d =6.6, the sc(001) film with 10000

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1 Deagnetization factor epenence of energy of thick ferroagnetic fil P. Saaraekara Departent of Phyic, Univerity of Peraeniya, Peraeniya, Sri Lanka. Abtract Secon orer perturbe Heienberg Hailtonian wa eploye to tuy the variation of energy of ferroagnetic thick fil with eagnetization factor. Uner the influence of eagnetization factor given by =6.6, the c(1) fil with 1 layer can be eaily oriente in.6 raian irection for the value of energy paraeter ue in thi report. Eay irection of thick fcc(1) fil with 1 layer wa eterine a.66 raian, when the eagnetization factor i given by =.6. The energy of c(1)thick fil i larger than that of fcc(1) thick fil. But the energy curve of fcc(1) thick fil i oother than that of c(1). 1. Introuction: Ferroagnetic ultra-thin fil have been previouly invetigate uing claical oel of Heienberg Hailtonian with liite nuber of ter [1]. A it i ifficult to unertaning the behavior of exchange aniotropy an it application in agnetic enor an eia technology, exchange aniotropy ha been extenively invetigate in recent pat []. Ferroagnetic fil are thoroughly tuie nowaay, ue to their potential application in agnetic eory evice an icrowave evice. Magnetic propertie of ferroagnetic thin fil have been invetigate uing Bloch pin wave theory [3]. Although the agnetization of oe thin fil i oriente in the plane of the fil ue to ipole interaction, the out of plane orientation i preferre at the urface ue to the broken yetry of uniaxial aniotropy energy. Alo two ienional Heienberg oel ha been ue to explain the agnetic aniotropy in the preence of ipole interaction [4]. Magnetic propertie of ferroagnetic thin fil with alternating uper layer have been tuie uing Iing oel [5]. Ultra-thin ferroagnetic fil with two an three layer have been tuie uing Heienberg Hailtonian with econ orer perturbation by u [6]. The angle between eay an har irection wa foun to be 9 for c(1), fcc(1) an bcc(1) 1

2 ferroagnetic lattice. Variable an contant econ an fourth orer aniotropy contant have been coniere for thee calculation [6]. The energy of oriente thick ferroagnetic fil ha been calculate for continuou an icrete variation for thickne by u [7]. Thick fil of bcc(1) ferroagnetic lattice up to 1 layer have been coniere for thee iulation. Alo the olution of claical Heienberg Hailtonian with econ orer perturbation wa foun for thick ferroagnetic fil up to 1 layer [8]. Stre inuce aniotropy i crucial in agnetic fil [9, 11]. The unperturbe, n orer perturbe an 3 r orer perturbe energy of pinel ferrite an ferroagnetic fil wa eterine by u [1, 1, 13, 14, 15, 16, 17].. Moel: Heienberg Hailtonian of any ferroagnetic fil can be given a following [6-8]. r r r r J r r r r S. Sn 3( S. n)( n. Sn) () z (4) z 4 S. Sn + ( ) D ( S ) D ( S 3 5 λ λ ), n n rn rn r r r [ H ( S / µ )]. S K Sin θ, H= -, n n where (or n),, J, Z, Φ,, θ n n (or θ n ), D (), D (4), H in, H out,, K are inice of layer, total nuber of layer, pin exchange interaction, nuber of nearet pin neighbor, contant arien fro partial uation of ipole interaction, trength of long range ipole interaction, aziuthal angle of pin, econ orer aniotropy, fourth orer aniotropy, in plane applie fiel, out of plane applie fiel, eagnetization factor an the tre inuce aniotropy factor, repectively. Beginning fro above equation, following equation ha been proven in one of our early report [8]. J 3 E( θ ) = [ Z + ( 1) Z1] + { Φ + ( 1) Φ1 }( + co θ ) 8 8 () 4 (4) (co θd + co θd in inθ out coθ + K µ 3 [ ( Φ 4 + Φ 1 ) + D () + D C (4) co θ ] ( )in θ 1 3 () (4) [ ( Φ + Φ1) + D + D co θ ] in θ (1) C 4 11 in θ )

3 Here C 11 = JZ1 Φ1(1 + 3co θ ) 4 + 4co () (in θ co θ ) D (4) θ (co θ 3in θ ) D in inθ out coθ + 4K in θ µ C = JZ1 Φ1(1 + 3co θ ) + 4co () (in θ co θ ) D (4) θ (co θ 3in θ ) D in inθ out coθ + 4K in θ µ All thee iulation will be carrie out for J = D () = H in = H out = K = 1an D (4) = Reult an icuion: For c(1) lattice, Z =4, Z 1 =1, Φ =9.336 an Φ 1 =.375 [1], 3-D graph of energy veru angle an i given in figure 1 for =1. Several energy iniu can be oberve at ifferent value of angle an eagnetization factor inicating that thee are the eay irection. At =6.6, one eay irection can be oberve at thi 3-D plot. The angle correponing to thi =6.6 can be foun uing energy curve given in figure. Angle correponing to energy iniu an axiu were foun to be.6 an.76 raian, repectively. 3

4 Figure 1: 3-D plot of energy veru angle an for c(1) thick ferroagnetic fil with 1 layer 4

5 Figure : Energy curve of c(1) fil at =6.6 with 1 layer For fcc(1) lattice with Z =4, Z 1 =4, Φ =9.336, an Φ 1 =1.494 [1], the 3-D plot of energy veru an angle i given in figure 3 for =1. Energy iniu an axiu oberve in thee graph are correponing to eay an har irection of thick fil. For exaple one eay irection can be foun at =.6. The angle correponing to thi eay irection can be foun fro figure 4. Thi energy curve i oother than the energy curve obtaine in previou cae. The energy iniu an axiu can be oberve at.66 an.55 raian, repectively. 5

6 Figure 3: 3-D graph of energy veru angle an for fcc(1) thick ferroagnetic fil with 1 layer Figure 4: Graph of energy veru angle for fcc(1) fil at =.6 6

7 3. Concluion: Many eay irection can be oberve at any value of eagnetization factor for c(1) an fcc(1) thick ferroagnetic fil with 1 layer. The c(1) fil with 1 layer can be eaily oriente in irection given by.6 raian uner influence of eagnetization factor =6.6. When the eagnetization factor i =.6, eay irection of thick fcc(1) fil with 1 layer wa foun to be.66 raian. Accoring to 3-D plot, the energy of c(1)thick fil i larger than that of fcc(1) thick fil. Accoring to -D plot, the energy curve of fcc(1) thick fil i oother than that of c(1). Although thi iulation wa carrie out for oe electe value of energy paraeter, thi ae iulation can be perfore for any value of energy paraeter. Reference: [1] K.D. Uael an A. Hucht,. Aniotropy of ultrathin ferroagnetic fil an the pin reorientation tranition. Phyical Review B 66, [] Davi Leeran, Ricaro Rairez an Miguel Kiwi, 4. Monte Carlo iulation of exchange bia of ferroagnetic thin fil on FeF (11). Phyical Review B 7(18), [3] Martin J. Klein an Robert S. Sith, Thin ferroagnetic fil. Phyical Review 81(3), [4] M. Dantziger, B. Glinann, S. Scheffler, B. Zierann an P.J. Jenen,. Inplane ipole coupling aniotropy of a quare ferroagnetic Heienberg onolayer. Phyical Review B 66(9), [5] M. Bentaleb,. El Aoua an M. Saber,. Magnetic propertie of the pin -1/ Iing Ferroagnetic thin fil with alternating uperlattice configuration. Chinee Journal of Phyic 4(3), 37. [6] P. Saaraekara, 6. Secon orer perturbation of Heienberg Hailtonian for non-oriente ultra-thin ferroagnetic fil. Electronic Journal of Theoretical Phyic 3(11),

8 [7] P. Saaraekara, 6. A olution of the Heienberg Hailtonian for oriente thick ferroagnetic fil. Chinee Journal of Phyic 44(5), [8] P. Saaraekara an S..P. De Silva, 7. Heienberg Hailtonian olution of thick ferroagnetic fil with econ orer perturbation. Chinee Journal of Phyic 45(-I), [9] P. Saaraekara,. Eay Axi Oriente Lithiu Mixe Ferrite Fil Depoite by the PLD Metho. Chinee Journal of Phyic 4(6), [1] P. Saaraekara, 7. Claical Heienberg Hailtonian olution of oriente pinel ferriagnetic thin fil. Electronic Journal of Theoretical Phyic 4(15), [11] P. Saaraekara, 3. A pule rf puttering etho for obtaining higher epoition rate. Chinee Journal of Phyic 41(1), [1] P. Saaraekara, 1. Deterination of energy of thick pinel ferrite fil uing Heienberg Hailtonian with econ orer perturbation. Georgian electronic cientific journal: Phyic 1(3), [13] P. Saaraekara, 11. Invetigation of Thir Orer Perturbe Heienberg Hailtonian of Thick Spinel Ferrite Fil. Inventi Rapi: Algorith Journal (1), 1-3. [14] P. Saaraekara an Willia A. Menoza, 11. Thir orer perturbe Heienberg Hailtonian of pinel ferrite ultra-thin fil. Georgian electronic cientific journal: Phyic 1(5), [15] P. Saaraekara an Willia A. Menoza, 1. Effect of thir orer perturbation on Heienberg Hailtonian for non-oriente ultra-thin ferroagnetic fil. Electronic Journal of Theoretical Phyic 7(4), [16] P. Saaraekara, M.K. Abeyratne an S. Dehipawalage, 9. Heienberg Hailtonian with Secon Orer Perturbation for Spinel Ferrite Thin Fil. Electronic Journal of Theoretical Phyic 6(), [17] P. Saaraekara an Uara Saparaau, 13. Eay axi orientation of Bariu hexa-ferrite fil a explaine by pin reorientation. Georgian electronic cientific journal: Phyic 1(9),