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

2. 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

3. 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 (1969)
M. Cardona and L. Ley: Photoemission in Solids 1, (Springer-Verlag, 1978)
Medium
Vacuum

4. 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

5. 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

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

7. 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

10. 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: 

11. 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, (2006)

12. 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:

13. 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:

14. 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?

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

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

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
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

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

23. 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

24. 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

25. 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

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

28. 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!

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

31.  = 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- 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

34. 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

35. 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 ( larger than Cs3Sb)
Before(I) and after (II) superficial oxidation
Photoemissive matrials, Sommer