NMR Studies of the Normal State of High Temperature Superconductors

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NMR Studies of the Normal State of High Temperature Superconductors

C. Berthier, M. Julien, M. Horvatić, Y. Berthier

To cite this version:

C. Berthier, M. Julien, M. Horvatić, Y. Berthier. NMR Studies of the Normal State of High Temperature Superconductors. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.2205-2236.

�10.1051/jp1:1996209�. �jpa-00247308�

NMR Studies of trie Normal State of High Temperature Superconductors

C. Berthier

(~'~~*),

^{M.H. Julien}

(~'*),

M. Horvatié

(~)

and Y. Berthier

(~)

(~) Laboratoire de

Spectrométrie Physique ("),

Université

Joseph Fourier,

Grenoble I, BP 87, 38402 Saint-Martin d'Hères

Cedex,

France

(~) Grenoble

High Magnetic

^Field

Laboratory,

Centre National de la Recherche

Scientifique

and Max Planck Institut für

Festkôrperforschung,

BP 166X, 38042 Grenoble Cedex 9, France

(Receiied

19 July1996, received in final form 29

August

1996, accepted 30

August 1996)

PACS.74.72.-h

High-Tc compound

PACS.76.60.-k Nuclear

magnetic

_resonance and relaxation

Abstract. We _review NMR and

NQR

studies

m trie normal state of

high-Tc superconducting

cuprates.

Emphasis

is

given

_on the three major contributions

brought by

these

techniques:

the _presence of

antiferromagnetic

fluctuations, trie possible

justification

of _a

single

_spin fluid

model,

and trie

opening

of _a

pseudogap

in trie spin excitations for underdoped compounds.

Some recent

developments

_are addressed and compared to theoretical models. In particular,

quantitative

analysis

of the nuclear relaxation _are detailed for

~~Cu,

~~Y _and ~~O _nuclei,

enabling

a companson with inelastic neutron scattering data, and

finally pointing

out to _a serious

difliculty

with the one-band approach.

1. Introduction

Besides the

appealing perspectives opened by

the

discovery

of _{a new} Mass of

superconduc-

tors

[Ii

~v-ith values of

Tc

above the temperature of

liquid nitrogen,

one of the

great

^merit of the

discovery

of the cuprate

high Tc superconductors (HTSC)

has been to

highlight

the

peculiar

role

played by

the

dimensionality

D

= 2 in solid state

physics.

Can _one

explain

the _{many un-} conventional

properties

of the cuprates,

including

the

high

value of

Tc,

_as natural consequences

of the _presence of Van-Hove

singularities

in the

density

of states _or

antiferromagnetic (AF)

fluctuations [3,

4],

ⁱⁿ a quasi two-dimensional

(2-D)

Fermi

hquid,

_or do

they

_{require a more} fundamental reexamination of low dimensional solid state

physics,

_i.e., the

possible

extension to 2-D

systems

^of new concepts

strictly

denved for _one dimensional

physics [5],

like

spin-charge

separation?

Ten years after the

discovery

of

HTSC,

these

questions

are still open, in

spite

of

a

huge

amount of

expenmental

work

involving

all

techniques

available _in sohd state

physics.

A

microscopic description

of the

quasi-particles

present ^{in the} normal state

(T

_>

Tc)

of these

strange

^metallic

compounds

_is still

highly controversial,

_{as is} the mechanism

responsible

for the formation of the

superconducting ground

state [6].

However,

it is

generally agreed

that the basic

ingredient

of the HTSC is

Cu02 Planes

with _{a square} lattice

arrangement

^{for Cu}

(*)

^Authors for correspondence

le-mail: [email protected]; [email protected]) (*)

UMR CNRS 5588

©

Les

Éditions

_de

Physique

1996

T

"

vnderdoped overdoped

2 % 16 % hales /

Cuoi unit

Fig. 1. Schematic phase

diagram

for noie doped cuprates. ^The _presence ^of a pseudogap phase is

supported by

_numerous experiments.

and O atoms,

separated by

functional blocks

playing

the role of _a

charge

reservoir- In the ab-

sence of hole

(or electron) doping,

the

Cu02 Planes

form _a

good representation

of _a

quasi-2D

Heisenberg antiferromagnet (La2Cu04, YBa2Cu306)

with _an

insulating

Néel

ground

state.

By introducing

holes _in the

Cu02 Planes

trie Néel

temperature

is

rapidly decreased,

before

producing

_{a new} "metalhc"

phase

with _a

superconducting ground

state I?i ^The value of the

superconducting

transition temperature first increases with the hole concentration _nh in the

Cu02 Plane (underdoped regime),

reaches _{a maximum} value for _an

optimal doping,

and then decreases with _a further increase of _nh

(overdoped regime)

_[8]- The full evolution

according

to this ideal

phase diagram (Fig. l)

is not

usually

observed in _a

single system.

For

example,

_a

variation _{in z} going from 0 to 1 in

YBa2Cu306+~

shows _an AF

phase

for _{x <}

0.4,

covers the

underdoped

_regime from 0.4 to 0.94 (T~ = 93

K),

and has _a small range of concentration in the

overdoped

_regime.

La2-~Sr~Cu04

_{as a} function of _x exhibits first _an AF

phase, separated

from the metallic _one

by

_a

spin-glass regime.

The

optimal doping

_is around _x

= nh = o-là and the

overdoped

_range is diflicult to obtain in stable form [9]- ^On the

contrary, T12Ba2Cu06+à only

exists in the

overdoped regime,

with T~ values

ranging

between _m 85 to 0 K.

Nowadays,

there

is more and _more

experimental

evidence that the

underdoped

and the

overdoped regimes

have

quite

^diflerent electronic

properties [10-12].

Indeed, this _was inferred _very

early [13,14],

from the

experimental

data obtained

by

Nuclear

Magnetic

Resonance

(NMR)

in various HTSC _sys-

tems- This

technique,

which

played

_a crucial role in the test of the

Bardeen-Cooper-Shriefler

(BCS) theory

for conventional

superconductors [là,16],

has been

extensively applied

to the

study

of the HTSC

during

the past decade and has

proved

to be _a

major

tool for

investigating

the low energy excitations in these

compounds-

The

goal

of this _paper is to review the current

knowledge

of NMR

experimental

data in the

cuprates

^and to discuss how

they put

constraints _on the various theoretical models

competing

for the

description

of the

quasi-partides

which _are

responsible

for the

superconducting ground

state. We shall limit ourselves to the norràal state

(T

_>

_T~)

of the metallic part of the

phase diagram.

For NMR studies of the

antiferromagnetic phase,

and that of the

superconducting

state, we refer the reader to other review papers

[17-22]

The _paper will be

organized

_as follows: _in Section

2,

~v-e review the NMR

background

_necessary to appreciate the contribution of NMR _{to our} better

understanding

of the

physical properties

of the cuprates- Section 3 will be devoted to the notion of

single spin fluid,

which has been

supported by

NMR, to the Mila-Rice Hamiltonian

commonly

used _to describe the Hamiltonian of the various nuclei in the cuprates, ^and to the

description

^of the

magnetic hyperfine couplings.

In Section

4,

we review the

experimental

results obtained

by

nuclear _spin lattice and

spin-spin

relaxation measurements, and discuss _current

interpretations

in the hterature, within the framework of the

nearly antiferromagnetic

Fermi

liquid,

and the quantum ^cntical regime of disordered 2D

Heisenberg antiferromagnets-

Section 5 is devoted

entirely

to the

question

of the

spin

_gap

in

underdoped _systems,

which _was first revealed

by

NMR

experiments,

and has become _a central issue in the

understanding

of electronic and

magnetic properties

of the normal _state of HTSC- In Section 6, we discuss the companson between NMR and inelastic _neutron

scattenng

data and ho~v. it affects the

interpretation

of the NMR data. Section 7

briefly

mentions the NMR data concerning the

problems

of

phase separation.

In the

concluding remarks,

_{we try}

to summarize the contribution of NMR to the

present understanding

of the

properties

of the normal state of HTSC, and _to stress upon the _open

questions

which should be addressed in _a

near future.

2. NMR and

NQR Background

Nudear _{spins in} solids _are submitted to various

couphngs

due to their environment, formed

by

locahzed and delocalized electrons in the structure. Their total Hamiltonian _can be _wntten

as:

~total

"

~zeeman

₊

~Q

+

~mhf

₊ ~nucle>-nucle>

(Î)

where the _vanous terms are

respectively

the Zeeman term, the

quadrupolar

term, the

magnetic hyperfine interaction,

and the nuclear spin-spin interaction.

NMR

usually

refers to the _case where the _main Hamiltonian _is the Zeeman

coupling

of the nuclear _spins I _to the external

magnetic

field H

=

Hok, 7izeeman

=

-~ih ~j _I]Ho,

_where

~i is

the

gyromagnetic

ratio of the isotope 1- In the absence of any other

perturbation,

i the Zeeman Hamiltonian grues rise to a

single

hne

spectrum,

at the Larmor

frequency

_{mn =}

2ir~iHo.

2-1- ELECTRIC FIELD GRADIENTS- For _spin I _>

1/2,

and non-cubic local

symmetry,

the

largest perturbation

to the Zeeman Hamiltonian _is

usually

the

quadrupole

Hamiltonian

7iQ,

which

couples

the

quadrupole

moment

Q

of the nuclei to the electric field

gradient (EFG)

due to the

surrounding

electronic

charge

distribution. Note that

Q

is

specific

to each

isotope.

The EFG tensor is _a second rank tensor which

usually strongly

reflects the local

symmetry

^of

the environment- If _we call

~[~

the

components

^of the EFG tensor

along

its

principal

_axes

la

=

X, Y, Z), 7iQ

can be

expressed

_as:

7iQ

₌

~~~

(Il ^~~~

^{~ ~~ + ~}

^II(

⁺

^I~ )1(2)

2 3 6

where vQ is called the

quadrupole frequency,

_~

=

~~~ ~~~

is the

asymmetry parameter Vzz

with the convention

jvzzl

>

[Vyyj

>

[Vxxl,

and vQ ^is related to the

largest component

of the

EFG _tensor

by

_vQ

=

~~~~~~

In _a

high magnetic field,

the

quadrupole perturbation (for 21(21-1)h

odd values of

21)

leads to a

splitting

of the _resonance into _a central line

(m

=

-1/2

₊₊

1/2

transition)

and satellites _on each side of

it,

the value of the

splitting being

related to both vQ ^{and the} orientation of the externat

magnetic

field with respect ^{to the}

principal

axis of the

EFG tensor. In the _case where vQ ^is

large,

the spectrum for unoriented

powder samples

is _a

complicated pattern [23, 24],

which

usually

inhibits useful studies-

Fortunately,

most of the

cuprate powder samples

_can be oriented with the

application

of _an external

magnetic field,

so that the c-axis

(perpendicular

to the

Cu02 Plane)

of each of the grains

(which

have to be

single-crystals)

is

aligned along

the field direction

[25]. Working

_on such oriented

powders

with

Ho II

c-axis _is

equivalent

to work _on

single crystals-

When

Ho

is

perpendicular

to the c-axis, two diflerent situations _are encountered: if there _{is no} detectable

asymmetry

^of the tensor in the

ab-plane (which

is the _case for the

planar copper)

the

spectrum

is

again

similar to that obtained in _a

single crystal. However,

when the

largest component

of the tensor lies in the

ab-plane (copper

in the chains of

YBa2Cu306+x, Planar oxygen)

_one obtains _a broad 2D

powder

_spectrum-

In the absence of both external

magnetic

field and spontaneous spin

polarization, 7iQ

be-

comes the main Hamiltonian and transitions _can be induced between its

eigenstates-

This

situation _is called Nuclear

Quadrupole

Resonance

(NQR),

_a

technique

which has been

widely

used in the

study

of cuprates,

namely

for Cu and La nuclei

(the quadrupole coupling

at the oxygen sites is too small _to allow _a direct detection in _zero external

magnetic field)-

One

advantage

of

NQR

is the

possibility

to follow the nuclear

spin-lattice

relaxation time

Ti

in _zero

field, avoiding complications

related to the presence of vortices _in the mixed

phase.

The

knowledge

of the

quadrupole

tensor for each

particular

nudear site

(when

it

exists)

is a

prerequisite

to

study

the

7imhf

Hamiltonian- Studies of the EFG tensor

provide

valuable information about the site

symmetry

and the

degree

^of disorder _in the local environment

[26, 27]

A recent

development

is worth

being

noted: thanks to the

increasing

number of NMR studies

in various

systems,

^it seems that all

compounds having

a

high (or potentially high)

T~ show

systematically relatively

_{low vQ} values for Cu in the

Cu02 Planes.

This

behavior,

first noted

by Zheng

et ai.

[28],

^has been

tentatively

ascribed to a

higher in-plane

Cu-O

hybridization.

Such _a conclusion _seems to be

supported by

recent NMR results in

Hg-based

_cuprates where the

buckling

of the

Cu02 Planes

is

negligible [29, 30]. However,

the

respective weights

of the electronic on-site and lattice contributions to the EFG _are not well

understood, making

this issue still controversial.We shall not discuss the EFG tensors _{in more} detail

here;

the reader is

referred to _review [19] or

specialized

_papers

[31,32], dealing

with this matter.

2.2. MAGNETIC HYPERFINE SHIFTS.

Here,

_we shall concentrate _on the

magnetic hyperfine

interaction. The net result of the Hamiltonian 7imhf can be considered _as the interaction of _a

nuclear

spin

with _a time

dependent

iocai

magnetic hyperfine

field

hL

due to the electronic

spins

and orbital motion. The static

(time-averaged)

_part

hL gives

rise to _a shift of the NMR line

proportional

to the

applied

external field, which is denoted _as the

Magnetic Hyperfine

Shift

(MHS

tensor K. Its components can be

decomposed

into _a

spin,

an orbital and _a

diamagnetic

part:

KaalT)

=

Klt"

₊

_K]S~

₊

KΩ ₁₃₎

where _a denotes the _x, y, z directions. In HTSC

compounds, ^K°~~

is

temperature independent

and K~~~ is

usually negligible-

Well below

T~,

^K~P~"

vanishes,

_as the

signature

of

singlet

_spin

pairing

and thus, the

remaining

MHS tensor is the orbital _one. In the

following,

_we shall

only

consider the MHS tensor of the _copper and oxygen nudei in the

Cu02 Plane (usually

referred to _as

Cu(2)

and

O(2,3)

in the _case of

YBa2Cu306+~),

and that of the

yttrium

when

present

in between the

Cu02 Planes. (In

ail NMR

studies,

the apex oxygen nudei have been found to be

quite weakly coupled

to the _spin

susceptibility,

and therefore results

concerning

this site will not be discussed

here.)

For copper and

yttrium,

the MHS tensors have _an axial

symmetry

and the components will be denoted

K~

and

Âab

for

Ho Parallel

_or

perpendicular

to the

Cu02 plane [33].

^{For the}

planar

_oxygen sites, the main

component

^{of the MHS} tensor lies

along

the Cu-O-Cu

bond;

thus the

in-plane

components are denoted _as

Kll along

this bond and

Ki perpendicular

to

it,

and the MHS tensor _is

usually expressed

_as

(Kc, Ki

and

K# ).

The

spin part

of the MHS tensor _can be

formally

related to the average

susceptibility

_per

site

through

the

relationship:

Kaa

"

Aaaxaa /ga~B ₁₄₎

where

Aaa

_is the

hyperfine

field per spin unit

[34],

ga is the Lande factor for the

electron,

~B the Bohr

magneton,

^and a is the direction of the

magnetic field,

which _we

always

_suppose

to be

along

_one of the

principal

_axis of the MHS tensor- This

simple

formula hides

actually

a rather

complex reality-

One

ambiguity

in the

cuprates

^{is whether} one should consider the electronic

spins

as localized _as in

magnetic

insulators [35] or as itinerant _as in transition metals for

example _[36].

In both cases, the

couphng

between the nudeus and the electronic

spin

density

is rather

local,

and

usually decomposed

in several components

corresponding

to the

spin polanzation

of the _various atomic _wave functions _on which the whole many

body

electronic wavefunction is

projected.

For

example,

_in the _case of the

planar

Cu

site,

^the

spin part

^{of the} MHS is

expected

to be

mainly

due to the

spin polanzation

of the on-site

d~2-y2

^orbital.

However,

after _an

analysis

of the

experimental

MHS tensor

[37, 38],

^of the

planar

_{copper in}

YBa2Cu307,

it was concluded that _an

important

contribution _was due to _a

super-transferred hyperfine

field

proportional

to the _spin

polarization

carried

by

the nearest

neighbors Cu(2).

through

the oxygen

O(2p)

and the

Cu(4s)

wave-functions

[39].

^{As far} as the MHS tensor of the

planar

_{oxygen is} concerned, _it has _to be

analyzed

in _terms of

spin polanzation

of the _oxygen

O(2p)

and

O(2s)

orbitals

[40-42] Consequently,

after removal of the

dipolar

contribution due to the spin

density

_on the

Cu(2) sites,

the MHS tensor of _{oxygen is} written as:

~~K~P"'(6)

₌

~~K]$"'

+

~~K]("'

^~~

°~~~

là

where ~~K;~~ and

~~Kax

_are related to the _spin

polanzation

of the

O(2s)

and

O(2p) orbitals, respectively,

and 6 is the

angle

bet~v.een the

magnetic

field

Ho

and the Cu-O-Cu bond in the

ab-plane-

In

equation là)

_we

neglected

the small deviation from the axial

symmetry

which is revealed

by expenmental data,

_so that

expenmentally,

_{~~Ki~~ and}

~~Kax

are defined _as:

~~

~~ ~~~ ^~~ _{_~}

17j~"~~ ~~l~ll~/3, ^~~~~~ ~~~~~~ ~~~~

~

~~~~~~~~~~

_~~~

Finally,

the

yttrium

MHS _{tensor is} dominated

by

the

core-polarization

of closed

yttrium

shells

by

the _spin

density

at the

Cu(2)

site

through

the

O(2p)

orbitals

[43].

^{Similar situation} can be found for other nuclei hke

~~Ca, ~°~Tl,

_or

_~99Hg

in other

systems,

^but we shall not comment on them here.

2.3. NUCLEAR SAIN LATTICE RELAXATION. The nuclear

spin-Îattice

relaxation rate

(NSL

RR),

which is the rate at which the nuclear

magnetization

relaxes to its thermal

equilibrium

value

along

the external

magnetic field,

is due to the fluctuations of the

magnetic hyperfine

field

hL (the

fluctuations of the EFG tensor have been shown to have _a

negligible

contribution

in the normal state, at least for the nuclei _species which have been

studied). According

to

Moriya _[44],

the NSLRR rate _can be written _as:

~iTiT~a~ ii~

~&~ ^{A~a i~~~} ~Îi,llii~~

^17~

where _i stands for the nuclear

species,

_a for the direction of

Ho (supposed

to be

parallel

to one of the

principal

_axes of

Aa,a,

and

x[,~, tensors), ,4a,a,(q)

is the Fourier transform of the

hyperfine

field defined above

[Aa,a, (q)

=

~j _Aa,a, (R~) exp(-iq R~)],

and

~[,~, (q, mn)

^is

Ri

the imaginary

part

^of

dynamic spin susceptibility

for trie wavevector q and nudear Larmor

frequency

_mn

(hwn

_m

0)

for the direction a'

perpendicular

to a-

Again,

the _use of this formula

in HTSC

compounds

hides _some

ambiguity, depending

_on whether _one considers the electronic

spins

as locahzed _or itinerant. For

example,

in the _case of the _oxygen relaxation, the _use of the

traditional formulae for

(Ti T)~~

in metals

[35, 36]

would lead _to consider the fluctuations of the

isotropic part

^of the

hyperfine

tensor and that of the axial _{one, as} incoherent. On the

contrary,

in the

picture

^of localized

spins

at the copper site,

coupled

to the oxygen nudei

through

_a transferred

hyperfine field,

_A,~~ and

Aax

act

coherently. Obviously,

both

descriptions

lead to

quite

^{diflerent values for the}

anisotropy

of the _oxygen NSLRR.

Expenmental

data

[45, 46],

favor the localized picture-

2.4. NUCLEAR SAIN-SAIN RELAXATION. In most solids. the Hamiltonian 7inucie>-nucie> is

restricted to the nuclear

dipole-dipole

interaction. It thus

gives

use to a

temperature indepen-

dent

spin-spin

relaxation rate which often has the form

exp(-t~/2(T~G)~)

where

1/(T2G)~

is

equal

to the second-moment of the

line-shape- By designating H[~

_as the

part

of 7inucie>-nucie>

which connects states of the _same energy,

1/(T2G)~

can be evaluated _{as a} trace:

~~~~~~

h~Trlj

However,

indirect nudear

spin-spin

interactions _can be mediated

by

the electronic

spin

_suscep-

tibility _[36].

The group of Slichter

[4î]

was the first to

point

out the importance of T2G to obtain information about the

antiferromagnetic exchange

between the electronic

spins- Formally,

this type ^{of indirect}

spin-spin

interaction _can be wntten _as:

( 2jgaa~~)2 ^~~~° (, ~~"~~> ~~~~~

~

~~~~~J

~ ~~~

where

t(R R')

is the Fourier transform of the real

part

^{of the electronic}

spin susceptibility

~(q)- Later,

it has been shown that this

quantity

was

temperature dependent [48, 49],

and that the nudear

dipole-dipole

contribution _~v.as

neghgible compared

to that of the indirect

interactions. The

study

of the

temperature dependence

of

T2G

^{has become} a

major

tool in NMR and

NQR

studies of

HTSC,

since due to the presence of _a transferred

hyperfine

interaction

in the these materials

(see

Sect.

3.2.), _T2G

is able to

provide

information about the static

part

of the

staggered susceptibility x'(q

m

QAF,

_~a

=

0).

3.

Magnetic Hyperfine Couplings

3.1- A SINGLE SAIN-FLUID- In the first few _years after the

discovery

of the

cuprates,

there has been _an intense effort to find the

siniplest

model Hamiltonian which would descnbe the low energy

properties

of the

Cu02 Planes. Very early,

Anderson [5]

proposed

_an effective Hamiltonian known _as the t-J

model,

which

corresponds

to a

single

band 2D Hubbard model

in the hmit of infinite on-site Coulomb

repulsion

U. Whether such _a

simple

model could

keep

the essential features of _{a more}

general

Hamiltonian

containing

Cu and O bands and

intra- and interatomic Coulomb interactions has been _a

long

debate- To

justify

the t-J

model,

Zhang

and Rice [50] ^have

proposed

that the holes introduced

by doping

the

Cu02 Planes

would delocalize onto the four _oxygen sites around _a

Cu~+

_{ion and form}

a local

singlet

with it. This

singlet

would then _move

through

the lattice of

Cu~+ ions,

in a similar way as a

hole, i-e., a

missing spin,

in _a

simple

_square

^Cu~+

ion lattice. The

ability

of the t-J model to

cover the whole

physics

of the

starting

two-band Hamiltonian has been criticized

by Emery

and Reiter

[51],

^{who daimed that the two-band} Hamiltonian

implies

the existence of _a second

spin degree

^{of freedoni related} to the

O(2p) band,

which cannot be accounted within the framework of the t-J model- This

question

has

polanzed

the

interpretation

of the

temperature dependence

of the

magnetic hyperfine

shifts in

high Tc superconductors. Early

measurements of the anisotropic part ^{of the MHS at} oxygen sites _in

YBa2Cu307 (40-42],

which is

directly

related to the _spin

polarization

of the

O(2p) orbitals,

have shown that most of the total

spin polarization (> 85%)

_was carried

by

the copper sites in these

compounds-

As

regards

the

temperature dependence

of the

àIHS,

it _was first

reported by

Alloul et ai. [52] ^{that in the}

YBa~Cu306+~ system,

the MHS of

yttrium

_was

scaling linearly

the

macroscopic .susceptibility.

Since the _main

coupling

of the

yttrium

to the

magnetic susceptibility

in the

Cu02 Plane

_occurs

through

the

polanzation

of the oxygen orbitals this _~vas

interpreted

_{as a}

proof

of the existence of _a

single

_spin fluid

[53]

ⁱⁿ

agreement

^{with the t-J model- Soon}

after,

it _was sho~i>n

by

Takigawa

et ai. [54] ^that

~~Kab. ~~K;~~,

and

~~I[ax

had the _same

temperature dependence

in

YBa2Cu3 06.63 (Fig. 2) Subsequently,

this

behavior,

1-e., the _same

T-dependence

on ail nudear sites

coupled

to the Cu _spin

polanzation,

_was observed in all the

cuprates

where it _was studied

jyBa~Cu4Os _[55], La~-~Sr~Cu04 [56]

^for

example). Strictly speaking,

this behavior

by

itself does not exdude the existence of _a

temperature independent susceptibility

due to an

O(2p)

band. For

example,

the T

dependence

of the

Knight

shift in Platinum metal _varies

linearly

with the macroscopic

susceptibility, regardless

^of the presence of _{an s-p} band giving use to _a T

independent

contribution [57] ^The

vahdity

of the

single spin-fluid picture

in HTSC thus relies

on the

precise knowledge

of the orbital shifts of the oxygen, copper, and

yttrium

nuclei- Let _us consider for

example

the axial

part

^{of the} oxygen MHS. There _are three

possible

contributions:

one

proportional

to the

polarization

of the

O(2p)

orbitals

by hybridization

with the

Cu(2) orbitals,

_one

proportional

to the

susceptibility (supposed

to be

temperature independent)

of

some

O(2p) band,

and _a

(temperature independent)

orbital contribution:

~~I(ax(T) ~~l~const

"

2ÎPA~~'~~'~

"

~~~~~~~~~~ ~~~°~~~

^~ ~

~

with

~~Kconst

"

À~~~~~~~~+ ~~I~ÎÎ~

O.25

O.6 O.05

'~ K~

36~

0 2 ^ab ,~

~ " ~

ii

K ^"

% O.04 ,~

M

0.5

£~

~p

~/ ^~" f.

_

~ Q-15

§

~

G

£

_~ ^O.l

^~ié

~f

^~ _O.03 ^{. '~K}

~ ^{~ ~}

17~

~

l

C ax

0.4 °'~

~ ~~

D

'~K il

~~~ ~ ~

~ 63

~

" $£

ab ^-

0.02

0.05

0

0 A 1où 200 300

à

T(K)

Fig.

2. Various components of trie Cu and O

magnetic hyperfine

shift ten80rs in YBa2Cu306

63.

From

Takigawa

et ai. [54j.

~~Kax(T

=

o)

₌

~~K][~ ~~Kax(T

=

0)

=

~~I[orb

T _<

Tc

It is clear that the

single spin

fluid

hypothesis

_can

only

be validated if

experimentally,

~~K~~n~t _"

~~Ii[[~-

It _can be _seen in reference

[54]

^that

although

_some deviation from this

equality

_seems

to _occur, it is well within the _error bars. and leads to _a

negligible

value of x~°~~~ However,

one must remember that the determination of the orbital shifts _in the

superconducting

state is diflicult. because of the corrections due to the finite

magnetization

_in the mixed state and

the

change

in the

lineshape

due _to the _presence of _{a vortex} lattice

[16].

3.2. THE MILA-RICE

HAMILTONIAN,

HYPERFINE COUPLING TENSORS _AND FORM FACT-

ORS. The

smgle spin

fluid picture, and the fact that most of the spin

density

_is borne

by

the copper

sites,

has led to the

following hyperfine Hamiltonian,

_in which the

only

_source of

coupling

_are the electronic _spins localized at the _{copper site:}

~mhf

^"~~

l~ ~j _~~~~(~~~+ ~j _{l~Si+à)+ ~~~~} ~j _~~IjCSj+à,

₊

_{~~i~k ~} ~~IkDSk+à"

~ à=1-4 j,à'=1-2 k,à"=1-S

(9)

where the tensor A

corresponds

to the Cu on-site

hyperfine

field tensor, ^B is the

(isotropic) supertransfered hyperfine

field from the 4 _nearest

neighbors

_copper

sites,

C is the

(anisotropic)

transferred

hyperfine couphng

to the

planar

_oxygen from _{its two nearest}

neighbors

_copper

sites,

and D the

supertransfered hyperfine

tensor to the

yttrium

from the

eight

nearest

neighbors

copper ^sites

(in

the _case of

YBa2Cu306+~

and related

compounds)

This

Hamiltonian, usually

quoted

_as the Mila-Rice _or the

Shastry-Mila

Rice Haniiltonian [58] is

commonly

used in the

literature to

quantitatively analyze

the NMR data. The first

step

in the

analysis

of NMR data

usually

_concerns the determination of the

hyperfine

field.

However,

this _is not obvious in the

cuprates

for several _reasons- In the _case of the

planar

_copper, there _are three

quantities

to determine:

A~,

-4ab and

B, although experimentally, only

A~ + 48 and

Aab

₊ 48 _can be obtained from

equation (4)-

The determination of -4~ +

48, Aab

₊

48, Ca,

and

D~

is

more accurate in

compounds

where the

susceptibility

_is

strongly

temperature

dependent,

that

is, in

underdoped compounds.

Indeed, few expenments ^that include

susceptibility

and MHS measurements on the _same

samples

_can be found in the literature

[42, 56,

59,

60].

^In

^addition,

it is not obvious what

susceptibility

to _use in

equation (4)

in

compounds

hke

YBa2Cu306+~

where

part

of the

susceptibility

is carned

by

the copper ^atoms located _on the chains-

Usually,

the Cu atomic

susceptibility

_is obtained

by dividing

the total

susceptibility by

2 _{+ z} and

assuming

that _copper in the chains and in the

planes

bear the _{same spin}

susceptibility.

This

assumption

has been used _as well for the determination of

Aab

+ 48 [59] ^and

Ca~ [42]

^The

anisotropy

of the static

susceptibility

is

usually

not taken into account. As far _as A~ + 48 is

concerned,

it turns out that

experimentally

for

Ho II c-axis,

the MHS is found

independent

of temperature ^{and have} identical values in the metallic and

superconducting

state within the

experimental

uncertainties in

the1"Ba2Cu306+~, YBa2Cu40s, La2-~Sr~Cu204 compounds.

Thus

A~

+ 48 _m 0 is _an usual

assumption

in all these

compounds-

The _same is not true for

compounds

hke

T12Ba2Cu06+à _(61], T12Ba2Ca2Cu3010 (62], B12Sr2CaCu20s (63, 64],

and

HgBa2Ca2Cu308 _{(29, 30]-} By assuming

that

A~

and

Aab

_are intrinsic

properties

of the

planar

Cu _ion and

taking

the current values for

l'Ba2Cu306+~,

tentative values of B have been extracted from the ratio

(A~

+

48)/(Aab

+

48). They

_are

systematically larger

than _in the YBCO and LASCO

compounds,

which may indicate _a

larger

value of the

exchange coupling

in these matenals

[29, 30, 61,62].

This behavior is

usually accompanied by

_a smaller value of the

planar

_copper

quadrupole coupling,

_as

already

mentioned

above,

a feature which has been attributed to a

larger hybridization

between

d~2-y2

and

O(2p)

orbitals

[28, 29]-

^Both

phenomena

_seem to be controlled

by

the

buckling

of the Cu-O-Cu

bonding.

Finally,

the ratio

2C~/(Aab

+

48)

has been determined in _some

compounds YBa2Cu306.63 (0.58) _[54], YBa2Cu40s (0.67) _[55], La2-~Sr~Cu04 (0.68) _[56], T12Ba2Cu06+à (0.30) _[61],

T12Ba2Ca2Cu3010 (0.50) _[62]

_as well _as

(8Dc

₊

Dd>p,c)IOC(-0-127)

in

YBa2Cu30663 (65].

Typical

values of the

hyperfine

^field constants _can be found in references

[22, 66,67].

It must be

emphasized

that these values _are

approxiniate:

in the most favorable _cases

they

_are not known with _{an accuracy} better than

20%.

The

knowledge

of these

coupling

is of

primary

importance ^for a

quantitative interpretation

of the nudear

spin-lattice

and

spin-spin

relaxation data. In the

hypothesis

of _a

single

_spin

degree

^of

freedom,

the evaluation

of1/Ti

and

1/T2G

involves _a summation _over the Brillouin

zone of the

product

of the

imaginary

or the real part of the

dynamic

_spin

susceptibility

located at the

planar

_copper sites square with the form factors

F(q)

=

jA(q)j~

These form factors

F(q)

_can be

expressed

_as:

~~fab(q)

"

(Aab

+

28(cos

_{q~a + cos}

_qyb)]~ (10a)

~~Fc(q)

= [A~ +

28(cosq~a

_{+ cos}

_qyb)]~ (lob)

~~F~(q)

=

2Cj~[1

₊

(cos

_q~a +

cosqyb)/2] (10c)

~9F~(q)

=

8D(~(l

₊

cos

q~a)(1

+ cos

qyb)(1

+ cos

qzd) (10d)

where d is the distance between the two

Cu02 Planes

_in YBCO

compounds.

Note that for _q

=

QAF

"

(ir la,

_ir

la),

which is the AF wavector of the

undoped

_{parent com-}

pounds

of

YBa2Cu306+~

and

La2-xSr~Cu204, ~~Fa(QAF)

and

~9Fn(QAF)

_{are zero,} whereas

63F~(Q

_AF ^and

~~fab(QAF) _Present

_a maximum

(Fig. 3).

Let _{us suppose} that AF fluctuations.

1~~Ac(()

^~

~~

~~Aa(1)

^~

~~

Il 00

~

ll.00

~

0

d

0

~

q

(*

_{qX ~}

qy) ^{q (*}

^{qX ~}

qy)

20 1.20

~~Aa(1)

^~

~~Aa(()

^~

°.°0

~

0.00

~

0

d

o

d

~ ~~ ~~ " ~Y~

q ("

qX ^"

qy)

Fig.

3. The

q-dependence

of the

hyperfine

form factor8 for

planar ~~Cu, planar

^~~O and ~~Y, denved from trie

Shastry-àfila-Rice

Hamiltonian. From Slichter et ai. in reference [18j.

reminiscent of the

long

_range order present ^{in the} parent

compounds,

are present ^{in the metallic}

phase, producing

a

peak

around

QAF

_in the

imaginary part

of the

dynamic spin susceptibility.

Obviously,

these AF fluctuations will

strongly

drive the _copper

NSLRR,

but their contribution

is

heavily

attenuated at the oxygen

site,

and still _{more so} at the

»ttrium

site. This

purely geometricÀ eflect,

known for fifteen _{years in}

magnetic

insulators

[68],

^{does not}

really depend

on whether the copper

spins

are localized _or not- On the

contrary,

the calculation of the

anisotropy

^{of the} NSLRR

using

the form factors defined above in

equation (10) fully

_assumes

that the electronic

spins

_can be considered _as localized-

Another important consequence of the

q-dependence

of the Fourier transform of the

hyperfine

field is its eflect _on the nudear

spin-spin

relaxation.

By taking

the Fourier transform of the

expression

_given

by equation (8),

it turns out that the

longitudinal component A~ +28(cos

_q~a+

cosqyb)

is much

larger

than the _{transverse one} for _q

=

QAF. This,

in addition to other

arguments [49] justifies neglecting

the transverse terms _in the

expression

of

T2G

for

Ho II

_c-

axis, which _can then be written _as

[69,

70]

(£j

=

) )j(( ^~j lAcclq)l~i~lq) (~j Acclq)l~~lq))

,

Ill)

~~

~

~

~ ~

Î

where _i stands for the copper isotope 63 _or 65 and _c~ for their natural abundance- Because of the

q-dependence

of

A~~,

which is maximum for _q

=

QAF

and close to _zero for _q

= 0, it is dear that in the _case where

~(q)

is

peaked

around

QAF. T2G

shows the

temperature dependence

of

Î(QAF).

4. Nuclear

Spin-Lattice

and

Spin-Spin

Relaxation in HTSC

4.1. NUCLEAR SAIN-LATTICE RELAXATION. Que of the most

striking

results obtained

by

NMR in

HTSC,

is the

completely

diflerent temperature

dependence

observed for the _copper and oxygen nudei in trie _same

Cu02 Plane.

This result is

quite general.

It is observed in all studied

.

La2-xSrxCu04 "Cu

_NQR

x=

0.24

° O.20

0.15 0.13

~o 0.lO

° O.075

lK

1

~

°~ ,a~

1

2

^é

~'

$m

"

~

~ o

Tc

-ioo o ioo

T

(K)

Fig.

4.

~~Cu

_{TIT in} La2-xSr~Cu04 for various Sr contents. From

Ohsugi

et ut. [71].

compounds, induding overdoped compounds, although

trie _more

underdoped

is trie

compound,

trie

higher

_is the _contrast- When

starting

from

high temperature

the Cu NSLRR divided

by

temperature,

~~(TIT)~~,

increases when T decreases- In the

La2-xsr~ Cu04 compounds,

which

can be studied _{on a}

larger temperature

_range than trie other famines. it has been

dearly

shown that

~~(TIT)~~

follows _a Curie-~veiss

behavior,

_or

alternatively ~~(TIT)

varies

hnearly

with T at

high temperature I?ii (Fig. 4)-

Still _more

striking

_are the results obtained

by

Imai et ai- who have shown that above _m î00 K,

I/~~Ti

_is constant and has the _same value within

5%,

for _x

ranging

from 0 to

o-là,

that is from the pure

antiferromagnetic parent compound

to the

optimally doped superconductor.

This result _seems to indicate that in the

high

temperature

hmit,

the motion of the holes _is not felt

by

the localized Cu

spins.

However, _a

slightly

diflerent

result has been

recently reported by

the ISSP _group

[73],

^{which daims that there is} a

20%

diflerence between the

high temperature

^{hmit of} x = 0 and _x

=

o-là,

which

they

attribute to the motion of holes. In all other HTSC

compounds,

but for

YBa2Cu40s (74],

^{Cu NSLRR}

measurements have been

performed

at

temperatures

below 350 K and the Curie-Weiss behavior

is not _as well defined. For

underdoped coinpounds, ~~(TIT)~~

_passes

through

_{a maximum} at a temperature ^T* m 150 K

(for YBCO)

_or

higher (for HgBa2Ca2Cu30s+à (29]),

^{that is} well

above

T~,

and then decreases with T, sometimes without _a noticeable

anomaly

at T

=

Tc.

For

overdoped compounds. ~~(TIT)~~

iiicreases

continuously

_as T is decreased down to T~

(with

some

exceptions

that _we shall comment

later),

and then falls

abruptly-

The situation _is

quite

^{diflerent for} the NSLRR of the

planar

_oxygen

(Figs.

5 and

6).

In the first

approximation, ~~(TIT)~~

follows trie _same temperature

dependence

as trie MHS

Iie.,

the

spin-susceptibility).

It decreases with T in the

underdoped

_regime and is

nearly

constant at

optimal doping.

In the

overdoped

_regime, ^the situation _is not _as

clear,

_{as in some cases} it

increases

shghtly

with

decreasing

T

[14, î5],

~vhereas _in others _it decreases

slightly [61].

o.3

(a) É'°

^~' ° ° ° o o

°'~

~a~

~ ^_ O.S

~

o ~

À

o,2

~

Ù.4 à

~' ~

à °

~ _À'~

^~ ~ _~ ~ ~ o

w à

~ ~

~ ~

?~

oi

?

O K~

[-

^°'~

_~

^~~ °

o

o Kyy - ~ °

~ A

K~

^{$ °°~} o H~llY

à ~ ~~~

Ù-1 ^° °

à H~ ^X C

(b)

O

o

~

°°~

T a

f~

0.3 (b)

11

_° §

Î

$

I

D

~

~ ~

É

o ~

j j

~~

°

Î+

_'

j

^°

~ O.l

o

~ ~

~

. '

. O K~ É

Ù-1

~o°~

o HllZ

° .

. K $

. . Ill _Y

~ "

d~ °

~

° ~

~~ ~oo 300 ° ~°° ~°° ~°° ~°°

Temperature (K) ~~~'~~~~~~~~~

~~~

Fig.

5. Left

panel: magnetic hyperfine

shift of

planar

^~~O for diflerent field orientations

in

(a)

YBai g2Sro o8Cu307 and

(b)

YBa2Cu306.52.

Right panel:

nuclear

spin-lattice

relaxation rate of

planar

~~O, for diflerent field orientations _in

(a)

YBai g2Sro o8Cu307 and

(bl

YBa2Cu306

52. From Horvatié et ai. [46].

~

0E7

Tfi ~ Tfi

' ~ '

(

~ .

~

~

f

~

QJ ~

. QJ

' j

b

~~~

_b

W W

' * ,

O

T [K]

Fig.

6. T-dependence of the planar

~~Cu

_and ~~O

spin-lattice

relaxation rates _in ~~Ba2Cu408.

From

Mangelschots

et ai. [55].

.,

'_

.

.. ^~ ~

~W

. .

~

~ ~

W

~

~

.

o

~°

° o

oÎ o

o

(

a ire _iso2oeo

(K)

Fig.

7.

Temperature dependence

of the ratio

~~Ti/~~Ti

From Tomeno et ai. [60].

In ail _cases,

however,

the ratio ^~~

(TIT)~~ /~~(TIT)~~

increases with

decreasmg T,

with _some

more or less evident

flattening

_near

T~.

Note that below

T~,

where the AF fluctuations

disappear

at low energy, the ratio ^~~

(TIT)

^~~

_/~~ (TIT)

^~~ faits clown

[60,

76]

(Fig. 7).

We shall _see later that

a similar

temperature dependence

is observed for ^~~

(T2G )~~,

at least above

Tc.

This saunas hke the

signature

^{of the} _presence ^{of AF}

fluctuations,

_{even m}

overdoped compounds [61j.

^The sites between the

Cu02 planes

in

bi-layers compounds,

such _as

yttrium [52j

^{in the YBCO}

family,

_or

occasionally

thallium _{as an}

impurity

_m

T12Ba2CaCu208 (î7j,

behave

quahtatively

and

nearly

quantitatively

_as the

planar

_oxygen.

Because the Fermi

liquid

character of the normal state of HTSC cuprates ^has been

strongly questioned,

there has been _an intense search for _a

Korringa relationship K~TIT

= const.

for those nuclei which _are less sensitive to AF

fluctuations,

hke oxygen and

yttrium.

The

quantitative comparison

between the spm

part

^of the MHS tensor and the NSLRR for oxygen

or

yttrium

nuclei _is net _an easy

task,

_smce it

requires

_precise

knowledge

of the orbital MHS

tensor. For most of the oxygen results obtained in the

underdoped

_regime,

plots

of

~~(TIT)~~

or

~~(TIT)~~/~

_~ersus

~~Kna

exhibit

a hnear

dependence

within the _errer

bars, simply leading

to different values for

~~Korb.

In the first _case, this leads to the relation

I(TIT

=

const.,

^which

can be

interpreted

_as

(TIT)~~ proportional

to

~(T) IF,

where

AIT

is _a

temperature mdependent averaged

correlation time. The other _case leads to the _more famihar

Korringa relationship, K~TIT

= const. A similar situation is found for the

yttrium

data

[52j.

^{However, in} spite ^of

some recent

interpretations

in terms of

Korringa

behavior _m the _case of

YBa2Cu408

_160,

78j,

the

vahdity

of the relation

KTIT

= const. is

commonly accepted, neglecting

clear deviations at

high temperature

observed at the

yttrium

site

[65, î9j. Indeed,

the ratio

~~(TIT)/~~(TIT)

has been found to increase

by 30%

between 300 and 100 K with

decreasing temperature,

in

spite

of the fact that

~~K(T)/~~K(T)

is _constant

[65j.

^This

unexpected temperature dependence

is diflicult to

explain

_usmg the Mila-Rice Hamiltoman. At the _same

time,

the

amsotropy

of NSLRR of the

planar

_oxygen has been found

T.dependent

in the normal state of the "60 K

phase"

of

YBa2Cu306+~ (45, 46],

which is aise

impossible

to

explam

within the framework of the Mila-Rice Hamiltoman

[80j.