Title: ANL THEORY INSTITUTE ON PRODUCTION OF BRIGHT ELECTRON BEAMS September 22-26, 2003, Argonne National Laboratory, Argonne, IL
1ANL THEORY INSTITUTE ON PRODUCTION OF BRIGHT
ELECTRON BEAMSSeptember 22-26, 2003, Argonne
National Laboratory, Argonne, IL
2ISSUES AND QUESTIONS
- QUESTIONS AND COMMENTS BY C. SINCLAIR 1
- Fundamental RD / theory question(s)
- What combination of achievable, external fields
results in the maximum charge density in 6-D
phase space (from a zero thermal emittance
source)? For a CW source, it is not obvious
whether DC or RF fields are best (particularly
for room temperature RF, where the fields are
limited by thermal considerations). For low duty
factor applications, the consensus appears to be
RF, but that must depend on the bunch charge. - How should emittance be measured, and what is
required to have a high quality measurement? - Regarding the relation between thermal emittance
and bunch duration at the cathode space charge
fields are reduced by making larger bunches and
emitting from a smaller area (which increases
longitudinal and transverse emittance) -
therefore, for a given bunch charge, what is the
optimal emitting area and bunch duration to
achieve bets final charge density in 6-D phase
space (the answer will depend on whether fields
are static or dynamic). - Application and requirements dictate
photocathode needs of low repetition rate, high
charge bunches differ from CW pulse trains of
lower bunch charges - A goal for progress in photoemission guns
develop reliable methods for generating uniformly
populated (transversely and longitudinally)
optical pulses to generate uniform charge
distributions from the cathode and result in
minimum emittance.
1 Emails to K. Jensen, and Kwang-Je, et al.,
September 2003
3OUTLINE
- AN INTRODUCTION TO
- ELECTRON EMISSION THEORY AND PROCESSES
- Nature Of The Emission Barrier
- Tunneling, Density, and Current Integral FN and
RLD Equations - Complications Semiconductors, Emission Near
Maximum - A Thermal - Field Emission Formula
- Photoemission Considerations
- Quantum Efficiency
- A Thermal - Photoemission Formula
- Laser Heating of the Electron Gas
- Laser Heating of the Electron Gas
- Time-dependent Model of Laser-induced Thermal
Photoemission - Dispenser Cathode Experiment
- Complicating Circumstances
- Field Enhancement
- Emission at the Barrier Maximum
4EMISSION BARRIER
- EMISSION FROM METALS
- Large Density of Electrons Exist In Conduction
Band (gt 60 Billion / µm3) - Very Small Fraction Contribute to Current (A/cm2
62 per µs per µm2)
The Largest Component of the Barrier Is Due to
the Exchange Correlation Potential
5DISTRIBUTION OF ELECTRONS IN METAL
- Electron Number Density r(m)
Electrons Incident On Surface Or Interface
Barrier Are Distributed In Energy According To A
1-D Thermalized Fermi Dirac Distribution
Characterized By The Chemical Potential and
Called The Supply Function
Zero Temp. (m(0 K) mo EF)
Density Does Not Change With Temperature, So µ
Must
6EXCHANGE-CORRELATION POTENTIAL
- The Density r of an Electron Gas Is
Kinetic Energy
Correlation (Potential) Energy Form due to Wigner
Exchange (Potential) Energy
7ELECTRON DENSITY
- Density and Wave Function
- Square Barrier (Vo mF Width L)
In the Limit of Large Vo and L
The Electron Density Near the Surface Barrier Is
Obtained From the Supply Function and the Wave
Function for Each k
? 1
? kxo
8DIPOLE COMPONENT
- Origin (xi) Of Background Differs From That Of
Electrons (x 0) To Preserve Global Charge
Neutrality
Tanh-Approx. Match r(xi), ?xr(xi)
Magnitude of Dipole (Tanh-model Qualitative,
Overestimates Magnitude)
9SIMPLE MODEL OF TUNNELING
- SQUARE BARRIER OF WIDTH L
- Match y(x) and ?xy(x) where V(x) Changes
Incident eikx
Vo
Transmitted t(k) eikx
Reflected r(k) eikx
kL q(E) Area under Curve kL ? q(E) T(EltVo)
exp2q(E) T(EgtVo) ? 1
10IMAGE CHARGE POTENTIAL
If We Let yk(x) R(x) expi S(x)
Slowly varying density and constant current
Q 0.36 eV-nm y 0.038 eV FMV/m1/2/FeV
11RLD FN SUMMARY part I
- Current Density Velocity X Transmission
Probability X Distribution
Field Emission
Thermionic Emission
12EMISSION EQUATIONS
- FOWLER NORDHEIM FIELD EMISSION
- Dimensionless Parameter n b /b is large
- RICHARDSON EQUATION THERMIONIC EMISSION
- Dimensionless Parameter n b /b is small
13RLD FN SUMMARY part II
- Field Emission
- Large of electrons with low transmission
probability - Relatively T-independent
- High Work Functions For Canonical Metals
- High Current Densities But Generally Small
Emission Sites
- Thermal Emission
- Small of electrons with near-unity transmission
probability - Exponential T-dependence
- Low Work Functions for Coated Materials
- Low Current Densities but Generally Large Areas
14BAND BENDING
- Poissons Equation (mo bulk)
Asymptotic Case bm 2
ZECA f(x) is the same as that which would exist
if no current was emitted.
Asymptotic Case bm 1
15FN AND RLD DOMAINS
- DOMAINS
- RLD Corrupted When Tunneling Near Barrier Max
Is Non-negligible
- FN Corrupted When Barrier Max Is Too Close to
Fermi Level or Slope of ln(T(E)) Exceeds ln(f(E)) - Maximum Field bf gt 6
- Minimum Field cfn lt 2b
Typical Operational Domain of Various
Cathodes Note Photocathodes Typically Run at
or Near 300 K and Surface Roughness Increases F
16ELECTRON DISTRIBUTION
- Electron Momentum Into Barrier Determines
Emission Probability - Finite Temperature (b 1/kBT)
- T(E) (b slope of -lnT(E))
0 K-like Regime
Maxwell Boltzmann Regime
Supply Function f(E) /nm2
Transmission Coeff T(E)
17TANH-WKB APPROXIMATION
- Form of Analytic T(E) Suggests a
Hyperbolic-tangent Approximation
Relationships (prime differentiation) Terms
Evaluated at Em (Max of J Integrand)
FN Linearization of WKB at E m RLD
Emission over Barrier n b/b 1/bkBT
WKB, FN, and RLD Are All Inadequate for n O(1)
Conditions
Estimate Of Em Is Required
18T(E) SYNTHESIS
- Combine Airy and WKB Results
19EVALUATION OF C(E)
- In Bohm (Quantum Potential) Form of Schrödingers
Equation - Where
- Steady State (?xj 0) Match y ?xy at x x
For Incident Plane Waves to Derive the
Transmission Coefficient
20GENERALIZED J(F,T) EQUATION
Nomenclature T(E) To / 1expb(EcE) x
b(Ec E) p b(Ec m) n b/b xp Integrand
Approximated l By Gaussian of Form Ng Ng
exp-l(x-xp)2
- Using the Tanh-form Best Fit of T(E)
- Separate N Into 3 Regions
- Field NA N(p,bmp)
- Intermediate NB N(-bfp,p)
- Thermal NC N(-8,-bfp)
- FN n 8 limit of NA
- RLD n 0 limit of NC
21RESULTS
- Current Integrand Behavior
- RLD Cut-off at Apex E Too Extreme
- FN Over-predicts T(E) at Apex
Variation of n-factor with Field Large Range of n
Obscures Differences 20 Error RLD F 0.22
FN F 1.7
22THERMAL PHOTO-CURRENT (I)
- Product/sum of Several Factors
- Electron Charge (q)
- Absorption Factor (1-R)
- of photons (Il/hw)
- Probability That Photo-Absorbing Electron in
Consequence Has Sufficient Energy to Surmount
Surface Barrier (Ratio of Us) 1 - Standard Thermal Emission Due to Laser Heating
(JRLD) - 1 The precise hypothesis which succeeds so
well in correlating the observed effect near the
threshold is that the photoelectric sensitivity
or number of electrons emitted per quantum of
light absorbed is to a first approximation
proportional to the number of electrons per unit
volume of the metal whose kinetic energy normal
to the surface augmented by hn is sufficient to
overcome the potential step at the surface. - R. H. Fowler, PR38, 45 (1931). (italics in
original)
lo 1064 nm
23QUANTUM EFFICIENCY (1D)
- DQ / DE Jl / Il Therefore, QE Governed by
Ub(hw-f)/U(bm)
ASYMPTOTIC LIMITS OF FOWLER FUNCTION (Richardson
Approximation Implicit in U(x) - must be modified
if tunneling becomes significant)
STANDARD PHOTOEMISSION (Photo-excited Electrons
Below Fermi Level)
THERMAL PHOTOEMISSION (PHOTO-EXCITED ELECTRONS IN
THERMAL TAIL)
1-D ASYMPTOTIC EXPRESSION FOR QUANTUM EFFICIENCY
24OUTLINE
- AN INTRODUCTION TO
- ELECTRON EMISSION THEORY AND PROCESSES
- Nature Of The Emission Barrier
- Tunneling, Density, and Current Integral FN and
RLD Equations - Complications Semiconductors, Emission Near
Maximum - A Thermal - Field Emission Formula
- Photoemission Considerations
- Quantum Efficiency
- A Thermal - Photoemission Formula
- Laser Heating of the Electron Gas
- Laser Heating of the Electron Gas
- Time-dependent Model of Laser-induced Thermal
Photoemission - Dispenser Cathode Experiment
- Complicating Circumstances
- Field Enhancement
- Emission at the Barrier Maximum
25LASER HEATING OF ELECTRON GAS
- Differential Eqs. Relating Electron (Te) to
Lattice Temperature (Ti)
Thermal Conductivity Relax. Time
26SPECIFIC HEAT APPROXIMATIONS
- Specific Heat Variation in Energy Density
With Temperature
High Temp (gt 300 K) Specific Heat Approximations
27TEMPERATURE VS. LASER POWER
- Steady State Formulation Valid When...
- Scattering Times O(0.01-0.1 ps)
- Experimental Laser Pulses O(ns)
Tmax
STEADY STATE ESTIMATE OF T( z 0 )
Assumption Te(z) Profile Is Exponentially
Decaying
28ABSORBED LASER POWER
- Laser Term G(z,t) Is a Product Of
- Reflection Coefficient
- Incident Laser Power
- Penetration Factor
- Absorbed Energy Factor
Fowler Function
29REFLECTION AND PENETRATION (Theory)
- Complex Dielectric Constant ? Optical Properties
Of Metals as function of l - Real index of refraction n
- Imag damping constant k
- Related to Reflectivity R and absorption length
of light in metal d.
r 1.68x1023 /cm3 so 3.6E5 (Ohm-cm)-1
- Plasma Frequencies
- r e- concentration
- so dc conductivity)
30REFLECTION AND PENETRATION (Practice)
- Complex Dielectric Constant ? Optical Properties
Of Metals as function of l - Real index of refraction n
- Imag damping constant k
- Related to Reflectivity R and absorption length
of light in metal d.
- Plasma Frequencies
- r e- concentration
- so dc conductivity)
31TIME EVOLUTION SIMULATION
LENGTH SCALES Cathode 1.0 cm Thermal
Distance 200 nm Laser 10 nm Ratio 100000000
- Diff Eqs. Governing T-Evolution
- Diffusion Equation
- Discretization (jtime, ispace)
- Standard Crank-Nicholson
TIME SCALES Engagement Time 10 s Laser
Pulse-to-Pulse 4 ns Lattice Therm (Ci/g) 8.4
ps Relaxation (RT) 128 fs ?t (600K ?x 20
nm) 0.4 fs Ratio 10000000
Straight-forward Numerical Program Not Feasible
To Model Multiple ns-Separated Laser Pulses
Incident On Cathode - Require Separate Time Scale
Models For Single / Multi pulses
32MACRO-TIME SCALE PRF HEATING (I)
- Bulk T Sum of Gaussian Laser Pulses Io
exp(-(t/dt)2) - Effect Sum of N Dirac-delta T- Pulses at
Surface - Magnitude of Temperature Rise Dictated by
Deposited Laser Energy DE - Temp. Depends on Location of Fixed Boundary L
surface
image
s 1/2 Evolution of surface temperature with
pulse number N s gt 1/2 Cooling of surface after
N pulses tks (ks)Dt
33MACRO-TIME SCALE PRF HEATING (II)
- Characteristic Temperature Tmax is temperature
at surface when T(x) profile is linear (asymptote
as N ? 8 at surface) - Ex 1 GW/cm2 A Pulse of DE 0.71 mJ with dt
40 ps Incident on 1 mm2 Area
65 mg NaCl
34TIME SIMULATION MATRIX EQUATIONS
- Matrix Equation
- Matricies
- Non-linearity
- Finite Difference Multi-point Algorithm Preserves
Stability - Boundary Tbc
- RHS reflect / LHS absorb
- Temp BC Given by Macro-scale Time Simulation and
Held Fixed
Matricies Require Evaluation of Ce(Te) / k(Te,Ti)
at Advanced Time Steps USE OF
PREDICTION/CORRECTION SCHEME Guess j1
elements - solve - use results in next guess
35VALIDATION OF TIME DEP. MODEL
- Numerical Evolution of Dirac-delta Pulse With
Reflecting RHS Boundary Compared to Analytical
Solution - BC RHS reflecting LHS absorbing
- PARAMETERS
- Dx 1 nm
- Dt 0.0368 fs
- Pulse Height 5 Kelvin
- Room Temperature background
- Discrete Coefficient 1000(non-iterative max
40)
Temperature Kelvin
CONCLUSION Numerical Model Functions Well
Compared to Analytical (Exact) Solution
36MICRO-TIME SCALE SINGLE PULSE
- Scandate Dispenser Cathode
- Field Enhancement 3.1
- Chemical Potential 18.08 eV
- Wavelength 1.064 microns
- F (W) 4.7 eV F (Ba) 1.8 eV
- Field 10 MV/m Theta 50
- Peak Intensity 710 Mw/cm2
- Reflectivity 58.5 d 22.7 nm
- Incident Gaussian Laser Pulse Bulk Boundary
Held Fixed At Temperature Dictated By Macropulse
Analysis
l 1064 nm
Gaussian y(t) exp-(t-to)/dt2 Laser
dt 10.00 ps to 0.00 ps
Resulting Temp. Distribution (Numerical Soln of
Time-dependent Heat Equations) ? Long Decay Time
Affects Subsequent Photoemission Long Thermal
Tail
Electron dt 8.36 ps to 0.93 ps
Current Distribution Is Wider
65 mg NaCl
37MICRO-TIME SCALE SINGLE PULSE
- Scandate Dispenser Cathode
- Field Enhancement 3.1
- Chemical Potential 18.08 eV
- Wavelength 0.266 microns
- F (W) 4.7 eV F (Ba) 1.8 eV
- Field 10 MV/m Theta 50
- Peak Intensity 710 Mw/cm2
- Reflectivity 46.2 d 8.65 nm
- Incident Gaussian Laser Pulse Bulk Boundary
Held Fixed At Temperature Dictated By Macropulse
Analysis
(T(t)-To)/(Tmax-To)
Go(t)/Go(0)
J(t)/J(0)
l 266 nm
Gaussian y(t) exp-(t-to)/dt2 Laser
dt 10.00 ps to 0.00 ps
Resulting Temp. Distribution (Numerical Soln of
Time-dependent Heat Equations) ? Long Decay Time
Affects Subsequent Photoemission Long Thermal
Tail
Electron dt 10.01 ps to 0.00 ps
Current Distribution Is Same
65 mg NaCl
38EXPERIMENTAL APPARATUS
- Scandate cathodes fabricated by Spectra-Mat Inc.
- Field between cathode and anode varied from 0-2.5
MV/m - Laser focused to circular spot on cathode with
FWHM area of approximately 0.3 cm2 - Q-switched NdYAG laser gave Gaussian pulses FWHM
4.5 ns
For n2,3,4 harmonics Electron emission
exhibited "normal" photoemission, i.e., emission
proportional to laser intensity independent of
field
39DISPENSER CATHODE
Surface Image
- Consequence Field Enhancement At Local
Emission Sites - (e.g., Hemisphere b 3)
0.1 mm2
Work Function 1.8 eV Partial Coverage of Surface
40CATHODE SURFACE
Interpore 6 µm Grain Size 4.5 µm Pore Diam.
3 µm (cf M. Green, Tech. Dig IEDM, (1987), p925
- the interaction between the barium adsorbate
and the substrate plays a crucial role in
determining the nature of te surface dipole,
which acts to lower the work function ... - 1 A. Shih, D. R. Mueller, L. A.
Hemstreet IEEE-TED36, 194 (1989) - 2 L. A. Hemstreet, S. R. Chubb, W. E. Pickett
PRB40, 3592 (1989)
Bulk vs Monolayer evaporation One Monolayer At
Most Exists On Cathode Surface
41THERMAL - PHOTOCURRENT (II)
- 1D Model Emitted Charge DQ Originates From Two
Sources - Bare Regions With Work Function of W (1-q,Fw)
- Ba-Covered Low Work Function Region (q,F)
- Photocurrent Has Two Components
- Thermal Emission From Laser-heated Electron Gas
- Direct Photoemission
Effect of Field F Schottky Barrier Lowering (Q
0.36 eV-nm)
42EXPERIMENTAL CONSIDERATIONS
- COMPLICATIONS TO THE 1-D MODEL
- LASER INTENSITY VARIATION
- Simulation Area (for FWHM Area 0.3 cm2)
implies ro 0.5249 cm. - MACROSCOPIC FIELD VARIATION
- Cathode 1.27 cm Diameter.Anode Tube With
1.27 cm ID / 2.54 cm ODAnonde-cathode Separation
0.4 Cm. POISSON 1 kV Anode 0.17 Mv/m _at_
center - TEMP VARIATION ACROSS SURFACE
- Electron Temperature Greatest Where Laser
Strongest (Center of Beam) for 1-D Theory,
"Effective" q lt Actual Coverage Factor
FWHM Values Area 0.3 cm2 Pulse 4.5 ns
43COMPARISON THEORY EXPERIMENT
- VARIATION PARAMETERS
- Anode Potential Va kV 0 -15
- Bulk Temperature To K 297-1040
- Deposited Energy mJ 4.87 - 22
- DIMENSIONLESS
- Field enhancement ba 4.0
- Decay Length Parameter n 26.8086
- Surface Coverage q 0.12 - 0.31
- Scattering Coeff. Ao (e-e) 32.0514
- Scattering Coeff. lo (e-ph) 0.0243
- Thermal e mass ratio (W) 1.2038
- DIMENSIONAL
- Chemical Potential (W) eV 18.08
- Work Function eV 1.80
- Penetration depth (W) nm 10.10
- Reflection 50.00
- Laser Wavelength nm 1064.00
- EXPERIMENTAL
- Anode Potential Va
- Bulk Temperature To
- Laser Intensity x pulse length DE
- SIMULATION ADJUSTABLABLES
- Field Enhancement Factor ba
- Thermal Decay Length Parameter n
- Surface Coverage Factor q
- MEASUREMENT / OUTPUT
- EMITTED CHARGE DQ
- Conversion Factors
- Potential/Field 1 kV 0.17 MV/m
- Energy/Intensity 1 mJ 0.2414 MW/cm2
- Current/Charge 1 nC 0.741 A/cm2
44QUANTUM EFFICIENCY WORK FUNCTION
- Prediction of Work Function from Exp. QE
simulation
asymptotic form
Experimental Data for QE for Au W QE
F eV Predict Au 7.54x10-6 4.72 -
4.78 4.69 W 3.49x10-5 4.63 - 5.25 4.52 Vario
us W faces have different F values shown are
for (100) and (111) faces, respectively. N.
A. Papadogiannis, S. D. Moustaïzis, J. of Phs. D
Appl. Phys. 34, 499 (2001).
- Simulation Assumptions
- Low Laser Intensity (Eliminates JRLD)
- No Field (Eliminates Schottky Barrier)
- UV Light (Over the Barrier Emission)
- Dt ratio taken as unity
45EXPERIMENTAL QUANTUM EFFICIENCY
- 2D AND TEMPORAL VARIATION
LIMIT Standard (phonon m gt barrier)
LIMIT Thermal-Photo (phonon m lt barrier)
46THERMAL-PHOTO EMISSION
- For Electrons Excited From Thermal Tail, Current
Density in Asymptotic Limit Appears As a
Shifted Richardson - Laue - Dushman Equation
And Should Therefore Be Linear on a Richardson
Plot As Function of 1/Tmax
47PROJECTED QUANTUM EFFICIENCY
Extension Simulation From Exp. Parameters to
Operational Parameters Intensity Restricted So
That Tmax 300K lt Melting Point of W
- Successive Approximations
- Present Cathode
- Smooth Surface, Increase Coverage
- Lower Wavelength to Ti-Saph (800 nm)
- Increase Field to Naval ApplicationsQE(118
MW/cm2 2.15) - Increase Field to Accelerator ApplicationsQE(118
MW/cm2) 5.52
48OUTLINE
- AN INTRODUCTION TO
- ELECTRON EMISSION THEORY AND PROCESSES
- Nature Of The Emission Barrier
- Tunneling, Density, and Current Integral FN and
RLD Equations - Complications Semiconductors, Emission Near
Maximum - A Thermal - Field Emission Formula
- Photoemission Considerations
- Quantum Efficiency
- A Thermal - Photoemission Formula
- Laser Heating of the Electron Gas
- Laser Heating of the Electron Gas
- Time-dependent Model of Laser-induced Thermal
Photoemission - Dispenser Cathode Experiment
- Complicating Circumstances
- Field Enhancement
- Emission at the Barrier Maximum
49SIMPLE MODEL OF 3-D
- Bump On Surface/ Distant Anode
Floating Sphere / Close Anode
r
a
V(a,q) 0 V(Da,0) Va
50LINE MODEL OF EMISSION TIP
- Emission Tip Close Anode (Diode) Crudely
Modeled As Line Charge Its Image About Anode
Axis With Parameters Adjusted Such That - V 0 ? Emitter Surface
Potential for Line Charge of Length L
Embedded Line Anode (La)
Free Line Charge Anode Image
Ratio(Free/Embed) (DL)/L
Emitter height L ao Apex Radius a (l
charge/length)
51FIELDS APEX AND SURFACE
- Hyperbolic Case (tip specified by bo)
- Ellipsoidal Case (tip specified by ao)
52FIELD ENHANCEMENT
- REGIONS OF CURVATURE ENHANCE FIELD Model Field
Enhancement Effect By - Line Charge of Length L In External Field Fo
- tip radius as, height zo length of line L
(zo - as)zo1/2
FIELD ENHANCEMENT
FIELD VARIATION ALONG SURFACE
53LASER ILLUMINATION OF NEEDLE
- Laser Heating of Electron Gas and Subsequent
Thermal-photo-field Emission Model Based on
Steady State ad hoc Linear Relation Between
Electron Temperature and Illumination - Affects Emittance of Photo-emission From
Protrusions
Hernandez-Garcia, Brau, Nucl. Inst. Meth. Phys.
A483 (2002) 273
K. L. Jensen, P. G. OShea, D. W. Feldman, 5th
Dir. Energy Symp. (Monterey, 11/12/02).
54ISSUES AND QUESTIONS
- QUESTIONS AND COMMENTS BY C. SINCLAIR 1
- Fundamental RD / theory question(s)
- What combination of achievable, external fields
results in the maximum charge density in 6-D
phase space (from a zero thermal emittance
source)? For a CW source, it is not obvious
whether DC or RF fields are best (particularly
for room temperature RF, where the fields are
limited by thermal considerations). For low duty
factor applications, the consensus appears to be
RF, but that must depend on the bunch charge. - How should emittance be measured, and what is
required to have a high quality measurement? - Regarding the relation between thermal emittance
and bunch duration at the cathode space charge
fields are reduced by making larger bunches and
emitting from a smaller area (which increases
longitudinal and transverse emittance) -
therefore, for a given bunch charge, what is the
optimal emitting area and bunch duration to
achieve bets final charge density in 6-D phase
space (the answer will depend on whether fields
are static or dynamic). - Application and requirements dictate
photocathode needs of low repetition rate, high
charge bunches differ from CW pulse trains of
lower bunch charges - A goal for progress in photoemission guns
develop reliable methods for generating uniformly
populated (transversely and longitudinally)
optical pulses to generate uniform charge
distributions from the cathode and result in
minimum emittance.
1 Emails to K. Jensen, and Kwang-Je, et al.,
September 2003