Using a nearest-neighbor tight-binding model we investigate quantum effects of plasmons on few-nanometer wide graphene nanoribbons, both for zigzag and armchair edge terminations. With insight from the Dirac description we find an emerging scale-invariant behavior that deviates from the classical model both for zigzag and armchair structures. The onset of the deviation can be related to the position of the lowest parabolic band in the band structure. Dirac theory is only valid in the parameter subspace where the scale invariance holds that relates narrow ribbons with high doping to wide ribbons with low doping. We also find that the edge states present in zigzag ribbons give rise to a blueshift of the plasmon, in contrast to earlier findings for graphene nanodisks and nanotriangles.
A graphene sheet gated with a ridged ground plane, creating a soft-boundary (SB) graphene nanoribbon, is considered. By adjusting the ridge parameters and bias voltage a channel can be created on the graphene which can guide TM surface plasmon polaritons (SPP). Two types of modes are found; fundemental and higher-order modes with no apparent cutoff frequency and with energy distributed over the created channel, and edge modes with energy concen-trated at the soft-boundary edge. Dispersion curves, electric near-field patterns, and current distributions of these modes are determined. Since the l…
Graphene plasmonics is a rapidly growing field with multiple potential applications. One of the standard ways to study plasmons in graphene is by fabricating an array of graphene nanoribbons where nanoribbon edges provide the efficient photon-plasmon coupling. We systematically analyze the problem of optical plasmonic response in such systems and demonstrate the purely geometric nature of the size quantization condition for graphene plasmons. Accurate numerical calculations allowed us to tabulate the universal geometric parameters of plasmon size quantization, which is expected to become usefu…
Doped graphene emerges as a strong contender for active plasmonic material in the mid-infrared wavelengths due to the versatile external-control of its permittivity-function and also its highly-compressed graphene surface plasmon (GSP) wavelength. In this paper, we design active plasmonic waveguide devices based on electrical-modulation of doped graphene nanoribbons (GNRs) on a voltage-gated inhomogeneous dielectric layer. We first develop figure-of-merit (FoM) formulae to characterize the performance of passive and active graphene nanoribbon waveguides. Based on the FoMs, we choose optimal GN…
We investigate higher-order plasmons in graphene nanoribbons, and present how electronic edge states and wavefunction fine structure influence the graphene plasmons. Based on nearest-neighbor tight-binding calculations, we find that a standing-wave model based on nonlocal bulk plasmon dispersion is surprisingly accurate for armchair ribbons of widths even down to a few nanometers, and we determine the corresponding phase shift upon edge reflection and an effective ribbon width. Wider zigzag ribbons exhibit a similar phase shift, whereas the standing-wave model describes few-nanometer zigzag ri…
Graphene plasmons have advantages over noble metal plasmons, such as high tunability and low loss. However, for graphene nanostructures smaller than 10 nm, little is known about their plasmons or whether a regular plasmonic behavior exists, despite their potential applications. Here, we present first-principles calculations of plasmon excitations in zigzag graphene nanoribbon segments. Regular plasmonic behavior is found: Only one plasmon mode exists in the low-energy regime (< 1.5 eV). The classical electrostatic scaling law still approximately holds when the width (W) is larger than about 1.…
A novel type of spaser with the net amplification of surface plasmons (SPs) in doped graphene nanoribbon is proposed. The plasmons in THz region can be generated in a dopped graphene nanoribbon due to nonradiative excitation by emitters like two level quantum dots located along a graphene nanoribbon. The minimal population inversion per unit area, needed for the net amplification of SPs in a doped graphene nanoribbon is obtained. The dependence of the minimal population inversion on the surface plasmon wavevector, graphene nanoribbon width, doping and damping parameters necessary for the ampli…
The main challenge to exploiting plasmons for gas vibrational mode sensing is the extremely weak infrared absorption of gas species. In this work, we explore the possibility of trapping free gas molecules via surface adsorption, optical, or electrostatic fields to enhance gas-plasmon interactions and to increase plasmon sensing ability. We discuss the relative strengths of these trapping forces and found gas adsorption in a typical nanoribbon array plasmonic setup produces measurable dips in optical extinction of magnitude 0.1 % for gas concentration of about parts per thousand level.
We calculate the polarization function of Dirac fermions in metallic armchair graphene nanoribbons for an arbitrary temperature and doping. We find that at finite temperatures due to the phase space redistribution among inter-band and intra-band electronic transitions in the conduction and valence bands, the full polarization function becomes independent of the temperature and the position of the chemical potential. As a result, for a given width of nanoribbons there exists a single plasmon mode, with the energy dispersion determined by the graphene's fine structure constant. In Coulomb-couple…
Interactions between localized plasmons in proximal nanostructures is a well-studied phenomenon. Here we explore plasmon plasmon interactions in connected extended systems. Such systems can now be easily produced using graphene. Specifically we employ the finite element method to study such interactions in graphene nanoribbon arrays with a periodically modulated electrochemical potential or number of layers. We find a rich variation in the resulting plasmonic resonances depending on the dimensions and the electrochemical potentials (doping) of the nanoribbon segments and the involvement of tra…
Plasmon resonance in nanopatterned single layer graphene nanoribbon (SL-GNR), double layer graphene nanoribbon (DL-GNR) and triple layer graphene nanoribbon (TL-GNR) structures is studied both experimentally and by numerical simulations. We use 'realistic' graphene samples in our experiments to identify the key bottle necks in both experiments and theoretical models. The existence of electrical tunable plasmons in such stacked multilayer GNRs was first experimentally verified by infrared microscopy. We find that the strength of the plasmonic resonance increases in DL-GNR when compared to SL-GN…
We have obtained and analyzed the electronic states, polarization function and the plasmon excitations for $α- \mathcal{T}_3$-based nanoribbons with armchair termination. The calculated plasmon dispersions strongly depend on the number of the atomic rows across the ribbon, and the presence of the energy gap between the valence and conduction bands which is also determined by the nanoribbon geometry. The bandgap was proven to have the strongest effect on both the plasmon dispersions and their Landau damping. We have also demonstrated that for a small electron doping the plasmon dispersions do n…
We report on nano-infrared (IR) imaging studies of confined plasmon modes inside patterned graphene nanoribbons (GNRs) fabricated with high-quality chemical-vapor-deposited (CVD) graphene on Al2O3 substrates. The confined geometry of these ribbons leads to distinct mode patterns and strong field enhancement, both of which evolve systematically with the ribbon width. In addition, spectroscopic nano-imaging in mid-infrared 850-1450 cm-1 allowed us to evaluate the effect of the substrate phonons on the plasmon damping. Furthermore, we observed edge plasmons: peculiar one-dimensional modes propaga…
Among their amazing properties, graphene and related low-dimensional materials show quantized charge-density fluctuations--known as plasmons--when exposed to photons or electrons of suitable energies. Graphene nanoribbons offer an enhanced tunability of these resonant modes, due to their geometrically controllable band gaps. The formidable effort made over recent years in developing graphene-based technologies is however weakened by a lack of predictive modeling approaches that draw upon available {\it ab initio} methods. An example of such a framework is presented here, focusing on narrow-wid…
We study the electromagnetic response and surface electromagnetic modes in a generic gapped Dirac material under pumping with circularly polarized light. The valley imbalance due to pumping leads to a net Berry curvature, giving rise to a finite transverse conductivity. We discuss the appearance of nonreciprocal chiral edge modes, their hybridization and waveguiding in a nanoribbon geometry, and giant polarization rotation in nanoribbon arrays.
We theoretically examine the role of Kerr nonlinearities for graphene plasmonics in nanostructures, specifically in nanoribbons. The nonlinear Kerr interaction is included semiclassically in the intraband approximation. The resulting electromagnetic problem is solved numerically by self-consistent iteration with linear steps using a real-space discretization. We derive a simple approximation for the resonance shifts in general graphene nanostructures, and obtain excellent agreement with numerics for moderately high field strengths. Near plasmonic resonances the nonlinearities are strongly enha…
The dispersion relations for plasmons in intrinsic and extrinsic semimetallic armchair graphene nanoribbons (acGNR) are calculated in the random phase approximation using the orthogonal p_z-orbital tight binding method. Our model predicts new plasmons for acGNR of odd atomic widths N=5,11,17,... Our model further predicts plasmons in acGNR of even atomic width N=2,8,14,... related to those found using a Dirac continuum model, but with different quantitative dispersion characteristics. We find that the dispersion of all plasmons in semimetallic acGNR depends strongly on the localization of the …
We report the plasmon dispersion characteristics of intrinsic and extrinsic armchair graphene nanoribbons of atomic width N = 5 using a p_z-orbital tight binding model with third-nearest-neighbor (3nn) coupling. The coupling parameters are obtained by fitting the 3nn dispersions to that of an extended Huckel theory. The resultant massive Dirac Fermion system has a band gap E_g \approx 64 meV. The extrinsic plasmon dispersion relation is found to approach a common dispersion curve as the chemical potential $μ$ increases, whereas the intrinsic plasmon dispersion relation is found to have both en…
We analyze the collective mode spectrum of graphene nanoribbons within the random phase approximation. In the undoped case, only metallic armchair nanoribbons support a propagating plasmon mode. Landau damping of this mode is shown to be suppressed through the chirality of the single particle wavefunctions. We argue that undoped zigzag nanoribbons should not support plasmon excitations because of a broad continuum of particle-hole excitations associated with surface states, into which collective modes may decay. Doped nanoribbons have properties similar to those of semiconductor nanowires, inc…
The effect of edge modification of armchair graphene nanoribbons (AGNRs) on the collective excitations are theoretically investigated. The tight-binding method is employed in conjunction with the dielectric function. Unconventional plasmon modes and their association with the flat bands of the specially designed AGNRs are thoroughly studied. We demonstrate the robust relationship between the novel collective excitations and both the type and period of the edge modification. Additionally, we reveal that the main features displayed in the (momentum, frequency)-phase diagrams for both single-part…
In this book chapter, we introduce different schemes to create quantum states of matter in engineered graphene nanoribbons. We will focus on the emergence of controllable magnetic interactions, topological quantum magnets, and the interplay of magnetism and superconductivity. We comment on the experimental signatures of those states stemming from their electronic and spin excitations, that can be observed with atomic resolution using scanning probe techniques.
We study the electronic band structure and optical conductivity of an array of tunnel-coupled array of graphene nanoribbons. We show that due to the coupling of electronic edge states for the zigzag nanoribbon structure, the Fermi surface can become a hyperbola similarly to the case of the layered metal-dielectric structures, where the hyperbolic isofrequency contours originate from the coupling of localized surface plasmon polaritons. Moreover, we show that for both types of the ribbon edge, the optical response of the structure can be characterized by a uniaxial conductivity tensor, having p…
An infinite sheet of graphene lying above a perturbed ground plane is studied. The perturbation is a two dimensional ridge, and a bias voltage is applied between the graphene and the ground plane, resulting in a graphene nanoribbonlike structure with a soft-boundary (SB) The spatial distribution of the graphene conductivity forming the soft-boundary is studied as a function of the ridge parameters and the bias voltage. The current distribution of the fundamental TM surface plasmon polariton (SPP) is considered. The effect of the ridge parameters and shape of the soft boundary on the current di…
Graphene is known to possess strong optical nonlinearity. Its nonlinear response can be further enhanced by graphene plasmons. Here, we report a novel nonlinear electro-absorption effect observed in nanostructured graphene due to excitation of graphene plasmons. We experimentally detect and theoretically explain enhanced nonlinear mixing of near-infrared and mid-infrared light in arrays of graphene nanoribbons. Strong compression of light by graphene plasmons implies that the effect is non-local in nature and orders of magnitude larger than the conventional local graphene nonlinearity. The eff…
Sub-wavelength graphene structures support localized plasmonic resonances in the terahertz and mid-infrared spectral regimes. The strong field confinement at the resonant frequency is predicted to significantly enhance the light-graphene interaction, which could enable nonlinear optics at low intensity in atomically thin, sub-wavelength devices. To date, the nonlinear response of graphene plasmons and their energy loss dynamics have not been experimentally studied. We measure and theoretically model the terahertz nonlinear response and energy relaxation dynamics of plasmons in graphene nanorib…
Graphene's optical properties in the infrared and terahertz can be tailored and enhanced by patterning graphene into periodic metamaterials with sub-wavelength feature sizes. Here we demonstrate polarization sensitive and gate tunable photodetection in graphene nanoribbon arrays. The long-lived hybrid plasmon-phonon modes utilized are coupled excitations of electron density oscillations and substrate (SiO2) surface polar phonons. Their excitation by s-polarization leads to an in-resonance photocurrent an order of magnitude larger than the photocurrent observed for p-polarization, which excites…
Optical excitation and subsequent decay of graphene plasmons can produce a significant increase in charge-carrier temperature. An efficient method to convert this temperature elevation into a measurable electrical signal at room temperature can enable important mid-infrared applications such as thermal sensing and imaging in ubiquitous mobile devices. However, as appealing as this goal might be, it is still unrealized due to the modest thermoelectric coefficient and weak temperature-dependence of carrier transport in graphene. Here, we demonstrate mid-infrared graphene detectors consisting of …
We present infrared transmission spectroscopy study of the inter-Landau-level excitations in quasi-neutral epitaxial graphene nanoribbon arrays. We observed a substantial deviation in energy of the $L_{0(-1)}$$\to$$L_{1(0)}$ transition from the characteristic square root magnetic-field dependence of two-dimensional graphene. This deviation arises from the formation of upper-hybrid mode between the Landau level transition and the plasmon resonance. In the quantum regime the hybrid mode exhibits a distinct dispersion relation, markedly different from that expected for conventional two-dimensiona…
We study plasmonic resonances in electrostatically gated graphene nanoribbons on silicon dioxide substrates. Absorption spectra are measured in the mid-far infrared and reveal multiple peaks, with width-dependent resonant frequencies. We calculate the dielectric function within the random phase approximation and show that the observed spectra can be explained by surface-plasmon-phonon-polariton modes, which arise from coupling of the graphene plasmon to three surface optical phonon modes in the silicon dioxide.
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 a highly unusual behavior with endpoints 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…
In the phenomenon of electromagnetically induced transparency1 (EIT) of a three-level atomic system, the linear susceptibility at the dipole-allowed transition is canceled through destructive interference of the direct transition and an indirect transition pathway involving a meta-stable level, enabled by optical pumping. EIT not only leads to light transmission at otherwise opaque atomic transition frequencies, but also results in the slowing of light group velocity and enhanced optical nonlinearity. In this letter, we report an analogous behavior, denoted as phonon-induced transparency (PIT)…
Work function-mediated charge transfer in graphene/$α$-RuCl$_3$ heterostructures has been proposed as a strategy for generating highly-doped 2D interfaces. In this geometry, graphene should become sufficiently doped to host surface and edge plasmon-polaritons (SPPs and EPPs, respectively). Characterization of the SPP and EPP behavior as a function of frequency and temperature can be used to simultaneously probe the magnitude of interlayer charge transfer while extracting the optical response of the interfacial doped $α$-RuCl$_3$. We accomplish this using scanning near-field optical microscopy …
We consider a quantum network of mid-infrared, graphene plasmons coupled to the hydrogen-like excited states of group-V donors in silicon. First, we show how to use plasmon-enhanced light-matter interactions to achieve single-shot spin readout of the donor qubits via optical excitation and electrical detection of the emitted plasmons. We then show how plasmons in high mobility graphene nanoribbons can be used to achieve high-fidelity, two-qubit gates and entanglement of distant Si donor qubits. The proposed device is readily compatible with existing technology and fabrication methods.
Using ab initio calculations we predict the existence of one-dimensional (1D), atomically confined plasmons at the edges of a zigzag MoS2 nanoribbon. The strongest plasmon originates from a metallic edge state localized on the sulfur dimers decorating the Mo edge of the ribbon. A detailed analysis of the dielectric function reveals that the observed deviations from the ideal 1D plasmon behavior result from single-particle transitions between the metallic edge state and the valence and conduction bands of the MoS2 sheet. The Mo and S edges of the ribbon are clearly distinguishable in calculated…
The canalization of terahertz surface plasmon polaritons using a modulated graphene monolayer is investigated for subwavelength imaging. An anisotropic surface conductivity formed by a set of parallel nanoribbons with alternating positive and negative imaginary conductivities is used to realize the canalization regime required for hyperlensing. The ribbons are narrow compared to the wavelength, and are created electronically by gating a graphene layer over a corrugated ground plane. Good quality canalization of surface plasmon polaritons is shown in the terahertz even in the presence of realis…
Graphene has emerged as a promising platform to bring nonlinear quantum optics to the nanoscale, where a large intrinsic optical nonlinearity enables long-lived and actively tunable plasmon polaritons to strongly interact. Here we theoretically study the collision between two counter-propagating plasmons in a graphene nanoribbon, where transversal subwavelength confinement endows propagating plasmons with %large effective masses a flat band dispersion that enhances their interaction. This scenario presents interesting possibilities towards the implementation of multi-mode polaritonic gates tha…
Visible surface plasmon resonances (SPRs) could be excited by TE wave polarization in one-dimensional photonic crystals (PCs) coated by a graphene layer under the Kretschmann configuration. In this work, the plasmonic Bloch wave properties beyond Dirac points in a one-dimensional graphene-based photonic sensing structure have been numerically studied. We demonstrate that emergent plasmonic dips in the reflectance spectra of the suggested photonic device in the visible region exhibit tunable characteristics upon modulation of the chemical potential and the hopping parameter. The sensitivity of …
Artifical superlattice (SL) potentials have been employed extensively for band structure engineering of two-dimensional (2D) Dirac electron gas in graphene. While such engineered electronic band structures can modify optical or plasmonic properties of graphene, an emergent polaritonic behavior beyond weak perturbative effects (e.g. anisotropic Drude weights) has not been reported. Here, we show that an extreme modulation of one-dimensional (1D) SL potentials in monolayer graphene deforms the underlying Dirac band dispersion and introduces ladder-like energy levels near the Fermi surface, which…
Stacking two-dimensional layered materials such as graphene and transitional metal dichalcogenides with nonzero interlayer twist angles has recently become attractive because of the emergence of novel physical properties. Stacking of one-dimensional nanomaterials offers the lateral stacking offset as an additional parameter for modulating the resulting material properties. Here, we report that the edge states of twisted bilayer zigzag graphene nanoribbons (TBZGNRs) can be tuned with both the twist angle and the stacking offset. Strong edge state variations in the stacking region are first reve…
Nonreciprocal plasmons in one-dimensional carbon nanostructures
Max-Born-Institut（马克斯·玻恩研究所）；Center for Nanophotonics, NWO Institute AMOLF（纳米光子学中心，NWO AMOLF研究所）；ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科学技术研究所）；ICREA-Institució Catalana de Recerca i Estudis Avançats（ICREA-加泰罗尼亚研究高级学院）；POLIMA—Center for Polariton-driven Light–Matter Interactions, University of Southern Denmark（POLIMA-极化激元驱动光物质相互作用中心，南丹麦大学）；Danish Institute for Advanced Study, University of Southern Denmark（丹麦高等研究院，南丹麦大学）
We analyze nonlinear optics schemes for generating pairs of quantum entangled plasmons in the terahertz-infrared range in graphene. We predict that high plasmonic field concentration and strong optical nonlinearity of monolayer graphene enables pair-generation rates much higher than those of conventional photonic sources. The first scheme we study is spontaneous parametric down conversion in a graphene nanoribbon. In this second-order nonlinear process a plasmon excited by an external pump splits into a pair of plasmons, of half the original frequency each, emitted in opposite directions. The …
Lateral heterostructures of two-dimensional materials may exhibit various intriguing emergent properties. Yet when specified to the orientationally aligned heterojunctions of zigzag graphene and hexagonal boron nitride (hBN) nanoribbons, realizations of the high expectations on their properties encounter two standing hurtles. First, the rapid accumulation of strain energy prevents large- scale fabrication. Secondly, the pronounced half-metallicity predicted for freestanding graphene nanoribbons is severely suppressed. By properly tailoring orientational misalignment between zigzag graphene and…
Edge states are ubiquitous for many condensed matter systems with multicomponent wave functions. For example, edge states play a crucial role in transport in zigzag graphene nanoribbons. Here, we report microscopic calculations of quantum plasmonics in doped graphene nanodisks with zigzag edges. We express the nanodisk conductivity $σ(ω)$ as a sum of the conventional bulk conductivity $σ_{\scriptscriptstyle\text{B}}(ω)$, and a novel term $σ_{\scriptscriptstyle\text{E}}(ω)$, corresponding to a coupling between the edge and bulk states. We show that the edge states give rise to a red-shift and b…
The two-dimensionality of graphene and other layered materials can be exploited to simplify the theoretical description of their plasmonic and polaritonic modes. We present an analytical theory that allows us to simulate these excitations in terms of plasmon wave functions (PWFs). Closed-form expressions are offered for their associated extinction spectra, involving only two real parameters for each plasmon mode and graphene morphology, which we calculate and tabulate once and for all. Classical and quantum-mechanical formulations of this PWF formalism are introduced, in excellent mutual agree…
The decay dynamics of excited carriers in graphene have attracted wide scientific attention, as the gapless Dirac electronic band structure opens up relaxation channels that are not allowed in conventional materials. We report Fermi-level-dependent mid-infrared emission in graphene originating from a previously unobserved decay channel: hot plasmons generated from optically excited carriers. The observed Fermi-level dependence rules out Planckian light emission mechanisms and is consistent with the calculated plasmon emission spectra in photoinverted graphene. Evidence for bright hot plasmon e…
Hybrid photonic-plasmonic nanostructures allow one to engineer coupling of quantum emitters and cavity modes accounting for the direct coherent and environment mediated dissipative pathways. Using generalized plasmonic Dicke model, we explore the non-equilibrium phase diagram with respect to these interactions. The analysis shows that their interplay results in the extension of the superradiant and regular lasing states to the dissipative coupling regime and an emergent lasing phase without population inversion having boundary with the superradiant and normal states. Calculated photon emission…
We report mid-infrared photocurrent spectra of graphene nanoribbon arrays on SiO2 dielectrics showing dual signatures of the substrate interaction. First, hybrid polaritonic modes of graphene plasmons and dielectric surface polar phonons produce a thermal photocurrent in graphene with spectral features that are tunable by gate voltage, nanoribbon width, and light polarization. Secondly, phonon-polaritons associated with the substrate are excited, which indirectly heat up the graphene leading to a graphene photocurrent with fixed spectral features. Models for other commonly used substrates show…
Spin-ordered electronic states in hydrogen-terminated zigzag nanographene give rise to magnetic quantum phenomena that have sparked renewed interest in carbon-based spintronics. Zigzag graphene nanoribbons (ZGNRs), quasi one-dimensional semiconducting strips of graphene featuring two parallel zigzag edges along the main axis of the ribbon, are predicted to host intrinsic electronic edge states that are ferromagnetically ordered along the edges of the ribbon and antiferromagnetically coupled across its width. Despite recent advances in the bottom-up synthesis of atomically-precise ZGNRs, their …
We review the transmission of Dirac electrons through a potential barrier in the presence of circularly polarized light. A different type of transmission is demonstrated and explained. Perfect transmission for nearly head-on collision in inffnite graphene is suppressed in gapped dressed states of electrons. We also present our results on enhanced mobility of hot Dirac electrons in nanoribbons and magnetoplasmons in graphene in the presence of the energy gap. The calculated carrier mobility for a graphene nanoribbon as a function of the bias field possesses a high threshold for entering the non…
We study a bipartite Kronig-Penney model with negative Dirac-delta potentials that may be used, amongst other models, to interpret plasmon propagation in nanoparticle arrays. Such a system can be mapped into a Su-Schrieffer-Heeger-like model however, in general, the overlap between 'atomic' wavefunctions of neighbouring sites is not negligible. In such a case, the edge states of the finite system, which retain their topological protection, appear to be either attenuated or amplified. This phenomenon, called "edge polarisation", is usually associated with an underlying non-Hermitian topology. B…
