April 2005 - Beam Physics Course
An Introduction to
Electron Emission Physics
And Applications
Kevin L. Jensen
Code 6843, ESTD Phone: 202-767-3114
Naval Research Laboratory Fax:
202-767-1280
Washington, DC 20375-5347EM:
kljensen@ieee.org
Donald W. Feldman, Patrick G. O’Shea
Nate Moody, David Demske, Matt Virgo
Inst. Res. El. & Appl. Phys, University of Maryland
College Park, MD 20742
Intro to Emission 1
SCOPE
What is about to happen:
 Introduction to Quantum Statistics,
Solid State Physics,
Quantum Mechanics & Transport
 Thermionic & Field Emission Theory
 Photoemission Theory & Practice
 Cathode Technology
Nature of the discussion
 Primarily Theoretical:
E. Rutherford “We haven't the money, so we've got to think.”
 Intended Audience:
 Nothing better to do
 Intermediate
 Frightened into incoherence
Intro to Emission 2
A NOTE ABOUT UNITS
In the equations of electron emission…




Length & time are short, small; fields & temperature high - annoying…
Work functions & photons energies, are usually expressed in eV
Properties of atoms are generally discussed
Hydrogen atom: characteristic units are pervasively useful
Name
MKSA
eV-fs-Å-q
Q
5.76789e-29 J m
3.59996 eV Å
q
1.6022e-19 C
1q
unit electron charge
m c2
8.18712e-14 J
510992 eV
Electron rest energy
a
1/137.032
1/137.032
Fine structure constant
ao
0.529161e-10 m
0.529161 Å
hc
3.16153e-26 J m
1973.24 eV Å
kB
1.3807e-23 J/K
(11604.5)-1 eV/K
q2
hc
a

4 0hc ao mc2
Ry 
Comment
Image Charge numerator
Bohr Radius
As a Date, The day US withdrew the last American
troops from Vietnam (March 29, 1973)
Boltzmann’s constant (actually named by Planck who
later regretted the generosity)
q2
1
2Q
 a 2 mc2 
8 0 ao 2
ao
1
Q  a hc
4
Intro to Emission 3
OUTLINE
The Basics
 Nearly-Free Electron Gas Model
 Barrier Models
 Quantum Mechanics & Phase Space
1-Dimensional Emission Analysis
 Thermionic Emission
 Field Emission
 Photoemission
Multidimensional Emission from Surfaces & Structures
 Field Emitter Arrays
 Dispenser Cathodes
 Photocathodes
Cathode Technology and Applications
 Vacuum Electronics, Space-Based Applications, Displays
 Operational Considerations
 Performance Regimes
 Operational Complications
Intro to Emission 4
PARTICLES IN A BOX
Consider a box containing N particles
(we will call them electrons later)
total # of Particles = N; Total Energy = E
Characterize particles by energy Ei
ni = # particles with energy Ei
wi = # ways to put ni particles in gi “states”
ni
N   i ni
E   i ni Ei
Ei
 
S   i Si   i k B ln wi
gi
Intro to Emission 5
STATISTICS
If the gas of particles is dilute, the issue of whether
particles can share the same box doesn’t come up:
therefore, gi boxes means gi possible locations to go,
but order within the box isn’t important
(Maxwell-Boltzman statistics)
However, if the gas is not dilute, it may matter whether
or not a state is occupied - if it does, and one state can
only hold one particle, then the statistics are different:
(Fermi-Dirac Statistics)
n
gi i
1
wi 

ni ! N !
wi 

gi !
“Correct
Boltzmann
Counting”

ni ! g i  ni !
The most probable state is found by maximizing W
(the sum over wi) with respect to ni
subject to the constraints of constant N and E
  
 

0   n   j ln w j  a N   j n j   E   j n j E j 
i 

 k B1 n Si  a   Ei
i
Intro to Emission 6
ENERGY DISTRIBUTION
Stirling’s Approximation:
n!  n ln(n)  n
Find:
 s=1
Fermi-Dirac
 s=0
Maxwell-Boltzmann
 s = -1
Bose-Einstein
 g



Si  k B
ln wi  k B ln  i  s 
ni
ni
 ni

 

ni
a   Ei
f (Ei )   s  e
gi

1
How to figure out a and 
A slide or two ago…

dE   i ni dEi  Ei dni

 E 
1
a
  i  ni i  dV 
dS

dni


i
i
i
V
k




B
 is CHEMICAL POTENTIAL, or change in
energy if one more particle is added
(alternately, energy of most energetic
particle at T = 0 for fermions)
…Thermodynamics
dE  TdS  PdV  dN
Fermi - Dirac Distribution Function

E   / kB T
f (E)  1 e

1
We shall retain  = 1/kBT in future
Intro to Emission 7
THE QUICK QUANTUM REFRESHER…
 Energy is a constant of the system
E   (t) Hφ  (t)   (0) Hφ  (0)
 The wave function at a future time
propagates from a past wave
function - the time arguments of
the propagators are additive
 (t1  t2 )  Uφ(t1 )Uφ(t2 )  (0)  Uφ(t1  t2 )  (0)
 Total particle number is conserved,
therefore, propagators are unitary
 (t)  (t)   (0) Uφ(t) Uφ(t)  (0)   (0)  (0)
 CONCLUSION: Schrödinger’s Eq:
φ / h  (0)
 (t)  exp iHt

 Basis States combine to form total
wave function of system
(t) (t)   dEf (E)  E (t)
2
Intro to Emission 8
ENERGY LEVELS IN A BOX
So what are the allowable states?
 Classical: whatever
 Quantum: don’t touch the sides
 2 l   2 m   2 n 
 x, y, z    lmn N lmn sin 
x  sin 
x  sin 
x
 L

 L

 L

2 ly
2 lz
2 lx
; ky 
; kz 
L
L
L
r
2 2 h2 2 2 2
h2 k 2
E l, m, n  
l x  l y  lz 
2
mL
2m
kx 
Energy
(1,1)


(2,2)
(3,3)
Intro to Emission 9
DENSITY OF PARTICLES
Transition to Continuum Limit

yp
Fp x  
dy
y x
0 1 e
Introduce Fermi Integral F1/2(x)
N
 2 f (Ei )
V
i
  
 2 2
Effective Density of
Conduction-Band
States @ RT
Nc= 0.028316 #/nm3
3
101

r
f E( k ) d 3k
4  m 

 2 h2 

3/ 2
 
F1/ 2 
 
Density of States:
D( E)dE  N c  E
# states between E & E+dE
1/ 2
Fermi Integral
  ,T 

F
(2/3) x^1/3
(№/4)^1/2 exp(-x)
2 3/ 2
x
3
0
10

-1
10
2
e x
p = 1/2
 dE
-2
10
-4
-2
0
2
4
x
Metals: Roughly 1 electron per atom:
Sodium @ RT:
•  = (1 e-/22.99 gram)x(0.9668 gram/cm3)
• Therefore:  = 3.14 eV
• (actual:  = 2.65 #/cm3,  = 3.23 eV)
Semiconductors: carriers due to doping
Doped Silicon @ RT:
•  = 1018 e-/cm3
• Therefore:  = -0.354 eV
Intro to Emission 10
CHEMICAL POTENTIAL & SUPPLY FUNCTION
Electron Number Density 
 m 
( )  2M c 
 2 h2 
3/2
2

F1/2 ( )
Zero Temperature (0 ˚K = o = EF)
1  2m 
( o )  2  2 o 
3  h 
3/2
kF3
 2
3
Current flows in one direction. It is
useful to consider A 1-D
“thermalized” Fermi Dirac distribution
characterized by the chemical
potential and called the “supply
function” to evaluate emission.
The supply function is obtained by
integrating over the transverse
momentum components
# of particles in box V does not change
with temperature, so  must:
  (T )   o ,T  0K 
2
4


1  
1   
(T )  o 1  
  80     
12




o
o
f (k) 

2
2 2


0
2 k dk
1  exp  (E||  E   )
m
ln 1  e (   E (k )) 
2
 h
Intro to Emission 11
FROM ONE ATOM TO TWO…
A bare charge in a sea of electrons is
screened by a factor depending on the
electron density (Thomas-Fermi Screening)
 Ex: e = 0.1
mole/cm3
kTF

4 0 r
-1
Atom 1
n=4
4 0 r
+
e
 kTF r
-1
Atom 1
n=3
n=4
n=5
Atom 2
Atoms 1&2
Bohr Levels
-3
1
2
3
Distance [Ѓ]
4
+
0
n=5
-3
 kTF r
Two Atoms
-2
n=3
-2
4 0 r
e
 1.7081 -1
q
One Atom
q
1/ 2
Energy [eV]
Energy [eV]
0
 q2

 D  
 o

q
1
2
3
Distance [Ѓ]
4
Intro to Emission 12
…TO AN ARRAY OF ATOMS
Electrons in a periodic array merge into regions where
energy value is allowed in Schrödinger’s Eq., or not:
Ev
Ec
 those permitted by Schrödinger’s Eq. called bands bands can overlap
Metal
 those not permitted are called “forbidden regions”
or Band Gaps
 A filled band does not allow current flow: in an insulator,
lower band filled, upper is not. In a metal, bands overlap
and partially filled. Semiconductors are insulators at 0 K
Ec
Ev
 Electrons in conduction band act free (i.e., no potential)
Conduction Band
Band Gap
Valence Band
Insulator
Ec
Ev
Ec
Ev
Semicon.
Intro to Emission 13
INTERACTION OF ELECTRONS
Energy of Electron Gas & Ions
 Electrons have kinetic energy and interact
with themselves (HelN)
ELECTRONS
Kinetic Energy
2 2
kF h k
h2 k 2
2
3
2

4

k
dk

o
 k 2m
5
2 3 0 2m
 Electrons interact with the ions (Vel-B)
1
V
 Self-interaction of background (VB)
Exchange Energy
 Their evaluation is… fascinating…
H N  H elN  Vel  B  VB
hk  
2
H elN   i1
N
Vel  B  
VB  
2m
2
N
2
N
1
4V 2

'
k1 k2
r r
k1k2 V̂ee k1  k2 k1k2
Correlation Energy
r r
a r  r '
q2 N e
 r r
4 0 i j 1 r  r '
r r
 a r  ri
q
re
r
d
r

(
r
)

r r
4 0 i1 
r  ri 
r r
a r  r '
q
r r e
r
r
d
r
d
r
'

(
r
)

(
r
')

r r 


8 0 i j 1
r r'


( )

m
2
3 2
 h ao
r r
Terms with k1k2 V̂ee k1  k2 k2 k1 & higher


…but what we find (if we did it) is that the energy
terms depend on the electron density.
Hohenberg & Kohn: “ALL aspects of system of
interacting electrons in ground state are
determined by charge density.”
Independent electrons move in an “effective
potential” emulating interaction with other eCorrelation (“stupidity”) Energy is the sum of a
heck of a lot of Feynman diagrams
Intro to Emission 14
AT THE SURFACE OF A METAL
Density of electrons goes from a region
where there are a lot (inside bulk) to where
there aren’t that many (vacuum)
F
Vxc

Exchange-Correlation Potential relates
change in density  to

V


Exc 

change in potential energy xc

Metal
Vacuum
Example using Wigner Approx to Corr. Energy: 15

4

rao
 3
1
 
3

 Q  5.35 eV
 ao
Calc: F=2.12 eV
Actual: F=2.3 eV
xc


2.444 2.335 r  5.859




2
  r
r

7.811

Exc
Metals
V (r) [eV]
 Consider Sodium (Na):
Electron density = 0.0438 mole/cm3
Chemical Potential = 3.23 eV
Cu
Au
Na
10
1019
#/cm3
5
0
1
10
r
100
…so Vxc gets most of barrier, but not all…
…and Na was a “good” metal…
Intro to Emission 15
AT THE SURFACE OF A SEMICONDUCTOR
How many electrons screen out a surface field?
qs
F
F
l 
2 o
2a hc
2
µo
 For metal densities at 1 GV/m:
 = 6.022x1022 / cm3 implies l = 0.00184 nm
Poisson’s Equation (o = bulk;   o  f)

 
100
Silicon @ RT
= 1018 #/cm3
o = -0.0861 eV
Fvac fs
2 N c K s   

3 o    
 8    2 
    1
 5   

Asymptotic Case Small Band Bending:  ≤ –2:
 
Fvac fs
2

2N c K s
o  
 
 
exp o exp fs  fs  1
-1
10
o
 
2
1/ 2
   [eV]
Asymptotic Case Large Band Bending:  » 1:
2
Fvac
ZECA: f(x) is the same as that which
would exist if no current was emitted.
N 2
 2
F  c
F1/ 2   o  f   F1/ 2 o
f
K s o 

µ
Ev
 For semiconductor densities at 100 MV/m
 = 1018 / cm3 implies l = 11.1 nm

c
Ec
Surface charge density s = q x bulk density  x width l
-2
10
-3
10
1
10
100
1000
4
10
Vacuum Field [MV/m]
Intro to Emission 16
OTHER CONTRIBUTIONS TO SURF. BARRIER
Electrons encounter barrier at surface
 Wave+Barrier = Quantum contributions
to barrier (Surface Dipole)

…and there’s the issue of the ion cores
(Approx: neglect what isn’t easily evaluated)
Density (Friedel Oscillations)
2
1 
f
(k)

(x)
dk
k
2 0

cos 
sin 
lim  x  o 1  3

3
 
2
3

(x) 

Not all electrons pointed at barrier:
2 2


0
2k dk

1  exp  (E||  E   )



m
ln 1  exp  (   E(k)) 
 h2 
Infinite Barrier:
Finite barrier
 k x  sin kx 
 k x  sin  k x  xo 

Vo  h2 / 2mxo2



1.2
0.9
(x)/o

2
 
  2k F x  xo 
 SUPPLY FUNCTION: Integrate over
transverse components of fFD(E)
f (k) 

0.6
Excess (+) charge
Excess (Р) charge
0.3
f 
2
Qk
9 2 F
0
-6
 5/ 4
-4
-2
0
2kF(xРxo) / 
Intro to Emission 17
IMAGE CHARGE APPROXIMATION
The Potential near the surface due to
Exchange-Correlation, dipole, etc. can
be modeled reasonably well using the
“Image Charge Approximation”
2x
Classical Argument: Force Between
Electron and Its Image Charge
F(x)  
q2
metal
Vacuum
4 o (2x)2
Energy to Remove Image Charge
Vi (x)   F( x )dx 
V(x) [eV]

No Image
Image
8
4
y = (4FQ)1/2/F
x
q2
Q


16 o x x
0
0
5
10
x [
]
15
20
Intro to Emission 18
BARRIER HEIGHT
Triangular Potential Barrier
k
2m
2
V( x)  Vo  Fx 
2
o
 fx

Potential
barrier
Schrödinger’s Equation Solution for High Vo
 k (x)  exp ikx   exp i(k  2 )x 
Phase Factor   k  k
2
2
o
Electron
Density
 ko 2
2m
 ko 2
2


k  k
2k
tan(2 )   2
f
2
 k
 2  2  k2
2
2


2m
2
3



k
k 
 (k)   O   
ko
 ko  
Electron Density Variation Primarily Due to
Barrier Height, Less by Barrier Details
(to Leading Order) for “Abrupt” Potentials
1 1
xo  xo  
ko ko
“Origin” Affected By
Barrier Height
Intro to Emission 19
ANALYTIC IMAGE CHARGE POTENTIAL
Given that barrier height affects origin, is it possible to retain Classical
Image Charge Eq.’s Simplicity (central to derivation of Emission Eqs.)?
Short answer: Yes…
Vanalytic x    T   Feff  F x  xo  
Long answer:
Q
x  xo
 Define Effective Work func.
10
 Account for ion origin not
coincident with electron origin

fi  o
0
 
 xi
x'
0
0

dx '' dx ' 
8
QkF3 xi2
3
 Introduce “ion” length scale
  
k F xo  

 Vmax 
1/ 2
3
;  xi  xo 
8k F
Potential [eV]
F eff T , F   F o  a oT  fi  2Fxo
Numerical
Analytic
Classical
8
6
4
Mo
Cs
(4 eV/nm)
(2 eV/nm)
2
0
-12
-8
-4
0
4
8
12
Position [Å]
Intro to Emission 20
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:
   f (x, k,t)dxdk
dn  f (x, k,t)dxdk  f x ', k ',t 'dx ' dk '  dn'
N  2
dx’
dk’
dk
to order O(dt)
1
x'  x 
dn’
dn
dx ' dk ' 
dx
hk
dt; hk '  hk  Fdt
m
x x ' x k '
k x ' k k '
dxdk  dxdk
Boltzmann Transport Equation
0


f x  dx, k  dk,t  dt  f (x, k,t)
dt
velocity & acceleration
  hk  F  
 

 f (x, k,t)  0
 t m x h k 
“Moments” give number density  and current density J: Continuity Equation

 1
 x,t  
t
t  2
 



   1
f (x, k,t)dk   
 x  2

 hk 

f
(x,
k,t)dk


J x,t

  m 
x


 
Intro to Emission 21
CURRENT - A QUANTUM APPROACH
Center discussion around states defined by
xφ x  x x ; kφ k  k k
  exp ikx 
x x '   x  x '; x k '   k  k '
Iφ 2   x x dx
 2   k k dk
x k  2
1/ 2
1/ 2


Heisenberg Representation:
operators O evolve, eigenstates don’t
Schrödinger Representation:
eigenstates evolve, operators don’t
OφH (t)  Uφ (t)OφSUφ(t)
φ / h Oφ exp  iHt
φ / h
 exp  iHt
 S






 φ
OH (t)   Uφ (t) OφSUφ(t)  Uφ (t)OφS  Uφ(t)
t
 t

 t


i φ φ
 H , OH (t) 

h
Relation from Heisenberg Uncertainty
 
 xφn , kφ  inxφn1  i xφ xφn




1/ 2
φ  xφkφ kφxφ i
 x,
φ
k
 
Heisenberg Uncertainty:
Consider H & the operator for density:
 h2 kφ2

Hφ  k (t)  
 V xφ   k (t)  E  k (t)
 2m


φ(t)   f E(k)  k (t)  k (t)
Then it follows that note: {A,B} = AB+BA


h2 φ2
ih2  φ
φ


 H , φ(t)  

 2m  k , φ(t)   2m xφ k, φ(t)


h
φ
j(t) 
φ(t), kφ
2m


φ t   φ
j t
t
xφ


Intro to Emission 22
CURRENT IN SCHRöDINGER REPRESENTATION
Consider a pure state φ(t)   (t)  (t)

Trivial Case: Plane waves
 (t)  exp(iEt / h) k

h
j(x,t)  x ĵ( x̂,t) x 
x ̂(t), k̂ x
2m
h

 † (x,t) x (x,t)   (x,t) x † (x,t)
2mi

x  (t)  exp(ikx  i t)

(x,t)  1
j(x,t)  hk / m
The form most often used in emission theory
Basis for FN & RLD Equations
Gaussian wave packet at t = 0:
 (x) 
 (x) 
 1

exp   k 2 x 2  iko x 
2
 2

k

 (x)  k 2 x  iko  (x)
x


1
2



2

 exp   k  ko / k   ikx dk
k 2
(x) 
exp  k 2 x 2 
2
hk
J (x)  o  (x)
m
Form of J(x): velocity x density
Intro to Emission 23
THE QUANTUM DISTRIBUTION FUNCTION
For the density operator, we considered: (x,t)  x φ(t) x
Wigner proposed a distribution function defined by



f x,k,t  2  e2iky x  y φ(t) x  y dy

Wigner Distribution function (WDF)
Time evolution follows from continuity equation:




h
i
f x,k,t  2  e2iky dy  x  y
 xφ φ(t), kφ x  y  x  y V xφ , φ(t)  x  y 

t
2m
h







A bit of work shows that:


hk 
f x, k,t  
f x, k,t   V x, k  k ' f x, k ',t dk '

t
m x
i  2iky
V x, k  
e
V (x  y)  V (x  y) dy
 h 


 







 x,t 
1
2
1
J x,t 
2
 



f (x,k,t)dk
 hk 
  m  f (x, k,t)dk

integrating both sides wrt k reproduces classical equations
Intro to Emission 24
WDF PROPERTIES

 


V x  y V x  y  
Taylor Expand V(x,k):
n0
2 y 2n1   
2n  1 !  x 


2n1

V x
It follows that for V(x) up to a quadratic in x, then WDF satisfies
same time evolution equation as BTE
Now, reconsider Gaussian Wave Packet:




t f x,k,t  

hk
 f x, k,t
m x


hk
f x,k,t  f  x  t,k,0
m




Note: this is special case of the constant field case,
i.e., V(x) = g x, case, for which:

hk
g 2
g 
f x,k,t  f  x  t 
t ,k  t,0
m
2m
h 



“trajectories” are same as classical trajectories

momentum (k/k)
2
 1
2
1  k  
2 2
f x, k,0 
exp  k x    
2  k  
k 2
 2

V(x)=0
1
0
-1
t = 0.0
t = 1.4
-1
0
position (x k)
1
The Schrodinger picture expansion of the
wave packet becomes, in the WDF framework,
a shearing of the ellipse
Intro to Emission 25
ANALYTICAL WDF MODEL: GAUSSIAN V(x)
How does V(x,k) behave?
Consider a solvable case
where V(x) is a Gaussian:
large x samples
f(x,k') near k
Sharp
x2 = 5.0



2
V x  Vo exp   x / x 






V (x  y)  V (x  y)  2Vo sinh 2xy / x 2 exp   x 2  y 2 / x 2 
 
 
2x
V x, k   1/ 2 Vo exp  x 2 k 2  sin 2kx
 h
small x samples
f(x,k') far from k
Broad
x2 = 0.1
Intro to Emission 26
ANALYTICAL WDF MODEL (II): GAUSSIAN V(x)
The behavior of V(x,k) signals the
transition from classical to
quantum behavior:
 Sharp: classical distribution
 
V x, k  
 
2x
2 2

V
exp
x
k  sin 2kx
o
1/ 2

 h
 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 =  or k = kF)
 Thermionic Emission:
x is very large - expect
classical description to be good
 Field Emission
kFx = O(2) implies
kF F
GV
 2  F  10
F
m
10
Energy [eV]
 If sin(kFx) does not “wiggle” much
over range x, QM important
Image Charge Potential
No Field
50 MV/m
4 GV/m

sin(k x)
8
6
4
F
0
20
40
60
Distance [Ѓ]
80
100
Intro to Emission 27
OUTLINE
The Basics
 Nearly-Free Electron Gas Model
 Barrier Models
 Quantum Mechanics & Phase Space
1-Dimensional Emission Analysis
 Thermionic Emission
 Field Emission
 Photoemission
Multidimensional Emission from Surfaces & Structures
 Field Emitter Arrays
 Dispenser Cathodes
 Photocathodes
Cathode Technology and Applications
 Vacuum Electronics, Space-Based Applications, Displays
 Operational Considerations
 Performance Regimes
 Operational Complications
Intro to Emission 28
RICHARDSON-LAUE-DUSHMAN EQ.
The RLD Equation describes Thermionic Emission
Electrons Incident on Surface Barrier & Classical Trajectory View is OK
 Therefore: If Energy < barrier height, no transmission
 Therefore: Emitted Electrons Must Have Energy > Emin    F  4QF
 Therefore: if f(k) is to be appreciable, T must be LARGE



q
hk
q
m
1  e   E  dE
f
k
dk

ln

2 k  m
2h E  h2 
min
min

qm
  E
 2 3  e  dE
2  h Emin
J (T , F ) 


qm
   F  4QF 
exp


2 2 h3 2
Maxwell Boltzmann
Richardson Constant
ARLD
2
B
2 3
qmk
Amp 1

 120.18
2 h
cm 2 K 2
Example:
Typical Parameters
• Work function 2.0 eV
• Temperature 1300 K
• Field
10 MV/m
J RLD  10.46
Amp
cm 2
Intro to Emission 29
THERMIONIC EMISSION DATA
The slope of current versus temperature on a RICHARDSON plot
produces a straight line, from which the slope gives the work function
Ex: J. A. Becker, Phys. Rev. 28, 341 (1926).
 Work function of clean W: 4.64 eV (Modern value: 4.6 eV)
 Work function of thoriated W: 3.25 eV (Modern value: 2.6 eV)
 so there are complications to the actual determination, such as coverage… [see Lulai]
J. A. Becker, Phys. Rev. 28, 341 (1926).
0
10-1
J. A. Becker, Phys. Rev. 28, 341 (1926).
-15
Thoriated W
-20
-2
10
Thoriated W
Slope = 3.25 eV
2
ln(J/T )
Current Density [A/cm2]
10
-3
10
-4
10
-25
Clean W
10-5
-30
-6
10
10-7
-35
1000
1400
1800
Temperature [Kelvin]
2200
Clean W
Slope = 4.64 eV
4
6
8
10
12
14
1/k T [1/eV]
Work function measurement for Thoriated Tungsten: <http://www.avs.org/PDF/Vossen-Lulai.pdf>
B
Intro to Emission 30
TUNNELING THEORY REFRESHER
Traditional Field Emission Theory: Extensive Use of Schrödinger’s Equation
Consider Simplest Analytically Solvable Tunneling Model: Square Barrier
 k (x)  exp ikx 
 Regions I & III:
 k (x)  exp  x 
 Region II:
k 2  2mE / h2
 2  2mVo  E / h2
Vo
eikx
Match  and d/dx at 0 and L
 At x = 0
1
1   1   1 1   a
 ik ik   r(k)       b
 At x = L
 e L
  e L
e L   a  eikL

 e L   b  ikeikL
TRANSMISSION COEFFICIENT

t k 

T k 
2 k
  
jtrans (k)
jinc (k)

 
2 k   
2
I
T(k)=|t(k)|2
2
2

2
 
 k sinh  L
2
2
0
II
+L
III
This is the “area” under
the potential maximum
but above E(k)
eikL
  
2 k
E(k)
r(k)e-ikx
eikL   t(k)
ike ikL   0 
2 k cosh  L  i  2  k 2 sinh  L
t(k)eikx
 2k 
   exp 2 L
 


Intro to Emission 31
FOWLER NORDHEIM EQUATION
The Fowler Nordheim Equation was originally derived for a triangular barrier
 Schrödinger’s Equation
k 2  2mE / h2
 

2
2

k

k

fx
o
 x 2
  k (x)  0

2

Vo
k  2mVo / h
2
o
2
f  2mF / h
2
 Airy Function Equation
 4 2
4k(ko2  k 2 )1/2
T (k) 
exp

ko  k 2
2

 3f
ko

I

3/2


Current
 
J (T , F ) 
kF
 
q /F
2

0

hk
T (k) k F2  k 2
m

 4
2
3
F
exp

2mF
 3hF

4 2 h   F

E(k)
ko2  k 2  fx
z
f 2/3
Same drill as with rectangular barrier…
but use Asymptotic Limit of Ai & Bi

t(k)eikx
r(k)e-ikx
 2


z
 z 2
 aAi z   bBi z  0
q hk
q
T k f k dk 

2 0 m
2
eikx

0
II
Vo/F
III
This is the “area” under
the potential maximum
but above E(k)
Action occurs near E = 
• Evaluate coefficient at 
• Linear expansion of
exponent about -E
Intro to Emission 32
WKB TRANSMISSION PROBABILITY
Schrödinger’s Equation

2
x

  (x)2  k (x)  0
12
Potential [eV]
V(x)
Wave Function (Bohm Approach)
and Associated Current
 k (x)  R(x)exp iS(x) 
h

jk (x)  R(x)   x S(x) 
m

2
4
x–
-2
-1
R4 jk2  R12x R  iR2x jk   2

T k 
Neglect for slowly
varying density
 k x   x 

x


exp i   x ' dx '
x
x+
0
1
2
Position [nm]
vanishes for
constant current
1/ 2

E(k)
0
Schrödinger Recast

h (x)  2m V (x)  E
8
jk (transmitted)
jk (incident)


x
2m
 exp 2 
V (x)  E dx 
2
x
h




“Area Under the Curve” Approach to WKB
Intro to Emission 33
IMAGE CHARGE WKB TERM
“Area Under the Curve” Approx:


2mFL G x / L 
cos( )sin( ) d
J(F) ≈ 7x105 A/cm2
2
h
Energy [eV]
T ( E)  exp 2 (E)
 ( E) 
3

2
G(s) 

2
0
6
2 ( E) 
 = 5.87 eV
F = 4.41 eV
2
F = 0.5 eV/Å
Q = 3.6 eV-Å
s  sin 2 ( )
FN Equation: Linearize (E) about
the chemical potential

 c fn   E
F
4
b fn 
2mF3 v y
3h
2
c fn 
2mFt y
hF



0
Elliptical Integral
functions v(y) & t(y)
4QF
F
v( y)  0.9369  y 2
y
t( y)  1.0566
µ
L
4
0
b fn
V(x) =   F - Fx - Q/x
8
5
10
Position [
]
15
20
Example:
Typical Parameters
• Work function 4.4 eV
• Temperature 300 K
• Field
5 GV/m
J FN  647000
Amp
cm 2
why the odd choice of v(y)? Perfect linearity on FN plot
Intro to Emission 34
FOWLER NORDHEIM EQUATION
Current Density Integral Has Three Contributions:
 Dominant Term: Tunneling due to Field
 Effects of Temperature
 Band Bending and/or small Fermi Level
(negligible except for semiconductors)
  /cfn of Order O(10) for Field Emission

Example:
• F = 4 GV/m
• T = 600 K
•  = 5.6023 eV
• 20930 Amp/cm2
• 1.142
• 2.108 x 10-12


qm
 b fn / F
c fn (   E )
( E)
J (T , F )  2 3 e
e
ln
1
e
dE

2  h
0
 b fn   c fn / 

J (T , F )  a fn exp    
 (1  c fn  )exp(c fn  ) 
 F   sin(c fn /  )

Field
a fn 
qm
;
16 2 hFt( y)2
Field-Thermal
bfn 
Semiconductor
4
2
2mF3 v( y); c fn 
2mFt( y)
3h
hF
Intro to Emission 35
FIELD EMISSION DATA
The slope of current versus voltage on a Fowler Nordheim plot
produces a straight line, from which the slope gives F3/2 / g
Ex: J. P. Barbour, W. W. Dolan, et al., Phys. Rev. 92, 45 (1953).
 Work function of clean W (4.6 eV) implies g factor = 4368 cm-1
 Work function of increasing coatings of Ba on W needle: [2] 3.38 eV [3] 2.93
0
0
Slope = 6.314
Slope = 3.771
Slope = 2.987
-5
-5
Clean W [1]
W+Some Ba [2]
W+More Ba [3]
2
ln(I/F )
ln(Current [A])
Pulsed Current
Field Enhancement
for F = 4.6 eV:
 = 4368 / cm
-10
2.93 eV
-10
3.38 eV
-15
-15
4.60 eV
Direct Current
-20
Units of I = Amps
-20
0
0.5
1
1.5
2
2.5
1000/Voltage [V]
3
3.5
1
2
3
4
5
6
7
1/(F [eV/Angstrom])
Modern Spindt-type field emitters: C. A. Spindt, et al, Chapter 4, Vacuum Microelectronics, W. Zhu (ed) (Wiley, 2001)
Intro to Emission 36
THERMIONIC VS FIELD EMISSION
9
The most widely used forms of:
 Field Emission: Fowler Nordheim (FN)
 Thermal Emission: Richardson-Laue-Dushman (RLD)
2 h

0 T
E  f E dE
8
Energy [eV]
J(F,T ) 
1
Field Emission:
Work Func = 4.6 eV
Field = 4 GV/m
Thermionic Emission:
Work Func = 1.8 eV
Field = 10 MV/m
7
6
High Temperature
Low Field
Low Temperature
High Field
0
Fowler Nordheim
Richardson
T (E)    E    f 
f (E) 
5
Fermi Level
m
exp     E 
 h2
J RLD (T )  ARLDT 2 exp f / k BT 
Arld  120.173
Amp
;Q  0.359991 eV-nm
Kelvin 2cm 2
Transmission Probability
Electron Supply
Emission Equation
2
4
Position [nm]

6


T (E)  exp   b fn / F  c fn   E  


m
f (E)  2   E    E 
h
J FN (F)  AF 2 exp B / F 
1.38072  10-6 Amp
A
exp 9.83624F-1/2
2
F
eV
eV
B  6.39952F3/2
nm
Constants for Work Function in eV, T in Kelvin, F in eV/nm


Intro to Emission 37
FN AND RLD DOMAIN OF VALIDITY
DOMAINS
102
 RLD: Corrupted When Tunneling
Contribution Is Non-negligible
4/3
Q1/ 3
 FN: Corrupted When Barrier
Maximum near  or cfn close to 
 Maximum Field: f > 6

1
F
F  6
4 Q

2
 Minimum Field: cfn < 2
F
4
h
2mF
Field [GV/m]
 2m 
F 

10h



FN (F=4.4 eV)
101
100
Field Emitter
Photocathode
-1
10
Thermionic
RLD (F=2 eV)
10-2
10-3
-4
10
300
700
1100
1500
1900
Temperature [K]
Typical Operational Domain of
Various Cathodes Compared to
Emission Equations
Intro to Emission 38
EMISSION DISTRIBUTION
Emission Distribution
Transmission Coefficient
101
10-3
10
-6
3 10
f(E)T(E)
T(E)
10
TFN(E)
Texact(E)
Twkb(E)
-1
-5
2 10-6
1600 K
-6
1 10
10-7
10
Vmax
µ(300K)
-9
3
4
5
6
7
8
0
300 K 600 K
4
4.5
5
5.5
6
6.5
7
E [eV]
E [eV]
Near Fermi Level, TFN(E) Is a
Good Approximation
For Typical Field Emission from
Metals such as Molybdenum,
f(E) dominates T(E) for E Large
7.5
Intro to Emission 39
THERMAL-FIELD ASSISTED PHOTOCURRENT
 Supply Function
X(F[GV/m],T[K])
0

 Transmission Coefficient T(E):
(b = slope of -ln[T(E)])

T E   To 1  exp b E  Ec 
Thermal
Ec   
b fn

10
10-4
 When b » : Richardson-LaueDushman Eq.
 When b ≈  : No simple analytic form
 Photocurrent: changes T(E) behavior
T(0.01,2000)
10-8
Fermi
-10
0 K-like
Regime
10
f(7,300)
f(0.01,2000)
-12
101.2
Ec    F  4QF
 When  » b: Fowler-Nordheim Eq.
T(7,300)
-6
10
1
Fc fn
Maxwell
Boltzmann
Regime
-2
7 GV/m
300 K
1
T(E) f(E) (norm.)
Field
10
T(E) & f(E)

m
f E  
ln 1  exp     E 
 h2
2 MV/m
1094 K
10 MV/m
2000 K
0.8
0.6
0.4
0.2
0
3
4
5
6
7
8
Energy [eV]
9
10
11
Intro to Emission 40
QUANTUM EFFICIENCY (3-D)
Quantum Efficiency is ratio of
total # of emitted electrons with
total # of incident photons
Lear* Approximation for temporal and spatial
behavior: Gaussian Laser Pulse gives Gaussian
Current Density such that time constants and area
factors approximately equal for both
1  
J   F  ,Te , t 2d dt
q  0
QE 
1  
I , t 2d dt



0
h
Photocurrent

T E  h ;   f E dE
q

0
J 
f 1  R I   

h
f E dE

Photocurrent J(F,T) depends on
0
1. Charge to Photon energy ratio (q/hf)
1
2
3
4
2. Scattering Factor f
Richardson Approximation:
3. Absorbed laser power (1-R) I
4. Photoexcited
e-

Escape Probability

0
T E  h ;   f E dE  

  f  h
ln 1  e (   E ) dE
 Richardson: T(E) = Step Function
 Fowler’s astounding approximation:
assume all e- directed at surface.

Fowler-Dubridge Formula (modified)

y
 Fowler Function U x    ln 1  e dy

x
* “Seek thine own ease.” King Lear, III.IV
U   h  f 
q
QE  f
1  R 
h
U  
Intro to Emission 41
FOWLER-DUBRIDGE EQUATION
Field significantly exaggerated to show detail
Photon energy: first four harmonics of Nd:YAG
U   h  f 
QE  f 1  R  
U  
Quantum Efficiency proportional to Fowler Factor
U(x), argument of which is proportional to the
square of the difference between photon energy &
barrier height for sufficiently energetic photons
“Fowler factor”

e x 1  beax 

U x    1
2
2
x
 ax
 x   e 1  be 
6
2
x  0 
x  0 

b  1   2 / 12
For metals
QE 



a  1  b  ln 2 / b
f 1  R 

2
 h  f 
2
h  f 
2
Fowler-Dubridge often
referred to in this way
16 #/cm22]
[1016
f(E) [10
T(E); f(E)
#/cm ]
T(E);
Fowler-Dubridge Formula… sort of
100  = 1064 nm
o
10-1
10-2
T(E)
T(E+4h)
-3
T(E+3h)
10
T(E+2h)
-4
10
T(E+h)
f(E)
-5
10
12
14
16
18
Energy [eV]
Example: Copper
• Wavelength
• Field
• R
• Work function
• Chemical potential
• Scattering Factor
• QE [%] (analytic)
• QE [%] (time-sim)
• QE [%] (exp)
20
22
266 nm
2.5 MV/m
33.6%
4.6 eV
7.0 eV
0.290
1.21E-2
1.31E-2
1.40E-2
Intro to Emission 42
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)

x
hk
 path to surface &
z   
; l k  
scattering length
cos  
m
1  m 
f  G 
2  hko 
 To leading order, k integral can be ignored
f 
k 


ko
 x z( ) 
f (k)dk  d  exp   
 dx
0
0

l
k
  




 x
f
(k)dk
d

exp
 ko
0 0     dx
 /2

1
2m E(k)  h 
h
ko: minimum k of e- that can escape after photo-absorption
: penetration of laser (wavelength dependent); : relaxation time
k
z()
Average
probability of
escape
argument < 1
 1  sin(y) 
cot y ln 
 cos(y) 

argument > 1
2 y 
  sin(y) 
G cos(y)  1 
G sec(y)  1 
Ex:
• 
• 
• 
• F
Copper:
= 12.6 nm
= 16.82 fs
= 7.0 eV
= 4.6 eV
2
cos(y)
f
= 0.371
= 0.290
Intro to Emission 43
OUTLINE
The Basics
 Nearly-Free Electron Gas Model
 Barrier Models
 Quantum Mechanics & Phase Space
1-Dimensional Emission Analysis
 Thermionic Emission
 Field Emission
 Photoemission
Multidimensional Emission from Surfaces & Structures
 Field Emitter Arrays
 Dispenser Cathodes
 Photocathodes
Cathode Technology and Applications




Vacuum Electronics, Space-Based Applications, Displays
Operational Considerations
Performance Regimes
Operational Complications
Intro to Emission 44
FIELD EMITTERS
anode
Vacuum
Ftip
Metal
gate
Field Enhancement provided by sharpened
metal or semiconductor structure
Close proximity gate provides extraction
field - large field enhancement possible
with small (50 - 200 V) gate voltage; gate
dimensions generally sub-micron.
base
Anode field collects electrons, but
generally does not measurably contribute
to the extraction field
Intro to Emission 45
COLD CATHODES
FEA
WBG
Comparable to Single
Tips
Operated @ 100 µA
Band Gap
Vacuum
Metal
Injection
Field Emitter Arrays: Materials such as
Molybdenum, Silicon, etc
WBG
Transport
Vacuum
Emission
Wide Bandgap Materials such as
Diamond, GaN, etc
Photos Courtesy of Capp Spindt (SRI)
Intro to Emission 46
REVIEW OF ORTHOGONAL COORDINATES
To transform from the (x,y,z) coordinate
system to the (a,,g) system, introduce
the “metrics” h defined by:
     
2
2
ha2  a x  a y  a z
2
and same for a replaced with  & g. In
terms of the metrics the Gradient and
Laplacian become
r
  aφha1a  φh1   gφhg1g
1
 
ha h hg
2
  h hg 

a   c.p.o.i.
a 

  ha

Why the trouble? The new coordinate
system may allow partial differential eq.
specifying potential to be separated into
ordinary differential equations.
c.p.o.i.: “cyclic permutation of indicies”
Spherical Coordinates
(spheres)
x  r sin( )cos( )
hr  1
y  r sin( )sin( )
h  r
z  r cos( )
h  r sin( )
r

1 
1

  rφ  φ
 φ
r
r 
r sin( ) 
2 
 


1
1
1
 r 2r  2
 sin( )  2 2
2
2 r
r
r sin( )
r sin ( )
Prolate Spheroidal Coordinates
(needles)
x  asinh(a )sin( )cos(g )
y  asinh(a )sin( )sin(g )
z  acosh(a )cos( )
ha  a cosh 2 a  sin 2 
h  a cosh 2 a  sin 2 
ha  acosh(a )sin( )
Intro to Emission 47
SIMPLE MODEL OF FIELD ENHANCEMENT
Bump On Surface &
Distant Anode
The All-Important Boundary Conditions:
 At the bump
V (a, )  0
 At the anode
V (a  D,0)  Va
D+a » a
r

It is an elementary problem in electrostatics to show that the potential
everywhere is given by:
a
  a3
V (r, )   Fo r cos( ) 1    
  r  
The Field on the bump (boss) is the
gradient with respect to r evaluated at
r = a of the potential
 

F a,  rV (r, ) r  a  3Fo cos 
Va
Fo 

Da



3
2
D  a  a3
Va 
Va 
a
1
D 
D 
Beta Factor Relation
Ftip  3Fo   gVa
g 
3
a
1
D 
D 
Intro to Emission 48
ANOTHER SIMPLE MODEL
Floating Sphere / Close Anode

D+a ≈ O(10a)
Same BC:
r
V (a, )  0

V (a  D,0)  Va

r'
Va
a
Define Fo to ensure sphere
potential is at zero
Fo 

(2D  a)Va
2aD
Beta Factor Relation
1
a 
 g   1
a  2D 
Potential and Field
 Fo a 2   Fo a 2 
V (r, )  Va  
 

r

  r (r, ) 

Va  4D 2  2a(2D  a) 2a 2 (D  a)(D  2a)
F a,   

1

cos


a  2D(2D  a)
D 2 (2D  a)3


Va 
a
a2
  1 

1

cos(

)

a  2D 4D 3


 

Big
 

Small
Intro to Emission 49
ELLIPSOIDAL MODEL OF NEEDLE / WIRE
Potential and Field Variation Along Emitter Surface Can be Obtained
from Prolate Spheroidal Coordinate System
tip radius as
z  L cosh(a )cos( )
  L sinh(a )sin( )
sinh 2 (a o )
as  L
cosh(a o )
Gradient to Evaluate F(a,)
v
a  

1
a x 2  a y 2  a z 2


a
F(ao,)
1/2 
1
sin 2 ( )  sinh 2 (a ) 
L
a
a
L
Potential in Ellipsoidal Coordinates
Fo
Qn(x) = Legendre Polynomial of 2nd Kind
 Q cosh(a ) cosh(a ) 
V a ,    Vo  Fo z  1


Q
cosh(
a
)
cosh(a o ) 


o
 1
ao
Intro to Emission 50
MODEL OF NEEDLE / WIRE, cont
Apex radius = 1 m
1
Legendre Polynomials of the Second Kind
n
n
n
n
n
n
n
L = 2 x 5 [m]

  1 
ln
1
2     1  
Field [eV/Ѓ]
Q1   
Field Along Surface of Emitter
F a o ,   
sinh a o cos  
sinh a o  sin  
2
2
0.1
1
2
4
6
8
10
Ftip
Fo
Ftip  
sinh 2 a o Q1 cosh(a o )
0.01
0
20
Asymptotic Limits: Let R be ratio of
major to minor ellipsoidal axis. The
height of the Ellipsoid is R2as
 2R 2  3
F0

Ftip   2 ln 2R   1

3Fo

=
=
=
=
=
=
60
 [degrees]
80
100
Emission Area
    J F d
barea Ftip 
1
J Ftip

 Ftip 
 2 a  0

 bFN  Ftip 
R   
2
s
R  1
R  coth a o 
40
0
FN
b

2mF3
v yo  3t yo ;
3h
yo 
4QFtip
F
Intro to Emission 51
APEX FIELD: SATURN MODEL
Simplest Analytical Model of a Triode Geometry
 
F as ,  3Fa cos( )  Fg  l
Where:
as = Apex Radius
ag = Gate Radius
a = Cone Angle
t = ar – ag
Fg = Qg/(rgas)
l
a 
2l  1  s  Pl cos(a ) Pl cos( )
 rg 



 

ar ag as
a

FIELD AT APEX
Ftip  Vgate
Fg r cos( )
r

 ag 
as ln  8 
 t 
tip
gate
anode
Intro to Emission 52
HYP./ELLIP. FIELD AND AREA FACTORS
0.8
General Formulae for Prolate
Spheroidal Geometries
F(a , o ) ellip  Ftip
sin 2 ( o )  sinh 2 (a )
cosh(a o )sin(  )

V

g
2
Ftip  
 tan (  o )
 ln kag / as
 as
k

ag 
1
86


 cot(  o )
54 
as 
 Ftip cos 2 (  o ) 
b area(Ftip )  2as2  o

2
b

F
sin
(

)
 fn
tip
o 
=16Þ
=24Þ
0.6
0.4
sin 2 (  )  sinh 2 (a o )
Gated Hyperbolic (FEA) Case
(hybrid theory - tip specified by o)

za,
F(a , o ) hyp  Ftip
sin( o )
=12Þ
Hyperboloids
a=0.5
Ellipsoids
0.2
-0.2
a=0.2
a=0.1
0
a,
0.2
Ellipsoidal (Needle) Case
(tip specified by ao)
Ftip 
Fo

sinh 2 (a o )Q1 cosh(a o )

 Ftip 
b area(Ftip )  2as2  o

b

F
 fn
tip 
Intro to Emission 53
THERMIONIC CATHODES
0K
Standard Barium
Dispenser Cathode
1200 K
Barium diffused over surface
Vacuum
Metal
Cathode matrix
Tungsten Particles
Impregnated Pores
6 µm
Heater
Intro to Emission 54
WORK FUNCTION AND COVERAGE
Here’s the problem:
If they’re measuring the same
thing, why don’t they get the
same result? They should.
Look at the theory more closely.
To the right: Cesium on Tungsten
Work Function [eV]
But what is really measured is
changes in work function with
changes in time, or deposition
depth or something else - the
“coverage” is inferred.
4
3
2
1
0
5
10
Time [min]
15
20
J. B. Taylor, I. Langmuir, Phys. Rev. 44, 423 (1933).
5
Work Function [eV]
Reported data claims to measure
changes in work function with
changes in degree of surface
coverage…
C-S Wang, J. Appl. Phys. 44, 1477 (1977)
5
4
3
2
1
0
0.2
0.4
0.6
Coverage [%]
0.8
Intro to Emission 55
GYFTOPOULOS-LEVINE THEORY
Coverings (e.g., Ba, Cs) on bulk (e.g., W) induces a change in
Work Function F by presence of dipoles and differences in
electronegativity
GL Theory* predicts F due to partial monolayer using
hard-sphere model of atoms (covalent radii)
Definition of terms


Work function (monolayer & bulk) f ,f
f
m
Covalent radii (monolayer & bulk) r ,r
C

Fractional coverage factor

Electronegativity Barrier

Dipole Moment of Adsorbed Atom
W


d  
W 
* E. P. Gyftopoulos, J. D. Levine, J. Appl. Phys. 33, 67 (1962)
J. D. Levine, E. P. Gyftopoulos, Surf. Sci 1, 171 (1964); ibid, p225; ibid p349

 
F  W   d 
Intro to Emission 56
ELECTRONEGATIVITY BARRIER


 
W   f f  fm  f f H 

H    n0 Cn n
H()  simplest polynomial satisfing:
3
 W0 = ff : the work function is equal to
electronegativity ff of bulk
C0  1
 ∂ W0 =0: …and the addition of a few
atoms doesn’t change that.
C1  0
 W1 = fm: the work function is equal to
electronegativity fm of adsorbate
 ∂ W1 =0 …and the subtraction of a few
atoms doesn’t change that.
 

H   1  2 1  
C2  C3  Π1
2C2  3C3  0

2
Intro to Emission 57
DIPOLE TERM
Pauling (paraphrased):
“Dipole moment of molecule A-B proportional
to difference in electronegativities (fA – fB)”
Assume true for site composed of 4 substrate
(hard sphere) atoms in rectangular array with
absorbed atom at apex.
Dipole moment per atom = M()





M   W   W 1  M   Mo H 
Top

cos   1
1
2g m R2
gm is number of substrate
atoms per unit area
 
M o  4 oro2 cos  fW  f B

ro  4.3653 Angstroms
Perspective
 R
Intro to Emission 58
DEPOLARIZATION EFFECT
Correction for “depolarizing effect” due to
other adsorbed atoms (other dipoles) turns
M into Me (“effective” dipole moment”)
Depolarizing field E

  
9
E  
g   M  

4
M   Me   M   E 
3/ 2
f
0
Dipole moment of adsorbed atom:

g f 
d  


3/ 2
9a
o 

1
g f
4 0

M 
 
e

M 

Me  
1
 
9a
g f
4 0
3/ 2
gf is number of adsorbate
atoms per unit area
Polarizability (a)

n = 1.00 for alkali metals,
1.65 for alkaline-earth

rC = covalent radius of adsorbate

rw = covalent radius of bulk
a  4 o nrC3
Intro to Emission 59
COVERAGE DEPENDENT WORK FUNCTION
g m [110]  2g m [100]
Express F in Terms of coverage ,
Covalent Radii rx, Dimensionless Factors
“f” and “w” (Act As “Atoms Per Cell”, Values of
which Depend on Crystal Face). G&L Argue That
General Surface Is “Bumpy [B]”
 alkali metal (n = 1)
 alkaline-earth metal (n = 1.65)
gm 
w
; gf 
2
 
2rC
f
 
2rW
2
g m [B]  3g m [100]
g f : g m  1 : 4 for Cs on W, Mo, Ta
g f : g m  1 : 2 for Ba, Sr, Th on W, etc.


Modified Gyftopolous-Levine Theory
 

 
F   f f  f f  f m  2 3  2 1  G 
2



r
2 W 
1    
w  R  

G  
3

 rC   
9n
f
1  n   1 
8

 R   

 ro 
r 
 C
2
 
3/ 2


f
W
C
 R
Hard Sphere Model of Surface Dipole
Intro to Emission 60
GYFTOPOLOUS-LEVINE MODEL PERFORMANCE
5
Phi (Wang)
Phi (Haas)
(Scale = 0.0889)
(Scale = 0.6858)
Work Function [eV]
Work Function [eV]
5
Phi (Taylor)
4
(Scale = 0.8698)
Theory
3
2
Phi (Longo)
4
(Scale = 1.000)
Theory
3
2
Ba Dispenser Cathode
Cs on Tungsten
1
0
0.2
0.4
0.6
Coverage 
0.8
LEAST SQUARES ANALYSIS:
Minimize Least Squares Difference
between Gyftopolous-Levine theory and
Exp. Data With Regard to:
 -experimental axis scale factor
 Monolayer work function value
 f coverage factor
1
1
0
0.2
0.4
0.6
Coverage 
0.8
1
• 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)
Intro to Emission 61
PHOTOCATHODES
Example:
Cs-dispenser Photocathode
K3Sb Layer
W plug w/ Cs
Al2O3 Potting
Band
Gap
Semiconductor
Vacuum
Metal
Incident Photon
Electron Excitation
Heater
Electron Emission
J e  kA/cm 2 
q
Je 
I QE  QE  1.2398
h 
I   MW/cm 2  [nm]
Intro to Emission 62
LASER HEATING & PHOTO-EMISSION
Laser Energy Transferred to Material
f(E)
 Photon Energy  Electron Excitations
 Hot Electrons thermalize with other eVia Electron-Electron Scattering.
1-D Supply Function
 Thermal FD e- Distribution thermalizes
With Lattice Via Electron-Phonon Scattering
 Long Laser Pulses: Photons
Encounter “Hot” Electron Distribution
Photoemission is Enhanced
“Ultrashort Laser-induced Electron Photoemission: a Method to
N A Papadogiannis, S D Moustaizis, J. Phys. D: Appl. Phys. 34, 499 (2001):

“cold”
Characterize
h
“hot”
Metallic
Vmax
E
Photocathodes”

“The duration of the laser pulse (450 fs) is relatively long compared to the electron–electron scattering time
for typical electron temperatures…”

“Thus, the electrons thermalize rapidly acquiring a Fermi–Dirac distribution and the refereed electron– electron
and electron–phonon scattering times concern the thermalized electrons.

“...a hot electron gas (a few thousand kelvin) requires about 0.5–2 ps (depending on the experimental conditions) to
relax again to its equilibrium state.
Intro to Emission 63
RELAXATION TIME & Electrical Conductivity
r
r
r F 
f k (t)  f  k  t 
h 

 
Subject (Fermi-Dirac) distribution fo(kx,ky,kz) =
f(k) to a linear potential (constant Electric field)
Electrons don’t continue accelerating in bulk they hit something after “relaxation” time and
start over. If field is turned off, system
exponentially relaxes to equilibrium. Combine
these equations:
r
r
 hk  r
2q
3
J F (k) 
d k  f k
3 
 m
2

 
 
3
 






t / 
r
r
r
f k  fo k
1 r r
 Fg k f o k
r
h
 k
r

r
f
E(
k
)



E
o
fo k  


k x
E
 k x  
r  
r
 hk x   F

d
k

k
f
k

o
 m h
3 



 k x
2
2q
        e
r
r
r
r
f k ,t  fo k  f F k  fo k
   h k   E   
2

x


 m 
 h2 k F

 hk x 
1  hk 

;

E




k

k
 m
F 
 m
3  m 




Electrical conductivity is defined as the ratio
of the Current Density with the Electric Field and it depends on the Relaxation Time
2
2




 q (  )k F3 
 F
JF  
F

s

2
 q 
 3 m 
q 2 (  ) o
s
m
Intro to Emission 64
RELAXATION TIME & Thermal Conductivity
r
If there is a gradient in temperature, that must be
r
r
r

1
hk r
r
accounted for as well: A gradient in temperature
f r , k ,t  Fg k f  g r f
t
h
m
will affect the spatial gradient term:
r
r
r
r
r

hk r
Again, consider time increments
r
r
r 
f r , k  f o r , k  Fg k f o   g  k T ( r )
fo
characteristic of Relaxation time:
h
m
T

 


 
we dealt with this
term previously
Like before, dfo/dT is sharply
peaked around Fermi Energy.
Blue {} is energy of electron gas.
Change of Energy with
Temperature is Specific Heat Cv(T):
can put in  because d/dT of its term is 0
THERMAL CONDUCTIVITY

This is the new
term to worry about
2
r  hk x  
r
r
2q  T  3
J T (k) 
3
  d k k  m  T f o k
x

2

 

 T    2
m
  ()  

3
 x  T  2


r 
d k  E(k)    f o k 
3 

 
3

2
K(T ) 
  CV T
3m
 
2
WIEDEMANN-FRANZ LAW
K  2  kB 
8  W-½ 


2.443

10

2
sT
3  q 
 Kelvin 
Intro to Emission 65
LASER HEATING OF ELECTRON GAS
Differential Eqs. Relating Electron (Te) to Lattice Temperature (Ti)


 
Te    (Te ,Ti ) Te   g Te  Ti  G z, t 
t
z 
z 
Laser Energy

Absorbed
by electrons
Ci Ti  g Te  Ti  Power transfer
to lattice
t
285.1 GW / K cm (W @ RT)
Ce
Electron & Lattice
Specific Heat
3
Thermal Conductivity
 Te ,Ti  
Ao kB2 electron-electron
Aee 
scattering
h
2o kB electron-lattice
Bep 
scattering
h
Relaxation Time
2
 Te ,Ti Ce Te 
3m
 Te ,Ti    AeeTe2  BepTi 
1
Ao and o = dimensionless parameters dictated by photo-cathode material
Deposited Laser Energy
Variation in Energy Density
with Temperature
 e z/   U   h  f  
G(z, t)  1  R  I  (t) 
1 

U  
   

Reflection
Penetration
Incident Laser Power [W/cm2]
Absorbed Energy
Ce Te  
g Te

7    
1 
    
40
e


2
Electrons
C T  
; Ci Ti  

E
T
3NkB
2

1  TD  
1 
 T  
20
i


Phonons
TD = Debye Temp
Intro to Emission 66
DIFFUSION
That last slide was painful. What did it mean?
This time, ignore the transfer of energy to the
lattice, and assume that the effect of the pulsed
laser is to add little Dirac-Delta pulses to the
sample. Now what is happening?
The pulses spread out, much like a wave
packet does in QM (the equations after all are
similar), but the addition of many pulses heats
things up.
One Pulse
Ce (T )tT   (T ,T )2xT  tT  D2xT
2
cm 2
D
 (T ,T )  138.1
3m
s
copper
@ RT
 x2 
T  To 
exp  

 4Dt 
4 Dot
C
C is related to temperature rise due to one pulse. Any
temperature profile can be thought of as the
summation of a bunch of “Dirac-Delta” pulses, and its
future profile therefore determined.
Effect of N pulses separated by t
T
x
t
t 
t 
 

Te x,t   To  2T SN  a  ,   SN  a  ,  

t 
t  
 
N
 a 
1/2
SN a , s    n  0 n  s  exp  
 n  s 
L2   x  
a 
1    1
4Do t   L  
2
L = Width of
cathode
Intro to Emission 67
INCREMENTAL TEMPERATURE RISE
What sort of temperature rise
numbers are we talking about?
Back of the envelope calculation:
if we have a slab of metal (say
copper) of a thickness equal to
the laser penetration depth, and
it absorbs one laser pulse, the
energy of which is uniformly
distributed over the slab, then
what is the temperature rise?
COPPER
Laser Penetration Depth (l)
Thermal Mass factor ()
Fermi Momentum
Incident Laser Intensity
N = Density / Atomic Weight
10 nm
1.375
1.355 1/Å
1 MW/cm2
0.141 moles/cm3
E
T 
g To  3NkB l
E  I o t
mk B2
g   2 kF
3h
t [ns]
E [mJ/cm2]
T [K]
0.001
0.001
0.282
0.05
0.05
14.1
1
This can’t
1 be right… 282
6
6
1692
Temperatures at higher t neglect diffusion into
bulk, which can be substantial - therefore, l should
become larger as t becomes larger
Intro to Emission 68
EXAMPLE: 400 nm on BaO-W Dispenser
Surface Temperature [K]
50.00
50.0
0.0491
3.100
0.0261
0.0218
0.0224
64.74
0.6075
0.0072
40
1150
30
Current
(normalized
to Laser)
1140
1130
20
10
1120
0
1200
60
800
40
400
20
2
Photocurrent [A/cm ]
1200
50
1000
T(0) [K]
Heating
Cooling
Tmax [K]
800
600
400
2
Thermal Current [A/cm ]
Temperature At Surface [K]
Field [MV/m]
Io [MW/cm2]
Area [cm2]
h*f [eV]
dE [mJ]
Scat Fac Max
Scat Fac BC
<theta> [%]
dQ [nC]
QE [%]
1160
Laser Intensity [MW/cm2]
Simulation of laser heating of surface and
subsequent emission
200
10-9
10-7
10-5
0.001
time [sec]
0.1
0
-15
-10
-5
0
5
time [ps]
10
0
15
Intro to Emission 69
SPECIFICATION OF SCATTERING TERMS
Data from CRC Handbook of Chemistry and Physics
(3rd Electronic Edition): Section 12
“SUM OF PARTIAL RESISTIVITIES”:
Total resistance to current flow is sum
of each kind of resistance;
resistance is inversely related to
scattering rate: (Matthiessen’s Law)
 1   ee1   ep1
1
h  1 
Te 

2
Ao  k BTe 
AeeTe
 ee 

log 10{K [W/m K]}
Heat Transfer in Solids Due to
Free Electrons & Phonons
3.2
2.8
2.4
2
Tungsten is complicated…
2
1
1
h  1 
Ti 

BepTi
2o  k BTi 
 ph 
Cu [W/m-K]
Cu Theory
Au [W/m-K]
Au Theory
Al [W /m-K]
Al Theory
W [W/m-K]
W Theory
1.5
Bep  Aee T 
HEAT CONDUCTIVITY
(Kinetic Theory of Gases)


 

2.5
3
log 10{Temperature [K]}

2
 Te ,Ti 
C T  Te ,Ti
3m e e
2
RT

Parameter
Au
7
-2
-1
Aee [10 K s ] 3.553
Bep [1011 K-1 s-1] 1.299
g
3m T,T 
W
57.86
18.41
Cu
4.044
1.859
Al
19.77
6.886
Intro to Emission 70
COUPLING OF LATTICE / ELECTRON TEMPERATURE
Transfer Of Electron Energy To Lattice:
For T > TD (400 K For W), Ci = Constant:
g
2
6
mv s2 Bep 
For Gaussian Te(t)
Temperature [K]

g
Ti (t)  Te  Ti   a Te  Ti 
t
Ci
Electron
density
[#/cm3]

440
Te (t)  Tbulk  Te (0)  Tbulk exp t /t 
2
surface
Electrons
Lattice
400
COPPER
360
320

(Laser: 10 ps FWHM;100 W/cm2)
-10
Ti (t)  Tbulk  a  ea (ts) Te (s)  Tbulk ds
t
0
10
20
30
time [ps]
40
50
60

Near Maximum:
T-TBULK
Ti (0)  Te (0)  at  Te (0)  Tbulk 
2
For t ≥ 10/a, Te(t) and Ti(t) and
equivalent to within 1%
Ex: Copper:
Gold:
Tungsten:
ELECTRONS
LATTICE
t ≥ 59.70 ps
t ≥ 209.5 ps
t ≥ 0.95 ps
x [µm]
time [ps]
Simulation using
time-dependent code
Intro to Emission 71
OUTLINE
The Basics
 Nearly-Free Electron Gas Model
 Barrier Models
 Quantum Mechanics & Phase Space
1-Dimensional Emission Analysis
 Thermionic Emission
 Field Emission
 Photoemission
Multidimensional Emission from Surfaces & Structures
 Field Emitter Arrays
 Dispenser Cathodes
 Photocathodes
Cathode Technology and Applications





Operational Considerations
Vacuum Electronics, Space-Based Applications, Displays
From One to Many: complications of array performance & statistics
Photocathodes: Performance and Issues
Unresolved Issues in Modeling and Simulation
Intro to Emission 72
BEAM ON / OFF ISSUES
Beam Blanking (Turn e-Beam Off): Imin ≈ 0.1% of Imax
 Reduction of kV-Voltage Swings Eases Demands on Solid State Power
MOSFET Driver Used to Control Grid
1
Thermionic Emission:
V(t)
I(V)
 Grid Voltage Vg ≈ 1–10 kV
 Min Voltage ≈ 1% Max Voltage
0.6
Field Emission
 Fowler Nordheim Current:
I(V) = A Vg2 Exp(–B/ Vg)
0.8
Amplitude
 Space Charge Limited Current:
I(V) = P Vg3/2
0.4
V(t)
I(V)
0.2
 Grid Voltage Vg ≈ 75V (B ≈ 8 Vg )
 Min Voltage ≈ 60% Max Voltage
0
0
90
180
t
270
360
Intro to Emission 73
PULSE REPETITION FREQUENCY (PRF)
FIELD EMITTER ARRAYS:
 10-ns Rise Time  Modulation @ 0.05 GHz.
 In Klystrode (DARPA/NASA/NRL VME Program),
Modulation @ 10 GHz From Ring Cathodes
 Demonstrated Operation @ 7 GHz of a Density
Modulated FEA-TWT (Whaley/Spindt)
1
Pulse to Pulse
9 µs
0.5
0
-5
Amplitude [a.u.]
RADAR SYSTEMS UNDER DEVELOPMENT
USING THERMIONIC EMITTERS:
 Required: PRFs of 100 kHz (100 ns rise time)
Desired: PRFs of 1 MHz (10 ns rise time)
 Present Gridded Thermionic Sources:
Pulse Rise Time Too Long: Larger Rise Times
Shorten Pulse-to-Pulse Time, Decreases
“Listen” Time Available for Return Signal
(Pulse-to-Pulse Separation); Emission Noise
Degrades Listening Window for Similar
Reasons.
Amplitude [a.u.]
100 kHz PRF Waveform
0
5
10
15
1
Pulse Length
1 µs
0.5
Rise Time
0.1 µs
0
0
0.5
1
Time [µs]
1.5
Intro to Emission 74
TRANSIT TIME & CUTOFF FREQUENCY
THERMIONIC
2.64 kV/cm
n/a
250 µm
104 ps
1.53 GHz
Quantity
Extraction Field Fo
Tip Field Ftip
Flight Length zg
Transit time t
Cut-off Frequency
FIELD EMITTER
20 kV/cm
0.5 V/Å
0.77 µm
0.096 ps
1667 GHz
Ftip = 0.5 V/Å; Vg = 54.5 V
zg = Vg √(2 / Fo Ftip)
Thermionic
Field Emitter Array
t
zg
zg
Fo
Fo

zg
0
Ftip z
m
dz; V (z) 
V ; s
2V (z)
Ftip z  Vg g
Ftip
2Fo
m
Vg
Analytic
Potential
on Saturn-like Model 
2
 s(sBased
t

1)

ln s  1  s 



Ftip


Intro to Emission 75
RF AMPLIFIER DEMANDS ON CATHODES
VPB CATHODE: Thermionic
 Increase Temperature to Increase Current
Density – Lifetime Decreases
 J ≤ 5 A/cm2:
Beam Convergence of 30-50:1 Required;
Exotic Devices Require >1000:1
 Large Beams & Sophisticated Gun Designs
with Highly Convergent Magnetic Fields
Required.
 Gridded Cathodes ≤ 2 GHz Modulation
 Velocity Modulation of Beam: Most of Circuit
In VPB-TWT Used for Bunching of Beam
prior to Power Extraction
Microwave Power Module
RF INPUT
270 V INPUT
Modulator
HV Power
Supply IPC
MMIC SSA
Vacuum Power
Booster – TWT
High Power
RF Output
Photo courtesy of Northrop Grumman Corporation
Intro to Emission 76
COLD CATHODE ADVANTAGES FOR RF
Vacuum Power Booster: Beam Bunching by Interaction With
RF Field (Velocity Modulation)
VPB
Cold-cathode VPB Beam
Bunching at Cathode (Density
Modulation)
FEA-VPB
VPB: Convergence of
Cathode Current Density
(≈ 1 A/cm2) by 10x Or More
FEA-VPB:
>500
A/cm2
Relaxes
Convergence, Simplifies B Profile,
Relaxes Machining Tolerances, Reduces
Beam Scalloping and Beam Interception
by Circuit (Helix)
Overall:
Power
Decrease in Weight, Volume,
Intro to Emission 77
FEA CATHODE FOR TWT
Photos courtesy of David Whaley (Northrop Grumman)
Operational 55 W TWT
TO5 Header
≈ 13 cm
Line / PPM
Stack
Encapsulated
Electron Gun
94.1 mA From Area of
Diameter ≤ 1 mm
With 50,000 Tips
D. Whaley, et al., IEEE-TOPS28, 727 (2000)
A
1.8 cm
C
B
A - Field Emitter Array
B - Gate Hold-Down Disk
C - Base/Gate Leads
Intro to Emission 78
ELECTRIC PROPULSION (EP)
Microscale Electric Propulsion Systems: Electron Emission
for Propellant Ionization and Ion Beam Neutralization
 Highly Efficient Spacecraft
Attitude Control and Solar
Pressure Drag Make-up for
Micro- up to Large (Inflatable)
Scale Spacecraft
 Highly Efficient / Precise
Spacecraft Repositioning &
Relative Position Maintenance
Space Physics Networks
Saturn Ring Observer
ARISE
ARISE
LISA
Figures Courtesy Of C. Marrese (JPL)
Intro to Emission 79
FEAs IN FLAT PANEL DISPLAYS
Photo courtesy of Alec Talin (Motorola, Tempe AZ)
Photo taken at MRS Spring 2001 Symposium D
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
Motorola 15" Diag HV Field Emission Display
VGA (640x480) Res W/ 8 Bit/color
Emission Current of 2 µA/color Pixel
250 Tips / 1 Color Sub-pixel
Candescent's High Voltage
Field Emission Display DVD
Demonstrated by Chris Curtin,
Candescent Technologies, San Jose, CA
Intro to Emission 80
CURRENT AND CURRENT DENSITY
Easier
Single Tip:
 SRI
101
ED Tether
Thermionic TWT
FEA-TWT (Northrop)
Twystrode (projected)
Klystrode (CPI)
Microtriode (NRL)
Space Applications
 ED Tethers
 Hall Thrusters
 Satellite Discharging
Display
 FEA Display (Motorola)
 CNT
 Diamond
Total Current [Amps]
RF Amplifiers





Harder
10
Display
TWT (NG)
-1
Hall Thruster
Twystrode
Therm. TWT
10
Triode
Klystrode (NRL)
(CPI)
-3
Sat.
Discharg.
10-5
Lithography
CNT
SRI
Test Single Tip
Station
Diamond
10-7
Graphite
10
-9
10-5
10-3
10-1
101
103
Current Density [A/cm2]
Intro to Emission 81
CURRENT PER TIP AND NUMBER
Easier
Single Tips Have Been
Driven Harder Than
Required By Any
Application (SRI)
Space Applications and
Display Per-tip
Performance
Requirements Not Large,
but Large Areas Required
101
Current / tip [A]
FEA Per-tip Performance
In rf Vac. Electronics More
Demanding But Require
Smaller Area
102
100
SRI
Single Tip
Harder
Twystrode
Triode (NRL)
Test Station
TWT (NG)
Klystrode
(CPI)
Lithography
10-1
Display
10-2
Hall Thruster
10-3
Sat.
Discharg.
-4
10
100
102
104
106
Number of Tips
ED Tether
108
1010
Intro to Emission 82
MODULATION AND PRESSURE
Easier
Modulation of Electron
Beam As for
RF Amplifiers
Limits Protection
Schemes That Can Be
Used to Mitigate Arcs
105
FEA Modulation [MHz]
Space-based Field Emitter
Applications Must Survive
in Environments Far More
Challenging Than Other
Applications.
Harder
103
10
Klystrode
Twystrode
Triode (NRL)
Display
1
Lithography
Therm. TWT
10-1
Sat.
Discharg. TWT (NG)
10-3
Test Station
10-5
ED/Hall
Max
ED / Hall
Vac. Sensor
SRI
Single Tip
10-7
10-5 10-4 10-3 10-2 10-1 100 101 102 103
Background Pressure [µTorr]
Intro to Emission 83
FIELD EMITTER ARRAY TIP SHARPNESS
 TEMs of Various Field Emitter
Tips Show Radii of Curvature
on the Order of 30–50 Å
 Surface Can Have Additional
“Structure” Giving Local Field
Enhancement Effects
Silicon
30 Å Radius
TEM
Photograph courtesy of M. Twigg (NRL)
50 Å Radius
Molybdenum
Silicon
Photograph courtesy of
W. D. Palmer (MCNC)
25 Å Radius
Photograph courtesy of M. Hollis (MIT-LL)
Intro to Emission 84
SINGLE EMITTER EMISSION PROFILE
FEEM
FIM
FEEM+FIM
Photographs Courtesy of Capp Spindt and Paul Schwoebel (SRI)
INTENSITY
 FEEM: Regions of HIGHER ß / LOWER F (Range of Values in Each Due To,
e.g., Presence of Adsorbates, Crystal Orientation, Grain Boundaries)
 FIM: Regions of HIGHER ß / HIGHER F.
 FEEM+FIM: F and ß Values Favorable to Both FEEM and FIM: (High ß and
moderate F); Very High ß May Cause Overlap Regardless of F.
EMITTER TIP:
 Images Obtained As Close Together in Time As Possible
 Single Spindt-type Mo Emitter; Emitters of This Class Give 100 µA/tip
Routinely.
Intro to Emission 85
THE FEA STAT / HYPER MODEL
Extraction of FEA Performance From Experimental Data for Spindt-type FEA
I array (Vg )  N tips f (Vg ,) a (Vg , , s )barea (Ftip )J FN (Ftip )
Parameters Adjusted Until
Theory = Exp. AFN and BFN
Red-Primary;
Green-Secondary
ag, T, F, , P
c, as, , s
FIXED PARAMETERS:
ag
Gate Radius
T
Temperature
F
Work Function
Ntips Number of Emitters
, P Work Function Parameter to
Account for Adsorbates, and
Pressure
Rarray Gate And/or Array Resistor
%
Percent Current Intercepted by Gate
Statistics
F
as, c
Characteristic
Area
ag, T, F
as, c
Current Density
ADJUSTED PARAMETERS:
as
Emission Site Radius
Exp & Theory Suggest ≈ 3-7 nm
c
Cone Angle,
Limited by SEM to 12˚ - 23˚ for Moly

Log-Normal Distribution
Mean Emission Site Radius Parameter
s
Log-Normal Distribution
Standard Deviation Parameter
Intro to Emission 86
STATISTICS
Emission Characteristics of Individual
Emitters Change from Site to Site Due to
 Differences In Field Enhancement Factor
as
 Changes in F Due to Adsorbates
Expression for Array Current Product of
# of Tips, a Tip Current Factor and
a Statistical Factor (two parts: a & f
a: Variation in Apex Radii
Log Normal Analysis (, s)
M
I array (Vg )   n j I tip (Vg ;a j )
j 1


 N tips  a Vg ,  , s I tip (Vg ;  ,F)
   
Itip (Vg )  f Vg , barea Ftip J F, Ftip

f: Variation in Work Function
Pressure Analysis (F, )
Intro to Emission 87
F - MODIFICATION BY ADSORBATES
Let P = pressure



 = coverage factor (0 ≤  ≤ 1)
ka = adsorption rate
kd = desorption rate
Recast: kd = ka Po
 

f P  Po  Pea / Po  P

In Equilibrium:
ka P(1   )  kd
Define:
f ( )  I tip ( ) / I tip (0)
 ea  (1   )
Where


 
a  F ln Itip
F(covered)  F(clean)  
For Molybdenum:
SSH/EM Suggests:
a ≈ 5.283 eV-1
Analysis of Exp. Data* Suggests:
Po ≈ 10–8 Torr;  ≈ 0.5 eV
Example:
f(0.1 µTorr) ≈ 0.156
*
Schwoebel, Spindt, et al., JVSTB19, 980 (2001)
Temple, Palmer, et al, JVSTA16, 1980 (1998)
Intro to Emission 88
DISTRIBUTION OF EMITTERS
15
In a Log-normal Distribution, a
Small Fraction of the Emitters Are
Responsible for Most of the Current


  
2
The Current Will Be Dominated by
the Smaller Emitters (%Current Is
Proportional to Integrand):

I array (Vg )  N tips  L(a; , s )Itip (a)da
0
Therefore I(V) Fluctuations Primarily
Depend
on
Fluctuations
Experienced by Sharpest Emitters

% Tips
L(ai ;  , s ) 
 1
exp  2  ln ai / 
2s ai
 2s 
1
% Tips
% Current
Array 1086
µ = 200 Å
s = 0.44
BFN ≈ 700 V
Iexp(90V) ≈ 35.5 µA
10
5
0
30
90
150
210
Tip Radius [
]
270
330
Mo FEA cathode developed at SRI. ZrC (F =
3.6 eV) deposited on FEA at Aptech. Height of
tips is ~1 µm; gate radius = 0.45 µm. Tip-to-tip
spacing = 4 µm. The # of tips = 50,000 in a 0.78
mm2 circular pattern area.
Intro to Emission 89
EXPERIMENTAL DATA (JPL)
Gate V
10
10-5
70
60
-6
Total I 10
Up
Down
50
50
100 150 200 250
Index Number
FN(U) = -12.36 - 415.1/V
FN(D) = -12.82 - 447.1/V
-18
-19
2
80
Current [Å]
Gate Voltage [V]
2
90
ln{I(V)/V [A/cm ]}
-17
-4
-20
-21
Up
Uncoated
Moly FEA
0.012
Down
0.016
0.02
1/(Gate Voltage [V])

Data divided into two regions: increasing voltage (UP) and decreasing (DOWN)

Data from running cathode prior to oxygen exposure to assess performance and
lifetime capabilities of Molybdenum field emitter arrays

Ntips = 50,000 tips, ag = 0.45 µm, tip to tip = 4 µm. FEAs made at SRI
Intro to Emission 90
ENFORCE LOG-NORMAL EQUILIBRIUM

Emitters for Which Ftip > Fcritical
Are Initially Removed, and
 and s Evaluated Afresh

As V Increases, Nanoprotrusion
Formation / Migration Increases
for The Larger Radii
Net Effect: As Voltage Increases:

() Shifts to Higher Values;
(s) Becomes Smaller

Conclusion: the Emitters
Become Both More Uniform and
Less Sharp
2
At Every Vg, The Emitters Which
Contribute To The Array Current
Are LN Distributed (Ntips ≤ Narray)
P = 0.18 µTorr
f ≈ 0.12
Fcrit = 0.67 eV/Å
-18
-19
2

ln{I(V)/V [A/cm ]}
Assumptions:
-20
-21
Experimental
Sputter (Numerical)
No Sputter (Numerical)
0.012
0.016
0.02
1/(Gate Voltage [V])
For Each Voltage Increase,
Tips > Critical Radius Removed,
LN Parameters Recalculated, and
Emitter Distribution Specified Anew
Intro to Emission 91
PHOTOINJECTORS & PHOTOCATHODES
• Bulk & Surface of Complex Materials Produced • Drive-laser Reliability <=> System Reliability:
Critical Components of Free Electron Lasers,
by Empirical Techniques; Short Lifetime,
UV Unsuitable for Hi-duty
Synchrotron
Light
Sources,
& X-rayCrystals
Sources
Complex Replacement Process.
• Non-linear
Decrease  by 2-4;
• Cathode Selection Influences Drive Laser
Efficiency Very Low for UV
Chosen (e.g., wavelength, spot bandwith, laser • Conversion by 2 From IR to Green ok:
energy, QE, etc.)
Seek High QE Photocathode in Visible
rf Klystron
Master
Oscillator
Drive laser
MW-class FEL Demands on Photocathode:
1 nC in 10-50 ps pulse (100 A Peak, 1 A Ave)
10 MV/m, Approx 10-8 Torr
Robust, Prompt, Operate At Longest 
Naval: Longevity & Reliability Paramount
Photocathode
Linac
0
1"
2"
3"
Scale
Intro to Emission 92
PHOTOCATHODES
DRIVE LASER
• Reliability <=> System Reliability: UV Unsuitable for Hi-duty
• Non-linear Crystals Decrease  by 2-4; Efficiency Very Low for UV
• Conversion by 2 From IR to Green ok: Seek High QE Photocathode in Visible
PHOTOCATHODE
 Bulk & Surface of Complex Materials Produced by Empirical Techniques; Short Lifetime,
Complex Replacement Process.
 Cathode Selection Influences Drive Laser Chosen (e.g., , spot bandwith, laser energy, QE)
METALLIC:
 High average power, drive laser w/ 5 - 500 µJ/pulse req.
 Rugged but require UV, have lower QE (≤ 0.01%).
 For low duty factor, low rep rate UV pulses
 Fast response time (fs-structure On Laser Appears on Beam)
DIRECT BAND-GAP P-TYPE SEMICONDUCTORS:
 Highest QE photocathodes
 alkali antimonides (Cs3Sb, K2CsSb); visible, PEA, RF gun
 alkali tellurides (Cs2Te, KCsTe) UV, PEA, RF gun
 Bulk III-V wCs + oxidant (O or F); IR - visible, NEA, DC guns
 Emission time is long (10-20 ps) for NEA sources: insufficiently responsive for pulse shaping.
 ALL chemically reactive: Easily poisoned by H20 & C02 (Protection at expense of QE);
“Harmless" H2 & CH4 damage by ion back bombardment (greater issue for DC guns)
Intro to Emission 93
QUANTUM EFFICIENCY OF VARIOUS CATHODES
J e  kA/cm 2 
q
Je 
I QE  QE  1.2398
h 
I   MW/cm 2  [nm]
General Rule of Thumb for QE
Photocathode
QE[%]
Good
Metallic
(Cu,Mg,Ag,
Niobium, etc)
0.001 0.01
•
•
•
•
•
•
Semiconductor
(Cs2Te, K2Te,
GaN, etc.)
5 - 30
• Photoelectrons have lower energy spread
(in principle) than metallic
• Works in Green
•
•
•
•
NEA (GaAs
family, GaP, etc)
10 - 60
• Widely used in PMT
• Source of polarized electrons (GaAs)
• Slow emission damps laser fluctuations
• Requires UHV: Problems in RF gun
(Exp: failure after 3 macropulses)
• Very long response time
Liquid Metal,
(Nb, NbN, etc.)
???
• Possible use in superconducting RF
photoinjector
• Little known in accelerator situation
• QE unknown - same as metallic?
“Needle” metal
cathodes
0.1 - 1.0
(?)
• High Brightness
• QE greatly enhanced over metal
• Table-top source
• Not extendable to shipboard FEL’s
• Lack of repair in situ (this may have
changed)
Dispenser
Cathodes
0.03 to 6
(dep. on
metal)
• Low work function / high QE
• self-repairing and robust
• mature technology base / off the shelf
• Recent innovation - qualification of
candidates underway
Easy to obtain/handle & Widely-used
Rugged, does not require UHV
QE constant for months
Fast response time O(fs)
Low dark currents
Allows for pulse shaping
Bad
• No systematic study of effective
cleaning & rejuvenating method,
especially in-situ at photoinjector
• Not indicated for high average power
• Beam tracks fluctuations in laser
• UV drive laser required
Requires UHV
Surface deteriorates with O2
response time >> metals
Initial QE has short lifetime
Intro to Emission 94
EMISSION NON-UNIFORMITY
Environmental Conditions Can
 Erode low work function coatings
 Deposit material that degrades performance
 Damage the surface (ion bombardment)
Re-cleaning / Reconditioning does not necessarily
restore original performance


31 Oct 01 – before 1st cleaning
QE
QE scans of LEUTL Photoinjector Mg Cathode
Courtesy of John W. Lewellen, Argonne National Lab
Details: images from APS photoinjector.
Blue = 2xYellow; pixels =10 micron^2; image = (300 pixels)^2
Operation: 6 Hz for 30 days (1.55E7 pulses total); macropulse = 1.5 s
5 Nov 2001 - after 1st cleaning
4 Dec 2001 - after 1st cleaning
10 Dec 2001 - after 2nd cleaning
Intro to Emission 95
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
Emitted Current [a.u.]
1.2
Mathematical Model (n = 2n/T)
1
 = 0.2 ps
0.8
 = 3.2 ps
 = 0.8 ps
 = 12.8 ps
0.6
0.4
0.2
0
0
I  t   I o t  T  t  n 0 cn cos  nt 
N
cn
  t  s  N
I e t  
I
s
exp




2
     
  

 n 0 1   n 
QE
t
5
10
15
20
25
30
time [ps]
 cos  nt   n sin  nt eT /  1 et /



eT /  1 et /



t T
t T
Intro to Emission 96
UMD EXPERIMENT: Cs ON W
GOALS:
• Investigate Basic (Low QE) Binary Systems in Preparation
for Dealing With More Complicated Ternary Systems.
• Prototype a Dispenser Photocathode Whose Low Work
Function Surface Coating Can Be Replenished.
• Validate & Support Predictive Theory Explaining Photoemission Process for Several Cesiated Metals (W, Ag).
EXPERIMENT:
• Evaporate Cs Onto Atomically Clean W (or Au) Surface.
• Find QE vs Cs Coverage (4 mW CW 405nm @ 1E-9 Torr)
• Measure Lifetime, Cesium Desorption Rate, and
Background Composition.
0.05
<Experiment>
Arrhenius
 E
QE  QEo exp  d
 k B
<Experiment>
Theory
0.04
 1 1  
T  T 

o

QE [%]
Cs
QE [%]
0.04
0.03
0.05
0.02
Oct 04
0.03
0.02
0.01
0.01
0
Ed=0.14 eV
W
0
300
Cs on W
407 nm @ 300 K
Presumed Error: ± 0.0035
-0.01
330
360
390
420
Temperature [Kelvin]
0
20
40
60
80
100
Coverage [%]Intro to Emission 97
QE OF Cs ON W: EXP. VS. THEORY
Assumptions and Conditions:
0.05
 Coverage Is Uniform
 Scale = 100%/(5.2 Angstroms)
 Compare averaged
experimental data to
theoretical calculation
 Field and Laser intensity low
enough so that Schottky
barrier lowering, field
enhancement, and heating
are negligible.
<Experiment>
Theory
0.04
QE [%]
 Scale factor between
Coverage (theory) and
Deposition thickness (exp)
taken as Atomic diameter:
Cs on W
407 nm @ 300 K
0.03
0.02
0.01
Feb
Oct 04
05
0
Presumed Error: ± 0.0007
0.0035
-0.01
0
20
40
60
80
100
Coverage [%]
Intro to Emission 98
QE OF Cs ON W, Ag: Predictions / Comparisons
0.1
6
Cs on Ag
Cs on W
5
0.08
QE [%]
QE [%]
QE [%]
0.06
0.04
Field [MV/m]
Lambda [A]
Area [cm2]
h*f [eV]
Io [MW/cm2]
T [Kelvin]
0.02
1.70000
2660.00
0.490874E-01
4.66106
0.100000
640.0
2
Field [MV/m]
Lambda [A]
Area [cm2]
h*f [eV]
Io [MW/cm2]
T [Kelvin]
1
1.70000
2660.00
0.490874E-01
4.66106
0.10000
640.000
Cs on W
0.3
QE [%]
0.02
QE [%]
QE [%]
QE [%]
3
0
0
0.03
0.01
0
QE [%]
4
Field [MV/m]
Lambda [A]
Area [cm2]
h*f [eV]
Io [MW/cm2]
T [Kelvin]
0
20
40
60
1.70000
4070.00
0.490874E-01
3.04629
0.100000
300.0
Coverage [%]
80
100
0.2
Field [MV/m]
Lambda [A]
Area [cm2]
h*f [eV]
Io [MW/cm2]
T [K]
0.1
0
0
20
40
60
1.70000
4070.00
0.490874E-01
3.04629
0.100000
300.0
Coverage [%]
80
100
Intro to Emission 99
SCANDATE DISPENSER CATHODE
Circle
FBlue
= 1.8
eV
Red Circle
F = 1.9 eV
1.7
1.8
1.9
Energy [eV]
2
2.1
Dispenser cathode
• Non-uniform emitting surface
depends upon T & environment
• Small changes in F produce
larger changes in thermal and
photoemission current
Image & Data courtesy of A. Shih, J. Yater (NRL)
Emission Map: Dark Areas =
Ave. Current Density > 10 A/cm2
Intro to Emission 100
PATCH MODEL
Variation can be geometric, adsorbateinduced, and/or coverage dependent:
 Let P = property dependent on surface
(e.g., work function) and macro variables
F and T (e.g., field, temperature)
 Define surface by regions indexed by (i,j)
 Macroscopic = sum over micro patches
1
Pi, j 
2
2
 P F
i, j
yj

xi

,Ti, j ; xi   cos f , y j   sin f df
0
 P F,T ; x, y d   

i, j

i , j
Pi, j di, j
d
1
HYPERBOLIC TANGENT VARIATION MODEL
Parameterize local (micro) variation by assuming
• Cylindrical symmetry
• Two parameters to control transition from
island-like to uniform distribution
Coverage
0.8
0.6
  
0.4
c = 0.5
 = 10
0.2
0
0
0.2
0.4
0.6
0.8
Radial Coordinate
1

   o
1  exp  




1
Intro to Emission 101
UMD EXPERIMENT: Dispenser Photocathodes
0.3
Experiment
Theory
0.25
Charge [nC]
PROGRAM: University of Maryland has 5-year JTO
funded program for R&D in FEL components and
technology. Task A (“Photocathode Development”) is
experimental program to develop & test robust
photocathodes capable of O(ps)-pulses with O(nC)
charge, suitable for high duty factor DC and RF guns.
A dispenser photocathode that can be self-annealed or
repaired, that operates with a visible drive-laser, and at
modestly elevated temperatures, is focus.
0.2
0.15
Q vs Field
T = 386 C
0.1
o
0.05
0
Laser In
E =20.9 mJ
0
0.5
Anode
Cathode
Ion
Pump
Window
1
Charge [nC]
Current Transformer
1
1.5
2
2.5
Field [MV/m]
3
Experiment
Theory
Q vs. Intensity
T = 386 C
o
0.1
F = 1.7 MV/m
0.01
12
16
20
24
2
Intensity [MW/cm ]
28
Intro to Emission 102
QE Measured (UMD), calculated (NRL), & in
literature for various dispenser cathodes
B-TYPE:
B. Leblond, NIMA317, 365 (1992)
UMD experimental data
M-TYPE
UMD experimental data
SCANDATE
UMD experimental data
description of theory and exp. conditions for UMD data is at:
http://fel2004.elettra.trieste.it/pls/fel2004/
Proceedings.html, paper TUPOS65 (proc. of FEL2004 Conf)
QUANTUM EFFICIENCY
QE PREDICTION & EXPERIMENT
Experiment
Theory
10-3
M-type
10-4
B-Type
Scandate
-5
10
250
300
350
400
450
500
550
QE Values for various metals (Au, Cu, Mg)
T. Srinivasan-Rao, et al. J. Appl. Phys. 69, 3291, (1990)
 Theory: All parameters taken from AIP
Handbook, 3rd Edition, CRC Tables, literature
 field enhancement: Mg = 7.0, Cu = 2.5, Au = 1.0
 Possibility of adsorbate contamination ignored
QUANTUM EFFICIENCY
W AVELENGTH [nm]
-3
10
Mg Exp
Cu Exp
Au Exp
Mg Theory
Cu Theory
Au Theory
-4
10
10-5
250
260
270
280
290
300
WAVELENGTH [nm]
Intro to Emission 103
OTHER FACTORS
I  
FACTORS AFFECTING EMISSION CURRENT
 t q  J T , F, h d
total emitted charge


QE 
total incident energy

 t  I R, , h d

d
 Differential surface area illuminated
d  2 d  2  dz 2
prolate spheroidal analysis
 Intensity on differential element
i
d
I surf    1  R   I  
2d 
d
index of ref & penetration
 Variation in illumination intensity
2
I    I o  exp    /   
dictated by experiment
(weak variation for small tips)
 Angular variation of reflection coefficient R:
determination of incidence angle
tan  
dz
tan    

prolate spheroidal analysis
d
tan a o 
 Electron Gas Temperature
T ,    Tbulk
2

 1  t  to  
 laser-material
 CI surf   exp  
  interaction & time
2
t

T



 dependent model
p
Intro to Emission 104
FIELD-ASSISTED PHOTOEMISSION FROM W
Tungsten needle:
 10 mm long with radius of curvature at apex = O(1 m)
 Laser Intensity of order O(100 MW/cm2) over O(10 ns)
and 4th harmonic of Nd:YAG ( = 266 nm)
0
-1
• Cathode to anode
separation ≈ 35 mm
-2
• Max Anode ≈ 33 kV
(Fo = 0.94 MV/m)
-3
• Match between prolate
spheroidal approx. &
actual tip is reasonable
o
z-z [micron]
Other Factors:
Photograph
courtesy of
C. A. Brau
Vanderbilt University
-4
-5
prol. spher.
0.5 m diam.
-6
0
1
2
3
4
rho [micron]
5
• Constraints of side
walls, temperature at
apex, etc. result in best
estimate of
as = 0.53 m
6
Intro to Emission 105
TUNGSTEN NEEDLE CATHODE
Reference Point:
Laser Illuminated W Needle Simulation
And Experimental Data†
 V(ref) = 17.0 kV
 F(ref) = 0.199 GV/m
†C.
Hernandez-Garcia, C. A. Brau
Nucl. Inst. Meth. Phys. Res. A483 (2002) 273–276
Simulation: Macro Q(ref):
2
 Q(266) = 0.528374 %
 Q(355) = 1.74e-03 %
 Current at Peak = 0.112 A
 Intensity = 32 MW/cm2
QE(V)/QE(ref)
Exp: Macro Q(ref) @ 266:
355 nm
 Gaussian Laser spot
50-100 microns (1/e) (depending
on ):
let  = 25 microns (radius)
Macro QE Estimation



200 hc 
0.112Amp
QE[%] 


MW  
q 0.266  m  
2 
  25  m   32.2 cm 2  
 0.0826 %
266 nm
1.6
1.2
Reference Point [17 kV]
0.8
0.4
0
Error Bars: ±20%
0
5
10
15
20
25
30
35
Anode Potential [kV]
355 comparison used same R, scat fac.,penetration
depth, etc. as 266 and is therefore only qualitative
Intro to Emission 106
SURFACE AND SEMICONDUCTOR THEORY
0
-4
V(eV)
ENERGY
Surface:
Interaction of BaO
on surface affects
barrier in manner
dependent on QM
effects
W
Clean W(001)
Potential
-8
Ba/O/W(001)
-12
Ba
-16
O
-20
-24
0
5
10
15
POSITION
z(bohr)


hk 
1
f 
f x, k;t    V x, k  k  f (x, k ;t)dk    f (x, k)  fo (x, k) QDF

t
m x

Empirical Scattering Term for Metals
Inadequate at Low T, Not Adequate for
Semiconductors
Scattering Using Quantum Distribution
Function good for metals, great for
semiconductors, but Scattering
Dependent on Carrier density, T, etc,
and evaluation for arbitrary conditions is
more complicated
Mass
density
 E  
Deformation
potential
Sound
Velocity
 h3vs2
2 mkBTk(E)
momentum
Temperature
Ex: Acoustic Phonon
 process



0
E 3/2 E  f (E)dE


0
E 3/2 f (E)dE
Intro to Emission 107
QUANTUM DISTRIBUTION FUNCTION
Steady State Solution
without Scattering to
Gaussian Potential Barrier
with incident electrons from
both boundaries for Copper
parameters

hk 
0
f x, k;t    V x, k  k  f (x, k ;t)dk 

m x

V (x)  Vo exp  (x  xo ) /  
2

Trajectory Representation
Distribution Function Phase Space
Intro to Emission 108
EMITTANCE & LOCAL CATHODE CONDITIONS
Time-dependent Photo-emission
Model Evaluates Current Density &
QE Via Presumed Equivalence
Between “Patches”
+ Rapid evaluation of <Average>
quantities
– Fails to predict macroscopic
emittance or give basis for
distribution
QE of surface likely to be random
on macroscopic scale
Patch Model
+ More realistic emission
distributions; mimics “hot
spots” and asymmetry
– Far greater numerical
complexity & simulation
Approach:
Propagate emitted surface
distribution away from cathode to
hand-off to PIC
Macro Model
Simulated QE maps of coated surface
Simulated Smoothing from propagation
Intro to Emission 109