Intraband Contributions Research Articles

Spin susceptibility of three-dimensional Dirac-Weyl semimetals

We theoretically study the spin susceptibility of Dirac semimetals using the linear response theory. The spin susceptibility is decomposed into an intraband contribution and an interband contribution. We obtain analytical expressions for the intraband and interband contributions of massless Dirac fermions. The spin susceptibility is independent of the Fermi energy while it depends on the cutoff energy, which is introduced to regularize the integration. We find that the cutoff energy is appropriately determined by comparing the results for the Wilson-Dirac lattice model, which approximates the massless Dirac Hamiltonian around the Dirac point. We also calculate the spin susceptibility of massive Dirac fermions for the model of topological insulators. We discuss the effect of the band inversion and the strength of spin-orbit coupling.

Temperature-dependent dielectric functions of bcc transition metals Cr, Mo, and W from ultraviolet to infrared regions: A theoretical and experimental study

In the bcc transition metals Cr, Mo, and W, the existence of the partially filled d bands makes interband transition occur at low photon frequencies and thus, it is difficult to differentiate it from intraband transition. Here, we present a thorough study on decomposing the intraband and interband contribution to finite temperature dielectric functions of these bcc transition metals by performing electron-phonon and electron-electron interaction calculations, as well as ellipsometry experiments. In this work, the Drude model and interband transition theory are applied to quantitatively describe the intraband and interband transition, respectively. To accurately determine intraband transition, the relevant parameters for the Drude model, such as plasma frequency and electron relaxation time, are calculated from first-principles. The electron-electron interaction within the many-body theory and electron-phonon interaction within the density functional perturbation theory are calculated to obtain the electron relaxation time and intraband dielectric function at finite temperature. As for interband transition, the spin-orbit coupling is included and it shows nontrivial influence on the interband dielectric function of Mo and W, especially at low frequencies. To verify theoretical calculations, ellipsometry experiments are performed to measure dielectric functions of Cr, Mo, and W over the temperature range of 300–700 K and energy range of 0.08–4.8 eV. The experimental results are then fitted by the Drude model, and it shows that the electron-phonon interaction rather than electron-electron interaction dominates the frequency dependence of the relaxation time for transition metals Cr, Mo, and W.

Optical properties of dense lithium in electride phases by first-principles calculations

The metal-semiconductor-metal transition in dense lithium is considered as an archetype of interplay between interstitial electron localization and delocalization induced by compression, which leads to exotic electride phases. In this work, the dynamic dielectric response and optical properties of the high-pressure electride phases of cI16, oC40 and oC24 in lithium spanning a wide pressure range from 40 to 200 GPa by first-principles calculations are reported. Both interband and intraband contribution to the dielectric function are deliberately treated with the linear response theory. One intraband and two interband plasmons in cI16 at 70 GPa induced by a structural distortion at 2.1, 4.1, and 7.7 eV are discovered, which make the reflectivity of this weak metallic phase abnormally lower than the insulating phase oC40 at the corresponding frequencies. More strikingly, oC24 as a reentrant metallic phase with higher conductivity becomes more transparent than oC40 in infrared and visible light range due to its unique electronic structure around Fermi surface. An intriguing reflectivity anisotropy in both oC40 and oC24 is predicted, with the former being strong enough for experimental detection within the spectrum up to 10 eV. The important role of interstitial localized electrons is highlighted, revealing diversity and rich physics in electrides.

Nonperturbative harmonic generation in graphene from intense midinfrared pulsed light

In solids, high harmonic radiation arises from the subcycle dynamics of electrons and holes under the action of an intense laser field. The strong-field regime opens new opportunities to understand and control carrier dynamics on ultrafast time scales, including the coherent dynamics of quasiparticles such as massless Dirac fermions. Here, we irradiate monolayer and few-layer graphene with intense infrared light to produce nonperturbative harmonics of the fundamental up to the seventh order. We find that the polarization dependence shows surprising agreement with gas-phase harmonics. Using a two-band model, we explore the nonlinear current due to electrons near the Dirac points, and we discuss the interplay between intraband and interband contributions to the harmonic spectrum. This interplay opens new opportunities to access ultrafast and strong-field physics of graphene.

Electrodynamics on Fermi Cyclides in Nodal Line Semimetals.

We study the frequency-dependent conductivity of nodal line semimetals (NLSMs), focusing on the effects of carrier density and energy dispersion on the nodal line. We find that the low-frequency conductivity has a rich spectral structure which can be understood using scaling rules derived from the geometry of their Dupin cyclide Fermi surfaces. We identify different frequency regimes, find scaling rules for the optical conductivity in each, and demonstrate them with numerical calculations of the inter- and intraband contributions to the optical conductivity using a low-energy model for a generic NLSM.

Electronic Structures and Optical Properties of Al, Cu and Ag in Bulk, Thin Film and Nanoparticle Forms

The electronic structures of aluminum, copper, and silver were investigated in bulk, thin-film and nanoparticle forms using first-principles band structure methods. The density of states (DOS) and partial density of states (PDOS) were also calculated. The calculations show the progression from continuous bands to subbands to discrete states as spatial confinement is imposed by going from bulk to thin film to particle forms. The choice of metals in this study allows for the comparison of the degree of localization of the d-electrons on inter-band transitions, which, in part, governs the optical properties of the metals. The associated optical properties described by the imaginary component of the dielectric function, ε2(ω), were also investigated. The interband contributions to ε2(ω) were calculated from the band structure, while the intraband contributions were calculated using the Drude theory for free electrons. Both contributions to ε2(ω)are needed to understand the optical properties of metals and to interpret their reflectance spectra. The interband transitions need to be considered to explain reflectivity at energies lower than the plasma frequencies, but not all interband transitions result in reflectance peaks since they are significantly weaker compared to the intraband contributions at lower energies. We have studied the dependence of the DOS, PDOS and ε2(ω) on the choice of exchange potentials by comparing results obtained using the generalized gradient approximation (GGA) and the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional. In copper, where ultraviolet photoelectron spectroscopy data are available, the HSE potential leads to better agreement with the experiment than when using the GGA potential, and HSE reproduces correctly the variation of binding energy of the d electrons as spatial confinement is imposed. The corresponding dependence of exchange potential for silver exists, but is less severe.

Electronic structures and optical properties of Al, Cu, and Ag in zero, two, and three dimensional structures

The electronic structures and associated optical properties of aluminum, copper, and silver were investigated in bulk, thin-film and nanoparticle forms using first-principles band structure methods. The calculations show the progression from continuous bands to subbands to discrete states as spatial confinement is imposed in one and three dimensions. The associated optical properties described by the imaginary component of the dielectric function, ε2(ω), were also investigated. The interband contributions to ε2(ω) were calculated from the band structure, while the intraband contributions were calculated using the Drude theory for free electrons. Both contributions to ε2(ω) are needed to understand the optical properties of metals and to interpret their reflectance spectra. The interband transitions need to be considered to explain reflectivity at energies lower than the plasma frequencies, but not all interband transitions result in reflectance peaks since they are significantly weaker compared to the intraband contributions at lower energies. We have studied the dependence of the density of states on the choice of exchange potentials. In copper, where ultraviolet photoelectron spectroscopy data are available, using a hybrid functional, Heyd-Scuseria-Ernzerhof, leads to better agreement with the experiment than when using the generalized gradient approximation potential, and it reproduces correctly the variation of binding energy of the d electrons going from bulk to thin film. The corresponding dependence exchange potential for silver is less severe.

Giant Optical Second Harmonic Generation in Two-Dimensional Multiferroics.

Nonlinear optical properties of materials such as second and higher order harmonic generation and electro-optic effect play pivotal roles in lasers, frequency conversion, electro-optic modulators, switches, and so forth. The strength of nonlinear optical responses highly depends on intrinsic crystal symmetry, transition dipole moments, specific optical excitation, and local environment. Using first-principles electronic structure theory, here we predict giant second harmonic generation (SHG) in recently discovered two-dimensional (2D) ferroelectric-ferroelastic multiferroics-group IV monochalcogenides (i.e., GeSe, GeS, SnSe, and SnS). Remarkably, the strength of SHG susceptibility in GeSe and SnSe monolayers is more than 1 order of magnitude higher than that in monolayer MoS2, and 2 orders of magnitude higher than that in monolayer hexagonal BN. Their extraordinary SHG is dominated by the large residual of two opposite intraband contributions in the SHG susceptibility. More importantly, the SHG polarization anisotropy is strongly correlated with the intrinsic ferroelastic and ferroelectric orders in group IV monochalcogenide monolayers. Our present findings provide a microscopic understanding of the large SHG susceptibility in 2D group IV monochalcogenide multiferroics from first-principles theory and open up a variety of new avenues for 2D ferroelectrics, multiferroics, and nonlinear optoelectronics, for example, realizing active electrical/optical/mechanical switching of ferroic orders in 2D multiferroics and in situ ultrafast optical characterization of local atomistic and electronic structures using noncontact noninvasive optical SHG techniques.

Linear and nonlinear optical response of one-dimensional semiconductors: finite-size and Franz–Keldysh effects

The dipole moment formalism for the optical response of finite electronic structures breaks down in infinite ones, for which a momentum-based method is better suited. Focusing on simple chain structures, we compare the linear and nonlinear optical response of finite and infinite one-dimensional semiconductors. This comparison is then extended to cases including strong electro-static fields breaking translational invariance. For large electro-static fields, highly non-perturbative Franz–Keldysh (FK) features are observed in both linear and nonlinear spectra. It is demonstrated that dipole and momentum formalisms agree in the limit of large structures provided the intraband momentum contributions are carefully treated. This convergence is established even in the presence of non-perturbative electro-static fields.

Theory of graphene saturable absorption

Saturable absorption is a non-perturbative nonlinear optical phenomenon that plays a pivotal role in the generation of ultrafast light pulses. Here we show that this effect emerges in graphene at unprecedentedly low light intensities, thus opening avenues to new nonlinear physics and applications in optical technology. Specifically, we theoretically investigate saturable absorption in extended graphene by developing a non-perturbative single-particle approach, describing conduction-electron dynamics in the atomically-thin material using the two-dimensional Dirac equation for massless Dirac fermions, which is recast in the form of generalized Bloch equations. By solving the electron dynamics non-perturbatively, we account for both interband and intraband contributions to the intensity-dependent saturated conductivity and conclude that the former dominates regardless of the intrinsic doping state of the material. The results are in excellent agreement with atomistic quantum-mechanical simulations including higher-band effects. Additionally, we find that the modulation depth of saturable absorption in graphene can be electrically manipulated through an externally applied gate voltage. Our results are relevant for the development of graphene-based optoelectronic devices, as well as for applications in mode-locking and random lasers.

High-harmonic generation in graphene: Interband response and the harmonic cutoff

We numerically examine the high-harmonic generation in graphene caused by an intense few-cycle terahertz laser field by solving the time-dependent Dirac equation. The observed spectra feature a complex interplay of interband and intraband electron dynamics. At high harmonic frequencies, we observe a plateau region with a cut-off frequency linearly proportional to the laser field. The linear dependence of the cutoff on the field resembles the behavior for electrons driven in bulk crystals but differs from the case of atoms or molecules. The unique features of the graphene band structure allow for a transparent decomposition of interband and intraband contributions and for an analytic estimate of the harmonic cut-off frequency in excellent agreement with the simulations.

Low-temperature semiconductor band-gap thermal shifts: T4 shifts from ordinary acoustic and T2 from piezoacoustic coupling

At low temperature T, the experimental gap of silicon decreases as E_g(T)=E_g(0)-AT^4. The main reason is electron-phonon renormalization. The physics behind the T^4-power law is more complex than has been realized. Renormalization by intraband scattering requires a careful non-adiabatic treatment in order to correctly include acoustic phonons and avoid divergences from piezoacoustic phonon interactions. The result is an unexpected low T term E_g(0)+A' T^p with positive coefficient A', and power p=4 for non-piezoelectric materials, and power p=2 for piezoelectric materials. The acoustic phonons in piezoelectric semiconductors generate a piezoelectric field, modifying the electron-phonon coupling. However, at higher T, when thermal acoustic phonons of energy hbar v_s q acquire energies comparable to the electronic intermediate state (higher than the band-edge state by hbar^2 q^2 /2m*), the low q and higher q intraband contributions to T^p rapidly cancel, giving little thermal effect. But there is an additional T-dependence from interband effects of acoustic phonons. This turns out to have power law T^4 for both non-piezoelectric and piezoelectric semiconductors. This term can have either sign, but usually reduces the size of gaps as T increases. It arises after cancellation of the T^2 terms that appear separately in Debye-Waller and Fan parts of the acoustic phonon interband renormalization. The cancellation occurs because of the acoustic sum rule.

Coupled multiple-mode theory for s± pairing mechanism in iron based superconductors.

We investigate the interplay between the magnetic and the superconducting degrees of freedom in unconventional multi-band superconductors such as iron pnictides. For this purpose a dynamical mode-mode coupling theory is developed based on the coupled Bethe-Salpeter equations. In order to investigate the region of the phase diagram not too far from the tetracritical point where the magnetic spin density wave, (SDW) and superconducting (SC) transition temperatures coincide, we also construct a Ginzburg-Landau functional including both SC and SDW fluctuations in a critical region above the transition temperatures. The fluctuation corrections tend to suppress the magnetic transition, but in the superconducting channel the intraband and interband contribution of the fluctuations nearly compensate each other.

Effective numbers of charge carriers in doped graphene: Generalized Fermi liquid approach

The single-band current-dipole Kubo formula for the dynamical conductivity of heavily doped graphene from Kup\v{c}i\'{c} [Phys. Rev. B 91, 205428 (2015)] is extended to a two-band model for conduction $\pi$ electrons in lightly doped graphene. Using a posteriori relaxation-time approximation in the two-band quantum transport equations, with two different relaxation rates and one quasi-particle lifetime, we explain a seemingly inconsistent dependence of the dc conductivity $\sigma^{\rm dc}_{\alpha \alpha}$ of ultraclean and dirty lightly doped graphene samples on electron doping, in a way consistent with the charge continuity equation. It is also shown that the intraband contribution to the effective number of conduction electrons in $\sigma^{\rm dc}_{\alpha \alpha}$ vanishes at $T=0$ K in the ultraclean regime, but it remains finite in the dirty regime. The present model is shown to be consistent with a picture in which the intraband and interband contributions to $\sigma^{\rm dc}_{\alpha \alpha}$ are characterized by two different mobilities of conduction electrons, the values of which are well below the widely accepted value of mobility in ultraclean graphene. The dispersions of Dirac and $\pi$ plasmon resonances are reexamined to show that the present, relatively simple expression for the dynamical conductivity tensor can be used to study simultaneously single-particle excitations in the dc and optical conductivity and collective excitations in energy loss spectroscopy experiments.

Optical Distinctions Between Weyl Semimetal TaAs and Dirac Semimetal Na3Bi: An Ab Initio Investigation

We present ab initio a study on linear and nonlinear optical properties of topological semimetal Tantalum arsenide and Sodium bismuthate. The real and imaginary part of the dielectric function in addition to the energy loss spectra of TaAs and Na3Bi have been calculated within random phase approximation (RPA); then, the electron–hole interaction is included by solving the Bethe–Salpeter equation for the electron–hole Green’s function. In spite of being in the single category of topological materials, we have found obvious distinction between linear optical responses of TaAs and Na3Bi at a high energy region where, in contrast to Na3Bi, Tantalum arsenide has excitonic peaks at 9 eV and 9.5 eV. It is remarkable that the excitonic effects in the high energy range of the spectrum are stronger than in the lower one. The dielectric function is overall red shifted compared with that of RPA approximation. The resulting static dielectric constants for Na3Bi are smaller than corresponding ones in TaAs. At a low energy region, the absorption intensity of TaAs is more than Na3Bi. The calculated second-order nonlinear optical susceptibilities χijk(2)(ω) show that Tantalum arsenide acts as a Weyl semimetal, and has high values of nonlinear responses in the low energy region which makes it promising candidate for the second harmonic generation in the terahertz frequency region. In the low energy regime, optical spectra are dominated by the 2ω intra-band contributions.

Ab initio computations of the linear and nonlinear optical properties of stable compounds in Al[sbnd]In[sbnd]N system

The frequency dependent linear and second order nonlinear susceptibilities of AlxIn1−xN have been studied by means of density functional theory within the random phase approximation. Predicted stable AlInN phases from our previous study were investigated. Scissor shift from earlier work was supplemented so as to activate the self-energy effects in response function calculations. By considering the many-body effects, optical calculations of semiconductors are improved. Apart from analyzing the real and imaginary optical spectra, calculations of refractive index were also carried out. Both low energy limit of real part and refractive index are increasing monotonically with a decrease in Al composition. Analyses of intraband and interband contributions towards nonlinear χ3332 (2ω, ω, ω) are too discussed. Our calculations show that |χ3332(0)| values for the stable alloys range from 0.18 to 120.98 pm/V.

New understandings of third harmonic generation coefficient of bulk silicon: Implementation and application of the full ab initio band structure sum-over-states method

We make a full ab initio band structure analysis of interband and intraband contributions for the third-order nonlinear optical susceptibilities of bulk silicon by implementing the Aversa and Sipe sum-over-states formulism. The band structure and momentum matrix elements were calculated by using the highly accurate all-electron full potential linearized augmented plane wave method within the local density approximation. The convergence tests including the scissor correction with different k-points meshes and empty states were performed. Both real and imaginary parts of susceptibility were directly calculated and checked by the Kramers–Kronig relation. The converged results are compared with other theoretical and experimental ones and in agreement with the recent ab initio real-time-based calculation. The nonlinear optical coefficient comes from three parts: the pure interband contribution (P_inter), the modulation of interband terms by intraband terms (P_mod), and the intraband contribution (J_intra). For each part, the origin of enhanced peaks is explored by tracing the sum-over-states process. The interband contribution is found to be dramatically modulated by the intraband contribution.

Partial Element Equivalent Circuit-Based Transient Analysis of Graphene-Based Interconnects

This paper presents a three-dimensional time-domain formulation of the partial element equivalent circuit (PEEC) method to model graphene-based interconnects. The frequency-dependent electrical conductivity is incorporated into the time-domain solver by resorting to a fitting process. It is proved that if only the intraband contribution to the electrical conductivity is considered, no additional unknowns are required with respect to a standard PEEC solver. If also the interband is to be modeled, auxiliaries unknowns are introduced to reproduce the dynamic of the material. Numerical examples are presented to validate and demonstrate the capabilities and advantages of the proposed approach. The comparison with a finite element method implementation and also with the frequency-domain PEEC solver shows a satisfactory agreement and a good speedup with respect to the frequency-domain counterpart.

Graphene-plasmon polaritons: From fundamental properties to potential applications

With the unique possibilities for controlling light in nanoscale devices, graphene plasmonics has opened new perspectives to the nanophotonics community with potential applications in metamaterials, modulators, photodetectors, and sensors. This paper briefly reviews the recent exciting progress in graphene plasmonics. We begin with a general description for optical properties of graphene, particularly focusing on the dispersion of graphene-plasmon polaritons. The dispersion relation of graphene-plasmon polaritons of spatially extended graphene is expressed in terms of the local response limit with intraband contribution. With this theoretical foundation of graphene-plasmon polaritons, we then discuss recent exciting progress, paying specific attention to the following topics: excitation of graphene plasmon polaritons, electron-phonon interactions in graphene on polar substrates, and tunable graphene plasmonics with applications in modulators and sensors. Finally, we seek to address some of the apparent challenges and promising perspectives of graphene plasmonics.

Graphene Based Surface Plasmon Polariton Modulator Controlled by Ferroelectric Domains in Lithium Niobate.

We proposed a ferroelectric domain controlled graphene based surface plasmon polariton modulator. Ferroelectricity-induced electronic and optical property tuning of graphene by domain in lithium niobate was theoretically investigated considering both interband and intraband contributions of surface conductivity. With the corrected Sellmeier equation of lithium niobate, the propagation of transverse magnetic mode surface plasmon polaritons in an air/graphene/lithium niobate structure was studied when monolayer graphene was tuned by down polarization direction ferroelectric domain with different polarization levels. The length of the ferroelectric domain was optimized to be 90 nm for a wavelength of 5.0 μm with signal extinction per unit 14.7 dB/μm, modulation depth 474.1 dB/μm and figure of merit 32.5. This work may promote the study of highly efficient modulators and other ultra-compact nonvolatile electronic and photonic devices in which two-dimensional materials and ferroelectric materials are combined.