Journal of Magnetism and Magnetic Materials 198}199 (1999) 468}470 Coercivity induced by random "eld at ferromagnetic and antiferromagnetic interfaces S. Zhang *, D.V. Dimitrov , G.C. Hadjipanayis , J.W. Cai , C.L. Chien Physics Department, New York University, 4 Washington Place, New York, NY 10003, USA Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA Department of Physics, Johns Hopkins University, Baltimore, MD 21218, USA Abstract In the presence of random "elds at an interface between a ferromagnetic and an antiferromagnetic layer, the domain walls in the ferromagnetic layer are pinned by local minimum energy. To move the domain walls, an applied magnetic "eld must be large enough to overcome statistically #uctuating energy. We have calculated this energy and found that the coercivity can be as large as a few kOe for a thin ferromagnetic layer. It is also found that the coercive "eld at low temperature scales as 1/t where t is the F layer thickness, and the coercive "eld decreases strongly with temper- ature. 1999 Elsevier Science B.V. All rights reserved. Keywords: Coercivity; Exchange bias The nature of the magnetic interaction at the interface theories predict domain formation in the antiferromag- between a ferromagnet and an antiferromagnet is a long netic layers and give the right order of magnitude of the debated issue. Experimentally, there are two distinct fea- exchange bias in comparing with experimental values. tures in the magnetic hysteresis loop. First, the hysteresis However, the above theories which are based on the loop is o!set from zero applied magnetic "eld if the &ideal' nature of the interfaces have two signi"cant draw- bilayer "lm (ferromagnetic and antiferromagnetic layers) backs. First, the structure of AF layer and F layer of is "eld cooled from temperature above the NeHel temper- experimental interfaces are usually not matched and far ature of the antiferromagnetic layer; this has been termed away from simple perfect uncompensated or compensate the exchange bias [1]. The second e!ect is that the interfaces. It seems from vast experiments that the ex- coercivity of the ferromagnetic layer is much larger than change bias does not require perfection of the interface. that without the underlying antiferromagnetic layer. The more serious problem is that the theories do not Up till now most of the theoretical studies have been address the phenomenon of the enhanced coercivity ob- focused on understanding the "rst phenomenon, the ex- served experimentally. change bias. Mauri et al. [2] considered a perfect &un- The present study follows the idea that experimental compensated' interface where the moments of the "rst interfaces are not perfect and interactions at the interface layer of the antiferromagnet in contact with ferromag- between F and AF layers are random as "rst proposed by netic layer are ferromagnetically aligned. Koon [3] inves- Malozemo! [4,5]. The presence of the random interac- tigated a &compensated' interface where there are no net tion leads to an energy term which competes with other moments on the "rst layer of the antiferromagnet. Both energies in the system. As a result, the antiferromagnetic layer breaks into domains with "nite sizes. Within each domain, the ferromagnetic layer receives a statistically * Corresponding author. Tel.: #1-212-9987724; fax: #1-212- net "eld from the antiferromagnetic layer, i.e., an ex- 9954016. change bias is induced by the random interaction [4,5]. E-mail address: shufeng.zhng@nyu.edu (S. Zhang) Here, we examine the role of this random interaction on 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 1 1 5 5 - X S. Zhang et al. / Journal of Magnetism and Magnetic Materials 198}199 (1999) 468}470 469 the coercivity, and determine the correlation between a a exchange bias and the enhancement of the coercivity in H"! hM! dx[g(x, y t 2¸t V >* #=/2) the ferromagnetic}antiferromagnetic bilayer system. V \* Let us consider a domain wall in the ferromagnetic #g(x, y !=/2)], (5) layer. The domain wall width and length are W and where we have used m((x, y L with its center coordinates x #=/2)"e(V, m( (x, y ! and y ; the thickness of =/2)"!e( the ferromagnetic layer is t, which is assumed to be small V at the domain wall boundaries, and de"ned g(x, y so that the magnetization does not vary along the direc- $=/2),hV(x, y $=/2)!hM with hM as the aver- age of the random "eld. The "rst term in Eq. (5) repres- tion perpendicular to the "lm plane. The domain size ents the exchange bias which has been calculated by L will be determined later. In a magnetic "eld H, the equilibrium position of the domain wall satis"es the Malozemo! [4,5]. hM+(A $K $/2M where A $ and following equation: K $ are the exchange sti!ness and anisotropy constant of the AF layer. It is the second term in Eq. (5) which is 2M H <" (y # y)! (y ), (1) related to the coercive "eld. As one changes the applied "eld, the wall moves back and forth according to the where the left-hand side represents the energy gain due to statistically #uctuating "eld g(x) until a critical "eld such external "eld as the domain wall moves along the y- that the second term reaches maximum. Further increas- direction by y and by <"¸t y in volume, M is the ing the "eld will lead to irreversible jumps of the wall. saturation magnetization per unit volume, the right hand Therefore, one can de"ne the coercivity by the maximum is the energy di!erence of the domain wall after and value of the second term in Eq. (5). As pointed out by before the move, and (y ) is the total energy of the wall, Ho!mann [6,7], the estimation of this quantity can be i.e., carried out by evaluating the statistical average of the random "eld, i.e., one de"nes the coercivity as (y )" (x,y)dxdy, (2) a H dx(g(x, y where the integration is limited to the wall region, "2¸t V >* #=/2) V \* "x!x "(¸/2 and "y!y "(=/2. The energy density of the wall (x, y) is assumed to be independent of z since the #g(x, y !=/2)) . (6) ferromagnetic layer is thin so that the magnetization is uniform along the z-direction. By taking the limit yP0, Since the random "eld is not correlated when they are Eqs. (1) and (2) result in separated by a distance of the domain wall width, the cross d (y term in Eq. (6) is averaged to zero, i.e., 1g(x, y #=/2)) 2M ) g(x, y H¸t" " dx[ (x, y dy V >* #=/2) !=/2)2"1g(x, y #=/2)2 ) 1g(x, y !=/2)2 V \* "0. Then we simplify the above equation to ! (x, y !=/2)]. (3) a H dx The wall energy density consists of number of energies in "(2¸t dx g(x)g(x#x ) , (7) the "lm; they are exchange energy, uniaxial anisotropy, demagnetization energy and random "eld energy from where we have dropped y $=/2 variable in the func- the antiferromagnetic "lm. The exchange energy and tion g. The random function g(x) is usually described uniaxial anisotropy are usually uniform within a "lm, by an auto correlation function, i.e., (1/¸) dxg(x) ) thus they do not contribute to the motion of the wall, g(x#x ),(J /M a ) f (x ), where J is the average Eq. (3). The demagnetization "eld varies spatially and it coupling energy of nearest-neighbour F and AF spins at gives rise to the coercivity. Here we are neglecting this, the interface while f (x ) represents the range of the cor- since this e!ect exists for "lms without the antiferromag- relation of the random function. As usual, f (x ) can be netic layer and we are interested in the enhancement of the assumed in the form of Gaussian or exponential, and we coercivity in the presence of the antiferromagnetic layer. take the latter form as an example, i.e., f (x )" We are focusing solely on the e!ect of random "elds on exp(!"x "/a ) where we have used white noise approxi- the coercive force below. mation for the random "eld so that the range of the Introducing a random "eld h(x, y) acting on the inter- correlation function is the order of monolayer thickness face of the bilayer, the energy density is written as a . With all these plausible simpli"cations, one can ex- press the coercivity, Eq. (7), as (x, y)"!M a m((x, y))h(x, y), (4) J where a H . (8) is the monolayer separation, and m((x, y) is the "M unit vector to represent the direction of the local mag- a t a ¸ netic moment. By placing Eq. (4) into Eq. (3), one has It remains to determine L, the domain size. 470 S. Zhang et al. / Journal of Magnetism and Magnetic Materials 198}199 (1999) 468}470 The domain size L can be derived from the minimiz- ation of the competing energies, 1 t E" z J !z J 2 $ a ¸ )a a ¸ , (9) where the "rst and second terms are the exchange energy and the energy from the random "eld per domain, z and z are the coordination numbers in the ferromagnetic layer and at the interface, and J$ is exchange constant of the ferromagnetic "lm. In writing down the "rst term, we assumed ¸;¸ square domains with linear variation of magnetization from one domain to the next, which is slightly di!erent from the circular domains assumed by Malozemo! [4,5], but the scaling relation that the ex- change energy goes as 1/¸ is generally valid. The second term comes from statistical average energy of N"¸ /a Fig. 1. The values of coercivity as function of NiFe thickness t of spins such that the root mean square of the random exchange-coupled NiFe(t)/CoO at 80 K and Ni Co O/ energy per site goes down as J NiFe(t) at 10 K showing the H /(N which results in the "A/tL dependence with n" second term in Eq. (9). Minimizing the above energy, one 1.51$0.05 and 1.427$0.05, respectively. "nds the domain size is zJ ¸" $ t. (10) many individual samples in which the thickness of NiFe z J was the only variable. In series (2), individual samples with 120 As of Ni It is noticed that the same line of reasoning has been Co O and di!erent NiFe thickness were grown and measured. In the thickness range of our applied to estimate the size of AF domains, where a sim- interest, we did not observe any microstructure di!er- ilar linear relation between the AF domain size and ences in each of the two sets of the samples. In both series, thickness of the AF layer was previously obtained [4,5]. while the coercivity of the uncoupled NiFe is small (a few By placing Eq. (10) into Eq. (8), we arrive at the scaling Oe at room temperature, and about 20 Oe at 10 K), the relation between the coercivity and the thickness of fer- coercivity of the exchange-coupled layers increases dra- romagnetic layers, matically for samples with small t to as much as 2 kOe. More importantly, H J has been observed to vary as H H "A/tL, and the exponent is n"1.51$0.05 at " ) M z J a T"80 K for the series (1) samples and 1.427$0.05 for a zJ$ t . (11) the series (2) samples, both in excellent agreement with To estimate the order of magnitude of the coercivity the theoretical prediction of Eq. (11) with n"1.5, see from Eq. (11), we need to determine J . J is expected to Fig. 1. depend on the interface roughness; therefore, the coerciv- ity will be di!erent for di!erent growth methods. Unfor- This work is supported by a MURI-ONR grant tunately, the independent measurement of J is currently N00014-96-1-1207 (S.Z.), a grant from NSF DMR- unavailable. Nevertheless, one may assume that J and 9307676 (D.V.D. and G.C.H.) and NSF MRSEC Pro- J$ are at the same order of magnitude. Within the mean gram No. 96-32526 (J.W.C. and C.L.C.). "eld approximation, J$ is a fraction of the Curie temper- ature. If we take J$"J "¹ /8 (where ¹ is the Curie temperature), z"8, z "4, and use the bulk magneti- zation of NiFe, we "nd that the coercivity is of the order References of 1 kOe for the thickness of t"50 As, which agrees with experiments well. A most important prediction of Eq. (11) [1] W.P. Meiklejohn, C.P. Bean, Phys. Rev. 102 (1956) 1413. is that of the coercivity scales as 1/t , which is quite in [2] C. Mauri, H.C. Siegmann, P.S. Bagus, E. Kay, J. Appl. Phys. contrast with exchange bias "eld which scales as 1/t. 62 (1987) 3034. To experimentally examine the magnitude and scaling [3] N.C. Koon, Phys. Rev. Lett. 78 (1997) 4865. [4] A.P. Malozemo!, Phys. Rev. B. 35 (1987) 3679. relation, Eq. (11), we have studied two series of samples: [5] A.P. Malozemo!, J. Appl. Phys. 63 (1988) 3876. (1) NiFe(t)/CoO/Si and (2) Ni Co O/NiFe(t)/MgO. [6] H. Ho!mann, Toshitaka, J. Magn. Magn. Mater. 128 (1993) In series (1), a wedged NiFe layer from 50 to 400 As was 395. grown on a uniform CoO layer of 250 As, resulted in [7] H. Ho!mann, IEEE Trans. Magn. 9 (1973) 17.