Journal of Magnetism and Magnetic Materials 191 (1999) 9-12 Origin of the force exciting domain wall deformation in a thin magnetic film I. Antonov , L. Vatskitchev *, M. Vatskitcheva Department of Physics, Medical Academy, 1431 Sofia, Bulgaria Faculty of Physics, Sofia University, 1164 Sofia, Bulgaria Received 30 April 1998; received in revised form 23 July 1998 Abstract The origin of the pinning force acting on Bloch or NeŽel domain walls in thin magnetic films with a uniaxial anisotropy is discussed. The energy changes are investigated assuming that initially the external magnetic field induces the bulge of the domain wall attended with the pinning of the wall edges. The critical values of the field when the wall moves to a new equilibrium state were determined. The `magnetic pressure' and the pinning force are calculated from the changes of the wall energy determined for both kinds of domain walls. These results provide a possibility to obtain the contribution of the exchange energy at the wall deformation and to evaluate the influence of this energy on the wall behavior around the magnetic defects. 1999 Elsevier Science B.V. All rights reserved. PACS: 75.70.Kw Keywords: Magnetic films; Magnetization; Domain wall; Pinning force; Magnetic pressure 1. Introduction density of the exchange energy for the elementary volume with a magnetization dM will be d The aim of this investigation is to describe the " !H density of the domain wall energy in a thin mag- ) dM"!(2A/M) M ) dM, where H and A are the exchange field and the exchange constant, netic film with a thickness t when the Bloch or NeŽel respectively, M wall changes its shape. We assume that the shape  is the saturation magnetization of the film material [2]. On the other hand, the force represents a spherical surface between two planes acting on an elementary area ds of the domain wall in an external magnetic field less than the critical is f" value for the concrete wall position (Fig. 1). Follow- ds"p ds. The `magnetic pressure' p pro- vokes the change of the radius of curvature 1/R ing the basic ideas in Refs. [1,2] we used the same until the internal field balances the pressure action. idealization in the case of Bloch type wall. Then the The surface density of the wall energy " #(1/R)j /j(1/R)#O is built from the energy of nondeformed state  which includes indepen- * Corresponding author. Fax: #359-2-9625276; e-mail: dent of R terms, the energy of deformed state (sec- lyuvats@phys.uni-sofia.bg. ond term) and the term of second order O. In 0304-8853/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 3 0 9 - 6 10 I. Antonov et al. / Journal of Magnetism and Magnetic Materials 191 (1999) 9-12 components of the exchange field at r"R, sin +1!t/2R and sin +1!(t/R) 4A H "!M 1 1 1 ; 1 # R 1 1!(t/2R) , 2 1  R!t , R .(4) If RPR, then the components HP0 and H P0. The second component of H in (4) is not equal to 0 and gives a contribution in . Fig. 1. Coordinate system for the bulged domain wall. From Eqs. (3) and (4) we can obtain the addi- tional `magnetic pressure' p "!H dM on the wall element ds, varying R in the exchange energy and excluding the energy of nondeformed Bloch spherical coordinates the expression for the entire wall exchange energy transforms as 4A tanh x 2A p dr "! @ "!H ) M"! m ) M, (1)  r cosh x M ?  where m"M/M tanh x  is the relative magnetization and #2A@ dr ? r sin cosh x jM jM 2M 2M M" 2 # ! ! cot r jr jr r r e 4A tanh x ! dr  @r cosh x jM jM M ? # 2 # ! r jr jr r sin e tanh x !2A@ dr (5) r sin cosh x jM jM M ? # 2 # ! r jr jr r sin e . (2) where a"R! /2, b"R# /2 and x" (R!r)/ . The sum of the second and the fourth term is equal to zero. The value of 1/R is nearly 2. Theoretical background constant at R . Then We determined the additional term of the ex- sinh x#1 p dr change field as a result of a deformed Bloch type "!4A@?r  cosh x wall. In this case the components of the magne- tization are 1 4A "4A@ dtanhx ! k, k"0.924. ? r R M"M (6)  0,$sech R!r , tanh R!r , (3) The expression (6) includes the part of the ex- where "(A/K is the Bloch wall width and change energy depending on 1/R which adds to K is the constant of an induced uniaxial anisot- ropy. From Eqs. (2) and (3) we determined the  for 180° domain wall. At k"1 our result is similar to the result of the other authors [2]. I. Antonov et al. / Journal of Magnetism and Magnetic Materials 191 (1999) 9-12 11 The force f acting on the wall element ds is The exchange field determining the pinning for- f"!(4A/R )k ds and the additional term in is ces in both cases is 1 M "! H ) M dr l 2@ H for Bloch wall, (11) ? "!2 R dr tanh x dr 2A "2A@ #2A@ " , M H l, for Ne&el wall. (12) ? r cosh x ? r R!  , "!2 R (7) The materials for thin magnetic films with where we suppose sin +1. In the presence of uniaxial magnetic anisotropy in the film plane is magnetic or other local defects it is possible that distinguished with the small characterizing length. 1/R&1/t, then the expression (7) gives a significant Therefore, the external magnetic field HM/  supplement to the exchange energy. will change the wall from plane to bulging shape. The behavior of the exchange energy for NeŽel The similar effect of a deformation can be observed domain wall can be treated similarly. In this case experimentally in the areas with structural and div MO0 and there are `magnetic charges' in the magnetic defects where the domain wall holds back film volume, which induce an augmentation in the during the motion. The radius of curvature of the magnetostatic energy of the domain wall. Besides domain wall is determined from Eqs. (11) and (12) the domain wall deformation increases the mag- as 1/R+ H/Ml ,. If the magnetic field interval netostatic energy. The pressure p for the initial part of the magnetization curve is "!H ) dM gives an auxiliary energy term similar to the expres- known, the average value of the radius can be sion (5). After integrating the procedure we could obtained as find f and changing with ,. M RM+  l ,. (13) H0  3. Results and discussion For typical values l"10\-10\ m, M" Using the characteristic length of the film mate- 0.5-1 T and H0 "6-8 A/m we obtain rial R"10\-10\ m. Introducing these data in Eq. (6) we determined the additional surface energy 4 of the order of 10\-10\ J/m. This value is l" (AK (8) a few orders less than M +10\ J/m and can be  neglected. But the contribution of will be essen- tial at R+t. and describing the domain wall width with the Analyzing the static of the deformed domain wall relevant expressions we found the pinning forces [3] we suppose the value of the parameter f and f, for Bloch and NeŽel domain wall, respec- "f/ tively: +0.1-0.2. Some concrete results are ob- tained at "0.147. The results of the other authors ( "0.116 for Ni M 4  Zn FeO and "0.1368 for f  l (AK YIG) are in the same interval [4,5]. To explain the "!k ds, where l , (9) " R M different value of the parameter as a result of the changes of the exchange energy must be taken into M f  l, account the second radius of curvature 1/R ,"!k ds, +1/t R (Fig. 2). Thus we can get similar expressions for f and 4 where l (A(K#(M/ )) where R is replaced with R ," . (10) M +t. In our calcu-  lations we used the analytical expression for Bloch 12 I. Antonov et al. / Journal of Magnetism and Magnetic Materials 191 (1999) 9-12 wall is a function of the rate between the wall width and the radius of curvature (Fig. 3):  ( )" +  R!   A K   " " , (14) R(1!( /R)) 1!  where " or " ,. The parameter " /R shows that the surface energy increases sharply at a small radius of the curvature. Fig. 2. Cross-section of the film in the case R The influence of the domain wall deformation on +t. the speed of wall motion is measured indirectly in Refs. [2,6]. To neglect this influence on the experi- mental results for nickel ferrite the thickness of the sample might be less than 100 m and the applied field should not exceed 20 A/m. Similar consider- ations are applied to YIG [8]. Acknowledgements We acknowledge the financial support from the National Fund for Scientific Research under Grant No. F-531/95. Fig. 3. The relative wall energy versus the parameter . References [1] H.L. Huang, J. Appl. Phys. 40 (1969) 855. and NeŽel walls but in the last case (t+ "10\) [2] T.H. O'Dell, The Dynamics of Magnetic Bubbles, Domain this is not justifiable [7]. In this case the pinning and Domain Walls, Macmillan, New York, 1981. [3] I.G. Antonov, L.P. Vatskitchev, Phys. Stat. Sol. (a) 153 force acting on the bulged wall may be explained (1996) 173. with the other terms of the wall energy. The result [4] M.A. Escobar, L.F. Magana, R. Valenzuela, J. Appl. Phys. for the additional energy due to 1/R 53 (1982) 2692.  may be used to evaluate the field around spherical magnetic [5] M.A. Escobar, R. Valenzuela, L.F. Magana, J. Appl. Phys. defects with a size of the order of thin film thickness 54 (1983) 5935. [6] J.K. Galt, Bell Syst. Tech. J. 33 (1954) 1023. when the exchange energy & . The relative [7] A. Aharoni, J. Appl. Phys. 47 (1976) 3329. energy of the deformed wall versus nondeformed [8] A. Harper, R.W. Teale, J. Phys. C 2 (1969) 1926.