1. One Step Photoemission from Ag(111)
- Siddharth Karkare
What role does band structure play in photoemission?
Can we use conservation of transverse momentum for better emittance?
Acknowledgements –
Weishi Wan (LBNL)
Jun Feng (LBNL)
Howard Padmore (LBNL)
T.-C. Chiang (UIUC) T. Miller (UIUC)
Answer question of what is the role of transverse momentum conservation in observed emittance. So far only disordered cathodes studied. No clear sign of conservation of transverse momentum or band structure strongly affecting the emittance. Moreover emittance always limited by kT

2. Why Ag(111)?
Easy to prepare – Simple ion bombardment and annealing to 4500C
Gives excellent LEED pattern and clean Auger spectrum
Very narrow transverse momentum spread from surface states as measured in ARPES experiments
Surface state peak
Intensity
F. Reinert et al., PRB, 63, (2001)
T. Miller et al., Surf. Sci. 376, 32 (1997)
Intense surface state peak in PES experiments – Could imply good QE

3. s/p polarized light
Cathode

4. Experimental measurements
Transmission of light
(1-Reflectivity)
QE in p polarized light
MTE
QE vs excess energy, …how they differ from dowells model
Vectorial photoelectric effect
MTE vs Excess energy
How does this relate to band structure

5. Ag(111) band structure
Vacuum Level
Fermi Level
Surface state
𝜙=4.47 eV
Projected band structure in direction transverse to surface
Zoom near Fermi level
ℏ𝜔=𝜙
ℏ𝜔=𝜙+35 meV
ℏ𝜔=𝜙+100 meV
ℏ𝜔=𝜙+60 meV
ℏ𝜔=𝜙+50 meV
𝐸 𝑧 = 𝐸 𝑣 +𝑉+ ℏ 2 𝑘 𝑧 2 2 𝑚 𝑖,𝑓 ∓ ℏ 4 𝑝 2 𝑘 𝑧 2 𝑚 𝑖,𝑓 + 𝑉 2 1/2
𝐸 𝑅 = ℏ 2 𝑘 𝑟 2 2 𝑚 𝑡
𝐸 𝑧 =−60 meV
𝐸 𝑅 = ℏ 2 𝑘 𝑟 2 2 𝑚 𝑠
Bulk states
Surface state
𝑬= 𝑬 𝒛 + 𝑬 𝑹
Nearly free electron model – sp bands
Conservation of energy
𝐸 𝑣𝑎𝑐𝑢𝑢𝑚 =𝐸+ℏ𝜔−𝜙
Conservation of transverse momentum
𝑘 𝑟𝑖 = 𝑘 𝑟𝑓
𝐸= ℏ 2 𝑘 𝑟 2 2 𝑚 𝑒 +𝜙−ℏ𝜔
Fermi Level
Surface state ℏ𝜔

6. One step photoemission model
Requires complex Green’s function based calculations…
However
Simplifies to transition between initial state and Time Reversed LEED state
Fermi Golden Rule with Time Reversed LEED states
𝑅=𝐾 𝑑 𝑘 𝑖 3 𝑑 𝑘 𝑓 𝜓 𝑓 𝐻 𝜓 𝑖 2 𝛿 𝐸 𝑓𝑖𝑛𝑎𝑙 − 𝐸 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 +ℏ𝜔−𝜙 𝐹( 𝐸 𝑖 )
Mahan, PRB, 2, 4334 (1970)

7. One step photoemission model
Fermi Golden Rule with Time Reversed LEED states
𝑅=𝐾 𝑑 𝑘 𝑖 3 𝑑 𝑘 𝑓 𝜓 𝑓 𝐻 𝜓 𝑖 2 𝛿 𝐸 𝑓𝑖𝑛𝑎𝑙 − 𝐸 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 +ℏ𝜔−𝜙 𝐹( 𝐸 𝑖 )
Initial bulk state (zoomed in)
Initial bulk state (zoomed out)
Initial wave function
𝝍 𝒊 = 𝝍 𝒊𝒛 𝒆 𝒊 𝒌 𝒊𝒙 𝒙 𝒆 𝒊 𝒌 𝒊𝒚 𝒚
Bulk states
𝜓 𝑖𝑧 =𝑁[ 𝑒 𝑖 𝑘 𝑖𝑧 +𝑝 𝑧 +Φ 𝑒 𝑖 𝑘 𝑖𝑧 −𝑝 𝑧 +
𝐶 1 𝑒 −𝑖 𝑘 𝑖𝑧 +𝑝 𝑧 +Φ 𝑒 −𝑖 𝑘 𝑖𝑧 −𝑝 𝑧 ], (𝑧< 𝑧 0 )
𝜓 𝑖𝑧 =𝑁 𝐶 2 𝑒 −𝜅(𝑧− 𝑧 0 ) , (𝑧> 𝑧 0 )
Surface states
Initial surface state
Final state
𝜓 𝑖𝑧 =𝑁 𝑒 𝑖𝑝𝑧 +Φ 𝑒 −𝑖𝑝𝑧 𝑒 𝜅 𝑠 𝑧 , (𝑧< 𝑧 0 )
𝜓 𝑖𝑧 =𝑁 𝐶 𝑆 𝑒 −𝜅(𝑧− 𝑧 0 ) , (𝑧> 𝑧 0 )
Final wave function
𝝍 𝒇 = 𝝍 𝒇𝒛 𝒆 𝒊 𝒌 𝒇𝒙 𝒙 𝒆 𝒊 𝒌 𝒇𝒚 𝒚
𝜓 𝑓𝑧 = 𝑡 𝑝𝑘 𝑒 𝑖( 𝑘 𝑓𝑧 + 𝑝)𝑧 +Φ 𝑒 𝑖( 𝑘 𝑓𝑧 − 𝑝)𝑧 𝑒 𝜅 𝑑 𝑧 , (𝑧< 𝑧 0 )
𝜓 𝑓𝑧 = 𝑒 𝑖 𝑘 𝑧 (𝑧− 𝑧 0 ) + r pk 𝑒 −𝑖 𝑘 𝑧 (𝑧− 𝑧 0 ) , (𝑧> 𝑧 0 )

8. One step photoemission model
Fermi Golden Rule with Time Reversed LEED states
𝑅=𝐾 𝑑 𝑘 𝑖 3 𝑑 𝑘 𝑓 𝜓 𝑓 𝐻 𝜓 𝑖 2 𝛿 𝐸 𝑓𝑖𝑛𝑎𝑙 − 𝐸 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 +ℏ𝜔−𝜙 𝐹( 𝐸 𝑖 )
𝐻=− 𝑒ℏ 𝑚 𝑒 𝑐 𝐴 ∙ 𝛻 + 𝑒ℏ 2𝑚 𝑒 𝑐 𝛻 ∙ 𝐴
Hamiltonian
=− 𝑒ℏ 𝐴 0 𝑚 𝑒 𝑐 𝑒 𝑧 ′ ℋ( 𝑧 ′ ) 𝑑 𝑙 𝜀 ∙ 𝛻 +𝐶 𝜀 𝑧 𝛿( 𝑧 ′ )
For p-polarized light
𝜓 𝑓 𝐻 𝜓 𝑖 2 = 𝐾 1 𝑇 𝑝 𝑰 𝒅 +𝐶 𝑰 𝒔 𝒔𝒊𝒏 𝜽 𝒕 ′ +𝑖 𝑘 𝑖𝑥 𝒄𝒐𝒔 𝜽 𝒕 ′ 𝑰 2 𝛿( 𝑘 𝑖𝑥 − 𝑘 𝑓𝑥 )𝛿( 𝑘 𝑖𝑦 − 𝑘 𝑓𝑦 )
For s-polarized light
𝜓 𝑓 𝐻 𝜓 𝑖 2 = 𝐾 1 𝑇 𝑠 𝑖 𝑘 𝑖𝑦 𝑰 2 𝛿( 𝑘 𝑖𝑥 − 𝑘 𝑓𝑥 )𝛿( 𝑘 𝑖𝑦 − 𝑘 𝑓𝑦 )
𝐼 𝑑 = 𝑑𝑧 𝜓 𝑓𝑧 ∗ 𝜕 𝜓 𝑖𝑧 𝜕𝑧
𝐼 𝑠 = 𝜓 𝑓𝑧 ∗ 𝑧 0 𝜓 𝑖𝑧 ( 𝑧 0 )
𝐼= 𝑑𝑧 𝜓 𝑓𝑧 ∗ 𝜓 𝑖𝑧
𝑰 𝒅 , 𝑰 𝒔 ≫𝐈
Finally
𝑅= 𝐾 3 𝑑 𝑘 𝑟 𝑑 𝑘 𝑧 …
𝐐𝐄= 𝑹 𝑭
𝐌𝐓𝐄= 𝑲 𝟒 𝒅 𝒌 𝒓 𝒌 𝒓 𝟐 𝒅 𝒌 𝒛 … 𝒅 𝒌 𝒓 𝒅 𝒌 𝒛 …

9. Comparison to experiments
Theory
60 deg
35 deg
0 deg
Experiment
Theory
Cathode
Grid
Screen

10. Conclusion Ag(111) surface states – Low emittance electron source
Demonstration of using band structure to enhance cathode performance
One step photoemission explains several photoemission phenomena
Possible strange effects within 100 meV of threshold

11. Decay length

12. With regular scattering
Mean free path ~7nm
Mean free path ~0.7nm