Journal of Magnetism and Magnetic Materials 195 (1999) 327}335 Spin con"gurations and magnetization reversal processes in ultrathin cubic ferromagnetic "lms with in-plane uniaxial anisotropy Ming-hui Yu*, Zhi-dong Zhang, Tong Zhao Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110015, People's Republic of China and International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110015, People's Republic of China Received 8 October 1998; received in revised form 11 January 1999 Abstract Based on a phenomenological model, spin structures either in the absence or in the presence of an external magnetic "eld have been studied for ultrathin cubic ferromagnetic "lms with an in-plane uniaxial anisotropy. Phase diagrams of the spin con"gurations in the absence of the magnetic "eld and of di!erent magnetization reversal processes have been given. The magnetization reversal processes in the ultrathin cubic ferromagnetic "lms have been found to depend sensitively on the competition among the energies of in-plane uniaxial and cubic anisotropies and of the domain wall pinning. 1999 Elsevier Science B.V. All rights reserved. PACS: 75.60.!d; 75.70.!i; 75.25.#z Keywords: Ultrathin "lms; Spin con"guration; In-plane uniaxial anisotropy; Magnetization reversal process; Domain wall pinning 1. Introduction and Wagner [1]. Magnetic anisotropies in ultra- thin "lms are strongly modi"ed, compared to those In recent years, ultrathin ferromagnetic "lms in bulk materials, due to the broken symmetry at have attracted a tremendous amount of attention. the interface. These anisotropies include shape, Magnetic anisotropies play an important role in surface, interface, and crystalline anisotropies, ultrathin ferromagnetic "lms. Anisotropy is neces- strain-induced magnetoelastic anisotropy, and an- sary in two-dimensional ferromagnets to obtain isotropies due to roughness, steps and atomic mix- long-range order, as proven rigorously by Mermin ing at the interface [2]. One of the most fascinating topics in this "eld is how to understand the unusual magnetization re- * Correspondence author. Tel.: #86-24-2384-3531-55857; versal processes which take place in some ultrathin fax: #86-24-2389-1320. cubic ferromagnetic "lms [3}12]. In these processes E-mail address: dygeng@imr.ac.cn (M. Yu) the in-plane spin con"guration changes abruptly at 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 9 ) 0 0 1 3 9 - 0 328 M. Yu et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 327}335 well-de"ned strengths and orientations of an ap- plied magnetic "eld, which is unexpected for ultrathin "lms with purely cubic magnetocrystal- line anisotropy. These peculiar magnetic switching processes were attributed to the occurrence of a weak in-plane uniaxial anisotropy which was superimposed on the strong cubic magnetocrystal- line anisotropy. Cowburn et al. [13,14] recently developed a simple phenomenological model, ex- plaining how so small a uniaxial anisotropy can signi"cantly in#uence the magnetization reversal processes. In their model a well-de"ned domain wall (DW) pinning energy is considered along with the anisotropy energy surface in order to determine the energetics of the reversal processes. A good agreement has been established between the model prediction and the experimental observation [10}14]. However, Cowburn et al. [13,14] focused their attention only on the behaviors of Fe ultra- Fig. 1. Schematic representation of the geometry of spin mo- ment M and applied magnetic "eld H in the plane of an ultrathin thin "lms, i.e., the case of a positive in-plane "lm. uniaxial anisotropy. In the present work, we extend their discussion to the whole system, including the situation with a negative in-plane cubic anisotropy *E which indeed corresponds to Co ultrathin "lms "2K *   cos 2 #2K cos 4 [15}18]. #MH cos( ! )'0. (3) 2. Model The solutions satisfying Eqs. (2) and (3) correspond to the local energy minima. There is a problem how The phenomenological model to be studied in to choose the resulting spin orientation among what follows is de"ned by the free energy [13,14] these solutions, if they are not sole. Departing from the Stoner}Wohlfarth's coherent rotation model E( )"K sin #K sin cos [19], one can take the resulting spin orientation !MH cos ( ! ), (1) always corresponding to the con"guration of abso- lute energy minimum which is determined by where K is the in-plane uniaxial anisotropy con- choosing between all the local energy minima stant, K is the in-plane cubic anisotropy constant, calculated for a given set of parameters [20]. Mag- and are the angles of the magnetization M and netization curves calculated in this case never show the applied "eld H with respect to the a-axis [1 0 0] hysteresis, as the energy wall between the di!erent direction, respectively, as shown in Fig. 1. The local minima is neglected during the spin reversal equilibrium state is found by minimizing Eq. (1) process. In real materials, however, the magneti- with respect to the angle . This involves the "rst zation reversal process also involves the nucleation and second partial derivatives of the free energy of domains and the propagation of domain walls with respect to the angle , [4,5,21,22]. Therefore, the energetics of domain * E formation and propagation are crucial in under- "K *  sin 2 #K sin 2 cos 2 standing the spin reversal process. Following the method developed in Refs. [13,14], a phenom- #MH sin( ! )"0, (2) enological constant is taken into account, which M. Yu et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 327}335 329 describes the pinning energy of a DW, and corres- ponds to the maximum pinning pressure that de- fects can exert on a DW as it propagates. Before the spin transition from one local energy minimum to another lower one, the energy advantage E between these two minima must be equal to the energy cost in propagating a DW of the relevant type. The activation energy needed to establish these walls is ignored, and we take care of only the drive energy involved in unpinning them so that they can sweep freely across the "lm. Setting "0 for any type of DW, we will return to the former case, i.e., comparing the absolute energy minimum directly. 3. Spin con5gurations at H"0 Fig. 2. Phase diagram of di!erent spin con"gurations of an In this section, we shall discuss the spin con"g- ultrathin ferromagnetic "lm with in-plane uniaxial and cubic urations in the absence of the applied magnetic "eld anisotropies in the absence of an external magnetic "eld. The for ultrathin ferromagnetic "lms with in-plane black arrows represent all possible directions of spin moment corresponding to the stablest energy minima. The blank arrows uniaxial and cubic anisotropies. One can easily "nd correspond to the metastable ones. the analytic solutions of Eqs. (2) and (3) in the absence of the magnetic "eld, i.e., H"0. takes easy a-axis, while subregion Ia like region III "03, 903, 1803, 2703 favors easy b-axis. It must be noted that the region K III has a biaxial anisotropy and four local energy '0 and!K(K(K, (4a) minima with spin orientations ,! , 1803! , "03, 1803 K or 1803# '0 and!K)K)K, (4b)  have the same value of the energy. In this case the uniaxial anisotropy K only changes "903, 2703 K the value of the cone angle, does not a!ect the (0 and K)K)!K, (4c) biaxial feature of the purely cubic anisotropy. The " ,! , 1803! , 1803#  magnetization reversal processes in those regions would be discussed below. "arcsin ((K#K)/2K) K(0 and K(K(!K. (4d) 4. Magnetization reversal processes Fig. 2 gives the phase diagram of the spin con"g- urations of the "lm system with zero applied "eld, In this section, magnetization reversal processes where regions I}IV correspond to solutions of the system studied will be discussed for three (4a)}(4d), respectively. Under di!erent conditions, di!erent cases in three subsections, respectively. three kinds of easy magnetization directions can be taken, namely, easy a-axis, easy b-axis and easy 4.1. K cone. Region I is divided into two subregions Ia '0,!K(K(K and Ib, because the absolute energy minima among 4.1.1. "K the four local energy minima in the two subregions ";K In regions Ia and Ib of Fig. 2 if "K di!ers from each other. Subregion Ib like region II ";K and H;K/M, the solutions of Eqs. (2) and (3) will 330 M. Yu et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 327}335 always approximately be in the set + "03, "903, "!903, "!1803,. These local energy min- ima can be found by substituting the relevant values of into Eq. (1): E    "!MH cos , (5a) E \   "MH cos , (5b) E    "K!MH sin , (5c) E \ "K#MH sin . (5d) The spin reversals between these four states will be mediated by the sweeping of 903 and 1803 DWs. In Ref. [13] it was proved experimentally that Fig. 3. Phase diagram of irreversible jumps expected during spin 3"2 3. When several possible jumps among reversal as a function of the applied "eld orientation and the these four states compete, we assume the one which ratio of the in-plane uniaxial anisotropy K to the pinning can occur at the lower critical "eld is taken as the energy 3 of a 903 DW. The spin states on increasing and one observed actually. A magnetic phase diagram decreasing the "eld are shown schematically by the down and up rows of the boxes, respectively. showing the number of irreversible jumps expected during the spin reversal was given by Cowburn et al. [14], only considering K'0. In Fig. 3, we zation reversal process in a strong cubic anisotropy extend this phase diagram to the case of K(0, system. Changing the sign of K and distinguish the region b from the region c,  can also lead to a similar e!ect while only the spin states and the where their spin states are distinct. The spin states relation with respect to the applied "eld orientation are shown schematically in the boxes. Two rows of are altered. This is because it does not matter for boxes are displayed in each region. The lower one a cubic system where the uniaxial anisotropy is demonstrates the spin jumps on increasing the ap- applied along the (1 0 0) or (0 1 0) axis. However, plied "eld, while the upper one corresponds to for the completeness of the phase diagram, we con- those when the applied "eld is decreased. It is worth sider the case of K noting that either 1-jump or 3-jump reversals ex- (0 in Fig. 3. If 3"0, i.e., K perience the same routes on increasing and decreas- / 3PR, one will see that only 1-jump or 3- jump reversals remain. No 2-jump reversal could be ing the external magnetic "eld, whereas 2-jump observed (no dashed lines in columns b}d of reversal undergoes di!erent ones. The critical "eld of these spin jumps can be solved analytically by setting the energy advantage E equal to the do- &&&&&&&&&&&&&&&&&&&&& main wall pinning energy between two spin stable Fig. 4. (a) Hysteresis loops of the magnetization either parallel or perpendicular to the applied "eld and the variation of the spin states. states during the magnetization reversal processes. The para- The hysteresis loops of the magnetization either meters used during the calculation are: K/M"3 Oe, parallel or perpendicular to the direction of the K/M"275 Oe, (a) "353, 3/M"3 Oe; (b) "353, 3/ applied "eld, and the variation of the angle with M 5 Oe; (c) "753, 3/M"4 Oe; (d) "753, 3/M" respect to the applied "eld in di!erent regions are 1.5 Oe; (e) "753, 3/M"0.5 Oe. The dashed lines represent the magnetization reversal processes for "0. (b) Hysteresis shown in Fig. 4a. The magnetization reversal pro- loops of the magnetization either parallel or perpendicular to cesses when "0 are shown as dashed lines for the applied "eld and the variation of the spin states during the comparison. Columns a}e correspond to the varied magnetization reversal processes. The parameters used during irreversible jumps in regions a}e in Fig. 3, respec- the calculation are: K/M"110 Oe, K/M"275 Oe, (a) tively. These irreversible jumps on the hysteresis "353, 3/M"150 Oe; (c) "803, 3/M"250 Oe; (d) "803, loops are markedly obvious. A very small uniaxial 3/M"100 Oe; (e) "803, 3/M"20 Oe. The dashed lines represent the magnetization reversal processes for anisotropy can signi"cantly in#uence the magneti- "0. M. Yu et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 327}335 331 332 M. Yu et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 327}335 Fig. 4a) if only the absolute energy minimum were magnetization reversal process. A coherent rota- considered. Thus the occurrence of the 2-jump tion model is requested to describe such a pro- reversals must involve the mechanism of domain cess in this case. The hysteresis loops possessing nucleation and propagation of the domain wall. the same shape as in columns (a) and (c) in Fig. 4b It is interesting to compare our numerical results have been frequently observed in Fe/GaAs(0 0 1) with the hysteresis loops observed experimentally [5,10,12]. in literature. The hysteresis loops in columns a}c of Fig. 4a are similar to those denoted as I}III (and 4.2. "K IV) in Fig. 3 of Ref. [13], observed in Ag/Fe/ "'"K" Ag(1 0 0). The hysteresis loops in column d seems Now, let us consider what will happen if to be of the same type as the Kerr signal in the right K column of Fig. 13 of Ref. [23]. The hysteresis loop 'K as in the region II of Fig. 2. As the strength of the uniaxial anisotropy is stronger than of 3-jumps in column e calculated for "753 is that of the cubic anisotropy, only two local energy similar to that observed by Cowburn et al. [14] in minima appear at zero applied "eld since the other a magnetic "eld at "513$33. This is because we two are eliminated. At low "eld there is nothing but have given an example of 3-jumps for between a 1-jump reversal mediated by the sweeping of the 453 and 903, where in a certain condition of K/ 3 1803 DW, irrespective of the direction of the ap- the hysteresis loops have the same characteristics. plied "eld. We need to increase the strength of the applied "eld in order to detect more jumps. Fig. 5 displays the simulation results for K 4.1.2. "K /K"1.1. " K The shape of the hysteresis loops resembles that in If the strength of the uniaxial anisotropy is close Fig. 4b. But one type of 2-jump reversal related to to that of the cubic anisotropy, a low "eld would be column c in Fig. 4b vanishes in Fig. 5. The reason is insu$cient to saturate the "lm. However, increas- the same as the above mentioned (in Section 4.1.2). ing the strength of the applied "eld will destroy the Further increasing the strength of the uniaxial an- approximate analytic solutions of (2) and (3), and isotropy, the critical "elds for the "rst and third force the local energy minima away from the crys- jumps in the case of 3-jump reversal will be grad- talline axes. In this case one has to develop a calcu- ually enhanced and "nally the jumps disappear at lation procedure to determine the stable spin state a certain point, where the 3-jump reversal turns at a certain magnetic "eld. Fig. 4b shows the calcu- into a 1-jump one. What happens in region III in lation results for K/K"0.4. The dashed lines in Fig. 2 is analogous to that in region II. Therefore, columns a and e correspond to the results cal- the in#uence of an in-plane uniaxial anisotropy on culated by considering only the absolute energy the jumps in the magnetization reversal processes minimum. It is similar to the case in Fig. 4a, but the of an ultrathin cubic ferromagnetic "lm would be "rst and third jumps of the 3-jump reversal in weakened, and the critical "elds would be greatly column e diverge from the crystalline axes. In this enhanced by increasing the strength of the uniaxial case if the propagation of the domain wall is taken anisotropy. into account the DW would no longer keep an angle of 903 or 1803. We assume that the pinning 4.3. K pressure of a DW is directly proportional to the (0, K(K(!K value of its angle . During the numerical calcu- In this case, in the absence of the external mag- lation, the phenomenological constant of a DW is netic "eld, the easy magnetization direction de"ned as F" 3) /180. Even if taking into ac- deviates from the crystallographic directions. The count the propagation of the domain wall, one type simulation results are di!erent from those in other of 2-jump reversal related to column b in Fig. 4a regions. No more jumps occur in the magnetization cannot be observed in Fig. 4b. This is due to the curves except for 2-jump spin reversal, as shown in limitation of the DWs pinning energy, beyond Fig. 6. The dashed lines also prove that in the case which the coherent rotation would dominate the of considering the absolute energy minima, only M. Yu et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 327}335 333 Fig. 5. Hysteresis loops of the magnetization either parallel or perpendicular to the applied "eld and the variation of the spin states during the magnetization reversal processes. The parameters used during the calculation are: K/M"300 Oe, K/M"275 Oe, (a) "403, 3/M"500 Oe; (d) "803, 3/M"200 Oe; (e) "803, 3/M"50 Oe. The dashed lines represent the magnetization reversal processes for "0. 1-jump reversal can take place no matter what kind of 2-jump spin reversal processes was found direction the magnetic "eld is applied. For experimentally by Diao et al. in Co/Cu/Co sand- K;"K" and K"!K/2, one obtains the cone wich structures [17]. angles "453 and 303, respectively. When a low "eld is applied, four states with spin orientations , ! , 1803! , 1803#  are the stable states. 5. Discussions and conclusion If the energy advantages between these states sat- isfy the density cost in unpinning a DW of one of In this work, based on the consideration of the the three types, F, 3\F and 3 which are domain process, we attain a clear cognizance of the directly proportional to the angle of the DW, the spin reversals in ultrathin ferromagnetic "lms with spin orientation will rotate from one state to an- in-plane uniaxial and cubic anisotropies. These re- other stabler one. It can be seen, from the four sults have certain signi"cance in both theoretical columns in Fig. 6 that only 2-jump spin reversal and experimental aspects. They provide a criterion occurs between these four states, regardless of the of the possible mechanism of spin reversal in ultra- values of the cone angle , the applied "eld ori- thin "lms, which is closely related with the spin entation and the ratio of K/ 3. They are con"gurations in the absence of the magnetic "eld. similar to the case of K"0 in Fig. 3 when no Recent advances in thin-"lm growth techniques uniaxial anisotropy is taken into account. This be- have enabled us to establish new phases of mag- havior can be attributed to the unique biaxial an- netic materials. Cobalt, which naturally occurs in isotropy of this system at zero applied "eld. This the hexagonal close-pack phase, has been grown in 334 M. Yu et al. / Journal of Magnetism and Magnetic Materials 195 (1999) 327}335 Fig. 6. Hysteresis loops of the magnetization either parallel or perpendicular to the applied "eld and the variation of the spin states during the magnetization reversal processes. The parameters used during the calculation are: (a) K/M"3 Oe, K/M"!275 Oe, "103, 3/M"1 Oe; (b) K/M"3 Oe, K/M"!275 Oe, "803, 3/M"1 Oe; (c) K/M"137.5 Oe, K/M"!275 Oe, "103, 3/M"3 Oe; (d) K/M"137.5 Oe, K/M"!275 Oe, "803, 3/M"3 Oe. The dashed lines represent the magneti- zation reversal processes for "0. metastable FCC and BCC phases as thin supported the thickness of the "lm [2,23]. One can obtain "lms. Generally, these ultrathin Co "lms have the cubic ferromagnetic "lms with di!erent values of easy magnetization direction along the 11 1 02 K axis, indicating that K  and K after accurately controlling the com- (0 [15}18,24}26]. On the position and the microstructure of the "lm and other hand, the easy magnetization direction of the substrate. According to the consequence of this Fe "lms is usually along the 11 0 02 axis, i.e., work, one can take a prior prediction of the pattern K'0 [10}14]. It is a common phenomenon in of the spin reversal in the "lms. ultrathin cubic ferromagnetic "lms that an addi- In conclusion, we have investigated the in#uence tional in-plane uniaxial anisotropy is introduced. of an in-plane uniaxial anisotropy on the magneti- It can be attributed to several mechanisms, zation reversal processes of ultrathin cubic fer- for example, the oblique incidence of deposition romagnetic "lms. The phase diagram of the spin particles [27,28], lattice mismatch strain-induced con"gurations of this system in the absence of the magnetoelastic e!ects [29,30], atomic steps on applied "eld has been derived. The spin reversals the interface [31}36], the internal oxidation along induced by the applied "eld via the absolute energy the oxygen chains on the surface of the sub- minimum or the propagation of the DW have been strate (AlO, MgO) [36,37], or dangling bands on studied numerically. Increasing the strength of the the arsenic/gallium rich surface [12]. 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