**Proteresis of Core-Shell Nanocrystals: Investigation through Theoretical Simulation and Experimental Analysis**

Jhong-Yi Ji and Sheng Yun Wu

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/62398

#### **Abstract**

A study of proteresis (inverted hysteresis) in core-shell nanocrystals is presented. A coreshell anisotropic energy (CSAE) model is established to describe the observed proteret‐ ic behavior in Ni/NiO core-shell nanocrystals. The magnetic compositions of core-shell nanocrystals can be selected for ferromagnetic, antiferromagnetic, or paramagnetic materials where the exchange intercoupling between them results in both a large effective anisotropic energy and intercoupling energy. Simulation of the magnetization of coreshell nanocrystals reveals the existence of an exchange in the intercoupling energy between the interface of the core and shell moments that, surprisingly, is tuneable in both hysteresis and proteresis. Observations have shown a distinct proteresis, which is related to the spinflip and exchange intercoupling energy between Ni and NiO. Our approach shows that the processing-dependent technology plays an important role when the grain size decreases to the order of nanometers and when the magnets are reduced from the single domain to core-shell domain. Integrated studies of process-dependent, theoretical modeling and core-shell nanocrystal fabrication technology will lead to more encourag‐ ing development in the overunity industry.

**Keywords:** proteresis, hysteresis, core-shell nanocrystals, exchange inter-coupling, spin flipping

## **1. Introduction**

Traditionally, the change in free energy that occurs at a constant temperature is attributed to the mechanical work by the system and the magnetic work by the applied magnetic field. The

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integral of the magnetic work gives the energy of the magnetization, which is always positive, because work must be done in the system in order for the magnetic field to establish the magnetization. Consequently, in an inverted hysteresis, the so-called proteretic system, the area of the inverted hysteresis loop ∮*HdM* represents the energy dissipation for each cycle. At the nanoscale, the various temperatures and size of the hysteresis *M*(*H*) loop influence the loop direction, such as whether it is counterclockwise (hysteresis) or clockwise (inverted hystere‐ sis). Interestingly, the modulation of hysteresis and inverted hysteresis breaks the conception of the magnetic system [1]. Recently, a clockwise *M*(*H*) loop has also been observed in CrO2/ Cr2O3 and Co/CoO core-shell particle systems. Zheng et al. [2] observed a series of inverted hysteresis loops between 261 and 292 K in CrO2 [ferromagnetic (FM)] core particles coated with a Cr2O3 [antiferromagnetic (AFM)] surface layer. Wei et al. [3] observed an inverted hysteresis loop in Co (FM) with a core size of 3–4 nm and a CoO (AFM) shell with a thickness of ~3 nm, only giving an example at a temperature *T*=70 K. When the Co core size is smaller than 3 nm and larger than 4 nm, the *M*(*H*) loop leads to an ordinary hysteresis loop. They provided a possible explanation that the proteretic behavior and its size effect dependence reflect the magnetic interaction between various anisotropies and exchange intercouplings in the Co and CoO phases [3]. In another work, Ohkoshi et al. [4] reported the observation of an inverted hysteresis loop that can be reasonably explained by competition between the spin-flip transi‐ tions and the uniaxial magnetic anisotropy. The competing anisotropy theory has been extended to systems involving competing cubic and uniaxial anisotropies [5] and two competing uniaxial anisotropies [6]. In the present work dealing with Ni/NiO core-shell nanoparticles, we ignore the influence of intracoupling to simplify the complex interactions and only discuss the intercoupling between Ni core and NiO shell. We design a core-shell anisotropic energy (CSAE) model, tuning the competition between the magnetic anisotropy and exchange intercoupling in the system, to describe the observation of hysteretic and proteretic behaviors in Ni/NiO coreshell nanoparticles. Recently, the tuning of the exchange intercoupling in core-shell magnetic nanocrystals has received intense attention because it allows the control of the interfaces in nanoscale building blocks and inverted hysteresis. The magnetic compositions of core-shell nanocrystals can be selected as FM, AFM, or paramagnetic (PM) materials, so that the ex‐ change intercoupling between them results in a large effective anisotropic energy [7] and intercoupling energy. We also integrate some magnetic models including the Langevin function, the CSAE, and the Stoner-Wohlfarth (SW) model, providing a convenient simulation for the description of the hysteresis and proteresis loops.

## **2. Stoner-Wohlfarth model**

The evolution of the magnetic theory model for the description of the hysteresis loop starts with the SW model [8], which was developed by Edmund Clifton Stoner and Erich Peter Wohlfarth in 1948. The basic hypotheses in this SW model include the following: (a) singledomain particles, (b) spherical geometry, (c) magnetic anisotropy along the easy axis (EA), and (d) non- or weak-interaction between small particles. Additionally, the sources of magnetic anisotropy can be divided into three parts: magnetocrystalline anisotropy, shape anisotropy, and exchange anisotropy [9]. In the first case, magnetocrystalline anisotropy, the preferential direction for magnetization, arises from the atomic structure of the crystal. In shape anisotropy, it originates from the nonuniform spherical particle shapes, which cause the demagnetizing field to be unequal in all directions, creating one or more EAs. Third, in exchange anisotropy, magnetization originates from the interaction between AFM and FM materials. The SW model is considered to relate to uniaxial magnetic anisotropy, which may originate from magneto‐ crystalline anisotropy and shape anisotropy. As a magnetic field is applied to a single-domain particle, the moment *M* of a single-domain particle is subjected to two competing forces: one is due to a uniaxial anisotropy characterized by *K* toward an EA and the other is due to an external magnetic field *Ha*, as shown in **Figure 1a**. The total energy is the sum of the anisotropic energy *E*A and the Zeeman energy *EZ* = −*M*⋅*Ha*. At absolute zero *T*=0 K, the energy is then expressed as

integral of the magnetic work gives the energy of the magnetization, which is always positive, because work must be done in the system in order for the magnetic field to establish the magnetization. Consequently, in an inverted hysteresis, the so-called proteretic system, the area of the inverted hysteresis loop ∮*HdM* represents the energy dissipation for each cycle. At the nanoscale, the various temperatures and size of the hysteresis *M*(*H*) loop influence the loop direction, such as whether it is counterclockwise (hysteresis) or clockwise (inverted hystere‐ sis). Interestingly, the modulation of hysteresis and inverted hysteresis breaks the conception of the magnetic system [1]. Recently, a clockwise *M*(*H*) loop has also been observed in CrO2/ Cr2O3 and Co/CoO core-shell particle systems. Zheng et al. [2] observed a series of inverted hysteresis loops between 261 and 292 K in CrO2 [ferromagnetic (FM)] core particles coated with a Cr2O3 [antiferromagnetic (AFM)] surface layer. Wei et al. [3] observed an inverted hysteresis loop in Co (FM) with a core size of 3–4 nm and a CoO (AFM) shell with a thickness of ~3 nm, only giving an example at a temperature *T*=70 K. When the Co core size is smaller than 3 nm and larger than 4 nm, the *M*(*H*) loop leads to an ordinary hysteresis loop. They provided a possible explanation that the proteretic behavior and its size effect dependence reflect the magnetic interaction between various anisotropies and exchange intercouplings in the Co and CoO phases [3]. In another work, Ohkoshi et al. [4] reported the observation of an inverted hysteresis loop that can be reasonably explained by competition between the spin-flip transi‐ tions and the uniaxial magnetic anisotropy. The competing anisotropy theory has been extended to systems involving competing cubic and uniaxial anisotropies [5] and two competing uniaxial anisotropies [6]. In the present work dealing with Ni/NiO core-shell nanoparticles, we ignore the influence of intracoupling to simplify the complex interactions and only discuss the intercoupling between Ni core and NiO shell. We design a core-shell anisotropic energy (CSAE) model, tuning the competition between the magnetic anisotropy and exchange intercoupling in the system, to describe the observation of hysteretic and proteretic behaviors in Ni/NiO coreshell nanoparticles. Recently, the tuning of the exchange intercoupling in core-shell magnetic nanocrystals has received intense attention because it allows the control of the interfaces in nanoscale building blocks and inverted hysteresis. The magnetic compositions of core-shell nanocrystals can be selected as FM, AFM, or paramagnetic (PM) materials, so that the ex‐ change intercoupling between them results in a large effective anisotropic energy [7] and intercoupling energy. We also integrate some magnetic models including the Langevin function, the CSAE, and the Stoner-Wohlfarth (SW) model, providing a convenient simulation for the

The evolution of the magnetic theory model for the description of the hysteresis loop starts with the SW model [8], which was developed by Edmund Clifton Stoner and Erich Peter Wohlfarth in 1948. The basic hypotheses in this SW model include the following: (a) singledomain particles, (b) spherical geometry, (c) magnetic anisotropy along the easy axis (EA), and (d) non- or weak-interaction between small particles. Additionally, the sources of magnetic anisotropy can be divided into three parts: magnetocrystalline anisotropy, shape anisotropy,

description of the hysteresis and proteresis loops.

**2. Stoner-Wohlfarth model**

148 Magnetic Materials

$$E = E\_{\mathbb{Z}} + E\_A = -MH\_a \cos \theta + K \sin^2(\theta - \varphi),\tag{1}$$

where *M* indicates the saturation magnetization, *θ* indicates the deviation of *M* from the field direction, *φ* indicates the deviation of the corresponding easy axis (EA) from the field direction, and *K* is the corresponding uniaxial magnetic anisotropy strengths. For the moment, we select an angle *θm*, which is oriented toward either a stable or a bistable state, giving the minimum energy *Em* and magnetic field *Hf* , respectively. The total energy of *E*(*θ*, *Ha*) may be derived by solving the equations

$$
\left(\frac{\partial E}{\partial \theta}\right)\_{\theta=\theta\_u} = 0 \quad \text{and} \quad \left(\frac{\partial^2 E}{\partial \theta^2}\right)\_{\theta=\theta\_u} > 0,\tag{2}
$$

Moreover, the angle of a magnetic moment dependent on total energy exhibits an energy barrier in a selective applied magnetic field *Ha*, such as *Ha*=0, *Ha*>0, or *Ha*>*Hf* , as shown in **Figure 1b**. When the field *Ha* is zero, the energy barrier exhibits a saddle shape, giving both stable solutions. When the field *Ha* increases, the energy barrier between the stable and bistable states decreases, implying that the field *Ha* provides Zeeman energy *EZ* to compete with the aniso‐ tropic energy *EA*. The spin direction in the bistable state gradually becomes more unstable. When the field *Ha* reaches *Hf* , the energy barrier between the stable and bistable states disappears. In this condition, the Zeeman energy *EZ* is larger than the anisotropic energy *EA*. The moment of a single-domain particle will then switch from spin-up to spin-down and orient itself toward the direction of the applied field, giving only one solution. As for the initial state of the magnetic moment *M*, it will make a choice between the stable and bistable states following a pattern like water flowing downhill. The ideal is to establish a loop process, which can be used to describe a transitional hysteresis loop. Its magnetization can readily be calcu‐ lated by evaluating *M*(*Ha*)=*Ms*cos*θm*. Thus, the SW model provides the basis for the description of the hysteresis loop, which can be further extended to a CSAE system.

**Figure 1.** (a) Schematic illustration of a single-domain particle with the magnetic moment (*M*) and EA under an ap‐ plied field (*Ha*). (b) Variation of the energy landscape with angle *θ* in a selective applied magnetic field, such as *Ha*=0, *Ha*>0, and *Ha*>*Hf* .

#### **3. Core-shell anisotropic energy model**

Neel [10] described the thermal fluctuations of the magnetic moment of a single-domain FM particle and its decay towards thermal equilibrium, neglecting the contribution of magnetic anisotropy. The further development of the thermal activated magnetization is described by the Brown model [11]. Fluctuations in the thermal energy *kBT* cause continual changes in the orientation of the moment of an individual nanoparticle. An ensemble of such nanoparticles maintains a distribution of orientation characteristics of statistical equilibrium. The thermal fluctuations are important in superparamagnetism, which can be described by a "Langevin function," providing a description of the thermal activated magnetization. A number of models have been established [12–14] to explain the phenomena of inverted hysteresis for germanium nanostructures [15] and hysteresis for Mackay icosahedral gold nanoparticles [16], using the competing anisotropy model, and based on the SW model [8]. In the case studied here, the general picture of the core-shell system established in the Ni core (FM) coated with a NiO shell (AFM) can be described as two nonidentical phases [17], as shown in **Figure 2a**. Usually, in the macroscopic view, most of the native oxide shells of transition metals (Co, Ni) are either AFM or FM. The magnetism on the FM core has been either neglected or reported to be effective only at low temperatures [18, 19].

**Figure 2.** (a and b) Schematic illustration of a core-shell particle with two interacting magnetic moments *m*1 and *m*2 and EAs EA1 and EA2 under an applied field *Ha*. They couple at an angle (*q*1-*q*2) and exchange intercoupling energy *EJ* between the surface and inner moments.

**Figure 1.** (a) Schematic illustration of a single-domain particle with the magnetic moment (*M*) and EA under an ap‐ plied field (*Ha*). (b) Variation of the energy landscape with angle *θ* in a selective applied magnetic field, such as *Ha*=0,

Neel [10] described the thermal fluctuations of the magnetic moment of a single-domain FM particle and its decay towards thermal equilibrium, neglecting the contribution of magnetic anisotropy. The further development of the thermal activated magnetization is described by the Brown model [11]. Fluctuations in the thermal energy *kBT* cause continual changes in the orientation of the moment of an individual nanoparticle. An ensemble of such nanoparticles maintains a distribution of orientation characteristics of statistical equilibrium. The thermal fluctuations are important in superparamagnetism, which can be described by a "Langevin function," providing a description of the thermal activated magnetization. A number of models have been established [12–14] to explain the phenomena of inverted hysteresis for germanium nanostructures [15] and hysteresis for Mackay icosahedral gold nanoparticles [16], using the competing anisotropy model, and based on the SW model [8]. In the case studied here, the general picture of the core-shell system established in the Ni core (FM) coated with a NiO shell (AFM) can be described as two nonidentical phases [17], as shown in **Figure 2a**. Usually, in the macroscopic view, most of the native oxide shells of transition metals (Co, Ni) are either AFM or FM. The magnetism on the FM core has been either neglected or reported

*Ha*>0, and *Ha*>*Hf*

150 Magnetic Materials

.

**3. Core-shell anisotropic energy model**

to be effective only at low temperatures [18, 19].

Because most of the AFM spins are arranged collinearly and the number of sublattices are the same, the resulting magnetization is zero and the coupling vanishes to the first order [20, 21]. However, in the present core-shell system, the existence of correlated magnetic exchange intercoupling between the core-shell moments can be treated as an "exchange spring," which can create an exchange field, inducing a canting of the AFM surface layer that "cannot" be neglected. The interpretation of the occurrence of exchange intercoupling between the surface and the core further indicates that magnetism develops not only in the core atoms but also extends into the surface ones. Treating the surface atoms and the inner atoms gives rise to a net surface macromoment *m*2 and a net inner macromoment *m*1 that will couple at angles (*θ*1 *θ*2) and will lead to the development of exchange intercoupling energy *EJ* between the surface and inner moments. We believe that the associated magnetism is indeed an intrinsic property of Ni/NiO core-shell nanoparticles. The extrinsic factors, such as the shape and temperature, are thus neglected in the considerations of the model. Conducting quantitative analysis of the intercoupling of the core-shell system, and neglecting the contribution of intracoupling, we can express the CSAE *E* triggered by various magnetic applied fields *Ha* as

$$\begin{split} E &= -m\_1 V\_1 H\_a \cos \theta\_1 - m\_2 V\_2 H\_a \cos \theta\_2 \\ &+ k\_1 V\_1 \sin^2(\theta\_1 - \phi\_1) + k\_2 V\_2 \sin^2(\theta\_2 - \phi\_2) + E\_J \cos(\theta\_1 - \theta\_2) \end{split} \tag{3}$$

where *V*1 and *V*2 indicate the volume of the Ni core and NiO shell for each nanoparticle, *θ*<sup>1</sup> and *θ2* indicate the deviations of *m*1 and *m*2 from the field direction, *φ<sup>1</sup>* and *φ2* indicate the deviations of the corresponding EAs from the field direction, and *k*1 and *k*2 are the corresponding uniaxial magnetic anisotropy strengths, as shown in **Figure 2b**. *EJ* is defined as *σ*ex*Vs*, where *σ*ex indicates the exchange intercoupling energy per area and *Vs* indicates the interfacial area between the Ni core and NiO shell. The magnetization can readily be calculated by solving *M*(*Ha*)=*m*1cos*θ*1m + *ms*cos*θ*2m, where *θ*1m and *θ*2m are the *θ*1 and *θ*2, respectively, that minimize *Em* at each *Ha*. Depending on the relative strength between *m*1 and *m*2 and between *k*1 and *k*2, the value of *EJ* and the bistable energy of *E*(*θ*1m,*θ*2m,*Ha*) may be derived by solving the following equations: *∂E/∂θ*1=*∂E*/*∂θ*1=0 [15, 22]. **Figure 3** shows an example of the dependence of the energy *E* of angles *θ*1 and *θ*2 on the selection of the applied magnetic field, such as *Ha*=0, *Ha*>0, or *Ha*>*Hf* , giving a two-dimensional energy barrier. Here, different colors are used to differentiate the energy intensity, with red indicating high energy and purple low energy. When the field *Ha* is zero, the energy barrier exhibits two whirlpools, indicating stable and bistable solutions. A two-dimensional and high-energy barrier appears between the two whirlpools. With an increase of the field *Ha*, the energy barrier between the stable and bistable states decreases, implying that the field *Ha* provides Zeeman energy to compete with the magnetic anisotropies and the exchange intercoupling. The magnetic directions of the core and shell moments in the bistable state gradually become more unstable. When the field *Ha* reaches *Hf* , the energy barrier between the stable and bistable states disappears. In this condition, the Zeeman energy is larger than the magnetic anisotropies and the exchange intercoupling. The core and shell moments will switch from bistable to stable states, and the core moment (FM) then becomes oriented toward the applied field direction, giving only one solution. The initial state of the core and shell moments *m*1 and *m*<sup>2</sup> can assume either a stable or bistable state following a pattern like water flowing downhill. Observe the energy barrier along the dashed line in **Figure 4**. The simplified energy barrier clearly exhibits stable and bistable states, which is similar to the SW model. The exchange intercoupling between the core and shell moments in the CSAE model also plays an important role on the *M*(*H*) loop process for the modulation of the hysteresis and inverted hysteresis loops. Huang et al. [15] reported the *M*(*H*) loop processes and the modu‐ lation of the magnetic anisotropies and exchange intercoupling describing proteresis.

**Figure 3.** Variation of the energy landscape with angles *θ*<sup>1</sup> and *θ*2 in a selective applied magnetic field, such as *Ha*=0, *Ha*>0, and *Ha*>*Hf* , giving a two-dimensional energy barrier.

**Figure 4.** Observation of the energy barrier along the dashed line (**Figure 3**) clearly shows the stable and bistable states in the one-dimensional energy barrier.

## **4. Spin dynamics**

,

, the energy barrier

the exchange intercoupling energy per area and *Vs* indicates the interfacial area between the Ni core and NiO shell. The magnetization can readily be calculated by solving *M*(*Ha*)=*m*1cos*θ*1m + *ms*cos*θ*2m, where *θ*1m and *θ*2m are the *θ*1 and *θ*2, respectively, that minimize *Em* at each *Ha*. Depending on the relative strength between *m*1 and *m*2 and between *k*1 and *k*2, the value of *EJ* and the bistable energy of *E*(*θ*1m,*θ*2m,*Ha*) may be derived by solving the following equations: *∂E/∂θ*1=*∂E*/*∂θ*1=0 [15, 22]. **Figure 3** shows an example of the dependence of the energy *E* of angles *θ*1 and *θ*2 on the selection of the applied magnetic field, such as *Ha*=0, *Ha*>0, or *Ha*>*Hf*

giving a two-dimensional energy barrier. Here, different colors are used to differentiate the energy intensity, with red indicating high energy and purple low energy. When the field *Ha* is zero, the energy barrier exhibits two whirlpools, indicating stable and bistable solutions. A two-dimensional and high-energy barrier appears between the two whirlpools. With an increase of the field *Ha*, the energy barrier between the stable and bistable states decreases, implying that the field *Ha* provides Zeeman energy to compete with the magnetic anisotropies and the exchange intercoupling. The magnetic directions of the core and shell moments in the

between the stable and bistable states disappears. In this condition, the Zeeman energy is larger than the magnetic anisotropies and the exchange intercoupling. The core and shell moments will switch from bistable to stable states, and the core moment (FM) then becomes oriented toward the applied field direction, giving only one solution. The initial state of the core and shell moments *m*1 and *m*<sup>2</sup> can assume either a stable or bistable state following a pattern like water flowing downhill. Observe the energy barrier along the dashed line in **Figure 4**. The simplified energy barrier clearly exhibits stable and bistable states, which is similar to the SW model. The exchange intercoupling between the core and shell moments in the CSAE model also plays an important role on the *M*(*H*) loop process for the modulation of the hysteresis and inverted hysteresis loops. Huang et al. [15] reported the *M*(*H*) loop processes and the modu‐

lation of the magnetic anisotropies and exchange intercoupling describing proteresis.

**Figure 3.** Variation of the energy landscape with angles *θ*<sup>1</sup> and *θ*2 in a selective applied magnetic field, such as *Ha*=0,

, giving a two-dimensional energy barrier.

*Ha*>0, and *Ha*>*Hf*

152 Magnetic Materials

bistable state gradually become more unstable. When the field *Ha* reaches *Hf*

In the Ni/NiO core-shell nanoparticle system, treating the inner Ni atoms and surface NiO atoms gives rise to a net inner macromoment *m*1 and a net surface macromoment *m*2. The strength of the exchange intercoupling *EJ* =*σ*ex*Vs* is between the two moments of *m*1 and *m*2. Moreover, the fluctuation in the thermal energy *kBT*≠0 causes continual changes in the orientation of the moment of an individual nanoparticle, giving a thermally activated mag‐ netization. Thus, to provide good simulation results, the magnetic model of the Ni/NiO coreshell nanoparticle system requires combining the CSAE model with the Langevin function. **Figure 5** shows an example of the field dependence of the magnetization. The inverted hysteresis for the 13.9(2) nm sample appears at *T*=150 K. The black open circles (experimental data) can be described by a magnetic model with a blue line (decreasing field) and a red line (increasing field). An examination of the bottom of **Figure 5** shows the process of spin dynamics from *Ha*=1 kOe to *Ha*=-1 kOe, indicating a decreasing field. In the field *Ha*=1 kOe, the Ni core moment *m*<sup>1</sup> is toward the field direction, showing coupling at an angle |*θ*1-*θ*2|=88.3° between *m*<sup>1</sup> (red line) and *m*<sup>2</sup> (blue line). When the field decreases to zero, the *m*1 magnetization along the vertical direction component decreases, and the coupling angle between *m*1 and *m*<sup>2</sup> increases to |*θ*1-*θ*2|=172.8°. As the field decreases to *Ha*=-1 kOe, the Ni core moment *m*1 moves toward the negative field direction, and the coupling angle decreases to |*θ*1-*θ*2|=115.6°. For an increasing field, the process of spin dynamics is similar to the results for a decreasing field, but the directions of *m*1 and *m*2 at *Ha*=1 kOe are not the same. The explanation is that one solution is still in a bistable state. Thus, the field *Ha* has a great impact on the coupling angle between *m*1 and *m*2. For the *M*(*H*) loop direction, the parameters in the simulation, *m*1*>m*2*, k*1*<k*2, and *EJ* ≠0, apply to the description of inverted hysteresis in the low field. In the higher field, the exchange intercoupling between *m*1 and *m*<sup>2</sup> will be destroyed by the Zeeman energy, which changes the *M*(*H*) loop direction to counterclockwise. A similar *M*(*H*) loop has been reported in a germa‐ nium nanostructure system [15]. The exchange intercoupling strength *EJ* between the core and shell plays an important role in guiding the direction of the *M*(*H*) loop. If there is no interaction between the core and shell moments, the *M*(*H*) loop only exhibits hysteresis. If the exchange intercoupling strength *EJ* is larger than the magnetic anisotropies *KV*, the binding energy between *m*1 and *m*<sup>2</sup> can be treated as a single nanoparticle system, giving rise to the hysteretic behavior. When the exchange intercoupling strength *EJ* is comparable to the magnetic aniso‐ tropy *K*1*V*1 or *K*2*V*2, the *M*(*H*) loop will be guided to be proteretic.

**Figure 5.** Field dependence of the magnetization, giving an example of inverted hysteresis for the 13.9(2) nm sample at *T*=150 K. The black circle (experimental data) can be described by a magnetic model with a blue line (decreasing field) and a red line (increasing field).

In the example illustrated in **Figure 5**, the parameters used in the simulation, *EJ* =*σ*ex*Vs*=3.4 meV, *K*1*V*1=0.74 meV, and *K*2*V*2=51 meV, give a proteretic loop, which corresponds to the above description. In terms of the spin dynamics, the angles *θ*1 and *θ*2 for each field provide spin processes that lead to an increasing field, as shown in **Figure 6a**, and a decreasing field as shown in **Figure 6b**. The CSAE simulation produces proteretic behavior in the 13.9(2) nm sample after solving the spin dynamics for the decreasing field and increasing field. This result suggests that the spin-orientation results in proteretic behavior. Applying the processdependent summary of angles *θ*1 and *θ*2, we consider a decreasing-field process below and above the blocking temperature *TB* for two selected samples [<*d*>=7.2(3) nm and 19.0(2) nm], as shown in **Figures 7** and **8**. **Figure 7** shows the spin processes for the decreasing field for angles *θ*1 and *θ*<sup>2</sup> in the 7.2(3) nm sample. Two examples of the field-dependent spin dynamics at temperatures below and above *TB*=20 K, *T*=2 K, and *T*=50 K, respectively, are given. The spin dynamics exhibit a spin-flip for the angle *θ*1 at *T*=2 K and spin-orientation at *T*=50 K for the angle *θ*1, implying a change of the Ni core moment *m*1. However, the phenomenon of spinorientation was not found in the 19.0(2) nm sample, as shown in **Figure 8**, where the temper‐ ature is above the blocking temperature. Based on the observed field-dependent spin dynamics for *T*=2 K to *T*=300 K, we give a summary of the region for spin-flip (white region) and spinorientation (gray region), which corresponds to the distinction between hysteresis and proteresis (see **Figure 9**). The phase diagram of the spin dynamics implies a limit of the proteretic behavior and spin-orientation below a mean size of <*d*>=15.8 nm, which indirectly indicates a limit to the magnetic instability in the Ni (FM) core particles coated with NiO (AFM) shells. It also reveals the reason why there is no proteresis when the mean size is large. Controlling the proteretic behavior requires a mean size below <*d*>=15.8 nm and temperature above *T*p. The proteretic behavior relates to the spin-orientation, which has a rotation along EA. This will cause the destruction of the magnetic record. The result helps us to understand the limitations of recording data for the Ni/NiO core-shell nanoparticles on the size effect, which will affect the development of the magnetic industry in the future.

**Figure 6.** Simulation results provide a possible solution to elaborate spin dynamics for (a) the increasing field and (b) the decreasing field in the Ni/NiO core-shell nanoparticles.

**Figure 5.** Field dependence of the magnetization, giving an example of inverted hysteresis for the 13.9(2) nm sample at *T*=150 K. The black circle (experimental data) can be described by a magnetic model with a blue line (decreasing field)

*K*1*V*1=0.74 meV, and *K*2*V*2=51 meV, give a proteretic loop, which corresponds to the above description. In terms of the spin dynamics, the angles *θ*1 and *θ*2 for each field provide spin processes that lead to an increasing field, as shown in **Figure 6a**, and a decreasing field as shown in **Figure 6b**. The CSAE simulation produces proteretic behavior in the 13.9(2) nm sample after solving the spin dynamics for the decreasing field and increasing field. This result suggests that the spin-orientation results in proteretic behavior. Applying the processdependent summary of angles *θ*1 and *θ*2, we consider a decreasing-field process below and above the blocking temperature *TB* for two selected samples [<*d*>=7.2(3) nm and 19.0(2) nm], as shown in **Figures 7** and **8**. **Figure 7** shows the spin processes for the decreasing field for angles *θ*1 and *θ*<sup>2</sup> in the 7.2(3) nm sample. Two examples of the field-dependent spin dynamics at temperatures below and above *TB*=20 K, *T*=2 K, and *T*=50 K, respectively, are given. The spin dynamics exhibit a spin-flip for the angle *θ*1 at *T*=2 K and spin-orientation at *T*=50 K for the angle *θ*1, implying a change of the Ni core moment *m*1. However, the phenomenon of spinorientation was not found in the 19.0(2) nm sample, as shown in **Figure 8**, where the temper‐ ature is above the blocking temperature. Based on the observed field-dependent spin dynamics for *T*=2 K to *T*=300 K, we give a summary of the region for spin-flip (white region) and spinorientation (gray region), which corresponds to the distinction between hysteresis and proteresis (see **Figure 9**). The phase diagram of the spin dynamics implies a limit of the

=*σ*ex*Vs*=3.4 meV,

In the example illustrated in **Figure 5**, the parameters used in the simulation, *EJ*

and a red line (increasing field).

154 Magnetic Materials

**Figure 7.** (a) Temperature dependency of magnetization. (b and c) Magnetic field-dependent summary of the angles *θ*<sup>1</sup> and *θ*2 for the decreasing-field process below (*T*=2 K) and above (*T*=50 K) the blocking temperature *TB* for <*d*>=7.2(3) nm sample.

**Figure 8.** (a) Temperature dependence of magnetization. (b and c) Magnetic field-dependent summary of the angles *θ*<sup>1</sup> and *θ*2 for the decreasing-field process below (*T*=50 K) and above (*T*=150 K) the blocking temperature *TB* for the <*d*>=19.0(2) nm sample.

**Figure 9.** Summary of the *M*(*H*) loop direction for hysteresis and proteresis at a series of temperatures for each sample and resulting spin-flip or spin-orientation.

## **5. Conclusion**

Analysis of the results leads to some interesting conclusions. First, the impact of the size effect on the temperature-dependent *M*(*H*) loop allows the modulation of hysteresis and proteresis, which breaks the concept of a magnetic system. The *M*(*H*) loop between hysteresis and proteresis can be distinguished through analysis of the difference in magnetization Δ*M*, giving a summary of *Tp*, which results in an appearance of the proteretic behavior. In addition, we design a CSAE model, tuning the competition between magnetic anisotropy and exchange intercoupling, to describe the observation of hysteretic and proteretic behaviors in Ni/NiO core-shell nanoparticles. The CSAE simulation results show how the spin dynamics in the hysteresis and proteresis loops give rise to a spin-flip and spin-orientation, respectively. Finally, the phase diagram of the spin dynamics suggests a limit to the proteretic behavior and spin-orientation below a mean size of <*d*>=15.8 nm, which indirectly indicates a limit to the magnetic instability of Ni (FM) core particles coating the NiO (AFM) shells. It also explains why no proteresis occurs when the mean particle size is large. Controlling the proteretic behavior requires a mean particle size below <*d*>=15.8 nm and temperature above *Tp*. Finally, the results of magnetic analysis help us to understand the limitations of recording data for the Ni/NiO core-shell nanoparticles on the size effect, which will have implications for the development of the magnetic industry in future.

## **Acknowledgements**

**Figure 8.** (a) Temperature dependence of magnetization. (b and c) Magnetic field-dependent summary of the angles *θ*<sup>1</sup> and *θ*2 for the decreasing-field process below (*T*=50 K) and above (*T*=150 K) the blocking temperature *TB* for the

**Figure 9.** Summary of the *M*(*H*) loop direction for hysteresis and proteresis at a series of temperatures for each sample

<*d*>=19.0(2) nm sample.

156 Magnetic Materials

and resulting spin-flip or spin-orientation.

We would like to thank the Ministry of Science and Technology (MOST) of the Republic of China for their financial support of this research through project numbers MOST-103-2112- M-259-005 and MOST-104-2112-M-259-001.

## **Author details**

Jhong-Yi Ji and Sheng Yun Wu\*

\*Address all correspondence to: sywu@mail.ndhu.edu.tw

Department of Physics, National Dong Hwa University, Taiwan

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