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Nernst effect as a probe of Nernst effect as a probe of electronic correlations

Kamran Behnia

Ecole Supérieure de Physique et de Chimie Industrielles Paris

The team in ESPCI

• Romain Bel
• Alexandre Pourret
• Hervé Aubin
• Marie-Aude Méasson
• Benoît Fauqué
• Aritra Banerjee

In collaboration with:

• Luis Balicas, NHFML (Tallahassee)
• Ilya Sheikin, GHMFL (Grenoble)
• Yakov Kopelevich (Campinas , Brazil)

p ( p , )

• Julien Levallois, Baptiste Vignolle & Cyril Proust, LNCMP (Toulouse)
• Koichi Izawa & Jacques Flouquet, CEA (Grenoble)
• Claie kikuchi & Louis Demoulin (Orsay)

Claie kikuchi & Louis Demoulin (Orsay)

Outline Outline

• On the magnitude of the Nernst effect in the zero-

g temperature limit

• Nernst effect as a probe of superconducting

fluctuations (the case of Nb0.15Si0.85)

• Nernst effect and quantum criticality (the case of

CeCoIn5)

• Nernst effect in the vicinity of quantum limit (the case
• f bismuth)

Thermoelectric coefficients Thermoelectric coefficients

• In presence of a thermal gradient,

electrons produce an electric field.

x

E r

• Seebeck and Nernst effect refer to the

longitudinal and the transverse components of this field.

B r

hot ld

components of this field.

JQ

y

E r

cold

T ∇ r

E S

x

− =

E S e N

y

−

] [ Ey − = ν

T S

x x

∇ =

T S e N

x y xy y

∇ = = =

] [ T B

x z∇

= ν

Set-up for monitoring thermal(κxx, κxy), thermo-electric (S, N) and electric (σ σ ) conductivity tensors and electric (σxx, σxy) conductivity tensors

Thermometers Heater SC wires

• 38
• 36
• 42
• 40

48

• 46
• 44

dV (nV)

• 52
• 50
• 48

9000 9060 9120 9180 9240 9300 9360 9420 9480 9540 9600 9660 9720

T(s)

20 mm

DC voltages of the order of 1 nV resolved!

Nernst effect in a single-band metal

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

B r

e-

JQ ≠ 0 ; Je= 0 ; Ey= 0

e-

Ey JQ T ∇ r In an ideally simple metal the Nernst effect vanishes! In an ideally simple metal, the Nernst effect vanishes! (« Sondheimer cancellation », 1948)

The Nernst coefficient is large when mobility is large and Fermi energy is small! mobility is large and Fermi energy is small!

1000

• 1T
• 1)

Bi PrFe4P12 ) (µV K

• 10

URu2Si2 e value CeRu2Si2 CeCoIn5 absolute

0.1

NbSe

2 2

ν (a 0.2 1 10 50 NbSe2 T(K)

Close-up on Sondheimer cancellation

T E J e ∇ − = r r r α σ T E T J Q

e

∇ − = r r r κ α

Je=0 Je 0 Boltzmann picture: If the Hall angle Θ does not depend on the position of the Fermi level

(See Oganessyan & Ussishkin, 2004)

If the Hall angle, ΘH, does not depend on the position of the Fermi level, then the Nernst signal vanishes!

Roughly, the Nernst coefficient tracks ωcτ / ΕF…

N ~ π2/3 k2 T/e ω τ / Ε N ~ π2/3 k2

BT/e ωcτ / ΕF

… and becomes large in clean semi-metals! g

Bismuth URu2Si2 PrFe4P12

2 2 4 12

n (per f.u.) 10-5 3 10-2 2 10-3

Nernst effect and superconducting fluctuations

• A. Pourret, H. Aubin, J. Lesueur, C. A. Marrache-Kikuchi,
• L. Bergé, L. Dumoulin and K. B.

Nature Phys. 2, 683 (2006) , PRB (2007)

Nernst effect in the vortex state

B r

Ey ∇T

• Thermal force on the vortex :

A superconducting vortex is:

∇T

Thermal force on the vortex :

F=-Sφ ∇T (Sφ : vortex entropy)

• The vortex moves

Th t l d t

A superconducting vortex is:

• A quantum of magnetic flux
• An entropy reservoir
• The movement leads to a

transverse voltage: Ey=vx Bz

py

• A topological defect

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

Wang, Li & Ong, ‘06 A finite Nernst signal in a wide temperature range above Tc

Nernst effect due to Gaussian fluctuations of the amplitude of the superconducting order parameter amplitude of the superconducting order parameter (Usshishkin, Sondhi & Huse, PRL 2002) In 2D: In 2D:

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

In two dimensions, the coherence length is the unique t ! parameter! Both the amplitude and the T-dependence of αxy is determined by ξ(T)! determined by ξ(T)!.

O i lt Our main result:

1. This theory is experimentally verified! 2. In a conventional dirty 2D superconductor, a signal due to fluctuating superconductivity can be signal due to fluctuating superconductivity can be resolved by Nernst measurements upto T~30Tc .

How does it work? How does it work?

Th t ib ti f th l t t i li ibl

• The contribution of the normal state is negligible
• Just above T

the agreement between theory and

• Just above Tc, the agreement between theory and

experimental data is better than 5 percent.

• For T>>Tc [Guinzburg-Landau approximation is no more

valid], the signal symmetrically depends on ξ and lB .This t l t d ti i i strongly supports a superconducting origin

Superconductivity in Nb0.15Si 0.85 thin films

1500 1200

d=125 A

900

d=250 A are(Ω)

300 600

d=500 A d 250 A

Rsqu

0 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 300

d=1000 A

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

T(K)

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

A simple dirty metal subjet to weak localization

1200

le~ a ~kF

• 1~ 0.7 nm

KF le ~1

800

)

KF le 1 Close to Mott-Ioffe limit

C R / 10 9 4

3 11 −

×

400

R(Ω)

Nb0.15Si0.85 d=12.5 nm

limit

C m RH / 10 9 . 4

3 11

× =

400

d 5 Tc=220 mK

ρ≅2 mΩcm

C m RH / 10 9 . 4

3 11 −

× =

50 100 150 200 250 300

T(K)

ρ

c

carrier density is large (1023 /cm3)

T(K)

A Nernst signal g persists when this dirty 2D y superconductor is warmed up well p above its critical temperature ! p

Deep into the normal state!

A signal distinct from the vortex signal A signal distinct from the vortex signal

The Nernst signal of the normal electrons is negligible!

Even at 6K : ν/T >> 285 ωcτ / εF ωcτ ∼10 −5 Resistivity + Hall + Seebeck εF ~104 K + Seebeck coefficients yield:

The link between ν and αxy

In our case: σxx > 103 σxy σSC < 10-1 σxx when T > 1.1 Tc σ 10 σxx when T 1.1 Tc

Therefore: α / B = ν σ = ν / R Therefore: αxy/ B = ν σxx= ν / Rsquare

Link to the superconducting coherence length Link to the superconducting coherence length

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

F

v l h 3 36 1 ξ =

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

c B F d

T k 2 36 . ε ξ =

T d d Amplitude T-dependence ε= (T-Tc/Tc)

The shortest link between data and vFle

F e

2

⎞ ⎛

B F

e k T v γ σ π γ κ

2

3 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = l

e e

e T γ γ ⎠ ⎝

Using specific heat and resistivity data, this yields:

1 2 5

10 35 . 4

− −

× = s m vFl

Coherence length above Tc 60

nm) B)1/2(n

sample 2 sample 1

10

αxy/B

10

9 10-7

2

= (5.9

0.1 1 10 2

ξ ξ ε ε = (T-Tc)/Tc

Amazing agreement for small ε!

• How about T>>Tc? Does the 6K signal

How about T Tc? Does the 6K signal come from SC fluctuations?

Let us examine the field dependence! dependence!

The ghost critical field g

Sample 2

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

A unique correlation length

Contour plot of ν=Ν/Β

F

v l h 3 36 1 ξ =

c B d

T k 2 36 . ε ξ =

The collapse

p

Α unique source for the Nernst signal In the window In the window Tc < T < 30Tc And 0< B < 5Bc2 No theory yet available!

Nernst effect and q ant m criticalit quantum criticality

• K. Izawa, K. B., Y. Matsuda, H. Shishido, R.Settai, Y. Onuki and J. Flouquet,
• Phys. Rev. Lett. 99, 147005 (2007)

The quantum critical point in CeCoIn5 The quantum critical point in CeCoIn5

Paglione et al. ,2003, Bianchi et al. 2003

Detected by resistivity and specific heat

Paglione 2003 Bianchi 2003 Bianchi 2003

Nernst effect in the vicinity of QCP

Nernst effect in the vicinity of QCP

Nernst effect in the vicinity of QCP

What is the origin of this enhancement ? A small EF ? A large Hall angle ? both?

Signatures of a small [vanishing?] EF near QCP

A from Paglione et al.2003 γ from Bianchi et al.2003

No anomaly in the field-dependence of the zero -T Hall coefficient !

Singh et al. 2007

logarithmic color plot of ν/T

Energy scales near QCP

Onset of T2 Min in L/L0

[Paglione 06] [Paglione 06]

min in q=S/C

Nernst effect in bismuth Nernst effect in bismuth across the quantum limit q

KB, M.-A. Méasson & Y. Kopelevich, Phys. Rev. Lett. 98, 166602 (2007) KB L Balicas & Y Kopelevich Science 317 1729 (2007) KB, L. Balicas & Y. Kopelevich, , Science 317, 1729 (2007)

A semi-metal with a tiny Fermi surface y

J.-P. Issi, Aust. J. Phys. (1979) y ( )

εFe= 27.6 meV

Fe

εg=15.3 meV

The tiny Fermi surface pockets

Liu & Allen, PRB 1995

trig Parabolic dispersion Perfect ellipsoid Ratio of axes: 3 Non-Parabolic [Dirac Fermion] Not a perfect ellipsoid Ratio of axes: 14 bisectrix Ratio of axes: 3 m3= 0.69 me m1=m2=0.06me EF= 15 meV Ratio of axes: 14 m3= 0.002 me m2=0.001me m1=0.26me EF 15 meV m1 0.26me EF= 27 meV

The Kadowaki-Woods ratio in bismuth exceptionally large! exceptionally large!

A=12 nΩcmK-2 A/γ2 =106 a0 Hartman, 1969 γ γ ~8 µJK-2mol-1 γ

Li, Taillefer et al. PRL 2004

Bi The KW ratio scales inversely with kF (See Hussey JPSJ2005, Kontani JPSJ2004)

Quantum oscillations in thermoelelctric coefficients

KB M A Mé & Y K l i h PRL 200 KB, M.A-Méasson & Y. Kopelevitch, PRL 2007

Quantum oscillations of the hole pocket hole pocket

T=0 28K ∆(1/H) =0.147 T-1 T=0.28K

Compare to 0 146T-1 Compare to 0.146T-1 According to SdH [Bompadre PRB ’01]

Giant quantum oscillations

When each Landau level crosses the chemical potential a Nernst signal is generated! potential a Nernst signal is generated!

« Quantum Nernst effect » , theory by Nakamura et al., Solid state comm.135,

510 (2005) for a 2 DEG

Hall data in a rotating magnetic field

Fauqué et al. 2008, unpublished

A second set of data with increased angular density

Quantum oscillations of the Hall response p

Fauqué et al. 2008, unpublished

Fragile plateaus… g p

Fauqué et al. 2008, unpublished

…and interplateau anomalies! p

What happens beyond the quantum limit?

The quantum limit (9T)

The quantum limit is accessible with a moderate magnetic field along trigonal

Ah=0.61nm-2 Ae=0.84nm-2 Barghava 1966 Edelman 1975 Schoenberg 1984

2π BQL= A ħ/e

g

QL

6.4 T for holes 8.9 T for electrons

It occurs at about 9 T!

Peaks beyond the quantum limit

KB, L. Balicas & Y. Kopelevich, Science 2007

These anomalies …

B-1/B-1

QL

KB L B li & Y K l i h S i 200

B /B

QL

KB, L. Balicas & Y. Kopelevitch, Science 2007

• Do not correspond to any obvious integer Landau level (either electrons or holes)
• Occur at fractional filling levels

A t i di i 1/B

• Are not periodic in 1/B
• Are concomittant with Hall anomalies

Fractional Quantum Hall effect?

• A property of a high-mobility and

interacting 2D electron system! D bi th lif ?

Stormer 1999

• Does bismuth qualify?

? C Is mobility high enough? Certainly yes

80 times higher than in 1983 GaAs/GaAsAl

Are interactions strong enough? Probably yes!

• n= 3 1017 cm-3
• m*=0.06-0.7 me
• Both comparable with GaAs
• (but εb=100 compared to 12)

Hartman 1968

Dimensionality is the most serious issue!

107

Dimensionality is the most serious issue!

A Field-induced quasi-1D conductor !

(A Luttinger liquid? Biagini Maslov Reizer Glazman EPL 2001) (A Luttinger liquid?, Biagini, Maslov, Reizer, Glazman EPL 2001)

Contrary to quasi-2D conductors, in ultraquantum bismuth ρxx >> ρzz !

B

2lB Is the system a set of quantum wires oriented along the magnetic field?

Summary

Nernst effect Nernst effect …

• Scales with the ratio of mobility to Fermi energy
• becomes large in clean semi-metals

It can be used to …

• detect superconducting fluctuations in a large

…detect superconducting fluctuations in a large temperature window!

• probe quantum criticality!
• …probe quantum criticality!
• … track QHE phenomena in a bulk solid!