1.Introduction

Magnetrons are widely used as sputtering sources for thin film deposition[1–3].In a magnetron discharge,electrons are trapped close to the cathode by the magnetic field,which leads to a more efficient use of electrons for ionizing inert or reactive gas particles,resulting in enhanced sputtering of the target[4–6].Since electrons are essential for controlling certain aspects of the plasmas,an understanding of their behavior,i.e.their transport and relaxation properties,is important in the design and optimization of magnetrons.

The theoretical analysis of electron transport in weakly ionized plasmas in the presence of electric and magnetic fields comes in three types:the hybrid model,particle-incell/Monte Carlo collision(PIC/MCC)and kinetic theory.Hybrid models treat fast electrons using MCC techniques and treat low-energy electrons using fluid models,which take advantage of the accuracy of MCC techniques and the computational Efficiency of the fluid models[7,8].The fluid parts of the hybrid models are based on Maxwellian electron energy distribution and have limitations;moreover,it is necessary to provide accurate electron transport and reaction coefficients.PIC/MCC techniques are time consuming,and occasionally cannot distinguish between small statistical fluctuations and intrinsic physical phenomena[3,9–17].In the present work we deal exclusively with kinetic theory.In the 2000s this problem was addressed by the JCU group in a series of literature outlining the multi-term approximation of the Boltzmann equation and obtaining electron transport properties in magnetrons[18–20].However,comparatively little attention has been paid to the experimental investigation of electron transport beyond the use of electric field probe arrays to study anomalous electron transport in high power impulse magnetron sputtering[21–23].

The purpose of this paper is to investigate theoretically the basic parameters influencing electron relaxation in Ar magnetron plasmas.We begin this paper with a description of the improved multi-term approximation of the Boltzmann equation in section 2.In section 3,the benchmark calculations of the Reid model and electron relaxation of the magnetron plasmas are presented.We conclude this paper with a discussion in section 4.

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2.Multi-term approximation of Boltzmann equation

2.1.Matrix equations

Electrons in neutral gases under the influence of spatially homogeneous electric and magnetic fields are described by the Boltzmann equations[18,24]:

where f(r,c,t)is the electron velocity distribution function,r and c denote the position and velocity coordinates,q and medenote the electron charge and mass,E is the electric field,Bis the magnetic flux density,J is the collision operator,and f0is the velocity distribution function for neutral atoms or molecules,and is usually assumed to be Maxwellian.It is noted that collisions between the electrons and ions are neglected as plasmas are weakly ionized in magnetrons.

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In order to obtain a unified hierarchy of coupled differential equations in the hydrodynamic and nonhydrodynamic regimes,we adjust the sequence of treatment of spatial dependence of the distribution function and representation of the speed dependence in term of Sonine polynomials,which are extended by the JCU group.In hydrodynamic and nonhydrodynamic regimes,the velocity distribution function can be represented in terms of Burnett polynomials[25,26]

where the bold type v denotes the indices(v,l,m),ω(α,c)is the　weight function,are　Burnett　functions,α2 = me/kBTb,and Tbis an arbitrary temperature.For a given l we have m = -l,-l + 1,…,l - 1,l,while v and l take on the value 0,1,2,…,∞.

In hydrodynamic regimes,the coefficients F(vlm;r;α,t)can be further decomposed in terms of the density gradient written in spherical tensor notation[27]

在材料进场后，合理安排材料的堆放以及使用，尽量避免材料的乱堆乱放以及浪费，设置专门的场地存放材料,并设立相关标识。同时对建筑施工涉及的各种型号、规格的材料做好详细记录,避免发生错用或混用现象；在材料堆放等待使用过程中，要不定期进行二级检测，避免因锈蚀等原因，造成材料质量下滑的状况发生，影响施工安全。

In hydrodynamic regimes,∂tn(r,t)can be expressed in this way,thanks to the equation of continuity[27]:

As the transport coefficients have many components,here we only give the components in the z(E)direction,and the other components can be derived similarly.

On substituting(2)–(4)into(1),the following matrix equations for the expansion coefficientsF　(νlm　;　ηkq　;　α,t )can be written as:

where　(l1m110|l2m2)isthe　Clebsch–Gordan　coefficient[27,　28],　the　reduced　integral　andare given in appendix III of[25]and the collision operatorJv2v1is given in section 2.2.

2.2.Collision operator

The conservative collision operator is a generalized form of Wang Chang et al[29,30]:

在形成A(n-1)′阵时，可以“按行消元，逐行规格化”方式，也可以“逐行规格化，按列消元”方式。这两种方式的计算结果和计算效率表面上看似乎一样[1]，但实际上后者计算效率更高。以“逐行规格化，按列消元”方式形成因子表时，其A(n-1)′阵中的各个元素的计算式如下：

The attachment collision operator is[29]

whereσA 　(j; cr,　　χ )is the differential cross section of the attachment of an electron to the neutral atoms or neutrals in internal state j.

表3给出了六个指数收益率序列的ARMA(p, q)-GJR-GARCH(m, n)模型估计结果。从表3来看，通过ARMA(p, q) - GJR-GARCH(m, n)模型，各收益率序列的自相关、条件异方差和杠杆效应得以消除，进而得到各个收益率序列的标准化残差序列η。

The ionization collision operator is

where σI(　j; cr,　　χ )is the differential cross section of the ionization and　B( 　j;c, 　c′)is the probability density function that divides the available momentum after ionization between the scattered and ejected electrons[29,31].

表5数据显示sig=0.000实验组和对照组平均值分别为4.68和1.01，说明两组差距显著，使用移动终端学习词汇的实验者对词汇学习的兴趣明显高于通过传统方式学习词汇的学习者。主要原因是这些学习软件的设计极具趣味性，有多种多样的练习方式，其中包括了真人发音，根据英文选汉语词义，根据汉语选英文单词，选词填空练习，每次练习都会积分升级，具有一定的游戏性质。相比较，传统的词汇学习，学习者只是记忆拼写，没有引发学习者兴趣的练习。

where

and m0denotes the neutral particle mass.We note that the self-adaptive Gauss–Kronrod rule is more efficient than the Gauss–Laguerre rule in evaluating the collision integrals as shown by formula(12)–(16).

2.3.Transport coefficients

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The bulk drift velocity in the z direction WBz(t)is given by

The bulk diffusion coefficient in the z direction DBzz(t)is given by

Note that the corresponding flux drift velocity WFz(t)and the flux diffusion coefficient DFzz(t)can be derived by removing the collision operator term from(20)and(21),respectively.

where pre-collision and post-collision velocities are denoted by(c,c0)and(cʹ,c0ʹ),where the subscript 0 refers to the neutral atoms or neutrals,cris the relative velocity,σC(jk;cr,χ )is the differential cross section of the conservative collision,j and k denote the initial and new internal state of the molecule,χ is the scattering angle and Ω is the solid angle.

The loss rate S(0)(t)is given by

The electron mean energy ε(t)and gradient energy parameter γ(t)are defined by

Substitution of(2)and(3)into(25)yields the electron mean energy ε(t)

不过，根策尔博士也在电子邮件中表示，要想做出比现有成就更高的发现，恐怕是颇为困难了。目前，对我们这个“从银河系乡下来的小男孩”来说，能把400万个太阳质量的物质压缩到体积半径不高于45天文单位的空间中去，已经是一个相当了不起且值得我们深入研究许久的壮举了。

Substitution of the expansion(2)into(6)–(8)yields:

The temperature tensor is defined by

The gradient energy parameter in the z direction γz(t)is given by

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The temperature in the z direction is given by

The electron energy flux is defined by

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The electron energy flux in the z direction qz(t)is given by

2.4.Numerical considerations

The numerical solution of matrix equation(5)proceeds after truncating(v,l)to(vmax,lmax)and the choice of the arbitrary temperature Tb.The optimal temperature Tb(opt)is the average temperature　of the　Maxwellian　distribution.Generally speaking,a large vmaxis indicative of the Significant departure of velocity distributions from a Maxwellian at Tb,and a large lmaxis indicative of a large deviation from spherical symmetry in the velocity space.

The procedure of the numerical solution is as follows:(1)the initial temperature Tb(0)is calculated based on the Einstein relationwhere DLis the longitudinal diffusion coefficient and μ is the electron mobility,which are derived from electron swarm experimental data.(2)Given l = 1 and Tb = Tb(0),v is incremented until the convergence tolerance of each transport coefficient is smaller than 5%,and v is recorded as vmax(0).(3)A new Tbis chosen based onwhere ρ is the proportion coefficient,ΔTbis the temperature increment,i is the number of iterations,and v of the ithiteration is recorded as vmax(i);(4)Tbis recorded as Tb(opt)when vmax(i)is smallest.(5)Given Tb = Tb(opt),v is incremented until the convergence tolerance of each transport coefficient is smaller than 5%,and l is incremented until the convergence tolerance of each transport coefficient is smaller than 1%.

where　is the electron number density and　is the spherical gradient operator.

3.Results and discussion

3.1.Benchmark calculations

In order to test the Efficiency of the improved multi-term approximation to obtain electron transport coefficients under the influence of Electricity and magnetism crossed at an arbitrary angle,the Reid model is considered.The details of the Reid model are as follows[32]:m0 = 4amu,T0 = 0,n0 = 1017cm-3,excitation energy εi = 0.2 eV,elastic collision cross section σel = 6 × 10-20m2,inelastic collision cross section

Figure 1.The electromagnetic field and velocity space coordinate system.

The electromagnetic field and velocity space coordinate systems are shown in figure 1,E/n0 = 12 Td,with the reduced magnetic flux density B/n0 = 0,500 Hx(1 Hx =10-27Tm3)and the electric and magnetic field crossed at angle ψ = π/2.The electron transport coefficients for the Reid model at E/n0 = 12Td and B/n0 = 0,500 Hx are shown in tables 1 and 2,respectively.The results in table 1 are compared with those of Ness et al(Ness)taken from[29],and the results in table 2 are compared with those of White et al(White)taken from[33].

It is observed in table 1 that four- figure convergence for all transport coefficients is achieved for lmax = 7,indicating that the distribution function is anisotropic in the velocity space.In　the　absence　of　nonconservative　collisions,WBz = WFzand DBzz = DFzz,which agree with the results of Ness et al[29].As is well known,the bulk transport coefficients have two components:one is due to the external electric fields,which are generally defined as the flux transport coefficients,and the other is due to the presence of nonconservative collision processes.It is found that our results have errors of less than 0.1%when compared with the results of Ness et al,and the calculation of the interaction integrals based on different rules may have resulted in the little discrepancy.

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From table 2 we observe that the bulk transport coefficients are also equivalent to the flux components in the presence of electric and magnetic fields when only conservative collisions are taken into account.When compared with theresults of White et al,the bulk drift velocity WBzand WBx,and the diffusion coefficient DBzzhave errors less than 0.1%,while the electron mean energy ε has errors on the order of 0.3%.Four- figure convergence for all transport coefficients is achieved for lmax = 2,indicating that the magnetic fields can reduce the anisotropy of the velocity distribution function.

Table 1.Convergence of the electron transport coefficients in the lmaxindex for the Reid model at E/n0 = 12 Td and B/n0 = 0 Hx.

WBz(104m s-1)　WFz(104m s-1)　n0DBzz(1023m-1s-1)　n0DFzz(1023m-1s-1)　　ε(eV)Ness-6.8380—　5.6880　　—0.2689 lmax = 1　-7.0281　-7.0281　5.0459　5.0459　0.2735 lmax = 2　-6.8164　-6.8164　5.7338　5.7338　0.2689 lmax = 3　-6.8356　-6.8356　5.6805　5.6805　0.2689 lmax = 4　-6.8323　-6.8323　5.6861　5.6861　0.2690 lmax = 5　-6.8333　-6.8333　5.6843　5.6843　0.2689 lmax = 6　-6.8330　-6.8330　5.6849　5.6849　0.2689 lmax = 7　-6.8331　-6.8331　5.6847　5.6847　0.2689 lmax = 8　-6.8331　-6.8331　5.6847　5.6847　0.2689

Table 2.Convergence of the electron transport coefficients in the lmaxindex for the Reid model at E/n0 = 12 Td and B/n0 = 500 Hx.

WBz(103m s-1)　WFz(103m s-1)　WBz(104m s-1)　n0DBzz(1023m-1s-1)　　ε(eV)White-4.1540—-2.3180　　—0.1123 lmax = 1　-4.1558　-4.1558　-2.3167　0.3828　0.1117 lmax = 2　-4.1562　-4.1562　-2.3177　0.3694　0.1120 lmax = 3　-4.1562　-4.1562　-2.3177　0.3694　0.1120

3.2.Electron relaxation in magnetrons

To investigate electron transport under the influence of the electric and magnetic fields crossed at arbitrary angles,Ar plasmas in high-power impulse magnetron sputtering devices are considered.The details of the Ar plasmas are given by:p0 = 0.53 Pa,T0 = 293 K,E/n0 = 100TdandB/n0 =200–20 000 Hx.We only consider four energy levels near 11.5eV for the electronic excited states of Ar plasmas:metastable　state　(11.62 eV),metastable state(11.72 eV)and(11.83eV).The excitation energy of these four states is assumed to be 11.5 eV,and the total excitation cross sections at 11.5eV are assumed to be the summation of the cross sections of these four states.The elastic and ionization cross sections are derived from[34].All cross sections are shown in figure 2;note that all cross sections are integral.

The temporal relaxation of transport coefficients for various magnetic fields is shown in figure 3.Because scaling with respect to different gas densities is very useful for the study of plasmas—i.e.it allows a direct comparison between experiments involving different gas pressures—we employ WBi,ε,Tii,qi,n0t,n0Dii,n0γiiand S(0)/n0 ≡ kionas plasma characteristics,which are generally regarded as invariant.

我坐在角落的位置，低头，流泪。对面是一对甜蜜的情侣，一个茶杯里插两个管，近得可以在低头的瞬间碰到彼此的鼻尖。

Figure 2.Collision cross section of Ar in the energy range of 0.1–1000 eV.

In figure 3(a)we show the temporal relaxation of WBxas a function of dimensionless time n0t for B/n0 = 200,500 and 1000 Hx.The oscillation amplitude increases with the increases of magnetic field.The starting time of the oscillation shifts to an earlier time,while the time required to achieve the steady states lags with the increase of the magnetic field.WBx in the steady states are-1.215 × 104,-3.187 × 104and-4.318 × 104m s-1for B/n0 = 200,500 and 1000 Hx,respectively.Note that after the damped period decay there is monotonic decay,which disagrees with White et al[20].The reason for this disagreement lies in the large difference in the collision cross sections.

In figure 3(b)we show the temporal relaxation of WBzas a function of the dimensionless time n0t for B/n0 = 200,500 and 1000 Hx.WBzis-6.023 × 104m s-1with no magnetic field.The same trends as figure 3(a)can be observed.The relaxation pro file is damped oscillatory,and after the damped periodic oscillation,the pro file is monotonic.The period of oscillation increases with the increases of the magnetic field.WBzin steady states are-5.872 × 104,-5.033 × 104and-3.376 × 104m s-1for B/n0 = 200,500 and 1000 Hx,respectively.

Figure 3.The temporal relaxation of transport coefficients for various magnetic fields.(a)The drift velocity WBx;(b)the drift velocity WBz;(c)the diffusion coefficient Dii;(d)the electron mean energy ε;(e)the reaction coefficient kion;(f)the temperature Tii;(g)the gradient energy parameter γii;(h)the electron energy flux qi.

In figure 3(c)we show the temporal relaxation of n0Diias a function of dimensionless time n0t for B/n0 = 200,500 and 1000 Hx.n0Dxx,n0Dyyand n0Dzzare 4.506,4.506 and 2.431 × 1024m-1s-1with no magnetic field.For B/n0 =200 Hx,n0Dxx,n0Dyyand n0Dzzare 4.915,4.608 and 2.330 × 1024m-1s-1,respectively;for B/n0 = 500 Hx,n0Dxx,n0Dyyand n0Dzzare 5.976,5.796 and 1.882 ×1024m-1s-1,respectively;for B/n0 = 1000 Hx,n0Dxx,n0Dyyand n0Dzzare 4.461,7.746 and 1.619 × 1024m-1s-1,respectively.n0Dxxand n0Dzzexhibit a transition from damped period decay to monotonic decay,while relaxation of n0Dyyis in general monotonic,which is in good agreement with White et al[20].Under the influence of a magnetic field,n0Dxxand n0Dyybecome larger than the quantities with no magnetic field,while n0Dzzbecomes smaller than the quantities with no magnetic field.However,with no magnetic field n0Dxx,n0Dyyand n0Dzzare all an order of magnitude larger than the quantities presented by White et al[20].Another unexpected phenomenon is that n0Dxxin the steady state does not increase as the magnetic field　increases(i.e.for B/n0 = 500 Hx,n0Dxxis the largest).

In figure 3(d)we show the temporal relaxation of ε as a function of dimensionless time n0t for B/n0 = 200,500 and 1000 Hx.The electron mean energy ε is 6.822eV with no magnetic field.The electron mean energies ε are 6.733,6.133 and 4.927eV for B/n0 = 200,500 and 1000 Hx,respectively.Relaxation of the electron mean energy ε is monotonic.The electron mean energy in the steady state decreases as the magnetic field increases.The reason for this is that the electron motion has changed due to the magnetic field and the electric field is unable to efficiently pump energy into the system.

In figure 3(e)we show the temporal relaxation of kionas a function of dimensionless time n0t for B/n0 = 200,500 and 1000 Hx.The electron reaction rate kionis 3.892 ×10-17m3s-1with no magnetic field.The electron mean energies ε are 3.782,3.183 and 1.982 × 10-17m3s-1for B/n0 = 200,500 and 1000 Hx,respectively.Relaxation of the electron reaction rate kionis monotonic.The electron reaction rate in the steady state decreases as the magnetic field increases,for the electron mean energy decreases with the increase of magnetic field and the ionization collision frequency decreases.

In figure 3(f)we show the temporal relaxation of Tiias a function of dimensionless time n0t for B/n0 = 200,500 and 1000 Hx.We observe that electron temperatures along the x,y and z direction are almost the same,meaning that the temperature is nearly isotropic.The electron temperature Tiiis 5.251 × 104K with no magnetic field.The electron temperatures Tiiare 5.184,4.726 and 3.802 × 104K for B/n0 = 200,500 and 1000 Hx,respectively.Relaxation of the electron temperature Tiiis monotonic.The electron temperature in a steady state decreases as the magnetic field increases for the same reason as that of the electron mean energy.

In figure 3(g)we show the temporal relaxation of n0γiias a function of dimensionless time n0t for B/n0 = 200,500 and 1000 Hx.n0γxx,n0γyyand n0γzzare 0,0 and-15.131 Jm-2 with no magnetic field.For B/n0 = 200 Hx,n0γxx,n0γyyand n0γzzare 6.131,0 and-14.802 Jm-2,respectively;for B/n0 = 500 Hx,n0γxx,n0γyyand n0γzzare 23.976,0 and-14.501Jm-2,respectively;for B/n0 = 1000 Hx,n0γxx,n0γyyand n0γzzare 41.892,0 and-23.142Jm-2,respectively.We find that the energy gradient parameter along the y(B)direction is always zero.This means that the electron energy in the y direction is uniform,for the magnetic field inhibits energy transportation in its own direction.Relaxation of n0γxx is monotonic,while relaxation of n0γzzis oscillatory in nature.The oscillatory amplitude of n0γzzincreases with the increase of the magnetic field.

In figure 3(h)we show the temporal relaxation of qias a function of the dimensionless time n0t for B/n0 = 200,500 and 1000 Hx.qx,qy,qzare-0.019,0 and-8.434 × 10-14Jms-1 with no magnetic field.For B/n0 = 200 Hx,qx,qyand qzare-1.084,0 and-8.221 × 10-14Jms-1,respectively;for B/n0 = 500 Hx,qx,qyand qzare-2.374,0 and-7.030 ×10-14Jm-2,respectively;for B/n0 = 1000 Hx,qx,qyand qzare-3.160,0 and-4.645 × 10-14Jms-1,respectively.We observe that the electron energy flux along the y direction is always zero,because there is no energy gradient in the y direction as shown in figure 3(g).qxand qzexhibit a transition from damped period decay to monotonic decay.

We should note that there are three distinct timescales for the Reid model which are not shown here:the gyro-period γ,the momentum relaxation time γmand the energy relaxation time γe,satisfying γ ≤ γm ≤ γe.However,for Ar plasmas in the presence of crossed electric and magnetic fields,only two distinct timescales are observed:the gyro-period γ and relaxation time γr.The reason for this may be that γmis of the same order as γe.

Another interesting phenomena is that when the magnetic field is switched on,the energy parameters such as ε,Tiiand n0γiichange slowest,WBz,qz,n0Diiand kionchange a little faster,and WBxand qxchange fastest.This means that the Lorentz force plays an important role in the initial relaxation of the plasmas.

4.Conclusions

In this study we have investigated the electron relaxation properties in magnetrons based on an improved multi-term approximation of the Boltzmann equation.We found that the two-term approximation is not accurate enough to obtain electron transport coefficients and a multi-term approximation is necessary.The magnetic fields can reduce the anisotropy of the velocity distribution function and a smaller lmaxis needed to achieve enough accuracy.For Ar plasmas in the presence of crossed electric and magnetic fields,only two distinct timescales are observed:the gyro-period γ and the relaxation time γr.The energy parameters such as ε,Tiiand n0γiichange slowest,WBz,qz,n0Diiand kionchange a little faster,and WBx and qxchange fastest.

路基沉降问题是当前高速工程中非常常见的病害问题，出现该问题之后，就会导致工程结构的稳定性下降，如果未能采取有效的措施来处理这一问题，就会导致工程无法正常进行，严重者会造成重大的滑塌事故，安全隐患比较大，建筑工程的施工也会受到很大影响。因此在高速公路的施工阶段，需要全面提升工程施工技术，有效解决所存在的路基沉降问题，大大提升了公路工程的建设施工速度。有了技术的提升就能够使公路工程的建设施工更具效果，沉降发生率也能够有效的降低，避免出现交通事故的问题，还能够促进我国公路事业的快速发展，具备非常高的经济效益与社会效益。

This work is supported by National Natural Science Foundation of China(grant No.51677024).

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