Photocathode Theory

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Photocathode Theory

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

2
Objectives

• Spicers 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 K2CsSb
• Concluding thoughts

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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

Vacuum level
F
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
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Step 1 Absorption and Excitation

• Fraction of light absorbed Iab/Iincident
  (1-R(?))
• Probability of electron excitation to energy E by
  a photon of energy h?
• Assumptions
• Medium thick enough to absorb all transmitted
  light
• Only energy conservation invoked, conservation of
  k vector is not an important selection rule

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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/
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Copper Density of States
DOS is mostly flat for h? lt 6 eV Past 6 eV, 3d
states affect emission
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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 gt Correction factor
  C(E,v,?) 1

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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

?
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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)
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Schottky Effect and Field Enhancement

• Schottky effect reduces work function
• Field enhancement
• Typically, ßeff is given as a value for a
  surface. In this case, the QE near threshold can
  be expressed as

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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, Gausss
  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
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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?
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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)

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Vacuum Arc deposited Nb Substrate Deuterium Lamp
w/ monochromator 2 nm FWHM bandwidth Phi measured
to be 3.91 V
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D. H. Dowell et al., Phys. Rev. ST-AB 9, 063502
(2006)
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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?

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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
F
h?
No States
Energy
Filled States
Laser
Medium
Vacuum
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Ettema and de Groot, Phys. Rev. B 66, 115102
(2002)
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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 Elt2Egap)
• Field does not penetrate into cathode
• Band bending at the surface can be ignored

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Parameters for K2CsSb Three Step Model

• e--phonon mean free path
• Energy transfer in each scattering event
• Number of particles
• Emission threshold (EgapEA)
• Cathode Thickness
• Substrate material
• Parameter estimates from
• Spicer and Herrea-Gomez, Modern Theory and
  Applications of Photocathodes, SLAC-PUB 6306

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Laser Propagation and Interference
Laser energy in media
Calculate the amplitude of the Poynting vector in
each media
563 nm
Vacuum
K2CsSb 200nm
Copper
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Data from Ghosh Varma, J. Appl. Phys. 48 4549
(1978)
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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!
• Spicers 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!
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DC results at 0.5 to 10 MV/m extrapolated to 0.5
GV/m
Dark current beta - 27
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? 3.72 eV _at_ 5MV/m
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Photoemission Results
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Schottky Effect
Slope and intercept at two wavelengths determine
F and ß uniquely
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Semiconductor photocathodes
Vacuum Level
Three step model still valid EgEvlt 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
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K2CsSb cathode
Properties Crystal structure Cubic Stoichiometry
211 Eg1 eV, Ev1.1 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