COMPREHENSIVE REVIEW OF HEAT TRANSFER IN THERMOELECTRIC MATERIALS AND DEVICES The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Tian, Zhiting, Sangyeop Lee, and Gang Chen. “COMPREHENSIVE REVIEW OF HEAT TRANSFER IN THERMOELECTRIC MATERIALS AND DEVICES.” Annual Review of Heat Transfer 17, no. N/A (2014): 425–483. As Published http://dx.doi.org/10.1615/ANNUALREVHEATTRANSFER.2014006932 Publisher Begell House Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/119206 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ A Comprehensive Review of Heat Transfer in Thermoelectric Materials and Devices Zhiting Tian, Sangyeop Lee and Gang Chen Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA 02139, USA Abstract Solid-state thermoelectric devices are currently used in applications ranging from thermocouple sensors to power generators in satellites, to portable air-conditioners and refrigerators. With the ever-rising demand throughout the world for energy consumption and CO2 reduction, thermoelectric energy conversion has been receiving intensified attention as a potential candidate for waste-heat harvesting as well as for power generation from renewable sources. Efficient thermoelectric energy conversion critically depends on the performance of thermoelectric materials and devices. In this review, we discuss heat transfer in thermoelectric materials and devices, especially phonon engineering to reduce the lattice thermal conductivity of thermoelectric materials, which requires a fundamental understanding of nanoscale heat conduction physics. Key words: Heat Transfer, Thermoelectric, Nanostructuring, Phonon Transport, Electron Transport, Thermal Conductivity Nomenclature Constant: = Boltzmann constant, ћ = reduced Planck constant, Symbols: A = cross-sectional area, m2 C = spectral volumetric specific heat, J m-3 Hz-1 K-1 D = density of states per unit volume per unit frequency interval, m-3 Hz-1 e = charge per carrier, C = Fermi level (i.e. electrochemical potential) relative to conduction band edge , J = band gap energy = Fermi-Dirac distribution = Bose-Einstein distribution I = electrical current, A -2 = electrical current density, A m -2 = heat flux, W m K = thermal conductance, W m-2 K-1 1 = transport coefficients = Lorenz number, W Ω K-2 n = carrier concentration m-3 Q = heat current, W R = electrical resistance, Ω S = Seebeck coefficient, V K-1 T = temperature, K v = velocity, m s-1 Z = figure of merit, K-1 Β = Thomson coefficient, V K-1 Π = Peltier coefficient, V η = efficiency κ = thermal conductivity, W m-1 K-1 Ф = electrochemical potential, V = coefficient of performance = electrical resistivity, Ω m σ = electrical conductivity, Ω-1 m-1 τ = lifetime, s ω = angular frequency, rad·Hz Subscript: C: cold side H: hot side i: inlet M: mean n: n-type o: outlet p: p-type Abbreviations: AMM = acoustic mismatch model BTE = Boltzmann transport equation COP = coefficient of performance DFT = density functional theory DMM = diffuse mismatch model EMA = effective medium approach EMD = equilibrium molecular dynamics MC = Monte Carlo MD = molecular dynamics MFP = mean free path NEMD = non-equilibrium molecular dynamics 1. Introduction Thermoelectric effects have long been known since the Seebeck effect and the Peltier effect were discovered in 1800s 1. The Seebeck effect describes the phenomenon that a 2 voltage is generated in a conductor or semiconductor subjected to a temperature gradient. This effect is the basis of thermocouples and can be applied to thermal to electrical energy conversion. The inverse process, in which an electrical current creates cooling or heat pumping at the junction between two dissimilar materials, is called the Peltier effect. The decade of 1950s saw extensive research and applications on Peltier refrigerators following the emergence of semiconductors and their alloys as thermoelectric materials 1. However, the research efforts waned as the efficiency of solid-state refrigerators could not compete with mechanical compression cycles. Starting in 1990s, interest in thermoelectrics renewed because of increased global energy demand and global warming 2,3 caused by excessive CO2 emissions . The maximum efficiency of a thermoelectric device for both thermoelectric power generation and cooling is determined by the dimensionless figure-of-merit, ZT, S 2 ZT T (1.1) where S is the Seebeck coefficient, is the electrical conductivity, S 2 is the power factor, and p e is the thermal conductivity which is composed of lattice (phononic) thermal conductivity p and electronic thermal conductivity e . The Seebeck coefficient is voltage generated per degree of temperature difference over a material. Fundamentally, it is a measure of the average entropy carried by a charge in the material. A large power factor means electrons are efficient in the heat-electricity conversion, while a small thermal conductivity is required to maintain a temperature gradient and reduce conduction heat losses. Achieving high ZT values requires a material simultaneously possessing a high Seebeck coefficient and a high electrical conductivity, but maintaining a low thermal conductivity, which is challenging as these requirements are often contradictory to each other in conventional materials 4. Recent years have witnessed impressive progresses in thermoelectric materials. There have been many reviews 3-24 emphasizing different aspects of thermoelectrics, including bulk thermoelectric materials 10,12, individual nanostructures 6,13,15,25, bulk nanostructures 7,8, and interfaces in bulk thermoelectric materials 16. In this paper, we intend to emphasize the heat transfer aspects of thermoelectric research at both materials and system levels. We will start with a brief summary of the basic principles of thermoelectric devices and materials (Sec. 2), followed by a summary of materials status (Sec. 3). We will then discuss heat conduction mechanisms in bulk materials (Sec.4), and strategies to reduce thermal conductivity in bulk materials (Sec.5) and using nanostructures (Sec.6). Heat transfer issues at device and system levels will be presented in Sec. 7. A shorter version of this paper will appear in the Journal of Heat Transfer due to page limit. This review is more comprehensive, including an introductory section on thermoelectric transport and device basics (Sec. 2), more detailed discussion on heat transfer in thermoelectric devices and systems (Sec. 7) and 16 more figures. 2. Thermoelectric Transport and Device Basics 3 2.1 Thermoelectric Effects and Transport Properties Thermoelectric effects dictate the coupling between heat and electricity. Fundamentally, charges in semiconductors and conductors carry heat with them. The coupled charge 26,27 current flux, Je, and heat current flux, Jq, can be expressed as d dT J e L11( ) L12( ) (2.1) dx dx d dT J q L21( ) L22( ) (2.2) dx dx for one-dimensional transport, where L11 is the electrical conductivity and is the electromotive force, i.e., the electrochemical potential divided by unit charge. In an isothermal conductor, the heat current, L J 21 J J (2.3) q L e e 11 is proportional to the electrical current and the proportionality constant is the Peltier coefficient, . Although the Peltier coefficient is an intrinsic material property, Peltier cooling and heating happens when two materials with different Peltier coefficients are joined together, as shown in Fig. 1, due to the imbalance of the Peltier heat flowing in and out of the junction. A control volume analysis shows that the cooling or heating, Q, happening at the junction is equal to Q ( )I (2.4) 2 1 where I is the electrical current, and subscripts 1 and 2 represent Peltier coefficients for the two materials. Under open voltage condition, Eq. (2.1) leads to the Seebeck coefficient d / dx V L S 12 (2.5) dT / dx T L 11 From the definition of the Seebeck coefficient, we can see that the Seebeck voltage V between two points on a homogenous material does not depend on the temperature profile. This fact is used widely in temperature measurements by thermocouples. The Seebeck and Peltier coefficients are related to each other through, TS , which is an example of the generalized Onsager reciprocity relation L21=TL12. 4 Fig. 1 Cooling or heating at the junctions of two materials occur because of the different Peltier coefficients of the two materials, which can be used as thermoelectric refrigerators or heat pumps Furthermore, one can eliminate the electrochemical potential in Eq. (2.1) and express the heat current as dT J q Je e (2.6) dx where the electronic contribution to thermal conductivity is L L L 12 21 (2.7) e 22 L 11 The above expression has not included the phonon contribution to thermal conductivity. However, as long as one can separate the electronic and phononic components, one can simply add an additional phononic term to Eq. (2.6) for heat flowing in a solid. Equation (2.6) can be used with the first law of thermodynamics, leading to the following equation that determines the temperature distribution in a thermoelectric material 28 2 dT d dT dS dT J e C p T J e (2.8) dt dx dx dT dx where Cp is the specific heat per unit volume at constant pressure. The above equation contains an extra term compared to the ordinary heat conduction equation. This is due to the Thomson effect 1, which suggests that distributed heating or cooling can occur even in the same solid because of the temperature dependence of Seebeck coefficient. The Thomson coefficient is defined by q dS c T (2.9) dT dT J e dx where q is the rate of cooling flux or heating per unit volume. 5 All three thermoelectric effects (Seebeck effect, Peltier effect, and Thomson effect) are intrinsically connected as they are simply different manifestations of the heat carried by charges. The Thomson effect is often neglected, although careful device simulation should take it into consideration. It can be seen from Eq.