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Low energy electron reflection from tungsten surfaces P. Tolias Division of Space and Plasma Physics, Association EUROfusion-VR, Royal Institute of Technology KTH, Stockholm, Sweden Theincidenceofverylowenergyelectronsonmetalsurfacesismainlydictatedbythephenomenon of quantum mechanical reflection at the metal interface. Low energy electron reflection is insignif- icant in higher energy regimes, where the more familiar secondary electron emission and electron backscattering processes are the dominant features of the electron-metal interaction. It is a highly 6 controversial subject that has mostly emerged during the last years. In this brief note we examine 1 the source of the controversy, present some basic theoretical considerations, recommend a dataset 0 ofreliable experimentalresultsforthereflectionoflowenergyelectronsfrom tungstensurfaces and 2 discussthesuppression ofreflected electrons byexternalmagnetic fieldsin thelight of applications n in fusion devices. a J Motivation. For relatively large incident electron ener- dictsconcreteexperimentalevidenceforcleanmetalsur- 8 gies Eienc & 50eV, electron emission from solids induced faces but also lacks a sound theoretical basis. For these ] by electron impact is mainly a consequence of backscat- reasonsithasbeenrecentlycriticizedonbothends[8–10]. h teringofprimariesandemissionofsecondaries. Recently, It is safe to state that the reflection coefficient cannot p reliable experimental results on electron backscattering reach values close to unity, unless plasma fundamentally - m and secondary electron emission from tokamak relevant changes the nature of electron-solid interactions or the materials have been reviewed and semi-empirical quasi- natureofthesurfaceinamannernotdocumentedatthe s a universalexpressionshavebeenproposedto describethe moment. l respective yields[1, 2]. We emphasize though that these Survey of theoretical results. We begin our investigation p . expressions are valid for Eienc & 50eV and that extrap- of low energy electron reflection with presenting some s olations below this energy range should not be carried theoretical considerations in order to demonstrate that c i out. However, in lack of an alternative option, such ex- the electron reflection coefficient does not reach unity s y trapolations are carried out in many fusion works. For but also in order to put emphasis to the intricate na- h this reason, in this work we shall review theoretical and ture of the problem. We assume that the metal surface p experimental results on electron induced electron emis- is planar and the electron incident velocity is normal to [ sion in the regime of very low incident energies. Before the surface. It is desirable to approximate the passage proceeding, it is essential to define the inner metal po- of an electron to the metal interior by a single-body de- 1 v tential that is the energy gained by a free electron when scription, where the one-electron Schrödinger equation 7 it penetrates a metal surface and is equal to the energy is solved for an effective one-dimensional potential. The 4 difference between the vacuum level and the bottom of simplest way to model the vacuum-metal interface is by 0 the valence band, V0 = Bw+Wf, where Wf is the work assuming a discontinuous square potential barrier, a po- 2 function and Bw is the valence band width. tential energy structure with a negative step of height 0 . For Eienc . 50eV, electron backscattering and sec- equaltothe inner metalpotentialV0 situatedatthe pla- 1 ondary electron emission from metals start becoming nar surface boundary x=0. 0 negligible. Even more important, at such a low energy 6 regime their semi-empirical description breaks down be- −V0, x≤0 1 V(x)= . : inglargelybasedonthecontinuousslowingdownapprox- ( 0, x≥0 v imation, since the incident electrons can only penetrate i X few atomiclayersofthe solidbeforethermalizing. More- The one-electron Schrödinger equation can be trivially over,whentheincidentenergybecomescomparablewith solvedforsuchapotentialenergy. However,the analytic r a the inner metal potentialthat acquires values within the solution yields a spurious 100% reflection at the limit of range ∼10−20eV for most metals, reflection of the in- verylowincidentenergy. Thisisaconsequenceoftheun- coming electron at the potential barrier becomes signifi- physicaldiscontinuityatthe interface[11]. Sucha model cant. This phenomenonis typically coinedas low energy neglectstheelectromagneticresponseofthemetaltothe electron reflection and it is quantified by the reflection incoming charge, which leads to a force acting on the coefficient ξe, the ratio of reflected to incident electrons. electron in a manner that guarantees that its potential There has been a small number of experimentalworks energy varies continuously. The next simplest way that claimingthatelectronreflectionfromtechnicalmetalsur- remedies this problem is to assume a continuous image faces at low incident energies can be significantly en- potentialbarrier,apotentialenergystructurethatfollows hanced, even reach 100% at the limit of zero incident the classical electrostatics image law. energy[3]. This argument has been iterated within the fusioncommunityinordertoexplainanumber ofexper- V(x)= −V0, x≤x0 , imental observations[4–7]. However, it not only contra- (−e2/(4x), x≥x0 2 where x0 is the displacement of the image charge plane with respect to the surface of the metal, given by x0 = e2/(4V0) in order to ensure continuity. The one-electron Schrödingerequationcanbeanalyticallysolvedforsucha potentialenergy[12]. Ityieldsareflectioncoefficientthat never exceeds 0.07 at the limit of zero incident energy and behaves as a monotonically decreasing function of the incident energy. The results are similar[11] for more involvedandmoreaccurateimagepotentialsofquantum mechanical origin[13–15]. However, such a model ne- glects the periodic nature of the potential in crystals. A more elaborate model that considers the metal interior is the continuous image-sinusoidal potential barrier that superimposes a sinusoidal modulation for x≤x0. FIG. 1: The elastic electron reflection coefficient Re(Eienc) −V0+V1sin[a(x−x0)], x≤x0 for normal electron incidence on monocrystalline tungsten - V(x)=(−e2/(4x), x≥x0 , Wfro(m11t0h)e-ewxiptheriemneerngtisesofinBtrhoensrhatnegineEetienacl=[230]−, K2h5aenV.etRaelsu[2lt4s] and Yakubovaet al[25]. where a and V1 are adjustable parameters. The one- electron Schrödinger equation can also be analytically solved for such a potential energy[16]. It yields a reflection coefficient that contains incident energy bands wherethe reflectionreaches100%. Thisisaconsequence of the so-far neglected inelastic component of the problem, since for instance the incoming electron can lose part of its incident total energy by interacting with the valence electrons of the metal[17]. Essentially,the problem is of many-body nature and cannot be described by a real effective potential. However, a complex potential barrier can be assumed, where the imaginary part de- scribes the inelastic aspects[18, 19]. The one-electron Schrödinger equation for such a potential no longer pre- servestheprobabilitycurrent,whichsimplyimpliesscat- tering of the electron to another energy[20]. There only exist numerical solutions for such model potentials. In FIG. 2: The total electron reflection coefficient ξe(Eienc) accordance with experiments, the reflection peaks are for normal electron incidence on monocrystalline tungsten - broadened and get suppressed far below unity[21, 22]. W(110)-withenergiesintherangeEienc=0−25eV.Results Moreover, reflection is no longer purely elastic but also from theexperimentsof Bronshtein et al[23], Khanet al[24] bears an inelastic component, again as measured in ex- and Yakubovaet al[25]. periments. Survey of experimental results. Clearly, the calcula- e e tion of the reflection coefficient ξe from first principles deviationsinthe measuredξe(Einc)andRe(Einc)[23–26] is a formidable task. In addition, the functional form -inwhatfollowsweshalldenotethetotalelectronreflec- of ξe(Eienc) is so highly non-monotonic and depends so tion coefficient by ξe and its elastic part by Re. strongly on the composition of the metal surface that The experimental results of Bronshtein et al[23] and simple empirical expressions are very hard to construct. Khan et al[24] are considered as the most reliable in Consequently, one has to resort to experimental results the literature. As seen in figure 1, in spite of the non- that are very sparse and contradicting[23–26]. Fortu- monotonic nature of the elastic electron reflection coeffi- nately, tungsten is the most studied material but almost cient, they correlate very well with each other. As seen exclusively as a single crystal (it is essential to have a in figure 2, they exhibit less than 25% deviations for the well-defined surface for reproducibility). We point out total electron reflection coefficient in the whole energy that very low energy experiments are notoriously hard range with the exception of a very narrow region below to perform. Apart from strict requirements in terms of 3eV. On the contrary, the results of Yakubova et al[25] surfaceconditions,detectordesign,beamcollimationand for Re strongly deviate in the whole range, while for ξe stability,oneneedstobeconfidentthatthecollectedelec- they start deviating below 7eV. Results from other au- troncurrentisindeedreflectedfromthemetaltargetand thorsexhibitmuchlargerdeviations[26–32],butthemea- isnotpartofthe incidentcurrentthatneverreachedthe suredvalueofξeforcleanmetalsurfacesisalwayssmaller target[10]. These difficulties are reflected in the strong than unity. 3 With the aid of figure 4, we can demonstrate that the extrapolation of secondary electron emission yield for- mulas to very low energies apart from being physically wrong can also lead to misleading results. We shall em- ploy the Young-Dekker formula for the tungsten values Emax = 600eV, δmax = 0.927 and the optimal exponent k = 1.38[2]. For Ee = 5eV the extrapolated yield is inc δ ≃ 0.02 in contrast to ξe ≃ 0.17, for Eienc = 10eV the extrapolated yield is δ ≃ 0.04 in contrast to ξe ≃ 0.3, for Eienc = 20eV it is δ ≃ 0.075 compared to ξe ≃ 0.4. For verylow energies,the difference betweenthe extrapolated and the experimental values can even reach one order of magnitude! Angle of incidence dependence. To our knowledge, there FIG. 3: The elastic electron reflection coefficient Re(Eienc) are no experiments available that study the dependence for normal incidence of electrons on tungsten with energies roughly in the range Eienc = 0−30eV. Results from the ex- oafboξveeatnhdeoRreetiocanltchoensaidnegrleatoiofnisn,coidneenccoeu.ldBaassseudmoenththaet perimentsofBronshteinetal[23]forpolycrystallinetungsten the problem is inherently one-dimensional. Therefore, and different tungsten single crystals, W(110) and W(112). only the normal velocity component would be relevant andaEe →Ee cos2θ mappingwouldbeviable,where inc inc θ is the incident angle measured from the normal to the surface. This is not valid for polycrystalline tungsten due to electrostatic patch effects[17]. The gen- eral consensus in the literature is that the work function of polycrystalline tungsten at room temperature is Wf ≃ 4.55eV[33–36]. However, the work function of monocrystalline tungsten exhibits strong variations for differentcrystallographicorientations[36],forinstancein the case of low index crystal faces W(110) → 5.30eV, W(100) → 4.58eV, W(111) → 4.40eV. Therefore, the work function deviations between different crystal faces can reach 0.9eV. This leads to a different surface potential on each mono-crystal region (polycrystalline tungsten surfaces are not strictly equipotential as stated by FIG. 4: The total electron reflection coefficient ξe(Eienc) elementary electromagnetic theory) and consequently to strongly localized patch fields that can be important for for normal incidence of electrons on tungsten with energies roughly in the range Eienc = 0−30eV. Results from the ex- surface effects such as low energy electron reflection. In perimentsofBronshteinetal[23]forpolycrystallinetungsten absence of experimental results, we shall assume that and different tungsten single crystals, W(110) and W(112). there is a weak angle of incidence dependence based on the argument of Stangeby that the omnipresent surface roughness diminishes the importance of such an ef- The experiments of Khan et al[24] span the energy fect[37]. range of 0−100eV for three different single crystal ori- Magnetic field suppression of electron reflection. Finally, entations and also study the effect of adsorbates. How- weemphasizethattheuseofexperimentalresultsforthe ever, we recommend the use of the results of Bronshtein low energy electron reflection coefficient is only appro- et al[23] for fusion applications, since these authors are priate for devices with magnetic field orientation nor- the only ones that also employed polycrystalline tung- mal to the plasma-facing-component (PFC). For such sten. The experimental data are plotted in figures 3 a magnetic field topology, the gyromotion of the re- and 4. It is worth mentioning that the investigations of emerging electrons cannot lead to their recapture. How- Bronshteinet al arethemostcomprehensive[23];lowen- ever, in tokamaks, magnetic field lines connect to the ergyelectronreflectionexperimentswerecarriedoutwith PFCs nearly tangentially. In devices such as ITER, also a large number of elemental metal surfaces (tungsten, duetothehighmagneticfieldstrength,thereflectedelec- nickel,aluminum,copper,molybdenum,silver,tantalum, tronscouldpromptlyreturntothePFCwithintheirfirst gold),semi-conductingsurfaces(silicon,germanium)but gyration[38–40]. The return fraction will be determined alsowithcompositesurfaces(oxides,salts). Independent bythecompetitionbetweentheLorentzforceandthere- of the nature of the surface studied, the measured value pellingelectrostaticsheathforce,thusitwillstronglyde- of ξe close to limit of zero incident energy was always pendontheenergydistributionofthereflectedelectrons. smaller than unity. Undoubtedly,lowenergyreflectionwillbesuppressedbut 4 the degree of suppression can only be quantified by sim- TsofthePFC[17],whichleadstoamostprobableenergy ulations (particle-in-cell codes), where the data quoted Ts/2<0.3eV. This implies a stronger Lorentz force and above can serve as an input. Two additional comments a more effective magnetic suppression of reflected elec- are inorder: First, since reflected electronsnearly retain trons[42–44]. Second,ithas neverbeenacknowledgedin theincidentelectronenergy,theycanbemuchmoreener- the literature that the return fraction from any emission geticthansecondaryelectronsorthermionicelectrons[1]. process will not be fully absorbed by the PFC but part SecondaryelectronscanbeassumedtofollowtheChung- of it will be reflected. This highlights the fact that with- Everhart energy distribution[41], which leads to a most out the aid of simulations it is neither possible to deter- probable energy Wf/3 ≃ 1.5eV. Thermionic electrons minethesteadystatecharacteristicsofthislocalelectron can be assumed to follow a Maxwellian energy distribu- population (energy distribution, number density) nor to tion with temperature equal to the surface temperature quantify its effect on the global sheath structure. [1] Tolias P 2014 Plasma Phys. Control. 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