Strong lattice anharmonicity exhibited by the high-energy optical phonons in thermoelectric material

High-quality doped and parent SnSe single crystals were synthesized by modified Bridgeman method, as described in previous studies [11, 12]. High-resolution time-of-flight neutron powder diffraction experiments were carried out at BL-09 SPICA of J-PARC. Single crystals of Na_0.003Sn_0.997Se_0.9S_0.1 with the weight of 2.5 g was grounded into powders and sealed into a V-Ni sample cell with the temperatures varied from 15 K to 410 K. Rietveld refinements of the neutron powder diffraction (NPD) data were conducted with the Z-Rietveld software [20]. Time-of-flight neutron total scattering measurements were carried out at the BL-21 NOVA of J-PARC. Single crystals of both Na_0.003Sn_0.997Se_0.9S_0.1 and Na_0.02Sn_0.98Se were ground into powders with the mass of 2.8 and 3.2 g, respectively. The samples were sealed into the V-Ni cell with a metallic O-ring, and the temperature varied from 50 to 600 K. Neutron PDF, G(r), was obtained by Fourier-transforming the structure function, S(Q) with the maximum Q value of 40 Å⁻¹. Such a large Q value corresponds to the d spacing of 0.16 Å, which is enough to acquire a high-resolution PDF. The PDF patterns were refined using the program PDFGUI [21]. High-resolution neutron scattering experiments were performed at the cold neutron time-of-flight chopper spectrometer BL-14 AMATERAS of J-PARC [22]. The single crystal of Na_0.003Sn_0.997Se_0.9S_0.1 with a total amount of 4.8 g was mounted onto the aluminum plate. The measurements were conducted in the high temperature stick for 100, 300, 450 and 600 K, using the incident beam energies of 7.74 and 15.16 meV. The consistency of the data at T = 300 K for the heating and cooling processes suggests that the sample degradation did not occur during the measurement. The data were reduced and visualized by using Utsusemi suite [23]. Raman scattering experiments were carried out under various temperatures using the LabRam Horiba HR800 UV spectrometer. The laser excitation with the wavelength of 532 nm was used for the measurement in the range of 20 to 200 cm⁻¹. The instrumental resolution was set to be better than 1 cm⁻¹.

The combination of ab initio simulation package (VASP) [24] and PHONOPY [25] was employed to calculate the frozen-phonon potentials. Starting from the experimental results, the fully-relaxed crystal structure was obtained by using the density functional theory as implemented in VASP with the plane-wave energy cutoff set to be 500 eV. The core electrons were described by the projector augmented-wave pseudopotentials [26], and the exchange function was chosen to be the local density approximation [27]. A 6 × 12 × 12 Monkhorst-Pack k-point mesh was adopted for unit-cell calculations. The crystal structure was relaxed until the Hellmann–Feynman force on each atom was smaller than 0.001 eV Å⁻¹. The relaxed lattice parameters, a = 11.304 Å, b = 4.121 Å and c = 4.296 Å, are well consistent with previous report [28]. Then a 2 × 4 × 4 supercell and the Γ-only K point were used for phonon-spectrum calculation, during which the force constants were generated based on finite-displacement method. Finally, the frozen-phonon potentials at Γ are obtained.

In our studies, we performed systematic measurements on single crystals of Na_0.003Sn_0.997Se_0.9S_0.1. SnSe and Na_0.02Sn_0.98Se were also studied to verify our findings in Na_0.003Sn_0.997Se_0.9S_0.1. The crystal structure of Na_0.003Sn_0.997Se_0.9S_0.1 with Pnma space group is depicted in figure 1(a). It is analogous to that of black phosphorus [29], possessing a puckered layered structure with Sn (Na)–Se (S) bi-layers stacking along the a-axis. The nearest Sn (Na)–Se (S) covalent bonds form zigzag and armchair atomic chains along the b and c axes, respectively, leading to anisotropic electronic structures and transport properties [5]. Inelastic neutron scattering is the most effective probe to visualize the phonon dynamics in 3D crystalline solids. Figure 1(b) shows phonon dispersions along the Γ-X, Γ-Y and Γ-Z directions at T = 100 K taken from Na_0.003Sn_0.997Se_0.9S_0.1. Two low-lying optical modes, labelled as TO1 and TO2, can be easily identified around the zone center Γ. In the Γ-X direction, the band crossing occurs between the two low-lying optical modes and acoustic phonon modes at the zone boundary. Compared to the phonon dispersions at T = 100 K, one can notice evident softening of both optical and acoustic phonons at T = 600 K (see figure 1(c)). Figure 1(d) displays the temperature-dependent of the energies of TO1 and TO2 at Q = (0, 1, 1). It is clearly shown that TO1 and TO2 exhibit a softening of 0.85 meV and 0.60 meV from 100 K to 600 K, respectively.

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Generally speaking, the increase of the phonon linewidth (or the scattering rate) is due to phonon–phonon scattering or electron–phonon coupling [30]. In the case of SnSe, first-principles calculations suggest that the scattering rate contributed by the phonon–phonon scattering is much more dominant [31], and the strength of electron–phonon coupling is considerably smaller than those evident electron–phonon coupling systems [32], for instance, cuprates [33] and manganites [34]. Moreover, previous angle-resolved photoemission spectroscopy measurements have not observed characteristic features of electron–phonon coupling, such as 'polaron satellite' or significant enhancement of the carrier effective mass [10, 35–39]. Therefore, we argue that the phonon–phonon scattering takes the dominant responsibility for the temperature dependence of the phonon linewidths. As shown in figure 1(e), the phonon linewidths of TO1 and TO2 increase by 0.28 meV and 0.15 meV from 100 K to 600 K, respectively. Interestingly, as will be shown later, a stronger linewidth broadening occurs in high-energy phonon modes, indicating that the phonon–phonon anharmonic scattering is crucial for the behavior of these modes.

Sn/Na form bonds with seven neighboring Se/S atoms, which are labelled as d₁, d₂, d₃, d₄, d₅, d₆ and d₇ (see figure 2(a)). Such a configuration gives rise to a highly distorted polyhedron structure surrounding Sn/Na site. To investigate how the atomic bonding length evolves as a function of temperature, we performed NPD measurements on Na_0.003Sn_0.997Se_0.9S_0.1 using two neutron diffraction instruments [see figure S3 (https://stacks.iop.org/NJP/22/083083/mmedia)], and the refinement results are illustrated in figure 2(b). We found that these Sn/Na–Se/S bonds show different temperature dependences. With temperature increasing, the variation of the nearest Sn/Na–Se/S bond d₁ is very weak, d₂, d₄, d₆ and d₇ increase mildly at high temperatures, d₃ monotonically shrinks, and d₅ displays a non-monotonic trend and becomes shorter at high temperatures. The detailed neutron diffraction patterns and the temperature dependence of lattice parameters are presented in figures S3 and S4, respectively.

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Figure 2(c) illustrates the temperature dependence of the isotropic atomic displacement parameters (ADPs) for Sn/Na and Se/S. In the Debye model, we use the following formula to fit the temperature-dependent ADPs [40, 41]:

$$\begin{equation}{U}_{\text{iso}}=\frac{3{\hslash }^{2}{T}^{2}}{m{k}_{\text{B}}{{\theta }_{\text{D}}}^{3}}{\int }_{0}^{T/{\theta }_{\text{D}}}\frac{x}{{\mathrm{e}}^{x}-1}\mathrm{d}x+\frac{3{\hslash }^{2}}{4\mathit{\text{m}}{k}_{\text{B}}{\theta }_{\text{D}}}+A\end{equation} \tag{ 3 }$$

where ℏ, k_B, θ_D, m and A are the reduced Planck constant, Boltzmann's constant, Debye temperature, average atomic mass and temperature-independent static disorder, respectively. The fitting result of A is ∼0 for both Sn/Na and Se/S, suggesting that we can ignore the static disorder in our samples. The Debye temperatures were determined to be 146(6) K and 204(9) K for Sn/Na and Se/S, respectively, being in excellent agreement with the values of 140 K for Sn and 195 K for Se in pristine SnSe [41].

In sharp contrast to the routine powder diffraction that provides average crystallography information, PDF analysis is a powerful technique to investigate local structures in crystals, where the peaks observed in the PDF spectra correspond to specific inter atomic distances [42]. Figure 2(d) shows the temperature dependence of the PDF of Na_0.003Sn_0.997Se_0.9S_0.1 over a wide range from 2 to 20 Å, in which the Pnma space group was adopted for the fitting. The good agreement between the model and the experimental data suggests that the doping of Na and S does not change the local structures of SnSe. As displayed in figure 2(b), at low temperatures, the bond length d₁ is very close to d₂, and correspondingly the first PDF peak at T = 50 K contains the contributions from both of the nearest and next nearest Sn/Na–Se/S pairs. Though d₄ is larger than d₃, the second PDF peak arise from the third and fourth nearest Sn/Na–Se/S pairs. And the PDF peaks centered ∼3.9 and 4.1 Å correspond to the nearest Se/S–Se/S pair and d₅, respectively. As for d₆ and d₇, they contribute to the PDF peak centered around 4.9 Å. Compared to the low-temperature PDF spectrum, the peaks broadened and asymmetric with the decline of peak intensity at high temperature. Such behavior has also been observed in Na_0.02Sn_0.98Se (see figure S5).

At low temperatures, the potential energy curve can be described as a parabola near the bottom of the well, which gives rise to the symmetric Gaussian profile of the PDF spectrum at low temperatures. With further increasing the temperatures, the potential energy curve follows the anharmonic model with a steeper slope on the left side with respect to the equilibrium position, which gives rise to the asymmetric profile of the line shape at high temperatures. In thermoelectric materials [43–46] and perovskite halides [47, 48] with strong anharmonicity, their PDF peaks exhibit non-Gaussian profiles at high temperatures. Furthermore, Li et al proposed that the asymmetry of the PDF peak could be regarded as a reference to evaluate the lattice anharmonicity [46]. Here, as shown in figure 2(e), the profile of the first PDF peak follows the Gaussian function at T = 50 K, while it becomes asymmetry at T = 300 K, though d₁ and d₂ remain almost the same in this temperature range. However, as shown in figure 2(d), we could notice that the neighboring peak with multi-components exhibits drastic changes in both peak intensity and shape as the temperature increases. Therefore, it is difficult to exclude the influence of the neighboring peak on the first PDF peak. As will be shown later, the evidence of anharmonicity could be found in following sections.

Considering that one-unit cell of SnSe comprises eight atoms, there are three acoustic (B_1u + B_2u + B_3u) and 21 optical phonon vibration modes (4A_g + 2A_u + 2B_1g + 3B_1u + 4B_2g + B_2u + 2B_3g+ 3B_3u) as predicted by group theory. Among them, 12 optical modes (4A_g + 2B_1g + 4B_2g + 2B_3g) are Raman active. Figure 3 illustrates the Raman spectra of Na_0.003Sn_0.997Se_0.9S_0.1 taken at various temperatures from 80 to 450 K. The Raman modes A_g ¹ (TO1 in figure 1), A_g ², B_3g, A_g ³ and A_g ⁴ can be clearly identified at T = 300 K, with the peak positions at 32.72, 71.33, 111.06, 132.78 and 150.06 cm⁻¹, respectively, being in excellent agreement with previously reports [38]. The missing of the TO2 mode in Raman spectra could be due to the weak intensity of the TO2 mode in our experimental configuration. With the increase of temperature, we observed a pronounced phonon softening. The low-lying optical phonon modes derived from INS are plotted as well in figures 3(b) and (c), which are in good consistent with Raman measurements.

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To understand the physical origin of the temperature dependence of the Raman shift, we adopted the model ω(T) = ω₀ + Δω_E (T) + Δω_A (T) to fit the Raman data [16--18], where ω₀ is the harmonic frequency at T = 0 K, Δω_E (T) corresponds to the shift of phonon frequencies caused by the quasiharmonic lattice thermal expansion, Δω_A (T) represent the shift induced by anharmonic multi-phonon scattering. Here, the shift induced by the lattice thermal-expansion could be expressed as [16, 18]:

$$\begin{equation}{\Delta}{\omega }_{E}={\omega }_{0}\left(\mathrm{exp}\left(-\gamma {\int }_{0}^{T}\alpha \left(T\right)\mathrm{d}T\right)-1\right).\end{equation} \tag{ 4 }$$

The isotherm mode Grüneisen parameters of A_g ², B_3g, A_g ³ and A_g ⁴ are 1.22, −0.24, −0.31 and 1.11, respectively [19]. Since the isotherm mode Grüneisen parameter of A_g ¹ is not reported in literatures, the calculated value of 2.83 was adopted [5]. For the anharmonic phonon–phonon scattering, both the three and four-phonon scattering processes were taken into consideration [49],

$$\begin{equation}{\Delta}{\omega }_{A}=A\left[1+\frac{2}{{\mathrm{e}}^{x}-1}\right]+B\left[1+\frac{3}{{\mathrm{e}}^{y}-1}+\frac{3}{{\left({\mathrm{e}}^{y}-1\right)}^{2}}\right]\end{equation} \tag{ 5 }$$

where $x=\frac{\hslash {\omega }_{0}}{2{k}_{\text{B}}T}$ and $y=\frac{\hslash {\omega }_{0}}{3{k}_{\text{B}}T}$. The coefficients A and B represent the individual contributions of three and four-phonon anharmonic scattering, respectively. Figures 3(b)–(f) show the fittings of the optical phonon modes in Raman spectra. Obviously, both the thermal-expansion effect and the intrinsic anharmonic scattering contribute to the phonon softening of A_g ¹, A_g ² and A_g ⁴ modes. However, as shown in figures 3(d) and (e), the thermal-expansion effect is relatively weak and the anharmonicity effect plays the key role for the considerable softening of the B_3g and A_g ³ modes.

Moreover, the linewidth is also helpful for the evaluation of the temperature-dependent behavior. In figures 3(g) and (h), the linewidths of the A_g ¹ and A_g ² modes show only slight changes with temperature in both Raman and INS experiments. Such behavior has also been observed in parent SnSe and Na_0.02Sn_0.98Se in our studies (see figure 5), which suggests the weak anharmonicity effect on these modes. In figure 3(i)–(k), the temperature dependence of the linewidths of B_3g, A_g ³ and A_g ⁴ modes are modeled with the following formula [49]:

$$\begin{equation}{\Gamma}={\Gamma}\left(0\right)+C\left[1+\frac{2}{{\mathrm{e}}^{x}-1}\right]+D\left[1+\frac{3}{{\mathrm{e}}^{y}-1}+\frac{3}{{\left({\mathrm{e}}^{y}-1\right)}^{2}}\right]\end{equation} \tag{ 6 }$$

where Γ(0) is the linewidth at T = 0 K. The detailed fitting parameters are summarized in table S1. In previous studies, three-phonon scattering was considered as the dominate phonon–phonon scattering process in SnSe system [50]. Based on our results, we found that the four-phonon process (black lines in figures 3(i) and (k)) is stronger than the three-phonon one (cyan lines in figures 3(i) and (k)) in B_3g and A_g ⁴ modes, indicating that the four-phonon scattering process is also crucial in determining the intrinsic low lattice thermal conductivity of SnSe.

In order to acquire more information on the anharmonicity effect of the phonon modes in SnSe, we performed first-principles calculations to obtain the frozen-phonon potentials for Raman modes, and the results are illustrated in figure 4. Though the potentials of the A_g ¹ and A_g ² modes can be well fitted with high order polynomials, their deviation from the quadratic potential is not significant. We thus expect a weak anharmoncity effect in A_g ¹ and A_g ², which is in line with our experimental results shown in figure 3. In contrast, B_3g, A_g ³ and A_g ⁴ modes largely deviate from the quadratic potential and higher order potentials have to be included to achieve a good fitting, suggesting they possess strong anharmonicity. In particular, the quartic potential is crucial to account for the phonon potentials of B_3g and A_g ⁴ modes.

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To obtain a better understanding of how the doping affects the lattice thermal conductivity, it is crucial to reveal the difference of linewidths of optical modes between pristine and doped SnSe. As shown in figure 5, compared to the pristine SnSe, the linewidths of Na-doped specimen remain basically the same, which suggests that Na dopant has very weak effect on the lattice thermal conductivity [6]. However, in contrast to pristine and Na-doped SnSe, our data in figures 5(b), (d) and (e) show that the linewidths of A_g ², A_g ³ and A_g ⁴ in Na_0.003Sn_0.997Se_0.9S_0.1 are evidently larger at low temperatures. With further increasing the temperature, the differences of the linewidths of A_g ² and A_g ³ between pristine SnSe and Na_0.003Sn_0.997Se_0.9S_0.1 become smaller. Such behavior is in line with the thermal transport measurement [12]. Our data indicates that mild sulfur doping could affect some particular modes of optical phonons and contribute to the reduction of the lattice thermal conductivity at low temperatures. In addition, though those high-energy optical phonon modes of pristine and Na doped SnSe show similar temperature dependence to those of Na_0.003Sn_0.997Se_0.9S_0.1, we found that the four-phonon process becomes the dominate phonon–phonon scattering process in A_g ³ at high temperatures (see figures S8 and S9), which further indicates that the four-phonon scattering process is important in the thermal conduction of SnSe system.

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