3b2v7 MAGMA : 8393 Prod:Type: com ED:Jolly=Bala pp:124ðcol:fig::NILÞ PAGN: Dorthy SCAN: Kalai ARTICLE IN PRESS 1 49 3 51 Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 5 53 7 Effect of roughness on the magnetic structure of ferro/ 55 9 antiferromagnetic interface 57 11 O.K. Dudko*, A.S. Kovalev 59 13 Institute for Low Temperature Physics and Engineering, 61103 Kharkov, Ukraine 61 15 63 Abstract 17 65 Spin structures at the ferro/antiferromagnetic interfaces perturbed by defects such as atomic high steps are analytically investigated. Atwo-dimensional model is proposed to describe the spin distribution formed on the 19 interfacial step at the domain wall. Acriterion of the domain wall configuration relative to the interface is found, 67 defined by the magnetic and geometrical characteristics of the interface and the magnet. r 2001 Published by Elsevier 21 Science B.V. 69 23 Keywords: Anisotropy-single ion; Domain wall-structure; Interface roughness; Layered Heisenberg systems 71 25 73 27 75 Layered magnetic structures and interfaces distribution at the perturbed interface expressed 29 between different magnetically ordered media have in terms of the material parameters of the magnet 77 aroused considerable interest in recent years due to can be used as a basis for analysis of the 31 their wide variety of surprising features and a observable physical effects, the formation of 79 multiplicity of technological uses. The roughness DWs may lead to, such as exchange bias and 33 of atomic high steps necessarily abundant on other related phenomena. 81 the interface involves severe consequences for the Consider classical Heisenberg FM/AFM system 35 magnetic order of the layered systems. The with atomic high step on the interface, taking into 83 intention of the present paper is to describe spin account a weak easy-axis anisotropy g along the x 37 structures at ferro/antiferromagnetic (FM/AFM) direction in the easy xz-plane (Fig. 1). As it will be 85 interfaces perturbed by defects such as steps. We seen from below, qualitative analysis of the 39 develop a model that allows one to obtain magnetic structure, we are interested in, is allowed 87 analytical expressions for the magnetic ordering under the assumption of equal anisotropy for FM 41 throughout the volume of the system and for the and AFM, however, the quantitative analysis 89 energy of the domain walls (DWs) of various would require one to differ anisotropy for the 43 configu 91 45 93 *Corresponding author. School of Chemistry, Sackler Faculty of Exact Sciences, Tel-Aviv University, Ramat Aviv, 47 699976 Tel UNCORRECTED PROOF rations. Information on the real spin two layers. At the exchange interaction through the interface JS under a critical value J spin ordering in FM and AFM is ideal, and collinear DW forms along one of the x half-axes. At J Aviv, Israel. Fax: +972-3-6409293. oJSoJnn the DW takes noncollinear form. As 95 E-mail address: dudko@post.tau.ac.il (O.K. Dudko). JS reaches the critical value Jnn; the DW is 0304-8853/01/$ - see front matter r 2001 Published by Elsevier Science B.V. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 7 9 7 - 1 MAGMA : 8393 ARTICLE IN PRESS 2 O.K. Dudko, A.S. Kovalev / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 1 where lA ¼ b JA=g and lF ¼ b JF=g are the 49 ``magnetic lengths'' in the half spaces. The values 3 zA and zF can be defined from the boundary 51 conditions (2) and are the functions of the 5 parameters JA; JF; JS; g: Using Eqs. (3) we obtain 53 the energy of the unit length of the DW along the 7 interface: 55 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi g g 9 E8E JFg þ JAg þ ðz 2 b F zAÞ 57 11 Fig. 1. DW caused by a step on FM/AFM interface given b=2 z b=2 þ z 2 59 single-ion anisotropy in the easy plane (xz). þ 2J F A S þ : ð4Þ 2lF 2lA 13 61 Energy (4) appears to be the function of the 15 repelled from the interface since the energy of the exchange integrals of FM, AFM and through the 63 DW in the layer is less than that at the interface. interface, depending also on the easy-axis aniso- 17 To find a criterion of the DW orientation and to tropy parameter. Common expression for the DW 65 determine the values J and Jnn; calculate the energy immediately follows from Eqs. (4) in the pffiffiffiffiffi 19 energy of the noncollinear DW at the step along case J 67 A ¼ JF ¼ JS: E0 ¼ 2 Jg; which agrees with the interface. that obtained by the direct calculation of the DW 21 From the energy of the magnetic interaction in energy in the homogeneous magnet. To compare 69 the spin chain along the z-axis at fixed x static the energies of variously configurated DW, con- 23 equations for spin deviations j in the chain can be sider the case of equal values of the exchange 71 derived. After variables substitution taking into parameters in FM and AFM: J JA ¼ JFaJS 25 account ``layered'' ordering in AFM, linearized (we use the assumption of equal exchange con- 73 equations take the form: stants in FM and AFM to obtain some qualitative 27 results. Note, that, for Fe/Cr, as an example, 75 g JAb2 q2j þ sinð2jÞ ¼ 0; J 29 qz2 2 Fe=JCrE2 while JFe=JFe2CrE10 and thus the assumption is valid to be a good approximation). 77 Then, the energy of the unit length of the DW 31 g J 79 Fb2 q2j þ sinð2jÞ ¼ 0; ð1Þ along the interface is qz2 2 33 pffiffiffiffiffi where b is lattice parameter along z direction, J g J 81 A E08ðJ; JS; gÞ ¼ 2 Jg þ 1 : ð5Þ and J 2 J F are the exchange constants in the z- S 35 directions in AFM and FM. Eqs. (1) is comple- 83 mented by the boundary conditions Comparing this expression with the energy of 37 the collinear DW in the plane of the interface 85 qj bJ Ecol ¼ 2JS; a critical value of the exchange inter- A ¼ J 39 qz S sinðj0 j1Þ; z¼ b=2 action through the interface JS can be found, at 87 which the transformation of the collinear DW into qj pffiffiffiffiffi 41 bJ the noncollinear DW occurs: Jn Jg: At J 89 F ¼ J S ¼ 12 S > qz S sinðj0 j1Þ: ð2Þ z¼þb=2 Jnn ¼ J the energy of the DW along the interface 43 exceeds the energy of the DW within the thickness 91 The solut 45 93 47 UNCORRECTED PROOF ions of Eqs. (1) describe the rotation of of the magnet, the DW is repelled from the spins in a chain along z at fixed x: interface and is oriented perpendicular to the j ¼ 2 arctan expððzA zÞ=lAÞ; ðzo0Þ; interface. It is easy to obtain the value of the Jnn pffiffiffiffiffiffiffiffiffiffiffi for J J : If the exchange para- 95 j ¼ 2 arctan expððz FaJA: Jn n ¼ FJA F zÞ=lFÞ; ðz > 0Þ; ð3Þ meters in FM and AFM differ, the DW at MAGMA : 8393 ARTICLE IN PRESS O.K. Dudko, A.S. Kovalev / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 3 1 JS > Jnn forms, obviously, in the magnet with the leads us to the following solution of the volume 49 smallest value of the exchange interaction. These problem (6): 3 conclusions as to DW orientation are in agreement J 51 S with the results of numerical calculations for Fe/ jðx; z > 0Þ ¼ pffiffiffiffiffiffiffiffiffiffi * 5 Cr multilayers presented in Ref. [1]. If the finite pa JFJF s 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 53 thickness of the FM and AFM layers and the finite Z þN ðx x0Þ2 z2 7 distance L between the steps on the interface are dx0K @ A 0 þ s2 s2 55 take into account, a prerequisite to the formation N x z 9 of the DW p a ffiffiffiffiffilong the interface is E8LoE>h; where sinðwðx0ÞÞsgnðx0Þ ð8Þ 57 E * > ¼ 2 Jg ( *J is the exchange integral along the 11 x-direction). The opposite inequality is the condi- (and the analogous expression for AFM half 59 tion of the DW formation perpendicular to the space), where Macdonald's function K0ðkÞ is the 13 interface. Green's function of the Klein­Gordon equation; 61 Analytical description of the nonuniform mag- w ¼ jjz¼þb=2 jjz¼ b=2 the functionqof relative ffiffiffiffiffiffiffiffiffiffi spin * 15 netization distribution caused by a monatomic deviation at the interface; sx ¼ a JF=g and sZ ¼ pffiffiffiffiffiffiffiffiffiffi 63 step at the FM/AFM interface can be provided in b JF=g are, respectively, the ``magnetic lengths'' 17 the framework of a simple 2D model proposed in along the x and z directions. From the expression 65 Ref. [2] for a system of AFM with the lattice (8) a 1D equation for the function wðxÞ follows. In 19 dislocation. Consider J the case of the equal exchange constants in FM S value on the interval J 67 oJ and AFM it takes the form SoJn n which corresponds to noncollinear DW 21 formation along the interface. For an equivalent 2J wðxÞ ¼ p S pffiffiffiffiffiffi 69 system of two FM half spaces in contact after pa J *J 23 corresponding variables change, long-wave equa- Z s 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 71 þN tions for the magnetization distribution take the ðx x0Þ2 b2 dx0K @ A 0 þ 25 form N s2x 4s2z 73 27 * g sinðwðx0ÞÞsgnðx0Þ: ð9Þ JAa2 q2j þ J sinð2jÞ ¼ 0; 75 qx2 Ab2 q2j qz2 2 Eq. (9) can be solved by the successive approx- 29 imations method. For the first approximation it 77 * g JFa2 q2j þ J sinð2jÞ ¼ 0; ð6Þ gives: qx2 Fb2 q2j qz2 2 Z 31 J N 79 S 1 where a is lattice parameter along the x direction, w1ðxÞ ¼ p sinðeÞ dpK0ðpÞ; ð10Þ 33 * Jn p J x=sx F and * JA are, respectively, the exchange integrals 81 in FM and AFM in x-direction. Nonlinear Eqs. (6) where e changes from p (at JS ¼ J ) to ðp=2Þ pffiffiffiffiffiffiffiffi 35 can be linearized by replacing single-ion aniso- g=J (at JS ¼ J). The function 83 tropy E Z N an ¼ gð1 cos2 jÞ=2 with the piecewise 1 37 parabolic function, which is possible when the IðxÞ ¼ dpK p 0ðpÞ 85 x=s exchange interaction in FM and AFM are of x 39 the same order of value. Since we are interested in can be estimated on the different intervals of the 87 the magnetization distribution over distances coordinate x values: 8 pffiffiffiffiffiffiffiffiffiffiffiffi 41 larger than atomic dimensions, replace an interface pffiffiffi > s 89 x=jxj expðx=sxÞ= p; x5 sx; > with a step by the ideal boundary, having reversed > > > > ðð1 jxj=s 43 the sign of > xÞ ðjxj=sxÞlnðjxj=sxÞÞ=p; 91 45 93 47 UNCORRECTED PROOF the exchange interaction through it on > > < one side of the step. Complementing the boundary s IE xoxo0; ð11Þ condition presenting the density of the effective > > ðð1 þ jxj=sxÞ ðjxj=sxÞlnðjxj=sxÞÞ=p; > forces acting at the interface > > > > 0oxosx; > > : pffiffiffiffiffiffiffiffiffiffi pffiffiffi 95 f7ðxÞ ¼ 7sgnðxÞJS sinðjjz¼þb=2 jjz¼ b=2Þ; ð7Þ 1 sx=x expð x=sxÞ= p; xb sx: MAGMA : 8393 ARTICLE IN PRESS 4 O.K. Dudko, A.S. Kovalev / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 1 The solution of the 2D problem can be restored parameters of the FM, AFM and through the 49 by substituting the solution of the 1D Eq. (9) into interface as well as by the thickness of the layers 3 expression (8): and geometry of the interface. The distribution of 51 1 J magnetization in the entire volume of the magnet jðx; z > 0ÞE S sinðeðJ 5 2 Jn SÞÞ containing the domain wall along the interface is 53 Z expressed in the terms of the magnetic and 7 1 N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi geometrical parameters of the system. Decrease 55 dpK ðp x=s Þ: p 0ð xÞ2 þ ðz=szÞ2 of the nonuniformity of the magnetization dis- 0 9 ð12Þ tribution into the depth of the magnets is 57 exponential, and the width of the domain wall is 11 At x ¼ 0 and zb !oZ it follows from Eqs. (12) pffiffiffiffiffiffiffiffiffiffi proportional to the exchange interaction in the 59 that jpðJS=JnÞsinðeðJSÞÞ z=sZ expð z=sZÞ: At magnets and inversely related to the anisotropy 13 large distances from the interface the system turns parameter. 61 to the ground state (Fig. 1). 15 In conclusion, a two-dimensional model is 63 presented for analytical description of the spin 17 structure at the FM/AFM interface with the References 65 atomic high step. The domain wall is necessarily 19 associated with the step on the interface. The [1] P. B.odeker, A. Hucht, J. Borchers, F. G.uthoff, A. Schreyer, H. Zabel, Phys. Rev. Let. 81 (1998) 914. 67 energy along with the orientation of the domain [2] O.K. Dudko, A.S. Kovalev, Low Temp. Phys. 24 (1998) 21 wall is dictated by the anisotropy and exchange 422. 69 23 71 25 73 27 75 29 77 31 79 33 81 35 83 37 85 39 87 41 89 43 91 45 93 47 UNCORRECTED PROOF 95