Local changes of work function near rough features on Cu surfaces operated under high external electric field

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1 Local changes of work function near rough features on Cu surfaces operated under high external electric field Flyura Djurabekova, 1, a) Avaz Ruzibaev, 1 Eero Holmström, 2, 3 Stefan Parviainen, 1 and Mikko Hakala 2 1) Helsinki Institute of Physics and Department of Physics, P.O. Box 43, FI University of Helsinki, Finland 2) Department of Physics, University of Helsinki, P.O. Box 64, FIN University of Helsinki, Finland 3) Department of Earth Sciences Faculty of Maths&Physical Sciences, UCL Earth Sciences, Gower Street London WC1E 6BT, UK (Dated: 10 January 2014) Metal surfaces operated under high electric fields produce sparks even if they are held in ultra high vacuum. In spite of extensive research on the topic of vacuum arcs, the mystery of vacuum arc origin still remains unresolved. The indications that the sparking rates depend on the material motivate the research on surface response to extremely high external electric fields. In the present work by means of density-functional theory calculations we analyze the redistribution of electron density on {100} Cu surfaces due to self-adatoms and in presence of high electric fields from -1 V/nm up to -2 V/nm (-1 to -2 GV/m, respectively). We also calculate the partial charge induced by the external field on a single adatom and a cluster of two adatoms in order to obtain reliable information on charge redistribution on surface atoms, which can serve as a benchmarking quantity for the assessment of the electric field effects on metal surfaces by means of molecular dynamics simulations. Furthermore, we investigate the modifications of work function around rough surface features, such as step edges and self-adatoms. PACS numbers: Mg,52.80.Vp,79.20.Rf,61.80.Jh Keywords: work function, Cu surface, high electric fields, DFT. I. INTRODUCTION The performance of accelerating radio-frequency (rf) cavities in linear colliders (for instance, the Compact Linear Collider (CLIC) under development at CERN 1 ) is limited by a critical value of the accelerating gradient of the applied rf field. If the applied field is higher than the critical value, the frequency of sparking events near metal surfaces of accelerating structures becomes intolerable because of the surface damage and the loss of particle bunches. This field limit, however, was found to have strong a surface material dependence, which shifts this problem from the domain of plasma physics to the real of materials science. If the surface is operated at an electrostatic (direct current dc) field, it also starts producing sparks even in the deep vacuum, when the value of the field is of order of hundreds of MV/m 2. In the present work we focus on understanding the mechanisms underlying the electric breakdowns believing that the insight gained in our studies will also be a significant step towards the understanding of mechanisms of more complex breakdown phenomenon observed near conductive surfaces in presence of rf fields. In practice, metal surfaces remain fairly rough even after thorough polishing, at least, on a nanoscale. Such rough features cause uneven redistribution of electron a) Electronic mail: flyura.djurabekova@helsinki.fi density shaping the potential energy wells and barriers near the surface defects even in the absence of an external field. Switching on the field affects the potential landscape, deepening or shallowing the potential minima and barriers. This will affect the surface diffusion of atoms, which might lead among the others to the formation of sharp tips on the surface. This is why, a clear understanding of charge redistribution on surface rough features is one of the most important steps to enable the prediction of surface evolution in the presence of high electric fields. On the other hand, there is a strong assumption of field emitters surface protrusions formed and destroyed under an external electric field 3. The magnitude of measured field emission current densities escaping from such protrusions as a function of the applied field j F E (E) presumes that the protrusions must be of extremely high aspect ratio (at least, β 100) 4. The credibility of such assumptions has been long debated 5, since no significantly elongated protrusions were experimentally observed so far. Since the dependence j F E (E) is conventionally considered within the classical Fowler-Nordheim theory 6, a strong effect of work function modification on the value of j F E (E) can also be expected. In the present work we aim to estimate the effect of local modification of the work function near point defects and step edges, which can be extrapolated to the larger surface defects. Conventional techniques to measure work functions, such as photon spectroscopy and thermionic emission 7,8, can give the work function value of a metal surface over

2 2 relatively large areas, which can contain the surface of different crystallites and number of different surface imperfections. In many cases this level of accuracy suffices the demands of industrial applications, however, the present-day miniaturization of electric devices up to the nanoscale embosses the need for higher precision of definition for the work function values. If the metal surface is operated in extreme conditions (for example, very high electric fields as in CLIC-related experiments), even slight local changes in the work function can be critical for the quality of machine performance. Recently, thanks to the latest developments of scanning tunneling microscopy (STM) 9,10, accurate experimental observations of local changes in work functions of solid surfaces caused by surface roughness and high electric fields became possible 11,12. These observations encourage us to investigate the local changes of electron density on rough features, which define the local changes of work function. The latter in turn can be strongly connected to the formation of individual field emitters expected to form on cathode surfaces at high electric fields 2,13. In this study we used the density functional theory (DFT) approach, which was successfully employed previously by other authors to calculate the work function modification due to molecules of surface contaminants However, in some demanding applications it is of great importance to know how the work function is modified due to intrinsic surface defects like self-adatoms and step edges, which will contribute in the re-arrangement of the surface topology under the electric field as well as contribute in explanation of a sudden rise of field emission currents on seemingly flat metal surfaces. Moreover, in ultra high vacuum (UHV) and in the presence of high electric fields the contaminant layer becomes insignificant after a short operation time. In spite of the fact that DFT methods are highly accurate and allow for account of effects of external electric fields, the simulation of an evolving extended surface with the number of different defects by using these methods is not yet feasible. However, it is essential to investigate the surface evolution in the presence of external electric fields, in order to pin the process, explaining the existence of individual field emitters. The most suitable contemporary technique for such study is molecular dynamics (MD). This technique, however, does not include electric field effects implicitly, since the electronic structure of atoms is not taken into consideration directly. We have recently attempted to introduce the electric field effect on a metal surface in MD simulations based on laws of classical electrodynamics (ED&MD code). The algorithm and the detailed description of the employed approach can be found in 22. This code assigns the partial charges to surface atoms estimated from the Gauss law in the limit of a small pillbox-shaped conductive surface with the surface charge density defined by an applied external field. Electric forces acting on the charged atoms, hence, are introduced into the classical MD algorithm. In this manner we can follow the dynamic evolution of the surface held under high electric fields. Since the verification of this approach against experiment is difficult, the calculations, which can take into account the electronic structure of atoms, can serve as a compromising benchmarking routine. In present work we report the values of the charges on the surface atoms for three cases: (i) flat surface, (ii) adatoms, and (iii) step edges. We also calculate the change of the work function of the surface in the corresponding cases, both in the absence of an external electric field and in its presence. In the Sect. II A and II B we discuss at first the features of the DFT method used in the present work, and then, we report and discuss the modification of surface properties (work function) due to single self-adatoms and step edges for Cu {100} obtained in the absence and in the presence of external electric field. II. COMPUTATIONAL METHODS In the present work we investigate properties of the surfaces with geometric imperfections using the Spanish Initiative for Electronic Structure with Thousands of Atoms (SIESTA) ab initio simulation package with the linear combination of atomic orbitals (LCAO) type basis functions and norm conserving pseudopotentials. For the exchange and correlation functionals we use the Perdew, Burke & Ernzerhof (PBE) scheme of Generalized gradient approximation (GGA) 26. In our study we favor SIESTA over the plane-wave DFT codes, since the LCAO basis sets allow the electron density to decrease naturally to zero in the vacuum; it is also featured by the well implemented models concerning the band offsets theory As do the majority of DFT codes SIESTA also uses the concept of nonlocal pseudopotentials developed by Kleinman and Bylander 30 and optimal mesh theories for integrals in real- and reciprocal-space as in 31. We model a Cu {100} surface by constructing supercells with 64, 192 and 256 atoms (all organized in eight atomic layer slabs according to the geometry along the <100> direction of the fcc structure z axis in our calculations) and a corresponding vacuum gap of the same size as the slab above the {100} surface. In the lateral directions (x and y axes) the slab is infinitely replicated via periodic boundaries. The periodic boundaries in the direction normal to the surface (z axis) leads to the infinite alternating of the slab and the vacuum gap. The atom positions in the entire slab are relaxed by using the conjugate gradient technique. Before performing the actual calculations we carried out convergence tests for the total energy and the lattice constant for bulk Cu. We found that the choices of energy cutoff of 265 Ry for all supercells and k-grid cutoff 2, 2 and 0.8 nm for the supercells 64, 192 and 256, respectively, were sufficient. The lattice constant and cohesive energy, were found to be in a good agreement (within 1% precision) with the

3 3 experimental values from32. However, the equilibrium lattice constant was found to increase by about nm upon the increase of the number of atoms from 64 to 256 in the slab. For all atoms we used the split type basis set with double ξ size for the wave functions. In the vacuum gap we also applied the surface dipole correction33 option, available in SIESTA23. The problem of fast decay of localized atomic orbitals from surface to vacuum is usually handled by either expanding the basis wave functions of surface atoms or by adding floating orbitals above the surface atoms in the vacuum region. It was shown in34 that addition of a shell of diffuse orbitals with cutoff radius 7-9 bohr in surface layer for Cu provides the best computational efficiency and sufficient accuracy for calculation of the work function value compared to the method of additional floating orbitals. This motivated our choice of adding the diffuse 5s orbitals with cutoff 9 bohr to all the surface atoms. We have calculated the local electronic charge distribution using the Mulliken charge density distribution analysis available in SIESTA35. Since SIESTA is based on the pseudopotential method, the electron density in the code includes information only on valence electrons. Moreover, constructing the surface with the finite vacuum gap results in some modification of electron density of the surface atoms as well. This is why in our simulations we follow only the relative change in charges per atom on the surface by comparing the calculated electron density on the corresponding atoms before and after the modification on the surface was made. The geometry applied in our calculations is illustrated in Fig.1a. The direction of the z axis is from the bottom of the slab to the vacuum, perpendicular to the surface. The value z = 0 is the coordinate of the atoms at the bottom layer of the slab before the surface was opened and relaxed. The slab (8 atomic layers) is terminated by the surface atoms, where the study was focused. The thickness of the vacuum gap equals to the four lattice parameters of the bulk Cu. While studying surfaces with adatoms, we have at first evaluated the values of surface and surface binding energies for the four-fold hollow adatom site (C1 ) and the bridging site C2, which are two possible adatom positions on the otherwise perfect {100} Cu surface (Fig. 1b). The obtained value of surface energy for the flat surface 2.13 J 36 m2 compares well to the value reported in. The slight difference can be explained by relaxation effects37. The surface binding energy for the adatom in C1 site has resulted in the value 2.68 ev, while in the site C2 it was only ev. The notably lower binding energy of an adatom in the C2 site allows for the focus of the study only on the C1 site as a stable site for an adatom on {100} Cu surface. The step edge defect on the surface was modelled as follows. We have used the 192 atoms slab with the surface consisting of 12 atoms: two rows of 6 atoms each (See Fig. 1c). We chose an elongated geometry of the slab to reduce the effect of periodic boundaries on the FIG. 1. Geometry used in the present calculations. a) A supercell of 64 atoms consists of a metal slab and a vacuum gap, equally thick. The z axis starts at the bottom of the metal slab. z = 0 is the z-coordinate of the bottom layer of the Cu slab before the surface was opened and relaxed. b) Positions of adatoms: C1 refers to the four-fold hollow fcc site; C2 is a bridging site. c) Step edge defect on the Cu (100) surface. For computational efficiency the slab is two layers thick in the x direction. surface in the direction perpendicular to the step edges. Six extra atom on the surface filled in the half of a surface layer forming a desired step edge defect in the < 100 > direction. For the calculations of work function and band offsets we use the method of profiling average electrostatic potential. This method was earlier developed in Refs.27,38 40 and further implemented for DFT calculations in Ref.29. Conventionally, the calculation of work functions of different materials had been done by using codes with plane wave basis functions such as VASP due to their straightforward output for electrostatic potential in all regions, while the calculations with local basis sets have dependence on the atomic basis functions41 and demand correct definition of potential in vacuum region. However, the latter can be useful for a more clear insight on intermediate macroscopic physical quantities involved in calculations, such as work function and charge density, assisting the interpretation of experimental results and bridging the phenomenological and fundamental descriptions of physical processes.

4 4 Averaged potential [ev] <V H > (flat surface) <V NA > (flat surface) <V H > (adatom) <V H > * (adatom) z [Bohr] FIG. 2. Distributions of the mean electrostatic V H (z) and neutral atom V NA (z) potentials for the supercells consisting of 64 Cu atoms for both cases of the flat surface and the surface with one adatom. V H indicates that the result is obtained with the surface dipole correction. The V H and V NA for flat surface are given after averaging by using the macroscopic averaging utility 42 in SIESTA code to indicate the potential mean values in the middle of the slab and in the vacuum region. Both curves V H for the adatom are given only after planar averaging to emphasize the potential difference due to the adatom added on the surface. Hartree potential in the undistorted bulk and, hence, all the eigenvalues of the one-particle Hamiltonian, as well as the Fermi level, are referred to it. In Fig. 2 we plot the distributions of the Hartree electrostatic and the neutral atom potentials, for the supercell comprising 64 atoms and the vacuum gap along the z axis. As it can be seen, the mean value of the Hartree potential in the middle of the slab differs significantly from that of the neutral atom potential due to the presence of the surface. When there is no surface both potentials coincide precisely. Since the formation of the surface does not affect the neutral atom potential inside of the slab, the mean value of this potential can be adopted as the reference level in SIESTA calculations 29. The mean Hartree potential Vi+e H can be re-written in two parts, separating the potential generated by the distorted electron density δve H. V H i+e = V NA i+e + δv H e (1) Since we have two well distinguished regions the slab and the vacuum the lineup term for all energy levels can be found as the difference between δve H in the vacuum and inside the slab, similarly to the case of an interface. Having in mind that Vi+e NA = 0 in vacuum, we write, A. Definition of vacuum level in SIESTA in the presence of surface The detailed description of the computational model realized in the SIESTA code is available elsewhere (see, e.g., 24,29 ). In this article we will revise, however, some of the considerations, which are of importance for our present calculations. The work function of any metal surface is calculated as the difference of the Fermi level and the vacuum level 7. These energy levels can be found by using a DFT code. In SIESTA, however, special attention must be paid to the definition of vacuum level, which is output in the SIESTA code with respect to the specific SIESTA zero-energy reference 29. The Hartree electrostatic potential V H ( r) a standard output of SIESTA is generated by electron density. The latter, in turn, consists of two independent parts ρ e ( r) = ρ atom e ( r) + δρ( r), where ρ atom e ( r) is the sum of the valence electron density localized on atoms in the undistorted bulk ρ atom e ( r) = N at k=1 ρatom e,k and δρ( r) is the distorted electron density due to the formation of surface. SIESTA combines the potential V H ( r) generated by the localized electron density with the local part of the pseudopotential V loc ( r) and outputs the so-called neutral atom potential, N at Vi+e NA = (Vi,k loc k=1 + V H,loc e,k ) which is needed to define the zero-energy reference. In SIESTA the latter is attributed to the mean value of the δve H = Vi+e NA slab + Vi+e H (2) where, Vi+e H = V i+e H vac Vi+e H slab. Since the lineup term is found as the difference of the distorted part of the mean Hartree potential in the vacuum and in the slab due to the flow out of electrons from the surface, it is, in fact, the surface barrier, which also defines the vacuum level. Assuming that the energy level inside the slab is zero, we can define the vacuum level as E vac = E 0 + δv H e (3) For the consistency we have to shift the Fermi level outputted in SIESTA, Ef SIEST A, also to the zero-energy level, from the reference level used in SIESTA at the mean total Hartree potential calculated for the bulk V H bulk or the V NA slab. We calculated the Fermi level for three different supercells with 64, 256 and 192 atoms. Again, for consistency, to avoid the dependence on the geometry of supercells, we define the Fermi level for all three cases with respect to E 0 = 0 ev. E f = E SIEST A f V NA slab (4) In this manner we obtain the consistent value of the Fermi level for all the three different supercells with 64, 256 and 192 atoms used in the present calculations, which is E f = 1.97 ± 0.01 ev.

5 5 Φ GGA (our calc.) 3.51 (4.898 ) Values obtained with different vacuum model parameters. The value was obtained for the surface with 8 atoms. A minor variation of this Φ LDA 41 Φ exp 46 parameter was observed due to relaxation of the lattice (the values 4.72 and 4.71 ev were obtained for surfaces with 32 atoms and 24 atoms, respectively.) TABLE I. Comparison of the work function values of Cu (100) surface (Φ, [ev]), calculated in the current work with SIESTA using GGA exchange and correlation term, LDA from 41 and experimental data 46. B. Calculation of work function near rough surface features Previous DFT calculations have shown that the work function is strongly related to the exchange part of total energy 43,44. In particular, the modification of work function relates to changes in the surface dipole barrier 44,45, whose value in metals is defined by the surface structure and by the presence of surface imperfections such as adatoms or step edges. Since the work function is understood as the minimal energy needed for an electron to be released into vacuum from the surface, it is found as the difference between the Fermi level and the value of electrostatic potential in the vacuum region 7. Using Eqs. (3) and (4) we find Φ = E vac E f = E 0 + δve H Ef SIEST A + V H bulk (5) Note that V H bulk is a Hartree potential calculated for the undistorted bulk, which, in practise, coincides with the neutral atom potential in the middle of the slab if the surface is present. When the system includes a surface irregularity such as an adatom or a step edge, the definition of vacuum level becomes less obvious. The potential difference on both sides of the slab due to the periodic boundaries gives a rise to an artificial electric field in the vacuum region. The surface dipole correction 33 available in SIESTA 23 allows to cancel out this field (Cf. Fig. 2, where both graphs with and without surface dipole correction are shown) and define clearly the vacuum level Vi+e H vac, which is used to determine δve H in Eq. 5 in case of the non-flat surfaces. III. RESULTS AND DISCUSSION A. Modification of work function due to adatoms and step edges The work function calculated for the flat {100} Cu surface is compared in Table I to the previous calculations from other theoretical studies 41 and the experimental value 46. Applying on surface atoms the additional diffuse orbitals for better description of long decay of the wave functions in vacuum, we obtained a value of the work function, which compares well with the experimental one 46. The same value is obtained consistently for all the calculated supercells within a small error bar (±0.03 ev, see text in the footnote of Table I. This gives us confidence that the obtained value of the work function of Cu {100} surface is sufficiently accurate to carry out the study of different effects, which can modify the value of work function. We perform such analysis in relative quantities. We analyze the modification of the work function due to adatoms and step edges on a Cu {100} surface. To estimate the local drop of the work function on an adatom compared to the value calculated for the flat surface, we calculate the work functions for the two supercells, which contain 64 and 256 atoms (8 and 32 atoms on the surface, respectively). The supercells are constructed in such a way that the only difference between them is the ratio of the surface areas (1:4). A single adatom on each surface is always positioned in the hollow fcc site C 1 (Fig. 1b). The results are presented in Table II in terms of relative drop of the work function (in %) compared to the value for the flat surface: Φ = (Φ flat Φ def ) Φ flat 100%, where Φ def stands for the work function of the surface with a defect (an adatom or a step edge) and Φ flat stands for the work function of the corresponding flat surface. The comparison of Φ found for the surfaces with an adatom (Table II) shows that the relative drop of the work function calculated for the bigger surface is about 4 times smaller than the one calculated for the surface with the smaller area. Since the work function is obtained as the averaged quantity over the entire surface of the slab as described in Section II B, the Φ parameter is expected to be 4 times smaller (following the surface area ratio) if the local change of the electrostatic potential near the adatom on the small surface is not affected by the periodic boundaries. The consistency of our result with this expectation confirms that the supercell with a surface layer comprising eight atoms is just sufficient for the calculation of the local drop Φ on an isolated adatom, since any smaller size of the supercell surface area will lead to the interaction of the adatoms via the periodic boundaries. This is why we can conclude that the found reduction of the work function calculated on the small (8 atoms) surface is very close to the local drop of the work function near the isolated adatom on the bigger (32 atoms) surface. In fact, the surface can be arbitrarily big. We also investigate the drop of the work function near the step edge (last column in Table II). For a better interpretation of the results we increase the dimension of the supercell perpendicular to the line of the step edge two times, making the calculations of this value for the supercell with 192 atoms. It is interesting to note that the drop of the work function on the step edge is also significant, however, it is apparent that the short-range bonding affects the value of the work function (6 nearest neighbors for the step edge atom versus 4 neighbors for

6 6 adatom (A small ) adatom (A large ) edge Φ 5.9% 1.5% 2.3% TABLE II. Modification of work function on the adatom in the hollow fcc lattice site and a step edge. A small/large indicate the small and large surfaces (A large /A small =4), respectively. By large surface we understand the surface less densely populated with adatoms. The numbers are relative change of the work function in per cent of the value of the work function calculated for the flat surface of the corresponding supercell. Averaged potential [ev] Flat surface (F ext =0) Flat surface Adatom on a small surface Adatom on a big surface the adatom), but does not define it entirely. B. Cu surface under high electric field In order to demonstrate the effect of electric field on the work function, which can be obtained from SIESTA calculations, we plot in Fig. 3 the combination of the four different cases: the flat surface without the external electric field (1) and three curves corresponding to the cases with the electric field F ext =-1 V/nm applied to the flat surface, the small surface with an adatom and the four times greater surface with one adatom. As can be clearly seen, there is a drop in the barrier (about 12% of the value of the work function without the field) near the surface due to the electric field (curves 1 and 2). The slope of the potential drop and hence, the thickness of the barrier, depends on the magnitude of the applied field. It is interesting to note that the presence of an adatom does not affect the height of the barrier, while it is affecting slightly the thickness of the barrier. This is important while interpreting the Fowler-Nordheim dependence of the current density on the value of the applied field, since in the derivation of the Fowler-Nordheim equation the geometric effects are taken into account only via the field enhancement. If the electric field is applied to the surface, the electron density is redistributed accordingly. Using the Mulliken analysis tool we calculate the partial charge induced on each surface atom based on the redistribution of electron density due to the external field. Employing the pillbox technique, we can apply the Gauss law 47 to estimate the surface charge density due to the external electric field in order to verify the current result. σ Gauss = ɛ 0 F n = qe /nm 2 (6) σ SIEST A = (q SA N SA )/A S = 0.12 q e /nm 2 (7) Here, F is the external electric field and n the normal unit vector directed outwards from the slab surface. q SA is the partial charge on surface atoms as calculated from SIESTA and N SA is the amount of atoms on the surface. A S refers to the surface area. The charges are measured in elementary charges q e z [Bohr] FIG. 3. Distribution of the electrostatic Hartree potential V H (z) in the presence of the electric field ( F ext = 1 V/nm) for the flat surface and two surfaces with one adatom. For comparison we also show the same distribution at the flat surface without an external electric field. The circle shows the slight decrease in the thickness of the barrier (a correspondence to the phenomenological Schottky-Nordheim barrier for flat surface) for electrons near the surface with an adatom. The smaller the area (a localized effect) the stronger the change in the barrier thickness. We see the good agreement between (6) and (7) only if we use in Eq.(6) the value of electric field, which corresponds to the resulting field in the SIESTA calculations and amounts to -2.1 V/nm (see the caption to Fig. 4, the first case). This feature of SIESTA code must be taken into account while studying the external electric field effects. The user-defined field is used in SIESTA to define the potential difference corresponding to the drop of potential over the entire size of the supercell at the beginning of calculations. This potential difference is introduced in the vacuum layer similarly to the use of the surface dipole correction mode and remains intact during the calculations. As a result of self-consistent calculations of electron density distribution in the presence of the external electric field, the internal field inside the slab due to the surface charges compensates the external field stabilizing the value of electrostatic potential within the slab. The fixed potential difference in the vacuum layer naturally leads to the increase of the field out of the slab, in the region above the surface, thus defining high value of the surface charge density. To verify this explanation we have performed similar calculations, but with different sizes of the vacuum region. The result presented in Fig. 4 reveals a slight decrease (see the caption to Fig. 4) of the high value of the field above the surface obtained for the supercell with the increasing vacuum region. The smallest vacuum gap is equal to the size of the metal slab. In the figure we used nanometers as dimensional units to facilitate the estimation of the magnitude of the slope of potential curve in the vacuum region. In the limit of infinitely large vac-

7 7 uum region, this slope will be very close to the input field, since the size of the metal slab, where the drop of potential is zero due to the surface charges, will be negligibly small compared to the size of the vacuum. Hence the metal slab will not affect the value of the field above the surface. In this limit the charges obtained on the metal surface will precisely correspond to the input electric field as expected from classical laws of electrodynamics. Averaged potential [ev] Vacuum region:1.46 nm Vacuum region: 1.83 nm Vacuum region: 2.19 nm z [nm] FIG. 4. Electrostatic potential V H (z) obtained for 3 different suppercells in the presence of input electric field F ext = 1 V/nm. These supercells comprise the same metal slab of four atomic layers and 8, 10 and 12 layers of vacuum region. The z-coordinate is given in nanometers for clarity to estimate the value of the field (the slopes in the vacuum region) in all cases. We obtain the slopes as -2.1±0.4, -1.9±0.4 and -1.7±0.4 ev/nm (divided by q e, the slope gives the value of the field) for all three cases, correspondingly. The value of the slope (thus actual resulting electric field defining the charges on the surfaces) strongly depends on the size of the vacuum region. The fluctuations in the vacuum region are left for eye guidance to indicate the size of the vacuum in each case. Previously we have attempted to estimate the partial charge induced on the surface atoms by the external electric field by developing the hybrid electrodynamicmolecular dynamic (ED&MD) approach 22. The atomic charges of order of fractions of an electron are dynamically modified following the change of the shape of the external electric field around the surface. This approach allows for the extension of MD algorithm and can be a valuable tool to simulate the evolution of metal surfaces held under high electric fields. However, the calculation of the charges is made based on the crude assumption of validity of the Gauss law on atomic level, while the shape of atoms is approximated as cuboids with the diagonal as large as the interatomic distance. In the present work we aim to validate our former approach by calculating the charges on adatoms due to the external field by applying the Mulliken charge distribution analysis tool on the calculated electron density and compare these results to the charges calculated by ED&MD code for the identical adatoms. This comparison is also summarized in Table III. We note that the feature of DFT calculations discussed in Single adatom Two adatoms DFT ED-MD DFT ED-MD Partial Charge,q e TABLE III. Comparison of fractions of unit charge on adatoms obtained by DFT and ED&MD 22 calculations for two cases of a single adatom and a cluster of two adatoms. Since the resulting electric field in SIESTA calculations near the conducting surface is twice as high as the input field (see the caption to Fig. 4 and the corresponding body text), we compare this value to the ED&MD results obtained by using the corresponding value of electric field ( F ED&MD =-2.1 V/nm). the previous paragraph is taken here into account by increasing the input field in ED&MD simulations to the corresponding value of the resulting field in SIESTA. As it is seen from the comparison, the obtained results are in satisfactory agreement. Some discrepancy is expected due to the nature of the approximations adopted in the ED&MD technique. IV. CONCLUSIONS In conclusion, we have studied the effect of electric field on the Cu surface by the DFT-GGA technique. To assess the behavior of the conducting surface under high electric field, we calculated the electron density from where we derived the change of work function due to the presence of an adatom as well as the charges accumulated on the adatoms due to the electron density transfer under high electric field. The satisfactory agreement with the estimation of the surface charge from the Gauss law for flat surfaces confirms the validity of the chosen technique. The change in the value of the work function indicates the strong influence of intrinsic defects on this parameter, which requires further investigation. Moreover, we were able to validate the approach which we developed previously for the dynamic simulation of the effect of electric field on extended metal surfaces. The calculation of the partial charge on the single adatom and the cluster of two adatoms agreed well with the ED&MD results, although the approaches applied in both calculations were fully independent. ACKNOWLEDGEMENTS We acknowledge financial support from the Academy of Finland MECBETH project. M.H. has been supported by the Research Funds of the University of Helsinki (No ). Grants of computer time from the Center for Scientific Computing in Espoo, Finland, are gratefully acknowledged. The authors acknowledge the useful discussions with Kai Nordlund on calculations of electric field effects by DFT methods.

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