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We show that if the solutions to the (2+1)-dimensional massless Dirac equation for a given 1D potential are known, then they can be used to obtain the eigenvalues and eigenfunctions for the same potential, orientated at an arbitrary angle, in a tilted anisotropic 2D Dirac material. This simple set of transformations enables all the exact and quasi-exact solutions associated with 1D quantum wells in graphene to be applied to the confinement problem in tilted Dirac materials such as borophene. We also show that smooth electron waveguides in tilted Dirac materials can be used to manipulate the degree of valley polarization of quasiparticles travelling along a particular direction of the channel. We examine the particular case of the hyperbolic secant potential to model realistic top-gated structures for valleytronic applications., 12 pages, 5 figures and 1 Supplementary Material

The double-well problem for the two-dimensional Dirac equation is solved for a family of quasi-one-dimensional potentials in terms of confluent Heun functions. We demonstrate that for a double well separated by a barrier, both the energy level splitting associated with the wavefunction overlap of well states, and the gap size of the avoided crossings associated with well and barrier state repulsion, can be controlled via the parameters of the potential. The transitions between the two states comprising a doublet, as well as transitions across the pseudo-gaps are strongly allowed, highly anisotropic, and for realistic graphene devices can be tuned to fall within the highly desirable terahertz frequency range., 26 pages, 6 figures

We use the robust nearest-neighbour tight-binding approximation to study on the same footing interband dipole transitions in narrow-bandgap carbon nanotubes and graphene nanoribbons. It is demonstrated that curvature effects in metallic single-walled carbon nanotubes and edge effects in gapless graphene nanoribbons not only open up bang gaps, which typically correspond to THz frequencies, but also result in a giant enhancement of the probability of optical transitions across these gaps. Moreover, the matrix element of the velocity operator for these transitions has a universal value (equal to the Fermi velocity in graphene) when the photon energy coincides with the band-gap energy. Upon increasing the excitation energy, the transition matrix element first rapidly decreases (for photon energies remaining in the THz range but exceeding two band gap energies it is reduced by three orders of magnitude), and thereafter it starts to increase proportionally to the photon frequency. A similar effect occurs in an armchair carbon nanotube with a band gap opened and controlled by a magnetic field applied along the nanotube axis. There is a direct correspondence between armchair graphene nanoribbons and single-walled zigzag carbon nanotubes. The described sharp photon-energy dependence of the transition matrix element together with the van Hove singularity at the band gap edge of the considered quasi-one-dimensional systems make them promising candidates for active elements of coherent THz radiation emitters. The effect of Pauli blocking of low-energy interband transitions caused by residual doping can be suppressed by creating a population inversion using high-frequency (optical) excitation., 22 pages, 6 figures

We outline a range of proposals on using non-simply-connected and helical nanostructures for terahertz device implementations. We show that an Aharonov-Bohm quantum ring system and a double-gated quantum ring system both permit control over the polarization properties of the associated terahertz radiation. In addition, we review the superlattice properties of a nanohelix in external electric fields, which reveals negative differential conductance and photogalvanic effects. We present several schemes utilizing carbon nanotubes, including population inversion proposals for both quasi-metallic nanotubes (via generation of optically active hot electrons by an electric field), and metallic nanotubes (by optical excitation across a magnetic field-induced band gap).

Interband dipole transitions are calculated in quasi-metallic single-walled carbon nan-otubes and armchair graphene nanoribbons. It is shown that the curvature effects for tubes and the edge effects for ribbons result not only in a small band gap opening, corresponding to THz frequencies, but also in a significant enhancement of the transition probability rate across the band gap. This makes these nanostructures perspective canditates for sources and detectors of THz radiation.

Analytic solutions of the quantum relativistic two-body problem are obtained for an interaction potential modeled as a one-dimensional smooth square well. Both stationary and moving pairs are considered and the limit of the {\delta}-function interaction is studied in depth. Our result can be utilized for understanding excitonic states in narrow-gap carbon nanotubes. We also show the existence of bound states within the gap for a pair of particles of the same charge., Comment: 14 pages and 6 figures

The results of the theoretical study of interband THz transitions in narrow-gap carbon nanotubes and graphene nanoribbons are reported. We consider dipole transitions across magnetically-induced gaps in armchair nanotubes, curvature-induced gaps in quasi-metallic nanotubes and edge-effect induced gaps in armchair nanoribbons. A giant enhancement of the transition matrix elements is discovered for all three types of nanostructures.

The diverse applications of terahertz radiation and its importance to fundamental science makes finding ways to generate, manipulate, and detect terahertz radiation one of the key areas of modern applied physics. One approach is to utilize carbon nanomaterials, in particular, single-wall carbon nanotubes and graphene. Their novel optical and electronic properties offer much promise to the field of terahertz science and technology. This article describes the past, current, and future of the terahertz science and technology of carbon nanotubes and graphene. We will review fundamental studies such as terahertz dynamic conductivity, terahertz nonlinearities, and ultrafast carrier dynamics as well as terahertz applications such as terahertz sources, detectors, modulators, antennas, and polarizers., 29 Pages, 11 figures

We show, via a solution of the quasi-one-dimensional two-body problem applied to a Dirac system, that excitonic effects suppress the one-dimensional Van Hove singularity in single-walled carbon nanotubes with narrow band gaps.

We analyze bound modes of two-dimensional massless Dirac fermions confined within a hyperbolic secant potential, which provides a good fit for potential profiles of existing top-gated graphene structures. We show that bound states of both positive and negative energies exist in the energy spectrum and that there is a threshold value of the characteristic potential strength for which the first mode appears. Analytical solutions are presented in several limited cases and supercriticality is discussed., 8 pages, 4 figures. Figure 2 removed, 2 additional figures and references added

We calculate the exciton binding energy in single-walled carbon nanotubes with narrow band gaps, accounting for the quasi-relativistic dispersion of electrons and holes. Exact analytical solutions of the quantum relativistic two-body problem are obtain for several limiting cases. We show that the binding energy scales with the band gap, and conclude on the basis of the data available for semiconductor nanotubes that there is no transition to an excitonic insulator in quasi-metallic nanotubes and that their THz applications are feasible., Comment: 11 pages, 3 figures. Several references and an additional appendix added

We present exact analytical solutions for the zero-energy modes of two-dimensional massless Dirac fermions fully confined within a smooth one-dimensional potential V(x)= - {\alpha}/cosh({\beta}x), which provides a good fit for potential profiles of existing top-gated graphene structures. We show that there is a threshold value of the characteristic potential strength {\alpha}/{\beta} for which the first mode appears, in striking contrast to the non-relativistic case. A simple relationship between the characteristic strength and the number of modes within the potential is found. An experimental setup is proposed for the observation of these modes. The proposed geometry could be utilized in future graphene-based devices with high on/off current ratios., Comment: 8 Pages including 4 figures, an appendix and two additional figures added, references updated

We propose and justify several schemes utilizing the unique electronic properties of different types of carbon nanotubes for a broad range of applications to terahertz (THz) optoelectronics, including THz generation by hot electrons in quasi-metallic nanotubes, frequency multiplication in chiral-nanotube-based superlattices controlled by a transverse electric field, and tunable THz radiation detection and optically-pumped emission in armchair nanotubes in a strong magnetic field.

We study interbanddipole transitions across curvature-induced narrow gaps in quasi-metallic single-walled carbon nanotubes. The curvature effects not only open a gap in the nanotube energy spectrum but also allow optical transitions, which happen to be in the highly- desired terahertz frequency range. Applying a magnetic field along the nanotube axis allows one to tune the frequency peaks in the spectral density of absorption.

We show analytically that the ability of Dirac materials to localize an electron in both a barrier and a well can be utilized to open a pseudo-gap in graphene's spectrum. By using narrow top-gates as guiding potentials, we demonstrate that graphene bipolar waveguides can create a non-monotonous one-dimensional dispersion along the electron waveguide, whose electrostatically controllable pseudo-band-gap is associated with strong terahertz transitions in a narrow frequency range., 21 pages and 3 figures

One of the most distinctive features of graphene is its huge inter-Landau-level splitting in experimentally attainable magnetic fields which results in the room-temperature quantum Hall effect. In this paper we calculate the longitudinal conductivity induced by two-phonon scattering in graphene in a quantizing magnetic field at elevated temperatures. It is concluded that the purely phonon-induced scattering, negligible for conventional semiconductor heterostructures under quantum Hall conditions, becomes comparable to the disorder-induced contribution to the dissipative conductivity of graphene in the quantum Hall regime., Comment: 8 pages, 1 figure, references added and discussed

Quasi-exact solutions to the quantum relativistic two-body problem are obtained for a one-dimensional Woods-Saxon-like potential. The quantised positive energy spectrum is obtained in the square well potential limit in terms of a set of simple transcendental equations. This potential is used to calculate excitonic states in narrow-gap single-walled carbon nanotubes and the binding energy is shown to scale with the band gap.