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Nernst effect as a probe of
superconducting fluctuations
Kamran Behnia
Ecole Supérieure de Physique et de Chimie Industrielles
Paris
• Luis Balicas, NHFML (Tallahassee, US)
• Ilya Sheikin & Arlei Antunes, GHMFL (Grenoble, FR)
• Baptiste Vignolle & Cyril Proust, LNCMP (Toulouse,FR)
• Yakov Kopelevich (Campinas, BR)
Samples
Claire Kikuchi, Laurent Bergé, Louis Demoulin (Orsay)
Jean-Paul Issi (Louvain-la-neuve)
Collaborators
ESPCI researchers:
Alexandre Pourret, Benoît Fauqué, Aritra Banerjee,
Zengwei Zhu, Huan Yang, Hervé Aubin
OUTLINE
• Introduction
• Superconducting fluctuations• Nernst signal of short-lived Cooper pairs
• Quantum ocillations across the
quantum limit• Nernst profiles in graphite and graphene
• Fractional states in bulk bismuth?
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
T
ESeN
x
y
xyy
][
TB
E
xz
y
hotcold
Set-up for monitoring thermal(kxx, kxy), thermo-electric (S, N) and electric (sxx, sxy) conductivity tensors
20 mm
Thermometers
Heater SC wires
9000 9060 9120 9180 9240 9300 9360 9420 9480 9540 9600 9660 9720
-52
-50
-48
-46
-44
-42
-40
-38
-36
dV
(n
V)
T(s)
DC voltages of the order of 1 nV resolved!
Nernt response of normal electrons can be very large!
0.2 1 10 50
0.1
10
1000
NbSe2
CeRu2Si
2
CeCoIn5
URu2Si
2
PrFe4P
12
absolu
te v
alu
e)
(V
K-1T
-1)
T(K)
Bi
KB, M.A-Méasson & Y. Kopelevitch, PRL 2007
Semi-classical picture
F
HBxy
e
TkS
22
3
TETJ
TEJ
Q
e
k
s
22
xyxx
xyxxxxxy
xySss
ss
If shifting the Fermi level does not change the Hall angle,then there is no Nernst signal!
xy
xxH
s
s
ss 10 STEJe
Roughly, the Nernst coefficient tracks /EF
Bismuth URu2Si2 PrFe4P12
n (per f.u.) 10-5 3 10-2 2 10-3… and becomes large in
clean semi-metals!
~ 2/3 kB/e / EF
F
HBxy
e
TkS
22
3
KB, J. Phys.: Condens. Matter
21, 113101 (2009)
Nernst effect and
superconcucting fluctuations
Nernst effect in the vortex state
• Thermal force on the vortex :
F=-Sf T (Sf : 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
Vortex-like excitaions in the normal state of the underdoped cuprates?
A finite Nernst signal in a wide temperature range above Tc
Wang, Li & Ong, ‘06
Nernst effect due to Gaussian fluctuations of the amplitude 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!Both the amplitude and the T-dependence of xy is determined by x(T).
Magnetic length
Our main result!
1. This theory is experimentally verified!
2. In a conventional dirty 2D superconductor, a signal due to fluctuating superconductivity can be resolved by Nernst measurements at T>>Tc.
A. Pourret et al. Nature Phys. 2, 683 (2006); Phys. Rev. B. 76, 214504 (2007)
For a review see New J. Phys. , 11, 055071 (2009)
Superconductivity in Nb0.15Si 0.85 thin films
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
300
600
900
1200
1500
d=1000 A
d=500 A
d=250 A
d=125 A
T(K)
Rsquare(
)
The normal state is a simple dirty metal: le~a~ 1/kF !
A Nernst signal persists deep into the normal state!
A signal distinct from the vortex signal
The link between and xy
In our case:
sxx > 103 sxy
sSC < 10-1 sxx when T > 1.1 Tc
Therefore: xy/ B = sxx= / Rsquare
Link to the superconducting coherence length
yields
cB
Fd
Tk
v
2
336.0
1
x
This should be compared to the expression for a 2D dirty superconductor:
T-dependence= (T-Tc/Tc)
Amplitude
The shortest link between data and vFle
1251035.4 smvF
e
B
e
Fe
k
Tv
s
k2
3
Using specific heat and resistivity data, this yields:
Coherence length above Tc
0.1 1 102
10
60
x=
(5
.9 1
0-7
x
y/B
)1/2
(nm
)
= (T-Tc)/T
c
sample 2
sample 1
Satisfactory agreement for small !
The ghost critical field
Contour plot of N= -Ey /(dT/dx)
Sample 2
A unique correlation length
cB
Fd
Tk
v
2
336.0
1
x
Contour plot of N/B
Why does it work so well here?The Nernst signal of the normal electrons is negligible in this dirty superconductor!
In Nb0.15 Si0.85 mobility is small and Fermi energy is large!
Qunatum oscillations in
Nernst response
Quantum oscillations of thermoelelctric coefficients in Bi
KB, M.A-Méasson & Y. Kopelevitch, PRL 2007
Giant quantum oscillations
0.0 0.3 0.6 0.9
0.1
1
10
31
02
1
2
1
0
3 1.20 K
0.46 K
0.28 K
B-1 (T
-1)
Sxy(m
V/K
)1-
2-
3-
0+
1+
unidentified
Quantum oscillations in graphiteZhu et al., Nature Physics, Nov. 8 2009
1 105
10
80HOPG sample 2
Sxy (V
K-1)
B(T)
1.65 K
0.98K
0.77 K
0.55 K
0.34 K
0.29 K
40
100
1000
40001 10
4.2 K
HOPG sample 1
1.6 K
2.7 K
8 K
16 K
Quantum oscillations in Graphene
See also:
Wei et al., PRL’09
Checkelsky and Ong, arXiv: 0812.2866
When a Landau level meets the
Fermi level, Sxy vanishes!
Zuev, Chang & Kim, PRL’09
Theory for 2DEG
Oji, J. Phys. C ‘84
Jonson & Girvin, PRB ‘84
Empirical correlation between the Nernst profile and dimensionality!
3D
0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05
100
1000
Ne
rnst S
ign
al V
y (
nV
)
B-1 (T
-1)
nV, 312mK
nV, 475mk
nV, 760mK
nV, 910mK
0.00 0.15 0.30 0.45 0.60 0.75 0.90
10
100
0.343K
0.549K
0.678K
0.771K
0.805K
0.853K
0.976K
1.651K
Sxy (V
/K)
B-1(T
-1)
2D
bismuth
graphite
GaAs
graphene
A topological phase transition in 3D
For a review paper on topological
phase transitions, see:Blanter, Kaganov, Pantsulaya and
Varlamov
Phys. Rep. 245, 159 (1994).
Zhu et al., Nature Physics 09
What happens beyond the quantum limit?
The quantum limit (9T)
Peaks beyond the quantum limit
KB, L. Balicas & Y. Kopelevich, Science 2007
•Do not correspond to any obvious integer Landau
level
•Are not periodic in 1/B
•Are concomittant with Hall anomalies
Length scales and Nernst coefficient in Bi
Bismuth
T=1.5 K
A surprise at still higher fields!No more Landau level
crossing is expected!Fauqué et al. New J. Phys. 11 113012 (2009)
All these states are expected to deteriorate metallicity!
The hig-field anomaly looks like a LL crossing!
The T-dependence confirms a hole-ellipsoid origin!
No critical temperature!
Topological and symmetry-breaking phase transitions (Xiao-Gang Wen, Adv. Phys. 1995)
symmetry - breaking topological
•The ground state is a quantum crystal
•An order parameter
•A critical temperature
•The ground state is a quantum liquid
•No order parameter
•No critical temperature
Examples: SC, DW,… Examples: QHE (Integer and fractional)
Summary
Nernst effect is a sensitve probe of :
• superconducting fluctuations
• quantum oscillations
• 3D metal beyond the quantum limit
Electron spectrum in bismuth at high field
When the field is along trigonal and exceeding 11 T
Holes at their lowest LL; electrons at their lowest Zeeman-splitted LL!
But, is this true?
Electron spectrum in bismuth at high field
Sharlai
&
Mikitik
PRB
2009
The magnetic field displaces the Fermi Energy, in
order to preserve charge neutrality: nh=ne1+ne2+ne3
A very anisotropic field scale associated with electron pockets
Lu Li et al., Science 08
Alicea & Balents PRB 09
ExperimentSharalai &
Mikitik, PRB 09
Are the high-field Nernst peaks a result of small
misalignment?
• In case of perfect alignment, no anomaly beyond 10T is expected!
Sharalai & Mikitik, PRB 09
Are the high-field Nernst peaks a result of small
misalignment?
• But there is an arbitrary angle for which three high-field e- anomalies are expected!
Sharalai & Mikitik, PRB 09
Angular dependent Nernst effectH. Yang et al., unpublished
Angular dependent Nernst effect
ANGLE (DEGREES)
Fie
ld (
Tesla
s)
B. Fauqué, LNCMI-Grenoble
3 AM 10 /09/09
Angular dependent Nernst effect
Quasi- horizontal lines and quasi-vertical lines
0+e
1-e
?
?
When the field is aligned along trigonal
The temperature dependence confirms an e- ellipsoid origin!
HOPG and natural graphite
Thermopower and Nernst effect in graphite
Non-trivial Berry phase in graphite
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