Theoretical Study Of Inelastic Mean Free Path Of Electrons In Materials

The electron inelastic mean free path(IMFP)is the important parameter for describing the transport properties of electrons in materials and is also one of the most important parameters for quantitative surface analysis.It has important applications in electron spectroscopy and electron microscopy.IMFP is an essential parameter in the quantitative analysis of many surface analysis techniques such as Auger Electron Spectroscopy(AES)and X-ray Photoelectron Spectroscopy(XPS).For other surface characterization methods using electron beams as probes,IMFP is usually used to determine the surface sensitivity of these methods.In this paper,we first introduced the electron IMFP and its application in quantitative AES and XPS analysis.We also intro-duced three methods of obtaining IMFP,i.e.,prediction formula,theoretical calculation,and experimental determination.Finally,some problems in the experimental determination of IMFP and the motivation of this thesis have been discussed in detail.(Chapter 1)When electrons are transported through a material,they are scattered by the nucleus and the electronic cloud of the material.These collisions can be classified into two categories depending on whether energy loss occurs:elastic scattering and inelas-tic scattering.We introduced the Mott cross section,which used to describe the elastic scattering of electrons,and presented the relevant elastic scattering potential,which used for the calculation of Mott cross section.The conventional dielectric functional theory for describing electron inelastic scattering in an infinite dielectric material is presented,as well as a semi-classical model for describing semi-infinite dielectric materials and considering the surface excitation.Various extension methods of the momentum transfer-dependent energy loss function(ELF)from the optical limit to the(q,?)-plane have been presented and compared.The Monte Carlo simulation procedures for electron transport inside the material and near the surface have been presented.(Chapter 2)The extraction of the electron IMFP from reflection electron energy loss spec-troscopy(REELS)spectra becomes an important scheme for the determination of IMFP in recent years.This scheme is mainly divided into two steps:1)extraction of the optical ELF or optical constant from the REELS energy spectrum;2)calculation of the IMFP based on the obtained optical ELF.The Reverse Monte Carlo(RMC)method,which is developed in recent years,combines a Monte Carlo modeling of electron transportation for REELS spectrum simulation with a Markov chain Monte Carlo calculation of pa-rameterized ELF.After briefly introducing the principle of the RMC method,we further improved the RMC method and applied it to more materials,including three transition metal materials Cr,Co,and Pd,the heavy metal Ir,the light element C,and the semi-conductor materials Si and Ge:(Chapter 3)I.A more accurate elastic scattering potential is used for the calculation of elastic cross section of three transition metal materials Cr,Co,and Pd.Based on these elastic scattering cross sections,we use the RMC method to perform the quantitative analysis of the REELS spectra of Cr,Co,and Pd for a wide energy range of 0-200 eV and extract the highly accurate ELF.The accuracy of the present results has been verified by two sum rules,i.e.the ps-sum rule and the f-sum rule.We propose an RMS value of three f-sum rules,i.e.f-sum rules for ELF,the imaginary part of the dielectric function,?2 and the extinction coefficient k,to describe the difference of all three f-sum rules.The RMS value is more effective than a single f-summing rule for proving the accuracy of the ELF,dielectric function,and optical constant data in the low-energy range.It is expected to be as important a criterion for judging the accuracy of optical data as the summation rule.II.We apply the RMC method to the heavy metal material Ir and the light element amorphous carbon(a-C).The ELFs of Ir and a-C extracted from the REELS spectra are highly accurate,and the relative errors of two sum rules compared to the theoretical values or RMS values are small and much lower than those of results obtained by other methods.We suggested using the ELF of a-C rather than the REELS spectrum to determine the plasmon excitation energy and thus calculate the local density.Due to the presence of surface effects or multiple scattering effects,the plasmon excitation energy shown in the REELS spectrum is usually inaccurate.On the other hand,based on the present ELF,we provide a simple formula for predicting the ELF,dielectric function,optical constants,of a-C for various densities.The formula is expected to provide important data supporting for investigation on properties of a-C materials and the quantitative surface analysis of a-C.III.An FPA-Ritchie-Howie model was proposed for extension of the ELF from the optical limit into the(q,?)-plane,which uses the FPA method for the treatment of the bulk excitation,and uses the Ritchie-Howie method for the treatment of the surface ef-fect.The multiple scattering effects can be described accurately and the computational efficiency has been taken into account in the FPA-Ritchie-Howie model.It is of great importance for quantitative surface analysis of free-electron-like materials.Applying the FPA-Ritchie-Howie model,we presented an improved version of the RMC tech-nique to extract the ELF,dielectric constants,and optical constants of free-electron-like materials i.e.Si and Ge.The obtained results have been verified by using various sum rules,such as ps-sum rule,f-sum rule,inertial sum rule,dc-conductivity sum rule,and RMS deviation of three f-sum rule.Except for the f-sum rule,the relative errors of all the tests were less than 0.1%.The RMS deviation of three f-sum rules for the two materials were 0.036%and 0.010%,respectively.Such small RMS deviation indicated that the relatively large errors of the f-sum rule(0.6-1.2%)are mainly originated from the inaccuracy of the high-energy optical data which are from other sources.On the one hand,the currently obtained high-accuracy results provide important data support for analysis of Si and Ge,and prove the validity of the FPA-Ritchie-Howie model.?.Based on the ELF of Cr,Co,Pd,Ir,a-C,Si,and Ge extracted from the REELS energy spectra by using the RMC method,the electron IMFP for these materials has been calculated,which is an important supplement to the IMFP database.A large number of publications have shown that there are some features in low energy range of the curves of effective attenuation length,IMFP,or mean free path.Although some researchers believe that these features come from surface excitation.However,there is not any direct evidence to prove such a view unambiguously.Those effective attenuation length,IMFP or mean free path,in which the features were ob-served,usually contains contributions caused by elastic scattering effects or coupling between the substrate and thin film.A more efficient and accurate method is urgently needed to extract the pure electron IMFP without the influence of these effects.Based on the classical electron trajectory framework,we develop a ZT model to describe elec-tron transport in unsupported 2D materials.Based on ZT model,we re-analyzed the elastic transmissivity and the elastic reflectivity data reported in the literature for one-,two-,three-,and four-layer graphene and extracted the elastic mean free path(EMFP)and IMFP for graphene.Our analysis shows that the inherent notion of "the thickness of a single atomic layer is much smaller than the IMFP of electrons" is incorrect,at least for low-energy electrons.Our IMFP results show that the IMFP of one-layer graphene has no obvious characteristic features,while a step feature begins to appear at 5-15 eV of IMFP of two-layer graphene and it becomes a hollow feature with increasing thick-ness.Such a feature is considered to be owing to the out-of-plane mode of the ?+?plasmon.Our results show that the plasmon contribution may need to be considered for IMFP,providing an important approach for future studies.This work shows that the classical electron trajectory framework still works for revealing the physics picture of low-energy electron interaction with graphene,even for the transverse direction of monolayer graphene,which is the thinnest material.(Chapter 4)The experimental determination of IMFP is usually based on the analysis of the electron energy spectrum.Different experimental approaches to obtain IMFP are actually processes of analyzing the electron signal from different energy ranges in an electron energy spectrum.A large number of methods have been developed for the analysis of elastic peak electrons,secondary electrons,REELS spectra,and Auger electrons to obtain the IMFP of electrons.However,there is a vacancy that is never used——the backscattered electron background.Usually,researchers investigate differ-ent signal peaks,such as the elastic and Auger peaks or secondary electron signals,to obtain useful information.The backscattered electron background signal is usually considered"useless".We proposed a method to perform the quantitative analysis based on the useless signal,i.e.the backscattered electron background,to extract the useful information,i.e.electron IMFP.This scheme extends the experimental measurement of IMFP.It also provides an important analytical tool for the subsequent quantitative analysis of the backscattered electron background.The scheme is scalable in that the electron IMFP can be obtained from the backscattered electron energy spectrum,with the knowledge of chemical formula and density.(Chapter 5,subsection 5.1)In addition to improving existing experimental methods for the determination of IMFP and proposing new ideas for determining IMFP experimentally,we also applied the Monte Carlo method for quantitative analysis of the backscattered electron energy spectrum and the REELS spectrum.The experimental data of backscattering coefficients of two heavy metals,Mo and W,were carefully analyzed using the Monte Carlo method.We found that the main reason for the large deviations in the previous experimental backscattering coefficient data for Mo and W below a few keV comes from the presence of surface contaminations on the experimental samples.These surface contaminations make the experimental backscattering coefficients below a few keV smaller than the values of pure samples.We analyzed in detail the influence of the amorphous C,water,and PMMA with various thicknesses covered on the surface to the backscattering coefficients of Mo and W.In addition,an uncertainty analysis has been performed to confirm the reliability of the present calculations and conclusions.We present a theoretical recipe for the clear and individual separation of surface,bulk,and begrenzungs effect components in surface electron energy spectra.By using this method,one can analyze the contribution from different components in a REELS spectrum in detail.This recipe is the first method which focuses on the quantitative study of the begrenzungs effect.The quantitative analysis of the REELS spectrum of Si at the primary energy of 5 keV has been performed as an example to demonstrate the effectiveness of this method.Our work proves that single scattering is the main contribution to the surface excitation component of the REELS spectrum of Si due to the localization of the surface effect.The probability that an electron suffers no bulk excitation and only surface excitation will decrease rapidly with the increase of the number of surface excitations.The present analysis clearly shows that the final collision order of an electron depends on the trajectory due to the depth dependence of the surface effect.This work extends the quantitative analysis method of the REELS spectra into the more detailed and accurate realm.(Chapter 5,subsection 5.2;Chapter 6,subsection 6.1)The negative cross sections in particle transport studies have plagued scientists for many years.The emergence of negative cross sections indicated that the negative probability of particlesolid interactions must be considered in particle transport studies.We focus on the electronsolid interaction and analyze the reasons for the negative values in the differential inelastic scattering cross section(DIIMFP)when electrons transport through a surface.The appearance of these negative values all originates from the plasmon wake effect generated when electrons move near the surface.The wake potential generated by the response of the medium to the moving electrons is oscillatory.The wake potential does not make a major contribution inside the sample but is treated as a perturbation.However,in a vacuum,this oscillating wake potential will lead to an oscillating DIIMFP,which has negative values.More interestingly,we found that DIIMFP of begrenzungs term is also oscillatory,i.e.it has some positive values in some regions.This is completely different from the conventional perception.Our results suggest that the current understanding of the begrenzungs effect is too simplistic and one-sided,the begrenzungs effect cannot be simply understood as a decrease of the bulk excitations in the material.It has different influences in different regions of(z,?)-plane and manifests as an increase or decrease effect of bulk excitations.Therefore,we proposed a method for the sampling of negative probability in the Monte Carlo simulation.The traditional methods can not"correctly" sample the energy loss based on DIIMFP,and there are always some regions in the(z,?)-plane in which the inelastic collision cannot occur.Present negative probability sampling scheme accurately handles the negative values in DIIMFP.Although the influence of negative DIIMFP on the REELS spectra is relatively weak,present scheme can correctly describe the physical process of electronsurface interaction,which is important for the highprecision quantitative analysis of electronsurface interaction and the study of the mechanism of electronsurface interaction.(Chapter 6,subsection 6.2)...