Optical Properties of Graphene

The following sections are included:

• Preface
• Contents

In this Introductory Tutorial, we first present a basic overview of optical processes and optical transitions in graphene and compare them to optical transitions in other material systems, such as semiconductor quantum wells. We discuss basic aspects of the bandstructure of graphene, both close to the Dirac point and away from that point. This is followed by a text-book level introduction to the tight-binding model and its application to graphene. Analytical expressions for the Bloch wave functions and the electronic bandstructure close to the Dirac point will be derived. The Bloch wave functions will then be used as an input to the calculation of optical matrix elements. The derivation of the interband polarization will be performed within the frame work of many-particle systems described in second quantization. Finally, the linear optical absorption spectrum of graphene will be derived, using a transfer matrix method for the approximate solution of Maxwell’s equations.

This chapter summarizes recent theoretical work to determine the groundstate and the linear optical properties of quasi-two-dimensional materials with hexagonal lattice symmetry. The main ingredients of the fully relativistic tight-binding model analysis for graphene and transition metal dichalcogenites are summarized. Dirac-Bloch equations are derived and solved in the regimes of weak and strong Coulomb coupling. Whereas single-layer graphene always seems to be in the weak Coulomb-coupling regime, realistic parameter analysis of WSe₂ and WS₂ identifies these materials to be in the regime of strong Coulomb coupling. Consequently, they are predicted to exhibit an excitonic insulator groundstate with optically active p-excitons. On this basis, a unified understanding and excellent agreement of the theoretical predictions with a large variety of systematic experiments is obtained.

Raman spectroscopy of graphene is reviewed from a theoretical perspective. After an introduction of the building blocks (electronic band structure, phonon dispersion, electron-phonon interaction, electron-light coupling), Raman intensities are calculated using time-dependent perturbation theory. The analysis of the contributing terms allows for an intuitive understanding of the Raman peak positions and intensities. The Raman spectrum of pure graphene only displays two principle peaks. Yet, their variation as a function of internal and external parameters and the occurrence of secondary, defect-related peaks, conveys a lot of information about the system. Thus, Raman spectroscopy is used routinely to analyze layer number, defects, doping and strain of graphene samples. At the same time, it is an intriguing playground to study the optical properties of graphene.

We present a theoretical study on the ultrafast carrier dynamics in graphene including direct comparison to recent experimental pump-probe results. Based on graphene Bloch equations, we resolve the Coulomb- and phonon-driven dynamics of optically excited carriers in time, momentum, and angle. Our calculations reveal how the optically generated anisotropic non-equilibrium carrier distribution relaxes towards its initial thermal distribution. The main steps of this dynamics include a Coulomb-dominated ultrafast carrier thermalization on a timescale of tens of femtoseconds, a phonon-driven scattering in all momentum directions giving rise to an isotropic distribution in the first hundred femtosecond, and a slower phonon-assisted carrier cooling occurring on a picosecond timescale. We reveal the characteristic features of the carrier dynamics in the near infrared as well as below the optical phonon energy and in the terahertz region close to the Dirac point. In particular, we emphasize the crucial role of Auger scattering processes that give rise to a technologically promising carrier multiplication in graphene.

The third order optical nonlinearity of graphene is theoretically investigated using the semiconductor Bloch equations under the independent particle approximation. The electronic states are described by a two band tight binding model, and the scattering is described by phenomenological parameters. For nominally weak optical fields, perturbation theory at zero temperature gives analytic expressions for the third order conductivities under the linear dispersion approximation around the Dirac points. These expressions show many singularities and divergences, which indicates a complex nonlinear optical behavior of graphene. We illustrate the dependence of several nonlinear optical effects, such as third harmonic generation, Kerr effects and two photon absorption, and parametric frequency conversion, on the relaxation parameters and temperature. The analytic results from perturbation theory, with the approximation employed, are verified by numerical solutions. The latter also allow us to discuss the effects of saturation on the optical nonlinearity. The results of the independent particle theory presented here are smaller than the values extracted from experiments by at least two orders of magnitude.

This chapter introduces recent experimental studies on nonlinear optical properties of graphene, including third harmonic generation and saturable absorption. Basic principles, experimental designs, and examples of experimental results are presented.

Owing to the recent developments of the intense terahertz radiation technology, the optical response of graphene under pulsed and continuous wave intense terahertz fields are of great interest. Due to the exotic band structure, the dynamics of carriers excited by photon of meV energy in graphene are significantly different from that in conventional semiconductors. Strong nonlinear effects are found from experimental observations and theoretical calculations. The interplay between the interband transition and the intraband transition leads to novel transport and optical properties. In the presence of continuous wave terahertz fields, the electronic structure of graphene is found to be modified due to the appearance of the Floquet states. A dynamical energy gap near the Dirac points can be opened and controlled by the terahertz fields.

The peculiar electronic band structure makes graphene highly interesting for nonlinear terahertz (THz) spectroscopy. Interband transition dipoles around the K and K’ points are huge in the THz frequency range, leading to highly nonlinear effects at moderate amplitudes of THz fields. Moreover, intraband motion of electrons is intrinsically nonlinear due to the cone-like bandstructure. We discuss results from recent ultrafast THz experiments in the nonperturbative regime of light-matter interaction. Different methods including two-dimensional THz spectroscopy are applied. The generation of the third and fifth harmonic of the THz fundamental is observed in propagation experiments. Pump-probe studies with different photon energies of the pump pulses reveal induced THz absorption which decays on a time scale of a few picoseconds. THz photon echo signals are absent. A theoretical description of the experimental results requires to include combined interband/intraband transitions and currents while the decay of the pumpprobe transients is a hallmark of radiative coupling in multilayer graphene.

Engineering electromagnetic waves has been central to the development of modern optoelectronic devices. Global networks through the longrange communication, internet, and data transfer were enabled by the controlling the associated light. Although actively pursued electromagnetic wavelength for various optoelectronic applications is widely spanned from ultra-violet to far-infrared wavelength, it has been particularly difficult to manipulate the terahertz (THz) frequencies, which lie between microwave and infrared region. For the last two decades, worldwide researchers have found how to generate and detect THz waves, and after which optical technologies including THz spectroscopy, THz imaging, and THz sensing have been greatly advanced. Nonetheless, the small energy of THz photon (∼ 4.14 meV at 1 THz) and the optical characteristics of long wavelength (300 µm at 1 THz) have still been imposing some technical limits on the THz applications: sensitive, and yet optically thin structure is required to compensate the energy loss of THz propagation. Recent discoveries of graphene have witnessed many potential for manipulating THz-related optical phenomena. This is especially because graphene provides both high sensitivity and atomically-thin dimension with a high THz absorption feature. Another technical advance of changing some characteristics of THz waves is to employ artificial structures, called metamaterials, which have also been extensively investigated to manage both absorption and transmission characteristics. By employing graphene and metamaterials, we have developed very effective methods in manipulating the THz spectra. Active modulation of THz waves was derived by layered composition of graphene metamaterials, where the electrons are migrated from graphene to metamaterials and vice versa. Emphasis should be made on the fact that the electrostatic and optical to the graphene is at the heart of the THz wave control. In this Chapter, two representative works of the THz manipulation in graphene metamaterials are presented. By changing the structure of metamaterials, distinguished functionalities of THz optics, refractive index change, and nonlinear wave mixing are demonstrated.

In contrast to the previous chapters of this book, in the following sections the optical properties of graphene far away from the Dirac points will be treated. As has been shown earlier, at frequencies above the far-infrared region, the optical response of graphene is dominated by interband transitions between the valence and conduction band. The band structure of graphene exhibits a saddle point between two neighboring Dirac cones, so that the optical absorption will deviate from its constant value of πα for photon energies in the visible and ultraviolet regime…

This chapter focuses on ultrafast nonlinear spectroscopy at the ultraviolet M- or saddle-point in the electronic bandstructure of graphene where the position and dynamical evolution of its absorption peak are especially sensitive to electron-phonon interactions. Specifically, we explore how these absorption peak changes are caused by optically-induced modifications of the phonon temperature by way of several electron-phonon scattering processes. We present a detailed theoretical model for electron-phonon interactions based on the concept of deformation potentials. We also include a discussion of the phonon dispersion obtained from dynamical matrices. We derive the electronic self-energy to lowest order in the electron-phonon interaction Hamiltonian, then use it to calculate the interband susceptibility and the differential transmission spectrum. Using literature values for deformation potentials, we find good agreement between theory and experiment, indicating that this formalism provides a good understanding of the microscopic electron-phonon coupling processes that renormalize the electronic transitions close to the M-point and produce the observed differential transmission spectra.

We describe experiments where voltage controlled graphene supercapacitor devices are used as saturable absorbers to initiate femtosecond pulses from tunable solid-state lasers. The device architectures allow the control of the Fermi level and hence the overall insertion loss of these devices at low operating voltages on the order of a few volts. Hence, they can be readily incorporated into the resonator of low-gain solid-state lasers. After the background sections on laser fundamentals and ultrafast pulse generation, the second part of the Chapter focuses on the spectroscopic characteristics of graphene and use of graphene supercapacitor devices as saturable absorbers to initiate femtosecond pulses. We present comparative, experimental data taken with a multi-pass cavity, Cr⁴⁺:forsterite laser, where two different graphene-based supercapacitor device architectures were used to generate sub-100-fs pulses near 1250 nm.

In this chapter, we summarize the recent progress on graphene based optical modulators. Ability to control density of high mobility electrons on large area graphene surface enables realization of new type of electrooptical modulators in optoelectronics. Due to the low electronic density of states, accumulation of charges on graphene significantly shifts the Fermi energy up to 1 eV giving rise to profound optical effects in the infrared and visible spectra. On the other hand, graphene operates as a tunable Drude metal in long wavelengths such as THz and microwave. This unique broadband activity of graphene has stimulated a great deal of interest in graphene community due to its potential use in new optoelectronic devices. After discussing the electrically tunable optical properties of graphene, we highlight the key achievements in the field.

This chapter evaluates the potential of current state-of-the-art doped graphene as a replacement of commercially available Transparent Conducting Oxides (TCOs) in display and photovoltaic applications. The requirements of transparent conductors, namely optical transmittance and sheet resistance, are elaborated. The optical properties of the doped graphene and two most commonly used TCOs, Indium Tin Oxide (ITO) and Fluorine-doped Tin oxide (FTO) are illustrated…

The following sections are included:

• Color Plates
• Index