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

Objectives Spicer’s Three Step Model Field effects 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 K2CsSb Concluding thoughts

Three Step Model of Photoemission 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  due to applied field (Schottky Effect) Integrate product of probabilities over all electron energies capable of escape to obtain Quantum Efficiency Φ Φ’ Vacuum level Φ h Empty States Energy Filled States Laser Krolikowski and Spicer, Phys. Rev. 185 882 (1969) M. Cardona and L. Ley: Photoemission in Solids 1, (Springer-Verlag, 1978) Medium Vacuum

Step 1 – Absorption and Excitation Fraction of light absorbed: Iab/Iincident = (1-R(ν)) Probability of electron excitation to energy E by a photon of energy hν: Assumptions Medium thick enough to absorb all transmitted light Only energy conservation invoked, conservation of k vector is not an important selection rule

Density of States for Nb W.E. Pickett and P.B. Allen; Phy. Letters 48A, 91 (1974) Density of States for Nb Large number of empty conduction band states promotes unproductive absorption Density of States for Lead Lack of states below 1 eV limits unproductive absorption at higher photon energies NRL Electronic Structures Database http://cst-www.nrl.navy.mil/

Copper Density of States DOS is mostly flat for hν < 6 eV Past 6 eV, 3d states affect emission

Step 2 – Probability of reaching the surface w/o e--e- scattering 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

Step 3 - Escape Probability Criteria for escape: Requires electron trajectory to fall within a cone defined by angle: Fraction of electrons of energy E falling with the cone is given by: For small values of E-ET, this is the dominant factor in determining the emission. For these cases: This gives: 

EDC(E,hn)=(1-R(n))P(E,hn)T(E,hn)D(E) EDC and QE At this point, we have N(E,hn) - the Energy Distribution Curve of the emitted electrons: EDC(E,hn)=(1-R(n))P(E,hn)T(E,hn)D(E) To obtain the QE, integrate over all electron energies capable of escape: More Generally, including temperature: D. H. Dowell et al., Phys. Rev. ST-AB 9, 063502 (2006)

Schottky Effect and Field Enhancement Schottky effect reduces work function Field enhancement Typically, βeff is given as a value for a surface. In this case, the QE near threshold can be expressed as:

Field Enhancement Let us consider instead a field map across the surface, such that E(x,y)= (x,y)E0 For “infinite parallel plate” cathode, Gauss’s Law gives: In this case, the QE varies point-to-point. The integrated QE, assuming uniform illumination and reflectivity, is: Relating these expressions for the QE:

Field Enhancement Solving for effective field enhancement factor: Not Good – the field enhancement “factor” depends on wavelength In the case where , we obtain 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?

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)

Vacuum Arc deposited Nb Substrate Deuterium Lamp w/ monochromator 2 nm FWHM bandwidth Phi measured to be 3.91 V

D. H. Dowell et al., Phys. Rev. ST-AB 9, 063502 (2006)

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?

Three Step Model of Photoemission - Semiconductors 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 Empty States Vacuum level Φ h No States Energy Filled States Laser Medium Vacuum

Ettema and de Groot, Phys. Rev. B 66, 115102 (2002)

Assumptions for K2CsSb 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<2Egap) Field does not penetrate into cathode Band bending at the surface can be ignored

Parameters for K2CsSb Three Step Model e--phonon mean free path Energy transfer in each scattering event Number of particles Emission threshold (Egap+EA) Cathode Thickness Substrate material Parameter estimates from: Spicer and Herrea-Gomez, Modern Theory and Applications of Photocathodes, SLAC-PUB 6306

Laser Propagation and Interference Laser energy in media Calculate the amplitude of the Poynting vector in each media Not exponential decay 563 nm Vacuum K2CsSb 200nm Copper

Data from Ghosh & Varma, J. Appl. Phys. 48 4549 (1978)

Concluding Thoughts Thank You! 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!

DC results at 0.5 to 10 MV/m extrapolated to 0.5 GV/m Dark current beta - 27

 = 3.72 eV @ 5MV/m

Photoemission Results QE = 0.27% @ 213 nm for Arc Deposited 2.1 W required for 1 mA Electroplated Φ = 4.2 eV Expected Φ = 3.91 eV

Schottky Effect Φ Φ’ Φ’ (eV) = Φ- 3.7947*10-5E = Φ- 3.7947*10-5βE If field is enhanced near photoemission threshold Slope and intercept at two wavelengths determine Φ and β uniquely

Semiconductor photocathodes Vacuum Level Three step model still valid Eg+Ev< 2 eV Low e population in CB Band Bending Electronegative surface layer Conduction Band Ev e-n Vacuum Level Eg E Valence Band Medium Vacuum

K2CsSb cathode Properties Crystal structure: Cubic Stoichiometry: 2:1:1 Eg=1 eV, Ev=1.1 eV Max QE =0.3 Polarity of conduction: P High resistivity (100-1000 larger than Cs3Sb) Before(I) and after (II) superficial oxidation Photoemissive matrials, Sommer