Department of Physics University of Toronto

Low Temperature Thermal Transport Across the Cuprate Phase Diagram

Mike Sutherland

Louis TailleferRob HillCyril ProustFilip RonningMakariy Tanatar

R.Gagnon, H.ZhangD.Bonn, R.Liang, W.HardyP.Fournier, R.GreeneA.P.Mackenzie, D. Peets, S. Wakimoto

Christian LupienEtienne BoakninDave HawthornJ. PaglioneM. Chiao

Carrier concentration

superconductor

metal

magnetism

Tem

pera

ture

pseudogap

What questions can we address by studying low temperature thermal conductivity as a function of doping in the cuprates ?

How well does d-wave BCS theory describe the superconducting state ?

Is the superconductingorder parameter pure d-wave throughout the phase diagram?

How does the pseudogapinfluence the behaviour oflow-energy quasiparticles?

The density of states in a d-wave superconductor

impurity effects

Finite density of delocalised states at zero energy

density of states

impurity bandwidth

presence of nodes quasiparticles at low T

Linear density of states at low energy- governs all low temperature properties

clean limit

Fermi Liquid Theory of d-wave Nodal Quasiparticles

22

22

21

2F kvkvE

With:

node

2

1v

Fk

The quasiparticle excitation spectrum near the nodes takes the form of a ‘Dirac cone’ :

d-wave gap: = 0cos(2)

E

Thermal Conductivity Primer

lA

Q

Kinetic theory formulation:

cvl31κ

= electrons + phonons

eFe lTv 03

1 γκ

phsph lvT 0

331 βκ A

l

T

Q

κ

T

κ

2T

electrons ~ T

phonons ~ T3

0

d-wave BCS theory of thermal conductivity

F

F

v

v

v

v 2

2

2B0

d

n

3

k

T

κ

Electronic heat transport provided solely by quasiparticles

( T0, T<<)

A. Durst and P. A. Lee, Phys. Rev. B 62, 1270 (2000).M. J. Graf et al., Phys. Rev. B 53,15147 (1996).

This result is universal universal with respect to impurity concentration

Cooper pairs carry no heat

)(21

v 0

node

2 wavedkk FF

Δo from κ0/T

Optimally Doped Bi-2212

192

F

cmK

mW

T 20 15.0

cm/s105.2 7Fv

cm/s1023.1 62 v

2v

vF

Ding et al. PRB 54 (1996) R9678 Mesot et al. PRL 83 (1999) 840

ARPES:

Nodal quasiparticles in optimally doped Cuprates

FF kkwaved

0

node

2

21)(v

Weak Coupling BCS:

0 = 2.14kBTc= 17 meV

0 = 30 meV

Increase Coupling:

0 4kBTc

M.Chiao et al. PRB 62 3554 (2000)

doping dependence: vF

15 ms10x 5.2~ FX.J. Zhou et. al. Nature 423 398 ( 2003 )

LSCO(x)

Fk1

)(v

k

kEF

essentially doping independent

Nodal quasiparticles in overdoped Cuprates

How do we estimate hole concentration [p]?

overdoped Tl 2201

2max

)16.0(6.821 pT

T

c

c

Dopin

g

Tc=15 K sample: Proust et al., PRL 89 147003 (2002).

Tc = 15K

Tc = 27K

Tc = 89K

Tc = 85K

0 = 4kBTc

Other samples: Hawthorn et. al. to be published

underdoped YBCO

Nodal quasiparticles in underdoped Cuprates

vF/v2 as doping

0/T as doping

decr

ease

decr

ease

decr

ease

decr

ease simple BCS theory violated:

Δo does not follow ΔBCS !

Sutherland et al. PRB (2003)

The pseudogap in underdoped Cuprates

pseudogap is : (i) quasiparticle gap (ii) must have nodes

(iii) must have linear dispersion

Campuzano et. al. PRL 83 (1999) 3709Norman et. al. Nature 392, 157 (1998)White et. al. Phys. Rev. B. 54, R15669 (1996)Loeser et. al. Phys. Rev. B. 56, 14185 (1996)

T = 15 K

Underdoped La2-xSrxCuO4

0

100

200

300

400

0.05 0.1 0.15 0.2 0.25

Linear Term vs. Doping

linear term YBCOlinear term LSCO

0 /

T [ W

/K2cm

]

hole concentration [ p ]

0

10

20

30

40

0.02 0.04 0.06 0.08 0.1

LSCO: low dopings

0 /

T [ W

/K2 cm

]

hole concentration [ p ]

2/3(kBh/2n/d)

F

F

v

v

v

v 2

2

2B0

d

n

3

k

T

κ

Presence of static SDW order?

Large intrinsic crystalline disorder?

Summary and Outlook

doping dependence of superconducting gap maximum :

overdoped – optimal doped: 0 scales with Tc (BCS theory)

optimal doped – underdoped: 0 increases while Tc decreases(Failure BCS theory)

Question: What happens near the AF – SC boundary?

existence of nodes throughout the phase diagram:

no evidence for quantum phase transition to d+ix in the bulk

doping dependence: vF

essentially doping independent15 ms10x 5.2~ F

ARPES data Z.X.Shen

LSCO(x)

Fk1

)(v

k

kEF

Specular Reflection of Phonons

Specular reflection lph = f(T)

ph/T~ T, <2

R. O. Pohl* and B. Stritzker, PRB 25, 3608 (1982).

Sapphire

V3Si s-wave SC (thermal insulator, el =0)

Fit data to /T = o/T + BT o/T = 0

= 1.7