MoS₂ nano flakes with self-adaptive contacts for efficient thermoelectric energy harvesting

Abstract

We examine the potential of the low-dimensional material MoS₂ for the efficient conversion of waste heat to electricity via the Seebeck effect. Recently monolayer MoS₂ nano flakes with self-adaptive Mo₆S₆ contacts were formed, which take advantage of mechanical stability and chemical covalent bonding to the MoS₂. Here, we study the thermoelectric properties of these junctions by calculating their conductance, thermopower and thermal conductance due to both electrons and phonons. We show that thermoelectric figures of merit ZT as high as ∼2.8 are accessible in these junctions, independent of the flake size and shape, provided the Fermi energy is close to a band edge. We show that Nb dopants as substituents for Mo atoms can be used to tune the Fermi energy, and despite the associated inhomogeneous broadening, room temperature values as high as ZT ∼ 0.6 are accessible, increasing to 0.8 at 500 K.

---

Introduction

The quest for high-performance thermoelectric devices and materials, which convert waste heat to reusable electrical energy with high efficiency, is a long-sought-after goal of modern materials science.^1–4 A thermoelectric device or material is characterised by its thermoelectric figure of merit ZT = S²GT/κ, where S, G, T, κ represent the Seebeck coefficient (thermopower), electrical conductance, temperature and thermal conductance. The latter is given by κ = κ_e + κ_p, where κ_e and κ_p are the electron and phonon contributions to thermal conductance respectively.^5–7 Consequently an efficient thermoelectric device requires a large Seebeck coefficient and electrical conductance and simultaneously low thermal conductance. The Mott formula⁸ indicates that a large Seebeck coefficient can be obtained if the Fermi energy E_F happens to coincide with a steep slope of electron transmission coefficient T(E), which describes the passage of electrons of energy E from the source to the drain of a device. This is a good approximation provided T(E) remains linear in the scale of k_BT around Fermi energy where k_B is Boltzmann's constant.⁸

In the past couple of decades, although the thermoelectric performance of the bulk, thin films, and superlattices of Bi, Te, or Sb alloys materials have improved,⁹ they are not yet sufficiently efficient for future energy demands and furthermore, some of them are toxic with limited global supply.¹⁰ Recently the potential of the nanoscale devices composed of organic molecules or 2D materials sandwiched between metallic electrodes was recognized. Molecular-scale devices are particularly interesting, because their transport properties may be tailored by chemical modification of the active part of the device,^11–14 varying the contacting configuration,^15–17 device architecture^18–20 and the Fermi level alignment by gating or doping.^21–26 Despite the fact that the nanoelectronic systems are of great interest, it remains a challenge to identify suitable molecules and contacting strategies^27,28 which overcome inhomogeneous broadening and junction variability.

Two dimensional materials provide an alternative approach to thermoelectricity in low-dimensional systems. Graphene is the most widely explored 2D material, but is not suitable since the pristine graphene does not have a bandgap and the material has a high in-plane thermal conductance. As an alternative to graphene, transition metal chalcogenides such as monolayer molybdenum disulfide (MoS₂) may be attractive alternatives,^29,30 since their band gaps could be used to optimise thermoelectricity. Here we show that this is indeed the case for monolayer MoS₂, which has a direct bandgap of around 1.6 eV (see Fig. 1 in the ESI†).

Although some contacting strategies have been reported to characterize the electronic and thermoelectric properties of MoS₂,^20,31 the contacting to metallic electrodes remains a challenge. Recently, it was shown that using a focused electron beam, structures consisting of MoS₂ connected directly to Mo₆S₆ could be formed.³² Mo₆S₆ nanowires (see Fig. S2 in ESI†) can be used as electrodes, because they possesses metallic properties and a robust structural conformation.^33,34 Large scale production of MoS₂ nano flakes with self-adaptive Mo₆S₆ contacts is possible, since their electronic structures are insensitive to the shape of MoS₂ monolayers.³² Furthermore, it has been reported that parallel-wire bundles of Mo₆S₆ nanowires are metallic and mechanically stable.³⁵ Mo₆S₆ nanowires might therefore be used as a flexible nano wires in electronic devices, because bending does not significantly change their electronic properties.³⁶

Although these junctions have been realized, their potential as thermoelectric devices is unexplored. These devices are advantageous, because they provide new direct and covalent contacting possibilities to MoS₂ edges. Furthermore, MoS₂ has shown high thermopower and thermoelectric properties using other contacting strategies.^20,31 In what follows, we show that flexible routes to tailoring their thermoelectric properties such as doping or gating of both MoS₂ and Mo₆S₆ make such devices very attractive for future thermoelectricity.

Results and discussion

In this letter, we calculate electronic and vibrational properties of MoS₂ monolayers connected to Mo₆S₆ nanowire electrodes (Fig. 1) and show that high Seebeck coefficients and low phononic thermal conductances lead to thermoelectric figures of merit as high as ZT ∼ 2.8. We then demonstrate that thermoelectric properties of these junctions could be tuned by introducing Nb dopants into either MoS₂ nano flakes or Mo₆S₆ electrodes.

Fig. 1 (a) Schematic of Mo₆S₆ nanowire–Mo₂S–Mo₆S₆ nanowire junction. (b) The Mo₆S₆–Mo₂S–Mo₆S₆ junction used in the simulation: yellow atoms represent sulphur and green represents molybdenum.

Fig. 1 shows a thermoelectric device consisting of a central MoS₂ monolayer connected to Mo₆S₆ nanowires as current-carrying leads. The device geometry is similar to that obtained experimentally³² by steering a focused electron beam onto a single MoS₂ monolayer. After making holes in the monolayer, the remaining slab reconstructs to form self-adaptive Mo₆S₆ leads contacted to MoS₂ nano flakes.³²

Fig. 2 shows the electron transmission coefficient obtained from the density-functional-theory (DFT) mean field Hamiltonian of the converged ground state geometry of the device (see methods for more details) shown in the Fig. 1b. There exists a transmission gap of approximate 1.65 eV for electron energies E between −0.55 eV and 1.1 eV (relative to the DFT-predicted Fermi energy E^DFT_F). This is due to the gap of the pristine two-dimensional Mo₂S (see Fig. S1 in the ESI†) and is in good in agreement with previous studies.^30,37 It is known that a step-like transmission coefficient can lead to a high Seebeck coefficient, provided the Fermi energy E_F lies close to such steps.^8,38 In our case, the step like transmission functions around E_F − E^DFT_F = −0.55 eV and 1.1 eV are due to the valance and conduction bands of MoS₂.

Fig. 2 Electron transmission of the junction in Fig. 1.

From the transmission function in Fig. 2, the conductance, the Seebeck coefficient, the thermal conductance due to the electrons and electronic thermoelectric figure of merit can be obtained (see methods). Fig. 3a shows the electrical conductance for different Fermi energies at room temperature. Due to the thermal averaging of T(E) around −0.55 eV and 1.1 eV, two peaks are obtained around these energies. They also have high slope and therefore, a high Seebeck coefficient is also obtained around these energies (Fig. 3b). Fig. 3c and d show the electronic thermal conductance and the electronic contribution to the figure of merit ZT_e. Fig. 3d depicts the variation of ZT_e with the Fermi energy. There appear three large resonances up to 50, 100 and 80 around −0.5, 0.5 and 1 eV. In Fig. 3, the red solid curves are exact values obtained using eqn (2)–(5). The blue dashed curves show the prediction of the Wiedemann-Franz law, which is valid provided T(E) varies approximately linearly with E near E = E_F on the scale of k_BT. Although the two are in close agreement, the Wiedemann-Franz law tends to overestimate the thermal conductance due to electrons and therefore underestimate ZT_e. However, ZT_e neglects the contribution from phonons in the denominator and therefore to obtain the full ZT, we now compute the thermal conductance due to phonons.

Fig. 3 Electronic contribution to thermoelectric properties as a function of the pristine DFT-predicted Fermi energy at 300 K. (a) Electrical conductance G, (b) Seebeck coefficient S, (c) electronic contribution to thermal conductance κ_e and (d) electronic thermoelectric figure of merit ZT_e.

The results of our phonon transport calculation are shown in Fig. 4, where Fig. 4a shows the phonon transmission spectrum and Fig. 4b shows the corresponding phonon thermal conductance. Compared to the electronic thermal conductance within the gap, the phonon (0.044 nW K⁻¹) contribution to thermal conductance is much higher. Consequently, the total ZT is lower than ZT_e. As shown in Fig. 5, a value as high as ZT ∼ 2.8 at 300 K is obtained around E_F = −0.5 eV, which is higher than unity and higher than currently-reported values for other materials at room temperature.^39–41 Two peaks in the thermoelectric figure of merit curve at −0.5 eV and 1 eV are due to two step-like transmission features at −0.5 eV and 1 eV.

Fig. 4 Phonon transmission spectra (a) and the corresponding thermal conductance (b).

Fig. 5 (a) The thermoelectric figure of merit at 300 K for the structure shown in Fig. 1(b). (b)–(d) The thermoelectric figure of merit as a function of temperature for three particular Fermi energies (E_F − E^DFT_F = −0.2 eV, E_F − E^DFT_F = 0 eV, E_F − E^DFT_F = 0.2 eV) for the structure shown in Fig. 1(b). These calculations are valid provided the forces entering the phonon dynamical matrix are harmonic. Anharmonic effects are negligible, provided the temperature does not approach the melting point of the material. The melting point of MoS₂ is greater than 1400 K and therefore we have restricted our analysis to approximately half that value.

To demonstrate that the high ZT at E_F − E^DFT_F = −0.5 eV and 1 eV is due to the valence and conduction bands of MoS₂ and not an edge effect, we consider other junctions shown in Fig. S3 and S5 in the ESI,† obtained by increasing the size of the sample, changing the edge shape and applying periodic boundary conductions in transverse direction. Fig. S3–S5† show that their corresponding transmission coefficients possess two main features: step-like transmission features associated with the valence and conduction bands of the MoS₂ monolayer around E = −0.5 eV and 1 eV and sharp resonances with small width inside the gap. Local density of states calculations around the DFT Fermi energy (Fig. S4 in the ESI†) reveal that these resonances are due to localized states at the edges of MoS₂ and are sensitive to the shape of the edges and size of the flake. However, the features due to the valence and conduction bands of MoS₂ are resilient and are less dependent on the details of the junction. Furthermore, the transmission amplitude and slope is higher close to the conduction and valence band edges, which is promising for an efficient thermoelectric device. High ZT is therefore obtained regardless of the shape of the MoS₂ flake as shown in Fig. S8 in the ESI† at E_F − E^DFT_F = −0.5 eV and 1 eV. These results indicate that the higher ZT peak is mainly due to the valence band edge of the MoS₂ monolayer and the edge states do not play a significant role.

In reality, the Fermi energy may be determined by extrinsic factors such as doping. Indeed Nb atoms were used in the past to tune the Fermi energy of MoS₂ to form a p-type semiconductor.^21,23 To demonstrate that in the presence of dopants, a high ZT is accessible in these devices, we now show that in the both Mo₆S₆ electrodes and MoS₂ flakes, Nb doping can be used to shift the position of the valence band edge towards the Fermi energy. Fig. S9 of the ESI† shows results for six different dopant configurations. By replacing Mo atoms by Nb substituents in the monolayer MoS₂ flake, the Fermi level shifts towards the valence band and simultaneously new transmission peaks due to defect states near the valence band are formed. The precise value of ZT is sensitive to the positions and concentration of the dopants, because multiple scattering of electrons between dopants placed in the different locations, causes the transmission coefficient to change (Fig. S9 in the ESI†). In a real device consisting of many structures such Fig. 1b and Fig S9,† placed in parallel, the ensemble average of transport coefficients are relevant, as defined by the equations in section 10 of the ESI.†Fig. 6a shows the resulting ensemble averaged figure of merit ZT_avversus Fermi energy at one particular temperature and Fig. 6(b–d) show ZT_av as a function of temperature for three different Fermi energies. Fig. 6b shows that even after allowing for inhomogeneous broadening due to random locations of dopants, room temperature values of ZT_av = 0.6 are possible, increasing to 0.8 at 500 K.

Fig. 6 (a) The ensemble average of the total thermoelectric figure of merit at 300 K for the configurations with p-type dopants Nb replacing Mo atoms in ESI Fig. S9.† (b)–(d) The ensemble average of the total thermoelectric figure of merit as the function of temperature when E_F − E^DFT_F = −0.2 eV, E_F − E^DFT_F = 0 eV, E_F − E^DFT_F = 0.2 eV for the configurations with p-type dopants Nb replacing Mo atoms in ESI Fig. S9.†

Conclusion

In summary, we have studied electron and phonon transport through MoS₂ nano flakes with self-adaptive Mo₆S₆ contacts and demonstrated that large Seebeck coefficients and electrical conductance, combined with low thermal conductance due to electrons and phonons lead to large room temperature thermoelectric figures of merit up to 2.8. Even after allowing for inhomogeneous broadening due to random locations of dopants, room temperature values of ZT_av = 0.6 are possible, increasing to 0.8 at 500 K. This demonstrates that MoS₂ nano flakes are an attractive material for the design of high efficiency, nanoscale, thermoelectric energy harvesters and conversely for nanoscale Peltier cooling.

Computational methods

The structures of Mo₂S–Mo₆S₆ junction obtained in the experiments³² were optimized using the first-principles DFT code SIESTA⁴² with a double-ζ (DZ) basis set. Subsequently, the nanoscale Mo₆S₆–Mo₂S–Mo₆S₆ sandwich device was constructed. From the relaxed xyz coordinates of the system, sets of xyz coordinates were generated by displacing each atom in positive and negative x, y, and z directions by δq′ = 0.01 Å. The forces in three directions q_i = (x_i, y_i, z_i) on each atom were then calculated by DFT without geometry relaxation. These values of forces were combined with the method described in ref. 7 to calculate the dynamical matrix and thermal conductance due to phonons. Furthermore, the mean-field Hamiltonian and overlap matrices were extracted from the first-principles calculation and used to obtain the thermoelectric properties of the devices using Gollum.⁴³ The electron transmission coefficient T(E) as a function of energy is calculated through the formula:

T(E) = Tr[Γ_R(E)G^R(E)Γ_L(E)G^R†(E)] (1)

where Γ_L,R(E) = i(∑_L,R(E) − ∑^†_L,R(E))/2 is the anti-Hermitian part of self-energy which describes the broadening of electron transmission resonance; ∑_L,R(E) is the self-energy due to the left or right electrodes; G^R, G^R† are the retarded and advanced Green's function respectively. The thermoelectric properties can be obtained from:

G = G₀L₀ (2)

where

The electrical conductance G, Seebeck coefficient S and thermal conductance due to electrons κ_e can be combined to obtain electronic thermoelectric figure of merit ZT_e. By including the thermal conductance due to phonons κ_p, the total thermoelectric figure of merit ZT is calculated. T is the mean temperature (T₁ + T₂)/2, G₀ = 2e²/h is the conductance quantum; h is the Planck's constant; e is the charge of electron; f(E) = (1 + exp(E − E_F/k_BT))⁻¹ is the Fermi–Dirac probability distribution function and E_F is the Fermi energy. The relationship between ZT and the power efficiency is discussed in section 11 of the ESI.†

Author information

HS and CJL conceived and conducted the project. Q. W. and H. S. carried out the calculations. All authors took part in the discussions and writing the manuscript.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

H. S. acknowledges the Leverhulme Trust for Leverhulme Early Career Fellowship no. ECF-2017-186. Further support from the UK EPSRC is acknowledged, through grant no. EP/M014452/1, EP/P027156/1, EP/N017188/1 and EP/N03337X/1. Support from the European Commission is provided by the FET Open project 767187 – QuIET.

Notes and references

1. A. L. Moore and L. Shi, Emerging challenges and materials for thermal management of electronics, Mater. Today, 2014, 17, 163–174.
2. Y. Kim, W. Jeong, K. Kim, W. Lee and P. Reddy, Electrostatic control of thermoelectricity in molecular junctions, Nat. Nanotechnol., 2014, 9, 881–885.
3. J. R. Widawsky, P. Darancet, J. B. Neaton and L. Venkataraman, Simultaneous determination of conductance and thermopower of single molecule junctions, Nano Lett., 2012, 12, 354–358.
4. H. Sadeghi, S. Sangtarash and C. J. Lambert, Electron and heat transport in porphyrin-based single molecule transistors with electro-burnt graphene electrodes, Beilstein J. Nanotechnol., 2015, 6, 1413–1420.
5. P. Reddy, S. Jang, R. A. Segalman and A. Majumdar, Thermoelectricity in molecular junctions, Science, 2007, 315, 1568–1571.
6. C. A. Perroni, D. Ninno and V. Cataudella, Electron-vibration effects on the thermoelectric efficiency of molecular junctions, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 275303.
7. H. Sadeghi, S. Sangtarash and C. J. Lambert, Oligoyne Molecular Junctions for Efficient Room Temperature Thermoelectric Power Generation, Nano Lett., 2015, 15, 7467–7472.
8. C. J. Lambert, H. Sadeghi and Q. H. Al-galiby, Quantum-interference-enhanced thermoelectricity in single molecules and molecular films, C. R. Phys., 2016, 17, 1084–1095.
9. X. Zhang and L. Zhao, Thermoelectric materials: Energy conversion between heat and electricity, J. Mater., 2015, 1, 92–105.
10. L. E. Bell, Cooling, heating, generating power, and recovering waste heat with thermoelectric systems, Science, 2008, 321, 1457–1461.
11. Q. H. Al-Galiby, H. Sadeghi, L. A. Algharagholy, I. Grace and C. Lambert, Tuning the thermoelectric properties of metallo-porphyrins, Nanoscale, 2016, 8, 2428–2433.
12. M. Noori, H. Sadeghi and C. J. Lambert, High-performance thermoelectricity in edge-over-edge zinc-porphyrin molecular wires †, Nanoscale, 2017, 9, 5299–5304.
13. S. V. Aradhya and L. Venkataraman, Single-molecule junctions beyond electronic transport, Nat. Nanotechnol., 2013, 8, 399.
14. M. L. Perrin, R. Frisenda, M. Koole, J. S. Seldenthuis, J. A. C. Gil, H. Valkenier, J. C. Hummelen, N. Renaud, F. C. Grozema, J. M. Thijssen, D. Dulić and S. J. H. van der Zant, Large negative differential conductance in single-molecule break junctions, Nat. Nanotechnol., 2014, 9, 830–834.
15. F. Schwarz and E. Lörtscher, Break-junctions for investigating transport at the molecular scale, J. Phys.: Condens. Matter, 2014, 26, 474201.
16. P. Moreno-García, M. Gulcur, D. Z. Manrique, T. Pope, W. Hong, V. Kaliginedi, C. Huang, A. S. Batsanov, M. R. Bryce, C. Lambert and T. Wandlowski, Single-molecule conductance of functionalized oligoynes: Length dependence and junction evolution, J. Am. Chem. Soc., 2013, 135, 12228–12240.
17. H. Sadeghi, S. Sangtarash and C. J. Lambert, Robust molecular anchoring to graphene electrodes, Nano Lett., 2017, 17, 4611–4618.
18. M. Noori, H. Sadeghi, Q. Al-Galiby, S. W. D. Bailey and C. J. Lambert, High cross-plane thermoelectric performance of metallo-porphyrin molecular junctions, Phys. Chem. Chem. Phys., 2017, 19, 17356–17359.
19. Q. Wu, H. Sadeghi, V. M. García-Suárez, J. Ferrer and C. J. Lambert, Thermoelectricity in vertical graphene-C60-graphene architectures, Sci. Rep., 2017, 7, 11680.
20. H. Sadeghi, S. Sangtarash and C. J. Lambert, Cross-plane enhanced thermoelectricity and phonon suppression in graphene/MoS 2 van der Waals heterostructures, 2D Mater., 2017, 4, 15012.
21. K. Dolui, I. Rungger, C. Das Pemmaraju and S. Sanvito, Possible doping strategies for MoS2 monolayers: An ab initio study, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 075420.
22. Q. H. Al-Galiby, H. Sadeghi, D. Z. Manrique and C. J. Lambert, Tuning the Seebeck coefficient of naphthalenediimide by electrochemical gating and doping, Nanoscale, 2017, 9, 4819–4825.
23. S. Gemming, G. Seifert and I. Vilfan, Li doped Mo6S6 nanowires: Elastic and electronic properties, Phys. Status Solidi B, 2006, 243, 3320–3324.
24. M. Ghorbani-Asl, A. N. Enyashin, A. Kuc, G. Seifert and T. Heine, Defect-induced conductivity anisotropy in MoS 2 monolayers, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 245440.
25. H. Sadeghi, S. Sangtarash and C. J. Lambert, Enhanced thermoelectric efficiency of porous silicene nanoribbons, Sci. Rep., 2015, 5, 9514.
26. R. Gillen, J. Robertson and J. Maultzsch, Indirect doping effects from impurities in MoS2/h-BN heterostructures, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 75437.
27. W. Hong, D. Z. Manrique, P. Moreno-García, M. Gulcur, A. Mishchenko, C. J. Lambert, M. R. Bryce and T. Wandlowski, Single molecular conductance of tolanes: Experimental and theoretical study on the junction evolution dependent on the anchoring group, J. Am. Chem. Soc., 2012, 134, 2292–2304.
28. Q. Wu, X. Zheng, X. Shi, J. Lan, H. Hao and Z. Zeng, Electron transport enhanced by electrode surface reconstruction: a case study of C 60 -based molecular junctions, RSC Adv., 2014, 4, 44718–44725.
29. C. K. Gan, Theoretical study of thermoelectric properties of few-layer MoS2 and WSe2. Phys, Chem. Chem. Phys., 2014, 16, 10866–10874.
30. E. S. Kadantsev and P. Hawrylak, Electronic structure of a single MoS 2 monolayer, Solid State Commun., 2012, 152, 909–913.
31. J. Wu, H. Schmidt, K. K. Amara, X. Xu, G. Eda and B. Özyilmaz, Large thermoelectricity via variable range hopping in chemical vapor deposition grown single-layer MoS2, Nano Lett., 2014, 14, 2730–2734.
32. J. Lin, O. Cretu, W. Zhou, K. Suenaga, D. Prasai, K. I. Bolotin, N. T. Cuong, M. Otani, S. Okada, A. R. Lupini, J.-C. Idrobo, D. Caudel, A. Burger, N. J. Ghimire, J. Yan, D. G. Mandrus, S. J. Pennycook and S. T. Pantelides, Flexible metallic nanowires with self-adaptive contacts to semiconducting transition-metal dichalcogenide monolayers, Nat. Nanotechnol., 2014, 9, 436–442.
33. A. Leen, S. Nano and S. Facilities, Torsional Deformations in Sub-Nanometer MoS Interconnecting Wires, Nano Lett., 2015, 16, 1210–1217.
34. B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti and A. Kis, Single-layer MoS2 transistors, Nat. Nanotechnol., 2011, 6, 147–150.
35. J. Kibsgaard, A. Tuxen, M. Levisen, E. Levisen, S. Gemming, G. Seifert, J. V. Lauritsen and F. Besenbacher, Atomic-scale structure of Mo6S6 nanowires, Nano Lett., 2008, 8, 3928–3931.
36. I. Popov, S. Gemming, S. Okano, N. Ranjan and G. Seifert, Etectromechanical switch based on Mo6S6 nanowires, Nano Lett., 2008, 8, 4093–4097.
37. D. Costanzo, S. Jo, H. Berger and A. F. Morpurgo, Gate-induced superconductivity in atomically thin MoS2 crystals, Nat. Nanotechnol., 2016, 11, 339–344.
38. H. Sadeghi, S. Sangtarash and C. J. Lambert, Enhancing the thermoelectric figure of merit in engineered graphene nanoribbons, Beilstein J. Nanotechnol., 2015, 6, 1176–1182.
39. H. Liu, X. Shi, F. Xu, L. Zhang, W. Zhang, L. Chen, Q. Li, C. Uher, T. Day and G. J. Snyder, Copper ion liquid-like thermoelectrics, Nat. Mater., 2012, 11, 422–425.
40. L. Han, D. V. Christensen, A. Bhowmik, S. B. Simonsen, L. T. Hung, E. Abdellahi, Y. Z. Chen, N. V. Nong, S. Linderoth and N. Pryds, Scandium-doped zinc cadmium oxide as a new stable n-type oxide thermoelectric material, J. Mater. Chem. A, 2016, 4, 12221–12231.
41. R. Chetty, A. Bali and R. C. Mallik, Tetrahedrites as thermoelectric materials: an overview, J. Mater. Chem. C, 2015, 3, 12364–12378.
42. J. M. Soler, E. Artacho, J. D. Gale, A. Garcìa, J. Junquera, P. Ordejòn and D. Sànchez-Portal, SIESTA Method For Ab Initio Order-N Materials Simulation, J. Phys.: Condens. Matter, 2002, 14, 2745–2779.
43. J. Ferrer, C. J. Lambert, V. M. García-Suárez, D. Z. Manrique, D. Visontai, L. Oroszlany, R. Rodríguez-Ferradás, I. Grace, S. D. Bailey, K. Gillemot, H. Sadeghi and L. A. Algharagholy, GOLLUM: A next-generation simulation tool for electron, thermal and spin transport, New J. Phys., 2014, 16, 093029.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8nr01635f

---