Photocathode Theory John Smedley Thanks to Kevin Jensen (NRL), Dave Dowell and John Schmerge (SLAC)
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Transcript of 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
Energy
Medium Vacuum
Φ
Vacuum level
Three Step Model of Photoemission
Filled S
tatesE
mpty S
tates
h
1) Excitation of e- in metalReflection (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 overall electron energies capable of escape to obtain Quantum EfficiencyLaser
Φ
Φ’
Krolikowski and Spicer, Phys. Rev. 185 882 (1969)M. Cardona and L. Ley: Photoemission in Solids 1, (Springer-Verlag, 1978)
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
hE
E
f
f
dEhENEN
hENENhEP
')'()'(
)()(),(
Step 1 – Absorption and Excitation
W.E. Pickett and P.B. Allen; Phy. Letters 48A, 91 (1974)
Lead Density of States
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12eV
N/e
V
Efermi Threshold Energy
Nb Density of States
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 2 4 6 8 10 12
eV
N/e
V
Efermi Threshold Energy
NRL Electronic Structures Database
Density of States for NbLarge 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
http://cst-www.nrl.navy.mil/
Copper Density of StatesFong&Cohen, Phy. Rev. Letters, 24, p306 (1970)
0 2 4 6 8 10 12 14 16
Energy above the bottom of the Valance Band [eV]
N(E
)
Fermi Level Threshold Energy
DOS is mostly flat for hν < 6 eVPast 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
),,()()(1
)()(),,(
ECE
EET
phe
phe
kph
4
Electron Mean Free Path in Lead, Copper and Niobium
0
50
100
150
200
250
2 2.5 3 3.5 4 4.5 5 5.5 6
Electron Energy above Fermi Level (eV)
MF
P (
An
gst
rom
s)
e in Pb
e in Nb
e in Cu
Threshold Energy for Emission Pb Nb Cu
Electron and Photon Mean Free Path in Lead, Copper and Niobium
0
50
100
150
200
250
2 2.5 3 3.5 4 4.5 5 5.5 6
Electron Energy above Fermi Level (eV)
MF
P (
Ang
stro
ms)
e in Pb190 nm photon (Pb)e in Nb190 nm photon (Nb)e in Cu190 nm photon (Cu)
Threshold Energy for Emission Pb Nb Cu
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:
fT EEm
k
2
22
21
min )(cosE
E
k
k T
T
T
f
f
Eh
E
Eh
E
dEEDdEEDQE)(
)()()(
2)()( hQE
))(1(2
1)cos1(
2
1''sin
4
1)( 2
1
0
2
0 E
EddED T
f
f
Eh
E
dEEDETEPRQE
)(),(),())(1()(
EDC and QEAt 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:
More Generally, including temperature:
0
2
0
1
1
2
0
1
)(cos
)(cos)()())(1)((
),,()(cos)()())(1)((
))(1()( max
ddEFENEFENdE
dETdEFENEFENdE
RQE FE
ee
E
D. H. Dowell et al., Phys. Rev. ST-AB 9, 063502 (2006)
Schottky Effect and Field Enhancement• Schottky effect reduces work function
• Field enhancementTypically, βeff is given as a value for a surface. In this
case, the QE near threshold can be expressed as:
][107947.34
][][
5
0
Vmee
e
m
VeVschottkey
E
20 )( EhBQE eff
Field EnhancementLet 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:
1),(1
A
dxdyyxA
A
dxdyEyxhB
QE areaemission
20 )),((
Relating these expressions for the QE:
A
dxdyEyxh
Eh areaemission
eff
20
20
)),((
)(
Field EnhancementSolving for effective field enhancement factor:
2
0
2/12
00
02
)(
)),((
1
h
A
dxdyEyxh
Eareaemission
eff
Not Good – the field enhancement “factor” depends on wavelength
In the case where , we obtain
0 h 1),(1
areaemission
eff dxdyyxA
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-04
1.0E-03
1.0E-02
4.00 4.50 5.00 5.50 6.00 6.50 7.00
Photon energy (eV)
QE
Theory
Measurement
Vacuum Arc depositedNb SubstrateDeuterium Lamp w/ monochromator2 nm FWHM bandwidthPhi measured to be 3.91 V
Energy Distribution Curves
0.00E+00
5.00E-04
1.00E-03
1.50E-03
2.00E-03
2.50E-03
0.0 0.5 1.0 1.5 2.0 2.5 3.0Electron energy (eV)
Ele
ctro
ns
per
ph
oto
n p
er e
V
190 nm
200 nm
210 nm
220 nm
230 nm
240 nm
250 nm
260 nm
270 nm
280 nm
290 nm
Copper QE vs Photon Energy
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
4.0 4.5 5.0 5.5 6.0 6.5 7.0
Photon energy(eV)
QE
Theory
Dave's Data
D. H. Dowell et al., Phys. Rev. ST-AB 9, 063502 (2006)
Energy Distribution Curves - Copper
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Electron energy (eV)
Ele
ctro
ns
per
ph
oto
n p
er e
V
190 nm
200 nm
210 nm
220 nm
230 nm
240 nm
250 nm
260 nm
270 nm
280 nm
290 nm
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?
Energy
Medium Vacuum
Φ
Vacuum level
Three Step Model of Photoemission - Semiconductors
Filled S
tatesE
mpty S
tates
h
1) Excitation of e-
Reflection, Transmission, Interference
Energy distribution of excited e-
2) Transit to the Surfacee--phonon scatteringe--e- scatteringRandom Walk
3) Escape surface Overcome Workfunction
Need to account for Random Walk in cathode suggests Monte Carlo modeling
Laser
No S
tates
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
210-7 410-7 610-7 810-7 110-6
0.2
0.4
0.6
0.8
Vacuum K2CsSb200nm
Copper
563 nm
Laser energy in media
Not exponential decay
Calculate the amplitude of the Poynting vector in each media
QE vs Cathode Thickness
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
2 2.2 2.4 2.6 2.8 3 3.2 3.4
photon energy [eV]
QE
50 nm
200 nm
Experiment
20 nm
20 nm
10 nm
Data from Ghosh & Varma, J. Appl. Phys. 48 4549 (1978)
QE vs Mean Free Path
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40
photon energy [eV]
QE
Experiment
10 nm mfp
5 nm mfp
20 nm mfp
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
0
0.005
0.01
0.015
0.02
0.025
0 5000 10000 15000 20000 25000 30000 35000
Sqrt F (F in V/m)
Sq
rt Q
E
Theory, Beta = 1.2
Theory, Beta = 1
Theory, Beta = 2
Theory, Beta = 3
Data (80 Ohm, 1.19 mm)
Data (80 Ohm, 2.11 mm)
Data (20 Ohm, 2.11 mm)
Phi = 4.40
Filter = .187
Figure 5.15
Dark current beta - 27
DC results at 0.5 to 10 MV/m extrapolated to 0.5 GV/m
Photoemission Results
Expected Φ = 3.91 eV
QE = 0.27% @ 213 nm for Arc Deposited2.1 W required for 1 mA
ElectroplatedΦ = 4.2 eV
Schottky Effect
Φ
Φ’
Φ’ (eV) = Φ- 3.7947*10-5E
= Φ- 3.7947*10-5βE If field is enhanced
)E)(1( 0 hRQE near photoemission threshold
Slope and intercept at two wavelengths determine Φ and β uniquely
Semiconductor photocathodes
Valence Band
Conduction Band
Medium Vacuum
Eg
Ev
Three step model still valid
Eg+Ev< 2 eV
Low e population in CB
Band Bending
Electronegative surface layer
Vacuum Level
e-n Vacuum LevelE