Size effect in thermoelectric materials

www.nature.com/npjquantmats REVIEW ARTICLE OPEN Size effect in thermoelectric materials Jun Mao1,2,4, Zihang Liu1,3,4 and Zhifeng Ren1 Thermoelectric applications have attracted increasing interest recently due to its capability of converting waste heat into electricity without hazardous emissions. Materials with enhanced thermoelectric performance have been reported in recent two decades. The revival of research for thermoelectric materials began in early 1990s when the size effect is considered. Low-dimensional materials with exceptionally high thermoelectric figure of merit (ZT) have been presented, which broke the limit of ZT around unity. The idea of size effect in thermoelectric materials even inspired the later nanostructuring and band engineering strategies, which effectively enhanced the thermoelectric performance of bulk materials. In this overview, the size effect in low-dimensional thermoelectric materials is reviewed. We first discuss the quantum confinement effect on carriers, including the enhancement of electronic density of states, semimetal to semiconductor transition and carrier pocket engineering. Then, the effect of assumptions on theoretical calculations is presented. Finally, the effect of phonon confinement and interface scattering on lattice thermal conductivity is discussed. npj Quantum Materials (2016) 1, 16028; doi:10.1038/npjquantmats.2016.28; published online 9 December 2016 INTRODUCTION Thermoelectric materials are capable of converting heat into electricity and vice versa by utilising the Seebeck effect and Peltier effect, respectively.1–3 Thermoelectric energy conversion efficiency is determined by Carnot efficiency and the dimensionless figure of merit (ZT), which is defined as ZT = (S2σ/κ)T, where S is the Seebeck coefficient, σ the electrical conductivity, κ the thermal conductivity and T the absolute temperature. To achieve a higher ZT has always been the motivation for the research of thermoelectrics, however, due to strong coupling of thermoelectric parameters S, σ and κ, improving one normally leads to the deterioration of other two and finally yields negligible enhancement of ZT.4 Research of thermoelectrics advanced rapidly in 1950s, when the basic science of thermoelectrics became well established. During this period, Bi2Te3 compounds and its alloys had been discovered and reported to have the highest ZT around unity. Over the following four decades or so, there was no big development in the thermoelectric field, therefore ZT≈1 had been regarded as the benchmark for advanced thermoelectrics.5 The turning point happened in early 1990s, when Hicks and Dresselhaus6,7 pointed out that quantum mechanics could provide a new route of designing thermoelectric materials by reducing the dimensionality. Low-dimensional materials with exceptionally high ZT have been presented by different groups later, which broke the limit of unity.8–10 More importantly, the idea of quantum effect subsequently led to the significant progress of bulk thermoelectric materials via the motivated strategies of nanostructuring11 and band engineering.12–15 Hence, there is the new revival of research of thermoelectrics that is still going strong. In this overview, thermoelectric materials with size effect, specifically low-dimensional materials such as nanowires, nanotubes and superlattice thin films, are reviewed from the view point of quantum confinement effect on carriers and phonons. The enhancement of density of states, semimetal to semiconductor transition and carrier pocket engineering are discussed in regards of quantum confinement on carriers. Besides, the effect of assumptions on theoretical calculations is presented. Finally, the effect of phonon confinement and interface scattering on thermal conductivity of low-dimensional materials is discussed. Interested readers are also referred to other excellent reviews on the lowdimensional thermoelectric materials.16–20 QUANTUM CONFINEMENT EFFECT ON CARRIERS In low-dimensional materials, the characteristic length of materials in certain direction is comparable to the effective de Broglie wavelength of carriers. Therefore, the motion of carriers is restricted in certain directions, which means that carriers are placed in the potential wells with infinitely high walls. In this case, the electronic spectrum will be drastically changed and this is the so-called quantum size effect. Theoretical modelling In early models for the calculation of thermoelectric properties of 2D quantum well structures, it was assumed that electrons were in simple parabolic bands and occupied the lowest subband of quantum well. The electronic dispersion relations for a 2D system were given by
 _2 k 2x _2 k 2y _2 π 2 ε2D k x ; k y ¼ þ þ 2mx 2my 2mz d 2W ð1Þ where dW was the width of quantum well, and mx, my and mz were the effective mass tensor components of the constant energy surfaces. It was further assumed that the current flow was in x direction and that quantum confinement was in z direction. The 1 Department of Physics and Texas Center for Superconductivity, University of Houston, Houston, TX, USA; 2Department of Mechanical Engineering, University of Houston, Houston, TX, USA and 3National Key Laboratory for Precision Hot Processing of Metals and School of Materials Science and Engineering, Harbin Institute of Technology, Harbin, China. Correspondence: Z Ren () 4 These authors contributed equally to this work. Received 19 August 2016; revised 24 October 2016; accepted 3 November 2016 Published in partnership with Nanjing University Size effect in thermoelectric materials J Mao et al D. O. S. D. O. S. 2 E 3D Bulk Semiconductor E D. O. S. D. O. S. 2D Quantum Well 1D Quantum Wire E 0D Quantum Dot E Figure 1. Electronic density of states for (a) a bulk semiconductor, (b) a 2D quantum well, (c) a 1D nanowire or nanotube and (d) a 0D quantum dot. (Adapted with permission from ref. 20). corresponding relation used for a square 1D quantum wire was ε1D ðk x Þ ¼ _2 k 2x _2 π 2 _2 π 2 þ þ 2 2mx 2my d W 2mz d 2W ð2Þ where the current flow was along the x direction, and quantum confinement occurred in y and z directions. Solutions of Boltzmann’s equation were obtained for the thermoelectric parameters of both 2D and 1D systems.16 Quantum confinement effect on electronic density of states Dimensionality plays a fundamental role in controlling the properties of materials. When the dimension of materials decreases and approaches nanometre length scales, it is possible to cause marked change in electronic density of states as shown in Figure 1.20 New strategy of designing thermoelectric materials by controlling the dimensionality was first discussed by Hicks and Dresselhaus.6,7 The calculation showed that Bi2Te3 with quantum well (two-dimension) or quantum wire (one-dimension) structure can have the potential to reach a significantly high ZT (Figure 2). The maximum ZT increased monotonically with the decrease of characteristic length (thi (...truncated)