1.Introduction

Phenomena of dielectric charging due to electron beam irradiation bring lots of influences on many fields,such as scanning electron microscopy(SEM)imaging[1,2],electron beam lithography[3],electro-probe microanalyzers(EPMAs)[4]and even spaceships in aerospace[5].Although some of these charging phenomena have good advantages[1],the majority of charging phenomena play a negative role,with a deterioration in performance[6,7].Dielectric charging degenerates properties not only via altering their working E- field environment directly,such as disturbing the trajectories of electron beams in SEM and EPMA[8],but also by influencing material intrinsic properties,such as altering the dielectric constant under an electron beam in aerospace.In general,variation of the dielectric constant may result in operating frequency excursion and coupling coefficient diminution in radio frequency(RF)equipment[9].Considering variation of dielectric constant due to electron beam irradiation is dynamic,hence,in order to obtain the influence on equipment properties,it is necessary to figure out the characteristics of the dielectric constant resulting from a charging effect under electron beam irradiation.

Figure 1.Sketch of the relationship between E-beam irradiation and dielectric charging.

For a high-resistivity insulator,under the E-beam irradiation and the following secondary electron emission[10,11],the remaining charges accumulate inside the sample and evolve in a complicated way[12].The internal local electric field generated by those accumulated and evolved charges may still result in nonlinear polarization and dielectric constant variation of sample material.Additionally,the variation in the dielectric constant will also affect the charging evolution process conversely.So,in order to find out the characteristics of the dynamic charging effects on the dielectric constant,we should consider not only processes of charge accumulation and transport,but also self-consistent processes of charging and dielectric evolutions.Generally,research on charging under E-beam irradiation always treats the dielectric constant as a definite value[13,14].In fact,for the majority of dielectric materials,when the applied E- field is strong enough,its dielectric polarization is no longer linear with a higher E- field,and the dielectric constant also varies[15].Traditional research on material dielectrics is always via an electrode for generating a uniform higher E- field,which hardly gratifies the conditions due to E-beam irradiation[16,17].In addition,the mismatch between the equipment conditions and the environment of E-beam irradiation and material dielectric testing results in dynamic charging effects on the dielectric constant is not often obtained directly in experiment.Fortunately,a method using numerical simulation can help us to understand the dynamic characteristics of charging effects on the dielectric constant based on the connecting charging and dielectric evolution processes.

Accordingly,in this paper we propose a numerical simulation to figure out the dynamic characteristics of the negative charge effect on the dielectric constant due to E-beam irradiation from the perspective of microcosmic physical processes.Considering processes of elastic and inelastic scattering between the incident electrons and the material atoms,accumulations of kinds of local plasma charges,including electrons and holes,are obtained.Internal charging states are calculated here via connecting evolutions of internal free electrons and holes in the form of transport process with variations in the material dielectric constant under the force of a local E- field.Consequently,we also obtain and analyze the effects of material nonlinear susceptibility and primary electron energy with respect to dynamic charging states and the effective dielectric constant evolution.

2.Simulation method

Primary electrons(PEs)with a certain energy incident dielectric sample have a reaction with material atoms as shown in figure 1.Emitted secondary electrons(SE)will generate from the sample surface immediately[18].Considering the number of emitted secondary electrons is not equal to the number of incident primary electronsin most cases,some of the remaining electrons and holes accumulate and form a local plasma charging state.For the situation of PE energy larger than several keV,the number of emitted SEs is more than the number of incident PEs,which results in the sample being in a negative charging state.Here,considering negative charging states are much more remarkable than positive charging states,in this study we mainly focus on negative charging states and the PE energy is set to be 5–20 keV.

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2.1.Electron scattering

Figure 2.(a)Space structure of the PTFE molecule and composition of its unit.(b)Scattering trajectory of electrons in PTFE under E-beam irradiation.

In order to obtain charging characteristics in time division and space distribution,we still need to calculate the scattering process as well as the process of charge transport.Incident PEs collide with material atoms in the form of elastic scattering with the nucleus and inelastic scattering with extranuclear electrons.In the case of inelastic scattering,parts of the electron kinetic energy will be transferred for exciting a free SE.In general,scattering processes are calculated based on a scattering cross section.Considering the incident electron energies in this study are set be 5–20 keV,the elastic scattering process is suitable for calculation using the Rutherford model,while inelastic scattering is propitious for the fast secondary electron(FSE)model.

Here,ρis the material density,andZ,AandJare the average atomic number,atomic weigh and ionization energy,respectively.For PTFE,andare 7.68,15.9 and 0.0955,respectively.

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For elastic scattering,the total scattering cross sectionσT with variation of electron energy E satisfies[19]

Here,z denotes the material atomic number,andαrepresents the shielding factor of extranuclear electrons to the nucleus,Considering most of insulator materials are composed of several kinds of elements,the probability of electron elastic scatter with each elementPican be determined with each element number proportionCiand elastic crossσi,and described as

Under the strong reaction of the E- field,material dielectric polarization presents is no longer linear with this internal build electric field.The distribution of internal dielectric constant can also be expressed as a function of internal E- field.Hence,the distribution of dielectric constant εei(　z, t)may also evolve during the charging process,as shown in figure 8(a).Based on the variation of internal E- field E　in　(z　, t),εei(　z, t)of PTFE increases from 2.2 in regions of net charges,and expands towards the grounded bottom with irradiation continuous from t = 0.02 s to t = 0.8 s,presented as different color lines.With the purpose of describing material dielectric constant as a whole,we use an effective dielectric constantεeffvia considering the distribution of internalεei(　z, t)in deep H,

Different to elastic scattering,calculation of the inelastic scattering should consider not only the scattering cross but also the energy loss and the generation of SEs.Based on the FSE model,the inelastic scattering crossσincould be entirely described as

Here,the integral lower limitΩc,set as0.001e,is a single electric quantity.

Under the reaction of inelastic scattering between incident electrons and extranuclear electrons,parts of the incident electron energy may transfer for exciting a free SE.Hence,we use a novel Continuous Slowing Down Approximation(CDSA)model for simulating the energy loss and SE generation in inelastic scattering.The energy loss dE at distant dS obeys the Bethe law

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In order to simulate the generation of SE,the continuous energy loss still needs to be discretized for SE generation[20]

Remaining charges accumulated in the scattering process build an internal E- field and change the charge distribution in different forms.Under the force of the internal E- field and charge density gradient,free charges,including electrons and holes,evolve as forms of drifting and diffusion,respectively.Meanwhile,excited electron–hole pairs may neutralize in a probability related to densities of electrons and holes.Owing to defects of the materials,free electrons and holes may still be trapped by the trapping centers.Electron and holes distributionn( z, t)andh( z, t)in the processes of drifting,diffusion,neutralization　and　trap　satisfy　the　following equations(7)–(12).

In this simulation,we track each collision process between electrons and atoms,and each electron trajectory inside and outside the sample,and treat all outgoing electrons,including internal SEs and reflecting primary electrons from the sample surface,as SEs.

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2.2.Charge transport

where N is the atom number per volume andZfis the material free electron number.

Here,nfandhf,andntrapandhtrapare free and trapped charges(electrons and holes).r is the neutralized coefficient of the electron–hole pair.JnandJhare the electron and hole current in the sample,respectively.Charge mobilityμand diffusion coefficientD　satisfy the Nernst–Einstein law,andτeare the trap center density and trap time constant,respectively.

In addition,the internal built E- field still varies the material internal dielectric constantεeiwith nonlinear polarization.In addition,the internal potential distributionV( z, t)can still be calculated using the charge density distribution ρ (z, t)according to Poisson’s equation

Here,χ is the nonlinear susceptibility of the material.Ein is the internal E- field.

Since the scattering process is hundreds of times faster than the charge transport process,we simulate the charge transport process after the scattering process,alternately.Figure 3 shows the flow diagram of the total charging process simulated in this study.After initializing the relative parameters at the very beginningt=0,this simulation calculates the scattering process and transport process successively and circularly.When the charging state reaches saturation balance,calculations are shut down,and related data are outputted and saved.

Figure 3.Flow diagram of the charging process with numerical simulation.

3.Results and discussion

3.1.Charging transient

Under the E-beam irradiation,electron–atom scattering and electron emission,part of the electrons(including PEs and internal SEs)and holes remain inside the material with a distribution of analogous plasma.Figure 4 shows the charge(including electrons,holes and total charge)distribution inside the PTFE sample in the deep H and radial R directions.The primary electron energy is set to beEP　E　=10 keV.Since the charge quantities vary with incident current,ordinates of charges distributions in figure 4 are normalized.As shown in figure 4,we can find that distributions of electrons and holes denoted with red and blue dashed line are very close in both the deep H and radial R directions,and present as an analog plasma.As a whole,the total net charge presents as negative except for the small surface region rounding the irradiation point.The maximum point of charge density due to E-beam irradiation is under the surface,which is related to material density and primary electron energy.

Figure 4.Distribution of electron,hole and total charge in the deep H and radial R directions.

Figure 5.Transient of internal electrons,holes and net charge distribution.

Corresponding to the internal charge distribution,tje internal E- field and potential distribution still evolve with irradiation,as shown in figure 6.Since the internal E- field mainly increase along the net charge,the distribution of the internal E- fieldE　in　(z, t)also expands to the grounded bottom from t = 0.02s to t = 0.8 s,as shown in figure 6(a).With integrating the internal E- field in distance,analogous evolution　of internal potential distributions Vin　(z, t)from t = 0.02 s to t = 0.8 s are also expressed with different color lines,as shown in figure 6(b).The nternal potential increase basically begins at the net charges region as well.

As we mentioned above,the remaining charges still evolve in the form of transport due to the self-build internal fields.Figure 5 shows the transient of sample internal electrons,holes and net charges distribution under continuous E-beam irradiation with energy and current density of EP　E　=10 keV and　JP　E　=　5.0 nA cm-2,respectively.Here,the sample thickness dimension is set to be comparable with the scattering range,H=5μm.The PTFE film is supposed as being placed on a grounded bottom,and its charges transport process under a large area irradiation can be calculated via a one-dimensional degeneration.As shown in figure 5(a),from irradiation time t = 0.02 s to t = 0.2 s,internal electronsninand hole shinaccumulate synchronously,except in the rear-end region.This distinction just results from the charge transport as shown in figure 5(b).With the exception of accumulation,the internal net charges still moves towards the grounded bottom from t = 0.02s to t = 0.8 s,as expressed with different color lines.Considering the mobility of electrons is several tens of times larger than that of hole,the transport charges towards the grounded bottom is mainly composed of free electrons.

3.2.Charging balance

For the whole sample,the charging balance comes when the input charges are equal to the output charges.Internal charges transport towards the grounded bottom and form the leakage currentJL.According to the law of current conservation,the output currents,including leakage currentJL,SE currentJSE,charging　current　JC　and　PE　current　JPE,　satisfy JL　　+　JC　　+　JS　E　=JPE.Besides charging currentJC,when the charging state comes to balance,the output currentsJSE,JL should equal the input currentJPE　,JP　E　=　JS　E　+JL,as shown in figure 7(a).As soon as the internal charge transport reaches the bottom at t = 0.65s,the leakage current emerges and increases rapidly.Meanwhile,the charging　currentJC decreases until the charging states reaches balance,JC =0.When charging states comes to balance,t = 1.05 s,the input current keeps balance with the output currents,and the sample surface potentialVSand total net charge quantityQTremain stable,as shown in figure 7(b).

Figure 6.Transient of internal E- field and potential distribution.

Figure 7.Evolvution of currents and charging states during E-beam irradiation.

Figure 8.Characteristics of internal dielectric constant and the total effective dielectric constant.

Figure 9.Effects of nonlinear susceptibility on effective dielectric constant and surface potential.

For PTFE(polytetra fluoroethylene)focused on in this study,as shown in figure 2,its element number proportions of C and F are 1 and 2,respectively.

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where,in this situation,parametersαandβare 0.196 and 8.12,respectively.αis possibly related to the transport process,while βmay be decided by the initial scattering charging states.

3.3.Nonlinear susceptibility

The variation of dielectric constant due to E- field is essentially caused by nonlinear polarization,and is related to the nonlinear susceptibility χ directly.For many polymers including PTFE,different preparation methods always correspond to various nonlinear susceptibility χ values in a large range.Hence,in this study we still analyze the effect of the nonlinear susceptibility χ on the evolution of the dielectric constant and charging states in a range of 10−6–10−3.Figure 9(a)shows the transient of effective dielectric constant εeffduring the charging process with different nonlinear susceptibilities χ .The increase of nonlinear susceptibility χ obviously enhances the variation of εeffduring the charging process,but has little effect on the charging period.εeffalmost keeps a fixed value when χ is less than 10−6.In balance states,as shown in figure 9(b),both the balance effective dielectric constantεeff-tand the balance surface potentialVS-t increase withχ ,denoted by black and red dot-dash lines,respectively.εeff-tincreases from 2.28 to 4.64 with χ changing from 10−6to 10−3,VS-t,meanwhile,increases from−31.0V to−20.2V.In addition,VS-tpresents as linear with εeff-tin differentχ ,

Figure 8(b)is the evolution of the effective dielectric constant εeffduring the charging process.We find εeffof PTFE increases from the beginning 2.20 to the balance value 2.67.

3.4.Primary energy

The distribution,quantity and transportation of scattering charges are greatly related to the primary electron energy.In consequence,we study the dynamic characteristics of charging states and the dielectric constant under different energy PE irradiations.For the purpose of avoiding penetration of PEs at different primary energies,the sample thicknesses in this section are all changed to 10 μm.As shown in figure 10(a),the total net charge distributions correspond to PE energyEP　E　=5,10 and 20 keV,described with black,blue and red lines,respectively.The scattering charges due to a larger energy PE present as being more gentle but deeper.ForEP　E　=20 keV,scattering charges can reach as far as 5500 nm from the sample surface.As a result of reaching deeper,PEs with a larger energy excite more charges inside and produce fewer emitted SEs.SE yieldδwith PE energy EP　E　=20 keVdecreases　to　0.21　compared　withas shown in the inset to figure 10(a).

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Figure 10.Effects of primary energy on charging states and the dielectric constant.

PE energy affects the initial scattering charges,and also impacts the transport process and subsequently the evolution of the dielectric constant.Figure 10(b)shows the distribution in balance states of both internal net charges and internal dielectric constant under different energy PE,EP　E=5,10 and 20 keV.Affected by the scattering charges,the ultimate charge distributionqindue to a larger PE energy also goes deeper.Under the reaction of an internal E- field,the internal dielectric constant εeiincreases from the net charge region.This means thatεei varies with a larger PE energy delay and the corresponding εeff-tis less(3.47,3.41 and 2.70 corresponding to 5,10 and 20keV,respectively).Since the stronger charged region makes variation ofεeisharper,an earlier and more gentle variation may cross with the later and sharper one,as shown in figure 10(b),EP　E=5 and 10 keV.

Different from the thickness setting　(5μm)before section 3.4,the thicknesses in section 3.4 are set to be 10 μm for avoiding the penetration of higher energy electrons.On the one hand,the situation with a thicker sample and a lower PE energy cannot present the effect of scattering charging observably.On the other hand,a thinner sample cannot retain higher energy PEs,which results in fewer charges accumulating in the sample.So,under these two considerations,simulations in those two parts are set to be different sample thicknesses(5 μm for the former and 10 μm for the latter).

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Scattering process under E-beam irradiation is simulated with a Monte Carlo method in this study.Since each electrons’trajectories and collisions inside the sample are tracked,all emitted electrons,including true SEs and backscattering electrons can be treated as SEs.In addition,SE yields with PE energy 5keV,10keV and 20keV,as shown in figure 10(a),basically agree with the literature data[21].The following charge transport modes are also checked in former research[22].Under the reaction of a strong E- field,dielectric polarization no longer presents with an E- field.In spite of excitation not being due to E-beam irradiation,the nonlinear character of a dielectric peculiarity via an E- field has been verified in previous studies[23,24].In the space environment,properties of dielectric materials out of the protection cabin may deteriorate due to the charging effect under space E-beam irradiation,which also results in the related RF dielectric component being in trouble.Hence,in th future work,we will further research the influence of charging effects on RF dielectric components.

4.Conclusions

With establishing a synthetical electron scattering and charge transport numerical simulation mode,we have investigated the dynamic characteristics of the charging effect on the dielectric constant due to electron beam irradiation in this study.The numerical simulation was shown to be a useful approach to　complicated　microcosmicprocesses and　interactions between charging effects and material dielectric properties.Our simulation demonstrates that continuous accumulated charges in the scattering area form an analog plasma,which include electron–hole pairs,and enhance the internal charging intensity until saturation due to charge leakage.Under the strong reaction of an internal E- field on dielectric polarization,the internal dielectric constant increases with depth significantly in the non-scattering area,and the whole effective dielectric constant enhances with charging states in the dynamic.Material nonlinear susceptibility promotes the variation of the dielectric constant with charging,and the effective dielectric constant presents as a linear relationship with surface potential in the charging balance.Primary electrons with a larger energy come with deeper charging and reduce the effective dielectric constant with shrinking the range of the non-scattering area.This study is helpful for understanding the dynamic micro-mechanism of dielectric materials under electron irradiation and has significance in research in space fields.In future work,we will move forward to investigate charging effects on RF dielectric components.

因此只需测得主固结完成后的剪切模量GPrimary，土层形成的时间tc和该土层深度的围压P，就能得到现场不同深度剪切模量GField的值。将公式(16)所得到的值代入公式(10)即可得到不同深度下桩周负摩阻力的值。

This work was supported by National Natural Science Foundation of China(Grant Nos.U1537211 and 11675278)and the China Postdoctoral Science Foundation(Grant No.2016M602944XB).

借鉴大量国内外的监测数据，按照震动波的传播频率15 Hz，煤炮发生区域煤体振动峰值速度0.3～5.0 m/s，根据岩体属性计算得出P波在煤体传播速度大约Cp=3 500 m/s，计算出动载荷的量值为1.47～24.5 MPa。再加上可能产生的S波的叠加，这里取0(静载状态)、5、15、25、35 MPa 4种强度进行动载分析，假设作用时间 0.1 s，发生位置在巷道上方4 m。

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