Monte Carlo Scattering Corrections to Moments-Based Emission Models

Transcription

1 Photocathode Physics for Photoinjectors Workshop 8- October, Cornell University, Ithaca, NY Monte Carlo Scattering Corrections to Moments-Based Emission Models NRL: UMD: SAIC: K. L Jensen E. Montgomery, D. Feldman, P. O Shea J. J. Petillo Distribution Statement A: Approved for public release; distribution is unlimited Acknowledgments: J. Smedley (BNL), J. Harris, J. Lewellen (NPS) N. Moody (LANL), D. Dowell (SLAC) We gratefully acknowledge support from: Joint Technology Office, Office of Naval Research P3 Workshop, Cornell, NY of 6

2 OUTLINE Demands on electron sources for high performance FELs include high currents (ka/cm^ peak, A/cm^ average) and low emittance. Implies constraints on E distribution, temporal characteristics of bunches, Quantum Efficiency (for photoemission) Monte Carlo based models developed to augment emission modules for the beam simulation particle-in-cell (PIC) code MICHELLE. Transport of charge through semiconductor materials followed by emission into vacuum - Augments Moments-based models of QE (photocathodes) Scattering determines how bunches evolve under band bending, temporal characteristics, and phase space distribution Revised code: faster, x more electrons than previous, more accurate Show how Moments is augmented to include scattered electrons, and what changes are entailed by semiconductor photocathodes. Examine temperature rise and impact P3 Workshop, Cornell, NY of 6

3 PHOTOEMISSION PHYSICS A Moments-Based Method is used to treat Metals & Semiconductor Photocathodes M n = π " 3 m $ # ' & 3/ Simple DOS Averages of powers of momentum k are known as Moments of the distribution function = k n f(x,k) dk E / π / ( de sinθ dθ m + ) ( E + ω)sin θ, * - Transverse momentum uses sin; J calc would use cos for moment n/ { } f λ cosθ, p( ω) D (E + ω)cos θ This is transmission probability for surface barrier This is scattering loss factor for bulk transport ( 5 3) 5 * f FD (E) f FD (E + ω) Θ ω + E E g This is initial & final state occupation factor Quantum Efficiency: Ratio of Current (kz ) Moments QE = { R( ω) } ε n,rms = mc This is absorption Emittance: Ratio of Transverse Energy (kρ ) Moments # mc " ρ c k = k ρ + k ω x k x $ & M ( k z ) M ( k z ) D=, fλ = M M xk x # $ & k ( k ) ρ c ρ ( ω φ) metal ( ω E a E g ) v semi ( ω φ) $ # & " 3mc / leading order (metal) Metals p(e) large & fλ cosθ/p: therefore, emittance indep. of p. Semiconductors larger ε due to p small, but D also has impact The Fatal Approximation Commonly made approximation is electron scattering prevents emission While good for metals w/ big barriers, it is bad when (a) barriers are small or (b) extended to semiconductors P3 Workshop, Cornell, NY 3 of 6

4 MOMENTS-BASED QE: SEMICONDUCTORS Leading Order Approximation with χ = ω ( E g + E a ) QE = ( R( ω) ) limqe χ ω E g E a ( R( ω) ) p o + E de x dxd Δ " # Ex $ f x,e E λ a /E ω E E # x dx & g $ '( de ( + E a / χ) QE B spicer + gχ 3/ Semiconductors: e-e scattering not allowed unless final states unoccupied & in conduction band (creates magic window) magic window semi Quantum Efficiency [] 4 8 Cs 3 Sb E A =.3 ev E G =.6 ev Spicer (3K) UMD exp 4 Theory Theory x. Theory x.3 Lasers Photon Energy [ev] Quantum Efficiency... Data courtesy of I. Bazarov, Appl. Phys. Lett. 99, 5 () CsK Sb E A =.7 ev E G =. ev 53 nm 45 nm LANL Michelato, et al. Basarov, et al. Theory Scaled Theory Lasers "Theory": semiconductor parameters borrowed from Cs 3 Sb if unavailable Photon Energy [ev] P3 Workshop, Cornell, NY 4 of 6

5 MOMENTS-BASED SCATTERING FACTOR In Polar Coordinates, Velocity of e- at angle θ to normal Assume Any Scattering Event Is Fatal To Emission ( Fatal Approximation ) Ratio of penetration depth to distance between scattering events = δ ( ω ) l ( E) p E mδ ω = k ( E)τ E Matthiessen s Rule: τ total j = τ j f λ ( cosθ,e) k θ ω Ea Fraction Of Photoexcited Electrons Surviving Transport Back To Surface f λ ( cosθ, p) = exp x δ x $ # &dx " l ( E)cosθ exp x $ # &dx " δ = cosθ cosθ + p y.5 Example: Cs3Sb-like δ = 7 nm v/c =.8 τ = 3 fs p.36 Example: Cu-like δ =.6 nm v/c =.675 τ =.6 fs p.38 Scattering Fraction As Used In Modified Fowler Dubridge Eq (/y Acts As Cosine Of Escape Cone Angle) F λ ( y) = xf λ ( x, p) dx /y = p ln# y + p " + yp $ &+ y for semiconductors, measure E w.r.t. Ea IF τ scales as /k, then p is constant ( y )( yp y ) E E a y p p o F λ (y).4 p =.3 o.3 p =.5 o. p =. o p =. o. p = 4. o y p o =. P3 Workshop, Cornell, NY 5 of 6

6 MONTE CARLO AND TRANSPORT Calculation of Emittance (and QE) requires distribution function f(x,k). MC provides it. NAIVE EXPECTATIONS: In Metals: e-e collisions syphons energy off primary (target electron at Fermi level) quickly so Fatal approximation expected to be good u Metals: collision in rest frame of target electron shares the energy of primary with daughters If barrier small, sharing energy may not kill emission In Semiconductors: e-ph take fraction of ev from primary, so Fatal Approximation is suspect v' θ' E o sin u' E o cos Photoemission (Photocathodes) electrons start with energy > barrier height spaced according to exponential decay of Photon number with depth into metal (Optical penetration depth) Primary energy loss mechanism: Electron-Electron scattering Secondary Emission (Diamond Amplifier) (Jensen et al,, JAP8, 4459 ()): primary with energy Eo generates Np Secondaries N p = Eo/ΔE E = µ +φ + r( ω φ) velocity perpendicular to incident K. Murata, D. Kyser Adv. Elect. El. Phys 69, 75 (987) primary photoexcited secondaries spaced as z j+ z j ΔE ( z E) z=z all N j p secondaries appear to be generated simultaneously to a good approximation Primary energy loss mechanism: Optical Phonon P3 Workshop, Cornell, NY 6 of 6

7 SCATTERING CONSIDERATIONS Electron-Electron Scattering Energy & Direction Lab Frame B electron frame E a + E b = E a + E b conservation of Energy v a + v b = v a + v a conservation of momentum # u u = u + v " = u o cosθ ( ˆx sinθ + ẑ cosθ) " $# v = u o sinθ ( ˆx " cosθ + ẑ sinθ ) Random Angles: azimuthal φ (R commutes with R(φ)) polar θ (subject to final state constraints) Energy / E b B electron at Fermi level: final products > Fermi level, therefore, only some θ post-collision states are allowed. forbidden inc A inc B Scat A Scat B φ R θ u = R v a ( v a " v b ) -v b v' Incident (photoexcited) electron Target (Fermi level) electron Move to rest frame of Target Rotate so va-vb is along z axis θ' u' E o sin E o cos ˆ z Dominant Scattering Rates (metals) Electron-Electron Theta [deg] qo is screening factor 4 K (" τ ee = s + ΔE " $ 'γ k + F * $ '- α π mc (k B T ) )* # π k B T & # q o &,- " γ(x) = x3 4 tan x + x tan x + x $ + x ' $ + x ' # & Acoustic (metals&semi) π 3 4 ρv ' τ ac s )" T $ ' W 4mΞ k ( E) ( k B T ) # 5, Θ $ + ) ( &, *) # Θ& " T -) W_: Bloch-Grünnisen Function Collision randomly changes direction, removes phonon energy kbθd from electron P3 Workshop, Cornell, NY 7 of 6

8 BARE COPPER 66 NM Fatal Non-Fatal each time step.4 fs 4 frames = 6 fs Width = nm # initial =,, Count Fatal Approximation works well for metals with large work function Approximately 9 of electrons removed per scattering event # of Emitted Electrons 4 3 Emitted Electron Energy Distribution Total Cu N(n) = N o αn N o = 9579 # emitted Fit α =. 3 Transverse # Scatterings n= n= n=3 Longitudinal Energy [ev] P3 Workshop, Cornell, NY 8 of 6

9 CESIATED COPPER 66 NM Fatal each time step. fs 6 frames = 6 fs Width = nm # Initial = 5,, Fatal Approximation works well for metals with large work function Approximately 9 of electrons removed after 3 scattering events # of Emitted Electrons N o = 899 c =.5 c = 3.7 # emitted Fit = N o exp N n c n n + c Emitted Electron Energy Distribution # Scatterings Total Transverse Longitudinal 5 Cs-Cu n= n= n=3 Non-Fatal Count Energy [ev] P3 Workshop, Cornell, NY 9 of 6

10 MC FIX TO QE MOMENTS - METALS MONTE CARLO modifications to Moments-Based Photoemission depends on size of emission barrier Collision Both not emitted A emitted B not emitted Both Emitted After e-e scattering, primary e shares energy with target e. If... hf barrier: shared energy means neither electron emitted hf.5 x barrier: more energetic e- may still emit hf > x barrier: both electrons can be emitted To Fix Moments QE: Define R = ratio of all e- emitted / all unscattered e- emitted Multiply Moments QE by R(x) to estimate QE where x = (hf - φ)/hf = ratio of energy above barrier to photon energy Fermi Sea of Electrons Vacuum R(x) Φ = 4.5 ev Cu 4 Φ =.6 ev λ = 66 nm λ = 45 nm 3 Analytic Universal Curve-like behavior R(x) = +. x + 7. x Quantum Efficiency Dowell, et al. Fatal x 5 Non-Fatal x 5 R(x) = +. x + 7. x 7 Φ Dowell (Cu) = 4.3 ev Not such a big difference for bare metals with good conductivity like copper fatal = no scattering non-fatal = with scattering (hf - φ) / hf Wavelength [nm] P3 Workshop, Cornell, NY of 6

11 MC FIX TO QE MOMENTS - CESIATED METALS Comparison to Cs-W Surfaces: UMD Dispenser Photocathode Contamination present, Multiple crystal faces exposed, and variation in crystal face work function expected Photoemission Φ (from Samsonov, Handbook of Thermionic Properties ) {} = 5.85 ev, {} = 4.39 ev, {} = 4.56 ev* Acoustic scattering more important in W than in Cu QE(Exp) = (scaling factor) x QE(Theory) Moments-based w/o MC: Scaling factor =.9 Moments-based w/ MC: Scaling factor.46 accounts for crystal faces Better correspondence to QE peak variation with λ Better overall shape of QE(θ) near respective peaks R(x) Rx 375 Rx 45 Rx 53 Analytic 375+ Analytic 66 Rx 66 R(x) = x + 5 x * {} value is for thermionic emission ratio of Non-fatal to fatal emission calculated from Monte Carlo (hf - φ)/hf Cs:W R(x) = x + 5 x 4.5 Quantum Efficiency [] Experimental Fatal Approx Non Fatal Approx Cs:W Coverage [] scaled to match this point P3 Workshop, Cornell, NY of 6

12 SHELL VS SPHERE EMISSION PROCESSES Electrons that do no scatter before emission travel ballistically as expanding SHELL Electrons that scatter before emission travel as a diffusively expanding SPHERE Scattered electrons change time dependence by adding long time tail Simple Current model: passage of shell or sphere through surface boundary as a function of time Expanding SHELL time Q(t) = Q(t) = Q d Q r 4π R t v d t π cos ( R()/R(t )) Rdϕ Rsinθ dθ Diffusive SPHERE t / exp( z / 4Dt)dz t / exp( z / 4Dt)dz = Q r = Q ') d ( Erf # *) " t t + t o τ * t $ + ) &, -) time R( t) = v o t R t P3 Workshop, Cornell, NY of 6

13 SHELL VS SPHERE EMISSION CURRENT Charge to pass through surface is sum of Shell & Sphere Processes: I(t) = dq/dt t = Q o s +Q d I t t + t o τ * 4πt 3 / exp τ * t Current [#/fs] t -3/ 3748 I Shell I Sphere I Sum t << t >> Q shell (t) = Q r t t + t o Q sphere (t) = Q ') d ( Erf *) # " τ * t $ + ) &, -) Current [#/fs] I Shell I Sphere I Sum t << t >> Time [fs] t -3/ 3748 SHELL SPHERE. Time [fs] P3 Workshop, Cornell, NY 3 of 6

14 EXTENSION TO SEMICONDUCTORS Scattered electrons contribute more to QE Electrons photoexcited much further into semiconductors Rate of energy loss due to phonons (ΔE phonon energy) far slower than e-e collisions (ΔE (E-EF)/) Complication: electrons likely distributed in deposition so that shell model has range of times (ta to tb) rather than characteristic time (to) as for metals. Affects SHELL model EX: GaAs mobility = 85 cm /Volt-sec Implies τ =.34 ps. Assume ta τ. Then B(.4,.) = B(.55,) = B(.7,) / Full width at half maximum (FWHM) of D(x) is.8 Long tails persist: D() 7, D(38) = CONCLUDE: Semiconductor physics results in longer characteristic times for both the shell and sphere components Time response & rise/fall multiples of scattering time Long diffusive time tails due to D(x) and associated with NEA photocathodes mitigated by small but finite positive electron affinities (PEAs) of multialkali antimonide photocathodes Normalized Current I shell I sphere ( t) = Q shell t b t a t a t b t a # I p B t, t & b ( $ t a ' ( t ) I D t $ # s " τ & B( x, n) = D x " # ln n 3x = e 3/ $ # & e /x " ln# n + x " + x s ( t + s) ds $ & x( n ) $ ( + x) ( n + x) & Shell. Shell Shell Sphere. Time / τ P3 Workshop, Cornell, NY 4 of 6

15 MODELING OF QE FOR SEMICONDUCTORS Moments-Based Model of QE for SEMICONDUCTORS R = reflectivity DΔ = Transmission Probability, fλ = scattering factor QE = ( R( ω) ) ω E g E a E de x dxd Δ Ex f λ ( x, E) E a /E ω E E x dx g de Scattering factor: Temperature dependence of QE arises from dependence of scattering rate (inverse relaxation time) on T factor accounts for fraction of electrons lost due to scattering during emission: δ= laser penetration depth, v = electron velocity τ = scattering time f λ Code validated for Cs3Sb ( cosθ, p) = cosθ cosθ + δ ( ω ) v( E)τ E Extended to CsKSb Quantum Efficiency K 4K 5K 6K 7K 8K Fit3 Fit4 Fit5 Fit6 Fit7 Fit8 QE(T)/QE(3) -.65*x+.*x x = (T-3)/ Photon Energy [ev] CsK Sb Simulation: QE declines with T because scattering rates increase (i.e., τ(e) decreases) Does Cathode heat up during few seconds of illumination? YES T t C v ( T ) dt dt = ΔE πr L = T o + 4K { t [ sec] } Assumptions: CsKSb is a thin (tens of nm) layer on a Mo disk of thickness L. cm w/ radius r =.5 cm. Energy / pulse is.5 mj Laser power is nm QE(3K) = => ΔQ = nc Cv.65 J/K-cm for Moly T-rise during a laser pulse is small, but SUM of all pulses add up to a large effect QE T QE T o.65χ +.χ χ = T T o ΔT T o = 3K ΔT = 5K after 3 sec, QE drops to 6 of initial P3 Workshop, Cornell, NY 5 of 6

16 SUMMARY Monte Carlo Corrections to Moments Revised scattering operators for electron-electron and acoustic phonon Speed improvement, greater accuracy Correction factor to Moments Approach: R(x) exhibits universal behavior - appropriate for PIC simulations Emitted Electrons and scattering: No-Scatter electrons: fast time response - Shell Model Scattered electrons: long time tail - Diffusive Sphere Model Temperature dependence of scattering Affects Quantum Efficiency Simple model: cathode heats up over time, so QE changes over time Next: Extend MC techniques to......qe from semiconductors optical phonon replaces e-e as dominant loss mechanism...pic-ready simulation modules P3 Workshop, Cornell, NY 6 of 6