Transcript Slides - Cornell University

Models for Cathode Roughness
David H. Dowell1,2
in collaboration with
H. Padmore1, T. Vecchione1and J. F. Schmerge2 & W. Wan1
1Lawrence Berkeley Laboratory
2SLAC National Accelerator Laboratory
Presented at the
Photocathode Physics for Photoinjectors (P3) Workshop
Cornell University
Ithaca, NY
October 9, 2012
Motivation for Model Analysis of Photoemission
create
heuristic
model
Complete Physics Theory and/or Simulation:
Contains all the physics so it’s exact
Complicated and difficult to solve
Simplified Models:
Strips down physics to essentials
Applicable over limited variable regime
Focuses on just a few important phenomena
Simple expressions for QE, emittance, etc.
Straightforward to implement in simulation codes
Experimental Analysis:
Model formulae allow fast, easy data analysis
Testing of model assumptions
Aids experiment design
D. H. Dowell -- P3 Workshop
2
Emittances Near the Cathode
Intrinsic (aka Thermal) Emittance, e intrinsic :
Cathode’s material properties (EF ,fw , EG ,EA , m* ,…)
Cathode temperature, phonon spectrum
Laser photon energy, angle of incidence and polarization
Bunch Space Charge Emittance:
Large scale space charge forces across diameter and length of bunch
Image charge (cathode complex dielectric constant) effects space charge limit
Emittance compensation
Bunch shaping (beer-can, ellipsoid) to give linear sc-forces
Rough Surface Emittance:
Electron and electric field boundary conditions important
Surface angles washout the exit cone
Coherent surface modulations enhances surface plasmons
Three principle emittance effects:
Surface tilt washes out intrinsic transverse momentum > escape angle increases
Applied field near surface has transverse component due to surface tilt
Space charge from charge density modulation due to Ex surface modulation
D. H. Dowell -- P3 Workshop
3
Emittance Due to a Tilted Surface
z
z
px ,out
~
p x ,out
T
x
pout  2mE    EF  feff 
out
x
Vacuum
in
Continuity of transverse momentum at surface:
~
p x ,in  ~
p x ,out
Metal
pin  2mE   
p x ,in
sin  out
p
 in 
sin  in
pout
E  
E    E F  feff
p x ,in
Max angle of incidence:
p x ,out 
p out
2
sin  max 
E    EF  feff
E  
sin  out cos T  cos  out sin T 
Surface
momentum:
D. H. Dowell -- P3 Workshop
~
pout
cos  out sin T
2
4
Normalized Emittance:
p x2,out
p x2,out
en   x
1/ 2
mc

pin2

sin  in cos T  sin 2  max  sin 2  in sin T
2


2
2 p x2,out  pin2 sin 2  in cos 2 T  pin2 sin 2  max  pin2 sin 2  in
2
~
p x2.in  m  feff 
3
 (Intrinsic
2
pout

Emittance)2
EF  feff
m( EF   ) 
1

3
EF  

 sin

E  feff
 3  5 F

EF  

2
T
 4
  m  feff 
 3

Emittance due to intrinsic + tilt:
E F  feff
e intrinsic tilt

E F    
1 
2
2







f
cos
T

sin
T

1

eff

x
2
E F  
3mc 2 


Near Threshold:

e intrinsic tilt

x

  f   cos
3mc
eff
2
2
D. H. Dowell -- P3 Workshop
T  sin 2 T

E  feff
 3  5 F

E F  


5




  f   cos
e intrinsic tilt

x
3mc
eff
2
2
T  sin 2 T
Choose a functional form for the surface roughness:

z ( x)  an cos kn x
The first derivative of the surface height function gives the tilt angle, T:
T
dz
 an k n sin k n x 
dx
This leads to an awkward sine-function of a sine-function:
sin 2 T  sin 2 an k n sin k n x 
Which can be simplified by assuming the surface tilt is small, specifically ankn < 1, therefore
cos 2 T  1 and
sin 2 T  an2 k n2 sin 2 k n x 
1 2 2
an k n
2
Emittance due to intrinsic + cosine-modulated surface:
  feff
e intrinsictilt

x
3mc2
 an2 k n2 
1 

2


D. H. Dowell -- P3 Workshop
feff  fW  fSchottky
6
Emittance Due to an Applied Electric Field
The transverse momentum due to an applied electric field is given by the integral of the
acceleration along the electron’s trajectory, (x(t),z(t)).
[D. J. Bradley et al., J. Phys. D: Appl. Phys., Vol. 10, 1977, pp. 111-125.],
t
p x, field  eE0 an k n  sin k n x(t )e kn z (t ) dt
0
At large distances from the surface, z > a few n , the transverse field vanishes and
transverse momentum gained in the field becomes constant. The variance of this
transverse momentum, px,field , gives the emittance due to the applied field [See D. Xiang
et al., Proceedings of PAC07, pp. 1049-1051],
e field
 2 an2eE0

x
2n mc2
n 
2
kn
This is the emittance only due to the transverse component of the applied field. To
include cross terms between the intrinsic, tilt and applied field, sum all the transverse
momenta and compute the variance of the sum,
p x ,out 
D. H. Dowell -- P3 Workshop
pout
2
sin  out cos T  cos out sin T   p x, field
7
p x,out 
eE0 m
p

sin  in   in sin 2  max  sin 2  in an k n sin k n x0   an k n
sin k n x0
2
k
2
2


n
pin
Squaring and dropping terms proportional to sinknx since its average when computed
over one wavelength is zero gives,
p
2
x ,out
pin2
eE0 m
an k n sin k n x 2

sin 2  in 
2
2k n
 pin2
pin2
eE0 m
2

sin  max 
sin 2  in 
pin sin 2  max  sin 2  in
2
kn
 2

2
 a n k n sin k n x 

By numerical integration it can be shown that the third term inside the square brackets is
quite small with respect the other terms and can be ignored. The variance of the
transverse momentum is then
p
2
x ,out
2

an k n   m
eE0 m 
m


   feff  



f

eff


3
2 3
2k n 
And the total emittance becomes
e intrinsic tilt  field   x
D. H. Dowell -- P3 Workshop
  f  a k     f 


2
eff
3mc 2
n
n
2


eff
3mc 2
eE0 

2k n mc 2 
8
What is the Size of These Emittances?
AFM of LCLS cathode sample
LCLS cathode & gun parameters
an  17nm
n  10microns
~10 microns
~35 nm k n  0.628 / micron
an k n  0.011
  4.86eV
fW  4.8eV
Ea  57.5MV / m
feff ( E0 )  fW  fSchottky
fSchottky  e
e intrinsic tilt  field

x
  f  1  a k 
2
3mc
eff
2


e intrinsic
 0.48microns / mm  rms
x
e tilt
 3.7 10 3 microns / mm  rms
x
e field
 0.13microns / mm  rms
x
D. H. Dowell -- P3 Workshop
n n
2
eE0
 0.29eV
4e0
 eE0
2


a
k

n n
2
 4k n mc
e intrinsic tilt  field
 0.49 microns / mm  rms
x
9
Generally the roughness of a real surface is much more complicated than
a simple cosine-like function. The effect of all spatial frequencies can be
included by Fourier transforming the AFM image and then summing the
emittance at each spatial frequency in quadrature,
e
2
  e intrinsic
 tilt field an , k n 
16
N
e
14
12
n  n0
10
2
intrinsic tilt field
an , kn 
for N  n0  1
8
6
4
0
2
4
6
Spatial wavenumber (1/micron)
FFT of AFM image
Use model to compute
emittance vs. wavenumber
and sum emittances
Emittance ( microns/mm-rms)
0.35
2
0
D. H. Dowell -- P3 Workshop
Inverse sum from
high to low
wavenumber:
n 1
18
Amplitude of modulation
an (nm)
10 microns
N
2
sum
0.3
8
0.25
57.5 MV/m
0.2
0 V/m
0.15
0.1
0.05
0
0
2
4
6
8
Spatial wavenumber (1/micron)
10
10
Charge Density Modulations Produced by Surface Roughness
beam cross section ~6.5 mm from cathode
Electrons are focused and go through a
crossover a few mm from the cathode .
120
100
80
60
8000
40
7000
20
6000
0
0
5000
2
4
6
8
10
6
8
10
6
8
10
of x-y
x Mapping
(microns)
6538
4000
6537.5
3000
6537
2000
y (microns)
longitudinal position, y (microns)
modulation amplitude = 0.02 microns; spatial wavelength = 10 microns; emittance = 0.15007 microns
1000
0
-20
-15
-10
-5
0
5
10
15
20
6536.5
6536
6535.5
transverse position, x (microns)
6534.5
0
2
4
xMapping
(microns)
of x-y
62.7
62.65
62.6
62.55
y (microns)
Due to this crossover, space charge forces
and other effects such as Boersch-scattering
need to be investigated. Since this ‘surface
lens’ is non-linear it can also produce
geometric aberrations and increase the
emittance. Plus the time dependence of the
RF field which changes the focus with time.
D. H. Dowell
P3 Workshop
Rich
area-- of
study! at Cornell
6535
62.5
62.45
62.4
62.35
62.3
62.25
62.2
11
0
2
4
x (microns)
Additional Space Charge Considerations
The space charge emittance near the cathode is driven by:
-Roughness focusing the beam to produce density modulations near the cathode.
-Non-uniform emission due to patchwork of QE variations from different work
functions of grain and crystalline orientations as well as due to local contamination.
PEEM image of Cu cathode
~15 microns
compliments of H. Padmore, ALS-LBNL
D. H. Dowell -- P3 Workshop
12
QE and Charge Density Uniformity:
Estimating Space Charge Emittance Near the Cathode
e n,sc 
x
2f s
I
Io
emittance for 100%
modulation depth
ec
I0 is the characteristic current: I 0   17kA
re
1
Space Charge Emittance (microns/mm)
The emittance due to space charge expansion
of an initial modulation with spatial frequency
fs and total beam current, I , is
The spatial frequency, fs ,(modulations/radius)is
the number of vertical surface modulations or
waves across the radius of emission.
I=100 A
0.1
I=50 A
I=10 A
0.01
0
20
40
60
80
100
Spatial Freq (modulations/radius)
work in progress!
D. H. Dowell -- P3 Workshop
13
Other Roughness Models
M. Krasilnikov, ‘Impact of the cathode roughness
on the emittance of an electron beam,’
Proc. FEL 2006, Berlin, pp. 583-586.
Model developed to compare 2D and
3D roughness effects. Also numerically
computed the field emittance for an
isolated protrusion .
2D Surface
3D Surface
z  h cos kx
z  h cos kx cos ky
The 2D roughness emittance
e x2 D  2 x
eE0
mc2

1
h1 

1  2





3D rough surface emittance is
e x3D  2 x
eE0
h I ( )
mc2

2 
sin 2 X cos 2 Y
dX dY
where   kh and I ( ) 
4 2   1   2 sin 2 X cos 2 Y  cos 2 X sin 2 Y 
Found that for the same roughness parameters :
D. H. Dowell -- P3 Workshop
e x2 D
~ 2
3D
ex
14
Comparison of Roughness Models
Krasilnikov
Field only
e
microns/mm  rms 
x
Roughness only
Cathode field (MV/m)
Krasilnikov model
e
2D
x
 2 x
eE0
mc2

1
h1 

1  2

Current model




e intrinsic tilt  field

x
  f  1  a k 
2
3mc
eff
2


n n
2
 eE0
2


a
k

n
n
2
 4k n mc
The Krasilnikov model because of it’s assumptions it doesn’t give the
roughness/tilted surface emittance, instead it is more like the field emittance.
D. H. Dowell -- P3 Workshop
15
Other Roughness Models
S. Karkare and I. Bazarov, ‘Effect of nano-scale surface roughness on transverse energy
spread from GaAs photocathodes’, arXiv:1102.4764v1 [physics.acc-ph] 23 Feb 2011]
Numerical simulations of electron trajectories from a tilted surface with an applied
field to get the tilt and field emittances vs. photon wavelength. Roughness map from
AFM image of GaAs surface before and after heat cleaning and activation.
Simulations showed the tilt+field emittance overestimates
the data but adding narrow cone emission1(due to m*
effects) and phonon scattering can explain the expt. results.
1
Z. Liu et al., ‘Narrow come emission from negative electron
affinity photocathodes’, J. Vac. Sci. Technol. B 23(6), pp. 2758-2762.
D. H. Dowell -- P3 Workshop
16
Conclusions
e intrinsic tilt  field   x
  f  1  a k 
2
3mc
eff
2


n n
2
 eE0 an k n 2

2
2
 2k n mc
-The electric field term dominates the roughness emittance. Surface roughness most
important at high fields.
-The term given by the product of the excess energy and roughness shows that
decreasing the intrinsic emittance will relax the roughness required to achieve a
desired emittance (Karkare & Bazarov).
-The 2D emittance is approximately 40% higher than the 3D emittance for a surface
with the same roughness parameters for Krasilnikov.
-The surface field enhancement can strongly focus the beam to modulate the charge
density with high spatial frequency. This can generate geometric and space charge
emittance as well as slice emittance. The size of these emittances and the
downstream effects requires further study.
D. H. Dowell -- P3 Workshop
17