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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: kljensen@ieee.org. Donald W. Feldman, Patrick G. O’Shea

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April 2005 - Beam Physics Course

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  1. 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: 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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…

  8. 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

  9. 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

  10. 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

  11. 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:

  12. 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

  13. …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

  14. 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

  15. 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…

  16. 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:

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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>

  31. 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)

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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)

  37. 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

  38. 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

  39. 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

  40. 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)

  41. 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

  42. 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

  43. 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

  44. 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

  45. 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

  46. 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)

  47. 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”

  48. 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

  49. 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

  50. 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

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