Article pubs.acs.org/cm
Thermoelectric Response of Bulk and Monolayer MoSe2 and WSe2 S. Kumar and U. Schwingenschlögl* PSE Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia ABSTRACT: We study the thermoelectric properties of bulk and monolayer MoSe2 and WSe2 by first-principles calculations and semiclassical Boltzmann transport theory. The lattice thermal conductivity is calculated using the selfconsistent iterative approach as well as the single-mode relaxation time approximation. The acoustical and optical contributions to the lattice thermal conductivity are evaluated along with the influence of the phonon mean free path. The employed methodology enables a quantitative comparison of the thermoelectric properties of transition-metal dichalcogenides. In particular, WSe2 is found to be superior to MoSe2 for thermoelectric applications.
Scuseria, and Ernzerhof (HSE)24 is used with 25% Hartree−Fock exchange and μ = 0.4 Å−1. The van der Waals interaction is taken into account by the semiempirical correction of Grimme.25 Integrals in reciprocal space are calculated using a Γ-centered 8 × 8 × 2 k-mesh. All structures are relaxed until the forces on the atoms have declined to 0.01 eV/Å, enforcing an energy convergence of 10−6 eV. In order to avoid interaction with periodic images, for the monolayer systems, vacuum slabs of 15 Å thickness are added. A refined 24 × 24 × 6 k-mesh is used for calculating the density of states (DOS). The thermoelectric properties are obtained using semiclassical Boltzmann transport theory and the rigid band approach as implemented in the BoltzTraP code.26 We employ the constant scattering time approximation,27 which is valid if the scattering time does not vary strongly with the energy on a scale of kBT. This formalism has accurately described the thermoelectric properties of many materials.28−32 Dense 24 × 24 × 6 and 40 × 40 × 1 k-meshes are introduced for the bulk (employing the HSE functional) and monolayer (employing the PBE functional) systems, respectively, to enable accurate Fourier interpolation of the Kohn−Sham eigenvalues. Using the ShengBTE code, the lattice thermal conductivity is obtained from the Boltzmann transport equation for phonons33,34 with second- and third-order interatomic force constants as input.35−37 Harmonic phonons are calculated by the Phonopy code38 using a 3 × 3 × 2 (4 × 4 × 1) supercell with 3 × 3 × 3 (3 × 3 × 1) k-mesh for the bulk (monolayer). The third-order force constants are obtained by a finite displacement approach, in which two atoms in the supercell are displaced simultaneously and the forces on the remaining atoms are calculated. An interaction range of 4.5 Å is considered. Well-converged 15 × 15 × 15 and 60 × 60 × 1 q-meshes are used for the bulk and monolayer systems, respectively.
1. INTRODUCTION Transition-metal dichalcogenides (TMDCs) are well-known materials for a variety of applications, making use of the catalytic activity1 and photoelectrochemical properties,2 for example. Recently, two-dimensional TMDCs have received considerable attention in photovoltaics,3 molecular sensing,4 photoluminescence,5−7 and nanoscale field effect transistors.8−11 Because selenides are more stable and resistant to oxidation in humid atmosphere than sulfides,10 high-mobility p- and n-type field effect transistors have been fabricated from monolayer WSe2.12 In addition, the band gap of monolayer MoSe2 (1.55 eV) is suitable for single-junction solar cells and photoelectrochemical cells,13 and heterojunctions of monolayer MoSe2 and WSe2 show new optoelectronic properties.14 TMDCs have been investigated intensively because of high Seebeck coefficients and low thermal conductivities.15−19 In general, the efficiency of a thermoelectric material to convert heat into electrical energy (and vice versa) is characterized by the dimensionless figure of merit ZT = S2σT/κ, where S is the Seebeck coefficient, σ the electrical conductivity, and κ = κe + κlatt the thermal conductivity (consisting of electronic and lattice contributions). According to the Wiedemann−Franz relation, κe is proportional to σT, so that an optimization of the figure of merit has to make a compromise between σ and κe. From a theoretical perspective, any simulation of thermoelectric properties of TMDCs has to take care of an accurate treatment of κlatt because the materials experimentally show large variations.18−21 We achieve this requirement by calculating the lattice thermal conductivity within first-principles theory using the Boltzmann transport equation for phonons.
3. RESULTS AND DISCUSSION Bulk MoSe2 and WSe2 have a 2H-polytype structure with space group P63/mmc (194) at room temperature. The unit cell comprises two Se−(Mo/W)−Se sandwiches, coupled by van der Waals forces and thus separated by 6.5 Å, in which the Mo/W atoms have trigonal prismatic coordination. There is a lateral shift
2. THEORETICAL METHODS First-principles calculations of bulk and monolayer MoSe2 and WSe2 have been conducted using density functional theory as implemented in the Vienna ab initio simulation package.22 A plane wave cutoff of 450 eV and the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE)23 to the exchange−correlation functional are used. To correctly describe the band gap of the bulk compounds, the computationally expensive hybrid functional approach of Heyd, © XXXX American Chemical Society
Received: November 19, 2014 Revised: January 9, 2015
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DOI: 10.1021/cm504244b Chem. Mater. XXXX, XXX, XXX−XXX
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Chemistry of Materials
Figure 1. Atomic structure. The Mo/W and Se atoms are represented by gray and green colors, respectively. Figure 2. Electronic band structure and DOS.
between the two sandwiches, as demonstrated in Figure 1. The layered structures give rise to quasi two-dimensional characters of the bulk compounds and also pave the way to monolayer systems, which consist of a single Se−(Mo/W)−Se sandwich with space group P6̅m2 (187) (see Figure 1), having no inversion symmetry. A comparison of the calculated structural parameters of bulk and monolayer MoSe2 and WSe2 with experimental values is given in Table 1. An accurate description of the band gap is key for correctly modeling the thermoelectric properties. In contrast to the PBE functional, the HSE functional achieves values close to the experimental band gaps of bulk MoSe2 (1.09 eV) and WSe2 (1.13 eV). On the other hand, for the monolayer systems, the band gaps obtained by the PBE functional (MoSe2: 1.44 eV; WSe2: 1.56 eV) and the HSE functional (MoSe2: 1.62 eV; WSe2: 1.72 eV) are similar, and also the trends of the band structures are the same. Therefore, we discuss in the following the PBE results for the monolayer systems. Band structures along with densities of states are shown in Figure 2. The bulk compounds have an indirect band gap with the conduction band minimum near the midpoint of the line Γ−K and the valence band maximum at the Γ point, whereas the monolayer systems have a direct band gap at the K point, which is interesting for optical applications. To maximize the figure of merit, the power factor S2σ should be large and κ small. TMDCs typically have large Seebeck coefficients16 and small densities of states at the Fermi level; see Figure 2. The conductivity σ = μne can be enhanced by increasing the DOS at the Fermi level (and thus the concentration n of
available charge carriers) or the mobility μ (which depends inversely on the effective mass). To mimick the effect of doping (p- and n-type), we use the rigid band approach, which assumes that the electronic bands are not modified by the doping and only the Fermi level is shifted appropriately. The obtained Seebeck coefficients of 690 and 680 μV K−1 for pristine bulk MoSe2 and WSe2, respectively, are reasonable estimates of the roomtemperature experimental values of 900 and 800 μV K−1.15,16 The positive values are a consequence of the smaller electron effective masses at the conduction band minimum along the line K−Γ as compared to the hole effective masses at the Γ point (0.47me versus 0.79me in MoSe2 and 0.45me versus 0.65me in WSe2). The Seebeck coefficients of monolayers MoSe2 and WSe2 are found to be 427 and 350 μV K−1, respectively, where experimental results are not available. The values for the monolayer systems are smaller than those for the bulk systems because of the larger DOS near the valence band maximum; see Figure 2. In-plane and out-of-plane power factors of the bulk and monolayer systems under doping are shown in Figures 3a−h and 4a−d, and corresponding electronic thermal conductivities are given in Figures 3i−p and 4e−h. We address wide ranges of doping (1018−5 × 1020 cm−3) and temperature (300−1200 K), noting that the melting points of MoSe2 and WSe2 are 1473 and 1773 K, respectively.39,40 As expected, under doping both S2σ/τ and κe/τ (τ being the relaxation time) are enhanced; see Figures 3 and 4. An increasing temperature has the same effect. The relaxation time is calculated by comparing experimental values of the hole conductivity of nominally undoped samples16,17 with the calculated values of σ/τ at room temperature. For WSe2, the in-plane and out-of-plane relaxation times are found to be 1.6 × 10−13 and 1.4 × 10−15 s, whereas we obtain for MoSe2 values of 5.7 × 10−15 and 8.0 × 10−17 s. These values are also used in the following for other doping levels and temperatures. The fact that in TMDCs the in-plane electrical conductivity is typically two orders of magnitude larger than the out-of-plane electrical conductivity17 explains the two orders of magnitude difference in the relaxation times. The power factor is larger for n- than that for p-type doping (in-plane and out-of-plane; bulk and monolayer systems) and increases for higher doping.
Table 1. Structural Parameters and Band Gaps of Bulk and Monolayer MoSe2 and WSe2, Compared with the Available Experimental Results
bulk MoSe2 monolayer MoSe2 bulk WSe2 monolayer WSe2
this work experiment52 this work experiment13 this work experiment53 this work experiment54
a (Å)
c (Å)
z
Eg (eV)
3.298 3.299 3.323 3.299 3.312 3.282 3.320 3.286
13.023 12.938
0.623 0.621
12.872 12.960
0.621 0.621
1.09 1.09 1.44 1.55 1.13 1.20 1.56 1.64 B
DOI: 10.1021/cm504244b Chem. Mater. XXXX, XXX, XXX−XXX
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Figure 3. In-plane and out-of-plane thermoelectric power factors and electronic thermal conductivities of p- and n-type bulk MoSe2 and WSe2.
harmonic phonon dispersion shown in Figure 5 (along with the available experimental data for comparison41,42). The bulk compounds have 18 vibrational modes at the Brillouin zone center with the irreducible representation Γ = A1g + 2A2u + B1u + 2B2g + E1g + 2E1u + E2u + 2E2g, where A1g, E1g, and E2g are Ramanactive, E1u and A2u are infrared-active, and the remaining modes are optically inactive. The A1g mode represents the out-of-plane vibrations of the Se atoms, while the two E2g modes belong to the in-plane vibrations and motion of the two Se−(Mo/W)−Se sandwiches against each other.43 The higher frequency obtained for the E2g mode as compared to the A1g mode in bulk MoSe2 and the reversed order in bulk WSe2 agree with experimental findings.7 The monolayer systems have nine vibrational modes at the Brillouin zone center with the irreducible representation Γ =
The electronic thermal conductivity is addressed in Figures 3i−p and 4e−h. As expected, the values increase with the temperature and the doping. The lattice thermal conductivity is calculated using the self-consistent iterative approach as well as the single-mode relaxation time approximation (SMRTA), for which we have αβ κlatt =
ℏ2 kBT 2 ΩN
∑ vqα,jvqβ,jωq2,jnq,j(nq,j + 1)τq,j q, j
(1)
where Ω is the volume of the unit cell, j is the branch index, and N is the number of q points used to sample the reciprocal space. Moreover, ωq,j is the angular velocity of mode (q,j), vαq,j = ∂ωq,j/ ∂qα is the group velocity, and nq,j is the equilibrium population at temperature T. The angular velocities are obtained from the C
DOI: 10.1021/cm504244b Chem. Mater. XXXX, XXX, XXX−XXX
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Chemistry of Materials
Figure 5. Phonon dispersions of bulk and monolayer MoSe2 and WSe2 along with experimental Raman frequencies (green dots). Raman-active modes are labeled green, and other modes are labeled red. E′ is both Raman- and infrared-active.
Figure 4. Thermoelectric power factors and electronic thermal conductivities of monolayer MoSe2 and WSe2.
A′1 + 2A″2 + 2E′ + E″, where A′1 and E″ are Raman-active, A″2 is infrared-active, and E′ is both Raman- and infrared-active. The reversed order of the E′ and A1′ modes in the two monolayer systems confirms previous first-principles calculations.44,45 The calculated lattice thermal conductivities (Figures 6 and 7) take into account three-phonon scattering, involving normal and Umklapp scattering, isotopic scattering, and boundary scattering. Experiments find for MoS2 a value of κlatt = 34.5 ± 4 W/mK at room temperature for a sample size of 1 μm21 (which we use for defining the boundary scattering term). Normal processes do not result in resistance to heat flow, whereas Umklapp scattering, isotopic scattering, and boundary scattering are resistive and
Figure 6. Lattice thermal conductivities of bulk MoSe2 and WSe2. D
DOI: 10.1021/cm504244b Chem. Mater. XXXX, XXX, XXX−XXX
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bulk and Figure 7a,b for the monolayer systems. The SMRTA assumes that all phonon modes except for one are in thermal equilibrium, whereas the iterative approach assumes that all phonon modes are thermally disturbed and thus simultaneously contribute to the scattering. Processes involving more than three phonons are neglected owing to their small contributions.46 The SMRTA treats normal processes as resistive and thus works well only if Umklapp processes dominate.47 The difference between the two approaches declines with increasing temperature because Umklapp processes become more important; see Figures 6a,b and 7a,b. The lattice thermal conductivity of bulk MoSe2 and WSe2 is calculated as 2/3·κin‑plane + 1/3·κout‑plane . In both compounds, latt latt κin‑plane dominates (a bit less in WSe2); see the results in Figure latt 6c,d, which are obtained using the iterative approach. The thermal conduction through the van der Waals gap is poor. Besides the acoustical modes, only the three lowest optical modes (quasi-acoustical modes) give significant contributions to κlatt. In the monolayer systems, the acoustical modes dominate completely. Because the mass of W is larger than that of Mo, we have larger frequencies for MoSe2 than for WSe2. At 300 K, we obtain for bulk MoSe2 and WSe2 similar values for κlatt, whereas the value for monolayer MoSe2 is almost twice that of monolayer WSe2. This behavior is explained by the fact that the group velocities of the acoustical modes in the long-wavelength limit along Γ−M are on average 10% larger. Experiments find for bulk MoSe2 and WSe2 values of κ = 2.315 and 1−4 W/mK,18−20 respectively. Even if we assume that the lattice thermal conductivity dominates in the experimental values, there is an order of magnitude difference to our results, which probably is due to point defects or dislocations. Stacking faults (leading to
Figure 7. Lattice thermal conductivities of monolayer MoSe2 and WSe2.
therefore reduce κlatt. We observe that the SMRTA gives always smaller values than the iterative approach; see Figure 6a,b for the
Figure 8. Figure of merit of bulk MoSe2 and WSe2. E
DOI: 10.1021/cm504244b Chem. Mater. XXXX, XXX, XXX−XXX
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Chemistry of Materials localization of the vibrational modes20) may also reduce κout‑plane , latt leading to a decrease of κlatt. The cumulative lattice thermal conductivity c κlatt (Λ ) =
∑
4. CONCLUSION MoSe2 and WSe2 in bulk and monolayer form have been studied by first-principles calculations, and the thermoelectric properties have been analyzed using Boltzmann transport equations for the electrons and phonons. The inclusion of phononic contributions is essential for an accurate description of the materials. Both pand n-type doping have been addressed, employing the rigid band approximation. The iterative approach and the SMRTA have been compared and the differences explained. For bulk MoSe2 and WSe2, the three lowest optical modes are found to be of almost equal importance as compared to the acoustical modes for the lattice thermal conductivity, whereas in the monolayer systems, the acoustical modes clearly dominate. The almost doubled lattice thermal conductivity of monolayer MoSe2, as compared to monolayer WSe2, is due to higher phonon frequencies and group velocities. Consideration of the phonon mean free path suggests that nanostructuring can significantly reduce the lattice thermal conductivity. The thermoelectric response of WSe2 turns out to be clearly superior to that of MoSe2, and the origin of this difference is clarified on an atomic level. The high figures of merit of bulk and monolayer WSe2 in a wide temperature range are interesting from the application point of view.
αβ κlatt
vq, jτq, j
