PHYSICAL REVIEW E 101, 053307 (2020)
Dynamic coupling between particle-in-cell and atomistic simulations
Mihkel Veske ,* Andreas Kyritsakis, and Flyura Djurabekova Department of Physics and Helsinki Institute of Physics, University of Helsinki, P.O. Box 43 (Pietari Kalmin katu 2), 00014 Helsinki, Finland
Kyrre Ness Sjobak CERN, Geneva, Switzerland and Department of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway
Alvo Aabloo and Vahur Zadin Intelligent Materials and Systems Lab, Institute of Technology, University of Tartu, Nooruse 1, 50411 Tartu, Estonia
(Received 19 June 2019; revised manuscript received 25 February 2020; accepted 16 April 2020; published 18 May 2020)
We propose a method to directly couple molecular dynamics, the finite element method, and particle-in-cell techniques to simulate metal surface response to high electric fields. We use this method to simulate the evolution of a field-emitting tip under thermal runaway by fully including the three-dimensional space-charge effects. We also present a comparison of the runaway process between two tip geometries of different widths. The results show with high statistical significance that in the case of sufficiently narrow field emitters, the thermal runaway occurs in cycles where intensive neutral evaporation alternates with cooling periods. The comparison with previous works shows that the evaporation rate in the regime of intensive evaporation is sufficient to ignite a plasma arc above the simulated field emitters.
DOI: 10.1103/PhysRevE.101.053307
I. INTRODUCTION explosion and plasma formation. This phenomenological de- scription does not give an insightful physical understanding Field-emitting tips play a detrimental role in various vac- of the underlying processes and cannot aprioripredict the uum devices that require high electric fields, such as vacuum behavior of a specific system. Such empirical considerations interrupters [1], x-ray sources [2], fusion reactors [3], and have been common in the vacuum arcing theory since the particle accelerators [4]. When a field emitter is subjected to 1950s, as the modern computational tools necessary to de- currents exceeding a certain threshold, the emission becomes scribe the physical processes of such a complex phenomenon unstable and swiftly results in a vacuum arc [5,6]. The latter is as vacuum arcing were not available. Thus, there is a growing characterized by the ignition of a plasma in the vacuum gap, need for the development of theoretical concepts and compu- which drives a large current and converts the gap into a short tational tools that describe the vacuum arc ignition in a rigor- circuit. The ignition of an arc in vacuum (vacuum breakdown) ous quantitative manner and provide a deep understanding of can cause catastrophic failure in electron sources [5,7]. the physical processes that lead to it. Although vacuum arcs have been studied more than half This has changed recently, since various computational a century and it is well-known that they appear after intense methods have been employed to study vacuum arcs. In a field electron emission [6,8], the physical mechanisms that recent work [11], plasma formation in Cu cathodes was lead from field emission to plasma ignition without the pres- studied by means of particle-in-cell simulations. It was found ence of any ionizable gas are not understood yet. One hypoth- that plasma can build up in the vicinity of an intensively esis commonly used to explain the plasma build-up in vacuum field-emitting cathode, assuming that not only electrons but after intense field emission is the “explosive emission” sce- also neutral atoms are emitted from the cathode surface. nario [8–10]. According to it, when intensive field emission Moreover, it was shown that it is sufficient to supply about takes place, there is a critical current density beyond which 0.015 neutrals per electron to initiate the ionization avalanche heating is produced at a rate the emitter cannot dissipate, leading to the formation of a stable plasma. Recent multiscale which leads to heat accumulation, sufficient to cause instant atomistic simulations on Cu nanotips [12] revealed a thermal runaway process which causes field-assisted evaporation of Cu atoms and nanoclusters at a rate exceeding the threshold *mihkel.veske@helsinki.fi found in Ref. [11]. Thus, combining these two independently obtained results, one can consider intensively field-emitting Published by the American Physical Society under the terms of the tips as plausible sources of plasma, since they can supply both Creative Commons Attribution 4.0 International license. Further species, electrons and neutrals, at individual rates. distribution of this work must maintain attribution to the author(s) However, it is not clear yet whether a field-emitting tip and the published article’s title, journal citation, and DOI. is indeed able to produce sufficient densities to initiate the
2470-0045/2020/101(5)/053307(15) 053307-1 Published by the American Physical Society MIHKEL VESKE et al. PHYSICAL REVIEW E 101, 053307 (2020) ionization avalanche, since there is no technique which can address this issue concurrently. To answer this question, a fully coupled molecular dynamics (MD) and particle-in-cell (PIC) simulation technique must be developed. The MD part will describe the behavior of atoms under given conditions, such as heat generated by field emission currents and stress FIG. 1. The relation between different components in the pro- exerted by the field, while the PIC part will follow the posed multiscale-multiphysics model. dynamics of the particles above the surface. In this way, the space-charge buildup above the field-emitting tip can also be found directly without a need for simplifying assumptions. movement of the atoms is followed by an MD algorithm, In Ref. [12] we attempted to take into account the space- the movement of the emitted electrons is tracked using the charge effect by using a semiempirical approximate model, PIC method. FEM provides the tools for concurrent and self- as proposed in Ref. [13]. This model treats the emitter as consistent electric field and heat diffusion calculations, whose planar, using a universal correction factor to account for the output affects the force and velocity calculations in MD and three-dimensional (3D) nature of the geometry, which was the force exerted on the electrons in PIC. Such a methodology an external adjustable parameter for our simulations, with forms the conceptual basis for the full coupling between metal its value roughly approximated based on previous PIC cal- surface and plasma simulations. A full schematic of the model culations [14]. Although we obtained promising results with is provided in Appendix A. this model, the semiempirical correction does not capture the The model is based upon the previously developed frame- complex 3D distribution of the space charge developed around work FEMOCS [19]. In FEMOCS, an unstructured mesh is the dynamically varying tip geometry. Moreover, the accuracy dynamically built around surface atoms, the density of which of the calculations of the space charge affects directly the is controlled independently in all regions of the simulation value of the field emission currents and all the subsequent domain. The mesh consists of linear hexahedral elements that processes developing in the tip due to the field emission are built into tetrahedra that allow achieving satisfactory accu- process. racy while keeping the computational cost within reasonable In this paper, we present an approach of the coupled MD- limits. For more details about meshing, local solution extrac- PIC technique for a more accurate estimation of processes tion, and optimization features in FEMOCS, see Ref. [19]. contributing to plasma onset. In particular, we develop a In the following sections we provide more details about the method that directly couples PIC and MD simulations of the newly developed methodology for calculating electric field, metal surface response to the field aided by the use of the electrostatic forces, and velocity perturbation. finite element method (FEM). We use the model to simulate similar process as in Ref. [12], but including accurately the 3D space-charge distribution that builds up above the highly A. Electric field curved nanometric field-emitting tip. There are some attempts A common way to take into account an electric field within in the literature to simulate the evolution of the space charge a computational model is to define it as the negative gradient using an MD-like method [15,16] where all interelectron of the electrostatic potential that, in turn, can be calculated interactions are considered. For specific applications, such by solving the Poisson’s equation in the vacuum (domain 1 models may be sufficient, but for computational efficiency in Fig. 2) we prefer to use the PIC method. The latter provides a sufficiently reliable kinetic description of space charge or ∇· ( ∇ ) =−ρ, (1) plasma evolution and allows incorporating the complex nature of the atomic processes and plasma-surface interactions, while maintaining a feasible computational load as compared to the MD-like methods [11,17]. We also present a comparison of the runaway process between two tip geometries of different widths. That comparison takes us closer to obtaining an es- MD timation for the runaway occurrence probability in structures region observed in experiments [18]. The incorporation of the PIC method into our simulations forms a decisive step towards the direct coupling between atomistic and plasma simulations, in order to fully reveal the processes that take place during the initiation of a vacuum arc. Wide extension
Thin extension II. METHOD We present here a model that combines classical MD, FEM, PIC, and electron emission calculations (Fig. 1). The model allows us to assess the effect of different physical processes on the dynamic evolution of the atomic structure FIG. 2. Boundary regions and geometry of the simulation of metal surfaces exposed to high electric fields. While the domain, which consists of an atomistic region and an extension.
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FIG. 3. Electric field calculation by means of particle-in-cell method. where is the dielectric constant and ρ is the space-charge 1. Particle mover density. We solve Eq. (1) iteratively with the boundary condi- The propagation of SPs is performed by solving iteratively tions (BC) defined in the vacuum (see Fig. 2)as Newton’s equations of motion for each SP: ∇ · n = E0 on 1, r ˙ = v , (3) ∇ · n = 0on2, (2) q v ˙ = E , = 0on3, m where q is the charge-to-mass ratio of SPs. In Eq. (3), SPs are where E0 is the long-range applied electric field. The bound- m ary value problem (1)–(2) is solved in FEMOCS by means affected only by the electrostatic field, since the magnetic field of FEM. The derivation of weak and algebraic formulation of which is induced by the currents within the tip is negligible. Eqs. (1)–(2) is provided in Appendix B. We solve Eq. (3) numerically by using the leapfrog integration scheme [20]. In this method, the positions and the velocities of SPs are calculated as B. PIC method v + . = v + QE , (4a) To obtain the space-charge distribution ρ(r ) in the vacuum k 0 5 k k above the metal surface, which enters Eq. (1), we use the PIC r k+1 = r k + tv k+0.5, (4b) method [20–22]. Within PIC, the space charge is formed by a v + = v + . + QE + , (4c) set of superparticles (SPs), which is a common approach for k 1 k 0 5 k 1 = q t space-charge and plasma simulations. Each SP is a group of where the subscript indicates the time step, Q m 2 , and t charges of the same type, electrons or ions, that act within the is the PIC time step. Note that the calculation of v k+1 requires model as a single particle. In other words, the SPs represent a the value of the electric field at the time step k + 1. This is sampling of the continuous phase space occupied by the real obtained in the field solver between steps (4b) and (4c), using particles like electrons or ions. Such a grouping is justified by as input the positions r k+1. the fact that the electrostatic and electrodynamic interactions Since we have used periodic BCs for the electric field in depend on the charge-to-mass ratio instead of the charge or the the lateral directions, we apply the same type of symmetry mass of the particles. The size of each SP is determined by its for the SPs. This means that an SP that crosses the side weight wsp, which corresponds to the (fractional) number of boundaries 2, will reappear at the symmetric point at the particles it represents. The SP weight is determined indepen- opposite boundary of the box. Such a mapping, however, dently for each simulation, in order to achieve a sufficiently occurs extremely rarely, as the simulation box is chosen wide smooth charge distribution with a feasible number of SPs. enough to prevent particles reaching sides before hitting the The movement of SPs is affected by the global electric top boundary 1 where they are absorbed. For that reason, field, which includes the long-range electrostatic interactions the contribution of mirror images of the simulation box to the between particles. The short-range interactions with nearby total SP flux is negligible. Note that the top boundary with a SPs are taken into account by means of binary Monte Carlo Neumann BC does not represent a physical anode electrode, collisions (see Appendix F for details). This makes it possi- but rather the uniform far field behavior of the gap between ble to significantly enhance the computational efficiency and two macroscopically flat electrodes. Thus the box is chosen simulate large systems for long times. Furthermore, the binary in our simulation to be sufficiently high so that the electric collision technique can be used to consider the interactions of field becomes uniform and the space charge negligible, which different types of particle species in a plasma, such as impact motivates the SP removal at that side. ionizations. Our current model, however, includes only the Next to the top boundary 1, the cathode boundary 3 electrons, ignoring the presence of ions as well as electron- also acts as a particle absorber. On 3, the absorbed particles neutral interactions. Within this approximation, we can study may cause secondary effects, such as secondary and higher the processes at the early stage of plasma formation, which order emission, plasma recycling, impurity sputtering, etc. is the focus of the current work. However, the model can Since these effects are beyond the scope of the current model be extended further if later stages of plasma evolution are of and almost all electrons are removed at the top boundary, we interest. In the following four sections, we give an overview leave their implementation for future work. Some attempts to of the various stages of the PIC model, which are summarized take into account the above-mentioned effects can be found in Fig. 3. elsewhere [23–27].
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2. Particle weighting them an initial velocity An important part of any PIC simulation, both conceptually q 1 and computationally, is mapping of the electric field to the v 0 = Et + R (9) m 2 position of SPs. In our model, we first locate the finite element that surrounds the particle and then use the shape functions of and displacement that element to interpolate the electric field at the position of = the SP. The search for the element is carried out in a similar r v0 tR. (10) manner as during the particle injection (see Appendix E), but 1 We add a term 2 in Eq. (9) to accelerate the particle by half instead of the three barycentric coordinates, here we need a time step and initialize the velocity for proper leapfrog four, i.e., one for each tetrahedral vertex. stepping. It is needed, because to enhance computational To solve Eq. (1) by solving a matrix equation (B7), an efficiency, we inject SPs between particle mover steps of (4a) inverse operation needs to be performed since the SP charge and (4b) that were described in Sec. II B 1. The additional is required in the right-hand side (RHS) fi of Eq. (B7). This, random number R provides a fractional push by a small however, cannot be done immediately, as the SP in PIC is random fraction of the previous time step, preventing the par- assigned a discrete charge q, while in Eq. (1) the charge ticles from forming artificial bunches [11]. This small initial ρ appears in a form of charge density . To overcome this issue, push enhances computational stability, without significantly δ we represent the charge of each SP by means of the function, affecting the results, as shown in Sec. III A. so that ρ(r ) = qiδ(r − r i ), (5) 4. Electron superparticle collisions i In addition to interaction with the field, every electron where δ(r − r i )istheDiracδ function and the summation experiences the effect of other charges present in the system. goes over the SPs that are located in the finite element where A naive approach to take into account the Coulomb forces ρ is evaluated. By doing such a substitution, the RHS of would be to calculate exactly the interaction between each pair Eq. (B7) becomes of particles within the Debye sphere. This, however, would 2 result in O(n ) complexity and is therefore not suitable for the qwsp large systems aimed in the present work. Instead, we applied fi = Ni(r j ) + NiE0 d, (6) j 1 a more reasonable Monte Carlo binary collision model [29]. More details about it can be found from Appendix F. where r j are the coordinates of the jth SP that is located in the element where f is computed. It is important to note that i C. Electrostatic forces to prevent self-forces, the same shape functions Ni areusedas during the mapping of the field to the position of the SPs and The charge induced by an electric field on the metal surface during the distribution of the charge inside an element. affects the dynamics of the atomic system due to additional electostatic interactions. To assess this effect, the value of the 3. Particle injection induced charge on the atoms must be estimated and introduced to the MD simulations where electrostatic interaction between In our simulations, the space charge is built up due to the atoms is included as a force perturbation. In this way the intensive electron emission. Using the field emission tool total force on an atom becomes F = F + F + F , where GETELEC [28], we calculate the current density J in the EAM L C e F is the force obtained from the interatomic potential. The centroid of each quadrangle that is located at the metal- EAM main perturbation is due to the Lorentz force vacuum boundary. Je determines the number of electron SPs, 1 nsp, that will be injected from a given quadrangle with area A FL (r ) = qE(r ). (11) at given time step: 2 In this equation, the term 1 originates from the Maxwell stress JeAt 2 nsp = , (7) tensor [30] and takes into account that only half of the charge ewsp is exposed to the field; the rest is shadowed by the material where e is the elementary charge. In general, nsp is a real domain. A less significant contribution to the total force F number that has integer and fractional parts. Since the number comes from the Coulomb interactions between the surface of injected SPs can be only integer, we use a uniformly charges distributed random number R ∈ [0, 1] to decide whether to 1 qiq j round nsp up or down: FC (r i ) = rˆij exp(−ξrij), (12) 4π 0 rij j=i n sp = floor(nsp)ifR nsp − floor(nsp), where qi is the charge of ith atom, rij the distance between n sp = ceil(nsp) otherwise. (8) the ith and jth atoms, andr ˆij the unit vector in the direction This discretization scheme guarantees that on average, the of r ij. The exponent in Eq. (12) describes the screening of the number of emitted SPs remains the same as given by Eq. (7). ith charge by the conduction electrons in metals and the value The emitted SPs are distributed uniformly on the quadrangle of the screening parameter ξ depends on the material and its surface. More details about it can be found in Appendix E. crystallographic orientation. In our case, ξ (Cu) = 0.6809 Å−1 After injection, the electric field E = E (r )isusedtogive as defined in Ref. [31].
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The surface charge can be estimated according to Gauss’s Voronoi facets law [31] exposed to vacuum Voronoi Q E · dA = , (13) cell 0 where Q is the charge inside a closed surface .Using Shift Support Eq. (13), one can discretize the continuous distribution of vector point surface charges on a metal surface to assign a partial charge to each atom on the surface corresponding to the value of the electric field applied at the atom position. However, it Triangles Bulk Surface is not trivial to decide how to identify the area which con- atom atom tributes to the charge on a given atom. In Refs. [12,31], a FIG. 4. Generation and usage of Voronoi cells for surface charge rectangular geometry of the grid points associated with each calculation. atom was suggested. Yet this approach is limited by a high computational cost for large-scale simulations. In the current work, we selected a different approach to calculate the surface (1) Copy surface atom as a new point p charge, which allowed for higher computational efficiency (2) Locate the triangle f where the projection of p ,in while maintaining satisfactory accuracy. This approach con- direction of f normal n , lies within f sists of two steps. First, we calculate the charge of the mesh (3) Move p in the direction of n for distance λ faces in the metal-to-vacuum boundary, hereafter referred to (4) Loop until each surface atom has a support point. as face charge: Empirical tests have shown that good results are obtained λ Qi = 0EiAi, i = 1, 2,...,nfaces. (14) if the length of the shift vector equals one lattice constant of the atomistic system. The coordinates of the surface atoms Notice that in Eq. (14) it is enough to use the scalar field and the support points are input in the TetGen [32] to generate and area, since E A due to the Dirichlet BC. By using a a Voronoi tesselation around these atoms. Note that the bulk ≡ − weight function wij w(r i r j ) (not to be confused with the atoms are in Fig. 4 shown for visual purposes only and are not PIC superparticle weight wsp), the face charge is distributed considered during the Voronoi tesselation generation. between the atoms as For each surface atom, we determine the Voronoi facets that are exposed to the vacuum, which are used in Eq. (13)to q j = wijQi, j = 1, 2,...,natoms. (15) i assess the partial charge associated with the atom. Assuming that the electric field in the location of the facets equals The total charge must remain conserved, therefore the weight approximately the field at the atom position, E ≈ E ,we function must satisfy i j approximate Eq. (13) for atoms in the regions with increased roughness as wij = 1 ∀ j. (16) i q j = 0E j · Ai, (18) As the exact mathematical form of wij can be chosen quite arbitrarily, we chose a form that is computationally cheap and i prevents singularities for close points: where the summation goes over the Voronoi facets of the jth exp(−|r i − r j|/rc ) atom that are exposed to vacuum. w = , ij −| − |/ (17) j exp( r i r j rc ) where the cutoff distance rc determines the range where the D. Heating charge Qi is distributed. The tests show that on flat regions the We have shown previously [12,33,34] that thermal effects results are almost independent of the exact value of rc until caused by field emission play a significant role in the evolution it exceeds the maximum edge length of surface triangles. For of nanotips under a high field. To take these effects into this reason in our simulations we use an empirical rc value of account, we calculate the electron emission current density J 0.1 nm. e and the Nottingham heat PN [35,36] on the emitting surface The face-charge method allows calculating an accurate by using our field emission tool GETELEC [28]. To calculate surface charge for atoms that are located on atomic planes. the volumetric resistive heating power density, we use Joule’s Charges in the regions with protruding features require further law as effort to achieve satisfactory accuracy. We improve the accu- racy by building the Voronoi tessellation around the surface P (T ) = σ (T )(∇ )2, (19) atoms in those regions as shown in Fig. 4. The facets of the J Voronoi cells are assumed to constitute the surface area for where σ (T ) is the electric conductivity and is the electric the given atom to estimate the charge on it. Before generation, potential obtained by solving the continuity equation in a however, additional support points need to be built above metal (domain in Fig. 2) the surface to limit the extension of the Voronoi cells into 2 the infinity. We build such support cloud by the following ∇· σ ∇ = , process: ( ) 0 (20)
053307-5 MIHKEL VESKE et al. PHYSICAL REVIEW E 101, 053307 (2020) which comes with the following boundary conditions: