Force between magnetic nanoplates with dipolar interactions

This study considers the dependence of the force caused by the dipolar interaction between small low-dimensional magnets such as single-molecule magnets and two-dimensional magnets on the distance between them within the framework of the dipolar Ising model with nearest-neighbor exchange interactions and long-range dipolar interactions. In particular, we focus on the rapid change in the force between ferromagnetic and antiferromagnetic plates, which arise from the transition of the spin states and explain that this behavior originates from the spin frustrations between magnetic plates. Furthermore, the size and temperature dependence of the interaction energy are investigated using a Monte Carlo simulation.


Introduction
Dipolar interactions play an important role in magnetic and dielectric materials, in addition to nearest-neighbor exchange interactions [1][2][3][4][5]. The dipolar Ising model is a simple model that includes these two interactions and has been used to explain various phase transitions, particularly in cases where short-and long-range interactions coexist [6,7]. For example, the phase transition of the rare-earth compound LiHoF 4 from a paramagnetic to a ferromagnetic state can be explained using the dipolar Ising model [8,9]. Furthermore, recent large-scale simulations [10,11] using the dipolar Ising model revealed the phase diagram with short-range magnetic exchange and long-range dipolar interactions.
We focus on the force between two magnetic nanoplates described by the dipolar Ising model. Among the forces related to dipolar interaction, the van der Waals force is one of the best known, and is a long-range attraction that is involved in surface tension and adhesion [12][13][14]. The magnitude of the van der Waals force between macroscopic objects is typically determined by their dielectric permittivity, which depend the interaction between dipoles in the objects. Similarly, the force between magnetic materials is determined by their magnetic permeability, which are determined by two contributions from short and long interactions such as the dipolar interaction between spins. Thus, essential future of the force between magnetic materials may be captured using the dipolar Ising model.
When two magnetic plates are located in parallel, a certain spin state in the plate depends not only on the spin states surrounding the spin but also the spins in the facing plate. If the plate is thick, the contributions from the spin states that surround the spin are typically large. Conversely, the contributions from the spins in the facing become relatively large as the thickness decreases. Recently, the van der Waals force has been used to combine two-dimensional plates, such as graphene and hexagonal boron nitride. Furthermore, single-molecule magnets [15] and two-dimensional magnets [16][17][18][19] have been developed. Therefore, understanding the force between these materials is important, and we investigated the force between two-dimensional magnetic plates with dipolar interactions [20].
The remainder of this paper is organized as follows. The dipolar Ising model, and calculations for the dipolar interaction between a single spin and spins on a one-dimensional lattice (spin chain) are introduced in section 2. The interaction energy between two parallel spin chains and change in force between spin chains with ferromagnetic and antiferromagnetic exchange interactions [21,22] are presented in section 3. Nano magnetic plates with spins located on the triangular lattice are introduced, and the relation between the rapid change in the interaction energy in two-dimensional spin systems caused by switching the ground state is considered in section 4. The impact of the dipolar interaction between spins in a single array on the interaction energy between arrays is explained in section 5. Finally, the conclusions in relation to frustrations between spins [23] and future problems are discussed in section 6.
2. Dipolar Ising model 2.1. Hamiltonian of the dipolar Ising model The one-and two-dimensional spin systems considered in this study are explained here. The configurations of spins are shown in figure 1. Two spin chains are located parallel to one other with a separation distance d AB , as shown in figure 1(a). The spins in chains A and B are at the same positions on the x-coordinate. Similarly, spin arrays are located in parallel with a separation distance d AB in two-dimensional systems, as shown in figure 1 The two-dimensional arrays are referred to as plates for the remainder of this paper. A cartesian coordinate system is introduced to indicate the spin position, whose z-axis is vertical to the plates. The dipole moment of the spin  p i at a position  r i is parallel to the z-axis and takes μ or −μ. We denote the sets of suffixes i of a spin in the lower A and upper B plates are denoted as Ω A and Ω B , respectively. The dipolar interaction between  p i and  p j is expressed as


The nearest-neighbor exchange between plates A and B is ignored in this study. In addition, it is assumed that the distance between nearest-neighbor spins a is the same for arrays A and B. Similarly, the nearest-neighbor exchange interactions μ are the same in arrays A and B. Accordingly, the unit length and energy is set to a and μ 2 /a 3 ≡U 0 , respectively. Using the unit length, the normalized distance between plates d is defined as d AB /a.

Interaction energy between a single spin and one-dimensional spin chain
The interaction energy is calculated between a single spin and one-dimensional spin chain arranged with a uniform space a before considering the interaction energy between plates. A single spin is set to (0, 0, d) in the normalized xyz-coordinate system, and the spin in the array is located at  r i = (i, 0, 0) for i = −n, −n + 1,K,n − 1, and n. We represent the spin states in spin chain A as S A ≡{σ −n , σ −n+1 , K, σ n }.
The Hamiltonian of a spin chain is expressed as The interaction energy between a single spin and spin chain at the normalized separation distance d is expressed as  The normalized dipolar interaction energy is defined by is the interaction energy for spin configuration Λ. We calculate the normalized dipolar interaction energies for the two spin configurations. The first configuration Λ F is S A = S F and σ B = 1. The second configuration Λ AF is S A = S AF and σ B = 1. Figure 2 shows U S int, F and U S int, AF as functions of the normalized separation distance d.
The interaction energy between a single spin and infinitely long spin array (n = ∞) is calculated using the Euler-Maclaurin formula. Cancelation of interaction energies for the antiferromagnetic state occurs among pairs between σ B and spins in the chain owing to the alternating sign of the spin. Thus, the absolute value of the interaction energy in an antiferromagnetic state is smaller than that in the ferromagnetic state at the same separation distance.
The interaction energy for a ferromagnetic state increases monotonously as the distance increases. This means that the force is attractive. However, the derivative of interaction energy of the spin chain for an antiferromagnetic state with n = 1 changes from positive to negative as the separation increases. Thus, the force changes from attractive to repulsive, and it is explained as follows. The spin state at the origin in spin chain A, which is directly below the single spin, is equal to σ B . This ferromagnetic state significantly contributes to the interaction energy. The interaction between spins at i = −1 and 1 is antiferromagnetic and reaches maximum at d = 3 2. Each contribution is smaller than that between σ 0 and σ B in sheet A. However, the number of pairs doubles. Thus, the antiferromagnetic interaction can exceed the ferromagnetic interaction, and the interaction energy becomes positive. The maximum repulsive force decreases by increasing when n increases from 1 to 3 because the difference between the total contributions from up and down spins to the dipolar interaction energy decreases as n increases.

Dependence of the interaction energy on the separation
The dipolar interaction energy is calculated between two spin chains on parallel lines labeled as A and B with the same lattice constant a, as shown in figure 1(a). The spins on lines A and B are located at the normalized positions {(i, 0, 0) : 0 i < n} and {(i, 0, d) : 0 i < n}, respectively. We denote s i A and s i B as the spin states at x = ia in chains A and B, respectively. The total number of spins is 2n, and is denoted as N. The Hamiltonian in each array is expressed in equation (7), and the Hamiltonian of the dipolar interaction between chains is expressed as The total Hamiltonian is expressed as For simplicity, |J AA | = |J BB | is assumed in the following calculations. The averaged interaction energy over thermal equilibrium states at temperature T is defined by where k B is Boltzmann constant. A dimensionless parameter β ≡μ 2 /a 3 k B T is used to consider the temperature dependence. In addition, the normalized nearest-neighbor exchange interactions J ≡J AA /(μ 2 /a 3 ) is introduced. The solid (β = 1) and dashed (β = ∞) lines in figure 3 show the normalized interaction energy per spin between spin chains, including three spins for the two spin systems. The first system is referred to as 'FF-chains', in which the normalized nearest-neighbor exchange interaction is J = 1 for both chains. The second system is referred to as 'AA-chains', in which J = −1 for all nearest spin pairs. The interaction energy of the FF-chains is smaller than that of the AA-chains, but they both increase in a similar way as the separation increases. The interaction energy weakens at a fixed separation with increasing temperature in both cases. The spin state of the FF-chain is S A = S B = S F or S A = S B = −S F at absolute zero. On the other hand, the spin state in AA-chains at absolute zero is S A = S B = S AF or S A = S B = −S AF [24]. An equality s j A = s j B is maintained in any cases for any j, and no frustration occurs. Thus, the similar separation dependence is observed for the interaction energy. Conversely, the separation dependence of the interaction energy between spin chains with ferromagnetic and antiferromagnetic interactions is considerably different, as explained in section 2.4.

Size dependence of interaction energy between spin chains
In contrast to the nearest-neighbor exchange interaction, the dipolar interaction is far-reaching. Accordingly, the interaction energy per spin strongly depends on the length of the spin chain. The different combinations of spin configurations rapidly increase as the number of spins increase, which it makes it difficult to calculate exactly the interaction energy for long spin chains at a finite temperature. Thus, a Monte Carlo simulation method is introduced [25].
There are many local minima of the energy landscape for spin chains with antiferromagnetic interactions. This implies that a long calculation time is required to obtain accurate interaction energies, particularly at low temperatures, and thus the convergence to the thermal equilibrium distribution must be carefully checked.
According to the Metropolis algorithm [26], the Markov process of spin states S Λ is described as a transition matrix whose elements are the transition probability from an old spin state S old to a new spin state S new , which is given by where S new is the spin state obtained by flipping the spin in S old . For example, S Λ represents an element of the set where λ denotes the eigenvalue of the transition matrix and c i,Λ are constant values determined from the initial probability distribution of the spin states. The largest eigenvalues of the transition matrix are always 1, and the second-largest eigenvalues determine the convergence to the equilibrium distribution for a large t [27]. Figure 4 shows the temperature dependence of the second-largest eigenvalue λ 2 of the transition matrix for the FF-chain with n = 3 and J = 1. As temperature decreases, λ 2 increases rapidly. Thus, the contribution of l 2 to the summation of the right-hand side of equation (15) decreases more slowly than the other contributions as the temperature decreases. Accordingly, this causes a slow convergence in the Monte Carlo simulation.
The dots in figure 3 show the interaction potential energy per spin between the spin chains with N = 100 and β = 1 obtained from Monte Carlo simulations. Here, the acceptance or rejection of the Metropolis algorithm is repeated 10 6 times and averaged for 100 random initial spin states. The potential energy of the FF-chain interactions decreases as the number of spins increase. However, the interaction energy increases in the case of AA-chains owing to the cancellation in the contributions of the parallel and antiparallel spin states.

Interaction energy between ferromagnetic and antiferromagnetic chains
The spin chains with ferromagnetic and antiferromagnetic interactions, which is referred to as FA-chains is also considered. The solid line in figure 5 shows the exact interaction energy of the FA-chains with n = 3, J AA = 1, and J BB = −1 at β = 50. The interaction energy rapidly increases near d = 0.102, unlike in figure 3. The lower dashed line labeled as C 0 represents the dipolar interaction energy between spin states S F and S F , which is a ground state for small separations. The interaction energy remained unchanged by inverting the direction of all spins. Thus,  the spin configuration of a pair −S F and −S F is also a ground state. If the dipolar interaction between spin chains is absent, the ground state in chain B is S AF or −S AF . The dipolar interactions vanishes for large separations. Thus, the spin configurations {1, − 1, 1} labeled as C ∞ becomes the ground state. The interaction energy of C ∞ is represented by the upper dashed line in figure 5, where the superscript in the inset indicates the sheet. Conversely, since the dipolar interaction between spins increases as their distance, the contribution of the  dipolar interaction between spin chains becomes dominant. Therefore, total energy can be decreased by aligning the spin direction, even if the inner energy of the antiferromagnetic energy increases. Consequently, the rapid change in the interaction energy between the chains is caused by a sudden change in the spin state.
The temperature dependence of the force between ferromagnetic and antiferromagnetic chains is shown in figure 6. The rapid change in force is moderated as the temperature increases. Conversely, the force changes discontinuously in the limit of T → 0.
The interaction energy between FA-chains for different number of spins n = 3, 4, and 5 at β = 50 are shown in figure 7. A two-level increase is observed for n = 4 and 5. Moreover, a new ground state C 1 appears that is neither C 0 nor C ∞ . For example, the ground state of FA-chains with n = 4 switches from C 0 to C 1 which is a combination of spin states {1, 1, 1, 1} and {1, 1, 1, − 1} near d = 1.02.

Interaction energy between two magnetic plates
The rapid changes in the interaction energy owing to the switching of the ground states is examined between two magnetic plates. The spins in a plate are located at a triangular lattice point with a lattice constant a, as shown in figure 1(b). The centers of plates A and B exist on the same line perpendicular to the plates. Here, the normalized nearest-neighbor exchange interactions of plates A and B are fixed at −1 and 1, respectively. To make it as simple as possible to grasp the competition between inner and outer interactions, the inner dipolar interaction in a sheet is ignored.
The solid line in figure 8(a) shows the dependence of the exact total energy on the distance between plates, including seven spins each at β = 10. The dashed lines labeled as C 0 and C ∞ represent the interaction energies of the ground states for small and large d, respectively. The spin configurations of C 0 and C ∞ are shown as insets in figure 8(a). Clear changes can be observed near d = 1.17 that are caused by switching the ground state from C 0 to C ∞ . The interaction energy per spin is shown in figure 8(b). Similarly to figure 5, a rapid change in the interaction energy occurs as the distance between the plates increases.
A two-level increase appears in the FA-chains by increasing the number of spins. Whether similar phenomena occurs for the magnetic plates is also examined. The interaction energy between magnetic plates, including 19 spins in each sheet at β = 1 obtained using the Monte Carlo simulation is shown in figure 9(a). A change in the ground state is important for considering the separation dependence of the interaction energy. However, finding the exact ground state becomes more difficult as the number of spins increases. Thus, the following two assumptions is introduced. First, the spin state in plates A and B is the same for small d. Second, all the spins in sheet A are up or down for a large d. The inset shows the spin configuration C 0.6 of the ground state at d = 0.6, which is determined using the first assumption, and C ∞ at d = ∞ , which is determined using the second assumption. It should be noted that there is ground states with the same energy, and the inset shows one of them.
The ground state changes depending on the distance, similar to figure 8(b). However, unlike figure 7, a pronounced multistep increase is not seen in figure 9(a). This is because the rapid change is suppressed by small β (i.e. high temperature), as shown in figure 6. In addition, there is another reason. Figure 9(b) shows that different ground states obtained from the assumption mentioned above exist. Although not all ground states are covered, the number of different ground states increases as the number of spins increases. Accordingly, the spin configuration of ground state gradually changes as the distance increases, and the interaction energy per spin increases more smoothly for large magnetic plates at a finite temperature. Consequently, the rapid change can be observed between small magnetic materials, and methods to measure the magnetic force acting single molecule has been developed [28].

Effect of the dipolar interaction in a sheet on interaction energy
The dipolar interaction is added between spins in a sheet, which has been neglected in previous sections. The exact total energy of the ground state of the magnetic plates, including seven spins per plate, is shown in figure 10(a) The ground states from spin configurations C 0 to C ∞ changes depending on the distance, as seen in inset in figure 10(a). The spin states in plates A and B are the same for C 0 and C 1 . Opposite-directed spin pairs exist in the ground state with an additional dipolar interaction for a small d. This is in contrast to the spin configuration C 0 in the ground state shown in figure 8. The interaction energy between the plates in the ground state change discontinuously at the distance where the ground state switches, as shown in figure 10(b). Although the total energy increases monotonously with the distance between plates, the interaction energy may be decreases at discontinuous point. The inset of figure 10(b) shows the interaction energy at β = 50 near d = 0.97, and indicates that the interaction energy decreases as the separation distance increases. Figure 11 shows the force between the magnetic plates at β = 50. Interestingly, the force becomes positive near d = 1. This means that the force can be repulsive. If the dipolar interaction between plates is large, i.e. μ 2 /a 3 ?J AA , J BB , the rapid change in the force is observed at larger distances.

Conclusion
Dipolar interactions can affect distant spins compared with the nearest-neighbor exchange interactions. Thus, the simplest model that can explain the force arisen by dipolar interaction is the dipolar Ising model. This study demonstrated that the force between two-dimensional spin arrays strongly depends on the spin state of each array using the dipolar Ising model. In particular, rapid changes in the force are demonstrated for a combination of ferromagnetic and antiferromagnetic array. These force changes originates from the frustration of the spin interaction.
All spins are in the same direction in the ground state of the ferromagnetic system. However, the interaction energy arising from the nearest-neighbor exchange interaction in the antiferromagnetic system decreases by increasing the opposite direction spin pairs. The dipolar interaction between plates decreases by aligning the direction of the spins in plates A and B. Aligning the spin pairs in opposite directions of plates A and B in order to reduce the dipole interaction energy between plates may increase the proximity exchange interaction energy of the antiferromagnetic system.
The interaction energy between arrays is dominant for small distances. Thus, all spin direction are aligned, even at the expense of increasing the nearest-neighbor exchange interaction in the antiferromagnetic system. Conversely, the spin state is determined only by the interactions between plates for large distances. A rapid change in the force occurs at a distance where the two contributions are balanced.
The van der Waals force is caused by the correlations in the fluctuating dipoles. Thus, the dielectric function, which depends on frequency, is necessary to consider the van der Waals force between dielectric solids more accurately. Accordingly, a kinetic dipolar Ising model [29,30] is needed in future to investigate the force acting on van der Waals structures. Two-dimensional van der Waals magnetic materials such as CrI 3 may be good materials to examine correlations between inner magnetic interactions in a plane and interactions between layers. In particular, coexisting ferromagnetic-antiferromagnetic state is observed in a twisted bilayer CrI 3 [31]. Thus, the frustrations considered in this study may be appeared.