**4. Ferromagnetic materials**

applied stress occur over the long length scales. After plastic deformation have been produced, high dislocation density regions, such as sub-grain boundaries, and low dislocation density regions, such as cell interiors, characterized by a different yield stress are formed in the system. Under mechanical loading, the internal stress of the high-dislocation density regions is higher than that in low-dislocation density regions. Upon unloading, the average internal stress is zero; at the same time, the stress in the high dislocation density regions is positive, while it is negative in the low dislocation density regions. As a consequence, during stress reversal (applied stress changes sign), plasticity occurs in the low dislocation density regions at a lower

**Figure 17.** Stress-strain curves of a NiTi SMAs under (a) a monotonic tension and compression and (b) tension-com‐

nder (a) a mono strain. After [78]

tonic tension an

nd compression a

and (b)

s in the ominant metals. cept of he long such as ed by a s of the Upon ocation . As a he low

tend to ry after defects rdering sionless

f metals reduces ved to play a do yclic deformed m LRIS). The conc ss occur over th density regions, s ors, characterize e internal stress density regions. n the high dislo density regions. ty occurs in th r yield stress.

nd solute atoms, crystal symmetr r symmetry of d he short-range or above the diffus

e yield stress of is effect is believ tonically and cy internal stress (L the applied stres gh dislocation d ch as cell interio nical loading, the w-dislocation d me, the stress in w dislocation d sign), plasticit e observed lower

as vacancies and comply with the hort-range order uring heating, th t temperatures a

.

non by which th applied.[79] Thi ur of both mono f a long range i stresses due to t een produced, hi sity regions, suc . Under mechan han that in low at the same tim ative in the low stress changes gives rise to the

a NiTi SMAs un ding within 4% s

o the phenomen very first stress -stress behaviou the presence of ns of the local s rmation have be dislocation dens d in the system. ns is higher th l stress is zero; while it is nega versal (applied lower stress and

t be present in th ge order symmet ged martensite (F metry of the mart uptly into a cub

In SMAs, the defects that might be present in the system, such as vacancies and solute atoms, tend to distribute following a short-range order symmetry, which will comply with the crystal symmetry after aging treatments.[80] In the aged martensite (Fig. 18a), the short-range order symmetry of defects conforms with the crystal symmetry of the martensite phase. During heating, the short-range ordering of defects does not change abruptly into a cubic symmetry at temperatures above the diffusionless martensitic transformation, because the diffusion of defects is a slow process (Fig. 18b). After aging, the short-range order symmetry of defects becomes conformed to the cubic crystal symmetry of the austenite parent phase (Fig. 18c). This phenomenon is referred to as the symmetry-conforming property of point defects in ferroe‐ lastic systems.[80] It should be mentioned that the short-range order of defects with martensitic symmetry can slightly distort the cubic lattice towards the martensitic symmetry, when the martensite is quickly heated up to the cubic phase. Such a lattice difference will produce a short range order-induced domain pattern in the cubic phase identical to the martensitic domain pattern. Thus, the symmetry property of point defects can give rise to the aginginduced two-way shape memory effect, namely the one observed in the aged martensitic phase

he system, such try, which will c Fig. 18a), the sh tensite phase. Du bic symmetry at

20

stress and gives rise to the observed lower yield stress.

strain curves of a ssion cyclic load

**hinger effect**  er effect refers to ite to that of the mmetrical strainosely related to that the variation fter plastic defor daries, and low stress are formed n density region average internal is positive, w during stress rev sity regions at a l **c mechanisms**  efects that might wing a short-rang s.[80] In the ag he crystal symm not change abr

pression cyclic loading within 4% strain. After [78].

**3.5. Microscopic mechanisms**

L b o s t m u m l b u c m

Liu *et al.* [78] h both monotonic of the martensit stress-strain curv the non-deform martensite twinn under stress. O martensite plate lattice defects, m boundaries. It w up to 4% may compression up mostly dislocati

have systematica and cyclic tensi e phase are diffe ves (Fig. 17). Tr med specimen t ning. Under tens On the other han es was observed mainly dislocati was suggested th y be predomina to 4%, the strai

ally studied tens ile/compressive erent under tens ransmission elec the martensite sion, the interfa nd, no migratio d in samples com ions, was obser hat in the marten antly related to in is likely cause

sion-compression loading. They fo ion and compres ctron microscopy variants are w ces between two on of the juncti mpressed up to rved inside both nsite phase the d o the migration ed by the genera

n behaviour of m ound that the def ssion, as evidenc y (TEM) observ well self-accom o variants are m ion planes betw 4% strain. Inste h the martensite deformation mec n of variant in ation and movem

martensitic NiTi formation mech ced by the asym vations revealed mmodated throug mobile and can m ween the neighb ead, a high den e twin bands an chanism under t nterfaces, while ment of lattice d

i under hanisms mmetric that in gh the migrate bouring nsity of nd twin tension under defects,

ons.

228 Ferroelectric Materials – Synthesis and Characterization

F t **3** T d r T L l s d h u d c d **3** I d a c o

Fig. 17. Stress-s tension-compres **3.4 The Bausch** The Bauschinge direction opposi role in the asym The effect is clo LRIS assumes t length scales. A sub-grain bound different yield s high-dislocation unloading, the a density regions consequence, d dislocation dens **3.5 Microscopic** In SMAs, the de distribute follow aging treatment conforms with th of defects does

of Au51Cd49 SMA.[80]

### **4.1. Distortions of M-H hysteresis loops**

In ferromagnetic materials, there is an intense direct exchange interaction between the magnetic moments of adjacent atoms due to the overlapping of atomic orbitals. The spins tend to align parallel each other and induce a spontaneous magnetization. The application of an alternating magnetic field produces a hysteretic magnetization-magnetic field (M-H) loop (Fig. 19). During the application of the very first magnetic field cycle, the ferromagnetic material becomes magnetized to the saturated state with a saturated magnetization (*M*s). By decreasing the magnetic field to zero, the magnetization decreases and shows a remanent magnetization (*M*r) at zero field. The magnetic field corresponding to zero magnetization represents the coercive magnetic field (*H*c).

In absence of biasing effects, the M-H hysteresis loops of ferromagnetic materials display certain symmetry characteristics with respect to both H- and M-axis. However, these can be lost consequently to the development of different type of biasing processes that give rise to the deformed and shifted M-H loops along both axes. In this context, two main mechanisms will be discussed; the *exchange bias effect* and the *coexistence of amorphous and crystalline phases.*

**Figure 19.** Magnetic M-H hysteresis loops of a ferromagnetic.

### **4.2. Exchange bias effect**

### *4.2.1. Shift of the M-H loop along the magnetic field axis*

The exchange bias effect can be observed in systems containing interfaces between a ferro‐ magnetic (FM) and an antiferromagnetic (AFM) phase, in which the Curie temperature (*T*c) of the FM is higher than the Neel temperature (*T*N) of the AFM.[81] When the FM-AFM system is cooled through the *T*N during the application of an external magnetic field (referred to as the cooling field *H*cf), a shift of the M-H hysteresis loop of the FM-AFM system along the magnetic field axis may appear. This defines the exchange bias field (*H*E).[82] The exchange bias phenomenon was firstly discovered by Meiklejohn and Bean in Co (ferromagnetic)-CoO (antiferromagnetic) system.[83] Later, the exchange bias has been observed in many other systems, including nanoparticles, inhomogeneous materials, single crystals and thin films containing FM-AFM, ferrimagnetic-FM and ferrimagnetic-AFM interfaces. The first intuitive mechanism proposed by Nogues *et al.* [84] to explain the exchange bias effect is schematically drawn in Fig. 20. In the range *T*N < *T* < *T*C, the cooling field induces the alignment of the FM spins along its direction (Fig. 20i), while at *T < TN*, the spins of the AFM phase arrange in an antiferromagnetic configuration (Fig. 20ii). Along the FM-AFM interface, the AFM spins tend to align ferromagnetically as they are influenced by those of the FM due to the FM exchange interaction (Fig. 20ii). When the magnetic field is reversed, the FM spins tend to reorient following the applied field, while the AFM spins remain unchanged due to the large AFM anisotropy (Figs. 20iii - 20v). The ferromagnetic interaction at the FM-AFM interface provides a strong restoring force on the FM spins reorientations, and thus a shift of the M-H loop is produced. Generally, when the interaction at the interface is ferromagnetic and the cooling field Hcf is applied along the positive direction, the exchange bias field *H*E is characterized as a negative exchange bias field and a shift of the M-H loop towards the negative direction is observed (Fig. 20).

**Figure 20.** Schematic of the exchange bias mechanisms in case of ferromagnetic interaction between the FM and AFM spins at the interface. After [82].

**4.2. Exchange bias effect**

observed (Fig. 20).

*4.2.1. Shift of the M-H loop along the magnetic field axis*

**Figure 19.** Magnetic M-H hysteresis loops of a ferromagnetic.

230 Ferroelectric Materials – Synthesis and Characterization

The exchange bias effect can be observed in systems containing interfaces between a ferro‐ magnetic (FM) and an antiferromagnetic (AFM) phase, in which the Curie temperature (*T*c) of the FM is higher than the Neel temperature (*T*N) of the AFM.[81] When the FM-AFM system is cooled through the *T*N during the application of an external magnetic field (referred to as the cooling field *H*cf), a shift of the M-H hysteresis loop of the FM-AFM system along the magnetic field axis may appear. This defines the exchange bias field (*H*E).[82] The exchange bias phenomenon was firstly discovered by Meiklejohn and Bean in Co (ferromagnetic)-CoO (antiferromagnetic) system.[83] Later, the exchange bias has been observed in many other systems, including nanoparticles, inhomogeneous materials, single crystals and thin films containing FM-AFM, ferrimagnetic-FM and ferrimagnetic-AFM interfaces. The first intuitive mechanism proposed by Nogues *et al.* [84] to explain the exchange bias effect is schematically drawn in Fig. 20. In the range *T*N < *T* < *T*C, the cooling field induces the alignment of the FM spins along its direction (Fig. 20i), while at *T < TN*, the spins of the AFM phase arrange in an antiferromagnetic configuration (Fig. 20ii). Along the FM-AFM interface, the AFM spins tend to align ferromagnetically as they are influenced by those of the FM due to the FM exchange interaction (Fig. 20ii). When the magnetic field is reversed, the FM spins tend to reorient following the applied field, while the AFM spins remain unchanged due to the large AFM anisotropy (Figs. 20iii - 20v). The ferromagnetic interaction at the FM-AFM interface provides a strong restoring force on the FM spins reorientations, and thus a shift of the M-H loop is produced. Generally, when the interaction at the interface is ferromagnetic and the cooling field Hcf is applied along the positive direction, the exchange bias field *H*E is characterized as a negative exchange bias field and a shift of the M-H loop towards the negative direction is However, it has been also understood that in some magnetic systems the interaction between the spins at the interfaces can be antiferromagnetic, meaning that below TN the AFM spins in the layer close to the interface are aligned antiparallel to the FM spins. In this case, the direction of the M-H loop shift depends on the strength of the cooling field. Nogues *et al.* [84] have found that in Fe (FM)-FeF2 (AFM) bilayers, the M-H hysteresis loops can shift along the direction of the cooling field, after a sufficiently large cooling field is applied (Fig. 21). It was proposed that in case of the antiferromagnetic interaction, when the system is cooled through *T*N, the spins of AFM at the interface tend to align antiparallel to those of FM. By applying a low cooling field along the positive direction, the FM spins will align along the positive direction, while the AFM spins will not switch. When the applied magnetic field is reversed, the FM spins will try to follow the field, while the AFM spins will oppose to that, and they will force the FM spins to be antiparallel to them, leading to a shift of the M-H loop towards left. However, when the positive cooling field is large enough, the AFM surface spins are forced to be parallel to the FM spins along the cooling field direction, and a positive exchange bias is developed. Therefore, when the applied field is reversed, the magnetization reversal is facilitated by the antiferromagnetic coupling of the spins at the interface, and a positive shift of the M-H loop is observed (Fig. 21).

In order to quantitatively describe the exchange bias effect, several phenomenological models have been proposed. The model developed by Meiklejohn [83] assumes that both the FM and the AFM are in single domain state, the FM-AFM interface is perfectly smooth, and the energy per unit interface area can be expressed as:

$$E = -HM\_{\rm PM}t\_{\rm PM}\cos\left(\theta - \beta\right) + K\_{\rm PM}t\_{\rm PM}\sin^2\left(\beta\right) + K\_{\rm AFM}t\_{\rm AFM}\sin^2\left(a\right) - I\_{\rm MT}\cos\left(\beta - a\right) \tag{8}$$

where *H* is the applied field, *M*FM is the saturation magnetization, *t*FM and *t*AFM are the thickness of FM and AFM layers respectively, *K*FM and *K*AFM are the anisotropy of FM and AFM layers

**Figure 21.** The exchange bias field as a function of the cooling field for FeF2-Fe bilayers with the FeF2 grown at different temperatures, square: 200°C, triangle: 250°C, and circle: 300°C. In-set: magnetization loops of the sample with FeF2 grown at 300°C for low (hollow circle) and high (solid circle) cooling field. After [84].

respectively, and *J*INT is the interface coupling constant. The terms *α*, *β* and *θ* represent the angles between *K*AFM and *M*AFM, *K*FM, and *M*FM, *K*FM and H, respectively.

By neglecting the FM anisotropy, which is much smaller than that of AFM, and by minimizing the energy with respect to *α* and *β*, the exchange bias is obtained as:

$$H\_E = \frac{J\_{\rm INT}}{M\_{\rm FM} t\_{\rm FM}} \tag{9}$$

However, the exchange bias *H*E calculated using the Eq. 9 is usually several orders of magni‐ tude larger than the value observed in the experiments. Malozemoff [85] proposed an exchange bias effect model based on the assumption of rough FM-AFM interfaces. A microscopically random exchange field at the interface due to the defects, roughness or lattice mismatch can give rise to a random field which produces a number of uncompensated spins at the interface, leading to the loop shift. It was assumed that the FM is in a single domain state, therefore, due to the presence of a random field the AFM system will split into domains in order to minimize the unidirectional anisotropy (i.e., one single stable configuration of FM spins). The model gives the following expression for the exchange bias field [85]:

$$H\_E = \frac{\mathfrak{D}z}{\pi^2 \mathcal{M}\_{\rm FM} t\_{\rm FM}} \sqrt{\frac{\mathcal{J}\_{\rm INT} K\_{\rm AF}}{a}} \tag{10}$$

where *z* is a constant in the order of unity related to the randomness degree of the interface and *a* is the lattice parameter of the FM lattice which was considered cubic. The exchange bias values estimated by this model are consistent with the experiments. However, the main drawback of the model is that the calculated bias depends on the defect concentration at the interface. Mauri *et. al.* [86] developed a model assuming that the domain walls develop in the AFM layer, which sets an upper limit on the exchange coupling energy in such a manner that it ultimately gives rise to a significantly smaller exchange bias than that provided by the Meiklejohn's model calculations [83]. Using a micromagnetic approach, Koon [87] has stated that the orientation of the FM spins should be perpendicular to the AFM magnetic easy axes in the ground state. Although this model can explain the coercivity enhancement in some systems, it fails to provide reasonable estimations of the exchange bias. Therefore, the devel‐ opment of more general models to more accurately quantify the exchange bias in different systems is still ongoing.

Another kind of asymmetry characterized by a sharp step in the upper branch of the hysteresis loop was reported in the exchange biased Co-CoO dot array system.[88] The hysteresis loops of the Co-CoO dot arrays with four different dot sizes are shown in Fig. 22. Together with a shift of the hysteresis loops along the H-axis due to the exchange bias, a peculiar anomaly can be observed in the upper branch of the M-H hysteresis loop. This loop distortion is considered to originate from the presence of an intermediate magnetization saturated state. Both loop features showed a strong dependence on the size of dots. The H-shift of the M-H loop reduces with decreasing the dot size due to a reduction of the exchange bias effect. The deformation in the upper branch of the M-H loop was attributed to a magnetostatic interaction between the dots. The interaction determines an intermediate saturation of the magnetization (indicated in Fig. 22 by an arrow), which hinders the magnetization reversal and lowers its rate. The stability of the intermediate magnetization is larger in the array with bigger dots (compare Figs. 22ad). For the dots with smaller size, the inter-dot magnetostatic interaction is weaker and the magnetization switching of each dot is less influenced by those of the surrounding dots. This determines a faster magnetization switching rate and an almost complete disappearance of the intermediate saturation magnetization (Fig. 22d).
