First-Principles Calculations of Thermoelectric Properties of IV–VI Chalcogenides 2D Materials

Introduction

Thermoelectric (TE) materials have attracted significant attention, especially in the past two decades, for their capacity to convert heat directly into electricity without the need of moving components (Ioffe, 1957; Sootsman et al., 2009; Vineis et al., 2010). They are expected to play important roles in the development of a more sustainable energy landscape worldwide (Kraemer et al., 2011; Barma et al., 2015; Favarel et al., 2015; Mehdizadeh Dehkordi et al., 2015; Moraes et al., 2015; Gayner and Kar, 2016), and there are also opportunities for using them for onboard power for wearable electronics (Leonov and Vullers, 2009; Kim et al., 2014; Du et al., 2015), sensors, or systems for disaster mitigations (Chen et al., 2016; Rais et al., 2016). However, the practical applications of TE devices has been largely impeded by the usually low energy conversion efficiency. The efficiency of TE materials depends on their dimensionless figure-of-merit ZT defined as ZT = S²σ T/k, where S is the Seebeck coefficient, σ is the electrical conductivity, T is the absolute temperature, and k is the total thermal conductivity, including the electronic and lattice contributions. Identifying TE materials with high values of ZT has been the research focus of this field for decades (Mahan and Sofo, 1996; Dresselhaus et al., 2007; Snyder and Toberer, 2008; Tian et al., 2013; Cahill et al., 2014; Liao and Chen, 2015). However, simultaneously optimizing these individual parameters to improve ZT is a challenging task since they are inter-correlated. There is a number of related reviews that describe different mechanisms to improve these properties and thus increase the ZT (Nolas et al., 1999; Snyder and Toberer, 2008; Sootsman et al., 2009; Heremans et al., 2012; Wang et al., 2013).

The discovery of 2D materials, exemplified by graphene (Novoselov et al., 2005; Geim and Novoselov, 2007; Geim, 2009; Jin et al., 2015) and transition metal dichalcogenides (TMDCs) (Gu and Yang, 2014; Wu and Luo, 2014; Yan et al., 2014; Jin et al., 2015; Kumar and Schwingenschlögl, 2015; Pu et al., 2016), has stimulated much research interests due to their unique structural, mechanical, optical, electrical, and thermal properties (Fiori et al., 2014; Zhang and Zhang, 2015; Singh and Hennig, 2016; Bernardi et al., 2017), and potential applications in photovoltaics (Britnell et al., 2013), transistors (Fang et al., 2012; Larentis et al., 2012; Liu et al., 2013; Li et al., 2014a), optoelectronics (Tritsaris et al., 2013), photodetector and molecular sensing (Wang et al., 2012), and wearable heating and cooling (Zhang and Zhang, 2017). Moreover, due to their intrinsic band gaps, TMDC monolayers have better potential in nanoelectronics applications compared to graphene (Guo and Zhang, 2016). In addition, TMDC materials have relatively high electrical properties such as Seebeck coefficient and electrical conductivity, positioning it as good candidates for TE applications. Although a large range of thermal conductivity have been reported for TMDCs, mostly with values higher than 30 W mK⁻¹, values for suspended monolayer in the order of 15 W mK⁻¹ or less have been observed recently (Adessi et al., 2017; Zhang and Zhang, 2017). It has been reported that the TE properties of 2D materials can also be unique because any modification to their environment or chemical functionalization, which play significant roles in thermal and electronic transport, can greatly impact their TE properties (Kim and Grossman, 2015).

In this work, we study and compare the electronic behavior and TE properties, including Seebeck coefficient, electrical conductivity, power factor, electronic thermal conductivity, and lattice thermal conductivity of a series of single-layer 2D materials belonging to the IV–VI Chalcogenides family. These include SnSe, GeSe, SnS, GeS, SnS₂, and SnS₂. We evaluate the importance of the spin orbit coupling (SOC) due its potential relevance in the electronic transport in TEs (Guan et al., 2015; Guo and Wang, 2017). To compute TE properties, we solve the semi-empirical Boltzmann transport model, through the BoltzTraP software (Madsen and Singh, 2006), and for thermal properties, we used first-principles lattice dynamics combined with the iterative solution of phonon Boltzmann transport equations (BTEs) using the ShengBTE software (Li et al., 2014b). For the electronic properties, three doping levels corresponding to 300 K carrier concentrations of 10¹⁸, 10¹⁹, and 10²⁰ cm⁻³ are studied. A temperature range from 300 to 800 K has been investigated. The results reveal a clear increasing trend of the temperature dependence of Seebeck coefficient (in absolute values), power factor, and ZT for all materials studied, be they p-type or n-type. Our calculations show that the compounds with orthorhombic crystal structures (i.e., GeS, GeSe, SnS, and SnSe) have much lower thermal conductivity values than those with hexagonal crystal structures (i.e., SnS₂ and SnSe₂), leading to a larger ZT values for the orthorhombic crystals. Based on the calculated ZT, we found that the most promising 2D TE materials at low and high temperature is SnSe at the three doping levels, respectively.

This paper is organized as follows: Section “Computational Details” includes a description of the computational details with emphasis in simulation processes used for different types of calculations. Section “Results and Discussion” includes the results and discussion of the TE properties obtained from the calculations. Section “Conclusion” outlines the main conclusions of this work.

Computational Details

The methodology used for our calculations is based on first-principles density functional theory (DFT) (Zhou et al., 2016) calculations and Boltzmann transport theory. The structures of the studied 2D materials are first optimized using DFT calculations and then the electron transport properties are calculated using DFT program Quantum Espresso (Giannozzi et al., 2009) and the semi-classical Boltzmann transport approach implemented in BoltzTraP (Madsen and Singh, 2006). The thermal conductivity values are calculated using the iterative solution of phonon BTE using the ShengBTE (Li et al., 2014b) code with force constants calculated from Quantum Espresso.

For all DFT calculations, the generalized gradient approximation of Perdew, Burke, and Ernzerhof are used for the exchange-correlation functional (Perdew et al., 1996). The kinetic energy cutoff of the wave functions is set to 50 Ry for all calculations. The Monkhorst–Pack k-mesh of 8 × 8 × 1 is used to sample the first Brillouin Zone. In the electronic band structure calculation, a finer mesh of 25 × 25 × 1 is used. To simulation monolayers, a large vacuum space of at least 10 Å is left in the z-direction to prevent the interactions between the layer and its periodic images in the cross-plane direction.

The optimized lattice parameters and their comparison to values reported in the literature are presented in Table 1.

TABLE 1

Table 1. Literature and our optimized lattice parameters for all the 2D materials studied.

Once the structures are optimized, we continue to calculate the electronic transport properties and the lattice thermal conductivity. The flow chart of these calculations, their corresponding input and output parameters and the programs used are schematically shown in Figure 1.

FIGURE 1

Figure 1. Schematic of the flow chart for TE properties and lattice thermal conductivity calculations.

To obtain the electronic transport properties, including electrical conductivity, Seebeck coefficient, and electronic thermal conductivity, the electron band structure and density of states are first calculated in the DFT framework. Then, the band structures are then input into the BoltzTraP package (Løvvik and Prytz, 2004; Madsen and Singh, 2006), which uses a Fourier expansion scheme to fit the band structures for transport property calculations. In the semi-classical transport theory, the Seebeck coefficient, electrical conductivity, and electronic thermal conductivity transport tensors are expressed as (Madsen and Singh, 2006):

$S_{\mathrm{\alpha }\mathrm{\beta }}(T,\mathrm{\mu })=\frac{1}{eT{\mathrm{\sigma }}_{\mathrm{\alpha }\mathrm{\beta }}(T,\mathrm{\mu })}\int {\mathrm{\sigma }}_{\mathrm{\alpha }\mathrm{\beta }}(\mathrm{\epsilon })(\mathrm{\epsilon }-\mathrm{\mu })\left[-\frac{\partial f_0(T,\mathrm{\epsilon },\mathrm{\mu })}{\partial \mathrm{\epsilon }}\right]d\mathrm{\epsilon }$ (1)

${\mathrm{\sigma }}_{\mathrm{\alpha }\mathrm{\beta }}(T,\mathrm{\mu })=\frac{1}{\mathrm{\Omega }}\int {\mathrm{\sigma }}_{\mathrm{\alpha }\mathrm{\beta }}(\mathrm{\epsilon })\left[-\frac{\partial f_0(T,\mathrm{\epsilon },\mathrm{\mu })}{\partial \mathrm{\epsilon }}\right]d\mathrm{\epsilon }$ (2)

$k_{\mathrm{\alpha }\mathrm{\beta }}^0(T,\mathrm{\mu })=\frac{1}{e^{2}T\mathrm{\Omega }}\int {\mathrm{\sigma }}_{\mathrm{\alpha }\mathrm{\beta }}(\mathrm{\epsilon }){(\mathrm{\epsilon }-\mathrm{\mu })}^2\left[-\frac{\partial f_0(T,\mathrm{\epsilon },\mathrm{\mu })}{\partial \mathrm{\epsilon }}\right]d\mathrm{\epsilon }$ (3)

where e is the electron charge, Ω is the reciprocal space volume, ε is the carrier energy, f is the Fermi distribution function, μ is the chemical potential, and T is the absolute temperature. It is noted here that k is the electronic thermal conductivity.

On the other hand, the conductivity tensor [σ_αβ(ε)] as a function of energy is expressed as:

${\mathrm{\sigma }}_{\mathrm{\alpha }\mathrm{\beta }}(\mathrm{\epsilon })=\frac{1}{N}{\displaystyle \sum _{i,k}}{\mathrm{\sigma }}_{\mathrm{\alpha }\mathrm{\beta }}(i,k)\frac{\mathrm{\delta }(\mathrm{\epsilon }-{\mathrm{\epsilon }}_{i,k})}{d\mathrm{\epsilon }}$ (4)

where N is the number of k-points in the reciprocal space.

The relaxation time τ depends, in principle, on both the wave vector and frequency. However, BoltzTraP treats the relaxation time to be a constant, which has been shown to be a reasonable assumption (Madsen and Singh, 2006). Our calculations for all TE properties were performed with a constant relaxation time approximation τ = 1.0 × 10⁻¹⁴ s, which is the value often chosen in similar calculations (Gao et al., 2005; Madsen and Singh, 2006; Yabuuchi et al., 2013; Ding et al., 2015).

To obtain the thermal conductivity, we calculate the harmonic force constants using the density functional perturbation theory as implemented in Quantum Espresso with a q-point grid size of 8 × 8 × 1. Then, a finite difference method based on supercell calculations (cell size of 4 × 4 × 1) is used to extract the cubic force constants. A cutoff radius of 5 Å is used for the cubic force constant calculation based on convergence test. These force constants are then fed into the ShengBTE program (Carrete et al., 2014) to calculate thermal conductivity based on an iterative solution of phonon BTE. A q-mesh of 30 × 30 × 1 is used for all thermal conductivity calculations.

We need to specially emphasize the choice of the thicknesses of the 2D materials used for calculating the electrical conductivity, electronic thermal conductivity, and lattice thermal conductivity. We used a thickness of 6.0 Å (the interlayer distance of bulk SnSe crystal) for all calculations to enable fair comparison of the properties among all the two materials. In 2D materials, thickness is not well defined, but the concept of electrical and thermal conductivity, which are 3D intensive property, require their quantities to be independent of the cross-sectional area of the material. However, since all electron and heat has to go through the single-layer structures, no matter how “thick” or “thin” the structure is, the same thickness should be used to remove any artifact in comparing the electron and thermal transport ability of different 2D materials when using the 3D property—electrical and thermal conductivity. Such a factor has been discussed in details in previous papers (Liu et al., 2017; Wu et al., 2017).

The 2D TE compounds simulated in this work have two kinds of crystal structures, including orthorhombic and hexagonal (Ding et al., 2015; Guo et al., 2015; Sun et al., 2015; Wang et al., 2015). Figure 2A shows the orthorhombic crystal structure for compounds such as SnSe, SnS, GeSe, and GeS in the Pnma-phase (#62) (Tritsaris et al., 2013; Ding et al., 2015; Guo et al., 2015; Wang et al., 2015), and Figure 2B shows the hexagonal crystal structure for compounds such as SnSe₂ and SnS₂ in the P3m1-phase (#164) (Sava et al., 2006; Bauer Pereira et al., 2013; Sun et al., 2015). For each simulation, only one layer was considered, where the number of atoms per primitive cell is 4 for the orthorhombic crystal structure and 3 for the hexagonal crystal structure.

FIGURE 2

Figure 2. Crystal structures of (A) orthorhombic and (B) hexagonal materials.

Results and Discussion

Band Structures

Since SOC has been reported to have the possibility of influencing the electronic transport in TEs (Guan et al., 2015; Guo and Wang, 2017), we computed the electronic structures for GeS, GeSe, SnS, SnSe, SnS₂, and SnSe₂ without and with SOC. The band structures are shown in Figure 3. As expected, all materials are found to be semiconductors. The results without and with SOC both show that all materials have indirect band gaps, except for GeSe, which has a direct band gap. The indirect band gap of GeS, SnS, and SnSe is between the valence band maxima (VBM) located along the Γ − X path and the conduction band minima (CBM) along the Y − Γ path. For SnS₂ and SnSe₂, the VBM is located along the K − M path and the CBM at the K-point. For GeSe, the direct band gap is located along the Γ − X path. The band gaps values (with and without SOC) are shown in Table 2. All reference band gaps are from numerical simulation with exception of Butt et al. (2012). After applying SOC, we can notice very minor effects on the electronic structure, decreasing slightly the band gaps in all TE materials. The computed band gaps are seen to agree reasonably with the literature values.

FIGURE 3

Figure 3. Band structures of TE materials (top row) without SOC and (bottom row) with SOC. TE, thermoelectric; SOC, spin orbit coupling.

TABLE 2

Table 2. Comparison of band gap energies (eV) with and without spin orbit coupling (SOC) for the 2D semiconductors.

TE Properties

We compute the TE properties of all materials considering three doping levels corresponding to 300 K carrier concentrations of 10¹⁸, 10¹⁹, and 10²⁰ cm⁻³, respectively. The TE properties are plotted at a temperature range from 300 to 800 K. The electronic properties of TE materials are influenced by SOC, resulting in some cases with better TE properties such as Seebeck coefficient and electrical conductivity (Guo, 2016; Guo and Zhang, 2016; Guo and Wang, 2017). Figure 4 shows the calculated Seebeck coefficient without and with SOC at different doping levels. The signs of the Seebeck coefficients indicate that SnSe, SnS, SnSe₂, and SnS₂ are n-type, while GeS and GeSe are p-type semiconductors. We can notice that for all cases, the influences of SOC in Seebeck coefficients are small. As a result, for the following text, we use cases without SOC for further study.

FIGURE 4

Figure 4. Seebeck coefficient as a function of temperature (A–C) without SOC and (D–F) with SOC at different doping levels. SOC, spin orbit coupling.

Figure 5 summarizes all the TE properties without SOC for the materials studied at the three levels of doping. The absolute values of Seebeck coefficients exhibit a decreasing trend with temperature at 10¹⁸ and 10¹⁹ cm⁻³, with the exception for SnS where the change is small, and for SnSe where the absolute values increase slightly at 10¹⁹ cm⁻³. At 10²⁰, an increasing trend for all TE materials in the Seebeck coefficient in absolute values is observed. Due to the interrelationship between carrier concentration and Seebeck coefficient (Snyder and Toberer, 2008), we notice that as doping level increases, the Seebeck coefficient decreases dramatically. We can see that for SnS and SnSe, the Seebeck coefficient at 300 K decreases more than six times when the doping level increases from 10¹⁸ to 10²⁰ cm⁻³. As shown in Figure 5 (second row), the electrical conductivity has the same increasing trend for all materials as a function temperature for all three doping levels. As temperature increase, more electrons are excited and thus the electrical conductivity increases.

FIGURE 5

Figure 5. TE properties as a function of temperature without SOC at different doping levels. TE, thermoelectric; SOC, spin orbit coupling.

After the calculations of the Seebeck coefficient and electrical conductivity, we can calculate the power factor (PF = S²σ). The PF for all compounds at the three doping levels as a function of temperature are shown in Figure 5 (third row). We can notice clearly the temperature dependence of PF, indicating that as the temperature increases, the PF for both p- and n-type semiconductors increases monotonically.

The temperature dependence of the electrical thermal conductivity (k_e) is shown to be relatively strong (Figure 5, fourth row). As the temperature increases, k_e for both p- and n-type semiconductors increases monotonically. In contrast to the PF, where we seek for a maximum value, for k_e we want to find a minimum value to maximize ZT. We can see that the minimum values of k_e are obtained at 10¹⁸ cm⁻³ level of carrier concentration. GeSe has a minimum value of 0.019 (W mK⁻¹) at lower temperature (300 K), and at higher temperature (800 K), GeSe and SnS both have a minimum value of 0.401 (W mK⁻¹). To validate and compare our results, we consider the reported experimental values of SnSe, because it is one of the most studied materials due to its high reported TE properties. From theoretical reports, the number of studies has been consistently increasing in the last decade, and several studies of 2D materials related to this field and presented in this work have been recently reported. For both experimental and theoretical studies, we found a fairly reasonable agreement of electrical properties with our reported values (Kumar and Schwingenschlögl, 2015; Sun et al., 2015; Zhang et al., 2015; Morales Ferreiro et al., 2016; Tyagi et al., 2016; Ding et al., 2017; Guo and Wang, 2017; Li et al., 2017; Shafique and Shin, 2017).

The lattice thermal conductivity, as we mentioned previously, are calculated by solving the BTE for phonons using the iterative method. The values are shown in Figure 6. We can see that as temperature increases, lattice thermal conductivity decreases monotonically. This is because as temperature increases, anharmonic phonon–phonon scattering becomes more intensive, leading to a thermal conductivity inversely proportional to the temperature (Balandin and Wang, 1998; Gang, 2005). The high temperature dependencies are ~Tⁿ, where n is −1.012, −1.003, −0.998, −1.036, −0.999, and −1.015 for GeS, GeSe, SnS, SnS₂, SnSe, and SnSe₂, respectively.

FIGURE 6

Figure 6. Lattice thermal conductivity of 2D TE materials as a function of temperature. TE, thermoelectric.

All the 2D materials studied here have relatively low lattice thermal conductivity compared with other 2D materials like graphene, MoSe₂, BN, WS₂, ZrS₂, HfS₂, and ZrSe₂ (Wu et al., 2017) making them promising candidates to have higher ZT. The lattice thermal conductivity values obtained in the present work are in good agreement with recently reported results (Qin et al., 2016; Shafique and Shin, 2017). If we consider the lattice thermal conductivity as a good approach to obtain high efficiency (ZT) in TE materials, we can expect, from our results (Figure 6), that the monolayer SnSe should be one of the best candidate for TE applications as we can see in previously reports for this material (Carrete et al., 2014; Zhao et al., 2014, 2016; Guan et al., 2015; Guo et al., 2015; Hong et al., 2015; Kutorasinski et al., 2015; Sassi et al., 2015; Wang et al., 2015; Chere et al., 2016; Leng et al., 2016; Morales Ferreiro et al., 2016; Popuri et al., 2016; Li et al., 2017).

Combining the all the calculated TE properties so far, we obtained the dimensionless figure-of-merit (ZT) as a function of temperature for different doping levels (Figure 5, fifth row). We notice that the ZT value can be significantly influenced by temperature. For all materials, ZT increases with temperature. Among all the 2D materials studied from 300 to 800 K here, the best ZT values are all obtained from SnSe disregard of doping levels.

Conclusion

In this study, DFT and the Boltzmann transport model are used to calculate the TE properties of monolayer materials belonging to the group IV–VI compounds, including SnSe, GeSe, SnS, GeS, SnSe₂, and SnS₂. We determine the electronic bands and Seebeck coefficient without and with SOC. We notice a small effect of SOC over the band gaps for all materials. Indirect band gaps from calculations without SOC of 0.83, 1.3, 1.58, 0.75, and 1.54 eV are obtained for SnSe, SnS, GeS, SnSe₂, and SnS₂, respectively. A direct band gaps of 1.25 eV is found for GeSe. We also obtain indirect band gaps from calculations with SOC of 0.74, 1.23, 1.54, 0.73, and 1.5 eV for SnSe, SnS, GeS, SnSe₂, and SnS₂, respectively, and a direct band gap of 1.21 eV for GeSe. The TE properties are calculated at different doping levels, and it was found the SnSe has the largest ZT disregard the doping level, which is related to the low lattice thermal conductivity of this material.

Author Contributions

The project was conceived by TL. The simulation was performed by JM-F. The data analysis was performed by JM-F and TL. The manuscript was written by JM-F and TL with editing and suggestions from DD-D and DC.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

We wish to thank Mechanical and Metallurgical Engineering Department of Pontificia Universidad Católica de Chile, the Physics Institute, the Research Center on Nanotechnology and Advanced Materials, CIEN-UC of Pontificia Universidad Católica de Chile, and finally thanks the support provided by the Chilean Council for Scientific and Technological Research (CONICYT). We would also like to thank the U.S. National Science Foundation grant 1433490.

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