High-temperature series expansion study

Kok-Kwei Pan ( 潘國貴 ) Physics Group, Center of General Education

Chang Gung University ( 長庚大學 )

No. 259, Wen-Hua 1st Road

Kwei-San, Tao-Yuan

Taiwan

Effect of easy-axis single-ion anisotropy on phase transitions of Heisenberg antifer

romagnetic films

Outline Motivation Magnetic thin films ( quasi-two-dimensional systems) of

fer unique opportunities for studying finite-size scaling effects on the critical behavior.

Single-ion anisotropy plays a major role in determining the magnetic behavior of the system .

High-temperature series expansion Results

Effect of easy-axis single-ion anisotropy on thickness-dependent Néel temperature TN(n)

The thickness dependence of Néel temperature TN(n) for n layers cubic lattice films

Conclusions

; N N

N

T ( ) - T (n) 1~ λ inverse of correlation length exponent

T ( ) n

1λ =

Hamiltonian of Heisenberg Antiferromagnet (HAF) with single-ion anisotropy

D

D Easy - axi

Easy - pla e

s

n

2z z z z z

ij i j i j i j i i ji,j i i A j

y yx xs sH = J S S S S S S D S h S h S

D single - ion anisotropy

Spin-1 Phase diagram (3D HAF with single-ion anisotropy )

D/J0

-Dc/J

zS 1

zS = 0

D

zS 1

zS = 0D

Heisenberg antiferromagnetNeel order with quantum fluctuations

Easy-axisEasy-plane

Ising Antiferromagnetic phase

Planar Antiferromagnetic phase

Quantum Paramagnetic phase

K. K. Pan, Phys. Rev. B 79 , 134414 (2009).

K. K. Pan, Phys. Lett. A 374 , 3225 (2010).

3

2zS

1

2zS =

2D

High-temperature series study

0 1H = H + H

222z z z z

0 i eff i j eff ji A j A

+1H = -D S - h S -D S + h S + NJz M

2

eff s+h = JzM h其中

z z1 ij i j ij i j i j

i,j i,j

+ ++ +H = J S M S M J S S S S

Cubic lattice film

∞

∞n-layers

Free energy and staggered susceptibility due to the quantum and thermal fluctuation correlations

Bκ T

其中

Staggered susceptibility of n interacting layers films

Free energy of n=2, 3, 4, 5 and 6 interacting layers films

two-rooted connected diagrams

K. K. Pan, Phys. Rev. B 71, 134524 (2005); Phys. Rev. B 64, 224401 (2001); Phys. Rev. B 59, 1168 (1999).

Analysis of the Series and Results

Ratio method D-log Pade approximant

Ratio method

Néel temperature and critical exponent n layers films

Three-dimensional ( 3D) bulk Néel temperature and critical exponent ( 3D)

Pseudocritical temperature of the n-layer and critical exponent ( 2D)

1 / [ r ( r - 1) ]1

/ 2

0.0 0.1 0.2 0.3 0.4 0.5

( a r

/ ar -

2 )1/

2

2.5

3.0

3.5

4.0

Fig. 1 ( K. K. Pan )

D/J=2.0

Bulk

n=2

n=3

n=4

n=6

n=5

Ratio plot of the high-temperature staggered susceptibility series of n-layer sc lattice films with easy-axis anisotropy D/J=2.0 for the spin-1 system.

1 / [ r ( r - 1) ]1

/ 2

0.0 0.1 0.2 0.3 0.4 0.5

( a r

/ ar -

2 )1/

2

3.0

3.5

4.0

4.5

5.0

D/J=10.0

Bulk

n=2

n=3

n=4

n=6

n=5

Ratio plot of the high-temperature staggered susceptibility series of n-layer sc lattice films with easy-axis anisotropy D/J=10.0 for the spin-1 system.

1 / [ r ( r - 1) ]1

/ 2

0.0 0.1 0.2 0.3 0.4 0.5

( a r

/ ar -

2 )1/

2

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

D/J=10.0

Bulk

n=2

n=3n=4

n=6

n=5

Ratio plot of the high-temperature staggered susceptibility series of n-layer sc lattice films with easy-axis anisotropy D/J=10.0 for the spin-3/2 system.

1 / [ r ( r - 1) ]1

/ 2

0.0 0.1 0.2 0.3 0.4 0.5

( a r

/ ar -

2 )1/

2

6.0

6.5

7.0

7.5

8.0

8.5

9.0 n=2 Col 43 vs Col 44

n=3 Col 46 vs Col 47

n=4 Col 49 vs Col 50

n=5 Col 52 vs Col 53

n=6 Col 55 vs Col 56

Bulk Col 58 vs Col 59

n=2 , TN/kT =6.419,

slope=2.465, =1.42n=3 , TN/kT =7.115,

slope=2.200, =1.34n=4 , TN/kT =7.509,

slope=1.803, =1.26n=5 , TN/kT =7.717,

slope=1.588, =1.22n=6 , TN/kT =7.830,

slope=1.508, =1.21Bulk , TN/kT =8.195,

slope=1.513, =1.19last 4 data

D/Jz=5.0

slope=( TN / J )( -1 )

Spin-1 Spin-3/2

( 3D)

( 2D)

( 3D)

( 2D)

( 2D) ( 2D)

( 3D)( 3D)

n ( number of layers)

1 2 3 4 5 6 7

kTN /J

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

Fig. 2 ( K. K. Pan ) Effect of easy-axis single-ion anisotropy on thickness-dependent Neel temperature

D/J=2.0

D/J=3.0

D/J=4.0

D/J=5.0

D/J=6.0D/J=7.0D/J=8.0D/J=9.0

D/J=10.0

Effect of easy-axis single-ion anisotropy on thickness-dependent Néel temperature ( S=3/2)

n (number of layers)

2 3 4 5 6 7 8 9

[TN( )-

TN(n

)] /

TN( )

0.04

0.05

0.06

0.070.080.09

0.15

0.40

0.50

0.10

D/J=2.0D/J=3.0D/J=4.0D/J=5.0D/J=6.0D/J=7.0D/J=8.0D/J=9.0D/J=10.0

Fig. 2 ( K. K. Pan )Log-log plot of of [- (n) ]/ versus n

for easy-axis anisotropy D/J=2.0 to D/J=10.0.

n (number of layers)2 3 4 5 6

[TN

( )-

TN

(n)]

/ T

N( )

0.040.050.060.070.080.09

0.15

0.400.50

0.10

slope λ 1.6

slope λ 1.57

3spin -

2

spin -1

The thickness dependence of Néel temperature TN(n) for n layers cubic lattice films

3D Ising

3D Ising

n-1/

0.0 0.1 0.2 0.3 0.4 0.5

[TN( )-

TN

(n)]

/ T

N( )

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

D/J=2.0D/J=3.0D/J=4.0D/J=5.0D/J=6.0D/J=7.0D/J=8.0D/J=9.0D/J=10.0

= 1.56

Fig. 3 ( K. K. Pan )

[ T- TN(n) ]/ versus 1/n with= 1.56

for easy-axis anisotropy D/J=2.0 to D/J=10.0.

n-1/

0.0 0.1 0.2 0.3 0.4 0.5

[TN( )-

TN

(n)]

/ T

N( )

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

= 1.56

3spin -

2

spin -1

3D Ising

3D Ising

N N

N

T ( ) - T (n) 1vs

T ( ) n

λ inverse of correlation length exponent

N N

N

T ( ) - T (n) 1vs

T ( ) n

λ inverse of correlation length exponent

Conclusions

The thickness dependence of Néel temperature TN(n) for n layers cubic lattice films with easy-axis anisotropy is described by a finite-size scaling relation with a shift exponent of

The obtained shift exponents for spin-1and spin-3/2 of the sc and bcc lattices with easy-axis anisotropy show good agreement with the finite-size scaling prediction for the 3D Ising universality class and the general universality principles of the spin independent shift exponents.

λ 1.56 ( 0.64

The thicknessdependence of the Neel temperature $T_{N}$ for the $n-$layers

cubic lattice films is described by a finite-size scaling relation with a

shift exponent of $\lambda \simeq 1.1 \pm 0.2$. Although the obtained shift exponents for spin-$1$ and spin-$\frac{3}{2}$ of the sc and bcc lattices are not accurate compared

with the expected value, they show good agreement with the

finite-size scaling prediction for the 3D Heisenberg universality class and the general

universality principles of the spin independent shift exponents.