Secondary Electron Emission from Surfaces with Small Structure A. R. Dzhanoev,1 F. Spahn,1 V. Yaroshenko,2 H. L¨uhr,2 and J. Schmidt3 1Universit¨atPotsdam, Karl-Liebknecht-Str. 24/25 Building 28, 14476 Potsdam-Golm, Germany 2GFZ, German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany 3University of Oulu, Astronomy and Space Physics, PL 3000, Oulu, Finland (Dated: May 3, 2016; http://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.125430) It is found that for objects possessing small surface structures with differing radii of curvature the secondary electron emission (SEE) yield may be significantly higher than for objects with smooth surfaces of the same material. The effect is highly pronounced for surface structures of nanometer scale, often providing a more than 100% increase of the SEE yield. The results also show that the SEE yield from surfaces with structure does not show an universal dependence on the energy of the primary, incident electrons as it is found for flat surfaces in experiments. We derive conditions for the applicability of the conventional formulation of SEE using the simplifying assumption of universal dependence. Our analysis provides a basis for studying low-energy electron emission from nano structured surfaces under a penetrating electron beam important in many technological applications. PACS numbers: 68.37.-d, 79.20.Hx, 94.05.-a INTRODUCTION use the simplifying assumption of SEE from a large, pla- nar sample [2{9, 11]. It has been suggested that the energy dependence of the SEE yield can be described by Secondary electron emission from solids by electron the Sternglass universal curve [6], when the yield is nor- bombardment has been subject of experimental and the- malized by the maximum yield and the primary electron oretical studies for many decades [1{24], with a wide vari- energy by the energy where the yield is maximized [2]. ety of applications including voltage contrast in scanning However a series of measurements of SEE covering a wide electron microscopy, micro channel plates, plasma display range of primary energies [1, 14] shows that the theory panels, electron beam inspection tools. In space, atten- of Sternglass fails to fit the experimental data at high tion has focused on SEE from spacecraft surfaces or small primary electron energy. The empirical formula devel- dust particles caused by auroral electrons or hot electrons oped by Draine and Salpeter [9] generally shows a better in planetary magnetospheres [25{28]. To date, theoret- agreement with experiments but overestimates the data. ical studies of SEE usually involve a slab model which Later Chow et al. [11] have modified the yield equation often gives a reasonable estimate of the SEE yield (but by Jonker [3] and derived the yield for secondary emission see also [9, 11, 12] and discussion below). In this paper, from a spherical dust grain immersed in a plasma envi- we show that the interplay between the penetration depth ronment. The influence of porosity on electron-induced of primary electrons, the escape depth of secondary elec- SEE was considered by Millet and Lafon [12]. There are trons and the size of surface structure (surface curvature) also some numerical models of SEE from dust grains [24]. can be the dominant mechanism for the SEE from small However all these approaches assume a smooth surface objects. Moreover the SEE from configurations involving of the object. The first attempt to include surface struc- nano structures are of fundamental importance for basic tures have been made by Nishimura et al. [19]. Their science as well as for applications [9{11] ranging from Monte Carlo simulations have shown that neglecting the astrophysics to technological processes. The detailed un- surface roughness may considerably underestimate the derstanding and proper interpretation of the SEE yield magnitude of the secondary electron yield. These results from either nano-scaled structures on surfaces or nano- are in a good agreement with the characteristics of low sized objects is strongly desired because such knowledge secondary and reflected primary electron emissions from is, for instance, important for scanning electron micro- textured surfaces measured by Wintucky et al. [17]. scope imaging of small objects and any charging pro- arXiv:1605.00637v1 [cond-mat.mes-hall] 2 May 2016 cesses where secondary electron currents are involved. The total electron yield is often written as a sum σ = r + η + δ of elastically (r) and inelastically (η) MODEL backscattered electrons (BSE) as well as true secondary electrons (SE) (δ). For incident electrons with energies In this paper, we study the effect of small spatial sur- in the range where the SEE dominates (typically above face structures of three-dimensional samples on the SEE 100 eV) elastically and inelastically backscattered pri- efficiency. We suggest an analytical expression for the maries (R = r + η) constitute only a small fraction of SEE yield accounting for the effect of surface curvature in the total yield [23]. In the literature, it is common to an approach generalizing existing elementary SEE mod- 2 els. We show that the presence of small structures on a Z φc l × exp − sin φdφ sample surface destroys the universal dependence of yield 0 λ on energy. Moreover, we find a significant growth of the electron yield. and average over the incidence angle θ Physically, the deviation from the universal behavior Z θc occurs because the curvature radii of individual surface hδ(E0; k)iθ = C δ(E0; k) sin θ cos θdθ: (2) structures can be significantly different from the sample's 0 overall radius of curvature. To highlight this effect we Here, following Jonker [3], we consider the case when sec- consider two elementary examples of surface structures ondary electrons are generated at a distance x from the like a single small spherical grain [Figure 1(a)] and a sin- entry of the primary electron into a sample and move to gle bump on a flat sample's surface [Figure 1(b)]. Surface the target's surface at an angle φ with the direction to the structures of more complex shape can be reduced to these nearest surface point. The Heaviside function Θ(∆ − x) elementary ones. For comparison we also give the SEE selects contributions to SEE only from primaries travel- characteristics in the case of a smooth surface [Figure ing on a straight path lying entirely in the target body. 1(c)]. Here, we deal with the case when the density of Here ∆(k; θ) is the linear dimension of the object along surface structures is low, i.e. when the distance between the path of the primary, labeled by the direction θ. The structural elements is much larger than their size. This surface curvature k is the function of the local curvatures excludes the effect of re-entrance of emitted electrons into χ of the object. The parameter λ denotes the mean free another part of the surface since the probability of this path of secondary electrons within the target, is the en- process becomes very small [19]. ergy necessary to produce one secondary electron, dE=dx is the energy loss of the primary per unit path length (a) and l(x; φ) is the distance that is necessary to reach the – ep − sample's surface from the point where secondaries es are generated. The normalization constant C, the distance θ l, and the limits of integration in Eq. (2) depend on the e– object geometry. Equations (1) and (2) are very general φ s and can be used to study the SEE from objects of arbi- trary shape. In order to be more specific we shall focus on the cases described by [Figures 1(a),(b),(c)]. For all types of surface structures that we consider C = 1. In equation (1) for a spherical dust grain [Figure 1(a)] the maximal possible penetration depth for incident direction (b) with angle θ to the surface normal vector is ∆ = 2a cos θ −1 n and R(E0) = (An) E0 is the projected range. The pa- rameters A and n depend on the projectile and target and are determined by experimental measurement of R(E0), giving n = 1:5 for electrons. In the case of a bump, [Fig- ure 1(b)] we have ∆ = a cos θ. In both cases we take a (c) semi-infinite slab as the underlying sample object. Fi- nally, in the case of Figure 1(c), a smooth surface, the maximum depth is always given by xm = R(E0). Note that when dealing with a smooth surface of radius of cur- vature r one can obtain from equation (1) an expression FIG. 1: (color online). A sample surface with radius of cur- for the secondary electron yield valid for a semi-infinite vature r possessing different structures. a) A spherical grain slab by taking the limit x=r ! 0. of radius a on a sample. The grain's curvature is determined We emphasize that despite the simplifications intro- by its radius viz. k = 1=a2; b) A bump of radius a on a duced by the model of SEE production described by sample; c) Smooth surface. Jonker [3], it does provide a useful approximation to ex- perimentally observed data. It was shown that the com- To describe SEE due to isotropic incidence of primary bination of this model with a Monte Carlo trajectory − electrons ep of energy E0 to the objects with varying simulation allows SEE and BSE yields to be calculated, surface curvature k we generalize a commonly used ex- simultaneously, with good accuracy [18]. pression [3, 6, 11] for the electron yield as To derive conditions for the applicability of the con- ventional formulation of SEE we examine how the sim- Z R(E0) 1 dE plifying assumption of universal dependence is expressed δ(E0; k) = − Θ [∆ − x] dx × (1) 0 dx in the framework of the current study.