Plasmon Branches Research Articles

Plasmon modes in double bilayer graphene heterostructures

We investigate the plasmon frequency and decay rate in a double bilayer graphene (DBLG) heterostructure made of two bilayer graphene (BLG) flakes separated by a spacer of thickness d. Calculations are carried out at zero temperature, within the random phase approximation, taking into account the inhomogeneity of the dielectric background of the system. Analytical expressions for plasmon frequencies are obtained by using the long wavelength expansions of the response and bare Coulomb interaction functions. We find that at long-wavelength limit, the acoustic (optical) plasmon mode shows the linear (square-root) energy dispersion and the inhomogeneity of dielectric background decreases the frequencies of both plasmon branches as in the case of DLG. The carrier density, spacer thickness and background dielectric constant affect remarkably on plasmon frequency and decay rate.

A theoretical study of collective plasmonic excitations in double-layer silicene at finite temperature

We explore the temperature-dependent plasmonic modes of an n-doped double-layer silicene system which is composed of two spatially separated single layers of silicene with a distance large enough to prevent interlayer electron tunneling. By applying an externally applied electric field, we numerically obtain the poles of the loss function within the so-called random phase approximation to investigate the effects of temperature and geometry on the plasmon branches in three different regimes: topological insulator, valley-spin polarized metal, and band insulator. Also, we present the finite-temperature numerical results along with the zero-temperature analytical ones to support a discussion of the distinct effects of the external electric field and temperature on plasmon dispersion. Our results show that at zero temperature both the acoustic and optical modes decrease when the applied electric field is increased and experience a discontinuity at the valley-spin polarized metal phase as the system transitions from a topological insulator to a band insulator. At finite temperature, the optical plasmons are damped around this discontinuity, and the acoustic modes may exhibit a continuous transition. Moreover, while the optical branch of plasmons changes non-monotonically and noticeably with temperature, the acoustic branch dispersion displays a negligible growth with temperature for all phases of silicene. Furthermore, our finite-temperature results indicate that the dependency of two plasmonic branches on the interlayer separation is not affected by temperature at long wavelengths; the acoustic mode energy varies slightly with an increase in the interlayer distance, whereas the optical mode remains unchanged.

Kinked plasmon dispersion in borophene-borophene and borophene-graphene double layers

We investigate the collective plasmon modes in double layer of two dimensional materials where either one or both of the layers have tilted Dirac cone. Consistent with quite generic hydrodynamic treatment, similar to double layer graphene systems we find two branches of plasmons. The in-phase oscillations of the two layers disperses as $\sqrt q$, while the out-of-phase mode disperses as $q$. When even one of the layers hosts tilted Dirac cone spectrum, the plasmonic kink which is a salient feature of a monolayer of tilted Dirac cone is inherited by both of these branches. In double layers composed of graphene (nontilted) and borophene (tilted) where the two layers have two different Fermi velocities, the velocity scale of plasmonic modes is set by the greater of the two. The kink always takes place when each plasmon mode crosses an energy scale $\omega_{\rm kink}$. When the two layers have different chemical potentials, there will be two such scales, and each of in-phase and out-of-phase modes develop two kinks. Moreover we find that an additional linearly dispersing overdamped mode of monolayer tilted Dirac cone system survives in the double layer system.

Tilt-induced kink in the plasmon dispersion of two-dimensional Dirac electrons

The list of two dimensional Dirac systems with a tilt in their Dirac cone spectrum is expanding, and now in addition to organic system $\alpha$(BEDT-TTF)$_2$I$_3$ includes the two dimensional $8Pmmn$-borophene sheet, which allows for controlled doping by the gate voltage. We analytically calculate the polarization function of tilted Dirac cone for arbitrary tilt parameter, $0\le \eta <1$, and arbitrary doping. This enables us to find two interesting plasmonic effects solely caused by the tilt: (i) In addition to the standard plasmon oscillations, strong enough tilt, induces an additional linearly dispersing overdamped branch of plasmons, which is strongly Landau damped due to overlap with a large density of intra-band free particle-hole (PH) excitations. (ii) There appears a kink in the plasmon dispersion for any non-zero tilt parameter. The kink appears when the plasmon branch enters the inter-band continuum of PH excitations. This kink becomes most manifest for wave vectors perpendicular to the tilt direction and fades away by approaching the tilt direction. Experimental measurement of the wave vector and energy of the plasmonic kink, when combined with our analytic formula for the kink energy scale, allows for a direct experimental measurement of the tilt parameter.

Collective excitations in a two-component one-dimensional massless Dirac plasma

We study spectra of long wavelength plasma oscillations in a system of two energy splitted one-dimensional (1D) massless Dirac fermion subbands coupled by spin-orbit interaction. Such a system may be formed by edge subbands in semiconducting transition metal dichalcogenide monolayers. Intrasubband transitions of massless Dirac fermions give rise to optical and acoustic gapless branches of intrasubband 1D plasmons. We reveal that the optical branch is of quantum character with group velocity being inverse proportional to square root of the Planck constant, whereas the acoustic branch is classical one with group velocity proportional to geometric mean of the edge subband velocities. Spin-orbit interaction, allowing intersubband transitions in the system, results in emergence of two branches of intersubband 1D plasmons: upper and lower ones. The upper and lower branches are gapped at small wave vectors and evolve with positive and negative group velocities, respectively, from energy splitting of the edge subbands at Fermi-level. The both intersubband branches adjoin intersubband single particle excitation continuum from above, while in case of the edge subbands with unequal velocities the lower one experiences Landau damping at small wave vectors. In addition, the lower branch, attaining zero frequency at a non-zero wave vector, alters its group velocity from negative to positive one.

Plasmon-phonon coupling in a valley-spin-polarized two-dimensional electron system: A theoretical study on monolayer silicene

We study the hybrid excitations due to the coupling between surface optical phonons of a polar insulator substrate and plasmons in the valley-spin-polarized metal phase of silicene under an exchange field. We perform the calculations within the generalized random-phase approximation where the plasmon-phonon coupling is taken into account by the long-range Fr$\ddot{\mathrm{o}}$hlich interaction. Our investigation on two hybridized plasmon branches in different spin and valley subbands shows distinct behavior compared to the uncoupled case. Interestingly, in one valley, it is found that while the high energy hybrid branch is totally damped in the spin-up state, it can be well-defined in the the spin-down state. Moreover, we show that the electron-phonon coupling is stronger in both spin-down subbands, regardless of valley index, due to their higher electron densities. In addition, we study the effects of electron-phonon coupling on the quasiparticle scattering rate of four distinct spin-valley locked subbands. The results of our calculations predict a general enhancement in the scattering rate for all subbands, and a jump in the case of spin-down states. This sharp increase associated to the damping of hybrid plasmon modes is almost absent in the uncoupled case. The results suggest an effective way for manipulating collective modes of valley-spin-polarized silicene which may become useful in future valleytronic and spintronic applications.

Plasmons in Dimensionally Mismatched Coulomb Coupled Graphene Systems.

We calculate the plasmon dispersion relation for Coulomb coupled metallic armchair graphene nanoribbons and doped monolayer graphene. The crossing of the plasmon curves, which occurs for uncoupled 1D and 2D systems, is split by the interlayer Coulomb coupling into a lower and an upper plasmon branch. The upper branch exhibits an unusual behavior with end points at finite q. Accordingly, the structure factor shows either a single or a double peak behavior, depending on the plasmon wavelength. The new plasmon structure is relevant to recent experiments, its properties can be controlled by varying the system parameters and be used in plasmonic applications.

Multicomponent plasmons in monolayer MoS2 with circularly polarized optical pumping

By making use of circularly polarized light and electrostatic gating, monolayer molybdenum disulfide (ML-MoS$_2$) can form a platform supporting multiple types of charge carriers. They can be discriminated by their spin, valley index or whether they're electrons or holes. We investigate the collective properties of those charge carriers and are able to identify new distinct plasmon modes. We analyze the corresponding dispersion relation, lifetime and oscillator strength, and calculate the phase relation between the oscillations in the different components of the plasmon modes. All platforms in ML-MoS$_2$ support a long-wavelength $\sqrt{q}$ plasmon branch at zero Kelvin. In addition to this, for an $n$-component system, $n-1$ distinct plasmon modes appear as acoustic modes with linear dispersion in the long-wavelength limit. These modes correspond to out-of-phase oscillations in the different Fermion liquids and have, although being damped, a relatively long lifetime. Additionally, we also find new distinct modes at large wave vector that are stronger damped by intra-band processes.

Graphene plasmons in the presence of adatoms

We theoretically investigate graphene plasmons in the presence of a low density of adatoms on the graphene surface. The adatoms can significantly modify the conductivity and plasmonic properties of graphene and may produce a level splitting with the plasmon mode, resulting in two plasmon branches. The high energy branch exhibits large losses and the low energy branch exhibits low losses. Our model may also be considered as a simple model for molecules on graphene and we show that graphene plasmons are sensitive to such changes in the environment. Our microscopic treatment of plasmons and adatoms shows the sensitivity of plasmons and highlights the potential of graphene plasmons for sensing purposes.

Doped Silicon Nanocrystal Plasmonics

Doped semiconductor nanocrystals represent an exciting new type of plasmonic material with optical resonances in the infrared. Unlike noble metal nanoparticles, the plasmon resonance can be tuned by altering the doping density. Recently, it has been shown that silicon nanocrystals can be doped using phosphorus and boron resulting in highly tunable infrared plasmon resonances. Due to the band structure of silicon, doping with phosphorus contributes light (transverse) and heavy (longitudinal) electrons, while boron contributes light and heavy holes and one would expect two distinct plasmon branches. Here we develop a classical hybridization theory and a full quantum mechanical TDLDA approach for two-component carrier plasmas and show that the interaction between the two plasmon branches results in strongly hybridized plasmon modes. The antibonding mode where the two components move in phase is bright and depends sensitively on the doping densities. The low energy bonding mode with opposite charge alignment ...

Temperature-dependent collective effects for silicene and germanene

We have numerically calculated electron exchange and correlation energies and dynamical polarization functions for recently discovered silicene, germanene and other buckled honeycomb lattices at various temperatures. We have compared the dependence of these energies on the chemical potential, field-induced gap and temperature and we have concluded that in many cases they behave qualitatively in a similar way, i.e. increasing with the doping, decreasing significantly at elevated temperatures, and displaying different dependences on the asymmetry gap at various temperatures. Furthermore, we have used the dynamical polarizability to study the ‘split’ plasmon branches in the buckled lattices and predicted a unique splitting, different from that in gapped graphene, for various energy gaps. Our results are crucial for stimulating electronic, transport and collective studies of silicene and germanene, as well as for enhancing silicene-based fabrication and technologies for photovoltaics and transistor devices.

Fermions on the low-buckled honey-comb structured lattice plane and classical Casimir–Polder force

We start with the well-known expression for the vacuum polarization and suitably modify it for 2[Formula: see text]1-dimensional spin–orbit coupled (SOC) fermions on the low-buckled honey-comb structured lattice plane described by the low-energy Liu–Yao–Feng–Ezawa (LYFE) model Hamiltonian involving the Dirac matrices in the chiral representation obeying the Clifford algebra. The silicene and germanene fit this description suitably. They have the Dirac cones similar to those of graphene and SOC is much stronger. The system could be normal or ferromagnetic in nature. The silicene turns into the latter type if there is exchange field arising due to the proximity coupling to a ferromagnet (FM) such as depositing Fe atoms to the silicene surface. For the silicene, we find that the many-body effects considerably change the bare Coulomb potential by way of the dependence of the Coulomb propagator on the real-spin, iso-spin and the potential due to an electric field applied perpendicular to the silicene plane. The computation aspect of the Casimir–Polder force (CPF) needs to be investigated in this paper. An important quantity in this process is the dielectric response function (DRF) of the material. The plasmon branch was obtained by finding the zeros of DRF in the long-wavelength limit. This leads to the plasmon frequencies. We find that the collective charge excitations at zero doping, i.e., intrinsic plasmons, in this system, are absent in the Dirac limit. The valley-spin-split intrinsic plasmons, however, come into being in the case of the massive Dirac particles with characteristic frequency close to 10 THz. Our scheme to calculate the Casimir–Polder interaction (CPI) of a micro-particle with a sheet involves replacing the dielectric constant of the sample in the CPI expression obtained on the basis of the Lifshitz theory by the static DRF obtained using the expressions for the polarization function we started with. Though the approach replaces a macroscopic constant by a microscopic quantity, it has the distinct advantage of the many-body effect inclusion seamlessly. We find the result that for the nontrivial susceptibility and polarizability values of the sheet and micro-particle, respectively, there is crossover between attractive and repulsive behavior. The transition depends only on these response functions apart from the ratio of the film thickness and the micro-particle separation ([Formula: see text]/[Formula: see text]) and temperature. Furthermore, there is a longitudinal electric field induced topological insulator (TI) to spin-valley-polarized metal (SVPM) transition in silicene, which is also referred to as the topological phase transition (TPT). The low-energy SVP carriers at TPT possess gapless (massless) and gapped (massive) energy spectra close to the two nodal points in the Brillouin zone with maximum spin-polarization. We find that the magnitude of the CPF at a given ratio of the film thickness and the separation between the micro-particle and the film are greater at TPT than at the TI and trivial insulator phases.

Frequency-dependent polarizability, plasmons, and screening in the two-dimensional pseudospin-1 dice lattice

We calculate the dynamic polarizability under the random phase approximation for the dice lattice. This two-dimensional system gives rise to massless Dirac fermions with pseudospin-1 in the low-energy quantum excitation spectrum, providing a Dirac-cone plus flat-band dispersion. Due to the presence of the flat band, the polarizability shows key differences to that of graphene (the pseudospin-1/2 Dirac material). We find that the plasmon branch is pinched in to a single point, $\omega_p=q=\mu$, independent of the background dielectric constant. Finally, screening effects are discussed with regard to impurities.

Faraday rotation, band splitting, and one-way propagation of plasmon waves on a nanoparticle chain

We calculate the dispersion relations of plasmonic waves propagating along a chain of semiconducting or metallic nanoparticles in the presence of both a static magnetic field B and a liquid crystalline host. The dispersion relations are obtained using the quasistatic approximation and a dipole-dipole approximation to treat the interaction between surface plasmons on different nanoparticles. For plasmons propagating along a particle chain in a nematic liquid crystalline host with both B and the director parallel to the chain, we find a small, but finite, Faraday rotation angle. For B perpendicular to the chain, but director still parallel to the chain, the field couples the longitudinal and one of the two transverse plasmonic branches. This coupling is shown to split the two branches at the zero field crossing by an amount proportional to |B|. In a cholesteric liquid crystal host and an applied magnetic field parallel to the chain, the dispersion relations for left- and right-moving waves are found to be different. For some frequencies, the plasmonic wave propagates only in one of the two directions.

Tunable surface plasmon instability leading to emission of radiation

We propose a new approach for energy conversion from a dc electric field to tunable terahertz emission based on hybrid semiconductors by combining two-dimensional (2D) crystalline layers and a thick conducting material with possible applications for chemical analysis, security scanning, medical (single-molecule) imaging, and telecommunications. The hybrid nano-structure may consist of a single or pair of sheets of graphene, silicene, or a 2D electron gas. When an electric current is passed through a 2D layer, we discover that two low-energy plasmon branches exhibit a characteristic loop in their dispersion before they merge into an unstable region beyond a critical wave vector qc. This finite qc gives rise to a wavenumber cutoff in the emission dispersion of the surface plasmon induced instability and emission of radiation (spiler). However, there is no instability for a single driven layer far from the conductor, and the instability of an isolated pair of 2D layers occurs without a wavenumber cutoff. The wavenumber cutoff is found to depend on the conductor electron density, layer separation, distances of layers from the conductor surface, and the driving-current strength.

Effective Coulomb interaction among electrons in cuprates

Analytical expressions for the charge susceptibility and permittivity of cuprates are obtained using the singlet-correlated conduction band model. The screening parameter caused by interband transitions is refined using experimental plasmon frequencies. A new branch of acoustic plasmons is predicted. The range of values in which acoustic plasmons do not experience Landau damping is determined for wave vectors in the Brillouin zone. Fourier images of the effective Coulomb interaction among charge carriers is calculated for different wave vectors.

Dynamical polarization function, plasmons, and screening in silicene and other buckled honeycomb lattices

We explore the dielectric properties of graphene-like two-dimensional Kane-Mele topological insulators manifest in buckled honeycomb lattices (ex. silicene, germanene, etc.). The effect of an onsite potential difference ($\Delta_z$) between sublattices is given particular attention. We present the results for the real and imaginary parts of the dynamical polarization function. We show that these results display features of three regimes (topological insulator, valley-spin polarized metal, and trivial band insulator) and may be used to extract information on the strength of the intrinsic spin-orbit coupling. We study the inverse dielectric function and provide numerical results for the plasmon branch. We discuss the behaviour of the plasmon as a function of sublattice potential difference and show that the behaviour of the plasmon branch as $\Delta_z$ is varied is dependent on the location of the chemical potential with respect to the gaps. The static polarization is discussed and numerical results for the screening of a charged impurity are provided. We observe a beating phenomenon in the effective potential which is dependent on $\Delta_z$.

Plasmons in spatially separated double-layer graphene nanoribbons

Motivated by innovative progresses in designing multi-layer graphene nanostructured materials in the laboratory, we theoretically investigate the Dirac plasmon modes of a spatially separated double-layer graphene nanoribbon system, made up of a vertically offset armchair and metallic graphene nanoribbon pair. We find striking features of the collective excitations in this novel Coulomb correlated system, where both nanoribbons are supposed to be either intrinsic (undoped/ungated) or extrinsic (doped/gated). In the former, it is shown the low-energy acoustical and the high-energy optical plasmon modes are tunable only by the inter-ribbon charge separation. In the later, the aforementioned plasmon branches are modified by the added doping factor. As a result, our model could be useful to examine the existence of a linear Landau-undamped low-energy acoustical plasmon mode tuned via the inter-ribbon charge separation as well as doping. This study might also be utilized for devising novel quantum optical waveguides based on the Coulomb coupled graphene nanoribbons.

Dielectric screening of surface states in a topological insulator

Hexagonal warping provides an anisotropy to the dispersion curves of the helical Dirac fermions that exist at the surface of a topological insulator. A subdominant quadratic in the momentum term leads to an asymmetry between conduction and valence band. A gap can also be opened through magnetic doping. We show how these various modifications to the Dirac spectrum change the polarization function of the surface states and employ our results to discuss their effect on the plasmons. In the long wavelength limit, the plasmon dispersion retains its square-root dependence on its momentum, $\mathbf{q}$, but its slope is modified and it can acquire a weak dependence on the direction of $\mathbf{q}$. Further, we find the existence of several plasmon branches, one of which is damped for all values of $\mathbf{q}$, and extract the plasmon scattering rate for a representative case.

Electrostatic waves in carbon nanotubes with an axial magnetic field

Based on a linearized hydrodynamic model and within the quasi-static approximation, the dispersion relation of electrostatic waves propagating through single-walled carbon nanotubes subject to an axial magnetic field is theoretically explored. In the classical limit, we obtain two main possible waves which in turn are divided into two branches, a low-frequency acoustical and a high-frequency optical plasmon branch. In the quantum case, we have found that the dispersion relation is substantially modified when the electron wavelength becomes large enough compared to the propagation wavelength of the electrostatic waves in the quantum plasma. We also show that the axial magnetic field manifest itself on the perturbed electron density through the quantum term and gives rise to the propagation of the electrostatic waves within the quantum plasma. As a result, the effect of the magnetic field is pronounced in the plasma dispersion relations in such a way that their curves approach to zero when the magnetic field is weak; and for the strong magnetic field, they asymptotically meet the constant lines.