Open access peer-reviewed chapter

Modal Phenomena of Surface and Bulk Polaritons in Magnetic- Semiconductor Superlattices

By Vladimir R. Tuz, Illia V. Fedorin and Volodymyr I. Fesenko

Submitted: June 20th 2017Reviewed: October 20th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.71837

Downloaded: 398

Abstract

We discuss peculiarities of bulk and surface polaritons propagating in a composite magnetic-semiconductor superlattice influenced by an external static magnetic field. Three particular configurations of magnetization, namely, the Voigt, polar, and Faraday geometries, are considered. In the long-wavelength limit, involving the effective medium theory, the proposed superlattice is described as an anisotropic uniform medium defined by the tensors of effective permittivity and effective permeability. The study is carried out in the frequency band where the characteristic resonant frequencies of underlying constitutive magnetic and semiconductor materials of the superlattice are different but closely spaced. The effects of mode crossing and anti-crossing in dispersion characteristics of both bulk and surface polaritons are revealed and explained with an assistance of the concept of Morse critical points from the catastrophe theory.

Keywords

  • electromagnetic theory
  • polaritons
  • magneto-optical materials
  • superlattices
  • metamaterials

1. Introduction

Surface polaritons are a special type of electromagnetic waves propagating along the interface of two partnering materials whose material functions (e.g., permittivities) have opposite signs that are typical for a metal-dielectric boundary [1]. These waves are strongly localized at the interface and penetrate into the surrounding space over a distance of wavelength order in a medium, and their amplitudes fall exponentially away from the surface. Observed strong confinement of electromagnetic field in small volumes beyond the diffraction limit leads to enormous increasing matter-field interaction, and it makes attractive using surface waves in the wide fields from the microwave and photonic devices to solar cells [2, 3]. Furthermore, surface electromagnetic waves are highly promising from physical point of view because from the character of their propagation, one can derive information about both interface quality and electromagnetic properties of partnering materials (such as permittivity and permeability). High sensitivity to the electromagnetic properties of media enables utilization of surface waves in the sensing applications, particularly in both chemical and biological systems [4]. Thus, studying characteristics of surface waves is essential in the physics of surfaces and optics; in the latter case, the research has led to the emergence of a new science—plasmonics.

Today plasmonics is a rapidly developing field characterized by enormous variety of possible practical applications. In many of them, an ability to control and guide surface waves is a crucial characteristic. Thereby in last decades, many efforts have been made to realize active tunable components for plasmonic integrated circuits such as switchers, active couplers, modulators, etc. In this regard, searching effective ways to control characteristics of plasmon-polariton propagation by utilizing external driving agents is a very important task. In particular, the nonlinear, thermo-optical, and electro-optical effects are proposed to be used in the tunable plasmonic devices for the control of plasmon-polariton propagation [5, 6, 7, 8]. In such devices, the tuning mechanism is conditioned by changing the permittivity of the dielectric medium due to applying external electric field or temperature control. At the same time, utilization of an external magnetic field as a driving agent to gain a control over polariton dispersion features is very promising, since it allows changing both permeability of magnetic materials (e.g., ferrites) and permittivity of conducting materials (e.g., metals or semiconductors). It is worth mentioning that the uniqueness of this controlling mechanism lies in the fact that the polariton properties depend not only on the magnitude of the magnetic field but also on its direction. An applied magnetic field also produces additional branches in spectra of magnetic plasmon-polariton resulting in the multiband propagation accompanied by nonreciprocal effects [9, 10, 11, 12, 13, 14, 15, 16]. Thus, a combination of plasmonic and magnetic functionalities opens a prospect toward active devices with an additional degrees of freedom in the control of plasmon-polariton properties, and such systems have already found a number of practical applications in integrated photonic devices for telecommunications (see, for instance [3, 8] and references therein).

In this framework, using superlattices (which typically consist of alternating layers of two partnering materials) that are capable to provide a combined plasmon and magnetic functionality instead of traditional plasmonic systems (in which the presence of a metal-dielectric interface is implied) has great prospects. Particularly, it conditioned by the fact that the superlattices demonstrate many exotic electronic and optical properties uncommon to the homogeneous (bulk) samples due to the presence of additional periodic potential, which period is greater than the original lattice constant [14]. The application of magnetic field to a superlattice leads to the so-called magneto-plasmon-polariton excitations. Properties of the magnetic polaritons in the superlattices of different kinds being under the action of an external static magnetic field have been studied by many authors during several last decades [10, 11, 12, 13, 14, 15, 16]. The problem is usually solved within two distinct considerations of gyroelectric media (e.g., semiconductors) with magneto-plasmons [10, 14] and gyromagnetic media (e.g., ferrites) with magnons [11, 12, 13, 16], which involve the medium characterization with either permittivity or permeability tensor having asymmetric off-diagonal parts. This distinction is governed by the fact that the resonant frequencies of permeability of magnetic materials usually lie in the microwave range, whereas characteristic frequencies of permittivity of semiconductors commonly are in the infrared range.

At the same time, it is evident that combining together magnetic and semiconductor materials into a single gyroelectromagnetic superlattice in which both permeability and permittivity simultaneously are tensor quantities allows additional possibilities in the control of polaritons using the magnetic field that are unattainable in convenient either gyromagnetic or gyroelectric media. Fortunately, it is possible to design heterostructures in which both characteristic resonant frequencies of semiconductor and magnetic materials are different but, nevertheless, closely spaced in the same frequency band. As a relevant example, the magnetic-semiconductor heterostructures proposed in [17, 18, 19, 20] can be mentioned that are able to exhibit a gyroelectromagnetic effect from gigahertz up to tens of terahertz [21]. Thus investigation of electromagnetic properties of such structures in view of their promising application as a part of plasmonic devices is a significant task.

This chapter is devoted to the discussion of dispersion peculiarities of both bulk and surface polaritons propagating in a finely-stratified magnetic-semiconductor superlattice influenced by an external static magnetic field. It is organized as follows. In Section 2, we formulate the problem to be solved and derive effective medium expressions suitable for calculation of the properties of modes under the long-wavelength approximation. Section 3 describes the problem solution in a general case assuming an arbitrary orientation of the external magnetic field with respect to the direction of wave propagation and interface of the structure. The discussion about manifestation of mode crossing and anti-crossing effects is presented in Section 4 involving a concept of the Morse critical points from the catastrophe theory. In Section 5, we reveal dispersion peculiarities of bulk and surface polaritons in the given superlattice for three particular cases of the vector orientation of the external magnetic field with respect to the superlattice’s interface and wave vector, namely we study the configurations where the external magnetic field is influenced in the Voigt, polar, and Faraday geometries. Finally, Section 6 summarizes the chapter.

2. Outline of problem

Thereby, in this chapter, we study dispersion features of surface and bulk polaritons propagating in a semi-infinite stack of identical composite double-layered slabs arranged along the y-axis that forms a superlattice (Figure 1). Each composite slab within the superlattice includes magnetic (with constitutive parameters εm,μ̂m) and semiconductor (with constitutive parameters ε̂s,μs) layers with thicknesses dmand ds, respectively. The stack possesses a periodic structure (with period L=dm+ds) that fills half-space y<0and adjoins a vacuum (ε0=μ0=1) occupying half-space y>0. Therefore, the superlattice’s interface lies in the xzplane, and along this plane, the system is considered to be infinite. The structure under investigation is influenced by an external static magnetic field Mthat lies in the yzplane and makes an angle θwith the y-axis. It is supposed that the strength of this field is high enough to form a homogeneous saturated state of magnetic as well as semiconductor subsystems. Finally, the wave vector kof the macroscopic electric field lies in the xzplane and makes an angle φwith the x-axis.

Figure 1.

Schematic representation of a magnetic-semiconductor superlattice influenced by an external static magnetic field M→ and a visual representation of the tangential electric field distribution of the surface polariton propagating over the interface between the given structure and free space.

In the general case, when no restrictions are imposed on characteristic dimensions (dm,ds,and L) of the superlattice compared to the wavelength of propagating modes, the transfer matrix formalism [22] is usually involved in order to reveal the dispersion features of polaritons. It implies a numerical solution of a canonical boundary value problem formulated for each layer within the period of superlattice and then performing a subsequent multiplication of the obtained transfer matrices to form a semi-infinite extent. On the other hand, when all characteristic dimensions of the superlattice satisfy the long-wavelength limit, i.e., they are all much smaller than the wavelength in the corresponding layer and period of structure (dmλ, dsλ, Lλ), homogenization procedures from the effective medium theory can be involved in order to derive dispersion characteristics in an explicit form [24, 25, 26, 27, 28] that is suitable for identifying the main features of interest. Therefore, further only the modes under the long-wavelength approximation are studied in this chapter, i.e., the structure is considered to be a finely-stratified one.

In order to obtain expressions for the tensors of effective permeability and permittivity of the superlattice in a general form, constitutive equations, B=μHand D=εE, for magnetic (0<z<dm) and semiconductor (dm<z<L) layers are represented as follow [26]:

Qνj=νgννjPνj,E1

where Qtakes values of the magnetic and electric flux densities Band D, respectively; Pis varied between the magnetic and electric field strengths Hand E; gis substituted for permeability and permittivity μand ε; the superscript jis introduced to distinguish between magnetic (mj) and semiconductor (sj) layers; and ννiterates over x, y, and z.

In the given structure geometry, the interfaces between adjacent layers within the superlattice lie in the xzplane, and they are normal to the y-axis; thus, the field components Pxj, Pzj,and Qyjare continuous at the interfaces. Therefore, the normal component Pyjcan be expressed from (1) in terms of the continuous components of the field as follow:

Pyj=gyxjgyyjPxj+1gyyjQyjgyzjgyyjPzj,E2

and substituted into equations for components Qxjand Qzj:

Qxj=gxxjgxyjgyxjgyyjPxj+gxyjgyyjQyj+gxzjgxyjgyzjgyyjPzjQzj=gzxjgzyjgyxjgyyjPxj+gzyjgyyjQyj+gzzjgzyjgyzjgyyjPzj.,E3

Relations (2) and (3) are then used for the field averaging [24].

With taking into account the long-wavelength limit, the fields Pjand Qjinside the layers are considered to be constant, and the averaged (Maxwell) fields Qand Pcan be determined by the equalities:

P=1LjPjdj,Q=1LjQjdj.E4

In view of the continuity of components Pxj, Pzj,and Qyj, it follows that

Px=Pxj,Pz=Pzj,Qy=Qyj,E5

and with using Eqs. (2) and (3), we can obtain the relations between the averaged field components in the next form:

Qx=αxxPx+γxyQy+αxzPz,Py=βyxPx+βyyQy+βyzPz,Qz=αzzPx+γzyQy+αzzPz.E6

Here, we used the following designations ανν=jgννjgνyjgyνj/gyyjδj, βyy=j1/gyyjδj,βyy=jgyνj/gyyjδj, γνy=jgνyj/gyyjδj, δj=dj/Lis filling factor, and ννiterates over xand z.

Expressing Qyfrom the second equation in system (6) and substituting it into the rest two equations, the constitutive equations for the flux densities of the effective medium Q=ĝeffPcan be derived, where ĝeffis a tensor quantity:

ĝeff=α˜xxγ˜xyα˜xzβ˜yxβ˜yyβ˜yzα˜zxγ˜zyα˜zz=g˜xxg˜xyg˜xzg˜yxg˜yyg˜yzg˜zxg˜zyg˜zz,E7

with components α˜νν=αννβyνγνy/βyy, β˜yy=1/β˜yy,β˜yν=βyν/βyy,and γ˜νy=γνy/βyy.

The expressions for tensor components of the underlying constitutive parameters of magnetic (μ̂mĝm) and semiconductor (ε̂sĝs) layers depend on the orientation of the external magnetic field Min the yzplane which is defined by the angle θin the form:

ĝj=g1iζg2iξg2iζg2ζ2g1+ξ2g3ζξg1g3iξg2ζξg1g3ξ2g1+ζ2g3,E8

where ζ=sinθand ξ=cosθ.

For magnetic layers [29], the components of tensor ĝmare g1=1+χ+, g2=Ω+iΩ, g3=1and χ=ω0ωmω02ω21b2D1, χ=ωωmbω02+ω21+b2D1, Ω=ωωmω02ω21+b2D1, Ω=2ω2ω0ωmbD1, and D=ω02ω21+b22+4ω02ωb2, where ω0is the Larmor frequency and bis a dimensionless damping constant.

For semiconductor layers [23], the components of tensor ĝsare g1=εl1ωp2ω+ωω+2ωc21,g2=εlωp2ωcωω+2ωc21, and g3=εl1ωp2ω+1, where εlis the part of permittivity attributed to the lattice, ωpis the plasma frequency, ωcis the cyclotron frequency, and νis the electron collision frequency in plasma.

Permittivity εmof the magnetic layers as well as permeability μsof the semiconductor layers are scalar quantities.

Hereinafter, we consider two specific orientations of the external magnetic field vector Mwith respect to the superlattice’s interface (see Figure 1), namely (i) the polar configuration in which θ=0and the vector Mis parallel to the surface normal (My) and (ii) θ=π/2and the vector Mis parallel to the surface plane (Mz), which is inherent in both the Voigt and Faraday configurations.

When My, then ζ=0, ξ=1, and tensor (8) is reduced to the form

ĝj=g10ig20g30ig20g1,E9

and for the components of tensor (7), we have following expressions: γ˜xy=γ˜zy=β˜yx=β˜yz=0, α˜xx=gxxmδm+gxxsδs, α˜zz=gzzmδm+gzzsδs, α˜xz=α˜zx=gzxmδm+gzxsδs, and β˜yy=gyymgyysτ, where τ=gyymδs+gyysδm1.

In the second case, when Mz, then ζ=1, ξ=0, and tensor (8) has the form

ĝj=g1ig20ig2g1000g3.E10

In this configuration, the components of tensor (7) can be written as follows: α˜xz=α˜zx=γ˜zy=β˜yz=0, α˜xx=gxxmδm+gxxsδs+gxymgxys2δmδsτ, α˜zz=gzzmδm+gzzsδs, γ˜xy=β˜yx=gxymgyysδm+gxysgyymδsτ, and β˜yy=gyymgyysτ.

For further reference, the dispersion curves of the tensor components of relative effective permeability μ̂effand relative effective permittivity ε̂effof the homogenized medium (with filling factors δm=δs=0.5) are presented in Figure 2. Figure 2(a) and (b) represents constitutive parameters for the polar configuration, whereas Figure 2(c) and (d) represents those for the Voigt and Faraday configurations of magnetization. For these calculations, we used typical constitutive parameters for magnetic and semiconductor materials. In particular, here we follow the results of paper [30], where a magnetic-semiconductor composite is made in the form of a barium-cobalt/doped-silicon superlattice for operating in the microwave part of spectrum. A distinct peculiarity of such a superlattice is that the characteristic resonant frequencies of the underlying constitutive magnetic and semiconductor materials are closely spaced within the same frequency band.

Figure 2.

Dispersion curves of the tensor components of (a, c) relative effective permeability μ̂eff and (b, d) relative effective permittivity ε̂eff of the homogenized medium. Panels (a) and (b) correspond to the polar geometry and panels (c) and (d) correspond to the Voigt and Faraday geometries. For the magnetic constitutive layers, under saturation magnetization of 2930 G, parameters are f0=ω0/2π=3.9 GHz, fm=ωm/2π=8.2 GHz, b=0, and εm=5.5; for the semiconductor constitutive layers, parameters are fp=ωp/2π=5.5 GHz, fc=ωc/2π=4.5 GHz, ν=0, εl=1.0, and μs=1.0. Filling factors are δm=δs=0.5.

From Figure 2, one can conclude that in both the Voigt and Faraday geometries, the next relations between the components of effective tensor (7) hold g˜xxg˜yyg˜zzand g˜xy=g˜yx0, so it means that the obtained homogenized medium is a biaxial bigyrotropic crystal [1]. In the polar geometry, it is a uniaxial bigyrotropic crystal, and the following relations between tensor components are met: g˜xx=g˜zzg˜yyand g˜xz=g˜zx0.

To sum up, with an assistance of the homogenization procedures from the effective medium theory, the superlattice under study is approximately represented as a uniaxial or biaxial anisotropic uniform medium when an external static magnetic field Mis directed along or orthogonal to the structure periodicity, respectively. In the latter case, the first optical axis of the biaxial medium is directed along the structure periodicity, whereas the second one coincides with the direction of the external static magnetic field M.

3. General solution for bulk and surface polaritons

In order to obtain a general solution for both bulk and surface polaritons, we follow the approach developed in Ref. [10] where dispersion characteristics of polaritons in a uniaxial anisotropic dielectric medium have been studied. Here, we extend this approach to the case of a gyroelectromagnetic medium whose permittivity and permeability simultaneously are tensor quantities.

In a general form [26], the electric and magnetic field vectors Eand Hused here are represented as

Pj=pjexpikxx+kzzexpκy,E11

where a time factor expiωtis also supposed and omitted, and sign is related to the fields in the upper medium (y>0,j=0), while sign +is related to the fields in the composite medium (y<0,j=1), respectively, which provide required wave attenuation along the y-axis.

From a pair of the curl Maxwell’s equations ×E=ik0Band ×H=ik0D, in a standard way, we arrive at the following equation for the macroscopic field:

××Pjk02ς̂jPj=0,E12

where k0=ω/cis the free-space wavenumber and ς̂jis introduced as the product of μ̂jand ε̂jmade in the appropriate order.

For the upper medium (j=0), direct substitution of expression (11) with P0and corresponding constitutive parameters (ς̂νν0=1for ν=νand ς̂νν0=0for νν; here and further subscripts νand νare substituted to iterate over indexes of the tensor components x, y, and zin Cartesian coordinates) into Eq. (12) gives us the relation with respect to κ0:

κ02=k2k02,E13

where k2=kx2+kz2.

For the composite medium (j=1), substitution of (11) with P1and ς̂1into (12) and subsequent elimination of Py1yield us the following system of two homogeneous algebraic equations for the rest two components of P1:

AxzPx1+BxzPz1=0,BzxPx1+AzxPz1=0,E14

where Anmand Bnmare functions of κderived in the form [10]:

Anmκ=km2k02ςnn1κ2ϰy2+kn2κ2+iknκςny1+ςyn1k02k04ςny1ςyn1,Bnmκ=knkm+ςnm1ϰy2+knkmκ2+knςym1+kmςny1k02k04ςny1ςym1,E15

with ϰy2=k2k02ςyy1, and subscripts mand niterate over indexes xand z.

In order to find a nontrivial solution of system (15), we set its determinant of coefficients to zero. After disclosure of the determinant, we obtain an equation of the fourth degree with respect to κ:

ςyy1κ4+aκ3+bκ2++d=0,E16

whose coefficients a, b, c, and dare

a=ikx(ςyz(1)+ςzy(1))+ikz(ςyx(1)+ςxy(1)),b=k02[ςyy(1)(ςzz(1)+ςxx(1))ςzy(1)ςyz(1)ςxy(1)ςyx(1)][k2ςyy(1)+kz2ςzz(1)+kx2ςxx(1)+kxkz(ςzx(1)+ςxz(1))],c=ik02{kz[ςxy(1)ςzx(1)+ςyx(1)ςxz(1)ςxx(1)(ςyz(1)+ςzy(1))]+kx[ςyz(1)ςzx(1)+ςzy(1)ςxz(1)ςzz(1)(ςyx(1)+ςxy(1))]}ik2[kz(ςyz(1)+ςzy(1))+kx(ςyx(1)+ςxy(1))],d=k2[kz2ςzz(1)+kx2ςxx(1)+kxkz(ςzx(1)+ςxz(1))]+k02k2[ςzx(1)ςxz(1)ςzz(1)ςxx(1)]+k02{kx2(ςxy(1)ςyx(1)ςyy(1)ςxx(1))+kz2(ςzy(1)ςyz(1)ςyy(1)ςzz(1))+kxkz[ςxy(1)ςyz(1)+ςzy(1)ςyx(1)ςyy(1)(ςzx(1)+ςxz(1))]}+k04[ςxx(1)ςyy(1)ςzz(1)ςyy(1)ςzx(1)ςxz(1)+ςxy(1)(ςyz(1)ςzx(1)ςzz(1)ςyx(1))+ςzy(1)(ςxz(1)ςyx(1)ςyz(1)ςxx(1))].E17

The dispersion relation for bulk polaritons is then obtained straightforwardly from (16) by putting κ=0inside it.

In order to find the dispersion law of surface polaritons from four roots of (16), two physically correct ones must be selected. In general, two such roots are required to satisfy the electromagnetic boundary conditions at the surface of the composite medium. We define these roots as κ1and κ2and then following [10] introduce the quantities Kw(w=1,2) in the form:

Px1κw=KwAzxκw,Py1κw=KwCκw,Pz1κw=KwBzxκw,E18

where Cκw=1/ϰy2ikxκwk02ςyx1Azxκw+ikzκwk02ςyz1Bzxκw.

In (18), unknown quantities Kwneed to be determined from the boundary conditions.

Taking into consideration that two appropriate roots κ1and κ2of (16) are selected, the components of the field P1can be rewritten as the linear superposition of two terms:

Px1=w=1,2KwAzxκwexpκwy,Py1=w=1,2KwCκwexpκwy,Pz1=w=1,2KwBzxκwexpκwy,E19

where y<0and the factor expikxx+kzzωtis omitted.

Involving a pair of the divergent Maxwell’s equations ·B=0and ·D=0in the form

·Qj=·ĝjPj=0,E20

where gis substituted for permeability μand permittivity εand Qis substituted for the magnetic Band electric Dflux densities; one can immediately obtain the relations between the field components in the upper (y>0) medium as follows:

Qy0=ig0/κ0kxPx0+kzPz0.E21

The boundary conditions at the interface require the continuity of the tangential components of Eand Has well as the normal components of Dand B, i.e., in our notations these components are Px, Pz,and Qy, respectively. Thus, application of the boundary conditions together with (21) gives us the next set of equations:

ig0/κ0kxPx0+kzPz0=g˜yxw=1,2KwAzxκw+g˜yxw=1,2KwCκwg˜yzw=1,2KwBzxκw,Px0=w=1,2KwAzxκw,Pz0=w=1,2KwBzxκw,kzPx0kxPz0=kzw=1,2KwAzxκw+kxw=1,2KwBzxκw.E22

The system of Eq. (22) has a nontrivial solution only if its determinant vanishes. Applying this condition gives us the required dispersion equation for surface polaritons.

Finally, the amplitudes K1and K2can be found by solving set of linear homogeneous Eq. (22). They are

K1=kxAzxκ2+kzBzxκ2κ0+κ2,K2=kxBzxκ1+kzAzxκ1κ0+κ1.E23

Here, the problem is considered to be formally solved, and the dispersion relations are derived in a general form for both bulk and surface polaritons.

4. Theory of Morse critical points: mode coupling phenomena

Further, for brevity, obtained dispersion equations for the bulk and surface polaritons are denoted in the form:

Dkk0=0.E24

Numerical solution of Eq. (24) gives a set of dispersion curves k0kof polaritons, which can contain both regular and singular (critical) points. The regular points draw dispersion curves of a classical form possessing either normal or anomalous dispersion line, at which a small variation in k0results in a smooth changing in the form of the curves. Besides, some situations are possible when a slight variation in k0leads to a very sharp (catastrophic) changing in the form of dispersion curves. Such singularities (extreme states) can be accompanied by mutual coupling phenomena of modes, which are further of our interest.

From the mathematical point of view the found extreme states in dispersion curves exist in the region where the differential Dkk0of dispersion Eq. (24) vanishes (see, e.g., Figure 3). These extreme states can be carefully identified and studied involving the approach based on the theory of the Morse critical points from the catastrophe theory [31, 32, 33, 34, 35, 36]. This treatment has been originally applied to study open waveguides and resonators [31, 32], and later it has been extended to more complex waveguide structures [33, 34, 35]. From viewpoint of this theory, the presence of the Morse critical points is generally defined by a set of nonlinear differential equations written in the form [33]:

Dkkk0kmk0m=Dk0kk0kmk0m=0,H=h11h22h12h21kmk0m0,E25

Figure 3.

Sketch of the band diagrams presenting different kinds of interaction between two neighboring modes: (a) weak interaction, (b) intermediate interaction, (c) strong interaction, and (d) accidental degeneracy. Areas where the critical points exist are pointed by circles.

where kmk0mare coordinates in the kk0plane of a particular m-th Morse critical point, the subscripts kand k0near the letter Ddefine corresponding partial derivatives /kand /k0, and His the Hessian determinant with elements:

h11=Dkkkk0,h12=Dkk0kk0,h21=Dk0kkk0,h22=Dk0k0kk0.E26

The type of each extreme state defined by set of Eq. (25) can be uniquely identified from the sign of the Hessian determinant [35]. For instance, when H<0, the corresponding Morse critical point represents a saddle point, which occurs in the region of a modal coupling (the anti-crossing effect), whereas in the case of degeneracy, when H=0, it is a non-isolated critical point (the crossing effect). In the case when H>0, the Morse critical point defines either a local minimum or maximum (this case is not considered here). In what follows we distinguish found critical points by circles for both crossing and anti-crossing effects, as shown in Figure 3.

In general, when conditions (25) are met, the type of interacting modes in the vicinity of the corresponding Morse critical point can be defined as follows [33]:

Codirectional forward:h12/h11<0,h22/h11>0;E27
Codirectional backward:h12/h11>0,h22/h11>0;E28
Contradirectional:h12/h11>0,h22/h11<0.E29

The strength of modes interaction within the found extreme states in the region of their coupling can be identified considering the classification introduced in paper [37], which concerns on the mode behaviors appearing in axial waveguides. In particular, (i) a weak interaction takes place when frequency band gap between dispersion curves of interacting modes is high enough (Figure 3a), (ii) an intermediate interaction of modes leads to formation of very flattened parts in the dispersion curves (Figure 3b), (iii) a strong interaction appears when the repulsion between modes is strong enough resulting in formation of dispersion curve having anomalous dispersion line (Figure 3c), and (iv) an accidental degeneracy arises when two dispersion branches are merged within the critical point which leads to nonzero group velocities at k=0(Figure 3d).

The strong interaction with forming negative-slope region in dispersion curve of one of interacting modes (as it is depicted in Figure 3c and d) leads to some unusual effects [37]. First, zero group velocity (vg=k0/k) can appear at a nonzero value of k. The flattened region of the dispersion curve around that extreme point (Figure 3c) is very useful for applications in nonlinear optics, where a small group velocity is suitable for enhancing nonlinear effects, while the phase-matching criterion can still be satisfied because the wave vector is nonzero. Second, when interacting modes are exactly degenerated, then their group velocities appear to be nonzero and roughly constant values as k0(Figure 3d). From viewpoint of practical applications, this situation is useful, because regular modes in the vicinity of k=0have extremely small value of their group velocity and thus become unusable. Since both dispersion and losses are inversely proportional to vg,they diverge when k0.

At the same time, the intermediate interaction with forming extremely flattened part in the mode dispersion curve (see Figure 3b) leads to strong divergence in density of states, and it can be utilized in designs of low-threshold lasers [37].

5. Dispersion features of bulk and surface polaritons for three particular orientations of magnetization

Since further our goal is to elucidate the dispersion laws of the bulk and surface polaritons (which are in fact eigenwaves), we are interested in real solutions of Eq. (24). In order to find the real solutions, the absence of losses in constitutive parameters of the underlying layers is supposed.

We consider three particular orientations of the external magnetic field Mwith respect to the superlattice’s interface (the xzplane) and to the wave vector k, namely, (i) the polar geometry in which the external magnetic field is applied perpendicular to both the direction of wave propagation (Mk) and structure’s interface (My) as shown in Figure 4a; (ii) the Voigt geometry in which the external magnetic field is applied parallel to the structure’s interface, and it is perpendicular to the direction of the wave propagation, so Mzand Mkas presented in Figure 4b; (iii) the Faraday geometry in which the external magnetic field is applied parallel to both the direction of wave propagation and structure’s interface, i.e., Mzand Mkas presented in Figure 4c.

Figure 4.

Three particular orientations of the external magnetic field vector M→ with respect to the superlattice’s interface and wave vector; (a) polar geometry, M→∥y,M→⊥k→; (b) Voigt geometry, M→∥z,M→⊥k→; and (c) Faraday geometry M→∥z,M→∥k→.

With respect to the problem of polaritons, in any kind of an unbounded gyrotropic medium, there are two distinct eigenwaves (the bulk waves), whereas the surface waves split apart only for some particular configurations (e.g., for the Voigt geometry), and generally, the field has all six components. Such surface waves are classified as hybrid EH modes and HE modes, and these modes appear as some superposition of longitudinal and transverse waves. By analogy with [38], we classify hybrid modes depending on the magnitude ratio between the longitudinal electric and magnetic field components (Pxand Pzcomponents for the polar and Faraday geometries, respectively). For instance, for the polar configuration, it is supposed that the wave has the EH type if Ex>Hxand the HE type if Hx>Ex. Contrariwise, in the Faraday geometry, we stipulate that the wave has the EH type if Ez>Hzand the HE type if Hz>Ez. In the Voigt geometry, the waves appear as transverse electric (TE) and transverse magnetic (TM) modes, where each of them has three field components.

The polariton dispersion relations for these specific cases of magnetization can be obtained using general results derived in Section 3 with application of appropriate boundary and initial conditions.

When an external static magnetic field is influenced in the Voigt geometry (Mz,Mk), solution of Eq. (16) for both bulk and surface waves splits apart into two independent equations for distinct polarizations [10], namely, TE modes with field components HxHyEzand TM modes with field components ExEyHz. Therefore, regions of existence of bulk polaritons are uniquely determined by solutions of two separated equations related to the TE and the TM modes as follows [26, 38]:

kx2k02εzzμvμyyμxx1=0,E30
kx2k02μzzεvεyyεxx1=0,E31

where εv=εxx+εxy2/εyyand μv=μxx+μxy2/μyyare the Voigt relative permittivity and permeability, respectively.

Under such magnetization, the dispersion equation for the surface polaritons at the interface between vacuum and the given structure has the form [26]:

κ1gv+κ2g0+ikxg0g˜xyg˜yy1=0,E32

where substitutions μgand εgare related to the TE and TM modes, respectively.

We should note that in two particular cases of the gyroelectric and gyromagnetic superlattices, dispersion relation (32) coincides with Eq. (33) of Ref. [10] and Eq. (21) of Ref. [15], respectively, which verifies the obtained solution.

It follows from Figure 4b that in the Voigt geometry, the magnetic field vector in the TM mode has components 00Hz, and it is parallel to the external magnetic field M, which results in the absence of its interaction with the magnetic subsystem [39, 40, 41]. Thus, hereinafter dispersion features only of the TE mode are of interest for which the dispersion equation for surface polaritons (32) can be rewritten as

κ1μv+κ2μ0+ikxμ0μxyμyy1=0.E33

Importantly, since the dispersion equation consists of a term which is linearly depended on kx(the last term in (33)), the spectral characteristics of the surface polaritons in the structure under study possess the nonreciprocal nature, i.e., k0kxk0kx.

A complete set of dispersion curves obtained from solution of Eq. (30) that outlines the passbands of the TE bulk polaritons as a function of the filling factor δmis presented in Figure 5a. One can see that behaviors of the TE bulk polaritons are quite trivial in the overwhelming majority of structure’s configurations, namely, there are two isolated passbands separated by a forbidden band. The upper passband is bounded laterally by the light line, and its lower limit is restricted by the line at which μv=0. The bottom passband starts on the line where εzz=0and then approaches the asymptotic frequency (kx) at which μxx=0(see Eq. (25)).

Figure 5.

(a) A set of dispersion curves of the TE bulk polaritons for different filling factors δm for the structure being in the Voigt geometry and (b) manifestation of crossing effect in dispersion curves of the TE bulk polaritons at the particular value of filling factor δm=0.132. Parameters of magnetic constitutive layers are shown in Figure 2. For the semiconductor constitutive layers, parameters are fp=ωp/2π=10.5 GHz, fc=ωc/2π=9.5 GHz, ν=0, εl=1.0, and μs=1.0.

In this study, we are mainly interested in those curves of the set which have greatly sloping branches and exhibit the closest approaching each other (i.e., they manifest the anti-crossing effect) or have a crossing point, since such dispersion behaviors correspond to the existence of the Morse critical points. Hereinafter, the areas of interest in which these extreme states exist are denoted in figures by orange circles.

It follows from Figure 5a that in the Voigt geometry, both the anti-crossing (H<0) and crossing (H=0) effects with forming the bottom branch characterized by the anomalous dispersion (i.e., the strong mode interaction occurs) can be achieved in the composite structure with a predominant impact of the semiconductor subsystem (further we stipulate that the composite system has a predominant impact of either magnetic subsystem or semiconductor subsystem if δmδsor δsδm, respectively). Moreover, since condition (29) is met near the critical point, the bulk waves appear to be contradirectional.

The crossing effect is found to be at kx0(accidentally degenerate modes) for the particular configuration of the structure, when δm=0.132(Figure 5b). Remarkably, such an extreme state corresponds to a particular frequency where εzzand μvsimultaneously acquire zero [26].

In the polar geometry (My,Mk), composite medium under study is a uniaxial crystal whose optical axis is directed along the y-axis. In this case, bulk polaritons split onto two waves with field components ExHyEzand HxEyHz, respectively [40], and their continua is outlined by two dispersion relations:

kx2k02μyyεyy=0,E34
kx2k02μvεv1μxzεxz/μxxεxx1=0,E35

where εv=εxx+εxz2/εxxand μv=μxx+μxz2/μxxare introduced as the effective bulk permeability and permittivity, respectively.

Hereinafter, we distinguish these two kinds of waves as ordinary and extraordinary bulk polaritons, respectively (note: such a definition is common in the plasma physics [43]).

In order to elucidate the dispersion features of hybrid surface polaritons, the initial problem is decomposed into two particular solutions with respect to the vector H(EH modes) and vector E(HE modes) [10, 42]. In this way, the dispersion equation for surface polaritons is derived in the form:

κ02g˜xxg01κ12+κ1κ2+κ22ϰz2+ϰy2ςxzg˜xzςyyg˜xx1+κ0κ1+κ2κ1κ2+ϰy2g˜xxgvg0ςyy1+g˜xzςxz1ϰz4ϰz2κ1+κ2+κ12κ22+ϰy2ϰz2+κ1κ2ςxzg˜zzςyyg˜xz1=0,E36

where ϰν2=kx2k02ςννand two distinct substitutions εvvg˜vv, ε0g0, εvgvand εvvg˜vv, μ0g0, μvgvcorrespond to the problem resolving with respect to the vector Eand vector H, respectively; εvand μvare the same shown in Eqs. (34) and (35).

For two particular cases of the gyroelectric and gyromagnetic superlattices, dispersion relation (36) coincides with Eq. (23) of Ref. [10] and Eq. (21) of Ref. [13], respectively, which verifies the obtained solution.

Complete sets of dispersion curves calculated from the solution of Eqs. (34) and (35) that outline the passbands of both ordinary (blue curves) and extraordinary (red curves) bulk polaritons as functions of the filling factor δmare presented in Figure 6a and b. Moreover, in order to discuss the observed crossing and anti-crossing effects more clearly, dispersion curves of both ordinary and extraordinary bulk polaritons are plotted in the k0kxplane at the particular values of filling factor δmas shown in Figure 6cf.

Figure 6.

Complete sets of dispersion curves of (a) ordinary (blue curves) and (b) extraordinary (red curves) bulk polaritons for different filling factor δm for the structure being in the polar geometry. Manifestation of (c) crossing and (d–f) anti-crossing effects in dispersion curves of bulk polaritons at the particular value of filling factor δm, (c) δm=0.267, (d) δm=0.99, (e) δm=0.05, and (f) δm=0.99. All structure constitutive parameters are shown in Figure 2.

From Figure 6a one can conclude that there are two isolated areas of existence of the ordinary bulk polaritons. The upper passband starts at the frequency where εyy=0, while the bottom passband is bounded above by the asymptotic line where εyy. Remarkably, the anti-crossing effect (H<0) between dispersion curves which restrict upper and bottom passbands of ordinary bulk polaritons can be observed in the composite structure with a predominant impact of the magnetic subsystem (i.e., δmδs) as presented in Figure 6a and d. Coupled modes exhibit an intermediate interaction for which an appearance of the flattened branches in dispersion curves is peculiar (see, for instance, Figure 3b). Besides, condition (27) is met, so these waves are codirectional forward in the area nearly the critical point.

As shown in Figure 6b, there are two separated areas of existence of extraordinary bulk polaritons for each particular filling factor δm. Moreover, there are two possible combinations of conditions for their passbands, which depend strictly on the value of filling factor δm[42]. Therefore, the particular critical filling factor is denoted here as δcat which εv=μv=0at kx=0, and it is considered as a separation point between these two combinations. Thus, the overall range of values δmcan be separated onto two subranges 0δm<δcand δc<δm1, respectively.

The value of the critical filling factor δcdepends on the constitutive parameters of magnetic and semiconductor layers, and for the structure under study, it is δс=0.267. Such critical state in the effective parameters of the superlattice leads to the absence of the forbidden band between the upper and bottom passbands as shown in Figure 6c. Remarkably, these dispersion branches appear to be coupled exactly at the frequency where εv=μv=0, i.e., the crossing effect (H=0) is found to be at kx0. Besides, such coupled waves are contradirectional, and they are described by condition (29).

At the same time, for all present values of filling factor δmfrom the first subrange 0δm<δc, the lower limits of both the upper and bottom passbands are restricted by the lines at which εv=0, as well as the bottom passband is bounded above by the asymptotic line where εv. The upper passband is outlined by the set of following conditions [40]: εv>0,μv>0,εxzμxz/εxxμxx<1. The bottom passband exists when the set of following conditions holds [42]: εv>0,μv<0and εxzμxz/εxxμxx>1or εv>0,μv>0and εxzμxz/εxxμxx<1.

For all other values of filling factor δm(i.e., δc<δm1), the conditions for the upper passband are the same (i.e., εv>0, μv>0, εxzμxz/εxxμxx<1), whereas its lower limit is at the line where μv=0. The bottom passband exists when either the set of conditions εv<0,μv>0and εxzμxz/εxxμxx>1or εv>0,μv>0and εxzμxz/εxxμxx<1is satisfied. For this band, the lower and upper limits are restricted by the lines at which μv=0and μv, respectively.

From Figure 6a and b, one can conclude that the upper passband possesses typical behaviors in the k0kxplane, whereas the width and position of the bottom passbands are defined by the corresponding resonant frequencies of effective bulk permeability μvand effective bulk permittivity εv. In fact, these constitutive parameters are multipliers of the numerator of Eq. (35), whereas the denominator of this equation originates a singularity at the asymptotic line where 1εxzμxz/εxxμxx. This asymptotic line splits the bottom passbands onto two separated sub-passbands which corresponds to the structures with a predominant impact of semiconductor and magnetic subsystems, respectively (Figure 6e and f). Note, the intermediate interaction with forming flattened region in dispersion curves between bulk modes from these two sub-bands is observed nearly the Morse critical points (H<0). Such coupled waves are contradirectional since condition (29) is met.

In the Faraday geometry (Mz, Mk), the superlattice is characterized by the biaxial anisotropy, for which, as already mentioned, the first anisotropy axis is associated with the structure periodicity (so it is directed along the y-axis), while the second anisotropy axis is a result of the external static magnetic field influence (so it is directed along the z-axis). For this geometry, the bulk polaritons in the structure under study appear as right-handed and left-handed elliptically polarized waves [10, 12, 44], and their passbands are outlined by curves governed by the following dispersion law [44]:

ϰx2ϰy2k04ςxyςyx=0,E37

where ϰν2=kz2k02ςνν.

The surface polaritons are hybrid EH and HE waves, and their dispersion relation can be written in the form [44]:

κ2+κ0g˜zzκ22ςyyϰy2ςzzκ1ψκ02kz2+κ0κ12g˜xyςyyςzzk02ρ+kz2g˜xyκ1+κ0g˜zzκ12ςyyϰy2ςzzκ2ψκ02kz2+κ0κ22g˜xyςyyςzzk02ρ+kz2g˜xy=0.E38

In Eq. (38), two distinct substitutions εvvg˜vv, εvgvand μvvg˜vv, μvgvcorrespond to the problem resolving with respect to the vector Eand vector H, respectively; εv=εxx+εxy2/εyy, μv=μxx+μxy2/μyy, ψ=gvg˜yyςyx; ρ=g˜yyςyxg˜xyςyy; and the constant g0=1is omitted.

For the semiconductor superlattice, the dispersion Eq. (38) agrees with Eq. (38) of Ref. [10], while in the case of magnetic superlattice, it coincides with Eq. (13) of Ref. [12], which verifies the obtained solution.

Complete sets of dispersion curves that outline the bands of existence of the bulk polaritons (see Eq. (37)) as functions of filling factor δmare presented in Figure 7a and b for right-handed (blue curves) and left-handed (red curves) elliptical polarizations. From these figures, one can conclude that there is a pair of corresponding sets of dispersion curves separated by a forbidden band for the bulk polaritons of each polarization. The dispersion curves of the right-handed elliptically polarized bulk waves demonstrate quite trivial behaviors, and they completely inherit characteristics of the right-handed circularly polarized waves of the corresponding reference semiconductor or magnetic medium (see, for instance [44]). Contrariwise, the dispersion features of the left-handed elliptically polarized waves are much more complicated being strongly dependent on filling factor δmand resonant frequencies of constitutive parameters of both semiconductor and magnetic underlying materials. Therefore, in what follows we are interested only in the consideration of dispersion features of the bulk polaritons having the left-handed polarization.

Figure 7.

(a, b) Complete sets of dispersion curves of both right-handed (blue curves) and left-handed (red curves) elliptically polarized bulk polaritons for different filling factor δm for the structure being in the Faraday geometry. Manifestation of (c, e) anti-crossing and (d) crossing effects in dispersion curves of bulk polaritons at the particular value of filling factor δm; (c) δm=0.15; (d) δm=0.1; and (e) δm=0.95. All structure constitutive parameters are shown in Figure 2.

One can conclude that the dispersion characteristics of the left-handed elliptically polarized bulk waves of the given gyroelectromagnetic structure are different from those ones of both convenient gyroelectric and gyromagnetic media. Indeed, in contrast to the characteristics of the left-handed circularly polarized waves of the corresponding reference medium whose passband has no discontinuity, the passband of the left-handed elliptically polarized bulk polaritons of the superlattice is separated into two distinct areas. This separation appears nearly the frequency at which the resonances of the functions εxyand μxyoccur for the composite structures with predominant impact of the magnetic and semiconductor subsystems, respectively. Also, we should note that the exchange by the critical conditions for the asymptotic lines (at which kz) between the bottom passbands of left-handed and right-handed elliptically polarized bulk polaritons appears at the particular frequency where εxyand μxysimultaneously tend to infinity.

The dispersion curves of bulk waves, which have left-handed polarization, demonstrate significant variation of their slope having subsequent branches with normal and anomalous dispersion that possess approaching at some points (extreme states) as depicted in Figure 7c and e. The features of these dispersion curves in the vicinity of the critical points are different for the composite structure which has predominant impact either semiconductor (i.e., δsδm) or magnetic (i.e., δmδs) subsystem. The corresponding modes are contradirectional (i.e., condition (29) is met), and they demonstrate strong interaction nearly the Morse critical point (the anti-crossing effect with H<0) in the case when δsδm. The modes acquire an intermediate interaction in the superlattice with predominant impact of the magnetic subsystem as depicted in Figure 7e.

Moreover, a particular extreme state in dispersion curves is found out where the upper branches of the left-handed and right-handed elliptically polarized bulk polaritons merge with each other (see Figure 6d) and crossing point (H=0) occurs at kz0. Likewise to the polar geometry, such extreme state corresponds to the particular superlattice configuration where εv=μv=0at kz=0. So, the interacting waves are degenerated at this point and possess the contradirectional propagation.

We should note that in [37] some unusual and counterintuitive consequences of such behaviors of the dispersion curves (e.g., backward wave propagation, reversed Doppler shift, reversed Cherenkov radiation, atypical singularities in the density of states, etc.) for the TE and TM modes of an axially uniform waveguide are discussed, and it is emphasized that these effects are of considerable significance for practical applications.

Finally, in order to obtain the dispersion curves k0kof surface polaritons in both the polar and Faraday configurations, characteristic Eqs. (36) and (38) are solved numerically [42, 44]. Here, as previously, the problem is decomposed into two particular solutions with respect to the vector H(EH waves) and vector E(HE waves) [10, 12]. Note, as evident from Eqs. (36) and (38), kxand kzappear only in even powers, so the dispersion curves of the surface waves appear to be reciprocal.

For both initial problem considerations, dispersion Eqs. (36) and (38) have four roots from which two physically correct ones (denoted here as κ1and κ2) are selected [10] ensuring they correspond to the attenuating waves. In papers [10] and [12], it was noted that depending upon the position in the k0kwplane, in the non-dissipative system, the following combinations between two roots κ1and κ2may arise: (i) both roots are real and positive (bona fide surface modes); (ii) one root is real and the other is pure imaginary or vice versa (pseudosurface modes); (iii) both roots are complex in which case they are conjugate (generalized surface modes); and (iv) both roots are pure imaginary (the propagation is forbidden). In our study we consider only bona fide surface mode. Nevertheless, we should note that in both the polar and Faraday geometries, pseudosurface waves (which attenuate only on one side of the surface) can also be supported [10, 45].

Among all possible appearance of dispersion curves of the surface polaritons, we are only interested in those ones which manifest the crossing or anti-crossing effect. The search of their existence implies solving an optimization problem, where for the crossing effect the degeneracy point should be found. For the anti-crossing effect, the critical points are defined from calculation of the first Dkk0and second Dkk0partial derivatives of Eq. (36) or (38) with respect to kand k0for both EH and HE modes. During the solution of this problem, the period and constitutive parameters of the underlying materials of the superlattice are fixed, and the search for the effect manifestation is realized by altering the filling factors δmand δswithin the period. The found crossing points are depicted in Figure 8a and b by green circles. We should note that in both configurations, the degeneracy points for EH (red line) and HE (blue line) surface polaritons can be obtained only for the case of composite structure with a predominant impact of the semiconductor subsystem. In particular, this state is found out to be at the values of filling factor δm=0.27and δm=0.15for the polar and Faraday configurations, respectively. Remarkable, in the Faraday geometry, both dispersion curves possess an anomalous dispersion line, namely, they start on the light line and fall just to the right of the light line, and then they flatten out and approach an asymptotic limit for large values of k. Contrariwise to the Faraday geometry, in the polar geometry, both dispersion curves demonstrate normal dispersion.

Figure 8.

Manifestation of the crossing effect in dispersion curves of hybrid EH (red line) and HE (blue line) surface polaritons; (a) polar configuration (δm=0.27) and (b) Faraday configuration (δm=0.15). All structure constitutive parameters are shown in Figure 2.

6. Conclusions

To conclude, in this chapter, we have studied dispersion features of both bulk and surface polaritons in a magnetic-semiconductor superlattice influenced by an external static magnetic field. The investigation was carried out under an assumption that all characteristic dimensions of the given superlattice satisfy the long-wavelength limit; thus, the homogenization procedures from the effective medium theory was involved, and the superlattice was represented as a gyroelectromagnetic uniform medium characterized by the tensors of effective permeability and effective permittivity.

The general theory of polaritons in the gyroelectromagnetic medium whose permittivity and permeability simultaneously are tensor quantities was developed. Three particular configurations of the magnetization, namely, the Voigt, polar, and Faraday geometries, were discussed in detail.

The crossing and anti-crossing effects in the dispersion curves of both surface and bulk polaritons have been identified and investigated with an assistance of the analytical theory about the Morse critical points.

We argue that the discussed dispersion features of polaritons identified in the magnetic-semiconductor superlattice under study have a fundamental nature and are common to different types of waves and waveguide systems.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Vladimir R. Tuz, Illia V. Fedorin and Volodymyr I. Fesenko (December 20th 2017). Modal Phenomena of Surface and Bulk Polaritons in Magnetic- Semiconductor Superlattices, Surface Waves - New Trends and Developments, Farzad Ebrahimi, IntechOpen, DOI: 10.5772/intechopen.71837. Available from:

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