Photocathode Theory
John Smedley
Thanks to Kevin Jensen (NRL),
Dave Dowell and John Schmerge (SLAC)
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 K2CsSb
• Concluding thoughts
Three Step Model of Photoemission
1) Excitation of e- in metal
Reflection (angle dependence)
Energy distribution of excited e-
2) Transit to the Surface
Φ
Φ’
e--e- scattering
Direction of travel
3) Escape surface
Φ
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
Filled States
Energy
Empty States
h
Vacuum level
Medium
Vacuum
Krolikowski and Spicer, Phys. Rev. 185 882 (1969)
M. Cardona and L. Ley: Photoemission in Solids 1,
(Springer-Verlag, 1978)
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:
P( E , h ) 
N ( E ) N ( E  h )
E f  h
 N ( E ' ) N ( E 'h )dE '
Ef
Assumptions
– Medium thick enough to absorb all transmitted light
– Only energy conservation invoked, conservation of k
vector is not an important selection rule
W.E. Pickett and P.B. Allen; Phy. Letters 48A, 91 (1974)
N/eV
Nb Density of States
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Efermi
Threshold Energy
Density of States for Nb
Large number of empty
conduction band states
promotes unproductive
absorption
0
2
4
6
8
10
12
eV
Lead Density of States
1.2
Density of States for Lead
0.8
N/eV
Lack of states below 1 eV limits
unproductive absorption at
higher photon energies
Threshold Energy
Efermi
1
0.6
0.4
0.2
NRL Electronic Structures Database
http://cst-www.nrl.navy.mil/
0
0
2
4
6
eV
8
10
12
Copper Density of States
Fong&Cohen, Phy. Rev. Letters, 24, p306 (1970)
Fermi Level
Threshold Energy
N(E)
DOS is mostly flat for hν < 6 eV
Past 6 eV, 3d states affect emission
0
2
4
6
8
10
12
Energy above the bottom of the Valance Band [eV]
14
16
Step 2 – Probability of reaching the
surface w/o e--e- scattering
e ( E )  ph ( )
T ( E , , ) 
C ( E , ,  )
1  e ( E )  ph ( )
 ph


4k
• 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
Electron Mean Free Path in Lead, Copper and Niobium
Threshold Energy for Emission
Pb
Nb Cu
250
MFP (Angstroms)
e in Pb
200
e in Nb
e in Cu
150
100
50
0
2
2.5
3
3.5
4
4.5
5
Electron Energy above Fermi Level (eV)
5.5
6
Electron and Photon Mean Free Path in Lead, Copper and Niobium
Threshold Energy for Emission
Pb
Nb Cu
MFP (Angstroms)
250
200
150
e in Pb
190 nm photon (Pb)
e in Nb
190 nm photon (Nb)
e in Cu
190 nm photon (Cu)
100
50
0
2
2.5
3
3.5
4
4.5
5
Electron Energy above Fermi Level (eV)
5.5
6
Step 3 - Escape Probability
•
•
Criteria for escape:
•
•

Requires electron trajectory to fall
within a cone defined by angle:
cos  
•
 2 k2
 ET  E f  
2m
kmin
E 1
 ( T ) 2
E
k
Fraction of electrons of energy E
falling with the cone is given by:
2
1 
1
1
ET 12
D( E ) 
sin  ' d '  d  (1  cos )  (1  ( ) )

4 0
2
2
E
0
For small values of E-ET, this is
the dominant factor in determining
the emission. For these cases:
This gives:
QE ( ) 
h  E f
( h  )  ET
E f
ET
D( E )dE   D( E )dE


QE ( )  (h   ) 2
EDC and QE
At this point, we have N(E,h) - the Energy Distribution Curve
of the emitted electrons:
EDC(E,h)=(1-R())P(E,h)T(E,h)D(E)
To obtain the QE, integrate over all electron energies capable
of escape:
h  E f
QE ( )  (1  R( ))
P( E, )T ( E, ) D( E )dE


E f
More Generally, including temperature:

QE ( )  (1  R( ))

dE N ( E   )(1  F ( E   )) N ( E ) F ( E )
E F   
d (cos  )T


e e
cos

 dE
2
1
( E ,  , )  d
max ( E )
0
1
2
1
0
N ( E   )(1  F ( E   )) N ( E ) F ( E )  d (cos  )  d
0
D. H. Dowell et al., Phys. Rev. ST-AB 9, 063502 (2006)
Schottky Effect and Field Enhancement
• Schottky effect reduces work function
V
 schottkey[eV ]   E[ ]
m
 e
e
40
 3.7947  10 5 [e Vm ]
• Field enhancement
Typically, βeff is given as a value for a surface. In this
case, the QE near threshold can be expressed as:
QE  B(h   0    eff E ) 2
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:
1
 ( x, y )dxdy  1

AA
In this case, the QE varies point-to-point. The integrated
QE, assuming uniform illumination and reflectivity, is:
B
QE 
2
(
h






(
x
,
y
)
E
)
dxdy
0

emission
area
A
Relating these expressions for the QE:
2
(
h






(
x
,
y
)
E
)
dxdy
0

(h   0    eff E ) 
2
emission
area
A
Field Enhancement
Solving for effective field enhancement factor:
 eff
1/ 2
2



   (h  0    ( x, y ) E0 ) dxdy 



1   emission
area
 2 
  (h  0 ) 
 E0  
A


 





2
Not Good – the field enhancement “factor” depends on wavelength
In the case where h  0 , we obtain  eff 
1
 ( x, y )dxdy  1

A emission
area
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)
Lead QE vs Photon energy
1.0E-02
QE
Theory
Measurement
1.0E-03
Vacuum Arc deposited
Nb Substrate
Deuterium Lamp w/ monochromator
2 nm FWHM bandwidth
Phi measured to be 3.91 V
1.0E-04
4.00
4.50
5.00
5.50
6.00
Photon energy (eV)
6.50
7.00
Energy Distribution Curves
Electrons per photon per eV
2.50E-03
190 nm
2.00E-03
200 nm
210 nm
220 nm
230 nm
1.50E-03
240 nm
250 nm
260 nm
270 nm
1.00E-03
280 nm
290 nm
5.00E-04
0.00E+00
0.0
0.5
1.0
1.5
2.0
Electron energy (eV)
2.5
3.0
Copper QE vs Photon Energy
1.E-02
QE
1.E-03
1.E-04
Theory
Dave's Data
1.E-05
D. H. Dowell et al., Phys. Rev. ST-AB 9, 063502 (2006)
1.E-06
4.0
4.5
5.0
5.5
Photon energy(eV)
6.0
6.5
7.0
Energy Distribution Curves - Copper
Electrons per photon per eV
1.2E-03
190 nm
200 nm
210 nm
220 nm
230 nm
240 nm
250 nm
260 nm
270 nm
280 nm
290 nm
1.0E-03
8.0E-04
6.0E-04
4.0E-04
2.0E-04
0.0E+00
0.0
0.5
1.0
1.5
Electron energy (eV)
2.0
2.5
3.0
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
1) Excitation of eEmpty States
Reflection, Transmission,
Interference
Energy distribution of excited e-
h
Vacuum level 2) Transit to the Surface
3) Escape surface
Overcome Workfunction
Filled States
Energy
No States
Φ
e--phonon scattering
e--e- scattering
Random Walk
Need to account for Random Walk in
cathode suggests Monte Carlo
modeling
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
0.8
Calculate the amplitude of
the Poynting vector in each
media
0.6
Not exponential decay
0.4
563 nm
0.2
2 10
Vacuum
-7
4 10
K2CsSb Copper
200nm
-7
6 10
-7
8 10
-7
1 10
-6
QE
QE vs Cathode Thickness
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
50 nm
200 nm
Experiment
20 nm
20 nm
10 nm
2
2.2
2.4
2.6
2.8
photon energy [eV]
Data from Ghosh & Varma, J. Appl. Phys. 48 4549 (1978)
3
3.2
3.4
QE
QE vs Mean Free Path
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
2.00
Experiment
10 nm mfp
5 nm mfp
20 nm mfp
2.20
2.40
2.60
2.80
photon energy [eV]
3.00
3.20
3.40
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!
Sqrt QE vs Sqrt F, KrF on Cu
Figure 5.15
0.025
Phi = 4.40
Filter = .187
0.02
DC results at 0.5 to 10 MV/m extrapolated to 0.5 GV/m
Sqrt QE
0.015
0.01
Theory, Beta = 1.2
Theory, Beta = 1
Theory, Beta = 2
0.005
Theory, Beta = 3
Data (80 Ohm, 1.19 mm)
Data (80 Ohm, 2.11 mm)
Dark current beta - 27
Data (20 Ohm, 2.11 mm)
0
0
5000
10000
15000
20000
Sqrt F (F in V/m)
25000
30000
35000
 = 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
QE  (1  R)(h  0   E ) near photoemission threshold
Slope and intercept at two wavelengths determine Φ and β uniquely
Semiconductor photocathodes
Vacuum
Level
Three step model still valid
Conduction Band
Ev
Eg+Ev< 2 eV
Low e population in CB
E
e-n Vacuum
Band Bending
Level
Eg
Electronegative surface layer
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
Before(I) and after (II)
superficial oxidation
Photoemissive matrials, Sommer
High resistivity (100-1000
larger than Cs3Sb)