Temperature Dependence of the Magnetic Hysteresis Curves in Magnetic Multilayers

Abstract

We performed a theoretical study of the magnetic hysteresis curves for an Fe₃O₄-based magnetic multilayer. Such a systems are known as the spin valves in the spintronics field. To investigate the temperature dependence of the magnetization process in such a system, we extend the retarded trace method and plot the magnetic hysteresis curve by computer simulation. Our results show very good agreement with the experimental data.

1. Introduction

Fe₃O₄ (magnetite) is a promising candidate as a spintronics device^1–3) material, and it has been used as a magnetic material for a long time. According to first-principles calculations of Fe₃O₄, the high spin polarization is expected high magnetoresistance ratio (MR ratio)⁴⁾. In a previous paper⁵⁾, it was conducted a theoretical study about the experimental data of the magnetic multilayers of Fe₃O₄ material shown in Ref. 6). In these systems, iwe reported results derived from magnetic hysteresis curves as a function of various parameters by computer simulation using the retarded trace method^5,7–9). However, theoretical consideration of the temperature dependence of these hysteresis curves has not been reported. The aim of this paper is to extend the retarded trace method to calculate the temperature dependence of the magnetic hysteresis curves. This paper is organized as follows. We will first explain the model Hamiltonian in magnetic multilayer Fe₃O₄ materials. In section 3, our retarded trace method is extended to calculate the temperature dependence of the magnetic hysteresis curves. In section 4, we report the simulation results, and in section 5, we make a comparison between the results of our simulation and the experimental results. The last section comprises a summary and discussion.

2. The Model Hamiltonian in the Magnetic Multilayers

In this section, we describe theoretical model in magnetic multilayer of the Fe₃O₄ materials. The magnetic multilayers used for these investigations are ferromagnetic/nonmagnetic/ferromagnetic/antiferromagnetic junction structures, which are spin valves^10–12). Figure 1 shows a model of the Fe₃O₄/MgO/Fe₃O₄/NiO magnetic multilayers, where each Fe₃O₄ layer has ferromagnetic (or antiferromagnetic) interactions via the MgO layer.

Fig. 1

Schematic diagram of Fe₃O₄/MgO/Fe₃O₄/NiO magnetic multilayer film system.

The Hamiltonian of this system, $\mathcal{H}$⁵⁾, using the analogy of the Ising model of ferrimagnets, is given as follows:   

\[\mathcal{H} = \mathcal{H}_A + \mathcal{H}_B + \mathcal{H}_{AB} + {H_{Zee}},\](1)

where $\mathcal{H}_A \equiv -\frac{J_{AA}}{2} \sum\limits_{i,j} {S_z}^A(i) {S_z}^A(j)$, $\mathcal{H}_B \equiv -\frac{J_{BB}}{2} \sum\limits_{i,j} {S_z}^B(i) {S_z}^B(j)$, $\mathcal{H}_{Zee} \equiv \sum\limits_i [{S_z}^A(i) + {S_z}^B(i)] h_0(i)$, and $\mathcal{H}_{AB} \equiv J_{AB} \sum\limits_{\langle i,j \rangle} [{S_z}^A(i) {S_z}^B(j) + {S_z}^B(i) {S_z}^A(j)]$ represent the exchange interaction energy in the A layer, the exchange interaction energy in the B layer, the Zeeman energy, and the exchange interaction between the A and B layers, respectively. Here, $J_{AA} > 0$, $J_{BB} > 0$, $J_{AB} > 0$, ${S_z}^A(i)$, ${S_z}^B(i)$, and $h_0(i) \equiv \mu_B g H_Z(i)$ represent the exchange interaction in the A layer, the exchange interaction in the B layer, the exchange interaction between A and B sites, the z component of the spins of the ith site in the A layer, the z component of the spins of the ith site in the B layer, and the applied magnetic field of the ith site (where, $\mu_B$, g and H_z are the g-factor, Bohr magneton, and z component of the magnetic field respecttively), respectively. Where, given the mean field $m_A$ of the spins ${S_z}^A(i)$ in A layer and the mean field $m_B$ of the spins ${S_z}^B(i)$ in B layer, the Hamiltonian $\mathcal{H}_{MF}$ of the mean field approximation, from the total spin Hamiltonian, $\mathcal{H}$ of eq. (1) is given by $\mathcal{H}_{MF} \equiv {\mathcal{H}_{MF}}^A + {\mathcal{H}_{MF}}^B + {\mathcal{H}_{MF}}^{AB}$. Where, ${\mathcal{H}_{MF}}^A \equiv z_A J_{AA} N {m_A}^2/2$, ${\mathcal{H}_{MF}}^B \equiv z_B J_{BB} N {m_B}^2/2$, and ${\mathcal{H}_{MF}}^{AB} \equiv -z_{AB} J_{AB} N_{AB} m_A m_B - {\bar{h}_{\mathit{eff}}}^A \sum\limits_j^N {S_z}^A(j) - {\bar{h}_{\mathit{eff}}}^B \sum\limits_j^N {S_z}^B(j)$ are the mean-field energies in the A layers, the mean field energies in the B layers, and the mean field energy between the A layer and B layer, respectively. Where, ${\bar{h}_{\mathit{eff}}}^A \equiv h_0 + z_A J_{AA} m_A - z_{AB} J_{AB} m_B$ and ${\bar{h}_{\mathit{eff}}}^B \equiv h_0 + z_B J_{BB} m_B - z_{BA} J_{BA} m_A$ are the effective applied magnetic fields in the A layer and the B layer, respectively. Further, $z_A$, $z_B$, $z_{AB}$ and $z_{BA}$ are parameters of effective number of the nearest neighbors sites in the A layer and the B layer, from the A layer to the B layer, and from the B layer to the A layer, respectively. The relationship between the mean field partition function $Z(\beta)$ and the mean field free energy $F_{MF}(\beta)$ in this system, which depends on the inverse temperature $\beta \equiv 1/k_B T$ (Where, T and k_B are temperature and Boltzmann factor, respectively.), is as follows:   

\[-\beta F_{MF}(\beta, h_0) \equiv \ln Z(\beta) = -\beta (F_A + F_B + F_{AB}),\](2)

where $-\beta F_a(\beta, h_0) \equiv N \{ -\beta z_a J_{aa} {m_a}^2/2 + \ln [\sinh [ (2S^a + 1) \beta {\bar{h}_{\mathit{eff}}}{}^a/2 ]/\sinh (\beta {\bar{h}_{\mathit{eff}}}{}^a/2) ] \}\ (a = A,B)$ and $-\beta F_{AB} \equiv \beta z_{AB} J_{AB} N_{AB} m_A m_B$ are the mean field free energies in the A layer, the B layer, and A from the B layer, respectively. The mean field free energy of eq. (2) is the starting point of our analysis method for simulating the magnetic hysteresis curves by the retarded trace method, which is described in the next section.

3. Theory of the Retarded Trace Method

In our previous paper⁵⁾, we discussed the basic theory of the retarded trace method. The local magnetization $m$, in magnetic multilayers of Fe₃O₄ is given by $-\partial F/\partial h_0 \equiv m = m_A + m_B$. The mean fields $m_A$ and $m_B$ in the A and B layers can be considered as the local magnetizations in the A and B layers, respectively. These are defined by $m_A = S_A B_S (\beta {\bar{h}_{\mathit{eff}}}{}^A S_A)$ and $m_B = S_B B_S (\beta {\bar{h}_{\mathit{eff}}}{}^B S_B)$ respectively, where $B_S(x)$ is the Brillouin function. Our aim is to plot a hysteresis curve from the numerical calculation. Therefore, $m_A$ and $m_B$ in a static self-consistent equation must be extended to a dynamic self-consistent equation because of the change that occurs upon application of a magnetic field, $h_0(t)$. In our previous paper, in conjunction with the Barkhausen jumps, we assumed the approximate relationship between the change of the applied magnetic field $\Delta h_0(t)$, and the time interval, $\Delta t$, was $h_0(t + \Delta t) \simeq h_0(t) + \Delta h_0(t)$. Its difference width $\Delta h_0(t)$ of the applied magnetic field is involve time interval $\Delta t$, it is related to the coercivity, $H_C$, which is determined from the experimental results. In accordance with the hypothesis that the change in the magnetic field on the right side of the equation is reflected in the delay time on the left side. This hypothesis was described on the basis of Hove's phenomenological dynamical theory in our previous paper. We introduce the pair of dynamic self-consistent equations⁵⁾ for each of the terms $m_a(\beta, h(t))\ (a = A,B)$ as follows:   

\[ \begin{split} m_A[h_A(t_{n+1},\beta)] \simeq {} & S_A B_{S_A} \{ b(T) [q_{AA} m_A (\beta, h(t_n)) \\ & {} + q_{AB} m_B (\beta, h(t_n)) + ch_A (t_n, \beta)] \}, \end{split} \](3)

  

\[ \begin{split} m_B[h_B(t_{n+1}, \beta)] \simeq {} & S_B B_{S_B} \{ b(T) [q_{BB} m_B (\beta, h(t_n)) \\ & {} + q_{BA} m_A (\beta, h(t_n)) + ch_B(t_n, \beta) ] \}, \end{split} \](4)

Where, $b(T) \equiv z_{AB} J_{AB} \beta$, $q_{AA} \equiv z_A J_A/z_{AB} J_{AB}$, $q_{BB} \equiv z_B J_B/z_{BA} J_{BA}$, $q_{AB} = q_{BA} = 1$, and c are the product of interlayer interaction and inverse temperature, interaction within A layer normalized by interlayer interaction, interaction within B layer normalized by interlayer interaction, self normalized interlayer interaction, and an arbitrary scale parameter of applied magnetic field.

and $h_a(t_n, \beta)\ (a = A,B)$ are the effective magnetic fields, defined by $h_a(t_n, \beta) \equiv {\bar{h}_{\mathit{eff}}}{}^a(t_n, \beta) - [z_a J_{aa} m_a(t_n, \beta) - z_{a\bar{a}} J_{a\bar{a}} m_{\bar{a}} (t_n, \beta)]$. Here, we have assumed that the approximate relationship between the change of the effective magnetic fields, $\Delta h_a(t_n, \beta)\ (a = A,B)$, and the time interval, $\Delta t$, is as follows:   

\[h_a(t_{n+1}, \beta) \simeq h_a(t_n, \beta) + \Delta h_a(t_n, \beta),\](5)

  

\[\Delta h_a(t_n, \beta) \equiv [1 + Q_a^0(\beta) \beta] \Delta h_0.\](6)

Here, $Q_a^0(\beta) \equiv z_a J_{aa} {\kappa_a}^0(\beta, \varepsilon) - z_{a\bar{a}} J_{a\bar{a}} {\kappa_{\bar{a}}}^0(\beta, \varepsilon)$ is a coefficient that depends on β, where ${\kappa_a}^0(\beta, \varepsilon) \equiv 1/2\{ (2S_a + 1)^2 [1 - \cosh [(2S_a + 1)\beta h_0] \cosh (2\varepsilon)]/[\cosh [(2S_a + 1) \beta h_0] - \cosh (2\varepsilon)]^2 - [1 - \cosh (\beta h_0) \cosh (2\varepsilon)]/[\cosh (\beta h_0) - \cosh (2\varepsilon)]^2\}$. In the derivation of eqs. (5) and (6) and in accordance with Refs. 13) and 14), instead of the usual definition $\coth x = \cosh x/\sinh x$, we used $\coth (x,\varepsilon) \equiv \sinh (2x)/[\cosh (2x) - \cosh (2\varepsilon)]$^13,14) (ε is an arbitrary cut-off parameter). Based on our previous study, a method for deriving the hysteresis curve from eqs. (3)–(6) will be referred to as the retarded trace method, and is dependent on the temperature.

4. Numerical Simulation Results

In this section, we show the simulation results of the hysteresis curve at each temperature calculated by the extended retarded trace method. Figure 2 shows a comparison between the simulation and experimental results at T = 300 K in the Fe₃O₄ multilayes. Figure 2(a) shows the numerical simulation results of the hysteresis curve under the following parameters: T = 300 K, $q_{AA} = q_{BB} = 0.56$, $S^B/S^A = -0.2$, $2zS = 2.8 \times 10^{-16}$, $c = 5 \times 10^{16}$, $J_{AB} = 5 \times 10^2$ erg, and $\Delta H \simeq 115.9$ Oe. Figure 2(b) shows the experimental results¹²⁾ of the magnetic hysteresis curve measured along the [100] direction at T = 300 K for the Fe₃O₄/MgO/Fe₃O₄/NiO multilayers. The coercivity values of the simulation and experimental curves at T = 300 K were 71.43 Oe and 49.18 Oe, respectively.

Fig. 2

Comparison of the magnetic hysteresis curve for (a) the numerical simulation results and (b) the experimental results at 300 K for the Fe₃O₄ multilayers.

Figure 3 shows a comparison between the simulation and experimental results at T = 200 K in the Fe₃O₄ multilayers. Figure 3(a) shows the numerical simulation results of the hysteresis curve under the following parameters: T = 200 K, $q_{AA} = q_{BB} = 0.56$, $S^B/S^A = -0.2$, $2zS = 2.8 \times 10^{-16}$, $c = 5 \times 10^{16}$, $J_{AB} = 3 \times 10^2$ erg, and $\Delta H \simeq 152.2$ Oe. Figure 3(b) shows the experimental results¹²⁾ of the magnetic hysteresis curve measured along the [100] direction at T = 200 K for the Fe₃O₄/MgO/Fe₃O₄/NiO multilayers. The coercivity values of the simulation and experimental curves at T = 200 K were 207.1 Oe and 105.8 Oe, respectively. Therefore, the coercivity values of the simulation and experiment curves at T = 200 K were 2.90 times and 2.15 times higher, respectively, than the corresponding coercivity values at T = 300 K.

Fig. 3

Comparison of the magnetic hysteresis curve for (a) the numerical simulation results and (b) the experimental results at 200 K for the Fe₃O₄ multilayers.

Figure 4 shows a comparison between simulation and experimental results at T = 100 K for the Fe₃O₄ multilayer film. Figure 4(a) shows the numerical simulation results of the hysteresis curve under the following parameters: T = 100 K, $q_{AA} = q_{BB} = 0.56$, $S^B/S^A = -0.2$, $2zS = 2.8 \times 10^{-16}$, $c = 5 \times 10^{16}$, $J_{AB} = 1.25 \times 10^2$ erg, and $\Delta H \simeq 478.3$ Oe. Figure 4(b) shows the experimental results¹²⁾ of magnetic hysteresis curve measured along the [100] direction at T = 100 K for the Fe₃O₄/MgO/Fe₃O₄/NiO multilayers. The coercivity values of the simulation and experimental results at T = 100 K were 826.4 Oe and 894.7 Oe, respectively. Therefore, the coercivity values of the simulation and experiment curves at T = 100 K were 11.57 times and 18.19 times higher, respectively, than the corresponding coercivity values at T = 300 K.

Fig. 4

Comparison of the magnetic hysteresis curve for (a) the numerical simulation results and (b) the experimental results at 100 K for the Fe₃O₄ multilayers.

5. Discussion and Summary

First, we compare the results of simulation and experiment about the shape of hysteresis curve. The simulation results shown in Fig. 2(a) show a double loop that is constricted near the center, which is characteristic of the hysteresis curve of the experimental results at T = 300 K, shown in Fig. 2(b). The simulation results shown in Fig. 3(a) also show constriction near the center, which is characteristic of the hysteresis curve in the experimental results at T = 200 K, shown in Fig. 3(b).

The simulation results shown in Fig. 4(a) show no constriction near the center for large coercivities of the hysteresis curve in the experimental results at T = 100 K, shown in Fig. 4(b).

As shown in Figs. 2–4, the simulation results under parameters of $q_{AA} = q_{BB} = 0.56$, $S^B/S^A = -0.2$, $2zS = 2.8 \times 10^{-16}$, and $c = 5 \times 10^{16}$ show very good agreement with the experimental data. Thus, differences in the shapes of these magnetic hysteresis curves are characterized by three parameters, i.e., temperature, $T$, interlayer coupling, $J_{AB}$, and the step of the external magnetic field, $\Delta H$. The value of $J_{AB}(J_1)$ with T = 300 K was used from the value in Ref. 12). Regarding $J_{AB}$ values in 200 K and 100 K, at each temperature, it was decided by selecting the value of $J_{AB}$ that resembles the shape of the magnetization curve is shown in the Ref. 12).

The value of $J_{AB}$ obtained from the shape of the magnetization curve at 200 K was $J_{AB} = 3 \times 10^2$ erg. This result is not a bad value because the value of $J_{AB}$ shown in the graph of Fig. 1(e) in Ref. 12) is $J_{AB} = 4 \times 10^2$ erg. On the other hand, the value of $J_{AB}$ obtained from the shape of the magnetization curve at 100 K was $J_{AB} = 1.25 \times 10^2$ erg. This result is different from our theoretical value and sign because the value of the experimental result of $J_{AB}$ shown in the graph of Fig. 1(e) in Ref. 12) is $J_{AB} = -8 \times 10^2$ erg. However, the shape of the magnetization curve was very well reproduced. Secondly, we will compare simulation and experiment about the coercivity. The steps of the external magnetic field, $\Delta H$, are related to the coercivity. As described in our previous paper⁵⁾, the physical cause of the coercivity is related to the delay of the response of the “Barkhausen jump”^15–19) by the applied magnetic field. From the temperature dependence of the step of the effective magnetic field that was introduced by eq. (5), it is possible to understand the causes of the increased coercive force with decreasing temperature.

We summarize this paper as follows. We have developed a “retarded trace method” in Fe3O4-based magnetic multilayers. One of the results is that the temperature dependence of the coercive force was obtained from the simulation. Another result is that it was able to explain delicate shape change of magnetization curve according to temperature change. We suggested one method to determine parameters such as interlayer coupling from the relationship with the parameters that determine the shape of the magnetization curve.

Acknowledgments

I would like to express my deepest gratitude to Prof. Mohri, whose comments and suggestions were extremely valuable throughout the course of my study. Special thanks also to Prof. Uchikawa and Prof. Tamaki, for their invaluable comments and encouragement.

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