## 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 *superlattice* (Figure 1). Each composite slab within the superlattice includes magnetic (with constitutive parameters

In the general case, when no restrictions are imposed on characteristic dimensions (*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 (*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,

where

In the given structure geometry, the interfaces between adjacent layers within the superlattice lie in the

and substituted into equations for components

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

With taking into account the long-wavelength limit, the fields *averaged* (Maxwell) fields

In view of the continuity of components

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

Here, we used the following designations

Expressing

with components

The expressions for tensor components of the underlying constitutive parameters of magnetic (

where

For magnetic layers [29], the components of tensor

For semiconductor layers [23], the components of tensor

Permittivity

Hereinafter, we consider two specific orientations of the external magnetic field vector

When

and for the components of tensor (7), we have following expressions:

In the second case, when

In this configuration, the components of tensor (7) can be written as follows:

For further reference, the dispersion curves of the tensor components of relative effective permeability

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 *biaxial* bigyrotropic crystal [1]. In the polar geometry, it is a *uniaxial* bigyrotropic crystal, and the following relations between tensor components are met:

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

## 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

where a time factor

From a pair of the curl Maxwell’s equations

where

For the upper medium (

where

For the composite medium (

where

with

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

whose coefficients

The dispersion relation for *bulk* polaritons is then obtained straightforwardly from (16) by putting

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

where

In (18), unknown quantities

Taking into consideration that two appropriate roots

where

Involving a pair of the divergent Maxwell’s equations

where

The boundary conditions at the interface require the continuity of the tangential components of

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

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:

Numerical solution of Eq. (24) gives a set of dispersion curves

From the mathematical point of view the found extreme states in dispersion curves exist in the region where the differential

where

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 *anti-crossing* effect), whereas in the case of degeneracy, when *crossing* effect). In the case when

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]:

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

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 (

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 *polar geometry* in which the external magnetic field is applied perpendicular to both the direction of wave propagation (*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 *Faraday geometry* in which the external magnetic field is applied parallel to both the direction of wave propagation and structure’s interface, i.e.,

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 (

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 (

where

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

where substitutions

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

Importantly, since the dispersion equation consists of a term which is linearly depended on

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

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 (

The crossing effect is found to be at

In the polar geometry (

where

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

where

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

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

As shown in Figure 6b, there are two separated areas of existence of extraordinary bulk polaritons for each particular filling factor

The value of the critical filling factor

At the same time, for all present values of filling factor

For all other values of filling factor

From Figure 6a and b, one can conclude that the upper passband possesses typical behaviors in the

In the Faraday geometry (

where

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

In Eq. (38), two distinct substitutions

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

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

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.,

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 (

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

For both initial problem considerations, dispersion Eqs. (36) and (38) have four roots from which two physically correct ones (denoted here as

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

## 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.