Magnetization Reversal in Magnetic Bilayer Systems

Li-Peng Jin(金礼鹏), Yong-Jun Liu(刘拥军)$^{\hyperlink{s}{**}}$

doi:10.1088/0256-307X/36/6/067504

⇦Chinese Physics Letters, 2019, Vol. 36, No. 6, Article code 067504⇨ Magnetization Reversal in Magnetic Bilayer Systems * Li-Peng Jin (金礼鹏), Yong-Jun Liu (刘拥军)** Affiliations School of Physical Science and Technology, Yangzhou University, Yangzhou 225002 Received 3 March 2019, online 18 May 2019 ^*Supported by the National Natural Science Foundation of China under Grant Nos 11647316 and 11847313.
^**Corresponding author. Email: yjliu@yzu.edu.cn Citation Text: Jin L P and Liu Y J 2019 Chin. Phys. Lett. 36 067504 Abstract Magnetization reversal in magnetic soft/hard bilayer systems is studied analytically by means of a variational method for magnetic energies in a continuum model. The demagnetization curve is involved with nonlinear equations, and the solution is given implicitly in the form of Jacobi functions, which is valid for the total reversal process. Based on the non-trivial solutions, hysteresis loops, as well as the maximum energy product $(BH)_{\max}$ versus thicknesses of soft/hard layers are obtained. With regard to $(BH)_{\max}$, improvement of the remanence competes with loss of coercive force. As a result, an optimum condition exists. For a given thickness of the hard layer, the optimum condition at which the largest $(BH)_{\max}$ could be achieved is discussed, which is slightly different from previous works. DOI:10.1088/0256-307X/36/6/067504 PACS:75.70.Cn, 75.60.Jk © 2019 Chinese Physics Society Article Text Large $(BH)_{\max}$ defined as the maximal value of energy product $(BH)=-\mu_0H(M+H)$ over the demagnetization curve in the second quadrant is crucial to practical applications, such as electric motors, wind turbine generators, power generators and future ultrahigh density magnetic storage.[1–7] To achieve high-performance magnets with large $(BH)_{\max}$, one way is to find new types of hard magnetic materials, but with slow progress.[8] Another way is to take advantage of both soft and hard magnets, i.e., a composite magnetic material which comprises a hard part to provide large coercivity and a soft part to provide high remanence,[9–15] which is the so-called spring-exchange mechanism. By applying a magnetic field in the $z$-direction, a bilayer system constituting an infinitely large, magnetically soft and hard layer in the $xy$-plane shows different behaviors. Magnetic moments in the soft layer begin to realign with the field direction, and the magnetization in the hard layer remains unchanged. Hence, the magnetic reversal process of the soft layer is obstructed at the interface, and a non-collinear magnetic configuration forms. To gain a deep insight into the magnetization reversal of the soft/hard bilayer magnets, we have derived analytical equations for the magnetization configuration under perpendicular external field using the micromagnetic energy functional. Different from previous works,[16–26] we present the detailed magnetic reversal implicitly in the form of Jacobi functions,[27–30] which are valid to the whole process, as a result, hysteresis loops as well as the maximum energy product $(BH)_{\max}$ can be safely obtained. As the thickness of the soft layer increases, there is a competition between remanence enhancement and loss of coercivity. The optimal conditions for obtaining permanent magnets with the largest $(BH)_{\max}$ are discussed. We study a magnetic system with exchange-coupled soft/hard layers, where the easy axis is perpendicular to the layers. We take the $z$ axis in the direction of easy axis and the origin at the interface as shown in Fig. 1.

Fig. 1. Geometry of the bilayer system and notation used in this study.

The angle $\theta$ measured from the negative direction of the $z$ axis is used to describe configurations of magnetization which is considered to be uniform in the $x$ and $y$ directions. Here $\theta$ is supposed to be continuous at the interface between the layers, which corresponds to a strong interlayer exchange coupling. The thicknesses of soft and hard magnets are denoted by $a$ and $b$, respectively. The total magnetic energy per unit area is expressed as[31,32] $$\begin{align} \gamma =\,& \int_0^a \Big[A_{\rm s} \Big(\frac{d\theta}{dz}\Big)^2-K_{\rm s} \cos^2\theta +\mu_0H M_{\rm s} \cos\theta\Big]dz \\ &+ \int_{-b}^0 \Big[A_{\rm h}\Big(\frac{d\theta}{dz}\Big)^2-K_{\rm h} \cos^2\theta +\mu_0H M_{\rm h} \cos\theta\Big]dz,~~ \tag {1} \end{align} $$ where the first term in the integral kernel is the ferromagnetic exchange-coupling energy, the second term is the anisotropy energy, the last term is the Zeeman energy, and $A$, $K$ and $M$ stand for the exchange stiffness constant, the anisotropy constant and the saturation magnetization, with subscripts s and h standing for soft and hard magnets, respectively. The applied field $H$ is in the positive direction of the $z$ axis, and the normalized component of the magnetization in the $z$ direction is given by $$ \bar{M} =\frac{ \int_0^a M_{\rm s}\cos\theta dz+\int_{-b}^0 M_{\rm h}\cos\theta dz }{a+b},~~ \tag {2} $$ with the remanence of the bilayer system of $M_{\rm ab}=(aM_{\rm s}+bM_{\rm h})/(a+b)$. By applying the variational method to the energy in Eq. (1), we obtain the following differential equations for the function $\theta(z)$, $$ \begin{cases} \!\! K_{\rm s}\frac{d\sin^2\theta}{d\theta}+\mu_0HM_{\rm s}\frac{d\cos\theta}{d\theta}=2 A_{\rm s} \frac{d^2\theta}{dz^2},& 0\le z\le a,\\\!\! K_{\rm h}\frac{d\sin^2\theta}{d\theta}+\mu_0HM_{\rm h}\frac{d\cos\theta}{d\theta}=2 A_{\rm h} \frac{d^2\theta}{dz^2},& -b\le z\le 0, \end{cases}~~ \tag {3} $$ under three boundary conditions $$ \begin{cases} \!\! \frac{d\theta}{dz}|_{z=a}=\frac{d\theta}{dz}|_{z=-b}=0,\\\!\! A_{\rm s}\frac{d\theta}{dz}|_{z=0^+}=A_{\rm h}\frac{d\theta}{dz}|_{z=0^-}. \end{cases}~~ \tag {4} $$ These three boundary conditions imply that the exchange interaction at the interface is very strong and the magnetic moments change very slowly at the top of the soft layer and the bottom of the hard layer. By multiplying $d\theta/dz$ to the two equations in Eq. (3) and integrating them from $z=a$ and $z= -b$ respectively, we arrive at $$ \begin{cases}\!\! A_{\rm s} (\frac{d\theta}{dz})^2=\,&K_{\rm s}(\cos^2 \theta_a -\cos^2 \theta)\\\!\! &+\mu_0HM_{\rm s} (\cos\theta-\cos\theta_a), \\\!\! A_{\rm h} (\frac{d\theta}{dz})^2 =\,& K_{\rm h}(\cos^2 \theta_{-b} -\cos^2 \theta)\\\!\! &+\mu_0HM_{\rm h}(\cos\theta-\cos\theta_{-b}), \end{cases}~~ \tag {5} $$ where we have used the first two boundary conditions in Eq. (4). The above equations always have two trivial solutions: $\theta\equiv 0$ and $\theta\equiv\pi$. For the configuration satisfying the condition $d\theta/dz>0$ in $-b < z < a$, there exits a non-trivial solution $$\begin{alignat}{1} \!\!|!\!\!\!\tan\theta_z/2=\begin{cases}\!\! {\rm cn}[\hat{a}-\hat{z},r]\tan\theta_a/2,& (0\le z\le a)\\\!\! \frac{\tan\theta_{-b}/2}{{\rm dn}[\tilde{b}+\tilde{z}, q]},& (-b\le z\le 0), \end{cases}~~ \tag {6} \end{alignat} $$ with Jacobian elliptic function ${\rm cn}[\hat{x},r]$ and ${\rm dn}[\tilde{x},q]$, where $$\begin{align} \hat{x} =\,& x\sqrt{K_{\rm s}/A_{\rm s}} \sqrt{(h_{\rm s}-2\cos\theta_a)/2},\\ r^2 =\,& (1-\cos\theta_a) (h_{\rm s}+1-\cos\theta_a)/(h_{\rm s}-2\cos\theta_a),\\ \tilde{x} =\,& \frac{x}{2}\sqrt{K_{\rm h}/A_{\rm h}} \sqrt{(1+\cos\theta_{-b})(1+\cos\theta_{-b}-h_{\rm h})},\\ q^2 =\,& 2(2\cos\theta_{-b}-h_{\rm h})/(1+\cos\theta_{-b})\\ &\cdot(1+\cos\theta_{-b}-h_{\rm h}),~~ \tag {7} \end{align} $$ with $\hat{x}$ and $\tilde{x}$ being the 'field dependent' scaled length for soft and hard layers, respectively. The external field $H$ enters into the above equations through the parameters $h_{\rm s(h)}=\mu_0HM_{\rm s(h)}/K_{\rm s(h)}$. The equations are meaningless for $h_{\rm h}>2$, which happens to be the predicted coercive force for the hard layer. Under such strong applied field, the magnetization should be the saturated along the direction of field. Integrating the above non-trivial magnetization distribution in $[-b,a]$, Eq. (2) becomes $$\begin{align} \bar{M}=\,&\frac{1}{a+b}\Big\{ M_{\rm s} \sqrt{2A_{\rm s}/K_{\rm s}} \frac{\cos\theta_a+1}{\sqrt{h_{\rm s}-2\cos\theta_a}}\\ &\times {\it \Pi} (\lambda,\sin^2\theta_a/2,r )\\ &+ M_{\rm h}\sqrt{A_{\rm h}/K_{\rm h}}\sqrt{\frac{1+\cos\theta_{-b}}{\cos\theta_{-b}+1-h_{\rm h}}}\\ &\times \Big[\sec^2(\theta_{-b}/2) F(\kappa,q)- \tan^2(\theta_{-b}/2)\\ &\times {\it \Pi}\Big(\kappa,\frac{2\cos\theta_{-b}-h_{\rm h}}{\cos\theta_{-b}+1-h_{\rm h}},q\Big)\Big]\\ &-(aM_{\rm s}+bM_{\rm h})\Big\},~~ \tag {8} \end{align} $$ where ${\it \Pi}$ and $F$ are the elliptic integral of the third and first kind, $r$ and $q$ are defined in Eq. (7), $\lambda=\arccos({\rm cn}[\hat{a},r])$ and $\kappa=\arcsin({\rm sn}[\tilde{b},q])$ for soft and hard layers, respectively. After obtaining the non-trivial magnetization distribution, the third boundary condition in Eq. (4) can be reduced to a nonlinear equation for $\theta_a$ and $\theta_{-b}$, $$\begin{align} \sqrt{A_sK_{\rm s}}\sqrt{2(h_{\rm s}-2\cos\theta_a)}{\frac{ {\rm sn}[\hat{a},r] {\rm dn}[\hat{a},r]}{ {\rm cn}[\hat{a},r]}}=\sqrt{A_hK_{\rm h}} \\ \times \frac{ 2(2\cos\theta_{-b}-h_{\rm h})}{\sqrt{(1+\cos\theta_{-b})(1+\cos\theta_{-b}-h_{\rm h})}} \frac{ {\rm sn}[\tilde{b},q]{\rm cn}[\tilde{b},q]}{ {\rm dn}[\hat{b},q]}.~~ \tag {9} \end{align} $$ Although Eq. (9) can give an implicitly analytical solution of magnetic configuration for any given $H$, $a$ and $b$, one should notice that boundary conditions in Eq. (4) require a very slow change of magnetic moments near both ends of the bilayer system and a rigid coupling at the soft/hard interface. It can be concluded that, as the applied field increases, the magnetic moment at the top of the hard layer (near the interface) begins to deviate while it hardly changes at the bottom. To satisfy this condition, $b$ should at least be of the same order of magnitude as the domain-wall length $\pi\sqrt{A_{\rm h}/K_{\rm h}}$. Hence, we only show the results in the region of $b\geq 10$ nm in the following. The main results in the present work are Eqs. (8) and (9), the implicit analytical solution of the magnetic moment reversal process, namely, we obtain a function $\theta(K_{\rm s},M_{\rm s},A_{\rm s},K_{\rm h},M_{\rm h},A_{\rm h},H)$ describing magnetic reversal for the positive magnetic parameters. One should note that the solution is valid for any $0 < K_{\rm s} < K_{\rm h}$, even for a combination of hard and 'not so hard' magnetic materials. For demonstrating, we choose $\alpha$-Fe and FePt corresponding to soft and hard magnets as an example in the following. The material properties are listed in Table 1.

**Table 1.** Micromagnetic properties of $\alpha$-Fe and FePt.[33]
	$K$ (MJ/m$^3$)	$M$ (MA/m)	$A$ (pJ/m)
$\alpha$-Fe	0.048	1.71	25
FePt	7.0	1.15	10

For a soft/hard bilayer demagnetized from the saturated state, the nucleation field is the point at which the soft layer starts to deviate at the top surface, which can be derived analytically. The initial magnetized structure $\theta(z)=0$ is the stable configuration without external field and remains metastable for small positive fields because the configuration is trapped by an energy barrier.[31,34] As the external field increases, $\theta(z)=0$ becomes unstable, and a non-trivial metastable state would originate from it. At the bifurcation point, the energy barrier is zero, i.e., the difference between the energies associated with these two states is zero. This field strength is defined as the nucleation field, above which a non-uniform magnetization contribution is nucleated and metastable. By setting $\theta_a=\theta_{-b} =0$ in Eq. (9), the explicit expression for the nucleation field $H_{\rm n}$ could be obtained as $$ H_{\rm n}= \frac{2A_sK_{\rm s}\tan^2\hat{a}_0+2A_hK_{\rm h}\tanh^2\tilde{b}_0}{A_sM_{\rm s}\tan^2\hat{a}_0+A_hM_{\rm h}\tanh^2\tilde{b}_0},~~ \tag {10} $$ where $\hat{a}_0=a\sqrt{K_{\rm s}/A_{\rm s}}\sqrt{\mu_0 M_sH_{\rm n}/2K_{\rm s}-1}$ and $\tilde{b}_0=b\sqrt{K_{\rm h}/A_{\rm h}}\sqrt{1-\mu_0 M_hH_{\rm n}/2K_{\rm h}}$ are the field-dependent scaled lengths at nucleation field. Equation (10) corresponds to an implicit equation for the nucleation field, which decreases with $a$ and increases with $b$. The corresponding plots of $H_{\rm n}$ versus $a$ and $b$ are given in Fig. 2.

Fig. 2. The nucleation fields versus thicknesses for $\alpha$-Fe and FePt bilayer systems.

One can see that the nucleation field is more sensitive to $a$ from Fig. 2. For a thick soft layer, the system always has small nucleation field even with large $b$ for $\tanh\tilde{b}\ll \tan\hat{a}$ under large $a$. For fixed $a$, the contribution of the hard layer to nucleation field would converge rapidly to the limit as $b$ increases. The reason lies in $\tanh\tilde{b}$ could be safely replaced by 1 for large $b$, then Eq. (10) is independent of $b$. For a system with imperfections, it is impossible to restore the hardness even with infinite hard parts. After the nucleation field, the non-uniform metastable states appear. By solving Eq. (9) with Eq. (6), $\theta_a$ could be obtained numerically. As a result, the magnetization distribution in the soft and hard layers could be obtained and $\bar{M}$ can be calculated with Eq. (8). As the applied field exceeds a certain point, there will be no non-trivial solutions of those equations, which indicates that there is no local energy extremum, thus the metastable state disappears and an irreversible jump to $\theta\equiv\pi$ happens. That field strength is defined as the reversal field $H_{\rm r}$, corresponding to the coercive force of the system. The hysteresis loops are calculated numerically through the obtained $\bar{M}$ under given fields between nucleation ($\bar{M}=M_{\rm ab}$) and reversal ($\bar{M}=-M_{\rm ab}$) point, and the inverse demagnetization process is easily obtained from symmetry.

Fig. 3. (a) Hysteresis loops for $\alpha$-Fe/FePt bilayer system with various thicknesses of the soft layer (the hard layer thickness is fixed at 20 nm). (b) The magnetization distribution at reversal point along the $z$ axis for various thicknesses of the $\alpha$-Fe layer (the FePt layer is fixed at 20 nm).

The hysteresis loops in Fig. 3(a) show the $M$–$H$ loops for various $a$ under $b=20$ nm. For the thin soft layer ($a=2$ nm), the hysteresis loop is nearly rectangular. The remanence $M_{\rm ab}$ is dominated by $M_{\rm h}$, and the reversal field is roughly equivalent to the nucleation field $\sim$2.43 MA/m, while it is far away from the predicted value $2K_{\rm h}/\mu_0M_{\rm h} \sim 9.74$ MA/m for the hard layer, indicating that imperfections would greatly reduce the hardness of magnets.[31] As the thickness of the soft layer increases, the remanence $M_{\rm ab}$ and the nucleation field converge to $M_{\rm s}$ and $2K_{\rm s}/\mu_0M_{\rm s}$ gradually, while the shape of the hysteresis loop loses its rectangularity, i.e., the reversal field deviates greatly from the nucleation point. After the nucleation field, the total magnetization undergoes a rapid drop. One can see that the reversal field occurs even after $\bar{M}$ changes sign for the thick soft layer ($a=20$ nm). Moreover, the reversal field has a lower limit with the increase of the soft layer. To clarify the change brought by various thicknesses of the soft layer, we plot the corresponding spatial magnetization distributions just before the reversal point in Fig. 3(b). We can see that the common location in the non-uniform distribution begins at the top surface of the soft layer and shrinks as it propagates into the hard layer, while for thick $a$, the top surface is almost nucleated at $\theta=\pi$. Since in the whole reversal, the soft layer always acts as an imperfection and triggers the nucleation. For a thin soft layer this nucleation would induce the irreversible jump of the whole system quickly, while the nucleation can be pinned at the interface for a thick soft layer in contrast, and an almost entire inversion would happen at the top surface. As a result, a $\pi$ domain wall can be formed for a thick soft layer. Thus the phenomena related to the total magnetization reversal change from nucleation to pinning as the soft layer increases. A general formula for the nucleation field can be derived analytically, while the formula for the pinning field can only be derived in the extreme case.[30] When both $a$ and $b$ are infinite, a $\pi$ domain wall can be formed before the total reversal takes place, i.e., $\theta_{a}=\pi$ and $\theta_{-b} =0$. Connecting with the third boundary condition (4), Eq. (5) turns to an implicit equation for $\theta_0$ at the interface, and the critical point can be found as follows: $$ H_{\rm r}=\frac{2(A_hK_{\rm h}-A_sK_{\rm s})}{(\sqrt{A_sM_{\rm s}}+\sqrt{A_hM_{\rm h}})^2}, ~a,b\rightarrow\infty,~~ \tag {11} $$ which has already been discussed.[30] The performance of a permanent magnet is described by the maximal value of energy product $(BH)=-\mu_0H(M+H)$ over the demagnetization curve in the second quadrant. For the Stoner–Wohlfarth model[34] which has an ideal rectangular hysteresis loop, $(BH)$ increases with coercive force $H_{\rm c}$ and remanent magnetization $M_{\rm r}$, while it can never exceed $\mu_0M_{\rm r}^2/4$ due to a parabolic dependence of applied field.[33] For the bilayer case, the magnetization remains $M_{\rm ab}$ until the applied field is less than $-H_{\rm n}$, then decreases with fields. Thus, for the case $H_{\rm n}\le M_{\rm ab}/2$, $(BH)_{\max}$ can be easily obtained as $\mu_0M_{\rm ab}^2/4$, while for $H_{\rm n}>M_{\rm ab}/2$, we need to find $(BH)_{\max}$ numerically. In practice, we only need to evaluate demagnetization curves from $H=-H_{\rm n}$ to the intersection of the straight line $H=-M(H)/2$ and hysteresis loops, because the energy product on the rest of demagnetization curves should always be less than that on the intersection. Figure 4(a) shows the calculated energy products for various $a$ and $b$, from which one can see that the maximum energy product $(BH)_{\max}$ is also more sensitive to the soft layer thickness $a$. For a given $b$, $(BH)_{\max}$ firstly increases with $a$, reflecting that the contribution of remanence enhancement takes advantage of the coercivity loss with the addition of the soft phase. Further increasing $a$ leads to a fast drop of $(BH)_{\max}$, because the slight gain of remanence cannot compensate for the significant loss of coercivity.

Fig. 4. (a) The value of $(BH)_{\max}$ versus thickness, and (b) $(BH)_{\max}$ versus thickness of the soft layer $a$ under fixed $b=20$ nm. Here $a_{\rm c}=6.906$ nm is the thickness satisfying $H_{\rm n}(a;b)=M_{\rm ab}/2$, while the actual largest value of $(BH)_{\max}$ appears at $a=7.199$ nm.

From the above discussion, we can obtain that the coercivity decreases but the remanence increases with the increase of the soft layer thickness. The remanence enhancement and loss of coercivity compete with each other as contributing to the energy product, hence the optimum $a$ for the largest energy product could exist for a given $b$. The optimal condition which has been suggested as following in previous researches[16,17] $H_{\rm n}=M_{\rm ab}/2$. However, we find that our numerical results slightly deviate from this prediction. In Fig. 4(b), we plot $(BH)_{\max}$ versus the soft layer thickness $a$ under the fixed hard layer thickness $b=20$ nm. We find that the actual largest value of $(BH)_{\max}$ appears at $a=7.199$ nm, while the thickness $a_{\rm c}$ satisfying $H_{\rm n}(a;b)=M_{\rm ab}/2$ equals 6.906 nm. For a fixed thickness $b$ of the hard layer, as long as the condition of $H_{\rm n}\le M_{\rm ab}/2$ is satisfied, $(BH)_{\max}$ equals $\mu_0M_{\rm ab}^2/4$ and monotonously increases with $a$. Supposing that $a_{\rm c}$ is the thickness of the soft layer which happens to satisfy $H_{\rm n}(a;b)=M_{\rm ab}/2$, then $(BH)_{\max}$ indeed has a peak value at $a=a_{\rm c}$ when $a$ is in the interval of $[0,a_{\rm c}]$. However, the peak value cannot be the largest $(BH)_{\max}$ for the fixed $b$. When $H_{\rm n} \le M_{\rm ab}/2$, we cannot obtain an analytical expression for $(BH)_{\max}$, the lower limit for $(BH)_{\max}$ could be obtained easily, $(BH)_{\rm max}\ge \mu_0(M_{\rm ab}-H_{\rm n})H_{\rm n}= f(a;b)$, which happens to be the exact $(BH)_{\max}$ at $a=a_{\rm c}$. The derivative of $f(a;b)$ to $a$ reads the expression $$ \frac{\partial f(a;b)}{\partial a}=\mu_0\Big(\frac{\partial H_{\rm n}}{\partial a} (M_{\rm ab}-2H_{\rm n})+ \frac{d M_{\rm ab}}{da} H_{\rm n}\Big).~~ \tag {12} $$ It can be easily deduced that the derivative at $a=a_{\rm c}$ is positive, thus a slight increase of $a$ would enhance $f(a;b)$ and the maximum value of $f(a;b)$ does not locate at $a=a_{\rm c}$. Since $f(a_{\rm c};b)$ happens to be the exact $(BH)_{\max}$ at $a=a_{\rm c}$, the largest $(BH)_{\max}$ should not occur at $a=a_{\rm c}$ for a fixed $b$. From the above discussion, we suggest that $H_{\rm n}=M_{\rm ab}/2$ is not the optimal condition for the largest $(BH)_{\max}$, which is consistent with our numerical results. However, the optimal condition cannot be given with an explicit expression, and the value of $(BH)_{\max}$ at $H_{\rm n}=M_{\rm ab}/2$ may be a rough approximation. The above discussion is straightforward for $b$, and the plot of $(BH)_{\max}$ in Fig. 4(a) gives a practical region where a giant energy product may be achieved within the upper limit of thickness $a$. In conclusion, magnetization reversal for a soft/hard bilayer has been investigated systematically based on a one-dimensional micromagnetic model. With an external magnetic field, soft and hard layers show different responses. Considering a rigidly coupled boundary condition, the magnetic reversal process of the soft layer is hindered at the interface, forming a spring-like magnetic configuration. The total reversal process begins with the nucleation at the top surface of the soft layer, followed by the reversible deviation from the soft layer to the hard layer and finally an irreversible reversal of the whole system occurs. This procedure can be described by nonlinear equations, which is analytically solved with an implicit expression in the present work. It is found that the thickness of the soft layer plays a much more important role in the magnetic reversal process. With increasing $a$, the factor governing coercivity switches from nucleation to pinning, and the nucleation field drops rapidly. The exchange-spring mechanism can take advantage of both soft and hard magnetic materials, i.e., large remanence and coercivity. However, as the thickness of the soft layer increases, there is a competition between remanence enhancement and loss of coercivity. We have discussed the optimum condition for the largest $(BH)_{\max}$ with a given thickness of the hard layer. Unlike previous research, our results suggest that the optimal thickness of the soft layer is slightly larger than the thickness satisfying $H_{\rm n}(a;b)=M_{\rm ab}/2$. However, the optimal condition cannot be given with an explicit expression, and our results provide possible ranges of $a$ and $b$ for a giant energy product. Lastly, it should be mentioned that there are still some drawbacks in the present work. For example, in real systems, a non-uniform distribution of the anisotropy axis is typically present and the demagnetization factor induced by dipole interaction should be considered, which is beyond the scope of the present work and need further numerical simulations. 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