Phase Transitions, Vol. 79, No. 12, December 2006, 1123–1133
Multiferroically composed exchange bias systems PAVEL BORISOV*, ANDREAS HOCHSTRAT, XI CHEN and WOLFGANG KLEEMANN Angewandte Physik, Universita¨t Duisburg-Essen, D-47048 Duisburg, Germany (Received 6 October 2006; in final form 9 October 2006) Magnetoelectric (ME) antiferromagnetic Cr2O3, being exchange coupled to a ferromagnetic multilayer (Pt/Co/Pt)n, n 1, is treated as a multiphase multiferroic material with sophisticated multifunctional properties. It is shown that the exchange bias (EB) of the ferromagnetic hysteresis loop cannot only be controlled by the magnetic freezing field, Hfr, but additionally also by an external electric freezing field, Efr, via the ME effect of Cr2O3. Either a gradual shift or complete switching from HEB to þHEB on the magnetic field axis are possible. Here the influence of Efr at constant Hfr on both the coercive field, Hc, and the antiferromagnetic domain wall susceptibility, , and on the temperature dependence of the ME controlled quantities Hc(T ) and HEB(T ) is reported. Keywords: Magnetoelectricity; Exchange bias; Coercive field; Susceptibility; Domain walls
1. Introduction Magnetoelectric (ME) multiferroics [1] are most promising materials for multifunctional applications and technology. By definition their electric polarization P can be controlled by a magnetic field, while the magnetization M is sensitive to an external electric field. Both modes are highly welcome in new applications for microor spinelectronics [2]. In order to realize this technology, two main streams are pursued in materials research and development since a couple of years. The traditional route aims at searching applicable single phase materials, which contain the desired properties by natural crystallographic and magnetic symmetry [3]. The materials engineering route tries to develop multiphase (composite) multiferroics, where the desired ME properties are mediated by a secondary interaction, e.g. magnetostriction (of a ferromagnetic [FM] component) and piezoelectricity (of a ferroelectric component) [4]. The cross-coupled information is transported mechanically from one component to the other via their interfaces. Alternatively, and probably less prone to fatigue, the authors propose that interfacial coupling may also be performed by quantum-mechanical interaction, e.g. by spin-spin exchange coupling between appropriate ferroic components. This route can yield solutions for *Corresponding author. Email:
[email protected] Phase Transitions ISSN 0141-1594 print/ISSN 1029-0338 online ß 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/01411590601067318
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sophisticated spintronic applications as will be shown in this article on the example of a multiferroically composed exchange bias (EB) system. An EB system consists of a FM and an antiferromagnetic (AF) layer closely attached to each other. Exchange coupling transfers a unidirectional anisotropy from the AF onto the FM layer, whose hysteresis loop becomes shifted along the magnetic field axis by the so-called EB field, HEB [5]. The present EB system consists of the ME antiferromagnet Cr2O3, which is exchange coupled to a FM multilayer (Pt/Co/Pt)n, n 1, containing an ultrathin ferromagnetic (FM) film of cobalt [6]. Recently it has been demonstrated that the FM hysteresis loop, M(0H), of the (Pt/Co/Pt)n, n 1, multilayer cannot only be shifted via the linear ME effect by a fraction of 1 mT along the magnetic field axis [7], but can also be switched from negative to positive EB fields and vice versa [8, 9]. Applications in ME random access memories (MERAM [10]) and logic devices based on magnetoelectric XOR cells (MEXOR [10]) have been envisaged [8], alternatively to other novel multiferroically based devices [12, 13]. In this article the ME system Cr2O3 and its ability to shift a FM hysteresis loop by virtue of the so-called EB effect within a multiferroically composed exchange coupled multilayer is presented (section 2). Then the complete ME switching effect of the EB field, HEB, which becomes possible by antiferromagnetic (AF) single domaining of Cr2O3 via ‘‘ME cooling’’ to below the Ne´el temperature TN ¼ 307 K in simultaneously applied parallel or antiparallel magnetic and electric fields is presented. The important properties, like the coercive field Hc of the FM component and the magnetic susceptibility of the entire multiferroic stack under switching conditions, are investigated (section 3). Thereafter, the temperature dependence of the EB controlled crucial properties, Hc(T ) and HEB(T ), which become easily accessible by magneto-optic Kerr effect measurements are discussed (section 4).
2. Electric shift of a ferromagnetic hysteresis loop Since the discovery of the ME effect in Cr2O3 [14] this system has remained the best understood and most thoroughly investigated ME material with relatively simple magnetic structure. The ME coefficient of Cr2O3 is fairly small (max ¼ Mz/Ez 3 106 A V1 at T 250 K [15]) when compared to that of more recently investigated materials (e.g. TbPO4: max ¼ Mz/Ez 3 105 A V1 at T 1.9 K [16]). That is why Cr2O3 is sometimes referred to as a so-called classic ME material, historically important, but not relevant for modern applications. However, one has to be aware that most of the so-called giant ME features are found only at low temperatures, e.g. at T < TN(TbPO4) 2.2 K, while the ME effect in Cr2O3 maximizes in an extremely convenient temperature range close to room temperature, viz. below TN(Cr2O3) 307 K. In this article it will be shown that the classical ME, but non-multiferroic material Cr2O3 promises to revive as the decisive component of an EB system with multiferroic properties. By definition EB systems show two kinds of ferroic order, which are usually antiferromagnetic (AF) and FM, respectively. According to Meiklejohn and Bean [5] the EB field is given by the expression 0 HEB ¼
JSAF SFM , MFM tFM
ð1Þ
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which describes the phenomenological coupling J between the FM and AF interface magnetizations SFM and SAF, respectively, while tFM and MFM are the thickness and the saturation magnetization of the FM layer. When using Cr2O3 as the AF component, one may encounter two different situations. For instance, in the uncompensated (0001) plane either SAF is due to spins of one of the sublattices, SAF ¼ Ld, where L is the sublattice magnetization and d is the distance between neighbouring atomic planes containing Cr3þ-ions with the same spin direction, or SAF ¼ M(E)d/2 is the electrically induced magnetic extra moment due to the bulk ME effect. With M(E) ¼ E and L ¼ /2, where is the density of the Cr3þ-ions and their magnetic moment an ideal relative value of the magneto-electrically induced change of the EB field can be calculated HEB ðE Þ SAF MðE Þ ¼ ¼ ¼ E: ð2Þ HEB ðE ¼ 0Þ L SAF ¼ 4.1 1028 m3, (250 K) Inserting (250 K) ¼ max 3 106 A V1, 23 2 6 1 (T ¼ 0) 3B ¼ 2.8 10 Am , E ¼ 10 V m [15] and assuming that the real value of SAF is approximately 100 times smaller than the calculated one [17] the authors estimate HEB/HEB 104. Obviously the direct control of EB via application of an electric field on Cr2O3 appears as a very tiny effect. However, for thin film geometries, one might envisage electric fields close to breakdown, E ¼ 108 V m1, and thus enhance the EB effect of Cr2O3 by a factor of 100. Another chance for future applications lies in the use of materials with larger ME coupling. It must be noted, that our above estimate considers the bulk ME effect, from which it can deviate drastically at the interface [18].
3. ME switching of a ferromagnetic hysteresis loop Cooling of Cr2O3 through the Ne´el temperature, TN, in simultaneously applied magnetic and electric freezing fields, 0Hfr and Efr, respectively, historically called ‘‘ME annealing’’ [21] (henceforth referred to as ‘‘ME cooling’’), can yield an AF single domain-state. The reason for this is the relevance of ME coupling energy, WME ¼ ME EH,
ð3Þ
which helps selecting one out of the two possible AF domains via the sign of the ME susceptibility ME [21]. Hence, due to the bilinearity involved in equation (3) the probability of growing one of these domains depends on the sign of the field product EfrHfr. Thus cooling in the same magnetic, but in different electric freezing fields can be used to control the AF domain selection electrically and thus the sign of the hysteresis loop shift of the attached FM layer. This idea was recently realized on a Co/Pt multilayer-system, which was epitaxially grown on a Cr2O3 single-crystal [8]. Because of its uncompensated (0001) plane occupied with out-of-plane oriented magnetic interface moments, Cr2O3 is an appropriate candidate for the design of perpendicular EB systems [19, 20]. To this end the conjunction with perpendicularly anisotropic Co/Pt multilayers on top of Cr2O3 crystal slabs or epitaxial layers are chosen. In such systems we apply magnetic as well as the electric freezing fields along
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M/MS
0.5
1
2 3
0.0 –0.5 –1.0 –0.2
0.0
0.2
µ0H (mT)
Figure 1. Normalized hysteresis curves of Cr2O3(0001)/Pt(0.5 nm)/[Co(0.3 nm)/Pt(1.5 nm)]3/ Pt(1.5 nm) measured with p-MOKE at T ¼ 298 K after cooling the sample from T ¼ 350 K to 298 K in the simultaneous presence of a magnetic field 0Hfr ¼ 0.3 T and an electric field Efr ¼ 500 (curve 1, blue down triangles), 26.4 (2, black circles), and 500 kV m1 (3, red up triangles), respectively. The lines are to guide the eyes.
the c axis, which is parallel to the (111) direction of the epitaxially grown Co/Pt overlayers. The c axis also represents the direction of the maximum ME effect in Cr2O3 [14]. Figure 1 shows three hysteresis loops, which were measured in zero electric field by means of the polar magneto-optic Kerr effect (p-MOKE) after cooling from 350 to 298 K in the same magnetic field 0Hfr ¼ 0.3 T, but different electric fields, Efr ¼ 500 kV m1 (curve 1, blue down triangles), Efr ¼ 26.4 kV m1 (2, black open circles) and Efr ¼ 500 kV m1 (3, red up triangles), respectively. The change of the sign of the electric freezing field from 500 to þ500 kV m1 obviously leads to the switching of the EB field as defined by the average value of the coercive fields from 0HEB ¼ 20.3 to þ18.4 mT. On the other hand, cooling in a low electric field, Efr ¼ 26.4 kV m1, results in a nearly vanishing EB field, 0HEB ¼ 0.6 mT. The switching of the EB field can be explained by the correlation between the uncompensated moments in the AF interface and the domain structure in Cr2O3. Obviously, the reversal of the AF vector L by 180 during the transition from one AF single-domain state to the opposite one causes a corresponding 180 rotation of the SAF vector, thus causing the extreme change of the EB field. EB and coercive fields derived from magnetic hysteresis measurements for other combinations of electric and magnetic freezing fields are shown in figure 2 [8]. This time three values of the magnetic freezing field, 0Hfr, have been chosen as familyparameter for ME cooling procedures from 350 to 298 K. EB and coercive fields are obtained from hysteresis loops at T ¼ 298 K and plotted in dependence on Efr. The monotonic shift of the zeros, E0, as defined by the condition 0HEB(E0) ¼ 0 to smaller values with increasing 0Hfr values (see figure 2b for details) has been attributed previously [8] to the competition between different energies involving the AF interface spins during the cooling process, viz. the ME, the exchange and the Zeeman energy. In this article another aspect of the FM hysteresis loops arising after ME cooling is considered, namely the close coincidence of the peaks of the coercive field, 0Hc, in the vicinity of E0 (figure 2c) as is clearly seen when comparing with the zeros of 0HEB in figure 2(b). According to our above theoretical consideration [8] all
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Efr (kV/m)
Figure 2. EB fields, 0HEB (a, b), and coercive fields, 0HC (c), obtained from hysteresis loops after field cooling the sample from T ¼ 350 to 298 K in the simultaneous presence of variant Efr and 0Hfr ¼ 0.1 T (blue open circles), 0.3 T (black squares) and 0.6 T (red solid circles), respectively. Panel (b) shows the same dependences in the reduced field range 5 < Efr < 90 kV m1. The lines are to guide the eyes.
magnetic energy contributions are expected to compensate at Efr ¼ E0, where the AF system undergoes a field-driven phase transition between þL and L. In this situation a multi-domain state is expected to be established close to the FM-AF interface. This induces an increase of coercivity due to a ‘‘spin-drag’’ effect wellknown from conventional EB systems at the temperature-driven phase transition in the vicinity of TN [17]. A similar effect was found by Leighton et al. [22] in the EB system MnF2/Fe. Its EB field shows a strong dependence on the magnetic freezing field Hfr and can be switched from negative to positive values at increasing Hfr due to the competition between negative AF exchange and Zeeman interaction at the interface. The authors discovered a close coincidence between the vanishing of the EB field and the peaking coercivity. The effect was explained by the formation of a strongly ‘‘frustrated’’ magnetic interface state, which produces additional pinning centers for FM domain walls and therefore leads to an increase in coercivity. In order to underpin our hypothesis of an AF multi-domain state at the interface its ac susceptibility was measured by superconducting quantum interference device (SQUID) susceptometry after ME cooling under the same conditions as used for the hysteresis measurements shown in figure 2. Basically enhanced susceptibility due to AF domain walls in the vicinity of E0 is expected. The existence of AF domain walls in Cr2O3 was first proposed by Astrov and Al’shin [23] in order to explain an observed non-diagonal ME effect in a fine-grained polycrystalline Cr2O3 specimen. The AF vector in the domain walls was proposed to be perpendicular to that inside the AF domains (figure 3), hence, the spins are oriented similarly as in a so-called spin-flop configuration [4]. Here it should be noticed that our values of 0Hfr 0.6 T are large enough to produce single-domain states in conjunction with proper values of Efr [21], but do not exceed the lower limit for the spin-flop transition in Cr2O3 at 298 K [24]. In its multidomain state Cr2O3 should have a sizeable fraction of spins, which lie in AF domain walls and have non-vanishing projections on the basal plane normal to the c axis [23]. Therefore, ac susceptibility measurements under a magnetic
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Simplified spin structure of two AF domains and an intercalated wall in Cr2O3.
field along the c axis should reveal appreciable signals from ? thus exceeding k, which characterizes the single-domain state. The sample was cooled down from 350 to 298 K under simultaneously applied magnetic field 0Hfr ¼ 300 mT and variant electric fields Efr. The ac susceptibility was measured at 298 K and zero electric field with a driving amplitude 0H ¼ 0.4 mT and a frequency f ¼ 1 kHz. Since we were interested only in the AF response, the ferromagnet was saturated during all measurements by a constant magnetic field, 0H ¼ 0.3 T. Just prior to the ac susceptibility measurements the direction of the applied magnetic field was 5 times reversed in order to overcome probable training effects [25]. Although weak uncompensated magnetic moments being coupled to AF domain walls are commonly proposed, parasitic ferromagnetism can probably be neglected in Cr2O3 [26, 27] and will not be discussed in the following. As shown in figure 4 the apparent ac susceptibility, , measured at f ¼ 1 kHz is by 0.16% larger for Efr ¼ 0 as compared to the values measured at Efr ¼ 500 kV m1. Please note that denotes an ac moment which contains some volume information, ¼ Veff, to be discussed below. Although the effect is very weak, it is reproducible after several field cooling procedures. Since hysteresis loop measurements do not show any significant changes of the coercivity for these extreme values (figure 2c), it is proposed this susceptibility maximum to refer to AF domains walls in the Cr2O3 bulk which grow due to pinning at intrinsic defects as usual [4, 23] for EfrHfr ¼ 0, but vanish in the presence of single-domaining ME cooling, EfrHfr 6¼ 0. Less trivial and much more interesting is the observation of a second susceptibility peak at Efr 27 kV m1 as marked by an arrow in figure 5 (open circles). This field coincides with E0 27 kV m1 for vanishing EB field in figure 2(b) and with Emax 27 kV m1 for maximum coercivity in figures 2(c) and 5 (solid circles) obtained under the same parallel magnetic field, 0Hfr ¼ 0.3 T. Obviously a significant AF domain wall pattern appears, which causes the uncompensated pinned interface moment to be equal to zero (HEB ¼ 0) and the wall susceptibility to maximize. The volume ratio of spins related to the domain walls can be estimated from the results of susceptibility measurements (figure 4). It is defined as the ratio of the total volume of all domain walls to the total volume Vtotal. In the single-domain state all spins are collinear to the applied magnetic field, hence, the measured susceptibility signal SD can be described as SD ¼ k Vtotal :
ð4Þ
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300
600
Efr (kV/m) Figure 4. ac susceptibility measured by SQUID at 298 K and f ¼ 1 kHz after cooling the sample from T ¼ 350 to 298 K in the simultaneous presence of 0Hfr ¼ 0.3 T and variant values of Efr. The lines are to guide the eyes.
170 2.268
x 2.266 168
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µ0HC (mT)
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2.262 0
20
Efr (kV/m) Figure 5. ac Susceptibility measured by SQUID at 298K after cooling the sample from T ¼ 350 to 298 K in the simultaneous presence of 0Hfr ¼ 0.3 T and variant electric freezing fields within the interval 25 Efr þ 35 kV m1 (red open circles). For comparison the corresponding dependence of 0HC (Efr) for 0Hfr ¼ 0.3 T from the figure 2(c) is shown (black squares). The lines are to guide the eyes.
In the multi-domain state there is a mixture from two spin fractions with spins parallel (k) and perpendicular (?) to the applied field. Therefore the total signal MD is given by ð5Þ MD ¼ ð1 Þk þ ? Vtotal : Excluding the volume Vtotal from both equations the volume fraction is obtained " # ðMD =SD 1Þ : ð6Þ ¼ ? =k 1
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Under the assumption that the susceptibility ratio at the frequency f ¼ 1 kHz is the same as in the static case as determined at T ¼ 298 K [28], ?/k 1.11, setting MD ¼ (Efr ¼ 0) and SD ¼ ½ðþ500kV=mÞ ð500kV=mÞ=2 one obtains 0.015 0.003. Let us finally remark that the complete reversal of the AF single-domain state in Cr2O3 can also be achieved isothermally at T < TN, without any annealing procedure, as was shown by Martin and Anderson [29]. To this purpose a magnetic and an electric field must be applied collinearly with the product of both field strengths, HE, being markedly larger than the product of the freezing fields, HfrEfr, as employed in ME cooling. Clearly, the best candidates for this procedure will be thin film systems, which are also expected to fulfill all criteria for a new class of ME-controlled magnetic random access memories, MERAM [10, 11]. They will be operated by pure voltage control and thus promise to have much lower power consumption than conventional current controlled MRAMs.
4. Temperature dependence of a ME switched EB system In order to increase the EB switching effect we measured another series of samples with reduced Co/Pt thickness, Cr2O3(0001)/Pt(0.5 nm)/Co(0.35 nm)/Pt(3 nm) [9, 11]. According to the Meiklejohn–Bean formula, equation (1), the EB field increases with decreasing FM layer thickness. Additionally a decrease of the FM coercivity is expected [17]. Both of these factors are the driving forces for the complete ME switching, which is shown in figure 6. It is seen that Hc decreases from 150 (figure 1) to 50 mT, at a certain expense, however, of the squareness of the hysteresis loop. The temperature dependence of the two hysteresis loops prepared by proper ME cooling with HfrEfr > 0 and HfrEfr < 0, respectively are studied, and show the corresponding data HEB and Hc versus T in figure 7. Very clearly, jHEBj decreases as T increases and vanishes sharply at Tc ¼ 307.5 K. Obviously this behaviour is related to the decrease of the AF order parameter, L, which is expected to obey a power law, L / (Tc T) with the critical exponent . Fitting both sets of data
1.0
M / MS
0.5 Hfr
Hfr
Efr
Efr
0.0 −0.5 −1.0 −0.2
0.0 µ0H (T)
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Figure 6. Normalized hysteresis curves of Cr2O3(0001)/Pt(0.5 nm)/Co(0.35 nm)/Pt(3 nm) measured with p-MOKE at T ¼ 297 K after cooling the sample from T ¼ 350 to 297 K in the simultaneous presence of 0Hfr ¼ 0.3 T and Efr ¼ 500 kV m1 (blue open circles), and Efr ¼ 500 kV m1 (red solid circles), respectively. The lines are to guide the eyes.
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Figure 7. EB field 0HEB (red open circles) and coercivity 0HC (black solid circles) vs. temperature for a Cr2O3(0001)/Pt(0.5 nm)/Co(0.35 nm)/Pt(3 nm) after ME cooling to below TN in the simultaneous presence of 0Hfr ¼ 0.3 T and Efr ¼ 500 kV m1 (a), and Efr ¼ 500 kV m1 (b), respectively. The solid black lines are to guide the eyes. Best fits of the 0HEB data to a power law are shown by blue lines.
points of HEB between 294 and 307.5 K, i.e. up to reduced temperatures " ¼ 1 T/Tc ¼ 0.044 (dashed lines), yields ¼ 0.361 0.029 on the average. Within errors this value agrees with the critical exponent of the 3D Heisenberg model, ¼ 0.3646 0.0012 [30]. This is, indeed, expected for Cr2O3 owing to its relatively low anisotropy when compared to other uniaxial antiferromagnets [31, 32]. Thus, the AF order and the EB effect seem to be strongly correlated. Deviations are found only in the immediate vicinity of TN, where the exchange coupling at the FM–AF interface starts to establish and the EB field cannot yet follow the AF order parameter. As a matter of fact, however, the ME cooled AF Cr2O3 seems to differ from usual EB systems, where HEB becomes non-zero only below the blocking temperature Tb < TN, i.e. only at a finite distance form the bulk AF ordering temperature. This result sheds some light onto the exchange coupling mechanism of the FM layer to the AF single domain of Cr2O3, which is probably a bulk rather than an interfacial phenomenon [8]. The temperature dependence of coercive field Hc shows a sharp peak at TN (figure 7). Its appearance can be explained as follows [17]. Due to the AF–FM exchange interaction a certain amount of AF spins at the interface is irreversibly ‘‘dragged’’ during every FM spin reversal process. Thus by lowering the AF anisotropy in the vicinity of TN the coercivity reaches its maximum. Since AF order breaks down above TN, Hc starts to decrease after temperature exceeds TN. A simplified expression for HC in an EB systems was derived by Scholten et al. [33], HC ¼
2D þ J2int AF =tFM , 1 þ Jint AF =tFM
ð7Þ
where D, Jint, AF and tFM denote the FM anisotropy constant, the AF–FM exchange constant, the AF volume susceptibility, and the thickness of FM layer, respectively. This expression reveals the temperature dependence of Hc via that of the AF susceptibility, AF, whose peak at TN dominates provided that D 1. It is noticed that equation (7) stresses, again, the close relationship between AF and Hc as observed previously under ME conditions (see section 3; figure 5).
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5. Conclusion In summary, a new type of EB systems based on the classical ME material Cr2O3 is presented, which the authors propose to call a ‘‘multiferroically composed EB system’’. By ME control of the AF component a critical polydomain situation is found on the border line between the two possible single domain states, where both the coercive field and the ac susceptibility maximize. While the coercivity is due to an enhanced spin drag effect, the susceptibility peak is due to the increased concentration of flopped spins in the domain walls. The volume ratio of spins residing in AF domain walls and in the bulk of the multi-domain state, respectively, is estimated to be in the 1% regime. Further, the temperature dependences of EB field and of the FM coercive field are analysed in the vicinity of the transition point TN. Strong correlations between the AF order parameter and the EB field, and between ME and thermally induced peaks of the FM coercivity are discussed.
Acknowledgements Thanks are due to Peter M. Hauck, Cornell University, for valuable assistance, to the Deutsche Forschungsgemeinschaft via Sonderforschungsbereich 491 (‘‘Magnetische Heteroschichten: Spinstruktur und Spintransport’’) and Graduiertenkolleg 277 (‘‘Struktur und Dynamik heterogener Systeme’’), and to the Konrad–Krieger–Stiftung for financial support.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
W. Eerenstein, N.D. Mathur and J.F. Scott, Nature 442 759 (2006). N. Hur, S. Park, P.A. Sharma, et al., Nature 429 392 (2004). M. Fiebig, Phase Transit., 79 (2006). For reviews see: M. Fiebig, J. Phys. D 38 R123 (2005); L. Mitoseriu and V. Buscaglia, Phase Transit., 79 (2006). W.H. Meiklejohn and C.P. Bean, Phys. Rev. 102 1413 (1956); 105 904 (1957). A. Hochstrat, Ch. Binek, Xi. Chen, et al., J. Magn. Magn. Mater. 272–276 325 (2004). Ch. Binek, A. Hochstrat, X. Chen, et al., J. Appl. Phys. 97 10C514 (2005). P. Borisov, A. Hochstrat, X. Chen, et al., Phys. Rev. Lett. 94 117203 (2005). P. Borisov, T. Eimu¨ller, A. Fraile-Rodrigue´z, et al., J. Magn. Magn. Mater. (2006). International patent application PCT/EP2006/002892, DOI: 10.1016/j.jmmm.2006.11. 150. X. Chen, A. Hochstrat, P. Borisov, et al., Appl. Phys. Lett. 89 202508 (2006). V. Laukhin, V. Skumryev, X. Martı´ , et al., Phys. Rev. Lett. 97 227201 (2006). M. Gajek, M. Bibes, S. Fusil, et al., Available online at: http://arxiv.org/cond-mat/ 0606444 (2006). D.N. Astrov, JETP 11 780 (1960); 13 729 (1961). T.H. O’Dell, The Electrodynamics of Magneto-Electric Media (North-Holland, Amsterdam, 1970). G.T. Rado, J.M. Ferrari and W.G. Maisch, Phys. Rev. B 29 4041 (1984). For a review see J. Nogue´s and I. Schuller, J. Magn. Magn. Mater. 192 203 (1999). F. Rohr, M. Ba¨umer, H.-J. Freund, et al., Surf Sci. 372 L291 (1997); Surf. Sci. 389 391 (1997) – erratum. B. Kagerer, Ch. Binek and W. Kleemann, J. Magn. Magn. Mater. 217 139 (2000).
Multiferroically composed exchange bias systems
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[20] S. Maat, K. Takano, S.S.P. Parkin, et al., Phys. Rev. Lett. 87 087202 (2001). [21] S. Shtrikman and D. Treves, Phys. Rev. 130 986 (1963); C.A. Brown, Magneto–electric domains in single crystal chromium oxide, PhD thesis, Imperial College, London, 1969. [22] C. Leighton, J. Nogue´s, B.J. Jo¨nsson-A˚kerman, et al., Phys. Rev. Lett. 84 3466 (2000). [23] D.N. Astrov and B.I. Al’shin, Sov. Phys. JETP 24 18 (1967). [24] S. Foner and S.-L. Hou, J. Appl. Phys. Suppl. 33 1289 (1962). [25] A. Hochstrat, Ch. Binek and W. Kleemann, Phys. Rev. B 66 092409 (2002). [26] Y.-Y. Li, Phys. Rev. 101 1450 (1956). [27] A.A. Khalfina and M.A. Shamsutdinov, Phys. Met. Metall. 92 28 (2001). [28] T.R. McGuire, E.J. Scott and F.H. Grannis, Phys. Rev. 102 1000 (1956). [29] T.J. Martin, Phys. Lett. 17 83 (1965); T.J. Martin and J.C. Anderson, IEEE Trans. Magn. 2 446 (1966). [30] J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. Lett. 39 95 (1977). [31] Y. Shapira and C.C. Becerra, Phys. Rev. B 16 4920 (1977). [32] M. Marinelli, F. Mercuri, U. Zammit, et al., Phys. Rev. B 49 4356 (1994). [33] G. Scholten, K.D. Usadel and U. Nowak, Phys. Rev. B 71 064413 (2005).
