1. 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 antiferromagnetic films

2. Outline Motivation Magnetic thin films ( quasi-two-dimensional systems) offer 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 T N (n) The thickness dependence of Néel temperature T N (n) for n layers cubic lattice films Conclusions

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

4. Spin-1 Phase diagram (3D HAF with single-ion anisotropy ) D/J 0 -Dc/J D D Heisenberg antiferromagnet Neel 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). 2D

5. High-temperature series study

6. Cubic lattice film ∞ ∞ n-layers

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

9. 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)

10. Spin-1 Spin-3/2 ( 3D) ( 2D) ( 3D) ( 2D) ( 3D)

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

12. The thickness dependence of Néel temperature T N (n) for n layers cubic lattice films 3D Ising

14. Conclusions The thickness dependence of Néel temperature T N (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.

16. The thickness dependence 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.