on low-lying phonons to clarify the anharmonic nature of SnS [21].
In this work, we performed a comprehensive investigation of lattice dynamics of SnS by means of INS measurements together with first-principles calculations. Optical phonons are revealed to contribute 58% to the lattice thermal conductivity along the a-axis at room temperature and is further supported by our high-energy INS experiments manifesting as dispersive phonon spectrum in intermediate and high-energy regions. In contrast to acoustic phonons, optical modes were observed to exhibit more striking softening and broadening characteristics, indicating that the optical modes are responsible for the reduction of the lattice thermal conductivity upon heating.

Sample synthesis
High-quality single crystals of SnS were fabricated by a modified Bridgeman method, as described in detail in a prior study [17]. The synthesized single crystal ingot has a typical size of 10 mm in diameter and 30 mm in length and crystallizes in an orthorhombic Pnma structure below 850 K [22].

INS measurements
Single crystal INS data were attained at 4D space access neutron spectrometer BL01 4SEASONS [23] and cold-neutron disk-chopper spectrometer BL14 AMATERALS [24] of Japan Proton Accelerator Research Complex (J-PARC). For the experiments at AMATERAS, single crystals were aligned with [H,0,L] and [0,K,L] in the scattering planes and multi-incident neutron energies with 16.3 and 7.74 meV were utilized [25]. In the case of [H,0,L] orientation, we performed the experiments at T = 100, 200, 300, 400 and 500 K, and for [0,K,L] orientation, we carried out the experiments at T = 300 K. The energy resolutions at the elastic plane are 0.6 and 0.21 meV for Ei = 16.3, 7.74 meV, respectively. For the experiments at 4SEASONS, we adopted the same single crystal as that of AMATERAS with the crystal was mounted in the [0,K,L] scattering plane and measured at T = 100, 300 and 500 K with E i = 45, 26.6, 17.6 and 12.5 meV. The energy resolutions at the elastic plane are 0.45, 0.66, 0.95 and 2.1 meV for E i = 12.5, 17.6, 26.6 and 45 meV. All data are visualized by using the UTSUSEMI software developed at J-PARC [26]. The phonon linewidths were derived by fitting the one-dimensional cut of the spectra using the DAVE software package [27]. A single crystal with a total weight of 11.4 g was ground into powders and enclosed in the thin-walled aluminum can. The INS experiment on powder sample was conducted at 4SEASONS with E i = 55 and 33.4 meV at T = 100, 200, 300, 400 and 500 K. The empty aluminum can was measured at each temperature to subtract the background originating from aluminum constituting sample environment.

Dynamical structure factor calculations
To understand the vibrational properties of SnS, we performed first-principles phonon calculations using the density functional perturbation theory (DFPT) [28] as implemented in the QUANTUM ESPRESSO code [29,30]. The calculations were performed using the plane-wave pseudopotential method and the generalized gradient approximation for the exchange-correlation functional in the Perdew-Burke-Ernzerhof parametrization [31]. Pseudopotentials and energy cutoffs of plane-wave basis were chosen based on the results of convergence tests provided in the standard solid-state pseudopotential library [32], in which the precision and performance of publicly available pseudopotential libraries are extensively tested to facilitate the optimal choice of pseudopotentials. In the present work, we employed ultra-soft pseudopotentials from the GBRV library [33] with cutoffs of 70 Ry and 560 Ry for the expansion of the wave functions and charge densities, respectively. The Brillouin zone integration was performed over a 12 × 12 × 12 k mesh. Dynamical matrices were computed on 6 × 6 × 6 meshes in q space, which were then interpolated to determine the full phonon dispersion. From the calculated phonon eigenvalues and eigenvectors, we computed the dynamical structure factor S(Q, E) using the coherent one-phonon scattering formula [34,35]. We performed two sets of DFPT phonon calculations using the experimental and fully-relaxed structural parameters. A comparison with the neutron scattering data shows that the phonon dispersion of SnS is better reproduced when the experimental structure is used. We found the same trend in the phonon calculation of iron-based superconductors [36]. If not otherwise stated, we present the results of DFPT phonon calculations obtained using the experimental crystal structure.

Lattice thermal conductivity calculation
The projector augmented plane wave method with local-density approximation for the exchange-correlation functional [37,38] was adopted for the structural relaxation, as implemented in Vienna ab initio simulation package [39]. The cutoff energy for the plane-wave basis and the force convergence were respectively set to 500 eV and 0.0001 eV Å −1 with 6 × 12 × 12 Γ centered K-points grids. The fully relaxed lattice parameters for SnS are a = 10.97 Å, b = 3.95 Å and c = 4.19 Å, which are in good agreement with a recent work [21]. To compute the lattice thermal conductivity, we need to obtain the second and third-order interatomic force constants (IFCs). The second-order IFCs are extracted by PHONOPY software [40] with 2 × 3 × 3, 3 × 3 × 3 and 4 × 4 × 4 supercells respectively. By comparison, we found that second-order IFCs derived from 3 × 3 × 3 supercell is enough to acquire the lattice thermal conductivity. The third-order IFCs were extracted using 2 × 3 × 3 supercell with a truncation radius of 6.12 Å. The lattice thermal conductivity was then calculated by solving the phonon Boltzmann transport equation using ShengBTE Package [41]. While we calculated the dynamical structure factor and the lattice thermal conductivity using different computational conditions, both results agree very well with the experiments. An interesting future direction is to verify whether DFT calculations with different implementations reach the same level of precision, irrespective of the choice of the codes, exchange-correlation functionals, and pseudopotential libraries.

Results and discussion
The thermal resistance is correlated to collisions of the phonon with other phonons (intrinsic scattering processes) or crystalline defects (extrinsic scattering processes). In single-crystalline materials, the finite lattice thermal conductivity is predominantly triggered by the intrinsic phonon scattering processes, namely Umklapp and normal processes. Generally speaking, the normal process will not directly contribute to the thermal resistance and the Umklapp process is regarded as the leading phonon scattering process in heat conduction at high temperatures [42]. Based on the assumption that the optical phonons do not participate in the heat transport, at the temperature where the Umklapp process dominates, the lattice thermal conductivity can be evaluated by the following expression proposed by Slack [43,44]: HereM is the average atomic mass, n is the number of the atoms in the primitive unit-cell, θ a is the Debye temperature of the acoustic phonon, defined as 1 , where LA and TA stand for the longitudinal and transverse acoustic phonon modes vibrating parallel or perpendicular to the direction of the propagation, respectively, δ 3 is the volume per atom, γ is the acoustic phonon mode averaged Grüneisen parameter, and A = 2.43×10 −6 In spite of such model is an empirical model, it has been widely exploited to evaluate the lattice thermal conductivity in various materials [43,45]. Furthermore, the lattice thermal conductivity of SnS decays non-monotonically above room temperature [16][17][18]21], which is characteristic of Umklapp phonon scattering. Accordingly, it is instructive to adopt Slack model to qualitatively judge whether acoustic phonons dominate heat transport of SnS. For SnS, the lattice constants of a, b and c are 11.19, 3.98 and 4.33 Å at T = 300 K [22], γ is reported to possess a value of 2.67 [16] and θ a can be experimentally determined from the phonon spectrum by means of INS measurements.
To begin, we present in figure 1 the experimental phonon dispersion of SnS obtained using the time-of-flight neutron spectrometers BL01 4SEASONS and BL14 AMATERAS of J-PARC. As can be seen, the phonon intensity varies in a complicated way from one Brillouin zone to another. Therefore, careful calculations of the intensity distribution and a clear picture of mode polarizations are necessary; otherwise, it is difficult to identify the mode characters only from the experimental information. To understand the experimental results, we performed first-principles phonon calculations using DFPT. For a head-to-head comparison between theory and experiment, we present the calculated phonon intensity maps on the right  figure 1(c)). The band top of LA mode (6.3 meV) is lower than that of TA mode along [0,2,L] (7.3 meV), which is attributed to the avoided crossing between the LA and low-lying optical mode. As a result, the highest energies for LA and TA modes are 11.2 and 7.5 meV, respectively, yielding a value of 95 K for θ a . Appling the acquired parameters to equation (1), the lattice thermal conductivity is estimated to 0.51 W m −1 K −1 at T = 300 K, which is considerably smaller than the reported value of 2.3 W m −1 K −1 from prior thermal transport measurements [16,17,21]. The contradiction between the Slack model and experiment highlights the necessity for an in-depth investigation of the relative contribution of optical and acoustic modes to the lattice thermal conductivity of SnS. In fact, a direct evaluation of the lattice thermal conductivity stemming from the optical phonons is a daunting task due to the complication of obtaining the lifetime and dispersions of all-optical phonons in the entire Brillouin zone, but we can quantitatively compute the acoustic phonon contributions via carrying out high-resolution INS experiment on single crystalline specimen. According to the Boltzmann transport theory under the relaxation time approximation, the lattice thermal conductivity of a specific phonon branch can be calculated by the following equation [46,47]: For simplicity, the cuboid approximation is assumed, and equation (2) could be rewritten as: Here, v j (q) and τ j (q) denote the group velocity and lifetime of the specific phonon mode at wave-vector q, respectively, and the mode heat capacity c j (q) is defined as: Here in this work, we will focus on evaluating the contribution of the acoustic phonons to the lattice thermal conductivity along the a-axis, because it is elusive to accurately determine the linewidth of the LA modes along [0,K,0] and [0,0,L] due to the steepness of those branches. As plotted in figures 2(a)-(c), the phonon dispersions are fitted by the second-order polynomial function, and the group velocity can be derived from the gradient of the fitted polynomial function. The phonon lifetime can be achieved from the measured linewidth by means of the energy-time uncertainty relation τ = ℏ/FWHM [48][49][50], where FWHM is the phonon linewidth (full width at half-maximum in the energy of the phonon peak). Owing to the high resolution of the instrument, it enables us to acquire the linewidth of the acoustic phonons using the Lorentzian function convoluted with the instrument resolution. In figures 2(d)-(f), the linewidth of the To further elucidate the role of the optical phonon in heat transport of SnS, we calculated the lattice thermal conductivity as a function of temperature. It is worth noting that we attempted to obtain the lattice thermal conductivity based on the experimental structure, unfortunately, we failed to achieve accurate values. Consequently, we calculated the lattice thermal conductivity using the fully relaxed structure. Figure 2(g) shows the calculated lattice thermal conductivity with respect to the temperature along the a-axis (see figure S1 of the supplementary material for b and c axes). In comparison, the experimental lattice thermal conductivity from previous studies are also plotted [16,21]. Obviously, our calculated data reproduces the experiment one well. Specifically, the calculated lattice thermal conductivity at T = 300 K along the a-axis is 1.7 W m −1 K −1 , being excellent consistence with experimental data with 1.6 W m −1 K −1 along the same direction [16,21]. Figure 2(h) displays the individual contributions of optical and acoustic modes to the lattice thermal conductivity at various temperatures. At T = 300 K, the optical phonon contributes approximately 50% to the lattice thermal conductivity along the a-axis, which is in a reasonable agreement with the value derived from our INS experiments. Despite that our calculation shows a good consistence with prior theoretical studies concerning both the group velocity and lifetime [20,21] (see figure  S2 of the supplementary material), there still have some discrepancies between the experiment and calculations. For instance, the calculated phonon velocity for LA mode is 3300 m s −1 along Γ-X direction, but it is smaller than our experimental data with about 4700 m s −1 for the same mode. On the other hand, our calculated lifetime is evidently larger than the experimental ones. As proof, the calculated linewidth of the LA mode at q = 0.13 Å −1 is around one order of magnitude smaller than that of the experimental one (see figure S3 of the supplementary material). Such discrepancy reflects the significance of the present work that we shall experimentally determine the individual contribution of the phonon mode.
To gain more insights into the significance of optical phonons, their band characters are explored using E i = 26.6 (figures 3(a)-(c)) and 45 meV (figures 3(d)-(f)) at 4SEASONS. Here, we defined three different phonon energy regions as low (<10 meV), intermediate (10-20 meV) and high (>20 meV). To identify the nature of the intermediate and high-energy optical modes, as shown in figure 3, theoretically calculated dynamical structure factors are compared to the experimentally observed phonon excitations. The calculated optical phonon dispersions are in impressive agreement with our experiment data, especially the sharp characteristics of the optical modes are well captured by our calculation. A noticeable feature shown in figure 3 is that the low INS intensity is observed in the high-energy region, which is due to the fact that high-energy optical modes are dominated by the collective vibrations of the lighter sulfur atoms [19] associated with a smaller coherent neutron scattering length compared to that of Sn atom. As plotted in figures 3(b)-(c), the low-lying and intermediate optical phonons disperse as steep as those of acoustic phonons with energies extending up to 16 and 14 along [0,K,2] and [0,0,L] , respectively. With moving toward higher energy, it can be identified from figure 3(e) that a sizable gap from ∼16 to 20 meV, leading to a saturated cumulative lattice thermal conductivity among this energy range shown in figure S1 of the supplementary material. Indeed, optical phonons with flat dispersion are expected to be observed in the high-energy region. Strikingly, above the gap, a well-defined dispersion is discernible with energy spanning from 20 to 27 meV (see figures 3(d)-(f)). Accordingly, the increment of the cumulative lattice thermal conductivity at high-energy shown in figure S1 of the supplementary material should be ascribed to the large group velocity of some particular optical phonon modes.
As a rule of thumb, an appropriate approach to suppress the lattice thermal conductivity is to enhance the phonon scattering by introducing extrinsic phonon scattering sources including point defects, dislocations, and grain boundaries [51][52][53][54][55][56]. It is noteworthy that these scatter sources take effect in different energy ranges. The grain boundary has been shown to scatter the low-energy acoustic phonons, the dislocations will scatter the mid-energy phonons, and the point defect is thought to be effective in scattering the high-energy optical phonons [53][54][55]. In general, grain boundaries are frequently absent in single-crystal materials, and dislocations are difficult to achieve owing to the deformation during the plastic bending of the materials [56]. According to aforementioned results, high-energy optical phonons are believed to be crucial in heat conduction of SnS, and they are primarily dominated by the vibration of sulfur atoms [19]. Consequently, introducing point defects, such as substituting sulfur ions should be an efficient and convenient way to scatter high-frequency optical phonons to minimize the lattice thermal conductivity of single-crystalline SnS. Prior experimental studies on single crystalline SnS show that the lattice thermal conductivity will not be apparently reduced through mild sodium [17] or thallium doping [57] to substitute tin ions. By contrast, He et al reported that the lattice thermal conductivity of SnS reduces from 3.0 to 1.7 W m −1 K −1 at T = 300 K by 9% Se substitution [18], which should be interpreted in terms of the enhanced phonon scattering of some particular modes of optical phonons [58] as well as the reduction of the velocity of optical phonons via replacing a heavier atom [18]. Those results further support our standpoint that optical phonons are of paramount importance to the heat conduction of SnS.
Considering that the lattice thermal conductivity decreases at elevated temperature, we thus turn to the inspection of how optical and acoustic phonons change with increasing the temperature. As presented in figures 4(a)-(c), it is evidenced that the fitted phonon dispersions for LA and TA modes along Γ-X direction do not show drastic softening characteristics when the temperature varies from 100 to 500 K. For instance, at q = 0.13 Å −1 , these fits reveal that the group velocities of LA mode along [H,0,0] and TA mode along [H,0,2] vary from 3264 and 2143 m s −1 at T = 100 K to 3214 and 2015 m s −1 at T = 500 K. In particular, the phonon dispersion curve of the TA mode along [H,2,0] at T = 100 K is nearly identical to that of T = 500 K. Figures 4(d)-(f) display the temperature dependence of the phonon linewidths. From the canonical point of view, the phonon linewidth generally broadens with increasing the temperature due to the enhanced phonon scattering. The averaged linewidth of TA mode along [H,0,2] increases from 0.25 to 0.45 meV when the temperature is raised from 100 to 500 K. In the case of the TA mode along [H,2,0], the linewidth at T = 500 K shows strong momentum dependence, leading to the linewidth alters from 0.41 to 0.94 meV as temperature increases from 100 to 500 K at the zone boundary. On the contrary, the linewidths of LA mode are essentially independent of the temperature, which indicates that LA mode is not the major factor of the decline of lattice thermal conductivity of SnS along the a-axis among this measured temperature range. Figures 5(a) and (b) depict the temperature dependence of the zone-centered optical phonons on single crystals. In contrast to a weak softening tendency of those acoustic phonons shown in figure 4, the low-lying and intermediate optical phonons are found to shift toward lower energy by approximately 0.5 meV as temperature varies from 100 to 500 K. Moreover, the high-energy optical modes at T = 500 K exhibit a considerable softening of 1.3 meV with respect to that at T = 100 K, as plotted in figure 5(c). To further investigate the phonon softening behavior of optical modes throughout the entire Brillouin zone, we have performed the temperature dependence INS experiments on an 11.4 g powder sample of SnS. Figure 5(d) plots the momentum-integrated neutron scattering intensity as a function of energy transfer over 1.5 Å ⩽ Q ⩽ 6.5 Å by using E i = 33.4 meV. We note here that the decline of the phonon intensity with increasing the temperature is related to the reduction of the Debye-Waller factor caused by the enhanced thermal atomic motion. At T = 100 K, two main peaks centered at 7.2 and 12.6 meV are resolved, respectively. The first peak contains primarily low-lying optical modes, and the second peak corresponds to the intermediate optical phonons. Given that the Bose factor is prone to diverge when energy approaches  elastic line, the peak centered at 0.6 meV is an artifact induced by dividing the Bose factor. Notably, the optical phonon peaks exhibit evident softening characteristics. As depicted in figure 5(e), the behavior of high-energy phonons was analyzed by integrating the powder averaged data over 2 Å ⩽ Q ⩽ 8 Å using E i = 55 meV. The optical mode originated peak centered around 26 meV is observed to have pronounced phonon softening. To quantify the phonon softening of the optical modes, the peak energies from Gaussian fittings are plotted in figure 5(f). It can be noticed that low-lying and intermediate and high energy optical phonons respectively shift by 0.30, 0.32 and 1.45 meV, as temperature increases from 100 to 500 K (see figure 5(f)). Besides, optical phonon modes are found to exhibit more evident broadening than those low-lying acoustic phonons (see figures 5(d) and (e)). These results signify that optical phonons host stronger reduction in both lifetime and velocity, and simultaneously demonstrate that the optical phonons make a substantial contribution to heat conduction of SnS.

Conclusions
In summary, the lattice thermal conductivity of SnS derived from the Slack model shows a large deviation from the experimental data. Undoubtedly, the optical modes should be considered to account for this discrepancy in SnS, and our INS results suggest that acoustic phonons take part in 42% of the heat conduction along the a-axis. Furthermore, we performed high-energy INS measurements to elucidate the thermodynamic behavior of the optical modes. The slopes of low-energy optical phonons are comparable to those of acoustic phonons. Most strikingly, the optical phonons above 20 meV are found to be dispersive. Besides, we have also examined how phonons evolve as a function of temperature. In contrast to acoustic modes, optical modes are found to show very strong softening and broadening, especially at high energy regions, which is suggestive of the importance of optical modes in the reduction of the lattice thermal conductivity as temperature increases. This work presents unambiguous experimental evidence that optical modes are indispensable in determining the lattice thermal conduction in SnS, which is beyond the conventional cognition that heat conduction is dominated by the acoustic phonons. Our findings also offer prospective ways to further suppress the thermal conductivity by lowering the velocity or enhancing the scattering rates of optical phonons, which is beneficial for further optimizing the high thermoelectric performance in SnS.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.