Spin gap and magnetic excitations in the cuprate superconductors

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Spin gap and magnetic excitations in the cuprate

superconductors

G. Stemmann, C. Pépin, M. Lavagna

To cite this version:

G. Stemmann, C. Pépin, M. Lavagna. Spin gap and magnetic excitations in the cuprate

superconduc-tors. Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical

Society, 1994, 50 (6), pp.4075 - 4085. �10.1103/PhysRevB.50.4075�. �hal-01896247�

PHYSICAL REVIEW

B

VOLUME 50, NUMBER 6 1AUGUST 1994-II

Spin

gap and

magnetic

excitations

in the cuprate superconductors

G.

Stemmann,

C.

Pepin, and

M.

Lavagna'

Commissariat al'Energie Atomique, Centre d'Etudes Xucleaires deGrenoble,

17ruedesMartyrs, 38054Grenoble Cedex 9,France

(Received 16March 1994)

Weanalyze the spectrum ofmagnetic excitations as observed byneutron diffraction and NMR

experi-ments in YBa2Cu30&+„,in the framework ofthe single-band t-t'-J model in which the

next-nearest-neighbor hopping term has been introduced in order tofit the shape ofthe Fermi surface revealed by

photoemission. Within the slave-boson approach, we have aswell examined the d-wave superconducting

state, and the singlet-resonating-valence-bond phase appropriate todescribe the normal state ofheavily

doped systems. Our calculations show asmooth evolution ofthe spectrum from one phase to the other,

with the existence ofaspin gap in the Frequency dependence of

y"

(Q,co). The value ofthe threshold of

excitations EG is found toincrease with doping, while the characteristic temperature scale T atwhich

the spin gap opens exhibits a regular decrease, reaching T, only in the overdoped regime. This very

atypical combined variation of Ez and

T

with doping results from strong-correlation effects in the

presence ofthe realistic band structure considered here. We point out that the presence ofa resonance

in the spectrum

y"

(Q,co)isin good agreement with the neutron-difFraction results obtained at x

=0.

92

and 1.

0.

This resonance isanalyzed as aKohn anomaly ofthe second kind in the Cooper channel.

Fi-nally, we examine the evolution ofthe Knight shift and of y"(q,co)at any q,allowing one to study the

magnetic correlation length gas a function ofdoping, frequency, and temperature.

I.

INTRODUCTION: EXPERIMENTS

AND PHENOMENOLOGY

In the past few years,

a

very intensive effort has been developed to study the spin excitations in high-T,

super-conductors

(I.

a2

„Sr„Cu04

and

YBa2Cu30&+„

for

in-stance). When undoped, all these systems behave as Mott

insulators at low temperatures,

a

fact which can be viewed as a probe for the existence

of

strong-correlation

efFects (Mott localization). With doping, the systems

un-dergo an insulating-metallic transition (typically at

x

=0.

4 in

YBa2Cu30s+„compounds)

with the onset

of

superconductivity at low temperatures. The comparison

of

theoretical results for the dynamic spin susceptibility with experimental results for the spectrum

of

magnetic

excitations constitutes an important test for candidate models explaining the physics

of

high-T,

superconduc-tors. ' Clearly, a general description

of

the physics

of

Cu02 planes relevant for the spin excitations can only be done within multiband models. However, as has been argued, ' the low-lying excitations

of

these systems can

be modeled within effective single-band models. Before

presenting a theoretical analysis

of

the dynamical suscep-tibility, we would like

to

summarize the essential results obtained by neutron-diffraction and nuclear magnetic res-onance experiments.

(1) At low doping, the insulating phase exhibits long-range antiferromagnetic order. The systematic study

of

the spin-wave spectrum realized by neutron-diffraction experiments brings strong arguments in favor

of

a

description in terms

of

a

bidimensional spin-—,Heisenberg

model with

J

of

order 2000

K.

(2)In the metallic phase

(5)

_5,

), long-range antiferro-magnetic order is destroyed but the neutron-scattering

signal keeps its maximum

of

amplitude around the vector

Q=(n,

u}

in yttrium compounds, ' and around the four incommensurate vectors q

=

(n

+5q,

m) and

q=(m,

m.

_+5q

_{}in lanthanum compounds.} ' _This feature is associated with short-range magnetic correlations respec-tively antiferromagnetic and incommensurable. The

magnetic correlation length g can be directly determined from the q width

of

the signal around its maximum. In

the case

of

YBazCu30&+„

that we will essentially discuss in this paper,

g/a

(where

a

isthe in-plane lattice

parame-ter) decreases with doping passing from

2.

5at

x

=0.

51to

0.

8 at

x=1.

0.

' Quite strikingly, the magnetic

correla-tion length does not exhibit any significant dependence

on temperature and frequency except in the vicinity

of

the resonance outlined further. This last point

concern-ing the dependence

of

g with doping, frequency, and tem-perature will be systematically discussed in the study presented here.

(3) Valuable information can be extracted from the fre-quency dependence

of

the magnetic form factor

S(Q,

cu)

at low temperatures. In yttrium compounds, there has been observed a depression

of

intensity at low frequencies with a Snite cutoff

of

order 50meV. The question

con-cerning the existence or not

of

a gap in the spectrum

of

spin excitations

S(Q,

co) has been extensively discussed

the last few years. Some authors'

'

report the forma-tion

of

a gap the value

of

which continuously varies with doping:

EG=4,

16, 28, and 26 meV at, respectively,

x

=0.

51,

0.

69,

0.

92, and

1.

0.

The corresponding super-conducting temperature

T,

of

order 47, 60,

91,

and 89

K

exhibits

a

maximum at a critical concentration. This behavior is at the origin

of

the terminology introduced, with the distinction between heavily doped (here

x

=0.

51,

0.

69, and

0.

92), and overdoped systems (here 0163-1829/94/50(6)/4075(11)/$06. 00 50 4075

1994

The American Physical Society

4076

6.

STEMMANN, C.PEPIN, AND M.LAVAGNA 50

x

=

1.0).

The difference between these two regimes isalso reflected in the temperature scale

T

at which the gap opens. Experimentally,

T

is determined from the posi-tion

of

the maximum

of

S(Q,

coo) (coo

«Ea

)with

temper-ature.

T

islarger than

T„

in heavily doped systems: it is, respectively,

of

order 150, 150,and 130

K

at

x

=0.

51,

0.

69,and

0.

92.

Hence comes the appellation

of

pseudo-gap or spin gap in this case. Oppositely, the gap disap-pears immediately at

T,

(T

=

T,

) in the overdoped re-gime. The fact that in this series

of

experiments

T

de-creases with doping while EG increases may surprise one

at first sight. Moreover, EG is found to be rather small

compared tothe superconducting gap reaching avalue

of

3.5T,

only in the overdoped regime. These last two points will be thoroughly discussed in the framework

of

the strong-coupling limit considered in this paper. Gen-erally, a large controversy still exists on the existence

of

the spin gap. The measurements are in fact particularly delicate due to the smallness

of

the signal itself especially at high doping when b,_{q becomes large, and also due} to

the difficulty

of

extracting the magnetic contribution.

Another experiment realized at

x

=0.

6(Ref. 14) leads to

a gap

of

10 meV, properly interpolating between the

values reported in the previous experiment at

x=0.

51

and

0.

69.

However, other studies performed at different concentrations have led to negative answers as regards the formation

of

the spin gap in spite

of

the natural argu-ments in favor

of

its existence coming from the low-temperature behavior

of

the nuclear relaxation time on

63Cu_{measured by NMR (see further).}

(4) Well established now is the presence

of

a

reso-nance'

'

in

S(Q,

ei) in the superconducting phase

of

YBa2Cu306+, typically at 41 meV for

x

=0.

92 or

1.

0

when

T=5

K.

As has been pointed out before, this surstructure is associated with an enhancement

of

the magnetic correlation length in its vicinity. The origin

of

the resonance has been addressed in arecent paper on the weak-coupling limit

of

the bidimensional t-t Hubbard model: it has then been analyzed as a manifestation

of

axial superconductity.

It

will be one

of

the purposes

of

this paper to examine to what extent this interpretation

can stand in the strong-coupling limit

of

interest for high-T, superconductors.

Nuclear magnetic resonance experiments offer a com-plementary way

of

studying the spin excitations. The

nu-clear relaxation rate is found to be very different

depend-ing on the nuclei considered, reflecting the different q-dependent hyperfine coupling involved in each case. '

(

_Ti

T)

' on Cu for instance does not follow a Korringa

law, but instead exhibits a maximum with

T

at the same temperature

T

as mentioned before.

It

is

common-lyaccepted that the decrease

of

(Ti T)

' at low temper-atures is related to the opening

of

a spin gap in

S(Q,

co}. Oppositely, the relaxation times on

Y

and '

0

nuclei

in-volving diferent _q filtering have difFerent temperature

behavior. Both

of

them show a slight decrease when

lowering the temperature starting from we11 above

T,

in

heavily doped systems. Again, the overdoped case is

par-ticular since the decrease observed starts directly at

T, .

In a nonconventional way,

(T,

T)

' on

Y

and '

0

nu-clei vary linearly with the Knight shift instead

of

the

quadratic law

of

conventional Fermi liquids. A11 those last-quoted quantities increase with doping.

The schematization

of

the experimental results is represented in

Fig.

1(a). The characteristic doping dependence

of

T,

(with the presence

of

a maximum, separating the heavily doped to overdoped regime}, com-bined with the variation

of

T

(defined either froin neutron-diffraction or NMR experiments) allow one to

define the different phases

of

this phenomenological-type

phase diagram. The superconducting phase below

T,

borders the so-called spin gap phase for heavily doped systems at

T,

(

T

&

T

.

As has been outlined by various authors in the last few years, there exists a striking resemblance between this diagram and that coming from heuristic arguments developed in the

t-J

model,

postulat-ing the existence

of

a singlet-resonating-valence-bond

(RVB) state above

T,

. Originally, the

RVB

state has

been variationally introduced through its wave function

g„vu=PG

~BCS),built by letting the Gutzwiller-Jastrow

projector act on the ~BCS) state. Later, the

RVB

state

receives a natural description within the slave-boson

rep-resentation' in which the particle

f;+

isrepresented by

a composite

of

two operators c;+e, (the spinon c,+

obey-ing Fermi statistics, and the holon e, Bose statistics).

Two characteristic temperatures arise from the

mean-field treatment

of

the model: the Bose-Einstein

tempera-ture

of

condensation

of

holons TzE, and the pairing tem-perature

of

spinons

T„v~.

The schematic doping depen-dence

of

both temperatures is reported in

Fig.

1(b}. Note

Sp

(a)

FIG.

1. Schematic phase diagram for YBa&Cu306+„. (a)

Sketch ofexperimental results. (b) Sketch oftheoretical results

50 SPIN GAP AND MAGNETIC EXCITATIONS INTHECUPRATE.

. .

that, strictly speaking, TnE should be zero whatever the

doping is, resulting from general arguments on Bose

con-densation in two dimensions (2D). The result is changed

if

one allows for additional coupling between layers with

a

variation

of

TaE with doping as represented in

Fig.

1(b).

The superconducting state is obtained when simultane-ously spinons arepaired and holons condensed, such that

T,

is given by TiiE below

5„TRvB

above

_5,

and then

ex-hibits the nonmonotonic behavior quoted before. The

RVB

state corresponds in the slave-boson representation

to

T~E&

T

&

T„»

for which pairing

of

spinons does not

transpose into pairing

of

physical particles since the

con-densate

of

holons has lost its macroscopic occupation.

It

has recently been demonstrated that the effect

of

gauge-field fiuctuations would transforin the different lines

of

transition

of

this phase diagram into simple cross-overs.

'

High-resolution angle-resolved photoemission experi-ments "' performed in YBa2Cuio6+, reveal

a

shape

of

the Fermi surface Xzin contradiction with the "diamond shape" arising from the tight-binding mean-field descrip-tion

of

the

t-J

model ' _{on a square lattice (n}

_&1)

_in

which the electronic transfer t is restricted to nearest-neighbor (NN) sites. The observed Fermi surface isfound

to

be rotated

of

45'

compared tothe diamond shape, and

centered around the point

S(ir,

m)

of

the Brillouin zone instead

of

I

(0,

0).

A simple way

to

reproduce a shape

of

the Fermi surface containing such features istointroduce the additional effect

of

next-nearest-neighbor (NNN) hopping term

t'.

' _{Ashas been pointed out in previous}

works, the consideration

of

such terms does not only affect the shape

of

the Fermi surface, but also gives the correct dependence

of

the Knight shift with doping, and the right sign

of

the Hail coefficient. This makes the

t'%0

model

a

better starting point from aperturbative point

of

view. As for the spin-excitations, previous studies developed in the weak-coupling limit

of

the

t-t'

Hubbard model concluded that there is no gap in the frequency dependence

of

y"(Q,

to) forthe current regiine

of

interest

(4t'&

_@

&0)

as long as pairing effects are not considered

(4t'

locates the position

of

the Van Hove singularity in

the density

of

states). Let us remind that the conclusions

reached for the

t'=0

case ' ' _{were radically}

difFerent with the existence

of

a spin gap

of

value 2~p~

an-alyzed as a dynamic nesting property. On the other

hand, when pairing effects are introduced at

t'%0,

with

for instance a d-wave symmetry

of

the order parameter, we have shown in the same paper that the frequency dependence

of

g"(Q,

co) gains a gap with a very charac-teristic evolution

of

the threshold

of

excitations EG with doping. In addition to the gap, the model leads to the

prediction

of

aresonance in clear analogy with the exper-imental results obtained in

YBazCu306+„at

T

&

_T,

.

The

I

resonance has been analyzed as a Kohn anomaly

of

the second kind in the Cooper channel and is typical

of

axial superconductivity.

Motivated by the striking resemblance existing be-tween the spectrum

of

magnetic excitations in the super-conducting state

of

the weak-coupling limit, and the spectrum experimentally observed in the normal phase

of

heavily doped

YBazCu306+„,

we propose to examine in this paper the strong-coupling limit

of

the

t-t'-J

model with the idea toextend the pairing effects to the

singlet-RVB

state above

T,

.

This problem already addressed in

some recent works, will be considered here closer to

the systematics that we developed earlier in the weak-coupling limit. Wewill successively examine the question concerning the existence

of

the spin gap and

of

the reso-nance, the evolution

of

the magnetic correlation length with doping, frequency, and temperature, the tempera-ture dependence

of

the Knight shift.

For

the reasons presented in

Ref.

49,we have examined here the case

of

d-wave symmetry

of

the order parameter. Since our study concludes in this case in the formation

of

a spin

gap, we will show how itis possible to reconciliate the

ap-parently contradictory variation

of

EG and

T

with dop-ing. The whole set

of

our predictions will give strong support when compared to experimental results to the establishment

of

an axial superconductivity the effects

of

which extend to the singlet-RVB state at

T)

T,

in the heavily doped regime. ~

II.

SADDLE-POINT EQUATIONS IN THEd-WAVE

SUPERCONDUCTING AND SINGLET-RVB STATE

OF THE

t-t'-J

MODEL

The low-lying excitations in the Cu02 planes are

be-lieved to be described by a generalized

t-t'-J

model in which

J

isthe Anderson superexchange coupling between neighboring Cu spins.

t, t'

represent, respectively, the nearest-neighbor and next-nearest-neighbor transfer in-tegrals

of

the Zhang and

Rice

singlets constituted after

doping by both Cu and

0

spins. In the slave-boson rep-resentation, the t-t

-J

Hamiltonian is written as

H=

—

t

g

c;+c

e;e+

—

t'

_g

c,

+c

e;e++J

g

S;

S.

&i,j& &i,

j

&' _&i,

j

&

in which the spin S, is noted:

S;

=

g

.c;+~ c;

.

As usual, the local constraint

e;+e;+g

c;+c;

=1

isenforced

at each site by time-independent Lagrange multipliers A,

;.

Using the Feynman variational principle, one can find

an upper bound

to

the free energy according to

E&FO+(H

Ho)tt

in which

(H

H—

_o)tt

_{is the ave—}

r-0 0

age performed on the trial Hamiltonian Ho taken as

&i

j

& &i,

j

&' &i

j

& &i

j

&'

+

_g

b, ,

(c,

tc~i

c;tcj.

+t

)+H.

c.

—

+

"gA.,_; e; e,.

+

gc;+c;

1

—

p

gc;+c,

.

—

—

X(1

—

5)

(i,

j

&

4078

6.

STEMMANN, C.PEPIN, AND M.LAVAGNA

Zo= Jdk,

;Dc

De, exp

—

I

dr

_gc,

+t},

c,

IV

+e,

+By;+Ho

.

(3)

At the mean-field level, the bond variables

F

'

and

8

'

are considered real and both time and bond independent (s-wave symmetry), while

_6;,

(also real and static} is as-sumed to have d-wave symmetry:

_5;,

=

b,o(

—

1

P,

with

t)

=2

(or 1)depending on whether horizontal (or vertical} bonds are involved. _yk

=

(cosk„+

cosk»)/2 for NN

(yk

=cosk„cosk„

for NNN), and ak

=(cosk„—

cosk» )/2

are, respectively, associated

to

these two types

of

sym-metries. A,_, is supposed site independent. The

saddle-point equations are

1

3J

F=

—

_g

_ykns(rlk)+

_B,

k 8t (4a)

F'=

~

&

k

r'kna(r)k»

(4b)

B'

'=

_g

y'k' tanh(Psk

/2),

2F-k

1=3J

—

1

_g

ak2 1

tanh(psk/2),

k k (4c) (4d} 1 1

—

5=

—

g

1—

k tanh(Psk /2

),

(4e}

5=

—

1

_y

na(gk)

k in which 1

ns(E)=

and

nF(E)=

exp

E

—

1 &k

=

Aa+~'k

1

exp(PE)

+

1

The free energy Fo can be calculated from

Fo=

—

ks

T(lnZo) in which the partition function Fo is

expressed as a functional integral over coherent states

of

Fermi and Bosefields:

straint which is now only globally fulfilled at the

mean-field leve1.

A considerable simplification occurs at low tempera-tures, when the bosons are either partially condensed

(T&

TnE), or occupy the first excited levels

(T~

_TsE

with y'k'

=

I in these states). In both cases,

(I/X)sky'k'nit(gk)

just equals

5.

When simultaneously

Ao is finite, the first case corresponds to the

supercon-ducting state, and the second

to

the singlet-RVB state

ac-cording to the terminology introduced in the Introduc-tion.

%e

have numerically solved the system

of

saddle-point equations in these two cases, and reported the re-sults in Figs. 2—

4.

At this point, let us make two

com-ments.

First, the brute force resolution

of

the saddle-point equations leads to a sharp decrease

of

the renormaliza-tion parameters

F

and

F'

when doping is lowered. This

feature

rejects

the underlying Brinkman-Rice transition which occurs as doping goes

to

zero. As outlined in pre-vious works, '

'

' this transition has unphysical effects

with a sharp increase

of

the susceptibility at low doping, and huge values

of

the effective masses which seem not to be observed experimentally. One way out already

sug-gested in

Ref.

41 is to push the approach

of

the

Brinkman-Rice transition to very low doping, and intro-duce saturation effects in the decrease

of

F

and

F'

when

lowering the doping as represented in

Fig.

2.

The second comment has

to

do with the validity

of

a rigid-band model in which the different

parameters-t, t',

and

J

—

are kept fixed, whatever the doping is. A realistic description

of

the cuprate superconductors would require the consideration

of

more general multi-band models to account for the different electronic states. In this scheme, the determination

of

the low-lying

excita-tions allows one to map the model into effective single-band models, the parameters

of

which can be systemati-cally determined. Recent works have shown asignificant linear decrease

of

the effective superexchange coupling

J

with doping (cf.

Ref.

52). Our calculations take account

of

this effect, assuming a linear decrease

of

J/t

from

0.

23 to

0.

20 when

_p

evolves from the heavily doped to the overdoped regime [cases (i) and (ii)

of Fig.

4(a)j. Given

that, the results coming from the numerical resolution

of

the saddle-point equations are reported in

Fig.

3 for the

4=4+~

—

_v gk

= —

zyktF

zykr

F

k zak~O & Ik 1k gk

= —

ZyktB

—

Zy'k

t'B' .

60 I I I i I ~ ~ i ~ ~ / ~ I / ~ _j I ~

Equations (4a)

—

(4c) give the renormalization

of

boson and fermion electronic transfer integrals resulting from strong-correlation effects. Equation (4d) is the equivalent

of

the gap equation

of

the

BCS

theory in its strong-coupling version since the coef6cient ak

rejects

the for-mation

of

Cooper pairs along nearest-neighbor bonds. Equation (4e) expresses the conservation

of

the average number

of

particles, and

Eq.

(4f) corresponds to the

con-20

0 I ~ I I I ~ I I I ~ I I ~ I I I I ~ I

7 9 11 13 15

i)t

i

[meV]

19

50 SPINGAP AND MAGNETIC EXCITATIONS INTHE CUPRATE.

. .

20 ~ ~ ~ ~ ~ ~ ~ ~ ~ _t ~ ~ r ~ ~ ~ ~ ~ ~ ~ ~ ~ a a ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 15 8 10

5-

4 Q ~ a 0.01 0.02 0.03 0.04 Q ~ a ~ ~ 0 I ~ 80

T

[K]

I a ~ I a 160 240

FIG.

3. Chemical potential ~p~ as a function ofhole doping

doping dependence

of

the chemical potential, in

Fig.

4(a} for the temperature dependence

of

b,_oat various doping, and in

Fig.

4(b)for the doping dependence

of

TRvz

The.

results are in agreement with the heuristic phase diagram discussed in the Introduction. Note that our numerical results give b,

o=2.

3TRvn with only small doping depen-dence corrections. This value to be compared to

bc=1.

7T, of

the weak-coupling

BCS

theory, essentially results in the effect

of

ak in the gap equation (formation

of

Cooper pairs on bonds).

III.

SPINDYNAMICS

The dynamic form factor

S(q,

ro)is related tothe imag-inary part

of

the susceptibility

y"(q,

ro) through the fluctuation-dissipation theorem:

S(q,

ro)

=

—

y"(q,

ro)

.

nl

—

_exp

—

.

_co

The dynamic susceptibility isgiven by the analytic

con-tinuation i

to„~r0+i

5

of

_y(q,

i

_to„)

defined interms

of

the

correlation function

of

the physical particles:

4 0y ~ ~ \ g ~ ~ ~ t ~ \ ~ J ~ ~ ~ / a ~ ~ / ~ ~ ~ J ~ ~ 2 180 120 60 a I ~ a a I ~ ~ a I ~ ~ ~ I a a, I a a a I a a a 7 9 11 13 15 17 19 I_p,I

[meV]

21

FIG.

4. (a)Pairing parameter b,oas a function oftemperature

obtained for doping concentrations corresponding, respectively,

to heavily doped and overdoped systems. The corresponding

sets of parameters are (i)

t=62

meU, t

= —

0.8t,

p= —

7.05

meV,

ho=13.

19 meU (heavily doped) and (ii)

t=74

meU,

t

=

0St,

p=

——

.

20.63 meV,

60=5.

64 meV (overdoped}. (b}

Temperature TR» vschemical potential.

2r

g(q,

iro„)=2

f

f

dq.

_exp[i(ro„q

—

q.

r)](T,[f+

(~)

f,

(r)

fo+~(0)

fo

(0)])

.

In the slave-boson representation, a straightforward calculation (benefiting from the constraints) gives that the

correlation function

of

physical particles reduces to that

of

only spinons:

(&,

[f,

+(~)f,

-(q)fo-((})fo ((})])

=(T,

[c,

+

(1)c

—(q')c (O)c

(0)]&

.

In the random-phase approximation, the renormalized dynamic susceptibility is then given by

Xc(q

i~.

}

y(q,

i

to„)

=

(7)

1+

[2Jyqyc(q,

i

co„}]

in which the bare susceptibility _yo(q,

iso„)

is derived from the mean-field treatment presented in

Sec.

II.

The latter is written as yo(q,

in)„)

=

g

—

1+

(k+q

4

k

4

~k+q ~k

Fk+,

kk ek+q ak kkfk+

+~k~k+

F(sk

)+

F(sk+

~kk+q '~n &k+q ~k

rkrk+

+~k~k+

1 _nF(sk} nF(ek+ &k~k+q

~~n+~k+q+~k

1 (krak+

+~k~k+

nF(sk+ } F(ek }

+g

—

1+

k ~k~k+q ~~n

~k+q+~k

4080

6.

STEMMANN, C.PEPIN, AND M.LAVAGNA 50

The strong-coupling eSects make themselves felt through the renormalization

of

various parameters

enter-ing these analytical expressions. We refer to Sec.

II

for t e determination

of

the renormalized parameters, which are solutions

of

the saddle-point equations in,

respective-ly, the superconducting and the singlet-RVB phases. e have sketched in

Fig.

5(a) the spectrum

of

individu-al excitations

ck=C'"

involved in both cases. Let us

note the formation

of

four "ellipses" around nodes

ek

=0)

at the intersection

of

the Fermi surfaceace, and o

f

e rst bisectors along which the gap vanishes. This

feature is characteristic

of

an axial pairing (d-wave sym-metry in the present case) in which the _gap parameter vanishes at points on the Fermi surface.

Far

from the

Fermi surface, one notes the presence

of

closed and open

orbits, respectively, centered around the points 1

(0,

0)

and

S(n.

,n.)

of

the Brillouin zone. The corresponding den-sity

of

excitations is represented in Fig. 5(b)with a very

characteristic triple-peak structure. The low-energy part coming from the ellipse contribution islinear as expected from topological arguments in the case

of

axial pairingng in'

A. Results for

y"(Q,

u)

1. T=O

At zt zero temperature, we have reported in Figs. 6(a} and 6(b) the frequency dependence

of

y"(Q,

ro) that we

obtained at two different values

of

doping. Figure 6(a}, corresponding to the overdoped case, shows the presence

of

agap

of

value EG

=

37meV followed by a resonance at

co+

=47

meV. The resolution

of

the gap equation in this

case gives

_{T„vn=120 K.}

Figure 6(b), corresponding to

the heavily doped regime is characterized by a lower

value

of

the gap

of

order EG

=14

meV even though the

characteristic temperature _TRv& is larger,

of

order

T„va

=210

K.

Apart from the high-energy part

of

the

spectrum, which drags much too far compared to the

cutoff experimentally observed (of order 50 meV), our

predictions concerning the value

of

the spin gap and the position

of

the resonance are in good agreement with the

measurements performed in

YBa2Cu306+„by

neutron-diffraction techniques. As emphasized by other

au-5 s / ~ j I I ' I ' I ' ( I ' l

2.

0.5 ~ I I I ~ ~ I ~ I I s I 0 10

I,

I,

I s I s I 20 30 (o

[meV]

40 50 1.2 1'5 I I I I ~ I I ~ I I I I I I I I I I 0.8 0.6

,

.

A

«i 0.2 0 I s I I s s I 0.

5—

0.4 0.8 tt)I.&1 1.2

FIG.

5. (a)Equienergetics ez=

C'"

in the presence of1-wave

pairin g.

.

Note the

f

ormation offour superconducting ellipses

around the nodes ofa&=0. The parameters chosen are t

=75

meV,

t'= —

0.45t,

p=

—

125 meV,

50=3.

75 meV. (b) Density

ofstates ofindividual excitations vs energy for the same set of

parameters asm (a). Note the linear behavior atsmall

frequen-cies due to the superconducting ellipses rejecting the d-wave

symmetry ofthe order parameter Al,

.

(b)

0 I I I s I s

0 10 20 30

(o

[meV

J

40 50

FIG.6. Spectral weight

y"(Q,

co)vsfrequency aI. The choice

ofparameters isthe same asinFig.4(a). (a)Overdoped systems.

Note the presence ofagap followed by a step and aresonance.

50 SPIN GAP AND MAGNETIC EXCITATIONS INTHECUPRATE.

. .

4081

thors, we expect that self-energy corrections neglected

in our mean-field approach would depress the energy tail

and restore the correct order

of

magnitude for the cutoff.

More work is probably required for a proper description

of

the high-energy contribution. Nevertheless, the results

at low frequencies show very interesting features when

compared

to

experiments, which we would like to

com-ment on.

(1) The value

of

the spin gap EG is found to increase with doping. This result may surprise one at first sight, since conjointly, the characteristic pairing temperature TavB is decreasing. In fact, this very atypical behavior results in the interplay

of

the anisotropy

of

the Fermi

sur-face XF and

of

the anisotropy

of

the pairing parameter

5k.

At zero temperature, and for co

&0,

the only contri-bution to

y"(Q,

co)comes from the first term in

Eq.

(8), expressing the creation

of

a pair

of

excitations from a

Cooper-pair-breaking mechanism.

y"(Q,

co)isthen given

by the density

of

pairs

of

excitations through 5(to

—

_sk+&

—

sk) weighted by the coherence factor. We

refer

to

our previous paper on the weak-coupling limit

for the precise determination

of

EG according to

_p.

For

30 l ~ I I I I 25— 20—

15—

10—

5—

0 100

o

40-s

30—

20—

50 y i I I I ~ co =2.5meV 0 200

T

[K]

300 I I I I I I I I l ~ ~ I 400

260

p)

=4t'+,

&

_p

&

p~=

2b,_o

t'

10—

we have established EG

=860'I/p/4t'

(50/2t—

')

Coming from the displacement

of

the minimum

of

(ek+sk+&)

from A(

—

m/2,

—

n/2)

to

B(0,

n)

whe—n p,

increases from _IM„to

_p,

₂ (cf.

Fig.

11

of

Ref.

20},one gets acharacteristic increase

of

EG from 2~@~(independent

of

b,_o since

h„vanishes

at the point A ), to

2+(4t'

—

p)

+(460)

[feeling the full effect

of

bo when

the minimum

of (sk+sk+&)

reaches the point

B].

This

geometrical effect isat the origin

of

the opposite variation

of

EG and

T„vB

(i.e., b,o), which we get as a function

of

doping.

(2}The existence

of

a resonance in the highly doped

case at ~&

=47

meV reminds one

of

the surstructure ob-served in YBa2Cu306 92 and YBa2Cu307 at Np

=41

meV.

'2's

In our scheme, the resonance arises as a Kohn

anomaly

of

the second kind in the Cooper channel. This

singularity due toquasi6at parts

of

the equienergetics line

(e„+e„+&)=Cste,

can be viewed as a signature

of

axial superconductivity for which the pairing parameter van-ishes at points

of

the Fermi surface (d-wave in the case that we consider here}. The resonance appears as soon as

p~4t'+260/t',

and is progressively shaded off when

_p

becomes larger than

260/t'.

We will analyze further the change

of

the magnitude correlation length which itis

as-sociated

to.

0

0 200 300

T

[K]

FIG.

7. Temperature dependence of

g

(Qcoo)/a)p for

coo=2.5meV for the same choice ofparameters as in Fig. 4(a).

(a) Overdoped systems. (b) Heavily doped systems. Note the

presence of an anomaly with a maximum at

T

( Tt:&Tm & TRv~). 400 200 0.75 -150 0.5 U -₁₀₀B 0.25 -50

and 7(b} the temperature dependence

of

y"(Q,

coo) at

t00&&EG for two values

of

doping. Note the presence

of

a maximum which isreminiscent

of

the Hebel-Schlichter anomaly, somehow washed out by the anisotropic effects.

The position

of

the maximum provides the temperature scale

T

at which the spin gap opens.

It

isinteresting to

gather the doping dependence that we get for both

T

and EG (cf.

Fig.

8). The apparently contradictory

2. Finite temperature

Our calculations have been as well pursued at finite temperature. The efFect

of

temperature is

to

fill up the spin gap in the regime

of

doping

4t'&@&0,

and

to

shade

ofF the resonance. Following the procedure currently

used by experimentalists, we have reported in Figs. 7(a)

0

7 11 13 15 17

I_p.I

[me

V]

I

19 210

FIG.

8. Combined variation ofthe spin gap EG measured in

units SLOand the temperatures

T

at which the spin gap 611s

4082

G.

STEMMANN, C.PEPIN, AND M.

I.

AVAGNA 50

behavior

of

EG and

T

observed in neutron-diffraction experiments receives a natural explanation here in the strong-coupling limit. This feature was out

of

scope

of

our previous weak-coupling calculations in which

T

=T,

at high doping

(p=4t'),

but

T

(T,

at lower doping ()tt&

4t').

Our study here shows that itisessential

to include self-consistent effects inherent to the strong-coupling regime to inverse this tendency (through the t)

dependence

of

the pairing temperature TRva ),and obtain a decrease

of

T

with doping, reaching

T =TRv&=T,

at high doping. (0,&

.

2 m,o) (0,& (P,rr

(b)

0.+

.

2 ,0) 0.&

.

2 ,0)

.

2 ,0)

FIG.

10. Three-dimensional representation ofyo'(Q,coo) in

the

_(q„,

q~ )plane at zero temperature for the same setof

param-eters as in Fig.4(a) (heavily doped systems) and coo=40 meV.

Note the smaller qwidth compared tothose shown in Fig.

9.

B. y"(q,

co)atany q

The method that we have developed allows one to ex-plore as well the magnetic excitations at any value

of

the

momentum q. The results are reported in a tridimension-al representation

g"(q,

coo)=

_f(q)

in Figs. 9 and 10 for

the same choices

of

doping as before. We find that the signal is always peaked around the antiferromagnetic

vec-tor

g=(m,

m_{), with} a _qwidening which evolves with dop-ing and frequency. By approximating the response to a Lorentzian curve, it is possible

to

evaluate _{the q width} Aq

of

the signal, and hence extract the magnetic correlation

length

g=(Aq)

'.

The corresponding quantity is

rela-tively anisotropic, depending on the direction involved.

Table

I

recapitulizes the values

of

b,q as a function

of

doping (at given too &EG ),while Table

II

gives the results

for b,q as a function

of

frequency (overdoped regime).

The results are comparable to the experimental data with

acharacteristic reduction

of

_{g with doping.}

It

is quite re-markable that our approach also predicts the expected enhancement

of

the magnetic correlation length around the resonance pointed out before. This constitutes anoth-er argument in favor

of

the interpretation

of

the reso-nance in terms

of

a Kohn anomaly in the Cooper chan-nel. Figure 11 reports the similar study at finite

tempera-ture. The response remains peaked at

Q=(n.

,n)with a _q

widening which does not practically vary with

tempera-ture until

T

except in the close vicinity

of

the

reso-nance. Again these predictions at finite temperature are in agreement with experiments.

FIG. 9. Three-dimensional representation ofyo'(Q,coo)in the

(q„,

q~ )plane atzero temperature for the same setofparameters

asin Fig.4(a) (overdoped systems). Note that the peaks around

q

=

(0,0) containing only logarithmic spectral weight are

suppressed after RPA renormalization. (a) coo=40 meU. (b)

coo=47 meV (value of the resonance frequency). (c) coo=50

meV. cg)=40 meV hq[2n/a] g/a g/a(exp. ) Heavily doped 0.080 3.9 2.5 (x

=0.

51) 0verdoped 0.156 2.0 0.8 (x

=0.

92) TABLE

I.

b,q and magnetic correlation length g at a given

frequency of40meV for the two doping regimes ofinterest [cf.

Fig. 4(a)]. The experimental values reported for comparison are

50 SPIN GAP AND MAGNETIC EXCITATIONS INTHECUPRATE.

. .

4083 Overdoped b,q[2m/a ] g/a f/(exp

).

co&=40 meV 0.156 2 0.8 co&

=47

meV 0.125 2.5 1.36 co2=50 meV 0.140 2.2

TABLE

II.

hq and magnetic correlation length _{g for}

different frequencies around the resonance in the overdoped

re-gime. The experimental values reported for comparison are

drawn from Ref. 17. Note the enhancement of the magnetic

correlation length atthe resonance frequency. Oll

s

CO ll CZ' C.Knight shift 100 200

T [K]

300

We have calculated the susceptibility

y'(0,

0}

as a func-tion

of

temperature for the same values

of

doping. The

results are reported in

Fig.

12. Let us note the charac-teristic reduction

of y'(0,

0)

immediately below

_T„va.

As

in the weak-coupling regime,

y'(0, 0)

follows a Yosida

law with alinear variation atvery low temperatures,

typi-cal

of

an axial superconductivity (existence

of

nodes).

Note that the strong-coupling limit studied here leads to drastic changes with a decrease

of

y'(0,

0)

starting at

TRva

)

T,

in the heavily doped regime.

IV. CONCLUSION

Let us first summarize the results obtained in this

pa-per for the strong-coupling limit expressed in the t

t'

J--model. The model appropriate for the description

of

the

Zhang-Rice singlets in the Cu02 layers includes

next-nearest-neighbor hopping terms in order

to

account for the realistic shape

of

the Fermi surface observed in

angle-resolved photoemission experiments. Within the slave-boson representation, the resolution

of

the saddle-point equations allows one

to

determine the phase dia-gram (

T, 5) of

the model

[cf.

Figs. 1(b) and 4(b)]. Relying on theoretical arguments in deriving an eff'ective single-band model from

a

general inultiband description, we

show that it is essential to broaden the approach

to

a

FIG.

11. Three-dimensional representation ofy~(Q,c00) in

the

_(q„,q„)

plane at

T=93

K

and coo=40meV for

t=74

meV;

t

= —

0.

8t,

p=

—

20.6 meV,

60=4.

5 meV (overdoped case).

Note the larger qwidth compared tothose shown in Fig. 9

cor-responding to T=O

K.

FIG.

12. Uniform static susceptibility y'(0, 0)with d-wave

pairing as a function oftemperature for heavily doped (i) and

overdoped systems (ii)[same parameters as inFig. 4(a)].

nonrigid band scheme in which the coupling parameter

J

is bound

to

vary with doping.

We have systematically studied the spectrum

of

spin

excitations for two phases

of

the diagram: the supercon-ducting phase, and the singlet-RVB phase appropriate for the description

of

the normal state

of

the heavily doped regime. Our calculations show a smooth evolution with temperature from one phase

to

the other.

y"(Q,

co}is found to have a very different frequency behavior

accord-ing to the value

of

the doping considered. We have

shown the existence

of

a spin gap whatever the doping is, with a characteristic increase

of

the threshold

of

excita-tions EG with doping, reaching the value

2(450)

only at

high doping when

_p

gets closer to

4t'

Simult.aneously, the temperature scale

T

at which the spin gap opens [determined from the position

of

the maximum

of

g"(Q, coo)=f(T)

at

c00«EG]

decreases with doping.

This very atypical feature concerning the combined vari-ation

of

EG and

T

with doping (cf.

Fig.

8), effectively observed experimentally, ischaracteristic

of

axial pairing effects in presence

of

the realistic electronic structure considered here. The whole set

of

our predictions in qualitative, almost quantitative agreement with experi-mental results brings strong support

to

the establishment

of

an axial superconductivity the effects

of

which extend

to the singlet-RVB state above

_T,

.

In addition to the above-quoted behavior

of

EG and

T

with doping, let us mention the following.

(1) The existence

of

a resonance at higher frequency in

the codependence

of

y"(Q,

co)in agreement with the mea-surements made at

x

=0.

92 or

1.

0

in

YBazCu306+„at

low temperatures. This resonance has been analyzed as a Kohn anomaly in the Cooper channel. it is remarkable

that our calculations also predict an enhancement

of

the magnetic correlation length g in the vicinity

of

this surstructure exactly as neutron-diSraction _experiments

see.

(2) The doping, frequency, and temperature depen-dence

of

the _q widening

of

the signal around the antifer-romagnetic wave vector. Typically, the corresponding magnetic correlation length g is found to decrease with

doping without any significant variation with

4084 G.STEMMANN, C.PEPIN, AND M.LAVAGNA 50 (3) The temperature dependence

of

the Knight shift

showing clear differentiation between heavily doped and

overdoped regimes.

Despite all the interesting features that we get for the low-frequency part

of

the spectrum, we would like to out-line the serious discrepancies found at higher values

of

the frequencies with notably a response which drags much too high in the frequency scale. We think that this drawback has to do with the choice

of

the effective single-band models.

It

is likely that these models, ap-propriate for the description

of

the low-lying excitations

of

more realistic multiband situations, poorly describe the higher part

of

the frequency scale.

It

would also be

in-teresting to see what will be the effects

of

the self-energy

corrections neglected until there.

Finally, to complete our study, it would remain to ex-amine the temperature behavior

of

the nuclear relaxation rates on the different sites. Apriori, this study should not

meet any diSculty. We expect reasonable results for

both

_(T,

T)

'on Cu, and

(T,

T)

' on '

0

or

Y,

since the former is likely tobehave as

y"(Q,

ro) quoted before, and the latter asthe Knight shift previously described.

For

the future, itwould be interesting

to

go beyond the

2D limit considered here, and consider the additional coupling between layers. There are some arguments

com-ing from neutron-diffraction experiments' which make us believe that these effects are bound

to

play an

impor-tant role in heavily doped systems. Typically, the

magnet-ic form factor

y"(q,

co}exhibits a modulation along the

wave vector q

=

( m,tr,_q,}resulting from interlayer

cou-pling. The modulation shades offin the overdoped case,

meaning that the bilayers become entirely decorrelated in this regime. Oppositely, the modulation is clearly present

at lower values

of

doping indicating significant

correla-tions among bilayers. The study

of

the consequences

of

such quasitridimensional effects is under investigation.

ACKNOWLEDGMENTS

We acknowledge the European Union

(E.

U.) for

pro-viding financial support under Contract No.

ERB

4050

PL

920925. One

of

the authors

(G.S.

) was supported by

E.

U.

Grant No.

SCI

B'/915056.

This work has benefited from stimulating discussions with

L.

-P.

Regnault and

1.

Rossat-Mignod on neutron-diffraction experiments, and

with

C.

Berthier and

M.

Horvatic on NMR results. We

would also like

to

thank

P.

Burlet,

K.

Fukuyama,

D.

Grempel,

H.

Kohno,

K.

Maki,

S. V.

Maleyev,

R.

S.

Mar-kiewicz,

F.

Onufrieva, A.

M.

Tremblay, and

H.

Won for

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