Th The t e trav avel o of hea eat i in solids Kamr mran Behni nia Ec Ecole Supérieure de e Physi sique et et de e Chimie Industrielle lles
Outline 1. Thermal conductivity 2. Thermoelectricity: Seebeck effect 3. Thermoelectricity: Nernst effect
Electric conduction = σ J e E Electric field Electric conductivity Electric current In an insulator σ =0
Electric conduction → = ρ E J e Electric field Electric resistivity Electric field In a superconductor ρ =0
Thermal conduction Thermal gradient is a vector from cold to hot Heat flows from hot to cold → = − κ ∇ J q T Thermal gradient Thermal conductivity Thermal current There is no thermal insulator or superconductor
Kinetic theory of gases κ = 1/3 C v l mean-free-path Thermal conductivity Heat capacity velocity cm -2 ) J q q (W c κ = J = J q /( /(- ∇ T) T) cold hot WK -1 cm cm -1 ] [WK ∇ T ( T (K cm cm -1 )
Heat conduction in insulators Phonon- defect Phonon- scattering phonon scattering Only phonons carry heat!
Heat conduction in insulators Phonons are both carriers and scatterers As T increases, there are more carriers and more scattering centers!
Heat conduction in metals Electron- Electron- phonon defect scattering scattering Car arri riers rs:electrons & phonons Scattere rers rs: phonons, deffects (and electrons) Electrons are dominant carriers of heat!
The he b best conductor a at room tempera rature! A l lot ot of of p phon onons which d do o not ot stro rongly i intera ract!
In the zero temperature limit • Mean-free-path attains its maximum value and then: κ ph ∝ T 3 (phonons are bosons) κ e ∝T (electrons are fermions) In principle, one can separate the two contributions!
Example Taillef lefer er et al., PR PRL ‘ L ‘97
Thermal conductivity of superconductors • Above T c a superconductor is a metal (mobile electrons carry heat!) • Below T c , mobile electrons condensate in a macroscopic quantum state: electronic heat carriers vanish! • A superconductor can be assimilated to a thermal insulator
Mesauring thermal conductivity Temperature captors: Resistive thermometers or thermocouples!
Mesauring thermal conductivity • A compact set-up: diameter can be reduced to 10 mm thermometers • RuO 2 or Cernox thermometers heater Cold finger sample • Calibration in magnetic field
Heat and charge current in a solid = σ − α ∇ J E T e = β − κ ∇ J E ' T Q β = α T Kelvin relation, (1860) Onsager relation (1930) Four vectors Three tensors J e : charge current density J Q : heat current density σ electric conductivity E : electric field κ thermal conductivity DT : thermal gradient α thermoelectric conductivity
Thermoelectricity timeline • 1821 : The Seebeck effect Metal A T+ ∆ T T ∆ V Metal B A Thermocouple : voltage generated by temperature difference
Thermoelectricity timeline • 1834: The Pelletier effect Metal A T+ ∆ T T+ T J e Metal B Electric current generates a temperature difference
Thermoelectricity timeline • 1854: Kelvin shows the link between Seebeck and Pelletier effects S in volts per Kelvin Π in volts William Thomson (Lord Kelvin)
Thermoelectricity timeline • 1886: Nernst and Ettingshausen discover the Nernst-Ettingshausen effect Walther Nernst st Nobe obel Prize ze for or chemist stry 1920 1920
Seebeck and Peltier coefficients Seebeck effect: An electric field created by a thermal current − E = x S ∇ E T x → ∇ T
Seebeck and Peltier coefficients Peltier effect: A thermal gradient created by an electric current J Π = Q J e J e J Q Π = ST The Kelvin relation :
Nernst and Ettingshausen coefficients B ∇ Nernst coefficient T E y E = y N − ∇ T x
Nernst and Ettingshausen coefficients B Ettingshausen coefficient ∇ J e T ε = ∇ y T/J e
Thermoelectricity timeline • 1924: Brigman deduces a link between Nernst and Ettingshausen coefficients Percy W. Bridgman Nobel Prize for Physics 1946
Nernst and Ettingshausen coefficients B Ettingshausen coefficient ∇ J e T ε = ∇ y T/J e Thermal conductivity N = ε κ / T Bridgman relation
Thermoelectricity timeline • 1931: Onsager reciprocal relations Lars Onsager Nobel Prize for Chemistry 1968
Thermoelectricity timeline • 1948: Callen shows that Brigman relation (as well as Kelvin relation) can be deduced from Onsager reciprocal relations. Herbert Callen
The thermodynamic origin of thermoelectricity (Callen 1948) energy extensive intensive
The thermodynamic origin of thermoelectricity (Callen 1948) An incremental increase in energy while keeping the intensive parameters constant!
The thermodynamic origin of thermoelectricity (Callen 1948) Let us imagine that the volume remains constant!
The thermodynamic origin of thermoelectricity (Callen 1948) Entropy flow Particle flow A flow of energy in conditions of constant T and µ
The thermodynamic origin of thermoelectricity (Callen 1948) Conservation of energy implies : Conservation of particle number implies : But entropy is NOT conserved :
The thermodynamic origin of thermoelectricity (Callen 1948) Entropy Particle flow driven by flow driven potential gradient by a temperature gradient Entropy produced locally
The thermodynamic origin of thermoelectricity (Callen 1948) A gradient in temperature generates a gradient in chemical potential
The Seebeck coefficient Experimentally, the Seebeck coefficient is the most directly accessible thermoelectric coefficient. Just measure the voltage difference and the temperature difference in absence of charge current = σ − α ∇ = J e E T 0 Therefore
The Seebeck coefficient Two strictly equivalent definitions [Callen 1948] Entropy flow due to charge current Entropy flow due to thermal gradient The Seebeck coefficient is a measure of entropy per charge carrier.
In a highly degenerate Fermi gaz k B T<< ε F The Wiedemann-Franz law: The ratio is heat to charge conductivity is fixed.
In a highly degenerate Fermi gaz k B T<< ε F The Mott formula
Seebeck coefficient of the free electron gas In the Boltzmann picture thermopower is linked to electric conductivity [the Mott formula]: This yields: transport Thermodynamic For a free electron gas, with τ = τ 0 ξ , this becomes:
Room temperature Seebeck coefficient of metals Magnitude is often ok! Sign is sometimes wrong!
Seeb eebec eck coe oefficient i in r real m metals ls i is n not ot feature urele less!
Phonon Dragg In presence of phonons, electrons feel a pressure because of electron-phonon collisions. This pressure is set by phonon energy density. p = 1/3 U(T) Phonon pressure gradient cold hot
Phonon Dragg In presence of phonons, electrons feel a pressure because of electron-phonon collisions. This pressure is set by phonon energy density. p = 1/3 U(T) Phonon pressure gradient cold hot
Phonon Dragg hot cold Flow of heat-carrying phonons Lattice specific heat C g = α Phonon dragg thermopower L S Ne e - -ph coupling (0< α <1 ) Carrier density
Order of magnitude of phonon dragg (T/ Θ D ) 3 C = α L Frequency of ph-e - scattering S / S β g diff events C e e - per atom T/T F
Heavy Fermion metals have a very large electronic specific heat π 2 = ε 2 C Tk N ( ) e B F 3 A s A smal all Fer ermi en ener ergy w with th a a lar arge k F : Th The m e mas ass i is lar arge! • Because of the large density of states at Fermi energy, the electronic specific heat is enhanced • But the phonon specific heat is like other metals. • Phonon dragg becomes negligible at low T C = α L S / S β g diff C e
Heavy Fermi liquids π 2 γ = ε 2 ( ) k B N • Enhanced specific heat F 3 χ = µ B N ε 2 ( ) • Enhanced Pauli Susceptibility F ρ = ρ 0 +A T 2 • Enhanced T 2- resistivity ∝ ε 2 ( ) A N F
Fermi liquid ratios π χ 2 2 k = B The Wilson ratio R µ γ W 2 3 B
Fermi liquid ratios The Kadowaki-Woods ratio A KW = γ 2
Thermopower and specific heat In a free electron gas : Thermopower is a measure of specific heat per carrier The dimensionless ratio: is equal to –1 (+1) for free electrons (holes) [if one assumes a constant mean-free-path , then ξ =1/2 and q=2/3]
Heavy electrons in the T=0 limit Replotting two-decades-old data! Extrapolated to T=0, the extracted S/T yields a q close to unity!
Another plot linking two distinct signatures of electron correlation Behni nia, , Flouq uque uet, J Jac accard rd JPC PCM ‘ ‘04 04
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