ANL 1
Numerical Modeling of Field-Enhanced Photoemission from Metals and Coated Materials
Numerical Modeling of Field-Enhanced Photoemission from Metals and Coated Materials
ARGONNE NATIONAL LABORATORY DECEMBER 6, 2005 Argonne, IL
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We gratefully acknowledge:FUNDING by Joint Technology Office & Office of Naval Research
INTERACTIONS with (alph). S. Biedron , C. Bohn, C. Cahay, D. Dimitrov, D. Dowell, J. Lewellen, J. Petillo, J. Smedley
UMD:
D. W. FeldmanN. A. Moody
P. G. O’Shea
NRL:
K. L. JensenJ. L. ShawJ. E. Yater
ANL Beams and Applications SeminarHost: John Lewellen ASD
Bldg. 401, Room A1100 Tuesday, 1:30 pm
ANL 2
OUTLINE
Electron Sources Photocathodes and Photocathode Issues
The Dispenser Photocathode Concept
Electron Emission Fundamentals
1st Generation Emission Model & Usage
Next Generation Components & Application Bare Metals
Cesiated Surfaces & Gyftopoulos-Levine Model
Quantum Distribution Function
Quantum Effects on Barrier & Scattering
Conclusion
ANL 3
Thermionic
ELECTRON EMISSION
The Manner in Which Electrons Are Extracted Dictates the Technological Gambits Invoked
Metal
Field
HeatWave Labshttp://www.cathode.com/c_cathode.htm
HeatWave Labshttp://www.cathode.com/c_cathode.htm
ANL-BNL-JLAB Gun: Ilan Ben-Zviwww.agsrhichome.bnl.gov/eCool/MAC_Ilan.pdf
ANL-BNL-JLAB Gun: Ilan Ben-Zviwww.agsrhichome.bnl.gov/eCool/MAC_Ilan.pdf
Photo
Courtesy of C. A. Spindtwww.sri.com/psd/microsys/vacuum/
Courtesy of C. A. Spindtwww.sri.com/psd/microsys/vacuum/
ANL 4
PHOTOINJECTORS & PHOTOCATHODESCritical Components of Free Electron Lasers, Synchrotron Light & X-ray Sources
LinacScale
2" 3"0 1"
Drive laser
Photo-cathode
rf Klystron MasterOscillator
High Power FEL Demands on Photocathode:CHARGE PER BUNCH: 0.1 - 1 nC in 10-50 ps pulseFIELD: 10 - 100 MV/m in pressure of 10-8 Torr (approx)OPERATION: Robust, Prompt, Operate At Longest LIFETIME: Longevity & Reliability Paramount
ANL 5
PHOTOCATHODE RESPONSE TIME
Pulse Shaping Optimal Shape for emittance:
beer-can (disk-like) profile
Laser Fluctuations occur (esp. for higher harmonics of drive laser)
Fast response: laser hash reproduced
Slow response: beer-can profile degraded
Optimal: 1 ps response time
Mathematical Model (n = 2n/T)
I t( )=Ioθ t( )θ T −t( ) cncosnt( )n=0
N∑
Ie t( )=QEτ
I s( )ex −t−sτ
⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥−∞
t
∫ ds∝cn
1+ nτ( )2
cosnt( )+nτ sin nt( )( )eT /τ −1⎡⎣ ⎤⎦e−t/τ t <T
eT /τ −1( )e−t/τ t≥T
⎧⎨⎪
⎩⎪n=0
N
∑
Formulation based on model of J. Lewellen
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-5 0 5 10 15 20 25 30 35time [ps]
IdealMetal25.6 ps6.4 ps3.2 ps0.8 ps
ANL 6
CATHODE + LASER NONUNIFORMITY
Laser
Cathode Emittance Is Important (esp. in Gun) Pulse Shape Can Result in Reduction of Emittance
Prediction of Photocathode / Drive Laser Combos & Beam Crucial to Design of Larger Systems
Realistic Initial Distributions Needed: Advanced Codes (Michelle, Argus, Magic, Mafia, Vorpal, Etc.) Need Adequate Emission Models, Photoemission Studies Sparse, Emission Distributions Are Unknown
Laser Nonuniformity
Work FunctionNonuniformity
Laser Non-uniformity + work function non-uniformity(sim)
Combined Nonuniformity
ANL 7
IDENTIFY factors that affect QE (e.g., laser, environment, photocathode material)DEVELOP a custom-engineered controlled porosity photo-dispenser cathode
PROGRAM OBJECTIVE
A Cs-dispenser Photocathode Concept
monolayer
W plug w/ Cs
Al2O3 Potting
Heater
Maryland DispenserCathode
Generic Parameter Range
Assumptions:
• Charge = 1 nC
• Wavelength = 266 nm
• Pulse FWHM = 10 ps
Example Cases of Io(QE)
For Radius of 0.3 cm
• Io(1%) = 0.165 MW/cm2
• Io(0.01%) = 16.5 MW/cm2
For Radius of 0.1 cm
• Io(1%) = 1.48 MW/cm2
• Io(0.01%) = 148 MW/cm2
Io =
hq
ΔQΔtFWHMr2
⎛
⎝⎜⎞
⎠⎟1
QE
Metal
TopView
CsO
SideView
Interpore ≈ 6 µm; Grain Size≈ 4.5 µm; Pore Diam. ≈ 3 µm
Conventional Dispenser
Controlled Porosity
ANL 8
0
2
4
6
8
1 2 3 4 5 6 7 8Grain Index
DISPENSER CATHODE POROSITY ANALYSIS
Central Questions:
Pore-to-Pore
separation &
Grain Size
Desorption &
migration rates of
low work function coating over surface
80
160
240
320
400
480
0 100 200 300 400 500 600 700Pixel X
10 μ = 143 m pixel
SEMS: Nathan Moody, UMD
L(x;μ,s 2 ) ≡1
2sxexp −
12s 2 ln x / μ( )( )
2⎡⎣
⎤⎦
⎧⎨⎩
⎫⎬⎭
Pore to Pore: Log-Normal Distribution
Grain Size: Effective Diameter
dA =14D2∫
0
1
2
3
4
5
6
7
0 16 32 48 64 80 96 112 128 144 160SEPARATION (pixel)
Pore-to-PoreLogNorm(x)
bin size = 8 pixelsLog-Normal Parametersμ = 35.3 ; pixelss = 0.786
- - :Mean pore to pore35.3 (10 x μ / 143 ) = 2.47 m pixel μm
ANL 9
Electron Number Density
Zero Temperature (μ(0 ˚K) = μo = EF)
N does not change with T so μ must:
μ,T( ) =N
V= 2 fFD (Ei )
i∑
⇒2
2π( )3 1+ exp β μ − E(
rk )( )⎡⎣ ⎤⎦( )
−1
d 3k∫
STATISTICAL MECHANICS OF ELECTRON GASElectrons Incident On Barrier Are
Distributed In Energy According To A 1-D
“Thermalized” Fermi Dirac Distribution
Characterized By The Chemical Potential
and Called The “Supply Function”
f(k) obtained by integrating over the
transverse momentum components
(μ o ) =
1
3π 2
2mμ o
h2
⎛⎝⎜
⎞⎠⎟
3/2
=kF
3
3π 2
f (k) =2
2 2
2k⊥dk⊥
1+ exp β(E||+ E⊥ −μ)( )0
∞
∫
=m
βh2 ln 1+ exp β(μ −E||)( )⎡⎣ ⎤⎦μ(T ) = μ o 1−1
12
π
βμ o
⎛
⎝⎜⎞
⎠⎟
2
−1
80
π
βμ o
⎛
⎝⎜⎞
⎠⎟
4
+…⎡
⎣⎢⎢
⎤
⎦⎥⎥
BAND BENDINGMetal vs. Semiconductor
Ec
μoFvac
Metal
Ec
Ev
μo
Fvac
c
μ
Semiconductor
ANL 10
CURRENT - A CLASSICAL APPROACH
f(x,k,t) is the probability a particle is at position x with momentum hk at time t
Conservation of particle number:
N = 2( )
−1f (x,k,t)dxdk∫∫
dn = f (x,k,t)dxdk= f x',k',t'( )dx'dk'=dn'
x'=x+hkm
dt; hk'=hk+ Fdtto order O(dt)
dk
dk’
dx
dx’
dndn’
dx 'dk ' =∂xx' ∂xk'
∂kx' ∂kk'dxdk=dxdk
0 =
f x+ dx,k+ dk,t+ dt( )− f (x,k,t)
dt⇒
∂∂t
+hkm
∂∂x
+Fh
∂∂k
⎧⎨⎩
⎫⎬⎭
f (x,k,t) =0
∂∂t
x,t( ) =∂∂t
12
f (x,k,t)dk−∞
∞
∫⎡
⎣⎢
⎤
⎦⎥=
∂∂x
12
hkm
⎛
⎝⎜⎞
⎠⎟f (x,k,t)dk
−∞
∞
∫⎡
⎣⎢
⎤
⎦⎥=−
∂∂x
J x,t( )
Boltzmann Transport Equation
“Moments” give number density and current density J: Continuity Equation
velocity & acceleration
ANL 11
CURRENT IN SCHRöDINGER REPRESENTATION
Consider a pure state
j(x, t) = x j(x,t) x =h2m
x (t), k{ } x
=h
2miψ † (x,t)∂xψ (x,t)−ψ (x,t)∂xψ
† (x,t){ }
The form most often used in emission theoryBasis for FN & RLD Equations
H ψ k(t) =h2k2
2m+V x( )
⎛
⎝⎜
⎞
⎠⎟ ψ k(t) =E ψ k(t)
(t) = f E(k)( ) ψ k(t)∑ ψ k(t)
∂∂t
t( ) = H , (t)⎡⎣ ⎤⎦=ih2
2m∂∂x
k, (t){ } =−∂∂x
j t( )
Schrödinger’s Equation
Time-dependence of Operators governed by commutators with Hamiltonian
Simple Case: Gaussian
x ψ (0) =Δk
2exp −
12Δk2x2 + ikox
⎡
⎣⎢
⎤
⎦⎥
(x) =Δk2
2exp −Δk2x2⎡⎣ ⎤⎦
J (x) =hko
m(x)
jk (x, t) = t(k) 2 hkm
J (x,t) =12
hkm
T(k) f(k)0
∞
∫
ψ k (x) → t(k)ψ k(x)→
T (k) =jtrans(k)jinc(k)
ANL 12
The most widely used forms of: Field Emission: Fowler Nordheim (FN) Thermal Emission: Richardson-Laue-Dushman (RLD)
THERMIONIC VS FIELD EMISSION
Fowler Nordheim
T (E) ≈ex − bfn / F( )+ cfn μ −E( )( )⎡⎣
⎤⎦
f(E)=mh2 μ −E( )Θ μ −E( )
J FN(F)=AF 2 ex −B / F( )
Richardson
T (E) =Θ E − μ +φ( )⎡⎣ ⎤⎦
f (E) =m
βh2 exp β μ −E( )⎡⎣ ⎤⎦
J RLD(T ) =ARLDT2 exp −φ / kBT[ ]
J(F,T )= 1
2hT E( ) f E( )dE0
∞∫
5
6
7
8
9
0 2 4 6Position [nm]
Field Emission:Work Func = 4.6 eV
Field = 4 GV/m
Thermionic Emission:Work Func = 1.8 eV
Field = 10 MV/m
Fermi Level
High TemperatureLow Field
Low TemperatureHigh Field
Transmission Probability
Electron Supply
Emission Equation
A
rld=120.173
Amp
Kelvin2cm2;Q=0.359991 eV-nm
A =1.38072×10-6
Amp
eV 2exp 9.83624-1/2( )
B=6.399523/2 eVnm
Constants for Work Function in eV, T in Kelvin, F in eV/nm
ANL 13
10-12
10-10
10-8
10-6
10-4
10-2
100
A GENERAL THERMAL-FIELD EQUATION
MaxwellBoltzmann
Regime
0 K-like Regime
Supply Function
General Transmission Coefficient
Define Slope Ratio:
n « 1: Richardson-Laue-Dushman Eqn » 1: Fowler Nordheim Equation
f E( ) =
mβTh2 ln 1+ exp βT μ −E( )⎡⎣ ⎤⎦{ }
T E( ) ≈To 1+ exp βF Eo −E( )⎡⎣ ⎤⎦{ }−1
Eo =μ +bfn
Fcfn
Eo =μ +− 4QF
Field
Thermalf(7,300)
f(0.01,2000)
T(0.01,2000)T(7,300)
X(F[GV/m],T[K])
Fermi
0
0.2
0.4
0.6
0.8
1
1.2
3 4 5 6 7 8 9 10 11Energy [eV]
7 GV/m300 K
2 MV/m1094 K
10 MV/m2000 K
n =βT / βFn =βT / βF
JFN ⇒ ARLD kBβF( )−2
1+ 2
6βF
βT
⎛
⎝⎜⎞
⎠⎟
2
+74
βF
βT
⎛
⎝⎜⎞
⎠⎟
4⎛
⎝⎜
⎞
⎠⎟exp βF (μ −Eo)( )
J RLD ⇒ ARLD kBβT( )−2
1+ 2
6βT
βF
⎛
⎝⎜⎞
⎠⎟
2
+74
βT
βF
⎛
⎝⎜⎞
⎠⎟
4⎛
⎝⎜
⎞
⎠⎟exp βT (μ −Eo)( )
JFN ⇒ ARLD kBβF( )−2
1+ 2
6βF
βT
⎛
⎝⎜⎞
⎠⎟
2
+74
βF
βT
⎛
⎝⎜⎞
⎠⎟
4⎛
⎝⎜
⎞
⎠⎟exp βF (μ −Eo)( )
J RLD ⇒ ARLD kBβT( )−2
1+ 2
6βT
βF
⎛
⎝⎜⎞
⎠⎟
2
+74
βT
βF
⎛
⎝⎜⎞
⎠⎟
4⎛
⎝⎜
⎞
⎠⎟exp βT (μ −Eo)( )
ANL 14
QDF
INTEGRATED SCATTERING & EMISSION MODEL
Quantum Distribution Function (QDF) Simulation Simultaneously Relates Scattering (τ), Barrier Emission (), Thermal () and Density Effects
k
h
z( )
f & U(x)Scattering &
Electron Transport & Emission
Coverage-Dependent Work Function
-24
-20
-16
-12
-8
-4
0
0 5 10 15
z(bohr)
Clean W(001)Potential
-24
-20
-16
-12
-8
-4
0
0 5 10 15
z(bohr)
Ba/O/W(001)
POSITION (x/ao)
EN
ER
GY
W c
ores
W c
ores
Hemstreet, et al. PRB40, 3592 (1989)
W
Ba O
& Relaxation Time &
Thermal Model2
4
6
8
10
12
Literatureln(Tac)ln(Tee)Theory Au
ANL 15
EXP. VALIDATED PHOTOEMISSION MODEL
GOAL Predict Quantum Efficiency From Laser & Material Parameters Analyze Experimental Results from UMD, NRL, Colleagues in FEL Program Emission Model For Beam Code for NRL, SAIC, Tech-X, NIU, Colleagues
GOAL Predict Quantum Efficiency From Laser & Material Parameters Analyze Experimental Results from UMD, NRL, Colleagues in FEL Program Emission Model For Beam Code for NRL, SAIC, Tech-X, NIU, Colleagues
QE =q
hf 1−R()( )PFD h( )
PFD h( ) =U β h −+ 4QF( )⎡
⎣⎤⎦
U βμ[ ]
COMPONENTS:
Work function variation with coating θ)
Gyftopolous-Levine theory
Thermal & Material; laser β, R(), f
Transient heating & heat diffusion
Simple Photoemission Model U-function
Revised Fowler-Dubridge Model
Quantum effects @ hi F & T U, f
barrier due to e- density; scattering
Transmission through barriers U
Relate barrier to emission probability
Photocurrent depends on
• Scattering Factor: f
• Absorbed laser power: (1-R) I
• Escape Probability: U-terms
First Generation Beam Code Model
Next Generation Beam Code Model
ANL 16
DETERMINATION OF R[%] &
Algorithm:
Spline-fit experimental optical data (e.g., CRC, AIP Handbook) for index of refraction (n), damping constant (k)
Designate incident angle = θ Use Equations to determine
Reflectance R[%] and penetration depth of laser for given wavelength
Rs =a2 +b2 −2acosθ + cos2 θa2 +b2 + 2acosθ + cos2 θ
Rp =Rs
a2 +b2 −2asinθ tanθ + sin2 θ tan2 θa2 +b2 + 2asinθ tanθ + sin2 θ tan2 θ
2a2 = n2 −k2 −sin2 θ( )2+ (2nk)2⎡
⎣⎤⎦
1/2
+ n2 −k2 −sin2 θ( )
2b2 = n2 −k2 −sin2 θ( )2+ (2nk)2⎡
⎣⎤⎦
1/2
− n2 −k2 −sin2 θ( )
R =12
Rs + Rp( ); =
4k
0.1
1
10
100
0.1 1 10
n(W)k(W)n(Cu)k(Cu)n(Au)k(Au)
Wavelength [micron]
10
100
0.1 1 10
delta(W)%R(W)delta(Cu)%R(Cu)delta(Au)%R(Au)
[ ]Wavelength micron
…other metals in database
Consider W, Cu, Au…
ANL 17
POST-ABSORPTION SCATTERING FACTOR
Factor (f ) governing proportion of electrons emitted after absorbing a photon: Photon absorbed by an electron at depth x Electron Energy augmented by photon, but
direction of propagation distributed over sphere Probability of escape depends upon electron
path length to surface and probability of collision (assume any collision prevents escape) path to surface &
scattering length
To leading order, k integral can be ignored z θ( ) =
xcos θ( )
; l k( ) =hkτm
f =
f(k)dkko
∞
∫ dθ0
/2
∫ exp −x−
z(θ)l k( )
⎛
⎝⎜⎞
⎠⎟0
∞
∫ dx
f(k)dk dθ0
∫ko
∞
∫ exp −x
⎛⎝⎜
⎞⎠⎟0
∞
∫ dx
k =1h
2m E(k) +h( )
ko: minimum k of e- that can escape after photo-absorption : penetration of laser (wavelength dependent); : relaxation time
θk
h
z(θ)
Average probability of
escape
f ≈12
Gmhkoτ
⎛
⎝⎜⎞
⎠⎟
G cos(y)[ ] =1−2
cot y( ) ln1+ sin(y)
cos(y)⎛⎝⎜
⎞⎠⎟
G sec(y)[ ] =1−2
ysin(y)
⎛⎝⎜
⎞⎠⎟
argument < 1
argument > 1
Ex: Copper: = 266 nm = 12.9 nmτ() = 0.85 fsμ = 7.0 eV• F = 4.3 eV
sec(y) = 7.7f = 0.038
ANL 18
U β(h −φ)( )⇒ U β(h −φ)( )+
2
12βb
⎛⎝⎜
⎞⎠⎟
2
QM contributes for photon E near barrier height, large fields,and cold temperatures
Copper * = 266 nm; F = 5 MV/m; β = 3• R = 33.6%, = 4.3 eV, EF = 7.0 eV• QE [%] (analytic) 1.31E-2• QE [%] (time-sim) 1.36E-2• QE [%] (exp) 1.40E-2
QM-EXTENSION OF FOWLER-DUBRIDGE EQ.
RLD-based Fowler Dubridge Model
• U(x): depends on thermal distribution and barrier μ+ for emission probability
• Reflectivity R and Scattering Factor f depends on material & relaxation time
QE ≈
f 1−R( )μ2 h −φ( )2
QE = f 1−R( )U h −φ( ) / kBT⎡⎣ ⎤⎦
U μ / kBT⎡⎣ ⎤⎦
U x( )= 2
12+ ln 1+ ez( )dz
0
x
∫“Fowler factor”
0
20
40
60
3 4 5 6 7
1064 (x100)532355266
Energy [eV]
0
20
40
60
3 4 5 6 7Energy [eV]
J =
12h
T(E) f(E)dE0
∞
∫ TRLD (E) =Θ E −μ −φ+h( )
TQM (E) = 1+ exp[b Ec(h)−E( ){ }
−1
coating = 2.0 eV
Exp data: T. Srinivasan-Rao, et al., JAP69, 3291 (1991).
ANL 19
SCATTERING & Electrical / Thermal Conductivity
If an electric field F (or temperature gradient T) is removed, then distribution “relaxes” back to equilibrium after a “relaxation time” τ
f x,t( ) ≈ fo x,k( ) + fF x,k( )− fo x,k( ){ }exp −t / τ (k)( )
f
rr ,
rk( )= fo
r ,k( )+
τh
Fg
∇k fo −τ
hkmg∇rT(
r )( )
∂∂T
fo
Distribution for Fermi-Dirac is approximately constant except near Fermi Energy μ
WIEDEMANN-FRANZ LAW
sT
= 2
3kB
q
⎛
⎝⎜⎞
⎠⎟
2
=2.443×10−8 W-ΩKelvin2
⎡
⎣⎢
⎤
⎦⎥
Thermal Conductivity
Electrical Conductivity
Specific Heat
rJ
F(k) =
2q
2( )3
d3khrkm
⎛
⎝⎜⎞
⎠⎟∫ frk( )
=2q
2( )3
d3khkx
m
⎛
⎝⎜⎞
⎠⎟Fhτ
rk( )
⎧⎨⎩
⎫⎬⎭
∫∂∂kx
fo
rk( )
=qτ (μ)kF
3
3 2m
⎛
⎝⎜
⎞
⎠⎟ F ≈s
Fq
⎛
⎝⎜⎞
⎠⎟ Field
rJ
T(k) =
2q
2( )3
∂T∂x
⎛
⎝⎜⎞
⎠⎟d3kτ
k( )
hkx
m
⎛
⎝⎜⎞
⎠⎟
2
∫∂∂T
fo
k( )
=m3τ(μ)
∂T∂x
⎛
⎝⎜⎞
⎠⎟∂∂T
2
2( )3
d3k E(k)−μ⎡⎣ ⎤⎦∫ fo
k( )
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
=2μ3m
τ μ( )CV T( )∂T∂x
=(T)∂T∂x Temp
Electric Field Temperature
ANL 20
Diffusion mimics the temporal spread of
Dirac-Delta-like pulses with Do
Do acts as Length2 / time
Length ≈ O(laser penetration depth)
Model captures physics…
as long as there is an estimate for to
LASER HEATING OF ELECTRON GASDifferential Eqs. Relating Electron to Lattice Temp
Electron & LatticeSpecific Heat Laser Energy
AbsorbedPower transfer by electrons to lattice
285.1 GW / K cm3 (W @ RT)
ΔT =1−R( ) IoΔt
DoΔt Ce To( ) +Ci To( )( )
Do ≈2 / to
ΔT =1−R( ) IoΔt
DoΔt Ce To( ) +Ci To( )( )
Do ≈2 / to
∂t y = Do∂x2y ⇒ y(x, t) =
exp -x2 / 4Dot( )⎡⎣ ⎤⎦4π Dot
Ce
∂∂t
Te =∂∂z
(Te,Ti )∂∂z
Te
⎛
⎝⎜⎞
⎠⎟- g Te -Ti( ) +G z,t( )
Ci
∂∂t
Ti =g Te -Ti( )
100 101 102 103
Intensity [MW/cm2]
Copper Δ = 6 t ps t
o = 1.495 ps
10 -1
10 0
10 1
10 2
10 3
10 -3 10 -2 10 -1 10 0
.Time Dep
Model
[ ]Pulse Width ns
Copper
Iμax
= 100 /MW cm2
to = 1.673 ps
ANL 21
PHOTOEMISSION MODULES IN BEAM CODEGoal: modules for 3D RF gun / beam codes for the analysis of beam generation and transport. Present model: high-T scattering operator with T evaluated using Delta-diffusion model as function of laser intensity for
copper; probability of emission factor based on Fowler Dubridge but without QM Next generation to include all-temperature scattering, QM, metal & coating library
SIMULATION CODE: MICHELLE (SAIC)• Photoemission from laser-illuminated Cu hemisphere
Using 1st generation photoemission model• J. Petillo, et al., 8th DEPS, Lihue, HI (2005)
Hemisphere unit cell model
SIMULATION CODE: VORPAL (TECH-X)
• 3D visualization of photo emitted electron particles (white dots) following the beam emission and its evolution at different times from simulations with steady-state photocathode model.
• Cu photocathode at left boundary. Front of laser pulse has reached the photocathode and emission of the electron beam has started.
• D. Dimitrov, et al., 8th DEPS, Lihue, HI (2005)
VORPAL Cu Photocathode Beam Emission and Evolution
ANL 22
-2
0
2
4
6
8
10
12
14
0 1 2 3 4 5 6 7
ln(Tau [fs])ln(Tac)ln(Tee)ln(TOTL)Room TempLiq. Nitrogen
ln(Temperature [Kelvin])
Cu
THEORETICAL EVALUATION OF SCATTERING
Scattering in metals due to collisions with lattice (acoustic phonons & defects) and e-e collisions. If mechanisms independent, then:
τ ac =h3vs
2
mkBTkFΞ2
TD / T( )5
ℜ TD / T( )
ℜ x( )=s5ds
es −1( ) 1−e−s( )0
x
∫
For T > TD (Debye Temp), then τac goes as T, but at low T, goes as T^5. Scattering cross section is also related to
• deformation potential Ξ (related to stress on lattice)
• sound velocity vs
τee =4hKs
2
a 2mc2(kBT)2
1+ΔEkBT
⎛
⎝⎜⎞
⎠⎟
2⎛
⎝⎜⎜
⎞
⎠⎟⎟g
2kF
qo
⎛
⎝⎜⎞
⎠⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
−1
g(x) =x3
4tan−1 x+
x1+ x2
−tan−1 xy( )
y
⎛
⎝⎜
⎞
⎠⎟
y(x) = 2 + x2
Ag
Pb
Au
W
AgCuAuWPb
Phonons:
Electron-Electron:e-e scattering in simple metals is not simple. Model of Lugovskoy & Bray [1] depends on electron energy above Fermi Level (ΔE) and Thomas-Fermi Screening Wave Number qo (depends on electron density) & dielectric Ks
[1] A. V. Lugovskoy, I. Bray, J Phys D: Appl. Phys. 31, L78 (1998)
Mathiessens Rule τ
−1 =τee−1 +τ ac
−1
ANL 23
10-7
10-6
10-5
10-4
10-3
10-2
200 220 240 260 280 300
Exp (Smedley)TheoryTheory (Extrapolated n,k)
Wavelenth [nm]
Lead
200 220 240 260 280 300
Exp (Dowell)
Theory x 0.7
Wavelength [nm]
Copper
EXPERIMENT VS. THEORY (BULK METALS)
Field Enhancement 3.0 Macroscopic field 1.0 MV/m Work Function 3.97 eV Temperature 300 K Data & Image
Courtesy of J. SMEDLEY (BNL)
BNLBNL SLACSLAC
Field Enhancement 1.0Macro field (MV/m) 0.01Work Function 4.31 eVTemperature 300 KData Courtesy of D. DOWELL (SLAC)
ANL 24
EXPERIMENT VS. THEORY (PART II)
Measured (UMD), calculated (NRL), & literature for various DISPENSER CATHODES(1st Generation model used)B-TYPE:
B. Leblond, NIMA317, 365 (1992)UMD experimental data
M-TYPEUMD experimental data
SCANDATEUMD experimental data
10-5
10-4
10-3
250 300 350 400 450 500 550
WAVELENGTH [nm]
Scandate
B-Type
M-type
ExperimentTheory
0.0
0.5
1.0
1.5
0 20 40 60 80 100
2% Any70˚ p70˚ sTheory
Laser [µJ]
Field = 39 MV/mBeta = 1t = 16 psA = π(0.39 mm)^2 = 266 nm
, ., 341,379 (1994)Rosenzweig et al NIMA
How the comparison is made:
• Time-Dependent Thermal Photoemission Model using τee(E=μ+hn) ran for each incident laser intensity
• Laser pulses were Gaussian in time
• Total energy and charge emitted evaluated via integration over Gaussian pulse (Laser) and Emitted charge profile (electron)
• Use of library values for Copper only (no adjustable constants)
ANL 25
COVERAGE DEPENDENT WORK FUNCTION
Gyftopolous-Levine Theory relates Work Function to coverage factor. Dependent upon Covalent Radii rx, Factors “f” and “w” (Act As “Atoms Per Cell”,
Values of which Depend on Crystal Face). General Surface = “Bumpy [B]”
alkali metal (n = 1)
alkaline-earth metal (n = 1.65)
gm ≡w
2rC( )2; g f ≡
f
2rW( )2
gm[110] = 2gm[100]
gm[B] = 3gm[100]
g f :gm =1: 4 for Cs on W, Mo, Ta
g f :gm =1: 2 for Ba, Sr, Th on W, etc.
θ( )=φ f − φ f −φm( )θ 2 3−2θ( ) 1−G θ( ){ }
G θ( )=
rorC
⎛
⎝⎜⎞
⎠⎟
2
1−2w
rWR
⎛
⎝⎜⎞
⎠⎟
2⎛
⎝⎜⎜
⎞
⎠⎟⎟
1+ nrCR
⎛
⎝⎜⎞
⎠⎟
3⎛
⎝⎜⎜
⎞
⎠⎟⎟
1+9n8
fθ( )3/ 2⎛
⎝⎜⎞
⎠⎟
fθ
Modified Gyftopolous-Levine Theory
β R
Hard Sphere Model of Surface Dipole
W
C
ANL 26
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
Phi (Wang)(Scale = 0.0889)Phi (Taylor)(Scale = 0.8698)Theory
Coverage θ
Cs on Tungsten1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
Phi (Haas)(Scale = 0.6858)Phi (Longo)(Scale = 1.000)Theory
Coverage θ
Ba Dispenser Cathode
GYFTOPOLOUS-LEVINE MODEL PERFORMANCE
LEAST SQUARES ANALYSIS:Minimize Difference between GL theory & Exp. Data With Regard to scale factor, monolayer work function value, and f coverage factor
C-S Wang, J. Appl. Phys. 44, 1477 (1977)
J. B. Taylor, I. Langmuir, Phys. Rev. 44, 423 (1933).
R. T. Longo, E. A. Adler, L. R. Falce, Tech. Dig. of Int'l. El. Dev. Meeting 1984, 12.2 (1984).
G. A. Haas, A. Shih, C. R. K. Marrian, Applications of Surface Science 16, 139 (1983)
• Tightly constrained parameter variation• Unique determination of theory based on
experimental values• Predictive ability from basic experiments
ANL 27
A n o d e / C a t h o d e
A s s e m b l y
D e p o s i t i o n S e n s o r
C e s i u m S o u r c e
C e s i u m E v a p o r a t i o n
2 0 0 L i t e r / s e c
I o n P u m p
4 0 L i t e r / s e c
I o n P u m p
T o D e p o s i t i o n
M o n i t o r C o n t r o l l e r
F i n e M e t e r i n g
V a l v e
E l e c t r i c a l
F e e d t h r o u g h s
L a s e r W i n d o w
V i e w p o r t W i n d o w
F o u r W a y C r o s s
U H V C h a m b e r
7 0 . 9 9 c m .
EXPERIMENTAL MEASUREMENTS @ UMD
• Test and evaluation chamber @ 2E-10 Torr
• Surface preparation using H-ion beam
• Surface deposition of various coatings
• Deposition monitor (+/- 0.01nm thickness)
• Femto-ampere current measurements (for QE)
• Solid state CW lasers on single-axis robot
• QE as function of time, temp, coverage, wavelength, and laser intensity (AUTOMATED)
ANL 28
0
0.01
0.02
0.03
20 40 60 80 100Theta [%]
QE OF Cs ON W: EXP. VS. THEORYCONDITIONS:
Coverage Is Uniform
Field & Laser intensity low: Schottky barrier lowering & heating negligible
For 407 nm
Cs deposited rapidly: Exp measured mass by a “depth” factor. Therefore: Scale factor = (100%/Atomic diameter)
For 532 nm and 655 nm
Cesium deposited slowly: accumulation rate affected by desorption (scale a)
Peak value affected by residual cesium left on W (center xmax)
Presence of O (hard to remove) affects work function variation
To account for effects:
θ x( ) =a xexp −xmax( ) +θmax
ExperimentTheory
405 nm: x 1 (theory x 1.40)
532 nm: x 4 (theory x 1.40)
655 nm: x 35 (theory x 0.84)
ANL 29
0
0.04
0.08
0.12
0.16
0 20 40 60 80 100COVERAGE [%]
25 K
77 K
300 K
CS ON W @ 405 nm
Io = 10 MW/cm2
F = 10 MV/mβ = 3
600 K
QE OF Cs ON W: EFFECT OF TEMPERATURE
Consider “typical” conditions of:
Laser Intensity10 MW/cm2
Field at cathode 10 MV/m
Field enhancement 2 (generic)
Pulse length10 ps
Relaxation Time Is Temperature-dependent: Impact of Operating Photocathode at Lower Temperatures Is to Raise QE (all other things being equal)
ΔQ 266 nm( ) =QE ×717 nCUnder these conditions
For r = 0.1 cm2:
ANL 30
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 20 40 60 80 100
TheoryTheory x 0.85Experiment AExperiment B
Coverage [%]
Cs on Ag = 405 nμ = 0.0174 MV/μ
min = 1.6 eV
a = 5.3 Angsτoμs
QE OF Cs ON Ag: EXP. VS. THEORY
Cs ON Ag Exp: INITIAL
COMPARISON METHODS for data taken week of Aug 8 2005
Experimental conditions same as for Cs-W for 532 and 655 nm exp.
Used same scale a as Cs on W
Shift factor aligns peaks for each experiment
Data taken by ANNE BALTER
ANL 31
HYPOTHESES OF EXP-THEORY DIFFERENCESActual surfaces differ from theory models because of Surface geometry (altered by cleaning?)
Reflectivity changes with exposed face (inc. angle)
Crystal Faces can have different work functions (e.g., Cu(111) = 4.86, Cu(110) = 5.61, etc)
Contaminants (e.g., Carbon-based ≈ 5.5 eV) & possibility of only sub-area contributing
Solid Silver 500x
Solid Tungsten 500x
Nathan Moody (UMD)
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 50 100 150 200
Horizontal (micron)
Surface of Silver
Angle = 88.607 deg
θ
θo
Shadows model: fraction of surface illuminated (f)
f =tanθo
tanθ + tanθo
ANL 32
0
1
2
-2 0 2Distance- +
IMPACT OF COATINGS ON WORK FUNCTION
Conventional View: charged atoms at surface & image charge = dipole
QM View: Electron penetration of barrier determined by height and width
Exchange-Correlation & Poisson’s Eq: Changes in density = changes in V(x)
0
2
4
6
8
10
-12 -9 -6 -3 0 3
Electrons
Lattice
Position [Angstroms]
+
_Cu ParametersF = 1 GV/m
-24
-20
-16
-12
-8
-4
0
0 5 10 15
z(bohr)
Clean W(001)Potential
W
Ba O
L. A. Hemstreet, et al. PRB40, 3592 (1989)
W c
ore
s
W c
ore
s
VBa
(x) =qi
4Q
x−rBa+
exp − x−rBa+
{ } Δφ≈
292
QkF
Friedel Oscillations
o 1+ 3cos ζ( )ζ 2
−3sin ζ( )ζ 3
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
ζ =2kF x−xo( )
tanh model
ExCorr+
Poisson-24
-20
-16
-12
-8
-4
0
0 5 10 15
z(bohr)
Ba/O/W(001)
EN
ER
GY
POSITION (x/ao)
ANL 33
0
5
10
15
1 10 100r[]
Al
Fe
Cs
BaAg
Au
Cu
W
Mg
2 18ESi
EXCHANGE CORRELATION ENERGY OF e-
Aspects of system of interacting e- in ground state determined by density
Exch.-Corr. Potential: change in gives rise to change in V = Fermi Level= electron density ao = Bohr radius
H N =hk( )
2
2m+
q2
4e0
e−a rr−r
r '
rr −
rr 'i< j=1
N
∑i=1
N∑ −q2
4e0
drr∫
e−a rr−r
ri
rr −
rrii=1
N
∑ +(rr ) +
q2
8e0
drr∫ d
rr '
e−a rr−r
r '
rr −
rr 'i< j=1
N
∑ +(rr )+(
rr ')
1
V
h2k 2
2mk∑ ⇒2
2( )3
h2k2
2m0
kF∫ 4k2dk=35μo Kinetic
1
4V 2k
1k
2V
ee
rk
1-
rk
2( ) k1k2
(-)
k1k2
'∑ ⇒m
3h2ao
μ2Exchange
Terms with k
1k
2V
ee
rk
1−
rk2( ) k2k1 & higher Correlation
KineticEnergy
e-einteraction
e-latticeinteraction
self-interaction ofbackground
V
xc=−
∂∂
Eexch + Ecorr( )
Vxc
Metal Vacuum
Typical Metals
1
=43 rao( )
3
ANL 34
OTHER CONTRIBUTIONS TO SURF. BARRIER
Simple Theory: Wave function penetration creates dipole & gives Friedel Oscillations
(x) =12
f (k)sin2 ζ / 2⎡⎣ ⎤⎦dk0
∞
∫
limβ→ ∞
x( ) =o 1+ 3cos ζ( )ζ 2
−3sin ζ( )ζ 3
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
ζ =2kF x−xo( ); xo2 ∼2mVo / h
2
0
0.02
0.04
0.06
0.08
-12 -9 -6 -3 0 3
Electrons
Background
Position [Angstroms]
Copper ParametersNo BaField = 1 eV/nm
origin of background dictatedby global charge neutrality
+
_
Why bother looking at Friedel Oscillations? For two good reasons:
• Friedel Density profile has analytic V(x) sol’n
• Metal + Ba Density profile can be decomposed into Friedel component + Gaussian add-on: enables an analytic solution to Poisson Eq… or at least a very easily solved solution
How?
• Approximations exist for location of background positive charge
• Poisson’s Eq. easily solved with Friedel Density where = 2kF(xi – xo)
∂xφ( )−∞
∞
∫ dx=− x ∂x2φ( )dx
−∞
∞
∫ =−q2
eo
x−xo( ) e(x) −i (x)( )dx−∞
∞
∫
=q2o
4eokF2
3dds
sin(s)s
⎛
⎝⎜⎞
⎠⎟−∞
0
∫ ds− sΔ
0
∫ ds⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪=
QkF
1632−3 2( )
Dipole due to electron-lattice
difference
ζ ⇒ 2kF x−xo( )
Δφ⇒QkF
1632−3 2( )
Lattice origin in relation to electron
x
i=xo −
38kF
ANL 35
f(x,k,t): quantum phase space distribution acts
like probability distribution function
Contours mimic classical trajectories;
when potentials are smooth and slowly varying, they are
classical trajectories
QUANTUM DISTRIBUTION FUNCTION
Heisenberg Representation
∂∂t
(t) =ih
H , (t)⎡⎣ ⎤⎦
H ψ k(t) =h2k2
2m+V x( )
⎛
⎝⎜
⎞
⎠⎟ ψ k(t) =E ψ k(t)
(t) = f E(k)( ) ψ k(t)∑ ψ k(t)
f x,k,t( ) =2 e2iky x+ y (t) x−y
−∞
∞
∫ dyWigner Distribution function (WDF)
∂∂t
f x,k,t( ) =−hkm
∂∂x
f x,k,t( )
+ V x,k−k'( ) f x,k',t( )dk'−∞
∞
∫V x,k( ) =−
ih
e2iky
−∞
∞
∫ V(x+ y) −V(x−y){ } dy
x,t( ) =12
f (x,k,t)dk−∞
∞
∫
J x,t( ) =12
hkm
⎛
⎝⎜⎞
⎠⎟f (x,k,t)dk
−∞
∞
∫
integrate both sides with respect to momentum (k) get density and current density
vacuum metal
vacuum metal
Copper parametersField = 1 eV/nm
ANL 36
ANALYTICAL WDF MODEL: GAUSSIAN V(x)
How does V(x,k) behave? Consider a solvable case where V(x) is a Gaussian:
V x( ) =Voexp − x / Δx( )2⎡
⎣⎢⎤⎦⎥
V(x+ y) −V(x−y) =−2Vosinh 2xy / Δx2( )exp − x2 + y2( ) / Δx2⎡⎣
⎤⎦
SharpΔx2 = 5.0
BroadΔx2 = 0.1
V x,k( ) =−
2Δx 1/ 2h
Voexp −Δx2k2⎡⎣ ⎤⎦sin kx( ) small x samples f(x,k') far from k
large x samples f(x,k') near k
ANL 37
ANALYTICAL WDF MODEL (II): GAUSSIAN V(x)
The behavior of V(x,k) signals the transition from classical to quantum behavior:
Sharp: classical distribution
Broad: quantum effects
Can V(x,k) give a feel for when thermionic or field emission dominates?
Consider most energetic electron appreciably present(corresponds to E = μ or k = kF)
If sin(kFx) does not “wiggle” muchover range Δx, QM important
Thermionic Emission:Δx is very large - expectclassical description to be good
Field EmissionkFΔx = O(2) implies
V x,k( ) =−
2Δx 1/ 2h
Voexp −Δx2k2⎡⎣ ⎤⎦sin kx( )
4
6
8
10
0 20 40 60 80 100
No Field50 MV/m4 GV/mμ
(sin kF)x
Δisτance [Å]
Image Charge Potential
kF
F=2 ⇒ F ≈3
GVm
ANL 38
ELECTRON DENSITY TO EMISSION BARRIER
1. Numerically solve QDF to Obtain Electron Density
2. Render Density in Terms of Friedel Components
3. Evaluate Potential From Exchange-Correlation Relation & Poisson’s Equation from the Friedel Representation
4. Change in Barrier seen to be due to shift in F and xo terms
-12 -8 -4 0 4
BaO
F =0.629319
xo = +0.143379 Å0
0.02
0.04
0.06
0.08
0.1
-12 -8 -4 0 4
[#/ 3]Rho Ang
[#/ 3]Friedel Ang
Posiτion [Angsτoμs]
No BaO
= 1 /F eV nm
F = 0.598473
xo = 0.055413 Å
-4
-2
0
2
4
6
8
-1 -0.5 0 0.5
NumericalFriedelGaussian
[ ]x nm
Gauss(x)=9.785exp−(x+0.17357)/0.11387⎡⎣⎤⎦2⎧⎨⎪⎩⎪⎫⎬⎪⎭⎪
ANL 39
EMISSION BARRIER TO TRANSMISSION PROB.
Modified Airy Function Approach:
V(x) = Vo + F(x-xn):
V(x) V(xn): N Linear Segments at xn
ψ, ∂xψ Matched at xn, xn+1, etc.
0
4
8
12
-1 -0.5 0 0.5
ExCor
Field
Ion
Total
Position [nm]
No BaO
-1 -0.5 0 0.5
BaO
0
0.2
0.4
0.6
0.8
1
8 9 10 11 12
No BaO
BaO
μ+
[ ]Electron Energy eV
= 2 eV = 4.6 eV
M (x) =
Zi(z) Zi(z)∂xZi(z) ∂xZi(z)
⎡
⎣⎢
⎤
⎦⎥;
rζn =
tn(k)
rn(k)
⎛
⎝⎜
⎞
⎠⎟
M x
n−( ) •
rζn−1 k( ) =M xn
+( ) •rζn k( )
z(x) = f −2/3 v± f (x−xn) −k2 ; v=
2mVo
h2; f =
2mF
h2
Zi = Linear combinations of Airy Functions Ai(z), Bi(z)
ANL 40
CONCLUSION
Components of the Photocathode Program
Analysis of Coated & Bare Metals (extend to semiconductors)
Development Custom Engineered Controlled Porosity Photocathodes
Creation of Photoemission Models Validated By Exp. for Beam Codes
Theory Components included in photoemission code
Work function dependence on coverage & components; local variation
Spatial & Time Dependence of Temperature for laser & material parameters
Fundamental models of scattering, photoemission, QE & Barrier
Validation by bare and coated metal QE (macro) measurements
Status of Modeling Effort
Integrated Simulation Model Framework Without Recourse (Insofar As Possible) to “Fit” Parameters for “Library” Metals Using Quantum Distribution Function, Emission Theory, Coatings Theory
Photoemission Modules Appropriate for Beam Simulation Code (1st Generation Model distributed) From Integrated Simulation Model
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