Appearance of a transverse component in dc electric field with respect to the applied current is investigated in periodic multilayer composite structures made of nanometer-to-micrometer scale alternating layers of two different homogeneous and isotropic conducting materials. Dependence of the transverse electric field on geometrical orientation of the layers is examined using the coordinate transformation approach. Electric field bending angle as a function of the layers’ resistivity ratio is studied in detail. It is shown that both the direction and the magnitude of the field can be changed using orientation angle of the layers as a tuning parameter.

1. Introduction

Recent realization of metamaterial devices based on cloaking and concentration effects for a static magnetic field [1–5] became one of the most up-to-date applications of transformation electrodynamics [6–9]. Analogous methods have been widely used to design devices of various type controlling manipulation of heat [10–13], acoustic [14], and material [15] waves. In [16], an analytical design of a multilayered composite structure allowing redirection (bending) of dc magnetic induction vector with respect to the applied external magnetic field has been presented. The aim of this paper is an investigation of dc electric field redirection relative to the direction of the applied electric current in stratified structures. Using coordinate transformation approach analogous to that in [16], we will show the possibility for the appearance of transverse (with respect to the current) component of the field in such a periodic multilayer structure. The ratio of the transverse and longitudinal components is very sensitive to geometrical orientation of the alternating isotropic layers. It means that both the direction and magnitude of the field can be changed using orientation angle of the layers as a tuning parameter. We hope that geometrically tunable anisotropic resistivity in multilayered structures predicted in this paper can find use in various devices based on the control and manipulations of dc electric field.

2. Model and Theory

Consider a periodic multilayer structure consisting of alternating layers of two different materials with isotropic resistivity ρ1 and ρ2, static dielectric permittivity ε1, ε2, and thickness l1, l2 in micrometer-to-nanometer scale, as both l1 and l2 are much larger than the lattice constant of the constituent components of the medium. The structure in the presence of dc electric current can be considered as an effective anisotropic homogeneous medium (a metamaterial) with electric field vector (1)E=ρ^efJ,where J is the applied electric current density, ρ^ef=σ^ef-1 is the effective resistivity tensor, and σ^ef is the effective conductivity tensor of the composite structure. In the coordinate system xyz with the x-axis perpendicular to the plane of the layers, tensor ρ^ef and effective permittivity tensor ε^ef have the form(2)ρ^ef=ρxx000ρyy000ρzz,ε^ef=εxx000εyy000εzz,where(3a)ρxx=ρ1l1+ρ2l2d,ρyy=ρzz=ρ1ρ2dρ1l2+ρ2l1,(3b)εxx=ε1ε2dε1l2+ε2l1,εyy=εzz=ε1l1+ε2l2d.and d=l1+l2 is the period of the structure. Note that parallel to the plane of the layers components ρyy and ρzz are always less than the transverse component ρxx.

An additional anisotropy in the resistivity and permittivity can be introduced rotating the layers in the structure around the z-axis on an angle φ (see Figure 1). Such a rotation can be realized using coordinate transformations [16](4)x′=xcosφ+ysinφ,y′=-xsinφ+ycosφ,z′=z,where x′y′z′ is the reference coordinate system fixed to the sample of rectangular parallelepiped form while xyz system of the coordinate axes is connected with the layers which are arranged in the direction perpendicular to the x-axis. The angle φ is assumed to be positive if the rotation is in counterclockwise direction.

Cross section of the multilayer composite in the x′y′-plane with layers rotated on angle φ around the z-axis. θ is the bending angle of dc electric field. The x-axis is perpendicular to the plane of the layers.

A static electric field in a source free space is described by Maxwell equations [17](5)curlE=0,divJ=0.

To keep (5) form-invariant under transformations (4), the effective conductivity and resistivity tensors should be transformed as(6)σ′^=A^σ^efAT^detA^,ρ′^=A^ρ^efAT^detA^,respectively, where A^ is the Jacobian transformation matrix(7)A^=cosφsinφ0-sinφcosφ0001and AT^=A-1^ is the transpose of A^. Then for the effective resistivity tensor in the coordinate system x′y′z′ we obtain(8)ρ′^=ρxxcos2φ+ρyysin2φρyy-ρxxsinφcosφ0ρyy-ρxxsinφcosφρxxsin2φ+ρyycos2φ000ρzz.

Let the static current of density J0 be applied along the x′ axis. Then from (3a) and (8) one can conclude that the electric field vector E/=ρ′^J0 inside the composite is bent with respect to J0 on the angle θ, where (9)tanθ=Ey′Ex′=-βtanφ1+β+tan2φ,(10)β≡l1l2ρ1-ρ22ρ1ρ2d2.

The negative sign in (9) indicates that, for a positive angle φ, the vector E/ is bending in clockwise direction (see Figure 1). Transversal component of the field is described by(11)Ey′=ρyx′J0=-12βρyyJ0sin2φ,and for a given value of the current density it can easily be tuned by changing the layers’ orientation angle φ. It is important to note that not only the direction, but also the magnitude of the field is a function of the angle φ:(12)E′=J0ρxx2cos2φ+ρyy2sin2φ1/2.

Consequently, E′ decreases monotonically with increasing φ from E′=ρxxJ0 at φ=0 to the value E′=ρyyJ0 at φ=π/2. It is easy to see that for any value of φ, in the range 0<φ<π/2, the absolute value of the bending angle is less than φ, and that with increasing φθ initially increases from zero and runs up to the maximum value at (13)φmax=tan-11+β1/2,given by(14)θmax=tan-11+β-1/2β2,and then it decreases monotonically and vanishes at φ=π/2. For all possible values of parameter β, φmax is always larger than π/4.

Note also that the bending angle θ is an odd function of φ: θ-φ=-θφ. It means that using samples with opposite rotated configuration of the layers, we can redirect the electric field vector upwards or downwards with respect to the applied current direction.

The dependence of θ on the thickness ratio l1/l2 at given values of φ, ρ1, and ρ2 is analogous to the behavior of the function θφ: θ increases with increasing l1/l2, has a maximum at l1/l2=1, and then decreases monotonically.

3. Electric Field Vector Redirection Details

For simplicity, we will further restrict ourselves to the consideration of the case when different layers of the structure have the same thickness, that is, when the function θl1/l2 has a maximum, and assume that ρ1>ρ2. Then (9) can be rewritten in the form(15)tanθ=-tanφ1+4γ1-γ-2/cos2φ,where γ=ρ2/ρ1. One can see that at a given value of the angle φ, the bigger θ is, the less γ is (i.e., the larger the contrast between resistivity values of individual layers is). We have to note that this conclusion is true not only at l1=l2 but also for any value of the ratio l1/l2. Note also that, with decreasing γ, φmax [see (13)] is shifted monotonically to the higher values while as a function of l1/l2 it has a maximum given by tanφmax=(ρ1+ρ2)/2(ρ1ρ2)1/2 at l1=l2.

In Figure 2, the dependence of the bending angle |θ| on the angle φ (in degrees) is demonstrated for five different values of the ratio γ, in the region φ≤π/2. Note that in the case when the second layer in a periodic unit of the multilayer structure is made from a perfect electrical conductor (ρ2→0), the absolute value of the bending angle coincides with the layers’ orientation angle φ. It may be interesting to note also that in the opposite case when one of the layers in the periodic unit is an ideal isolating material (ρ1→∞), the bending angle has the same absolute value: θ=-φ. In both these limiting cases, the parameter γ→0.

Dependence of the electric field bending angle |θ| on the layers’ orientation angle φ at five different values of the resistivity ratio γ=ρ2/ρ1.

4. Conclusions

The possibility of redirection of dc electric field with respect to the applied current is studied in periodic multilayer structures consisting of alternating layers of two isotropic materials with different resistivity values. It is shown that not only the direction but also the magnitude of the field can be changed using the layers’ orientation angle φ as a tuning parameter.

In this paper, we have found effective resistivity tensor of the composite structure using method of effective anisotropic homogeneous medium. Further, using coordinate transformation approach analogous to that in transformation optics, an analytical expression for the electric field bending angle θ has been found for arbitrary orientation of the layers with respect to the applied static current. It is shown that both the magnitude and sign of θ are very sensitive to the orientation of the layers. Dependence of θ on the layers’ orientation angle φ is examined in detail for different values of the resistivity ratio. It is shown that with increasing φ, the absolute value of the bending angle increases from zero and reaches a maximum at φ=φmax (which is always larger than π/4) and then decreases monotonically and vanishes at φ=π/2. Unlike that, with increasing φ the magnitude of the field decreases monotonically. In two opposite particular cases at which the resistivity ratio ρ2/ρ1≪1, the bending angle θ=-φ. For a given value of the thickness ratio l1/l2 and a given orientation of the layers relative to the applied current, the larger the bending angle is the bigger the contrast between resistivity values of the alternating layers is.

The possibility of the tunable field bending in multilayered structures by changing φ can find widespread applications in various devices based on the control and manipulation of dc electric field such as solar cells, batteries, multilayered thin-film sensors, and flat-panel displays and in computer chips.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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