modeling the evaporation rate of cesium off tungsten based controlled porosity dispenser photocathodes

Modeling the evaporation rate of cesium off tungsten based controlled porosity dispenser photocathodes Z Pan and K L Jensen Citation: AIP Advances 3, 042105 (2013); doi: 10.1063/1.4800700 View online: http://dx.doi.org/10.1063/1.4800700 View Table of Contents: http://aip.scitation.org/toc/adv/3/4 Published by the American Institute of Physics AIP ADVANCES 3, 042105 (2013) Modeling the evaporation rate of cesium off tungsten based controlled porosity dispenser photocathodes Z Pan1,2 and K L Jensen2 Physics Department University of Maryland, College Park, MD 20742, USA Code 6843, Naval Research Laboratory, Washington, DC 20375, USA (Received January 2013; accepted 25 March 2013; published online April 2013) The evaporation of cesium from a tungsten surface is modeled using an effective onedimensional potential well representation of the binding energy The model accounts for both local and global interactions of cesium with the surface metal as well as with other cesium atoms The theory is compared with the data of Taylor and Langmuir [Phys Rev 44, 423 (1933)] comparing evaporation rates to sub-monolayer surface coverage of cesium, gives good agreement, and reproduces the nonlinear behavior of evaporation with varying coverage and temperature Copyright 2013 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4800700] The development of robust, long life, high efficiency photoemitters is critically needed for applications demanding high brightness electron sources.1 To that end, a cesium (Cs) based dispenser photocathode is under development.2–4 The determination of an optimal in situ rejuvenation procedure to restore quantum efficiency degradation due to lost Cs motivates the modeling of evaporation and surface diffusion High efficiency photocathodes generally rely on a thin coating of Cs to lower the work function and enhance emission.5 However, Cs loss severely shortness the cathode lifetime and is a concern in high gradient accelerators A sintered wire controlled porosity dispenser (CPD) cathode can actively dispense fresh Cs to the surface during operation through a periodic array of pores leading from the surface down to a cesium reservoir6 to replace Cs even as it is being lost.7 Previously, the prediction of the surface coverage during operation of a CPD was based on evaporation models using empirical fits and adjustable fitting parameters.8 In this work, a model of evaporation without fitting parameters is presented and shown to agree with experimental data Although applied to Cs evaporation from tungsten, the model can be extended to Cs evaporation off other substrates Cesium atoms occupy well defined adsorption sites at the surface,9 modeled by a binding potential characteristic of the surface and the interactions present The evaporation rate is a product of the frequency at which the Cs atom oscillates, the probability that it has sufficient energy to overcome the binding potential, and the local surface density of atoms Evaporation per unit area per unit time is then given by: E evap = Pσ θ τ (1) where τ is the characteristic evaporation time, P is the probability that an atom can overcome the binding potential and evaporate, σ is the number density per unit area of available binding sites, and θ is the fraction of those sites which are occupied (aka the coverage) An atom at an adsorption site can occupy discrete bound energy states If the cesium atoms are in thermal equilibrium with the rest of the solid lattice, the probability of a particular energy state E is proportional to e−E/k B T P is then given by: P= 2158-3226/2013/3(4)/042105/5 −E/k B T f r ee e n −E i /k B T + −E/k B T i=0 e f r ee e 3, 042105-1 (2) C Author(s) 2013 042105-2 Z Pan and K L Jensen AIP Advances 3, 042105 (2013) FIG Schematic diagram of the effective potential well binding the cesium atom to a tungsten surface where kB is the Boltzmann constant and T is the temperature To apply Eq (2) to calculate P, the bound state energies En and free state energies Efree require determination by solving Schrăodingers equation Ignoring surface roughness and contamination, the binding sites on a given crystal face are indistinguishable Since the motion of an atom along the surface plane does not contribute to evaporation, the problem becomes an effective one dimensional binding potential in the zˆ direction (perpendicular to the surface), as indicated in Fig Although a realistic binding potential is undoubtedly more complex, the square well approximation gives good agreement to the evaporation data The well width w is less than a nm (here, Å, or the covalent diameter of a Cs atom), and is set in a larger bounded region of length L w so that Efree is discrete as well The well depth parameter V0 is separable into contributions arising from different sources: (i) Coulomb contribution, VC , arising from the partial transfer of charge ±F(θ )e between the cesium atoms and the surface (ii) a Van der Waals contribution, VL J , accounting for nearest neighbor cesium interactions which are approximated by the semi-empirical Lenard Jones potential10 (iii) a Thermodynamic contribution, Vμ , accounting for the change in surface free energy and entropy as coverage is varied from to monolayer, and (iv) a Covalent contribution, VC V , due to the partial sharing of the valence band electrons between substrate and surface atoms.9 Therefore, the potential energy V0 is given by V0 = VC + VL J + Vμ + VC V (3) Each are discussed in turn The coulomb contribution to the well depth VC , is VC = δ − F(θ ) + F(θ )2 V f + i N (i, θ )Vnn (4) i=0 where δ accounts for global interactions from charged cesium atoms farther away (because the electrostatic force is long range, it would be in error to consider only nearest neighbor interactions), −F(θ ) + F(θ )2 V f ensures charge neutrality upon evaporation, where is the work function, and V f is the ionization potential of cesium Finally, the last term is the weighted average over the i electrostatic potential Vnn that a partially charged cesium atom on the surface experiences from “i” nearest neighbors, where N(i, θ ) is the probability that a cesium atom will have i nearest neighbors 042105-3 Z Pan and K L Jensen AIP Advances 3, 042105 (2013) for a given θ : N(i, θ ) is given by the Bethe-Peierls approximation for nearest neighbor atoms.11 The total number of nearest neighbors sites is dependent on the crystal structure of the substrate as well as the crystal face cut of the surface For tungsten, the crystal structure is body centered cubic There are a total of nearest neighbor sites for the [001] and [011] crystal face cuts of a tungsten surface i is given by Let ±F(θ )e be the charge of the cesium atoms and their images, then Vnn i Vnn = −k (F(θ )e)2 +i · d k (F(θ )e)2 k (F(θ )e)2 − √ a d + a2 (5) where a is the nearest neighbor separation distance, d is the distance between the cesium atom and its corresponding image, e is the fundamental charge unit, and k is the Coulomb constant The charge ±F(θ )e can be derived from knowing the strength of the dipole moment M produced by each cesium atom at the surface The factor F(θ ) is given by F(θ ) = 4k (xs − xCs ) G(θ ) M = ed e R(1 + α/(4π R )) (6) where xs and xCs are the relative electronegativities of the substrate atom and the adsorbed cesium atom, respectively, R is the covalent radius of the cesium atom, α is the polarizability of cesium, and Gyftopolous structure factor G(θ ) accounts for the electronegativity variation with coverage at the surface.12 In Eq (4), the long range electrostatic interactions, represented by δ, are approximated by δ≈ 2π θ (7) a βT kθ σ e2 where is the interaction energy between nearest neighbor cesium atoms, and β T = 1/kB T Eq (7) is obtained from representing the interactions as a sum over the average number of occupied adsorption sites for a given θ , then approximating the summation by an integral The interactions between Cs atoms far away is treated as a “screened” coulomb interaction e−k0 r where the damping factor k0 is from the Debye-Hăuckel approximation13 and is T k e2 The second component to V0 uses a Lenard Jones (LJ) potential, which is an empirical formula for the short range energy of interaction between atoms due to van der Waals dispersion forces as well as Pauli repulsion The LJ parameters are from Ref 14 The net contribution from the weighted average of nearest neighbor cesium interactions is given by VL J = i N (i, θ )L (8) i=0 where L is the LJ potential (e.g Eq 1.3 of Ref 10) The third term in Eq (3) is the thermodynamic contribution Vμ to the well depth and is Vμ = k B T θ ∂(μ/k B T ) ∂ log θ (9) T,A The proportionality factor ∂(μ/kT)/∂(log θ ) is from the Darken equation relating the change in the relative “order” of the surface θ / δθ to the chemical potential μ set up by the cesium atoms on the surface.15 Since systems naturally tend towards states of higher disorder, the effect of Vμ is to lower the well depth V0 as the coverage approaches a monolayer (θ → 1) For a 2D Langmuir layer, the thermodynamic contribution Vμ is then Vμ = k B T θ 1−θ (10) The final contribution to VC V arises from the covalent bond formed between the adsorbed cesium and the substrate due to the partial sharing of the valence band electrons The covalent contribution should be independent of θ and T and will be approximated as a constant to be determined from experimental data All terms in Eq (3) to evaluate V0 are now in place: with V0 in Eq (3) determined Z Pan and K L Jensen 10 25 10 20 10 15 10 10 AIP Advances 3, 042105 (2013) Evap Rate [atoms / cm s] 042105-4 Taylor - Langmuir extrapolated data 800 K 700 K 10 10 600 K 500 K Evaporation of Cs off W 10 -5 0.2 0.4 0.6 0.8 Coverage FIG Comparison to experiment, gray region represents the data Taylor and Langmuir had reported in their paper.16 as a function of θ and T, Eq (2) is applied to determine P Combined with τ and σ , Eq (1) then gives the evaporation rate per unit area per unit time The [001] crystal face of tungsten was used and σ taken to be σ = × 1014 sites/cm2 as per Ref 12 For τ , it is assumed that a bound cesium atom will undergo random energy changes from collisions with the well walls for each period, making 1/τ the frequency of oscillation averaged over all available energy states Assuming Boltzmann statistics, the average cesium frequency of oscillation is approximated as ν = n E i −E i /k B T i=0 2π e n −E i /k B T i=0 e (11) where Ei /2π is the quantum mechanical oscillation frequency in a bound state with energy Ei Fig shows the comparison between the evaporation rates calculated using the theory with experimental data (and its extrapolation by Taylor and Langmuir) for cesium evaporation off tungsten.16 It is seen that the theory performs well in capturing the qualitative behavior of evaporation over a wide range of coverages and temperatures It can also be noticed that the evaporation rate can vary over several orders of magnitude as coverage at the surface is varied from to for the range of temperatures considered Quantitatively, the evaporation values predicted by the model agree to within 10% for the range of data presented in the figure At full monolayer cesium coverages, the model has a singularity and diverges to infinity due to the assumptions in the model However, as shown in Fig 2, good agreement for coverages close to a monolayer up to 0.98 are found In conclusion, a model for cesium evaporation off tungsten was developed as part of a program to optimize and predict the performance of CPD photocathodes The model captures the nonlinear dependence of evaporation on surface coverage of cesium and the temperature even using the simplifying assumption of a flat well to model the bound states No adjustable parameters for the 042105-5 Z Pan and K L Jensen AIP Advances 3, 042105 (2013) physical model for the evaporation process were required The methodology can be extended to other substrates as well ACKNOWLEDGMENTS We thank the Office of Naval Research and the Joint Technology Office for their support, and E Montgomery and D Feldman for useful discussions P G O’Shea and H P Freund, Science 292, 1853 (2001) Moody, K Jensen, D Feldman, P OShea, and E Montgomery, Appl Phys Lett 90, 114108 (2007) E J Montgomery, D W Feldman, P Oshea, Z Pan, N Sennett, K L Jensen, and N A Moody, Journal of Directed Energy 3, 66 (2008) D Dowell, I Bazarov, B Dunham, K Harkay, C Hernandez-Garcia, R Legg, H Padmore, T Rao, J Smedley, and W Wan, Nucl Instr and Meth A 622, 685 (2010) A H Sommer, J Vac Sci Technol 1, 119 (1983) R L Ives, L R Falce, G Miram, and G Collins, IEEE Trans Plasma Sci 38, 1345 (2010) E Montgomery, Z Pan, J Leung, D Feldman, P OShea, and K Jensen, Advanced Accelerator Concepts 1086, 599 (2009) Z Pan, K L Jensen, and P G O’Shea, Appl Phys Lett 100 (2012) J Levine and E Gyftopoulos, Surface Science 1, 171 (1964) 10 J.-P Hansen and I R McDonald, Theory of simple liquids (Academic Press, London; New York, 1976) p 11 D A Reed and G Ehrlich, Surface Science 102, 588 (1981) 12 E P Gyftopoulos and J D Levine, J Appl Phys 33, 67 (1962) 13 A L Fetter and J D Walecka, Quantum theory of many-particle systems (McGraw-Hill, San Francisco, 1971) 14 M H Ghatee and H Shams-Abadi, The Journal of Physical Chemistry B 105, 702 (2001) 15 R Gomer, Rep Prog Phys 53, 917 (1990) 16 J B Taylor and I Langmuir, Phys Rev 44, 423 (1933) N  ... 3, 042105 (2013) Modeling the evaporation rate of cesium off tungsten based controlled porosity dispenser photocathodes Z Pan1,2 and K L Jensen2 Physics Department University of Maryland, College... predict the performance of CPD photocathodes The model captures the nonlinear dependence of evaporation on surface coverage of cesium and the temperature even using the simplifying assumption of a... characteristic of the surface and the interactions present The evaporation rate is a product of the frequency at which the Cs atom oscillates, the probability that it has sufficient energy to overcome the