Nanotechnology, Seebeck coefficient, figure of merit ZT, Two dimensional materials

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Guided by the attached article, write a literature review. Discuss recent research(4 articles that are not discussed in the attached word document) that address seebeck effect, peltier effect, figure of merit (ZT). Organize your work like the one in the attached word document.

1 Literature review There has been an increasing wave of research on 2D materials, with a good number of researchers focussing on thermoelectric properties of 2D single element materials and 2D heterostructure materials. In the 1990s, the thermoelectric properties of a large number of nanomaterials have been widely recognized, and some recent scientists have studied this feature more deeply 1–9. A large number of theoretical and experimental studies have described the thermal and thermoelectric properties of graphene/h-BN heterostructure10. 1.1 2009 The measurement techniques for in-plane Seebeck and four-probe conductivity were first demonstrated on monolayer graphene samples in 200911–13.Anomalous Seebeck (Sxx = ΔVxx/ΔTxx) and Nernst (Sxy = ΔVxy/ΔTxx) signals were observed, in violation of the well-accepted Mott relation for metallic samples. This was especially seen to be true close to the charge neutrality point (CNP) away from the highly doped degenerate limit, where electron-hole puddles were purported to be present. In this resistive state, the carrier density is strongly inhomogeneous and also the effect of impurities was found to be nontrivial. The manifestation of this on the thermoelectric power (Seebeck and Nernst) was explained well using an effective medium theory14. 1.2 2015 Wang et al. used the first-principles density functional calculations combined with the nonequilibrium Green’s function to simulate the thermal transport and thermoelectric properties of the graphene/h-BN heterostructure in the AB stacking mode 15. They observed that thermal transport characteristics of graphene in the heterostructure are lower than that of pure graphene, but the Seebeck coefficient is enhanced due to the opening of the graphene band gap. The figure of merit, ZT of the graphene/h-BN superlattice is larger than that of pure graphene, with 44% enhancement15. From the formula 𝑍𝑇 = (𝑆^2 𝜎)𝑇/𝑘 , it can be seen that all of the factors that affect ZT are enhanced and lead to the enhancement of ZT itself. 1.3 2015 The Seebeck measurements across a graphene/h-BN/graphene heterostructurehas been determined by inducing a temperature gradient between the bottom and top graphene layers and measuring the corresponding thermoelectric voltage (∆V) across the heterostructure16. A temperature gradient (∆T) of 39 K and thermoelectric voltage (∆V) of almost 4 mV is observed in the device, which results in a Seebeck coefficient of -99.3 μV/K, a power factor (𝑆 2 𝜎) of 1.51 × 10–15 W/K2 , and a thermoelectric figure of merit of ZT = 1.05 × 10–6 for the graphene/h-BN/graphene heterostructure. While the overall thermoelectric energy conversion efficiency of this device is small, the relatively large Seebeck coefficient and temperature drops observed here indicate that the I–V characteristics of 2D heterostructures can contain an appreciable thermoelectric component. 16. 1.4 2016 The thermal and thermoelectric properties of the most structurally stable AB configuration among the heterostructures formed by vertical stacking of graphene and h-BN have been revealed in recent studies 15,17. Molecular dynamics simulations have been performed to study the interfacial thermal resistance (ITR) of a graphene/h-BN bilayer system as well as its one-dimensional counterpart, a concentric CNT/BNNT double-walled nanotube, based on the lumped capacity model. The calculated ITR is in an order of magnitude of 10-7-10-6 Km2/W and it monotonically decreases with temperature and interlayer/intertube coupling strength17. 1.5 2016 Recent experiments where the graphene was processed to be extremely clean with low defect density on an atomically smooth, high-k hexagonal boron nitride (h-BN) dielectric indeed verified predictions that electron-electron scattering can dominate at high temperatures. This enabled the exploration of rich physics, where strong inelastic collisions between electrons (e–e interactions) dominate, resulting in a violation of the Mott Relation18. Further, near the CNP, a plasma of electron–holes (unlike charge puddles shown previously in defect-dominated samples) manifested in large Seebeck values18,19 as well as a violation of the Wiedemann– Franz Law20. Studies have show that the thermoelectric performance of graphene can be significantly improved by using hexagonal boron nitride (hBN) substrates instead of SiO219. The power factor times temperature, reached values as high as 10.35 W·m−1·K−1. higher values of PF3D were reported, as large as 34.5 mW m−1 K−2 (converted from a 10.35 W m−1 K−1 (Figure 3c) powerfactor × temperature (PFT) value at 300 K).19. These large values are similar to the report of violation of the Mott Relation in clean samples and can be ascribed to hydrody- namic electrons as well18. 1.6 2017 Researchers have used the Molecular Dynamic (MD) simulation method to assess the thermal conductivity of single-layer graphene based on a multilayer h-BN substrate 21. They have demonstrated that bulk hexagonal boron nitride (h-BN) is a more appealing substrate to achieve high performance heat dissipation in supported graphene. Notable length dependence and high thermal conductivity have been observed in single layer graphene on h-BN substrate. At room temperature, the thermal conductivity of h-BN-supported SLG is as high as 1347.3 ± 20.5 Wm−1 K−1, which is about 77% of that for the suspended case, and is more than twice that of the SiO2-supported SLG. 1.7 2017 The thermoelectric figures of merit of pristine two-dimensional materials are predicted to be significantly less than unity22, making them uncompetitive as thermoelectric materials. Since the inplane transverse and longitudinal phonons are effectively filtered out from contributing to crossplane transport because they do not substantially alter the tunneling matrix elements, theoretical calculations predict enhanced cross-plane thermoelectric properties in van der Waals heterojunctions, including high ZT factors at room temperature 22. These researchers have analysed the thermoelectric performance of monolayer molybdenum disulphide (MoS2) sandwiched between two graphene monolayers. They observed that CP ZT can be as high as ~2.8 for the graphene/MoS2/graphene heterostructure. 1.8 2017 Researchers have used quantum transport and molecular dynamics (MD) simulations to calculate the electronic and thermal properties of polycrystalline graphene-hBN heterostructures 23. They have estimated the thermoelectric conversion ratio and found that it remains far too low to be useful for energy harvesting applications. The upper value of the figure of merit, 𝑍𝑇 = 𝑆 2 𝜎𝑇 𝑘 has been found which is quite small. For 40nm average grain size and 20% hBN, ZT~1x10-4 for a carrier concentration n=5x1012 cm-2 , which is quite small. Even for energies near the edge of the gap, where the see beck coefficient should be maximised, the value of ZT only reaches ~1x10-2. 1.9 2017 Other researchers observed that an effective “inter-layer phonon drag” determines the Seebeck coefficient (S) across the van der Waals gap formed in twisted bilayer graphene (tBLG)24. They have demonstrated that the cross-plane thermoelectric transport is driven by the scattering of electrons and interlayer layer breathing phonon modes, which thus represents a unique “phonon drag” effect across atomic distances. They calculated the cross-plane thermoelectric power-factor (S2Gcp), by combining the experimentally observed magnitudes of see beck coefficient, S and interlayer conductance, Gcp. They observed that, the maximum effective PFT = TS2Gcp/d, where d ≈ 0.4 nm is the van der Waals distance, increases with temperature, and can be as high as ≈ 0.3 Wm−1K−1 at room temperature. 1.10 LAST In summary, several researchers have used theoretical and experimental approaches to study the thermoelectric properties of graphene based 2D heterostructures. However, more study is needed to fully optimise the thermoelectric performance of these materials. Besides, there is need to explore the performance of other 2D heterostructures. 2 1. Reference Dresselhaus, M. S. et al. New directions for low-dimensional thermoelectric materials. Adv. Mater. 19, 1043–1053 (2007). 2. Shuai, J. et al. Recent progress and future challenges on thermoelectric Zintl materials. Mater. Today Phys. 1, 74–95 (2017). 3. Zhao, H. et al. High thermoelectric performance of MgAgSb-based materials. Nano Energy 7, 97–103 (2014). 4. Chang, C. & Zhao, L.-D. Anharmoncity and low thermal conductivity in thermoelectrics. Mater. Today Phys. 4, 50–57 (2018). 5. Liu, Z. et al. Tellurium doped n -type Zintl Zr 3 Ni 3 Sb 4 thermoelectric materials: Balance between carrier-scattering mechanism and bipolar effect. Mater. Today Phys. 2, 54–61 (2017). 6. Takaki, H. et al. Thermoelectric properties of a magnetic semiconductor CuFeS 2. Mater. Today Phys. 3, 85–92 (2017). 7. Mao, J. et al. Anomalous electrical conductivity of n-type Te-doped Mg 3.2 Sb 1.5 Bi 0.5. Mater. Today Phys. 3, 1–6 (2017). 8. He, R. et al. Improved thermoelectric performance of n-type half-Heusler MCo 1-x Ni x Sb (M = Hf, Zr). Mater. Today Phys. 1, 24–30 (2017). 9. Brull, S. Un modèle ES–BGK pour des mélanges de gaz. Comptes Rendus Math. 351, 775– 779 (2013). 10. Wang, J. et al. The thermal and thermoelectric properties of in-plane C-BN hybrid structures and graphene / h-BN van der Waals heterostructures. Mater. Today Phys. 5, 29–57 (2018). 11. Checkelsky, J. G. & Ong, N. P. Thermopower and Nernst effect in graphene in a magnetic field. Phys. Rev. B - Condens. Matter Mater. Phys. 80, 1–4 (2009). 12. Zuev, Y. M., Chang, W. & Kim, P. Thermoelectric and magnetothermoelectric transport measurements of graphene. Phys. Rev. Lett. 102, 1–4 (2009). 13. Wei, P., Bao, W., Pu, Y., Lau, C. N. & Shi, J. Anomalous thermoelectric transport of dirac particles in graphene. Phys. Rev. Lett. 102, 1–4 (2009). 14. Hwang, E. H., Rossi, E. & Das Sarma, S. Theory of thermopower in two-dimensional graphene. Phys. Rev. B - Condens. Matter Mater. Phys. 80, 1–5 (2009). 15. Wang, X.-M. & Lu, S.-S. First-Principles Study of the Transport Properties of GrapheneHexagonal Boron Nitride Superlattice. J. Nanosci. Nanotechnol. 15, 3025–3028 (2015). 16. Chen, C. C., Li, Z., Shi, L. & Cronin, S. B. Thermoelectric transport across graphene/hexagonal boron nitride/graphene heterostructures. Nano Res. 8, 666–672 (2015). 17. Li, T., Tang, Z., Huang, Z. & Yu, J. Interfacial thermal resistance of 2D and 1D carbon/hexagonal boron nitride van der Waals heterostructures. Carbon N. Y. 105, 566–571 (2016). 18. Ghahari, F. et al. Enhanced Thermoelectric Power in Graphene : Violation of the Mott Relation by Inelastic Scattering. 136802, 1–5 (2016). 19. Duan, J. et al. High thermoelectricpower factor in graphene/hBN devices. Proc. Natl. Acad. Sci. 113, 14272–14276 (2016). 20. Lewis, G. S. & Hering, S. V. 2 1 . 0343, 1–8 (1988). 21. Zhang, Z., Hu, S., Chen, J. & Li, B. Hexagonal boron nitride: A promising substrate for graphene with high heat dissipation. Nanotechnology 28, (2017). 22. Sadeghi, H., Sangtarash, S. & Lambert, C. J. Cross-plane enhanced thermoelectricity and phonon suppression in graphene/MoS2van der Waals heterostructures. 2D Mater. 4, 1–8 (2017). 23. Barrios-Vargas, J. E. et al. Electrical and Thermal Transport in Coplanar Polycrystalline Graphene-hBN Heterostructures. Nano Lett. 17, 1660–1664 (2017). 24. Mahapatra, P. S., Sarkar, K., Krishnamurthy, H. R., Mukerjee, S. & Ghosh, A. Seebeck Coefficient of a Single van der Waals Junction in Twisted Bilayer Graphene. Nano Lett. 17, 6822–6827 (2017).
Front. Energy REVIEW ARTICLE Kai-Xuan CHEN, Min-Shan LI, Dong-Chuan MO, Shu-Shen LYU Nanostructural thermoelectric materials and their performance © Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract In this review, an attempt was made to introduce the traditional concepts and materials in thermoelectric application and the recent development in searching high-performance thermoelectric materials. Due to the use of nanostructural engineering, thermoelectric materials with a high figure of merit are designed, leading to their blooming application in the energy field. One dimensional nanotubes and nanoribbons, two-dimensional planner structures, nanocomposites, and heterostructures were summarized. In addition, the state-of-the-art theoretical calculation in the prediction of thermoelectric materials was also reviewed, including the molecular dynamics (MD), Boltzmann transport equation, and non-equilibrium Green’s function. The combination of experimental fabrication and first-principles prediction significantly promotes the discovery of new promising candidates in the thermoelectric field. Keywords nanostructural, low-dimensional, thermoelectric material, figure of merit, first-principles 1 Introduction Due to their ability of directly converting waste heat into electricity and vice versa, thermoelectric materials [1–6] have attracted a lot of attention in recent years. It has been considered as one of the effective ways to resolve the severe energy problem using thermoelectric materials. Compared with other methods such as lithium ion Received Aug. 17, 2017; accepted Nov. 12, 2017; online Feb. 10, 2018 ✉), Shu-Shen LYU Kai-Xuan CHEN, Min-Shan LI, Dong-Chuan MO ( ✉ ( ) School of Chemical Engineering and Technology, Sun Yat-sen University, Guangzhou 510275, China; Guangdong Engineering Technology Research Centre for Advanced Thermal Control Material and System Integration (ATCMSI), Guangzhou 510275, China E-mail:; batteries, solar cells, and supercapacitor, it possesses the advantages of having no moving components, no leakage, and being environmentally friendly [7,8].Generally, the energy conversion efficiency is adopted to evaluate the conversion performance. As is known, the Carnot coefficient hC is considered as the highest reversible energy conversion efficiency under certain circumstances, which is defined as ηC ¼ ΔT T –T ¼ h c, Th Th (1) where DT = Th – Tc denotes the temperature gradient between the heat and cold terminals, which are defined as Th and Tc, respectively. As in the thermoelectric field, the energy generation efficiency of thermoelectric material, hTE, can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ZT – 1 ΔT ηTE ¼ ⋅ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , (2) Th 1 þ ZT þ Tc =Th where ZT is the so-called thermoelectric figure of merit, which is determined by the intrinsic properties of a certain material. As is known from Eq. (2), in order to obtain a high energy generation efficiency (i.e. to approach the Carnot coefficient), ZT should be large enough, ideally exceeding 3 or even 4, as shown in Fig. 1. The thermoelectric figure of merit [9,10] is defined according to S 2 T , κ (3) κ ¼ κel þ κph , (4) ZT ¼ where s, S, and k are the electrical conductivity, Seebeck coefficient, and thermal conductivity, respectively. The thermal conductivity can be contributed by both electrons and phonons, denoted as kel and kph, respectively [11]. As shown in the definition of ZT, it would require a high electrical conductivity and Seebeck coefficient as well as a 2 Front. Energy 2 Enhancing thermoelectric performance using nanostructural engineering Tables 1 and 2 summarize the ZT obtained in experimental samples in recent years based on Bi2Te3 (working temperature usually at 200–500 K) and IV–VI family (working temperature usually at 600–900 K), respectively. Using nanostructural engineering, the highest ZT of these families can even exceed 2, as illustrated in Fig. 3. For instances, in the work of Venkatasubramanian et al. [15], a figure of merit of 2.4 can be observed in p-type Bi2Te3/ Sb2Te3 at 300 K. Fig. 1 Thermoelectric efficiency (hTE) plotted for different temperature gradient between the hot side and the cold side where the cold side is assumed to be at room temperature (300 K) low thermal conductivity to obtain high-performance thermoelectric materials. However, these transport coefficients are inter-related. It is difficult to alter one parameter without significantly affecting the other transport coefficients. For example, as the carrier concentration increases, both electrical conductivity and thermal conductivity increase but the opposite is true of the Seebeck coefficient, as shown in Fig. 2. Fig. 3 Development of ZT in the past several decades (The reference number of the original data are denoted in the center of the symbol. For Bi2Te3 series, the references sorted by years are Refs. [16–27]. The references for Zn4Sb3, PbTe and SiGe series are Refs. [28–31], [32–39], and [34,40–44], respectively.) Fig. 2 Variation of thermoelectric factors against carrier concentration Due to the inter-relationship among thermoelectric factors, ZT has been around for many years during the last century. Two strategies to design high-performance thermoelectric materials are proposed. The first one is the framework of phonon-glass electron-crystal (PGEC), in which glass-like thermal properties and crystal-like electrical properties are required in one system [12] and only small progress has been made. The other strategy, as pointed out by Hicks and Dressalhaus [13,14], is nanostructural engineering, in which the quantum confinement can weaken the inter-relationship among the transport coefficients and many progresses have been made. Nanostructured engineering can enhance the thermoelectric performance by lowering the thermal conductivity. The reason comes from the stronger phonon scattering induced by the increasing nano-surfaces and interfaces in nano-size samples. It should be mentioned that the effect of nanostructures on reducing thermal conductivity is significant at low temperatures. At a high temperature range, the contribution of short wave phonons to the lattice thermal conductivity will be dominant and therefore point defects will have much stronger effects on the lattice thermal conductivity than nanostructures. 2.1 Nanograin/interface effect Both Boukai et al. [68] and Hochbaum et al. [69] investigated the efficient thermoelectric performance of silicon nanowires by varying the nanowire size and their studies indicated that the improved efficiency originates from phonon effects (a three-dimensional to one-dimensional crossover of the phonons participating in phonon drag). Poudel et al. [25] observed a ZT of 1.4 at 373 K in Kai-Xuan CHEN et al. Nanostructural thermoelectric materials 3 Table 1 Thermoelectric materials based on Bi2Te3 Reference Year Samples ZT Temperature/K Chung et al. [22] 2000 p-Type CsBi4Te6 0.8 225 Venkatasubramanian et al. [15] 2001 p-Type Bi2Te3/Sb2Te3 and n-type Bi2Te3/Bi2Te2.83Se0.17 2.4/1.4 300 Polvani et al. [45] 2001 p-Type Bi0.5Sb1.5Te3 alloy under 2GPa >2 300 Sidorenko et al. [46] 2001 Crystals of Bi-Sb alloy under a magnetic field >1 100 Chung et al. [23] 2004 p-Type CsBi4Te6 0.82 225 Zhao et al. [47] 2005 n-Type Bi2Te3 nanocomposite 1.25 420 Tang et al. [48] 2007 Bi2Te3 bulk with layered nanostructure 1.35 300 Cao et al. [49] 2008 Bi2Te3/Sb2Te3 bulk nanocomposites 1.47 440 Poudel et al. [25] 2008 p-Type nanocrystalline BiSbTe bulk alloy 1.2/1.4/0.8 300/373/523 Yan et al. [50] 2010 n-Type Bi2Te2.7Se0.3 1.04 400 Zhang et al. [51] 2011 n-Type Bi2Te3 ultrathin nanowires 0.96 380 Guo et al. [52] 2013 Tl9BiTe6 1/0.86 450/560 Hong et al. [53] 2016 BixSb2 – xTe3 nanoplates 1.2 300 Pan and Li [54] 2016 n-Type Bi2(TeSe)3 alloys 1.1 473 Dharmaiah et al. [55] 2016 p-Type 25%Bi2Te3–75%Sb2Te3 alloys 1.23 350 Table 2 Thermoelectric materials based on IV–VI family Reference Year Samples ZT Temperature/K Hsu et al. [56] 2004 n-Type AgPbmSbTe2m (m = 10/18) 2.2 800 Wang et al. [57] 2006 Ag0.8Pb22SbTe20 1.37 673 Johnsen et al. [58] 2011 PbS 0.94 710 Pei et al. [59] 2011 Polycrystalline Pb0.98Na0.02Te12xSex 1.8 800 Zhang et al. [60,61] 2013 Al0.03PbTe 1.2 700 Zhao et al. [62] 2014 SnSe 2.6 923 Yamini et al. [64] 2014 n-Type (PbTe)0.75(PbS)0.15(PbSe)0.1 composites 1.1 800 Lu et al. [65] 2015 (Ge0.8Pb0.2)0.9Mn0.1Te 1.3 720 Zhao et al. [11] 2016 Hole-doped single-crystal SnSe 2.0 773 Zhao et al. [66] 2016 Counter-doped SnTe with strained endotaxial SrTe 1.2 823 Li et al. [67] 2017 p-Type SnSe doped with Zn 0.96 873 the p-type nanocrystalline BiSbTe bulk alloy and found that the improvement results from the low thermal conductivity were caused by grain boundaries and defects. Miao et al. [70] concluded that the hollow structure of titanate nanotubes was responsible for the ultralow thermal conductivity, which greatly enhanced their thermoelectric properties. Li et al. [71] synthesized SnTe particles with controlled sizes from micro-scale to nano-scale and found that the ZT of the specimen using 165-nm-sized nanoparticles was about 2.3 times that of the SnTe bulk samples due to the enhanced phonon scattering. Yang et al. [72] found that the lattice thermal conductivity of nanoscale three dimensional Si phononic crystals was decreased by 500 times compared with that of porous Si, which led to a 26 times increase in ZT. He et al. [73] showed that YbAl3 from the nanometer to mesoscopic scales can effectively scatter phonons and remarkably decrease the lattice thermal conductivity, which produced a 74% increase in ZT. 2.2 Nano particles/inclusions effect Zhao et al. [74] achieved dual control of phonon- and electron-transport properties by embedding nanoparticles of a soft magnetic material in a thermoelectric matrix and thereby improved the thermoelectric performance of the resulting nanocomposites. Pei el al. [75] found that PbTe with nanoscale Ag2Te precipitates and La doping had a low lattice thermal conductivity. In the work of Johnsen et al. [58], the nanostructuring in (PbS)1–x(PbTe)x samples led to substantial decreases in lattice thermal conductivity 4 Front. Energy relative to pristine PbS. Gahtori et al. [76] reported a ZT of 2.1 at 973 K in Cu2Se with different nanoscale dimensional defect features, in which the low thermal conductivity origined from the enhanced low-to-high wavelength phonon scattering by different kinds of defects. Ahmad et al. [77] reported a ZT of 1.81 at 1100 K in p-type SiGe alloys since YSi2 nanoinclusions formed coherent interfaces with SiGe matrix and facilitated reduction in the grainsize of SiGe, which greatly reduced the thermal conductivity. In addition, many other researchers conducted a lot of research in reducing the thermal conductivity in the systems of grapheme [63,78], carbon nanotubes [61,79], graphynenanoribbons [80,81], PbTe family [37,64,82,83], and Bi2Te3 nanowires [51]. The electrical properties can be tuned as well to enhance the thermoelectric properties. The electrical conductivity can usually be increased by enhancing the electron mobility or altering the electronic structures. Nanostructuring can enhance the density of states near Fermi level via quantum confinement, and therefore, increase the thermopower, which provides a way to decouple the thermopower and electrical conductivity [84]. For example, Ginting et al. [85] synthesized composites with nano-inclusions of ntype (PbTe0.93 – xSe0.07Clx)0.93(PbS)0.07 while the composites with nano-inclusions enhanced the Seebeck coefficient in a dilute Cl- doped compound and led to a ZT of 1.52 at 700 K. 3 Theoretical calculation in low-dimensional thermoelectric materials In recent years, the research into thermoelectric materials has achieved a great success in the combination of theoretical calculations [86]. Based on the start-of-the-art density functional theory, several computational methods have been employed in the prediction of high-performance thermoelectric materials and exploration of the enhancement mechanism. In the heat transport field, there are typically three kinds of methods which are used to study the thermal properties. This section will concentrate on the theoretical method of thermal transport properties using the molecular dynamics (MD) [87,88], Boltzmann transport equation [89–91], and non-equilibrium Green’s function method [92–94], respectively. Generally, these methods adopt the potential data from the first-principles density functional theory. 3.1 3.1.1 Theoretical methods Molecular dynamics Based on Newton’s laws of motion, MD can be used in the study of heat transport. Equilibrium MD adopts the GreenKubo formula to calculate the thermal conductivity, as shown below. 1 καβ ! 1 ¼ hJ α ðtÞ⋅J β ðtÞidt, VkB T 2 (5) 0 where καβ, V, kB, and T are the thermal conductivity tensor, volume, Boltzmann constant, and absolute temperature, respectively. Jα and Jβ are the heat flow along the α and β direction, respectively. Non-equilibrium MD obtains the thermal conductivity of the system according to Fourier’s law. By introducing a gradient of temperature or heat flow density into the system, the heat flow can be calculated as Jα ¼ – 3.1.2 X ∂T : ∂xβ β (6) Boltzmann transport equation The phonon (lattice) thermal conductivity can be calculated by employing the phonon Boltzmann transport equation with relaxation time approximation (RTA) according to the following formula. 1X κph,αβ ¼ C ν ν τ , (7) V l ph,l lα lβ lα where α and β denote the components of the second-order tensor kph. In addition, Cl, ν, and t denote the phonon mode volumetric specific heat, group velocity, and phonon lifetime, respectively [83]. The RTA phonon lifetime can be computed according to the Matthiessen rule where phonon-phonon scattering (tph), isotope scattering (tiso), and boundary scattering (tb) are fully taken into consideration [89]. 1 1 1 1 ¼ ph þ iso þ b : τl τl τl τl (8) The electronic transport coefficients can be computed based on the Boltzmann transport equation. The Seebeck coefficient S and electrical conductivity s can be calculated by   ekB ∂f0 ε– ðεÞ S¼ dε – , (9)  ∂ε kB T ! ¼e where ðεÞ ¼ X k 2  ∂f0 dε – ðεÞ, ∂ε !  (10) νk νk τ k denotes the transport distribu- tion; m, ε, e, and kB are the chemical potential, electron energy, unit charge, and Boltzmann constant, respectively. In addition, f0 is the Fermi distribution function, ν is the group velocity, and t is the relaxation time at the k state [95,96]. Kai-Xuan CHEN et al. Nanostructural thermoelectric materials 3.1.3 Non-equilibrium Green’s function 5 f ðE,,T Þ ¼ When the transport scale is smaller than the mean free path (MFP), the ballistic regime is valid. In such a condition, the non-equilibrium Green’s function method can be effectively used in the study of transport properties. Generally, a typical model of the NEGF can be described as the “left lead-conductor-right lead” (LCR) configuration, as demonstrated in Fig. 4. The conductor can be set to be exactly the same as the leads (in the case of perfect crystal) or totally different from the leads (in the case of interface). 1 E– e kB T , where f(E, m, T) is the Fermi-Dirac distribution function and m, T, and h are the chemical potential, absolute temperature, and Planck constant, respectively. According to Eqs. (17)–(19), the electronic conductance s, Seebeck coefficient S, and the electronic thermal conductance kel, can be derived, respectively.  ¼ q2 L0 , S¼ (17) 1 L  1, L0 qT (18)   1 L21 κel ¼  L2 – : L0 T Fig. 4 A typical model of non-equilibrium Green’s function To investigate the thermoelectric properties, both electronic and phononic transport should be studied. For ballistic electronic transport, the retarded Green’s function of the central conductor is G ¼ r ½ES C – H C – Σ rL – Σ rR  – 1 , (19) To obtain the ZT, the phononic thermal conductance has to be calculated, which can be determined from the phonon transport part. The calculation of phonon transmittance is similar to that of the electron transmittance in the electronic transport part which has been described above. By replacing the Hamiltonian matrix and electron energy E with the interatomic force constant (FC) matrix and phonon frequency ω, the phonon transmittance T(ω) can be calculated as G r ¼ ½ðω þ iηÞ2 – F C – Σ rL – Σ rR  – 1 , (20) TðωÞ ¼ T rðGr Γ L Ga Γ R Þ: (21) (11) where E, HC, and SC are the electron energy, Hamiltonian and overlap matrix, respectively; and the self-energy term Sr can be obtained as Σ rL ¼ H yLC grL H LC , Σ rR ¼ H CR grR H yCR , (16) þ1 After that, the phononic thermal conductance kph can be calculated by (12) where gr is the retarded surface Green’s function from semi-infinite lead. Besides, HLC (HCR) denotes the Hamiltonian matrix between the left (right) leads and the central conductor. The electronic transmittance matrix T(E), which is significant in electronic transport, can be obtained by κph ðT Þ ¼ ` 2π Þ dω, !0 T ðωÞω ∂gðω,T ∂T 1 gðω,T Þ ¼ 1 `ω e kB T , (22) (23) –1 Γ β ¼ iðΣ rβ – Σ aβ Þ, β ¼ L, R, (13) where g(w, T) is the Bose-Einstein distribution function, while ` and kB denote the reduced Planck constant and Boltzmann constant, respectively. TðEÞ ¼ T rðGr Γ L Ga Γ R Þ, Ga ¼ ðG r Þy : (14) 3.2 For convenience, Lorenz functions Ln are introduced to calculate the thermoelectric factors Ln ð,T Þ ¼ ! 2 dETðEÞ  ðE – Þn h   ∂f ðE,,T Þ  – , ∂E (15) Ballistic thermoelectric transport In the past few years, the research group in Guangdong Engineering Technology Research Centre for Advanced Thermal Control Material and System Integration in Sun Yat-sen University has paid a lot of attention to the theoretical study of ballistic thermoelectric transport by using the density functional theory and non-equilibrium Green’s function method. The emphasis has been laid on the two-dimensional systems including graphyne, transi- 6 Front. Energy tion metal dichalcogenides (TMDs) and the VA group family. 3.2.1 Graphyne Shortly after the discovery of graphene, graphyne, another member of the carbon family, attracted researchers’ attention. In Ref. [97], the thermoelectric transport properties of graphyne nanotubes were investigated by using the non-equilibrium Green’s function method, as implemented in the density functional based tight binding framework. Figure 5 shows the previous study with regard to the thermoelectric properties of new emerging twodimensional materials. Figure 5(a) reveals that both the band gap and thermoelectric figure of merit ZT of graphyne nanotubes show a damped oscillation as the tube diameter increases. In addition, by introducing hydrogenation, the thermoelectric performance is reduced. The thermoelectric performance of graphyne nanotubes is much better than that of graphene according to the theoretical calculation. Fig. 5 3.2.2 TMDs Of all newly proposed two-dimensional materials, TMDs are believed to possess the lowest thermal conductivity. Therefore, these TMDs families were studied as new thermoelectric materials in Ref. [93]. Four kinds of monolayer TMDs (MoS2, MoSe2, WS2, WSe2) were investigated. It was discovered that the monolayer WSe2 harbored the highest thermoelectric figure of merit (ZT = 0.91) at room temperature. The nanotubes that scrolled from these monolayer TMDs were also investigated as a comparison. A degeneration in thermoelectric performance was observed from monolayers to nanotubes, as shown in Fig. 5(b).Following this research, in Ref. [98] the thermoelectric properties of WSe2 nanoribbons were studies since monolayer WSe2 was found to exhibit the most excellent thermoelectric performance. The ZT of WSe2 nanoribbons was higher than that of monolayer WSe2 in both armchair and zigzag ribbons. The highest value of 2.2 could be observed in armchair WSe2 nanoribbons, as depicted in Fig. 5(c). Previous study on the thermoelectric properties of two-dimensional materials (a) Graphyne nanotubes; (b) TMDs; (c) WSe2 nanoribbons; (d) buckled antimonene (Adapted with permission from Ref. [97],Copyright (2013) American Chemical Society; Ref. [93], Copyright (2015) American Chemical Society; Ref. [98], Copyright (2016) Royal Society of Chemistry and Ref. [94], Copyright (2017) American Chemical Society.) Kai-Xuan CHEN et al. Nanostructural thermoelectric materials 3.2.3 VA group family In the framework of 2D systems, the buckled and puckered systems which consist of the VA elements (denoted as arsenene, antimonene and bismuthene) were also studied. The first-principles calculation indicated that buckled antimonene harbored a thermoelectric figure of merit ZT of 2.15 at room temperature. The ZT could even be enhanced to 2.9 at a3% biaxial tensile strain. This is probably the highest value that has ever been reported in pristine 2D materials, as can be seen from Fig. 5(d). The enhancement mainly results from both tuning the electronic structures and reducing the thermal conductance [94]. 4 Low-dimensional thermoelectric materials As pointed out in Section 2, nanostructural engineering is now acting as an effective strategy to enhance the thermoelectric performance [99,100]. Lots of low-dimensional materials have been adopted and studied. Here, the item of low dimensional means the size of the shape or crystal structure. These families can be summarized according to the diverse dimensionality, as are introduced in the following subsections. It should be mentioned that low-dimensional materials are difficult to fabricate precisely due to the high requirement in nanoscale samples. Therefore, many of these results have not yet been realized in experiments. However, the theoretical calculation can provide a valuable guidance to the future research and as the development of nanostructuring technology proceeds, nanoscale materials with atomic accuracy may be prepared. 7 experimental study in carbon nanotubes [105,106], silicon nanowire [68,69], titanate nanotubes [70], Bi2Te3 nanowires [51,107] and the theoretical study in carbon nanotubes [79,92,106,108], InSe nanotubes [109], phosphorene nanoribbons [96], graphyene nanoribbons and nanotubes [80,97], and transition metal dichalcogenide nanoribbons and nanotubes [93,95,98]. 4.2 Two-dimensional mono- and few-layers materials As for two-dimensional systems, much work has been devoted to the study of TMDs. Huang et al. [110,111], Chen et al. [93] and Tahir and Schwingensch lögl [112] studied the monolayer TMDs and found that the monolayer TMDs exhibited a great potential in thermoelectric performance. Huang et al. [111] and Wickramaratne et al. [113] studied the layer dependence of few-layer TMDs and discovered that thermal conductance per thickness approached bulk as the thickness increased. In the work of Lee et al. [114], layer mixing was predicted to be a promising way of improving the thermoelectric properties. Bhattacharyya et al. [115] and Guo [116] studied the stain effect on the thermoelectric performance of TMDs. They observed that the electronic structures of the TMDs family were sensitive to the applied strain and the thermoelectric figure of merit could be tuned by the strain level. Many theoretical workers also devoted their effort to the design of high-performance two-dimensional thermoelectric candidates. Table 3 summarizes the calculated figure of merit for many new emerging two-dimensional materials. It can be seen that the TMDs and the VA group families are both promising candidates for thermoelectric application. Table 3 Theoretical prediction of maximum room-temperature ZT for typical two-dimensional materials 4.1 One-dimensional nanotubes and nanoribbons Much research based on the one-dimensional nanoribbons and nanotubes has been conducted. As is known, the thermoelectric application of graphene has been hampered due to its zero bandgap in the electronic structure. Considerable research has been carried out to open the bandgap of graphene, such as grapheme nanoribbons and nanotubes [101]. Sevinçli et al. [63,102] showed that by geometrical structuring and isotope cluster engineering, the thermal conductance of grapheme nanoribbons could be reduced by 98.8% and the thermoelectric figure of merit could be as high as 3.25 at 800 K. Chang et al. [103] studied the grapheme nanoribbons perforated with an array of nanopores which exhibited a high ZT of ~5 at room temperature. Yeo et al. [104] studied the thermoelectric performance of strained grapheme nanoribbons and discovered that the tensile strain increased the ZT value of certain armchair grapheme nanoribbons. More research focusing on the thermoelectric performance of other family are presented, such as the 2D monolayer ZTmax at RT Graphene 0.0094 [117] Graphyne 0.157 [117] Silicene 0.36 [118] Germanene 0.41 [118] MoS2 0.75 [93], 0.58 [110], 1.35 [113] MoSe2 0.88 [93], 1.39 [113] WS2 0.72 [93], 1.52 [113] WSe2 0.91 [93], 1.88 [113] Black phosphorene > 0.6 [119], 1.44 [120] Puckered arsenene 0.85 [121] Buckled antimonene 2.15 [94] 4.3 Nanocomposites and heterostructures Nanocomposites and heterostructures are also adopted as candidates of thermoelectric materials. Zhang et al. [61] fabricated the Bi2Te3-Te micro-nanoheterostructure and 8 Front. Energy discovered a ZT of ~0.4 at room temperature for such samples, with an enhancement of 40% compared to those without nanoscale heterostructures. Carrete et al. [122] theoretically investigated the thermoelectric transport properties of hybrid thiophene/SiGe superlattices and found that the ZT was twice as those in bulk SiGe. Savelli et al. [123] fabricated the Ti-based silicide quantum dot superlattices by reduced-pressure chemical vapor deposition and observed a trifold increase in the power factor compared with SiGe thin films. Duan et al. [124] observed that the thermoelectric performance of graphene could be significantly enhanced in graphene/hBNvdW device. Yin et al. prepared the Mg2Si1 – xSnx/SiC nano-composites and obtained a ZT of 1.2 at 750 K owing to the introduction of SiC nano-additives. In the work of Luo et al. [125], ZnX acted as a nanoscalehetero structure barrier blocking in CuInTe2, which led to an enhanced Seebeck coefficient and reduced thermal conductivity, with a ZT of 1.52. Yin et al. [126] prepared Mg2Si1 – xSnx/SiC nano-composites with a maximum ZT value of ~1.20 at 750 K. 5 Perspective on the design of highperformance thermoelectric materials With the advancement of nanotstructural technology, the fabrication of nanoscale thermoelectric materials is feasible. In this review, the importance of nanostructural engineering in the design of high-performance thermoelectric materialsis mainly emphasized. High ZT materials can be achieved in nanoscale family due to the quantum confinement effect. The theoretical prediction regarding nanostructural systems has been conducted in recent years, and time will tell whether these systems exhibit excellent performance in experimental testing. Promising thermoelectric candidates may be discovered in the systems which possess heavy elements. Another suggestion for the design of high-performance thermoelectric materials would be the lowering of the dimensionality. For instance, preparing the half-Heusler alloys in nanoscale films or composites with nanoparticle additives might be an effective way to achieve this goal. In addition, research into thermoelectric materials regarding spin needs our attention. The spin Seebeck effect may provide a new opportunity in this field. Moreover, lots of topological insulators are found to be excellent thermoelectric candidates and the mechanism is still unclear. The fast development in state-of-the-art first-principles method in the study of electron and phonon transport has also provided another strategy regarding material design. It is believed that by combining the non-equilibrium Green’s function (adopted in nanoscale systems) and the Boltzmann transport equation (adopted in mesoscale systems), one may effectively predict the properties of newly proposed thermoelectric materials. That would signifi- cantly enhance the explosive growth in excellent thermoelectric candidates. 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