Electrodynamic boundary conditions for planar arrays of thin magnetic elements

The progress of microwave technology is impossible without the development of novel man-made metamaterials having designed-to-order physical properties.^1–3 A good example of such man-made novel magnetic metamaterials is the arrays of dipolarly coupled magnetic elements.^4–6 In contrast to continuous magnetic films, the microwave properties of a magnetic array could be varied by changing the shape and separation of the arrays' elements, and/or by switching the magnetic ground state of the array.^5,6

A direct numerical solution of almost any electrodynamic problem for such an array is rather complicated as the dipolar magnetic fields near the magnetic elements are spatially non-uniform, and the elements are much smaller than the wavelength of the electromagnetic wave interacting with the array. The numerical solution of such a problem, typically, requires a very fine mesh in the computational process and is, therefore, prohibitively time consuming.

The way out of this situation is to use the fact that magnetic elements of the array are much thinner than the wavelength of the electromagnetic (EM) wave, and to derive an approximate electrodynamic boundary conditions connecting electric and magnetic fields at the opposite sides of the array. This approximate general approach was successfully used in the past. For example, Leontovich derived a boundary condition for a thick conductive non-magnetic metal layer,⁷ which was later extended to the cases of anisotropic and inhomogeneous metals.⁸ Similar boundary conditions were obtained for infinitesimally thin non-magnetic conducting planes⁹ and spheres.¹⁰ 

A problem of an EM wave interaction with thin magnetic films¹¹ and structures¹² has also been considered previously, but it was done in the approximation of “effective medium,” where the internal magnetic permeability was used. Unfortunately, that approach cannot be applied to magnetic metamaterials (or element arrays), since the dynamic magnetization and the dipolar magnetic fields are non-uniform across the metamaterial.

In our current work, we derive approximate electrodynamic boundary conditions describing the change of the magnetic and electric fields at the boundary of a planar thin metamaterial array. The derived boundary conditions (8) show that the discontinuities of the electric and magnetic fields taking place at the array depend only on the array thickness and on its average dynamic (or microwave) magnetization of the array. The average microwave magnetization can be found via an external magnetic susceptibility tensor $χ̂$ of the array. The tensor $χ̂$⁠, in its turn, characterizes the collective spin-wave (SW) modes of the array⁵ and has poles at the eigen-frequencies of these collective spin wave modes.

It is important to note that in our derivation we do not make any assumptions about the particular form of the tensor $χ̂$⁠, enabling one to solve practically any electrodynamic problem involving a magnetic metamaterial of any type. At the same time, in the simplest and most practically important cases of a ferromagnetic (FM) and chessboard-antiferromagnetic (CAFM) ground states of the array the expressions for the external susceptibility tensor $χ̂$ of the array can be obtained analytically (see below).

To illustrate the use of the derived approximate electrodynamic boundary conditions, we consider below the problem of oblique scattering of a plane EM wave on an array of cylindrical magnetic nano-elements.

Let us consider an infinite array of cylindrical magnetic nano-elements distributed in a plane, as shown in Fig. 1. Our goal is to relate the electric and magnetic fields existing at the top and bottom sides of the array. Typically, for a continuous magnetic film an effective magnetic permeability of the film material can relate the magnetic field to a magnetic induction.^11,12 However, this technique cannot be used due to the spatial non-uniformity of the involved fields. Thus, we will divide the magnetic field existing near the array's surface into two parts

where $hdip(r)$ is the dipolar field generated by the individual elements of the array and $hext(r)$ is the remaining magnetic field which we will call here and below an external magnetic field. The dipolar field can be found by integrating the magnetostatic Green function^13–15 

where $m(r)$ is the array's dynamic magnetization.

Using expressions (1) and (2), one can get the following system of Maxwell equations for an array of magnetic nanoelements:

Here we neglected the influence of eddy currents, assuming that the array's elements are electrically disconnected and also are too small to support significant eddy currents.^12,16 The external field $hext(r)$⁠, in contrast to the total magnetic field $h(r)$⁠, is continuous everywhere in space. This property of the external magnetic field $hext(r)$ allows us to find the material equation relating the dynamic magnetization of the array $m(r)$ to this external dynamic magnetic field.

In the following, we assume that the wavelength of the EM radiation acting on the array is much larger than the geometric sizes of the array's elements. This allows us to use the following two approximations: (i) the array is infinitesimally thin and (ii) the dynamic magnetization of the array can be averaged in the array plane. Assuming linear magnetic response of the array, we can write the following material equation:

where $m¯$ is the homogeneous (averaged) part of magnetization, $ρ$ is the radius-vector in the array plane, d is the array thickness, $δ(z)$ is a delta function, and $χ̂$ is the array external susceptibility tensor to be introduced in the text below. Here, we also assume that the array is periodic (possible containing several sublattices), i.e., the tensor $χ̂$ does not depend on $ρ$⁠. Later we will extend our theory to a case of a quasi-periodic array.

To find the dipolar magnetic field at the top and bottom sides of the array, we introduce the in-plane wave-vector $k$ and expand the dynamic magnetization of the array, as well as electric and magnetic fields, in a Fourier series of the form

where $c$ means either dynamic homogeneous magnetization $m¯$ of the array or the dynamic magnetic $h$ or electric $e$ field.

Substituting the Fourier expansion (5) for $m(r′)$ in (2) and integrating over the $z′$ coordinate, we get¹⁵ 

where $m¯k=χ̂·hext,k|z=0, hdip,k±=hdip,k|z=±0$⁠, the sign $⊗$ denotes a Kronecker product, and $ẑ$ is a unit vector normal to the array (see Fig. 1). The dipolar field $hdip$ is not the same at the different sides of the array: $hdip,k+≠hdip,k−$ for $k≠0$⁠.

The value of the dynamic magnetization $m¯$ can be found using the expressions for $hdip,k±$ and $hext,k$ in (1) and summing over all the elements at the both sides of the array

Then, using expressions for $hdip,k±$ given by (6) in (1) and taking a difference of fields existing at the different sides of the array, we obtain the following boundary condition for the magnetic field:

To derive the boundary condition for the electric field, we integrate the Maxwell equations (3) over a surface perpendicular to the array and take a limit of the vanishing area of this surface.¹³ As a result, we get the following boundary condition:

where ω is the frequency the EM radiation.

Equations (8) are the central result of this paper. These equations relate the discontinuities of the electric and magnetic fields taking place at the surfaces of an array of dipolarly coupled magnetic nanoelements to the spatially averaged dynamic magnetization $m¯$ of the array and the array's thickness d. The dynamic magnetization $m¯$ is found via the external susceptibility tensor $χ̂$ of the array. The tensor $χ̂$ is determined by the collective spin wave modes of the array, and the expressions for this tensor in simple cases can be calculated analytically (see (12) and (13)).

We did not make any assumptions about the particular form of the external susceptibility tensor $χ̂$⁠. One can easily show that the derived boundary conditions (8) satisfy the Maxwell equations (3) for any form of the tensor $χ̂$⁠. Also, if the tensor $χ̂$ is Hermitian, it is possible to show that the energy for an EM wave passing through a nano-element array is conserved.

Although, it is implied that the considered array is infinite, the boundary conditions are still valid for arrays with the in-plane sizes that are substantially larger than $2π/k$⁠. For arrays having smaller in-plane sizes, the situation becomes more complicated, and the influence of the array's edges should be considered explicitly.

The magnetization dynamics of dipolarly coupled arrays of magnetic nano-elements have been studied previously (see, e.g., Ref. 5). In particular, it was shown that the arrays of coupled magnetic nano-elements support collective SW modes and that these modes determine how the arrays interact with external magnetic fields. Since collective SW modes in nano-element arrays are much slower than the electromagnetic waves in vacuum, we do not consider here any retardation effects.

The net dynamic homogeneous magnetization of an array can be represented as an averaged superposition of all the SW modes with a zero SW wave-vector supported by the array

where $m0,ν(r)$ is the amplitude of the νth SW mode, f is the magnetic material filling fraction, defined as $f=∑jVj/V$⁠, V_j is the volume of the jth element, and V is the total volume of the array, including spaces between the elements.

The partial tensor $χ̂ν$ corresponding to a particular collective SW mode of the array is determined by the spatial vector profile $mν,i$ and the resonance frequency $ων$ of this SW mode¹⁷ 

where $ωM=γμ0Ms$⁠, M_s is the saturation magnetization of the magnetic material from which the elements are made, γ is the gyromagnetic ratio, N is the number of the elements in the array, $Γν$ is the damping factor of the SW mode, $Aν=i∑jmν,j*·μj×mν,j$ is the norm of the SW mode, and $μj$ is a unit-vector pointing in the direction of the equilibrium magnetization in the jth array's element. Comparing (4) and (9), we define the total external susceptibility tensor of the array as

The tensor $χ̂$ contains all the information about the internal structure of the array (lattice symmetry, elements shape, ground state, etc.) needed to solve any particular electrodynamic problem involving the array. In the simplest (and most practically important) cases, the expressions for the tensor $χ̂$ can be obtained analytically. In particular, for an array consisting of identical magnetic elements arranged in a square lattice and magnetized normally to the plane of the array, the total susceptibility tensor $χ̂$ can be expressed as

where $ωFMR$ is the FMR frequency of the array,¹⁸ $Γ=αGω$⁠, and α_G is the Gilbert damping constant.

Another practically interesting case is a case of an array of identical magnetic nano-pillars having perpendicular magnetic (or/and shape) anisotropy. The ground state of such an array, corresponding to the minimum dipolar energy, is a so-called chessboard-antiferromagnetic (CAFM) ground state,^5,6 in this case the net magnetization of the array is zero, and the total susceptibility tensor $χ̂$ has a diagonal form

where $Î$ is an identity matrix, $ωAFMR$ is the frequency of the antiferromagnetic resonance,¹⁹ and $a≈1$ is a constant which depends on the elements' shape and the lattice constant, and can be evaluated numerically.

The external magnetic susceptibility tensor $χ̂$ differs from the material susceptibility tensor $χ̂mat$ which relates the magnetization to the internal magnetic field in a uniform magnetic medium.²⁰ Also it differs from the volume-averaged susceptibility tensor¹² in the cases when dipolar coupling between the array's elements is significant.

Although we derived the boundary conditions (8) assuming that the array is periodic, they can also be applied in the cases when the array is not spatially uniform, but when its parameters, e.g., the element shape or the internal magnetic field vary only slightly on the distances of the order of the array's lattice constant.²¹ To apply our boundary conditions (8) in such a case, it is necessary to replace in (8) the SW mode wave vector $k$ by the in-plane gradient operator $∇ρ$ and to calculate the local tensors $χ̂(ρ)$ in (11).

To illustrate the application of the above derived boundary conditions (8), we consider below the problem of oblique EM wave incidence on an array of magnetic nanowires.²² The array consists of vertically oriented identical cylindrical nanowires having the length of $d=30 μm$⁠, radius of 60 nm, and arranged in a square lattice with the lattice constant of 200 nm. It is assumed that the wires are made of Permalloy (Py) having the saturation magnetization of $Ms=800 kA/m$ and the Gilbert damping constant of $αG=0.01$⁠. We consider the array to be in a ferromagnetic ground state, when all the wires have the same direction of the magnetic moment, and the array is saturated with the out-of-plane magnetic field of 50 mT. For these parameters of the array, the FMR frequency is $ωFMR/2π≈3.5 GHz$ (wavelength in vacuum $λ0≈8.5 cm$⁠). The susceptibility tensor of the array in this case is calculated using expression (12).

We consider the interaction of a nano-wire array with an incident plane EM wave. The dynamic electric field in such an EM wave is defined as

where $p̂$ and $ŝi$ are the unit vectors corresponding to the p and s polarizations,¹³ $Ai$ and $Bi$ are the complex amplitudes for the corresponding polarizations, $k0$ is the wave-vector of the incident wave (⁠$k0=2π/λ0$⁠), and the incidence angle is $θ=arccos(ẑ·k0/k0)$ (see Fig. 1).

Matching the electric and magnetic fields of the incident, reflected, and transmitted waves with the help of the boundary conditions (8), one can relate the complex amplitudes of the reflected wave with the amplitudes of the incident wave in Fresnel-like formulae

Fig. 2(a) shows the intensities of the reflected and transmitted waves for a linearly polarized incident wave in the case of normal (θ = 0) incidence. Although the length of the nano-wires is substantially smaller than the EM wave wavelength (⁠$d/λ0≈3×10−4$⁠), the reflection of the EM wave from the array is significant (2%) for the frequencies that are close to the FMR frequency of the array. In contrast to continuous conducive thin films,¹¹ an array of magnetic nano-wires is practically transparent for the EM radiation having frequencies that are far from the FMR frequency the array.

In general, the s and p polarizations in the reflected wave are not independent, because of the non-diagonal components of the $χ̂$ tensor. In the case when the frequency of the incident EM wave is exactly equal to the FMR frequency of the array (⁠$ω=ωFMR$⁠), we can obtain a relatively simple expression for the reflected intensity

The analysis of this expression shows that the reflection of an EM wave from the array is strongly dependent on the polarization of the incident wave. Fig. 2(b) shows the intensity of the wave reflected from the array of nanowires as a function of the angle of incidence for waves having right and left circular polarizations. The array is practically transparent for the wave having left circular polarization the array, while for the wave having right circular polarization the array has a maximum possible reflection. For relatively large angles of incidence, the reflection of the both polarizations increases with the increase of the angle of incidence and can be significant even for very thin array's elements.

In general, the EM waves reflected from and transmitted through an array of nanowires are elliptically polarized. To characterize the ellipticity, we use here the third Stokes parameter, defined as $S3=−2Im (ArBr*)$⁠. Using the expressions (15), one may find a simple expression for the ratio of the third Stokes parameter to the intensity of the reflected wave

The calculated values of the Stokes parameter normalized by the intensity of the reflected wave for different angles of incidence are shown in Fig. 2(c). It is important to note that the polarization of the reflected wave depends only on the angle of incidence and does not depend on the array's thickness or on the polarization of the incident wave.

In conclusion, we derived approximate electrodynamic boundary conditions describing the interaction of an incident EM wave with a planar array of thin magnetic elements. The boundary conditions describe the discontinuities that dynamic electric and magnetic fields experience at a thin array of magnetic nanoelements using the external susceptibility tensor of the array, which can be calculated from the previously developed theory of the collective spin-wave modes supported by the array. These boundary conditions enable one to solve any electrodynamic problem involving an array of interacting magnetic nanoelements. As an example of an application of the derived boundary conditions, we considered the problem of scattering of a plane EM wave from an array of identical cylindrical magnetic nanowires. We derived Fresnel-like formulae for the amplitudes of the reflected waves for the arrays in a FM. We demonstrated that the reflection of a resonant EM wave can be significant even for an array which is much thinner than the wavelength of the incident EM wave. For the case of a perpendicular magnetized array in the FM ground state, we showed that the reflected wave is elliptically polarized, and the ellipticity depends only on the incident angle and does not depend on the polarization of the incident wave and the particular parameters of the array.

This work was supported in part by the Grants Nos. DMR-1015175 and ECCS-1305586 from the National Science Foundation of the USA, by the contract from the U.S. Army TARDEC, RDECOM, and by the DARPA grant “Coherent Information Transduction between Photons, Magnons, and Electric Charge Carriers”. I.L. and S.N. acknowledge the Russian Scientific Foundation, Grant No. 14-19-00760 for financial support.