Pressure induced enhancement of anharmonicity and optimization of thermoelectric properties in n-type SnS

Abstract

Pressure serves as a powerful approach to regulating the thermal conductivity of materials. By applying pressure, one can alter the lattice symmetry, atomic spacing, and phonon scattering mechanisms, thereby exerting a profound influence on thermal transport properties. SnS, sharing the same crystal structure as SnSe, has often been overlooked due to its higher lattice thermal conductivity. While extensive efforts have been dedicated to enhancing the power factor of SnS through doping, its thermal transport properties remain underexplored, limiting its potential as a thermoelectric material. In this study, we investigated the impact of pressure modulation on the thermoelectric performance of SnS. Remarkably, the application of negative pressure significantly enhanced its thermal transport characteristics, leading to a reduction in the lattice thermal conductivity (${\kappa }_L$) along the $a$ axis to 0.23 W/(m·K) at 800 K, on par with that of SnSe. Despite the negligible improvement in carrier mobility under negative pressure, the electronic transport properties were preserved within an acceptable range. Most notably, a maximum ZT value of 2.7 was achieved along the $a$ axis at 800 K, marking a substantial advancement in the thermoelectric performance of n-type SnS.

Graphical abstract

Keywords

hydrostatic pressure / thermoelectric materials / n-type SnS / anharmonicity / lattice thermal conductivity

1 Introduction

With the intensification of the energy crisis, thermoelectric materials (TE) have received widespread attention which can directly convert thermal energy into electrical energy without generating unusable heat, effectively improving energy utilization efficiency [1-3]. The thermoelectric conversion efficiency is mainly judged by the figure of merit $(ZT)$, which is evaluated as $ZT=S^2\sigma T/({\kappa }_e+{\kappa }_L)$, where $S$, $\sigma$, ${\kappa }_e$, ${\kappa }_L$, and $T$ are the Seebeck coefficient, electrical conductivity, electrical thermal conductivity, lattice thermal conductivity, and Kelvin temperature, respectively [4-7]. With the phonon-glass electron single crystal (PGEC) concept, we can clearly see that the ideal thermoelectric material should have the phonon transport characteristics like glass, and at the same time have crystal-like electron transport characteristics [8-10], in other words, it needs to have a high $PF$ and a low $\kappa$. To enhance the $ZT$ value of a material, several strategies can be employed, including (i) resonant doping [11, 12]; (ii) band structural engineering and defect engineering [13, 14]; (iii) forming composite materials [15, 16]. In addition, thermoelectric properties of a material also can be changed when different pressure conditions are applied [17-19]. For example, according previous research for CoP${}_3$, under hydrostatic pressures, the newly predicted $Pnma$ phase of CoP${}_3$ has a super high Seebeck coefficient that is 6.5 times that of the ambient $Im$−3 phase [20]. And for Bi${}_2$Te${}_3$, by applying negative pressure, the Bi${}_2$Te${}_3$ exhibits a power factor of 1.32 mW·m⁻¹·K${}^{-2}$ and a $ZT$ of 0.76 at 300 K [21].

In recent years, the thermoelectric properties of IV−VI compounds have attracted much attention due to their unique structures, such as PbTe [22, 23], GeTe [24], SnTe [25], and SnSe [26]. Among them, SnSe is a layered structure with weak interlayer interactions, resulting in a very low interlayer lattice thermal conductivity, which ultimately enables it to achieve high thermoelectric properties [27]. Considering the isoelectronic nature, similar crystal structures and comparable vibrational properties of SnS, along with the higher abundance of S compared to Se, SnS holds significant potential and research value as a thermoelectric material. In comparison to SnSe, SnS displays suboptimal thermoelectric properties [28]. Therefore, researchers have resorted to doping as a method to improve the $ZT$ value. For instance, Na doping in SnS has improved the electrical conductivity and Seebeck coefficient of p-type SnS, leading to a $ZT$ value of 0.75 at 873 K [29]. Furthermore, Cu doped SnS exhibits an enhanced $ZT$ within the temperature range of 300−500 K [30] and Br doped SnS achieves a $ZT$ of 0.4 at room temperature and a conversion efficiency of 5% at elevated temperatures [31]. While doping has achieved a modest enhancement of the $ZT$ in SnS, its performance remains inferior to other high-performance thermoelectric materials. To further improve thermoelectric performance, it is necessary to reduce the lattice thermal conductivity while maintaining its electrical transport properties. Therefore, we investigated hydrostatic pressure as a potential strategy to enhance the thermoelectric properties of SnS.

In this work, we combined semi-classical Boltzmann transport theory with the application of density functional theory to study the thermoelectric properties of SnS. We found that, due to the reduction of phonon group velocity and enhancement of anharmonicity, the lattice thermal conductivity along the $a$ axis decreased to 0.23 W/(mK) at 800 K after the application of negative pressure. Finally, due to the strong anisotropy of SnS, the $ZT$ values increased by 11.2%, 30%, and 16.7% along the $a$, $b$, and $c$ axes, respectively. Moreover, the maximum value of $ZT$ for n-type SnS reaches 2.7 at elevated temperatures, exceeding the reported peak $ZT$ value of 2.6 for SnSe [32]. Our research results indicate that the application of pressure effectively achieves a significant improvement in thermoelectric performance, opening up a new avenue for improving the thermoelectric properties of SnS.

2 Computational method

We used the projector augmented-wave (PAW) method implemented in the Vienna ab initio simulation package (VASP) to perform all the density functional theory (DFT) calculations in the following sections [33-35]. We choose a k-point grid of 6 × 12 × 12 with the Monkhorst Pack scheme and set a high cutoff energy of 500 eV for the plane-wave basis. The energy convergence and force convergence standards of ${10}^{-8}$ eV and ${10}^{-4}$ eV/Å were tested and adopted to ensure the accuracy and stability of the calculation results. The Perdew−Burke−Ernzerhof (PBE) of the generalized gradient approximation (GGA) method was applied to describe the exchange-correlation effects, which is very common for the prediction of electronic properties [36, 37]. Moreover, considering the layered structure of SnS and van der Waals interactions between its layers, van der Waals corrections were incorporated into the property calculations, improving the reliability of the results [38].

The lattice thermal conductivity ${\kappa }_L$ was calculated by using the phonon Boltzmann transport equation [39] which was expressed as

(1)${\kappa }_L={\displaystyle \frac{1}{3V{N}_q}}{\displaystyle \sum _{q\gamma }{C}_{q\gamma }{v}_{q\gamma }^2{\tau }_{q\gamma },}$

where V is the volume of the unit cell, $\gamma$ is phonon mode index, $q$ is wave vector, $C_{q\gamma }$ is the heat capacity, $N_q$ is the number of discrete $k$-points, ${\tau }_{q\gamma }$ is the phonon relaxation time and $v_{q\gamma }$ is the group velocity. In the study of the thermal transport characteristics of SnS, we applied the “frozen phonon” technique, integrated within Phonopy [40], to create a set of 2 × 2 × 3 supercells with atomic displacements. This approach allowed us to derive the harmonic properties, such as the second-order interatomic forces (second-order IFCs), phonon dispersion relations, and the eigenvectors corresponding to each vibrational mode. We employed the thirdorder.py script to construct a series of 2 × 2 × 3 SnS supercells, from which we obtained the anharmonic third-order interatomic force constants (IFCs). Subsequently, these third-order IFCs, along with the previously calculated second-order IFCs, were utilized as inputs for the ShengBTE code. This software iteratively solves the Peierls-Boltzmann transport equation to determine the lattice thermal conductivity [41].

In the analysis of electrical transport properties, we utilized the AMSET code [42, 43] to compute the carrier lifetime — the inverse of the scattering rate — by applying Fermi’s golden rule. This rule formulates the scattering rate as follows:

(2)${\tau }_{a\mathit{k}\to b\mathit{k}+\mathit{q}}^{-1}={\displaystyle \frac{2\pi }{\hslash }}⟨b\mathit{k}+\mathit{q}\left|g_{ab}\right|a\mathit{k}{⟩}^2\delta \left(E_{a\mathit{k}}-E_{b\mathit{k}+\mathit{q}}\right),$

where $a\mathit{k}$ is the initial state of the electron, $b\mathit{k}+\mathit{q}$ is the final state, $\delta$ is the Dirac function, $g_{ab}$ represents the electron−phonon coupling matrix elements for different scattering mechanisms, including acoustic phonon deformation potential (ADP) scattering, the polar optical phonon (POP) scattering, and the ionized impurity (IMP) scattering,

(3)$g^{ADP}(k,q)={\left[{\displaystyle \frac{k_{B}T{\epsilon }_d^2}{B}}\right]}^{\left(\frac{1}{2}\right)}⟨{\psi }_{nk}|{\psi }_{mk+q}⟩,$

(4)$g^{IMP}(k,q)=\left[{\displaystyle \frac{e^2{n}_{ii}}{{\epsilon }_s^2}}\right]{\displaystyle \frac{⟨{\psi }_{nk}|{\psi }_{mk+q}⟩}{|q{|}^2+{\beta }^2}},$

(5)$g^{POP}(k,q)=\left[{\displaystyle \frac{\hslash {\omega }_{po}}{2}}\left({\displaystyle \frac{1}{{\epsilon }_{\mathrm{\infty }}}}-{\displaystyle \frac{1}{{\epsilon }_s}}\right)\right]{\displaystyle \frac{⟨{\psi }_{nk}\mid {\psi }_{mk+q}⟩}{|q|}},$

where T is the temperature of the material, $k_B$ is the Boltzmann constant, B is the bulk modulus, $\hslash$ is the reduced Planck constant, ${\omega }_{po}$ is an effective optical phonon frequency, ${\epsilon }_{\mathrm{\infty }}$ is the high-frequency dielectric constant, ${\epsilon }_s$ is the static dielectric constant, e is the electron charge, $\beta$ is the inverse screening length calculated from the density of states, $n_{ii}$ is the concentration of ionized impurities. The final characteristic relaxation time is following Matthiessen’s rule, which is followed by

(6)${\displaystyle \frac{1}{\tau }}={\displaystyle \frac{1}{{\tau }^{ADP}}}+{\displaystyle \frac{1}{{\tau }^{IMP}}}+{\displaystyle \frac{1}{{\tau }^{POP}}}.$

Once we have determined the scattering, we can calculate the conductivity on a 51 × 145 × 135 grids. The Seebeck coefficient (S) and the electronic thermal conductivity (${\kappa }_e$) are derived using the linearized Boltzmann transport equation.

3 Results and discussion

3.1 Crystal structure

SnS has a orthogonal crystal structure ($Pnma,\mathrm{N}\mathrm{o}.62$) and a phase transition temperature of 873 K as shown in Fig.1(a). Each Sn atom bonds with three S atoms, forming an SnS${}_3$ tetrahedron. Furthermore, it is evident that SnS exhibits layered stacking along the $a$ axis, where different layers are connected by relatively weak van der Waals forces. Along the $b$ axis, it presents a zigzag chain of atoms, while along the $c$ axis, it displays an accordion-like arrangement of atomic chains. The unique structural features of SnS originate from the presence of a lone pair electrons on Sn atoms [as illustrated by the blue isosurface in Fig.1(b)]. Due to the electrostatic repulsion between lone pairs, the adjacent SnS${}_3$ tetrahedron must be positioned as far away as possible to accommodate these lone pairs. Thus the orientations of neighboring SnS${}_3$ tetrahedron are always interlocked, ultimately leading to the unique structure of SnS. This phenomenon result in strong anisotropy. Similar to SnSe, SnS also has lone pair of electrons, which are known to lead to elevated anharmonicity of the material [44, 45]. These unique properties make SnS a strong candidate for thermoelectric materials. The relaxed lattice constants are $a$ = 11.26 Å, $b$ = 4.02 Å, and $c$ = 4.27 Å for SnS (as depicted in Tab.1), which are in good agreement with experimental results and some previous ab initio calculations [46, 47]. Tab.1 depicts the variation of SnS lattice constants under different pressures. Undoubtedly, the lattice constant along the $a$ axis shows the largest variation, which is largely due to SnS being layered in the $a$ axis direction.

Fig.1 (a) Crystal structure of the $1\times 2\times 2$ SnS supercell. (b) The projection of crystal structure along the $c$ direction.

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Tab.1 The calculated bulk modulus $B$ (GPa), shear modulus $G$ (GPa), Pugh’s ratio ($B/G$), Young’s modulus $E$ (GPa), average wave velocity $\nu$${}_a$ (m/s) and lattice constants at −1 GPa, 0 GPa, and 1 GPa.

Pressure (GPa)	B	G	E	B/G	$\nu$${}_a$	$a$	$b$	$c$
−1	29.2	46.49	18.83	1.55	2148.83	11.39	4.03	4.36
0	36.3	56.58	22.8	1.59	2327.95	11.26	4.02	4.27
1	42.39	65.42	26.32	1.61	2468.34	11.16	4.01	4.21

Mechanical characterization plays a vital role in our understanding of material, and Tab.1 also lists the various mechanical properties of SnS. Through first-principles calculations, we obtained the elastic constants of SnS under different pressures, and all of them satisfy the relevant mechanical stability criteria $(C_{11}>0,$ $C_{11}{C}_{22}>C_{12}^2,C_{11}{C}_{22}{C}_{33}+2C_{12}{C}_{13}{C}_{23}-C_{11}{C}_{23}^2-C_{22}{C}_{13}^2-$$C_{33}{C}_{12}^2>0,C_{44}>0,C_{55}>0,C_{66}>0)$, confirming its mechanical stability. The thermal stability was tested by performing ab initio molecular dynamic (AIMD) simulation with 10 ps at 300, 500, and 800 K. The total energy of SnS fluctuate in a narrow range with time evolution for all cases (Fig. S1, Supporting information). As result, the SnS at different pressure can still maintain their initial crystal state at high temperature, which enhance their application value. The bulk modulus $B$ reflects resistance to fracture, while the shear modulus $G$ signifies resistance to plastic deformation. The elastic moduli $(B,E,G)$ decreases at −1 GPa, while they increase 1 GPa. If the toughness index $B/G$ is greater than 2.67, it means that the material is characterized as rigid, if it is less than 2.67 means that the material is brittle. Notably, SnS exhibits brittle characteristics. The average acoustic phonon velocity $({\nu }_a)$ can be quantitatively calculated by $B$ and $G$ [48], which is followed by

(7)$v_a={\left({\displaystyle \frac{1}{3}}\right)}^{-\frac{1}{3}}{\left\{2{\left({\displaystyle \frac{G}{\rho }}\right)}^{-\frac{3}{2}}+{\left({\displaystyle \frac{B+{\displaystyle \frac{4}{3}}G}{\rho }}\right)}^{-\frac{3}{2}}\right\}}^{-\frac{1}{3}}.$

The effect of pressure on the average wave velocity is shown in Tab.1. When negative pressure is applied, the velocity is significantly lower than the velocity at 0 GPa. The average velocity is usually an important parameter for predicting the lattice thermal conductivity of a material; therefore, based on the data shown in Tab.1, we can tentatively speculate that the lattice thermal conductivity of SnS decreases significantly upon the application of negative pressure [49].

3.2 Lattice vibration and electronic properties under pressure

Fig.2(a) illustrates the phonon dispersion curves of SnS under various pressures. The absence of imaginary frequencies within these curves confirms the dynamic stability of SnS. Furthermore, as shown in Fig. S2 (Supporting Information), the low frequency phonons are mainly contributed by Sn, which has a larger atomic mass, and the high frequency phonons are mainly contributed by S atoms. The phonon group velocity plays an important role in estimating the lattice thermal conductivity [50]. In order to gain a further understanding of the thermal transport properties of SnS, we determine the phonon group velocity by extracting it from the perfect dispersion curves [51]. As depicted in Fig.2(b), there is a discernible trend of decreasing phonon group velocity with the decrease of pressure. The reduction of the phonon group velocity will have a significant effect on the lattice thermal conductivity of SnS.

Fig.2 (a) Phonon dispersion, (b) phonon group velocity, (c) atomic displacement parameter and (d) Grüneisen parameter for SnS at different pressures. The insets represent the vibrational directions and vibrational amplitudes of the TA and LA phonon modes at the $\mathrm{\Gamma }$ point.

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Furthermore, we plot the trend of the atomic displacement parameter [as shown in Fig.2(c)]. Atomic displacement parameters (ADPs) evaluate the mean-square displacement of atoms in a solid around their equilibrium positions and reflect the strength of chemical bonds. In the $a$ direction, the ADPs of Sn atoms are larger than those of S atoms over the studied range, indicating that Sn atoms are weakly bond with adjacent atomic environments in the crystal lattice. At −1 GPa, the ADPs of both Sn and S atoms are significantly larger than the other pressures, suggesting a significant enhancement of the lattice anharmonicity of SnS. Moreover, the ADPs of the $b$ and $c$ axes have the same trend with pressure as that of the $a$ axis (shown in Fig. S3). By comparing the ADPs of the $a$, $b$, and $c$ axes, we find that the $a$ axis has the largest value, which may induce the low lattice thermal conductivity of the $a$ axis due to the strong anharmonic vibrations. To gain deeper insights into the lattice inharmonicity of SnS, we conducted a calculation of the Grüneisen parameter for SnS, as depicted in Fig.2(d). Our findings reveal a pronounced increase in the Grüneisen parameter for both the transverse acoustic (TA) and longitudinal acoustic (LA) phonon modes. This enhancement is associated with a reduction in lattice thermal conductivity. The inset shows the atomic vibrational directions and vibrational amplitudes of TA and LA at the $\mathrm{\Gamma }$ point, and it is clear that the vibrational amplitude of the Sn atoms is larger than that of the S atoms, which is consistent with the previously analyse of ADPs.

In addition to the effect of pressure on lattice vibrations, it also strongly affects the electronic structure. We mainly explore the electrical transport properties of n-type SnS. Excitingly, through our calculations, we find that the conduction band changes significantly under different pressures. After applying negative pressure, the band width becomes narrower, which will further affect the effective mass of the carriers, thereby influencing the power factor of n-type SnS. Fig.3(a) demonstrates that, at −1 GPa, the valence band maximum (VBM) is located on the $\mathrm{\Gamma }$−Z path, while the conduction band minimum (CBM) is positioned on the $\mathrm{\Gamma }$−Y path, and the second conduction band minimum (CBM1) is situated at the $\mathrm{\Gamma }$ point, with a band gap of 1.39 eV. Fig.3(b) shows that at 0 GPa, the VBM and the CBM are located in the same position as at −1 GPa, with a band gap of 1.25 eV which is consistent with other calculations [52, 53]. Fig.3(c) shows that at 1 GPa the CBM shifts to the $\mathrm{\Gamma }$ point, while CBM1 is located on the $\mathrm{\Gamma }$−Z path and the band gap decreases to 1.11 eV. It is evident that the band degeneracy is enhanced at 1 GPa, which will improve the electrical transport properties [54]. Figure S4 shows that at a pressure of 1 GPa, the energy isosurface at 0.05 eV above the CBM is significantly enlarged, which will also effectively enhance the electrical transport properties of SnS.

Fig.3 The electronic band structure of SnS at (a) −1 GPa, (b) 0 GPa and (c) 1 GPa. (d) The calculated negative crystal orbital Hamilton population (−COHP) at different pressures. The inset show the partial −COHP projected of SnS.

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Furthermore, we calculated the crystal orbital Hamiltonian population (COHP) of SnS. As depicted in Fig.3(d), SnS exhibits antibonding states near the VBM with minimal change under pressure. The inset demonstrates that the antibonding states are predominantly composed of S 3p and Sn 5s. The existence of antibonding states implies that SnS possesses significant anharmonicity [55]. In Fig. S5, the partial density of states (PDOS) under pressure reveals that the VBM is comprised of Sn 5p, Sn 5s and S 3p states, whereas the CBM is primarily dominated by Sn 5p with a minor contribution from S 3p, with pressure exerting no significant influence on these contributions.

3.3 Lattice thermal conductivity analysis

The calculated ${\kappa }_L$ at different pressures is presented in Fig.4. At ambient pressure and room temperature, our calculated lattice thermal conductivities (${\kappa }_L$) along the $a$, $b$, and $c$ axes are 1.0, 2.8, and 1.8 W/(mK), respectively. These values are close to the experimental data [56, 57], thereby validating the accuracy of our method. As shown in Fig.4, the lattice thermal conductivity of SnS is extremely low and exhibits anisotropy, which coincides with previous lattice structures analyses. The lattice thermal conductivity decreases with increasing temperature. Excitingly, we found that the ${\kappa }_L$ on the $a$, $b$ and $c$ axes of SnS were reduced by 42.9%, 38% and 42.7%, respectively, when negative pressure was applied. Moreover, the lattice thermal conductivity of $a$ axis is only 0.23 W/(mK) at −1 GPa and 800 K, which is comparable to the level of SnSe [58]. Undoubtedly, this proves the effective modulation of pressure on the lattice thermal conductivity of SnS.

Fig.4 Lattice thermal conductivity ${\kappa }_L$ at different pressure along the (a) $a$ axis, (c) $b$ axis and (e) $c$ axis. The spectral/cumulative thermal conductivity ${\kappa }_L(\omega )$ along (b) a axis, (d) b axis and (f) c axis. In the spectral/cumulative thermal conductivity diagram the solid line represents 300 K and the dashed line represents 800 K.

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As can be seen from the spectral/cumulative thermal conductivity illustrated in Fig.4(b), (d) and (f), SnS has a characteristic with other IV−VI semiconductor compounds, where the lattice thermal conductivity (${\kappa }_L$) is predominantly governed by optical phonons. Under ambient pressure, the contributions of optical phonons to the thermal conductivity along the $a$ axis, $b$ axis, and $c$ axis are 67%, 71%, and 63%, respectively. This indicates that optical phonons may have relatively high group velocities or longer mean free paths. Upon application of −1 GPa pressure, the contributions of optical phonons to the thermal conductivity along the $a$ axis, $b$ axis, and $c$ axis are reduced by 3%, 5%, and 7%, respectively.

Particularly, in comparison with the $b$ axis and $c$ axis, the high-frequency optical phonons (150−300 cm⁻¹) along the $a$ axis make a notably significant contribution to the thermal conductivity, accounting for 27%. This phenomenon can be elucidated by the phonon dispersion structure. As shown in Fig.2(a), the high symmetry directions $\mathrm{\Gamma }$−X, $\mathrm{\Gamma }$−Y, and $\mathrm{\Gamma }$−Z correspond to the $a$ axis, $b$ axis and $c$ axis of SnS, respectively. Analysis reveals that the high-frequency optical phonons (150−300 cm⁻¹) in the $\mathrm{\Gamma }$−X direction exhibit steeper slopes, implying higher group velocities. Therefore, the impact of these phonons on thermal conductivity is exceptionally pronounced. Under the pressure of −1 GPa, the contribution of the high-frequency optical phonons along the $a$ axis to ${\kappa }_L$ increases to 37%. This is likely due to the minimal change in the group velocities of phonons in the 150−300 cm⁻¹ frequency range along the $a$ axis under negative pressure, which remain relatively high.

The scattering phase space and the scattering rates are key parameters that influence thermal transport properties [59, 60]. Accordingly, we have plotted the weighted phase space at different pressures [Fig.5(a) and (b)] and scattering rates [Fig.5(c) and (d)]. In Fig.5(a), we have analyzed the weighted phase space at 300 K and observed that it reaches its maximum value under a pressure of −1 GPa within the scope of our study. This finding indicates a significant increase of the probability for three phonon scattering. In contrast, at 1 GPa, the weighted phase space exhibits minimal variation, suggesting that the increasement of pressure has a marginal effect on phonon scattering. Additionally, Fig.5(b) illustrates the variation of the weighted phase space with pressure at 800 K. By comparing the weighted phase spaces at these two temperature regimes, we have noted an increase in the weighted phase space for SnS as the temperature is elevated to 800 K.

Fig.5 (a) The weighted phase space $W_{ph}$ as a function of frequency at 300 K. (b) The weighted phase space $W_{ph}$ as a function of frequency at 800 K. (c) The three phonon scattering rates as a function of frequency at 300 K. (d) The three phonon scattering rates as a function of frequency at 800 K.

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Fig.5(c) depictes the scattering rates of SnS under different pressures at 300 K. It is evident that the scattering rate of SnS at −1 GPa is considerably higher than those observed at 0 GPa and 1 GPa, consistent with our analysis of the weighted phase space. This phenomenon stems from the increase of the lattice constant under negative pressure and the weakening of interatomic interactions. Consequently, the increased scattering rate leads to a reduction of the mean free path, thereby diminishing the lattice thermal conductivity. Furthermore, as depicted in Fig.5(d), there is a significant increase of the three phonon scattering at 800 K, indicating that SnS exhibits an extremely low lattice thermal conductivity at elevated temperatures. From the figures, we can also deduce that the scattering rate for high frequency phonons (150−300 cm⁻¹) is significantly higher than that for low and medium frequency phonons (0−150 cm⁻¹). This implies that the high-frequency phonons have shorter lifetimes. Therefore, despite their relatively high group velocities, their contribution to the lattice thermal conductivity is still quite limited.

3.4 Electrical transport properties under pressure

In order to thoroughly investigate the electrical transport characteristics of n-type SnS under different pressures, we present the variation of electrical transport parameters with pressure in Fig.6. Fig.6(a) depicts the relationship between electrical conductivity ($\sigma$) and carrier concentration along the $a$ axis direction for different pressure conditions. The findings reveal that at a pressure of 1 GPa, there is a significant enhancement of $\sigma$, and as the carrier concentration increases, the $\sigma$ under different pressures tends to converge. For the $b$ axis and $c$ axis, Fig.6(d) and (g) demonstrate that the $\sigma$ reaches a maximum at 0 GPa. To elucidate these phenomena, we have computed the effective masses of CBM and CBM1 along the $a$, $b$ and $c$ axes. Data from Tab.2 indicate that along the $a$ axis direction, the effective mass of CBM carriers under −1 GPa, 0 GPa and 1 GPa are 0.145$m_0$, 0.131$m_0$, and 0.073$m_0$, respectively. At 1 GPa, the effective mass of carriers is the lowest, which accounts for the highest conductivity at this pressure. For the $b$ axis and $c$ axis, the effective mass of carriers is the lowest at 0 GPa, hence the conductivity achieves its maximum value under ambient condition. At high concentrations, the relative contribution of electrons from the CBM1 increases, which leads to a gradual convergence of electrical conductivity.

Tab.2 The calculated carrier effective masses $m^{\ast }$ for the CBM and the CBM1 along the $a$, $b$ and $c$ axes at −1, 0 and 1 GPa. where CBM represents the conduction band minimum, CBM1 represent second conduction band minimum, and $m_0$ represents the mass of the free electron.

Pressure	Carrier	$m^{\ast }(m_0)$
		a	b	c
−1 GPa	CBM	0.145	0.129	0.157
	CBM1	0.087	1.033	1.540
0 GPa	CBM	0.131	0.119	0.147
	CBM1	0.079	0.945	1.787
1 GPa	CBM	0.073	0.891	1.939
	CBM1	0.132	0.123	0.148

Fig.6 Calculated electrical conductivity $\sigma$, Seebeck coefficient $|S|$ and power factor PF of each axis of SnS at −1, 0 and 1 GPa. The solid and dashed lines represent data at 300 K and 800 K, respectively.

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To further understand the phenomenon, we also plotted the trends of ADP mobility, IMP mobility and POP mobility with pressure for different axes [61, 62] in Fig. S6. A smaller partial carrier mobility implies a larger scattering rate. Therefore, from Fig. S6, we can observe that the POP scattering should be the decisive mechanism for the carrier mobility. Through the POP scattering rate along $b$ axis and $c$ axis, we find that the scattering rate is the smallest under ambient pressure, this may be the key factor leading to the maximum value of $\sigma$ at 0 GPa. Ultimately, We can conclude that the variation of $\sigma$ with pressure is caused because of the combined effect of POP scattering rate and carrier effective mass. Furthermore, we also show the carrier mobility $\mu$ of different axes of SnS under different pressures in Fig. S7. It is well known that the conductivity $\sigma$ is directly proportional to the carrier mobility $\mu$ according to the equation $\sigma =ne\mu$, where $n$ represents the carrier concentration, $e$ represents the electron mass, $\mu$ represents the carrier mobility [63, 64]. We can analyze that the mobility of SnS is consistent with the trend of electrical conductivity changes with pressure.

In contrast to the trend observed for conductivity $\sigma$, the Seebeck coefficient |S| diminishes as the carrier concentration increases [As depicted in Fig.6(b), (e) and (h)]. In Fig.6(b), there is an enhancement of |S| at 300 K and 1 GPa. In many cases, the Seebeck coefficient and electrical conductivity do not increase simultaneously. Interestingly, the pressure induces enhanced Seebeck coefficient and electrical conductivity at 300 K along $a$ axis, a phenomenon very favourable to the improvement of power factor. A similar phenomenon occurs in SnSe alloyed via Pb [65]. This phenomenon can be attributed to an increase in the degeneracy of the electronic bands at 1 GPa. The relationship between $PF$ and carrier concentration along $a$, $b$ and $c$ axes are represented in Fig.6(c), (f) and (i). We can find that when negative pressure is applied, the $PF$ of the $a$, $b$ and $c$ axes decreases by 15.9%, 3.2% and 6% at 800 K, respectively. Although the PF has decreased, it remains at a relatively high value, and the significant reduction of ${\kappa }_L$ ultimately leads to a substantial enhancement in the thermoelectric performance of n-type SnS.

3.5 Dimensionless figure of merit

By combining these transport properties, we can calculate $ZT$ values at various pressures and carrier concentrations. As illustrated in Fig.7, where the carrier concentrations ranging from ${10}^{18}$ to ${10}^{21}$ cm${}^{-3}$ and temperature are 300 K and 800 K. Fig.7(a) demonstrates the trend of $ZT$ values of n-type SnS along the $a$ axis at different pressure. We can analyse and obtain that at −1 GPa, its $ZT$ achieves an 11.6% increase at 800 K. This is attributed to the fact that SnS shows a huge decrease in its lattice thermal conductivity ${\kappa }_L$. Fig.7(b) and (c) depict the trend of $ZT$ values along $b$ axis and $c$ axis, respectively, we can analyse that at 800 K, the $ZT$ value of $b$ axis increases by 30% and $c$ axis increases by 16.7%. Notably, the $ZT$ of the n-type SnS at −1 GPa achieves the largest enhancement along the $b$ axis, which is due to that the reduction of ${\kappa }_L$ in the three axes is not much different, but the $PF$ has the smallest reduction in the $b$ axis, which finally results in this phenomenon. At ambient pressure, we captured an optimal $ZT$ of 2.41 at a carrier concentration of −2.21 × 10${}^{19}$ cm${}^{-3}$ and a temperature of 800 K along $a$ axis. When at −1 GPa, the $ZT$ reaches 2.7 at 800 K with a carrier concentration of −2.21 × 10${}^{19}$ cm${}^{-3}$ and reached an average $ZT$ of 1.67 in the $a$ axis direction over the entire concentration range at 800 K. Undoubtedly, the thermoelectric properties achieve a net enhancement in the present working range. Our research results show that the ZT value of SnS has been greatly enhanced under negative pressure. Compared with previous studies on In${}_4$Se${}_3$[66] and Sb${}_2$Te${}_3$[67] under negative pressure, the improvement is even more significant.

Fig.7 The thermoelectric figure of merit $ZT$ as a function of carrier doping concentration of (a) $a$ axis, (b) $b$ axis and (c) $c$ axis at −1, 0, and 1 GPa.

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4 Conclusion

In conclusion, we have systematically investigated the effects of different pressure on the thermal transport and electrical transport properties of n-type SnS. The unique layered structure of SnS allows for significant differences in electrical and thermal transport in all directions, which ultimately results in thermoelectric properties exhibit significant anisotropy. Our investigation reveals that at −1 GPa, the lattice constant of SnS increases, particularly in the interlayer direction. Subsequently, we analyzed the anharmonicity of SnS through its ADPs and Grüneisen parameter. The results conclusively demonstrate a significant enhancement in the anharmonicity of SnS, which ultimately lead to a reduction in the lattice thermal conductivity. The electrical transport properties of SnS decrease slightly. However, due to the significant decrease in the lattice thermal conductivity, a net increase of 11.6%, 30% and 16.7% for the $ZT$ values of n-type SnS was achieved in the $a$, $b$ and $c$ axes at 800 K. Moreover, a maximum $ZT$ value of 2.7 was obtained, which surpassed that of SnSe. Our work shows that pressure can significantly optimise the thermoelectric properties of SnS by reducing the lattice thermal conductivity and provide new directions for the development of practical thermoelectric applications of SnS.

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Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

The authors thank the National Natural Science Foundation of China (No. 52072188), the Program for Science and Technology Innovation Team in Zhejiang (No. 2021R01004), and acknowledge the Institute of High Pressure Physics of Ningbo University for its computational resources.

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