In this book chapter, we review some of the progress made in nanoplasmonics and related optoelectronics phenomena in the field of two-dimensional (2D) materials and the recent 3D Weyl semimetals. We give a brief overview of plasmonics for three-dimensional (3DEG) and two-dimensional electron gases and draw comparisons with graphene, 3D topological insulators, 3D Weyl semimetals, and nanoplasmonics in nanogeometries. We discuss the decay of plasmons into electron-hole pairs and the subsequent thermalization and cooling of the hot carriers. We present our recent results in the fields of plasmonics in different nanostructures made of noble metals, such as Silver, and plasmonics in Dirac systems such as graphene and 3D topological insulators. We show a possibility of dynamically shifting the plasmon resonances in hybrid metal-semiconductor nanostructures. Plasmonics in 3D topological insulator and 3D Weyl semimetals have been least explored in nanoplasmonics although it can provide a variety of interesting physical phenomena involving spin plasmonics and chirality. Due to the inherent large spin-orbit coupling, locked spin-momentum oscillations can exist under special conditions and in the presence of an external laser field. We explore symmetric and antisymmetric modes in a slab of 3D TIs and present their dependences on the thickness of the slab.
- surface plasmon polaritons
- Dirac fermions
Electromagnetic properties of metal-dielectric interfaces have attracted a vast amount of research efforts. Ever since, the work of Mie  for small particles and Ritchie  for flat interfaces, a wide variety of scientists ranging from physicist, chemists, material scientists to biologists have explored plasmonics-based phenomena and their potential applications in practical life. In nanostructures, under the right circumstances, light waves propagating at metal-dielectric interface excite collective modes of electrons at the metal surface, resulting in the generation of charge density waves called surface plasmons (SPs), which can be divided into localized modes called localized surface plasmons (LSPs) and propagating modes called surface plasmon polaritons (SPPs), propagating along the interface like ripples across the surface of water with an effective wavelength much less than that of incident electromagnetic wave. Free electrons respond collectively by oscillating in resonance with the light waves. In optics, scientists have investigated methods to use plasmonics for concentrating, channeling, and changing the phase of light using subwavelength metallic structures. This would lead to miniaturized plasmonic circuits with length scales much smaller than those in current use [3–13]. A creatively engineered metal-dielectric interface can generate surface plasmons with the same frequency as the outside electromagnetic waves but with several times shorter wavelength. This interesting phenomenon can be utilized in a way in which surface plasmons carry information in microprocessors faster than current electronic transistors . Plasmonics holds promise for a higher information density than conventional electronics . While this proposed application needs still to be proven feasible, metallic nanostructures much smaller than the wavelength of light have already been successfully built for amplifying signals in surface enhanced Raman spectroscopy (SERS), providing a powerful method to detect a single molecule [16–27]. Plasmonically enhanced electric fields are already being used for sensing biomolecules [28–36]. The efficient heating property of plasmonic fields can be used for photothermal cancer treatment [37, 38] and also for thermally assisted magnetic recording [39–41]. Plasmonic lasers are able to achieve ultrafast dynamics with sub-wavelength mode confinement [42–50].
Metallic nanoparticles can be made in different sizes and shapes, and the distance among them can be controlled as well. These parameters can be used to tune the plasmon resonance frequency [51–62]. The use of optically excited plasmons as a tunable frequency source that can be mixed with a laser through Raman scattering enables dynamical shifting of the wavelength of light in a controlled manner . Fluegel et al.  used a continuous laser beam of a few microwatts power to excite carriers in a carefully engineered narrow GaAs quantum well. These photogenerated carriers are selectively passed through a thick barrier of AlAs into a wide GaAs quantum well in which the two-dimensional electron gas (2DEG) supports high charge density waves with collective electron motion normal to the layer. A signal laser beam operating at a different wavelength undergoes inelastic Raman scattering from the plasmon-phonon (longitudinal optical phonon) modes in the two-dimensional (2D) quantum well. The result is that a signal with 13–15 nm redshifted frequency is generated. Plasmonics can also be exploited in optical tweezers to confine nanoparticles to small dimensions . Grigorenko et al. [65, 66] have made electromagnetically coupled gold pillars. A trapping beam simultaneously excites gap plasmon (GP) modes in the gap between the pillars when they are used as a substrate, resulting in an enhanced field at the trapping site.
In a metallic nanoshell containing a core of dielectric material, due to different dielectric environments in the core and outside of the nanoshell, plasmons are excited in the inner and outer surface of the metallic nanoshell, which can be tuned by varying the ratio of inner to the outer radii of the nanoshell and by altering the dielectric materials [67, 68]. Figure 1 shows that the plasmon resonance energy can be tuned over a wide range by varying the material and/or shaping parameters. Figure 1 compares the plasmon resonance wavelengths of gold, silver, graphene, and topological insulator nanostructures. For a particle diameter much smaller than the wavelength of light, the light-matter interaction leads to an oscillating homogeneous polarization of the particle volume, resulting in an oscillating dipole field. For spherical Au and Ag nanoparticles, the dipole plasmon resonance occurs in the visible part of the spectrum. In particle ensembles, additional shifts are noticed due to electromagnetic coupling between LSP modes. For example, in case of a dimer, plasmons can be viewed as bonding and antibonding combinations, in analogy to molecular orbitals, i.e., hybridization of the individual nanoparticle LSPs occurs, giving rise to resonance shifts following the 1/
Nevertheless, we proposed a method of guiding electromagnetic waves along a chain below the diffraction limit in a controlled manner using a chain of nanoshells [72, 73]. The dynamic control over the plasmon resonances and their coupling gave rise to the idea of an optically controlled plasmonic switch. In this chapter, we present a concept of dynamical control over the plasmon resonances that can be obtained by controlling the dielectric environment of the LSPs using a pump probe technique. In Ref. , we show that a shift up to 125 nm can be achieved in an Ag core-TiO2 coated nanostructure.
In addition to that, we also review plasmonics in Dirac systems. Like in the case of a two-dimensional electron gas (2DEG), the surface plasmon mode in 2D materials such as graphene is tunable by changing the gate voltage through shifting the Fermi energy. Exciting surface plasmons give rise to light absorption enhancement in graphene, which can be utilized for photodetectors based on surface plasmon polaritons , optical switching of infrared plasmon polariton , and THz plasmonic lasing . Plasmonics in Dirac systems show interesting features due to massless electrons around the Dirac nodes. In particular, we focus on surface plasmons in graphene, in Bismuth-based 3D topological insulators (3DTIs), and in 3D Weyl semimetals. Graphene, 3DTIs, and 3D Weyl semimetals are interesting due to their special electronic and optical properties arising from the linear dispersion relation around the Dirac cones in the Brillouin zone . Around these points, energy dispersion of electrons can be described by a low energy Dirac Hamiltonian: HG(
Due to the excitation of surface plasmons, it is now possible to engineer the behavior of light on nanometer length scales and to increase the light-matter interaction [79–82]. This interaction is an outcome of the near-field enhancement close to the metal surface, which also leads to plasmon damping through radiative decay and through nonradiative decay inside the material, due to Landau damping, i.e., creation of electron-hole pairs via interband or intraband transitions, electron-phonon interaction, and boundary effects. The intraband transition happens in the conduction band and the interband transition occurs between other bands (such as the d-band) and the conduction band, as shown schematically in Figure 2a [83–85].
The radiative decay part of plasmon damping is due to the direct photon emission by coherent electron oscillation. As the size of the nanoparticle increases, the radiative decay of the plasmon is more significant. For larger nanoparticle elements, the radiative decay component is the main reason of plasmon resonance broadening and weakening of the dipole strength. In contrast, decreasing the size of the nanoparticle lets the nonradiative component dominate the plasmon decay. For applications in information technology, a slow dephasing of optical polarization by electron oscillation is essential, which is characterized by the dephasing time (
This book chapter is organized as follows: in Section 2, we discuss the SP resonances in hybrid metal-semiconductor nanostructures. By altering the dielectric environment of nanostructures dynamically using pump-probe techniques, we show that it is possible to shift the SP resonance wavelength. In Section 3, we discuss in detail the plasmon excitations and their damping pathways in a three-dimensional electron gas (3DEG). We discuss both the radiative and nonradiative damping mechanisms of SPs in 3DEGs. In Section 4, we give a brief overview of plasmons in a two-dimensional electron gas (2DEG). In Section 5, we present the size dependent properties of the SPs in nanostructures. Graphene plasmonics and losses are discussed in Section 6. Section 7 is dedicated to the description of the SPs in Dirac systems. We focus on the SPs in 3DTI materials and 3D Weyl semimetals, and we discuss graphene plasmons as a limiting case of the 3DTI plasmons in the limit where the thickness of the 3DTI slab
2. Surface plasmon resonances in metal nanostructures
A nanoparticle shows tunable optical properties under controlled variation of its geometry. In a pure Ag spherical nanoparticle in vacuum, for example, the plasmon resonance occurs at 320 nm. These plasmon modes are shifted if the nanoparticle is coated with dielectric materials. It has been shown that with increasing shell thickness, the local electric field enhancement factor peak increases and redshifts for
The quasi-static approximation provides a good estimate for a nanoparticle size of around 1/10 or smaller of the incident light wavelength. For larger nanoparticles, due to the finite speed of light, retardation effects lead to a redshift of the plasmon resonance . In Ref. , authors have found an analytical expression for a spheroid that takes into account the depolarization factors and that gives a good approximation for nanogeometries of size up to 150 nm. Figure 4 shows our results for the local field enhancement in the presence of an Ag nanocube. As expected from electrostatics, the largest enhancement occurs at the vertices of the Ag nanocube.
The optical resonances of a nanoshell exhibit enhanced sensitivity to its local dielectric environment relative to the solid nanoparticle, as shown in Figure 5. For a particle diameter less than the wavelength of light, the light-matter interaction leads to an oscillating homogeneous polarization of the particle volume, resulting in a dipole field. Figure 5a and b shows the dependence of plasmon resonances on the shell thickness and the size of a hybrid metal-semiconductor nanostructure of 15 nm diameter with an Ag core coated by TiO2 shell obtained in the quasi-static approximation. The expressions are given in Ref. .
Plasmon resonances in a nanoshell can be tuned dynamically by letting a pump laser pulse of energy equal to the band gap or above generate electron-hole pairs in a semiconducting material surrounding the nanoshell. A probe laser pulse at a plasmon resonance frequency is used to excite plasmons on the metal surface. The generation of free electron-hole pairs alters the dielectric function of the surrounding semiconducting material. Due to the reduced dielectric function caused by the excitation of the electron-hole pairs, the excitation of surface plasmons by a probe pulse requires a higher energy. The frequency of the probe pulse is smaller than that of the pump pulse ensuring that no excitons are excited in the semiconductor during the probing. The change in the dielectric function of the surrounding medium due to the pump pulse can be calculated using Fermi’s golden rule:
Figure 6 shows shifts in the resonance peak of the surface plasmons occurring at around 620 nm before the generation of excitons in a nanoshell structure (as shown in inset of Figure 5a) with diameter of 15 nm. After the pump pulse, depending on the density of the excitations, the plasmon resonance peaks are excited by the probe pulse shift. The larger the density of excited free electron-hole pairs in the semiconductors, the larger is the blueshift of the plasmon resonance peak. For a density of excitation of 5 × 1021 cm−3, a resonance shift of up to 125 nm can be achieved.
3. Plasmon excitation and damping for a three-dimensional electron gas (3DEG)
In this section, we discuss the plasmon excitation and their damping pathways in 3D materials made of metal. The plasmonic damping pathways in 3D materials include radiative decay, Landau damping, and resistive loss, as depicted in Figure 7. During Landau damping, plasmon quasi-particles lose their energy by exciting hot electron-hole pairs via direct interband or phonon/geometry-assisted intraband transitions. In the case of geometry-assisted intraband transition, the translational symmetry is broken due to electric field confinement or boundaries of the material . In the case of resistive loss, single carriers, electrons or holes that are the building blocks of the plasmon quasi-particle, are kicked out of the phase-coherent collective plasma oscillation through electron-electron or electron-phonon scattering, giving rise to plasmon damping. In 2D materials, the plasmons follow similar damping pathways. Figure 7 shows the stages of the plasmon decay, the initial nonequilibrium configuration after the excitation, the thermalization, and the cooling of the hot carriers. Some of these damping pathways can be used to inject hot carriers into other materials. For example, at a metal-semiconductor interface, hot electron-hole pairs can be separated by means of the Schottky barrier for the purpose of energy harvesting. In metal-graphene or metal-MoS2 junctions, the surface plasmons can generate hot electron-hole pairs, thereby injecting electrons/holes into n/p-doped 2D materials, giving rise to hot carrier-induced doping  or even insulator-to-metal phase transitions .
The initial distribution of the hot carriers can be estimated using the jellium model for metal nanoparticles and nanoshells [95, 96], but this approach cannot explain the material dependence of this process because the specific band structure of the metal is completely neglected. In order to capture the material properties, it is necessary to combine FDTD calculations for obtaining the plasmon modes with
The plasmon resonance frequency and dispersion can be obtained by evaluating the dynamic polarizability in the presence of the carrier-carrier Coulomb interaction. The dynamic polarizability in the random phase approximation (RPA) is given by
For the simplest case, when only the carrier-carrier Coulomb interaction is present, we can derive the dynamical plasmon dispersion relation following standard textbooks . The first step is to calculate noninteracting dynamical polarizability.
where Ω is the volume of the sample, is the equilibrium electron density, and
Using the equation for 3D metals, one obtains
Consequently, the dynamical polarizability in the RPA and long wavelength regime is
which yields the bulk plasmon dispersion relation for 3D metals, i.e.,
The same result can be obtained by solving . Note that the slope of the parabolic dependence at is
This difference in slopes between the 3D bulk plasmon and the photon-like surface plasmon polariton is clearly visible in Figure 8 (see below).
It is well known that in 3D metals Landau damping occurs when the plasmon resonance energy enters the electron-hole continuum, which is determined by the condition
Since and , the electron-hole continuum is given by the gray shaded area in Figure 8. The plasmon can decay into an electron-hole pair when the plasmon dispersion curve enters the electron-hole continuum limit. This decay corresponds to intraband Landau damping.
The presence of a planar boundary for a 3D metal adds a new mode known as surface plasmon, which propagates at the metal-dielectric interface. Since the electron charge density of a metal leaks outside the interface into the dielectric in the order of Å, a macroscopic description based on Maxwell’s equations is sufficient to understand the surface plasmon qualitatively. Taking the boundary conditions into account, the dispersion relation of the surface plasmon is determined by Ritchie and Eldridge 
in the Drude model, where . This biquadratic equation can be solved analytically, yielding
Since , there are only two physical solutions, which are drawn in Figure 8. While the upper branch corresponds to the photon-like plasmon polariton, the lower branch represents the plasmon-like surface plasmon polariton (SPP) with the following asymptotics:
which in the nonretarded regime reduces to Ritchie’s equation ,
for wave vectors
To estimate the plasmon decay rate, the band structure of the materials should be calculated by means of DFT to find out the exact quasi-particle orbitals and energies. Different electronic structures can be used to calculate the band structure of noble metals. To estimate the decay rate, the electronic states and energies of the metals resulting from PBEsol +
In nanoconfined structures, because of lack of translational symmetry, the crystal momentum
In metals, plasmons decay is not only in the ultraviolet and visible spectrum but also in the infrared and microwave regimes [105, 106]. Due to the conservation of momentum in infinite crystal lattices, the direct interband transition induced by plasmon decay is only possible for energies larger than the band gap energy. However, for energies below the visible spectrum, typically phonon-assisted and surface-assisted intraband electron-hole pair generations are able to bypass this selection rule .
The plasmon decay rate is related to the imaginary part of the dielectric tensor , i.e.,
Let us consider a 3D semi-infinite metal slab extending in the negative
The decay rate of plasmon as a function of frequency can be calculated by substituting the experimental data for the complex dielectric function measured by ellipsometry. Within the random phase approximation (RPA), the nonradiative decay rate induced by direct interband transition is 
where are momentum matrix elements between the quasi-particle orbitals and .
where is the energy of a phonon with wave vector
Excitation of surface plasmon generates a strong field confinement on the surface with the exponential decay length inside the metal, which creates a Lorentzian distribution for the momentum of the plasmon in the
where is the bulk plasma frequency of the metal, is the Fermi velocity, and
Numerical studies based on the free electron jellium model show that in nanostructures, due to the localization of electronic states and the nonconservation of the crystal momentum, intraband transitions are enhanced [95, 96, 109]. Using Fermi’s golden rule together with the free electron eigen states and the dipole field profile, the nanoconfinement contribution is
In the Landau damping theory, the lowest-order processes consisting of direct, phonon-assisted, and surface/geometry-assisted electron-hole pair excitation contribute to the decay of plasmons [102, 110]. Higher-order processes leading to the excitation of many electron-hole pairs or many phonons are suppressed due to the phase-space factors at small energies . Only at large energies, the higher-order processes become significant. Being completely different from Landau damping, another source of plasmon damping is the resistive loss in the metal, which can be calculated by means of the linearized Boltzmann equation in the relaxation time approximation , giving
4. Plasmons in a two-dimensional electron gas (2DEG)
where and are the dielectric constants of the semiconductor and oxide layers, respectively. The oxide layer has a thickness of
the 2DEG plasmon frequency exhibits a dependence. Note that the electron-hole continuum in a 2DEG is determined by a similar formula to Eq. (16). Therefore the gray shaded area in Figure 8 has similar forms for 3DEGs and 2DEGs.
5. Static geometry of metallic objects and environment
When the size of the nanoparticle is much smaller than the wavelength of the incident light, the particle exhibits a dipolar oscillation mode (Fröhlich mode). As the diameter of the nanoparticle is increased, the electrostatic limit is not a good approximation anymore and the multipolar oscillation modes start to appear. Excitation of these modes gives rise to the broadening of the resonance , as seen from Figure 9b. The LSP resonance redshifts with increasing diameter of the sphere, which is due to retardation effects . Decreasing the size of nanoparticle less than mean free path of electrons moves the material band structure and dielectric function away from the bulk properties and increases the surface scattering that gives rise to broadening of the absorption spectrum , as shown in Figure 9a. The internal field enhancement of an illuminated spherical nanoparticle is
is satisfied. The shape factor for the axis parallel to the polarization of the incident light (e.g., the axis
For the core-shell spherical nanoparticles the resonance condition is given by
where is the core fill fraction and
6. Plasmon theory for graphene
Graphene is a two-dimensional (2D) material comprised of a single layer of carbon atoms in a honeycomb lattice. It has unique electrical, optical, and mechanical properties due to its tunable band dispersion relation and atomic thickness. Because of its unique band structure graphene possesses a very high mobility and a fast carrier relaxation time [117–121], making it an attractive candidate for ultrafast electronics and optoelectronics. Exciting surface plasmons on graphene is a distinct technique to increase absorption with low damping rate. The surface plasmon couples the electromagnetic (EM) wave to the conductive medium, giving rise to direct absorption of light by monolayer graphene and providing the opportunity of electrical tunability of the plasmon resonance frequency, high degree of electric field confinement, and low plasmon damping rate [122–125]. The increased light-matter interaction results in an enhanced spontaneous emission rate close to the nanostructure edges [126, 127]. Recent experiments have achieved an absorption of 90% in the mid-IR range by connecting graphene with high carrier mobility to a silicon diffractive grating  and designing graphene nanoribbons [128, 129], nanodisks , and antidot array  theoretically. These high carrier mobilities can be achieved only for mechanically exfoliated graphene. Exciting plasmon on CVD-grown monolayer graphene with lower mobility than the mechanically exfoliated one reduces the absorbance to 19% and 28% for graphene nanoribbons [99, 127, 131] and nanodisks [132, 133], respectively. We show in experiments that the coupling of a patterned CVD-grown graphene sheet to an optical cavity amplifies the excited LSPs and enhances the light absorption to a current world record of 45% . We also show that the theoretically achievable enhancement is 60% for a square lattice of holes .
The electric current of graphene in the interaction picture is given by
For infrared and THz radiation, the Fermi energy can be tuned to become much larger than the incident photon energy, and therefore due to Pauli blocking there are only intraband transitions. According ot the Boltzmann equation and under the relaxation time approximation, the carrier distribution in the presence of a constant electric field with x-polarization is given by
In the absence of the external electric field, the net electric current is zero, so the summation of the first part of Eq. (43) in the electric current is zero. Since is so narrow around the Fermi surface, only the wave vectors near the Fermi energy contribute to the integration, and
In the presense of an oscillating electric field, the relaxation time is a complex function and the intraband conductivity is given by
The dielectric function of graphene can be obtained via its AC conductivity by means of 
The bandstructure of graphene is linear in the tight-binding approximation, as shown in Figure 11b. The dispersion relation for the TM mode in the geometry depicted in Figure 11a, which consists of graphene surrounded by dielectrics with constants
The larger the
There are two different approaches to obtain the dispersion relation of the surface plasmon. In the semiclassical approximation, the Drude-like conductivity is used to obtain the plasmon dispersion relation [122, 138]
and the plasmon loss
Another approach that works for both intraband and interband regimes is based on the selfconsistent linear response theory, known as random phase approximation (RPA) along with the relaxation time approximation defined by Mermin . The dispersion relation of the plasmon can be obtained by solving
with the complex wave vector . Considering the Coulomb interaction of electrons in graphene and the medium dielectric, first and second terms of Eq. (4), one obtains the plasmon dispersion relation
and the plasmon loss relation 
The plasmon losses , wave localization , and the group velocity of the graphene plasmon are calculated by means of the semiclassical approximation and RPA, as shown in Figure 12 for the Fermi level of and relaxation time of . Below the interband regime, the plasmon loss, the wave localization, and the group velocity calculated by RPA are in very good agreement with the semiclassical approach.
For 2D materials such as graphene, the amount of plasmon loss can be calculated by using the effective dielectric function of the material. The dynamical polarizability
determines several important quantities, such as the effective electron-electron interaction, plasmon spectra, phonon spectra, and Friedel oscillations. are the Matsubara frequencies,
where are the spin and valley degeneracy, is the Fermi distribution function, and is the graphene energy. The band overlap of the wavefunctions
is a specific property of graphene, where
where the + and – signs denote the intra and interband transitions, respectively. Integration over φ and k gives the retarded polarizability or charge-charge correlation function
The two functions and are defined by
Where . For and in the long wavelength limit
In this condition and for intraband transitions ( )
As a result, reduces to
If , then
where is the average of dielectric constants of graphene’s environment. The collective oscillation modes of the electrons can be obtained by solving . The extinction function is identified by , or for the plasmonic structure coupled to an optical cavity , where and
The loss function represents the amount of energy dissipated by exciting the plasmon coupled to the substrate and graphene optical phonons. The collective oscillation modes of the electrons can be obtained by solving . Considering the first two terms in Eq. (71) gives the plasmon dispersion relation of graphene
where the graphene fine structure constant is given by
with being the fine structure constant. In the case of graphene on a SiO2 substrate, for air and for SiO2, which yield . For suspended graphene, and therefore . Thus, it is possible to tune
It is also interesting to note that in contrast to the dependence of the bulk plasmon resonance frequency in Eq. (20) and the dependence of the 2DEG plasmon, the graphene plasmon exhibits a dependence.
In order to show the results for , we define two regimes for , i.e.,
and the real part is obtained by the Kramers-Kronig relation, yielding
Different regions are shown in Figure 13. As we mentioned, the plasmon dispersion relation is determined by , where
The solution of the first part of Eq. (78) exists only for , which is valid only for finite graphene doping and . Interestingly, a plasmon does not decay if , which is the case in region 1B shown in Figure 13. For the finite doping , the acoustic phonon at long wavelength is inside the region 1A in Figure 13.
7. Surface plasmon resonances in 3D topological insulators
So far, we have discussed nanoplasmonics in 3D metals and graphene and their damping mechanisms. Now we focus on the RPA theory of nanoplasmonics in 3D topological insulator (TI) materials. In particular, we are going to identify the graphene plasmons as a limiting case of 3DTI plasmons in the case when the thickness of the 3DTI slab becomes atomically thin. Bismuth selenide (Bi2Se3) is the prime example of a 3DTI material that has a rhombohedral crystal structure and consists of five atomic layers arranged along the
Let us consider a slab of a 3DTI material of thickness d > 5 nm, which is large enough to suppress any overlap of the single-electron states between the top and the bottom layers. In contrast, long-range Coulomb interaction exists and couples the opposite surfaces as in 2D electron plasmas.
We consider a Hamiltonian that describes the properties of collective oscillations on the top and the bottom surfaces of a 3DTI material as 
where indicates the top (bottom) surface, are spin indices,
where is a 2 × 2 matrix. is a 2 × 2 matrix whose diagonal and off diagonal elements are the two-dimensional Fourier transforms of Coulomb potentials and can be obtained by solving Poisson’s equation . Similar to the RPA dielectric function for graphene shown in Eq. (71), the equivalent equation for the RPA dielectric function in 3DTIs is
Compared to graphene, the main difference is that is a 2 × 2 matrix accounting for the intrasurface and intersurface interactions. Here we write these expressions as
for the intrasurface interaction, and
for the intersurface interaction.
The response function χ provides important information about the collective states that are excited at small transferred momentum. The collective mode frequencies of the system can be obtained by solving . In the region , collective modes of oscillations are undamped. Beyond that limit, such modes are not observed because the energy of the modes is transferred to the particle-hole excitations. In Section 6, we derived the dynamical polarizability for the case of a graphene sheet. For with , the linear response function can be written as 
where . Using the relation for the relative permittivity , the linear response function given by Eq. (84) is obtained from Eq. (61) in the long wavelength limit. This charge-charge response function describes explicitly the dispersion of the collective modes exist on the two sides of a slab geometry of a 3DTI slab. Using Eq. (80), the pole of can be solved with the help of Eq. (35). The potential υ(
In the limit of small thickness of the slab, when the condition
and the symmetric plasmon-like mode as
where is the density polarization and is the total Fermi wave vector. is the fine structure constant for the Dirac system. It is to be noted that indices 1 and 2 indicate two different surfaces. For equal Fermi energy at the top and the bottom surfaces, the density polarization reduces to zero, resulting in an equilibrium situation. When the thickness of the slab tends to zero, we recover only the antisymmetric mode, which is the mode obtained for a sheet of graphene. As the slab thickness goes to infinity, the two surfaces interact weakly and the intersurface potential falls rapidly as the thickness increases. In this case, we obtain that the two antisymmetric modes each correspond to a single sheet of graphene. Using a series expansion of the frequency, it is possible to obtain a more accurate solution for the symmetric mode, which is derived elsewhere . The solution is given by
For a size of the slab smaller than the mode wavelength, i.e., , the antisymmetric mode is independent of thickness, and the symmetric mode has a
The topological surface states are extended inside the bulk with a localization length given by where
Surface plasmons in 3DTI surfaces do not consist only of charge density waves but are also accompanied by spin density waves due to inherent nature of spin-momentum locking. This can be qualitatively understood by calculating the surface current in terms of the spin and charge quantity by . The continuity equation shows that the charge density
where . The induced charge density is calculated from the response function . Spins are inherently attached to the momentum as accounted by the complete response function given by Eq. (80). The symmetric and antisymmetric modes are purely spin-like and purely charge-like for a slab of 3DTI. This can be seen in the following expressions in the limit of as
for the antisymmetric mode, and
for the symmetric mode, where and , and the indices
It has been observed that the phonon modes in Bi2Se3 have frequencies of around 2 THz. Since the phonon modes overlap with the plasmon modes in energy space, there is a possibility of mode hybridization, similar to the case of graphene. In a micro-ring structure of Bi2Se3, there are bonding (lower) and antibonding (upper) plasmon modes . The antibonding plasmon mode frequency is much larger than the phonon mode frequency, and the bonding plasmon mode overlaps with the phonon mode, resulting in hybridization in two branches with an interaction frequency of around 0.35 THz.
Recently, in Ref. , authors investigated the possibility of obtaining SPPs in Weyl semimetals. In Weyl semimetals such as TaAs, NbAs, YbMnBi2, and Eu2Ir2O7 [150–152] the valence and the conduction band touch in isolated points of the Brillouin zone close to the chemical potential, and their dispersion is described by an equation similar to the one for Dirac metals. A pair of Weyl nodes appears with opposite chiralities with a distance of
In this book chapter, we present descriptions of the plasmonic properties of metal nanostructures of different geometry, their size dependence, and applications in modern nanotechnology. We show dynamic control over the plasmon resonances where a shift up to 125 nm at a density of 1022 cm−3 can be achieved using a pump-probe technique. This provides the opportunity to utilize plasmonics in modern information processing devices. In addition to plasmonics in 3D metal nanostructures, we present a description of graphene and 3DTI plasmonics using classical and quantum perspectives. Using the RPA theory, we obtain symmetric and antisymmetric modes in a slab of 3DTI, which reduces to a graphene plasmonics in the limit of zero thickness of the slab. Surface plasmon (SP) damping mechanisms are interesting due to their potential applications for enhanced current density that comes from SPs nonradiative damping in nanostructures smaller than the skin depth. We present a quantum theory of SPs damping in metals and layered materials like graphene.
There are several potential applications of the nanoplasmonics in graphene and 3DTI. Graphene and 3DTI are potential candidates for nanospasers that utilize Dirac fermions, unlike the massive electrons or holes in the originally proposed spasing scheme by Bergman and Stockman in 2003 . The spaser is a nanoplasmonic counterpart of a laser, where photons are not emitted. In Ref. , the authors have proposed a scheme of nanospasing using a sheet of graphene with an electrically pumped cascaded quantum well structure working as a gain medium. For the range of Fermi energy, it was shown that spasing could be potentially obtained for a plasmon relaxation time of several femtoseconds in the mid-IR range. A similar scheme with an optically pumped nanospaser for a slab 10 nm of 3DTI was proposed in Ref. . It is advantageous to use a 3DTI, such as Bi2S3, as a nanospaser due to the possibility of using its bulk as a gain medium and the surface as a medium that supports SPPs. This configuration avoids the use of a separate gain medium to provide the feedback for the SPPs. Therefore, a 3DTI nanospaser can be truly nanoscopic and can be used for various applications in physics, chemistry, and biology.
We acknowledge the support provided by NSF grant CCF-1514089 and DARPA grant HR0011-16-1-0003.
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