The calculation of the magnonic spectra using the plane-wave method has limitations, the origin of which lies in the formulation of the effective magnetic field term in the equation of motion (the Landau-Lifshitz equation) for composite media. According to ideas of the plane-wave method the system dynamics is described in terms of plane waves (a superposition of a number of plane waves), which are continuous functions and propagate throughout the medium. Since in magnonic crystals the sought-for superposition of plane waves represents the dynamic magnetization, the magnetic boundary conditions on the interfaces between constituent materials should be inherent in the Landau-Lifshitz equations. In this paper we present the derivation of the two expressions for the exchange field known from the literature. We start from the Heisenberg model and use a linear approximation and take into account the spacial dependence of saturation magnetization and exchange constant present in magnetic composites. We discuss the magnetic boundary conditions included in the presented formulations of the exchange field and elucidate their effect on spin-wave modes and their spectra in one- and two-dimensional planar magnonic crystals from plane-wave calculations.
For the first time the exchange effects were discovered independently by W. Heisenberg and P.A.M. Dirac in 1926. They proposed the energy operator (Hamiltonian) for the exchange interaction between two particles with spins
The derivation of an exchange field in a uniform ferromagnetic material from the microscopic Heisenberg Hamiltonian (
On the interface between two ferromagnetic materials, the boundary conditions (BCs) on dynamical component of the magnetization vector should be imposed. Such boundary conditions were proposed by Hoffman, then developed, and investigated by other authors [
Composites with a periodic arrangement of two (or more) different materials are extensively studied from many years. In the past, structures with periodicity in one dimension were investigated, these are multilayered structures which found many applications, for example, as a Bragg mirror or in GMR devices [
Semiconductor periodic heterostructures (SHs) allow to tailor the electron and heat transport in nanoscale [
In this study we will show in details the derivation from the microscopic model different forms of the exchange field used for SW calculations in MCs. Then we will analyze differences in SW spectra in one- (1D) and two-dimensional (2D) thin films of MCs calculated with PWM for three different expressions of the exchange field. We will discuss the boundary condition implemented in each formulation. The paper consists of five sections. In Section
We split the derivation of the expression of the exchange field in inhomogeneous materials into two steps. First, in Section
We start our calculations from the Heisenberg Hamiltonian
The discrete lattice of spins. The angle between neighboring spins:
Let us assume that the angle
Having the continuous function of a position vector in hand, we can expand a unit vector
For a homogeneous material the length of the spins is preserved,
Equation (
To define energy density,
To calculate the total exchange energy,
The SW can be regarded as coherent precession of the magnetization vector around its equilibrium direction. Based on this observation most SW calculation are performed in linear approximation. This approximation was already used once in our paper, it is in (
In linear approximation the magnetization vector
The exchange energy, (
In the following we will make further assumptions to obtain another expression for the exchange field. We can write that
To summarize, we have derived two different formulas for the exchange energy in linear approximation, which are equivalent in the case of homogeneous material. These are
Exchange field can be derived from the exchange energy functionals (
For Form I of the exchange energy functional
After this assumption the exchange magnetic field in the Form I will be obtained solely from the second term in (
For exchange energy written in the Form II as defined in (
We have shown the derivation of two different expressions for the exchange field in nonuniform ferromagnetic materials in linear approximation. These are
We can also add to this list the exchange field in Form III, which is derived directly from the exchange energy functional (
From the parameters introduced just above:
It is worth to note at this moment that differential operators in the definition of the exchange field, (
Those three different formulas for the exchange field will be investigated for the calculation of the magnonic band structure in thin plates of 1D and 2D Mcs with PWM.
The PWM is a useful tool used for study systems with discrete translational symmetry, including electronic, photonic, phononic and magnonic crystals [
We will consider slabs of 1D or 2D MCs (Figure
Structure of a 1D MC (a) and 2D MC (b) considered in this manuscript. 2D MC is formed by cylindrical dots A arranged in a square lattice immersed into a ferromagnetic matrix B. The external magnetic field
1D MC
2D MC
In PWM calculations we shall consider a saturated magnetization in the whole magnonic crystal. This allows us to use linear approximation and a global coordinate system in which the
Using the linear approximation, we derive the following system of equations for
In MCs the material parameters, namely,
To solve the LL equation we will use Bloch’s theorem, which asserts that a solution of a differential equation with periodic coefficients can be represented as a product of a plane-wave envelope function and a periodic function. For dynamical components of the magnetization vector and its normalized values, those are
In the next step we perform the Fourier transformation to map the periodic functions
We need formulas for the static and dynamic demagnetizing fields,
The substitution of the (
The submatrices in (
We solve the system of (
All three forms of the exchange field were used in literature in calculations of the magnonic band structure in MCs, but to our best knowledge there is missed their detailed derivations, and in this paper we would like to fill this gap. In the first paper devoted to MCs by Vasseur et al. [
In this section we will present results of our calculations performed with PWM for three different forms of the exchange field as defined in (
We chose for our study a 1D MC consisting of Co and permalloy (Py) stripes of equal width 250 nm. The thickness of the film is 20 nm and the length is assumed infinite. Our choice is motivated by recently published papers presenting experimental and theoretical results [
We assume values of material parameters (spontaneous magnetization and exchange constant for cobalt and permalloy) equal to those presented in the experimental paper [
Magnonic band structures calculated for three different definitions of the exchange field (as defined in (
The dispersion relations calculated with the three different forms of the exchange field are overlapping almost perfectly (see Figure
To observe effects connected with various definitions of the exchange field in nonuniform materials, the decrease of the role of magnetostatic interaction is necessary. This can be done by decreasing a lattice constant. In Figure
Magnonic band structure and profiles of SWs calculated for three different definitions of the exchange field for the 1D MC with the lattice constant of 30 nm and thickness 4 nm (schematically shown in the inset)—for the case of dominating exchange interaction. (a) Magnonic band structure with the Brillouin zone border marked by dashed gray line. In (b) an amplitude of the dynamical component of the magnetization vector,
The differences connected with various definition of the exchange field start to be visible already near the BZ border and for higher modes also at the BZ center. The most essential difference can be observed between SW dispersions calculated according to Form I and Forms II and III. For the Form I (dashed line), the magnonic gap is absent between 1st and 2nd band, while for the other two formulations of
In Figures
Let us consider a thin film of a 2D MC composed of ferromagnetic circular dots in the square lattice and immersed in other ferromagnetic material. First, we will calculate the SW spectrum for Co dots in Py matrix in the exchange interaction dominating regime. This is obtained by assuming small lattice constant
(a) Magnonic band structure calculated for three different definitions of the exchange field for 2D MC. The MC is composed of Co dots in square lattice and immersed in Py matrix. The film has a thickness of 4 nm, the lattice constant is 30 nm, and diameter of dots is 14 nm. (b) Modulus of the dynamical component of the magnetization vector calculated for the wave vector from the center of the BZ. The profiles shown are calculated for Form II and Form I of the exchange field.
We observed similar dependences as for the case of 1D MC in Figure
We discussed so far PWM results for various definitions of the exchange field performed for MC consisting of two materials: Co and Py only. We showed also that the different expressions for the exchange field are important only for small lattice constant. In Figures
Magnonic band structure calculated for various forms of the exchange field. MCs composed of Fe and YIG were studied with a periodicity in (a) 1D and (b) 2D. The lattice constant is 30 nm and the film thickness 4 nm in both cases. The external magnetic field is applied along
Now we can try to answer the question for the physical reasons of different solutions found with From I, II and III of the exchange field. We have seen significantly different results obtained from Form II, and Form I for Co/Py MC, while the solutions obtained from Forms II and III are close to each other. It is different to Fe/YIG MC where the solution for Form III is much closer to the solution of Form I. Those effects shall be related to BCs for dynamic components of the magnetization vector implemented in various formulations. We can obtain the BC implemented in the differential equation, in our case LL equation, by integrating them over the interface [
We presented derivations of the two different expressions for the exchange field used in literature for SW calculations in magnonic crystals with pointing at the surface terms neglected in each case. We compared these formulas with the definition of the exchange field used for SW calculations in a uniform ferromagnetic material. Numerical calculations with PWM were performed to study the influence of these different expressions on the magnonic band structure and profiles of SWs in 1D and 2D planar magnonic crystals. We found that for a large lattice constant the magnonic band structure is independent of the formulation used. It is because the magnetostatic interaction dominates over the exchange one. The situation changed for small lattice constants where in dependence on the form of the exchange field used in calculations the magnonic gap can be present or absent in magnonic band structure. By numerical calculations we showed that various formulations of the exchange field have strong relation to the boundary conditions at the interfaces between two ferromagnetic materials. Further investigation is necessary to elucidate the proper form of the exchange field which fulfill the physical boundary conditions on interfaces imposed on dynamic components of the magnetization vector in magnonic crystals.
The authors acknowledge the financial support from the European Community’s Seventh Framework Programme FP7/2007-2013 (Grant Agreement no. 228673 for MAGNONICS). The calculations presented in this paper were performed in Poznan Supercomputing and Networking Center.