Photocathode Theory. John Smedley. Thanks to Kevin Jensen (NRL),

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1 Photocathode Theory John Smedley Thanks to Kevin Jensen (NRL), Dave Dowell and John Schmerge (SLAC)

2 Objectives Spicer s Three Step Model Overview Application to metals Comparison to data (Pb and Cu) Field effects Schottky effect Field enhancement Three Step Model for Semiconductors Numerical implementation Comparison for K 2 CsSb Concluding thoughts

3 Three Step Model of Photoemission Energy Empty States Filled States hν Vacuum level Φ Φ Φ Laser 1) Excitation of e in metal Reflection (angle dependence) Energy distribution of excited e 2) Transit to the Surface e e scattering Direction of travel 3) Escape surface Overcome Workfunction Reduction of Φ due to applied field (Schottky Effect) Integrate product of probabilities over all electron energies capable of escape to obtain Quantum Efficiency Medium Vacuum Krolikowski and Spicer, Phys. Rev (1969) M. Cardona and L. Ley: Photoemission in Solids 1, (Springer Verlag, 1978)

4 Step 1 Absorption and Excitation Fraction of light absorbed: I ab /I incident = (1 R(ν)) Probability of electron excitation to energy E by a photon of energy hν: P E, hν = E hν f E f N E N E hν N E ' N E ' hν de ' Assumptions Medium thick enough to absorb all transmitted light Only energy conservation invoked, conservation of k vector is not an important selection rule

5 N b D ensity of S tates W.E. Pickett and P.B. Allen; Phy. Letters 48A, 91 (1974) N /ev E fe rm i ev T hreshold Energy Density of States for Nb Large number of empty conduction band states promotes unproductive absorption Lead D ensity of S tates 1.2 Density of States for Lead Lack of states below 1 ev limits unproductive absorption at higher photon energies N /ev E fermi Threshold E nergy NRL Electronic Structures Database ev

6 Copper Density of States Fong&Cohen, Phy. Rev. Letters, 24, p306 (1970) Ferm i Level Threshold E nergy DOS is mostly flat for hν < 6 ev Past 6 ev, 3d states affect emission N(E) Energy above the bottom of the Valance Band [ev]

7 Step 2 Probability of reaching the surface w/o T E, ν, θ = e e scattering λ e E / λ p h ν 1 λ e E / λ p h ν C E, ν, θ λ p h = λ 4 πk e mean free path can be calculated Extrapolation from measured values From excited electron lifetime (2 photon PE spectroscopy) Comparison to similar materials Assumptions Energy loss dominated by e e scattering Only unscattered electrons can escape Electrons must be incident on the surface at nearly normal incidence => Correction factor C(E,v,θ) = 1

8 250 E lectron M ean Free P ath in Lead, C opper and N iobium T hreshold E nergy for E m ission P b N b C u e in P b M FP (A ngstrom s) e in N b e in C u E lectron E nergy above Ferm i Level (ev )

9 M FP (A ngstrom s) E lectron and P hoton M ean Free P ath in Lead, C opper and N iobium T hreshold E nergy for E m ission Pb N b C u e in P b 190 nm photon (P b) e in N b 190 nm photon (N b) e in C u 190 nm photon (C u) E lectron E nergy above Ferm i Level (ev )

10 Step 3 Escape Probability Criteria for escape: ħ 2 2 k 2 m E =E φ T f Requires electron trajectory to fall within a cone defined by angle: k min cos θ= k = E T E 2 Fraction of electrons of energy E falling with the cone is given by: D E = 1 θ 2 π sin θ' dθ' 4 π 0 0 For small values of E E T, this is the dominant factor in determining the emission. For these cases: This gives: 1 QE ν hν E f D E de = φ E f QE ν hν φ 2 d = cos θ = E T E E T θ 1 hν φ E T D E de 2

11 EDC and QE At this point, we have N(E,hν) the Energy Distribution Curve of the emitted electrons: EDC(E,hν)=(1 R(ν))P(E,hν)T(E,hν)D(E) To obtain the QE, integrate over all electron energies capable of escape: QE ν = 1 R ν hν E f φ E f More Generally, including temperature: P E, ν T E, ν D E de QE ω = 1 R ω E φ ħ ω F 1 de N E ħ ω 1 F E ħ ω N E F E cos θ max E de 0 D. H. Dowell et al., Phys. Rev. ST AB 9, (2006) 1 N E ħ ω 1 F E ħ ω N E F E 1 2 π d cos θ T e e E, ω, θ dφ 0 2 π d cos θ dφ 0

12 Schottky Effect and Field Enhancement Schottky effect reduces work function Δφ schottkey [ ev ]=α E [ V m ] α=e e 4 πε 0 = [ e Vm] Field enhancement Typically, β eff is given as a value for a surface. In this case, the QE near threshold can be expressed as: QE=B hν φ 0 α β eff E 2

13 Field Enhancement Let us consider instead a field map across the surface, such that E(x,y)= β(x,y)e 0 For infinite parallel plate cathode, Gauss s Law gives: 1 β x, y dxdy=1 A A In this case, the QE varies point to point. The integrated QE, assuming uniform illumination and reflectivity, is: Relating these expressions for the QE: area emission A area emission A

14 Field Enhancement Solving for effective field enhancement factor: area emission A 1/ 2 hν φ 0 2 Not Good the field enhancement factor depends on wavelength In the case where, we hν obtain =φ 0 area emission =1 Local variation of reflectivity, and non uniform illumination, could lead to an increase in beta Clearly, the field enhancement concept is very different for photoemission (as compared to field emission). Perhaps we should use a different symbol?

15 Implementation of Model Material parameters needed Density of States Workfunction (preferably measured) Complex index of refraction e mfp at one energy, or hot electron lifetime Optional surface profile to calculate beta Numerical methods First two steps are computationally intensive, but do not depend on phi only need o be done once per wavelength (Mathematica) Last step and QE in Excel (allows easy access to EDCs, modification of phi) No free parameters (use the measured phi)

16 Lead QE vs Photon energy 1.0E 02 Theory M easurem ent QE 1.0E E 04 Vacuum Arc deposited Nb Substrate Deuterium Lamp w/ monochromator 2 nm FWHM bandwidth Phi measured to be 3.91 V Photon energy (ev)

17 Energy Distribution Curves 2.50E 03 Electrons per photon per ev 2.00E E E E nm 200 nm 210 nm 220 nm 230 nm 240 nm 250 nm 260 nm 270 nm 280 nm 290 nm 0.00E Electron energy (ev)

18 Copper QE vs Photon Energy 1.E 02 1.E 03 QE 1.E 04 Theory 1.E 05 D ave's D ata 1.E 06 D. H. Dowell et al., Phys. Rev. ST AB 9, (2006) Photon energy(ev)

19 Energy Distribution Curves Copper Electrons per photon per ev 1.2E E E E E E nm 200 nm 210 nm 220 nm 230 nm 240 nm 250 nm 260 nm 270 nm 280 nm 290 nm 0.0E Electron energy (ev)

20 Improvements Consider momentum selection rules Take electron heating into account Photon energy spread (bandwidth) Consider once scattered electrons (Spicer does this) Expand model to allow spatial variation Reflectivity Field Workfuncion?

21 Three Step Model of Photoemission Semiconductors Energy Empty States No States Filled States hν Vacuum level Φ Laser 1) Excitation of e Reflection, Transmission, Interference Energy distribution of excited e 2) Transit to the Surface e phonon scattering e e scattering Random Walk 3) Escape surface Overcome Workfunction Need to account for Random Walk in cathode suggests Monte Carlo modeling Medium Vacuum

22 Ettema and de Groot, Phys. Rev. B 66, (2002)

23 Assumptions for K 2 CsSb Three Step Model 1D Monte Carlo (implemented in Mathematica) e phonon mean free path (mfp) is constant Energy transfer in each scattering event is equal to the mean energy transfer Every electron scatters after 1 mfp Each scattering event randomizes e direction of travel Every electron that reaches the surface with energy sufficient to escape escapes Cathode and substrate surfaces are optically smooth e e scattering is ignored (strictly valid only for E<2E gap ) Field does not penetrate into cathode Band bending at the surface can be ignored

24 Parameters for K 2 CsSb Three Step Model e phonon mean free path Energy transfer in each scattering event Number of particles Emission threshold (E gap +E A ) Cathode Thickness Substrate material Parameter estimates from: Spicer and Herrea Gomez, Modern Theory and Applications of Photocathodes, SLAC PUB 6306

25 Laser Propagation and Interference Calculate the amplitude of the Poynting vector in each media Laser energy in media Not exponential decay nm 0.2 Vacuum K 2 CsSb 200nm Copper

26 QE QE vs Cathode Thickness 50 nm 200 nm E xperim ent 20 nm 20 nm 10 nm photon energy [ev] Data from Ghosh & Varma, J. Appl. Phys (1978)

27 QE vs Mean Free Path QE Experim ent 10 nm m fp 5 nm m fp 20 nm m fp photon energy [ev]

28 Concluding Thoughts As much as possible, it is best to link models to measured parameters, rather than fitting Ideally, measured from the same cathode Whenever possible, QE should be measured as a function of wavelength. Energy Distribution Curves would be wonderful! Spicer s Three Step model well describes photoemission from most metals tested so far The model provides the QE and EDCs, and a Monte Carlo implementation will provide temporal response The Schottky effect describes the field dependence of the QE for metals (up to 0.5 GV/m). Effect on QE strongest near threshold. Field enhancement for a normal (not needle, grating) cathode should have little effect on average QE, though it may affect a QE map A program to characterize cathodes is needed, especially for semiconductors (time for Light Sources to help us) Thank You!

29 Sqrt QE vs Sqrt F, KrF on Cu F igure 5.15 P hi = 4.40 F ilter = DC results at 0.5 to 10 MV/m extrapolated to 0.5 GV/m Sqrt QE Dark current beta 27 Theory, B eta = 1.2 Theory, B eta = 1 Theory, B eta = 2 Theory, B eta = 3 Data (80 O hm, 1.19 m m ) Data (80 O hm, 2.11 m m ) Data (20 O hm, 2.11 m m ) Sqrt F (F in V/m)

31 Φ = MV/m

32 Photoemission Results QE = 213 nm for Arc Deposited 2.1 W required for 1 ma Electroplated Φ = 4.2 ev Expected Φ = 3.91 ev

33 Schottky Effect Φ Φ Φ (ev) = Φ *10 5 E 5 = Φ *10 βe If field is enhanced QE= 1 R hν φ 0 α βe near photoemission threshold Slope and intercept at two wavelengths determine Φ and β uniquely

34 Semiconductor photocathodes Vacuum Level Three step model still valid Conduction Band E v E g +E v < 2 ev E E g e n Vacuum Level Low e population in CB Band Bending Valence Band Electronegative surface layer Medium Vacuum

35 K 2 CsSb cathode Properties Crystal structure: Cubic Stoichiometry: 2:1:1 E g =1 ev, E v =1.1 ev Max QE =0.3 Polarity of conduction: P Before(I) and after (II) superficial oxidation High resistivity ( larger than Cs 3 Sb) Photoemissive matrials, Sommer

RF Gun Photo-Emission Model for Metal Cathodes Including Time Dependent Emission

RF Gun Photo-Emission Model for Metal Cathodes Including Time Dependent Emission SLAC-PUB-117 February 6 (A) J. F. SCHMERGE, J. E. CLENDENIN, D. H. DOWELL, AND S. M. GIERMAN SLAC, Stanford University,