Phonon-polaritonics: enabling powerful capabilities for infrared photonics

3.1 What are PhPs?

The native out-of-phase vibrational modes of the lattice (optical phonons) can strongly couple to impinging IR EM fields (photons) for certain types of solids. Such strong coupling can be triggered only in solids comprising oppositely charged atoms in their lattice basis so that the incident electric field can drive their out-of-phase vibrational modes; also known in the spectroscopy community as IR active phonons [58]. As a result of such strong coupling, an EM wave going through such solid would have mixed wave dispersion characteristics, which are neither “photon-like” nor “phonon-like” [23]. We depict the wave dispersion of these “mixed modes,” known as PhPs, in Figure 3 (red solid line). Polar covalent semiconductors can be seen as effectively comprising two separate oppositely charged ions [59] and therefore also support IR active phonons and thus PhP modes.

Figure 3:

PhP dispersion relation.

The frequency ω versus wave vector q for strongly coupled lattice optical phonons (long-dashed horizontal line at Ω_free) with the radiation field (slanted solid lines) signifying mixed phonon-photon modes (red lines). The asymptote at the lower branch (short-dashed horizontal line) designates the frequency of the TO phonon, ω_TO. The yellow shaded regions represents the PhP gap or reststrahlen band (region of no mixed phonon-photon modes). Conversely, a slab of material in this frequency region is nearly perfectly reflecting. [See reflectance, R (green line).] ε_∞ and c represent the high frequency permittivity of the material and speed of light, respectively.

3.2 A microscopic model for the macroscopic PhP photonic responses

To understand the strong phonon-photon coupling phenomenon and its effect on the macroscopic optical response of ionic or ionic-like solids, we consider a basic, simple cubic diatomic lattice model. This comprises two oppositely charged ions with charges +Q, −Q and respective masses M₊, M₋, as depicted in Figure 4A. These charges correspond to effective charges in real materials as the concept of a perfect ionic medium is an idealized model [59]. We consider an EM wave incident with electric field E_inc along the x-direction, launched from atop the structure of Figure 4A along the +y-direction. In order to satisfy the EM boundary conditions at the interface the wave-vector of the coupled radiation field-phonon wave inside the ionic medium, q, must also lie in the y-direction [60]. Also, from Maxwell’s equations it follows that the displacement vector D should be divergence-free inside the ionic material, implying that q·D=0 [60]. Assuming linear and isotropic materials:

Figure 4:

The ionic lattice and its response under the local electric field in the immovable-ion and rigid-ion models.

(A) A simple cubic ionic lattice of lattice constant, a, comprising positive ions of mass M₊ and charge +Q and negative ions of mass M₋ and charge –Q. (B) Dipole moment contributions, p₊ and p₋ of each of the deformable ion types assuming they are immovable. The displaced electron clouds around the ionic cores under the influence of the local electric field E_loc are shown. (C) The ions moving as rigid, non-deformable ions, under the influence of the local electric field E_loc and elastic restoring forces coming from the extended and compressed “inter-ionic springs”, F_ex and F_com. A diatomic chain of the ions at certain depth, y, of the cubic lattice is depicted, where the positive (negative) ions are displaced from their equilibrium positions by u₊ (u₋). The equilibrium positions for each ion type are designated with respective faint colored spheres. The bottom panel, is a zoomed depiction of the section within the cyan dashed lines, with the relevant forces in each kind of ions depicted. Damping forces, are also present (designated as F_damp).

where P is the polarization, defined as the dipolar moment per unit volume, ε is the relative permittivity of the material and ε₀ is the vacuum permittivity. It follows from all the above, that D, P, and E are along the x-direction inside the ionic material and normal to the wave-vector q of the PhP wave.

In order to understand the phonon-photon strong coupling we must understand how the macroscopic electric field, E, and polarization, P, correlate with the microscopic properties and motion of the atoms, as well as with the microscopic local electric field E_loc. In lattices of tetrahedral symmetry, the local microscopic field at the center of both types of ions [61], [62] is given by:

This local electric field, E_loc, is the superposition of the impinging EM wave, and the long-range electrostatic field of the rest of the ions in the lattice. The macroscopic polarization in the ionic crystal, P, will be a superposition of the polarization resulting from the dipole moments of each kind of the two polarizable ions and the polarization from the dipole moments emanating from the oppositely-charged ions as they move. In particular, for the former contribution, we assume immovable but deformable ions under the influence of the local electric field E_loc. The local electric field shifts the electron cloud around the ionic cores in each ion type (see Figure 4B). The contributing dipole moments of each ion type (p₋ and p₊) yield a total macroscopic polarization of:

where N represents the number of positive-negative ion pairs per unit volume, and α₊, α₋ the respective polarizabilities of the positively-charged and negatively-charged ions. We have set α=α₊+α₋ to represent the total polarizability of the oppositely-charged ion pair.

Conversely, for the latter contribution, i.e. for the polarization from the dipole moments emanating from the oppositely-charged ions as they move, we consider the oppositely-charged ions as rigid ions moving under the influence of the local electric field E_loc and the restoring forces from the inter-atomic potential. By rigid ions we mean that the ions move with the electron cloud tight around them [23], i.e. neither of the ions is polarizable (see Figure 4C). Then a net dipole moment in the elementary cell can be obtained only when there is a net displacement w of the positive ions with respect to the negative ions, yielding a total macroscopic polarization of:

Accounting for both types of contributions to the microscopic dipole moments, the total polarization, P, would be:

As the macroscopic polarization and field are parallel to each other [see Eq. (1)], i.e. E | |P, it follows from Eq. (2) that the microscopic local field, E_loc, is parallel to both of them. Then Eq. (5) implies that the relative displacement between the positive and negative ions, w, must also remain parallel to all fields. So, we have that E_inc | |E_loc | |E | |P | |w, with all perpendicular to the wave vector q. In other words, w·q=0, meaning that the impinging EM can drive only the transverse phonon vibrational waves [19], [24].

Eqs. (2) and (5) can be also used to express both the local electric field, E_loc, and the polarization, P in term of the macroscopic field E. We get:

and

So, all displacements and fields are along the x-direction, while the PhP mode propagates along the y-direction. If the incoming EM field is a harmonic field of frequency ω, so will be the macroscopic field inside the ionic medium, which still must satisfy Maxwell’s equations. As Eqs. (2), (6) and (7) must be obeyed at any depth of the lattice, the macroscopic field, E, the local field, E_loc, the polarization, P, and the relative displacement, w, all would follow the same waveform [24], i.e. we have:

with E₀, E_loc,0, P₀ and w₀ representing their respective amplitudes.

To understand further the motion of the ions in the lattice (in the rigid-ion model of Figure 4C), we take E_loc at a certain depth y to be constant, i.e. we ignore any spatial variations at the inter-ion scale [24], [61]. Thus, it suffices to focus our attention on two particular ions at certain depth y, one positive and one negative [e.g. see ions designated with (1) and (2) in Figure 4C]. The short-range forces between adjacent ions can be approximated as restoring forces [23], similar to the elastic force between two masses connected with a spring. A damping force proportional to the velocity of the ion is also taken, to account for dissipation in realistic lattices with the assumption that the more massive ion experiences a proportionally stronger damping force. We take u₊, u₋ to represent the respective displacements along the x-direction of the positive and negative ions in the chain from their equilibrium position. We also take k to represent the elastic constant of the restoring force on the ions, and γ to be a constant related to the strength of the damping force. Then the the equation of motion becomes [23], [62]:

for the positive ions, and

for the negative ions.

In order to understand better how the incident field drives the motion of the two kinds of ions in the lattice we temporarily ignore the damping term in Eqs. (9) and (10). Then, by adding these and integrating over time (assuming at t=0 the ions are at their equilibrium position) we get that:

Eq. (11) signifies that the motion of the positive and negative ions will be out-of-phase with u₊u₋<0. Also, if w₀ is the maximum value of the relative displacement, w, along the x-direction, i.e. w=u+−u−=w0ei(qy−ωt), we would have that:

and

The PhP wave is a slow varying wave in the y-direction. Meaning, the wave would need to cross several atoms typically of the order of a 1000 or more for a full cycle. In Figure 5, we depict a snapshot of the transverse PhP wave.

Figure 5:

The transverse PhP wave.

The displacement of the ions is along the electric field of the impinging EM wave and perpendicular to the propagation direction, y. Because this is a slow-varying wave in the atomic scale, the ions along the y-direction are depicted only every N lattice planes, with N being of the order of a 1000 atoms or more. The displacements of the positive and negative ions are always out-of-phase (optical-phonon mode) with a respective ratio of −M₋/M₊ in the absence of damping. This out-of-phase phonon mode is activated by an impinging IR EM wave that displaces the two oppositely-charged ions in opposite directions; hence this mode is also known as IR-active optical phonon.

We follow through to determine the amplitude of the relative ion displacement, w₀, needed to determine the macroscopic polarization and thus the effective photonic response of the ionic medium. It can be shown, by subtracting Eq. (10) from Eq. (9), after multiplying them respectively, with M₊ and M₋, and using Eq. (7) to express the local field in terms of the macroscopic field, that:

with M being the reduced mass of the the ion-pair of Figure 4C given by:

After setting,

and using Eq. (8) into Eq. (14) we obtain:

We first consider a limiting case: when the driving field is a high frequency field. For such a case, the atomic polarizability from the electron displacement dominates the permittivity response; the fields change too fast for the ions to respond (the atoms having much larger inertia in comparison to electrons). This means that the displacement, w, in Eq. (6) can be taken to be zero. We can see that also from Eq. (17) as w₀→0 as ω→∞. Then from Eqs. (1) and (6) we get the familiar Clausius-Mossotti relation [19] for the high-frequency permittivity, ε_∞:

Eq. (18) allows simplifying (16) to:

Now we consider another limiting case: the driving field being a static field, i.e. ω=0. Then from Eqs. (17) along with Eqs. (1) and (6) we get for the static permittivity ε_s:

Conversely, we get for the permittivity response [with the successful application of Eqs. (1), (6), (17) and (20)]:^[6]

To understand the physical meaning of the frequency ω_R in Eq. (21) we assume for a moment an undamped system, i.e. a system with γ=0; then as ω→ω_R, Eq. (17) implies that the amplitude of oscillations becomes infinite. Essentially, ω_R, represents the frequency of the transverse optical (TO) phonon vibrations that are driven by a static field [63]. We will designate such frequency as ω_TO and simply refer to it as TO phonon frequency, i.e. ω_R≡ω_TO. The TO phonon frequency is somewhat different from the free vibration optical phonon mode, Ω_free, of a diatomic chain which for q=0 [19] is:

Then, with the use of Eqs. (16), (18) and (22), it can be shown that:

which means the TO phonon frequency of modes driven by a static electric field is lower than the free vibrational optical phonon modes of a diatomic chain (both are designated with horizontal long-dashed and short-dashed lines, respectively in Figure 3).

Continuing, we can observe that at ω=εsε∞ωTO, the permittivity, ε, becomes zero (for the undamped system). Previously, we stated that only TO phonons couple to the radiation field. Indeed, the divergence-free nature of the displacement field D, mandates εq·E to be zero which requires the wave vector q to be perpendicular to the driving field E, i.e. requires transverse modes. However, when the permittivity, ε, is exactly zero, the divergence-free condition for the displacement field D may be satisfied without requiring transverse modes, meaning longitudinal modes may be possible in this case. This is the reason why the frequency at which the permittivity, ε becomes zero for the undamped system, is called the longitudinal optical (LO) phonon frequency [19], ω_LO. Hence we obtain the famous Lyddane-Sachs-Teller (LST) relation [19], [23]:

With the use of the LST relation, we are able to obtain from Eq. (21) the familiar expression for the dielectric function of PhP media:

Eq. (25) encompasses the basic photonic responses that result from strong phonon-photon coupling as derived from a classical microscopic picture for the motion of the ions and the associated electron cloud. The particular parameters of a certain material are extracted from experimental reflectivity and/or ellipsometry spectra (see Section 4). Very recently Rivera et al. [64] showed that a quantum mechanical model leads to an effective permittivity response that is essentially equivalent to that of Eqs. (21) and (25).^[7] The quantum model [64], which encompasses treatment of more complex unit cells with multiple-atom basis and tensorial Born effective charges, provides a pathway to first-principle predictions of PhP permittivities, which is particularly important for emergent materials where experimental data may not be available.

3.3 The PhP gap and the reststrahlen band

Figure 3 depicts the coupled phonon-radiation field wave dispersion, ω versus q, i.e. the PhP dispersion (red solid lines). This is determined by ω=cε(ω)q, where ε(ω) is the permittivity response of Eq. (25). The two mixed phonon-photon mode branches are strongly modified from both the phonon dispersion (long-dashed horizontal line) and the high-frequency photon dispersion (slanted solid line, with ω=cε∞q). The phonon-mode dispersion appears flat as it represents phonons with a near-zero phonon wave vector, q. As EM waves are extremely slow varying in the atomic scale it is only the near-zero-q phonons which may couple to EM waves. The frequency region where no mixed phonon-photon modes (yellow-shaded region) is known as the PhP gap signifying forbidden EM propagation; almost all impinging light would get reflected from the material [see Figure 3 (right panel) showing the reflectance from a respective slab with thickness about a half the impinging light’s wavelength].

Alternatively, the PhP gap is called the reststrahlen band; a term coming from German for the spectrum of “residual radiation”. Historically, this term emerged [65] from experiments where a pulsed signal is reflected back and forth between slabs made of a PhP material. While the radiation comprising most of the pulse’s frequency spectrum transmits through the slabs and escapes the set-up, the part of the pulse with frequencies within the PhP gap nearly-perfectly reflects on the surfaces of these slabs, thereby remaining trapped in the slit area between them. These “residual” radiation/rays [66] exit eventually through the slit as a narrow-band beam after multiple bounces. Set-ups like these had been used in the past to create narrowband/quasi-monochromatic sources from broadband thermal sources [65].

The permittivity is negative, i.e. Re(ε)<0, within the PhP gap/reststrahlen band. In the sections that follow we review interesting photonic phenomena and properties within but also around the vicinity of the reststrahlen band; we will be referring to this broader spectral region as the PhP regime.

3.4 Extensions to the basic PhP permittivity model for a broader class of material

The basic permittivity model of Eq. (25) covers successfully the photonic response of some materials but it may be too crude for others where the assumptions made in Section 3.2 fail by a large degree. For example, in the microscopic model the damping was simply associated with a single force proportional to the velocity of the ions. However, there are different distinct mechanisms causing damping of the PhP mode. Mechanisms that can cause PhP damping are scattering at crystal defects [67], as well as decay of the phonon-part of the PhP into acoustic phonons [67], [68], [69], [70]; the latter is caused by the presence of anharmonic terms in the interatomic forces [71]. Damping can also occur because of other coupling to low frequency phonon excitations aided by strain or defects [67], [70], [72].

Schwarz and Maier [70] have asserted that damping caused by the presence of defects and the anharmonicity in the ionic potential can be lumped together in the decay constant γ of Eq. (25). However, according to [70] coupling to low frequency phonon excitations adds up a number of Lorentzian oscillatory terms in the denominator of Eq. (25) with associated poles at the zeros of ωn2−ω2−iωγn, where ω_n and γ_n are the resonant frequencies and damping constants of the low-frequency phonon excitations. In materials where multiple of these damping mechanisms are dominant, as for example, in lithium niobate [73], other authors [67], [73], [74] have considered a frequency-dependent decay constant in the model of Eq. (25), γ(ω); a causal model [60] however requires γ(ω) to be itself complex valued [74]. The treatment of [70] offers more insight to the actual physical mechanisms of damping, beyond considering ad-hoc parameters that improve the agreement between the model-predicted PhPs and experimental results [74].

Furthermore, depending on the lattice symmetry multiple optical phonon branches may exist also. If these are well-separated in frequency, Eq. (25) serves as a good model in the neighborhood of the TO phonon frequency of interest. By well-separated we mean that there should not be a significant spectral overlap between adjacent TO phonon resonances, when taking into account their respective broadening as determined by the decay constant, γ_l, associated with the respective phonon resonance, ω_TO,l. This however, is not true for many materials for which in order to construct the photonic response N independent harmonic oscillators should be considered. Each oscillator corresponds to one of the relevant TO phonon resonances ω_TO,l with damping γ_l with a corresponding LO phonon frequency ω_LO,l, that is independent of the other oscillators [75]. So, the full permittivity response would be in such case:

Not all the oscillators contribute equally; the strength of each oscillator, S_l, is determined by [76]:

implying that a certain oscillator makes a stronger contribution the larger the relative difference between the respective LO and TO phonon frequencies.

Eq. (26) can be approximated to the simpler expression [75], [77]:

Note here that, the independent oscillator model of Eq. (26) would have led to the same decay constant, γ_l, appearing in both the numerator and denominator of Eq. (28) [75], [78]. However, Gervais and Piriou [78] have asserted that a model that permits a different broadening of the LO and TO phonon resonance, – or in other words a different decay constant in the numerator, γ_LO,l and in the denominator, γ_TO,l respectively –, better captures the permittivity response for materials where the anharmonic decay channel of the PhP is strong [75], [78]. There are some constraints in the permissible values for the decay constants, mandated by the passivity condition for the permittivity model which requires the imaginary part of the permittivity function to be positive or zero, i.e. Im[ε(ω)]≥0 [79]. In particular [75], [79]:

4.1 Ionic crystals (insulators)

Ionic crystals are the quintessential example of media with a PhP response. Table 1 outlines the permittivity model parameters of Eq. (26) for some characteristic material of this class. The TO and LO phonon frequencies are accompanied by their corresponding free-space wavelengths, λ_TO and λ_LO. The representation of the reststrahlen band in terms of free-space wavelength is helpful for gauging the relevant feature-size ranges for platforms functional in this spectrum. We will therefore include such representation for all materials in the subsequent tables, 1 through 5 that follow.^[8] While the subject of this review is how to utilize the reststrahlen band for photonic applications, we remark in passing that for some applications the reststrahlen band may need to be avoided. For example, CaF₂ and BaF₂ are popular as IR transparent materials, in which cases they are utilized for spectral regions away from their reststrahlen band.

4.2 Semiconductors: semi-insulating versus doped materials

4.2.1 Isotropic semiconductors

Heteropolar and polar covalent semiconductors comprise effective cation-anion pairs; so they may also exhibit IR active phonons, and hence PhPs. However, unlike the case of the insulating ionic crystals of Table 1, in general their permittivity response is not fully captured by the models of Section 3; these would be only appropriate for semi-insulating or undoped semiconductors. The full permittivity response of semiconductors must include the contribution of the free carriers typically showing a Drude response [19] to the macroscopic electric field. It can be reasonably assumed that the free carriers do not contribute to the local field experienced by the ions.^[9] So the semiconductor’s full permittivity response, ε_sem, can be considered as the cumulative result of the electric-field induced motion of the ions and their bound electrons as if free carriers were not there and then the electric-field induced motion of the free carriers as if the ions and their bound electrons were immovable. In other words:

(30)εsem=εion+εDrude,

where the ionic contribution ε_ion has the same expression as one of the models of Section 3 as appropriate per case; ε_Drude represents the Drude permittivity response of the free carriers, i.e.:

(31)εDrude=−ωp2ω(ω+iγp),

where ω_p represents the plasma frequency^[10] given by [20], [59], [83]:

(32)ωp=(Neme∗+Nhmh∗)e2ε0.

For generality the contribution of both types of carriers, electrons and holes, is taken [84] in Eq. (32) with respective carrier densities N_e and N_h and effective masses me∗ and mh∗;e is the electron charge, ε₀ the vacuum permittivity and γ_p the decay-rate/damping constant of the free-carriers motion.^[11] The Drude response should be suitably adapted to account for cases where contributions from multiple valleys or sub-bands are relevant (e.g. see [20]).

In Table 2 we show some examples of widely used semiconductor material and the parameters relevant to their PhP permittivity response, ε_ion. The parameters refer to the model of Eq. (26) (with N=1) except for the case of AlN that requires the model of Eq. (28) (with N=1). In Figure 6A, we plot the real part of the ionic permittivity response but without the bound electron contribution, i.e. Re(ε_ion–ε_∞), for the materials and model parameters of Table 2. We observe the PhP responses of the different semiconductors cover a wide IR spectral range from about 10 μm to 60 μm. We also show in Figure 6B the related skin depth, δ, scaled with the free-space wavelength λ₀ along with the skin-depth value for Au (solid orange line), – multiplied with a factor of a hundred for an easier comparison. We observe high PhP related skin-depth values for all semiconductors, much higher than the skin-depth of Au. Thus, semiconductors in the reststrahlen band are ideal material for strong IR light-matter interactions.

Table 2:

Ionic permittivity model parameters of widely-used semiconductors.

Sem.a	ε∞	ωTO ×1012 (rad/s)	ωLO×1012 (rad/s)	λTO (μm)	λLO (μm)	γTO×1012 (rad/s)	γLO×1012 (rad/s)
InP	9.61	57.21	64.99	32.93	28.99	0.66	0.66
InSb	15.7	33.91	36.17	55.56	52.08	0.57	0.57
GaAs	11.0	50.48	55.02	37.32	34.23	0.45	0.45
GaP	9.09	68.94	75.87	27.32	24.83	0.21	0.21
SiC*	6.7	149.37	182.52	12.61	10.32	0.90	0.90
ZnSe	5.4	39.00	46.34	48.31	40.65	0.70	0.70
AlNb	4.53	126.20	171.79	14.93	10.96	1.13	1.70
GaNb	5.35	105.48	140.52	17.86	13.40	3.20	3.20

1. ^aParameters are for N=1 and the model of Eq. (26) (if γ_LO=γ_TO=γ) or the model Eq. (28); they are taken from: [34] for InP; [85] for InSb; [86] for GaAs; [74] for GaP; [87], [88] for SiC (*6H/α-II); [89] for ZnSe; [90] for AlN; [91] for GaN.

2. ^bPlease refer also to Table 3 for AlN and GaN.

Figure 6:

PhP permittivity and skin depth of widely-used semiconductors.

(A) Ionic response of the permittivity (real part) without the bound-electron contribution, i.e. Re(ε_ion–ε_∞), versus the free-space wavelength, λ₀, for different semiconductor material (for AlN and GaN which are anisotropic the response for electric field ⊥ to the c-axis is shown). (B) Corresponding skin depth, δ, scaled with the free-space wavelength, λ₀ (shown only within the respective reststrahlen bands with Re(ε_ion)<0). The bold orange line represents the respective value for Au, multiplied by a factor of 100 to aid the comparison.

The examples of Table 2 and associated models capture the photonic responses of most semiconductors. Semiconductor alloys tend to show the vibrational resonances of their constituents typically leading to photonic responses that can be described with the multi-pole models of Eqs. (26) and (28) [20]. For small gap or zero-gap semiconductors, as for example, HgTe, the full picture is more involved [92], [93] as the intra- and inter-band transitions spectrally overlap with the PhP regime and contribute to the permittivity response; these fall out of the scope of phenomena and applications in this review. Furthermore, some semiconductors show directional dependent responses that we discuss later.

4.3 Other anisotropic PhP material

Aside from the semiconductors of Section 4.2.2, other widely used materials in photonics with an anisotropic PhP response are quartz [104], sapphire [79] and vanadium oxide (VO₂) [105], [106]. Their permittivity model parameters are given in Table 4, where we observe these materials possess multiple PhP resonances. The anisotropy in TO phonon frequency underpins the existence of extreme, hyperbolic, photonic responses as we discuss in more detail in Section 5.2 and drives a strong interest towards materials with this property. Furthermore, VO₂ has additional interest as a phase-transition material [107] offering tunability by switching on or off PhP responses (see Section 8.5).

One particular material, hexagonal boron nitride (hBN), has received a recent heightened attention for IR applications because of its anisotropic PhP responses [108], [109], [110], [111], [112], [113], [114], [115], [116], [117], [118], [119], [120], [121], [122], [123], [124], [125]. While in principle the exotic hyperbolic modes exist in all uniaxial PhP material, what sets hBN apart is the degree of TO-phonon-frequency anisotropy leading to a much broader bandwidth where these modes are available. hBN is a van der Waals (vdW) crystal (see Figure 7A). The much looser connection between the hexagonal-lattice planar sheets with weaker vdW forces significantly red-shifts the out-of-plane IR-active phonon frequency [see Eq. (22)] and accordingly the TO phonon frequency [see Eq. (23)]. We show hBNs in-plane (⊥ to the c-axis) and out-of-plane (|| to the c-axis) IR active phonons in Figure 7B and C, respectively, designated with Mulliken’s notation [127] that is common in phonon/Raman spectroscopy communities. In Table 5 we list hBN’s PhP permittivity model parameters. Indeed, we observe a tremendous anisotropy between the in-plane (⊥ to the c-axis) and out-of plane (|| to the c-axis) PhP responses of hBN. To quantify such anisotropy in the observed PhP responses we introduce the PhP anisotropy factor, f_anis, defined as:

Figure 7:

Structure and IR active phonon modes of hBN.

(A) The hBN vdW crystal (atomic spacings taken from [126]). (B) [(C)] The IR-active in-plane (E_1u) [out-of-plane (A_2u)] phonon modes for hBN.

where ω_TO,⊥, ω_TO,|| are the respective TO-phonon frequencies for electric field perpendicular and parallel to the c-axis. hBN has a PhP anisotropy factor, f_anis, of more than 50% compared to only ~10% and ~5% f_anis for AlN and GaN, respectively. Large f_anis values indicate a wider spectral band where indefinite uniaxial photonic responses [25] and associated exotic hyperbolic modes can be available as we discuss in Section 5.2.1.

While the model parameters of the Tables 1–4 are typically obtained by fitting to experimental reflectivity spectra, for hBN this process would be highly non-trivial. Available experimental data highly depend on the polycrystalline nature of the samples, leading to substantial overestimation of the decay constants, as well as on domain off-plane orientation defects [126], leading to mixing between the on- and off-plane responses [126]. As a result the older experimentally extracted model parameters [126] may not be appropriate for single crystalline hBN systems of current interest [108]. It appears that similar but different permittivity parameters have been used in different works with no clear consensus. We choose to indicate in Table 5 the model parameters that have been adopted by many current works [109], [110], [111], [112], following the ones of Dai et al. [108]; i.e. ω_TO, ω_LO, from [126], ε_∞ values from the ab-initio calculations of [129] and γ, from the observed decay of exotic guided modes in hBN by Dai et al. [108].

Another emergent vdW material for PhP-based photonics is α-MoO₃ (see Figure 8A). Due to its crystal structure this material has a very rich vibrational response. We focus our discussion here on the main three IR active modes relevant to photonic applications in the free-space wavelength range around 10 μm–20 μm [128]. The vdW planes connect same-species atoms, O-atoms in this case; so the out-of-plane PhP response comes only from the asymmetric stretching vibration of the Mo-O(1) bond (B_2u mode) (see Figure 8B). This is not too different from the B_3u asymmetric-stretching mode of the Mo-O(3) bond; so both are spectrally very close. However, the O(2) forms two bonds of significantly unequal length with the adjacent Mo atoms [133], which significantly red-shifts the associated IR active mode (mode B_1u) with respect to the B_2u and B_3u modes. As a result α-MoO₃ has a biaxial PhP response (see also Section 5.2.2) with a large in-plane anisotropy (plane ⊥ to the stacking direction [010]). To be specific, an analogous PhP anisotropy factor, f_anis−IP, can be defined for the in-plane (IP) PhP modes as:

Figure 8:

Structure and IR active phonon modes of α-MoO₃.

(A) The unit cell of the α-MoO₃ structure is designated with the green-dotted box. The blue-dashed lines enclose one α-MoO₃ layer that interacts with the adjacent layers with weaker vdW forces. (B) Magnified part of an α-MoO₃ layer: the three in-equivalent O atoms and associated asymmetric stretching vibrations are indicated. [Figure reprinted (adapted) with permission from [130]. Copyright (2013) American Chemical Society.] Note: The stretching modes designation has been added in accordance with the modes reported in [131]. The notation of the in-plane directions has been altered to be consistent with [128] and [132].

where x and z define the two in-plane symmetry directions of α-MoO₃, [100] and [001]. The in-plane PhP anisotropy factor for α-MoO₃, f_anis−IP, is about 40% indicating the possibility for broad spectral availability of indefinite biaxial photonic responses [25] that we discuss in Section 5.2.2.

Thus, hBN and α-MoO₃ are PhP material of high current interest because they exhibit an extreme PhP anisotropy, out-of-plane and in-plane, respectively; this anisotropy leads to unusual photonic responses that we discuss in more detail in Section 5.2. But this tells only part of the story. The weaker cross-plane vdW forces also allow for straightforward fabrication of few layer hBN [108] or α-MoO₃ thereby unlocking extreme possibilities for molding EM waves with PhPs in atomic scale materials. In the section that follows, we go beyond the examples of hBN and α-MoO₃ and discuss the importance and implications of 2D and quasi-2D PhP materials for phonon-polaritonics.

4.4 PhPs in the “flatland”: 2D and quasi-2D PhP materials

2D materials have broken boundaries in material science and are a subject of intense research efforts both for their appealing electronic [134] and photonic [135], [136] properties. Going a step further, Geim and Grigorieva [137] envisioned that 2D material could be used as building-block “legos” that can be joined together plane-by-plane to make materials with new properties. These super-2D materials are structurally stable as the material planes are held together with vdW forces in the absence of dangling bonds; hence their name vdW heterostructures [134], [137]. When it comes to PhPs these vdW heterostructures can provide an avenue to make atomic-level designer PhP material with tailored functional spectral ranges of their IR-active phonons (see schematic of Figure 9). To the best of our knowledge these atomic-level designer PhP materials, comprising dissimilar adjacent vdW planes, are uncharted territory at present. Their possibility has only been envisioned by Caldwell and Novoselov in [142] and briefly discussed in [143], where they were referred to as “crystalline hybrids”.

Figure 9:

Atomic-level designer PhP materials.

A basic conception of a vdW heterostructure involving two fictitious atomic layer materials. The in-plane IR-active phonons emanate from the lattice vibrations within each atomic layer (depicted as springs along the cyan and orange planes). Quantum effects could cause counter-intuitive and completely unexpected responses [138], [139], [140], [141]. In addition, new cross-plane IR-active phonons (vertical springs) could emerge with the appropriate choices of the atomic material layers.

One could argue that for these atomic-level designer materials the effective in-plane PhP responses might bear some similarities to the case of semiconductor alloys, especially as the intra-plane bonds are much stronger than the inter-plane forces. Nevertheless, interlayer charge transfer effects as well as the charged-phonon effect [138], [139], [140], [141] that depend on the particular materials involved may cause completely unexpected responses. The resulting effective permittivity could be very different from the one hypothesized from simpler classical arguments as above. In addition, new IR-active phonons for an electric field perpendicular to the atomic planes could emerge, as a result of possible new IR-activated lattice vibrations in the cross-plane direction. Most certainly, these vdW hetero-composites open exciting bottom-up design routes to new materials with different PhP spectral ranges. vdW hetero-composites could form a very promising research direction for on-demand IR light control with atomically-thin materials.

Although vdW hetero-composites have not been studied in the context of creating new effective photonic responses, vdW crystals supporting IR-active phonons have been interfaced with other 2D material with the purpose of achieving strong coupling between their respective native EM modes (e.g. see [123]). The underlying physics principles of these systems are to tune/modulate the EM wave propagation properties by a strong interaction between the EM wave modes supported in each sub-system. As these do not relate to changing inherent material properties we will review these phenomena in detail in Section 9 where we discuss hybrid platforms. The vast possibilities of these hybrid platforms can have a strong impact on future photonic applications. This has stimulated a strong interest in the community for realizing 2D/quasi-2D PhP materials.

Many of the research efforts have mainly focused on hBN. Monolayer [124], [144] and few layer hBN [108], [122], [125] have been reported and studied for its PhP-related hyperbolic responses by various groups [see also Sections 6.1.2 and 6.1.4]. Detailed studies of atomic-scale (few-layer) hBN [108], [122] have shown that exotic hyperbolic-type of guided modes survive even for atomic-scale thickness [albeit with a modified guided wave dispersion, ω(k)]. This fact provides evidence that the strong TO phonon anisotropy, in the respective planar and c-axis directions, is maintained for atomically-thin hBN [108]. Indeed, [108] reports an excellent agreement between theoretical predictions and experimental results on the PhP guided-mode even for sub-5 nm thicknesses. Furthermore, Raman spectra for the similar in-plane Raman-active phonon mode in monolayer and few-layer hBN have confirmed a small shift only for the optical phonon frequency [144]. These results in combination imply that the TO phonon frequency for the in-plane IR-active mode (Figure 7B) is fairly robust to finite-size effects.

This is likely not the case for the TO phonon mode across vdW planes of hBN, where the atoms oscillate along the c-axis as depicted in Figure 7C. In the extreme case of monolayer hBN, no out-of-plane vibration modes would be possible. Thus, the TO phonon mode along the c-axis completely disappears [123] in hBN monolayers along with the associated extraordinary hyperbolic photonic responses (see also Section 5.2.1). The significant interest in mid-IR photonics with atomically-thin hBN calls for further experimental evaluation of finite-size effects when thicknesses approach the atomic scale. Characterization of the in-plane and out-of-plane TO-phonon modes and their associated photonic responses for atomically-thin hBN will be extremely important in elucidating such effects.

As an insulator [145], hBN may have a limited potential for achieving PhP-based structures or devices with actively tunable responses. Semiconductor 2D/quasi-2D PhP materials would be highly desirable for tunable IR devices. Atomically-thin semiconductors that have been most frequently discussed as 2D PhP material, at least theoretically [146], [147], [148], are the transition metal dichalcogenides (TMDs). TMDs are also vdW crystals allowing their realization in a monolayer or few-layer form via mechanical exfoliation [149]. Semiconducting TMD monolayers are typically in the 1H phase (hexagonal/trigonal prismatic arrangements) [149], [150] as depicted in Figure 10A, which is the standard thermodynamically stable phase for sulfides and selenides of group VI TMDs [150]. Although tellurides of group VI TMDs are typically semi-metallic following octahedral (trigonal anti-prismatic) arrangements [150], a recent work has shown that it is possible to mechanically exfoliate MoTe₂ monolayers onto Si/SiO₂ substrates in the semiconducting hexagonal phase [151].

Figure 10:

Structure and IR active phonons of TMDs.

(A) A monolayer TMD in hexagonal (trigonal prismatic) arrangement (1H-phase). M denotes the transition metal atoms and X the chalcogenide atoms. (B) [(C)] Side view of the monolayer TMD indicating the phonon vibrations induced by an electric field ⊥ [||] to the c-axis, i.e. the in-plane (out-of-plane) IR-active phonon modes (E′) [(A″2)].

What is remarkable about these 2D semiconducting TMDs is that even in the monolayer form they may support extraordinary hyperbolic modes, in stark contrast to monolayer hBN which loses the hyperbolic photonic response in its monolayer form. The monolayer-TMD hyperbolic photonic response is a direct consequence of the trigonal prismatic atomic arrangement, allowing out-of-phase vibrations between the transition metal and chalcogenide atoms not only along the atomic planes (see Figure 10B) but also in the direction perpendicular to the atomic planes, i.e. along the c-axis (see Figure 10C). The bandwidth of such hyperbolic photonic response depends on the spectral separation of the in- and out-of plane IR-active phonons. To the best of our knowledge, experimental characterization for the permittivity responses of monolayer TMDs in the mid/far-IR is not yet available. However, first-principle calculations based on density functional theory (DFT) can predict the resonant frequencies/wavelengths for the IR-active phonon modes of Figure 10B and C [146], [147], [148], which are related to the TO phonon frequency, ω_TO [see Eq. (23)].

We list in Table 6 the IR-active phonon mode wavelengths for certain semiconducting monolayer TMDs (1H-phase), calculated from DFT at the Γ point [146], [147], [148], i.e. at zero phonon wave vector, q~0 (see discussion in Section 3). We observe from Table 6 that the typical anisotropy between the in-plane and out-of-plane IR-active phonon frequencies is about 20%. Although not exactly equal, the anisotropy between the in-plane and out-of-plane IR-active phonon frequencies relates to the PhP anisotropy factor, f_anis, of Eq. (33) (see also footnote c in Table 6). Thus, a more moderate f_anis of about 20% is expected for TMDs, much smaller than that of hBN, which has an f_anis of about 50%, but still much larger than that of the traditional semiconductors of Table 3, which have an f_anis of about 10% or less. Such significant PhP anisotropy factor makes hexagonal 2D TMDs excellent media for hyperbolic photonic responses with the added advantage of gate tunability, which is not available in hBN. The PhP modes listed in Table 6 would be of course lossy, just like in hBN and other semiconductors, because of the relaxation to the low-lying phonon bands [146]. The actual PhP resonances in experimental samples may show a slight spectral shift depending on the isotopic purity of the TMD samples which affects the mass of the ionic oscillators [152].

In reviewing 2D materials as PhP media, special attention should be given to graphene, the first ever 2D material, discovered in 2004 [153]. Since graphene’s discovery there have been numerous reports of its unique electronic properties and vastly tunable photonic response. However, graphene has not received much attention for its potential as a PhP material. Although graphene is comprised of only carbon atoms, it can actually support optical phonon modes [154]. This is because graphene has two atoms within the hexagonal Bravais-lattice unit cell which can vibrate out-of-phase. However, because of the lack of charge asymmetry between the two atoms of the unit-cell basis, it is not possible for EM waves to trigger their out-of-phase vibrational modes. In other words, monolayer graphene supports optical phonon modes, which however are not IR active; hence PhPs cannot be supported on monolayer graphene.

The situation is a bit different in Bernal-stacked bilayer graphene (AB stacking) (see Figure 11). The structural asymmetry of the AB stacking, together with electron hopping between the two layers, breaks the equivalence between the A and B sub-lattice carbon atoms, leading to a dipole moment in each Bravais lattice unit cell. This dipole moment would enable coupling to incident light, i.e. it would make the optical phonon-mode IR active. Indeed, Kuzmenko et al. [138] observed a PhP response for Bernal-stacked bilayer graphene. However, the experimentally observed PhP showed a much higher oscillator strength, which in turn implied a three-orders-of-magnitude higher dipole moment from what would be expected from classical arguments [139].^[12] Also, the observed optical response, as manifest in the measured conductivity, exhibits a spectral profile that is characteristic of Fano resonances [156].

Figure 11:

Structure and IR active phonons of Bernal-stacked graphene (AB stacking).

The IR active lattice vibration mode (E_u mode) is shown. The effective dipole moment between sub-lattice atoms, A and B is acquired by a combined effect of inter-layer charge-transfer and phonon interaction with electron-hole pairs (charged-phonon effect) [140], [141]. The atomic spacings are taken from [155].

This IR active response can be strongly tuned by gating the Bernal-stacked bilayer graphene [138], [139]. These curious results of [138] were interpreted within the quantum mechanical framework of charged-phonon theory adapted for the graphene-bilayer system [140], [141]. In particular, based on this theory the “bare” optical phonon mode via interactions with the allowed electron-hole excitations acquires a large dipole moment. Hence, this “dressed” phonon has a much stronger effective dipole moment [139], [140], [141], [157] than the one predicted by the asymmetry alone, thus leading to a prominent IR active phonon mode observed in related experiments [138], [157], [158], [159]. In other words, this “dressed” phonon effect gives rise to a PhP with much higher oscillator strength. At the same time, the phonon part of this discrete PhP mode interacts with the electronic continuum via electron-phonon scattering, which leads to a Fano profile for the conductivity response [138], [139], [140], [141], [157].

Similar effects can occur also in trilayer graphene [160] with the ABC-stacking demonstrating a much stronger tunability with the gate voltage. Other explored 2D PhP materials include AB-stacked black-phosphorus-carbide (b-PC) [161], with IR-active phonons at frequencies similar to the IR-active spectrum graphene. 2D b-PC shows in-plane optical anisotropy and a Fano spectral response; the latter does not appear to be fully understood at present and could be related to similar effects that were observed in bi-/tri-layer graphene.

We discussed previously α-MoO₃, a vdW transition metal oxide (TMO) [162], that also exhibits in-plane anisotropic PhPs [128], [132]; α-MoO₃ with atomic-scale thickness has been recently realized and studied for its PhP responses [162]. This emergent PhP material is expected to attract increasing interest as an atomically-thin bi-axial material [96] with extreme in-plane anisotropy and IR activity covering a spectral range between ~10 μm and 20 μm [128]. Finally, phosphorene, i.e. a monolayer of black phosphorus has been shown to support IR-active phonons at wavelengths around 20 μm [163].

Summarizing, IR-active phonons in 2D materials, and associated PhP photonic responses, cover a large part of the IR spectrum between free-space wavelengths of 6 μm and 45 μm. In Figure 12 we outline such spectral coverage in IR activity with representative 2D materials discussed in this section. The 2D material are arranged from right to left in order of decreasing wavelength of their IR-active optical phonons.

Figure 12:

Atomically-thin PhP materials.

The PhP responses of the examples shown fully cover the spectral regime between 6 μm and 45 μm (free-space wavelength). The material schematics are placed in order of decreasing wavelength (from right to left) of the associated IR active optical phonons (see top arrow; note: wavelength scale is not linear/proportional). [*] The α-MoO₃ schematic is adapted from [164] via CC BY 4.0 license (see link at https://creativecommons.org/licenses/by/4.0/).

5.1 Photonic responses in isotropic PhP material

To visualize the photonic responses in the PhP regime we consider a general fictitious material, with permittivity following the model of Eq. (26) that broadly covers many PhP materials. We consider realistic model parameters within the range of the parameters for the semiconductors of Table 2. In particular, we take ω_LO=1.3ω_TO, ε_∞=6, and γ=0.008ω_TO (N=1). We plot in Figure 13 the real and imaginary part of the permittivity, versus the frequency, ω, that has been scaled with the TO frequency, ω_TO. We can identify three distinct types of photonic responses that underpin distinct design principles for platforms relevant to IR applications.

Figure 13:

Photonic responses of isotropic PhP media.

A general example of the permittivity response, ε(ω), from the model of Eq. (26) (with N=1) is shown representing a fictitious material yet with realistic parameters within the range of widely-used semiconductor material. The blue-shaded region indicates the metallic spectral regime with Re(ε)<0. Within this spectral regime, the blue rectangle designates the regime where the PhP permittivity takes values typical of visible-light plasmonics with noble metals. The yellow-shaded region designates the spectral regime of high permittivity, and so high refractive index (HRI). The ENZ point at ω=ω_LO is also shown.

5.2 Photonic responses in anisotropic PhP material

As we discussed in Section 4 some material, e.g. AlN, GaN, hBN, sapphire, insulating-phase VO₂, semiconducting TMDs, α-MoO₃ etc., have anisotropic PhP responses, with hBN dominating recent attention [108], [109], [110], [111], [112], [113], [114], [115], [116], [117], [118], [119], [120], [121], [122], [123], [124], [125], [173]. hBN owes such increased attention to its being a vdW crystal that not only leads to an extreme PhP anisotropy but also brings such anisotropy in atomically-thin structures. α-MoO₃ is another emergent PhP material attracting significant recent interest [128], [132], [162] due to its inherent in-plane PhP anisotropy that enables a biaxial response. Also a vdW crystal, α-MoO₃ can be made into 2D/quasi-2D structures. Next, we discuss how these anisotropic PhP responses, of uniaxial or biaxial type, translate to distinct types of supported photonic modes by these material.

5.2.1 Uniaxial PhP material

Most anisotropic PhP material would be of uniaxial type, with different permittivity responses, ε_|| and ε_⊥, for an electric field parallel and perpendicular to the c-axis; thus the c-axis is also the optical axis. Propagating EM modes through uniaxial PhP media have been studied since the 1960s [174]. In analogy to bulk EM modes supported in uniaxial crystals [96] these modes were called ordinary PhPs (O-PhPs) and extraordinary PhPs (EO-PhPs) with respective wave dispersion relations:

(38)kx2+ky2+kz2=ε⊥ω2c2,

and

(39)kx2+ky2ε||+kz2ε⊥=ω2c2.

It was not however until the rise of the metamaterials field, and in particular until after the conception of hyperbolic metamaterials [47], [48], that the truly “extraordinary” properties of the EO-PhPs were uncovered. Metamaterials spurred the design of man-made EM meta-atoms yielding vast and unusual effective EM responses not readily available in nature, including media with permittivity of opposite sign along different directions (indefinite media) [25]. Later, in 2010, da Silva et al. [175], noticed that these curious indefinite permittivity responses are actually available in natural quartz by virtue of its anisotropic PhP response. In particular, within a distinct frequency regime, quartz has ε_||ε_⊥<0, causing a hyperbolic dispersion form for the EO-PhPs (Eq. 39) and in turn negative refraction [176], [177], a phenomenon which up to that point had been associated only with man-made artificial metamaterials [178]. An example from [179] of such EO-PhP-enabled negative refraction in natural crystals is shown in Figure 14; this negative refraction also enables superfocusing effects [176], [179] similar to those reported with artificial hyperbolic metamaterials [47], [48].

Figure 14:

PhP media as natural “metamaterials”.

Negative light refraction at a MgF2 slab enabled by its indefinite permittivity response of ε_||ε_⊥ <0 emanating from anisotropic PhPs. (After [179]; reprinted from [179]. Copyright 2013 with permission from Elsevier.)

Generally an arbitrary EM wave incident on an anisotropic uniaxial crystal with a general interface cut couples to both ordinary and extraordinary modes [80], [96]; for PhP crystals this would mean both O-PhPs and EO-PhPs would simultaneously couple leading to birefringent PhP propagation [174]. However, in most frequently encountered situations, the interface plane is perpendicular to the c-axis. For this particular interface cut, the O-PhP and EO-PhP modes decouple with O-PhP modes becoming accessible only by S-polarized waves (left panel of Figure 15A) and EO-PhP modes becoming accessible only by P-polarized waves (right panel of Figure 15A) [80], [96].

Figure 15:

Photonic responses for uniaxial PhP media.

(A) S- and P-polarized waves incident on a uniaxial PhP medium with interface cut ⊥ to the c-axis. The anisotropic permittivity tensor is also indicated. (B) S-and P-polarized waves couple respectively to the ordinary (O) and extraordinary (EO) waves. The tables designates the possible distinct photonic wave mode responses, arising in different frequency regimes determined by the respective TO and LO phonon frequencies in the directions || and ⊥ to the c–axis. (C) Graphical representation in wave-vector space, (k_x, k_y, k_z), of the characteristic wave modes outlined in (B) (EFS/surfaces of wave normals). These are determined by the respective dispersion relations for the O-waves and EO-waves (equations in boxes) for the applicable signs of the permittivity values in each case (material loss is neglected). The gray planes represents the plane of incidence, as seen also in (A). The modes designated as metallic in (B) are not drawn here; these refer to highly lossy wave solutions similar to plane waves in metals.

In Figure 15B we summarize the corresponding wave dispersion forms for these cases, determined from Eqs. (36) and (37) as applicable per case; we consider all the possible permittivity value combinations available throughout the PhP regime with widely used materials. We depict in Figure 15C the graphical representations^[13] of the distinct possible wave dispersion forms. Both types of the hyperbolic dispersion forms, type I and type II (both seen in Figure 15C), are available for different frequency ranges. These are the familiar dispersion forms observed in hyperbolic metamaterials [47], [48], [80], [182]. Aside from negative refraction and superfocusing effects reported in [175], [176], [179] these PhP-enabled hyperbolic dispersions in natural PhP materials can bring to the IR many of the other exotic effects associated with hyperbolic metamaterials, such us dark-field imaging or a Purcell enhancement [48], [182] from IR quantum dots [183].

5.2.2 Biaxial PhP material

Certain materials may additionally exhibit in-plane anisotropy, i.e. the PhP response is also anisotropic in the plane perpendicular to the stacking direction, as for example, in α-MoO₃ (see also Figure 8 and Table 5). The photonic responses of biaxial media is much richer and a lot more complex than that of uniaxial media. Similar to uniaxial media, the wave dispersion of waves propagating in a non-magnetic biaxial medium, is determined by Maxwell’s equations mandating that the supported modes must satisfy Det[A]=0, with [A] being a 3×3 matrix with elements:

(40)Aii=εiω2c2−∑j=13(1−δij)kj2,

with i=1, 2, 3 and

(41)Aij=kikj, with i≠j,

and i, j=1, 2, 3; δ_ij is Kronecker’s delta and the running indices correspond to the Cartesian directions as: 1↔x, 2↔y, and 3↔z.

However, unlike the case of uniaxial media, the wave dispersion relations, ω(k), resulting from solving the condition above cannot be represented by simple expressions like Eqs. (38) and (39). Given the vast possibilities in biaxial PhP responses, it goes beyond the scope of this review to identify all possible photonic mode forms that are supported. To give the reader a flavor for the unconventional possibilities with PhP biaxial media we showcase an example in Figure 16 that corresponds to ε_x<0, ε_y>0, ε_z>0 with ε_y≠ε_z. This case is entirely possible with α-MoO₃ in the spectral range λ₀ ~ 11–12 μm (free-space wavelength). The corresponding depicted equi-frequency surface (EFS) shows an unusual shape resembling type I hyperbolic dispersion joined with an ellipsoidal dispersion. This unusual EFS has been recently reported for α-MoO₃ (see [128]). The EFS of Figure 16 depicts all available modes in the entire wave vector space for a particular frequency, ω. However, the associated electric field corresponding to a certain wave vector k at a certain frequency, ω, must also be examined to determine whether the mode is or is not accessible by a certain light excitation.

Figure 16:

Example of an indefinite photonic response of a biaxial PhP medium.

The indicated crystallographic directions, associated with the x, y, z directions correspond to the case of α-MoO₃ (see also Figure 8). The EFS/surface of wave normals depicts a general possible case with in-plane indefinite permittivity response, i.e. with ε_xε_z<0. In the depicted case: ε_x<0, ε_y>0, ε_z>0 with ε_y≠ε_z.

In Section 5 we reviewed the different types of photonic responses enabled by PhPs. In the following section we review the geometries which in synergy with PhP responses lead to confined EM modes which form the backbone of many of the diverse reported IR PhP-based capabilities and phenomena.

6.1 Confined PhP modes at planar interfaces and ultrathin films

6.1.1 Ordinary surface phonon-polaritons (O-SPhPs)

Localized quasi-bound EM modes^[14] may be supported at the surface of PhP materials, with Re(ε)<0. These modes share similar characteristics to SPPs [38] and are so analogously called surface phonon polaritons (SPhPs) [18], [184].^[15] The dispersion relation for SPhPs, relating their lateral wave vector k_x, with their frequency ω, is the same as for SPPs given by [38]:

(42)kx=ε(ω)εdε(ω)+εd ωc,

with c being the speed of light, ε(ω) the complex frequency-dependent permittivity response of an isotropic PhP medium and ε_d the permittivity of the dielectric material interfacing with the PhP medium.^[16]

SPhPs are supported within the PhP gap/reststrahlen band where Re(ε)<0. The black-solid line in Figure 17A is a graphical representation of the dispersion relation of Eq. (42) for an example SPhP mode supported at the interface between air and the PhP material of Figure 13. The frequency ω is scaled with the TO phonon frequency, ω_TO, and the real part of the wave vector is represented with the dimensionless quantity β=Re(k_x)c/ω; thus the vertical brown-dashed line at β=1 represents the lightline. Furthermore, the electric-field landscape (x-component) for the mode designated with the green rectangle, shown in the upper part of Figure 17, suggests a well-confined surface mode around the interface of significant propagation range.

Figure 17:

Surface phonon polaritons (SPhPs) at plane interfaces.

(A) SPhP dispersion (black solid line) at the interface of air and the PhP medium of Figure 13. Re(k_x) is represented with the dimensionless parameter β after scaling its value with the free-space wave vector ω/c. The upper panel shows the x-component of the electric-field for the mode designated with the green rectangle. (Note, λ_TO=2πc/ω_TO.) The SPhP dispersion is compared with the SPP dispersion at an air-Au planar interface at two different frequency regions; around telecomm wavelengths (red-dashed line) and in the green-blue part of the visible spectrum (cyan line). (B) Figure of merit (FOM) [see Eq. (43)] for the SPhP mode (black-solid line) compared to the FOMs of the SPP mode at the two frequency regimes that were shown in A.

The quality of confinement and propagation range are typically in competition for surface waves at interfaces of lossy materials, with Re(ε)<0 [186], [187]. Different quantifiable measures, i.e. figures of merit (FOM), to capture the performance of these confined surface waves have been hitherto introduced. Here we consider for the SPhP a FOM with the intent to capture simultaneously both the confinement quality and propagation sustainability of the mode, given by:

(43)FOM=|Im(ky)|Im(kx).

The defined FOM represents the ratio of the distance the surface wave travels along the interface until its electric field strength drops by a factor of 1/e over the perpendicular-to-the-interface distance at which the electric field strength drops by a factor of 1/e inside the dielectric medium. The FOM value from Eq. (43) should be used in conjunction with the dispersion relation to characterize the overall quality of an SPhP mode. For example, dispersion regions which are very close to or above the lightline certainly signify a weak confinement to the interface; thus any relatively high FOM values in these region only signify a large propagation range. The FOM for the SPhP of Figure 17A is shown in Figure 17B with a black solid line.

To obtain a sense of how SPhPs relate to SPPs, we also include in Figure 17A and B the dispersion relation and FOM-values respectively of an SPP mode supported at the air-Au interface (see also Figure 2A), where realistic optical parameters are used for Au (see Figure 1). To make such comparison, first the SPP’s frequency ω is scaled with the plasma frequency, ω_p. As we discussed in Section 2, SPPs in the mid- and longwavelength IR are essentially grazing waves loosely confined at the interface. So we find it useful to compare the IR SPhPs with their SPPs counterparts but around telecomm and visible frequencies. For this reason, the SPP’s ω/ω_p value is shown in Figure 17A after being multiplied by a suitable factor yielding, respectively, the red-dashed curves, representing the SPP mode around telecom frequencies, and the cyan curve, representing the SPP mode around visible frequencies.

It appears the SPPs on the surface of Au are generally closer to the lightline in comparison to SPhPs. As with respect to FOM values, the telecom-wavelength SPP modes show a high FOM- but that would mean only a low propagation loss as confinement is ultra-weak for the telecom SPP mode which essentially follows the lightline. The more confined visible SPP mode is clearly outperformed by its SPhP counterpart for most of the reststrahlen band range. This superior performance of the SPhPs modes, in terms of strong confinement and low propagation loss, occurs around the middle of the reststrahlen band away from the TO and LO phonon frequencies. Actually, it is interesting to observe that unlike its SPP counterpart, the SPhP mode runs through a broader range of β values,^[17] even crossing through the lightline into β values less than 1 at frequencies approaching the LO phonon frequency, ω_LO [Foteinopoulou S, unpublished]. It is worth mentioning that such SPhP dispersion crossing into the above-the-lightline region was first experimentally observed in the 1970s for the case of α-quartz [189], [190]. However, as the FOM shown in Figure 17B attests, high performing SPhPs are associated with the portion of the dispersion curve below the lightline and around the middle of the reststrahlen band.

SPhPs have been utilized to enhance near-field heat transfer [191], [192], [193], [194] (see also Figure 18A), as well as to enhance SHG signals [195] (see Figure 18B). Theoretical calculations have also shown [196] that SPhPs can enhance the inherent thermal conductivity of a material. SPhPs substrates may enhance coupling with biomolecular vibrational modes and are thus relevant to SEIRA spectroscopy [197]. Moreover, when SPhPs are accessed with an attenuated total internal reflection (ATR)-type prism [1] ultra-sharp peaks can emerge in the reflectance spectra that can be used for the IR analog [198] of surface plasmon resonance (SPR) sensors [199]. Pairing up SPhP modes with other physical systems such as phase-change materials [107], [200], [201], 2D materials with gate-tunable properties [135], [136], or plasmonic [2], [186], [187] and dielectric [15] waveguides, as well as IR active media such as quantum dots [183] and quantum wells including strained superlattices [202], unleashes many more possibilities for harnessing IR light pertinent to integrated and/or tunable infrared devices, IR imaging systems [37], [203] and quantum-well IR photodetectors [202], [204]. We discuss more examples of these hybrid systems in Section 9.

Figure 18:

Examples of SPhP-enabled phenomena.

(A) SPhP-enabled enhanced near-field heat transfer: (Left) heat transfers from a hot to a cold quartz plate, separated by a distance, d; (Right) Observation of a ∝1/d² scaling law, a signature of near-field heat transfer facilitated by the SPhP. (Reprinted with modification with permission from [194] (see link at https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.175901). Copyright 2018 American Physical Society.) (B) SPhP-enabled enhanced second harmonic generation (SHG): (Left) The fundamental beam couples via an ATR prism to a SPhP mode supported at the air-SiC interface. (Right) The coupling to SPhP manifests itself as a dip in the reflectivity spectrum (red line), giving rise to a SHG signal that out-couples on the other side of the prism (blue line). (Reprinted (adapted) with permission from [195]. Copyright 2017 American Chemical Society.)

6.1.2 Extraordinary surface phonon-polaritons (EO-SPhPs)

The indefinite anisotropic PhP responses, as discussed in Section 5.2, unlock possibilities for unprecedented types of confined EM modes at PhP material interfaces; this is an unfamiliar territory with no analogue in plasmonics where metals have isotropic negative permittivity responses. The most trivial case involves a uniaxial PhP material in a regime where the permittivity is negative both along and in the normal plane to the c-axis. In such a case, assuming the interface is along the normal to c-axis plane, a minor adjustment to Eq. (42) accounts for such anisotropy and determines the dispersion for the SPhP modes; this can be seen, for example, in [205], where SPhPs are excited on the surface of 4H-SiC for a frequency within the associated reststrahlen bands of both the parallel and normal to the c-axis directions [97].

However, the in-plane anisotropy present in certain bi-axial PhP material, such as α-MoO₃, introduces entirely new avenues for confining EM waves at PhP-material surfaces beyond the familiar SPPs, such as Dyakonov surface waves [206]. Dyakonov type surface waves are also supported in uniaxial hyperbolic PhP material for an interface cut along a plane that includes the optical axis [207]. Actually, very recently such atypical surface confined waves have been theoretically predicted and experimentally observed on the edge face of an hBN-crystal flake excited by the near-field emanating from a scattering-scanning near-field optical microscopy s-SNOM^[18] tip (see mode designated as hyperbolic surface polariton (HSP) in Figure 19 after [209]). Alfaro-Mozaz et al. [210] also report antenna behavior for an hBN sub-micron strip that is enabled by hybridization of the HSP modes between the edge faces that lead to sharp resonances of a quality factor, Q, of around 100. Moreover, the guided modes on the hBN strips have shown to couple strongly with the molecular vibrations of organic semiconductors making this system ideal for SEIRA spectroscopy [211].

Figure 19:

Two types of confined modes at an hBN flake with ε_x=ε_y<0 and ε_z>0.

(I) SPhPs confined on the edge-face-air boundary, i.e. || to the xz-plane; these emanate from the indefinite, hyperbolic, dispersion on the xz-plane and were termed as hyperbolic surface plasmons (HSPs) in [209]. (II) Hyperbolic GPhPs, emanating from type II hyperbolic dispersion within the hBN flake; these propagate along the y-direction yielding a fringe pattern on the top face (and bottom face not shown here) and were termed as hyperbolic polaritons (HPs) in [209]. The different distance between the field maxima at the edge and top face as indicated manifests the co-existence of these two distinct modes. [Reprinted (adapted) with permission from [209]. Copyright (2016) American Chemical Society.]

6.2 Particle PhPs

Light incident on plasmonic nanoparticles may give rise to highly localized resonant modes that have been termed as particle plasmons [41], LSPs [40] or Mie plasmons [42], [43], because of the Mie-type light scattering involved [44]. Nanoantennae [41], high-Q Fano resonances for biosensing [218], aperture-less near-field optical microscopy [219] and spasers [220], etc. form only an indicative list of a plethora of fantastic effects and applications enabled by particle plasmons. Conversely, localized modes “molded” around micron-sized PhP particles can be the basis for shaping IR light for targeted IR applications; analogously we will refer to these modes as particle PhPs.

However, unlike particle plasmons, particle PhPs may arise from distinctively different material responses which are available in the PhP regime; these can range from plasmonic-like responses characteristic of noble metals in the visible spectrum to HRI responses (see Section 5.1 and Figure 13). Furthermore, indefinite uniaxial or biaxial responses (see Section 5.2 and Figures 15 and 16) are also available with PhP material. These responses do not have any counterpart in dielectrics outside the PhP regime or metals. Thus, in the following we will be referring to particle PhPs as ordinary if connected to plasmonic-type or HRI isotropic photonic responses and as extraordinary if connected to indefinite anisotropic [25] PhP photonic responses as discussed in Section 5.2.

6.2.1 Ordinary particle PhPs

Just like the case of plasmonic nanoparticles, the resonant response of PhP microparticles to incoming light can be captured by the extinction efficiency, Q_ext [221], which is the extinction cross-section [44] normalized to the particle’s cross-sectional area. The extinction efficiency involves the particle’s response both in terms of scattering and absorption and so gives a more complete picture for the resonant response of particles comprising lossy photonic materials, such as PhP materials. The extinction efficiency sums up the effect of all possible resonant orders, that may be excited at the microparticle (multipoles). Assuming a plane wave of angular frequency, ω, is incident on a microsphere of radius R immersed in air (vacuum), the extinction efficiency, Q_ext, is equal to [44], [221], [222], [223]:

(45)Qext=2c2ω2R2∑n=1∞(2n+1)Re(an+bn),

with n=1 representing the dipolar resonant order, n=2 the quadrupolar resonant order and so on.

For each order the mode is a superposition of two characteristically different field distributions each having a strength represented by the coefficients a_n and b_n, determined by the EM boundary conditions at the sphere’s surface.^[19] Actually, a_n represents the strength of a field distribution having an electric field with a strong radial component (implying a strong surface charge density) and a magnetic field that is purely tangential. Conversely, the coefficient b_n represents the strength of a field distribution having a magnetic field with a strong radial component and an electric field that is purely tangential. Accordingly, a_n and b_n have been associated with an electric- and magnetic-type resonant response, respectively [223].

We plot in Figure 20A the extinction efficiency for an example case. We do so in order to showcase the existence of distinct types of ordinary particle PhPs emanating from the different isotropic photonic responses in the PhP regime: plasmonic-like and HRI responses, respectively. The particular example we consider here involves a microsphere made of the PhP material of Figure 13, with radius, R, equal to 0.065 λ_TO, with λ_TO being the transverse-optical phonon free-space wavelength. We observe two resonances, hence two particle PhPs: one within the plasmonic permittivity regime, that we will be referring to as the type A particle PhP, and one within the HRI regime, that we will be referring to as the type B particle PhP.

Figure 20:

Ordinary particle PhPs.

(A) Extinction efficiency calculated from Mie theory [44] for a sphere with the permittivity of Figure 13 and radius equal to 0.065λ_TO where λ_TO=2πc/ω_TO. The dashed vertical lines designate the two particle PhP resonances: type A, i.e. an LSPhP mode and type B that is a high-refractive-index (HRI) Mie PhP mode. The two top insets show the permittivity of the microsphere around the respective spectral ranges of the two resonances. (B) Electric-field intensity, |E |², and streamlines for the HRI-Mie PhP having a mixed electric-dipole (ED) and magnetic-dipole (MD) character. (C) Electric-field intensity, |E |², and streamlines for the LSPhs of electric-dipole (ED) character.

The type A ordinary particle PhP is similar to the LSP; and hence is typically referred to as the localized surface phonon polariton (LSPhPs) [224], [225], [226], [227]. Indeed, its dominant response comes from the dipolar electric-type term, a₁, yielding an electric field landscape shown in Figure 20C that resembles the familiar dipolar response of plasmonic nanospheres in the visible spectrum (see, e.g. LSP1 mode of Figure 2E). The observed LSPhP resonance is excited near the Fröhlich condition [1], [222].^[20] This is expected given the deep-subwavelength size of the microsphere. Just like in the case of LSPs, as the microsphere becomes larger the electric dipolar resonance would occur increasingly further from the Fröhlich condition and an electric quadrupolar resonance would emerge. Eventually, for microsphere sizes of the order of the wavelength the near-field response to the incoming EM field involves multiple orders and both electric- and magnetic-type field distributions.

The type B ordinary particle PhP shown in Figure 20B resembles the Mie-resonant responses of dielectric particles [221]. For this particular example, the electric field involves contributions both from the electric dipole (ED) term, a₁, and the magnetic dipole (MD) term, b₁.^[21] Therefore the type B particle PhPs may be referred to as HRI-Mie PhPs [228], in order to distinguish these from LSPhPs; however unlike pure dielectric Mie modes, these type B particle PhPs have a lossy character. In fact, the high extinction efficiency that is observed in Figure 20A is in large part a result of a high absorption efficiency, as opposed to a high scattering efficiency. HRI-Mie PhPs have not, thus far, been widely investigated.

Actually, to-date it is mostly type A particle PhPs, i.e. LSPhPs, that have been widely studied. Theoretical reports of LSPhs involve infinite or finite-height cylinders with circular, elliptical and rectangular cross-section cylinders [100], [229], [230] as well as disks [225]. The existence of LSPhs has been experimentally verified; in fact LSPhPs have been spectroscopically characterized and/or imaged in sub-micron pillars [226], [231], microdisks [224] (see Figure 21A) or nanoparticle clusters [227]. The field enhancements in LSPhP excitations may be utilized for thermal emission [86], [98], [100], [232], enhancing non-linear phenomena such as second harmonic generation, for example, see [226] and in sensing platforms like SEIRA [197]. LSPhP properties can be further tuned by harnessing their collective interactions in a 2D array lattice [233], just like with lattice LSPs [218]. Finally, particle PhPs in SiO₂ microspheres have been shown to enable enhanced near-field heat transfer from the microsphere to a SiO₂ plane by coupling to the respective plane SPhPs [234]; likely both LSPhPs and HRI Mie PhPs contribute to this phenomenon in this case.

Figure 21:

Experimentally observed particle PhPs.

(A) LSPhPs [Type A ordinary (O) particle PhPs] in GaN microdisks resting on a SiC substrate. The two dipolar modes depicted are excited at different free-space wavelengths and have enhanced fields at the top or bottom of the microdisk, respectively. This is because of the different corresponding dielectric environments affecting the resonance conditions. (Adapted from [224] with the permission of AIP publishing, see link at https://aip.scitation.org/doi/10.1063/1.4929502.) (B) EO particle PhPs at hBN sub-micron cones observed at: 7 μm. The unconventional photonic interference inside the hBN nanocone (top panels) stemming from hBN’s hyperbolic dispersion yields characteristic hot rings on its surface at deep-subwavelength distances, which were experimentally observed with s-SNOM (bottom images) (scale bars are 300 nm). [Reprinted (adapted) with permission from [113]. Copyright 2016 American Chemical Society.] (C) EO particle PhPs at an α-MoO₃ microdisk at: ~11 μm. The observed deep-subwavelength fringe pattern (top panel) comes from the underlying indefinite biaxial permittivity response. The associated unconventional photonic dispersion (similar to the one of Figure 16) manifests itself in the associated Fourier transform (bottom panel). (Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature-Nature [132]. Copyright 2018.)

6.3 New physics with particle PhPs: LO-TO phonon mixing and atomic-scale thermal emitters

Both O-PhPs and EO-PhPs are expected to be major protagonists in conceiving new platforms to shape, divert, absorb or emit IR-light as mandated by specific applications. Very importantly, these systems may also unveil new PhP physics. For example, although LO phonon oscillations are known to be inaccessible by IR light in planar geometries [19] it has been very recently reported [236] that microparticle resonators comprising certain anisotropic long-cell polytypes of a PhP material, like 4H-SiC, may enable making zone-folded LO phonons accessible with IR light excitations. In turn, this enables new hybridized longitudinal-transverse PhP excitations. This recent study will inspire new investigations for the understanding, control and realization of these hybrid excitations in other PhP material systems. As Gubbin et al. report [236], these mixed excitations pave the way for the realization of electrically pumped mid-IR emitters. Furthermore, particle PhPs can be brought down to the atomic-scale in low-dimensional particles in which cases atomistic models would be required to capture their PhP responses. Recently, Venkataram et al. [237] developed an atomistic theory to capture PhP-mediated emission and heat-transfer effects between elongated low-dimensional molecules and macroscopic objects that cannot be interpreted with classical models.

7.1 PhP-based metamaterials

7.1.1 Hyperbolic PhP metamaterials

While PhP media can be “natural metamaterials” with hyperbolic photonic responses of type I or type II, as we discussed in Section 6.2, it is certainly worthwhile to look at the metamaterial patterning route to a hyperbolic response with PhP material. This way a broader parameter space and spectral availability of the hyperbolic photonic dispersion can be accessed and tailored by design at will. The associated building-block units should comprise a PhP material in its reststrahlen band and should be deeply sub-wavelength, with a size of the order of λ₀/100, with λ₀ being the free-space wavelength [80], [238], [239]. When these elemental units are put in periodic arrangements they respond as “meta-atoms” to incoming IR light making a new material with effective photonic properties. The extremely, deep-subwavelength size for the meta-atom is a requirement for a high quality effective hyperbolic response [80] that sustains up to large lateral wave vectors [238].

Typical arrangements reported with PhP material involve either periodic-rod [80], [240], [241] or lamellar structures [238], [239], [242] (Figure 22A). At such deep-subwavelength dimensions it can be assumed the electric-field is quasi-static within each constituent. Then the respective effective permittivity responses parallel and perpendicular to the optical axis, ε_|| and ε_⊥, can be obtained by established mixing formulae [48], [80], [238], [240] that take into account the EM boundary conditions that apply at the two different electric field orientations. In particular, for the case of periodic rods [80], [240]:

Figure 22:

PhP metamaterials: Representative characteristic structures with indicative building-block sizes with respect to the free-space wavelength, λ₀.

(A) Type I and Type II PhP hyperbolic metamaterials (see [80], [240], [241], [242]). (B) Biaxial indefinite PhP metamaterial from natural PhP hyperbolic crystals, e.g. hBN (see [243]). (C) Magnetic PhP metamaterial (see [181], [244], [245]).

(46a)ε||=fε1+(1−f)ε2

(46b)ε⊥=ε2(1+f)ε1+(1−f)ε2(1−f)ε1+(1+f)ε2,

and for the case of periodic lamellae [48], [182], [238]:

(47a)ε||=ε1ε2fε2+(1−f)ε1

(47b)ε⊥=fε1+(1−f)ε2,

where ε₁, ε₂ are the isotropic permittivities of the two material constituents and f is the volumetric ratio of constituent ‘1’ with respect to the volume of the unit meta-atom. The optical-axis direction of the effective metamaterial for both systems has been designated in Figure 22A.

An indefinite response of Re(ε_||)· Re(ε_⊥)<0 is needed to obtain hyperbolic photonic responses of either type. For this to be possible the constituents should have an opposite permittivity, i.e. Re(ε₁)· Re(ε₂)<0. One possibility is for one constituent to be a PhP material operating within its reststrahlen band with the other constituent being a dielectric. Alternatively, both constituents can be a PhP material with one operating within and the other outside their respective reststrahlen bands. In principle, in a dielectric-PhP material composite both type I and type II hyperbolic responses are accessible at different spectral bands [80], [240]. When both constituents are PhP materials, several spectral bands of type I and type II hyperbolic behavior may emerge; this behavior has been reported for self-organized eutectic ionic crystal mixtures such as NaCl-LiF and NaCl-KCl [80], [241].

PhP-based hyperbolic metamaterials can be instrumental in enabling new components for table-top IR optics. For example, [80] reports that PhP-based hyperbolic metamaterials can function as both polarization filters and converters. Furthermore, the IR lensing and beam-steering capabilities reported for natural PhP hyperbolic crystals [175], [176], [177], [179] can be extended to PhP hyperbolic metamaterials while adding at-will design flexibility over the operational bandwidth and steering angles. Recently, dark-field imaging in the optical spectrum was shown to be possible with a type-I/typeII hyperbolic metamaterial heterostructure [246]. This capability can be extended to the IR with corresponding judiciously designed heterostructures of either two PhP hyperbolic metamaterials or a PhP hyperbolic metamaterial and a natural PhP hyberbolic crystal and can benefit IR imaging systems [37], [203]. Finally, [242] reports that PhP hyperbolic metamaterial may exhibit enhanced super-planckian thermal emission.

7.2 PhP-based mesophotonic systems

We use the term “mesophotonic systems” here to describe systems in the sub-wavelength regime where geometric optics fails but that may not respond with effective photonic responses as metamaterials. Typical feature/building-block sizes in this class of system are ~λ₀/2, with λ₀ being the free-space wavelength. We outline below couple of characteristic platforms in this class that have utilized PhP material.

7.2.1 Particle-PhP arrays: metasurfaces

One-dimensional (1D) or 2D arrays of periodic or more general arrangements of particle PhP resonators can be envisioned constituting essentially PhP-based metasurfaces [see, e.g. depiction in Figure 23A]. This line of research has not been widely pursued to the best of our knowledge but is very promising for sculpting, diverting, absorbing/emitting IR light and biosensing methods like SEIRA [197]. For example, one could envision an LSPhP-based platforms for IR beam steering operating analogously to LSP-based ultra-refractive metasurfaces [253]. Then, established design routes of all-dielectric Mie-based metasurfaces [228], [254], such as reflection suppression relying on the lattice Kerker effect [255], can inspire new Mie-PhP-based platforms for near-perfect absorption/emission [256]. Furthermore, new extraordinary particle PhPs set a pathway for unexplored metasurface platforms that have no analogue in IR metallic [26], plasmonic [253] or all-dielectric [228], [255] metasurfaces. Collective phenomena analogous to lattice LSPs [218] will be important to all particle PhP-based metasurfaces.

Figure 23:

Examples of PhP-based mesophotonic platforms.

(A) PhP analogues of metasurfaces and EOT platforms (e.g. see [87], [100]). (B) PC systems: (left panel) A PC made of PhP-material rods with overlapping photonic and PhP band gaps that exhibits extreme field re-configurability within an ultra-narrow spectral window around the TO phonon frequency (see [252]). (Middle panel) An ultra-thin PhP-material layer atop a PhP-material-dielectric Bragg reflector comprising only one unit cell acts like a frequency-selective near-perfect absorber (see [101]). (Right panel) A photonic hypercrystal: A 1D periodic system comprising alternating layers of a PhP natural hyperbolic crystal (e.g. sapphire) and a dielectric yields new types of low-loss ultra-confined interface states (see [238], [239]). Note that indicative feature/unit-cell size-ranges with respect to the free-space wavelength, λ₀, for the stated targeted operation are given in all examples.

Currently, specific 1D [86], [98], [100] and 2D [232] arrangements of unit or composite particle-PhP resonators have primarily been researched for harnessing enhanced and/or highly-directional thermal emission. As an example we discuss the platform of [100] seen in Figure 24. LSPhPs in rectangular cross-section rods arranged in a 1D array of micropyramids have been utilized to achieve polarization selective near-perfect absorption that covers about half the SiC reststrahlen band. The micropyramid array responds like moth eyes with a near-zero albedo [257] and thus near-perfect absorption when light impinges from the tip side of the pyramid, sequentially coupling from the smaller cross-section to the larger cross-section rods. By virtue of the directional Kirchoff’s law [100] this structure then behaves as a near-unidirectional emitter. The phenomenon can be relevant for highly efficient polarized spectrally selective globar thermal sources which can benefit MIR biomedical imaging [258]. LSPhP-based absorption/emission engineering [86], [98], [100] is promising for many applications including IR-imaging bolometers [259], radiative cooling [10], [11] and optothermal devices [260].

Figure 24:

Extreme absorption/emission control with LSPhPs.

(A) TM-polarized light gets near-perfectly absorbed with a SiC micropyramid array in the reststrahlen band of SiC by virtue of a cascaded coupling effect of the LSPhPs from the tip-to-the base SiC blocks. (B) Corresponding highly-asymmetric emissivity, e, of the micropyramid array. A value of 2.0 for the asymmetry factor defined in the inset implies a perfect uni-directional emission (C) Structural size details for the micropyramid array design (zoomed around the middle micropyramid). (After [100]).

8.1 Doping-tunable PhP permittivity in semiconductors: the LOPC effect

The carriers in semiconductors provide a tuning “knob” for the ionic permittivity response seen in Figure 6 via their Drude contribution to the full permittivity [see Eqs. (30) and (31)]. For a significant carrier concentration, ωp/ε∞ can approach or become larger than the TO phonon frequency, ω_TO [271]. While the TO phonon frequency, ω_TO remains unaffected by the presence of free carriers, as these do not influence the physical properties of the IR activated lattice vibrations, this is not true for the LO phonon frequency, ω_LO. With carriers present, there are two possible frequencies, where the permittivity can become zero thus implying that the original LO optical phonon frequency ω_LO splits into two branches: ω_LO+ (blue shifted) and ω_LO− (red shifted). This splitting of the LO phonon frequency into two branches [271] has been termed as the longitudinal optical phonon-plasmon coupling (LOPC) effect.

The LOPC effect arises from the ionic and carrier motion being coupled together via the macroscopic electric field. Although known since the 1960s [82], the significance of the LOPC effect for IR light-matter interactions was realized only recently when utilized in planar [272] or microstructured platforms [83], [84] for tunable IR light confinement, enhancement and absorption. To get a feeling for the impact of the LOPC effect on the photonic responses of semiconductors around their reststrahlen band we showcase a demonstrating example in Figure 25 depicting the full permittivity, ε_sem, of InP for different carrier concentrations (assuming room temperature).

Figure 25:

The LOPC effect in InP and its cost in optical loss.

(Top panels) The full permittivity, ε_sem (real part), versus the free-space wavelength λ₀, for three different plasma frequencies, ω_p of the free carriers is shown. The vertical blue (red) lines annotated with LO⁺ (LO⁻), depict the blue (red)-shifted LO phonon frequencies. The shaded regions highlight spectral ranges with Re(ε_sem) between −5 and 0 which is of interest for plasmonic-type effects and certain metamaterials. (Bottom panels) The imaginary part of ε_sem is plotted within the shaded regions in the top panels, scaled with a reference value, Im(ε_ref) equal to the imaginary part of the ionic-only permittivity, ε_ion at the frequency where the respective real part is equal to −2.5.

Doping may be enabled via photoinjection, i.e carrier generation with optical pulsed or continuous wave fields with frequencies above the semiconductor band gap [273]. This has been utilized by [84] to dynamically tune particle PhPs in arrays of InP and 4H-SiC sub-micron pillars. In the semimetals and semiconductors of the 2D/quasi-2D PhP material of Section 4.4, tunability via the LOPC can be achieved by applying a gate voltage. Note that some materials demonstrate a stronger phonon-plasmon coupling leading to cross-coupled damping coefficients; the permittivity response of these materials is different from the independent plasmon and phonon responses of Eqs. (30) and (31) (see [274]).

8.2 Tunable PhPs with strong optical fields: unlocking non-linear phonon-polaritonics

Another interesting optical-fields-based mechanism has been very recently discovered, unlocking new avenues for tunable phenomena in PhP materials with optical pulses [275]. Specifically, Cartella et al. [275] point out that the linear ionic oscillator model (see Section 3) would be valid only for small-to-moderate displacements. A theoretical model developed by the authors of [275] indicates that the high-frequency permittivity and the ionic effective charge Q (Born-effective charge) may acquire a term that is quadratic on the displacement, u [275] under the influence of strong optical fields. Their hypothesis is supported by experimental evidence showing that the PhPs in SiC can act as a type of a parametric oscillator yielding gain in SiC’s reflectance spectrum when excited by a strong pump beam. This very recent study sets the foundations for phonon-enabled IR non-linear photonics opening a whole new range of phenomena and capabilities in IR photonics.

8.3 Electric-field tunable PhP responses: the case of ferroelectrics

Discovered in 1921 [276], ferroelectrics represent a special class of dielectrics with intricate photonic responses stemming from a special acentric crystal structure that however possesses an axis of symmetry [277]. Ferroelectric materials exhibit a residual polarization even in the absence of an applied electric field; with application of an electric field the electric polarization follows a hysteresis similar to the magnetic polarization hysteresis of ferromagnetic materials [19]. Ferroelectric materials lose their curious responses for temperatures above a critical temperature, T_C, known as the Curie temperature [19], [23], [277], [278], [279]. The Curie point, T_C, of transition to the ferroelectric phase, can be tuned with the application of stress [280] or even material thickness [281], and can be very high, approaching the melting point for many ferroelectric materials [277].

The explosion and progress in the field of 2D and quasi-2D materials helped also bring ferroelectrics down to the scale of a few nanometers, despite that being initially thought of as impossible [282]. Examples of 2D/quasi-2D ferroelectrics that have been experimentally realized are SnTe, CuInP₂S₆ (CIPS) as well as α-In₂Se₃ [282]. Moreover, first principle atomistic calculations revealed additional ferroelectric material that can be stable in 2D form such as SbN [283], BiP [283] and β-GeSe [284]. β-GeSe can be versatile for a range of applications, as it also exhibits strong tunability under strain, which can bring its Curie temperature down to the room temperature range.

Ferroelectrics are very interesting from the point of view of PhPs. Going back to the microscopic model of Section 3.2, we see from Eq. (19) that when:

the PhP frequency ω_R=ω_TO becomes zero while simultaneously the static dielectric constant, ε_s, becomes infinite thus still obeying the LST relation [Eq. (24)] [278], [279]. This unstable PhP behavior is known as polarization catastrophe and signifies the transition to the ferroelectric state [19], [278], [285]. Accordingly, at the transition point a small change in the electric field yields a giant change in the polarization [23]. Following thermodynamic arguments (Devonshire model) it can be shown that in the ferroelectric state, i.e. for temperatures below the Curie temperature T_C, the low-frequency PhP mode (also known as soft-mode) increases in frequency, i.e. “hardens” with the application of a static electric field. Note, however, that the Curie transition temperature itself may increase with the application of a static electric field [286].

Clearly, the PhP (ionic) photonic responses of ferroelectric material are strongly tunable with a static external electric field. Thus ferroelectrics are desirable components for IR tunable devices; especially those with a Curie point at or above room temperature. Ferro-electric material with DC controlled ionic permittivity responses have been recently utilized for tuning the defect mode of a 1D photonic crystal cavity [287]. Moreover, tunable near-field radiative heat transfer has been demonstrated [191] that relies on the electric-field tunability of the associated surface PhP modes (see Section 6.1.1) of the ferroelectric material.

8.4 Magnetic-field tunable photonic responses in the PhP regime: the case of n-InSb

The cyclotron motion of the carrier electrons [23] under a static magnetic field causes a directional change in the full permittivity response of n-doped semiconductors (see [288], [289]). The phenomenon becomes relevant to tuning photonic responses in the PhP regime only when the cyclotron frequency, ωc=eB/me*, with B= |B | being the magnetic field strength, is in the range of the TO phonon frequency, ω_TO for relatively low magnetic fields. Not many semiconductors would meet that requirement. Given its low electron effective mass, n-doped InSb has been predominantly studied for its magnetic-field tunable photonic responses in its PhP regime from the 1970s [85] until today (see indicative example in Figure 26A).^[22] Recent related research includes tunable non-reciprocal plasmonic-type devices functional at THz frequencies for photonic isolators [289], [291], THz sensors [292], THz tunable polarization conversion [289] as well as magnetic-field tunable near-field radiative heat transfer pertinent to ultrafast control of thermal management [293], [294].

Figure 26:

Tunable photonic responses in the PhP regime.

(A) A static magnetic field induces anisotropy with large off-diagonal terms in the permittivity of n-doped InSb [85]. (B) Phase-change enabled temperature tunability in VO₂: I and II depict the anisotropic room-temperature permittivity in monoclinic VO₂ showing multiple PhP resonances (see Table 4 and [105]). Panel III shows the real and imaginary parts of the VO₂ permittivity (dotted-green and dotted-blue lines) in the rutile metallic phase at 360 °K calculated here based on the experimental data and analysis of [290] [see note 24]. The black-solid and red-dashed lines depict the respective values from a matching simple Drude model considered here. [The VO₂-crystal-structure insets are reproduced (adapted) from [106] through Creative Commons Attribution License, see link at https://creativecommons.org/licenses/by/3.0/].

8.5 Turning PhP photonic responses on and off by phase transitions

VO₂ has been attracting increasing attention as a tunable photonic material owing to its transition from an insulating monoclinic to a metallic rutile phase at the temperature of 68°C [106], [107]. VO₂ exhibits multiple PhP oscillations in its insulating phase (see Table 4). When heated beyond the transition temperature, these PhP responses completely disappear and VO₂ responds with a metallic Drude-like permittivity.^[23]

In Figure 26B-I and 26B-II we plot the room-temperature permittivity of VO₂ in the PhP regime for electric fields parallel and perpendicular, respectively, to its crystallographic direction, α_M (ionic permittivity model parameters are from [105] shown in Table 4). Indeed, we observe from Figure 26B-III that the permittivity changes dramatically past the phase-transition point at a temperature of 87°C where VO₂ is in the rutile metallic phase.^[24] Based on permittivity values extracted from the Basov-group experimental data [290], we also considered here a standard fitting Drude model that captures well the permittivity in the metallic rutile phase for the frequency range relevant to the PhP regime of the insulating phase. This Drude model’s parameters are ω_p=15.67×10¹⁴ rad/s and γ_p=4.92×10¹⁴ rad/s; the associated real (imaginary) part of the permittivity is represented with the black-solid (red-dashed) lines in Figure 26B-III.

The ionic-to-Drude permittivity transition in VO₂ has been utilized to tune the near-field radiative heat transfer with temperature [192] as well as to device a microsecond switchable thermal antenna [295]. Both effects rely on switching on/off surface or particle PhPs in VO₂. As other phase-change material, like GST [200], [201], are gaining interest for photonics in the last years, new PhP switching capabilities beyond VO₂ may come to light. In fact, there is recent on-going research on characterizing IR active phonons in GST and other phase-transitioning alloys [296], [297]. Temperature enabled volatile PhP control in phase-change material would be highly attractive for applications pertinent to thermal homeostasis capability [298] or for thermal camouflage [299]. Non-volatile PhP switching is pertinent to reconfigurable elements for integrated IR photonics. Furthermore, recent research that enabled gate-controlled non-volatile phase-transition in VO₂ [300] via hydrogen ion infiltration offers potentially exciting pathways towards gate-tunable PhP-based IR photonics with VO₂.

Of course phase-transition enabled tunable devices can also be envisioned by interfacing PhP material not undergoing a phase transition with other phase-change material not possessing PhPs in the spectrum of interest. We will review the latter type of systems when we discuss hybrid platforms in the following section.