Th The t e trav avel o of hea eat i in solids Kamr mran Behni - - PowerPoint PPT Presentation

Th The t e trav avel o

• f 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

E Je   σ =

Electric current Electric field Electric conductivity

In an insulator σ =0

Electric conduction

e

J E  ρ =

→

Electric field Electric field Electric resistivity

In a superconductor ρ =0

Thermal conduction

→

∇ − = T Jq κ 

Thermal current Thermal gradient Thermal conductivity

There is no thermal insulator or superconductor

Thermal gradient is a vector from cold to hot Heat flows from hot to cold

Kinetic theory of gases

κ = 1/3 C v l

Thermal conductivity Heat capacity velocity mean-free-path

hot cold

Jq

q (W c

cm-2) ∇T ( T (K cm cm-1)

κ= J = Jq /( /(-∇T) T) [WK WK-1cm cm-1]

Heat conduction in insulators

Only phonons carry heat!

Phonon- phonon scattering Phonon- defect scattering

Heat conduction in insulators

As T increases, there are more carriers and more scattering centers!

Phonons are both carriers and scatterers

Heat conduction in metals

Electrons are dominant carriers of heat!

Electron- phonon scattering Electron- defect scattering

Car arri riers rs:electrons & phonons Scattere rers rs: phonons, deffects (and electrons)

The he b best conductor a at room tempera rature!

A l lot

• t of
• f p

phon

• nons

which d do

• not
• t

stro rongly i intera ract!

In the zero temperature limit

• Mean-free-path attains its maximum value

and then: κph ∝T3 (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 Tc a superconductor is a metal

(mobile electrons carry heat!)

• Below Tc, 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

• r thermocouples!

Mesauring thermal conductivity

thermometers heater sample

Cold finger

• A compact set-up:

diameter can be reduced to 10 mm

• RuO2 or Cernox

thermometers

• Calibration in

magnetic field

Heat and charge current in a solid

T E J T E J

Q e

∇ − = ∇ − =       ' κ β α σ

Three tensors σ electric conductivity κ thermal conductivity α thermoelectric conductivity Four vectors Je : charge current density JQ : heat current density E : electric field DT : thermal gradient

T α β =

Kelvin relation, (1860) Onsager relation (1930)

Thermoelectricity timeline

• 1821 : The Seebeck effect

Metal A Metal B ∆V

T

T+∆T A Thermocouple : voltage generated by temperature difference

Thermoelectricity timeline

• 1834: The Pelletier effect

Metal A Metal B

T Je

Electric current generates a temperature difference

T+ T+∆T

Thermoelectricity timeline

• 1854: Kelvin shows the link between Seebeck

and Pelletier effects

Π in volts S in volts per Kelvin William Thomson (Lord Kelvin)

Thermoelectricity timeline

• 1886: Nernst and Ettingshausen discover

the Nernst-Ettingshausen effect

Walther Nernst st Nobe

• bel Prize

ze for

• r chemist

stry 1920 1920

Seebeck and Peltier coefficients

T E S

x x

∇ − =

E T

→

∇

Seebeck effect: An electric field created by a thermal current

Seebeck and Peltier coefficients

Je

Q

J

Peltier effect: A thermal gradient created by an electric current

e Q

J J = Π

ST = Π

The Kelvin relation :

Nernst and Ettingshausen coefficients

Nernst coefficient

Ey T ∇ 

B 

T E N

x y

∇ − =

Nernst and Ettingshausen coefficients

Ettingshausen coefficient ε = ∇yT/Je

Je T ∇ 

B 

Thermoelectricity timeline

• 1924: Brigman deduces a link between

Nernst and Ettingshausen coefficients

Percy W. Bridgman Nobel Prize for Physics 1946

Nernst and Ettingshausen coefficients N = ε κ / T

Ettingshausen coefficient ε = ∇yT/Je

Je T ∇ 

B 

Thermal conductivity

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 intensive extensive

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)

A flow of energy in conditions of constant T and µ

Particle flow Entropy flow

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 flow driven by a temperature gradient Entropy produced locally Particle flow driven by potential gradient

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

= ∇ − = T E Je    α σ

Therefore

The Seebeck coefficient Two strictly equivalent definitions

Entropy flow due to charge current Entropy flow due to thermal gradient

The Seebeck coefficient is a measure of entropy per charge carrier.

[Callen 1948]

In a highly degenerate Fermi gaz

kBT<< ε F The Wiedemann-Franz law:

The ratio is heat to charge conductivity is fixed.

In a highly degenerate Fermi gaz

kBT<< 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 =τ0

ξ, this becomes:

Room temperature Seebeck coefficient of metals

Sign is sometimes wrong! Magnitude is often ok!

Seeb eebec eck coe

• efficient i

in r real m metals ls i is n not

• t feature

urele less!

p = 1/3 U(T)

Phonon Dragg

hot cold

Phonon pressure gradient 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 Dragg

hot cold

Phonon pressure gradient In presence of phonons, electrons feel a pressure because of electron-phonon collisions. This pressure is set by phonon energy density.

Phonon Dragg

hot cold

Flow of heat-carrying phonons

α Ne C S

L g =

Phonon dragg thermopower

Lattice specific heat Carrier density e- -ph coupling (0<α<1)

Order of magnitude of phonon dragg

α β

e L diff g

C C S S = /

e- per atom (T/ΘD)3 T/TF

Frequency of ph-e- scattering events

Heavy Fermion metals have a very large electronic specific heat

• 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

) ( 3

2 2 F B e

N Tk C ε π =

α β

e L diff g

C C S S = /

A s A smal all Fer ermi en ener ergy w with th a a lar arge kF: Th The m e mas ass i is lar arge!

Heavy Fermi liquids

• Enhanced specific heat
• Enhanced Pauli Susceptibility
• Enhanced T2-resistivity

) ( 3

2 2 F BN

k ε π γ =

) (

2 F BN ε

µ χ =

ρ=ρ0 +A T2

2

) (

F

N A ε ∝

Fermi liquid ratios

The Wilson ratio

γ χ µ π

2 2 2

3

B B W

k R =

Fermi liquid ratios

The Kadowaki-Woods ratio

2

γ A KW =

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

Thermoelectric coefficients

• In presence of a thermal gradient,

electrons produce an electric field.

• Seebeck and Nernst effect refer to the

longitudinal and the transverse components of this field.

JQ T ∇ 

x

E 

y

E 

B 

T E S

x x

∇ − =

T E S N

x y xy

∇ − = =

] [ T B E

x z y

∇ − = ν

hot cold

Ner ernst ef effect ect in n a a si single-ban and m metal al

Absence of charge current leads to a counterflow of hot and cold electrons:

T ∇ 

B 

e- e-

JQ ≠ 0 ; Je= 0 ; Ey= 0 In an n an ideally simple m meta tal, t the N Nernst e effect t vani anishes! (« (« Sond ndheimer c canc ncellat ation », 1 1948) 48) E

y

JQ

There is a measurable Nernst effect in a number of remarkable metals

T E T J T E J

Q e

∇ − = ∇ − =       κ α α σ

2 2

xy xx xy xx xx xy

N σ σ σ α σ α + − =

Nernst coefficient :

Wh What do does « Sondhe dheimer ca cance cellation » » me mean?

If shifting the Fermi level does not change the Hall angle, then there is no Nernst signal!

xx xy H

σ σ = Θ

A large diffusive component in the zero-temperature limit!

Roughly, the Nernst coefficient tracks µ/ ΕF…

F B H B

e k T eB k T

F

ε µ π ν ε π ν

ε 2 2 2 2

3 3 ≈ ⇒ ∂ Θ ∂ =

Recipe for a large diffusive Nernst response:

• High mobility
• Small Fermi energy
• Ambipolarity

Nernst effect in the vortex state

• Thermal force on the vortex :

F=-Sφ ∇T (Sφ : vortex entropy)

• The vortex moves
• The movement leads to a

transverse voltage: Ey=vx Bz

A superconducting vortex is:

• A quantum of magnetic flux
• An entropy reservoir
• A topological defect

B 

Ey ∇T

A superconductor

• Perfect conductor

Resistivity is zero

• Perfect diamagnet

The magnetic field is totally expulsed A A macr crosco copic m c manifes estati tation o

• f quantu

tum m mechan anics ics! The s e super erco conducti cting o

• rder

er par arameter is is a a wave ave-fun unction: amplitude phase

A superconducting vortex

• Superfluid density vanishes in the core of the vortex

(coherence length ξ).

• Magnetic field penetrates inside the vortex (penetration depth

λ). In a type II superconductor λ>ξ

A vortex is a quantum of magnetic flux

Flux quantum At a given field B, the vortex density is B/φ0

A vortex is an entropy reservoir

The vortex core is in the normal state. Since the entropy of the normal state exceeds the entropy of the superconducting state, the vortex has more entropy than the surrounding superfluid.

Nernst effect in conventional superconductors

A Nernst signal emerges in an immediate temperature range Huebener 1967

Ne Nernst e effect i in op

• ptimally

ly-dop

• ped Y

YBCO

(Ri, Huebener et al. 1994)

A finite signal in the vortex liquid state!

A positive Nernst signal survives above Tc

Wang, Li and Ong, 2006 T(K) The fluctuating tail becomes longer in the underdoped regime

Vortex-like excitations in the normal state of the underdoped cuprates?

A finite Nernst signal in a wide temperature range above Tc Wang, Li and Ong, 2006

Preformed Cooper pairs in the pseudogap state?

Two distinct temperature scales for superconductivity: T* as the onset of phase fluctuating Cooper pairs Tc as the onset of Phase coherence Nature 1995

But Nernst coefficient of real metals can be large!

Recipe for a large diffusive Nernst response:

• High mobility
• Small Fermi energy

ν~ (π2/3) k2

BT/e µ / ΕF

A small Fermi surface Highly mobile electrons Only seen in YBCO

The Ne Nernst coe coeffici cien ent i in YBCO

The background signal is negative! Wang, Li and Ong, 2006

The small electron pocket is the source of a negative Nernst signal in YBCO! Within a factor

• f 2 of what is

expected for the electron pocket! Chang et al., 2010 YBCO y =6.67 Tc=66 K

The Nernst effect caused by Gaussian fluctuations of the superconducting order parameter

The Ginzburg-Landau theory

Ψ is the order parameter Free energy

• Above Tc <ψ>=0. Instantaneously and locally ψ is

not necessarily zero.

• In a finite length-time window, ψ can be finite.
• There are short-lived, local Cooper pairs in the

normal state.

Nernst effect due to Gaussian fluctuations of the superconducting order parameter (Usshishkin, Sondhi & Huse, 2002)

Quantum of thermo-electric conductance (21 nA/K)

In 2D: In two dimensions, the coherence length is the unique parameter!

Magnetic length

Experiment

This theory is experimentally verified in a conventional 2D superconductor! The normal state :

• is dirty
• has a large carrier density
• is uncorrelated

Therefore the the Normal state Nernst signal is negligible!

Superconductivity in Nb0.15Si 0.85 thin films

The normal state is a simple dirty metal: le~a~ 1/kF !

50 100 150 200 250 300 400 800 1200

R(Ω ) T(K)

Nb0.15Si0.85 d=12.5 nm Tc=220 mK ρ≅2 mΩcm

Nernst effect across the resistive transition

• A large vortex

signal below Tc

• A long

tail above Tc

A signal distinct from the vortex signal

Deep into the normal state!

The Nernst signal of the normal electrons is negligible!

Even at 6K : The expected normal state contribution is three orders of magnitude smaller than ν/T !

µ ~10 -5 T εF ~104 K

Comparison with theory

0.02 0.1 1 4 1E-5 1E-4 1E-3 0.01

α

xy/B (Sample1) (B=0)

α

xy/B (Sample2) (B=0)

α

xy/B (USH)

α

xy/B(µA/TK)

ε=ln(T/TC) ∝ ξ

• 2

d

2 d 2 2 B xy

.ξ 6π e k         = 

USH

B α B R

SC XY square

α ν =

Experiment: Theory: The coherence length is the single parameter here

Estimating the superconducting coherence length

c B F d

T k v   2 3 36 . 1 ε ξ =

This should be compared to the expression for a 2D dirty superconductor: T-dependence Amplitude

e B e F

e k T v γ σ π γ κ

2

3       = = 

Using specific heat and resistivity data, this yields:

1 2 5

10 35 . 4

− −

× = s m vF

The ghost critical field

Contour plot of N= -Ey /(dT/dx)

A unique correlation length

c B F d

T k v   2 3 36 . 1 ε ξ =

Contour plot of the Nernst coefficient ν=N/B

Ho How d do the f fluctuat ating g Cooper p pairs g gene nerat ate a a Nerns nst s t signa gnal?

cold hot cold hot

dT/dx Ey B

• Above Tc, the lifetime
• f the Cooper pairs

decrease with increasing temperature

• Therefore, those pairs

which travel from the hot side along the cold side live longer!