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.
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