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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 20375-5347 EM: [email protected] Donald W. Feldman, Patrick G. O’Shea

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April 2005 beam physics course
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 20375-5347 EM: [email protected]

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


Scope
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


A note about units
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


Outline
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


Particles in a box
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”

Ei


Statistics
STATISTICS

“Correct

Boltzmann

Counting”

  • 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)

  • The most probable state is found by maximizing W (the sum over wi) with respect to nisubject to the constraints of constant N and E


Energy distribution
ENERGY DISTRIBUTION

…Thermodynamics

Fermi - Dirac Distribution Function

  • Stirling’s Approximation:

  • Find:

    • s = 1 Fermi-Dirac

    • s = 0 Maxwell-Boltzmann

    • s = -1 Bose-Einstein

  • How to figure out a and 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)

    We shall retain b = 1/kBT in future

A slide or two ago…


The quick quantum refresher
THE QUICK QUANTUM REFRESHER…

  • Energy is a constant of the system

  • The wave function at a future time propagates from a past wave function - the time arguments of the propagators are additive

  • Total particle number is conserved, therefore, propagators are unitary

  • CONCLUSION: Schrödinger’s Eq:

  • Basis States combine to form total wave function of system


Energy levels in a box
ENERGY LEVELS IN A BOX

(2,2)

(3,3)

(1,1)

  • So what are the allowable states?

    • Classical: whatever

    • Quantum: don’t touch the sides

Energy


Density of particles
DENSITY OF PARTICLES

  • Transition to Continuum Limit

  • Introduce Fermi Integral F1/2(x)

Effective Density of Conduction-Band States @ RTNc= 0.028316 #/nm3

p = 1/2

Density of States:

# states between E & E+dE

  • Metals: Roughly 1 electron per atom:

  • Sodium @ RT:

    • r = (1 e-/22.99 gram)x(0.9668 gram/cm3)

    • Therefore: m = 3.14 eV

    • (actual: r = 2.65 #/cm3, m = 3.23 eV)

  • Semiconductors: carriers due to doping

  • Doped Silicon @ RT:

    • r = 1018 e-/cm3

    • Therefore: m = -0.354 eV


Chemical potential supply function
CHEMICAL POTENTIAL & SUPPLY FUNCTION

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

  • Electron Number Density r(m)

  • Zero Temperature (m(0 ˚K) = mo = EF)

  • # of particles in box V does not change with temperature, so m must:


From one atom to two
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: re = 0.1 mole/cm3

Two Atoms

One Atom

Bohr Levels


To an array of atoms
…TO AN ARRAY OF ATOMS

Ev

Ec

Metal

Ec

Ev

Insulator

Ec

Ev

Semicon.

  • Electrons in a periodic array merge into regions where energy value is allowed in Schrödinger’s Eq., or not:

    • those permitted by Schrödinger’s Eq. called bands - bands can overlap

    • 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

    • Electrons in conduction band act free (i.e., no potential)

Conduction Band

Ec

Band Gap

Ev

Valence Band


Interaction of electrons
INTERACTION OF ELECTRONS

Kinetic Energy

Exchange Energy

Correlation Energy

ELECTRONS

  • Energy of Electron Gas & Ions

    • Electrons have kinetic energy and interact with themselves (HelN)

    • Electrons interact with the ions (Vel-B)

    • Self-interaction of background (VB)

    • Their evaluation is… fascinating…

  • …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 e-

  • Correlation (“stupidity”) Energy is the sum of a heck of a lot of Feynman diagrams


At the surface of a metal
AT THE SURFACE OF A METAL

F

Vxc

m

Metals

Cu

Au

Na

1019

#/cm3

Metal

Vacuum

  • Density of electrons goes from a region where there are a lot (inside bulk) to where there aren’t that many (vacuum)

  • Exchange-Correlation Potential relates change in density r to change in potential energy

  • Example using Wigner Approx to Corr. Energy:

    • Consider Sodium (Na):Electron density = 0.0438 mole/cm3Chemical Potential = 3.23 eV

Calc: F=2.12 eV

Actual: F=2.3 eV

…so Vxc gets most of barrier, but not all…

…and Na was a “good” metal…


At the surface of a semiconductor
AT THE SURFACE OF A SEMICONDUCTOR

Ec

c

µo

µ

Fvac

Ev

  • Silicon @ RT

  • = 1018 #/cm3

    mo = -0.0861 eV

  • How many electrons screen out a surface field? Surface charge density s = q x bulk density r x width l

    • For metal densities at 1 GV/m: r = 6.022x1022 / cm3 implies l = 0.00184 nm

    • For semiconductor densities at 100 MV/mr = 1018 / cm3 implies l = 11.1 nm

Poisson’s Equation (mo = bulk; m = mo + f)

ZECA: f(x) is the same as that which would exist if no current was emitted.

Asymptotic Case Large Band Bending: bm » 1:

Asymptotic Case Small Band Bending: bm ≤ –2:


Other contributions to surf barrier
OTHER CONTRIBUTIONS TO SURF. BARRIER

Density (Friedel Oscillations)

  • Electrons encounter barrier at surface

    • Wave+Barrier = Quantum contributionsto barrier (Surface Dipole)

    • …and there’s the issue of the ion cores(Approx: neglect what isn’t easily evaluated)

  • Not all electrons pointed at barrier:

    • SUPPLY FUNCTION: Integrate over transverse components of fFD(E)

  • Infinite Barrier:

  • Finite barrier


Image charge approximation
IMAGE CHARGE APPROXIMATION

y = (4FQ)1/2/F

metal

Vacuum

2x

  • The Potential near the surface due to Exchange-Correlation, dipole, etc. can be modeled reasonably well using the “Image Charge Approximation”

  • Classical Argument: Force Between Electron and Its Image Charge

  • Energy to Remove Image Charge


Barrier height
BARRIER HEIGHT

Potential barrier

  • Triangular Potential Barrier

  • Schrödinger’s Equation Solution for High Vo

  • Phase Factor

  • Electron Density Variation Primarily Due to Barrier Height, Less by Barrier Details (to Leading Order) for “Abrupt” Potentials

Electron Density

“Origin” Affected By

Barrier Height


Analytic image charge potential
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…

  • Long answer:

    • Define Effective Work func.

    • Account for ion origin not coincident with electron origin

    • Introduce “ion” length scale


Current a classical approach
CURRENT - A CLASSICAL APPROACH

dx’

dn’

dk’

dk

dn

dx

  • f(x,k,t) is the probability a particle is at position x with momentum hk at time t

  • Conservation of particle number:

to order O(dt)

Boltzmann Transport Equation

velocity & acceleration

“Moments” give number density r and current density J: Continuity Equation


Current a quantum approach
CURRENT - A QUANTUM APPROACH

Heisenberg Uncertainty:

  • Center discussion around states defined by

Relation from Heisenberg Uncertainty

Consider H & the operator for density:

Heisenberg Representation: operators O evolve, eigenstates don’t

Schrödinger Representation:

eigenstates evolve, operators don’t

Then it follows that note: {A,B} = AB+BA


Current in schr dinger representation
CURRENT IN SCHRöDINGER REPRESENTATION

Trivial Case: Plane waves

The form most often used in emission theory

Basis for FN & RLD Equations

  • Consider a pure state

Gaussian wave packet at t = 0:

Form of J(x): velocity x density


The quantum distribution function
THE QUANTUM DISTRIBUTION FUNCTION

Wigner Distribution function (WDF)

  • For the density operator, we considered:

  • Wigner proposed a distribution function defined by

Time evolution follows from continuity equation:

A bit of work shows that:

integrating both sides wrt k reproduces classical equations


Wdf properties
WDF PROPERTIES

  • Taylor Expand V(x,k):

  • 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:

V(x)=0

t = 0.0

t = 1.4

Note: this is special case of the constant field case, i.e., V(x) = g x, case, for which:

The Schrodinger picture expansion of the wave packet becomes, in the WDF framework, a shearing of the ellipse

“trajectories” are same as classical trajectories


Analytical wdf model gaussian v x
ANALYTICAL WDF MODEL: GAUSSIAN V(x)

  • How does V(x,k) behave? Consider a solvable case where V(x) is a Gaussian:

large Dx samples f(x,k') near k

small Dx samples f(x,k') far from k

Broad

Dx2 = 0.1

Sharp

Dx2 = 5.0


Analytical wdf model ii gaussian v x
ANALYTICAL WDF MODEL (II): GAUSSIAN V(x)

  • The behavior of V(x,k) signals the transition from classical to quantum behavior:

    • Sharp: classical distribution

    • Broad: quantum effects

  • Can V(x,k) give a feel for when thermionic or field emission dominates?

    • Consider most energetic electron appreciably present(corresponds to E = m or k = kF)

    • If sin(kFx) does not “wiggle” muchover range Dx, QM important

    • Thermionic Emission:Dx is very large - expectclassical description to be good

    • Field EmissionkFDx = O(2p) implies

Image Charge Potential


Outline1
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


Richardson laue dushman eq
RICHARDSON-LAUE-DUSHMAN EQ.

  • Example:Typical Parameters

    • Work function 2.0 eV

    • Temperature 1300 K

    • Field 10 MV/m

  • 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 >

    • Therefore: if f(k) is to be appreciable, T must be LARGE

Maxwell Boltzmann

Richardson Constant


Thermionic emission data
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]

Work function measurement for Thoriated Tungsten: <http://www.avs.org/PDF/Vossen-Lulai.pdf>


Tunneling theory refresher
TUNNELING THEORY REFRESHER

Vo

eikx

t(k)eikx

E(k)

r(k)e-ikx

0

+L

I

II

III

  • Traditional Field Emission Theory: Extensive Use of Schrödinger’s Equation

  • Consider Simplest Analytically Solvable Tunneling Model: Square Barrier

    • Regions I & III:

    • Region II:

  • Match y and dy/dx at 0 and L

    • At x = 0

    • At x = L

  • TRANSMISSION COEFFICIENT T(k)=|t(k)|2

This is the “area” under the potential maximum but above E(k)


Fowler nordheim equation
FOWLER NORDHEIM EQUATION

Vo

eikx

t(k)eikx

E(k)

r(k)e-ikx

0

Vo/F

I

II

III

  • The Fowler Nordheim Equation was originally derived for a triangular barrier

    • Schrödinger’s Equation

    • Airy Function Equation

  • Same drill as with rectangular barrier… but use Asymptotic Limit of Ai & Bi

  • Current

This is the “area” under the potential maximum but above E(k)

  • Action occurs near E = m

    • Evaluate coefficient at m

    • Linear expansion of exponent about m-E


Wkb transmission probability
WKB TRANSMISSION PROBABILITY

V(x)

E(k)

x–

x+

vanishes for constant current

Neglect for slowly varying density

“Area Under the Curve” Approach to WKB

  • Schrödinger’s Equation

  • Wave Function (Bohm Approach) and Associated Current

  • Schrödinger Recast


Image charge wkb term
IMAGE CHARGE WKB TERM

V(x) = m + F- Fx - Q/x

J(F) ≈ 7x105 A/cm2

µ

L

m = 5.87 eV

F = 4.41 eV

F = 0.5 eV/Å

Q = 3.6 eV-Å

Elliptical Integral functions v(y) & t(y)

  • “Area Under the Curve” Approx:

FN Equation: Linearize q(E) about the chemical potential

  • Example:Typical Parameters

    • Work function 4.4 eV

    • Temperature 300 K

    • Field 5 GV/m

why the odd choice of v(y)? Perfect linearity on FN plot


Fowler nordheim equation1
FOWLER NORDHEIM EQUATION

Field

Field-Thermal

Semiconductor

  • Example:

    • F = 4 GV/m

    • T = 600 K

    • m = 5.6023 eV

    • 20930 Amp/cm2

    • 1.142

    • 2.108 x 10-12

  • 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)

    • b /cfnof Order O(10) for Field Emission


Field emission data
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 / bg

  • Ex: J. P. Barbour, W. W. Dolan, et al., Phys. Rev. 92, 45 (1953).

    • Work function of clean W (4.6 eV) implies bg factor = 4368 cm-1

    • Work function of increasing coatings of Ba on W needle: [2] 3.38 eV [3] 2.93

2.93 eV

3.38 eV

4.60 eV

Modern Spindt-type field emitters: C. A. Spindt, et al, Chapter 4, Vacuum Microelectronics, W. Zhu (ed) (Wiley, 2001)


Thermionic vs field emission
THERMIONIC VS FIELD EMISSION

Richardson

Fowler Nordheim

  • The most widely used forms of:

    • Field Emission: Fowler Nordheim (FN)

    • Thermal Emission: Richardson-Laue-Dushman (RLD)

High Temperature

Low Field

Low Temperature

High Field

Transmission Probability

Electron Supply

Emission Equation

Constants for Work Function in eV, T in Kelvin, F in eV/nm


Fn and rld domain of validity
FN AND RLD DOMAIN OF VALIDITY

FN (F=4.4 eV)

Field Emitter

Photocathode

Thermionic

RLD (F=2 eV)

  • DOMAINS

    • RLD: Corrupted When Tunneling Contribution Is Non-negligible

  • FN: Corrupted When Barrier Maximum near m or cfn close to b

    • Maximum Field: bf > 6

    • Minimum Field: cfn< 2b

Typical Operational Domain of Various Cathodes Compared to Emission Equations


Emission distribution
EMISSION DISTRIBUTION

TFN(E)

Texact(E)

1600 K

Vmax

µ(300K)

600 K

  • Emission Distribution

  • Transmission Coefficient

Twkb(E)

300 K

  • For Typical Field Emission from Metals such as Molybdenum, f(E) dominates T(E) for E Large

  • Near Fermi Level, TFN(E) Is a Good Approximation


Thermal field assisted photocurrent
THERMAL-FIELD ASSISTED PHOTOCURRENT

Field

Thermal

X(F[GV/m],T[K])

  • Supply Function

  • Transmission Coefficient T(E): (b = slope of -ln[T(E)])

  • When b » b: Fowler-Nordheim Eq.

  • When b » b: Richardson-Laue-Dushman Eq.

  • When b ≈ b : No simple analytic form

  • Photocurrent: changes T(E) behavior

Maxwell

Boltzmann

Regime

T(7,300)

T(0.01,2000)

Fermi

f(0.01,2000)

0 K-like Regime

f(7,300)


Quantum efficiency 3 d
QUANTUM EFFICIENCY (3-D)

Photocurrent

1

2

3

4

Richardson Approximation:

Fowler-Dubridge Formula (modified)

  • 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

  • Photocurrent Jl(F,T) depends on

    • Charge to Photon energy ratio (q/hf)

    • Scattering Factor fl

    • Absorbed laser power (1-R) Il

    • Photoexcited e- Escape Probability

      • Richardson: T(E) = Step Function

      • Fowler’s astounding approximation: assume all e- directed at surface.

      • Fowler Function

* “Seek thine own ease.” King Lear, III.IV


Fowler dubridge equation
FOWLER-DUBRIDGE EQUATION

Field significantly exaggerated to show detail

Photon energy: first four harmonics of Nd:YAG

Fowler-Dubridge Formula… sort of

lo = 1064 nm

T(E); f(E) [1016 #/cm2]

“Fowler factor”

Fowler-Dubridge often referred to in this way

  • 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

  • Example: Copper

  • Wavelength 266 nm

  • Field 2.5 MV/m

  • R 33.6%

  • Work function 4.6 eV

  • Chemical potential 7.0 eV

  • Scattering Factor 0.290

  • QE [%] (analytic) 1.21E-2

  • QE [%] (time-sim) 1.31E-2

  • QE [%] (exp) 1.40E-2

For metals


Post absorption scattering factor
POST-ABSORPTION SCATTERING FACTOR

k

z(q)

q

Average probability of escape

argument < 1

argument > 1

  • Ex: Copper:

  • d = 12.6 nm

  • t = 16.82 fs

  • m = 7.0 eV

  • F = 4.6 eV

cos(y) = 0.371

fl = 0.290

  • Factor (fl) governing proportion of electrons emitted after absorbing a photon:

    • Photon absorbed by an electron at depth x

    • Electron Energy augmented by photon, but direction of propagation distributed over sphere

    • Probability of escape depends upon electron path length to surface and probability of collision (assume any collision prevents escape)

      • path to surface &scattering length

    • To leading order, k integral can be ignored

ko: minimum k of e- that can escape after photo-absorption

d: penetration of laser (wavelength dependent); t: relaxation time


Outline2
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


Field emitters
FIELD EMITTERS

anode

Ftip

gate

Vacuum

Metal

base

  • 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.

  • Anode field collects electrons, but generally does not measurably contribute to the extraction field


Cold cathodes
COLD CATHODES

Band Gap

Vacuum

Metal

Injection

WBG

Transport

Vacuum

Emission

Field Emitter Arrays: Materials such as

Molybdenum, Silicon, etc

Wide Bandgap Materials such as Diamond, GaN, etc

FEA

WBG

Comparable to Single Tips

Operated @ 100 µA

Photos Courtesy of Capp Spindt (SRI)


Review of orthogonal coordinates
REVIEW OF ORTHOGONAL COORDINATES

Spherical Coordinates

(spheres)

  • To transform from the (x,y,z) coordinate system to the (a,b,g) system, introduce the “metrics” h defined by:

  • and same for a replaced with b & g. In terms of the metrics the Gradient and Laplacian become

  • Why the trouble? The new coordinate system may allow partial differential eq. specifying potential to be separated into ordinary differential equations.

Prolate Spheroidal Coordinates (needles)

c.p.o.i.: “cyclic permutation of indicies”


Simple model of field enhancement
SIMPLE MODEL OF FIELD ENHANCEMENT

D+a » a

r

q

Va

a

Beta Factor Relation

Bump On Surface &

Distant Anode

  • The All-Important Boundary Conditions:

    • At the bump

    • At the anode

  • It is an elementary problem in electro-statics to show that the potential everywhere is given by:

  • The Field on the bump (boss) is the gradient with respect to r evaluated at r = a of the potential


Another simple model
ANOTHER SIMPLE MODEL

D+a ≈ O(10a)

r'

q

Va

Beta Factor Relation

  • Floating Sphere / Close Anode

    • Same BC:

    • Define Fo to ensure sphere potential is at zero

    • Potential and Field

r

a

Big

Small


Ellipsoidal model of needle wire
ELLIPSOIDAL MODEL OF NEEDLE / WIRE

tip radius as

b

F(ao,b)

a

L

ao

  • Potential and Field Variation Along Emitter Surface Can be Obtained from Prolate Spheroidal Coordinate System

Gradient to Evaluate F(a,b)

Potential in Ellipsoidal Coordinates

Qn(x) = Legendre Polynomial of 2nd Kind

Fo


Model of needle wire cont
MODEL OF NEEDLE / WIRE, cont

Emission Area

Apex radius = 1 mm

  • Legendre Polynomials of the Second Kind

Field Along Surface of Emitter

Asymptotic Limits: Let R be ratio of major to minor ellipsoidal axis. The height of the Ellipsoid is R2as


Apex field saturn model
APEX FIELD: SATURN MODEL

ar ag as

a

q

r

  • Simplest Analytical Model of a Triode Geometry

  • Where:

  • as = Apex Radius

  • ag = Gate Radius

  • a = Cone Angle

  • t = ar – ag

  • Fg = Qg/(rgas)

  • FIELD AT APEX

tipgateanode


Hyp ellip field and area factors
HYP./ELLIP. FIELD AND AREA FACTORS

Gated Hyperbolic (FEA) Case(hybrid theory - tip specified by bo)

Ellipsoidal (Needle) Case(tip specified by ao)

  • General Formulae for Prolate Spheroidal Geometries


Thermionic cathodes
THERMIONIC CATHODES

0 K

1200 K

Barium diffused over surface

Vacuum

Tungsten Particles

Metal

Cathode matrix

Impregnated Pores

6 µm

Heater

  • Standard Barium Dispenser Cathode


Work function and coverage
WORK FUNCTION AND COVERAGE

  • Here’s the problem:

  • Reported data claims to measure changes in work function with changes in degree of surface coverage…

  • But what is really measured is changes in work function with changes in time, or deposition depth or something else - the “coverage” is inferred.

  • 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


Gyftopoulos levine theory
GYFTOPOULOS-LEVINE THEORY

  • Coverings (e.g., Ba, Cs) on bulk (e.g., W) induces a change in Work Function F(q) by presence of dipoles and differences in electronegativity

  • GL Theory* predicts F(q) due to partial monolayer using hard-sphere model of atoms (covalent radii)

  • Definition of terms

    • Work function (monolayer & bulk)

    • Covalent radii (monolayer & bulk)

    • Fractional coverage factor

    • Electronegativity Barrier

    • Dipole Moment of Adsorbed Atom

* 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


Electronegativity barrier
ELECTRONEGATIVITY BARRIER

  • H(q) =simplest polynomial satisfing:

    • W(0) = ff : the work function is equal to electronegativity ff of bulk

    • ∂q W(0) =0: …and the addition of a few atoms doesn’t change that.

    • W(1) = fm: the work function is equal to electronegativity fm of adsorbate

    • ∂q W(1) =0 …and the subtraction of a few atoms doesn’t change that.


Dipole term
DIPOLE TERM

Top

Perspective

R

b

  • 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(q)

gm is number of substrate atoms per unit area


Depolarization effect
DEPOLARIZATION EFFECT

  • Correction for “depolarizing effect” due to other adsorbed atoms (other dipoles) turns M into Me (“effective” dipole moment”)

  • Depolarizing field E(q)

  • Dipole moment of adsorbed atom:

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


Coverage dependent work function
COVERAGE DEPENDENT WORK FUNCTION

Modified Gyftopolous-Levine Theory

W

C

R

b

Hard Sphere Model of Surface Dipole

  • Express F in Terms of coverage q, 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)


Gyftopolous levine model performance
GYFTOPOLOUS-LEVINE MODEL PERFORMANCE

  • LEAST SQUARES ANALYSIS:Minimize Least Squares Difference between Gyftopolous-Levine theory and Exp. Data With Regard to:

    • q-experimental axis scale factor

    • Monolayer work function value

    • f coverage factor

  • C-S Wang, J. Appl. Phys. 44, 1477 (1977)

  • J. B. Taylor, I. Langmuir, Phys. Rev. 44, 423 (1933).

  • R. T. Longo, E. A. Adler, L. R. Falce, Tech. Dig. of Int'l. El. Dev. Meeting 1984, 12.2 (1984).

  • G. A. Haas, A. Shih, C. R. K. Marrian, Applications of Surface Science 16, 139 (1983)


Photocathodes
PHOTOCATHODES

Band

Gap

Vacuum

K3Sb Layer

W plug w/ Cs

Semi-

conductor

Metal

Incident Photon

Al2O3 Potting

Electron Excitation

Heater

Electron Emission

Example:

Cs-dispenser Photocathode


Laser heating photo emission
LASER HEATING & PHOTO-EMISSION

“hot”

f(E)

  • Laser Energy Transferred to Material

    • Photon Energy  Electron Excitations

    • Hot Electrons thermalize with other e-Via Electron-Electron Scattering.

    • Thermal FD e- Distribution thermalizes With Lattice Via Electron-Phonon Scattering

    • Long Laser Pulses: Photons Encounter “Hot” Electron DistributionPhotoemission is Enhanced

  • “Ultrashort Laser-induced Electron Photoemission: a Method to Characterize Metallic Photocathodes”N A Papadogiannis, S D Moustaizis, J. Phys. D: Appl. Phys. 34, 499 (2001):

    • “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.

1-D Supply Function

m

Vmax

hw

E

“cold”


Relaxation time electrical conductivity
RELAXATION TIME & Electrical Conductivity

  • 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:

Electrical conductivity is defined as the ratio of the Current Density with the Electric Field - and it depends on the Relaxation Time


Relaxation time thermal conductivity
RELAXATION TIME & Thermal Conductivity

we dealt with this term previously

This is the new term to worry about

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 m because d/dT of its term is 0

THERMAL CONDUCTIVITY

WIEDEMANN-FRANZ LAW

  • If there is a gradient in temperature, that must be accounted for as well: A gradient in temperature will affect the spatial gradient term:

  • Again, consider time incrementscharacteristic of Relaxation time:


Laser heating of electron gas
LASER HEATING OF ELECTRON GAS

Electron & Lattice

Specific Heat

Laser Energy

Absorbed

Power transfer by electrons to lattice

285.1 GW / K cm3 (W @ RT)

Thermal Conductivity

Relaxation Time

electron-electron

scattering

electron-lattice

scattering

Ao and lo = dimensionless parameters dictated by photo-cathode material

Reflection

Penetration

Absorbed Energy

Incident Laser Power [W/cm2]

Electrons

Phonons

TD = Debye Temp

  • Differential Eqs. Relating Electron (Te) to Lattice Temperature (Ti)

Deposited Laser Energy

Variation in Energy Density with Temperature


Diffusion
DIFFUSION

One Pulse

T

Effect of N pulses separated by Dt

x

t

L = Width of cathode

  • 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.

copper

@ RT

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.


Incremental temperature rise
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?

This can’t be right…

COPPER

Laser Penetration Depth (l) 10 nm

Thermal Mass factor (h) 1.375

Fermi Momentum 1.355 1/Å

Incident Laser Intensity 1 MW/cm2

N = Density / Atomic Weight 0.141 moles/cm3

Temperatures at higher Dt neglect diffusion into bulk, which can be substantial - therefore, l should become larger as Dt becomes larger


Example 400 nm on bao w dispenser
EXAMPLE: 400 nm on BaO-W Dispenser

  • Simulation of laser heating of surface and subsequent emission

  • Field [MV/m] 50.00

  • Io [MW/cm2] 50.0

  • Area [cm2] 0.0491

  • h*f [eV] 3.100

  • dE [mJ] 0.0261

  • Scat Fac Max 0.0218

  • Scat Fac BC 0.0224

  • <theta> [%] 64.74

  • dQ [nC] 0.6075

  • QE [%] 0.0072


Specification of scattering terms
SPECIFICATION OF SCATTERING TERMS

Data from CRC Handbook of Chemistry and Physics (3rd Electronic Edition): Section 12

Tungsten is complicated…

Parameter Au W Cu Al

Aee [107 K-2 s-1] 3.553 57.86 4.044 19.77

Bep [1011 K-1 s-1] 1.299 18.41 1.859 6.886

  • Heat Transfer in Solids Due to Free Electrons & Phonons

  • “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)

  • HEAT CONDUCTIVITY(Kinetic Theory of Gases)


Coupling of lattice electron temperature
COUPLING OF LATTICE / ELECTRON TEMPERATURE

surface

COPPER

Electron

density

[#/cm3]

(Laser: 10 ps FWHM;100 W/cm2)

T-TBULK

ELECTRONS

LATTICE

x [µm]

time [ps]

  • Transfer Of Electron Energy To Lattice: For T > TD (400 K For W), Ci = Constant:

  • For Gaussian Te(t)

  • Near Maximum:

For Dt ≥ 10/a, Te(t) and Ti(t) and equivalent to within 1%

Ex: Copper: Dt ≥ 59.70 ps

Gold: Dt ≥ 209.5 ps

Tungsten: Dt ≥ 0.95 ps

Simulation using time-dependent code


Outline3
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


Beam on off issues
BEAM ON / OFF ISSUES

V(wt)

I(V)

  • Field Emission

    • Fowler Nordheim Current: I(V) = A Vg2 Exp(–B/ Vg)

    • Grid Voltage Vg ≈ 75V (B ≈ 8 Vg )

    • Min Voltage ≈ 60% Max Voltage

V(wt)

I(V)

  • 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

  • Thermionic Emission:

    • Space Charge Limited Current:I(V) = P Vg3/2

    • Grid Voltage Vg ≈ 1–10 kV

    • Min Voltage ≈ 1% Max Voltage


Pulse repetition frequency prf
PULSE REPETITION FREQUENCY (PRF)

100 kHz PRF Waveform

  • 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.

  • 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)


Transit time cutoff frequency
TRANSIT TIME & CUTOFF FREQUENCY

Thermionic

zg

zg

Fo

Fo

  • THERMIONIC Quantity FIELD EMITTER

  • 2.64 kV/cm Extraction Field Fo 20 kV/cm

  • n/a Tip Field Ftip 0.5 V/Å

  • 250 µm Flight Length zg 0.77 µm

  • 104 psTransit time t0.096 ps

  • 1.53 GHzCut-off Frequency1667 GHz

Ftip = 0.5 V/Å; Vg = 54.5 Vzg = Vg √(2 / FoFtip)

Field Emitter Array

Analytic Potential Based on Saturn-like Model


Rf amplifier demands on cathodes
RF AMPLIFIER DEMANDS ON CATHODES

RF INPUT

270 V INPUT

Modulator

HV Power Supply IPC

MMIC SSA

Vacuum Power Booster – TWT

High Power RF Output

Microwave Power Module

  • 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

Photo courtesy of Northrop Grumman Corporation


Cold cathode advantages for rf
COLD CATHODE ADVANTAGES FOR RF

VPB

FEA-VPB

FEA-VPB: >500 A/cm2 Relaxes Convergence, Simplifies B Profile, Relaxes Machining Tolerances, Reduces Beam Scalloping and Beam Interception by Circuit (Helix)

Overall: Decrease in Weight, Volume, Power

VPB: Convergence of Cathode Current Density (≈ 1 A/cm2) by 10x Or More

  • Vacuum Power Booster: Beam Bunching by Interaction With RF Field (Velocity Modulation)

  • Cold-cathode VPB Beam Bunching at Cathode (Density Modulation)


Fea cathode for twt
FEA CATHODE FOR TWT

≈ 13 cm

  • Line / PPM

  • Stack

A

Encapsulated

Electron Gun

1.8 cm

C

B

A - Field Emitter Array

B - Gate Hold-Down Disk

C - Base/Gate Leads

Photos courtesy of David Whaley (Northrop Grumman)

  • 94.1 mA From Area of

  • Diameter ≤ 1 mm

  • With 50,000 Tips

  • D. Whaley, et al., IEEE-TOPS28, 727 (2000)

Operational 55 W TWT

TO5 Header


Electric propulsion ep
ELECTRIC PROPULSION (EP)

Figures Courtesy Of C. Marrese (JPL)

  • 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

ARISE

ARISE

Saturn Ring Observer

LISA


Feas in flat panel displays
FEAs IN FLAT PANEL DISPLAYS

Photo courtesy of Alec Talin (Motorola, Tempe AZ)

Photo taken at MRS Spring 2001 Symposium D

  • 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


Current and current density
CURRENT AND CURRENT DENSITY

Easier Harder

  • Single Tip:

    • SRI

  • RF Amplifiers

    • 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


Current per tip and number
CURRENT PER TIP AND NUMBER

Easier Harder

  • Single Tips Have Been Driven Harder Than Required By Any Application (SRI)

  • FEA Per-tip Performance In rf Vac. Electronics More Demanding But Require Smaller Area

  • Space Applications and Display Per-tip Performance Requirements Not Large, but Large Areas Required


Modulation and pressure
MODULATION AND PRESSURE

Easier Harder

  • Space-based Field Emitter Applications Must Survive in Environments Far More Challenging Than Other Applications.

  • Modulation of Electron Beam As for RF AmplifiersLimits Protection Schemes That Can Be Used to Mitigate Arcs


Field emitter array tip sharpness
FIELD EMITTER ARRAY TIP SHARPNESS

50 Å Radius

Molybdenum

25 Å Radius

  • 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)

Silicon

Photograph courtesy of

W. D. Palmer (MCNC)

Photograph courtesy of M. Hollis (MIT-LL)


Single emitter emission profile
SINGLE EMITTER EMISSION PROFILE

FIM

FEEM

FEEM+FIM

  • 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.

Photographs Courtesy of Capp Spindt and Paul Schwoebel (SRI)


The fea stat hyper model
THE FEA STAT / HYPER MODEL

ag, T, F, D, P

bc,as, m, s

F

as, bc

ag, T, F

as, bc

Characteristic Area

Statistics

Current Density

  • Extraction of FEA Performance From Experimental Data for Spindt-type FEA

Parameters Adjusted Until Theory = Exp. AFN and BFN

Red-Primary;Green-Secondary

FIXED PARAMETERS:

ag Gate Radius

T Temperature

F Work Function

Ntips Number of Emitters

D, P Work Function Parameter to Account for Adsorbates, and Pressure

Rarray Gate And/or Array Resistor

% Percent Current Intercepted by Gate

ADJUSTED PARAMETERS:

as Emission Site Radius Exp & Theory Suggest ≈ 3-7 nm

bc Cone Angle, Limited by SEM to 12˚ - 23˚ for Moly

m Log-Normal DistributionMean Emission Site Radius Parameter

s Log-Normal Distribution Standard Deviation Parameter


Statistics1
STATISTICS

  • Emission Characteristics of Individual Emitters Change from Site to Site Due to

    • Differences In Field Enhancement Factor

    • Changes in F Due to Adsorbates

  • Expression for Array Current Product of # of Tips, a Tip Current Factor and a Statistical Factor (two parts: Sa & Sf)

as

Sa: Variation in Apex Radii

Log Normal Analysis (m, s)

Sf: Variation in Work Function

Pressure Analysis (F, D)


F modification by adsorbates
F - MODIFICATION BY ADSORBATES

Recast: kd = kaPo

For Molybdenum:

SSH/EM Suggests:

a ≈ 5.283 eV-1

Analysis of Exp. Data* Suggests:

Po ≈ 10–8 Torr; D ≈ 0.5 eV

Example:

Sf(0.1 µTorr) ≈ 0.156

* Schwoebel, Spindt, et al., JVSTB19, 980 (2001)

Temple, Palmer, et al, JVSTA16, 1980 (1998)

  • Let P = pressure

    • q = coverage factor (0 ≤ q ≤ 1)

    • ka = adsorption rate

    • kd = desorption rate

  • In Equilibrium:

  • Define:

  • Where


Distribution of emitters
DISTRIBUTION OF EMITTERS

  • In a Log-normal Distribution, a Small Fraction of the Emitters Are Responsible for Most of the Current

Array 1086

µ = 200 Å

s = 0.44

BFN ≈ 700 V

Iexp(90V) ≈ 35.5 µA

  • The Current Will Be Dominated by the Smaller Emitters (%Current Is Proportional to Integrand):

  • 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.

  • Therefore I(V) Fluctuations Primarily Depend on Fluctuations Experienced by Sharpest Emitters


Experimental data jpl
EXPERIMENTAL DATA (JPL)

  • 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

Uncoated Moly FEA


Enforce log normal equilibrium
ENFORCE LOG-NORMAL EQUILIBRIUM

  • Assumptions:

    • At Every Vg, The Emitters Which Contribute To The Array Current Are LN Distributed (Ntips ≤ Narray)

    • Emitters for Which Ftip > Fcritical Are Initially Removed, and m and s Evaluated Afresh

    • As V Increases, Nanoprotrusion Formation / Migration Increases for The Larger Radii

  • Net Effect: As Voltage Increases:

    • (m) Shifts to Higher Values; (s) Becomes Smaller

    • Conclusion: the Emitters Become Both More Uniform and Less Sharp

P = 0.18 µTorr

S(f) ≈ 0.12

Fcrit = 0.67 eV/Å

For Each Voltage Increase, Tips > Critical Radius Removed, LN Parameters Recalculated, and Emitter Distribution Specified Anew


Photoinjectors photocathodes
PHOTOINJECTORS & PHOTOCATHODES

  • Bulk & Surface of Complex Materials Produced by Empirical Techniques; Short Lifetime, Complex Replacement Process.

  • Cathode Selection Influences Drive Laser Chosen (e.g., wavelength, spot bandwith, laser energy, QE, etc.)

  • Drive-laser Reliability <=> System Reliability: UV Unsuitable for Hi-duty

  • Non-linear Crystals Decrease l by 2-4; Efficiency Very Low for UV

  • Conversion by 2 From IR to Green ok: Seek High QE Photocathode in Visible

rf Klystron

Master

Oscillator

Drive laser

Photo-

cathode

Linac

  • Critical Components of Free Electron Lasers, Synchrotron Light Sources, & X-ray Sources

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 l

Naval: Longevity & Reliability Paramount


Photocathodes1
PHOTOCATHODES

DRIVE LASER

  • Reliability <=> System Reliability: UV Unsuitable for Hi-duty

  • Non-linear Crystals Decrease l 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., l, 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)


Quantum efficiency of various cathodes
QUANTUM EFFICIENCY OF VARIOUS CATHODES

  • General Rule of Thumb for QE


Emission non uniformity
EMISSION NON-UNIFORMITY

QE

31 Oct 01 – before 1st cleaning

5 Nov 2001 - after 1st cleaning

4 Dec 2001 - after 1st cleaning

10 Dec 2001 - after 2nd cleaning

  • 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

    • QE scans of LEUTL Photoinjector Mg CathodeCourtesy of John W. Lewellen, Argonne National Lab

    • Details: images from APS photoinjector. Blue = 2xYellow; pixels =10 micron^2; image = (300 pixels)^2Operation: 6 Hz for 30 days (1.55E7 pulses total); macropulse = 1.5 ms


Photocathode response time
PHOTOCATHODE RESPONSE TIME

t = 0.2 ps

  • Pulse Shaping

    • Optimal Shape for emittance: beer-can (disk-like) profile

    • Laser Fluctuations occur (esp. for higher harmonics of drive laser)

    • Fast response: laser hash reproduced

    • Slow response: beer-can profile degraded

    • Optimal: 1 ps response time

  • Mathematical Model (wn = 2pn/T)

t = 0.8 ps

t = 3.2 ps

t = 12.8 ps


Umd experiment cs on w
UMD EXPERIMENT: Cs ON W

Cs

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 Photo-emission 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.

Oct 04

Cs on W

407 nm @ 300 K

Ed=0.14 eV


Qe of cs on w exp vs theory
QE OF Cs ON W: EXP. VS. THEORY

  • Assumptions and Conditions:

    • Coverage Is Uniform

    • Scale factor between Coverage (theory) and Deposition thickness (exp) taken as Atomic diameter:

      • 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.

Cs on W

407 nm @ 300 K

Oct 04

Feb 05



Scandate dispenser cathode
SCANDATE DISPENSER CATHODE

  • Emission Map: Dark Areas = Ave. Current Density > 10 A/cm2

<F> = 1.8 eV

<F> = 1.9 eV

  • 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)


Patch model
PATCH MODEL

= S

  • Variation can be geometric, adsorbate-induced, 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

  • HYPERBOLIC TANGENT VARIATION MODEL

  • Parameterize local (micro) variation by assuming

    • Cylindrical symmetry

    • Two parameters to control transition from island-like to uniform distribution

rc = 0.5

Dr = 10


Umd experiment dispenser photocathodes
UMD EXPERIMENT: Dispenser Photocathodes

Laser In

Anode

Ion

Pump

Current Transformer

Window

Cathode

  • 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.


Qe prediction experiment
QE PREDICTION & EXPERIMENT

Experiment

Theory

M-type

B-Type

Scandate

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)

  • 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


Other factors
OTHER FACTORS

r

total emitted charge

total incident energy

dr

qi

dW

  • FACTORS AFFECTING EMISSION CURRENT

    • Differential surface area illuminated

    • Intensity on differential element

    • Variation in illumination intensity

    • Angular variation of reflection coefficient R: determination of incidence angle

    • Electron Gas Temperature

prolate spheroidal analysis

index of ref & penetration

dictated by experiment

(weak variation for small tips)

prolate spheroidal analysis

laser-material interaction & time dependent model


Field assisted photoemission from w
FIELD-ASSISTED PHOTOEMISSION FROM W

  • Tungsten needle:

    • 10 mm long with radius of curvature at apex = O(1 mm)

    • Laser Intensity of order O(100 MW/cm2) over O(10 ns) and 4th harmonic of Nd:YAG (l = 266 nm)

  • Other Factors:

  • Cathode to anode separation ≈ 35 mm

  • Max Anode ≈ 33 kV(Fo = 0.94 MV/m)

  • Match between prolate spheroidal approx. & actual tip is reasonable

  • Constraints of side walls, temperature at apex, etc. result in best estimate of as = 0.53 mm

Photograph

courtesy of

C. A. Brau

Vanderbilt University


Tungsten needle cathode
TUNGSTEN NEEDLE CATHODE

Macro QE Estimation

Laser Illuminated W Needle Simulation

And Experimental Data††C. Hernandez-Garcia, C. A. BrauNucl. Inst. Meth. Phys. Res. A483 (2002) 273–276

  • Reference Point:

    • V(ref) = 17.0 kV

    • F(ref) = 0.199 GV/m

  • Simulation: Macro Q(ref):

    • Q(266) = 0.528374 %

    • Q(355) = 1.74e-03 %

  • Exp: Macro Q(ref) @ 266:

    • Current at Peak = 0.112 A

    • Intensity = 32 MW/cm2

    • Gaussian Laser spot 50-100 microns (1/e) (depending on l):let Dr = 25 microns (radius)

355 nm

266 nm

355 comparison used same R, scat fac.,penetration depth, etc. as 266 and is therefore only qualitative


Surface and semiconductor theory
SURFACE AND SEMICONDUCTOR THEORY

Clean W(001)

Potential

Ba/O/W(001)

Sound

Velocity

Mass

density

Deformation

potential

momentum

Temperature

W

Ba

O

  • Surface: Interaction of BaO on surface affects barrier in manner dependent on QM effects

ENERGY

POSITION

QDF

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

Ex: Acoustic Phonon


Quantum distribution function
QUANTUM DISTRIBUTION FUNCTION

  • Steady State Solution without Scattering to Gaussian Potential Barrier with incident electrons from both boundaries for Copper parameters

Trajectory Representation

Distribution Function Phase Space


Emittance local cathode conditions
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

    • 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

Patch Model

Macro Model

Simulated Smoothing from propagation

Simulated QE maps of coated surface


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