Monte Carlo for spin crossover compounds

(1)

EPJ Web of Conferences 14, 02004 (2011) DOI: 10.1051/epjconf/20111402004

This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which

“ Fundamentals of Thermodynamic Modelling

of Materials ”

November 15-19, 2010 INSTN – CEA Saclay, France

Organized by

Bo SUNDMAN [email protected]

Constantin MEIS [email protected]

PROFESSOR & TOPIC

Jorge LINARES

Université Versailles - UVSQ

Monte Carlo for

spin crossover

compounds

(2)

1

Monte Carlo simulations for 1D- and 2D spin crossover compounds using the atom phonon

coupling model

J. LINARES1

A. Gindulescu1,2_{, A. Rotaru}1,2,3 _{, M. Paez}1_,

C. Chong4_{, J.. Nasser}4

(1) GEMaC, CNRS-UMR 8635, UVSQ, 78035 Versailles Cedex, France (2) University of Suceava, Romania

(3) “AlexandruIoanCuza”University, 700506, Iasi, Romania

(4) LISV- UVSQ, 78035 Versailles Cedex, France

2

A

B

(3)

3 Fe(btr)₂(NCS)₂,H₂O

Fe-N_NCS= 2.125(3) Å

Fe-N_btr= 2.188(2) Å 2.180(3) Å

N

(btr = 4,4’-bis-1,2,4-triazole)

S. Pillet et al., Eur. Phys. J. B, 2004,

38, 541.

4 NCS

NCS

btr btr

Fe Fe(II)

d6

[Fe(btr)₂(NCS)₂]H₂O

Spin crossover compounds

NCS NCS

btr btr

Fe

Adiabat

ic

ener

gy

r (Fe-ligand)

(4)

5 NCS

NCS

btr btr

Fe Fe(II)

d6

[Fe(btr)₂(NCS)₂]H₂O

NCS NCS

btr btr

Fe

Adi

ab

at

ic

en

er

gy

r (Fe-ligand)

6

Spin crossover compounds

Adi

ab

at

ic

en

er

gy

r (Fe-ligand)

t

_2g

e

_g

e

_g

(5)

7

Low spin state

High spin state

t

_2g

e

_g

e

_g

t

_2g

Diamagnetic S=0 Paramagnetic S=2

Different colours

Different volumes

Different vibronic proprieties

8

g : degeneracy _Adi

ab

at

ic

en

er

gy

r (Fe-ligand)

0,00 0,25 0,50 0,75 1,00

nH

S

T (K)

g=1

0,0 0,2 0,4 0,6 0,8 1,0

g=15

t

_2g

e

_g

S=0 g=1

e

_g

t

_2g

S=2

(6)

9

Adi

ab

at

ic

en

er

gy

r (Fe-ligand)

0,00 0,25 0,50 0,75 1,00

nH

S

T (K)

g=1

0,0 0,2 0,4 0,6 0,8 1,0

g=15

0,00 0,25 0,50 0,75 1,00

T2

nH

S

T (K) T1

0,00 0,25 0,50 0,75 1,00

nH

S

Pressure

P1 P2

10

Adi

ab

at

ic

en

er

gy

r (Fe-ligand)

0,00 0,25 0,50 0,75 1,00

T2

nH

S

T (K) T1

0,00 0,25 0,50 0,75 1,00

nH

S

Pressure

P1 P2

0,00

0,25 0,50 0,75 1,00

T2

nH

S

(7)

11 Light Induced Excited Spin State Trapping (LIESST)

= 820 nm invers

~ 514 nm direct

S. Decurtins, P. Gütlich, K. Hasselbach, A. Hauser, H. Spiering, Chem. Phys. Lett.,

105 (1984) 1.

A. Hauser, Chem. Phys. Lett.,124

(1986) 543.

12

Adi

ab

at

ic

en

er

gy

r (Fe-ligand)

0,00 0,25 0,50 0,75 1,00

T2

nH

S

T (K) T1

0,00 0,25 0,50 0,75 1,00

nH

S

Pressure

P1 P2

0,00

0,25 0,50 0,75 1,00

T2

nH

S

(8)

13 Fe(btr)₂(NCS)₂,H₂O: Cell parameters during the thermal transition

●Isostructural transition : no space group change

●Abrupt changes a, b, c: Δa/a = +2.2%, Δb/b = -4.1%, Δc/c = -2.8%, ΔV= -91(9) Å3

S. PILLET et al Eur. phys J. B. 2004, 38, 541

1.0 0.8 0.6 0.4 0.2 0.0

H

igh

Spin

frac

tion

250 200

150 100

50

Temperature(K)

Fe(ptz6)(BF4)2

(9)

15

-1 +1

2 H

'

i j

(i) i,j

J

2

i

H

2 ln

kT

g

H

J ????

Adiabat

ic

ener

gy

r (Fe-ligand)

BS

r rHS

Ising-like model:

phenomenological model

16

MONTECARLO - METROPOLIS

'

i j

(i) i,j

J

2

i

H

Square Lattice _{tab[size][size]}

tab[i][j] in -tab[i][j]

(10)

17

Atom-phonon coupling model

phonon

H

Hspin

H Hspin

ˆi

2

HH

C

LH

C

LL

C

LL LH HH

C C C

Elastic constant values of the « spring » linking two atoms depends on their electronic states

J. Nasser, K. Boukheddaden, J. Linares, Eur. Phys. J. B. 39 (2004) 219-227

18

N

i

i i i

P e q

E

1

2 1 ,

2 1

i i

i

u

q

1

, 1 1 1

2 2

ˆ ˆ ˆ ˆ

+

4 4 4

LL LH HH LH LL LL LH HH

i i i i i i

C C C C C C C C

e

Atom-phonon coupling model

HH

C

LH

C

LL

C

phonon c p

(11)

19

8

2

_LH _HH

LL o

C

J

2 1

V

E

_p

_o

2 1 8 2 i N i HH LH LL o q C C C V

N i i i i

o

q

h

V

1 2 2 1

1

ˆ

N i i i i o

q

J

V

1 1 2

2

ˆ

Exchange energy Atom-phonon coupling model

20

Thermal hysteresis

ˆ

_i

m

1

ˆ ˆ

_i _i

s

1

exp( )sinh( )

( , ) r

J h

m f s m

B

2

2exp( 2

)

1 J

( , )

s

f s m

A B

max r

T

t

LL HH

C

x

C

LL

C

HH

C

LL

C

HH

y

2

0 0

J

y

h

m

(12)

21

Thermal hysteresis

ˆ

_i

m

1

ˆ ˆ

_i _i

s

1

exp( )sinh( )

( , ) r

J h

m f s m

B

2

2exp( 2

)

1 J

( , )

s

f s m

A B

max r

T

t

LL HH

C

x

C

LL

C

HH

C

LL

C

HH

y

LH

2

0 0

J

y

h

m

0,00 0,05 0,10 0,15 0,20 0,25 0,0 0,2 0,4 0,6 0,8 1,0 nH S Reduced temperature x=0,4 y=0,7 r=5 =0,55 N=2000 22

(13)

23

)

,

(

T

n

K

n

dt

dn

HL

) exp( ) , ( T k E k n T K B a HL HS o

a

E

phonons o H E - 2 * *exp T B dn _N n k

dt k T

thermal relaxation a E HS E 0 E Fe-ligand En erg y 0,00 0,25 0,50 0,75 1,00 T2 nH S T (K) T1 24 relaxation

[Fe_0.5Zn_0.5(btr)₂(NCS)₂]H₂O,

0 10000 20000 30000 40000 50000 60000

0,0 0,2 0,4 0,6 0,8 1,0

55 K 50 K 45 K n_HS

temps (s)

Experimental results Simulation

0 10000 20000 30000 40000 50000 60000

0,0 0,2 0,4 0,6 0,8 1,0 x=0.38 =0.6 N=2000

T_r=0.042

Tr=0.034

Tr=0.038

n_HS temps (s) 0,0 0,2 0,4 0,6 0,8 1,0 Superposition

0 10000 20000 30000 40000 50000 60000 0,0 0,2 0,4 0,6 0,8 1,0 55 K 50 K 45 K n_HS temps (s)

A. Rotaru, J. Linares

(14)

25 0,00 0,05 0,10 0,15 0,20 0,25

0,0 0,2 0,4 0,6 0,8 1,0

nH

S

Reduced temperature

N=2000

0,00 0,05 0,10 0,15 0,20 0,25 0,0

0,2 0,4 0,6 0,8 1,0

N=16

nH

S

0,00 0,05 0,10 0,15 0,20 0,25 0,0

0,2 0,4 0,6 0,8 1,0

N=50

nH

S

Size effect

0,00 0,05 0,10 0,15 0,20 0,25 0,0

0,2 0,4 0,6 0,8 1,0

N=14

nH

S

26

Size effect

0 500 1000 1500 2000 0,000

0,005 0,010 0,015 0,020

H

y

s

te

re

s

is

c

y

c

le

w

id

th

(15)

27

Metastables states at low temperature à 1D: X=(HS-HS elastic constant)/(LS-LS elastic constant)

0,00 0,05 0,10 0,15 0,20 0,25 0,30 -1,0

-0,5 0,0 0,5 1,0

x=0.21 y=0.0 dégé=5 pdelta=0.75 size=500

m

t

phonon

H

spin

H

i

spin

H

ˆ

(16)

1 1, 1 1, 2 2 1

, 1 1 , 1 1

1 1, 2 2 1 1, 1

(

)

(

)

(

)

(

))

.

(

)

(

)

i i i i i i i i i

i i i i i i i i i

i i i i i i i i i

mu

C

u

C

u

mu

C

u

C

u

mu

C

u

C

u

,

i t j j

u

A e

²

i t

.

j j

u

A e

C

_i-1,i

C

_i+1,i

---O---O---O

---u

_i-1

u

_i

u

_i+1

1, 1,2 1,2 1,

1, 2 1, 1, 2 1,

, 1 , 1 , 1 , 1

1, 1, 1, 2 1, 2

,1 , 1 , 1 ,1

² 0 0

²

0. ²

²

0 0 ²

N N

i i i i i i i i

i i i i i i i i

i i i i i i i i

N N N N N N

C C C C

m m m m

C C C C

m m m m

C C C C

m m m m

C C C C

m m m m

C C C C

m m m m

(17)

some results: (no hysteresis)

0,0 0,2 0,4 0,6 0,8 1,0 1,2 0,0

0,2 0,4 0,6 0,8 1,0

x=0.6 x= 1.

n

h

s

t

x=0.2

0,0 0,2 0,4 0,6 0,8 1,0 1,2 0,0

0,2 0,4 0,6 0,8 1,0

y = 0.

y = 0.5

y = - 0.5

n

h

s

t

y = 2.

15 LS deg

HS

deg

eneracy eneracy r

δ=0.606, y=0, N=200

y=0 x=0.2

With additional long-range interaction

phonon i

i G H

H

2

δ=0.606, y=0, N=200 ₀_.₂₁

max

w G g

x = 0.15 (hysteresis) x = 0.20 (hysteresis) x= 0.30

0,05 0,10 0,15 0,20 0,25 0,30 0,0

0,2 0,4 0,6 0,8 1,0

n

h

s

t

15 LS deg

HS deg

(18)

33

Monte Carlo –Metropolis simulations for 2D spin crossover systems

2D square lattice of spin crossover compounds:

nearest as well as next-nearest interactions have been taken into account

The dynamical equation for each displacement’s atom

and for a perpendicular wave is given by:

2 1

, I I

u

m_i _j

i j i j

j i j

i u u e u u e u u e u u

e

I₁ ₁_, _/_, ₁_, _, _, ₁_/_, _, ₁ _, ₁_, _/ _, ₁_, _, _, ₁_/ _, _, ₁ _,

u

_i _j

u

_i _j

u

_i _j

u

_i _j

u

_i _j

K

I

₂

₁_, ₁

4

_,

34

Then we construct the dynamical matrix of the system.

The eigenvalues of this matrix are the phonon frequencies.

(19)

35

0,015 0,020 0,025 0,030

-1,0 -0,5 0,0 0,5 1,0

<

>

(a)

0,015 0,020 0,025 0,030

-1,0 -0,5 0,0 0,5 1,0

<

>

The high spin fraction evolution in function of the reduced temperature,

obtained for 2D system using: , x=0.1, r=15, , K=0.01 for a) y=0.02, b) y=0.10

36

Summary

-atom-coupling model can help to understand the physical origin of the interaction

-Atom-coupling model :

- thermal hysteresis 1D + long range int. - thermal relaxation

-Size effet

(20)

37