The wave impedance of an atomically thin crystal

I propose an expression for the electromagnetic wave impedance of a two-dimensional atomic crystal, and I deduce the Fresnel coefficients in terms of this quantity. It is widely known that a two-dimensional crystal can absorb light, if its conductivity is different from zero. It is less emphasized that they can also store a certain amount of electromagnetic energy. The concept of impedance is useful to quantify this point

Role of the radiation-reaction electric field in the optical response of two-dimensional crystals

A classical theory of a radiating two-dimensional crystal is proposed and an expression for the radiative-reaction electric field is derived. This field plays an essential role in connecting the microscopic electromagnetic fields acting on each dipole of the crystal to the macroscopic one, via the boundary conditions for the system. The expression of the radiative-reaction electric field coincides with the macroscopic electric field radiating from the crystal and, summed to the incident electric field, generates the total macroscopic electric field.Comment: Two-dimensional crystal, metasurface, local field, radiative-reaction, Fresnel, boundary conditio

Picosecond cathodoluminescence

We have developed an original time resolved cathodoluminescence (TRCL) set-up with temporal performances similar to those of conventional time resolved optical techniques, keeping the capability to get structural information through the secondary electron mode of an electron microscope (SEM). This system allows for performing ultrafast spectroscopy on nanostructures leading insight into phenomena like transport, carrier capture and carrier recombination. A traditional TRCL is based on the use of a SEM column equipped with an electrostatic beam blanking unit and a photon counting system for the detection. Electron pulses of rise and decay times ~ 200 ps, varying width from 1ns to 1µs and a repetition rate up to 1 MHz are used for the excitation. Due to the long duration of the pulse compared to characteristic relaxation times, the system under investigation is set in a quasi-equilibrium state before the study of the luminescence decay. This leads to a non-straightforward interpretation of the temporal luminescence profile; moreover resolution is limited to 250 ps. In order to overcome such limitations, we have replaced the thermionic electron gun of a SEM with a home-made ultrafast electron gun. Femtosecond mode-locked laser pulses are focused on a metal photocathode to create electron bunches. An extraction electrode and an anode accelerate photoelectrons up to 30 kV. The electron lens systems of the microscope column focus the photoelectron beam on the sample. Luminescence emitted as a consequence of the probe beam excitation is spectrally analyzed with a monochromator and it is then collected with a streak camera for temporal analysis. This set up (picoCL) has achieved unprecedented combined spatial and temporal resolutions. The test for the spatial resolution is carried out on gold particles grown on a carbon substrate: a sample currently used to test commercial SEMs resolution. We prove that the ultrafast electron gun brightness is high enough to focus electrons on a probe diameter of 50 nm still having enough current to obtain a secondary electron image of the sample. The temporal width Δt of the electron pulses is measured by an indirect method. We compare the time resolved photoluminescence response (200 fs laser pulses excitation) of a semiconductor sample with that obtained with the picoCL. A Δt (FWHM) = 12 ± 1 ps is found. As a first study with the picoCL we investigate the time resolved luminescence emission from quantum structures located in InGaAs/AlGaAs tetrahedral pyramids. An In0.10Ga0.90As quantum dot (QD) formed just below the top of the pyramid is connected to several types of low-dimensional barriers: InGaAs quantum wires (QWRs) on the edges of the pyramid, InGaAs quantum wells (QWs) on the (111)A facets and segregated AlGaAs vertical quantum wire (VQWR) and AlGaAs vertical quantum wells (VQWs) formed at the centre and at the pyramid edges. PicoCL is successful in identifying the spectral features of the different nanostructures. Indeed the rise and decay times of their luminescence emissions vary strongly with the location of the excitation point on the pyramid. The intricate and complex carrier transport among the different quantum structures is enlightened: our results suggest the scenario that after excitation on the facet or on the edge of the pyramid, carriers diffuse towards the central structures (QD and VQWR) via the QWR. According to these findings we model the carrier diffusion along the QWR and fit our experimental data. A carrier mobility of 1300 cm2/Vs is found

Measurement of the surface susceptibility and the surface conductivity of atomically thin MoS2\rm MoS_2 by spectroscopic ellipsometry

We show how to correctly extract from the ellipsometric data the surface susceptibility and the surface conductivity that describe the optical properties of monolayer $\rm MoS_2$. Theoretically, these parameters stem from modelling a single-layer two-dimensional crystal as a surface current, a truly two-dimensional model. Currently experimental practice is to consider this model equivalent to a homogeneous slab with an effective thickness given by the interlayer spacing of the exfoliating bulk material. We prove that the error in the evaluation of the surface susceptibility of monolayer $\rm MoS_2$, owing to the use of the slab model, is at least 10% or greater, a significant discrepancy in the determination of the optical properties of this material.Comment: Keywords: Ellipsometry, graphene, MoS2, two dimensional crystals, optical contrast, absorption, transition metal dichalcogenide monolayer

Optical beam shifts in graphene and single-layer boron-nitride

Optical beam shifts from a free-standing two-dimensional atomic crystal are investigated. In contrast to a three-dimensional crystal the magnitude of the Goos-H$\rm \ddot{a}$nchen shift depends on the surface susceptibility of the crystal and not on the wavelength of the incident light beam. The surface conductivity of the atomically thin crystal is less important in this context because it enters in the expression of the shifts only as a second order parameter. In analogy to a three-dimensional crystal the magnitudes of the Imbert-Fedorov shift and of the angular shifts depend respectively on the wavelength and on the square of the beam angular aperture.Comment: Keywords: Goos Hanchen, Imbert Fedorov, graphene, monolayer, Boron Nitride, spin Hall effect of light, metasurface

Optical response of a bilayer crystal

We extend the recently developed classical theory for the optical response of a single-layer crystal to bilayers. We account for the interaction between the two atomic planes and the multiple reflections inside the crystals. We show how to define a global susceptibility meaningful for the bilayer crystal and how its expression varies compared to the single-layer case. We compute both the local and the macroscopic fields which allow us for a direct comparison with experimental data.Comment: Keywords: 2D Crystals, metasurfaces, optics, susceptibility, local fields, radiation-reaction fiel

Role of spatial coherence in the goos-hanchen shift

Amongst the various nonspecular phenomena that affect the reflection of a beam of light by a smooth surface, the Goos-Hanchen (GH) shift is certainly the most investigated one. Experimental studies of this effect have been performed so far only with fully spatial coherent beams. �� 2013 IEEE