Magnetic properties of Cobalt films described by second order perturbed Heisenberg Hamiltonian Abstract I. INTRODUCTION

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1 Magnetic propertie of Cobalt fil decribed by econd order perturbed Heienberg Hailtonian P. Saaraekara and Aila D. Ariyaratne Departent of Phyic, Univerity of Peradeniya, Peradeniya, Sri Lanka Abtract Second order perturbed Heienberg Hailtonian wa eployed to vetigate the agnetic propertie of hexagonal Cobalt fil. Initially the nuber of nearet neighbor and the contant arien fro the partial uation of the dipole teraction of the tructure of cobalt were calculated ug oe pecial algorith. Miization of the energy difference between the eay and hard direction of a eory device i very iportant. When the energy difference between the eay and hard direction i ignificantly all, the agnetic oent a eory device can be quickly rotated between eay and hard direction under the fluence of a all agnetic field. The thickne of a cobalt fil correpondg to thi iu energy difference calculated ug thi theoretical odel agree with oe experiental data of cobalt baed agnetic eory device. I. INTRODUCTION Th fil of cobalt baed aterial fd potential application eory dic and torage device [1]. But any theoretical vetigation related to the ferroagnetic cobalt th fil have not been perfored by any reearcher. The agnetic propertie of ferroagnetic th and thick fil with iple cubic (c), body centered cubic (bcc) and face centered cubic (fcc) have been explaed ug oriented, econd order perturbed and third order perturbed Heienberg Hailtonian by u previouly [-4, 8]. The eay and hard direction have been detered each cae. Accordg to thoe tudie, eay and hard direction of ferroagnetic th fil depend on agnetic exchange teraction, dipole teraction, econd and fourth order aniotropy, deagnetization factor, agnetic field and tre duced aniotropy. In all above cae, c-axi of the lattice wa aued to be perpendicular to the ubtrate. In addition, the ferroagnetic 1

2 propertie of Fe and Ni have been explaed ug a iilar odel by oe other reearcher [5, 6]. Due to the coplexne of the hexagonal cloed packed (hcp) tructure, the deteration of contant arien fro the partial uation of dipole teraction were coplicated copared to deteration of thoe of c, fcc and bcc lattice. So an algorith ha been eployed to evaluate thee contant. Thi ae trategy ha been applied to detere thee contant of Nickel ferrite by u previouly [7, 11, 1, 13, 14]. Accordg to experiental data, the tre duced aniotropy of ferrite fil i coniderable [9, 1, 15]. II. THE MODEL The Heienberg Hailtonian of any ferroagnetic fil can be baically expreed a followg. r r r r J r r r r S. Sn 3( S. n)( n. Sn) S. Sn + ( ) 5, n n rn rn r r r [ H ( N S / µ )]. S K S θ () z H=- Dλ ( S ) (4) z 4 D S 3 λ ( ), n d n For a thick ferroagnetic fil, the olution of above equation can be given a, J 3 E( θ ) = [ NZ + ( N 1) Z1] + { N Φ + ( N 1) Φ1 }( + co θ ) 8 8 () 4 (4) N d N (co θd + co θd θ coθ + K µ 3 [ ( Φ 4 + Φ 1 ) + D () + D C (4) co θ ] ( N ) θ θ ) 1 3 () (4) [ ( Φ + Φ1) + D + D co θ ] θ () C 4 11 In above equation; N, J, Z, Φ,, θ, D () n n, D (4), H, H, N d, K are total nuber of layer, p exchange teraction, nuber of nearet p neighbor, contant arien fro partial uation of dipole teraction, trength of long range dipole teraction, aziuthal angle of p, econd order aniotropy, (1)

3 fourth order aniotropy, plane applied field, of plane applied field, deagnetization factor and the tre duced aniotropy factor, repectively. Here C 11 and C are given by, C 11 = JZ1 Φ1(1 + 3co θ ) 4 + 4co () ( θ co θ ) D (4) N d θ (co θ 3 θ ) D θ coθ + 4K θ µ C = JZ1 Φ1(1 + 3co θ ) + 4co () ( θ co θ ) D (4) N d θ (co θ 3 θ ) D θ coθ + 4K θ µ III. RESULTS AND DISCUSSION The diagra of conventional unit cell of cobalt with lattice paraeter (a and c) i given figure 1. The c/a ratio for Cobalt i 1.6. Fig. 1: Conventional unit cell of cobalt 3

4 Fd the tranlational vector a, b, c for the tructure Select a lattice pot 1 t layer with poition vector n a a + n b b + n c c Select a lattice pot nd layer with poition vector n a a + n b b + n c c Calculate the ditance d between the two lattice pot. d= (n a -n a )a + (n b -n b )b + (n c -n c )c Add 1 to Z Ye I d <= 1? No Generate another lattice pot on nd layer. I.e. value for n a and n b Fig. : Algorith to calculate Z 4

5 Fd the tranlational vector a, b, c Fd the poition vector of a lattice pot on the nd layer ( n a a + n b b + n c c ) n 1 =-5, n =-5 r = (n a a + n b b + n c c) + n 1 a + n b Let R = r r rˆ = R W_tep= 1 R Fd W_tep atrix 3 1 ra r ˆ ar r ˆ ar b c r ˆ br 1 rb r rˆ b a c r ˆ cr r ˆ cr 1 r a b c W = W + W_tep Increent n by 1 Increent n 1 by 1 Ye I n =5? Set n = -5 No I n 1 =5? No End Ye Fig. 3: Algorith to calculate Φ 5

6 Then the algorith given figure wa ipleented to evaluate the nuber of nearet neighbor a cobalt fil with p layer parallel to the ubtrate. The nd algorith given figure 3 wa applied to calculate the contant arien fro the partial uation of dipole teraction. Fally followg value were found for hcp lattice. The nuber of nearet neighbor one lattice plane=z =6 Nuber of nearet neighbor between two adjacent lattice plane=z 1 =3 Contant arien due to the partial uation of dipole teraction one layer=φ =11.34 Contant arien due to the partial uation of dipole teraction between two adjacent layer=φ 1 =.41 () Sce the experiental value of D, D, H, H, K, J and have not been eaured for cobalt th fil by any reearcher yet, the iulation were carried () (4) for a reaonable et of D, D, H, H, K, J and value a given below. (4) J = ( ) ( 4) D H H Nd K D = = = = = 1 and µ = 5 The graph of E(θ ) veru angle i plotted Figure 4. A dicated, the eay direction i found at an angle of ab 4 and the hard direction occur at an angle of ab 14. For a aterial with a iple tructure, the angle between the eay and hard direction i 9. But the angle between eay and hard direction i 1 thi cae due to the coplexne of the tructure of cobalt. The Figure 5 how the variation of the energy difference between the eay direction and the hard direction agat the nuber of layer for a Cobalt th fil. It could be oberved fro thi graph that the energy difference i a iu for a fil of 5 layer. Therefore, a hard dik drive would require a le aount of energy to tore data if the agnetic fil i yntheized with the nuber of layer beg the above region. Thi theoretical reult agree with that of odern hard 6

7 dik. The optiu experiental reult for Co baed agnetic eory device have been obtaed for a th fil with the ae nuber of layer by oe other reearcher [1]. Fig. 4: The graph of E(θ ) veru θ for a Cobalt th fil with 5 layer Fig. 5: The variation of the energy difference between the eay and the hard direction agat the nuber of layer N 7

8 3-D plot of E(θ ) veru angle and nuber of layer i given figure 6. The angle and nuber of layer correpondg to eay and hard direction can be detered ug thi plot. The difference between the axiu and iu energie i really all around N=5 accordg to thi graph too. So the energy required to rotate fro eay to hard direction (crytal aniotropy) i ignificantly all for fil with 5 layer. 4 x 1 19 E(θ)/ angle θ(radian) 4 N Fig. 6: 3-D plot of IV. CONCLUSION E(θ ) veru angle (θ) and nuber of layer (N) Value of nuber of nearet neighbor and the contant arien fro the partial uation of dipole teraction calculated ug the algorith given figure and 3 are Z =6, Z 1 =3, Φ =11.34 and =Φ 1 =.41 for cobalt th fil. Thi iulation wa carried for a elected et of value of energy paraeter order to tudy the variation of total agnetic energy with angle (θ) and the 8

9 nuber of layer (N). Accordg to the energy curve, the energy difference between the eay and hard direction can be iized at N=5. Thi nuber of layer (N=5) i approxiately equal to the thickne of cobalt fil yntheized for agnetic eory application by oe other reearcher [1]. Thi iplie that the agnetic oent of a cobalt baed eory device can be eaily rotated between eay and hard direction, when the nuber of layer i 5. REFERENCES 1. U-Hwang Lee et al., 6. Teplated ynthei of nanotructured cobalt th Fil for potential terabit agnetic recordg. NANO 1(1), P. Saaraekara, 6. Second order perturbation of Heienberg Hailtonian for non-oriented ultra-th ferroagnetic fil. Electronic Journal of Theoretical Phyic 3(11), P. Saaraekara and Willia A. Mendoza, 1. Effect of third order perturbation on Heienberg Hailtonian for non-oriented ultra-th ferroagnetic fil. Electronic Journal of Theoretical Phyic 7(4), P. Saaraekara and S.N.P. De Silva, 7. Heienberg Hailtonian olution of thick ferroagnetic fil with econd order perturbation. Chee Journal of Phyic 45(-I), K.D. Uadel and A. Hucht,. Aniotropy of ultrath ferroagnetic fil and the p reorientation tranition. Phyical Review B , A. Hucht and K.D. Uadel, Theory of the p reorientation tranition ultra-th ferroagnetic fil. Journal of Magneti and Magnetic Material 3, P. Saaraekara, 7. Claical Heienberg Hailtonian olution of oriented pel ferriagnetic th fil. Electronic Journal of Theoretical Phyic 4(15), P. Saaraekara, 6. A olution of the Heienberg Hailtonian for oriented thick ferroagnetic fil. Chee Journal of Phyic 44(5), P. Saaraekara,. Eay Axi Oriented Lithiu Mixed Ferrite Fil Depoited by the PLD Method. Chee Journal of Phyic 4(6),

10 1. P. Saaraekara, 3. A puled rf putterg ethod for obtag higher depoition rate. Chee Journal of Phyic 41(1), P. Saaraekara, 1. Deteration of energy of thick pel ferrite fil ug Heienberg Hailtonian with econd order perturbation. Georgian electronic cientific journal: Phyic 1(3), P. Saaraekara, 11. Invetigation of Third Order Perturbed Heienberg Hailtonian of Thick Spel Ferrite Fil. Inventi Rapid: Algorith Journal (1), P. Saaraekara and Willia A. Mendoza, 11. Third order perturbed Heienberg Hailtonian of pel ferrite ultra-th fil. Georgian electronic cientific journal: Phyic 1(5), P. Saaraekara, M.K. Abeyratne and S. Dehipawalage, 9. Heienberg Hailtonian with Second Order Perturbation for Spel Ferrite Th Fil. Electronic Journal of Theoretical Phyic 6(), P. Saaraekara and Udara Saparaadu, 13. Eay axi orientation of Bariu hexa-ferrite fil a explaed by p reorientation. Georgian electronic cientific journal: Phyic 1(9),