Ecole Supérieure de Physique et de Chimie...

The travel of heat in solids

Kamran Behnia

Ecole Supérieure de Physique et de Chimie Industrielles

Outline

1. Thermal conductivity 2. Thermoelectricity: Seebeck effect 3. Thermoelectricity: Nernst effect

Electric conduction

EJe

σ=

Electric current

Electric field

Electric conductivity

In an insulator σ =0

Electric conduction

eJE

ρ=→

Electric field

Electric field

Electric resistivity

In a superconductor ρ =0

Thermal conduction

→

∇−= TJq 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


Heat capacity velocity

mean-free-path

hot cold

Jq (W cm-2)

∇T (K cm-1)

κ= Jq /(-∇T) [WK-1cm-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

Carriers:electrons & phonons Scatterers: phonons, deffects (and electrons)

The best conductor at room temperature!

A lot of phonons which do not strongly interact!

In the zero temperature limit

• Mean-free-path attains its maximum value and then: κph ∝T3 (phonons are bosons) κe ∝T (electrons are fermions)

In principle, one can separate the two contributions!

Example

Taillefer et al., PRL ‘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 or 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

TEJ

TEJ

Q

e

∇−=

∇−=

'κβ

ασ

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

Tαβ = 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

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 Nobel Prize for chemistry 1920

Seebeck and Peltier coefficients

TESx

x

∇−

= E

T→

∇

Seebeck effect: An electric field created by a thermal current

Seebeck and Peltier coefficients

Je

QJ

Peltier effect: A thermal gradient created by an electric current

e

Q

JJ

=Π

ST=ΠThe Kelvin relation :

Nernst and Ettingshausen coefficients

Nernst coefficient Ey T∇

B

TE

Nx

y

∇−=


Ettingshausen coefficient ε = ∇yT/Je

Je T∇

B

Thermoelectricity timeline • 1924: Brigman deduces a link between


Percy W. Bridgman Nobel Prize for Physics 1946


N = ε κ / T

Ettingshausen coefficient ε = ∇yT/Je

Je T∇

B


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


An incremental increase in energy while keeping the intensive parameters constant!


Let us imagine that the volume remains constant!


A flow of energy in conditions of constant T and µ

Particle flow

Entropy flow


Conservation of energy implies :

Conservation of particle number implies :

But entropy is NOT conserved :


Entropy flow driven by a temperature gradient

Entropy produced locally

Particle flow driven by potential gradient


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

0=∇−= TEJe

ασ

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 =τ0ξ, this becomes:

Room temperature Seebeck coefficient of metals

Sign is sometimes wrong!

Magnitude is often ok!

Seebeck coefficient in real metals is not featureless!

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

αNeCS L

g =Phonon dragg thermopower

Lattice specific heat

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

Order of magnitude of phonon dragg

αβe

Ldiffg C

CSS =/

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

22

FBe NTkC επ=

αβe

Ldiffg C

CSS =/

A small Fermi energy with a large kF: The mass is large!

Heavy Fermi liquids

• Enhanced specific heat • Enhanced Pauli Susceptibility

• Enhanced T2-resistivity

)(3

22

FB Nk επγ =

)(2FB N εµχ =

ρ=ρ0 +A T2

2)( FNA ε∝

Fermi liquid ratios

The Wilson ratio γχ

µπ

2

22

3 B

BW

kR =

Fermi liquid ratios

The Kadowaki-Woods ratio

2γAKW =

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

Behnia, Flouquet, Jaccard JPCM ‘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∇

xE

yE

B

TESx

x∇−=

TE

SNx

yxy ∇

−== ][ TB

Exz

y

∇−=ν

hot cold

Nernst effect in a single-band metal

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 ideally simple metal, the Nernst effect vanishes! (« Sondheimer cancellation », 1948)

Ey

JQ

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

TETJ

TEJ

Q

e

∇−=

∇−=

καασ

22xyxx

xyxxxxxyNσσ

σασα

+

−= Nernst coefficient :

What does « Sondheimer cancellation » mean?

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

xx

xyH σ

σ=Θ

A large diffusive component in the zero-temperature limit!

Roughly, the Nernst coefficient tracks µ/ ΕF…

F

BHB

ek

TeBk

TF

εµπν

επν

ε

2222

33≈⇒

∂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 macroscopic manifestation of quantum mechanics!

The superconducting order parameter is a wave-function:

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

Nernst effect in optimally-doped 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


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) k2BT/e µ / ΕF

A small Fermi surface Highly mobile electrons

Only seen in YBCO

The Nernst coefficient in YBCO

The background signal is negative!


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

Within a factor of 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 !

0 50 100 150 200 250 3000

400

800

1200

R(Ω

)

T(K)

Nb0.15Si0.85

d=12.5 nmTc=220 mKρ≅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

αxy/B (Sample1) (B=0) αxy/B (Sample2) (B=0) αxy/B (USH)

α xy/B

(µA

/TK

)

ε=ln(T/TC) ∝ ξ-2d

2d2

2Bxy .ξ

6πek

=

USHBα

BR

SCXY

square

αν=

Experiment:

Theory:

The coherence length is the single parameter here

Estimating the superconducting coherence length

cB

Fd Tk

v

2336.01

εξ =

This should be compared to the expression for a 2D dirty superconductor:

T-dependence

Amplitude

e

B

eF e

kT

vγσπ

γκ 2

3

==

Using specific heat and resistivity data, this yields:

1251035.4 −−×= smvF

The ghost critical field

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

A unique correlation length

cB

Fd Tk

v

2336.01

εξ =

Contour plot of the Nernst coefficient ν=N/B

How do the fluctuating Cooper pairs generate a Nernst signal?

cold hot

cold hot

dT/dxEy B

•Above Tc, the lifetime of the Cooper pairs decrease with increasing temperature

•Therefore, those pairs which travel from the hot side along the cold side live longer!

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