Magnetization Process in Bilayer Honeycomb Spin Lattice

Abstract

The magnetization process of the AA-stacked bilayer honeycomb spin lattice in the out-of-plane applied field is examined in the framework of the Ising model and mean field approximation. Competition between different kinds (antiferromagnetic: AF, or ferromagnetic: FM) of intra-layer and inter-layer spin exchange couplings could lead to the first order magnetization process, characterized by sharp jumps in magnetization curves occurring in critical external fields (spin-flop and spin-flip fields). There is only the spin-flip field at very low temperature and its magnitude depends not only on the intra-layer frustration level but also exchange coupling between spin layers. During the magnetization process, the initial AF bilayer honeycomb spin lattice may undergo ferrimagnetic or week-ferromagnetic states before reaching full ferromagnetic saturation state by the spin-flip field. The results of this work can be applied for an explanation of the spin-flip phenomenon registered in the AF bilayer CrI₃.

Fig. 1 AA-stacking bilayer honeycomb spin lattice in the external magnetic field perpendicular to the layer surfaces.

1. Introduction

Due to their novel fundamental features and potential usage in contemporary nano-scale technologies, two-dimensional (2D) layered magnetic materials, including monolayer and bilayer systems, are intriguing scientific objects.¹⁾ The work²⁾ reviews the magnetic behavior of a few typical 2D-systems and associated hetero-structures. It was shown that the magnetic structure of the layered crystals depends strongly on the signs of the nearest neighbor (NN) intra-layer J and NN interlayer J_P exchange parameters. There are some van der Waals material systems,²⁾ which have the intralayer ferromagnetic (FM) NN exchange coupling J much larger in magnitude than the interlayer antiferromagnetic (AF) exchange coupling J_P. For example, the CrI₃ bilayer is layered antiferromagnetic, while each spin plane has a FM order because of J > 0 (FM exchange) and J_P < 0 (AF exchange). CrI₃ bilayer has almost zero magnetization at zero magnetic field and presents spin-flip transition into FM state for the magnetic fields larger than a critical value ∼0.7 T (see Ref. 3)). It is well known that a system with random competing FM and AF exchange interactions could exhibit the first-order magnetization process^4,5) (FOMP) with step-like anomalies. The aim of the present research is to study the finite temperature magnetization process of the bilayer antiferromagnetic honeycomb spin lattice using the simple Ising model and the mean field approximation (MFA). This analysis helps us better understand the magnetization curve with the application of an out-of-plane field perpendicular to the honeycomb spin lattice thin film surfaces.

2. Ising Model and Mean Field Approximation

2.1 Hamiltonian of the Ising model for the bilayer honeycomb spin lattice

Consider a bilayer honeycomb bipartite spin lattice having 4N spins. Each layer has 2N spins distributed evenly in two triangular spin sub-lattices and is denoted by the index n (n = 1, 2). The position of a spin in every ν sub-lattice (ν = a, b) of a monolayer is characterized by two-dimensional lattice vectors j. Three nearest neighbors of a-spin at site j are b-spins defined by vectors $\boldsymbol{\varDelta}:\boldsymbol{\varDelta}_{1} = \frac{a_{0}}{2}(3,\sqrt{3} )$, $\boldsymbol{\varDelta}_{2} = \frac{a_{0}}{2}(3, - \sqrt{3} )$, Δ₃ = a₀(−1, 0), where a₀ is the length of the hexagonal edge (Fig. 1). Three NN of a b-spin at site j are a-spins positioned by −Δ. The six next nearest neighbors (NNN) of ν-spin are the same ν-spins and its position is denoted as Δ′ (not shown in Fig. 1). Each spin belonging to the ν-sub-lattice of the single layer arranges above its equivalent on the other, or AA-stacking. Ising model Hamiltonian for the honeycomb bilayer spin system in the external magnetic field is written as   

\begin{align} H &= -h_{0}\sum\nolimits_{n\nu \boldsymbol{j}}s_{n\nu \boldsymbol{j}} - \frac{J}{2}\sum\nolimits_{n\boldsymbol{j},\boldsymbol{\varDelta}}(s_{na\boldsymbol{j}}s_{b\boldsymbol{j} + \boldsymbol{\varDelta}} + s_{nb\boldsymbol{j}}s_{a\boldsymbol{j} - \boldsymbol{\varDelta}}) \\ &\quad - \frac{J'}{2} \sum\nolimits_{n\boldsymbol{j},\boldsymbol{\varDelta}'}(s_{na\boldsymbol{j}}s_{a\boldsymbol{j} + \boldsymbol{\varDelta}'} + s_{nb\boldsymbol{j}}s_{b\boldsymbol{j} + \boldsymbol{\varDelta}'}) \\ &\quad - J_{P}\sum\nolimits_{\nu\boldsymbol{j}}s_{1\nu\boldsymbol{j}}s_{2\nu\boldsymbol{j}} \end{align} (1)

Fig. 1

AA-stacking bilayer honeycomb spin lattice in the external magnetic field perpendicular to the layer surfaces.

Here h₀ is the external (out-of-plane) magnetic field perpendicular to the spin layers and given in an energy unit. The spins of the bilayer spin lattice direct themselves along the z direction and are denoted by s_nνj in the crystallographic xyz frame. This frame has the z-axis (x-axis) perpendicular (in-plane) to the honeycomb spin lattice planes. J and J′ are the in-plane NN and NNN exchange couplings between spins. J_P is the exchange coupling between the NN spin pair at different layers but belonging to the same sub-lattice. We consider the case realized in the van der Waals structures, when the magnitude of the intra-layer NN exchange coupling J is sufficiently larger than that of the NNN intra-layer J′ and the NN inter-layer J_P exchange parameters.

2.2 Mean field approximation

For simplicity, we adopt the mean field approximation (MFA). MFA Hamiltonian H₀ can be obtained by reducing the product of two spin operators in the second, third, and fourth sums of (1) to linear terms: $s_{n\nu \boldsymbol{j}}s_{n'\nu '\boldsymbol{j'}} \Rightarrow s_{n\nu \boldsymbol{j}}m_{n'\nu '} + s_{n'\nu '\boldsymbol{j'}}m_{n\nu } - m_{n\nu }m_{n'\nu '}$. One gets   

\begin{align} H_{0} &= 3NJ \sum\nolimits_{n}m_{na}m_{nb} + NJ_{P}\sum\nolimits_{\nu = a,b}m_{1\nu}m_{2\nu} \\ &\quad - \sum\nolimits_{n\nu \boldsymbol{j}}h_{n\nu}s_{n\nu \boldsymbol{j}}, \end{align} (2)

  

\begin{align} &h_{1a(1b)} = h_{0} + 3(Jm_{1b(1a)} + 2J'm_{1a(1b)}) + J_{P}m_{2a(2b)},\\ &h_{2a(2b)} = h_{0} + 3(Jm_{2b(2a)} + 2J'm_{2a(2b)}) + J_{P}m_{1a(1b)}. \end{align} (3)

Here m_nν is the thermodynamic average of the spin moment per site m_nν = ⟨s_nνj⟩. Brackets mean the thermodynamic average ⟨…⟩ = Tr[exp(−βH₀)…]/Tr[exp(−βH₀)] and β = 1/(k_BT). In the formula (3), h_nν plays the role of the total field exerted on the spin at nνj site, which is composed of the external h₀ and internal molecular fields. It is easy to obtain the expression for the free energy $f = - \frac{1}{4N\beta }\textit{ln}\,Tr(e^{ - \beta H_{0}})$ and magnetic moment per spin from (2)   

\begin{align} f &= \frac{3}{4}\sum\nolimits_{n}[Jm_{na}m_{nb} + J'(m_{na}^{2} + m_{nb}^{2})] \\ &\quad + \frac{J_{P}}{4}\sum\nolimits_{\nu}m_{1\nu}m_{2\nu} - \frac{1}{4\beta}\sum\nolimits_{n\nu}\textit{ln}\left\{\cfrac{sh\biggl[\biggl(s + \cfrac{1}{2}\biggr)y_{n\nu}\biggr]}{sh\biggl(\cfrac{y_{n\nu}}{2}\biggr)}\right\}, \end{align} (4)

  

\begin{equation} m_{n\nu} = b_{s}(y_{n\nu}). \end{equation} (5)

The Brillouin function b_s(y) figured in the eq. (5) has the following form   

\begin{equation} b_{s}(y) = (s + 1/2)cth[(s + 1/2)y] - \frac{1}{2}cth(y/2). \end{equation} (6)

In the next part, we consider possible anti-parallel orientations of the spins in the bilayer spin lattice, which may be not only AF but also ferrimagnetic (FERI) orders in present of the external magnetic field. In the numerical calculation, all quantities are non-unit, and the unit of energy is the magnitude of the NN exchange energy |J|. That means temperature $\tau = \frac{k_{B}T}{|J|} = \beta^{ - 1}$, and field $h \equiv \frac{h}{|J|}$.

3. Magnetization Process

The different bilayer spin lattice orders can be analyzed in the MFA using eqs. (3), (4), and (5).

3.1 Ferromagnetic spin configuration in each layer but AF order in bilayer

Firstly, we study the case when the spins in each layer are ferromagnetically ordered, and the average sub-lattice spin moments of each layer are equal   

\begin{align} & m_{1a} = m_{1b} = m_{1}, m_{2a} = m_{2b} = m_{2},\\ & m_{1} = b_{s}(y_{1});m_{2} = b_{s}(y_{2}). \end{align} (7)

The magnetic fields exerting on the spin at the first and the second layers and the free energy per spin are   

\begin{align} h_{1} &= h_{0} + 3(J + 2J')m_{1} + J_{P}m_{2}, h_{2} \\ &= h_{0} + 3(J + 2J')m_{2} + J_{P}m_{1}, \end{align} (8)

  

\begin{align} f &= \frac{3}{4}(J + 2J')(m_{1}^{2} + m_{2}^{2}) + \frac{J_{P}}{2}m_{1}m_{2} \\ &\quad - \frac{1}{2\beta}\sum\nolimits_{n}\textit{ln}\left\{ \cfrac{sh\biggl[\biggl(s + \cfrac{1}{2}\biggr)y_{n}\biggr]}{sh\biggl(\cfrac{y_{n}}{2}\biggr)}\right\}, \end{align} (9)

  

\begin{align}y_{n} = \beta h_{n}\end{align}

It is clear that, the intra-layer NNN interaction J′ gives an additional contribution to the molecular field of the same layer spins. The first and the second derivatives of free energy with respect to the magnetic field give us the spin moment m and the susceptibility χ of the bilayer   

\begin{equation} m = -\frac{\partial f}{\partial h_{0}} = \frac{1}{2}(m_{1} + m_{2}), \end{equation} (10)

  

\begin{equation} \chi = -\frac{\partial^{2}f}{\partial h_{0}^{2}} = \frac{\beta}{2}\left\{\frac{b_{s}^{\prime}(y_{1}) + b_{s}^{\prime}(y_{2}) - 2\beta[3(J + 2J') - J_{p}]b_{s}^{\prime}(y_{1})b_{s}^{\prime}(y_{2})}{1 - 3(J + 2J')\beta[b_{s}^{\prime}(y_{1}) + b_{s}^{\prime}(y_{2})] + \beta^{2}[9(J + 2J')^{2} - J_{p}^{2}]b_{s}^{\prime}(y_{1})b_{s}^{\prime}(y_{2})}\right\}. \end{equation} (11)

The first derivative of the Brillouin function presented in (11) is written as   

\begin{equation} b_{s}^{\prime}(y) = \frac{1}{4sh^{2}(y/2)} - \cfrac{\biggl(s + \cfrac{1}{2}\biggr)^{2}}{sh^{2}\biggl[\biggl(s + \cfrac{1}{2}\biggr)y\biggr]}. \end{equation} (12)

We consider the AF solution of the eqs. (7), (8), (9) (m₂ = −m₁) at a certain temperature, which satisfies the extreme condition of free energy (10) (11) in the zero external magnetic field (h₀ = 0). When the field is turned on, m₂ may differ m₁ in magnitude and direction.

Figure 2(a) shows the temperature dependence of the spin moment and the susceptibility in the zero field. The starting state of the spin bilayer is AF observed in zero field when two ferromagnetic layers have opposite spin moments (m₁ = −m₂). If a small field (h₀ = 0.2) is activated, the FERI spin configuration (m₁ > 0, m₂ < 0, m ≠ 0) is induced. The spins of each layer have opposite directions and different magnitudes and lead to the weak ferromagnetic behavior with net spin moment m per site shown in Fig. 2(b).

Fig. 2

Temperature dependence of the net spin moment m, and the susceptibility χ in the zero (a) and h₀ = 0.2 magnetic field (b). Here the parameters are: J = 1, J′ = 0.1, J_P = −0.2, s = 3/2. The layer spin moments m₁, m₂ are also shown in (a).

Figure 3 shows the field dependence of the free energy in the out of plane magnetic field for three cases J′ = 0.1, 0, −0.1, respectively. The starting state of the spin bilayers is AF (m₂ = −m₁) at the zero magnetic field. We notice that, the free energy of the antiparallel layer spin configuration changes discontinuously in the critical field h₀₁ and h₀₂. In the magnetic field h₀ ≤ h₀₂, FERI state occurs. The FERI state changes to a saturated FM state when the magnetic field is larger than the critical magnetic field h₀₂. The saturated FM state corresponds to the linear part of the free energy curve with the greatest slope. The small NNN intra-layer exchange coupling J′ only shifts the free energy by a constant amount. Shape and magnitude of the free energy depend strongly on the value of the interaction parameters and temperature.

Fig. 3

Field dependence of the free energy of the bilayer honeycomb spin film for three cases when the intra-layer NNN exchange J′ = 0.1, 0, −0.1 respectively. Temperature is τ = 1.2 and other parameters are: J = 1, J_P = −0.2, s = 3/2 for all curves.

Figure 4(a) exposes the magnetization process of the antiparallel layer spin configuration in the out-of-plane magnetic field, which exhibits clearly the FOMP with anomalies at the critical fields, h₀₁ and h₀₂. The critical fields, h₀₁ and h₀₂ could be interpreted as the “spin flop” and “spin flip” fields as in the 3D spin model,⁶⁾ respectively. Different from the rotation 3D spin model where the spin moment only changes direction, the anisotropic Ising model shows the shortening of the layer spin moment m₂ oriented opposite to the magnetic field direction. The positive NNN intra-layer exchange couplings J′ = 0.1 has the effect of decreasing the critical fields at a given temperature τ = 1.2. Frustration occurs in the case when J = 1 and J′ = −0.1 and shows the opposite effect, which is increasing the critical fields at a given temperature. Figure 4(b) shows a change of magnetization curves with decreasing temperature. At very low temperature, only the spin-flip process occurs when the spin system changes directly from the AF state to the saturation FM state.

Fig. 4

Magnetization process of the AF bilayer honeycomb spin lattice exhibits steps at the critical field h₀₁, h₀₂, which are shifted with varying J′ (a); Change of magnetization curves with varying temperature when J′ = 0 (b). Spin s = 3/2, J = 1 for all curves and other parameters are shown in figures.

Figure 5 shows the FOMP occurs at low temperature τ = 0.01 in the weak frustrating bilayer honeycomb spin lattice (|J′|/J < 1). There is only the spin flip field and its magnitude decreases with increasing frustration as in Fig. 5(a) or with decreasing the magnitude of the AF exchange coupling J_P between bilayer spins as in Fig. 5(b). By the similarity, the FOMP described in Fig. 5 can be used to illustrate the experimental magnetization curve of CrI₃ (see Ref. 3)). In this work, the spin-flip phenomenon is registered at low temperature and the CrI₃ bilayer is believed to be the Ising antiferromagnet with two ferromagnetic layers oriented oppositely due to AF interlayer exchange coupling.

Fig. 5

Low temperature magnetization curves of the AF bilayer honeycomb spin lattice with variation of the NNN J′ (a) and the interlayer J_P (b) exchange couplings. The values of the parameters are: τ = 0.01, J = 1, s = 3/2 for all curves.

3.2 Sub-lattice spin moments have the same magnitude but opposite direction

In this case, the sub-lattice spin moments of the two layers are equal (m_1a = m_2a = m_a, m_1b = m_2b = m_b). The magnitude of both spin layer moment is the same (m₁ = m₂ = m). The magnetic fields acting on the sub-lattices and the free energy are now   

\begin{align} &h_{a} = h_{0} + 3J\,m_{b} + (J_{p} + 6J')m_{a},\\ &h_{b} = h_{0} + 3J\,m_{a} + (J_{p} + 6J')m_{b}, \end{align} (13)

  

\begin{align} f &= \frac{3J}{2}m_{a}m_{b} + \frac{1}{4}(J_{p} + 6J')(m_{a}^{2} + m_{b}^{2}) \\ &\quad - \frac{1}{2\beta}\sum\nolimits_{\nu}\textit{ln}\left\{\cfrac{sh\biggl[\biggl(s + \cfrac{1}{2}\biggr)y_{\nu}\biggr]}{sh\biggl(\cfrac{y_{\nu}}{2}\biggr)}\right\}. \end{align} (14)

Similar to (10) and (11) the spin moment m and the susceptibility χ of the spin bilayer are given by   

\begin{equation} m = -\frac{\partial f}{\partial h_{0}} = \frac{1}{2}(m_{a} + m_{b}), \end{equation} (15)

  

\begin{equation} \chi = -\frac{\partial^{2}f}{\partial h_{0}^{2}} = \frac{\beta}{2}\left\{\frac{b_{s}^{\prime}(y_{a}) + b_{s}^{\prime}(y_{b}) + 2\beta[3J - (J_{p} + 6J')]b_{s}^{\prime}(y_{a})b_{s}^{\prime}(y_{b})}{1 - \beta(J_{p} + 6J')[b_{s}^{\prime}(y_{a}) + b_{s}^{\prime}(y_{b})] - \beta^{2}[9J^{2} - (J_{p} + 6J')^{2}]b_{s}^{\prime}(y_{a})b_{s}^{\prime}(y_{b})}\right\}. \end{equation} (16)

Figure 6(a) shows the dependence of the free energy on the out of plane magnetic field when J′ = 0.15, 0, −0.15, and the honeycomb spin bilayer is on AF order without an external field. The corresponding magnetization curves for the total spin moment are shown in Fig. 6(b). We observe FOMP when J′ = 0, 0.15 and the temperature is τ = 0.2. In the intra-layer frustration case (J′ = −0.15, J = −1), the bilayer spin system transforms smoothly from the AF state into a saturated FM state with no jumps. The small frustration could destroy FOMP at nonzero temperature, and an evolution of the magnetization process with varying temperature is exhibited in Fig. 7.

Fig. 6

Field dependence of the free energy of a spin bilayer system with initial AF state when J′ = 0.15, 0, −0.15 (a). Magnetization curves in the field perpendicular to the layer spin planes (b). Temperature τ = 0.2 and the other parameters are J_P = 0.2, J = −1.0, s = 3/2.

Fig. 7

Magnetization process of the bilayer honeycomb spin lattice at different temperatures. Parameters are given as: J = −1.0, J_P = 0.2, s = 3/2, and J′ = 0.15 (a), J′ = −0.15 (b).

Figure 7(a) illustrates a change of the magnetization curves with increasing temperature when frustration is absent. Two steps of FOMP are clear at low temperature, smeared at increasing temperatures and joined to one at τ = 3. For higher temperatures τ > 3, FOMP is not clear. Unlike case 3.1, the AF state in each spin honeycomb layer gradually transitions to a weak FM state with a constant spin moment in the range h₀₁ ≤ h₀ < h₀₂, and suddenly changes to a saturated FM state in the spin-flip field. Temperature is in the range of 0.1 to 1. No FOMP in the temperature interval 0.1 ≤ τ ≤ 3.0 is observed in Fig. 7(b) for the frustration case.

Magnetization process at the ground state is illustrated in Fig. 8 with a variation of the interlayer exchange coupling J_P. Figure 8(a) shows that only the spin-flip field exists at zero temperature and that the magnitude of this field decreases as the FM interlayer exchange coupling J_P decreases. The similar decrease of the spin-flip field is also seen in the case of the intra-layer frustration (Fig. 8(b)).

Fig. 8

Variation of magnetization process of the AF spin bilayer at zero temperature and different FM interlayer exchange coupling. Case (a) and (b) shows FOMP when J′ = 0 and J′ = −0.15.

In general, frustrating ground-state phenomena in bilayer spin systems are difficult to describe with great accuracy by the MFA method because it ignores fluctuations. Different AF phases including collinear AF phase may exist at zero temperature in AA-stacked honeycomb bilayer lattice as shown by the authors of Ref. 7) using the frustrated spin 1/2 Heisenberg model and the coupled cluster method implemented to very high orders. However, the simple Ising model and MFA used here provides a basis understanding of the magnetization process at finite temperature in “more classical” spin 3/2 AA-stacked honeycomb bilayer lattice with the AF order.

4. Conclusion

The magnetization process of the AA-stacked bilayer honeycomb spin lattice is investigated using the Ising model in the out-of-plane magnetic field and the mean field approximation. It is shown that FOMP can occur at low temperatures and characterizing by the spin–flop and the spin-flip fields even in the case of small intra-layer frustration. Intermediate states of the AF bilayer honeycomb spin lattice in the magnetization process may be FERI or weak FM state depending on the exchange couplings. The obtained result can be used to interpret the spin-flip phenomenon observed experimentally in CrI₃.

Acknowledgments

The authors thank the National Project ĐTĐL. CN-27/23 for support.

REFERENCES

• 1)   K.F.  Mak,  J.  Shan and  D.C.  Ralph: Nat. Rev. Phys. 1 (2019) 646–661. doi:10.1038/s42254-019-0110-y
• 2)   M.  Gibertini,  M.  Koperski,  A.F.  Morpurgo and  K.S.  Novoselov: Nat. Nanotechnol. 14 (2019) 408–419. doi:10.1038/s41565-019-0438-6
• 3)   B.  Huang,  G.  Clark,  E.  Navarro-Moratalla,  D.R.  Klein,  R.  Cheng,  K.L.  Seyler,  D.  Zhong,  E.  Schmidgall,  M.A.  McGuire,  D.H.  Cobden,  W.  Yao,  D.  Xiao,  P.  Jarillo-Herrero and  X.  Xu: Nature 546 (2017) 270–273. doi:10.1038/nature22391
• 4)   G.H.  Bach,  O.K.T.  Nguyen,  C.V.  Nguyen and  C.T.  Bach: Mater. Trans. 56 (2015) 1320–1322. doi:10.2320/matertrans.MA201507
• 5)   B.T.  Cong: J. Magn. Magn. Mater. 117 (1992) 126–132. doi:10.1016/0304-8853(92)90300-D
• 6)  M. Getzlaff: Fundamental of Magnetism, (Springer-Verlag, Berlin, Heidelberg, 2008) pp. 100–102.
• 7)   P.H.Y.  Li and  R.F.  Bishop: J. Magn. Magn. Mater. 555 (2022) 169307. doi:10.1016/j.jmmm.2022.169307