Normal State Properties of High-Temperature Superconductors and the Marginal Fermi Liquid

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Normal State Properties of High-Temperature Superconductors and the Marginal Fermi Liquid

Elihu Abrahams

To cite this version:

Elihu Abrahams. Normal State Properties of High-Temperature Superconductors and the Marginal Fermi Liquid. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.2191-2203. �10.1051/jp1:1996204�.

�jpa-00247307�

J.

Phj.s.

I France 6

(1996)

2191-2203 DECEMBER1996, PAGE 2191

Normal State Properties of High-Temperature Superconductors

and trie Marginal Fermi Liquid

Elihu Abrahams

(*)

Serin

Physics Laboratory, Rutgers University, Piscataway,

NJ 08854, USA

(Receiied

6 June 1996, received in final form 25 June 1996,

accepted

1

July1996)

PACS.75.25.+z

Spin

arrangements _in

magnetically

ordered materials

(including

neutron and

spin-polanzed

electron studies,

synchrotron-source

_x-ray

scattering, etc.)

PACS.74.72.-h

High

Tc

compounds

PACS.71.27.+a

Strongly

correlated systems; heavy fermions

Abstract. Trie

expenmentally

observed normal state thermodynarnic ^and transport _prop-

erties of cuprate

superconductors

_are

umversally quite

unusual and _so for

unexplained.

Some of these will be reviewed and the

challenges

for _an overall theoretical picture will be discussed. It wiII be described how, _{in some cases,} the

Marginal

Fermi

Liquid hypothesis

_may be

apphed.

1. Introduction

A decade has

passed

_since the

original report

^of

high-temperature superconductivity

_in lan- thanum

cuprate ("2-1-4") by

Bednorz and Müller

iii.

In addition to his seminal contributions to the

physics

of low-dimensional

metals,

I.F.

Schegolev participated

in

important

work _on the

high-temperature cuprate superconductors during

this

period.

In

particular,

_{we can} cite his NMR

experiments

on

yttrium

banum _copper oxide

(1-2-3)

and _on thallium copper oxide

(2-2-0-1)

_[2] and his

analysis

of electrical transport ^and structure [3]. ^{Since the}

original

dis- covery

by

Bednorz and

àlüller,

there have been

significant

achievements _in

synthesizing

_new

high-Tc compounds.

In addition to

identifying

and

descnbing

the

properties

of _{new supercon-}

ducting materials,

there has been _{an enormous} effort to improve the

quality

of the

specimens

used to

investigate

their

physical properties-

Successes in the latter

enterprise

^has meant that

in the last few _years,

widespread unanimity

has

developed

_on aIniost all of the

experimental properties

of the

high-temperature cuprate superconductors- Thus,

while it

appeared

_quite

early

that in several

experiments

some transport and

thermodynamic properties

of

high-Tc

su-

perconductors

_were unusual, it is at the

present

^time

universally recognized

that all the normal state

properties

_are

sufliciently

anomalous that the matenals _are to be descnbed _as

"strange metals,"

which is to say that

they

fall outside the Mass of normal "Fermi

hquids"

to which conventional

superconductors

and

ordinary

metals

belong

_[4].

Thus,

the electrical conductiv-

ity

[5], ^the

magnetoresistance [6],

^{the Hall eflect} [6,

7],

^the

optical conductivity [8],

^{the thermal}

transport

[9],

^{the electronic} spin

susceptibility

and the nuclear

magnetic

relaxation rate

[loi,

the

angle-resolved photoemission spectrum il Ii,

,

all exhibit unusual behavior.

(*)

e-mail:

abrahamstÙphysics.rutgers.edu

©

Les

Éditions

_de

_Physique

₁₉₉₆

2192 JOURNAL DE

PHYSIQUE

I N°12

The references above

give

reviews of the relevant

experiments.

While there _{is consensus}

on the

data, agreement

on the

underlying

mechanism

responsible

for both the normal _state behavior and the

superconductivity

is still

lacking.

It is _{a comnion view} that _an understand- ing of the normal state will facilitate the

explanation

of the inechanism

responsible

for

high

superconducting

transition

temperatures, consequently

trie normal state is _a

subject

of intense

interest. No

attempt

is made in this _paper to review either all the

expenments

or the several

theoretical

approaches

for the normal state. For discussion

here,

I have selected _a number of

key

observed

properties

which _are

strongly

indicative of non-Fermi

liquid behavior,

which

seem to be universal from

compound

to

compound

and which any

comprehensive theory

of the normal state of the cuprates must address. After

describing

these

experimental

results, I review the

"Marginal

Fermi

Liquid" (MFL) phenomenological approach

and discuss how it

might

account for two of these features.

The paper is

organized

as follows: We first discuss _sonie of the evidence for non-Fermi

hquid

behavior _in the normal state

properties, especially _transport,

of

high-Tc superconductors.

We then introduce the MFL

phenomenology-

The

application

of the MFL

approach

to the

anisotropic resistivity

and to the

optical conductivity

concludes the _paper.

2. Non-Fermi

Liquid

Behavior

2.1. IN-PLANE RESISTIVITY. The normal-state

transport properties

which _we wish to

discuss _are the

in-plane resistivity

_pab, the _c-axis

resistivity

_p~, the Hall constant

RH,

the Hall

conductivity

and

angle

_a~y,

6H,

the transverse

magnetoresistance Ap/p,

and

finally,

the

in-plane optical conductivity a(w).

We shall _see that each of these

quantities

has anomalous behavior _{as a} function of

temperature (or frequency)

with power laws

having exponents

which

are quite ^{diflerent from those of} a Fermi

hquid.

For

"undoped"

lanthanum

cuprate, La2Cu04,

for

example,

there _{is one} hole _per unit cell in the CUO

planes.

If the inter-electron interaction _were

weak,

the

compound

would be metallic

an

ordinary

Fermi

hquid. However,

the

ground

state is _an

antiferromagnetic

Mott insulator.

This _is

already

_an indication of unusual

behavior,

^due to

strong interpartide

interactions and

narrow bands. Trie addition of strontium _on lanthanum sites in

La2Cu04

increases the number

of holes in the

planes

and leads to metallic behavior

la "doped

Mott

insulator")- Thus,

the matenal

La2-~Sr~Cu04 (2-1-4)

is

superconducting

for 0.05

/

z < 0.25. The _maximum Tc

occurs for _{x m}

o-là, "optimal doping." Samples doped

at levels less

("underdoped")

_{or more}

("overdoped")

than

optimal

can still show

superconductivity

but several of the normal state

properties

have

important diflerences,

_{as we} shall _see.

The

ubiquitous

hnear temperature

dependence

of the

in-plane resistivity

_[5j is

perhaps

the best-known of the anomalous power laws. It is seen, for

example,

in

2-1-4,

ⁱⁿ the

yttrium

banum _copper

oxides, YBa2Cu306+~

and

YBa2Cu408 (1-2-3

and

2-4-8),

and in the bismuth copper

oxides, B12Sr2CaCu20s

and

B12Sr2Cu06 (2-2-1-2

and

2-2-0-1)

_at

_optimum doping.

At the _same time, it is trie _case that deviations from hnear-T behavior _occur for

samples

which

are not

optimally doped.

In the region of

optimum doping,

the hnear

resistivity

is seen over a

temperature

_range

extending

for several hundreds of

degrees

above the transition. A _most

striking example

_is

2-2-0-1,

_in ~v-hich the

resistivity

is

quite

linear from _a T~ of about 7 K to 700 Il

[12].

^{~vhile such} a behavior could arise when the

8cattering

is

pnmanly

from _a bosonic

excitation, for

example phonon(s) _[13]- having

_a charactenstic _{energy smaller than}

kBT~,

the fact that the

scattenng

rates are

quite

similar from matenal to

material,

_even

though

the

T~'s

are

substantially diflerent,

_is indicative that the

scattering

_is

probably

due to _an electronic mechanism which is intrinsic to the copper-oxygen

planes.

This _{view is} further

supported by

many

expenmental

observations

showing

that the

scattenng

rate decreases

sharply

_{as one goes}

N°12 NORMAL STATE IN

HIGH-T~;

MARGINAL FERMI

LIQUID

2193

1/~

T

Fig.

1.

Transport scattering

rate. Note the precipitou8

drop

at Tc.

into the

superconducting

state

[14].

^{This is shown}

schematically

_in

Figure

1.

Thus,

when _a gap opens in the electronic spectrum, ^the scatterers

disappear, indicating

that

they

_are excitations of the electrons themselves. The _mean free

paths

are not

short,

of order 100 to 200

À _just

above T~. If the temperature

dependence

of the normal-state

resistivity

_p =

(m/ne~)(1/T)

is attributed

entirely

to the

scattenng rate,

^then the

magnitude of1/T(T)

is several times

kBT/h,

whatever the Tc- ^In contrast, _{as is} well-known

[15],

electron-electron

scattenng

in _a Fermi

hquid gives

the

quite

diflerent

scattenng

rate of order

(kBT)~ /hEF,

determined

by

the restrictions _on

phase

_space which _are

imposed by

the Pauh

pnnciple.

Because the

magnitude

of the

scattenng

rate

land

_even the

resistivity)

does not vary

substantially

from

high-Tc superconductor

to

high- Tc superconductor,

there _seems to be _no doubt that the electron

scattering

mechanism is the

same for ail the

cuprate superconductors.

There _are several

published

microscopic

explanations

of the T-linear

resistivity- Scattenng by antiferromagnetic spin

fluctuations

[16], scattenng by phonons

when the Fermi surface _is

very close _{to a} Van Hove

singularity

in the

density

of states

[lî],

and

scattering by

constraint gauge fluctuations in models with

spin-charge separation

[18] are

examples.

A discussion of these theones would

necessarily

_{require a} critical sumniary and _as mentioned

earher,

_we do not propose ^to do that in this _paper.

However,

in _a later section _we shall descnbe the

phenomenological marginal

Fermi

liquid approach

to the

expenmental

data.

2.2. HALL EFFECT AND MAGNETORESISTANCE.

Next,

_we discuss the Hall efiect. The

standard relations [19] are:

OExy ^" OE~~LdCT, p~y #

RHB, (Î)

where _~a~

=

eB/mc.

When _{~a~r «} 1, the Hall

angle

and Hall constant are given

by

COtÙH ^~~~ j~ ^LdcT

a

$,

^{H OE p~y m}

^j~~

~~ ~ OEx~

The

expenmental

results for _a

vanety

of

high-T~ superconductors

_[20] for the Hall

angle

_are summanzed

by

cot 6H

" a +

bT~, (3)

where _{a is a} small _constant

depending

_upon

impunty

concentration. Since _{a~~ cc} T for small

fields,

and if a18 small

enough.

this is consistent with _a Hall constant

decreasing

_as

1/T,

_{as is}

observed

[î]. Usually,

1-e- for _a Fermi

liquid,

the

cotangent

^{of the Hall}

angle

_is determined

by

2194 JOURNAL DE

PHYSIQUE

I N°12

the relaxation rate, as in

equation (2).

It would

then, contrary

to

experiment,

be linear in T since

1/r

is.

An

interpretation

of the

expenmental

results _was

suggested by

Anderson

[21].

^He

argued,

on

the basis of

spin-charge separation,

that in _a

magnetic

field there _are two diflerent relaxation

times,

_{one, rt>,} for the

longitudinal transport ii-e- a~~)

and

another,

_rH, for the transverse relaxation in _a

magnetic

field _so that in the above

equations

_{w~r - w~rH.} Then if _we assume, besides the observed _{rtr cc}

1/T,

that the _new relaxation time _{rH cc}

1/T~, independent

of mag- netic field, and _use these in the standard formulae [6], we

find,

in

agreement

with

expenment:

cot 6H cc

T~,

_{a~y cc rtrrH cc}

^T~~,

_{p~y cc rtr cc}

^T~~- (4)

The

assumption

is that when the

magnetic

field is

perpendicular

to the _copper oxide

planes,

the relaxation of the second-order distortion of the Fermi

distribution,

due to the electric and

magnetic

fields, relaxes with _rH, while the first-order distortion

(due

to the electric field

alone)

relaxes with _rtr. This

phenomenology,

which involves two diflerent relaxation

times,

is outside the usual Fermi

liquid

transport

theory.

There have been several

attempts

to realize _a situation with two relaxation times. A par- ticular

phenomenology

has

recently

been

suggested by Coleman,

Schofield and Tsvelik

[22].

They

introduce diflerent relaxation times for currents which _{are even} and odd under

charge conjugation

and discuss the

resulting

transport

equation- Kotliar, Sengupta

and Varma [23]

investigated

the consequences of

introducing

a

skew-scattering

contribution in the collision

integral

of the Boltzmann

equation-

The Anderson

suggestion

that the Lorentz force term be

multiplied by

_{rHrtr (24] can} be derived

directly

from the Boltzmann

equation

with _a

generalized

collision term as follows: The

steady-state

Boltzmann

equation

for

spatially

uniform electric and

magnetic

fields

E,

B is [25]

~~ _~k

~

+

)~~~

~

~~ ÎÎ Î

~~~~

' ~~~

where the distribution function

fk

will be wntten _as

fi

_{+ gk.} The collision term has the form

jk

~6)

~~~~

m

[

_II~kk~19k

_9~'~

As usual

[25],

we look for _a solution _in the form

ô

fk

~~

~Î~~

^'~' ~~~

To make the point, ^{it is} suflicient,

though

not necessary, in what follows to take _a

spherical (circular

_in two

dimensions)

Fermi surface _so that _vk

=

hk/m-

In

addition,

_{we assume} elastic

scattering only

so that in the collision term, _u

= u' and

Ivkki

_cc

W(cos6),

where 6 is the

angle

between k and k'. The Boltzmann

equation

then

simplifies

to

v E +

(wc

x

A)

_v

=

/

_dû

_iV(cos

9) iv v') _{A, (8)}

where _wc is _a vector

parallel

to B with

magnitude equal

to the

cyclotron frequency eB/mc.

For

simphcity,

we treat the _case of two

dimensions,

_appropnate for the cuprates- ^The

angle

between _v and v' _is 6; ^{let the}

angle

between _v and E be

ç§- Then _we define

velocity

_components

N°12 NORMAL STATE IN HIGH-T~; MARGINAL FERMI

LIQUID

2195

parallel ("longitudinal")

and

perpendicular ("transverse")

to the electric field induced shift of the Fermi surface. Thus

~i v( =

u[cos

_ç§(1

cos6)

sin6 sin

ç§]

ut

u(

=

~[sin

_{ç§(1 cos}

6)

sin 6 _{cos ç§],}

where _v is the Fermi

velocity.

The

purely phenomenological assumption

_now, is that

longitu-

dinal

(transverse) components

relax

according

to diflerent

scattering

rates

Wi(t~(cos6).

^The

collision term _is then

~[(l /rtr)Ai

_cos

ç§ +

(1/rH)At

sin

dl, (9)

where

1/rtr(Hj

"

Î d6%(tj Il cos6)

and

Ai(t)

^{is the} component of A which is

parallel (per- pendicular)

_to the electric field- A natural choice for A is A

=

aiE

+

at(wc

_x

E)-

Then the Boltzmann

equation

is satisfied with _ai

" rtr

Ill

₊

w)rtrTH)

and _at

" airtr (26] ^The deviation

of the distribution function from

equilibrium

is then

à

/k lTtrv

E + _TtrTHV

jwc

X

E)1

~~

~S _Ii jio)

+

~?TtrTH)

The

conductivity

tensor follows

immediately

_as

"

1+ÎrtrrH -uÎrH ~Î _~'

~~~~

where _{ao cc rtr.} This formulation _is consistent with the

expenmental

results

if1/rtr

_cc T and

1/rH

_cc

^T~.

As

satisfactory

_as this

phenomenological approach

_may be for the Hall

eflect,

it

presents

a

difliculty

for the

magnetoresistance

which is also unusual in the normal state. Of _course, very close _to

Tc,

the

resistivity

decreases below the linear-in-T law due to

superconducting

fluctuations

("paraconductivity")

This _is

suppressed by

_a

magnetic

field which results in _a

positive magnetoresistance. However,

here _we discuss

only

the behavior at temperatures ^above this

magnetoparaconductivity

_regime.

Experimentally,

_in 1-2-3 there is _a

positive magnetore-

sistance which _{varies as}

1/T~,

thus _as

cot~ 6H.

In

2-1-4,

the

positive magnetoresistance

has _a

dependence

which is consistent with

cot~

_6H

provided

cot 6H has the behavior _a +

bT~-

This is the behavior

actually

observed in Hall

angle

measurements _on

2-1-4,

^{which show} a

positive intercept,

attributed to

impunty scattering,

_{on a}

cot6H

_us.

^T~ plot [î]-

^A calculation of the

magnetoresistance

on the basis of the two-relaxation time

assumption

^{discussed above}

gives

the result

~~

~

=

~~~~~~~ ~~~ ⁽¹²⁾

p 1+ ~a~r~

Of _course, for _an

isotropic

Fermi surface and _a

single

relaxation

time,

the

magnetoresistance

vanishes

[26].

^For an

anisotropic

Fermi surface and _a

single

relaxation rate, the

magnetore-

sistance _is

positive

and

proportional

to the _square of the relaxation

time,

at small

magnetic

fields- For _a Fermi

liquid

with the observed T-hnear

transport scattering rate,

we would _ex-

pect Ap/p

_cc

_r/~

_cc

T~~, _contrary

_to the

experimental

result another indication of non-Fermi

liquid

behavior- In the

present phenomenology,

when the two relaxation rates have diflerent

temperature dependences,

_a

change

_in

sign

of the

magnetoresistance

is

predicted

at the tem-

perature

at which the two rates _are

equal.

It is therefore diflicult to reconcile this

simple analysis

with the

above-quoted expenmental results,

if _one

adopts

the

temperature depen-

dences

1/rH

_cc T~ for1-2-3 and

1/rH

_{cc a} +

bT~

_for

2-1-4,

which _are consistent with the Hall eflect.

However,

the behavior of the

magnetoresistance

_may be

essentially

determined

by

the

2196 JOtÎRNAL DE

PHYSIQUE

I N°12

Table I- Power iaws _m

charge transport

High-T~

Fermi Fermi

liquid

experiment liquid

with

1/r

_cc T

pab T

1/r

_cc

TP,

_p / 2 T

cot 6H

T~ 1/r

T

1/RH

_cc

1/p~y

T const const

~ip/p

_> 0

T~~ r~ T~~

shape

of the Fermi surface and its

proximity

to van Hove

singularities

so that this issue _re- mains unresolved at present. ^As an aside, _{we can} remark that the

positive sign

of the measured

magnetoresistance

is contrary to what _one would

expect

if the transport was dominated

by

carriers

scattering

from

spin

fluctuations.

It is instructive to summarize the unconventional

in-plane transport

in Table I- The first column

gives

^the

experimental

_power laws for the

in-plane resistivity,

_pab, the

cotangent

of the Hall

angle,

cot

6H:

the Hall constant,

RH,

and the

magnetoresistance, /lp/p-

The second grues the Fermi

liquid prediction

for _a conventional

[15]

electron-electron

scattering

rate

varying

as

T~-

_{The third column is the}

prediction

for _a Fermi

liquid

with _an unconventional

scattering

rate

varying

_as ^T. It is _seen that all the

experimental

_power laws _are anonialous.

2.3. PSEUDOGAP BEHAVIOR. The unconventional

transport properties

_are

quite

universal and _{are a}

challenge

for any

microscopic picture

of the normal state of the

cuprate

supercon-

ductors- At the _same

time,

there _{is an}

interesting regularity

which is _seen in _a number of

experiments

on diflerent

compounds, especially

in the

underdoped regime-

^There _appears to be _a characteristic temperature ^T* >

Tc,

a function of hole concentration, above which difler- ent

properties

all

change

their behavior- For

example _I?i,

in

La2-~Sr~Cu04 (2-1-4),

the Hall

constant becomes

essentially temperature independent

above T*, which is about 500 Il _at op-

timal

doping ix

_m

o-là)

and _even

higher

for

underdoped samples.

^A

comparable temperature scale,

with the _same

doping dependence,

is _8een in the electron

spin susceptibility

_as measured either from the bulk

susceptibility

_or from the NMR

Knight

shift. In

magnetic

measurements on 2-1-4, T* _{appears as} the

temperature

at which the temperature

slope

of the

susceptibil- ity, d~/dT, changes sign

from

positive

to

negative

_as the temperature ^is increased. A similar set of T*'s is _seen in transport,

magnetic

_resonance,

thermodynamic-

and neutron

scattenng experiments

on the

yttrium compounds,

both 1-2-3 and 2-4-8

(1"Ba2Cu408).

In

general,

T*

decreases with

doping

and there is _no condusive evidence of its existence at all _in

overdoped

materials- A summary of

expenmental

results is

given

_in

Figure

2- This

regularity,

that all these determinations of T* fall _{on a}

single

_{curve as a} function of hole

doping

_z, as shown in the

figure,

_was first

pointed

out

by Hwang

et ai. I?i ^{and sumnianzed}

by Batlogg

et ai.

[27].

This behavior _is sometimes ascribed to the

development,

at

T*,

of _a

"spin gap" (actually

_a

pseudogap)

_in the

spin

and

charge

excitation

spectra

_[28]

2.4. DUT-OF-PLANE RESISTIVITY. The transport

properties perpendicular

to the CUO

planes,

the ^'~ _c-axis

transport,"

also exhibit

puzzling

features

là, 29].

^The

anisotropy

of the

resistivity

has been studied

intensively

from the

beginning

of

high-Tc

research and _a

particu- larly

dear

example

for

single-crystal

1-2-3 was given

early by Schegolev

et ai-

[3].

^For

samples

at

optimum doping

or

less,

the _c-axis

resistivity invanably

shows _an

upward

curvature _{as a}

N°12 NORMAL STATE IN

HIGH-T~;

MARGINAL FERMI

LIQUID

219î

m 214 Hall Eflect

~

. 214 Resistivily

A 214NMR

. .

~ 214

+ . . * A

Susceptibility '

+ n 123 Resistivity

+ .

.

.~

_. O 123NMR

~ .

+

* _m

n e

n

~

~

n

+ .

Ôo

A m . . .

0 O.05 D.1 0.15 0.2 0.25 0.3 0.35

Hole Concentration, _X

Fig.

2.

Doping dependence

of character18tic

p8eudogap

temperature.

function of

temperature,

and

indeed,

for

underdoped samples,

_a

negative temperature

coeffi- cient of resistance in the region

just

above

Tc.

In

general, dp/dT

< 0 in all _cases

except

for

overdoped 2-1-4, optimally doped

1-2-3 and thallium 2-2-1-2.

In summary,

then,

the

in-plane resistivity

_{pab appears} metallic while the

out-of-plane

_re-

sistivity

_{pc appears}

semiconducting-

This is sometimes considered to be

paradoxical

and has also been viewed _as evidence of non-Fermi

liquid

behavior

[30].

^{One of the} main issues is

whether _{one can} have _a metal-insulator transition in _one direction and not in the other

"anisotropic

locahzation" _as has been

recently proposed [31,32]-

Within the

scaling theory

of

localization,

for _an

anisotropic

Fermi

liquid,

it is known [33] ^{that this} is not

possible.

The observed behavior is of _course cut off at low temperature

by

the

superconducting

tran- sition which is not

anisotropic-

^Therefore it is not dear whether the _true low

temperature

behavior is metalhc extended states _in all

directions, insulating

localized states _in all

directions,

_{or a new} state unforeseen in the

scahng theory

of locahzation [33]

It appears ^to have been

generally

overlooked that _a

negative temperature

coefficient of resistance

(dp/dT

_<

0)

is not

necessanly

_an indication of

insulating

behavior at the lowest temperatures. ^It was

pointed

out in 1980

by Imry

[34] ^{that close} to the Anderson metal- insulator transition, b~lt _on the metaihc side, the charactenstic correlation

length ( gets large

and that at not too low

temperature

^{the inelastic}

scattenng

mean free

path iin

_<

(.

Then the

temperature dependent resistivity p(T)

_cc

1,n(T)

and

consequently dp/dT

_< 0- In

fact, Imry proposed

an

interpolation

formula for the

conductivity:

~ ~~~~~

In

~ ~

2198 JOURNAL DE

PHYSIQUE

I N°12

Fig.

3. Feynman

graph

for MFL self energy. The _wavy Iine is the anomalous MFL collective mode.

The localization

length (

_can be

highly anisotropic

in

magnitude,

but

according

to the

scaling theory [33],

^{it becomes infinite for all directions at the} same value of disorder. This

point

of view has been

recently

revived and

apphed

to the

high-Tc superconductors by

Zambetaki et ai.

[35],

^who argue that the

cuprate superconductors

_are

highly anisotropic

metals which

exhibit _a

negative temperature

coefficient of _c-axis resistance close to

T~.

Another

explanation

was

given

earlier

by

Kotliar et ai.

[36]. They argued

that the non-Fermi

hquid

character of the normal state enhances the

impunty scattering

to such _a

degree

that for moderate amounts of

disorder,

the zero-temperature ^{state is}

insulating

in all directions.

The observed behavior above the

superconducting

transition is then _a consequence of _a

highly anisotropic

localization

length.

This will be discussed further in the next section.

A successful

theory

of the

high-temperature superconductors

must account for the normal

state

properties

which _are,

indeed,

more

non-Fermi-liquid

like than the

properties

^{of the} su-

perconducting

state- The anomalous behaviors of the Hall eflect and

magnetoresistance,

the existence of the _crossover

temperature

^{T* and the}

anisotropy

of the

transport

remain central issues for any

theory

of the normal _state of the

cuprate superconductors-

3.

Marginal

Fermi

Liquid Phenomenology

The idea of _a

"marginal

Fermi

liquid (MFL)"

_was introduced in _a series of _papers [37] ^several

years ago. The basic

assumption

is that there exists _an anomalous

charge

_{or spin response}

(or both)

of _a metalhc

system.

The

hypothesis

is that _{over a} wide _range of momentum, the energy scale for the

low-energy partide-hole

excitations _is set

by

the temperature- ^The

spin- spin

_or

density-density

_response function _was assumed to be of the

"glassy" form,

with

spectral

distribution

~~

~~~'~~

'~

-ÎÎÎÎÎ~~'

~a~ >

~ÎÎÎ

~~~~

The upper

cut-off,

_~a~,

plays

_a role _in the

analysis

of the

expenmental

results-

Typically,

for

a Fermi

liquid _[15],

Zm

P(q,~a)

r~

~a/vfq

for _~a < ufq so that the charactenstic

scale,

for q

in the

major part

^of the Brillouin _{zone, ie-} of order

kF,

is the Fermi energy

uFkF/2.

The MFL

susceptibility,

_{in contrast,} has _no characteristic _energy

scale,

other than the

temperature- Scattenng

from this anomalous mode grues a contribution to the retarded

single-partide

self energy

(see Fig. 3)

which is

L~(~a)

₌

Li(~a)

+

iL2(~a)

₌

À~alog

^~

i~~z, _(là)

~a~ 2

where _x

=

max(jwj,T)

and is _a

coupling

constant. The term

"marginal"

arises because the

quasipartide

residue _at the Fermi surface which is given

by

Z

=

(1- ôLi/ôw)~~

r~

log(w/~a~)j~~

vanishes

loganthmically

at low

temperature

and

frequency.

This result has _a number of

expenmentally significant

consequences. In

particular, _transport properties

_are

strongly

aflected

by

the

imaginary

part,

L2

of the self energy. Just _{as in a} Fermi

liquid,

_we

expect

^{that the}

temperature dependence

of the

resistivity,

for

scattering independent

N°12 NORMAL STATE IN

HIGH-T~;

MARGINAL FERMI

LIQUID

2199

of momentum

transfer,

_is that of

L2.

Thus for _a

MFL,

the

resistivity

is

linearly proportional

to

temperature,

as is observed at

optimum doping

_in most

cuprate superconductors.

The MFL

phenomenology

also leads to _a natural

explanation

of the

resistivity anisotropy

discussed at the end of the

previous

section. The

argument

_[36] rests on the renormalization of the

impurity scattenng potential

which arises because of the interactions which lead to MFL

behavior. The scalar vertex which renormalizes this

coupling

can be related to the

density

response function _as follows:

By Kramers-Kronig,

the real part ^{of the} response function _is obtained from

equation (14)

_as

Re

P(q, w)

_r~

log

^~~

(16)

z

Therefore,

the _zero

frequency

_response is

P(q,0)

_cc

log(wc/T)

_cc

1/Z.

At the _same

time,

this

can be related to the _zero

frequency

scalar vertex

À(q, 0)

_since

P(q, 0)

=

À(q, 0) ~j _Ç(k

_{+ q,}

_{w)Ç(k, w), (17)}

k,w

where

Ç(k,w)

is the

single partiale propagator.

Thus _we condude that for _a range of _q not near zero that the scalar vertex for the MFL is enhanced _as

À(q, 0)

_cc

1/Z-

It is

just

the scalar vertex

À(q, 0)

which renormalizes the elastic

scattenng impunty potential V(q).

It then follows that the

impurity scattenng

rate,

given by

1/Tik)

+~

~j lÀ(q,0)l~iq)l~çlk

₊

_{q,0), (18)}

grows ^at

sufliciently

low

temperature, leading

to

unitary scattering

and the

possibility

of la- calization.

It is

only

when the inelastic

length

exceeds the localization

length

that the diflerence between metal and insulator becomes

apparent [38].

^{If the} localization

length

is

highly anisotropic,

we

have the

following picture:

^{As the}

temperature

is

lowered,

the short localization

length

in the _c- direction becomes shorter than the inelastic

length

and _an upturn in the

resistivity

is

expected

(insulating behavior). However,

the localization

length

in the

ab-plane

is much

larger

than the inelastic

length

and the

in-plane transport

_appears metalhc until much lower

temperature,

below the

superconducting

transition.

Thus,

while influence of disorder is

yet

to be

fully understood,

it appears

[3î]

^that disorder

is

reieuant,

in the _sense that the correlations which lead to MFL behavior _are such that _as the

temperature

is

lowered,

the effective

scattenng

rate mcreases which _can

ultimately

lead to

localization, barring

the intervention of

superconductivity.

There is _{now some} evidence

obtained

by high magnetic

field

expenments

[39] ^which

supports

^this

picture.

Another test of the

hypothesis

is obtained

by comparison

to

experimental

results _on the

opti-

cal

conductivity

_[8] of electrons _moving in the copper oxide

planes

of

high-Tc superconductors.

For _a Fermi

liquid,

the effective Drude

conductivity

has the form

~2

~ T(Ld)

aju~)

_{= ai} +

ia2 (~~)

"

#

m*(u~) li

_iu~Tl~d)1 ^'

where _{wp is} the band _mass

plasma frequency

and _we have

generalized

to the situation of

frequency dependent

relaxation rate and effective _mass. There have been _vanous

essentially

unsuccessful

attempts [40]

to fit this form _to

expenment.

The

conductivity

data

invariably

show _a

high frequency

tail which falls off _even slower than

1lu

which would be the behavior

expected

from the

generalized

Drude form with

1/r

_{cc ~a.} Such _{an ~a}

dependence of1/r

_is

2200 JOURNAL DE

PHYSIQUE

I N°12

sooo

~

j~~

-1)

3000

2000

iooo

1000

tÔ (Cm ~~ )

Fig.

4. MFL

theory (solid Iine)

and expenment

(dots)

for the R

= Mai

la2

as a function of

frequency (in

^cm~~

).

consistent with the observed T-linear behavior of the de

resistivity

and the MFL form of the self energy in

equation (15).

For the

general

_case of _a

moment~lm-independent

self energy such _as in the MFL

picture,

the

conductivity

is

given by

_[41]

~~~~ ~ÎÎ lu ÎÎ

~~~~

Î~~ ^LR(x

+ ~a)

LA(x)

_v

LR(x)

~(x

w)

Î

' ~~~~

where the

superscripts

indicate retarded and advanced

[sgn(Zm L~)

=

-sgn(Zm L~)

_{< 0]}

sectors for the self _energy.

As _an

example

of the _success of the MFL

phenomenology

for transport, we discuss the fit [41]

to the

optical conductivity

data of Baraduc et ai.

[42]

^{It is convenient} to define the ratio

R =

1 (21)

which is

independent

of the

plasma frequency

and which in Drude

theory, equation (19),

is

simply

the relaxation rate

1/r.

For _a

MFL,

this ratio may be calculated

using

the MFL self energy;

equation (là)

_in the _expression for the

conductivity, equation (20)

The

comparison

of the

theory (sohd line)

and the expenment

(dots)

is shown _in

Figure

4. The data _was taken at _room

temperature

and _we have chosen _a cutofl of 8000

cm~~

_and

a

couphng

constant of o-à- With the _same parameters and _a

plasma frequency

of 24000

cm~~,

_we find _an

equally

excellent fit to ai

lu).

We show this _in

Figure

5.

Although

the

temperature (cf

200

cm~~)

_is

small

compared

to the

frequencies,

it is necessary ^to indude it to achieve such _a

good

fit to the

N°12 NORMAL STATE IN

HIGH-T~;

MARGINAL FERMI

LIQUID

2201

8.5

8

.

ln(ô~)

^{7 5}

7

6.5

6.5 7 7.5 8 8.5

In(m)

Fig.

5. MFL

theory (sohd Iine)

^and experiment

(dots)

for trie real part ^{of trie} optical conductivity

as a function of

frequency (in cm~~).

data. A recent fit to this _same data

using

_a

Luttinger-Iiquid-Iike

fractional _power law behavior has been made

by

Anderson

[43]. However,

the

upward

curvature in the

plot

for the ratio R is trot

obtained,

_{nor are}

temperature

^{eflects induded-}

The successful fit _to the

optical conductivity

of the 1-2-3

samples

of reference [14] encourages further

development

of the MFL

phenomenology-

4.

Summary

Recent

experimental

results _on the normal state of

high-Tc superconductors

have demonstrated

a number of

non-Fermi-liquid

like

eflects, especially

in the

transport properties- Furthermore,

there _are similarities from

property

to

property

^{and from material} to material which

indicate,

near

optimal doping

and

below,

a characteristic modification of the electronic

spectrum

as the

temperature

^{decreases- Neither this feature} nor the unusual _power laws _{seen in} transport fall within Fermi

hquid

behavior- Furthermore, _no

complete

theoretical

explanation

has been

given

for all these behaviors-

From _a

phenomenological point

^of view, the

marginal

Fermi

hquid hypothesis

leads to _a

description

of _some transport

properties

which _are in remarkable accord with

expenment, although

some transport

phenomena

do _not

yet

_appear to fall within the framework _as

presently

constructed.

2202 JOURNAL DE

PHYSIQUE

I N°12

Acknowledgments

The author is indebted to _a number of

expenmental colleagues

for their

insights

into the

regulanties

summarized in Section 2- B.

Batlogg

and N-P-

Ong

_are

especially responsible

for

originating

such _an

approach-

This work has been

partially supported by

NSF

grant

^DMR

92-2190î.

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P.W.,

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For

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electron-

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resistivity

and the

high

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_see Littlewood

P-B-,

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also Girvin S-M- and Jonson

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G.,

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^{See for}

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Thomas

G.A-,

Orenstein

J., Rapkine D.H., Capizzi M.,

Milhs

A.J.,

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Schneemeyer

L.F. and Waszczak

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Z- and Collins

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Reu. Lett. 65

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801-

[4l]

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[42]

^Baraduc

C.,

El Azrak A. and

Bontemps

N-, J-

S~lpercond~lctmity

8

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[43]

^Anderson

P.W., cond-mat/9506140.