Download The travel of heat and charge in solids

The travel of heat and charge
g
in solids
K
Kamran
Behnia
B h i
Ecole Supérieure de Physique et de Chimie Industrielles
Paris, FRANCE
Contents
I. Introduction
•
•
•
Kinetic picture of heat travel in metals and insulators
Phonons and electrons
Superconductors (conventional and unconventional)
II Transport Equations
II.
•
•
•
The three conductivity tensors
The Wiedemann-Franz law
The Bridgman relation
III. Electronic correlations
•
•
•
T-square Resistivity
Enhanced linear specific heat
Seebeck coefficient
•
•
•
correlated electrons
superconducting vortices
short-lived Cooper pairs
IV. Nernst effect
ff
Kinetic theory
y of gases
g
 = 1/3 C v l
mean-free-path
Thermal conductivity
Heat capacity
velocity
Jq (W cm-2)
hot
∇T (K cm-1)
cold
= Jq /(-∇T)
[WK-1cm-1]
Heat conduction in insulators
Phonondefect
scattering
Phononphonon
h
scattering
Only phonons carry heat!
Heat conduction in insulators
Phonons are both
carriers and scatterers
As T increases,
increases
• More carriers !
• More scattering centers!
Heat conduction in metals
Electrondefect
scattering
Electronphonon
h
scattering
Carriers:electrons &
phonons
Scatterers: phonons,
deffects (and electrons)
Electrons are dominant carriers of heat!
The best conductor at room temperature!
Graphene at room T
Non-interacting
phonons !
In the zero temperature limit
• Mean-free-path
p attains its maximum value
and then:
ph ∝T3 (phonons
( h
are b
bosons))
 ∝T ((electrons are fermions))
e
In principle, one can separate
the two contributions!
Example
Taillefer et al., 1997
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
In conventional superconductors
p
• Electron thermal conductivity decreases
exponentially
ti ll
• Phonon thermal conductivity increases due
to a diminished electron scattering
At finite temperature, the temperature dependence of thermal
conductivityy in a superconductor
p
is complex
p !
Example: niobium
A phonon peak,
whenever the
phonon
h
meanfree-path can
increase below
Tc!
Kes et al.
J. Low Temp. Phys.
(1974)
normal
superconducting
e- component
Unconventional superconductors
• The order parameter of the unconventional
superconductors is less symmetric than the
Fermi surface of the mother metal.
• Th
The gap function
f
i can vanish
i h along
l
particular orientations (nodes).
• Nodal quasi-particles
quasi particles can carry heat!
Effect of an unconventional
superconducting
d ti ttransition
iti on th
thermall
transport
• The electronic thermal transport does NOT
decrease exponentially
• It can even increase below Tc (due to an
increase in the electronic mean-free-path
 = 1/3 C v l
Scattering
g events are restricted in an
unconventional superconductor
s-wave
d-wave
Heat conduction in YBCO
Aubin 1997
The increase in thermal conductivity below Tc is due to electrons!
Enhanced electronic thermal conductivity in
other unconventional superconductors
CeCoIn5 (Seyfarth et al., 2008)
Thermal transport in the zero-temperature
limit
• A finite linear term in thermal conductivity
• A residual normal fluid of nodal quasi-particles
= 00T +...+
+ + bT3
Example: heavy-fermion superconductor CePt3Si
Izawa et al.,, PRL ‘05
A superconductor with
no inversion symmetry and
yet with nodal quasiparticles!
Universal thermal conductivity
• In a d-wave superconductor, in a first
approximation 0 is independent of impurity
concentration
Decrease the mean-free-path
Impurities:
Enhance the specific heat
These two cancel out in a subset of unconventional superconductors
superconductors.
A TALE OF TWO VELOCITIES!
(
(Durst
, Lee ’99))
E it ti spectrum
Excitation
t
in
i
the vicinity of a node:
E(k)
= (k2 + k2)1/2
= (vF2k12 + v22k22)1/2
Fermi velocity :vF = dk/dk1
Gap velocity: v2= dk/dk2
00/T =(nkB2/3ħ) (vF / v2)
Experimental observation of
universal thermal conductivity
A 30-fold
30 f ld d
decrease iin mean-free-path
f
th iin Z
Zn-doped
d
d YBCO
leaves 0 unchanged (Taillefer et al., PRL 1997)
Universal conductivity in other
unconventional superconductors
0 independent of impurity concentration
checked in:
•
•
Bi2Sr2CaCu2O8 (Nakamae et al. , 2001)
Sr2RuO4 ((Suzuki et al. , 2004))
0 of the right order of magnitude found in:
•
•
•
•
κ-(BEDT-TTF)2Cu(NCS)2 (Belin et al.,
al 1998)
CePt3Si (Izawa et al., 2005)
URu2Si2 (Kasahara et al., 2007)
CeCoIn5 (Seyfarth et al., 2008)
Mesauring thermal conductivity
Temperature captors:
Resistive thermometers
or thermocouples!
Part II- transport equations
Heat and charge current in a solid


Je   E


J Q    T



J e   E   T



J Q   E   T
 T
 T
Kelvin relation, (1860) Onsager relation (1930)
Four vectors
Je : charge current density
JQ : heat
h t currentt density
d
it
E : electric field
DT : thermal gradient
Three tensors
 electric conductivity
 thermal conductivity
 thermoelectric conductivity
Definition of thermoelectric
coefficients
•
In p
presence of a thermal g
gradient,,
electrons produce an electric field.
•
Seebeck and Nernst effect refer to the
longitudinal and the transverse
components of this field.

Ex

B
hot

Ey
JQ
cold
ld

T

E
x
S
xT
N  S xy 
 Ey
 xT
[  Ey ]
BzxT
Link to the Peltier tensor: 
Seebeck coefficient
Nernst coefficient

S
 xx
xx
  N/B
   
xy
xx
 
xx
2
2
xx
xy
Experimentally,
p
y what is measured is S and 
xy
Seebeck and Peltier coefficients
Peltier effect:
effect A thermal gradient created by
b an electric ccurrent
rrent

JQ
Je
Je
JQ
Seebeck and Peltier coefficients
Seebeck effect:
effect An electric field created b
by a thermal ccurrent
rrent
 Ex
S
 xT
E

T
The Kelvin relation :
  ST
Nernst and Ettingshausen coefficients

B
N
Nernst
t coefficient
ffi i t
S xy 
 Ey
 xT

T Ey
Nernst and Ettingshausen
g
coefficients

B
Ettingshausen
Etti
h
coefficient
ffi i t
 = yT/Je

T
Je
Nernst and Ettingshausen
g
coefficients
Sxy=   / T
Bridgman
g
relationship,
p
Thermal conductivity
Ettingshausen
Etti
h
coefficient
ffi i t
[ = yT/Je]

B

T
Je
Nernst and Ettingshausen
g
coefficients
Sxy=   / T
Bridgman
g
relationship,
p
Sommerfeld and Frank, Rev. Mod. Phys., 1931

B
Thermal conductivity
Ettingshausen
Etti
h
coefficient
ffi i t
[ = yT/Je]

T
Qy =  y T = Je 
Je
Qy
Nernst and Ettingshausen
g
coefficients
Sxy=   / T
Bridgman
g
relationship,
p
Sommerfeld and Frank, Rev. Mod. Phys., 1931
Thermal conductivity

B
Ettingshausen
Etti
h
coefficient
ffi i t
[ = yT/Je]
Qy =  y T = Je 

T
Je
Qy
Energy cost : (Qy /T) y T =  Je  y T / T
Can b
C
be provided
id d by
b an Electric
El t i field
fi ld Ex
JeEx = k Je  y T / T
Ex / y T =   / T
The Semi-classic
Semi classic picture



J e   E   T



J Q   T E   T

 k

2
3

T
e

2
B
 k
2
2
3 e
B
2
T
F
The Wiedemann-Franz law!
The Wiedemann
Wiedemann-Franz
Franz law
• Strictly valid in absence of inelastic
scattering
tt i
• A robust signature of a Fermi liquid
(expected to break down in case of spin-charge separation)
• Even in Fermi liquids a finite downward
d i ti is
deviation
i expected
t d in
i presence off
inelastic scattering
Carriers of charge are carriers of heat
The ratio of two conductivities is
li k d to
linked
t the
th ratio
ti off the
th ttwo quanta
t
of charge and entropy!
  2 k 2B

T
3 e2
Can be derived using the kinetic equation!
C
2
3
k B2TN ( F )
1
 2 k 2B
  Cel vF  
T
2
3
3 e
1
3
  e 2 N ( F )vF  e
The effect of an electric field and a thermal
gradient on a Fermi surface
T. Ziman
El
Electrons
andd phonons
h
The electric field displaces the Fermi surface, but a thermal
gradient makes it fuzzier in the hotter end!
The Wiedemann-Franz law is recovered only at
T 0 and
T=
d att high
hi h temperatures!
t
t
!
At high temperatures vertical scattering beomes marginal because of the
thermal broadening of the Fermi surface
III Correlated
III.
C
l d electrons
l
Landau theory of Fermi liquids
• Why band theory is successful in spite of its
neglect of electronic interactions?
• Interaction electrons can be mapped to noninteracting quasi-particles
quasi particles with the same spin and
charge
• Normalized mass of quasi-particles contains all
information on the magnitude
g
of the interactions
Heavy Fermi liquids
• Intermetalics containing a 4f (Ce, Yb…)
or 5f (U, Pu,…) element.
• High temperature: A Fermi sea and
localised spins
• Low
L
t
temperature:
t
H
Heavy
quasi-particles
i
ti l
as a result of hybridization of f electrons
and conduction electtrons
Heavy Fermi liquids
• Enhanced specific heat
• Enhanced Pauli Susceptibility
p
y
• Enhanced T2-resistivity


2
3
k N ( F )
2
B
   N ( F )
2
B
=0 +A
A T2
A  N ( F )
2
Fermi liquid ratios
The Wilson ratio
 2 k 2B 
RW 
3  B2 
Fermi liquid ratios
The Kadowaki
Kadowaki-Woods
Woods ratio
KW 
A
2
Thermoelectric response
hot
cold
 hot cold
1.0
O
Occupation
F
0.5
A finite thermal gradient!
A finite electric field!
But no charge current!
Response to a gradient in
chemical potential!
0.0
Energy
Thermoelectric response
hot
cold
Carriers with a charge, q, and an entropy, Sex will suffer two forces:
•Electric force: F= E q
•Thermal force: F= Sex ∇T
 hot  cold
1.0
F
Thermopower measures entropy
per [charged] carrier.
S = (kB T/ EF)/ e
Occupatiion
S = E/ ∇T = Sex /q
0.5
0.0
Energy
Seebeck coefficient of the free electron gas
In the Boltzmann picture thermopower is linked to electric conductivity:
This yields:
t
transport
t
Th
Thermodynamic
d
i
For a free electron gas, with =0, this becomes:
Thermopower and specific heat
In a free electron gas (with =0):
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 =1/2 and q=2/3]
Thermoelectricity in real metals
• Even in simplest metals,
the free-electron-gas
free electron gas
picture does not work at
finite temperature !
Structure in the thermopower of
AlKali metals (MacDonald 1961) !
Phonon Dragg
h t
hot
cold
ld
Flow of heat-carrying phonons
•If electrons and phonons exchange energy then
electrons would be dragged by phonons
Phonon dragg thermopower
Carrier density
Sg 
L tti specific
Lattice
ifi heat
h t
CL

Ne
e- - ph coupling (0<<1)
Order of magnitude of phonon dragg
(T/D)3
S g / S diff 
T/TF
CL

Ce 
Frequency of ph-e- scattering
events
e- per atom
MacDonald 1961
•
•
Expected to vanish in the
zero-temperature limit
Becomes negligible when
Ce>> CL
Heavy electrons in the T=0 limit
Two-decades-old data replotted!
Extrapolated to T=0, data yields a q close to unity!
Another plot linking two distinct signatures
of electron correlation
KB, D. Jaccard,
J Flouquet
J.
Fl
t 2004
Data since 2004

S/T
q
UPt3
430 2.5
0.6
NpPd5Al2
200 -1.3
-0.65
CeNi2Al3
30
0.9
PrFe4As12
340 -1
0.3
YbRh2Si2
750 -6.7
-0.8
FeTe
34
-0.3
-0.9
MgB2
3
0.04
-1.3
1.3
SrRu03
30
0.24
0.8
SrRh2O4
10
0.22
2.2
0.27
In semi-metals q is large!
•URu
URu2Si2 (q=-11)
(q= 11)
•PrFe4P12 (q=-58)
•PrRu4P12 (q=-43)
•CeNiSn (q=107)
( 107)
•Bi0.96Sb0.04 (q=10 )
4
In these systems the FS occupies a small fraction (~1/2q) of the
BZ.
Theory on correlation between S and 
Miyake & Kohno, JPSJ (2005)
In both unitary and Born limits, q ~ 1
Zlatic, Monnier, Freericks & Becker, PRB 2007
(Single-impurity Anderson model)
Paul & Kotliar, PRB 2001
Near a QCP both expected to diverge logarithmically
Haule and Kotliar, CorrelatedThermoelectricity workshop (2008)
DMFT
Measuring the Fermi energy of an
electronic system
• Resistivity:
• Specfific heat
• Thermopower
h 1
A 2 a 2
e F

2
3
2
B
k n
1
F
S  kB 1

T
3 e F
2
Electronic correlations…
correlations
• Do not enhance or diminish the absolute
value of  or  components in the T
T=00 limit
• But, they do enhance the magnitude of ,
the thermoelectric response
IV Nernst
IV.
N
t effect
ff t
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.

Ex

B
hot

Ey
JQ
cold
ld

T

E
x
S
xT
N  e y  S xy 
 Ey
 xT
[ 
 Ey
B z  xT
]
Nernst effect in a single-band metal
Absence of charge current leads to a counterflow of hot and cold
electrons:
JQ  0 ; Je= 0 ; Ey= 0

B
e-
JQ
e-

T
In an ideally simple metal
metal, the Nernst effect vanishes!
(« Sondheimer cancellation », 1948)
Ey
Nernst coefficient in remarkable metals!
A large diffusive component in the zero-temperature limit!
Close-up on Sondheimer cancellation



J e   E    T



JQ   T E    T
Je=0
0
Boltzmann picture:
If the Hall angle
angle, H, does not depend on the position of the Fermi level
level,
then the Nernst signal vanishes!
Roughly, the Nernst coefficient tracks cF…
N ~ 2/3 k2BT/e c / F
Recipe for a large diffusive Nernst response:
•High mobility
•Small Fermi energy
gy
•Ambipolarity
In graphite at 20 K and 1T
 ~3mcm
Sxy ~3mVK‐1,, 
Sxy2/ = 3000 W K‐2cm‐1
Nernst effect as a probe of quantum criticality
The case of CeCoIn5
Izawa et al. 2007
Paglione et al. ,2003, Bianchi et al. 2003
logarithmic color plot of /T
Nernst effect directly reveals the quantum critical point!
Nernst effect in the vortex state

B
Ey
T
A superconducting vortex is:
• A quantum of magnetic flux
• An entropy
py reservoir
• A topological defect
•
Thermal force on the vortex :
•
•
The vortex moves
Th movementt leads
The
l d tto a
transverse voltage: Ey=vx Bz
F=-S T (S : vortex entropy)
Nernst effect in optimally-doped YBCO
The Nernst coefficient is finite only
in the vortex liquid state!
(Ri, Huebner et al. 1994)
Vortex-like excitations in the normal state of the
underdoped cuprates?
Ong et collaborators 2006
A finite Nernst signal in a wide temperature range above Tc
A small Fermi surface
Highly mobile electrons
The smallll electron
Th
l t
pocket generates a
sizeable negative
Nernst signal!
Within a factor of 2 of
what
h t is
i expected
t d for
f
the normal state!
Nernst effect due to Gaussian fluctuations of the
amplitude of the superconducting order parameter
(Usshishkin, Sondhi & Huse, 2002)
In 2D:
Magnetic length
Quantum of thermo-electric conductance (21 nA/K)
In two dimensions, the coherence length is the unique
p
parameter!
The coherence length
g above Tc
How do the fluctuating Cooper pairs generate a Nernst signal?
cold
hot
•Above Tc, the lifetime of
the Cooper pairs
d
decrease
with
ith iincreasing
i
temperature
cold
h t
hot
dT/dx
Ey
B
•Therefore, those pairs
which travel from the hot
side along the cold side
live longer!
Superconductivity in Nb0.15Si 0.85 thin films
1500
d=125 A
1200
900
800
d=250 A
R()
Rsquare()
1200
600
d12.5 nm
Tc=220 mK
 mcm
400
d=500 A
300
Nb0.15Si0.85
d=1000 A
0
00
0.0
05
0.5
10
1.0
15
1.5
20
2.0
T(K)
25
2.5
30
3.0
35
3.5
40
4.0
0
0
50
100
150
T(K)
200
The normal state is a simple dirty metal: le~a~ 1/kF !
250
300
Nernst effect across the resistive transition
• A large vortex
signal below Tc
Pourret 2006
•A long
g
tail
above Tc
A signal distinct from the vortex signal
Deep into the normal state!
Pourret 2006
Can this signal come from normal electrons?
The Nernst signal of the normal electrons is negligible!
 ~10
10 -55 T
F ~104 K
Even at 6K :
The expected normal
state contribution is three
orders of magnitude
smaller than /T !
Comparison with theory
Experiment:
Rsquare

SC
 XY
B
Theory:
0.01
xy/B(A
A/TK)

1E-3
xy/B (Sample1) (B=0)
1E-4
xy/B (Sample2) (B=0)
xy/B (USH)
1E-5
 xy
B
USH
 k Be 2  2
.ξ
 
2  d
 6π 
0.02
0.1
1
-2
ln(T/TC) d
Satisfactory agreement close to Tc
4
Pourret 2008
The g
ghost critical field
Sample 2
Contour plot of N= -Ey /(dT/dx)
A unique correlation length
Contour plot of the Nernst
coefficient 
=N/B
N/B
Pourret 2008
1
3 v F 
d 
0.36
2 k BTc
