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WS2 As an Excellent High-Temperature Thermoelectric Material Appala Naidu Gandi and Udo Schwingenschlögl* Physical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia ABSTRACT: The potential of WS2 as a thermoelectric material is assessed. The electronic contribution to the thermoelectric properties is calculated within the constant relaxation time approximation from the electronic band structure, whereas the lattice contribution is evaluated using self-consistently calculated phonon lifetimes. In addition, the dependence of the lattice thermal conductivity on the mean free path of the phonons is determined.

band gap (∼1.34 eV) semiconductors. 26 W and Mo dichalcogenides satisfy this requirement and thus have been widely studied.27−31 Limited by the band gap, WS2, for example, can only convert photons with wavelengths below 800 nm32 so that only a part of the solar radiation is converted into electricity, while the near-infrared radiation results in heat generation33 (∼42% of the total solar radiation34). With the use of a thermoelectric generator with the heat source operating at approximately 1000 K, the remaining solar energy could be harvested.34 The out-of-plane thermal conductivity of a 62 nm WSe2 thinfilm is 30 times smaller than the bulk value and 6 times smaller than the minimum predicted by an Einstein model at 300 K.35−37 Exfoliation and restacking of WS2 at 300 K reduces the thermal conductivity by 50%.38 In addition, Coleman et al.39 and Suh et al.40 have reported enhancement of the thermoelectric properties when carbon nanotubes are added to WS2. Most theoretical studies are restricted to the evaluation of the electronic contribution to the thermal conductivity because of difficulties in the handling the lattice thermal conductivity.41 Huang et al. have calculated the thermoelectric potential of few layer transition metal dichalcogenides MX2 (M = Mo, W; X = S, Se) using a ballistic transport model,42,43 and Wickramaratne et al. have used the Landauer formalism for addressing the figure of merit.44 However, Huang et al. did not consider anharmonic terms in the evaluation of the phonon band structure, whereas Wickramaratne et al. have employed the MoS2 lattice thermal conductivity from ref 45 for all studied dichalcogenides. Until recently, few methods were available for calculating the lattice thermal conductivity. Either assumptions are made for the relaxation times of the phonons46−48 or the electronic part of the thermal conductivity is calculated using the Wiedemann−Franz law and subtracted from the measured electrical conductivity (the less accurate indirect method49). The method developed in

1. INTRODUCTION The increasing per capital energy consumption, along with an increasing world population, causes rapid depletion of fossil fuel resources. Renewable energy sources would be a self-sustaining solution, while effective usage of the fossil fuel resources can help filling the short-term gap between supply and demand. Various thermoelectric materials have been developed to serve this purpose by converting waste heat from exhaust gases into electricity, making use of the Seebeck effect (a temperature gradient induces a potential difference). The efficiency of a thermoelectric material is measured by the figure of merit (ZT), which depends on the Seebeck coefficient (S), electrical conductivity (σ), and thermal conductivity (κ): ZT = S2σT/κ.1 The thermal conductivity comprises electronic (κel) and lattice (κlat) contributions. In general, the figure of merit can be enhanced either by increasing the power factor (S2σ) or by decreasing the thermal conductivity. Heavily doped semiconductors often have good thermoelectric properties.2 This materials class includes skutterudites,3,4 clatharates,5 zintl phases,6 complex oxides,7 and chalcogenides.8,9 Various physical and chemical principles have been used to optimize the ZT value. For example, introduction of rattler atoms into open structures like clatharates10 and skutturides11 results in low-frequency phonons near the acoustic branches, thus increasing the phonon scattering and reducing the thermal conductivity. Creation of superlattices,12,13 nanostructuring,14−17 grain refinement,18,19 and microstructure control20,21 are other successful phonon engineering methods. Traditional thermoelectric materials have been studied extensively using these methods, and the figure of merit has steadily increased but now seems to approach saturation. Two-dimensional materials are gaining more and more importance following the successful application of graphene in diverse fields.22−24 Transition metal dichalcogenides have layered structures and thus can be thinned down to few layer systems, with applications in digital electronics and optoelectronics.23−25 Moreover, efficient solar cells require narrow © XXXX American Chemical Society

Received: September 22, 2014 Revised: November 2, 2014

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dx.doi.org/10.1021/cm503487n | Chem. Mater. XXXX, XXX, XXX−XXX

Chemistry of Materials

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energy are the force constants of various orders. Truncating the expansion at second order results in a harmonic approximation, for which the solutions of the equation of motion are the normal mode polarization vectors and frequencies. In the harmonic approximation, each normal mode vibrates independently of the others, leading to an infinite lattice thermal conductivity. Therefore, higher order force constants are required. Still, the solutions from the harmonic approximation are valuable inputs for solving the Boltzmann transport equation for phonons.66 We use a direct method67 based on a 3 × 3 × 2 supercell for calculating the harmonic dispersion relation68 and a 6 × 6 × 2 k-point mesh for the Brillouin zone integration while evaluating the total energies and forces. The second-order force constants obtained from this approach describe the short-range interaction. Thus, the dynamical matrices constructed from these force constants do not accurately predict the splitting of the longitudinal and transversal optical phonon branches in polar materials,69 as it is due to long-range dipole−dipole interactions. A nonanalytical correction has been proposed in ref 70. The additional contribution to the dynamical matrix is a function of the Born effective charges and the high frequency static dielectric tensor,71 which is calculated using density functional perturbation theory as implemented in VASP.72,73 The thermal conductivity is lowered by Umklapp scattering processes (the phonon momentum is changed by a reciprocal lattice vector, whereas the energy is conserved). Three phonon scattering events are dominating at room temperature, while events involving four or more phonons can play a role at elevated temperatures.74 However, such a complex problem is beyond the scope of state-of-the-art ab initio methods. Therefore, we restrict our considerations to three phonon scattering, using a finite difference scheme to calculate the third order force constants in real space.75 In this scheme, a supercell is constructed such that there is only negligible interaction between atoms in the center and at the boundary. Four or more phonon scattering events increase the resistance at high temperature so that the thermal conductivity evaluated considering only three phonon scattering events is an upper limit. Accordingly, the figure of merit is a lower limit. In our 3 × 3 × 2 supercell, we consider atoms up to fourth nearest neighbors (one at a time) in displacing them simultaneously with a given atom. The induced forces are calculated using the Hellmann−Feynman theorem76 as implemented in VASP (6 × 6 × 2 k-point mesh), and the third-order force constants are obtained from these forces and displacements following the finite difference method. The phonon distribution at a specific temperature is given by the Bose−Einstein distribution.77 A temperature gradient across a material results in a heat flow due to the diffusion of phonons from hot to cold. However, these phonons undergo scattering as they pass through the material, which alters the phonon distribution. Under Umklapp scattering the direction is also changed. Boltzmann’s transport equation states that the overall rate of change of the phonon distribution must vanish in steady state.78 Practically, a linear version63 of the equation is used, in which the derivative of the phonon distribution with respect to the temperature is replaced with a derivative of the equilibrium Bose−Einstein distribution.79 In single crystals, the phonons are scattered mainly by other phonons. The scattering cross sections are evaluated from the third-order force constants and polarization vectors80 and the scattering rates for various events are obtained from these cross sections, phonon distribution functions, and mode frequencies, considering the energy and momentum conservation. Phonon scattering times are obtained from the scattering rates. Because each scattering event involves up to three phonons, the scattering time of one of them depends on the scattering times of the others.81 Therefore, the scattering times have to be calculated self-consistently. The thermal conductivity is as an integral over the Brillouin zone involving the phonon lifetimes along with the group velocities, frequencies, and the equilibrium distribution function.82 We use the ShengBTE code for solving the linear Boltzmann transport equation numerically.50 This is a major improvement as compared to the relaxation time approximation for the evaluation of the lattice thermal conductivity. A 33 × 33 × 8 k-point mesh is used in the Brillouin zone integration and Gaussian functions approximate the Dirac delta distributions arising from the energy conservation conditions.

ref 50 can accurately predict the lattice thermal conductivity without any assumption about phonon lifetimes, including the dependence on the phonon mean free path, thus being very useful for phonon engineering by nanostructuring. In the following, we employ this method for evaluating the potential of the transition metal dichalcogenides MX2 (M = Mo, W; X = S, Se) in thermoelectrics, using WS2 as an example.

2. METHODOLOGY Density functional theory based on the Vienna Ab-initio Simulation Package (VASP)51 is used with plane wave energies up to 500 eV in the expansion of the electronic wave function. The generalized gradient approximation in the Perdew−Burke−Ernzerhof (PBE) flavor is employed for the exchange−correlation functional,52 and the W 5d4 and 6s2 electrons, as well as the S 3s2 and 3p4 electrons, are considered as valence electrons. Moreover, the tetrahedron method with Blöchl corrections is employed.53 In studying thermoelectric properties, an accurate band structure is of crucial importance because the Seebeck coefficient, electrical conductivity, and electronic contribution to the thermal conductivity depend on it. Within the PBE scheme, we employ a 42 × 42 × 12 k-point mesh and afterward shift the conduction states to higher energy to reproduce the experimental band gap. Using a 30 × 30 × 9 k-point mesh, an accurate band gap is directly achieved by the HSE06 hybrid functional,54 which models the short-range exchange energy of the electrons by fractions of Fock exchange and PBE exchange.55 Because the 2H-WS2 structure under study has hexagonal symmetry,56 Γ-centered k-meshes of 24 × 24 × 6 k-points are used for the Brillouin zone integrations in the structural relaxation. Spin-polarized calculations result in zero magnetic moment, in agreement with the nonmagnetic nature of bulk 2H-WS2, and therefore can be avoided. 2H-WS2 consists of slabs of W layers sandwiched between S layers. Bonding within the slabs is covalent, whereas bonding between the slabs is due to van der Waals forces. Standard density functionals do not include the van der Waals interaction, while the semiempirical DFT-D3 method by Grimme57 adds a van der Waals dispersion contribution, which depends on the geometry and exchange−correlation functional to the total energy. This method is used in the following. The electronic transport problem is solved by semiclassical Boltzmann transport theory within the constant relaxation time approximation for the electron−electron scattering processes. We use BoltzTraP58 for evaluating the transport coefficients. This code fits an analytical function to the ab initio electronic band structure by writing it as Fourier expansion,59,60 maintaining the symmetry of the crystal.61 Extra expansion coefficients guarantee smooth interpolation between data points.62 The transport coefficients at temperature T and Fermi level μ are calculated from the obtained analytical expression of the conductivity tensor σ̅αβ(ε), using the Fermi−Dirac distribution function f 0, as

⎡ ∂f (T , ε , μ) ⎤ 0 ⎥dε ∂ε ⎦

σαβ(T , μ) =

1 Ω

Sαβ(T , μ) =

1 eT Ωσαβ(T , μ)

el καβ (T , μ) =

1 e TΩ 2

∫ σαβ̅ (ε)⎢⎣−

(1)

⎡ ∂f (T , ε , μ) ⎤ 0 ⎥dε ∂ε ⎦ (2)

∫ σαβ̅ (ε)(ε − μ)⎢⎣−

⎡ ∂f (T , ε , μ) ⎤ 0 ⎥dε ∂ε ⎦

∫ σαβ̅ (ε)(ε − μ)2 ⎢⎣−

(3)

where Ω is the volume of the unit cell, and e is the charge of an electron. Specifically, we employ a mesh with 8100 k-points (455 in the irreducible Brillouin zone) to obtain the ab initio band structure and one with 40500 k-points for calculating the Fourier expansion. Heat conduction in semiconductors and insulators happens through lattice vibrations and is determined mainly by phonon−phonon scattering at high temperatures.63 The lattice vibrations are modeled assuming wavelike atomic displacements64 and solving the atomic equation of motion.65 Moreover, the Hamiltonian is expanded in the atomic displacements from the equilibrium positions, where the expansion coefficients of the potential B



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Table 1. Structural Parameters PBE HSE experiment

a (Å)

c (Å)

z

Eg (eV)

3.166 3.166 3.15583

12.431 12.431 12.3583

0.623 0.623 0.62256

1.0 1.4 1.486,87

Table 2. Effective Masses Evaluated from the HSE Electronic Band Structure electron hole

3. RESULTS AND DISCUSSION The structural parameters of relaxed WS2 are presented in Table 1 and show close agreement with the experimental values.56,83 The electronic band structure calculated using the PBE functional and considering the van der Waals interaction is shown in Figure 1a. We find an indirect band gap of 1.0 eV between the path Λ (Γ ↔ K) and the high symmetry point Γ, as expected from the literature,44,84,85 while the experimental band gap is 1.4 eV.86 This underestimation is critical for calculating the electronic contribution to the thermoelectric properties. The experimental band gap can be achieved by the HSE hybrid functional with 17%

effective mass (me)

location

path

0.55 0.61 0.61 0.61

Λ Λ Γ Γ

Λ→K Λ→Γ Γ→K Γ→L

Fock exchange, which is used in the following (Figure 1b). Electron and hole effective masses calculated at the conduction band minimum and valence band maximum are listed in Table 2. The xx and zz tensor components of the Seebeck coefficient as a function of the carrier concentration are shown in Figure 2. Single-crystal WS2 thermopower measurements in the temperature range of 303−423 K have found a p-type state with a carrier concentration of 1.17 × 1016 cm−3.88 The experimental Seebeck coefficient of 885 μV/K at 308 K agrees well with our

Figure 1. Electronic band structures calculated including the van der Waals interaction: (a) PBE and (b) HSE.

Figure 2. Seebeck coefficient as a function of the carrier concentration. C



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Figure 3. Electrical conductivity as a function of the carrier concentration.

Figure 4. Electronic contribution to the thermal conductivity as a function of the carrier concentration.

calculated value of 815 μV/K for the same hole concentration and temperature. This agreement is due to the accurate representation of the band structure by the HSE functional. The electrical conductivity evaluated from the band structure is addressed in Figure 3, and the electronic contribution to the thermal conductivity is addressed in Figure 4. The electrical conductivity from experiments on a WS2 polycrystal at 300 K with hole concentration 1.4 × 1017 cm−3 was reported to be 240 Ω−1m−1.89 We use the average value of the diagonal components of σ/τ at the experimental carrier concentration and

the experimental conductivity to determine the relaxation time τ = 3.72 × 10−14 s. This is the only experimental input used in our ab initio approach. The diagonal components of the high-frequency dielectric tensor are presented in Table 3, together with the calculated Born effective charges. The harmonic phonon dispersion relation obtained from the force constants and Table 3 is shown in Figure 5. There is excellent agreement with the bulk phonon dispersion reported by Molina-Sànchez et al.90 Results from Raman spectroscopy91,92 and inelastic neutron scattering93,94 D



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of the real acoustic branches at most k-vectors and thus carry a similar amount of heat. For example, midway between Γ and M, we obtain for the acoustic branches group velocities of 2.6 × 103, 2.4 × 103, and 2.7 × 103 m/s, whereas those of the quasi-acoustic branches are 2.8 × 103, 2.3 × 103, and 3.1 × 103 m/s. The lattice contributions to the thermal conductivity at different temperatures are addressed in Figure 6. The points represent the calculated values, while the line is obtained by fitting κlat ∝ 1/T, which is valid for anharmonic phonon−phonon interactions.63 Reflecting a large anisotropy, the ratio of the inplane and out-of-plane values is 32 at 300 K and 33 at 1500 K because the in-plane bonding is predominantly of strong covalent type and the out-of-plane bonding of weak van der Waals type. The order of magnitude of the out-of-plane values equals that of the thermal conductivity (≈ 2 W/mK96), as suggested for a good thermoelectric material. The contributions of the acoustic and quasi-acoustic branches (highlighted in Figure 6) sum to almost the total thermal conductivity of WS2. Branches 1−3 and branches 4−6 contribute similarly to the in-plane values, whereas branches 1−3 dominate the out-ofplane values. Comparison to the experiment is difficult because measurements always contain other phonon scattering events (such as boundary and impurity scattering). For this reason, the calculated lattice thermal conductivity always exceeds the experimental value. In addition, the experimental values include the electronic contributions. Therefore, we plot the sum of the electronic and lattice contributions in Figure 7. The out-of-plane thermal conductivity of a p-type WS2 thin-film measured at 300 K is reported to be 2.12 W/mK38 (without giving the hole concentration of the sample). Therefore, we estimate the hole concentration by comparing the Seebeck coefficient at 300 K (477 μV/K38) to Szz in Figure 2, as 5.7 × 1017 cm−3. Our total thermal conductivity at this hole concentration is 3.32 W/mK (Figure 7). As grains in polycrystalline materials are randomly oriented, a phonon approaching the grain boundary faces a different atomic arrangement and thus undergoes scattering.63 Such boundary scattering dominates in nanostructures, whereas phonon− phonon scattering dominates as long as the grain size exceeds the largest mean free path of the phonons. To demonstrate that the thermal conductivity decreases under boundary scattering, we plot the cumulative lattice thermal conductivity (kclat) as a function of the mean free path at 300 K in Figure 8a. At a given

Table 3. Diagonal Components of the High-Frequency Dielectric Tensor and Born Effective Charge in Electrons ε∞ Z*(W) Z*(S)

xx = yy

zz

13.68 −0.55 0.27

5.89 −0.40 0.20

Figure 5. Harmonic phonon dispersion.

(dots in Figure 5) show fair agreement. The differences (notably at the zone boundaries) could have two reasons: anharmonicity is relevant for the phonon dispersion, or there is a significant temperature dependence as the inelastic neutron scattering was carried out at 343 K. Layered compounds can show quasi-acoustic phonon branches (very-low-frequency optic branches).95 In our 2H structure, the three acoustic modes at the Γ point correspond to translations of the crystal as a whole in the three dimensions of space. All three atoms in a S−W−S slab (per unit cell) vibrate in phase with the three atoms in the next slab. For the optic 95 (2) modes with B(2) 2g and E2g symmetries three atoms in one slab vibrate in phase, while the three in the adjacent slab vibrate 180° out of phase. Because the only restoring force of these normal modes is the weak van der Waals interaction, they have very low frequencies and thus overlap with the acoustic modes (being well separated from the other optic modes) in Figure 5. These quasi-acoustic branches are important for the lattice thermal conductivity because they have group velocities similar to those

Figure 6. Lattice contribution to the thermal conductivity as a function of the temperature. E



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Figure 7. Total thermal conductivity as a function of temperature.

Figure 8. Cumulative lattice thermal conductivity as a function of the mean free path: (a) 300 K and (b) 1500 K.

are observed at typical carrier concentrations (1019−1020 cm−3) at which a narrow-gap semiconductor has the largest power factor.2,96 Because such carrier concentrations have been reported experimentally for both n-type87 and p-type97 WS2, it should be possible to achieve an excellent high temperature thermoelectric performance in WS2. The experimental ZT value of 4.5 × 10−5 for a polycrystalline p-type sample at 300 K40 agrees well with the calculated average value of 9.5 × 10−5 at the experimental hole concentration (2.28 × 1015 cm−3).40 Such a low value is of little use. Remembering from the previous paragraph that ZTzz is the largest tensor component, it is interesting to realize that it is possible to achieve films oriented along the z-direction by

mean free path this is the sum of the thermal conductivities due to all phonons with less or equal mean free paths. Figure 8a suggests that klat zz can be reduced to 50% by reducing the thickness of a WS2 film to 220 nm. Figure 8 paves the way to phonon engineering. Turning to the figure of merit, Figure 9 shows peaks at 7.3 × 1019 cm−3 electron concentration and 2.8 × 1019 cm−3 hole concentration, corresponding to out-of-plane ZT values of 0.90 and 0.77, respectively, at 1500 K. Maximal in-plane ZT values of 0.70 and 0.40 are observed at 3.5 × 1020 cm−3 electron concentration and 1.8 × 1020 cm−3 hole concentration, respectively. The low in-plane ZT values are due to the high inplane thermal conductivity. The maximal out-of-plane ZT values F



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Figure 9. Figure of merit as a function of the carrier concentration.

Figure 10. Electrical conductivity as a function of the carrier concentration evaluated from the PBE band structure.

We have also evaluated the thermoelectric properties from the PBE band structure after shifting the conduction states to match the experimental band gap. The electrical conductivity as a function of the carrier concentration in Figure 10 agrees well with the HSE result in Figure 3. We determine a relaxation time of τ = 4.36 × 10−14 s and show the figure of merit as a function of the carrier concentration in Figure 11. Comparison to the HSE results in Figure 9 demonstrates that complex systems can be studied by computationally less expensive PBE calculations once the band gap is adjusted.

exfoliation. Even polycrystalline samples with optimized carrier concentration would reach ZT = 0.76 (n-type) and ZT = 0.52 (p-type) at 1500 K (average of diagonal components). The cumulative lattice thermal conductivity at 1500 K gives an estimation of the dimensions below which an enhancement can be expected (Figure 8b). Films with thicknesses less than 403 nm would exceed the bulk ZT value and nanostructuring or grain refinement below 232 nm is expected to reduce the in-plane thermal conductivity. G



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Figure 11. Figure of merit as a function of the carrier concentration evaluated from the PBE band structure.

4. CONCLUSIONS We have used density functional theory for calculating the electronic band structure, phonon band structure, and interatomic force constants of WS2 in order to study the thermoelectric properties. Because a precise band structure (in particular band gap) is a critical input for an accurate prediction of the electronic contribution to the thermoelectric properties, we have employed the HSE hybrid functional. Starting from the electronic band structure, we have solved the semiclassical Boltzmann transport equation for the electrons within the constant relaxation time approximation. The obtained Seebeck coefficients show very good agreement with experimental values. We have established the relaxation time of the electrons by comparing the electrical conductivity to experimental values, and the lattice thermal conductivity has been evaluated from self-consistently calculated phonon lifetimes. Most of the lattice thermal conductivity (>97%) is due to the acoustic and quasiacoustic phonon branches. Except for the Seebeck coefficient, all thermoelectric quantities show a strong anisotropy. For the lattice thermal conductivity, the ratios of the in-plane and out-ofplane values exceed 30. Maximal ZT values of 0.90 and 0.77 for n-type and p-type WS2, respectively, are found at 1500 K and are comparable to the best thermoelectric materials currently used in devices.96 The required carrier concentrations should be well accessible. The large ZTzz component calls for z-axis orientation, which can be achieved by exfoliation techniques. In particular, exfoliated films with optimized carrier concentrations can be used for harvesting energy from the low-frequency region of the solar spectrum. The performance can be further improved by nanostructuring and grain refinement below the largest mean free path of the phonons. The theoretical approach demonstrated in this study is suggested as routine to be implemented in the future design of thermoelectric materials on the basis of the well-described important lattice contributions herein. Overcoming the constant relaxation time approximation for the electrons by a model that describes the electron relaxation as

a function of the temperature and carrier concentration could further improve the accuracy.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST). Computational resources were provided by the Supercomputing Laboratory of KAUST.

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REFERENCES

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