Geometry and Electron Emission Analyses of Random Surfaces
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Universidade Federal da Bahia Instituto de F´ısica Geometry and Electron Emission Analyses of Random Surfaces presented by Caio Porto de Castro accepted on the recommendation of Prof. Dr. Thiago Albuquerque de Assis, supervisor Prof. Dr. Roberto Fernandes Silva Andrade, co-supervisor A thesis submitted as a partial requirement to be assigned the degree of DOCTOR OF PHYSICS of Federal University of Bahia October 2017 Acknowledgements I would like to thank my supervisor Prof. Dr. Thiago Albuquerque de Assis, Prof. Dr. Caio M´arioCastro de Castilho, Prof. Dr. Roberto Fernandes Silva Andrade and Prof. Dr. Hans J¨urgenHerrmann. Also, thanks my family and friends. Because all of them, this thesis could became a reality. Finally, thanks to the government funding agencies CAPES, the postgraduate of physics of the UFBA and CNPq. i ii Dedication To my parents, Charles and Jesseni. iii `Although each of us is unrepeatable, we are not even exclusive, although we are unique.' `Embora cada um de n´osseja irrepet´ıvel, n´osn~aosomos nem exclusivos, embora sejamos ´unicos. Mario Sergio Cortella iv Abstract By Fourier transform we generate correlated random surfaces that was used as a tool to study two different fields, statistical physics in terms of the Schramm/Stochastic Loewner Evolution (SLE) and Cold Field Electron Emission (CFE). We have shown that critical exponents of percolating systems are not affected by the form of the distribution of the Fourier coefficient and phase and Fourier phase correlations. Our results of extensive numerical SLE test, indicate that the full perimeters of percolating clusters of correlated surfaces, with negative Hurst exponent, are statistically equivalent, in the scaling limit, to SLE curves. From the perspective of CFE, a more general criterion for detecting and interpreting nonorthodox field emission is proposed and can be applied to any distribution of local field enhancement factors in conducting large-area field emitters (LAFEs). We show that morphology changes on LAFEs can lead to saturation of the Fowler-Nordheim (FN) plot. Finally, we emphasize that the linear behavior of the FN plot does not guarantee that the emission phenomenon is orthodox. v Resumo Atrav´esda transformada de Fourier n´osgeramos superf´ıcies aleat´oriascorrelacionadas que foram utilizadas para estudarmos dois diferentes campos da F´ısica,F´ısicaEstat´ısticano ^ambito da evolu¸c~aode Schramm-Loewner (SLE) e emiss~aode el´etronspor efeito campo el´etrico(CFE). Primeiro verificamos que expoentes cr´ıticosde sistemas de percola¸c~aoem surperf´ıcies correla- cionadas n~aoapresentaram depend^enciacom as distribui¸c~oesdos coeficientes e fase de Fourier, bem como devido a correla¸c~oesna fase. Nossos resultados, ap´osextensivos testes num´ericos de SLE, indicaram que o per´ımetrocompleto do aglomerado de percola¸c~aode sistemas cor- relacionados, com expoente de Hurst negativo, s~aoestatisticamente equivalentes, no limite de escala, a curvas SLE. Na perspectiva de CFE, propomos um crit´eriomais geral para detec- tar e interpretar emiss~oesn~ao-ortodoxas, o qual pode ser aplicado para qualquer distribui¸c~ao de fatores de amplifica¸c~aoem emissores de ampla area de emiss~ao(LAFEs). Mostramos que mudan¸casmorfol´ogicasem LAFEs podem levar `asatura¸c~aono gr´aficode Fowler-Nordheim (FN). Por fim, enfatizamos que um comportamento linear do gr´aficode FN n~aogarante que o fen^omenode emiss~ao´eortodoxo. vi Contents Acknowledgements i 1 Introduction 1 2 Random Surfaces 5 2.1 Introduction . 6 2.2 Methods . 8 2.2.1 Fourier Filtering Method . 8 2.2.2 Rank Method . 10 2.3 Clusters and perimeters . 10 2.4 Results and Discussion . 11 2.5 Conclusions . 17 3 Schramm-Loewner Evolution Theory 19 3.1 Conformal Mappings . 20 3.2 The Riemann Mapping Theorem . 21 3.3 Loewner Differential Equation . 22 3.4 Schramm-Loewner Evolution . 23 vii viii CONTENTS 3.5 Numerical Method . 26 3.5.1 Zipper Algorithm . 27 3.6 Results . 29 4 Fowler-Nordheim Theory 35 4.1 General Physical Assumptions . 35 4.2 Local Current Density Type Equation . 36 4.2.1 Energy-space Diagrams . 37 4.2.2 Gamow Exponent - Approximation of the Escape Probability . 41 4.2.3 Emission Current Density Equation . 43 4.2.4 Approximations for the Special Elliptic correction functions . 46 4.3 Large Area Field Emitters . 49 4.4 The Orthodox Emission Hypothesis . 52 5 Large-area field emitters: the dependence between area of emission and the applied field 56 5.1 The dependency of the formal area of emission on the applied field and LAFE model . 57 5.2 Results and generalization for scaled barrier field extraction . 60 5.3 Discussion of the discrepancies between the slope characterization parameter, FN aprox β , and the approximated FEF, γC , extracted from simulated FN plot data at high applied field regime . 65 5.4 Conclusions . 66 6 Degradation of a large area field emitter: from saturation in Fowler-Nordheim plots to unorthodox field electron emission 68 6.1 Introduction . 69 6.2 Method . 71 6.3 Results and Discussion . 73 6.4 Conclusions . 77 7 Final Conclusions and Remarks 78 Bibliography 80 ix x List of Tables 4.1 The dependence of FR, η, θ and f values on work function φ[126], considered in this thesis. 54 FN aprx 5.1 Values of γC extracted from the distribution ρ(γ), β , γC and the slopes SF extracted from the FN plots shown in Fig.5.1 (a). The results are presented for the linear region of the FN plot in the limit of a high applied field, for various work functions. 66 xi xii List of Figures 2.1 Schematic picture of the percolating cluster (light gray) connecting the top of the square with the bottom. The white region corresponds to sites that do not belong to the percolating cluster (unoccupied sites and other clusters) and the black line is the external (complete) perimeter. This map was generated by OriginPro 2016 (64-bit) Sr1 b9.3.1.273 (http://www.originlab.com/). 11 2.2 a) Illustration of the rules used to compute the fractal dimension of the complete and accessible perimeters with the yardstick method. Suppose the sticks start to follow the coast from the bottom. The green circle shows the region of coast accessible to a particular stick. The X's represent the next possible starting points of that particular stick. If the closest point along the coast (red X) is always chosen as the next starting point, we obtain the complete perimeter. If, on the other hand, the most distant point (blue X) is chosen, then we obtain the accessible perimeter. Here, points such as the green X, which are neither the closest nor the most distant from the center of the circle, are disregarded. b) Paths made by sticks of equal sizes of the complete (blue sticks) and accessible (red sticks) perimeters. 12 2.3 Probability density function of ju(q)j (left) and φ(q) (right) in the case of the graphene sheet (red squares) and the vorticity field (black circles). The red 2 −c2juj and black curves in left panel are best fits for f(juj) / c1juj exp and the Gaussian function, respectively. 12 xiii xiv LIST OF FIGURES 2.4 Fractal dimension of the complete and accessible perimeters as a function of H, for a) H < 0 and b) H > 0, and different φ(q) distributions. In a), the black lines are conjectures proposed by Schrenk K. J. et al. [47]. All values are averages over at least 104 samples and error bars are defined by the variance of the distribution. 13 2.5 Surface map with inversion symmetry with respect to the center. This symmetry of the surface results from the use of a Gaussian distribution φ(q) with a small variance σ = 0:001. 14 2.6 Maps of phase correlated surfaces. Panels a), b), c), and d) show examples of surfaces with H = 0:5 and Hphase = −0:9; −0:2; 0:1; and 0:4 respectively. The arrows serve as a guide to show the linear translation of the random surface due to correlations introduced between the Fourier phases. 15 2.7 Scale analysis of the convergence of the percolation threshold pc;H . For the square lattice, the site percolation threshold pc for uncorrelated surfaces is pc ' 0:592746. The black lines serve as guides to the eye with slope H = −1/νH [74, 75, 47]. 16 2.8 Fractal dimension df of the percolation cluster and critical exponent ratio γH /νH as a function of the Hurst exponent H for surfaces with different distributions of u(q). The black lines are conjectures proposed by Schrenk K. J. et al. [47] based on the hyperscaling relation [55]. All values are averages over at least 104 realizations and error bars are defined as the variance of the distribution of their values. 17 3.1 An example of hull K in the upper hall plane together and the conformal map −1 gK : H ! H n K and its inverse gK : H n K ! H[82] . 21 3.2 An example of chordal trace in a upper half plane H = fz : Imfzg ≥ 0g with γ0 = 0 and γ1 = 1. ................................. 22 LIST OF FIGURES xv 3.3 Scheme of the evolution of the Loewner equation. For each time step there is a conformal map gt that maps the upper half plane minus the hull Kt, related to the path γt, to the upper half plane itself, gt : H n γ[0;t] ! H. The point on the boundary of H where the trace is mapped is the driving function ζt. 23 3.4 Illustrative examples of the SLE phases.