charge transport in correlated metals – i i.a the basics i...

Charge transport in correlated metals – I

I.a The basics

I.b Charge transport in the cuprates – the SCES poster-child

Nigel Hussey High Field Magnet Laboratory Radboud University, Nijmegen

ICTP workshop, Trieste, 12th July 2015

Charge transport in correlated metals – II

II.a Bad metals

II.b Kadowaki-Woods ratio The link between transport and thermodynamics

Nigel Hussey High Field Magnet Laboratory Radboud University, Nijmegen


Outline

• Introduction

• Boltzmann transport theory

• Introduction to the cuprates

• Charge transport in the cuprates


Introduction

• Often the first thing to be measured, but the last to be understood…

• “What scatters may also pair” Hence, electrical resistivity is a powerful, albeit coarse, probe of superconductivity

What makes dc transport measurements such an important probe of correlated electron systems?

Gunnarsson et al., RMP 75 1085 (03)

Drude model

Drude assumed that electrons were scattered by random collisions with the immobile ion cores. He assumed a mean free time τ between collisions so that after a time t, the number of electrons having survived without collisions was If the electric field E has been present for this time t, then an unscattered electron will have achieved a drift velocity of v = (-eEt/m) and have travelled a distance This gives for the total electronic transport in the direction of the applied field which is equivalent to n0 electrons having mean drift velocity for time τ. Finally, for a metal containing n electrons / m3, the current density with the corresponding electrical conductivity

( ) ( )τtntn −= exp0

( )mteEx 22=

( )

−=−

−=

∫∫

∞∞

meEnttt

meEnt

tnx

20

0

20

0

dexp2

ddd ττ

τ

2τ3 ( )meEv τ−=

( )m

EnenevEj τσ2

=−==

mne τσ

2

=

Despite the crude approximations and wildly incorrect assumptions, the Drude expression serves as an excellent, practical way to form simple pictures and rough estimates of properties whose deeper comprehension may require analysis of real complexity. How come? Well, some of it is fortuitous. Drude then estimated the velocity from the kinetic equation From this he extracted a mean-free-path of ~1 nm, i.e. the approximate interatomic spacing! Of course, while this seemed reasonable to Drude, it was way off the mark... The workability of the Drude model reflects the fact that two of the fundamental assumptions (the action due to the Lorentz force and the exponential decay in n(t)) are also found to be equally applicable to Bloch waves and fermionic quasiparticles.

( ) sec10 1cm2300 -14×≈⇒Ω≈ τµρ

/sm105232

21 ≈⇒= vTkmv B

Drude model

( ) ( ) ( ) ( ) [ ]( )BvEffpp×+−=+−= ettt

tt

τdd( ) ( ) ( ) ( ) ( )( )2dd d1d tOtttttt ++−=+ fpp τ

Drude model

Hall effect ac conductivity In steady state, current is independent of t Seek steady state solution of the form px and py thus satisfy Thus where ωc = eB/m (B//z) Thus Current density Hall field is determined by condition jy = 0 where Hence, Hall coefficient

τω

τω

yxcy

xycx

ppeE

ppeE

−+−=

−−−=

0

0

×+−−= BpEpp

me

t τdd

yxcy

xycx

jjE

jjE

+−=

+=

τωστωσ

0

0

xxc

y jneBjE

−=

−=⇒

0στω

neBjE

Rx

yH

1−==

( )tet

Epp−−=

τdd

( ) ( ) ( ) tit ωω −= expRe pp

( )ωτ

σωσi−

=1

0

( ) ( ) ( ) ( )( ) ( ) ( )ωωσ

ωτωωω 0

02

1E

Epj =−

=−

=i

mnem

ne

( ) ( ) ( )ωτωωω 0Epp ei −−=−

( ) ( ) ( ) tit ωω −= expRe 0EE

( )tfk ,r

(i) Carriers move in and out of the region r

(ii) The k-vector will be changed by external fields

(iii) Carriers are scattered

kκk v

rr fftt

f k ∇⋅−=∂∂

⋅∂∂

−=

∂∂

diff

kk

kkk

∂∂

⋅−=∂∂

⋅∂∂

−=

∂∂ kk ff

ttf

field

[ ]( )BvEk k ×+=

ewhere

( ) ( ) ( )τ

0kk

kkkkk kkk ff,Qfffftf −

−=′′−−−=

∂∂

∫ ′′ d11coll

Local concentration of carriers “occupancy” in the state k in the neighbourhood of the point r in space and time t

relaxation time approximation

Distribution functions

Boltzmann equation – 3 key points

0collfielddiff

=

∂∂

+

∂∂

+

∂∂

tf

tf

tf kkk

Note that this is steady state, not equilibrium state fk

0 For electrons We are most concerned with small departures from fk

0

( ) 1/exp1

+−=

Tkf

Bµεk

0k

∂∂

∝−=ε

0k0

kkkfffg

Fermi distribution

∂∂

∝−=ε

0k0

kkkfffg

f k0 (ε)

µ or εF

- (dfk 0/dε)

ε

1

0

0collfielddiff

=

∂∂

+

∂∂

+

∂∂

tf

tf

tf kkk

⇒−= 0kkk ffg

( ) ( )

k

0k

0k

k

0kk

0k

k

0k εµε

µεµε ∂

∂−=

∂∂

∂∂−

=∂∂

⇒+−

=fff

TTf

Tkf

B

and 1/exp

1

[ ]( )

[ ]( )k

BvEr

v

kBvEv

kk

kκ

k

kk

kkκ

∂∂

⋅×++∂

∂⋅+

∂∂

−=

∂∂

⋅×+−

∇∂∂

+∇∂∂

⋅−

gegtf

fefTTf

coll

000

µµ

[ ]( )coll

∂∂

−=∂∂

⋅×+−

∇∂∂

+∇∂∂

⋅−⇒tffefT

Tf kk

kkk

κ kBvEv

µ

µ

[ ]( )coll

∂∂

−=∂∂

⋅×+−∇⋅−⇒tffef kk

kkκ kBvEv

Boltzmann equation – all aboard the Chain Rule!

Assume “infinite homogeneous medium”, constant temperature and zero magnetic field; and for simplicity, let us make the phenomenological relaxation time approximation Hence

( )τ

kkkkk gt

gt

fgtf

−=∂

∂=

∂+∂

=

∂∂ 0

coll

( ) ( ) τtegtg −= 0kk

Evκk

k ⋅

∂∂

−=⇒ τε

efg k0

coll

0

∂∂

−=⋅

∂∂

−⇒tfefk k

κk

Evε

Linearized Boltzmann equation

( ) [ ]k

Bvr

vEv kκkk

κk ∂

∂⋅×+

∂∂

⋅+

∂∂

−=

∇−+∇

−⋅

∂∂

− kkk gegtf

eeT

Tf

coll

0 1 µµεε X X X X

( ) teefg ttt

k ′⋅

∂∂

−= ′−−

∞−∫ d

0τ

εEvκ

kk

Boltzmann vs. Drude

( ) ( ) τtegtg −= 0kk

0collfielddiff

=

∂∂

+

∂∂

+

∂∂

tf

tf

tf kkk

( ) ( ) [ ]( )BvEpp×+−−= et

tt

τdd

fielddiff

∂∂

+∂

−=

∂∂

−tfg

tf kkk

τ

[ ]( )k

BE kk

kk

∂∂

⋅×+−∂

−=

∂∂

−fveg

tf

τdiff

c.f. Drude ( ) τtentn −= 0

Thus, despite their extreme starting points, both models consider a steady state solution involving forces acting on charged particles or wave packets with a distribution of velocities.

Current response

A n electrons/unit vol.

v

v

( ) jijii kgekfn EvJ kk σππ

==⇒= ∫∫ 33

33 d

41d

22

( )0d 30 =∫ kfei kv

vvxJ neAneAAI === ˆ

Similarly for thermal current ( ) jijii Tkg ∇−=−= ∫ κµεπ

33 d

41

kvJ

jijii kge EvJ k σπ

== ∫ 33 d

41

∫

∂∂

−⋅=⇒ kfe jiij3

02

3 d 4

1ε

τπ

σ vv

and Eκk ⋅

∂∂

−= vefg τε

0

Consider spherical Fermi surface

d3k

kF

dϕ dkr

dθ kF dθ

∫∫∫∫∫∫

∂∂

−=

∂∂

−=

∂∂

−ππ

ϕϕθεεε 0

2

0

003

0

d sindd dd d FFrFr kkkfSkfkf

dc electrical conductivity


Recall where cosγ is the angle between vF and kF = 1 for isotropic 3D FS And for most metals, behaves as a δ-function. Hence, Note that this is the same surface integral that appears in the expression for the electronic density of states

rF kdcosd1 γεε vv kkkk

=⇒∇=

( )( ) ∫∫ ∫∫ ∫ =−=

∂∂

−γ

εεδε cos

dd d d d 0

F

FFrFnFr

SSkSkfv

k

∂∂

−ε

0f

( )( )∫ ∇

=kn

Fn

Sgεπ

ε 14d

3


Hence,

θϕϕτπ

σπ π

d d sin

4

2

0 0

23

23D ∫ ∫

⋅= F

F

jiij k

ve vv

ϕθθϕτπ

ϕθϕτπ

σπ ππ π

∫ ∫∫ ∫ ==2

0 0

2323

22

0 0

23

2

d dcossin4

d d sin 4 FFF

F

xxxx vkek

vvve

22

2

0

32

0

23

22

3

d sind cos4

Fxx

FFxx

ke

vke

πσ

ϕϕθθπ

τσππ

=⇒

=⇒ ∫∫π 4/3

c.f. Drude result: ( )

22

22

2

323

3

2

33*34

22

* FF

FFF kekvek

mek

mne

πτ

πτπ

πτσ =

=

== QED

( )θϕ cossinFx vv =

Assume, isotropic, cylindrical Fermi surface

1cos =⇒ γ

FFF

xx kc

ec

vkeπ

ϕϕπ

τσπ

2d cos

2

22

0

22

2

==⇒ ∫

c.f. Drude result: ( )

F

F

FFF k

ce

kve

ck

mek

cmne

πτ

πτππ

πτσ

22*2

22

*

22222

3

2

=

=

== QED

π

Fxx kc

eπ

σ2

2

=

cπ2

vF || kF

|kF |

∫∫⋅

=−

ππ

π

ϕγ

τπ

σ2

03

22D d

cosd

4 FF

jic

czij k

vke vv

In-plane conductivity for quasi-2D metal

ϕγ

ϕτππ

ϕγ

τπ

σπππ

π

d cos

cos24

d cos

d4

2

0

2

3

22

03

2

∫∫∫

==

−

FFF

F

xxc

czxx

vkc

ekv

vvke

Conductivity in a magnetic field

In metals, the effect of a magnetic field is usually to deviate the trajectory of the carriers from their electric-field induced path. The two major corresponding changes to the electrical conductivity tensor are: (i) the well-known Hall effect emerges as a result of cross terms (σxy etc...) (ii) the longitudinal (diagonal) conductivity also changes - magnetoresistance The Hall effect can be of either sign, depending on the majority or most mobile carrier type. The magnetoresistance, on the other hand, is almost always positive. Why? A definition of magnetoresistance The increase in the longitudinal resistance caused by the additional scattering all mobile carriers experience per unit length in the direction of the applied electric field in the presence of an applied magnetic field.

-e

B

L

X X X X X

Conductivity in a magnetic field

Within the relaxation time approximation; Continuous series Hence e.g. To calculate σxy

(1), simply work through with the applied magnetic field H//z and the electric field E//y and calculate the response Jx with the first-order Jones-Zener equation.

[ ]τε

kkk ggefe −=∂∂

⋅×+

∂∂

⋅⇒k

BvvE kk

κ

0

( ) [ ]

∂∂

−⋅

∂∂

⋅×−=k

κk vEk

Bvε

ττ 0k

nn

kfeeg

( ) [ ] kfeveev kj

n

in

ij3

0

3 d 4

1∫

∂∂

−

∂∂

⋅×−=k

k kBv

εττ

πσ

( ) [ ]k

Bvr

vEv kκkk

κk ∂

∂⋅×+

∂∂

⋅+

∂∂

−=

∇−+∇

−⋅

∂∂

− kkk gegtf

eeT

Tf

coll

0 1 µµεε X X X

Interlude: Taking into account the shape of quasi-2D FS

γ ϕ−γ

ϕ

dϕ

dkF

kF dϕ

γ γ

( )( )[ ]

∂∂

=⇒

= − ϕ

ϕγ

ϕγ F

F

F kk

k lntand

dtan 1

The relation between vF and kF

For B//c: where and Finally, NB (i) σxy

(1) probes the variation of the mean-free-path around the Fermi surface. NB (ii) σxy

(1) in a quasi-2D metal does NOT depend on the carrier density

( ) [ ] kfeveev kyxxy

30

31 d

41

∫

∂∂

−

∂∂

⋅×−=k

k kBv

εττ

πσ

B vk

[ ]//k

BvF ∂∂

=∂∂

⋅×k

Bvk

ϕγ

∂∂

=∂

∂

Fkkcos

//

∫∫∫−

=

∂∂

−ππ

π

ϕγε

2

0

30

dcos1dd F

F

c

cz kkkf

v

( ) ( ) ( )[ ]∫∫ −∂∂

−−

=∂∂−

=ππ

ϕγϕτϕ

γϕτπ

ϕϕ

ππ

σ2

022

32

023

31 dsincos

2d2

4 FFy

xxy vvc

Bec

Be

Hall conductivity in a quasi-2D conductor

//k∂∂

Thus σxx(2) as expected, is negative and scales with B2

Consider isotropic cylinder:

( ) [ ] [ ]

( ) ( )[ ]∫

∫

−∂∂

∂∂

−=

∂∂

−

∂∂

⋅×−

∂∂

⋅×−=

π

ϕγϕϕ

γϕ

γϕπ

ετττ

πσ

2

032

24

30

32

dcoscoscos2

d 4

1

F

kxxxx

kcBe

kfeveeevk

kk kBv

kBv

( )

( ) ( )222

222

23

324

0

2 22

τωππσ

σc

FFFxx

xx

kBe

kec

ckBe

−=−

=⋅−

=⇒

In-plane magnetoresistance in a quasi-2D metal

=

=

F

F

c

keBvmeB

*ω

( ) ( )FF

xx ckBe

ckBe

3

3242

02

2

32

3242

2dcoscos

2

πϕ

ϕϕϕ

πσ

π

−=∂

∂= ∫

In-plane magnetoresistance in a quasi-2D metal

Magnetoresistance MR is zero in strictly isotropic system even though magnetoconductance is always finite

2

0

)1(

)0(

)2(

0

−−=

∆

xx

xy

xx

xx

xx

xx

σσ

σσ

ρρ

( )

( )

( ) ( ) 0

22

22

22

2

22

232

0

=−+=

−+=

⋅−+=

∆

τωτω

τω

ππ

τωρρ

cc

Fc

Fc

xx

xx

keB

kec

cBe

( ) ( )c

Bec

Bec

Bexy 2

232

0

222

232

022

231

2dcos

2dsincos

2

πϕϕ

πϕ

ϕϕϕ

πσ

ππ

==∂

∂−= ∫∫ π

Fxx kc

eπ

σ2

2

=

Kohler’s rule

If the only effect of a change of temperature or of a change of purity of the metal is to alter τtr(k) to λτtr(k), where λ is not a function of k, then ∆ρ/ρ is unchanged if B is changed to B/λ. Since , , the product ∆ρ.ρ (= ∆ρ/ρ . ρ2) is independent of τtr and a plot of ∆ρ/ρ versus (B/ρ)2 is expected to fall on a straight line with a slope that is independent of T (provided the carrier concentration remains constant). Kohler’s rule is obeyed in a large number of standard metals, including those with two types of carriers, provided that changes in temperature or purity simply alter τtr(k) by the same factor. Deviations from Kohler’s rule imply a change in carrier content, possibly a change in dimensionality, or most likely, a change in the variation of τtr(k) around the Fermi surface.

Narduzzo et al., PRL 98, 146601 (07)

( )2τωρρ c∝∆

• Their transition temperatures are anomalously high

• Their superconducting order parameter is unconventional (d-wave)

• The superconductivity emerges out of a highly correlated insulating state

• Their normal metallic state is unlike anything that has been seen before

Tab 10~ ααρ +

Martin et al., Phys. Rev. B (1990)

Bi2+xSr2-xCuO6+δ

Ando et al., Phys. Rev. B (1999)

Bi2+xSr2-xLaxCuO6

Why are high temperature superconductors interesting?

Zaanen Senthil Schmalian Vojta Randeria

“The metallic state at optimal doping embodies the enlightenment. Rather than being complicated, this ‘bad’ metal shows a sacred simplicity – symbolized, for example, by its linear resistivity…”

“Most mysterious is the strange optimally doped metal. There simply do not seem to be (m)any good ideas on how to think about it. Yet it is empircally characterized by simply stated laws.”

“The biggest mystery is the linear resistivity at optimal doping.”

“The difficulties lie with the normal state, featuring phenomena like the apparent linear-in-temperature resistivity at optimal doping.”

“However, many questions about the precise nature of the transition from the superconductor to the Mott insulator, the possibilities of various competing/coexisting ordered states at low doping and, most particularly, the description of the non-Fermi-liquid normal states remain unclear at the present time.”

High temperature superconductivity and Nobel Laureates

J. G. Bednorz (1987) K. A. Muller (1987)

A. A. Abrikosov (2003) P. W. Anderson (1977) P. G. de Gennes (1991)

A. W. Geim (2010) V. L. Ginzburg (2003) A. J. Heeger (2000) H. Kroemer (2000)

R. W. Laughlin (1998) A. J. Leggett (2003) N. F. Mott (1977)

J. R. Schrieffer (1972) F. Wilczek (2004)

The key structural element, the copper-oxide plaquette, appears either in a square planar, or an octahedral arrangement. Hence, the Jahn-Teller effect is not necessarily important in all cuprates.

Crystal and electronic structure

Tl2Ba2CuO6+δ La2-xSrxCuO4

copper

oxygen

apical oxygen

La/Sr

Electronic configuration of Cu 1s2 2s2 2p6 3s2 3p6 3d9 4s2 Electronic configuration of Cu2+ 1s2 2s2 2p6 3s2 3p6 3d9 = 1 carrier/unit cell Electronic configuration of Cu(2-x)+ 1s2 2s2 2p6 3s2 3p6 3d9-x = Hole-doped CuO2 plane

Crystal and electronic structure

(La3+)2Cu2+(O2-)4

(La3+)2-x(Sr2+)xCu(2-x)+(O2-)4

t2g orbitals eg orbitals

At half filling

Mott insulator 3d9 3d9 3d8 3d10 Charge transfer insulator 3d9 3d10 L

+U

O Cu

+U

+∆pd

~ 7-8 eV

+∆pd

Beyond half filling…

Doping away from half-filling strongly frustrates the spin background – leads to a strong suppression of the AFM ordering temperature. Frustration is significantly greater for spin exchange along the coordination axis than diagonally to it. Inherent in-plane anisotropy within the CuO2 plane.

“Nodal-anti-nodal dichotomy” O Cu

+t’

Phase diagram

AFM suppressed with the addition of only a few percent of holes. Superconductivity emerges at a hole concentration p ~ 0.05. Maximum Tc occurs at an optimal doping of p = 0.16. Superconductivity vanishes again around p > 0.30. Dip in Tc also seen around p = 0.125 – so-called 1/8-anomaly. Normal state phase diagram dominated by strange metallic phase and the pseudogap.

Phase diagram

AFM suppressed with the addition of only a few percent of holes. Superconductivity emerges at a hole concentration p ~ 0.05. Maximum Tc occurs at an optimal doping of p = 0.16. Superconductivity vanishes again around p > 0.30. Dip in Tc also seen around p = 0.125 – so-called 1/8-anomaly. Normal state phase diagram dominated by strange metallic phase and the pseudogap.

In underdoped cuprates, possibly an additional temperature scale Tf due to extended region of phase fluctuating superconductivity

Phase diagram – evolution of in-plane resistivity

At low doping, ρab(T) is non-metallic at low T. Superconductivity develops out of insulating ground state.

Note almost exponential decrease in ρab(300) with increased doping, showing how the mobility of doped holes improves with doping.

Optimally doped cuprates

Fxx kc

eπ

σ2

2=

Yoshida et al., PRB 61 R15035 (99)

YBa2Cu3O7 Tl2Ba2CuO6+δ

Tyler + Mackenzie, PhysicaC 282 1185 (1997)

Negative intercepts imply that T-linear resistivity cannot continue to absolute zero – there has to be a crossover to a

higher exponent of T at the lowest temperature

1) T-linear resistivity is observed only in a narrow composition range near optimal doping; the sharp crossover to supralinear resistivity on the OD side being more suggestive of electron correlation effects than phonons.

2) The absence of resistivity saturation in OP La2-xSrxCuO4 up to 1000K argues

against a dominant e-ph mechanism.

3) The frequency dependence of 1/τtr, extracted from extended Drude analysis of the in-plane optical conductivity, is inconsistent with an electron-boson scattering response due to phonons. In particular, Γ(ω) does not saturate at frequencies corresponding to typical phonon energies in HTC.

4) It has proved extremely problematic to explain the quadratic T-dependence of the inverse Hall angle cotθH(T) in a scenario based solely on e-ph scattering.

The problem(s) with phonons

,

Strong T-dependence over a very wide temperature and doping range. What does it signify?

Hall effect in hole-doped cuprates

,

Hwang et al., PRL 72 2636 (94)

Marked, almost 1/T increase in RH(T) could indicate a reduction in carrier number with decreasing temperature (Recall that RH = 1/ne) However, this increase in RH(T) is occurring in range above pseudogap temperature.

Strong T-dependence over a very wide temperature and doping range. What does it signify?

,


Hwang et al., PRL 72 2636 (94)

Moreover, in the underdoped regime, below T = T*, RH(T) begins to decrease with T in the pseudogap regime. This suggests that RH(T) does not in any way reflect the variation of n(T). Cuprates are also largely single-band.

,

Ando + Murayama, PRB 60 R6991 (94)


Variation in ρab(T) and 1/RH(T) with Zn-doping in Y123 does not appear to be correlated with one another…

However when data plotted as cotθH(T), it shows a very simple T2 dependence.

The inverse Hall angle

,

Chien et al., PRL 67 2088 (91)

Variation in ρab(T) and 1/RH(T) with Zn-doping in Y123 does not appear to be correlated with one another…

However when data plotted as cotθH(T), it shows a very simple T2 dependence.

The inverse Hall angle

,

2Hxx

xyxy

BBR

σσρ

==

xyσ

xxσHθ

ab

xxxy

xx

xx

xy

BR

BR

ρθ

σσσθ

σσ

θ

HH

HH

H

cot

cot

tan

=⇒

==⇒

=

This so-called separation of lifetimes is most apparent at optimal doping, where ρab(T) ~ T and cotθH(T) ~ T2. Such remarkable transport behaviour is a hallmark of the cuprates that has engaged some of the greatest condensed matter thinkers.

The separation of lifetimes

,

Tyler + Mackenzie, Physica C 282 1185 (97)

Ono + Ando, PRB 75 024515 (07)

The quasi-2D heavy-fermion compound CeCoIn5 shows a similar phenomenon of separation of lifetimes between Tc and Tcoh ~ 20 K.

We are not alone…

,

Nakajima et al., JPSJ 75 023705 (06)

1) The two-lifetime model of Anderson and co-workers

Transport theories – the “Big Three”

[ ]

( )

( )

( ) ( )22

2H

2

)0(

)1(

)0(

)2(

2

HH)1(H

2

H

30H3

3)(

1

1cot1

11

d4

BTAT

BTATT

TTT

kfvk

Bvve

xx

xy

xx

xx

ab

ab

tr

tr

xy

xx

trab

tr

trj

n

kin

ij

+∝∝

−−=

∆⇒

+∝∝∝=⇒∝

∝∝⇒∝

∂∂

−

∂∂

×−= ∫

τσσ

σσ

ρρ

ττττ

σσθ

τ

τρ

τ

εττ

πσ

Kohler’s rule violated all the way up to room temperature

Violation of Kohler’s rule

Harris et al., PRL 75 1391 (95)

T-dependence of in-plane MR extremely well described by Anderson’s two-lifetime model

The non-integer inverse Hall angle

,

Konstantinovic et al., PRB 62 R11989 (00)

Ando + Murayama, PRB 60 R6991 (99)

This non-integer exponent in the inverse Hall angle and its doping-dependence are very hard to capture within this model

2) The marginal Fermi-liquid model of Varma and co-workers

The anisotropic T-independent elastic scattering term γ(k) was argued to originate from small-angle scattering off impurities located out of the plane. ⇒ scattering rate anisotropy reflects that of DOS, i.e. ⇒ Violation of isotropic-l approximation and strong variation in RH(T) and Unfortunately, this model struggles to account for both the in-plane MR and the evolution of the exponents in OD cuprates.

( )kγλτ

+∝ Ttr

1

Cu O La, Sr

( )2

H1cot

∝

tr

Tτ

θ

( ) ( )ϕϕτ Fv

11∝

Carter+Schofield, PRB 66 241102(R) (02)


3) Anisotropic single lifetime models

3a) Nearly AFM FL model 3b) Cold spots model 3c) Anisotropic scattering rate saturation model

TTfc

∝∝

1;1 2\

( ) ( )Fτ

ϕϕ 12sin20 +Γ∝Γ

Carrington et al., PRL 69 2855 (92) Stojkovic+Pines, PRL 76 811 (96)

Ioffe+Millis, PRB 58 11631(98)

Hussey, EPJB 31 495 (03) Hussey, JPCM 20 123201 (08)

( ) ( ) ( )

maxidealeff

22

210ideal

1112sin

Γ+

Γ=

Γ

Γ+Γ+Γ=Γ TTϕϕϕ


Altogether now!

( ) ( ) ( )

maxidealeff

22

210ideal

1112sin

Γ+

Γ=

Γ

Γ+Γ+Γ=Γ TTϕϕϕ

( )kγλτ

+∝ Ttr

1MFL Two-lifetime

( ) ( )Fτ

ϕϕ 12sin20 +Γ∝Γ

Cold Spots model NAFL

2

H

1;1 TTtr

∝∝ττ

21;1 TTcf

∝∝

charge transport in correlated metals – i i.a the basics i...

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