Magnetic properties of antiferromagnetic two-singlet systems. I. Theoretical phase diagram

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Magnetic properties of antiferromagnetic two-singlet systems. I. Theoretical phase diagram

R. Bidaux, A. Gavignet-Tillard, J. Hammann

To cite this version:

R. Bidaux, A. Gavignet-Tillard, J. Hammann. Magnetic properties of antiferromagnetic two- singlet systems. I. Theoretical phase diagram. Journal de Physique, 1973, 34 (1), pp.19-26.

�10.1051/jphys:0197300340101900�. �jpa-00207351�

MAGNETIC PROPERTIES OF ANTIFERROMAGNETIC

TWO-SINGLET SYSTEMS. I. THEORETICAL PHASE DIAGRAM

R.

BIDAUX,

A. GAVIGNET-TILLARD and J. HAMMANN Service de

Physique

^{du Solide et} de Résonance

Magnétique

Centre d’Etudes Nucléaires de

Saclay

BP n°

2, 91, Gif-sur-Yvette,

^France

(Reçu

^{le 7}

juin 1972)

Résumé. ²⁰¹⁴ Le

diagramme

^de

phase

^de

l’antiferromagnétisme

^{dans un}

système

^{à deux}

singulets

est étudié dans le cadre de

l’approximation

^de

champ

moléculaire. Différentes interactions donnent pour la transition

métamagnétique

^{un réseau de} courbes d’aimantation dont on donne les _expres- sions

analytiques

^ainsi que les

propriétés principales (champ seuil,

température

critique, etc...),

^et

amènent à reconsidérer _pour les ions ^« non-de-Kramers » le

problème

^de

l’énergie

libre. L’influence de l’écartement des deux niveaux

d’énergie

^sur la transition

métamagnétique

^montre

qu’il

^existe

une

analogie

^{entre un}

système

^{à moment}

magnétique

induit à 0 °K et un doublet de Kramers à

température

^{non nulle.}

Abstract. ²⁰¹⁴ The

phase diagram

^{of an}

antiferromagnetic two-singlet

system is studied within the frame of the molecular-field

approximation.

Different interactions

give

^{for the}

metamagnetic

transition a net of

magnetization

^{curves whose}

analytical expressions

^are

given ;

their main _pro-

perties

^are considered

(threshold field,

^critical temperature,

etc...) along

^{with the}

problem

^{of the}

free _energy of « non-Kramers » ions. The influence of the

splitting

^{of the} energy levels _upon the

metamagnetic

transition suggests ^an

analogy

^{between an} induced-moment system ^{at 0 °K and a} Kramers doublet at non-zero temperature.

Classification Physics abstracts :

17.68

1. Introduction. ^- The

magnetic properties

^{of rare-}

earth

(RE) compounds

^{where the}

crystal-field ground-

state of the RE ion is a

singlet

^{have been studied}

extensively recently

^both

theoretically

^and

experi- mentally [1]-[14] ;

the main feature of these « non-

Kramers » ions is that

large enough magnetic

^inter-

actions can induce an order

through

^the

mixing

^{of the}

ground

^state with the upper levels.

The case of two

singlets

^well

separated

^{from upper}

levels is of

particular

interest from two

points

^{of view :}

a)

^a very

simple

^{model is}

appropriate

^and

b)

^a

compari-

son with the

properties

^{of «} Kramers » ions with ^a fundamental doublet can be made and the effects of the _energy

splitting

^{L1 of the two levels can then be}

analysed

^{in detail.}

The theoretical works so far have been oriented toward the

study

^{of the}

magnetic

transition itself when different

approximations

^are used like the molecular- field

approximation (MFA)

^or the random

phase approximation [8], [9], [10].

^In

^fact,

^{a very}

important thing

^{is to} understand what interactions and what mechanisms are involved in the

magnetic systems

studied. From this

point

^of

^view,

^the

phase diagram

of an

antiferromagnet

^{at 0 OK is} very

important :

the value of the threshold field of ^a

metamagnet gives precious

informations about the

interactions,

^{even in a}

simple approximation

^{like the MFA which is valid near}

the absolute zero.

We intend here to examine in detail the

phase

^dia-

gram in ^an external

magnetic

^{field of a}

two-singlet antiferromagnet

^{in the}

MFA ;

^{for this we shall}

study

the molecular-field

equations giving

^the

magnetiza-

tion and determine the free _energy of such a

magnetic system.

^{A later} paper, which we shall refer to as II from now on, will be devoted to the

interpretation

^of

extensive

experimental

^{data on Terbium Aluminium}

Garnet

(TbAIG),

for which the

two-singlet

^{model is}

expected

^{to be}

appropriate [15]-[16],

and which will

provide

^an excellent illustration of the

present

^theore- tical results.

We shall first review

briefly

^the

properties

^{of the two-}

singlet

model that are

already

known. Then in section

3,

we shall turn our attention to the

free-energy

^{of our}

particular

induced-moment

system.

In section

4,

^the two-sublattice « non-Kramers »

antiferromagnet

^will

be studied

through

^its

magnetization

^{versus field curves}

as a function of

temperature

and of the interaction

parameters.

^{In section 5 a}

general

method for the determination of the main

properties

^{of its}

phase diagram

^{will be}

proposed,

and the role of d will be

emphasized.

^{As a}

conclusion,

^{we shall} outline how the

previous

^{results can be used in the}

description

^of

more

complex systems,

like TbAIG.

2.

Two-singlet

^{model. -} 2. 1 TIME REVERSAL OPE- RATOR. ^-

Let p

>

and

> be the kets corres-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197300340101900

20

ponding

^{to the two}

singlets,

^{and let us call T the time}

reversal

operator (and ^T+

^its

conjugate) ; ( p

> and

1

q > ^can be assumed to form a real

basis,

^{i. e.}

If p

^{is the}

magnetic

^moment

operator (in

^Bohr

magneton units),

^then

which leads to

and

(oc

^stands

for x, y

^{or z,}

assuming

^{that a set} of orthonor- mal ^axes

Oxyz

has been chosen at the RE

site).

Assume that two

kets p’

>

and q’

> ^are deduced

from 1 p

>

and

>

by

^an

orthogonal transformation,

it is then

straightforward

^{to show that}

p’ l poe 1 q’ > = ima .

The above results have been reviewed in detail

by

^Griffiths

[11] and

^are

quite important :

^the

magnetic

moment ^m of the ^rare earth

ion,

^{if it}

exists,

^will

point along

^a well defined

crystallographic

^{axis that}

depends

upon the space

spanned by { ^p >, q

>

},

^{but not}

upon the

particular

basis chosen within this _space.

Let us illustrate this

by

^an

example :

^{assume that we}

are at a site of orthorhombic

symmetry (group

^D

2).

Then

by inspecting

^a character table of D 2

(see

Tinkham

[18])

^{it is easy to} determine that the

magnetic

moment ^m will

necessarily

^be

parallel (if

^it

exists)

to one of the local axes of orthorhombic

symmetry.

2.2 MAGNETIZATION. - Let us label Oz the ani-

sotropy

axis. We have then

If a

magnetic

^{field h is}

applied

^{at the RE}

site,

^the

Hamiltonian is

(X,

⁼

two-singlet crystal-field Hamiltonian).

^This

Hamiltonian reduces to the

diagonalization

^{of a}

2 x 2 matrix. The _energy

splitting

^becomes

and the

magnetic moment m_,

^{at a}

given temperature

^{is :}

(k

^{= Boltzmann}

constant) .

On

figure 1, mZ/ms

^versus

ms hz/Ais plotted

^for

different values of

77 J.

^For

TIA

^{= 0 it} looks like a

Brillouin function where A

plays

the role of a non- zero

temperature.

^{We shall encounter and}

emphasize

this behaviour in all the results to follow.

FIG. 1. ^- Reduced magnetization ^versus reduced field for different values of the ratio KTIA, ^{the field} being applied along

the anisotropy ^direction.

2.3 CONDITION FOR A TRANSITION TO A MAGNETIC ORDER. ^- A

magnetic

^order appears if the interactions

are

large enough [2], [3].

^{Let us}

study

what is the condi- tion for a

magnetic ordering

^{in the}

present

^case within the molecular-field

approximation.

If all the RE ions have the same

magnetic

^moment

(in magnitude)

^{in no external}

magnetic ^field,

^the

effective field

along

^{Oz is :}

Then at

temperature

^T

leading

^{to a critical}

temperature given by

and the condition for the existence of the

ordering

^is

obviously

3. Free _energy of a

two-singlet system :

^molecular-

field treatment. ^- The

anisotropy

direction in the

two-singlet

^model

implies metamagnetic properties

when the stable structure at 0 OK in the absence of a

magnetic

^{field is}

antiferromagnetic.

^When

studying

the behaviour of a

metamagnet

^{in a}

magnetic field,

it is essential to have in hand the

expression

^{of the free}

energy of the

system,

since it may be used in the calcu- lation of the threshold field. It will be _necessary in the

interpretation

^of

specific

^heat measurements.

In order to calculate the free _energy in the MFA

we can use

Bogoliubov’s

variational

principle [19].

3.1 FERROMAGNETIC CASE. ^- Consider ^a

crystal containing

^N rare-earth ions

(with

^two fundamental

singlets separated by A),

^with interactions

leading

^to

ferromagnetic ordering.

The presence of the aniso-

tropy

direction Oz at a

particular

ion site in the two-

singlet

^{model leads to an}

Ising-like

hamiltonian. The external

magnetic

^{field H can be assumed}

parallel

^to

Oz without loss of

generality.

^Then the hamiltonian JC is

where

lij

is the interaction factor and

Rk

^{is the}

crystal-

field Hamiltonian for the ion i

(Hic

^is

^diagonal

^{in the}

space

spanned by p

>

and 1 q > ).

The trial Hamiltonian in the

Bogoliubov

^{method is}

then

(in

^the

MFA).

h

being

^{the effective field}

along

^Oz.

If F is the free energy of the

system, Bogoliubov’s inequality

^infers

that

with

The

symbol > 0

^{refers to an}

expectation

^value

in the space

spanned by

^the

eigenkets

^of

Ro

^and

T w T , v -1

Using

^the variational

principle,

^{we can minimize F}

with

respect

^{to the} molecular field h :

This

gives

the molecular field

equation

where m ⁼

uzi

> is determined

by

^the

implicit equation

with

If we

replace

^{h in}

F by

^its

expression,

^and

explicit Zo, we get

and the energy U ^at 0 OK is :

or, in an alternative form :

These results for

F

and U are

interesting ;

^the energy U does not have the usual form and is rather a « ma-

gneto-crystal

^{field »} energy,

resulting

^{from the addi-}

tion,

^{to a}

strictly magnetic part,

^{of a}

« magneto-

electrostatic » term

Then U looks like a free energy, L1

playing again

^the

role of a

temperature

and its factor

being analogous

^to

an

entropic

^term.

It is also trivial to check that

which is the

expected

^{result for a doublet.}

3.2 CASE OF A TWO-SUBLATTICE ANTIFERROMAGNET.

- Let _mi and m2 be the

magnetic

^moments of the ions

belonging

^to sublattices 1 and 2

resptctively ;

^a

treatment

analogous

^{to the one}

developed

^{for a}

ferromagnet gives

^{two effective fields}

and the free energy

F is

with

22

If no external

magnetic

^{field H is}

applied

^{to the}

system

^then

and

Then U becomes

The molecular-field treatment ^can be

adapted

^to

more

complex

^cases very

easily.

4.

Metamagnetic properties

^{of a} two-sublattice

antiferromagnet.

^{- Let} us assume a two-sublattice

metamagnet.

^{If the}

magnetic

^{moments of the two}

sublattices are mi and _m2 we have then :

with

tanh

The

problem

^of

solving

^{these two} simultaneous

equations

^{can be}

simplified

^{if we} introduce dimension- less variables and

parameters :

=

ferromagnetic

^order

parameter

=

antiferromagnetic

^order

parameter .

Then

F(H)

^becomes

£(h)

^{and the}

system

^{to be solved} is :

It is convenient at this

point

^{to use :}

BF = j

^{+ 1 =}

ferromagnetic

molecular-field cons- tant

BAF ^{= j - l}

⁼

antiferromagnetic

molecular-field constant .

Let us look first at the solutions obtained at T ⁼ 0 OK. On

figure

^{2 are}

plotted

^{curves of}

m/ms

versus h

for

diffèrent values of the ration

f3 APl f3F’

{3F being

^{constant. The saturated}

paramagnetic phase depends

upon

f3F only.

^{We see that for a}

large enough

value of the

ratio,

^the

metamagnetic

transition is ^a first order one. The threshold

field hs

^{can be deter-}

mined

by

^the calculation of the free

energies

^{in the AF}

(antiferromagnetic) phase

and in the P

(saturated paramagnetic) phase,

^or

by

^{the use} of the Maxwell rule.

When

f3 AFI f3F decreases,

the transition _goes from first to second

order,

^and

finally

^vanishes.

Let us

point

^{out that even}

though

^{we are at zero}

temperature,

^the

magnetization

^{in the AF}

phase

^{is not}

null. This is

again

^a

temperature-like

effect of L1.

The effect of the

temperature

^{on the}

metamagnetic

transition is shown ^on

figure

^{3. If we start at T = 0 OK}

.u5 . 1 -)3

FIG. 2. ^- m/ms ^versus the reduced external magnetic ^field

for different interactions.

with a first order transition and increase T the transi- tion _goes from first to second order. The

phase

^dia-

gram of this transition is

reported

^on

figure

^4.

The

simple examples of figures

^{2 and 3} show that the interactions as well as the

temperature

^{influence the} order of the

metamagnetic

transition. A few

questions

arise

naturally :

^{what values of}

f3 AF

^and

f3F

^{will lead to}

a first or a second order transition ? How can the critical

temperature (at

^{which we} go from first to

second

order)

^be calculated ? Can the threshold field be evaluated ?

Therefore a

general study

^{of these}

properties

^must

be undertaken.

FIG. 3. ^- m/ms ^versus the reduced external magnetic ^{field at}

different temperatures. The interaction constants are

flF ⁼ 0.25, PAF ^{= 0.75.}

FIG. 4. ^- Reduced threshold field versus T/TN. ^{AF refers to}

the antiferromagnetic phase, ^{and P to the saturated} parama-

gnetic phase. The interaction constants are 8F ^{= 0.25 and} PAF ^{= 0.75.}

5. Phase

diagram

^of two-sublattice

antiferromagnet.

- We propose ^to deal with a two-sublattice

system

^of the form

where

hl

^and

h2

^{are the} reduced effective molecular fields

acting

upon sublattices 1 and 2

respectively (see

^section

2).

^{At this}

point, f,(h)

^can be any

analytic

function. Such

symmetric expressions imply only

^that

the sublattices 1 and 2 are

magnetically equivalent

and that the

magnetic

^{state of each} sublattice is characterized

by

^{one and}

only

^one observable

(m 1

and _m2

respectively).

It is

straightforward

^{to show that}

In order to see how

(4) gives m(h)

^{for a}

given t,

^we

can solve the

equations

for ^a

given

^{value of v.} We find then two solutions ul ⁼

hl

^and u, ⁼

h2.

^In the cartesian space defined

by

^u

and v,

^we

get

^{then two}

points Ml (h 1 ; v)

^and

M2(h2 ; v). The"middle of Ml M2

^is

M[(Ai

⁺

^{h2)/2 ;} v],

otherwise

’

If u and ^v are the unit vectors on axes u

and v,

^{it can}

be shown

easily

^that

This is an

important result ;

the coordinates of M

are

(h, m/mS)

^{in terms of the basis vectors that are}

between brackets : this

system

^of coordinates is determined

uniquely by

the interactions

(the

^{h axis and}

the

asymptote of gt(u) being parallel)

^{so that the whole}

net of

magnetization

^{curves is available for all t}

without

manipulations.

S .1 APPLICATION TO THE TWO-SINGLET CASE. ^- We have now

(with

^{t = 2}

kT/d).

Figure

^{5 shows a}

family

^{of curves}

f,(u)

for different values

of, j

^{and 1}

being given.

Let us discard first the case for which

gt(u)

^decreases

monotonously.

^This

corresponds

^{to a}

paramagnetic

state of the

system,

^the eq.

(5) having

^then

only

^one

root in u ; this root is u ⁼ 0 for v ⁼ 0. The limit value of t is when

gt(u)

^{starts with a zero}

slope

^{at u =}

0, defining

^{so the Néel reduced}

temperature tN

^{as :}

which is

simply

eq.

(2).

24

FIG. 5. ^- Representation ^of

v ⁼ gt(u) ^{= 2 u tanh}

(JI

^{+ 4}

U2/t)/ JI

^{+ 4 U2 - u/( j - 1)}

for different values of t.

From now on, let assume t

tN.

^{The locus of}

r , 1 1 1

is characterized

by

^three

regions (see Fig. 6) :

-

C2

^and

C3 correspond

^{to a}

paramagnetic

^{or a}

ferromagnetic

^{state :}

hl = h2

^and m i ⁼ m2 ;

- C1 corresponds

^{to an}

antiferromagnetic (or

rather

metamagnetic)

^{state. That is}

h 1

>

h2

^and

mi > m2.

m(h)

^is

everywhere

^an

analytic

^function.

FIG. 6. ^- Representation ^{of v =} gt=o(u) (dotted ^line plus ^curve C3). Ci corresponds ^{to the locus of} points ^{M = middle of the} intersections ^of gt=o(u) with horizontal lines. When M reaches A, ^then hl ⁼ h2 and M follows C3. The h axis is parallel ^{to the} asymptotes. The orientation of the m axis depends upon the

interactions.

The direction of the h axis is

parallel

^to the asymp- totes

of gt(u). According

^{to the values} of the interaction

constants,

^we may have different situations :

a)

¹ >

0 ;

^{the m axis} lies below the u axis. Free energy

considerations

show

easily

^{that M will}

always

lie on

C3 (see Fig. 6),

^the

system being

then ferro-

magnetic.

^{For h =}

0, ithere isla spontaneous

magne- tization.

b)

^{If 1 =}

0,

^{the h} axis coincides with the v axis and the Maxwell rule

applies

^{for h =}

0,

^{the two shaded}

surfaces being equal (see Fig. 6).

c)

^{If 1}

0,

the m axis lies above the u axis. Then for

a

given t,

^{the free} energy is first minimized if M is on

CI,

as

long

^{as h}

hs,

^{then on curve}

C3

^{for h} >

hs,

^which

is the definition of the threshold field. The

magnetiza-

tion m will

undergo

^a

jump

^{if M leaves}

Cl

^before

reaching A (see Fig. 3).

^{This is true for t}

tcR, tCR being

the

temperature

^{for which the}

slope

^of

Cl

^{is infinite}

in the

m(h) representation.

5.2 DETERMINATION OF

hs.

^{- When h =}

hs,

^the

line

parallel

^to the m axis leaves between the

magnetiza-

tion curve and itself

equal

^areas

(Maxwell Rule).

^This

condition is met with a reasonable _accuracy

by

^the

line

parallel

^to the m axis

passing

^at

point

^{1 on}

figure

⁶

(1

⁼ middle of

OA). h.,

is obtained when

solving

The _{root uo}

being

uo ⁼

At zero

temperature (and

^this threshold field is a

good physical information),

^{this can be done}

formally

and

When the transition becomes ^a second order _one, the

antiferromagnetic

^order

parameter

^{becomes null}

at

point A (see Fig. 6)

^{and then}

hs

⁼

hA .hs

^{can then be}

determined

numerically.

5.3 CRITICAL POINT AND CRITICAL TEMPERATURE. ^-

Let us call

tCR

the critical

temperature

^{for which the}

metamagnetic

transition goes from first to second order. This is characterized

by

^{the fact that}

dm/dh

is infinite at

point A (see Fig. 3),

^{i. e. in the}

(u, v)

coordinates the

slope

of the m axis is

equal

^{to the}

slope

^of

Cl

^{at A. To calculate this}

slope,

^{we can go}

back ^to the definition of

Cl,

^{locus of}

points M,

^and

use the fact that if a function

y(x)

^{can be}

expanded

about ^one of its extrema as

(Fig. 7)

then the middle M of the horizontal intersection is located on a curve whose

slope

^{at the}

stop point

^is

FIG. 7. ^- Representations ^{of curves} Ci, C2 ^and C3 ^of figure ³

near point A ^with (x, y) axis such that x is tangent ^to C2-C3.

given by -

²

A2/B. Unfortunately,

the coordinates of A are determined

by

the condition

This does not

yield

^an

explicit expression

^of

u(A)

and a desk

computer

is needed.

tc,

^is thus determined

by

^{the condition}

At this

point,

it is worthwhile to

point

^{out the}

importance

^{of a correct}

scaling

^of

temperature

^when

comparing experimental

^{results to} theoretical

predic-

tions :

contrary

^to the Kramers doublet _case, in the

two-singlet

^{model there is no linear} relation between the Néel

temperature

^and the energy of the antiferro-

magnetic

^{state at 0 OK. If it is assumed that the}

invariant

quantity

should be the ratio of A to the energy of the

antiferromagnetic interactions,

^{then the}

temperatures

^{should be}

adjusted

^{via tanh}

(d/2 kT),

since tanh

(d/2 kTN)

^is

proportional to 4 /( j - 1).

^The-

refore a

satisfactory

result would be for instance :

5.4 ABSENCE OF A FIRST ORDER METAMAGNETIC TRANSITION. ^- Just as in the case of the Kramers doublet

[20],

^a first order transition from the antiferro to the saturated

paramagnetic

^state may become

impossible

^{if the}

ferromagnetic coupling

^becomes

negative, making

collective effects ineffective in the mechanism of

spin

^reversal

(this

condition is

simply j

^{0 if 4}

= 0).

A sufficient condition for the

magnetization

^to

undergo

^no

discontinuity

^{at any}

temperature

^{is to}

prescribe

the absence of

discontinuity

^{at t =}

0 ;

ⁱⁿ other words

C2

should have a

positive slope

^at

point A

in the

(h, m) system

of coordinates. When t ⁼

0,

eq.

(7)

^can be written

explicitly.

^{Therefore we}

get

^the condition

Figure

^{8 shows in the}

( j, 1) plane

the frontiers of the

regions

^{where there is a first or a second order transition}

(when

^it

exists).

^{Let us}

point

^{out that}

for j

^0.325

the transition will

always

^{be a second order one}

whatever the value of

( j - 1) (compatible

^{with a tran-}

sition)

may be.

FiG. 8. ^- Phase diagram ^at temperature ^{T = 0 °K as a} function of the interactions. In the shaded _zone, there is no ordering ^at

0 OK. F refers to a ferromagnetic state, ^{AF to an} antiferroma- gnetic one, the metamagnetic transition at 0 °K being ^{either a}

first order or a second order one.

It is _easy to see on

figure

^{8 what is the} influence of L1 upon the order of the transition. If we start from a

point

^{in the} first order zone and increase

A,

^{we follow}

a

straight

^line

passing

^{at the}

origin. Therefore,

^we

cross

successively

^the critical line and the transition line. A acts as a

temperature

^{as usual.}

6. Conclusion. ^- The results of the

present study

are in

agreement

^with

experimental

^{results on induced-}

moment

systems [12]-[14]

^{and may}

help

^{with the}

determination of the interactions

responsible

^{for the}

magnetic ordering. They

^can

easily

^{be extended tomore}

complex two-singlet systems

^like

TbAIG,

^{which forms} below 1.35,DK six

antiferromagnetic sublattices, a-a’, j8-j6’

^and

y-y’

^with

magnetic

^moments

lying along

^direc-

tions

[001], [100]

^and

[010] respectively [15]-[16]-[17].

When an external

magnetic

^{field H is}

applied along [111 ],

^TbAIG behaves like a two-sublattice

antiferromagnet.

^{If H is}

large

^and

parallel

^to

[001]

^or

[110],

^some sublattices are

polarized

^{and the others}

are left under the influence of their own interactions.

They

^may

undergo

^a

reordering

^{or their}

magnetic

moment may

vanish, depending

upon the fulfillment

26

of the transition condition

(2’).

^{If it}

happens

^{that the}

transition is

always

^of the AF --> P

type (i.

^{e. no}

reordering

^{of the}

uncoupled sublattices),

^{the order of}

the

metamagnetic

transition _may

change

^{with the}

direction in which H is

applied,

^{for this is in a} way

equivalent

^to

changing

the interactions in the molecular- field

equations.

^This may lead to

interesting phase

diagrams

^{at 0 OK.}

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J.,

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