On Landauer versus Boltzmann and full band versus effective mass evaluation of thermoelectric transport coefficients

Abstract

The Landauer approach to diffusive transport is mathematically related to the solution of the Boltzmann transport equation, and expressions for the thermoelectric parameters in both formalisms are presented. Quantum mechanical and semiclassical techniques to obtain from a full description of the bandstructure, E(k), the number of conducting channels in the Landauer approach or the transport distribution in the Boltzmann solution are developed and compared. Thermoelectric transport coefficients are evaluated from an atomistic level, full band description of a crystal. Several example calculations for representative bulk materials are presented, and the full band results are related to the more common effective mass formalism. Finally, given a full E(k) for a crystal, a procedure to extract an accurate, effective mass level description is presented. 1) IntroductionMuch experimental and theoretical effort has been directed at improving the thermoelectric (TE) figure of merit, ZT = S 2 GT K , where T is the temperature, S is the 2 Seebeck coefficient, G is the electrical conductance, and K is the thermal conductance, which is the sum of the electronic contribution, K e , and the lattice thermal conductance, K l . Careful tradeoffs are needed to obtain high ZT. Recent experimental reports of high ZT 1-8 are attributed to suppressing the lattice thermal conductivity, and now the question of whether the electronic performance can be enhanced is being asked 9-11 . New materials 1,12-15 , new structures (e.g. nanowires [16][17][18][19][20][21][22][23][24][25][26] , quantum wells 2,27,28 , superlattices 10,11,17,[29][30][31][32][33][34][35][36] , and nanocomposites 3,4,37-39 ), and strain engineering 29,40-42 , which has been so successful for enhancing the electronic performance of nanotransistors, are all being explored. To address these opportunities, thermoelectric coefficients must be related to an accurate description of the electronic structure of the material.Thermoelectric parameters are usually evaluated by solving the Boltzmann Transport Equation (BTE) 43 . For low temperature thermoelectrics in mesoscopic structures, the Landauer approach is commonly used 44,45 . The Landauer approach applies to high temperature diffusive samples as well, and it provides an alternative formulation that can be insightful 46 . One objective for this paper is to discuss the mathematical relation between the Landauer and Boltzmann approaches.In both the Landauer and Boltzmann approaches the thermoelectric parameters are related to the electronic structure of the material. The effective mass approach is widelyused to analyze experiments and to design devices. For more complex materials, full band treatments (ab initio or empirical tight binding) have been used 12,47-56 . It is still not clear, however, exactly how full band treatments relate to effective mass level treatments

Electronic and thermoelectric properties of few-layer transition metal dichalcogenides

“…[34], the transport distribution, Ξ(E), arising from the Boltzmann transport equation is related to the above quantities by Ξ(E) = 2 h T (E)M(E). The density of modes M(E) can be defined as 17,32 …”

Gate-tunable and thickness-dependent electronic and thermoelectric transport in few-layer MoS2

“…The in-plane thermoelectric properties of single-and double-layer MoS 2 are calculated using the Landauer transport formalism, which is equivalent to solving the Boltzmann equation in the case of diffusive transport 26,[40][41][42][43] . Here, we will briefly describe our approach to calculate the Seebeck coefficient and electrical conductivity using the full band dispersions obtained from the first-principles density functional theory (DFT).…”

Thermoelectric properties of Bi2Te3 atomic quintuple thin films

“…[27] When phonon scattering is dominant, the mean free path can be written as λ(E) = λ 0 , a constant. [20] The density of modes M(E) can be expressed as [20,27] M(E) =…”

“…[24] In this study, the objective is to compare the TE parameters between the 3D bulk and the 2D film, hence we have chosen the Landauer approach. Within the Landauer formalism in the linear response regime, the electronic conductivity (σ), thermal conductivity for zero electric current (κ e ), and the Seebeck coefficient (S) are expressed as [20] …”