Journal of Magnetism and Magnetic Materials 177-181 (1998) 195-196 Internal structure of a domain wall in ultrathin magnetic film Vladimir L. Sobolev* Department of Physics, Montana State University, Bozeman, MT 59717, USA Abstract Micromagnetic analysis of a domain wall internal structure in ultrathin magnetic film is carried out. The case when the film thickness is much smaller than the exchange length (the effective width of a Bloch line) is considered. It is shown that in this limiting case the angle of the domain wall twisting is very small and the domain wall has the quasi-Bloch structure. 1998 Elsevier Science B.V. All rights reserved. Keywords: Domain wall - structure; Bloch lines; Thin films - thickness The internal domain wall (DW) structure in thin mag- to the analysis of stability of domain structure; but the netic films was studied in detail [1, 2] in the approxi- DW internal structure has not been studied so far. mation that the film thickness t is much greater than This presentation is a brief report of micromagnetic the exchange length l "(A/2 M) (A is the in- consideration of a DW structure in ultrathin magnetic homogeneous exchange interaction constant, 4 M is the film with tl . saturation magnetization). The exchange length repres- The following geometry of the problem is considered. ents the characteristic value of the effective width The axis of uniaxial anisotropy (the 0z axis) is normal to "(A/2 M) of the elements of DW internal struc- the film surface. The DW plane is the x0z plane. The total ture which are called Bloch lines [1, 2]. Films of sub- energy density is the sum of contributions from in- stituted yttrium-iron garnet that were an object of homogeneous exchange, uniaxial anisotropy, and mag- intensive investigation for last 20 years, were thick netic-dipole interaction. enough to provide the fulfillment of the condition The static Landau-Lifshitz equation in our case has tl " along with another one t , where the form "(A/K) is the effective width of the Bloch DW, K is the uniaxial anisotropy constant. Introduction of vertical mV mW!mV mW" \[mWh K V !mVh K W ], and horizontal Bloch lines (the elements of the DW m internal structure) played a key role in theoretical ex- X mV!mV mX! \ mXmV" \[mVh K X !mXh K V ]. planation of peculiarities of the DW dynamics in thin (1) films of substituted yttrium-iron garnet [1, 2]. Here m"M · M\ A different situation, namely, tl  ; the condition (mV#mW#mX)"1 is  is realized in ultra- taken into account; h K "H thin magnetic films that attracted a great deal of atten- K/(4 M), HK is the demag- netization field which is determined by magnetostatic tion recently [3, 4], in connection with the giant equations. The boundary conditions for Eq. (1) may be magnetoresistance phenomenon observed in magnetic written as follows: multilayers. There exist several experimental studies of domain structure and DW dynamics in ultrathin films jm jm m X! n(m · n) !m W"0, [5-8]. All theoretical considerations [3, 4] were devoted W  jn X  jn jm jm m X V V  ! n(m·n) "0, jn !mX  jn * Tel.: #1 406 994 7836; fax: #1 406 994 4452; e-mail: jm jm m W!m V"0 at z"$t/2. (2) sobolev@physics.montana.edu. V jn W jn 0304-8853/98/$19.00 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 3 3 1 - 4 196 V.L. Sobolev / Journal of Magnetism and Magnetic Materials 177-181 (1998) 195-196 Here n is the unit vector along the anisotropy axis; the motion of the DW and precession of magnetization in- constants  and  characterize the exchange and aniso- side domains. The solutions of Eq. (1) may be represent- tropy at the film surface; the components of the vector ed as expansions with respect to the modes of spectrum of m have to satisfy the conditions mV, mWP0, mXP$1 at Winter's operator. The following result for the contribu- yPGR. Eq. (2) follow from the equation of motion of tion of the translational mode localized in the vicinity of magnetic moments on the surface and the fulfillment of the DW to the ( , ) angle can be obtained: the exchange boundary condition [9] for solutions of Eq. (1) on the surface.  ( , )+!  ( I)sin ( ), To solve system (1) with conditions (2) the polar coor- where " /(2(Q)1; I" / ; and dinates for vector m"+sin sin , cos , sin cos , with the polar axis oriented along the DW normal were ( I)"[(1# I)ln(1# I)!(1! I)ln(1! I) introduced. Using the representation !4 ln(2 I)!2 I]. " # ( , ), " ( )# ( , ), The value of  ( , ) at y"0 and z"$t/2 describes the DW twisting. As one can see this contribution is really where "y/ , "z/ ; ( ) and  describe the very small, Bloch DW, sin ( )"1/(cosh ), and " /2, Eq. (1) may be written as  (y"0, z"$t/2)+$ . [OK! \] ( , )" \h K The following result can be obtained for the contribution W ( , ), of the precessional mode: OK ( , )" \ sin ( )h K X ( , ). (3)  ("y"(t, z"t/2)+ Q\[ln 4! " "/(2 )]. The operator OK is determined by the expression The contribution of the translational mode to ( , ) is 1 j cos 2 OK" # ( ). equal to zero because h  j X K (y) is an odd function and the j  j  !  precessional contribution is given by the expression For solution of the magnetostatic problem the compo-  (y, z"t/2)+! Q\[1#ln(2 / )]. nents of the demagnetization field may be represented in Thus, the contributions to the DW twisting in ultrathin the form magnetic films caused by the translational and precessional sinh(k ) modes of the DW spectrum are negligibly small. It is easy h K W ( , )"!2  dk exp+ik( !i ), , to show that the standard variational procedure [1] for \ sinh( k/2) calculation of the DW twisting gives the same results. cosh(k ) h K The result obtained above allows to suppose that the X ( , )"2 i dk exp+ik( !i ), , (4) \ sinh( k/2) DW dynamic in ultrathin magnetic film should corre- where "t/2 spond to the one of a Bloch DW at least in the case of the is the main small parameter of the prob- lem. We consider the case when the material quality stationary motion. factor Q"K/(2 M) is much bigger than unity. This condition was used for the theoretical analysis of a DW References internal structure in thin yttrium-iron garnet films [1, 2] whereas samples that were used for applications had, as [1] A.P. Molozemoff, J.C. Slonczewski, Magnetic Domain a rule, very moderate values of Q*1. The material Walls in Bubble Materials, Academic Press, New York, quality factor in ultrathin metallic magnetic films is also 1979. Q*1 because the surface anisotropy that rotates mag- [2] F.H. de Leeuw, R. van der Doel, U. Enz, Rep. Prog. Phys. netization from the film plane is caused by magnetic- 43 (1980) 690. [3] J.A.C. Bland, B. Heinrich, Ultrathin Magnetic Structures I, dipole interaction. Springer, Berlin, 1994. Eqs. (3) and (4) represent the complete formulation of [4] B. Heinrich, J.A.C. Bland, Ultrathin Magnetic Structures the problem under consideration. The fulfillment of the II, Springer, Berlin, 1994. conditions [5] M. Ruhring, A. Hubert, J. Magn. Magn. Mater. 121 (1993) 330. j j "0, "0 [6] G. Bochi et al., Phys. Rev. Lett. 75 (1995) 1839. jz X!R jz X!R [7] A. Kirilyuk, J. Ferre, J. Pommier, D. Renard, J. Magn. at the film surfaces is important for the analysis of the Magn. Mater. 121 (1993) 536. [8] A. Kirilyuk, J. Ferre, D. Renard, IEEE Trans. Magn. 29 problem in the case t . The part of the operator OK that (1993) 2518. is given by the expression !j/j #cos 2 ( ) is the [9] G.T. Rado, J.R. Weertman, Phys. Rev. 94 (1954) 1386. well-known Winter operator [10, 11]. Its spectrum con- [10] J.M. Winter, Phys. Rev. 124 (1961) 452. tains two modes which correspond to translational [11] A.A. Thiele, Phys. Rev. B 7 (1973) 391.