Transfer-matrix and Green-function quantum-mechanical theory of electronic field emission applied to the simulation of diffraction by a carbon fiber in the Fresnel projection microscope

J. Phys.: Condens. Matter

2010

We study the transmission coefficient of one-dimensional barriers that are relevant to field-emission problems. We compare, in particular, the results provided by the simple Jeffreys-Wentzel-Kramers-Brillouin (JWKB) approximation, the continued-fraction technique and the transfer-matrix methodology for the electronic transmission through square, triangular and Schottky-Nordheim barriers (the Schottky-Nordheim barrier is often used in models of field emission from flat metals). For conditions that are typical of field emission (Fermi energy of 10 eV, work function of 4.5 eV and field strength of 5 V nm(-1)), it is shown that the simple JWKB approximation must be completed by an effective prefactor P(eff) in order to match the exact quantum-mechanical result. This prefactor takes typical values around 3.4 for square barriers, 1.8 for triangular barriers and 0.84 for the Schottky-Nordheim barrier. For fields F between 1 and 10 V nm(-1) and for work functions φ between 1 and 5 eV, the prefactor P(eff) to consider in the case of the Schottky-Nordheim barrier actually ranges between 0.28 and 0.98. This study hence demonstrates that the Fowler-Nordheim equation (in its standard form that accounts for the image interaction and that actually relies on the simple JWKB approximation) overestimates the current emitted from a flat metal by a factor that may be of the order of 2-3 for the conditions considered in this work. The study thus confirms Forbes's opinion that this prefactor should be reintegrated in field-emission theories.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A comparative study of the electron transmission through one-dimensional barriers relevant to field-emission problems

J. Phys.: Condens. Matter

2010

“…Our approach of the problem was based on both the transfer-matrix and Green's functions formalisms. [9][10][11][12][13][14] This approach enables direct-scattering simulations [15][16][17] with reduced storage requirements. However, due to the fact that the potential energy remains explicitly present in the Green's functions formulation, this latter will be used in this inverse scattering approach.…”

Section: Introductionmentioning

confidence: 99%

Inverse electronic scattering by Green’s functions and singular values decomposition

Vigneron

2000

Phys. Rev. B

Self Cite

An inverse scattering technique is developed to enable a sample reconstruction from the diffraction figures obtained by electronic projection microscopy. In its Green's functions formulation, this technique takes account of all orders of diffraction by performing an iterative reconstruction of the wave function on the observation screen. This scattered wave function is then backpropagated to the sample to determine the potential-energy distribution, which is assumed real valued. The method relies on the use of singular values decomposition techniques, thus providing the best least-squares solutions and enabling a reduction of noise. The technique is applied to the analysis of a two-dimensional nanometric sample that is observed in Fresnel conditions with an electronic energy of 25 eV. The algorithm turns out to provide results with a mean relative error of the order of 5% and to be very stable against random noise.

“…Our approach to the problem was based on both the transfer-matrix and Green's-functions formalisms. [9][10][11][12][13][14] This approach enables direct scattering simulations [15][16][17] with reduced storage requirements.…”

Section: Introductionmentioning

confidence: 99%

Electronic diffraction tomography by Green’s functions and singular values decompositions

2001

Phys. Rev. B

Self Cite

An inverse scattering technique is developed to enable a three-dimensional sample reconstruction from the diffraction figures obtained for different sample orientations by electronic projection microscopy, thus performing a diffraction tomography. In its Green's-functions formulation, this technique takes account of all orders of diffraction by performing an iterative reconstruction of the wave function on the observation screen and in the sample. In a final step, these quantities enable a reconstruction of the potential-energy distribution, which is assumed real valued. The method relies on the use of singular values decomposition techniques, thus providing the best least-squares solutions and enabling a reduction of noise. The technique is applied to the analysis of a three-dimensional nanometric sample that is observed in Fresnel conditions with an electron energy of 40 eV. The algorithm turns out to provide results with a mean relative error around 3% and to be stable against random noise.