kevin l. jensen - psec.uchicago.edu · kevin l. jensen naval research laboratory, 4555 overlook...
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Kevin L. JensenNaval Research Laboratory,
4555 Overlook Ave. SW, Washington DC 20375 USA
:
Funding: We gratefully acknowledge support provided by
Naval Research Laboratory, Office of Naval Research, & Joint Technology Office
Colleagues: We are indebted to too many to name, but in particular:
• D. Feldman, P. O’Shea, E. Montgomery, Z. Pan (UMD), J. Yater, J. Shaw (NRL), M. Kordesch (Ohio)
• N. Moody, D. Nguyen (LANL), J. Lewellen (NPS), D. Dowell, J. Schmerge (SLAC)
• J. Smedley (BNL), C. Brau (Vanderbilt), Y. Y. Lau (UM), J. Petillo (SAIC), S. Biedron (ANL)
http://psec.uchicago.edu/photocathodeConference/photocathodeIndex.html
1st Workshop on Photo-cathodes
We Shall Discuss Electron Emission (Principally
Photoemission) With A Focus On:
1. The Canonical Equations
2. Photoemission From Metals & Semiconductors: How Terms Are Calculated,
Typical Values And Estimates, Comparison Of Models
• Absorption & Reflection,
• Transport To Surface
• Emission
3. Complications: Emittance, Diffusion, Evaporation
4. Other Issues: Geometry, Dark Current, Space Charge
2
1st Workshop on Photo-cathodes
Equation Formula
3
• PhotoemissionFowler-DubridgeL.A. DuBridgePhysical Review 43, 0727 (1933).
• Thermal EmissionRichardson-Laue-DushmanC. Herring, And M. Nichols, Reviews Of Modern Physics 21, 185 (1949).
• Field EmissionFowler NordheimE.L. Murphy, And R.H. Good, Physical Review 102, 1464 (1956).
JFN F( ) = AFNF2 exp
B 3/2
F
JRLD T( ) = ARLDT2 exp
kBT
JFD F( ) ( )
2
1st Workshop on Photo-cathodes
Simple Restatement of Obvious:
• QE = # of electrons emitted (proportional to charge emitted Q)
# of photons absorbed (proportional to energy absorbed E)
• RULE OF THUMB:
4
QE =Q / q
E /
J / q
Io /
If Emission Is Prompt, Then Emitted Current Has Same Temporal Shape As Absorbed Laser Power:
Common Factors Of Pulse Duration ( t) Drop Out
Left Discussing CURRENT DENSITY And LASER INTENSITY
Electrons Transport To Surface Subject To Collisions
(Look At Scattering)
# That Escape Depends On Impact Of Surface Barrier If Present
(Look At Emission Probability)
# Of Photons Absorbed Depends On Reflectivity Of Surface
(Look At Reflectivity)
Photo-excitation Depth Depends On How Deeply Photon Penetrated
(Look At Dielectric Constant)
QE %[ ] =123.98J A/cm2
Io W/cm2 μm[ ]
1st Workshop on Photo-cathodes
• Spicer’s Model For Semiconductors (This Version Looks Different From Spicer, But Is Same)
• p: quasi-empirical, argued to be 3/2
• B: (Escape)x(Transport) = B exp(- x)
• g: absorption factor “over” + “under” barrier terms
• Vo: Band gap Eg + Electron Affinity Ea
5
QEB
1+ g E Vo( )p
0.0
0.050
0.10
0.15
0.20
0.25
1.5 2 2.5 3 3.5 4
Spicer
Fit (Spicer)
Taft
Fit (Taft)
Qu
an
tum
Eff
icie
ncy
hf [eV]
Taft Spicerp 1.5 1.5B 0.397 0.1863g 1.701 0.6992Vo 1.876 2.038
Originals: W.E. Spicer. "Photoemissive, Photoconductive, and Optical Absorption Studies of Alkali-antimony Compounds." Physical Review 112, 114 (1958).
E.A. Taft, and H.R. Philipp. "Structure in the Energy Distribution of Photoelectrons From K3Sb and Cs3Sb." Physical Review 115, 1583 (1959).
For Metals: C.N. Berglund, and W.E. Spicer. "I - Photoemission Studies of Copper and Silver: Theory." Physical Review 136, A1030 (1964).
Modern Usage: D.H. Dowell, F.K. King, R.E. Kirby, J.F. Schmerge, J.M. Smedley. "In Situ Cleaning of Metal Cathodes Using a Hydrogen Ion Beam." Physical Review Special Topics Accelerators and Beams 9, 063502 (2006).
QE =q1 R( )( )F ,( )PFD ( )
abso
rptio
n
scat
terin
g loss
estra
nsm
ission
• Fowler-Dubridge Model For Metals• L.A. DuBridge. "Theory of the Energy Distribution of Photoelectrons." Physical Review 43, 0727 (1933).
• K.L. Jensen, D.W. Feldman, N.A. Moody, and P.G. O'Shea. "A Photoemission Model for Low Work Function Coated Metal Surfaces and Its Experimental Validation." J. Appl. Phys. 99, 124905 (2006).
QE PFD ( ) ( )2+
kBT( )2
6
• (Modified)
T TemperatureWork Function
1st Workshop on Photo-cathodes
THREE STEP MODEL OF PHOTOEMISSIONABSORPTION of light in bulk material and photo-excitation of electrons
• reflectivity R( )
• laser penetration depth ( )
TRANSPORT of photo-excited electrons to surface subject to scattering f (cos ,E)
• electron energy
• scattering rates (relaxation times)
EMISSION probability D(E)
• Metal: Chemical Potential μ, Work Function (work function measured from Fermi level)
• Semiconductor: barrier height Ea, band gap Eg
(Electron affinity measured from conduction band minimum)
6
D E,Ea ,Eg ,F( )
Semiconductor
D E,μ, ,F( )
f cos ,E( )
k
( )
R( )Metal
1st Workshop on Photo-cathodes
10-1
100
101
102
0 2 4
n[W]k[W]n[Cu]k[Cu]n[Mg]k[Mg]
n,
k
Photon Energy [eV]
0
0.2
0.4
0.6
0.8
1
0 2 4
R(W)
R(Cu)
R(Mg)
R [
%]
For METALS, spline fitting of readily available n, k data works well
• k = extinction coefficient
• n = index of refraction
• Off-normal reflectivity related to normal values
7
0
= n ik( )2
R ( ) =n 1( )
2+ k2
n+1( )2+ k2
( ) =4 k
=c
2k
n2 k2 K + Ko K( )T2
T2 2( )
2T2( )2+ o T( )
2
2nk Ko K( ) o T3
2T2( )2+ o T( )
2
For SEMICONDUCTORS... A Drude-Lorentz model makes up for incomplete n,k data
• Ko, K = static & high freq. dielectric const
• o = damping term
• T = transverse optical phonon
• Some semiconductors may require multiple T
Cs3Sb
0
1
2
3
4
1 2 3 4
n Wallisk Wallisn Theok Theo
Photon Energy [eV]
n,
k
0
10
20
30
40
1 2 3 4
Wallis
Theory
multialkali
R [
%]
1st Workshop on Photo-cathodes
10-6
10-4
10-2
100
102
104
106
10-5
10-4
10-3
10-2
10-1
100
101
-3 -2 -1 0 1 2 3
(x)
x5 W
– (5,1
/x)
ln(x)
x/4
120x5
x5W
(x)
x3/4
x3 / 8
After Photon Absorption, Electrons Migrate And Scatter...
• ... Off Of Each Other (metals; Semiconductors If E > “magic Window” (Off Of Valence Electron If Final State Allowed)
• ... Off Of Lattice Vibrations (Phonons - Primarily Acoustic For Single Atom Material, Polar Optical If Multicomponent)
Metals: Primary Mechanism Is Electron-Electron, But Acoustic Phonon Can Contribute
8
ac
3 vs2
4m 2k kBT( )T
4
W 5,T
1
= 13.1 fs Cu,E = μ + , = 266nm,T = 300K( )
ee=
4 Ks
2 E( )2
fs
2 mc21+ E μ( )
2
2kF
qo
1
= 2.61 fs Cu,E = μ + , = 266nm,T = 300K( )
General
• = 1/kBT (To = RT)
• kB = Boltzmann’s constant
• ao = Bohr Radius
• fs = Fine structure constant
• m = e– mass (eff. or rest)
• E = Electron energy
Metal-Specific
• μ = Fermi level
• qo = Thomas Fermi Screening
• Ks = Dielectric constant
Semiconductor-Specific
• = Debye Temperature
• = Deformation Potential
• = Mass density
• vs = sound velocity
• hk = Momentum 2 (2mE)1/2
Fermi Level for Cu
• 1/ ee = AT2
• 1/ ac = AT
(x) =x3
4arctan x( ) +
x
1+ x2
arctan x 2+ x2( )2+ x2
; qo
2=
4kF
Ksa
o
1st Workshop on Photo-cathodes
0
0.08
0.16
0.24
0.32
10 100
Alpha SemiPhysical SemiCs
3Sb
Eg /
R
mo / m
AlP
ZnS
ZnO
CdSZnSe
ZnTe
CdSe
GaP
AlSbCdTe
InP
GaAs
GaSb
InAsInSb
R = 13.6057 eV
mc2 = 510999 eV
Cs3Sb
An Alpha - Semiconductor Model Can Provide Needed Parameters (e.g., Electron Effective Mass) If Such Quantities Are Unknown / Ill-defined... And Even If They’re Not...
Also, Gives Forms Of Polar Optical And Ionized Impurity Scattering That Are Related To, But Different Than, Small Electron Energy Representations Found In Transport / Scattering Tomes
9
Eg
R=m
mo
Alpha Semiconductor Model Restricts Upper Limit On Electron Velocity In Semiconductor To Half Of
Product Of Fine Structure Constant With Speed Of Light; Implies Relationship Between Band Gap
Energy And Electron Effective Mass Of:
1
pop q ,E( )= 2 q
1
Ko
1
K2n q( ) +1( )
E
Eg
u( ) =16u2 +18u + 3
3 1+ 2u( ) u u +1( )
1
ii E( )
4 2
fsm2c
Ni
Ko2
ln 2( )kBT
Eg2
E E + Eg( )
General
• = 1/kBT (To = Room T)
• kB = Boltzmann’s constant
• Ko,K = Static & High Freq. Dielectric Const.
• fs = Fine structure constant
• m = electron mass (eff.)
• E = Electron energy
• Ni = ionized impurity concentration
• n(x) = Bose-Einstein Distribution 1/[ex – 1]
Scattering for Cs3Sb: Typical Values at RT in fs
• Polar Optical on the order of 26.6
• Ionized Impurity on the order of 4395.0
• Acoustic Phonon on the order of 694.0
For u small, (u) 1/u1/2
1st Workshop on Photo-cathodes
In Polar Coordinates, Velocity of e- at angle to normalAssume Any Scattering Event Is Fatal To Emission
Matthiessen’s Rule:
Ratio of penetration depth to distance between events
Fraction Of Surviving Electrons
Weighted Scattering Fraction (e.g. MFD Eq.)(1/y Acts As Cosine Of Escape Cone Angle)
10
f cos ,E( )
k
p E( ) =( )
l E( )=
m ( )
k E( ) E( )
f cos , p( ) =
expx x
l E( )cosdx
0
expxdx
0
=cos
cos + p y( )
total1
= j1
j
F = xf x, p( )1/ y
1dx
= p2 lny 1+ p( )
1+ yp+1
2y21 y( ) 2yp y 1( )
Example: Cs3Sb-like
• = 27 nm
• v/c = 0.8%
• = 31 fsp 0.36
Example: Cu-like
• = 12.6 nm
• v/c = 0.675%
• = 2.6 fsp 2.38
Ea
0
0.1
0.2
0.3
0.4
0.5
1 2 3 4 5
F(y
)
y
po = 0.1
po = 0.3
po = 0.5
po = 1.0
po = 2.0
po = 4.0
E Eay2
p po
• for semiconductors, measure E w.r.t. Ea
• IF scales as 1/k, then p is constant
1st Workshop on Photo-cathodes
“Moments” Method Of Calculating QE Is Via Solutions To Schrödinger’s Eq.
• CLASSICAL: f(x,k,t) Is Distribution Function Where x & k Are Conjugate Coordinates => Boltzmann’s Eq.
• Integration Of kn • f(x,k) = Moments: Continuity Eq. Relates 1st (Density) & 2nd (Current Density) Moment:
11
f x + dx, k + dk, t + dt( ) f (x, k, t)
dt t+
k
m x+
F
kf (x, k, t) = 0
tx, t( ) =
t
1
2f (x, k, t) dk =
x
1
2
k
mf (x, k, t) dk =
xJ x, t( )
velocity acceleration
• QUANTUM MECHANICS
tˆ(t) =
iH , ˆ(t) =
2m xk, ˆ(t){ } =
xj t( ) j(x,t) =
2mx ˆ(t), k{ } x =
2mi
†
x x
†{ }“pure state” form
Tsu-Esaki-like formula
J(F,T ) =q
2
k
mD k( ) f k( )
0
dk
• Velocity
• Transmission Probability
• Supply Function
ˆ(t) = f
FDE
k( ) k(t)
k(t) x( ) = 2( )
3
dk dk fFD
E(k)( ) kx( )
2
Mixed state form
f(x,k) Approximated By Product Of Supply Function f(k) X Probability Of Transmission D(k) Past Barrier
D k( )jtrans
k( )jinc
k( )
E k( )2
2mk 2
+ k 2( )
f k( ) = 2( )2
fFD
E k,k( )( ) dk
Transmission Probability = ratio of transmitted to incident
current density for given k
Commonly assumed that energy is parabolic in
momentum k
Supply Function
1st Workshop on Photo-cathodes
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
D(E
)
E [eV]
Surface Barrier Subject To Applied Field Does Not Entail That D(E) Is A Step Function
• Triangular Barrier (Fowler-Nordheim Potential)Reasonable If No Image Charge Modification
• Exact Solution: Airy Functions
• Approximation: JWKB Method (As Commonly Used)
D(E) Calculation Requires Auxiliary TermsObserve Definitions Work For E > Ea Too
12
E Ea +E Ea( )
2+4
25
2F2
2m
2/3 1/2
E < Ea( ) = 2 k(x)dx =2 2
3mF
2m2Ea E( )
3/2
D E( ) =4 E E Ea +
2 E E Ea ++ E Ea +
+ E( )e (E )
D E,μ, ,F( )
D E,Ea ,Eg ,F( )
F
Ea
E
•Even For Small Ea, D(E) Is NOT A Step Function
•To Extend To Metals ( Image Charge ), Append v(y) to (E)
Ea = 0.5 eVF = 100 MV/m
D> E( )4 E E Ea( )
E Ea( )1/2
+ E1/22
D< E( ) =4
Ea
E Ea E( )e (E )
JWKB “area under curve”
1st Workshop on Photo-cathodes
Common Forms Of Emission Equations Obtained From One Formulation Using Energy Slope Terms (field) F, (temperature) T & Expansion Point Eo1D Approach To Evaluation (E Is “forward” Energy)
13
JT ARLD kB T( )2 T
F
exp T Eo μ( )
JF ARLD kB F( )2 F
T
exp F Eo μ( )
QEMFD = 1 R( )( )F( )
2+ 2 (2) T
2+ F
2( )2 2μ( )
Limit (n = 0) Richardson (Thermal) Eq
Limit (n = ) Fowler-Nordheim (Field) Eq
Energy Augmented By Photon (s<0) Fowler-Dubridge Equation
ARLDT2 exp T[ ]
AFNF2 exp
BFNF
( )2
10-3
10-2
10-1
100
101
0 0.4 0.8 1.2 1.6
ExactAprox
Diffe
ren
ce
x
J(F,T ) =e
2
m
T2
ln 1+ exp T μ E( ){ }1+ exp F Eo E( ){ }0
dE = ARLDT2N T
F
, F Eo μ( )
• Numerator: Supply Function With T = 1/kBT
• Denominator: Kemble Form Of D(k) With
N n, s( ) = n21
ne s
+ n( ) e ns=1
2n2s2 + 2( ) n2 +1 N n, s( )
x( ) 1+ 1 21 2 j( ) 2 j( ) x2 j
j=1
When F & T Become Comparable, Can Be Large
(x) = Riemann zeta function
E( ) = F Eo E( )
1st Workshop on Photo-cathodes
DEFINE the “Moments” function Mn by generalizing distribution function approachmetals - final state may be occupied (blue)
semiconductors - final state unoccupied & in conduction band (creates “magic” window)
To Calculate Emittance, Swap Forward Momentum With Transverse:
14
Mn= 2( )
3 2m2
3/ 2
E1/ 2 dE sin d2m
2E cos2
n / 2
0
/ 2
0
D (E + )cos2{ } f cos , p ( ) fFD
(E) 1 fFD
(E + ){ }
QE = Ratio Of Emitted M1 To Possible M1 (3D: E Is Total Energy; Semiconductor Shown)
QE = 1 R( )( )E xf x,E( )D E +( ) x2 dx
Ea /E
1dE
Ea
Eg
2 E xdx0
1dE
0
Eg
Leading Order (FD) Approximation: Ignore cos Dependence In D (i.e., take x = 1)
This form will lead to the comparison with the Spicer Model form
QE0 =1 R( )( )
2 Eg( )2
ED E +( ) xf x,E( )dxEa /E
1dE
Ea
Eg
This Term Is Same As Scattering Factor In MFD
“magic” window
1st Workshop on Photo-cathodes
Most Adaptable Model For Particle-In-Cell (PIC) Codes Modeling Beams Is QE0
• How Does QE0 Compare To QE?
• How Does QE0 Compare To Spicer 3-Step Model?
15
lim0G y( )
1
po +1( ) 1+Ea
2
In The Limit Of Small (asymptote)
QE0 =1
21 R( )( )G 1+
Ea
PIC - Ready Formulation
Comparison To Spicer 3-Step Model
• Interpretation Of B, g Altered
• Power Of And Dependence Changed
QEspicer
B
1+ g 3/2
0
0.2
0.4
0.6
0 1 2 3
Exact
Spicer Fit
Asymptote
G(y
)
0
0.04
0.08
0.12
2 2.4 2.8 3.2 3.6
QE0
QE0 - QE
QE
Quantu
m E
ffic
iency
Photon E [eV]
Eg = 1.6 eV
Ea = 0.3 eV
T = 300 K
F x( ) = sf s,Eax2( )ds
1/ x
1G y( ) =
8
y4x3F x( )dx
1
yRecall: Define:
= Eg + Ea( )
Define:
Spicer Parametric Fit Is Good QE0 Works Well - Good For PIC
1st Workshop on Photo-cathodes
Theory Of Photoemission From Cesium Antimonide Using An Alpha-semiconductor Model
Kevin L. Jensen, Barbara L. Jensen, Eric J. Montgomery, Donald W. Feldman, Patrick G. O'Shea, and Nathan Moody
J. Appl. Phys. 104, 044907 (2008); DOI:10.1063/1.2967826
A model of photoemission from cesium antimonide (Cs3Sb) that does not rely on adjustable parameters is proposed and compared to the experimental data of Spicer [Phys. Rev. 112, 114 (1958)] and Taft and Philipp [Phys. Rev. 115, 1583 (1959)]. It relies on the following components for the evaluation of all relevant parameters: (i) a multidimensional evaluation of the escape probability from a step-function surface barrier, (ii) scattering rates determined using a recently developed alpha-semiconductor model, and (iii) evaluation of the complex refractive index using a harmonic oscillator model for the evaluation of reflectivity and extinction coefficient.
16
10-3
10-2
10-1
0 20 40 60 80 100
375
405
532
655
808
Quantu
m E
ffic
iency [
%]
Coverage %
QE in %
0
0.04
0.08
0.12
0.16
1.6 2 2.4 2.8 3.2 3.6 4
300 K
77 K
QE0
QE
QE0 + QE
x 1.15
Qu
an
tum
Eff
icie
ncy
Photon E [eV]
Exp Data:W.E. Spicer
PR112, 114 (1958)Figure 8
QE not in %
Photoemission From Metals And Cesiated Surfaces
Kevin L. Jensen, N. A. Moody, D. W. Feldman, E. J. Montgomery, and P. G. O'Shea
J. Appl. Phys. 102, 074902 (2007); DOI:10.1063/1.2786028
A model of photoemission from coated surfaces is significantly modified by first providing a better account of the electron scattering relaxation time that is used throughout the theory, and second by implementing a distribution function based approach (“Moments”) to the emission probability. The latter allows for the evaluation of the emittance and brightness of the electron beam at the photocathode surface. Differences with the Fowler-Dubridge model are discussed. The impact of the scattering model and the Moments approach on the estimation of quantum efficiency from metal surfaces, either bare or partially covered with cesium, are compared to experiment. The estimation of emittance and brightness is made for typical conditions, and the derivation of their asymptotic limits is given. The adaptation of the models for beam simulation codes is briefly discussed.
1st Workshop on Photo-cathodes
THERMAL EMISSION
No photons
Uniform emission
Richardson Approx.
No Scattering
Maxwell-Boltzmann f(x,k)
No “final state” issues
17
= 0
2 x2 =2
= c2
D k( ) = E(k) μ( )
f x, p( ) =1
D(k) f (k) exp T E(k) μ( ){ }
1 fFD E( ) 1
Mn= 2( )
3 2m2
3/2
E1/2 dE sin d2m
2E +( )sin2
n/2
0
/2
0D (E + )cos2{ } f cos , p( )
fFD
(E) 1 fFD
(E + )( )
+ E Eg( )
n,rms (thermal) = mcx2 kx
2
=mc
c
2
M 2
2M 0
1/2
=c
2
kBT
mc2
1/2
Transverse Moments (metal & semiconductors)
Explanation of addition of photon energy to E for kn term:
• Schrödinger’s Eq.: E of e- in vacuum measured wrt conduction band min decreased by barrier height BUT
• Continuity of & x means k is conserved*
* See also: D.H. Dowell, J.F. Schmerge, SLAC-PUB-13535 (2009) they show eq. (klj) didn’t include hf in k so small by [μ/(μ+hf)]1/2
kvacuum = ksemiconductor =2m2 E +( )
1/2
sin
THEREFORE:
PHOTO-EMISSION
Photons
Uniform emission
JWKB Approx.
Scattering
Schottky Lowering
> 0
2 x2 =2
= c2
D k( ) = DJWKB E,F( )
f x, p( ) = x / x + p E( )( )
= q2F / 4 0
n,rms (photo) = mcc
2
M 2
2M 0
1/2
c
2
( )3mc2
1/2
leading order (metal)
Note: for metals, p large & f cos /p: therefore, emittance indep. of p. Semiconductors larger due to p small, but D behavior also has impact
1st Workshop on Photo-cathodes 18
0 0.2 0.4 0.6 0.8 1
WangTaylor
Gyfto Lev
2
3
4
5
0 0.2 0.4 0.6 0.8 1
Longo-1Longo-2Haas
SchmidtGyfto Lev
Work
Function [
eV
]
Coverage
Cs on WBa on W
related to of surface via Gyftopoulos-Levine Theory Experiments at UMD to understand relation of QE to coating & rejuvenation methods
• Work function depends on crystal face, bulk material, and alkali (or alkali earth) coating
• Work function minimizes at sub-monolayer
• Coatings on metals simplest - increasingly more complex semiconductor surfaces under study
• With evap/diff, possibility of uniform coverage at desired factor (e.g., dispenser photocathode)
Evap of Cs on W shows power-law dependence on coverage of form c(T) x n
BUT c & n change depending on
• Yellow: c exp(-68.1 + 5.19/kBT) n = 18
• Blue: c exp(-39.6 + 3.04/kBT) n = 10
850K800K
750K700K
0.1
1
10
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Evap R
ate
x10
-12 [
ato
ms/c
m2-s
ec]
Coverage
Data: Becker
Dash: c(T) ^10
Solid: c'(T) ^18Cs on W
J.A. Becker, Physical Review 28, 0343 (1926).
1st Workshop on Photo-cathodes 19
10 nm Cs on 200 nm Sb on SILICON
365 nm Hg, Heated by lamp, T~320-330KE = vo s jj=1
s0j=1
s j
Energy of coating atoms:
• vo = depth of well
• = interaction energy
• = # of nearest neighbors
• sj = 0,1 occupation factor
• = osc. frequency
• = hopping distance
• ( ) = hopping prob.
• P(E>vo) = Prob. evap
What’s happening:
These are PEEM images of squares of Cs (25 μm on a side) laid down on Sb, & heated. QE images: frames taken about 10 seconds apart; intensity adjusted for image (not held fixed)
Pjump
sisj
vo
Ej
D ( ) = ( ) 2 1
1+ Do =
3
Ro
voM
1/2
exp vo( )
1= e vo
1+ e vo( )1+ e vo( )
( ) = 0( )1+ e vo( )
3
1+ e vo( )4
Diffusion and evaporation of coatings: atoms as harmonic oscillators
• Diffusion D( ) = product of oscillation freq , jump length 2, and jump probability Pjump
• Both Jump prob & evap rate proportional to exp(- vo)
• Question: If atom only sees four ( ) nearest neighbors (microscopic view), how does it “know” what local coverage is (macroscopic view)?
• Answer: can be shown value of vo is related to
coverage, and therefore affects evaporation (through c(T)) & diffusion (through Pjump)
t Do2
1st Workshop on Photo-cathodes
POINT CHARGE MODEL
• Goal: Impact of surface features on emittance
• Serendipity: Dark current model (field emission if enhancement is high,
low, or both) is analytically tractable
• 3D, get trajectories, estimate emittance
20
V , z( ) Foa0Vn a0,z
a0
Vn , z( ) z + 2+ z2( )
1/2
+ j2+ z z j( )
2
( )1/2
j=1
n2+ z + z j( )
2
( )1/2
Vn 0, zn+1( ) 0 boundary conditions
dipole term
monopole term
PCM is dimensionless, scalable analytical method to get tip radii, field enhancement, total current
n = # chargesr = radii scaling
n = 1 2 3 4 5 6 7r = 0.75
monodipl
Background field Fo
radius of first charge ao
zn = ajj=0
n= r j 1a0j=0
n
tip height factor z
n r( ) = zVn 0, z( )z=zn+1
an r( ) =Vn , z( )2Vn , z( )
=0,z=zn+1
Field Enhancement
Apex Radius
& an all that is needed for analytical model of current from an apex: well-tested against real conical emitters
JFN F( ) = AF2 exp B / F( )
=8Q
9
2m
A =q
16 2 t yo( )2
2e6
4Q
B =4
32m 3
I F( )1
2a2
s
1/2
Erf p F( )s F( ) Erf s F( ){ }exp s F( )2( )J F( )
p x( ) =5B 4 2( )x
2B 2 1( )x
s x( ) =B 1( )x
2x 3B 2 3( )x
F = field at apex of emission site
100
101
102
103
10-5
10-4
10-3
10-2
10-1
100
2 4 6 8 10 12
0.40.71.0
0.4
0.7
1.0
n
dipole
an
n
10-4
10-2
100
102
104
1 3 5 7
4.5 eV3.6 eV2.0 eV
Pro
trusio
n C
urr
ent
[μ]
Apex Field [eV/nm]
atip
= 10 nm
1st Workshop on Photo-cathodes
• Copper Photoemission from Boss
• Bulk Temp = 300 Kelvin
• Wavelength = 266 nm
• Field = 100 MV/m
• Laser Intens = 160 MW/cm2
• 45 μm length (cath-anode)
• 12 μm center to center spacing
21
Cs3Sb:
• Outer: Cs-depleted el. affinity
• Inner: Cs-monolayer el. affinity
• Field = 10 MV/m
Emitted Charge Changes Fields On Surface That Affects Subsequent Emissions - Oscillations Induced By A Sudden Influx Of Charge Can Persist. Demonstration For Metal
(Cu) And Semiconductor (Cs3Sb) Using Particle-in-Cell (PIC) Code MICHELLE
Flat metal copper surface without & with hemispherical bump
Flat Sb surface with an inner square coated with Cs monolayer & depleted
1st Workshop on Photo-cathodes
What We Attempted
• Treatment Of Photoemission
• Spicer Model, Fowler-Dubridge, Moments-based Approach, And Their Relation
• Absorption - Dielectric Constant And Drude-Lorentz Model
• Transport - Scattering Mechanisms (electron-electron, Acoustic, Polar Optical, Ionized Impurity)
• Emission - Transmission Probability Through The Fowler-Nordheim Barrier
• General Thermal-Field-Photoemission Equation
• Comparison Of Theory To Experiments (UMD & NRL)
• Related Matters
• Emittance
• Evaporation And Desorption
• Dark Current Via Point Charge Model
• Space-Charge Induced Current Oscillations
22
What We Did