Talk 1. Quasi-particle charge and heat currentsin d-wave superconductor

Boulder School for Condensed Matter and Materials Physics 2008

N. P. Ong, Princeton University

Talk 3. Magnetization of vortex liquid state

Talk 2. The Nernst effect in vortex liquid state of cuprates

Boulder, July 7-11, 2008Supported by NSF-MRSEC, ONR

http://www.princeton.edu/~npo/

Talk 1

Quasi-particle charge and heat currentsin d-wave superconductor

1. Introduction : Charge and heat currentsHall effect, Nernst effect, Thermal Hall effect

2. The Hall effect in cuprates

3. Quasiparticles and Thermal Hall conductivity

N. P. Ong, Princeton University

Collaborators: Lu Li, Joe Checkelsky, Yayu Wang, Kapeel Krishana, Wei Li Lee, Yuexing Zhang, Jeff M. Harris, Y.F. Yan, P.W. Anderson

Boulder, July 7-11, 2008Supported by NSF-MRSEC, ONR

Boulder School for Condensed Matter and Materials Physics 2008

The Hall effect

J JE .ρt

=

yxyxxxx EEJ σσ +=

yyyxyxy EEJ σσ +=

Thermopower

T∇−

V )( TEJ x−∂+= ασ

σα

=⎟⎠⎞

⎜⎝⎛

∇=

=0JTES Seebeck

coef.

heater

bath

kkk gff =− 0

Boltzmann-equation expressions for currents

∑=k

ge kk vJ 2 kkk v)(J gk

h ∑ −= με2

Charge current

Thermoelectric current )(J T−∇= α

kk

k .E lr

efgε∂

∂−=

0

).()(k

kkk T

Tfg −∇

−∂∂

−= lrμε

ε

0

EJ σ=

charge heat

conductivity

Thermoelectric cond.

kkkk ϑε

σ 20

2 cosv2 l∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=k

fe

kkkkk )( ϑμε

εα 2

0

cosv2 l∑ −⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=k T

fe

Presence of temp. gradient

kk v

xh

ε∂∂

=∂∂ 0ffk

τk

kv.x

k.k

gff kk −=∂∂

+∂∂ &

Quasi-particle excitations in normal state of Fermi liquid

particlehole

vacancy

εk

kεF

holeparticle

hk+ = c-k

A spin-down vacancy at –k translates to a spin-up hole excitation at k

Charge and heat currents in the “excitation” representation

o

Large left hole current

Large right electroncurrent

Large vacancy population

E T∇−

Large vacancy population

Both mass (entropy)currents flow to right

Charge currents nearly cancel.Difference is the Peltier charge current

Mass currents nearly cancel.Difference is the Peltier heat current

E~J α=h )(J T−∇= α

EJ σ= )(J Teh −∇= κ

k k

E(k) E(k)

Charge currents add Mass currents add

The Nernst effect (quasiparticles)

).(E.J T−∇+= ασtt

).(.E T−∇−= αρtt

( ) )( TE xyxyxy −∂+−= αρρα

Open boundaries, so set J = 0.

)(k

.Bvk

kkxyxy T

fe ll∂∂

×−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−= ∑ μεε

α0

22

εθπ

∂∂

=∇

≡eTk

TE

e ByN

22

3||

Off-diag. Peltier cond.

Measured Nernst signal

Generally, very small because of cancellation between αxy and σxy

Wang et al. PRB ‘01

The 2D Hall conductivity σxy

τkk k.

kgf

−=∂∂ &

kkk gff =− 0

lr

lr

×= ∫ dBexy 213σ

Boltzmann Eq.

Eq. of motion

Hall current in 2nd order

Area swept out in ell-space!

Gauss mapping to …

NPO, PRB (1991)

B)vE(.kk

k ×+∂∂

−= efg τ

τε yy eEefeJ v

0

).(k

.Bvk

k lr

∂∂

×⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−= ∑

( ) yxxyfBe llk

k

k .t̂ ∇⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−= ∑ εσ

03

The 2D Hall conductivity σxy

∫ ×= lr

lrr

d21A

0

22φ

σ A.B)/( hexy =

swept area

σxy is the area swept out by mfp (for arb. anisotropy)

Harris, Yan, NPO, PRB ‘92

Temp. dependences of Hall coef. and Hall angle in YBCO

Hwang Batlogg et al., PRL ‘94 Ono, Komiya, Ando, PRB ‘07

Similar T dependence of RH seen in LaSrCuO

Kanigel, Campuzano et al. Nature Phys. 2007

Antinode node

ky

kx

Δ(φ)

φ

Ferm

i arc

leng

th

T/T*

NPO and Wang unpubl.

Model: Only states on Fermi archave long mean free path

NPO and Wang unpubl.

y = 6.95

6.63

6.7

6.8

Fits to T dependence of RH in YBCO

Fits to cotθ and resistivity ρ

Vortex in cuprates

CuO2 layers

2D vortex pancake

ξ

Gap Δ(r) vanishes in core

|Ψ| = Δ

Vortex in Niobium

Js

superfluidelectrons

ξ

= 500 A

Js

b(r)Normal core

H

Vortex motion in type II superconductor

Applied supercurrent Js exerts magnus force on vortex core

Velocity gives induced E-field in core (Faraday effect)Current enters core and dissipates (damping viscosity)

Motion of vortices generates observedE-field

Consequence of Josephson equation

Tilt angle of velocity gives negative vortex Hall effect

In clean limit, vortex v is || - Js

E = B x v

(Bardeen Stephen, Nozieres Vinen)

ηρρ /02

Φ== BHH

cNxx

0Φ×=r

sM JF

Vortex Hall current

Vortex Hall σxy is negative.Appearance is abrupt

Sxy

Nxyxy σσσ +=

Quasiparticle and vortex Hallconductivities are additive

22yxxx

yxxy ρρ

ρσ

+=

Invert matrix

~H ~ -1/H

Harris et al. PRB ‘95

Thermal Hall conductivity of quasi-particles in cuprates

Problem: Separate the QP current from vortex currents?

K. Krishana, Yuexing Zhang, J. M. Harris, NPO

κxx vs. T in 90-K YBCO (twinned and untwinned)

Is peak from QP or phonons?

Monitor thermal currents.

Excitations of an s-wave superconductor

remove

Energy cost Ek = [ ξk2 + Δ2 ]1/2

Ek

ξk

quasiparticles

Δ

QP’s cost energy, but increase entropy S (lower free energy F)

γk = uk c k + vk c-k

Heat transport in low-Tc superconductors

JQ = κtot (- T)

Condensate does not carry heat (zero entropy)

κtot

T

Tc

0

Pb

Heat-current density

κtot = κel + κph

electrons phonons

T > Tc , κtot mostly electronic

Below Tc , QPs carry heat

T << Tc , QP population 0

Phonons are long-lived

κph

Jericho, Odoni, Ott, PRB ‘85

Dirac-like spectrum of QP at nodes

Quasiparticle dispersion E vs. k is linear.

Δ(φ) = Δ0 cos 2φ

Quasiparticles in a d-wave superconductor

gap max. (antinode)

node

Δ(φ) = Δ0 cos 2φ

Ek

ξk

Δ0 antinode

Ek

ξk

node

fk

εk - μ

0 kF-kF

ptcl mass currenthole mass current

Q-ptcl mass current

Q-hole mass current

- T

Heat current in normal and superconducting states

Fermi Surface

Εk

0 ξkξ−k

nodalQP’s

holeexcitations

electrons

Separating electronic and phonon κ

in 93-K YBa2 Cu3 O7

κa = κe + κph

vortex Dirac quasiparticle

H

Quasiparticles in the CuO2 plane

Hall thermal current from asymmetric scattering of QP by vortex

heater

asymmetric scattering of QP by vortex

scattering of phonons:no asymmetry

warm

cool

heaterbath

Doppler shift

vs

counter-moving qp

co-moving

vs

counter-moving qp

co-moving

J = Js + Jqp ( Js || -Jqp )

In a supercurrent vs , energy of counter-moving QP’slowered.

QP excitations

Origin of QP Hall current

counter-moving

co-moving

vortex core

vs

Incident QP’s

Scattered QP’s flow against vs circulation

Doppler shift lowers energy of counter-moving QPs

σH

T/H1/2

Skew scattg cross-section (Durst,Vishwanath, Lee, PRL ‘03)

σH ~ T / H1/2

Thermal Hall Conductivity κxy in high-purity YBCO7(normal state)

Thermal Hall Conductivity κxyIn high-purity YBCO7

1) Hall signal much larger below Tc

2) Giant increase in initial slope 85 to 40 K

3) Strongly non-linear in H

Thermal Hall Conductivity κxy in high- purity YBCO7

(12.5 to 35 K)

Plot initial slope limB-0 κxy /Bvs. T.

Initial slope increases by 1000 between 85 and 30 K

Steep increase in QP mean-free-path ~ 120

QP mean-free-path l derived from Hall angle

Increases by 120 from 85 to 30 K

Abrupt increase at Tc (coherence effect?)

κxy = C0 (TH)1/2

F(x) ~ x

Simon-Lee scaling

κxy = T2 F( H1/2/T )

Calculated fits to Kxx and Kxy(Adam Durst, Ashvin Vishwanath, P.A. Lee, 2003)

κxx = ce vF l ~ T2T-1

κxy = κxx tan θ

= κxx nV σH l

~ T2T-1 . H .TH-1/2. T-1 ~ (TH)1/2

Durst, Vishwanath, Lee

Explains observation

κxy = C0 (TH)1/2

Quasi-particles are hole like in UD YBCO y=6.60

Summary

Below Tc , we observe

. 1000-fold increase in κxy (weak field)

. 200-fold increase in QP mfp l.(80 Angstrom to 2 microns)

. Giant anomaly in κtot is entirely from QP.

. Steep increase in mfp starts just below Tc

(conflicts with ARPES)

. Intriguing scaling behavior in κxy (Simon-Lee)

. No evidence (yet) for Landau quantization

References for Lect. 1 (website http://www.princeton.edu/~npo/)

1. N. P. Ong, Phys. Rev. B 43, 193 (1991).2. J.M. Harris, Y.F. Yan, and N.P. Ong, Phys. Rev. B 46, 14293 (1992).3. J.M. Harris, Y.F. Yan, O.K.C. Tsui, Y. Matsuda and N.P. Ong, Phys. Rev. Lett. 73, 1711 (1994).4. J. M. Harris, N. P. Ong, P. Matl, R. Gagnon, L. Taillefer, T. Kimura and K. Kitazawa, Phys. Rev. B 51, 12053 (1995).5. K. Krishana, J. M. Harris, and N. P. Ong, Phys. Rev. Lett. 75, 3529 (1995).6. Y. Zhang, N.P. Ong, Z.A. Xu, K. Krishana, R. Gagnon, and L. Taillefer, Phys. Rev. Lett. 84, 2219 (2000).7. Y. Zhang, N.P. Ong, P. W. Anderson, D. A. Bonn, R. X. Liang, and W. N. Hardy, Phys. Rev. Lett. 86, 890 (2001).8. Adam C. Durst, Ashvin Vishwanath, and Patrick A. Lee, Phys. Rev. Lett.

90, 187002 (2003).

Vortices in cuprates

CuO2 planes

2D vortex pancake

ξ

Gap amplitude vanishes in core

|Ψ| = Δ

A new length scale ξ* Cheap, fast vortices

ξ0

Js (r)

Δ(r) Δ(r)

Js (r)

ξ0

00

H* = φ0

2πξ*2

Is H* determined byclose-packing of fat vortices?

ξ*