ARGONNE NATIONAL LABORATORY DECEMBER 6 2005 Argonne IL
ARGONNE NATIONAL LABORATORY DECEMBER 6, 2005 Argonne, IL Numerical Modeling of Field-Enhanced Photoemission from Metals and Coated Materials ANL Beams and Applications Seminar Host: John Lewellen ASD Bldg. 401, Room A 1100 Tuesday, 1: 30 pm NRL: K. L. Jensen J. L. Shaw J. E. Yater UMD: D. W. Feldman N. A. Moody P. G. O’Shea We gratefully acknowledge: FUNDING by Joint Technology Office & Office of Naval Research INTERACTIONS with (alph). S. Biedron , C. Bohn, C. Cahay, D. Dimitrov, D. Dowell, J. Lewellen, J. Petillo, J. Smedley ANL 1
OUTLINE Electron Sources Photocathodes and Photocathode Issues The Dispenser Photocathode Concept Electron Emission Fundamentals 1 st Generation Emission Model & Usage Next Generation Components & Application Bare Metals Cesiated Surfaces & Gyftopoulos-Levine Model Quantum Distribution Function Quantum Effects on Barrier & Scattering Conclusion ANL 2
ELECTRON EMISSION The Manner in Which Electrons Are Extracted Dictates the Technological Gambits Invoked Metal Thermionic Heat. Wave Labs http: //www. cathode. com/c_cathode. htm Field Photo ANL-BNL-JLAB Gun: Ilan Ben-Zvi www. agsrhichome. bnl. gov/e. Cool/MAC_Ilan. pdf Courtesy of C. A. Spindt www. sri. com/psd/microsys/vacuum/ ANL 3
PHOTOINJECTORS & PHOTOCATHODES Critical Components of Free Electron Lasers, Synchrotron Light & X-ray Sources High Power FEL Demands on Photocathode: CHARGE PER BUNCH: 0. 1 - 1 n. C in 10 -50 ps pulse FIELD: 10 - 100 MV/m in pressure of 10 -8 Torr (approx) OPERATION: Robust, Prompt, Operate At Longest LIFETIME: Longevity & Reliability Paramount rf Klystron Master Oscillator Drive laser Photocathode Linac ANL 4
PHOTOCATHODE RESPONSE TIME Pulse Shaping Optimal Shape for emittance: beer-can (disk-like) profile Laser Fluctuations occur (esp. for higher harmonics of drive laser) Fast response: laser hash reproduced Slow response: beer-can profile degraded Optimal: 1 ps response time Mathematical Model (wn = 2 pn/T) Formulation based on model of J. Lewellen ANL 5
CATHODE + LASER NONUNIFORMITY Cathode Emittance Is Important (esp. in Gun) Pulse Shape Can Result in Reduction of Emittance Prediction of Photocathode / Drive Laser Combos Laser & Beam Crucial to Design of Larger Systems Realistic Initial Distributions Needed: Advanced Codes (Michelle, Argus, Magic, Mafia, Vorpal, Etc. ) Need Adequate Emission Models, Photoemission Studies Sparse, Emission Distributions Are Unknown Laser Non-uniformity + work function non-uniformity(sim) Laser Nonuniformity Combined Nonuniformity Work Function Nonuniformity ANL 6
PROGRAM OBJECTIVE IDENTIFY factors that affect QE (e. g. , laser, environment, photocathode material) DEVELOP a custom-engineered controlled porosity photo-dispenser cathode Interpore ≈ 6 µm; Grain Size≈ 4. 5 µm; Pore Diam. ≈ 3 µm Metal Top View Conventional Dispenser Generic Parameter Range Assumptions: • Charge = 1 n. C Cs O Side View • Wavelength = 266 nm • Pulse FWHM = 10 ps Example Cases of Io(QE) Controlled Porosity monolayer W plug w/ Cs Maryland Dispenser Cathode Al 2 O 3 Potting Heater For Radius of 0. 3 cm • Io(1%) = 0. 165 MW/cm 2 • Io(0. 01%) = 16. 5 MW/cm 2 For Radius of 0. 1 cm • Io(1%) = 1. 48 MW/cm 2 • Io(0. 01%) = 148 MW/cm 2 A Cs-dispenser Photocathode Concept ANL 7
DISPENSER CATHODE POROSITY ANALYSIS Central Questions: Pore to Pore: Log-Normal Distribution Pore-to-Pore separation & Grain Size Desorption & migration rates of low work function coating over surface Grain Size: Effective Diameter SEMS: Nathan Moody, UMD ANL 8
STATISTICAL MECHANICS OF ELECTRON GAS Electron Number Density r Electrons Incident On Barrier Are Distributed In Energy According To A 1 -D “Thermalized” Fermi Dirac Distribution Characterized By The Chemical Potential and Called The “Supply Function” Zero Temperature ( 0 ˚K = o = EF) f(k) obtained by integrating over the transverse momentum components N does not change with T so must: Metal Semiconductor Ec BAND BENDING Metal vs. Semiconductor o Ec Fvac o Fvac Ev ANL 9
CURRENT - A CLASSICAL APPROACH f(x, k, t) is the probability a particle is at position x with momentum hk at time t Conservation of particle number: dx’ dk dn to order O(dt) dn’ dx Boltzmann Transport Equation velocity & acceleration “Moments” give number density r and current density J: Continuity Equation ANL 10
CURRENT IN SCHRöDINGER REPRESENTATION Schrödinger’s Equation Simple Case: Gaussian Time-dependence of Operators governed by commutators with Hamiltonian Consider a pure state The form most often used in emission theory Basis for FN & RLD Equations ANL 11
THERMIONIC VS FIELD EMISSION The most widely used forms of: Field Emission: Fowler Nordheim (FN) Thermal Emission: Richardson-Laue-Dushman (RLD) High Temperature Low Field Low Temperature High Field Fowler Nordheim Richardson Transmission Probability Electron Supply Emission Equation Constants for Work Function in e. V, T in Kelvin, F in e. V/nm ANL 12
A GENERAL THERMAL-FIELD EQUATION Supply Function X(F[GV/m], T[K]) Maxwell Boltzmann Regime General Transmission Coefficient T(7, 300) Field Thermal 0 K-like Regime T(0. 01, 2000) Fermi f(7, 300) f(0. 01, 2000) Define Slope Ratio: n « 1: Richardson-Laue-Dushman Eq n » 1: Fowler Nordheim Equation ANL 13
INTEGRATED SCATTERING & EMISSION MODEL Quantum Distribution Function (QDF) Simulation Simultaneously Relates Scattering ( ), Barrier Emission ( ), Thermal ( ) and Density Effects z( ) & QDF Clean W(001) Potential W cores Ba/O/W(001) W cores ENERGY W f & U(x) Scattering & Electron Transport & Emission Relaxation Time & Thermal Model k POSITION (x/ao) Ba O Hemstreet, et al. PRB 40, 3592 (1989) Coverage-Dependent Work Function ANL 14
EXP. VALIDATED PHOTOEMISSION MODEL GOAL Predict Quantum Efficiency From Laser & Material Parameters Analyze Experimental Results from UMD, NRL, Colleagues in FEL Program Emission Model For Beam Code for NRL, SAIC, Tech-X, NIU, Colleagues COMPONENTS: First Generation Beam Code Model Work function variation with coating ) Gyftopolous-Levine theory Thermal & Material; laser , R( ), f Transient heating & heat diffusion Simple Photoemission Model Photocurrent depends on • Scattering Factor: f • Absorbed laser power: (1 -R) I • Escape Probability: U-terms U-function Revised Fowler-Dubridge Model Quantum effects @ hi F & T U, f barrier due to e- density; scattering Transmission through barriers U Relate barrier to emission probability Next Generation Beam Code Model ANL 15
DETERMINATION OF R[%] & d Algorithm: Spline-fit experimental optical data (e. g. , CRC, AIP Handbook) for index of refraction (n), damping constant (k) Designate incident angle = Use Equations to determine Reflectance R[%] and penetration depth of laser for given wavelength Consider W, Cu, Au… …other metals in database ANL 16
POST-ABSORPTION SCATTERING FACTOR Factor (f ) governing proportion of electrons emitted after absorbing a photon: k z( ) Photon absorbed by an electron at depth x Electron Energy augmented by photon, but direction of propagation distributed over sphere Probability of escape depends upon electron path length to surface and probability of collision (assume any collision prevents escape) path to surface & scattering length Average probability of escape To leading order, k integral can be ignored argument < 1 argument > 1 ko: minimum k of e- that can escape after photo-absorption : penetration of laser (wavelength dependent); : relaxation time Ex: Copper: • = 266 nm • = 12. 9 nm • = 0. 85 fs • = 7. 0 e. V • F = 4. 3 e. V sec(y) f = 7. 7 = 0. 038 ANL 17
QM-EXTENSION OF FOWLER-DUBRIDGE EQ. RLD-based Fowler Dubridge Model • U(x): depends on thermal distribution and barrier for emission probability • Reflectivity R and Scattering Factor fl depends on material & relaxation time coating = 2. 0 e. V “Fowler factor” Copper * • = 266 nm; F = 5 MV/m; = 3 • R = 33. 6%, = 4. 3 e. V, EF = 7. 0 e. V • QE [%] (analytic) 1. 31 E-2 • QE [%] (time-sim) 1. 36 E-2 • QE [%] (exp) 1. 40 E-2 QM contributes for photon E near barrier height, large fields, and cold temperatures Exp data: T. Srinivasan-Rao, et al. , JAP 69, 3291 (1991). ANL 18
SCATTERING & Electrical / Thermal Conductivity If an electric field F (or temperature gradient T) is removed, then distribution “relaxes” back to equilibrium after a “relaxation time” Electric Field Temperature Distribution for Fermi-Dirac is approximately constant except near Fermi Energy Field Electrical Conductivity Temp Specific Heat Thermal Conductivity WIEDEMANN-FRANZ LAW ANL 19
LASER HEATING OF ELECTRON GAS Differential Eqs. Relating Electron to Lattice Temp Electron & Lattice Specific Heat Power transfer by electrons to lattice 285. 1 GW / K cm 3 (W @ RT) Laser Energy Absorbed Diffusion mimics the temporal spread of Dirac-Delta-like pulses with Do Do acts as Length 2 / time Length ≈ O(laser penetration depth) Model captures physics… as long as there is an estimate for to ANL 20
PHOTOEMISSION MODULES IN BEAM CODE Goal: modules for 3 D RF gun / beam codes for the analysis of beam generation and transport. Present model: high-T scattering operator with T evaluated using Delta-diffusion model as function of laser intensity for copper; probability of emission factor based on Fowler Dubridge but without QM Next generation to include all-temperature scattering, QM, metal & coating library VORPAL Cu Photocathode Beam Emission and Evolution Hemisphere unit cell model SIMULATION CODE: VORPAL (TECH-X) • 3 D visualization of photo emitted electron particles (white dots) following the beam emission and its evolution at different times from simulations with steady-state photocathode model. SIMULATION CODE: MICHELLE (SAIC) • Photoemission from laser-illuminated Cu hemisphere Using 1 st generation photoemission model • J. Petillo, et al. , 8 th DEPS, Lihue, HI (2005) • Cu photocathode at left boundary. Front of laser pulse has reached the photocathode and emission of the electron beam has started. • D. Dimitrov, et al. , 8 th DEPS, Lihue, HI (2005) ANL 21
THEORETICAL EVALUATION OF SCATTERING Scattering in metals due to collisions with lattice (acoustic phonons & defects) and e-e collisions. If mechanisms independent, then: Mathiessens Rule Phonons: For T > TD (Debye Temp), then ac goes as T, but at low T, goes as T^5. Scattering cross section is also related to • deformation potential (related to stress on lattice) • sound velocity vs Electron-Electron: e-e scattering in simple metals is not simple. Model of Lugovskoy & Bray [1] depends on electron energy above Fermi Level ( E) and Thomas-Fermi Screening Wave Number qo (depends on electron density) & dielectric Ks Ag Cu Au W Pb [1] A. V. Lugovskoy, I. Bray, J Phys D: Appl. Phys. 31, L 78 (1998) ANL 22
EXPERIMENT VS. THEORY (BULK METALS) BNL Field Enhancement 3. 0 Macroscopic field 1. 0 MV/m Work Function 3. 97 e. V Temperature 300 K Data & Image Courtesy of J. SMEDLEY (BNL) SLAC Field Enhancement 1. 0 Macro field (MV/m) 0. 01 Work Function 4. 31 e. V Temperature 300 K Data Courtesy of D. DOWELL (SLAC) ANL 23
EXPERIMENT VS. THEORY (PART II) Measured (UMD), calculated (NRL), & literature for various DISPENSER CATHODES (1 st Generation model used) B-TYPE: B. Leblond, NIMA 317, 365 (1992) UMD experimental data M-TYPE UMD experimental data SCANDATE UMD experimental data Experiment Theory M-type How the comparison is made: • Time-Dependent Thermal Photoemission Model using ee(E= +h ) ran for each incident laser intensity B-Type • Laser pulses were Gaussian in time Scandate • Total energy and charge emitted evaluated via integration over Gaussian pulse (Laser) and Emitted charge profile (electron) • Use of library values for Copper only (no adjustable constants) ANL 24
COVERAGE DEPENDENT WORK FUNCTION Gyftopolous-Levine Theory relates Work Function to coverage factor. Dependent upon Covalent Radii rx, Factors “f” and “w” (Act As “Atoms Per Cell”, Values of which Depend on Crystal Face). General Surface = “Bumpy [B]” Modified Gyftopolous-Levine Theory alkali metal (n = 1) alkaline-earth metal (n = 1. 65) W C R Hard Sphere Model of Surface Dipole ANL 25
GYFTOPOLOUS-LEVINE MODEL PERFORMANCE LEAST SQUARES ANALYSIS: Minimize Difference between GL theory & Exp. Data With Regard to scale factor, monolayer work function value, and f coverage factor • Tightly constrained parameter variation • Unique determination of theory based on experimental values • Predictive ability from basic experiments C-S Wang, J. Appl. Phys. 44, 1477 (1977) J. B. Taylor, I. Langmuir, Phys. Rev. 44, 423 (1933). R. T. Longo, E. A. Adler, L. R. Falce, Tech. Dig. of Int'l. El. Dev. Meeting 1984, 12. 2 (1984). G. A. Haas, A. Shih, C. R. K. Marrian, Applications of Surface Science 16, 139 (1983) ANL 26
EXPERIMENTAL MEASUREMENTS @ UMD • Test and evaluation chamber @ 2 E-10 Torr • Surface preparation using H-ion beam • Surface deposition of various coatings • Deposition monitor (+/- 0. 01 nm thickness) • Femto-ampere current measurements (for QE) • Solid state CW lasers on single-axis robot • QE as function of time, temp, coverage, wavelength, and laser intensity (AUTOMATED) ANL 27
QE OF Cs ON W: EXP. VS. THEORY CONDITIONS: Coverage Is Uniform Field & Laser intensity low: Schottky barrier lowering & heating negligible 405 nm: 532 nm: 655 nm: x 1 (theory x 1. 40) x 4 (theory x 1. 40) x 35 (theory x 0. 84) For 407 nm Cs deposited rapidly: Exp measured mass by a “depth” factor. Therefore: Scale factor = (100%/Atomic diameter) For 532 nm and 655 nm Cesium deposited slowly: accumulation rate affected by desorption (scale ) Peak value affected by residual cesium left on W (center xmax) Presence of O (hard to remove) affects work function variation Experiment Theory To account for effects: ANL 28
QE OF Cs ON W: EFFECT OF TEMPERATURE Consider “typical” conditions of: Laser Intensity 10 MW/cm 2 Field at cathode 10 MV/m Field enhancement 2 (generic) Pulse length 10 ps Relaxation Time Is Temperaturedependent: Impact of Operating Photocathode at Lower Temperatures Is to Raise QE (all other things being equal) Under these conditions For r = 0. 1 cm 2: ANL 29
QE OF Cs ON Ag: EXP. VS. THEORY Cs ON Ag Exp: INITIAL COMPARISON METHODS for data taken week of Aug 8 2005 Experimental conditions same as for Cs-W for 532 and 655 nm exp. Used same scale as Cs on W Shift factor aligns peaks for each experiment Data taken by ANNE BALTER ANL 30
HYPOTHESES OF EXP-THEORY DIFFERENCES Actual surfaces differ from theory models because of Solid Tungsten 500 x Surface geometry (altered by cleaning? ) Reflectivity changes with exposed face (inc. angle) Crystal Faces can have different work functions (e. g. , Cu(111) = 4. 86, Cu(110) = 5. 61, etc) Contaminants (e. g. , Carbon-based ≈ 5. 5 e. V) & possibility of only sub-area contributing Shadows model: fraction of surface illuminated (f) Solid Silver 500 x Nathan Moody (UMD) ANL 31
IMPACT OF COATINGS ON WORK FUNCTION W - Ba + O Conventional View: charged atoms at surface & image charge = dipole QM View: Electron penetration of barrier determined by height and width Clean W(001) Potential W cores Ba/O/W(001) W cores Friedel Oscillations ENERGY Exchange-Correlation & Poisson’s Eq: Changes in density = changes in V(x) L. A. Hemstreet, et al. PRB 40, 3592 (1989) Ex. Corr + Poisson POSITION (x/ao) tanh model ANL 32
EXCHANGE CORRELATION ENERGY OF e- Kinetic Energy e-e interaction e-lattice interaction self-interaction of background Kinetic Aspects of system of interacting e- in ground state determined by density Exchange Exch. -Corr. Potential: change in r gives rise to change in V Correlation = Fermi Level r= electron density ao = Bohr radius Typical Metals Metal Vxc Vacuum ANL 33
OTHER CONTRIBUTIONS TO SURF. BARRIER Simple Theory: Wave function penetration creates dipole & gives Friedel Oscillations Why bother looking at Friedel Oscillations? For two good reasons: • Friedel Density profile has analytic V(x) sol’n • Metal + Ba Density profile can be decomposed into Friedel component + Gaussian add-on: enables an analytic solution to Poisson Eq… or at least a very easily solved solution How? • Approximations exist for location of background positive charge Lattice origin in relation to electron • Poisson’s Eq. easily solved with Friedel Density where = 2 k. F(xi – xo) Dipole due to electron-lattice difference ANL 34
QUANTUM DISTRIBUTION FUNCTION Heisenberg Representation vacuum metal Copper parameters Field = 1 e. V/nm f(x, k, t): quantum phase space distribution acts like probability distribution function Wigner Distribution function (WDF) vacuum metal Contours mimic classical trajectories; integrate both sides with respect to momentum (k) get density and current density when potentials are smooth and slowly varying, they are classical trajectories ANL 35
ANALYTICAL WDF MODEL: GAUSSIAN V(x) How does V(x, k) behave? Consider a solvable case where V(x) is a Gaussian: large x samples f(x, k') near k Sharp x 2 = 5. 0 small x samples f(x, k') far from k Broad x 2 = 0. 1 ANL 36
ANALYTICAL WDF MODEL (II): GAUSSIAN V(x) The behavior of V(x, k) signals the transition from classical to quantum behavior: Sharp: classical distribution Broad: quantum effects Can V(x, k) give a feel for when thermionic or field emission dominates? Consider most energetic electron appreciably present (corresponds to E = or k = k. F) Image Charge Potential If sin(k. Fx) does not “wiggle” much over range x, QM important Thermionic Emission: x is very large - expect classical description to be good Field Emission k. F x = O(2 p) implies ANL 37
ELECTRON DENSITY TO EMISSION BARRIER 1. Numerically solve QDF to Obtain Electron Density 2. Render Density in Terms of Friedel Components 3. Evaluate Potential From Exchange-Correlation Relation & Poisson’s Equation from the Friedel Representation 4. Change in Barrier seen to be due to shift in F and xo terms ANL 38
EMISSION BARRIER TO TRANSMISSION PROB. Modified Airy Function Approach: V(x) = Vo + F(x-xn): V(x) V(xn): N Linear Segments at xn , ∂x Matched at xn, xn+1, etc. Zi = Linear combinations of Airy Functions Ai(z), Bi(z) ANL 39
CONCLUSION Components of the Photocathode Program Analysis of Coated & Bare Metals (extend to semiconductors) Development Custom Engineered Controlled Porosity Photocathodes Creation of Photoemission Models Validated By Exp. for Beam Codes Theory Components included in photoemission code Work function dependence on coverage & components; local variation Spatial & Time Dependence of Temperature for laser & material parameters Fundamental models of scattering, photoemission, QE & Barrier Validation by bare and coated metal QE (macro) measurements Status of Modeling Effort Integrated Simulation Model Framework Without Recourse (Insofar As Possible) to “Fit” Parameters for “Library” Metals Using Quantum Distribution Function, Emission Theory, Coatings Theory Photoemission Modules Appropriate for Beam Simulation Code (1 st Generation Model distributed) From Integrated Simulation Model ANL 40
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