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

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

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

= 3.72 eV @ 5MV/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

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