# Thermoelectric Properties of Pnma and Rocksalt SnS and SnSe

^{*}

*Solids*2021)

## Abstract

**:**

## 1. Introduction

_{2}Te

_{3}($ZT$ ≃ 1 from 350–450 K) [2], due to its favourable electronic structure and intrinsically low lattice thermal conductivity. Engineered PbTe is a leading candidate for high-temperature applications ($ZT$ ≃ 2.2 with endotaxial nanostructuring with SrTe [9]), due to a “convergence” of the band structure leading to multiple band extrema at elevated temperatures and strongly anharmnonic lattice dynamics [10,11]. However, the environmental toxicity of Pb and the rarity of Te means that both materials are unsuitable for mass-produced TEGs and are restricted to niche applications. There has therefore been significant research effort devoted to exploring alloys such as Bi

_{2}(S, Se, Te)

_{3}[12,13] and Pb(S, Se, Te) [14,15], as well as other chalcogenide systems [16,17,18].

_{1−x}Se

_{x}) alloys have found that Se-rich alloys can potentially show higher performance than pure SnSe [34,35,36].

## 2. Computational Modelling

## 3. Results and Discussion

#### 3.1. Structure and Lattice Dynamics

_{2}S

_{3}preferentially adopts a low-symmetry orthorhombic Pnma phase while Bi

_{2}Se

_{3}and Bi

_{2}Te

_{3}are more stable in a rhombohedral $R\overline{3}m$ phase with the Bi cations in an octahedral bonding environment [13]. On the other hand, compression shortens the Sn–S bond lengths and introduces a barrier to the structural distortion, which causes the imaginary mode to harden and become real.

#### 3.2. Electronic Structure and Transport Properties

#### 3.3. Lattice Thermal Conductivity

#### 3.4. Thermoelectric Figure of Merit

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Representative structures of the Pnma (

**a**) and rocksalt (

**b**) phases of SnS and SnSe. These images were prepared using the VESTA software [62].

**Figure 2.**Calculated phonon dispersion and density of states (DoS) of Pnma SnS (

**a**), equilibrium and compressed rocksalt SnS (

**b**,

**c**), Pnma SnSe (

**d**), and rocksalt SnSe (

**e**). On each DoS plot, the total DoS is shown in black and projections onto the Sn and S/Se atoms are shown as blue and orange shaded areas.

**Figure 3.**Comparison of the calculated electrical conductivity $\sigma $ (

**a**,

**b**), Seebeck coefficient S (

**c**,

**d**), power factor ${S}^{2}\sigma $ (PF) (

**e**,

**f**), and electronic thermal conductivity ${\kappa}_{\mathrm{el}}$ (

**g**,

**h**) of the five structures examined in this work. For each structure, we show the scalar averages computed using Equations (9)–(11). The four properties are compared as a function of carrier concentration n for a fixed T = 800 K (

**a**,

**c**,

**e**,

**g**) and as a function of temperature for a fixed n = 2.15 × 10${}^{19}$ cm${}^{-3}$ (

**b**,

**d**,

**f**,

**h**).

**Figure 4.**Anisotropy in the electronic transport of SnSe. The plots show the diagonal $xx$, $yy$, and $zz$ components of the power factor ${S}^{2}\sigma $ (PF) (

**a**,

**b**) and electronic thermal conductivity ${\kappa}_{\mathrm{el}}$ (

**c**,

**d**) of the orthorhombic Pnma phase of SnSe together with the isotropic averages calculated using Equations (9)–(11). The isotropic averages of the rocksalt phase, for which the three diagonal components are equal, are also shown for comparison. As in Figure 3, both properties are shown as a function of carrier concentration n for a fixed T = 800 K (

**a**,

**c**) and as a function of temperature for a fixed n = 2.15 × 10${}^{19}$ cm${}^{-3}$ (

**b**,

**d**).

**Figure 5.**Calculated scattering rates ${\tau}_{\mathit{k}j}^{-1}$ as a function of energy ${\u03f5}_{\mathit{k}j}$ for the electronic states in Pnma (

**a**) and rocksalt SnSe (RS) (

**b**) for a carrier concentration n = 2.15 × 10${}^{19}$ cm${}^{-3}$ and temperature T = 800 K. The energy zero is set to $\u03f5$ = ${\u03f5}_{\mathrm{F}}$. Rates are shown separately for the three scattering mechanisms relevant to the two phases of SnSe, viz. acoustic deformation potential (ADP), polar optic phonon (POP), and ionised impurity (IMP) scattering.

**Figure 6.**Calculated lattice thermal conductivity ${\mathbf{\kappa}}_{\mathrm{latt}}$ as a function of temperature for Pnma SnS (

**a**), equilibrium and compressed rocksalt SnS (

**b**), Pnma SnSe (

**c**), and rocksalt SnSe (

**d**). For the Pnma phases, the principal ${\kappa}_{xx}$, ${\kappa}_{yy}$, and ${\kappa}_{zz}$ components of the ${\kappa}_{\mathrm{latt}}$ tensor, corresponding to transport along the a, b, and c axes, respectively, are shown together with the diagonal average ${\kappa}_{\mathrm{ave}}=\frac{1}{3}\left({\kappa}_{xx}+{\kappa}_{yy}+{\kappa}_{zz}\right)$. For the rocksalt phases, the three diagonal components and the average are equivalent by symmetry, so we only show ${\kappa}_{\mathrm{ave}}$. The data for Pnma SnS and SnSe is from [61], but the three principal components have been relabelled to match the orientation of the unit cells in this study.

**Figure 7.**Analysis of the lattice thermal conductivity ${\kappa}_{\mathrm{latt}}$ of the five structures examined in this work using the constant relaxation time approximation (CRTA) model defined in Equation (18). The three subplots compare the ${\kappa}_{\mathrm{latt}}$ (

**a**) to the harmonic term $\kappa /{\tau}^{\mathrm{CRTA}}$ (

**b**) and the weighted average lifetime ${\tau}^{\mathrm{CRTA}}$ (

**c**) as a function of temperature. For the two Pnma phases, we analyse the averaged ${\kappa}_{\mathrm{latt}}$ computed using Equation (17).

**Figure 8.**Thermoelectric figures of merit $ZT$ for the Pnma SnS (

**a**), equilibrium and compressed rocksalt SnS (

**b**,

**c**), Pnma SnSe (

**d**), and rocksalt SnSe (

**e**) calculated using Equation (1) as a function of carrier concentration n and temperature T.

**Table 1.**Optimised lattice parameters of the five structures examined in this work. Eq.—equilibrium; Comp.—compressed.

a (Å) | b (Å) | c (Å) | V (Å^{3}) | |
---|---|---|---|---|

SnS (Pnma) | 11.000 | 3.965 | 4.202 | 183.2 |

SnS (RS, Eq.) | 5.712 | - | - | 186.3 |

SnS (RS, Comp.) | 5.426 | - | - | 159.8 |

SnSe (Pnma) | 11.350 | 4.124 | 4.335 | 202.9 |

SnSe (RS) | 5.912 | - | - | 206.7 |

**Table 2.**Calculated thermal conductivities of the five structures examined in this work at T = 800 K. Each row lists the three diagonal components ${\kappa}_{xx}$, ${\kappa}_{yy}$, and ${\kappa}_{zz}$ of the ${\mathbf{\kappa}}_{\mathrm{latt}}$ tensor together with the diagonal average ${\kappa}_{\mathrm{ave}}=\frac{1}{3}\left({\kappa}_{xx}+{\kappa}_{yy}+{\kappa}_{zz}\right)$ and its decomposition into harmonic and lifetime components, ${\left(\kappa /{\tau}^{\mathrm{CRTA}}\right)}_{\mathrm{ave}}$ and ${\tau}^{\mathrm{CRTA}}$, according to Equation (18). The data for Pnma SnS and SnSe is from [61] with the ${\kappa}_{xx}$, ${\kappa}_{yy}$, and ${\kappa}_{zz}$ relabelled to be consistent with the orientation of the unit cells in this work.

$\mathit{\kappa}$ (W m^{−1} K^{−1}) | ${\left(\mathit{\kappa}/{\mathit{\tau}}^{\mathbf{CRTA}}\right)}_{\mathbf{ave}}$ | ${\mathit{\tau}}^{\mathbf{CRTA}}$ | ||||
---|---|---|---|---|---|---|

${\mathbf{\kappa}}_{\mathit{xx}}$ | ${\mathbf{\kappa}}_{\mathit{yy}}$ | ${\mathbf{\kappa}}_{\mathit{zz}}$ | ${\mathbf{\kappa}}_{\mathrm{ave}}$ | (W m^{−1} K^{−1} ps^{−1}) | (ps) | |

SnS (Pnma) | 0.508 | 1.196 | 0.708 | 0.804 | 0.755 | 1.065 |

SnS (RS, Eq.) | 0.606 | - | - | 0.606 | 7.699 | 0.079 |

SnS (RS, Comp.) | 3.142 | - | - | 3.142 | 5.512 | 0.570 |

SnSe (Pnma) | 0.354 | 0.814 | 0.613 | 0.593 | 0.380 | 1.561 |

SnSe (RS) | 1.665 | - | - | 1.665 | 1.578 | 1.055 |

**Table 3.**Predicted maximum thermoelectric figures of merit $ZT$ for the five systems examined in this work, extracted from Figure 8. For each system, we list the carrier concentration n and temperature T at which the maximum $ZT$ was obtained together with the properties from Equation (1), namely the electrical conductivity $\sigma $, the Seebeck coefficient S, the power factor ${S}^{2}\sigma $ (PF), the electronic and lattice contributions to the thermal conductivity ${\kappa}_{\mathrm{el}}$/${\kappa}_{\mathrm{latt}}$, and the total thermal conductivity ${\kappa}_{\mathrm{tot}}$.

n | T | $\mathit{ZT}$ | $\mathit{\sigma}$ | S | PF ${\mathit{S}}^{2}\mathit{\sigma}$ | $\mathit{\kappa}$ (W m^{−1} K^{−1}) | |||
---|---|---|---|---|---|---|---|---|---|

(cm^{−3}) | (K) | (S cm^{−1}) | ($\mathit{\mu}$V K^{−1}) | (mW m^{−1} K^{−2}) | ${\mathbf{\kappa}}_{\mathbf{el}}$ | ${\mathbf{\kappa}}_{\mathbf{latt}}$ | ${\mathbf{\kappa}}_{\mathbf{tot}}$ | ||

SnS (Pnma) | 4.64 × 10${}^{19}$ | 1000 | 1.75 | 252 | 272 | 1.87 | 0.43 | 0.64 | 1.07 |

SnS (RS, Eq.) | 10${}^{20}$ | 720 | 2.99 | 885 | 297 | 7.84 | 1.22 | 0.49 | 1.70 |

SnS (RS, Comp.) | 10${}^{20}$ | 800 | 1.48 | 999 | 303 | 9.19 | 1.82 | 2.51 | 4.33 |

SnSe (Pnma) | 4.64 × 10${}^{19}$ | 1000 | 2.81 | 348 | 274 | 2.62 | 0.46 | 0.47 | 0.93 |

SnSe (RS) | 10${}^{20}$ | 800 | 2.60 | 1196 | 302 | 10.90 | 1.69 | 1.33 | 3.02 |

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**MDPI and ACS Style**

Flitcroft, J.M.; Pallikara, I.; Skelton, J.M.
Thermoelectric Properties of *Pnma* and Rocksalt SnS and SnSe. *Solids* **2022**, *3*, 155-176.
https://doi.org/10.3390/solids3010011

**AMA Style**

Flitcroft JM, Pallikara I, Skelton JM.
Thermoelectric Properties of *Pnma* and Rocksalt SnS and SnSe. *Solids*. 2022; 3(1):155-176.
https://doi.org/10.3390/solids3010011

**Chicago/Turabian Style**

Flitcroft, Joseph M., Ioanna Pallikara, and Jonathan M. Skelton.
2022. "Thermoelectric Properties of *Pnma* and Rocksalt SnS and SnSe" *Solids* 3, no. 1: 155-176.
https://doi.org/10.3390/solids3010011