Effect of Anisotropy Asymmetry on the Switching. Behavior of Exchange Biased Bilayers

Transcription

1 Applied athematial Siene, Vol. 5, 0, no. 44, Effet of Aniotropy Aymmetry on the Swithing Behavior of Exhange Biaed Bilayer Congxiao Liu a, atthew E. Edward b, in Sun and J. C. Wang b a Department of athemati, Alabama A& Univerity, Normal, AL 3576, USA b Department of Phyi, Alabama A& Univerity, Normal, AL 3576, USA Department of athemati and the Center for aterial for Information Tehnology (INT, The Univerity of Alabama, Tualooa, AL 35487, USA Abtrat In the angular murement of ome exhange biaed bilayer, a jump of exhange bia i oberved around the F magnetization tabilized diretion. Thi magnitude abrupt hange reult from an aymmetri reveral of the F magnetization. In thi paper, we give a theoretial analyi on the withing hirality of the F magnetization preent in exhange biaed bilayer. Coherent rotation of the F magnetization i aumed. The analyi tart from a general diuion of a -d ingle domain partile, for whih the third derivative of the aniotropy energy play an important role in determining the behavior of magneti withing. In exhange biaed bilayer, a non-vanihing third derivative of the total aniotropy energy arie from the off-alignment of the unidiretional aniotropy and the uniaxial aniotropy. For thee ytem, reveral of the F magnetization on the ide of the uniaxial aniotropy attain a higher withing field than on the ide of the unidiretional aniotropy. Suh information i important for an experimental determination of the unidiretional aniotropy in exhange biaed bilayer. Keyword: Exhange bia, withing hirality, aniotropy off-alignment, oherent rotation

2 96 Congxiao Liu et al. Introdution In an exhange biaed bilayer [], a ferromagneti (F layer i oupled with an antiferromagneti (AF layer through the atomi exhange interation at their interfae []. A a reult, the hyterei loop of the F layer beome broader and i hifted along the field axi. The former i alled oerivity enhanement and the latter i alled exhange biaing. Unidiretional aniotropy and uniaxial aniotropy oexit in uh a ytem [3]. The unidiretional aniotropy ould be off-aligned with the uniaxial aniotropy in exhange biaed bilayer [4-7], lding to an aymmetry in the total aniotropy of the F layer. Without external field, the F magnetization i not tabilized along the unidiretional aniotropy, auing a diffiulty in determining thi aniotropy experimentally. A onequene of thi off-alignment of different aniotropy axe i the aymmetri angular dependene of exhange bia and oerivity [4,8,,4,7]. Furthermore, a jump of the exhange bia wa oberved around the F magnetization tabilized diretion [4,8,,7]. Fig. how imulated angular dependene of exhange bia H eb and oerivity H for a modeled F/AF ytem. Coherent rotation of the F magnetization i aumed. Hyterei loop were omputed every 0 degree for the applied field diretion. The uniaxial aniotropy ontant i half of the unidiretional aniotropy ontant and the two aniotropy axe are off-aligned by 30. The plotted urve are aymmetri and a jump of H eb i preent at 80, the F magnetization tabilized diretion. Rotational hyterei of Co/Fen thin film ha alo been oberved howing uh aymmetry effet [8]. To undertand thi anomaly and to determine the unidiretional aniotropy experimentally, it i nery to arry on a detailed analyi of magnetization withing in uh ytem. Experimental reult have uggeted a oherent rotation of the F magnetization in ome F/AF ytem [8,0,5,7,9]. Neverthele, effet of aniotropy ymmetry property on the withing of F magnetization i not diued in related literature. In thi paper, we give a theoretial analyi on the withing hirality in exhange biaed bilayer. The analyi i baed on a oherent rotation of the F magnetization. We tudy the evolution of the ytem energy, peifially, the reloation of the energy minimum. Our reult revl that the F magnetization withing field depend on the ene of rotation. The paper i divided into two part: in the firt part, magnetization withing i tudied for a -d ingle domain partile with an aymmetri aniotropy; in the eond part, the withing hirality of the F magnetization in exhange biaed bilayer i analyzed.

3 Effet of aniotropy aymmetry 97 Fig.. Simulated angular dependene of exhange bia H eb and oerivity H of an exhange biaed bilayer. There i an off-alignment of 30 between the unidiretional aniotropy and the uniaxial aniotropy. The uniaxial aniotropy ontant i half of the unidiretional aniotropy ontant. In the plot, diamond repreent H eb and quare repreent H.. agneti withing in the y diretion of a -d ingle domain partile with an aymmetri aniotropy Conider a magneti ingle domain partile with aniotropy energy denity E (, with the angle of the magnetization with repet to one of it y diretion. The diretion of = 0 i the initial diretion of the magnetization in the abene of external field. athematially E ( ha a loal minimum at = 0. We aume E( ha an nonzero third derivative at = 0. In exhange biaed bilayer, thi happen when different aniotropy axe are off-aligned. When an external field H i applied anti-parallel to the magnetization diretion, the free energy denity of the partile beome F( = E( + H o, where i the aturation magnetization. For a mall value of H, the magnetization tabilize at = 0 and the firt derivative of the free energy i zero. The eond

4 98 Congxiao Liu et al derivative of the free energy F"(0 i poitive but will dere with the inre of H. At ome ritial value H SW 0 of H, F "(0 dere to zero. The magnetization i then untable. H SW 0 an be olved from the ondition F "(0 = 0 and i given by E"(0 H SW 0 =. ( A the field H i inred further, F "(0 beome negative and = 0 now i a loation of energy maximum. Conequently, the magnetization will rotate to the diretion orreponding to the new energy minimum. There are two poibilitie: it an either with to the oppoite diretion immediately at H = H SW 0, or initially rotate ontinuouly and withe later at ome H > H SW 0. Obviouly, thee two ituation will reult in different withing field. We will fou on the latter ae. Let be the hift of the energy minimum. an be olved from the ondition F' ( = E'( H in = 0. ( athematially, i a funtion of the external field H. Generally, will depend on the expliit form of E (. However, when we onider withing of the magnetization, our interet i in the hange of the magnetization diretion. In other word, we do not need to olve for expliitly. Intd, we alulate the derivative of. For a bigger H, the magnetization will rotate further away from the original diretion, whih mn that (H i an inring funtion. Therefore (H ha an invere funtion that an be written a H = H (. Solving for H from (, we obtain E'( H =. (3 in We then take the derivative with repet to on both ide of (3, dh E"( in E'( o =. (4 d (in Our interet i in the initial behavior of the magnetization at H H w 0. If we aume an initial gradual rotation of the magnetization, will be mall when H i loe to H w0. That will allow u to expand equation (4 at = 0. In the following alulation we will adopt the big O notation. Namely, O (x refer to a term on the order of ~ x. For a onvergent erie, if x i uffiiently mall, all the term of higher order than ~ x an be aborbed in O (x. Alo note

5 Effet of aniotropy aymmetry 99 a ( n f ( x ( = = = x a + f ( x a f ( x a n = 0 ( f ( x + ( f ( x L = ( + O( x x +L = a a = ( + O( x = + O( x = x, (5 a a a a for any a 0 and f ( x <, where f (x i a funtion on the order of ~ x. Sine = 0 i an y diretion of the magnetization, the ondition F '(0 = 0 give E '(0 = 0. (6 Expand E" (, in, E' (, and o at = 0, (3 E"( = E"(0 + E (0, (7 3 in =, (8 (3 E (0 3 E'( = E'(0 + E"(0 +! (3 E (0 3 = E"(0 +, (9! 4 o =. (0! Uing (7, (8, (9, and (0, equation (4 i redued to (3 (3 E (0 3 E (0 dh = = 4 d + O( = (0 (3 E (0 ( = (3 E (3 (3 (3 E (0 E (0 E (0 = O( O( O( + = + = +. ( Taking the invere of ( and applying (5, we get d = ( 3. ( dh E (0

6 00 Congxiao Liu et al A wa pointed out rlier, i the loation of the new energy minimum and d provide information about the hift of the energy minimum. Sine we have dh (3 (3 (3 aumed E 0, we diu the two ae E > 0 and E (0 < 0 eparately: (3 d (a E > 0. In thi ae, ( implie 0 < < for a uffiiently mall. dh Thi indiate a gradual inre of with the initial inre of H, whih orrepond to a gradual ounterlokwie rotation of the F magnetization. (3 Converely, ( implie that in the ae of E (0 > 0, if the magnetization rotate ounterlokwie, it undergoe a gradual rotation before it withe to the oppoite diretion. Note that the field i applied anti-parallel to the magnetization diretion. The magnetization an rotate either ounterlokwie or lokwie. In the latter ae, it will with to the oppoite diretion immediately at H = H w 0. Thi i bue on thi ide, there i no energy minimum loated at the intermediate diretion. (3 (b E (0 < 0. From a imilar argument, if the magnetization rotate ounterlokwie upon reveral, it will with to the oppoite diretion immediately at H = H w 0. On the other hand, if the magnetization rotate lokwie, it will undergo an initial gradual rotation before it withe. (3 Thu in the ae of E 0, magneti withing field will depend on the ene of rotation. The hiral ymmetry of magneti withing i broken. Thi ituation i (3 illutrated in Fig.. Fig. (a how the ae of E (0 > 0 and Fig. (b how (3 the ae of E (0 < Swithing hirality in exhange biaed bilayer due to off-aligned aniotropy axe For an exhange biaed bilayer with off-aligned unidiretional and uniaxial aniotropy axe [8], we denote the y diretion of the unidiretional aniotropy by u.d., and the y axi of the uniaxial aniotropy by e.a.. Without external field, the F magnetization tabilize at ome intermediate diretion between u.d. and e.a., hown in Fig. 3. Thi table diretion i the referred y diretion of the magnetization diued in the previou etion. To ditinguih the two ene of rotation for the F magnetization, we all rotation on one ide the u.d. ide, that i, rotation of the F magnetization from the initial diretion toward u.d. (ee Fig. 3,

7 Effet of aniotropy aymmetry 0 agnetization diretion for H > H w 0 ounterlokwie rotation H agnetization diretion for H > H w 0 lokwie rotation Fig. (a Initial magnetization tabilized diretion ( H = 0, = 0 agnetization diretion for H > H w 0 ounterlokwie rotation H Initial magnetization tabilized diretion ( H = 0, = 0 agnetization diretion for H > H w 0 lokwie rotation Fig. (b Fig.. Swithing hirality of a ingle domain partile with an aymmetri E aniotropy. A the external field H i inred to the ritial value H = "(0 SW 0, (3 the magnetization rotate away from the y diretion. (a E (0 > 0. For a ounterlokwie rotation, the magnetization undergoe a gradual rotation before it withe. For a lokwie rotation, the magnetization withe immediately at (3 H = H SW 0. (b E (0 < 0. The ituation i revered. i the magnetization vetor and i the aturation magnetization, the magnitude of.

8 0 Congxiao Liu et al and the other ide the e.a. ide. Denote δ ud the angle of u.d., and δ the aute angle of e.a., both with repet to the F magnetization tabilized diretion (Fig. 3. The e.a. ide H δ e.a. u.d. F Initial F tabilized diretion ( H = 0, = 0 δ ud The u.d. ide Fig. 3. In thi exhange biaed bilayer, the unidiretional aniotropy (u.d. i offaligned the uniaxial aniotropy (e.a.. When a field i applied anti-parallel to the F magnetization tabilized diretion, a withing hirality of the F magnetization i preent. F i the F magnetization vetor. For a oherent rotation of the F magnetization, the total aniotropy energy denity of thi F/AF ytem i E( = K ud o( δ ud + K in ( δ, (3 where K ud and K are the orreponding aniotropy ontant for the unidiretional aniotropy and the uniaxial aniotropy, repetively. i the angle of the F magnetization. Sine = 0 i an y diretion of the F magnetization, ondition (6 give, E' (0 = K ud inδ ud K in(δ = 0. (4 Rewrite (4 a Kud inδ ud = K in(δ. (5 (3 Uing (3 and (5, we an implify E (0, (3 E (0 = K ud inδ + 4K in(δ = K ud in( δ + 4K in(δ = 3K in(δ. (6 ( 3 Sine δ 0, (6 indiate E ( 0 0. For an external field H applied antiparallel to the F magnetization, the analyi in the previou etion implie a broken

9 Effet of aniotropy aymmetry 03 hiral ymmetry of magneti withing. A H i inred, the F magnetization will with at different field depending on the ene of rotation. (3 Conider the F/AF ytem in Fig. 3. δ > 0. Then from (6, E (0 > 0. The reult in the previou etion (ae (a diued in the final part of etion indiate that if the F magnetization rotate ounterlokwie, it undergoe an initial gradual rotation before it withe. In the onfiguration of Fig. 3, a ounterlokwie rotation of the F magnetization i on the e.a. ide. Thu we onlude that if the F magnetization rotate on the e.a. ide, it undergoe an initial gradual rotation and then withe. On the other hand, if the F magnetization rotate on the u.d. ide, it withe immediately at H = H w 0. A higher withing field i attained on the e.a. ide. In the angular dependene murement of exhange bia, hyterei loop are mured at different in-plane angle. For a field deviated by a mall angle from the F magnetization tabilized diretion, we an ue the analyi above to explain the aymmetri withing of the F magnetization, a hown in Fig. 4. Shown in Fig. 4(a, the field i applied on the u.d. ide, that i, the poitive diretion of the field i on the u.d. ide. Upon reveral the F magnetization rotate on the e.a. ide. In Fig. 4(b, the field i applied on the e.a. ide. Upon reveral the F magnetization rotate on the u.d. ide. A diued rlier, a higher withing field i attained for the F magnetization to with on the e.a. ide, whih i the u.d. ide for the applied field. When the field diretion i hanged from the e.a. ide to the e.d. ide, or vie vera, an abrupt hange of exhange bia i preent. u.d. i on the ide of higher value of exhange bia and e.a. i on the ide of lower value of exhange bia. Suh information i important for an experimental determination of u.d. and e.a. in exhange biaed bilayer. It mut be pointed out that although the main purpoe of thi paper i to tudy the withing behavior of exhange biaed bilayer, reult in etion an be applied to more ompliated ytem, uh a the ae where three or more off-aligned aniotropie oexit in a ytem. 4. Conluion In ummary, we gave a theoretial analyi on the withing hirality of the F magnetization in exhange biaed bilayer, auming a oherent rotation of the F magnetization. For a ingle domain partile with an aymmetri aniotropy, the withing field depend on the ene of rotation for the magnetization. In exhange biaed bilayer, a non-vanihing third derivative of the total aniotropy energy arie from the off-alignment of the unidiretional aniotropy and the uniaxial aniotropy. Reveral of the F magnetization on the ide of the uniaxial aniotropy

10 04 Congxiao Liu et al attain a higher withing field than on the ide of the unidiretional aniotropy. Our reult explain the abrupt hange in the angular dependene of exhange bia. Suh information i important for an experimental determination of the unidiretional aniotropy in exhange biaed bilayer. F F magnetization diretion for H 0 H e.a. u.d. Initial F magnetization tabilized diretion ( H = 0, = 0 Fig. 4(a H e.a. u.d. Initial F magnetization tabilized diretion ( H = 0, = 0 F F magnetization diretion for H 0 Fig. 4(b Fig. 4. In the angular murement, the external field i applied off from the F magnetization tabilized diretion by a mall angle. (a applied field on the u.d. ide, the F magnetization rotate on the e.a. ide. (b applied field on the e.a. ide, the F magnetization rotate on the u.d. ide. F i the F magnetization vetor.

11 Effet of aniotropy aymmetry 05 Referene [] W. H. eiklejohn and C. P. Bn, New agneti Aniotropy, Phy. Rev. 0 ( [] See review, J. Nogué and I. K. Shuller, Exhange bia, J. agn. agn. ater. 9 (999 03; A. E. Berkowitz and K. Takano, Exhange aniotropy - a review, ibid. 00 (999 55; R. L. Stamp, ehanim for exhange bia, J. Phy. D 33 (000 R47;. kiwi, Exhange bia theory, J. agn. agn. ater. 34 (00 584; F. Radu and H. Zabel, agneti Heterotruture, Springer Trat in odern Phyi (Springer, Berlin, 007, Vol. 7, pp [3] Z. Qian, J.. Siverten and J.H. Judy, agneti behavior of NiFe/NiO bilayer, J. Appl. Phy. 83 ( [4] G. J. ankey, private ommuniation. [5] N. C. Koon, odelling of interation at ferro/antiferromagneti interfae, Phy. Rev. Lett. 78 ( [6] V. Ström, B. J. Jönon, and D. Dahlberg, Determination of exhange aniotropy by mn of a ueptometry in Co/CoO bilayer, J. Appl. Phy. 8 ( [7] T. J. oran, J. Nogué, D. Lederman, and I. K. Shuller, Perpendiular Coupling at Fe-FeF Interfae, Appl. Phy. Lett. 7 ( [8] H. Xi, and R.. White, Angular dependene of exhange aniotropy in Ni 8 Fe 9 /CrnPt x bilayer, J. Appl. Phy. 86 ( [9] D. Spenato, S. P. Pogoian, H. Le Gall, Aymmetri magnetization reveral in exhange-biaed polyrytalline F/AF bilayer, J. agn. agn. ater. 6 ( [0] S. H. Chang, A. Hoffmann, and. Grimdith, Interplay between exhange bia and uniaxial aniotropy in a ferromagneti/antiferromagneti exhangeoupled ytem, Phy. Rev. B 7 (

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