shell nanoparticles on the exchange bias of a hysteresis loop

Journal Pre-proofs Research articles Modeling the Effect of Temperature and Size of Core/Shell Nanoparticles on the Exchange Bias of a Hysteresis Loop S.V. Anisimov, L.L. Afremov, A.A. Petrov PII: DOI: Reference:

S0304-8853(19)31582-3 https://doi.org/10.1016/j.jmmm.2019.166366 MAGMA 166366

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Journal of Magnetism and Magnetic Materials

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6 May 2019 7 October 2019 27 December 2019

Please cite this article as: S.V. Anisimov, L.L. Afremov, A.A. Petrov, Modeling the Effect of Temperature and Size of Core/Shell Nanoparticles on the Exchange Bias of a Hysteresis Loop, Journal of Magnetism and Magnetic Materials (2019), doi: https://doi.org/10.1016/j.jmmm.2019.166366

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© 2019 Published by Elsevier B.V.

MODELING THE EFFECT OF TEMPERATURE AND SIZE OF CORE/SHELL NANOPARTICLES ON THE EXCHANGE BIAS OF A HYSTERESIS LOOP Anisimov S.V.1, Afremov L.L.2, Petrov A.A.3

Affiliation: Far Eastern Federal University, Russia, 690091, Vladivostok, Sukhanova st. 8; e-mail: 1 [email protected] 2 [email protected] 3 [email protected] Abstract The effect of temperature and size on the exchange bias field of a hysteresis loop was studied based on the model of two-phase core/shell nanoparticles. Using the random interaction field method, the temperature dependence of the interfacial exchange interaction constant was estimated. It is shown that with increasing temperature, the field of exchange bias as well as the constant of interphase exchange interaction fall exponentially. An increase in the inverse size of nanoparticles leads to a linear growth of the exchange bias field. Keywords: core/shell nanoparticles, magnetism, exchange bias, coercivity, magnetization.

Introduction The study of the exchange bias of a hysteresis loop, discovered more than 60 years ago [1], remains relevant today. Particular interest is associated both with practical applications and with the study of the nature of the exchange bias 𝐻𝐸 in nanoparticles in which 𝐻𝐸 has a number of features compared to bilayer films [2]. The development of technologies for the synthesis of core-shell nanoparticles allows the use of exchange bias in various areas, for example: when creating sensitive magnetic sensors or in biomedical applications such as drug delivery systems, hyperthermia and magnetic resonance imaging [3], as well as in manufacturing high-density recording devices and permanent magnets [4, 5]. The effect of exchange bias in nanoparticles manifests itself at low temperatures and is manifested in the displacement of the magnetic hysteresis loop [1, 6, 7]. An experimental study of the dependence of hysteresis characteristics, including exchange bias, on the temperature and size of nanoparticles was carried out for core/shell systems with different compositions of metals and their oxides. For example, in [7], the temperature dependence of the exchange bias field 𝐻𝐸 of Co/CoO nanoparticles with an average size of 19 nm was studied using static and dynamic methods. It was shown that with an increase in temperature and, hence, thermal fluctuations, 𝐻𝐸 decreases as expected. A similar temperature behavior of the exchange bias field of Co/CoO and Ni/NiO nanoparticles was obtained in [8–11], and in these studies it was shown that larger exchange bias fields correspond to smaller nanoparticles. Along with a decrease in 𝐻𝐸, as the temperature increases, the coercive field 𝐻𝑐 [6, 9–11] of Co/CoO and Ni/NiO nanoparticles also decreases. In addition, it was shown in [8–11] that smaller nanoparticles correspond to the larger exchange bias fields. Experimentally observed size dependence of 𝐻𝐸 is confirmed by the results of MonteCarlo simulations of the dependence of the exchange bias field of Co/CoO nanoparticles on core size and shell thickness [12]. The authors of [12] showed that a decrease in the size of the

ferromagnetic core leads to an increase in the field of exchange bias, while with an increase in the thickness of the AFM shell 𝐻𝐸 increase sharply. Note that the magnetostatic interaction between Fe/Fe3O4 nanoparticles reduces the exchange bias field without changing the nature of its temperature dependence [13]. The matrix surrounding nanoparticles has a significant effect on 𝐻𝐸. For example, Al2O3 [14] and Cr2O3 [15] matrices reduce the exchange field to almost zero, while in MgO matrix, the exchange bias field reaches enormous values. In addition, as atomistic modeling [16] shows, a change in the shape of the interface can lead to a sharp change in both coercive field and exchange bias field. Various experimental studies [17-23] show that the form of temperature dependence of the exchange bias field and coercive field is determined by many factors. For example, according to [17], nature of the change in 𝐻𝑐 and 𝐻𝐸 from the temperature of the NiO nanoparticles depends on a magnetic state of the surface layer. If the state of spin-glass is formed in a thin surface layer, then at temperatures below the spin-glass transition temperature TSG we have 𝐻c(𝑇)~Exp ( ― 𝑇/𝑇0) and 𝐻𝑐(𝑇)~(1 ― (𝑇/𝑇𝐵)1/2), 𝐻𝑒(𝑇) = 0 at TSG < Т < TB. The characteristic temperature T0 decreases with an increase in the size of a nanoparticle, while the blocking temperature TB increases. In addition, it was noted that the exchange bias field increases linearly with increasing nanoparticle size, and 𝐻𝑐 varies nonmonotonously, reaching a maximum at 4 nm. In [18] the exponential temperature dependence of the exchange bias field is associated with the presence of a spin-glass state. 𝛼 It was shown in [19] that in the system of two-layer Co/Co3O4 films 𝐻𝐸~(1 ― (𝑇 𝑇0) , with α < 1. The authors attribute this temperature dependence to the formation of a CoO layer at the interface.

The temperature dependence of the coercive field for systems of pure and oxidized Co nanoparticles with the formation of core/shell Co/CoO nanoparticles is given in [6]. Oxidation of the cobalt changes the character of the temperature dependence, which can be approximated using 𝑇

𝛼 ― 𝑇0 , 𝐻c~(1 ― (𝑇 𝑇0) (where α > 1), to a dependence approximated by the exponent 𝐻𝐸~e which is obviously related with the occurrence of exchange interaction at the interface between Co and CoO. A similar exponential dependence can be used to approximate the temperature change in the exchange bias field in Co/CoO nanoparticle systems [20-22].

A detailed study of the temperature and size dependences of 𝐻𝑐 and 𝐻𝐸 of the core/shell Ni/NiO ―

𝑇

𝑇

0 nanoparticles is presented in [9]. It shows that 𝐻𝐸(𝑇)~e . An anomalous behavior of Hc(T) is also noted, which consists in the fact that at T < T0 the coercive field has larger values for smaller particles. A similar size dependence of Hc is also observed in Co/CoO nanoparticles [13].

A theoretical interpretation of the temperature dependence of the exchange bias field of the ferromagnet/spin-glass system is rather consistently presented [22]. The authors of this work used a phenomenological model where RKKY interaction is performed between spin magnetic moments. This approach made it possible to describe the rapidly falling temperature dependence of 𝐻𝐸(𝑇) of the two-layer Co/CuMn system, including the region of negative values of the exchange bias field. The exponential temperature dependence of the exchange bias field and coercive field was obtained as a result of Monte-Carlo simulation of core/shell nanoparticles with a ferromagnetic core coated with a disordered ferrimagnetic shell [23]. The modelling results of the dependence of

𝐻𝐸 and 𝐻𝑐 on the temperature of the core/shell nanoparticles with an antiferromagnetic core and a ferromagnetic shell containing nonmagnetic defects [24]. In the same way, the interaction of the magnetic moments of atoms is described by the Heisenberg Hamiltonian. Monte-Carlo calculations showed that the temperature dependence of 𝐻𝐸 and 𝐻𝑐 can be approximated by a power-law dependence, which, according to the authors, is due to the presence of defects. Note that despite the coincidence of the model of interaction of magnetic atoms and the calculation method, the temperature dependences of the exchange bias field 𝐻𝐸(𝑇) and the coercive field 𝐻𝑐(𝑇 ) of the core/shell nanoparticles ferromagnet/ferrite and antiferromagnet/ferromagnet presented in [23] and [24] are different. In this work, within the framework of the two-phase nanoparticle model [25], we present a simple and clear phenomenological approach that we developed. It allows, from a single point of view, to simulate the temperature and size dependence of the hysteresis characteristics of core/shell nanoparticles with various combinations of magnetic phases, including the exchange bias field, without resorting to rather difficult Monte-Carlo calculations. As we show, the coercive field and exchange bias field substantially depend on the interaction between the magnetic phases of a core and a shell, and the hysteresis characteristics are determined by the constant of the interphase exchange interaction.

1. Model of ferromagnetic/antiferromagnetic nanoparticle We use the core/shell nanoparticle model, described in detail in reference [25]. For ease of reading this work, we present the main points of the model: 1. A uniformly magnetized antiferromagnetic (AF) nanoparticle in the shape of an ellipsoid of revolution with an elongation Q and a volume 𝑉 = 4𝜋𝑄𝐵3/3 contains a uniformly magnetized ellipsoidal ferromagnetic core (F) with a length of q and a volume of 𝑣 = 𝜀 𝑉 = 4𝜋𝑞𝑏3/3, the long axis of which coincides with the long axis of the nanoparticle and is parallel to the Oz axis. 2. We assume that the axes of the crystallographic anisotropy of the ferro- and antiferromagnet are parallel to the long axes of the nanoparticle and the core. 3. The spontaneous magnetization vectors of the ferromagnet 𝓜𝑠 and one of the sublattices of the antiferromagnet 𝓜0 are located in the yOz plane containing the long axes of the magnetic phases and constitute the angles 𝜗(F) and 𝜗(𝐴𝐹) with the axis Oz, respectively. We assume, that magnetic moments of antiferromagnetic sub-lattices are compensated. 4. An external magnetic field 𝐻 is applied along the Oz axis. 5. A number 𝑁0 of noninteracting core/shell nanoparticles uniformly distributed over a volume 𝑉0 is considered. It is believed that nanoparticles are distributed in size 𝑎 with probability 𝑓(𝑎)𝑑𝑎. Then, according to [25]:

∫𝜀 (𝑎){𝑛 (𝑡, 𝑎) ―𝑛 (𝑡, 𝑎) ― 𝑛 (𝑡, 𝑎) + 𝑛 (𝑡, 𝑎)}𝑓(𝑎)𝑑𝑎.

𝑀(𝑡) = сℳ𝑠

1

2

3

4

(1)

Here с = 𝑁0𝑉 𝑉0 is the volume concentration of core / shell nanoparticles and 𝑛𝑖(𝑡, 𝑎) determines the population vector 𝒏(𝑡) = {𝑛1(𝑡), 𝑛2(𝑡), 𝑛3(𝑡), 𝑛4(𝑡)} of the four magnetic states of the core/sell nanoparticles, the calculation of which is described in Appendix I. 2. Selection of the modeling parameters

The simulation of the size and temperature dependence of the exchange bias field 𝐻𝐸 and the coercive field 𝐻𝑐 was carried out using the following types of core/shell systems of Ni/NiO, Co/CoO and Fe/Fe3O4 nanoparticles. Since the magnetic moments of nanoparticles are subject to thermal fluctuations, according to (1), the magnetization of the system should depend on the time of its measurement 𝑡𝑒𝑥𝑝. In the calculations, it was assumed that 𝑡𝑒𝑥𝑝 = 1 s. For comparison with the experimental data, the geometrical characteristics of the nanoparticles studied in detail in [9, 11, 13] were used in the calculations. The hysteresis characteristics were determined using hysteresis loops, calculated using expression (1). When integrating (1), the law of lognormal 𝑓(𝑎) size distribution of nanoparticles a was used: 𝑓(𝑎) =

{

1

exp ― 𝑎 2𝜋𝜎2

}

(𝑙𝑛 𝑎 ― ln〈𝑎〉)2 2𝜎2

,

(2)

Size distribution parameters are listed in the Table 1. Table 1. Mean sizes 〈𝑎〉 and deviation 𝜎 of core/shell nanoparticles [9, 11, 13].

Material

Sample number

Mean size and deviation, nm

1

2

3

4

5

〈𝑎〉

9

11

12

14

24

𝜎

0.04

0.06

0.06

0.08

0.07

〈𝑎〉

9

11

23

𝜎

0.52

0.35

0.56

〈𝑎〉

6.3

𝜎

0.3

Fe/Fe3O4

Ni/NiO

Co/CoO

Moreover, it was assumed that 𝑎 = 2𝐵, and the shell thickness 𝑑 of cobalt oxide, nickel oxide, and magnetite oxide varies slightly within 2 nm, regardless of the size of nanoparticles [9, 11, 13]. In the simulation, we used the dependence of the spontaneous magnetization ℳ𝑠 (ℳ0) and crystallographic anisotropy of 𝐾𝑉 nanoparticles on temperature and their sizes 𝑎. To describe the dependence of ℳ𝑠 and 𝐾𝑉, the following relations were used [26, 27]:

(

ℳ𝑠(𝑎, 𝑇) = ℳ𝑠 𝑏𝑢𝑙𝑘 1 ―

𝑎

2 𝛼

)( ( ) )

6𝑎0

𝑇 1― 𝑇𝑐(𝑎)

, 𝐾𝑉𝑛(𝑇) = 𝐾𝑉𝑛(0)

(

ℳ𝑠(𝑎, 0)

𝑛(2𝑛 + 1)

)

ℳ𝑠(𝑎, 𝑇)

(3)

Here, 𝛼 was determined by approximating the experimental values of spontaneous magnetizations of Fe (𝛼 = 0.91 [28]), Co (𝛼 = 0.22 [using data, obtained by Bloch, Honda and Masumoto listed in 29]), Ni (𝛼 = 0.45 [30]) and Fe3O4 (𝛼 = 0.25 [31]) (see Appendix III). The dependence of the Curie temperature 𝑇𝑐(𝑎) on the size of the nanoparticles is determined by the scaling relation [32]:

1

(

𝑇𝑐(𝑎) = 𝑇𝑐 𝑏𝑢𝑙𝑘 1 ―

( )) 𝜉0 𝑎

𝜈

𝐾.

(4)

Experimental values of the above-mentioned constants are listed in the Table 2. Table 2. Experimental values of spontaneous magnetization ℳ𝑠 𝑏𝑢𝑙𝑘 (ℳ0) and Curie temperature 𝑇𝑐 𝑏𝑢𝑙𝑘 of macroscopic material, dimensional constant 𝑎0, length 𝜉0 at 𝑇 = 0 and critical index of spin-spin correlations 𝜈, crystallographic constant 𝐾𝑉 and surface anisotropy 𝐾𝑆 of various materials. 𝐾𝑉, 105

𝐾𝑆,

Material

ℳ𝑠 𝑏𝑢𝑙𝑘 emu/g;

𝑎0, nm

Ref

𝑇𝑐 𝑏𝑢𝑙𝑘, K

𝜉0, nm

𝜈

Ref

𝑁𝑖𝑂

38.4 (1)

0.11 (2)

[33]

523

1.4

0.5

[9]

28

[42]

𝑁𝑖

55

0.69 (2)

[30, 34]

627

0.67

0.7

[38]

0.5

[43]

𝐶𝑜𝑂

70.7 (1)

0,36 (2)

[35]

293

1.8

0.63

[39]

2700

[6]

0.6

[39]

𝐶𝑜

143

3.75 (2)

[36]

1400

2.2 (4)

0.62 (4)

[40]

39.8

[44,

0.2

[45]

2 10-2

[50]

𝑒𝑟𝑔 𝑐𝑚3

Ref

𝑒𝑟𝑔 𝑐𝑚2 1.3 0.87 (3)

Ref [48] [49]

45] 88.65

Fe3O4

2.26

[26]

550

0.51

0.82

[41]

1.08

[46]

(at 4К) 170.9

Fe

0.37

[37]

1043

0.11

0.97

[32]

4.8

[47]

0.9 10-3

[51]

(1) The magnetization values of the ℳ0 sublattices of NiO and CoO calculated on the basis of the experimental data of [52]. (2) The 𝑎0 values were calculated for Ni, NiO and CoO nanoparticles on the basis of the experimental data from [32], [33], [36] and [52], respectively. (3) The value of the constant 𝐾𝑆 is calculated using the value of the effective anisotropy constant 𝐾𝑒𝑓𝑓 from [49]. (4) The values of 𝜉0 and 𝜈 are calculated on the basis of data from [40].

When modeling the temperature dependence of the hysteresis characteristics of the nanoparticles, the exchange interaction constant 𝐴𝑖𝑛 was determined using the relation (5. AII) (see Appendix II):

( )

𝐴𝑖𝑛(𝑇) = 𝐴exp ―

𝑇 , 𝑇0

(5)

where the characteristic temperature 𝑇0 is related to the approximation parameter 𝑡0 as follows: 𝑇0 = 𝐽1𝑡0 𝑘 . 𝐵

The constants 𝐴 and 𝑇0 were chosen in such a way that one of the calculated values of 𝐻𝐸 was closest to the experimental one [9, 11, 13] (see Table 3). Table 3. Constant A and characteristic temperature 𝑇0 for Ni/NiO, Co/CoO, and Fe/Fe3O4 nanoparticles.

Material

Size, nm

Constant

Theoretical value of the constant

Experimental value of the constant

Ni/NiO Co/CoO Fe/Fe3O4

9 11 23 6,4 12

𝐴, 10 ―11 erg/cm 7,5 3,75 1,425 48.9 89.3

𝑇0 = 𝐽1𝑡0 𝑘𝐵, К 8,61 8,42 8,40 37,03 45.8

𝑇0, К 9,1 ± 0,7 33.8 (1) 47 (1)

[9] [11] [13]

(1) Calculated from experimental data from [11] and [13], in which the measurement error was not specified.

3. Modeling results The calculation of the temperature dependence of the exchange bias field 𝐻𝐸 and the coercive field 𝐻𝑐 of the Ni/NiO nanoparticle system carried out using relations (1) – (5) is presented in Fig. 1 and 2. From the figures it can be seen that an increase in temperature leads to a fall in 𝐻𝐸 and 𝐻𝑐. As expected, with the exception of points close to Θ~40 K, the calculated temperature dependence 𝐻𝐸 complies with the exponential law (see Fig. 1) with the characteristic temperature 𝑇0 = 9,1 ± 0,7 K, presented in Table 3.

Fig. 1. The dependence of the logarithm of the exchange bias field Log(𝐻𝐸) of Ni/NiO nanoparticles on temperature 𝑇 and particle size 𝑎. 𝐻𝐸 is measured in Oe. Dots mark experimental data [9].

Moreover, if the exchange bias field of nanoparticles of various sizes decreases to zero at almost the same temperature Θ, then the temperature at which the coercive field 𝐻𝑐 vanishes depends on the size of the nanoparticles. This is due to an increase in the height of potential barriers with increasing particle size: the larger particles go over to the superparamagnetic state (𝐻𝑐 = 0) at a higher temperature.

Fig. 2. Dependence of the coercive field 𝐻𝑐 of Ni/NiO nanoparticles on temperature 𝑇 and particle size 𝑎. Dots mark experimental data [9].

Note that the described temperature dependence of the coercive field and the exchange bias field is characteristic of metal/metal oxide core/shell nanoparticles, where the metal belongs to a transition group. An example is the temperature dependences of 𝐻𝑐 and 𝐻𝐸 of the systems of Co/CoO and Fe/Fe3O4 nanoparticles shown in Fig. 3 and 4.

Hc, Oe

450 400 350 300 250 20

40

60

80

T, K

Log HE 6 5 4 3 2 1 20

40

60

80

100

T, K

Fig. 3. Temperature dependences of the coercive field 𝐻𝑐 and the logarithm of the exchange bias field 𝐻𝐸 of Co/CoO nanoparticles of size 𝑎 = 6.3 nm. 𝐻𝐸 is measured in Oe. Dots show experimental data [11].

Hc, Oe 1200

1000 800 600 20

40

60

80

100

T, K

Log HE 4.0 3.5 3.0 2.5 2.0 1.5 40

60

80

100

T, K

Fig. 4. Temperature dependences of the coercive field 𝐻𝑐 and the exchange bias field 𝐻𝐸 of Fe/Fe3O4 nanoparticles of size 𝑎 = 12 nm. 𝐻𝐸 is measured in Oe. Dots show experimental results [13].

Figure 5 shows the size dependences of the coercive field calculated at T = 5 K and the exchange bias calculated at T = 5 and 20 K for Ni/NiO nanoparticles.

Hc,Oe 650 600 550 500 450 400 10

15

20

25

a, nm

HE , Oe 400

300 200 100

1 0.06 0.08 0.10 0.12 a

, nm

1

Fig. 5. Dependencies of the coercive field 𝐻𝑐 on size 𝑎 at temperature 𝑇 = 5 K and field of exchange bias 𝐻𝐸 of Ni/NiO nanoparticles on inverse size 1⁄𝑎 at temperatures 𝑇 = 5 and 20 K. Experimental results are shown by dots [9].

The exchange bias field values 𝐻𝐸 are described by the relation: 𝐻𝐸 = 𝐻𝐸0(𝑇) + ℎ(𝑇) 𝑎.

(6)

At a temperature of 5 K, we find 𝐻𝐸0 = ―219 Oe, ℎ = 5135 Oe nm, the values of which agree well with the experimental results (𝐻𝐸0 = ―210 Oe, ℎ = 4969 E nm) [9]. At a higher temperature (20 K), the constants in expression (6) decrease to 𝐻𝐸0 = ―34,5 Oe, ℎ = 953 Oe nm (see Fig. 5). 4. Discussion The exponential temperature dependence of the calculated exchange bias field noted above is due to the similar behavior of the interfacial exchange interaction constant, 𝐴𝑖𝑛(𝑇)~exp ( ― 𝑡 𝑡0) = exp( ― 𝑡 𝑡 ), wherein 𝑡 = 𝑘𝐵𝑇 𝐽 . The characteristic temperature 𝑇 = (𝐽1 𝑘 )𝑡 expressed 0

1

0

𝐵

0

through the approximation parameter 𝑡0is determined by the energy of the interphase exchange interaction 𝐽12 = 𝐽1𝑖12 (see Appendix II). Expressing the energy of the exchange interaction of nickel atoms 𝐽 = 𝐽 through the relative Curie temperature 𝑡 = 𝑘𝐵𝑇𝑐 𝐽 , we have: 𝑡 = 𝑡 𝑇0 𝑇 . 1

𝑁𝑖

𝑐

1

0

𝑐

с

Using the 𝑡𝑐 = 3,05 value calculated using equations (3.AII), (4.AII), experimental 𝑇0 values and assuming that the Curie temperature depends on the size of nanoparticles (see (4)), one can estimate the size dependence of the 𝑡0 parameter (see tab. 4). Table 4. The approximation parameter 𝑡0, the energy of the exchange interaction 𝐽𝑁𝑖 and the energy of the interphase exchange interaction 𝐽12 for Ni/NiO nanoparticles. Nanoparticle size 𝑎, nm

Approximation parameter 𝑡0

Energy of the exchange interaction 𝐽𝑁𝑖, 10 ―14 erg

Energy of the interphase exchange interaction 𝐽12, 10 ―16 erg

9 11 23

0,0454 0,0451 0,0446

2,77 2,79 2,82

0,886 0,891 0,902

The energy of the nickel exchange interaction can be estimated using the expression 𝑡𝑐 = 𝑘𝐵𝑇𝑐 𝐽 = 3,05: – 𝐽 = 𝑘𝐵𝑇𝑐 3,05 (see Table 4). The calculation carried out using relations 1 𝑁𝑖 (2.AII) - (4.AII) showed that the table values 𝑡0 and 𝑖12 = 3,1 ∙ 10 ―3 correspond to certain values of the interfacial exchange interaction energy 𝐽12. Presented in table. 4 are the energy values 𝐽𝑁𝑖 and 𝐽12, which agree with the results of the experiments 𝐽𝑁𝑖 𝑒𝑥𝑝 = 2,7 10 ―14 erg (see, for example, [53]) and 𝐽12 𝑒𝑥𝑝 = 0,06𝑎2𝑁𝑖 ≈ 0.8 10 ―16 erg [9]. A sharp decrease in 𝐴𝑖𝑛(𝑇), and hence the exchange bias field, is due to magnetic ordering processes in a two-sublattice system, in which the exchange energy of interaction between the 𝐽12 sublattices is 2-3 orders of magnitude lower than the exchange energies 𝐽1 = 𝐽𝑓𝑒𝑟𝑟𝑜 and 𝐽2 = 𝐽𝑎𝑛𝑡𝑖𝑓𝑒𝑟𝑟𝑜 in the sublattices. In case of Ni/NiO system, an exchange energy of nickel 𝐽1 is an order of magnitude larger than the interaction energy in the sublattice of nickel oxide 𝐽2, which also accelerates the decrease in the magnetic moment 𝑚2(𝑇) in it (see Appendix II). A type of temperature dependence of the exchange bias field is determined not only by the ratio of the exchange interaction constants noted above, but also by other factors. For example, the authors of [24] performed Monte-Carlo simulation of the FM/AFM core/shell nanoparticle system with defects in the interface. Calculations performed in the approximations of 𝐽12 = 𝐽1 and 𝐽2 = 𝐽1/5 showed that the defective interface increases the rate of fall of 𝐻𝐸. As the calculations show [23], the heterogeneous distribution of magnetic moments in the interface leads to the same effect, which is confirmed experimentally [54]. As noted earlier, with an increase in the temperature of the field of exchange bias of 𝐻𝐸, Ni/NiO nanoparticles of various sizes a disappear at a temperature of Θ~40 K. The independence of Θ, as well as the characteristic temperature 𝑇0 on the size 𝑎 of nanoparticles (see table 3), is due to the small change in the energy of the interphase exchange interaction 𝐽12 (see table 4). Let us pay attention to the “anomalous” temperature dependence of 𝐻𝑐 (see Fig. 2) in the region 𝑇 < Θ: the coercive field of smaller Ni/NiO nanoparticles is higher than the 𝐻𝑐 particles of larger sizes. At 𝑇 > Θ, the coercive field of larger nanoparticles 𝐻𝑐 is higher than of small nanoparticles, which agrees with the notion of an increase in the height of the potential barrier with an increase in their volume. The “anomalous” dependence of the coercive field 𝐻𝑐 on the size of nanoparticles at 𝑇 < Θ is associated with the interphase exchange interaction between the ferromagnetic and antiferromagnetic phases, which “complicates” the magnetization reversal of the ferromagnetic core. Since this interaction is “surface”, its effect on the internal ferromagnetic atoms should decrease with increasing size of the nanoparticles. As calculations show, it is this interaction that determines the fall of the potential barriers 𝐸𝑖1 of transitions to the first state with an increase in the size a of Ni/NiO nanoparticles at temperatures 𝑇 < Θ. Figure 6 shows the size dependence of the height of the smallest potential barrier 𝐸21 at 𝑇 < Θ (blue curve). At 𝑇 > Θ, the interphase exchange interaction constant is 𝐴𝑖𝑛(𝑇)→0, which leads to an increase in 𝐸21 with an increase in 𝑎 (see Fig. 6).

E21,10 0.20 0.15 0.10 0.05 0.00

13

erg

10

15

20

25

a,nm

Fig. 6. Dependence of the height of the smallest potential barrier 𝐸21 on the size 𝑎 of Ni/NiO nanoparticles at temperatures 𝑇 < Θ (solid line, 𝑇 = 5 К) and 𝑇 > Θ (dashed line, 𝑇 = 50 К).

Acknowledgement

This work was financially supported by the state task of the Ministry of Education and Science of the Russian Federation № 3.7383.2017/8.9

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Appendix I Magnetic states of ferromagnetic/antiferromagnetic nanoparticles Using the results of [25, Appendix I], the total energy of a nanoparticle can be represented as the sum of the energy of crystallographic anisotropy: 1 (𝐴𝐹) (𝐹) + (ℳ𝑠)2𝑘(𝐹) (1.𝐴I) 𝐸𝐴 = ― {(2ℳ0)2𝑘(𝐴𝐹) 𝐴 (1 ― 𝜀)cos 2𝜗 𝐴 𝜀cos 2𝜗 }𝑉, 4 the energy of demagnetizing field:

𝐸𝑚 = ―

(ℳ𝑠)2 4

𝜀𝑘𝑁cos 2𝜗(𝐹)𝑉,

(2.AI)

the energy of the exchange interaction across the border: 2𝐴𝑖𝑛 cos (𝜗(𝐴𝐹) ― 𝜗(𝐹))𝑠, 𝐸𝑒𝑥 = ― 𝛿

(3.AI)

the surface anisotropy energy (see Appendix I in [25]): 𝐸𝑆 = 𝐸𝑆0(𝑞) ―

―

{

(ℳ0)𝟐 2

(𝜋𝐵 𝜉(𝑄) ― 𝜋𝑏 𝜉(𝑞))𝑐𝑜𝑠2𝜃

𝑘(𝐴𝐹) 𝑠

2

2

(𝐴𝐹)

+

ℳ𝑠𝟐 2

}

2 (𝐹) V (4.AI) 𝑘(𝐹) 𝑠 𝜋𝑏 𝜉(𝑞)𝑐𝑜𝑠2𝜃

and the Zeeman’s energy:

𝐸𝐻 = ―𝐻𝜀ℳ𝑠cos 𝜗(𝐹)𝑉, (𝐴𝐹)

(5.AI)

(𝐹)

2 where 𝑘(𝐴𝐹) = 𝐾𝑎 (2ℳ0)2, 𝑘(𝐹) 𝐴 𝐴 = 𝐾𝑎 (ℳ𝑠) , 𝑘𝑁 = 𝑁𝑥(𝑞) ― 𝑁𝑧(𝑞) are dimensionless constants of crystallographic anisotropy and shape anisotropy of the antiferromagnetic and ferromagnetic phase, respectively, 𝑁𝑥,𝑧(𝑞) are the demagnetizing coefficients of the ferromagnet, depending on the elongation of the core 𝑞, 𝐾(𝐴𝐹,𝐹) are the phase anisotropy constants, 𝑠 is the area 𝑎 of the surface separating the ferromagnet and the antiferromagnet, 𝐴𝑖𝑛 is the constant of interphase exchange interaction, 𝛿 is the width of the transition region, which is on the order of the lattice constant. Thus, the total energy is: (ℳ𝑠)2 2𝐴𝑖𝑛 𝑠 2 (𝐴𝐹) (𝐴𝐹) cos (𝜗(𝐴𝐹) ― 𝜗(𝐹)) ― 𝐸 = { ― (ℳ0) 𝒦 cos 2𝜗 ― 𝒦(𝐹)cos 2𝜗(𝐹) ― 4 𝛿 𝑉

― 𝜀ℳ𝑠𝐻 cos 𝜗(𝐹)}𝑉, (6.AI) where effective anisotropy constants 𝒦(𝐴𝐹,

𝐹)

are: 3 𝑄𝐵 𝒦(𝐴𝐹) = (1 ― 𝜀)𝑘(𝐴𝐹) + 𝜉(𝑄) ― 𝑘(𝐴𝐹) 𝜀 𝜉(𝑞) , 𝐴 𝑆 2 𝑄𝐵 𝑞𝑏

(

𝒦(𝐹) = 𝜀 𝑘(𝐹) 𝐴 +

(

)

)

3 (𝐹) 𝑘 𝜉(𝑞) + 𝑘𝑁 . 2𝑏𝑞 𝑆

(7.AI) (8.AI)

Energy minimization (6.AI) shows that the nanoparticle can be in 4 magnetic states: the first is {𝜗(𝐴𝐹) = 0, 𝜗(𝐹) = 0} , the second is {𝜗(𝐴𝐹) = 0, 𝜗(𝐹) = 𝜋} , the third is {𝜗(𝐴𝐹) = 𝜋, 𝜗(𝐹) = 𝜋} and the fourth is {𝜗(𝐴𝐹) = 𝜋, 𝜗(𝐹) = 0}. According to [25], in a system of nanoparticles at temperature 𝑇 ≠ 0, the probabilities of filling each of the above states are described by the population vector (𝑡) = {𝑛1(𝑡), 𝑛2(𝑡), 𝑛3(𝑡), 𝑛4(𝑡)}. The population vector is determined using the matrix exponent: 𝑡

𝑵(𝑡) = exp (𝑾𝑡) ∙ 𝑵(𝑡 = 0) +

∫ 𝑒𝑥𝑝(𝑾 (𝑡 ― 𝜏))𝑑𝜏 ⋅ 𝑽, 0

(9.AI)

where 𝑵(𝑡) =

n1(𝑡) n2(𝑡)

( ) { n3(𝑡)

, 𝑊𝑖𝑘 =

4

―

∑𝑊

𝑖𝑗

𝑗≠𝑖

― 𝑊4𝑖,

𝑊𝑘𝑖 ― 𝑊4𝑖,

𝑖 = 𝑘, 𝑖 ≠ 𝑘,

,

𝑽=

W41 W42

( ) W43

,

(10.AI)

𝑊𝑖𝑘 = 𝑓0𝑒𝑥𝑝( ―𝐸𝑖𝑘 𝑘В𝑇) are matrix elements of the transition probability matrix from the 𝑖-th to is the potential barrier ― 𝐸(𝑚𝑖𝑛) 𝑘-th equilibrium state, 𝑓0 is the frequency factor, 𝐸𝑖𝑘 = 𝐸(𝑚𝑎𝑥) 𝑖𝑘 𝑖 (𝑚𝑎𝑥) height, and 𝐸𝑖𝑘 is the smallest of the maximum energy values that correspond to the transition of the magnetic moment from the 𝑖-th equilibrium state with energy 𝐸(𝑚𝑖𝑛) to the 𝑘-th equilibrium 𝑖 state. Expressions for potential barriers 𝐸𝑖𝑘 are presented in [25] (relations (1AII) - (10AII)). Appendix II Temperature dependence of interphase exchange interaction energy of core/shell nanoparticles We consider the interface between the core and the shell as a set of a monolayer of the shell and a monolayer of the nucleus with average values of the magnetic moments of atoms in the layers 𝜇1𝒎1(𝑇) and 𝜇2𝒎2(𝑇), respectively (where 𝒎(𝑇) is the relative magnetic moment of the atom). For simplicity, we assume that the concentration of magnetic atoms n in the layers is the same. Then the energy of the interphase exchange interaction can be estimated as follows: 𝐸𝑒𝑥 = ―𝛼 𝜇1𝜇2𝑛𝑠𝛿(𝒎1(𝑇),𝒎2(𝑇)) = ―𝛼 𝜇1𝜇2𝑛𝛿𝑚1(𝑇)𝑚2(𝑇)cos (𝜗(1) ― 𝜗(2)) 𝑠, (1.𝐴II) where 𝛼 is a temperature independent constant with the dimension of volume and 𝑠 and 𝛿 are the surface area and width of the interface, respectively. Comparing (1.II) with (3.I) we have: 𝛼𝜇1𝜇2 𝑛𝛿2 𝐴𝑖𝑛(𝑇) = 𝑚1(𝑇)𝑚2(𝑇) = 𝐴 𝑚1(𝑇)𝑚2(𝑇). (2.𝐴II) 2 To estimate 𝑚1(𝑇) and 𝑚2(𝑇), we use the random interaction field method (see, for example, [50, 55]). In the Ising model approximation, the equations for the average values of the magnetic moments of atoms in monolayers are: 𝑚1 =

1

𝑧1

𝑧12

∑∑

𝐶𝑧𝑘1𝐶𝑧𝑙 12(1 𝑧1 + 𝑧12 2 𝑘 = 0𝑙 = 0

+ 𝑚1)𝑘(1 ― 𝑚1)𝑧1 ― 𝑘(1 + 𝑚2)𝑙(1 ― 𝑚2)𝑧12 ― 𝑙

tanh

𝑚2 =

1

𝑧2

(2𝑘 ― 𝑧1) + (2𝑙 ― 𝑧12)𝑖12 𝑡

, (3.𝐴II)

𝑧21

∑∑

𝐶𝑧𝑘2𝐶𝑧𝑙 21(1 𝑧2 + 𝑧21 2 𝑘 = 0𝑙 = 0

+ 𝑚2)𝑘(1 ― 𝑚2)𝑧2 ― 𝑘(1 + 𝑚1)𝑙(1 ― 𝑚1)𝑧21 ― 𝑙

tanh

(2𝑘 ― 𝑧2)𝑖2 + (2𝑙 ― 𝑧21)𝑖21 𝑡

, (4.𝐴II)

Where 𝑖2 = 𝐽2 𝐽1, 𝑖12 = 𝐽12 𝐽1, 𝑖21 = 𝐽21 𝐽1, 𝐶𝑧𝑘 is the binomial coefficient, 𝛿(ℎ ― ℎ0) is the Dirac delta function, 𝑧1, 𝑧2, 𝐽1, 𝐽2 are the numbers of nearest neighbors and the exchange interaction energies in the layers and 𝑧12, 𝑧21, 𝐽12, 𝐽21 between the layers, 𝑡 = 𝑘𝐵𝑇 𝐽1 is the relative temperature. We assume that the first layer of the interface belongs to a ferromagnetic metal (for example, Ni, Co, Fe), and the second to the sublattice of an antiferromagnet.

In the case of Ni/NiO nanoparticles, the number of nearest neighbors in nickel is 𝑧1 = 4, and 𝑧12 = 𝑧21 = 1. The nickel oxide lattice has a monoclinic syngony with two periods a = 2.95 A and c = 7.23 A, and if the long side of the lattice with a period c = 7.23 A is located in the adjacent layer, then 𝑧2 = 2. To estimate 𝑖2, we use the fact that the exchange energy in the NiO sublattice is 𝐽2 = 16𝑘𝐵 erg and also that 𝐽1 = 𝐽𝑁𝑖 ≈ 200𝑘𝐵 erg [56]. Thus, 𝑖2 = 0,08, and 𝑖12 = 𝑖21 will be considered as a variable parameter. The temperature dependences of the relative mean moments 𝑚1(𝑇), 𝑚2(𝑇) and the relative interfacial interaction constant 𝐴𝑖𝑛(𝑇) 𝐴 calculated using relations (2.AII) - (4.AII) for different values of the energy of interphase exchange interaction 𝐽12 = 𝐽1𝑖12 are presented in Figures 1.AII and 2.AII. The dependence 𝐴𝑖𝑛(𝑇) is approximated by the exponent for 𝑖12 = 0,005: 𝑡 𝐴𝑖𝑛(𝑇) = 𝐴exp ― . (5.𝐴II) 𝑡0

( )

m 1.0 0.8 0.6 0.4 0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5

t

Fig. 1 AII. Dependence of the relative mean moments 𝑚1(𝑇), 𝑚2(𝑇) on the relative temperature 𝑡 = 𝑘𝐵𝑇 𝐽 and the relative energy of the exchange interaction between the atoms of the interface 𝑖 = 𝐽12 𝐽 . 12 𝑁𝑖 𝑁𝑖 𝑖 𝑖 𝑖 𝑚 (𝑇) = 0,005 = 0,05 = 0,25 The red line is 2 with 12 , the yellow line with 12 , and the green line with 12 . The blue line is 𝑚1(𝑇) with 𝑖12 = 0,005; 0,05; 0,25.

Ain A 1.0 0.8 0.6 0.4 0.2 0.0

0.1

0.2

0.3

0.4

0.5

t

Fig. 2 AII. Dependence of the relative interfacial coupling constant 𝐴𝑖𝑛(𝑇) 𝐴 on the relative temperature 𝑡 = 𝑘𝐵𝑇 𝐽 . The curve marked by dots is constructed using relations (2) - (4) with 𝑖 = 0,005, the solid 𝑁𝑖

12

curve is approximated by the exponent 𝐴𝑖𝑛(𝑇) 𝐴~ exp ( ― 𝑡 𝑡0).

Appendix III Temperature dependence of the spontaneous magnetization ℳ𝑠 of nanoparticles was approximated using the following relation: 2 𝛼

( ( ))

ℳ𝑠(𝑎, 𝑇) = ℳ𝑠 𝑏𝑢𝑙𝑘

𝑇 1― 𝑇𝑐(𝑎)

,

(1.𝐴III)

where 𝛼 = 0,91 for iron, 𝛼 = 0,22 for cobalt, 𝛼 = 0,45 for nickel and 𝛼 = 0,25 for magnetite.

Fig. 1 AIII. Dependence of the relative magnetic moments 𝑀𝑠(𝑇)/𝑀𝑠(0) on the relative temperature 𝑇/𝑇𝑐 of various materials. Squares mark the experimental results for Ni [30], triangles are for Co [29], diamonds are for Fe [28] and circles are for Fe3O4 [31].

None

Fig. 1. The dependence of the logarithm of the exchange bias field Log(𝐻𝐸) of Ni/NiO nanoparticles on temperature 𝑇 and particle size 𝑎. 𝐻𝐸 is measured in Oe. Dots mark experimental data [9].

Fig. 2. Dependence of the coercive field 𝐻𝑐 of Ni/NiO nanoparticles on temperature 𝑇 and particle size 𝑎. Dots mark experimental data [9].

Hc, Oe

450 400 350 300 250 20

40

60

80

T, K

Log HE 6 5 4 3 2 1 20

40

60

80

100

T, K

Fig. 3. Temperature dependences of the coercive field 𝐻𝑐 and the logarithm of the exchange bias field 𝐻𝐸 of Co/CoO nanoparticles of size 𝑎 = 6.3 nm. 𝐻𝐸 is measured in Oe. Dots show experimental data [11].

Hc, Oe 1200

1000 800 600 20

40

60

80

100

Log HE 4.0 3.5 3.0 2.5 2.0 1.5 40

60

80

100

T, K

T, K

Fig. 4. Temperature dependences of the coercive field 𝐻𝑐 and the exchange bias field 𝐻𝐸 of Fe/Fe3O4 nanoparticles of size 𝑎 = 12 nm. 𝐻𝐸 is measured in Oe. Dots show experimental results [13].

Hc,Oe 650 600 550 500 450 400 10

15

20

25

a, nm

HE , Oe 400

300 200 100

1 0.06 0.08 0.10 0.12 a

, nm

1

Fig. 5. Dependencies of the coercive field 𝐻𝑐 on size 𝑎 at temperature 𝑇 = 5 K and field of exchange bias 𝐻𝐸 of Ni/NiO nanoparticles on inverse size 1⁄𝑎 at temperatures 𝑇 = 5 and 20 K. Experimental results are shown by dots [9].

E21,10 0.20 0.15 0.10 0.05 0.00

13

erg

10

15

20

25

a,nm

Fig. 6. Dependence of the height of the smallest potential barrier 𝐸21 on the size 𝑎 of Ni/NiO nanoparticles at temperatures 𝑇 < Θ (solid line, 𝑇 = 5 К) and 𝑇 > Θ (dashed line, 𝑇 = 50 К).

m 1.0 0.8 0.6 0.4 0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5

t

Fig. 1 AII. Dependence of the relative mean moments 𝑚1(𝑇), 𝑚2(𝑇) on the relative temperature 𝑡 = 𝑘𝐵𝑇 𝐽 and the relative energy of the exchange interaction between the atoms of the interface 𝑖 = 𝐽12 𝐽 . 𝑁𝑖 12 𝑁𝑖 The red line is 𝑚2(𝑇) with 𝑖12 = 0,005, the yellow line with 𝑖12 = 0,05, and the green line with 𝑖12 = 0,25. The blue line is 𝑚1(𝑇) with 𝑖12 = 0,005; 0,05; 0,25.

Ain A 1.0 0.8 0.6 0.4 0.2 0.0

0.1

0.2

0.3

0.4

0.5

t

Fig. 2 AII. Dependence of the relative interfacial coupling constant 𝐴𝑖𝑛(𝑇) 𝐴 on the relative temperature 𝑡 = 𝑘𝐵𝑇 𝐽 . The curve marked by dots is constructed using relations (2) - (4) with 𝑖 = 0,005, the solid 𝑁𝑖

curve is approximated by the exponent 𝐴𝑖𝑛(𝑇) 𝐴~ exp ( ― 𝑡 𝑡0).

12

Fig. 1 AIII. Dependence of the relative magnetic moments 𝑀𝑠(𝑇)/𝑀𝑠(0) on the relative temperature 𝑇/𝑇𝑐 of various materials. Squares mark the experimental results for Ni [30], triangles are for Co [29], diamonds are for Fe [28] and circles are for Fe3O4 [31].

The modeling of the temperature and size dependencies of coercive field and exchange bias has been carried out It is shown, that exchange interaction constant fall exponentially with growth of temperature Anomalous size dependence of coercive field is explained at low temperatures

Table 1. Mean sizes 〈𝑎〉 and deviation 𝜎 of core/shell nanoparticles [9, 11, 13].

Material

Fe/Fe3O4

Ni/NiO

Co/CoO

Sample number

Mean size and deviation, nm

1

2

3

4

5

〈𝑎〉

9

11

12

14

24

𝜎

0.04

0.06

0.06

0.08

0.07

〈𝑎〉

9

11

23

𝜎

0.52

0.35

0.56

〈𝑎〉

6.3

𝜎

0.3

Table 2. Experimental values of spontaneous magnetization ℳ𝑠 𝑏𝑢𝑙𝑘 (ℳ0) and Curie temperature 𝑇𝑐 𝑏𝑢𝑙𝑘 of macroscopic material, dimensional constant 𝑎0, length 𝜉0 at 𝑇 = 0 and critical index of spin-spin correlations 𝜈, crystallographic constant 𝐾𝑉 and surface anisotropy 𝐾𝑆 of various materials. 𝐾𝑉, 105

𝐾𝑆,

Ref

𝑇𝑐 𝑏𝑢𝑙𝑘, K

𝜉0, nm

𝜈

Ref

0.11 (2)

[33]

523

1.4

0.5

[9]

28

[42]

55

0.69 (2)

[30, 34]

627

0.67

0.7

[38]

0.5

[43]

𝐶𝑜𝑂

70.7 (1)

0,36 (2)

[35]

293

1.8

0.63

[39]

2700

[6]

0.6

[39]

𝐶𝑜

143

3.75 (2)

[36]

1400

2.2 (4)

0.62 (4)

[40]

39.8

[44,

0.2

[45]

2 10-2

[50]

Material

ℳ𝑠 𝑏𝑢𝑙𝑘 emu/g;

𝑎0, nm

𝑁𝑖𝑂

38.4 (1)

𝑁𝑖

𝑒𝑟𝑔 𝑐𝑚3

Ref

𝑒𝑟𝑔 𝑐𝑚2 1.3 0.87 (3)

Ref [48] [49]

45]

Fe3O4

88.65

2.26

[26]

550

0.51

0.82

[41]

1.08

[46]

(at 4К)

Fe

170.9

0.37

[37]

1043

0.11

0.97

[32]

4.8

[47]

0.9 10-3

[51]

(1) The magnetization values of the ℳ0 sublattices of NiO and CoO calculated on the basis of the experimental data of [52]. (2) The 𝑎0 values were calculated for Ni, NiO and CoO nanoparticles on the basis of the experimental data from [32], [33], [36] and [52], respectively. (3) The value of the constant 𝐾𝑆 is calculated using the value of the effective anisotropy constant 𝐾𝑒𝑓𝑓 from [49]. (4) The values of 𝜉0 and 𝜈 are calculated on the basis of data from [40].

Table 3. Constant A and characteristic temperature 𝑇0 for Ni/NiO, Co/CoO, and Fe/Fe3O4 nanoparticles.

Material

Ni/NiO Co/CoO Fe/Fe3O4

Size, nm

Constant 𝐴, 10 ―11 erg/cm

9 11 23 6,4 12

7,5 3,75 1,425 48.9 89.3

Theoretical value of the constant 𝑇0 = 𝐽1𝑡0 𝑘𝐵, К 8,61 8,42 8,40 37,03 45.8

Experimental value of the constant 𝑇0, К 9,1 ± 0,7 33.8 (1) 47 (1)

[9] [11] [13]

(2) Calculated from experimental data from [11] and [13], in which the measurement error was not specified.

Table 4. The approximation parameter 𝑡0, the energy of the exchange interaction 𝐽𝑁𝑖 and the energy of the interphase exchange interaction 𝐽12 for Ni/NiO nanoparticles. Nanoparticle size 𝑎, nm

Approximation parameter 𝑡0

Energy of the exchange interaction 𝐽𝑁𝑖, 10 ―14 erg

Energy of the interphase exchange interaction 𝐽12, 10 ―16 erg

9 11 23

0,0454 0,0451 0,0446

2,77 2,79 2,82

0,886 0,891 0,902