Thermoelectricity  Thermoelectricity / thermoelectric effect electric field and a temperature gradient along the z direction of a conductor Let  : electrochemical potential (electrostatic + chemical)

The Seebeck Effect and Thermoelectric Power  Seebeck effect: used to produce an electrical power directly from a temperature difference.  Seebeck coefficient: induced thermoelectric voltage across the material of unit length per unit temperature difference. (thermopower or thermoelectric power) When no current flow, Seebeck coefficient [ W/K ]

Appendix B.8

for copper at T = 300 K and 600 K Experimental values are positive with 1.83 mV/K and 3.33 mV/K. The sign error is due to simplification used to evaluate  1. Seebeck coefficient positive for p -type semiconductors negative for n -type semiconductors

T2T2 V2V2 T1T1 V1V1 Thermoelectric voltage cannot be measured with the same type of wires because the electrostatic potentials would cancel each other. T2T2 T1T1 ∆V∆V T1T1 Type I (+) Type II (-)

The Peltier Effect and the Thomson Effect  Peltier effect: reverse of the Seebeck effect. a creation of a heat difference from an electric voltage. : Peltier coefficient

 Thomson effect: Heat can be released or absorbed when current flows in a material with a temperature gradient. It describes the heating or cooling of a current- carrying conductor with a temperature gradient. : Joule heating Energy received by a volume element : heat transfer due to temperature gradient : Thomson effect Thomson coefficient:

Thermoelectric Generation and Refrigeration TLTL THTH L ACAC ∆V∆V x Assumptions negligible contact resistances same length and cross-sectional area of all thermoelectric elements heat transfer by conduction only through thermoelectric elements Joule heating due to resistance of the thermoelectric element only thermal, electrical conductivities and Seebeck coefficient: independent of temperature. very small temperature difference btw. the two heat reservoirs Steady-state temperature distribution

boundary conditions

output power

figure of merit  thermal efficiency

Onsager’s Theorem and Irreversible Thermodynamics In thermodynamics, the Onsager reciprocal relations express the equality of certain relations between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.  T : heat diffusion and current  : current and heat flow both  T & E → driving force F j current and heat flow→ flux of physical quantity J i  ij : Onsager kinetic coefficient Onsager reciprocity relation 4 th law of thermodynamics

Classical Size Effect on Conductivities and Quantum Conductance Classical Size Effect Based on Geometric Consideration Classical Size Effect Based on BTE Quantum Conductance

Classical Size Effect Based on Geometric consideration Free-path reduction due to boundary scattering  Size dependence of the mean free path Knudsen number and thermal conductivity relations

in the ballistic transport limit : d <<  b f = df = d conductivity ratio : Matthiessen’s rule

 Consideration of free path duistribution When d <<  eff and all energy carriers originate from the boundary

Applicable for Kn > 5 Cannot be applied for small values of Kn since ln(Kn) becomes negative. for Kn < 1 for thin films

For the z-direction consideration applicable for Kn > 5

When d <<  eff and all energy carriers originate from the center

For circular wires, the conduction along a thin wire when d <<  eff and all energy carriers originate from the center

As Kn bigger, effect of boundary scattering more significant Paths with larger polar angles are more important for parallel conduction, whereas paths with smaller polar angles are more important for normal conduction. Reduction in thermal conductivity due to boundary scattering

Classical Size Effect Based on the BTE steady state electron movement in x, z directions temperature and electric field in x direction

under the assumption that f is not far from f 0 electron movement in a particular direction

Upward motion

: arbitrary function that accounts for the accommodation and scattering for perfect accommodation with inelastic and diffuse scattering

Downward motion similarly

current density for electric conduction without any temperature gradient and with

average current flux energy integral at the Fermi surface effective electrical conductivity

exponential integral function

asymptotic relations Thermal conductivity

 Electrical and thermal conductivities based on the BTE Assumption – Relaxation time approximation, under the local equilibrium conditions Temperature gradient & electric field in the x-direction only Finite thickness in the z-direction, the distribution function x-direction >> z-direction Temperature gradient & electric field, x-direction >> z-direction Replace General solution of the steady-state BTE under the relaxation time approximation : arbitrary function

 Electrical conductivities based on the BTE Assumption – Relaxation time approximation, under the local equilibrium conditions : arbitrary function that accounts for the accommodation and scattering. If perfect accommodation is assumed with inelastic and diffuse scattering, and electrical conduction without any temperature gradient,

 Electrical conductivities based on the BTE No temperature gradient, current density can be written as, (5.61a) : the effective electrical conductivity of the film : Properties at the Fermi surface

 Electrical conductivities based on the BTE No temperature gradient, current density can be written as, (5.61a) (5.62)

 Electrical conductivities based on the BTE No temperature gradient, current density can be written as, mth exponential integral for Kn << 1for Kn >> 1 Asymptotic relations Similar to the result of “electron originates from the center of the film” for Kn >> 1, (thinner film case) Because the derivation using the BTE presented earlier inherently assumed that the electrons are originated from the film rather than from the boundaries.

 Thermal conductivities based on the BTE Assumption – Relaxation time approximation, under the local equilibrium conditions If perfect accommodation is assumed with inelastic and diffuse scattering, and temperature gradient without any electric field

 Thermal conductivities based on the BTE No electric field, thermal conductivity can be written as, (5.65a)

 Thermoelectricity based on the BTE Assumption – Relaxation time approximation, under the local equilibrium conditions First-order approximation, L 12 and L 21 subjected to boundary scattering Seebeck coefficient along the film remains the same regardless of boundary scattering : specularity, represent the probability of scattering being elastic and specular

 Thermoelectricity based on the BTE For electronic transport, wavelength < 1nm, p=0 (as diffuse) For phonons, wavelength may vary, : Surface roughness In reality, p depends on the angle of incidence Kn > 0.1, the size effect may be significant.Kn > 10, boundary scattering dominate. size effect important boundary scattering dominate As temperature is lowered, size effect more significant.

 Conduction along a thin wire based on the BTE from reference 41,42 The asymptotic approximations, with about 1% accuracy for Kn < 0.6 for Kn > 1 If p=1, If p≠1, where,

Quantum conductance (1)  When the quantum confinement becomes significant, the relaxation time approximation used to solve the BTE is not applicable.  Electrical conductance of metallic materials and thermal conductance of dielectric materials.  0 Bulk solid, 3-D Quantum well, 2-D for n = 1,2,3,… Quantum well, 1-D For 3-D confined quantum dots, the energy levels are completely discrete; subsequently, the density of states becomes isolated delta functions.

 Transport phenomena in the quantum or ballistic regimes Landauer treated electrical current flow as transmission probability a) Electrical current flow through a narrow metallic channel due to different electrochemical potentials b) Heat transfer between two heat reservoirs through a narrow dielectric channel Ballistic transmission, absence of losses by scattering and reflection The net current flow : chemical potential

 Transport phenomena in the quantum or ballistic regimes There is no resistance or voltage drop associated with the channel itself. The voltage drops are associated with the perturbation at each end of the channel as it interacts with the reservoir. Transmission coefficient ξ 12, the actual distribution function,the electronic spin degeneracy

 Transport phenomena in the quantum or ballistic regimes For small potential difference, using the following approximation Ö given by scattering matrix based on Schr Ö dinger’s equation transmission coefficient between 0 and 1

 Ballistic thermal transport process Resembles electromagnetic radiation between two blackbodies separated by vacuum For a 1-D photon gas, Stefan-Boltzman law

 Ballistic thermal transport process