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6.1 Introduction

Proximity effects in general and those of magnetic materials in particular are cornerstones of modern condensed matter physics. The investigation of exchange-coupled magnetic thin films has enormous technological importance particularly for applications which utilize spin-dependent transport across magnetic interfaces [1–4]. Exchange bias (EB) is a prototypical magnetic proximity phenomenon widely used in modern magnetic field sensors and read heads exploiting giant magnetoresistance (GMR) and tunnel magnetoresistance (TMR) effects. Scalability of read head sizes toward smaller devices is a prerequisite for the ongoing exponential growth of the areal storage density in magnetic hard disk drives. Remarkably, this exponential growth known as Moore’s law [5] shows an even increased rate since 1997 when GMR-based read heads with exchange-biased electrodes were first introduced in hard drive technology.

The basic physics of the EB phenomenon is best studied at the interface of exchange-coupled ferromagnetic (FM) and antiferromagnetic (AF) heterostructures [6–12] . In the proximity of an AF pinning layer a FM film can experience an exchange-induced unidirectional anisotropy. The latter reflects its presence most prominently by a shift of the FM hysteresis along the magnetic field axis and is quantified by the amount \(\mu _0 H_{{\rm{EB}}}\) of the shift. The EB effect is initialized when field cooling an AF/FM heterosystem to below the blocking temperature, \(T_{\rm{B}}\), where AF order establishes at least on mesoscopic scales [13].

Microscopically the EB phenomenon depends on a large number of system-specific details like structural and magnetic interface roughness and anisotropy to name just a few. Therefore, a large number of theories have been proposed competing to explain the origin of the EB effect. This chapter will not attempt to give an overview of the microscopic theories or even favor a particular microscopic mechanism over another one. The interested reader is referred to a number of excellent review articles with emphasis on experimental and theoretical aspects, respectively [8–11, 14].

Despite the ongoing controversy about “the origin” of EB many of the macroscopic observations are satisfactorily summarized in the phenomenological description which goes back to Meiklejohn and Bean. It has often been stated that the Meiklejohn–Bean (MB) expression is an invalid oversimplification which overestimates the expected EB field typically by more than an order of magnitude. This overestimation arises when bulk properties and ideal interface conditions are naively assumed. However, when using the MB expression in its appropriate phenomenological sense and the non-trivial relation between macroscopic and microscopic parameters is carefully considered, the MB approach remains a useful description with even quantitative predictive power. State-of-the-art molecular beam epitaxial growth of EB heterostructures allows the fabrication of nearly ideal interfaces with no or negligible roughness on mesoscopic lateral length scales. In fact, such ideal interface regions follow precisely the MB prediction of vanishing EB for compensated AF interfaces [15]. Similar results are known already for nearly a decade now from experiments using artificial AF superlattices as virtually ideal pinning systems confirming the MB approach [16]. Therefore, it is meanwhile widely accepted that EB requires a net irreversible AF interface magnetization \(S_{\rm AF}\) coupling via exchange with the FM interface magnetization \(S_{\rm FM}\) [17]. It turns out to be the major challenge for microscopic theories to explain why \(S_{\rm AF}\) can be surprisingly large for antiferromagnets with compensated AF surfaces and can be surprisingly low in systems with uncompensated surfaces. The MB description does not address these questions. It is therefore not a flaw of the MB approach when unrealistic values for \(S_{\rm AF}\) for instance are used which consequently overestimate the EB fields. It is one of the major experimental and theoretical insights in recent years that only a fraction of the AF interface magnetization remains stationary during the FM magnetization reversal. It is this stationary or irreversible fraction \(S_{\rm AF}\) of the AF interface magnetization that should be used in the MB expression to estimate realistic EB field values.

Another potential flaw of the MB approach originating from the assumption of uniform rotation of the ferromagnet seems also significantly overrated. Intrinsic properties which determine for instance the absolute values of the coercive fields and the magnetization reversal process are not necessarily relevant for the EB field, only their asymmetry with respect to the magnetic zero field determines the EB field. The latter is therefore insensitive on microscopic and micromagnetic details as long as they do not affect the irreversible interface magnetization.

When considering the exchange coupling of strength J between \(S_{\rm AF}\) and the FM interface magnetization, \(S_{\rm FM}\), as phenomenological parameters the original MB approach describes in fact major features of the EB effect in a correct manner. In particular, the EB field, \(\mu _0 H_{{\rm{EB}}}\), quantifying the shift of the FM hysteresis loop along the magnetic field axis is in its simplest form expressed as

$$\mu_0 H_{\rm EB}=-\frac{JS_{\rm AF}S_{\rm FM}}{M_{\rm FM}t_{\rm FM}},$$
((6.1))

where \(M_{\rm FM}\) is the saturation magnetization of the FM film and \(t_{\rm FM}\) is its thickness. The latter inverse thickness dependence on the FM film has been confirmed in countless investigations and reflects the true interface nature of the effect. Experimentally this interface sensitivity implies that control over interface properties is crucial for systematic studies of the phenomenon.

Meiklejohn and Bean derived Eq. (6.1) from the Stoner–Wohlfarth type model where uniform rotation of FM magnetization is induced by a magnetic field applied along the easy AF and FM axes. In addition to the free Stoner–Wohlfarth ferromagnet, exchange coupling at the interface between the ferromagnet and a stationary AF pinning system of infinite anisotropy is the major ingredient of the simple MB model. Equation (6.1) is analytically derived from the MB energy expression. The most convenient and original MB approach contracts the Zeeman and the AF/FM coupling energy into an effective Zeeman term where the renormalized field is interpreted as the sum of the applied field and \(\mu _0 H_{{\rm{EB}}}\).

MB-type approaches have been studied for a variety of generalizations including finite anisotropy of the antiferromagnet, finite AF film thickness, and arbitrary orientation of the applied field with respect to the easy axes to name just a few [18].

In contrast to the usual analytic MB approach discussed elsewhere, Fig. 6.1 provides a graphical illustration associated with the evolution of the MB energy on variation of the magnetic field. Here, as an example, the applied magnetic field makes an angle of \(\alpha=\pi /6\) with respect to the easy axis of the Stoner–Wohlfarth-type ferromagnet with uniaxial anisotropy of energy density K. The displayed hysteresis loop, M/M s vs. h, represents the projection, \(M=M_{\rm{s}} \;\cos \;\left( {\alpha - \beta _{\min } (h)} \right)\), of the uniformly rotating magnetization vector M of magnitude M s onto the direction of the reduced magnetic field \(h=\mu _0 M_{\rm{s}} t_{{\rm{FM}}} H\). Rotation of M takes place in accordance with the evolution of the angle \(\beta _{\min }=\beta _{\min } (h)\). The latter follows a crossover from a global energy minimum into a local minimum on approaching the coercive fields \(h_{{\rm{c}}1}\) and \(h_{{\rm{c}}2}\), respectively. In addition to the hysteretic M/M s vs. h curve the evolution of the corresponding energy density

$$E/A=-h \cos(\alpha - \beta) + K \sin^2 \beta - JS_{\rm AF} S_{\rm FM} \cos \beta$$
((6.2))

is plotted with K=0.5, \(JS_{{\rm{AF}}} S_{{\rm{FM}}}=0.2\) (solid line) for selected fields \(h=\pm 2\) (upper right and lower left insets), and for the switching fields \(h=h_{{\rm{c}}1}\) (upper left inset) and \(h=h_{{\rm{c}}2}\) (lower right inset) where M changes sign, respectively. It is the asymmetry of the switching fields with respect to h=0 which determines the EB field \(h_{{\rm{EB}}}=\left( {h_{{\rm{c}}1} + h_{{\rm{c}}2} } \right)/2\). Full lines in the insets of Fig. 6.1 show the respective energy densities when exchange coupling between the ferromagnet and the AF pinning layer is included. The dashed curves show the corresponding energy densities of the same Stoner–Wohlfarth ferromagnet in the absence of coupling (J = 0). The comparison of these energy curves in particular at \(h_{{\rm{c}}1}\) and \(h_{{\rm{c}}2}\) illustrates the mechanism giving rise to the EB shift h EB.

Fig. 6.1
figure 6_1_150780_1_Enfigure 6_1_150780_1_En

Field dependence \(M/M_{\rm s}\) vs. h of an uniaxial anisotropic Stoner–Wohlfarth ferromagnet exchange coupled to the stationary and field-independent interface magnetization of an adjacent pinning system. The coupling gives rise to the exchange bias effect shifting the loop along the field axis by the amount\(\left| {h_{{\rm{EB}}} } \right|\)indicated by an arrow. Insets show the energy densities \(E/A\) vs. \(\beta\) for \(h=2\) (upper right), \(h\,=\,h_{{\rm{c}}1}\)(upper left), \(h=-2\) (lower left), and \(h\,=\,h_{{\rm{c}}2}\)(lower right) with (solid lines) and without (dashed lines) exchange coupling. Magnetization states of specific interest are marked by large dots on the \(M/M_{\rm s}\) vs. h curve. The corresponding minima in \(E/A\) vs. \(\beta\) are indicated in the energy densities by small dots. Pictograms show the easy axis of the ferromagnet of anisotropy energy density K and the relative orientation of the magnetization and the magnetic field. The former rotates uniformly with constant magnitude \(M_{\rm s}\) in contrast to h which is applied under a constant angle of \(\alpha=\pi /6\) with respect to the easy axis

Full size image

In the presence of the strong positive field h=2 the magnetization vector aligns virtually parallel to the magnetic field. Here, in a state close to positive saturation, the Zeeman energy controls the pronounced minimum in E/A vs. β where β measures the orientation of the magnetization with respect to the easy axis of the ferromagnet. The dominance of the Zeeman energy over the AF/FM coupling energy is reflected in the fact that the energy densities with (solid line) and without coupling (dashed line) virtually coincide. On lowering the magnetic field toward \(h\,=\,h_{{\rm{c}}1}\), antiparallel orientation of M with respect to the applied field becomes unstable. The instability is reflected by a saddle point in E/A vs. β. The dashed curve in the upper left diagram of Fig. 6.1 indicates that in the absence of interface coupling the instability and, hence, magnetization reversal set in at \(h_{{\rm{c}}1} (J=0)=h_{{\rm{c}}1} - h_{{\rm{EB}}}\). Finally, negative saturation at h= −2 is virtually achieved, corresponding to a pronounced minimum in E/A vs. β and parallel alignment of the magnetization vector with the applied field. With increasing magnetic field again antiparallel alignment of M with respect to h becomes unstable in accordance with a saddle point in E/A vs. β at \(h_{{\rm{c}}2}=h_{{\rm{c}}2} (J=0) + h_{{\rm{EB}}}\). This switching field is lowered by the amount \(\left| {h_{{\rm{EB}}} } \right|\) in comparison to the switching field of the unpinned Stoner–Wohlfarth ferromagnet [\(h_{{\rm{c}}2} (J=0)\)]. The dashed E/A vs. β curve in the lower right inset of Fig. 6.1 shows still a clear local minimum when the actual system with coupling evolves already into the saddle point instability.

The shift of the FM hysteresis loop along the magnetic field axis is often accompanied by an EB-induced loop broadening [19, 20]. This effect is not included in the MB description. The understanding of the loop broadening makes it necessary to consider the role of the loosely coupled majority fraction of AF interface spins which do not affect the EB field. The magnetic moment of these loose spins is not irreversible but follows to some extent the magnetization reversal of the top ferromagnet giving rise to a drag effect which broadens the FM hysteresis. More quantitatively based on mean-field arguments it has been predicted that the FM coercivity is related to the AF interface susceptibility [21]. Loosely coupled spins are in particular sensitive to either exchange or applied magnetic fields and, hence, increase the AF interface susceptibility and by that the FM coercivity. The coercivity enhancement accompanies the EB effect and is characteristically reduced when the blocking temperature T B of vanishing EB is approached from \(T<T_{\rm B}\). While loosely coupled moments flip easier when their AF neighboring spins lost long-range order, nevertheless a drag effect on the adjacent FM film is still present above the blocking temperature and even above the Néel temperature, \(T_{\rm N}\), of the pinning layer allowing for persistence of loop broadening above \(T_{\rm N}\).

In addition to the EB loop shift and broadening, a gradual degradation of the EB field can take place when cycling the heterostructure through consecutive hysteresis loops [22–32]. This aging phenomenon is known as training effect. It is quantified by the \(\mu _0 H_{{\rm{EB}}}\) vs. n-dependence, where n labels the number of loops cycled after initializing the EB via field cooling. EB and its accompanying training effect have been observed in various magnetic systems [8, 33–39]. The MB expression does not directly address the phenomenon of EB training. However, Eq. (6.1) correlates the bias field with the AF interface magnetization \(S_{\rm AF}\). The latter can and typically does change during successively cycled hysteresis loops of the FM top layer such that \(S_{{\rm{AF}}}=S_{{\rm{AF}}} (n)\) gives rise to an n-dependence in \(\mu _0 H_{{\rm{EB}}}\). This chapter dedicates a section on the recent progress in understanding aging phenomena in various exchange-coupled systems. Selected experiments evidence certain control over the aging effects through the initialization protocol that sets the magnetization state of the pinning system.

While many intrinsic details of the EB phenomenon are still a matter of scientific debate, extrinsic control of the EB has been added in recent years to the major research topics in this field. Among the various possible control mechanisms, electrically tuned EB appears most attractive for spintronic applications. Research on electrically controlled EB has been initiated by work on \({\rm Cr}_2{\rm O}_3/{\rm CoPt}\) heterostructures stimulating at the same time a revived interest in magnetoelectric materials. The same motivation, namely control of magnetism via electric fields, created recently a tremendous renewed interest in multiferroic materials where multiple ferroic order parameters (magnetic, electric, and elastic) can be simultaneously present and sometimes significantly couple. Conjugate magnetic and electric fields can be used to manipulate the respective cross-coupled order parameter. Switching of FM order by an electric field for instance promises significant impact in the design of future spintronic devices. Note, however, that the magnetoelectric effect does not require multiferroics and multiferroics do not necessarily show appreciable magnetoelectric effects. Actually in most cases they do not [40]. Nevertheless, some of the multiferroic materials promise potential for spintronic applications in addition to their more general exciting physical properties. Most recently studied multiferroics can be classified into single-phase and two-phase systems. Single-phase multiferroics are predicted to be rare [41]; however, many perovskite-type oxides have been successfully exploited to control magnetic order to some extent by electrical means and vice versa.

Next, various attempts of electric and magnetic field control of EB-related phenomena are introduced. Some of them are based on multiferroics and most are not.

6.2 Electrically Tuned Exchange Bias

6.2.1 Electrically Tuned Exchange Bias with Magnetoelectrics

Experiments on the perpendicular EB heterostructure \({\rm Cr}_2{\rm O}_3/{\rm CoPt}\) pioneered the field of electrically tuned EB [42–46]. Here the magnetoelectric \({\rm Cr}_2{\rm O}_3\) has a twofold role. It serves as an AF pinning layer allowing at the same time for electrical tuning of the AF interface magnetization. Exchange coupling of the latter with the FM interface magnetization of the perpendicular anisotropic CoPt gives rise to electrically tunable interface coupling energy and, hence, to electrically tunable EB fields. The AF \({\rm Cr}_2{\rm O}_3\) is a prototypical magnetoelectric material. Its intrinsic magnetoelectric effect sets in below the Néel temperature, T N= 307 K, together with AF long-range order when spatial and time inversion symmetry are broken, respectively, but build a symmetry operation in combination. AF magnetoelectrics do not depend on spontaneous ferroelectric polarization to achieve coupling between the magnetization and an electric field and are therefore complementary to the multiferroic materials discussed in Section 6.2.2.

In a magnetoelectric material an applied electric field induces a net magnetic moment which can be used to electrically manipulate the magnetic states of adjacent exchange-coupled FM films. More specifically, the linear magnetoelectric effect is characterized by a linear magnetic response \(M^i\) (electric response \(P^i\)) induced by the application of an electric field E j (magnetic field H j ) such that \(\mu _0 M^i=\alpha _{{\rm{me}}}^{ij} E_j\) and \(P^i=\alpha _{{\rm{em}}}^{ij} H_j\), where \(\alpha_{{\rm{em}}}^{\rm ij} = \alpha _{{\rm{me}}}^{\rm ji} = \alpha _{{\rm{em}}}^{\rm ji}\) are the tensors of magnetoelectric susceptibility and its transpose counterpart while i, j label the vector and tensor components, respectively [47, 48]. The magnetoelectric susceptibility tensor \(\alpha _{{\rm{me}}}^{ij}\) of \({\rm Cr}_2{\rm O}_3\) has diagonal structure with \(\alpha _{{\rm{me}}}^{11}=\alpha _{{\rm{me}}}^{22}=\alpha _ \bot\) and \(\alpha _{{\rm{me}}}^{33}=\alpha _\parallel \approx {\rm{4}}{\rm{.13 }}\) ps/m at its maximum value achieved close to T=263 K. This magnetoelectric susceptibility is small in comparison to some other magnetoelectric materials [49]. However, the AF nature of \({\rm Cr}_2{\rm O}_3\) and its appreciably high \(T_{\rm N}\) still make it a first choice candidate for an intrinsic magnetoelectric material when applications at or close to room temperature are envisioned.

Figure 6.2 shows the schematics of a \({\rm Cr}_2{\rm O}_3/{\rm CoPt}\) heterostructure for electrically tunable EB. Here the electric field orients along the (111) direction of \({\rm Cr}_2{\rm O}_3\) taking advantage of the parallel magnetoelectric susceptibility. Note that EB requires a non-zero projection of \(S_{\rm AF}\) on \(S_{{\rm FM }}\) making a top ferromagnet with perpendicular anisotropy mandatory when \(\alpha _\parallel\) is used.

Fig. 6.2
figure 6_2_150780_1_Enfigure 6_2_150780_1_En

Sketch of an electrically controlled EB heterosystem based on the magnetoelectric antiferromagnet \({\rm Cr}_2{\rm O}_3\) and a thin exchange-coupled CoPt multilayer. A voltage V applied across the top and bottom electrodes creates an electric field \(E=V/d\) where d is the thickness of the insulating \({\rm Cr}_2{\rm O}_3\) film. The E field induced the magnetization \(\Delta {\rm M} \infty \alpha_\parallel {\rm E}\) in \({\rm Cr}_2{\rm O}_3\). It contributes to the interface magnetization \(S_{\rm AF}\) and modifies the EB field. Arrows indicate the AF and FM spin structure in the \({\rm Cr}_2{\rm O}_3\) bottom and CoPt top layer, respectively. The right sketch depicts the crystal and spin structure of \({\rm Cr}_2{\rm O}_3\). The spin orientations in the four sublattices are shown for one of the two 180° domains. The \(S=3/2\) spins of the \({\rm Cr}^{3+}\) ions are aligned along the threefold symmetry axis. \({\rm O}^{2-}\) ions form distorted octahedrons (distortion not shown) surrounding the \({\rm Cr}^{3+}\) ions. Electrically induced displacement of the latter into crystal fields of alternating higher and lower strength gives rise to the single ion contribution of the magnetoelectric effect (see text)

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The current status of the description of the microscopic origin of the magnetoelectric effect in \({\rm Cr}_2{\rm O}_3\) has been summarized by Alexander and Shtrikman [50] and by Hornreich and Shtrikman [51]. Figure 6.2 shows a sketch of a section of the unit cell of \({\rm Cr}_2{\rm O}_3\). It indicates the A (spin up) and B (spin down) sites of the \({\rm Cr}^{3+}\) ions along the threefold rotation axis. In the presence of a positive applied electric field, the \({\rm Cr}^{3+}\) ion on the A site moves toward the upper triangle of \({\rm O}^{2-}\) ions. The corresponding ion on the B site moves in the same direction along the symmetry axis toward the smaller triangle of O2− ions. The latter triangle is shared among the two distorted octahedra of O2− ions, which surround the \({\rm Cr}^{3+}\) ions on the A and B sites. The asymmetry in the change of the crystal fields experienced by A and B ions gives rise to the single-ion contributions of the magnetoelectric effect. In terms of a spin Hamiltonian, the single-ion contributions reflect the changes of the Landé g-tensor and the single-ion anisotropy, which differ for the spins at position A and B in the presence of an electric field. In addition, the asymmetric displacement of the ions gives rise to different modifications of the exchange interaction between \({\rm Cr}^{3+}\) ions. The resulting electric field-induced change of the exchange integrals is known as the two-ion contribution to the magnetoelectric effect.

Electric control of the EB field can be approached at least in two very distinct ways. An electric field can be applied simultaneously with a magnetic field in order to magnetoelectrically anneal the AF pinning layer. Magnetoelectric annealing is a well-known procedure to bring a magnetoelectric antiferromagnet into a single domain state by cooling the system from \(T>T_{N}\) to below \(T_{\rm N}\) in the presence of E and B fields. By energetically favoring one of the AF 180° domains over the other, the magnetic state of the AF interface magnetization is selected too and at least partially irreversibly frozen in. In an EB system using a magnetoelectric pinning layer the interface magnetic state can be magnetoelectrically selected. This type of electric-controlled EB has been named magnetoelectric switching because it allows switching between positive and negative EB fields by controlling the sign of the product \(E \cdot B\) [39]. This pronounced effect is highly interesting from a fundamental point of view. Spintronic applications, however, favor an electric control of the EB field at constant \(T<T_{\rm N}\) where ideally \(T_{\rm N}\) is significantly above room temperature.

Recently, a reversible electrically induced shift of the magnetic hysteresis loops along the magnetic field axis has been achieved. The heterostructure follows the schema in Fig. 6.2 and is based for the first time on an all thin film c-\({\rm Al}_2{\rm O}_3/{\rm Pt} 5.7 {\rm nm}/{\rm Cr}_2{\rm O}_3\) 50 nm/Pt 0.5 nm/[Co 0.3 nm/Pt 1.5 nm]3/Pt 1.5 nm heterostructure.

Similar experiments have been performed earlier in \({\rm Cr}_2{\rm O}_3(111)/{\rm CoPt}\) heterostructures using bulk \({\rm Cr}_2{\rm O}_3\) single crystals as AF pinning systems with magnetoelectric properties [40, 41]. Note, however, that millimeter thick single crystals require orders of magnitude higher voltages to achieve the electric fields that can be realized in nanometer thin films by a few millivolts. It is important to stress that the electrically controlled EB shown in Fig. 6.3 resembles a global effect. Here the tunable exchange anisotropy affects the entire FM top electrode and the EB field can be reversibly changed back and forth with the applied electric field. Competing EB systems based on multiferroic pinning layers show significantly larger isothermal EB tuning effect but are limited so far to local or irreversible control [52, 53]. A variety of spintronic applications based on electrically controlled exchange bias have been proposed. Those concepts of spintronic applications using electrically controlled EB have been pioneered by the authors in Refs. [42, 43]. Here spin-dependent transport devices with electrically controlled resistance states have been suggested. In general, magnetoelectrically induced interface magnetization provides control of the pinning in exchange-biased GMR- or TMR-type structures. In the GMR-type structures the active magnetoelectric material is used as a tunable pinning bottom layer offering new versatility for logic and memory devices based on a change of resistance due to a change of the magnetic configurations. In TMR-type structures the conventional passive tunneling barrier can be replaced by a magnetoelectrically active barrier material [42]. The electric field-induced magnetization of the latter allows switching magnetic states of exchange-coupled layers by pure electrical means. Following these suggestions numerous variations have been proposed [54]; often the magnetoelectric antiferromagnet has been replaced by a multiferroic component like \({\rm BiFeO}_3\) [55–57].

Fig. 6.3
figure 6_3_150780_1_Enfigure 6_3_150780_1_En

The left frame shows normalized magnetic hysteresis loops of the all thin film \({\rm Cr}_2{\rm O}_3 (111)/{\rm CoPt}\) heterostructure measured by polar magneto-optical Kerr effect. Loop 1 is measured at T=300 K. Loops 2, 3, and 4 are measured after magnetoelectric annealing in \(\mu _0 H=- 0.4\,\;{\rm{T}}\) and V= −100 mV from T=300 K to T=250 K<\(T_{\rm N}\). The isothermal loops have then been measured at T=250 K in the presence of applied axial voltages V=−100 mV (2), +90 mV (3), and +100 mV (4), respectively. The inset in the left frame provides a detailed look onto the electrically controlled change of the negative coercive fields of loops 2, 3, and 4. The right frame depicts the systematic results of the electric field control of the exchange bias field. The line is a linear best fit to the \(\mu _0 H_{{\rm{EB}}}\) vs. V data

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6.2.2 Electrically Tuned Exchange Bias with Multiferroics

The above discussed attempts to achieve electrically tuned EB with the help of magnetoelectric antiferromagnets require further investigations and show clearly the need for an increase of the magnetoelectric response in order to accomplish appreciable electric tuning effects. In the case of magnetoelectric antiferromagnets significant efforts are made to further improve the structural and dielectric properties of \({\rm Cr}_2{\rm O}_3\) films allowing for higher applied electric fields and improved interface coupling [58].

Many research teams who currently focus on the use of multiferroic materials expect a qualitative progress from this class of materials. The latter show simultaneous presence of two or more ferroic order parameters like ferroelectric polarization and (anti)ferromagnetic order. This coexistence seems to make them obvious candidates for the quest of increased magnetoelectric response. The renewed excitement about multiferroics for spintronic applications is based on this hope. It is supported by a fundamental inequality which states that the square of the magnetoelectric susceptibility is limited by the product of the magnetic and the dielectric susceptibility [59]. Since multiferroics have spontaneous ferroelectric and ferromagnetic order the latter susceptibilities are high and their product limiting the magnetoelectric susceptibility α is high. However, to take advantage of this potential upper limit of α strong coupling between the ferroic order parameters is required. The coupling is, however, typically weak. Hence, the magnetoelectric susceptibility of most multiferroics is at room temperature or above disappointingly weak and it is not straightforward to exceed the magnetoelectric bulk susceptibility of, e.g., \({\rm Cr}_2{\rm O}_3\) which is of the order of 4 ps/m close to room temperature [60]. Nevertheless, in view of the potential applications there is an intense search for the ideal magnetoelectric single-phase multiferroic [61–69].

Currently there are at least two multiferroic single-phase materials under intense investigation for use in electrically controlled EB applications [44, 45]. Both \({\rm YMnO}_3\) and \({\rm BiFeO}_3\) order antiferromagnetically with Néel temperatures of 90 and 643 K, respectively. Despite their AF order the terminology multiferroics is still applied in the literature in a somewhat generalized sense.

Complete electrically induced suppression of EB has been achieved at T=2 K in an \({\rm YMnO}_3/{\rm Py}\) heterostructure when applying a voltage of 1.2 V across the c-axis of the hexagonal \({\rm YMnO}_3\) film of 90 nm thickness [44]. So far, the effect is, however, irreversible which means that a voltage of opposite sign does not recover any EB effect. Moreover, the limitation to low temperatures makes it an interesting system for proof of principle but little attractive for the envisioned applications.

This situation changes in part when \({\rm BiFeO}_3\) is used as an electrically controllable pinning layer [45]. In heterostructures of the type \({\rm BiFeO}_3\)/CoFe 10 times higher electric potential differences of ±12 V were applied across the 100−150 nm thick \({\rm BiFeO}_3\) films. The resulting electric fields achieved local magnetization reversal on a lateral length scale of about 2 μm. The limitation to local switching seems to originate from the characteristic domain pattern of the AF, ferroelectric, and ferroelastic order parameters. It remains to be shown if this limitation can be overcome. Subsequently, it is shown that magnetoelastic coupling alone is a very effective and simple way to achieve electrically controlled magnetization.

6.2.3 Piezomagnetically and Piezoelectrically Tuned Exchange Bias

As discussed above, one of the major challenges which microscopic theories of EB face is to explain the quantitative presence of net AF interface magnetization remaining stationary during the FM reversal. The details of the frozen interface magnetization in various systems are affected by a huge variety of crucial influences. Very likely this is one of the reasons why “the origin of EB” has not yet been found in terms of a universal mechanism despite countless attempts. It seems more likely that there is nothing like the universal microscopic mechanism. Many microscopic processes can lead to AF interface magnetization which couples with the adjacent ferromagnet giving rise to EB. For this reason even today a phenomenological description of the EB effect like the Meiklejohn–Bean approach is still valuable.

An exotic possible origin or at least contribution to EB is found for instance in the rutile structured antiferromagnet \({\rm FeF}_2\). The latter has been extensively studied as a prototypical EB pinning system. Compensated and uncompensated AF surfaces can be realized via (111) or (100) surfaces. The (111) surface has been studied at length in order to understand the origin of AF interface magnetization in the particularly puzzling case of compensated surfaces. In addition to the impact of surface roughness and defects which may create random field-type AF domains [70, 71] a more exotic mechanism of stress-induced piezomagnetism has been discussed as one of the possible contributions to the AF interface magnetization [72–75].

Piezomagnetism is known to create the vertical shift of the EB hysteresis loops in \({\rm FeF}_2(110)/{\rm Fe}\) [76]. Since correlation between the horizontal and vertical loop shifts is well known [77] it is straightforward to correlate piezomagnetism also with EB fields. The concept of an activation of the symmetry-allowed piezomagnetism in \({\rm FeF}_2\) via lattice mismatch-induced stress has been more systematically evidenced in an investigation of the EB heterostructure \({\rm Fe}_{0.6}{\rm Zn}_{0.4}{\rm F}_2/{\rm Fe}\). Here external sheer stress along the [110] direction activates a piezomoment within the (110) plane that contributes to the AF interface magnetization \(S_{\rm AF}\) and, hence, to the EB field [78].

More generally, stress-induced control of magnetic anisotropy which potentially includes also unidirectional anisotropy is a very promising tool to achieve electric control over magnetic states in very simple heterostructures. The use of magnetoelastic coupling for electrically controlled magnetism is well known from investigations of two-phase multiferroics which include artificially grown ferroelectric/ferromagnetic heterostructures such as \({\rm BaTiO}_3/{\rm La}_{0.66}{\rm Sr}_{0.33}{\rm MnO}_3\), \({\rm BiFeO}_3/{\rm SrTiO}_3\), nanopillar-embedded structures of \({\rm BaTiO}_3-{\rm CoFe}_2{\rm O}_4\) or \({\rm BiFeO}_3-{\rm CoFe}_2{\rm O}_4\), \({\rm Pb}({\rm Zr,Ti}){\rm O}_3/{\rm CoPd}\), and \({\rm Pb}({\rm Zr,Ti}){\rm O}_3/\hbox{terfenol-D}\) [79–84]. The magnetoelectric behavior of these composite materials resembles the product properties of the magnetoelastic effect of the ferromagnetic component and the piezoelectric effect of the (relaxor) ferroelectric [85]. Complex oxides have been some of the most favored materials.

Recently, however, reversible control of magnetism has been reported for a remarkably simple ferroelectric/FM heterostructure. A polycrystalline Fe thin film deposited on the c-plane of a \({\rm BaTiO}_3\)single crystal shows a large magnetization change in response to the ferroelectric switching and structural transitions of \({\rm BaTiO}_3\) controlled by applied electric fields and temperature, respectively [86]. Interface strain coupling is the primary mechanism altering the induced magnetic anisotropy. As a result, coercivity changes up to 120% occur between the various structural states of \({\rm BaTiO}_3\). Up to 20% coercivity change has been achieved via electrical control at room temperature. This is remarkable considering the fact that the change of the \({\rm BaTiO}_3\) lattice parameter within the c-plane is only a secondary effect of the far more pronounced piezoelectric deformation of the c-axis when the electric field is applied along the latter in the tetragonal phase of \({\rm BaTiO}_3\). Hence, much larger effects can still be expected. The concept of ferro- or piezoelectrically controlled magnetism can be straightforward generalized toward electrically controlled EB when taking advantage of the fact that the magnetoeleastic coupling constant of some antiferromagnets like \({\rm CoO}\) is even higher than the corresponding values for the simple 3d metals [87].

6.3 Magnetic Field Control of Exchange Bias

The easiest way to tune the EB field in common EB heterostructures is achieved via the initializing field-cooling process. It takes place in the presence of a freezing field, \(h_{\rm f}\), applied on cooling the heterostructure to below the Néel temperature, \(T_{\rm N}\), of the AF pinning layer. Here, the orientation of the freezing field can select either negative or positive EB fields [69]. Note, however, that the positive EB fields established in \(h_f <0\) have nothing in common with the effect known as positive EB. The latter is observed only when positive (negative) freezing fields give rise to positive (negative) EB fields. Positive EB is a rather unusual case but sometimes observed in systems where the interface exchange interaction is AF and, at the same time, the freezing field applied during the field-cooling procedure is strong enough to overcome the exchange interaction on cooling the system to below the blocking temperature [88–90].

Figure 6.4 illustrates this situation and clarifies a common misconception which confuses the presence of positive EB fields with a sufficient condition to identify the positive EB effect. The Zeeman energy of the interface magnetization in a freezing field favors parallel alignment of \(S_{\rm AF}\) with respect to \(S_{\rm FM}\). AF interface coupling in turn favors antiparallel alignment of \(S_{\rm AF}\) with respect to \(S_{\rm FM}\). The net effects of the competition between these two energies on the EB field has been experimentally and theoretically studied for instance in Ref. [91]. Inspection of Fig. 6.4 illustrates intuitively that the measurement of the EB field in a single freezing field is not an appropriate tool to determine the sign of the interface coupling, J. In fact, independent of the sign of J, field cooling allows setting the EB field at negative and positive values [92]. Next these details and the specific case of positive EB are discussed with the help of the spin structures displayed in Fig. 6.4.

Fig. 6.4
figure 6_4_150780_1_Enfigure 6_4_150780_1_En

Hysteresis loops of an ideal EB heterosystem with FM interface coupling \(J>0\) (upper left and right frames) and AF interface coupling \(J<0\) (lower left and right frames). For \(J >0\) (upper frames) negative (positive) EB fields \(h_{\rm EB}\) are indicated by arrows and achieved by field cooling in a freezing field \(h_{\rm f}>0\) (\(h_{\rm f}<0\)). For \(J<0\) (upper frames) field cooling in \(0<h_{\rm{f}}<\left| {JS_{{\rm{FM}}} } \right|\) creates a regular negative EB field while field cooling in \(h_{\rm{f}}>\left| {JS_{{\rm{FM}}} } \right|\) gives rise to a positive EB field which is the fingerprint of the positive EB effect. The frozen AF spin structure and the FM spin structure during the field-cooling process are depicted by arrows. The ideal interface is indicated by a solid line, AF and FM interface spins are marked by boxes (dashed lines).

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The two upper frames of Fig. 6.4 show sketches of hysteresis loops after field cooling an EB heterostructure with FM interface coupling \(J>0\) in positive (left upper frame) and negative (right upper frame) freezing fields \(h_{\rm f}\). When applying \(h_{\rm f}\) at \(T>T_{\rm N}\) no AF long-range order has established and pinning is absent. Hence, the FM top layer is free to align parallel to \(h_{\rm f}\) giving rise to \(S_{\rm FM}>0\) in \(h_{\rm f}>0\) and \(S_{\rm FM}<0\) in \(h_{\rm f}<0\). This state of \(S_{\rm FM}\) affects the orientation of the AF interface magnetization which establishes on cooling to below \(T_{\rm N}\). The coupling energy \(JS_{\rm AF}S_{\rm FM}\) together with the Zeeman energy control the orientation of \(S_{\rm AF}\). In the case \(J>0\) both the exchange interaction and the Zeeman energy favor parallel alignment of \(S_{\rm AF}\)and \(S_{\rm FM}\) such that \(JS_{\rm AF}>0\) for \(h_{\rm f}>0\) and \(JS_{\rm AF}<0\) for \(h_{\rm f}<0\). Since \(S_{\rm FM}\) follows the overall magnetization of the FM layer during a hysteresis loop it is the sign of the stationary product \(JS_{\rm AF}\) that determines the sign of the EB field in accordance with Eq. (6.1). Hence, in the case of \(J>0\) negative (positive) EB fields are achieved in positive (negative) freezing fields. Obviously, the positive EB field shown in the upper right frame has nothing in common with the phenomenon of positive EB.

The more complex scenario appears in the case of AF exchange coupling \(J>0\). Both of the two lower frames of Fig. 6.4 show sketches of hysteresis loops after field cooling an EB heterostructure with AF interface coupling in positive freezing fields. The lower left frame displays the situation of field cooling in a moderate magnetic field \(0<h_{\rm{f}}<\left| {JS_{{\rm{FM}}} } \right|\). Again, when applying \(h_{\rm f}\) at \(T>T_{\rm N}\) there is no pinning effect and the FM top layer aligns parallel to \(h_{\rm f}>0\) giving rise to \(S_{\rm FM}>0\). The coupling energy \(JS_{\rm AF}S_{\rm FM}\) favors now antiparallel alignment of \(S_{\rm AF}\) relative to \(S_{\rm FM}>0\). The product \(JS_{{\rm{FM}}}<0\) can be interpreted as an exchange field acting on the AF interface magnetization \(S_{\rm AF}\) on cooling. At the same time, \(S_{\rm AF}\) has potential or Zeeman energy in the applied freezing field \(h_{\rm f}>0\) which favors \({\rm S}_{\rm AF}>0\). However, as long as \(0<h_{\rm{f}}<\left| {JS_{{\rm{FM}}} } \right|\) is fulfilled, the interface exchange energy overcomes the Zeeman energy resulting in \({\rm S}_{\rm AF}<0\) and, hence, \(JS_{\rm AF}>0\) giving rise to a regular negative EB field despite \(J<0\).

The situation changes, however, in the case of large positive freezing fields \(h_{\rm{f}}>\left| {JS_{{\rm{FM}}} } \right|\). Now the Zeeman energy overcomes the AF interface coupling giving rise to a parallel alignment of \(S_{\rm AF}\) and \(S_{\rm FM}\) during the field-cooling process. Hence, \(JS_{\rm AF}<0\) results in a positive EB field in accordance with Eq. (6.1). The latter scenario displayed in the lower right frame of Fig. 6.4 describes the positive EB effect. Out of all situations displayed in Fig. 6.4, only here a positive freezing field gives rise to a positive EB field due to AF interface coupling. Of course one can repeat the arguments above for the analogous situation of negative EB fields when field cooling took place in negative freezing fields.

Tuning the EB field by cooling heterostructures in the presence of an applied magnetic field to below the blocking temperature is experimentally straightforward to do but not very attractive for most applications. Here, temperature is typically not a control parameter and the EB field needs to be tuned just with the applied magnetic field at about room temperature. For an AF system an applied homogeneous magnetic field affects the AF long-range order in many ways similarly as temperature does. Increased temperature and increased magnetic fields weaken or even completely suppress AF long-range order. It is therefore not surprising that even below \(T_{\rm B}\) strong magnetic fields typically of the order of several 10 T can set the EB field to a new value which is observed when FM hysteresis loops are measured in moderate magnetic fields significantly smaller than the set field [93]. Again, for applications, it is far more attractive to achieve such a resetting of the EB with set fields of the order of 1 T or below such that no high-field technology is needed.

Recently, \({\rm Cr}_2{\rm O}_3\) has been used as an AF pinning system in a magnetic trilayer structure having two \({\rm Cr}_2{\rm O}_3/\hbox{Fe EB}\) interfaces [94]. Here, similar to the magnetoelectrically switchable interface magnetization in \({\rm Cr}_2{\rm O}_3 (111)/{\rm CoPt}\) the AF surface magnetization in \({\rm Cr}_2{\rm O}_3 (111)/{\rm Fe}\) can be isothermally tuned by moderate applied magnetic fields.

Figure 6.5 shows two room temperature hysteresis loops of an Fe 10 nm/\({\rm Cr}_2{\rm O}_3 2.72 {\rm nm}/{\rm Fe} 10.8 {\rm nm}\) trilayer structure after subjecting the system to negative (open circles) and positive (solid circles) DC fields of 1 T magnitude, respectively. The sign of the EB field is controlled exclusively by the applied set field. This is reminiscent of the scenario of changing the sign of the EB field depending on the strength of the cooling fields [86]. Note, however, that here EB tuning takes place isothermally. The pinning magnetization is reversed by a magnetic field only without breaking and reestablishing AF long-range order during a field-cooling process. Competition between AF exchange coupling of the pinning and the Fe magnetizations with the Zeeman interaction energy of the pinning system determines the sign of the EB field. For set fields \(\mu_0H_{\rm DC} \le 0.2 {\rm T}\), AF coupling remains dominant over the Zeeman energy, resulting in regular (negative) EB. For larger \(\mu_0H_{\rm DC}\), the Zeeman interaction overcomes the AF exchange coupling which aligns the pinning moments parallel to larger \(\mu_0H_{\rm DC}\), leading to positive EB. The fact that very moderate magnetic fields are sufficient to isothermally tune the EB fields at room temperature promises applicability of this interesting effect in future spintronic devices.

Fig. 6.5
figure 6_5_150780_1_Enfigure 6_5_150780_1_En

Room temperature hysteresis loops of \({\rm Fe} 10 {\rm mn}/{\rm Cr}_2{\rm O}_3\) 2.72 nm Fe 10.8 nm trilayer after subjecting the system isothermally to \(\mu_0H_{\rm DC}=-1 {\rm T}\) (open circles) and μ0 H DC=1T (solid circles) DC magnetic fields. Lines are guide to the eye

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So far we considered various possibilities in tuning the EB field by electric, thermal, and/or magnetic means. It has been mentioned earlier in this chapter that the EB fields can change with consecutively cycled hysteresis loops. Subsequently more light is shed on the physics of the training effect and means to tune strength and other characteristics of the training effect are presented.

6.4 Training Effect in Exchange-Coupled Bilayers

6.4.1 Physical Background of Training Effects in Various Systems

In a general sense training at finite temperature is observed as a consequence of changes in the spin structure taking place in most casesFootnote 1 away from a nonequilibrium toward an equilibrium configuration. This training effect deviates from the well-known aging phenomena in magnetic systems [95–100] by the fact that the spin configurational changes in the pinning system are triggered by the magnetic hysteresis loop of the pinned magnetic component which is magnetically coupled with the pinning system. The latter is AF in regular EB systems but can be a spin glass or a hard ferromagnet with a coercivity significantly larger than the coercivity of the pinned system.

Figure 6.6 shows a comparison of the physical mechanism of the training effect in regular AF/FM EB systems and hard/soft all FM bilayers. In both cases there is a pinning and a pinned layer. It is the pinning layer that experiences spin configurational changes on successively cycling the pinned layer through its magnetic hysteresis loop. The left column of sketches displays a regular AF/FM EB system where the pinning layer is an antiferromagnet. The right column displays an all FM hard/soft bilayer system. Here the pinning layer is a hard ferromagnet (HL). A soft ferromagnetic layer (SL) is pinned via AF RKKY interaction to the HL. In both cases the pinning layer can be initialized into a domain state which deviates from its perfect long-range order. In the case of an AF pinning layer an AF domain state is created during the field-cooling procedure which, in compensated EB systems, induces magnetization in the AF pinning layer and with that AF interface magnetization \(S_{\rm AF}\) giving rise to EB in accordance with Eq. (6.1). Subsequently cycled hysteresis loops trigger spin configurational changes in the pinning layer which drive the antiferromagnet closer toward perfect long-range order characterized by the equilibrium AF order parameter \(\eta_{\rm e}\). The latter approach toward a new quasi-equilibrium spin configuration is accompanied by a decay of \(S_{\rm AF}\) with increasing number of loops, n. The degrading AF interface magnetization reduces the magnitude of the EB field. This process resembles the training effect. It is again in accordance with Eq. (6.1) when a static interface magnetization is generalized into \(S_{\rm AF}=S_{\rm AF}(n)\).

Fig. 6.6
figure 6_6_150780_1_Enfigure 6_6_150780_1_En

Comparison of the training effect in an AF/FM and a HL/SL heterostructure. The left column depicts three sketches of an AF/FM EB heterostructure after initializing EB (top), after the first (middle), and after a very large number of hysteresis loops (bottom). The nonequilibrium AF domain state carries magnetization to the interface (horizontal line). Neighboring spin pairs with non-compensating moment contributing to \(S_{\rm AF}\) are highlighted. The quasi-equilibrium state reflects the asymptotic approach of nearly perfect AF long-range order. \(S_{\rm AF}\) is reduced and so is the EB field. The right column depicts sketches of a HL/SL heterostructure after initializing a FM domain state (top), after the first SL hysteresis loop (middle), and after a very large number of hysteresis loops (bottom). The nonequilibrium FM domain state reduces the HL interface magnetization. The latter recovers on subsequent cycling when the domain state asymptotically approaches nearly perfect FM long-range order

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An analogous scenario takes place in the exchange-coupled HL/SL bilayers [3]. Here the initialization of the out-of-equilibrium domain state of the pinning layer is isothermally achieved by applying a negative (positive) set field after positive (negative) saturation of the HL. The sketches in the right column of Fig. 6.6 illustrate the HL domain state after its initializing in a negative set field (top). In this state the overall magnetization of the HL is reduced with respect to its saturation value and with it the HL interface magnetization. Therefore, the HL/SL coupling energy is reduced when the HL is in a FM domain state and the bias field characterizing the loop shift of the SL hysteresis is low. Subsequent SL hysteresis loops trigger spin configurational changes in the HL. The latter approaches asymptotically a quasi-equilibrium state of increased magnetization \(M_{\rm e}\). The increase in HL magnetization gives rise to an increase in its interface magnetization which increases the bias field.

Figure 6.7 displays the evolution of the free energy \(\Delta F\) of the AF (upper left) and the FM HL (upper right) pinning layer when they change from their respective nonequilibrium states with order parameters \(\eta\) and M into the quasi-equilibrium states \(\eta_{\rm e}\) and \(M_{\rm e}\). The pinning layer domain states associated with \(\eta\) and M and the final states associated with \(\eta _{\rm e}\) and \(M_{\rm e}\) are sketched, respectively. A harmonic approximation (dashed parabolas) of the Landau free energy landscape of the double well type catches already the essentials necessary to describe the triggered relaxation toward equilibrium quantitatively. The left bottom frame of Fig. 6.7 shows the evolution, \(\mu _0 H_{{\rm{EB}}}\) vs. n, of the EB field of the \({\rm Cr}_2{\rm O}_3/{\rm Fe}\) heterostructure introduced previously. The solid squares are experimental data obtained from the hysteresis loop shifts measured at T=300 K by SQUID magnetometry. Open circles connected with eye-guiding lines are results of a single parameter fit of Eq. (6.5). The inset shows the 1st (solid diamonds), the 2nd (open diamonds), and the 10th (stars) hysteresis loop. The right bottom frame of Fig. 6.7 displays the evolution, \(\mu _0 H_{\rm B}\)vs. n, of the bias field of the all FM HL/SL system CoPtCrB 15 nm (HL)/Ru 0.7 nm/CoCr 3 nm (SL) [3]. The solid squares are experimental data obtained from the SL hysteresis loop shifts measured at T=395 K by SQUID magnetometry. The line represents a single parameter best fit of Eq. (6.4). Note that only integer values of n have physical meaning. The inset shows the 1st (solid diamonds), 2nd (open diamonds), and the 15th (stars) hysteresis loop of the SL. Despite the intuitive similarities between these two training phenomena the details of the EB and bias field evolution show significant differences in particular for the transition n=1 to n=2 but also for the asymptotic behavior.

Fig. 6.7
figure 6_7_150780_1_Enfigure 6_7_150780_1_En

Free energy ΔF vs. η (upper left graph) and M (upper right graph) for AF and FM pinning systems, respectively. Arrows assign sketches of the spin and domain structures of AF and FM (non)equilibrium states (η and M) η e and M e to the free energy graphs. Dashed lines show harmonic approximations of the Landau free energy landscape. The bottom left frame shows \(\mu _0 H_{{\rm{EB}}}\)vs. n for \({\rm Cr}_2{\rm O}_3\)/Fe. The solid squares are experimental data measured at T= 300 K. Open circles connected with eye-guiding lines are results of a single parameter fit of Eq. (6.4). The inset shows the 1st (solid diamonds), 2nd (open diamonds), and the 10th (stars) hysteresis loop. The bottom right frame displays \(\mu _0 H_{\rm{B}}\)vs. n of CoPtCrB 15 nm/Ru 0.7 nm/CoCr 3 nm. Solid squares are experimental data measured at T= 395 K. The line represents a single parameter best fit of Eq. (6.3). The inset shows the 1st (solid diamonds), 2nd (open diamonds), and 15th (stars) hysteresis loop of the SL.

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Next we briefly sketch how these differences are reflected in the theoretical description. The latter has been introduced in Ref. [26] and is based on a discretized form of the Landau–Khalatnikov equationFootnote 2 which reads [26, 3]

$$\frac{S_{{\rm AF}/{\rm HL}} (n + 1) - S_{{\rm AF}/{\rm HL}} (n)}{\tau}=-\frac{1}{\xi} \frac{\partial \Delta F}{\partial S_{{\rm AF}/{\rm HL}}}$$
((6.3))

for AF or ferromagnetic HL pinning systems, respectively. Figure 6.7 shows the harmonic approximation in the vicinity of the equilibrium used to obtain the simplest expressions for the free energy entering Eq. (6.2). In the case of all FM HL/SL systems the order parameter M is proportional to the interface magnetization \(S_{\rm HL}\) which leads with the help of Eq. (6.3) straightforward to the explicit expression [3]

$$\mu_0H_{\rm B}(n)=(K + 1)^{n-1} \left \{ \mu_0 H_{\rm B} (n=1) - K \mu_0 H_{\rm B}^{\rm e} \left[ \frac{(K+1)^{n+1}-1}{K(K+1)^{n-1}} - (K+2) \right]\right \}, $$
((6.4))

where \(0<K<1\) is the fitting parameter that determines the rate of change of the bias field approaching the equilibrium value \(\mu _0 H_{\rm{B}}^{\rm{e}}\) with exponential asymptotic.

The relation between the AF order parameter η and the interface magnetization \(S_{\rm AF}\) is somewhat more involved [27]. With the help of the corresponding free energy expression one obtains the implicit sequence [26,27,29]

$$\mu_0 \big(H_{\rm EB} (n+1) - H_{\rm EB}(n)\big)=-\gamma\big(\mu_0\big(H_{\rm EB}(n) - H_{\rm EB}^{\rm e}\big)\big)^3,$$
((6.5))

where the fitting parameter γ controls the rate of change of the EB field approaching the equilibrium EB field \(\mu _0 H_{{\rm{EB}}}^{\rm{e}}\) with a \(1/\sqrt n\) asymptotic behavior. The latter has been successfully applied very early as an empirical expression first suggested in Ref. [21]. Its consistency with the asymptotic behavior of Eq. (6.5) is a strong confirmation of the theoretical approach sketched here.

6.4.2 Tuning the Training Effect

From the previous discussion it can be concluded that any means affecting the initial spin structure of the pinning layer will have an impact on the training behavior. Therefore, one has to expect that training characteristics can be tuned with the help of magnetic fields strong enough to perturb the AF pinning layer. Recently it has been shown in NiO/Ni bilayers that a training behavior following Eq. (6.5) can be perturbed by a large out-of-plane magnetic field of about 1 T such that subsequent training data show a substantially accelerated training effect [101]. Brems et al. showed that it is possible in CoO/Co bilayers to partially reinduce the untrained state when measuring a hysteresis loop with an in-plane external field perpendicular to the cooling field [102]. A simultaneous isothermal quenching of training and EB has been achieved in the \({\rm Fe}/{\rm Cr}_2{\rm O}_3/{\rm Fe}\) trilayer introduced above. After field cooling from T=395 K in \(\mu_0H=50 {\rm mT}\) down to T = 20 K the system shows a large EB and EB training effect. However, when subjecting the system to large in-plane DC magnetic field of 7 T EB bias and with it EB training is completely quenched [103].

The AF order parameter of the pinning systems in regular AF/FM EB heterostructures does not directly couple to homogeneous applied magnetic fields. Therefore, all of the effects described above require either large fields or sources for significant imbalance in the sublattice magnetizations such as thermal excitation or imperfections giving rise to deviation from perfect AF long-range order. In all FM heterostructures the situation is different. Here, homogeneous applied magnetic fields are conjugate to the FM order parameter of the pinning system and can directly influence the magnetic state of the latter. A systematic field-induced variation of the training behavior has been demonstrated and the characteristics of the training effect have been directly related to the field-controlled magnetization of the pinning layer [3].

Equation (6.3) shows that training is a triggered relaxation phenomenon. Hence one has to expect that the magnetic sweep rate of the hysteresis loops also affects the relaxation dynamics of the pinning system. In accordance with this intuitive qualitative picture a dynamic enhancement of the exchange bias training effect has been observed in the same all FM bilayer discussed above [28]. A similar enhancement has been observed in regular AF/FM heterostructures as well [26].

6.5 Conclusion

This chapter reviewed selected modern attempts to tune the characteristics of a number of effects associated with the exchange bias phenomenon. The most prominent is certainly the emergence of an exchange bias field in field-cooled antiferromagnetic/ferromagnetic thin film heterostructures. Control of the exchange bias field has been reviewed with special emphasis on electrically controlled exchange bias. Its successful integration in novel device architectures promises tremendous impact on future spintronic applications. The exchange bias training effect is a less investigated field with many open questions on a microscopic level. A simple but powerful phenomenological description of the training effect has been introduced and its universality has been evidenced. In accordance with this intuitive understanding various attempts to control the exchange bias and its training effect have been reported.