*4.2.2. Shift of the M-H loop along the magnetization axis*

respectively, and *J*INT is the interface coupling constant. The terms *α*, *β* and *θ* represent the

**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

By neglecting the FM anisotropy, which is much smaller than that of AFM, and by minimizing

*INT*

*FM FM*

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

*INT AF*

*FM FM <sup>z</sup> J K <sup>H</sup> M t a* <sup>2</sup>

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

2

*M t* <sup>=</sup> (9)

<sup>=</sup> (10)

angles between *K*AFM and *M*AFM, *K*FM, and *M*FM, *K*FM and H, respectively.

grown at 300°C for low (hollow circle) and high (solid circle) cooling field. After [84].

232 Ferroelectric Materials – Synthesis and Characterization

the energy with respect to *α* and *β*, the exchange bias is obtained as:

gives the following expression for the exchange bias field [85]:

*E*

p

*E*

*<sup>J</sup> <sup>H</sup>*

In some magnetic systems containing the FM-AFM interfaces, the M-H hysteresis loop can also shift along the magnetization axis. Generally, the M-shift of the M-H loop is caused by the presence of a pinned magnetization which cannot be switched by the applied external magnetic field. The pinned magnetization can be developed through different mechanisms, including: i) an incomplete spin reversal of the FM spins; ii) the presence of spin glass-like phases in FM material nearby the interface region; iii) the presence of uncompensated spins in the AFM layer, and iv) the spin canting in the AFM layer. An example of systems where a spin glass-like arrangement develops in the FM phase is represented by the composite made up of granular ferrimagnetic NiFe2O4 nanoparticles embedded in an antiferromagnetic NiO matrix.[89] Fig. 23a shows the M-H hysteresis loops of the system measured at 10 K after both the zero-field cooling (ZFC) and field cooling (FC) processes. In the latter process, the M-H hysteresis loop shows a field offset as well as a magnetization shift with the exchange bias linearly increasing with the vertical magnetization offset. The underlying mechanism of the switching asymmetry in these materials was suggested to originate from the spin glass-like

f the exchange b magnetostatic int e magnetization wers its rate. The ompare Figs. 22

bias effect. The teraction betwee (indicated in Fi e stability of the 2a-d). For the

deformation in t n the dots. The ig. 22 by an arr intermediate m dots with smal

the upper branch interaction deter row), which hind agnetization is l ller size, the in dot is less influ witching rate and . 22d).

h of the M-H loo rmines an interm ders the magnet larger in the arra nter-dot magnet uenced by those d an almost co

op was mediate tization ay with tostatic e of the omplete

magnetization sw gnetization (Fig

ines a faster m te saturation ma

ts. This determi f the intermediat

t a s r b

to a reduction o attributed to a m saturation of the reversal and low bigger dots (co

surrounding dot disappearance o

s d

F

Fig. 22. Hystere

esis loops of the

**he M-H loop alo**

**4.2.2 Shift of th**

**4**

w with permission n from Girgis *et a al*. [88]. Copyrig ght 2003, APS. **Figure 22.** Hysteresis loops of the Co-CoO square dot arrays with four different dot sizes. Reproduced with permission from Girgis *et al*. [88]. Copyright 2003, APS.

**ization axis** 

e dot arrays with

h four different

dot sizes. Repro

oduced

e Co-CoO square

**ong the magneti**

phase formed in NiFe2O4 due to the fine particle size and the consequent structural disorder at the interface regions. During cooling under a magnetic field in the temperature range Tirr < T < TC (Tirr is the temperature above which the exchange bias disappears), the FM spins align along the applied magnetic field, while the spins of the glass-like phase remain randomly oriented. As the temperature is lowered below Tirr, some of the net uncompensated spins of the glass-like phase also line-up along the field-cooling direction and produce a pinned magnetization state. During the field reversal, the latter does not follow the field, which is probably the reason for the vertical shift of the loop. At the same time, the frozen spins in the glass-like phase try to keep the spins of the ferrimagnetic NiFe2O4 along their original direction leading to a negative shift of M-H loop along the H-axis. I a p m r r A r a m l e In some magnet along the magne pinned magnetiz magnetization c reversal of the F region; iii) the p AFM layer. An represented by t an antiferromag measured at 10 latter process, th exchange bias li tic systems conta etization axis. G zation which ca can be develope FM spins; ii) the presence of unc example of syst the composite m gnetic NiO matr K after both th he M-H hysteres inearly increasin aining the FM-A Generally, the M nnot be switche ed through diff presence of spin compensated spi tems where a sp made up of granu rix.[89] Figure he zero-field coo sis loop shows a ng with the vert AFM interfaces, M-shift of the Med by the applie ferent mechanism n glass-like phas ins in the AFM in glass-like arra ular ferrimagneti 23a shows the oling (ZFC) and a field offset as tical magnetizati the M-H hystere H loop is cause d external magn ms, including: ses in FM materi layer, and iv) t angement develo ic NiFe2O4 nano M-H hysteresis d field cooling ( well as a magne ion offset. The u esis loop can als ed by the presen netic field. The i) an incomplet ial nearby the in the spin canting ops in the FM p oparticles embed s loops of the (FC) processes. etization shift w underlying mech so shift nce of a pinned te spin nterface g in the phase is dded in system In the with the hanism

The vertical shift of M-H hysteresis loops due to spin canting in the AFM layer was reported by Yuan *et al*. [90] for the Co/Ca2Ru0.98Fe0.02FeO4 (FM/AFM) heterostructure, where the Co layer was sputtered on the Ca2Ru0.98Fe0.02FeO4 single crystal. When the FM/AFM system was cooled down to 10 K in a magnetic field, a horizontal and a vertical shift of the M-H loop was observed. In particular, the shift along the H-axis is negative for a positive cooling field and positive for a negative cooling field. On the other hand, the shift along the M-axis is positive for a positive cooling field and negative for a negative cooling field (Fig. 23b). The Ca2Ru0.98Fe0.02FeO4 single crystal does not have uncompensated AFM spins, however, the magnetic moments of the Ru (Fe) ions in the B-site of the AFM oxide are canted and give rise to the net ferromagnetic o p of the switching phase formed in g asymmetry in n NiFe2O4 due t these materials to the fine parti 25 s was suggested cle size and the d to originate fro e consequent str om the spin gla ructural disorder ass-like r at the

n an antiferroma

agnetic

nge Tirr < T < TC lign along the a ed. As the temp hase also line-up the field revers

C (Tirr is applied perature p along sal, the

netic field in the as disappears), t e phase remain r ensated spins of ed magnetizatio bly the reason fo ase try to keep t ve shift of M-H

temperature ran the FM spins al randomly oriente the glass-like ph on state. During

les embedded in

N A C NiO matrix mea AIP Publishing Courtesy of S. J asured at 10 K. g LLC. (b) The . Yuan, after [90 Reproduced wi e M-H hysteres 0]. ith permission fr sis loops of Co from Tian *et al*. o/Ca2Ru0.98Fe0.02 [89]. Copyright 2FeO4 heterostr t 2008, ructure. **Figure 23.** (a) The M-H hysteresis loops of NiFe2O4 nanoparticles embedded in an antiferromagnetic NiO matrix meas‐ ured at 10 K. Reproduced with permission from Tian *et al*. [89]. Copyright 2008, AIP Publishing LLC. (b) The M-H hysteresis loops of Co/Ca2Ru0.98Fe0.02FeO4 heterostructure. Courtesy of S. J. Yuan, after [90].

e2O4 nanoparticl

is loops of NiFe

moments. In the first layer near the Co phase, these moments align parallel to those of the Co layer, but they cannot be readily switched when the magnetic field is reversed. As a conse‐ quence, the vertical shift of the M-H hysteresis loops is produced (Fig. 23b). Nogues *et al.* [91] studied the dependence of the exchange bias and magnetization shift on cooling field in the Fe-FeF2-Al system. They observed that for low cooling fields the vertical shift can be opposite to the cooling field suggesting that an antiferromagnetic coupling exists at the FM-AFM interface. Additionally, at low cooling fields, the vertical shift was negative when the thickness of the FeF2 layer was 200 nm and it was positive for the 100 nm layer. This behaviour was attributed to the roughness of the interface; smooth interface tends to induce a negative vertical shift, while rough interface usually leads to a positive shift. For large cooling fields, a positive vertical shift was observed in all cases. T *e* s t f f u A p w i m The vertical shif *et al*. [90] for sputtered on the 10 K in a magne the shift along t field. On the oth for a negative uncompensated AFM oxide are phase, these mo when the magne is produced (F magnetization s ft of M-H hyster the Co/Ca2Ru0 e Ca2Ru0.98Fe0.02 etic field, a horiz the H-axis is neg her hand, the shif cooling field AFM spins, how canted and give oments align pa etic field is reve ig. 23b). Nogu hift on cooling resis loops due to 0.98Fe0.02FeO4 (F 2FeO4 single cry zontal and a vert gative for a pos ft along the M-a (Fig. 23b). The wever, the magn e rise to the net arallel to those o rsed. As a conse ues *et al.* [91] field in the Feo spin canting in FM/AFM) heter ystal. When the F tical shift of the itive cooling fie axis is positive fo e Ca2Ru0.98Fe0.0 netic moments o ferromagnetic m of the Co layer, equence, the ver studied the de -FeF2-Al system n the AFM layer rostructure, whe FM/AFM system M-H loop was o eld and positive or a positive coo 02FeO4 single c of the Ru (Fe) io moments. In the , but they canno rtical shift of the pendence of th m. They observed r was reported by ere the Co laye m was cooled d observed. In part for a negative c oling field and ne crystal does no ons in the B-site first layer near ot be readily sw e M-H hysteresis he exchange bia d that for low c y Yuan er was own to rticular, cooling egative ot have e of the the Co witched s loops as and cooling

The vertical offsets of M-H hysteresis loops were also observed in magnetic systems without exchange bias. Watanabe *et al.* [92] reported on the shift of the magnetization hysteresis loop along the magnetization axis in the antiferromagnetic LaFeO3 when cooled in a magnetic field below the Neél temperature. The shift phenomenon was explained by the fact that the fieldcooled LaFeO3 always shows a weak ferromagnetism with a small remanent magnetization, which cannot be reversed by the external field. A vertically shifted M-H hysteresis loop was also observed in Co2VO4 at 4.2 K when cooled in a small magnetic field.[93] Due to the high anisotropy and magnetic hardness of cobalt in spinel lattices, the spins of cobalt ions align along the cooling field direction when cooled through the Curie temperature and, at low temperatures, they preserve the orientation producing the shift of M-H loop along the magnetization axis. f c n T n a e t fields the vertic coupling exists negative when t This behaviour negative vertica a positive vertic The vertic exchange bias. W the magnetizatio cal shift can be at the FM-AFM the thickness of was attributed t l shift, while rou al shift was obse cal offsets of M Watanabe *et al.* on axis in the an e opposite to th M interface. Add the FeF2 layer w to the roughness ugh interface usu erved in all cases -H hysteresis lo [92] reported o ntiferromagnetic 26 e cooling field ditionally, at low was 200 nm and s of the interfac ually leads to a p s. oops were also o on the shift of th c LaFeO3 when suggesting that w cooling fields, d it was positive ce; smooth inter positive shift. Fo observed in mag he magnetization cooled in a ma t an antiferroma , the vertical sh e for the 100 nm rface tends to in or large cooling gnetic systems w n hysteresis loop agnetic field bel agnetic hift was m layer. nduce a g fields, without p along low the

### **4.3. Effect of the amorphous-crystalline phases coexistence**

phase formed in NiFe2O4 due to the fine particle size and the consequent structural disorder at the interface regions. During cooling under a magnetic field in the temperature range Tirr < T < TC (Tirr is the temperature above which the exchange bias disappears), the FM spins align along the applied magnetic field, while the spins of the glass-like phase remain randomly oriented. As the temperature is lowered below Tirr, some of the net uncompensated spins of the glass-like phase also line-up along the field-cooling direction and produce a pinned magnetization state. During the field reversal, the latter does not follow the field, which is probably the reason for the vertical shift of the loop. At the same time, the frozen spins in the glass-like phase try to keep the spins of the ferrimagnetic NiFe2O4 along their original direction

**Figure 22.** Hysteresis loops of the Co-CoO square dot arrays with four different dot sizes. Reproduced with permission

**ization axis**  AFM interfaces, M-shift of the Med by the applie ferent mechanism n glass-like phas ins in the AFM in glass-like arra ular ferrimagneti 23a shows the oling (ZFC) and a field offset as tical magnetizati s was suggested cle size and the

e dot arrays with ght 2003, APS.

h four different

dot sizes. Repro

oduced

so shift nce of a pinned te spin nterface g in the phase is dded in system In the with the hanism ass-like r at the

esis loop can als ed by the presen netic field. The i) an incomplet ial nearby the in the spin canting ops in the FM p oparticles embed s loops of the (FC) processes. etization shift w underlying mech om the spin gla ructural disorder

the M-H hystere H loop is cause d external magn ms, including: ses in FM materi layer, and iv) t angement develo ic NiFe2O4 nano M-H hysteresis d field cooling ( well as a magne ion offset. The u d to originate fro e consequent str

e Co-CoO square *al*. [88]. Copyrig

**ong the magneti** aining the FM-A Generally, the M nnot be switche ed through diff presence of spin compensated spi tems where a sp made up of granu rix.[89] Figure he zero-field coo sis loop shows a ng with the vert these materials to the fine parti

The vertical shift of M-H hysteresis loops due to spin canting in the AFM layer was reported by Yuan *et al*. [90] for the Co/Ca2Ru0.98Fe0.02FeO4 (FM/AFM) heterostructure, where the Co layer was sputtered on the Ca2Ru0.98Fe0.02FeO4 single crystal. When the FM/AFM system was cooled down to 10 K in a magnetic field, a horizontal and a vertical shift of the M-H loop was observed. In particular, the shift along the H-axis is negative for a positive cooling field and positive for a negative cooling field. On the other hand, the shift along the M-axis is positive for a positive cooling field and negative for a negative cooling field (Fig. 23b). The Ca2Ru0.98Fe0.02FeO4 single crystal does not have uncompensated AFM spins, however, the magnetic moments of the Ru (Fe) ions in the B-site of the AFM oxide are canted and give rise to the net ferromagnetic

25

leading to a negative shift of M-H loop along the H-axis.

esis loops of the n from Girgis *et a*

**he M-H loop alo** tic systems conta etization axis. G zation which ca can be develope FM spins; ii) the presence of unc example of syst the composite m gnetic NiO matr K after both th he M-H hysteres inearly increasin g asymmetry in n NiFe2O4 due t

from Girgis *et al*. [88]. Copyright 2003, APS.

t a s r b i s d

to a reduction o attributed to a m saturation of the reversal and low bigger dots (co interaction is we surrounding dot disappearance o

f the exchange b magnetostatic int e magnetization wers its rate. The ompare Figs. 22 eaker and the m ts. This determi f the intermediat

234 Ferroelectric Materials – Synthesis and Characterization

bias effect. The teraction betwee (indicated in Fi e stability of the 2a-d). For the magnetization sw ines a faster m te saturation ma

deformation in t n the dots. The ig. 22 by an arr intermediate m dots with smal witching of each magnetization sw gnetization (Fig

the upper branch interaction deter row), which hind agnetization is l ller size, the in dot is less influ witching rate and . 22d).

h of the M-H loo rmines an interm ders the magnet larger in the arra nter-dot magnet uenced by those d an almost co

op was mediate tization ay with tostatic e of the omplete i t m i t l s a

interface regions the temperature magnetic field, w is lowered below the field-cooling latter does not f same time, the f along their origi

s. During coolin above which th while the spins w Tirr, some of t g direction and follow the field, frozen spins in th inal direction lea

ng under a magn he exchange bia of the glass-like the net uncompe produce a pinn which is probab he glass-like ph ading to a negativ

e M-H hysteresi

F

Fig. 23. (a) The

F w

Fig. 22. Hystere with permission

**4.2.2 Shift of th** In some magnet along the magne pinned magnetiz magnetization c reversal of the F region; iii) the p AFM layer. An represented by t an antiferromag measured at 10 latter process, th exchange bias li of the switching phase formed in

**4** I a p m r r A r a m l e o p

> A distinct shift of the M-H hysteresis loop was observed in magnetic systems subjected to a conventional annealing at a temperature below their crystallization temperature. Ohta *et al.*

 [94] studied the Fe5Co70Si10B15 amorphous alloys and reported that the as-quenched sample shows a symmetric hysteresis loop (Fig. 24a). After annealing at 420 °C, the hysteresis loop becomes pinched (Fig. 24b) and annealing under a magnetic field of 10 kOe leads to a signif‐ icant shift of the loop along the H-axis (Fig. 24c).

**Figure 24.** The magnetic induction (B) vs. magnetic field (H) hysteresis loops of amorphous alloys Fe5Co70Si10B15: (a) asquenched, (b) annealed and (c) annealed in a magnetic field. These images were published in Ohta *et al*. [94]. Copy‐ right 1980, Elsevier.

Kohmoto *et al.* [95] have reported on the loop shift of the Fe5Co70Si10B15 amorphous alloys annealed at 180 °C for different time. The shift along the field axis was found to increase with the increase of annealing time. Rivas *et al.* [96] observed the shifted M-H loops in the metallicglass Co66Si16B12Fe4Mo2 ribbons annealed at 510 °C and "pre-magnetized" under a magnetic field of 400 kA/m at room temperature. A large negative field offset was observed when the initial applied field was parallel to the pre-magnetizing field, and the field shift was positive as the initial applied field was antiparallel to the pre-magnetizing field. However, no shift of M-H hysteresis loops was observed when the magnetic field was applied perpendicular to the direction of the pre-magnetizing field. The results suggest that the shift of M-H hysteresis loops in the annealed pre-magnetized ribbons originate from the coexistence of the amorphous and crystalline phases. After annealing, the amorphous matrix contains partially crystallized particles which are much magnetically harder if compared to the amorphous matrix. When subjected to a relatively high pre-magnetizing field, the magnetic moments of the particles tend to align along the applied field direction. Due to the large magnetic anisotropy, the magnetic spins of the hard-magnetic particles remain unchanged under the ac applied magnetic field used for the reorientation of the magnetic moments in the soft matrix. Therefore, a strong unidirectional dipolar magnetic field is formed, which can exert a strong restoring force on the reorientations of magnetic moments of the soft matrix, and thus produces a shift of M-H hysteresis loops.

### **4.4. Magnetic aging**

The term "magnetic aging" indicates the time-dependent changes in the magnetization of ferromagnetic materials. These changes are also commonly referred to as "magnetic viscosity" or "magnetic after-effects". The typical relaxation time spans several orders of magnitude; it ranges from less than a second in superparamagnetic particles to millions of years in magnetic rocks.[97] Various mechanisms that contribute to the magnetic aging have been proposed, including i) thermal fluctuations, ii) the diffusion after-effect of ferrous ions and vacancies, iii) chemical modifications.[98] The magnetic aging originated from thermal fluctuations is driven by the thermally activated jumps over energy barriers of domain walls. This process deter‐ mines the time-, temperature- and magnetic field-dependence of the remanent magnetization. The diffusion of point defects, impurity atoms or electrons towards domain walls can cause the departure of the domain wall energy from a minimum which determines the displacement of the domain walls in a new equilibrium configuration. This type of magnetic aging mecha‐ nism is usually defined as "diffusion after-effect". It was first observed in iron containing carbon or nitrogen impurities.[98]
