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Dilution Effects in a Spin Crossover System, Modelled in

Terms of Direct and Indirect Intermolecular Interactions

H. Constant-Machado, J. Linares, F. Varret, J. Haasnoot, J. Martin, J.

Zarembowitch, A. Dworkin, Azzedine Bousseksou

To cite this version:

H. Constant-Machado, J. Linares, F. Varret, J. Haasnoot, J. Martin, et al.. Dilution Effects in a Spin

Crossover System, Modelled in Terms of Direct and Indirect Intermolecular Interactions. Journal de

Physique I, EDP Sciences, 1996, 6 (9), pp.1203-1216. �10.1051/jp1:1996124�. �jpa-00247241�

(2)

J. Phys. I Rance 6

(1996)

1203-1216 SEPTEMBERI996, PAGE 1203

Dilution

Elllects

in

a

Spin

Crossover

Syste~n,

Modelled

in

Ter~ns

of

Direct

and

Indirect

Inter~nolecular

Interactions

H. Constant-Machado (~~*), J. Linares

(~),

F- Varret

(~,**),

J-G- Haasnoot

(~),

J-P-Martin

(~),

J. Zarembowitch

(~),

A. Dworkin

(~)

and A. Bousseksou

(~)

(~) Laboratoire de Magnétisme et d'optique (***), Université de Versailles,

78035 Versailles Cedex, France

(~) Institute of Chernistry, Gorlaeus Laboratories, Leiden University, POB 9502,

2300 RA Leiden, The Netherlands

(~) Laboratoire de Chimie Inorganique (****), Université Parts Sud, 91405 Orsay Cedex, France

(~) Laboratoire de Chimie Physique des Matériaux Amorphes (*****), Université Paris Sud, 91405 Orsay Cedex, France

(~) Laboratoire de Chimie de

Coordination(******),

31077 Toulouse Cedex, France

(Received

21 July 1995, revised 27 February 1996, received in final form 7 May 1996, accepted

10

May1996)

PACS.31.70.Ks Molecular solids

PACS.33.45.+x Môssbauer spectra

PACS.64.60.-i General studies of phase transitions

Abstract. We have studied by ~7Fe Môssbauer spectroscopy and magnetic measurements

trie spin transition exhibited by the series:

[Fe~coi-~

(4,4'-bis-1,2,4-triazole)2 (NCS)2].H20,

con-sisting of a spin-crossover

iroil(II)

system diluted in a high-spin

cobalt(II)

matrix. We performed

Monte Carlo simulations of the thermal variation of the high-spin fraction, using a two-level Ising-type Hamiltonian. A good agreement with experiment is obtained by the introduction of an indirect interaction of the spin-crossover units via

the non-spin crossover unit, the origin of

which is discussed in steric terms.

1. Introduction

The

physical

nature of tue

cooperative

interactions

responsible

for tue

abrupt

transition in

spin-crossover solids is still a

subject

of discussion. So

far,

macroscopic

descriptions

m terms

of

regular

solutions

Iii

or in terms of elastic

theory

[2j bave succeeded in

explaining

most

of

trie characteristics of

spin

transitions: for

example

trie conversion curve

nHs(T),

where nHs is

tue

uigu-spm

fraction,

as well as /hH and

/hS,

wuicu

respectively

are tue

changes

m

enthalpy

and entropy associated with trie conversion of a mole of trie

spin-crossoùer

unit.

(*) On leave from Universidad Central de Venezuela

(**) Author for correspondence

(e-mail:

francois.varretilphysique.uvsq.fr)

(***) URA 1531 CNRS

(****) URA 420 CNRS

(*****) URA 1104 CNRS

(******) UPR 8241 CNRS

(3)

Here we focus on trie

microscopic

description

of trie

problem.

We

previously

introduced an

Ising-like

model [3j,

adapted

from

previous

works

[4,5j,

wl~icl~ was shown to be

equivalent,

in trie

mean-field

approach,

to trie

regular

solution model in trie

Bragg-

Williams approximation [5].

Also,

a

microscopic

counterpart of trie elastic model was

recently

proposed

[6], and was solved

for trie ID case.

Trie

Ising-like

model is a two-level model

including pl~enomenological Ising

interactions. Tl~e

two levels bave dilferent

energies

(gap /h)

and

degeneracies

(ratio

g

=

gHs/gLs)1

it is

formally

equivalent

to a true

Ising

system under a

temperature-dependent

field [7]. Tl~e Hamiltonian

is:

É=L)t-©J~jt.âj

il)

wl~ere â~ is tl~e

pseudo-spin

operator, tl~e

eigenvalues

of wl~icl~ are +1 in tl~e HS state, -1 in tl~e LS state; L sums over molecules and L' over pairs.

We stress on tl~e

pl~enomenological

cl~aracter of tl~e

Ising-hke

model. Tl~e interaction

pa-rameter bas not been

directly

measured in any case; a

single

attempt bas been made to

predict

its value

by

calculating

trie

energies

of a pair of molecules in dilferent spin states:

Molecu-lar Orbital calculations bave been

presented

in Bolvin's thesis [8] but

they

are not accurate

enough

to be

really

conclusive.

Furthermore,

it is not clear whether one can define a value of

trie interaction parameter between two

neighbouring

molecules,

irrespective

of trie spin states

of trie

surrounding

molecules- We bave undertaken trie

study

of a diluted system in order to

obtain some information

concerning

this diflicult

point.

From a

general point

of

view,

trie

theory

of diluted systems

was

developed

for

magnetic

systems,

leading

to trie concept of percolation [9j. Trie value of trie

pàrcolation

tl~resl~old was

found to be

extremely

sensitive to tl~e interaction range

(see

[10] for

exatnple)-

Concerning

diluted

spin-crossover

systems, a previous

adaptation

of tl~e

macroscopic

elastic

tl~eory

(infinite

range

interactions)

was

successful,

particularly

in tl~e case of

[(Fe~Zni-~) (2-pic)3jCl~-EtOH

Ill]

We use l~ere an

Ising-like

approacl~ solved

by

numerical

simulations,

in order to avoid

tl~e

infinite-range

approximation involved in tl~e mean-field approacl~. On tl~e otl~er l~and, it

bas been sl~own tl~at tl~e numerical simulations based on tl~e Monte Carlo

Metropolis

metl~od

better reproduce tl~e

sl~ape

of tl~e

l~ysteresis loops (12j,

wl~icl~ l~ere corresponds to an important

aspect of trie available information

(due

to trie

quasi-complete

character of trie

conversions).

Trie system

(Fe~coi-~(4,4'-bis-1,2,4-triazole)2l'NCS)2)

.H20,

under

study

l~ere, abbreviated

as

(Fe~j,

bas been studied over tl~e wl~ole composition range,

by

magnetic,

calorimetric and

Môssbauer measurements. Tl~e experimental data bave been

reported

in [13],

together

with

an interpretation in terms of a

regular

solution

model,

equivalent

to an

Ising-type

mean-field

approach-

An essential feature is trie decrease of trie width of trie

hysteresis

loop

on

decreasing

x; trie

loop

collapses

below a threshold value x~ m 0A. This value is refined in tl~e present

work,

tl~anks to new

samples;

ail tl~e data are summanzed in Section 2,

togetl~er

witl~ tl~e

calonmetric data wl~icl~ are re-examined. Section 3 is devoted to a brief recall of tl~e Monte

Carlo

Metropolis

technique,

Section 4 to tl~e results and discussion.

At tl~e

begmning

of tl~e present

work,

our idea was to

investigate

tl~e range of tl~e

inter-molecular interactions. Tl~e results were found to

depend

very httle on tl~e interaction range,

but to

sizeably

dilfer from tl~e

experimental

data. We

finally

found a convenient agreement

by

introducing

"indirect" interactions between two crossover units via a non-spin-crossover unit,

by purely

formal

analogy

to

magnetic

super-excl~ange

interactions. Tl~e

pl~ysical

meaning of

(4)

N°9 SPIN-CROSSOVER DILUTED SYSTEM 1205 T=140K T=50K T=77K T=154K T=140 x=1-Ù x=0.01 o 4 4 Veiocity(mws)

Fig. l. Selected Môssbauer spectra of [Fei.o]

lin

the heating mode) and [Feo_oi].

2.

Experimental

Data

In tl~e solid state

(Fe~coi-~(4,4'-bis-1,2,4-triazole)2(NCS)21.H20,

consists of

layers

of

six-coordinated metal ions linked to eacl~ otl~er in tl~e

equatorial

plane

tl~rougl~

tl~e Ni,

Ni'-bridging

triazole

ligands, basically giving

trie system a 2D-character. Trie iron sites

form,

at

first

approximation,

a square

planar

lattice.

Magnetic

and Môssbauer measurements [13] show

that tue thermal

hysteresis loop,

m 24 K wide for x = 1,

rapidly

narrows on

decreasing

x, and

collapses

at tue turesuold value x~ m 0.37. Tuese data are summarised in

Figures

1-4.

Selected 57Fe Môssbauer spectra for x = 1 and x = 0-01 are suown

in

Figure

1. At

uigu

temperature all Môssbauer spectra exuibit a

single

doublet witu

quadrupole

splitting

(2.80

2.85 mm

/s)

and isomer suift

il.04

1.05 mm

/s,

witu respect to metallic iron at room

temperature)

values

typical

for

uigu-spin (HS)

Fe~~ At lower temperature a new

peak

appears

witu a very small

quadrupole

splitting (0-11

o-là mm

/s)

and isomer suift

(0.52

0.54 mm

/s)

values

typical

for

low-spin

(LS)

Fe~~-Typical

conversion curves

mis(T)

are

presented

in

Figure

2. Tue

uysteresis

widtu /hTc is

defined as Tc Î -Tc Î, wuere tuese temperatures, as well as

Ti/2,

are

defined as tue

temper-atures for whicu tue spm transformation is ualf

complete

ii-e-

for nHs "

1/2

after correction

for tue residual HS and LS

values).

Trie whole set of data for these critical temperatures is

presented

in Table I.

We show in

Figure

3 tue Môssbauer data oi tue

spin

conversion m tue diluted system,

x = 0.01,

plotted

in Arruenius axes. To a first

approximation,

tue

plot

is

linear,

wuicu is

typical

for a spin

equilibrium

between

independent

spin-crossover

units; tue small deviation oi

(5)

nHs """'""""'""""""" x =1.o """"' x = 0.8 1 ~.. .. ... ... 0 x= 120 ~~

Fig. 2. Thermal hysterests loops of

[Fei],

(Feo.8], (Feo.5g], from magnetic data [13j.

(thermal

population

oi dilferent vibrational states in tue

LS,

HS states

(14j).

A linear regression

around tue

equilibrium

temperature

provides

accurate determinations oi

/hH,

/hS,

wuicu are

reported

in Table I.

In

Figure

4 we summanze tue variations oi

Ti/2,

Tc Î, Tc Î

versus x, wuicu iorm tue phase

diagram

oi tue system in tue

(T,x)

space. It is wortu

noting

tue quite linear

suape

of tue 2

branches for

x > x~. Linear regressions

give

for all branches:

for x < 0A

Ti/2

'~~ 78 + 68x

(2)

for x > 0A

Tc

Î'~~ 78.5 + 66z

Tc

Î'~~ 92 + 29x

(3)

Equations

(2, 3)

form tue borderlines of tue "idealized" expenmental

phase

diagram,

and

accurately

determme tue

collapse

value of tue

uysteresis loop:

xc

r~J 0.36.

In order to discuss tue present

data,

it is wortu

considering

an

"equihbrium

transition

tem-perature",

Teq, in-between

Tc Î

and

Tc Î,

corresponding

to equal free energies to tue HS and

LS

phases

(tuer

x

=

1/2).

At

Teq,

/hH =

T/hS,

giving

similar

tuermodynamical

properties

to

Teq(x

>

x~)

and

Ti/2(x

<

z~);

tuerefore tue

Teq(x)

line is

expected

to

exactly

follow tue

(6)

N°9 SPIN-CROSSOVER DILUTED SYSTEM ~ 1207

Ù-1

0.005 ~~~ 0.020

Fig. 3. Arrhenius plot of the spin equilibrium m [Feo.oi], from the Môssbauer data, and best linear fit: equilibrium constant Keq =

flflfi

versus

1/T-Table I.

Ezperimentai

data

for

the

eqmbrium

and transition temperatures, moiar

enthaipy

and entropy

changes

upon total conversion

of

ffej

and ratio

of

the deduced

effective

degenera-cies,

for

the system

ffe~coi-~ (4,4'-bis-1,2,$-triazoie

)2

(NOS)2).H20.

x

Î

Tc g

(K)

kJ mol~~ J K~~ mol~~ 75 2.90 38.5 0.05

§2.3

3.25 0.24 97 5.21 0.39 107 105 5.31 53.7 638 0.54 l14 107 6.43 59.9 1346 0.59 ils 107 7.60 61-8 1690 0-8 130 ils 7.72 69 3995 1.o 145 121 10.20 74.8 8021

took trie

followmg

empirical

relation

(derived

from numerical simulations

[12j):

~ 9Tc

1+7Tc

j4)

~~ 16

leading

to:

Teq(x)

= 84.4 + 49.8x

(5)

wuicu is suown m

Figure

4 as a dashed

line-As

expected,

tue Teq line follows tue

Ti/2

hne; tue latter consists of a linear variation down

to x m 0.2, witu an extra-decrease for tue most diluted

samples-A similar trend bas been

(7)

/ / / jç / / / / w ~ ~ ' . w ~ w

X

(8)

N°9 SPIN-CROSSOVER DILUTED SYSTEM 1209 ~ ~

f

~

O.O X (Fe

(9)

3. Monte Carlo

Metropolis

Simulations

The

Metropolis

algonthm

provides

trie most representative

configurations

of trie

thermody-namical system

througu

a Markov cuain of

elementary

transformations

yielding

a limit wuicu

corresponds

to tue canonical

equilibrium

distribution. For tuis purpose tue

equilibrium

distri-bution

P(1, T)

c~ e~~E~ is used in tue detailed balance

equation-Pli> T)Wli

~ J>

T)

"

PIJ,

T)WIJ

~ i,

T)

Ill)

For tue practical aspects, tue

Metropolis

algoritum

[17j is tue

following:

consider an initial

configuration

A~ witu energy E~- First a lattice site

j

is selected at random and tue energy

Ef is calculated wuen aj is

flipped.

Tuen tue transition

probability

w,~

--a~ = e~~(Ef~E~~ is

computed-

Next a random number between 0 and

unity

is

generated

to be

compared

witu tue

transition

probability-

If tue

probability

wa~--a~ is

larger

tuan tue random number tue new

configuration

is

accepted-

Otuerwise tue lattice site remains witu tue initial value aj. Tuis

procedure,

as

explained

in

[17j, obeys

tue detailed balance equation and

provides

tue fastest

convergence of iterative process. Tue

algoritum

continues

by

selecting

anotuer lattice site. To account for tue different

degeneracies

gHs, gLs, we bave introduced a pre-exponential

factor

[12j,

tuus

leading

to tue

completed

expression:

wa

--a =

~~~~

e~P(E~-E~j

j~~~

' ~ §«,

wuere tue pre-exponential factor g-a~

/ga~

takes tue values g for LS ~HS and

1/g

for HS ~LS.

For calculations tue square lattice was 100 x 100; 2000 Monte Carlo steps were

performed

for

reacuing

tue thermal

eqmhbrium,

and agam 2000 for

computing

tue average values.

Tem-perature steps were

typically

K-4. Results and Discussion

In

Figure

6 we show tue results obtained witu

only

nearest-neigubour

interactions. For 2D

interactions tue

computed

phase

diagram significantly departs

from tue experimental

diagram:

tue Tc

Î,

Tc

Î

borderlines are

strongly

curved;

tue

collapse

value of tue

uysteresis

loop,

x~ r~J

0.55,

is too

large.

In tue percolation

description,

one would expect tue

uysteresis

widtu

/hTc

to decrease faster and faster witu z on

approacuing

tue turesuold value. Tue computed

data follow

an opposite

)rend.

To improve tue model, we bave examined several possibilities:

ii)

3D interactions. Tuese case tue results reported in

Figure

6; tue computer turesuold

value is still z~ .r~J 0.55 and tue borderlines remain

curved;

(ii)

a

composition-dependent

interaction parameter

J(x) (as

basically

done in

[13j):

tue

needed

variation,

suown in

Figure

7, is

strongly

non-linear and tuerefore seems

unlikely,

because all measured

tuermodynamical

quantities vary

quasi-linearly

upon z;

(iii)

furtuer

interactions,

namely

second-nearest-neigubour

interactions: tue

results,

reported

in

Figure

8, do not

markedly

differ from tuose obtained witu

nearest-neigubour;

(iv)

indirect interactions

turougu

tue Co

units,

of

ferromagnetic

type

(J

>

0),

m order to

provide

an extra-contribution to

cooperativity

related to tue presence of Co units. Tue

patuways

of tuese interactions

(10)

N°9 SPIN-CROSSOVER DILUTED SYSTEM 1211 ~. ~

~

i + .O

X

(11)

m . ~ ~ . u ~ .

~

. 0.2 .3 X Fe x

(12)

N°9 SPIN-CROSSOVER DILUTED SYSTEM 1213 ~ ~ ~ o o o O

O.2 O.3 O.4 O.S O.fi O.7 O.8 O.9 1.O

X

(Fe

x CO

i,xÎ

Fig. 10. Numerical 2D simulations

(O),

including both direct Fe-Fe and indirect Fecofe

interac-tions; mput parameters are listed in Table II.

Table II. Interaction parameter values

(K) for

the numericai simulations.

direct Fecofe Fe

Figure

Lattice

(Comment)

lrst

120

6,8

(.)

2D

60 6

(+)

3D

100

(20)

8

(.)

2D

125 +28 10

(o)

2D

(good fit)

120

-5(~S~

+ 10(~S~ ii

(à)

2D

50

+16(~~~

+ 16(~~~ 11 2D

developed

for trie second- and

third-nearest-neighbour

interactions, was

easily

adapted

thanks to

simple

Kronecker functions. An excellent

agreement

was obtained with trie

experimental

phase

diagram,

as shown in

Figure

la. Parameter values are

given

in

(13)

iso

u

»

~

j

iÎé

*

O.2 O.3 O.4 O.S O.S O.7 O.8 O.9

X

[Fe

x CO

i_x]

Fig. Il. Numerical 2D simulations mcluding both direct Fe-Fe and indirect Fefefe interactions;

symbol definitions and input parameters are given m Table II.

Table III.

Computed

data

for

the short range

ejfect

around Go units and vacancies:

nHs,co(nHs,v)

is the ratio

of

Fe units

having

at ieast a Go

(vacancy)

unit in a

nearest-neighbour

position. The rate

of

Go and vacancies units was

jized

ta 10%.

nHS nHS,Co nHS,v

T(K)

0.9970 0.9965 0.9927 139

ii-à

above T~

Î)

0.0019 0.0039 0.0045 l13

ii-à

below T~

Î)

Such

uypotuetic

indirect interactions bave to be discussed at various levels:

ii)

Similar Fe-Fe-Fe interactions

migut

be introduced as well. We

investigated

two

limiting

cases: indirect interaction Fe-Fe-Fe

irrespective

of tue

spin

state of tue intermediate Fe~~,

and interactions of opposite signs for

Fe-Fe~S-Fe

and

Fe-Fe~S-Fe.

Results are suown in

Figure

11 and hsted in Table II- We observed in all cases tuat tue indirect Fe-Fe-Fe

interactions act in a way

exactly

similar to tue direct Fe-Fe interaction. In otuer

words,

it is not

possible

to determine

separately

tuese Fe-Fe and Fe-Fe-Fe interactions from tue

knowledge

of tue

phase

diagram-

Interestingly,

it appears tuat tue system is not

equally

sensitive to tue indirect interactions

turougu

tue LS and HS Fe units: for

exainple,

a

given

interaction

turougu

a LS unit is balanced

by

a smaller

(and

opposite)

interaction

turougu

a HS unit.

(ii)

On tue otuer

uand,

a

qualitative

explanation

for tue presence of sucu an indirect

inter-action

turougu

tue Co units cjn be

given

in terms of steric effects. Tue effect of a few Co atoms introduced into tue Fe lattice can be considered as

follows;

a

long-range

effect

(14)

N°9 SPIN-CROSSOVER DILUTED SYSTEM 1215

decreases tue

equilibrium

temperature

(as

suown in Sect-

2);

a

short-range

effect will

differ

according

to tue spin state of tue solid: in tue LS

phase,

around a

(bigger)

Co

unit,

a cluster of LS Fe units is

stabilized;

in tue HS

phase,

around a

(smaller)

Co unit,

a cluster of HS Fe units is also stabilized- Sucu an enuancement of tue

short-range

cor-relations around tue

impurity

unit is

puenomenologically

accounted for

by

tue indirect

"ferromagnetic-

type"

interaction. In Table III we selected

computed

data

comparing

tue

surroundings

of Co units and of vacancies, wuicu mimic Co units wuen tue indirect

interactions are switcued off; tue average mis value of Fe units

surrounding

tue Co units

departs

more from tue

equihbrium

value

il

/2)

tuan tuat of Fe units

surrounding

tue

vacancies: tue

expected

enuancement of tue

"ferromagnetic-type"

correlation is indeed obtained.

We do not discuss bene tue direct interactions in similar steric terms. Tuis remams an open

problem,

for which we bave

provided

here a further

piece

of information, 1-e- trie weak influence

of tue range assumed for tuese

phenomenological

interactions.

At last, we

briefly

discuss tue present

study

in terms of

geometrical

percolation:

indeed,

tue

introduction of Fe-Co-Fe interactions creates new

patus

for

propagating

tue interactions, and

lowers trie

hysteresis loop collapse value,

z~, in agreement with trie

expected

decrease oi trie

geometrical

percolation

threshold, xg- However, trie

computed

z~ values are almost unsensitive

to trie

dimensionality

of trie system, as well as to trie range oi trie Fe-Fe interactions, at variance

from xg values.

Obviously,

trie

relationship

between x~ and fg is not

tight,

and

certainly

deserves a further

investigation.

5. Conclusion

Tue effect of trie dilution of a spin-crossover system bas bçen studied in terms of trie two-level

model witu nearest

neigubours

interactions,

completed

by

an indirect interaction

occurring

via tue non spin-crossover

(Co)

atom- Tue data are

successfully

reproduced

and a steric

explanation

for tue onset of tuese indirect interactions bas been

given.

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Figure

Fig. l. Selected Môssbauer spectra of [Fei.o] lin the heating mode) and [Feo_oi].
Fig. 2. Thermal hysterests loops of [Fei], (Feo.8], (Feo.5g], from magnetic data [13j.
Fig. 3. Arrhenius plot of the spin equilibrium m [Feo.oi], from the Môssbauer data, and best linear fit: equilibrium constant Keq = flflfi versus
Fig. 10. Numerical 2D simulations (O), including both direct Fe-Fe and indirect Fecofe interac-
+2

Références

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