Stability of the atomic magnetic moments in La2-xSrxCuO4 and effect of their fluctuations on the electronic density of states

(1)

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Stability of the atomic magnetic moments in

La2-xSrxCuO4 and effect of their fluctuations on the electronic density of states

T. Hocquet, J.-P. Jardin, P. Germain, J. Labbé

To cite this version:

T. Hocquet, J.-P. Jardin, P. Germain, J. Labbé. Stability of the atomic magnetic moments in La2-

xSrxCuO4 and effect of their fluctuations on the electronic density of states. Journal de Physique I,

EDP Sciences, 1994, 4 (3), pp.423-439. �10.1051/jp1:1994149�. �jpa-00246918�

(2)

J. Phys. I Fianc.e 4 (1994) 423-439 MARCH 1994, PAGE 42~

Classification Physics Abstracts

75.10L 74.70v 74.20

Stability of the atomic magnetic moments in La~ ~Sr~Cu04 and

effect of their fluctuations

_on

the electronic density of states

T.

Hocquet (1),

J.-P. Jardin

(2),

P. Germain

(1)

and J. Labbé

(1)

il)

Laboratoire de

Physique

de la Matière Condensée de l'Ecole Normale

Supérieure,

24 _rue Lhomond. 75231 Paris Cedex 05, France

(2) Laboratoire PMTM (CNRS), Université Paris Nord, 93430 villetaneuse, France

(Received Jo September 1993,

accepted

29 November

1993)

Abstract. We calculate the band structure of the

antiferromagnetic phase

of La~ ~Sr~CUO~ m a

bidimensional itinerant electron model. Then, we discuss the

stability

of the

antiferromagnetic

phase and _we calculate the _size of the atomic magnetic moments with respect to the temperature and the

doping

ratio _x. We show that the

antiferromagnetic

solution is _m any case much _more stable than the

ferromagnetic

_one.

Finally,

_we _grue _a

description

of the

antiferromagnetic

fluctuations from which merges an

exphcit physical meaning

of the pseudo-gap ^found ⁱⁿ ^the electronic density of _states. Implications ^for superconductivity _are proposed.

l. Introduction.

It _is _a _common idea that the

superconductivity

^of the copper oxides _cannot be understood

mdependently

of their

magnetic properties.

Indeed, _spm fluctuations _are observed _even in the

superconducting phase,

and it is an

extremely

controverted

question

tu know whether

superconductivity

in these materais finds its

physical origm

_in these

fluctuations,

_or

simply

coexists with them.

In this paper, in view tu

clanfy

_a little this

question,

_we _come back tu a detailed theoretical

study

^of the

antiferromagnetic phase by

using a bidimensional itmerant electron model in the

CUO~ planes.

We concentrate our attention _on the

La~ ~Sr~CUO~ compounds,

^which ^have a

single-layered

structure of

CUO~ planes.

A simùar

study certainly

^would ^be ^valid ⁱⁿ ^other

oxides with

CUO~ planes,

but with

complications

due _tu their

mufti-layered

structure

and,

in

YBa2Cu30~,

tu trie CUO linear chains.

We calculate trie

antiferromagnetic

_gap and trie _size of trie _atomic

magnetic

moments as

functions of trie temperature and trie

doping

ratio _x, and _we discuss their stabilities. Then _we calculate the electromc

density

of states of trie

paramagnetic phase by taking

mto account trie fluctuations of the _size of the atomic

magnetic

moments. As _a result, we obtam _a

deep pseudo-

gap which is very similar tu that found

by Charfi-Kaddour,

Tarento and Hénlier

Ii by

_a

(3)

424 JOURNAL DE PHYSIQUE I N° 3

different method. From Dur

analysis,

the main feature of this

pseudo-gap

is

entirely

due tu trie fluctuations of the size of trie magnetic moments. Furthermore, our

explicit description

of the fluctuations indicates that the overall

density

of _states is

nothing

else than the addition of contributions

coming

from different parts ^of ^the

crystal

at a

given time,

and thus _it _cannot be used _tu calculate local

properties.

As _a matter of

fact,

_in the

paramagnetic phase,

near the transition fine,

only

the _most

probable

value of the atomic magnetic moments vanishes and, at a given time, _areas of the

crystal

^with non

vanishing

moments coexist with other _areas without

moment, where _a

superconducting

state can

locally

exist. From this

point

of

view,

une can

understand trie observed coexistence of

antiferromagnetism

and

superconductivity

without

having

tu assume this coexistence _at trie _same

place

in trie

crystal.

In section

2,

_we

give

_a detailed

analytical

calculation of trie band _structure of trie

antiferromagnetic phase.

In section

3,

trie

equihbrium properties

of the

antiferromagnetic phase

are deduced from trie

grand potential.

In section 4, _we _compare trie stabilities of trie

ferromagnetic

and

antiferromagnetic

solutions,

showing

that

ferromagnetic

fluctuations would be

entirely negligible.

In section

5,

_we

explicitly analyse

the effect of trie fluctuations of trie size of trie atomic

magnetic

moments on trie overall

density

of states, and we discuss its

physical

_meamng.

2. Band structure of the

antiferromagnetic phase.

In trie

La~ ~Sr~CUO~

type ^of

compounds,

^the ^Fermi ^level ^is

customarily

assumed tu lie in trie

antibonding

part ^of a band ansmg from

hybridization

between copper d~2_~,2 ^and oxygen p~ and _p~, orbitals inside

CUO~ planes [2-4].

A two-dimensional

tight-binding

model leads tu trie

following simplified dispersion

relation in trie

paramagnetic phase

E(k)=E~~-2t(cosk~a+cosk,a),

il)

where E~~ ^is the effective energy of _an electron

occupying

_{trie d~2}

_~,2 orbital _on _a copper site,

t is the effective transfer

integral

between _two _nearest

neighbounng

_copper sites

through

the p~ or p~, orbital of the intercalated oxygen atom, a is the square lattice parameter, ^and k~ and _k~ are the Cartesian components ^{of the} wave vector k

along

the _axes of trie square lattice.

Within this scheme,

only

the two-dimensional array of copper ^atoms m

CUO~ planes

is

considered.

The

simplest

_way to take into account the

repulsion

between the electrons in this

tight- binding

model _is to introduce the

single

band Hubbard Hamiltoman which, m the mean-field

approximation,

reduces to

H~t

1 CÎ«Cjoe+11(Eat+U(fin))fin

+

(Eat+U(fin))fin ~U(fin)(fin)1> (2)

1jw i

where

c)~ (resp.c~~)

is the creation

(annihilation)

operator ^for an electron with

spin

w on a copper ^site i,

(n~~)

is the expectation ^value ^of ^the ^number operator n~~ = c.)~ c~~, U is the Coulomb

repulsion

_parameter between two electrons with opposite spms on the same

copper site, and, m the first term, the summation must be extended

only

_on the _nearest

neighbounng

sites _i and

j.

Different types ^of

magnetic

^solutions can be discussed from trie model Hamùtoman

(2).

In trie present section, _we _are interested in trie

antiferromagnetic

solution. It will be

compared

to trie

ferromagnetic

solution _m _section 4.

In trie _case of

antiferromagnetic order,

trie copper lattice _is

split

mto two sublattices referred to as and

2,

and trie unit cell size is doubled as shown m

figure

I. Trie _new unit cell contains

two

unequivalent

_copper atoms

having

the same number of electrons but opposite magnetiza-

(4)

N° 3 ANTIFERROMAGNETIC STABILITY IN

La2-~Sr~CUO~

₄₂₅

~

i ,"""' """",

x

",,, _à

_,,"

~Î

",, _,,."

a

Fig. l.

Antiferromagnetic

unit cell (dashed fine), with the

magnetic

moments localised _on the copper sites.

tions _m and _m, with

n =

(n

ji +

(ni

_j

)

=

(n~

_t

)

₊

(n~

_j

)

,

(3a)

~ ~

l1 ) _l

_Î

)

~

21 21) (~~)

As _a result, the initial band

sphts

into two subbands. In the

tight-binding approximation,

the Bloch _wave function becomes

9'(«(r)

=

jj e'~'~"' jjAj~(q) 4 (r Rm

r~ ),

(4)

,,1 a

where q is trie _wave vector,

R~

is the

ongin

of trie mth

antiferromagnetic

unit cell, where r~ Iocates trie copper ^atom « =

I _or 2, _v

= ± will be trie subband

index,

and trie coefficients

A[~(q)

_are

determined,

_as well _as trie

eigenvalues E(

of trie Hamiltoman

H, by solvmg

trie

Schrüdinger equation.

One finds trie

following

two

dispersion

relations _:

E(

=

Eo

±

/A2

⁺ ⁶⁴ ^t~

^cos~ (qj

^~

^cos~ _(q~

^~ ,

(5)

2 2 2

which _are

independent

of the spm w, as

expected

for _an

antiferromagnetic

solution, and where A

=

U

mi, Eo

₌ E~~ + nU, ^b ₌ a

/

is trie

antiferromagnetic

lattice parameter, ^and 2

qj =

(k~

+

k~)/ À

_and

q~ = (k~ k~,)/

/

are the Cartesian components ^of ^the wave vector q

along

the axes of the

antiferromagnetic reciprocal

lattice.

When _m

=

0,

_equation

(5)

_comes down to

equation (1),

with

Eo

_in

place

of E~~, due to the introduction of the on-site electron interaction. But when _m #

0,

_a _gap _is

opened along

the constant energy hne the

equation

of which is

cos k~ a + cos k~, ^a ⁼ 0

(6)

The

density

of _states per copper atom,

including

trie two spm states, is

5l« (E j

=

£ jj s~j~ (E j

,

(7j

with

b 2

jj

^~ ^~

s~lu(E)

=

~

^d

q[Aj~(qj[ à(E-Ej). 18)

(5)

When

solving

trie

Schrüdinger equation,

_one finds

(Ali _(q)(~

₊

(Ait (q)[~

₌ i

independently

of _« and _v indices. Thus trie

density

of _states per atom is trie _same for trie two

copper atoms a = or 2 within _a _same unit

ceII,

and reduces _to

b 2

jj

5~(E)=ij(E)=d~~(E)= (-j _jj d~qô(E-E(). (9)

2 _w

~__

This Ieads to

i

[E-Eo[

"/ _~> ^(~°)

(E-Eo)~-~

Î

4 with

F

(E)

=

&i<(E)i &((E ^Eo)~ ^Ki<(E)i

,

(i i)

where

&[u]

= I for _u

m 0 _and 0 for _u

<

0,

f(E)

=

~

(E ^Eo

⁺

~j

(Eo

⁺

~ Ej

,

(12)

16t 2

and

~~~~~~~ 0

,

<(E)sin~u

^~~~~

m which _we recognise trie

complete elhptic integral

^{of trie} ^first ^kind as

only

(

(E

_m 0 bas _to be retamed in

expression II).

Trie total

density

of _states per copper atom is shown in

figure

2, _m trie

following

two _cases _:

(il

trie

paramagnetic

case

(Fig. 2a),

when _m

=

0,

and thus A

= 0,

Ieading

to

%~~~~(E)

= 2

d~1°~(E)

=

F1°~(E), (14)

w t

where

d~~°~(E)

is trie

density

of _states per atom and per spm, with

Fi°~(Ei

=

&1<~°~(E)i Ki<~°~(Eii

and

(~°~(E)= ~~(E-Eo+4t)(Eo+4t-E).

16t

It _is easy ^to show that d~~~~~(E) ^exhibits ^a

loganthmic

Van Hove

smgulanty

in trie middle

Eo

^of ^the band, ^this ^kind ^of

singularity being expected

^for a two-dimensional lattice _:

(ii)

the

antiferromagnetic

case

(Fig. 2b),

when _m #

0,

_m which the band is

split

into two

subbands

separated by

_a

non-vanishing

_gap A, trie previous

smgularity

at

Eo

in the

paramagnetic case

being split

mto two new

singulanties

at trie

edges

of trie band gap. Close _to

(6)

La~_~Sr~CUO~

427

E

flfpa~a(ll~

~a

E~ ^4t E~ E~⁺ ^4t

E

b

A _m imiu

flf(é~

E~+E~

E

E~-e~

E~ _E~ ^° E~

Fig.

2. Electronic density ^of states m the

paramagnetic

state (ai, ^and ⁱⁿ ^the antiferromagnetic state (b)

with _a gap ^A =

[m

U, Eo E + nU, _F~

=

,fi

_~~~

~, ~

l

~

~ 2 2 2

these

singulanties, d~(E)

_can be

approximated by

trie

following analytic expression

~

~~

~

/

~2

~~

~2

~~~~

~~ Î~ ^-

~ ^(E

~ÎÎ)2

~

Î Î

which in trie

previous paramagnetic

case would reduce to

d~~(E)=2fl°~(E)>

^Ln ^~~~

(16)

" t

ÎE~EOÎ

Since the band is

nearly

half-filled in these

compounds,

the Fermi level

E~

lies close _to _one of trie gap

edges.

^Thus we will _use in trie

following,

as a

good approximation

of

d~(E),

the

simplified

_version

(15) adequately

normahsed _tu unity

simply by changing

gr2 _into ₄

(1

+ 2 Ln 2

).

3.

Stability

of the

antiferromagnetic phase.

The

equilibrium

value of A

= m

U,

_as _a function of trie

temperature

^{T and} ^trie occupancy ^of trie

band,

which itself

depends

on trie

doping

ratio x m

La~ _,Sr~CUO~,

is that which minimises trie

grand potential D(T,

_p,

A),

for fixed values of T and of the chemical

potential

p. In the meanfield

approximation,

the

expression

of D

(T,

_p, A j, ^counted per copper atom, is

D(T,

_p,

A)

₌ _kB T

Îd~(E)Ln ^[1+ ^e~Pl~~~~]

^dE ^~₄

^(n~ ^-m~) ⁽¹⁷⁾

where

ks

_is the Boltzmann constant,

p

=

1/ks

T and trie

integration

_must be extended _over trie

two intervals for which A _w E

Eo

_w

~.

From section 2, %

(E)

is symmetn-

2 2

(7)

428 JOURNAL DE PHYSIQUE I N°

cal with respect to E

=

Eo

and normalised to

unity

for each _one of trie _two

subbands,

_as it includes trie two spm states. Thus expression

(17)

becomes

n(T, p,A)=

F2

_u

~ ~

=-p

+Eo-ksT p(F)Ln [2chpe+2chp(p -Eo)]de--(n

_-m

(18)

~~

4

with

e=E-Eo, p(e)=d~(E), ej=-),fi

_and

_e~=-)A.

As trie Fermi energy, or chemical

potential,

lies closes _to _one of trie

edges

of trie band gap, trie

grand potential depends only

_very little on trie _exact expression of _p

(e)

far from these

edges.

Thus, _we shall

replace

_p

(e) by

its

simplified expression (15)

extended to trie entire range of trie two

subbands,

but

adequately

normalised _as indicated _at trie end of section 2.

Then,

mtroducing

the dimensionless variable _a~

=

tilt,

as well _as the dimensionless parameters r =

ks

T/t, _q

= t/U, and a~~ =

(p Eo)/t,

X

=

mU/2 _t, and thus A

=

2 t

(X(,

one gets

n(r,

_7~~,

X)

t

=a~~-(j~Ln(~))Ln(2chfi+2ch)jda~+qX~-£ (19)

W o q

The chemical

potential

_p

(or

_{a~~) is} related to trie

doping

ratio _x

by

trie

following equation

fit1-X#-

(~

H T,~ ,

which leads to trie

equation

Sh

~~

4 j6

d~

~

(2°)

~2

^~~ _'Î

~

fi

~ '~F

C

~

~ ~

y

where _~~

depends

on x

through Eo

= E~~ ⁺

j

_nU.

By denvmg

_equation

(19)

with respect to X, for fixed values of T and _p, une gets

~~

=

2

tXg(r,

_~~,

X) (21)

ôX _T,

~

with

j16 ^~,/~~+X~

4~~

~

~''"PÎO ~/~~,fi~j~~'

^~~~~

r r

Equation (21)

vanishes either when X

=

0 _or when

g(T,

_~F, X

=

0.

(23)

(8)

La~_ôr~cuo~

429

For given values of _r and _x,

equations (20)

and

(23)

can be

numencally

solved tu determine

both ~~ and X. A solution with _a

non-vanishing

value of X will charactense _a stable

antiferromagnetic

state if the

corresponding

_extremum of D

(r,

_~~, X is _a minimum. An the

equation

of the frontier of trie

antiferromagnetic

domain _m trie

(x, ri plane

wùl be

g(r,

_~~,

0)

=

0,

with ~~ determined

by equation (20)

with X

= 0.

Now,

trie

stabihty

of trie paramagnetic state

(X

=

0) depends

_on trie

sign

of trie second denvative

lf) (X=0)=2tg(r, ~~,0). (24)

ôX _T.~

But it _is easy to show from

equations (20)

and

(22)

with X

=

0 that, for _a given value of r,

g(r,

_~~,

0) monotonously

_mcreases with _~r and

thus,

_as it vanishes _on trie frontier of trie

antiferromagnetic domain,

is

negative

inside this domain and

positive

outside

(Fig. 3).

This demonstrates that within trie

antiferromagnetic

demain D

(r,

_~~,

X)

bas _a local _maximum for X

= 0.

Thus,

_as

D(r, a~~,X)

_goes toward _an mfinite

positive

value when X

infinitely

increases, the previous solution of equations

(20)

and

(23)

with _a

non-vanishing

value of

X

corresponds

to an absolute _minimum of

D(r,

_a~~,

X),

and _so charactenses _a stable

antiferromagnetic

state.

o o.05 o-i o.15

~°'~

PARA

^°'~

M

o.1

~

_Xc ~

Fig. 3. Phase diagram _m the plane (x, r) showmg the transition fine between the antiferromagnetic (AF) and the paramagnetic (PARA) phases, calculated for _q

=

t/U

=

0.5.

The frontier between trie

antiferromagnetic

and trie paramagnetic

phase

m

figure

3 has been

numencally computed.

At zero

temperature (r

=

0),

equation

(20)

reduces to

x=

~~a~)(1+Ln ~~), ⁽²⁵⁾

iT 71F

with

a~/

=

fiwhen

x #

0,

but

a~/

₌ _a~~ ₌ 0 when _x

=

0.

Furthermore, when _r

=

0, the

equilibrium

^condition

(23)

becomes

~ iô

4 n

~

~~

_~

~

Î~j,fi _~'~

~~~~

The variation at _r ₌ 0 of the gap

(X

versus the

doping

^ratio x can be

numerically

calculated

(9)

from

equations (25)

and

(26).

The result is

plotted

in

figure

4. From

equation (26),

the critical value xc of _x for which X vanishes _at _r

=

o

corresponds, by

equation

(25),

to the value

a~/c

^of

a~/

the

expression

of which is

7~

)~

= )6

~-'fi

~~~

which, ^for _q »

0.05, roughly gives a~)c

=

16 _e~ ^~ "

à

0 0.02 o-M 0.06 0.08 o-1

x~

^0A

~0.3

0.2

(X(

o-1

o

~ x

X

~

Fig.

4. The variation of the reduced gap

[X[

₌

[m[

U/2 _t _versus

x at r = 0, calculated for t/U

= 0.5.

The gap X gets ^its maximum value

Xo

^when x = o and thus

a~)

₌ ^o.

Therefore,

for

r = o,

Xo

is determined

by

condition

(26)

with

a~)

=

o,

which leads _to _:

~

"~

~ ~~

Î _~~~~~ ^o ^)~ ) _~~

_~~~~

One _can subtract trie

vanishing

contribution from trie last

integral

_m

equation (28)

~

~~~~~

)

_dY

~

°

o + y

Then, looking

for _an approximate ^solution

Xo

« with trie remaming ternis of equation

(28),

one finds

Xo

~

32

e~'&

_«~q

+ o.33 ~~~

which, for q ^»

0.01, roughly

grues

Xo

_~ 32 e~ ^~ ^"

à

Thus trie

expression

for trie maximum

value mo =

(2 t/U) Xo

of trie magnetic moment per copper ^atom m trie

antiferromagnetic phase

at zero temperature ^is

roughly

mo _m 64

e-2«@

~

(30)

A

striking

feature of

expression (30)

is trie

exponential dependence

of trie last factor _on

ÀÎ

_instead _of

_t/U,

as wùl be found for trie

ferromagnetic

solution _m section 4.

To compare trie stabilities of trie

antiferromagnetic

and

ferromagnetic phases

at zero

(10)

La~_~Sr~CUO~

431

temperature, we will bave _to _use in section 4 the

following expression

of trie

grand potential

per copper atom in trie

antiferromagnetic phase

at _r ₌ o

~

~~'

_~~

~~

=

(l x)

_a~F

~

j~ fi

Ln ~~

da~ +

qX~ (31)

t gr

~j

^a~

The _curve in

figure 3,

which separates ^trie

antiferromagnetic

from trie

paramagnetic domam,

grues trie variation, versus the

doping

ratio _x, of trie critical temperature _r~ at which the _atomic

magnetic

moment on each copper site vanishes. As

long

as x remains much smaller than xc, r~

slowly

decreases _as _x _increases, with _a

vanishing slope

for _x

= 0. Trie value ro~ of _r~ for _x

= 0 is determined

by

^the condition

g(ro~,

_a~~

= 0, X

=

0)

= 0,

leading

to 2 gr~ _q

=

j~ ^~'~

^Ln ^~~ ^th _~ ^'~

⁽³²⁾

o 71 71 TO

m

Equation (32)

is

entirely

similar _to that

determining

the

superconducting

critical temperature m

trie _same

compounds,

which had been denved in _a

[4]. Solving

it

by

^trie same

method,

_ones finds

~°m " 18.14

e'~

~~~~

which,

for _q _»

ù-ù15, roughly gives

_ro

~ ^-

l 8.14 _e ^~ ^"

à.

The _curve in

figure

3,

exhibits,

above trie critical

point (r

₌

0,

_xc

),

_a

prominent

part which

actually

bas _a clear

physical meaning.

To understand

it,

_one

merely

bas to realise that trie

mstabihty

of trie

paramagnetic phase

arîses from the Bloch states

Iying just

below trie middle of trie band in this

phase

because trie energies of these _states _are decreased

by

trie

opening

^of an

antiferromagnetic

_gap when it

happens. New,

when _x becomes

Iarger

than xc

along

trie axis

r =

0 in

figure

3, trie Fermi Ievel _moves away from trie middle of he band and trie

paramagnetic phase

recovers its

stability. Then,

when

starting

from _a

paramagnetic

state with

r =

0 and _x

just

_a little

larger

than xc, the temperature r is increased with _x

kept

constant, the

smearing

of the Fermi Dirac distribution

partly repopulates

the states

lying just

below trie middle of trie band and, for

r

larger

than _a critical value

(which

of _course

depends

_on

x),

the

antiferromagnetic phase

_recovers its

stability.

But this

stability

of the

antiferromagnetic

phase

it lost

again

after _a further increase of _r

large enough

to excite _a

non-negligible

number of electrons mto trie _states above the

antiferromagnetic

_gap.

By expanding

expressions

(20)

and

(22)

with X

=

0 for _r much smaller than a~~, one obtains the

following approximate

equation

of trie _curve m

figure

3 in the immediate

neighbourhood

of the critical

point (r~

₌

0, xc)

12

y 2

x m xc ⁺

Ù j ₍₃₄₎

~ 'Î_FC

Equation (34),

which is valid for r~ ^« a~

)~ only, clearly

^shows ^that x staffs increasing from its initial value xc when r~ ^mcreases from _zero.

Trie numerical values of the parameters ^U ^and t in these

compounds

_are

extensively

controverted

by

different authors. From _our

results,

two

physical quantities depend

_on the

single

dimensionless parameter q = t/U. The first one is the _maximum value mo pB of the

magnetic

moment per copper atom, where _pB _is the Bohr magneton ^and mû is given

by

equation

(30).

The second _one is the cntical

doping

ratio xc ^at which the atomic moments _on copper vanish _at T

= 0, and which _can be calculated from trie

equations (25)

and

(27).

Another

(11)

mteresting physical

quantity ^is ^trie ^maximum ^value

To~

=

ro~t/k~,

which _occurs for

,r =

o,

of trie critical temperature

T~

at which trie atomic moments on copper vanish. with ro~

given by equation (33).

It _seems _to _exist _a

general

agreement ^for trie measured value mû =

0.4 in trie

compound La~CUO~ [5, 6]. By introducing

this value into

equation (30),

_one _gets t/U

>

0.475,

and then, from

equations (25)

and

(27),

one finds xc > 0.095.

Besides,

with for instance t ₌ 0.5

eV, equation (33)

_grues

To

~ ⁼

I 390 K

(a

direct numerical calculation from equation

(26)

leads to more exact values q w

0.496,

_xc

~ 0.097 and To

~ ^>

I 270

K).

It must be

emphasised

that the cntical values of the

doping

rate x and the

temperature

^T ^which ^bave ^been calculated in this

section are those for which trie _atomic

magnetic

moment goes ^to zero, and

they

_are

certamly

larger

than trie other critical vlues of _x _or

T,

which remain _to be calculated, for which trie

long-

range order between the non yen

vanishing

atomic moments

disappears.

From ibis

point

of

view, trie above calculated numencal values of xc and To

~

are not unreasonable, _as

compared

to trie extension of the

antiferromagnetic

domain in the measured

phase diagram

of these

compounds,

which appears to exist for,r smaller than

roughly

0.03 and T smaller than _a few hundred of kelvins.

Finally, by

using trie above numencal value of q, we bave calculated trie map of trie atomic

magnetic

moments for _a discrete set of values of _x and _r. Trie results _are shown _in

figure

5.

o

o.2

~

~~

x

o.2

~

o-1

0

~

Fig.

5. The map of the atomic magnetic moments for _a discrete _set of values of _x and

r calculated for t/U

= 0.5.

4.

Companson

with _a

ferromagnetic

solution.

In _a

ferromagnetic

state, the

penodicity

is the _same _as _m trie paramagnetic state, trie expectation ^value

(n~)

of the number operator n~~

being

site

independent,

^and thus also the magnetic moment m =

(ni ) (nj )

_per _copper site. In the Hartree-Fock

approximation,

the

dispersion

relation

(1)

has

simply

to be

replaced by

a new one

E~(k)

=

E(k

₊ U

(n_ ~) (35)

By mtroducmg

here agam the

expectation

value _n

=

(ni )

₊

(ni )

of the total number of electrons per copper

site, equation (35)

_can also be written _as

Eu (k)

=

E

(k)

₊

) _(n

wm

(36)

(12)

N° 3 ANTIFERROMAGNETIC STABILITY IN La~

_~Sr~Cu04

433

As _a result from the _new

dispersion

relation

(36),

the band is

split

into _two identical subbands with

opposite

_spms _w

= ±

,

shifted from trie band in the

paramagnetic

state for _one spm

by

trie two

opposite

_energies wUm. Trie

densky

of _state of trie subband with trie spm

2

w is

S)[(E)

=

~1°~(E

+ w

~"~

(37)

where

d~1°~(E)

is trie

density

of _state per copper atom and per spm in trie

paramagnetic

state.

Trie expression of trie

grand potential

_per _copper atom m a

ferromagnetic

state is

D'(T,

_p, _m

= kB ^T

jj S)[ (E )

Ln [1 ₊ e~ POE _~

~] dE

) _(n~ _{nt~ )} ₍₃₈₎

«

where the

integration

is extended from

Eo

⁴ t wUm _to

Eo

+ 4 t wUm.

By

_usmg the

2

density

of states which results from

equations (37)

and

(16),

and

by introducmg

the dimensionless variable _a~

=

(E-Eo)/t

^and ^trie dimensionless parameters r

=kBT/t,

X

=

mU/2 t, q =

t/U and a~~ =

(p Eo)/t,

_ones gets

~''~'=a-~j~~Ln(Ln(2ch~+2ch'~ _'~~)da~+

2gr _4 71 T T

~2

+

qX~ (39)

4 _q The chemical

potential

_p

(or

_{a~~) is} related to the

doping

ratio,r

by

trie condition

~~

ôD'j

ôH _T,,n

leading

to trie

followmg equation

~Jp ~J

+4 16

~~

~ d_a~ ~~~~

~

2 7r~

-

4

~~

Î'~

ch ~

+ ch

~~ ~

~ r

The

equùibrium

value of the

magnetic

moment m =

2 tX/U per copper atom is determined

by

the condition

(ôD'/ôX)~

~ ⁼

0, which leads to the condition

4gr~qX=sh~l~Ln

^~~

^~'~ ⁽⁴¹⁾

T ~ Î711 X 71F~71

~~

r

~~~

r

For given values of _r and _x, trie

equilibrium

values of X and _a~~ _can be

numencally

calculated from equations

(40)

and

(41).

A

ferromagnetic

state is characterised

by

trie _existence of a stable

solution with _a

non-vanishing

value of X.

At

first,

let _us compare the stabilities of the

ferromagnetic

and trie

antiferromagnetic phase

for _x

= o and _r

=

0. When _x

=

0, equation (40)

_grues _a~~

=

o

anyhow

is the value of

r. When _x

=

0 and

r = 0, _expression

(39)

reduces to

~

x~

_Ln x

~ ~

3 + 8 L~ ~

x~ ₍₄₂₎

~'~~j~'p~~~~~

2gr~ 4"~

(13)

Expression (42)

_gets its minimum when X is

equal

to

Xi

=

16 e~ ^~~ "~~

(43)

with the value

D '

(0, 0, Xi

₄

X(~

=

-j(1+4Ln2)-j. (44)

t gr 4gr

For _a _same value of q, these results have _to be

compared

with

Xo

and D

(0, 0, Xo)

in the

antiferromagnetic phase.

The value of

Xo

is given

by equation (29).

If _we

replace

q m the term

qX( by

its

expression resulting

from

equation (26),

with a~~ =

a~)

₌

0,

we get

n(o,

o,

Xo)

₄

j4

(1J

/~)~

16

t

~2

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

~2

~

fi

^~~ ~

~'~

~~~~

71 + o

The

mtegral

in

equation (45)

is

easùy

^calculated

by introducing

trie new variable _u defined

by

a~ =

Xo

^sh u. One finds

~ ~~~

~ ~

~ ~ ~~ ~

~~

~ 4

Î~ià

_~

_~~~~~

~°

^~~~~

The numerical evaluation of

equations (44)

and

(46),

with

Xo

^and

Xi

_given

by equations (43)

and

(29) respectively,

shows that trie

antiferromagnetic phase

is _more stable than the

ferromagnetic phase

_as far _as q is

langer

than the small critical value qo >

0.092,

and thus _as t is

larger

^than trie critical value 0.092 U. For q much

larger

than qo, as it must be assumed for

our itmerant model _to be valid, the minimum of trie

grand potential

for,r

= 0 and for

r = 0 is much lower _m trie

antiferromagnetic

than _m trie

ferromagnetic _phase,

with _a

larger

atomic

magnetic

moment. For _instance with q =

0.5,

_one finds

D

(0, 0, Xo)

D

(0, 0, 0)

mû = 2

qXo

~

> 0.4 and

> 2 _x 10~

m trie

antiferromagnetic phase,

and

D'(0, 0, Xi D'(0, 0, 0) mi

~

= 2

qX(

> 0.003 and

> 2.5 _x 10~

in trie

ferromagnetic phase,

with

D'(0, 0, 0)

=

4(1

+ 4 Ln 2

)Iv

_~.

As _m section

3,

the critical value

xi

of the

doping

_ratio _x for which the atomic

ferromagnetic

moment

m=2qX

would vamsh _at _zero temperature can

easùy

be calculated from

equations (10)

and

(41)

with _r

=

o. For _r

=

0,

equation (41)

reduces _to

~F ^+X ~~

4

gr~ qX

=

Ln da~

(47)

~~ -X

Î'Î

For much smaller values than a~~, as is the _case when _x

approaches

the critical value

xi,

the

mtegral

m

equation (47)

can be

expanded

with respect to

Xla~~. By retaining

trie first

(14)

N° 3 ANTIFERROMAGNETIC STABILITY IN La~

_~Sr~Cu04

435

non-vanishing

terms

only,

_one _gets

4 gr~

qX

> 2 X Ln ~~

+

~~

~

(48)

71F 3 a~~

Equation (48)

bas two solutions. Trie first _one _is X

=

0 and trie second _one is the

positive

solution of the

following equation

X~

= 6

a~j(2 ^gr~q

^Ln

~~

(49)

71F

It

exists,

and is then trie _more stable

solution,

_as for _as a~~ <

a~)(,

^with

a~)(

₌ ¹⁶

e", ₍₅₀₎

Besides

equation (40)

with _r

= 0 and X

=

0 leads to

xi

₌

'~~) 1'

^l ⁺ ^Ln

~(, (51)

a~~c

and

thus, by using expression (50)

of _~

il,

_to

x)

=

~~ ^(l

⁺ ²

gr~q) e". ₍₅₂₎

w

It _is

interesting

to compare trie cfitical value

xi

for trie

ferromagnetic solution, given by equation (52),

to the

corresponding quantity

_x~ for trie

antiferromagnetic

solution, which from

equations (27)

and

(25) roughly

is

equal

to

xcw

~~(l

+2gr

à)e~~"à. ₍₅₃₎

gr

It appears that for _q much

Iarger

than trie

already

found critical value qow

0.029,

xi

_is much smaller than xc. For

instance,

with q =

0.5,

_one finds

x[

m

0.01xc.

The critical temperature

r(~

at which the atomic

ferromagnetic

moment m = 2

qX

would

vanish for _x

= 0 is

given by equation (41)

with X

=

0 and ~p =

0, leading

to

2 gr~

qr(~

=

~

Ln ~~

~~

(54)

° ~ l ₊ ch

)

~o m

For _q much

larger

than the previous critical value qo, trie

approximate

solution of equation

(54)

is

easily

found _to be

r(~

w 18.14 e~ ^~"~~

(55)

The

comparison

of the

expression (55)

of

r(~

in the

ferromagnetic

state with the _ex-

pression

(33)

of _ro~ _m the

antiferromagnetic

state shows that, for _q much smaller than

qo, ro~ is much

Iarger

than

ri

~.

For instance, with q =

0.5,

_one finds ro~ > 200

ri

~.

The conclusion of this _section _is that,

although

the Stoner condition

Ufl~1°~(Ep)~

l is

automatically

satisfied when trie Fermi energy

E~

coincides with trie energy

Eo

at which trie

density

of state

d~1°1(E)

bas _an infinite

smgulanty,

trie

antiferromagnetic

state is much _more

(15)

stable than the

ferromagnetic

state as far _as t is much

larger

than 0.092

U,

whatever trie values of trie temperature and trie

doping

ratio _are. Thus, _an itinerant electron model is also able to

explain why

a

ferromagnetic phase

has _never been observed in these

compounds.

This follows from the fact that the

nesting

of trie two-dimensional Fermi surface for _a half-filled, _or

nearly

half-filled, subband,

which is the _cause of

antiferromagnetism

in such an itinerant model,

produces

a much stronger ^effect ^than ^that

resulting

from the _existence of _an infinite Van Hove

singularity

in trie middle of

Eo

of trie subband, which would

give

use to

ferromagnetism

if it

were not dominated

by antiferromagnetism.

Trie effect which would be

produced by

the

infinite

singularity

is _non dominant because this

singularity

is

only logarithmic,

and thus is _not very

Sharp.

5. ERect of the fluctuations _on the

density

of _states.

One of the effects of the fluctuations is _to introduce _an orientational disorder among the atomic

moments. This

important phenomenon

^has

already

been considered

by

several authors. In this

section _we deal with another

phenomenon consisting

in the fluctuations of the

magnitude

of each atomic moment around ils

equilibrium

value, and the

resulting

effect _on the electronic

density

of the _states in the paramagnetic

phase.

It is

always

a very

complicated

matter to calculate the effect of fluctuations, as all the

physical quantities

fluctuate around their

averaged

values.

But,

in order _to

simplify

the calculations, _we will _not take into _account the fluctuations of trie temperature ^T ^and ^the

chemical

potential

_p. For fixed T and _p, trie value of trie atomic magnetic moment

m ₌ 2

qX

which minimises trie

grand potential

is

only

the _most

probable

value, around which it fluctuates. So, all the other values of _m, between and ₊ 1, bave _some

probability

to exist

physically.

In _a

paramagnetic

state, m fluctuates around its

equilibrium

value, which is

equal

to zero.

But when this

paramagnetic

_state _is close

enough

to trie transition hne, trie correlation

length

is

expected

to be

large,

and thus _m _to

keep

a

nearly

constant

value, equal

_or not to ils

vanishing equilibrium value,

_over _a

large

distance. Then, at a given time, trie

crystal

can be

thought

of _as

being

made of _a

large

number of different extended _areas. The atomic

magnetic

moment

m ₌ 2

qX

varies from _one _area _to

another,

but it

keeps

the _same value _on all the atomic sites

within each _area, _to which _a local electronic

density

of _states

àj~~(X, E),

as given

by

equation (15),

can thus be

assigned.

In each _area, the electron number per copper ^site bas the local value

dE

(56)

nj~c(x)

₌ ^I

Xloc(X)

_"

°~'~~

i ₊

e~~~

^~~

In trie above equation, ^trie chemical

potential

_p is assumed to bave _a well defined uniform value,

mdependent

of X,

throughout

trie whole

crystal.

This allows _us _to _assume that trie

electrons _are

mstantaneously

redistributed between trie _vafious _areas of the

crystal,

_so _as _to

keep

_a uniform value of _p, _even if trie chemical composition is

homogeneous.

When

calculating

the total number of electrons in the whole

crystal

for fixed values of T and _p, the statistical

weight

of trie contribution ansmg from all trie areas with _a given value of X must be calculated from trie

grand potential D(T,

_p,

X),

and is

equal

to

P~ ~(X)

₌

poe~P'~l~.~~~~~~l~.~~°~' (57)

with

I

=

'

~-PiniT,~,xi-n(T.~,

oj) dx_

P0 _1/2

q