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

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, 115415 (2001) T. Miller et al., Surf. Sci. 376, 32 (1997) Intense surface state peak in PES experiments – Could imply good QE

s/p polarized light Cathode

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

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

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 3 𝜓 𝑓 𝐻 𝜓 𝑖 2 𝛿 𝐸 𝑓𝑖𝑛𝑎𝑙 − 𝐸 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 +ℏ𝜔−𝜙 𝐹( 𝐸 𝑖 ) Mahan, PRB, 2, 4334 (1970)

One step photoemission model Fermi Golden Rule with Time Reversed LEED states 𝑅=𝐾 𝑑 𝑘 𝑖 3 𝑑 𝑘 𝑓 3 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 )

One step photoemission model Fermi Golden Rule with Time Reversed LEED states 𝑅=𝐾 𝑑 𝑘 𝑖 3 𝑑 𝑘 𝑓 3 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 2 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 2 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 …

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

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

Decay length

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