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Alexandre Mayer

Summary of my research activities

Here follows a summary of my research activities since the beginning of my PhD in 1995. The different thematics are presented in a chronological order. The story gives the context of all my publications.

Feel free to skip to the end for the most interesting parts. :-)

1. The Fresnel projection microscope

The Fresnel projection microscope consists of an electronic point source (a field-emission nanotip), which is approached to a sample by a distance sufficiently small to produce Fresnel diffraction patterns on a distant screen. The technique thus provides images that are correlated with a geometric projection of the sample from the source, with typical magnifications between 105 and 106 and a resolution limit that can be as small as 0.5 nm.

To simulate this instrument, I developed a complete methodology [5, 12] by transfer matrices [1, 2] and Green's functions [3] to solve the corresponding Schrödinger equation in a full-order approach. The technique provides total-energy distributions and current densities on the anode and on the projection screen, by taking account of the whole three-dimensional potential-energy distribution between the cathode and the anode and by considering the contribution of every incident state in the metal that supports the field-emission tip. My key contribution to the transfer-matrix methodology was the control of the numerical instabilities [4] and the reduction of the computational requirements (essentially by taking advantage of Cn symmetries [11]). Another methodological aspect of my work was the extension of the transfer-matrix formalism to non-square matrices [8]. This extension is useful when the potential energy in the region of incidence is different from that in the region of transmission. The technique in its final form enables the potential energy to have an imaginary component (describing absorption processes) [9] and can take account of magnetic-field distributions [10].

This methodology was used to study of the field-emission properties of nanotips [1, 8, 6] and simulate the observation of carbon fibers [5, 9, 7] and a C60 molecule [13]. This methodology was able to reproduce the Fraunhofer/Fresnel observation modes, describe the "sucking-in" effects that result from the fields excerted by the sample, study the observability of atomic corrugations in the sample and predict effects associated with absorption and stationary states in the sample.

For this part of my work, collaborations with Michel Devel and Christophe Adessi (Besançon, France), Vu Thien Binh (Lyon, France), Pavel Dorozhkin (Moscow, Russia) and Hitoshi Nejoh (Tsukuba, Japan) were established.

2. Photon-stimulated field emission

The transfer-matrix methodology was extended here to include an oscillating barrier [14, 15, 19, 18]. This additional contribution to the potential energy is representative of an electromagnetic radiation and gives rise to the absorption or emission of photons in the simulations. The technique can then simulate photon-stimulated field emission, which is the intermediate general situation between usual field emission (no radiation field) and the pure photoelectric effect (no extraction field). The methodology takes account of the three-dimensional aspects of the static and oscillating parts of the potential energy and describes the quanta exchanges between the electrons and the radiation field.

This technique was used to study photon-stimulated field emission from conical [14] or elliptical [15, 19, 18] tungsten tips. The effect of a dielectric coating was also considered [18]. The photon-stimulated part of the emission is proportional to the power-flux density of the radiation. The shape and extension of the corresponding contribution in the total-energy distribution depend on how the photon-excited electrons can escape the emitter (by a tunneling effect if the photon energy is smaller than the height of the potential barrier or by ballistic motion in the other case). The current enhancement resulting from the oscillating field shows a maximum when represented as a function of the radiation wavelength. This enhanced emission is due to the photon-excited electrons having stationary states above the surface barrier (thus enhancing the coupling with electrons at their intrinsic energy level).

This part of my work was achieved in collaboration with Mark Hagmann (Miami, Florida) and Marwan Mousa (Al-Karak, Jordan).

3. Inverse electronic scattering

For the purpose of improving the extraction of information from images obtained by electronic projection microscopy and enabling a "tomographic" reconstruction if several positions/orientations of the sample are considered, a specific reconstruction technique was developed.

A first full-order reconstruction scheme was developed within the Green's-functions formalism [16]. The idea is to solve iteratively a set of linear equations that relate the screen intensities to the potential energy in the sample (by a singular values decomposition to get the "best least-squares solutions"). After the reconstruction of the wave function in the sample and on the screen, a final reconstruction of the sample potential energy is provided. When the sample has a complex three-dimensional structure, the technique can take account of several projections (associated with different sample orientations) [17]. This approach was demonstrated to be efficient and stable against random noise.

In order to reduce the storage space requirement associated with a three-dimensional representation of the sample, this reconstruction scheme was reformulated within the Fresnel-Kirchhoff theory, which describes the sample as a two-dimensional mask [20]. This formalism is well suited for thick samples since the screen intensities then come essentially from the electrons that travel outside the sample (those meeting the sample being absorbed or reflected). In these conditions, this second approach turns out to provide better reconstructions than those obtained within the Green's-functions formalism. An extension was developed to find automatically the parameters of the incident wave [22], i.e. its amplitude and the source-sample distance. The determination of this distance is made more efficient by enabling the procedure to treat simultaneously the two diffraction patterns obtained before and after a lateral shift (of known amplitude) of the sample.

4. Field emission from carbon nanotubes

I applied here the transfer-matrix methodology to the simulation of field emission from carbon nanotubes [21, 23]. For these calculations, the carbon atoms are represented using the Bachelet et al. pseudopotential. Band-structure effects are obtained by repeating periodically a basic unit of the nanotubes in a region preceding that containing the fields. This technique enables the computed density of states to reproduce the band-gap of semiconducting nanotubes and the constant plateau of the metallic ones. The band structure of the nanotubes can be extracted from the transfer matrices associated with a single unit of the structure [25].

The nanotubes considered were essentially the metallic (5,5) [25, 24] and the semiconducting (10,0) [26]. As expected, the field-emission properties of the metallic (5,5) are better than those of the semiconducting (10,0). The total-energy distribution of the field-emitted electrons exhibit peaks, which are related to standing waves in the body of the structure, van Hove singularities in the density of states or localized states at the apex of closed structures. The emission rate of open single-wall nanotubes is higher than that of closed ones. An hydrogen-saturation of the dangling bonds of the open (5,5) nanotube improves the emission by two orders of magnitude [24].

Due to its poorer field-emission properties, the current enhancement resulting from a photonic stimulation is more pronounced for the (10,0) nanotube than for the (5,5) [27, 29]. The efficiency of the photon-stimulation process depends on the power flux density of the radiation, on how the photon energy compares with the band-gap and the height of the surface barrier and on final-state effects. A maximal current enhancement is found at a given photon energy, this observation meeting that reported for tungsten emitters [19].

By comparing the field-emission properties of the (5,5), (10,10) and (15,15) single-wall structures with those of the multi-wall (5,5)@(10,10)@(15,15), it was found that multi-wall structures are better emitters than their single-wall components (due to field penetration being more pronounced in multi-wall structures) [28]. I also investigated the field-emission properties of bundles of carbon nanotubes [36].

Finally, by comparing the field-emission properties of a single carbon atom, a half C60 molecule on a flat metallic substrate or on top of a metallic cylinder, and a closed (5,5) nanotube, it was shown that the field amplification factor, as deduced from a Fowler-Nordheim analysis of the I-V characteristics, should not be interpreted literally or used as only factor of consideration for explaining the field-emission properties of nanometric emitters [30].

This part of my work was achieved with Nicholas Miskovsky and Paul Cutler in the context of a postdoc at the Pennsylvania State University (University Park, Pennsylvania).

5. Transport properties of carbon nanotubes

Back from my postdoc, I started studying the transport properties of carbon nanotubes [31, 32]. The objective was to determine to what extend the layers of multiwall carbon nanotubes could be considered as independent for the conduction of current. I considered in this context multiwall carbon nanotubes with semiconducting or metallic layers and I put these nanotubes in conditions in which electrons would be injected in a single layer. The study aimed at determining the distance these electrons would cross before spreading significantly to neighboring layers [33].

For an appropriate modeling of the transport properties of carbon nanotubes, it was necessary to develop a pseudopotential for the representation of carbon atoms. This pseudopotential was build to reproduce the bands of pi electrons in isolated graphene sheets and in simple hexagonal graphite [38, 117]. A stable algorithm was developed in this context to compute band structures by the transfer-matrix methodology [34, 102]. My pseudopotential turned out to be surprisingly useful for the study of electronic transport in graphene and 2-D materials [39, 48, 82, 83, 108].

I acknowledge here collaboration with Philippe Lambin, Peter Vancso and Geza Mark.

6. Transport and field-emission properties of semiconducting materials

In a continuing collaboration with PennState, I studied transport and field emission properties of semiconducting materials. The methodology used in this context consists in solving iteratively the Poisson equation as well as the conservation equations for electrons and holes [35]. The objective of this study is to determine if cooling can be achieved by the circulation of currents in this type of materials. This is the case for example if the mean energy of the electrons that leave the semiconductor is larger than the mean energy of the electrons that replace them [37, 57, 58, 63, 68, 69, 77, 81, 85].

A result of this study was to determine the role of surface charges in the field emission from semiconductors [40]. There was indeed a controversy on that issue and some believed surface charges could simply make field emission from semiconductors impossible, because they would screen the external field completely. Others simply ignored these surface charges and assumed that the field in the semiconductor was just the external field divided by the dielectric constant. We found the right answer to that question.

I finally studied various systems that combine metallic and semiconducting materials [44, 45, 46]. I also developed a finite-difference method to compute the Green's function of periodic materials [47].

This part of my work is achieved in collaboration with Nicholas Miskovsky and Paul Cutler from the Pennsylvania State University (University Park, Pennsylvania) and Moon Chung from the University of Ulsan (Korea).

7. Polarization properties of carbon nanotubes, fullerenes, hydrocarbons and silver clusters

A model commonly used to compute molecular polarizabilities consists in associating with each atom of the structure considered a dipole. This model fails at describing charge transfers both (i) inside the structure considered and (ii) between this structure and a metallic substrate (this is however essential for the modeling of field emission). I extended this model by associating with each atom of the structure considered both a net electric charge and a dipole [42, 43]. This enabled the polarizability of metallic nanotubes and fullerenes to be calculated by taking account of these charge displacements.

This charge-dipole model for the representation of atomic structures was combined with a dielectric-function model for the representation of their surrounding. This enabled the study of the electrostatic forces that act on nanotubes placed in the vicinity of metallic protrusions [43]. The numerical stability of the charge-dipole model was finally improved by using normalized propagators for the description of the electrostatic interactions between the charges and the dipoles [49, 50, 51]. This model was used to compute the polarization properties of fullerenes with various shapes and defects [52, 61] or embedded in a composite material [60, 74]. It was also used to determine the inner shell charging of multiwall carbon nanotubes [55, 73].

The next step was to extend the model to structures that contain different types of atoms, like the alkanes, the alkenes and the aromatic molecules [53]. This work was immediately generalized to deal with the case in which the external field is oscillating [54]. The idea consists in relating the time variations of the atomic charges to the currents that flow through the bonds of the structure considered. This enables the frequency-dependent polarizability of these structures to be calculated. The agreement with reference data obtained using time-dependent density functional theory turns out to be excellent.

Compared to models that only use dipoles for the representation of materials, this charge-dipole interaction model provides electric fields that keep meaningful down to the atomic level. This motivated an application to silver clusters [64, 66]. The modeling of surface-enhanced Raman scattering for small molecules that are placed in the vicinity of silver clusters requires indeed that level of description. We found again an excellent agreement between the results of this charge-dipole interaction model and reference data achieved using time-dependent density functional theory.

I finally wrote a user-friendly software (DCDA) that implements this model.

I acknowledge collaborations for this part of my work with Per-Olof Astrand from the University of Trondheim (Norway) and George Schatz from Northwestern University (Evanston, Illinois).

8. Theoretical modeling of field electron emission

Some of my recent activities have focussed on checking the validity of the various approximations that support the standard Fowler-Nordheim equation. This equation provides the electronic current emitted by metals when subject to external electric fields. This equation accounts for the electric field and the work function. The Murphy-Good equation includes in addition to the standard Fowler-Nordheim equation the effects of the temperature. These equations however discard any influence of the Fermi energy, which is related to the density of electrons in the emitter.

It was appropriate at this point of my work to confront the results provided by standard equations in the theory of field emission with those provided by my transfer-matrix methodology [70]. This confrontation lead to the determination of a correction factor to use with the standard Fowler-Nordheim equation in order to get the exact result [71, 75]. I also investigated the accuracy of Simmons' theory for the current density obtained in flat metal-vacuum-metal junctions [97, 100].

I developed in this context a Field Emission calculator that provides exact values for the field emission achieved from a metal.

I acknowledge here Richard Forbes (University of Surrey) for always constructive suggestions.

9. Rectification of infrared and optical frequencies using geometrically asymmetric metal-vacuum-metal junctions

In the context of a collaboration with PennState (Brock Weiss, Nicholas Miskovsky, Paul Cutler), I studied the rectification properties of metal-vacuum-metal junctions in which one of the metals is essentially flat while the other is extended by a sharp tip. Because of this geometrical asymmetry, these junctions act as rectifiers when subject to oscillating potentials. This rectification holds up to the point where the frequency becomes so high that the electric field changes sign before electrons have been able to cross the junction. With a gap distance between the two metals of the order of one nanometer, this device can however work up to frequencies in the visible domain [62, 79, 80, 84, 94].

In order to achieve a quantum mechanical modeling of these junctions, by taking account of three-dimensional aspects of the problem, I used a transfer-matrix methodology [56]. This methodology was extended in order to deal with the time-dependence of the potential barrier [59, 65, 67]. This enables our approach to this problem to be exact and one can confront the quantum-mechanical results provided by the transfer-matrix technique with those provided by classical models of these junctions [76, 78]. This confrontation reveals the limits of classical concepts. It also provides a convenient framework to analyze the results provided by the transfer-matrix technique.

I keep collaborating with Peter Lerner (Device Consultants) on subjects related to the coherence of light [115, 118].

10. Genetic algorithms & Machine Learning

Most problems in physics involve at some point the optimization of parameters in order to get the best agreement with experimental data or in order to optimize the efficiency of the system considered. Genetic algorithms constitute a smart approach to this problem. The idea consists in working with a population of individuals. Each individual is representative of a given set of physical parameters and therefore of a given value for the objective function we seek at optimizing. The initial population usually consists of random individuals. The best individuals are then selected. They generate new individuals for the next generation. Mutations in the coding of parameters are finally introduced. When repeated from generation to generation, this evolutionary strategy makes it possible to determine the global optimum of a problem [95, 96, 99].

I applied genetic algorithms for the determination of a local pseudopotential representative of monovacancies in graphene, for the optimization of light-emitting diodes [87, 88, 91], for the optimization of photovoltaic solar cells [86, 90, 92, 93, 104, 105, 112], for the optimization of solar thermal collectors [87, 89] and for the optimization of metamaterial superabsorbers [98, 103, 110]. A multi-objective genetic algorithm was also developed. My codes are written in Fortran 90, MATLAB and Python. They are easily adaptable to any problem. The Fortran 90 code was parallelized using OpenMP. A special version for the CECI and Tier-1 supercalculators was also developed. It enables individual jobs to be responsible for each evaluation of the fitness. This multi-agent programming model enables a very efficient parallelization of the algorithm.

From genetic algorithms to neural networks and artificial intelligence, there was only one step ! I take regularly on-line courses on these topics and machine learning techniques are now part of my portfolio. Training agents to play Atari games or transfering the style of Vincent van Gogh on my camera pictures is of course a lot of fun. These techniques are also useful for problems in physics. I contributed to the identification of historical parchments [101] and the conservation of cultural heritage [116] with machine learning techniques. I interact on a regular basis now with the team of Benoit Frenay (Faculty of Computer Science, UNamur) on machine learning problems [106]. My Master & PhD students develop smart evolutionary algorithms [107, 111]. A beautiful extension of my PhD work is to apply the Fourier-Bessel representations that I used for solving Schrödinger's equation to rotation-invariant image classification [109] ! I teach finally a class on deep neural networks to physicists.

Press release: Where Life Meets Light: Bio-Inspired Photonics, by Valerie C. Coffey (Optics & Photonics News, April 2015)

Last update: October 2023.