Today’s quote:

1. ψ Today’s quote: “Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world. In our endeavor to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. If he is ingenious he may form some picture of a mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility of the meaning of such a comparison.” A. Einstein, 1938 (from Gary Zukav The Dancing Wu Li Masters)

2. ψ The wave function describes the state of motion of a particle …The 4f wave functions (Mark Winter…the orbitron)

3. |ψ|2 … the probability density …The 4f electron density

4. Ψ Electron transmission across a potential barrier Δ(A)Δ(B) ≥ (|ψ|[A,B]|ψ |)/2 A and B are observables, or measurable properties (energy, position, momentum, spin, etc.) ΔEΔt ≥ ħ (E = energy, t = time) ΔpΔx ≥ ħ (p = momentum, x = distance, or position) Vothe potential barrier Uncertainty of electron energies ΔE = pΔp/m, ≈ (ΔP)2/m (or position) E electron energy X

5. Electron gun Cathode Anode Alignment coils Lenses condensers objective Objective aperture assembly Electron Optics Two essential components: • Electron source (gun) • Focusing system (lenses) • Add scanning apparatus for imaging sample

6. Current and Voltage Voltage = electrical potential (volts) consider as the speed or energy of electrons SEMs 1-50 kV (or keV) Current = number of electrons/unit time (amps) 1 coulomb ~ 6 x1018 electrons 1 amp = 1 coulomb/sec SEMs typically operate in the picoamp (10-12A) to nanoamp (10-9A) range (final beam current at sample) so at 1nA ~ 9X109 electrons/sec

7. Current and Voltage Carl Zeiss EVO50 Cameca SX50 (Read) Filament heating current Gun emission current Beam current At sample Beam voltage Filament heating current Beam current Control (condensers) (HV set)

8. Electron Guns Purpose: Provide source of electrons Large, stable current in small beam Located at the top of the column Topics: 1) Thermionic emission 2) Tungsten cathode 3) LaB6 / CeB6 cathodes 4) Field emission and Schottky sources

9. Thermionic emission Work function of metal: Energy required to elevate an electron from the metal to vacuum Metal Vacuum E The work function is the solid-state electronic analog of the ionization potential (binding energy) – the amount of energy required to liberate an electron from a particular energy level… Ew E Ef x 2-5 Volts in metals Interface

10. Heat electrons to overcome work function Thermionic emission Work function of metal: Energy required to elevate an electron from the metal to vacuum Metal Vacuum Ef = Fermi level highest energy state in conduction band in this case E = Work necessary to remove electron to infinity from lowest state in metal Ew = Work function Ew = E – Ef E Ew E Ef x Interface

11. Wehnelt cylinder surrounds filament and has small opening at base Biased negatively between 0 and -2500V relative to the cathode Equipotentials = field lines Emitted electrons are drawn toward anode by applied potential (usually +15kV in probe) converge to crossover - attempt to follow the highest potential gradient (perpendicular to field lines) Forms first lens Self – biased electron gun Filament heating supply Filament (Cathode) Bias Resistor Wehnelt Cylinder Vbias=ieRbias - High voltage supply - 0 + + Equipotentials d0 Anode α0 Beam current ib

12. Cathode current density (emission current density) Richardson Law: Jc = AcT2exp(-Ew/kT) in A/cm2 Ac = material dependant constant T = emission temperature k = Botzmann’s constant For W: T = 2700K Ew = 4.5ev Jc = 3.4 A/cm2 Improve current density? Use cathode material of lower Ew Emitted electrons repelled by Wehnelt Column lenses produce demagnified image of the gun crossover to give the final beam spot at the sample

13. Biasing of electron gun and saturation Variable bias resistor in series with negative side of HV power supply and filament Apply current to heat filament negative voltage will be applied across Wehnelt cylinder Change in resistance produces directly related change in negative bias voltage Major effect: Field topology Change in constant field lines near cathode Field topology also affected by filament-Wehnelt distance Filament heating supply Filament Bias Resistor Wehnelt Cylinder Vbias=ieRbias - High voltage supply - 0 + + d0 Equipotentials Anode α0 Beam current ib

14. Low bias negative field gradient weak Focusing action weak Emitted e- see only + field from anode = high emission current Produces large crossover size Poor brightness Down column toward anode Cathode tip High bias negative field gradient strong Focusing action strong Emitted e- see only - field from Wehnelt = return to filament Emission current → 0

15. An optimum bias setting exists in conjunction with the filament – Wehnelt distance for maximum brightness Bias and distance are adjustable parameters on most instruments 200 Emission current Brightness 200 Optimum bias voltage Emission Current (mA) 200 -300 -400 -500 Bias Voltage (V)

16. Saturation Carl Zeiss EVO50 Cameca SX50 Filament heating current Gun emission current Filament heating current

17. Saturation Want a well regulated beam current 200 Operating filament current 100 Emission Current (mA) 50 0 2.0 4.0 Filament Current (A) Increase if – heat filament to overcome Ew of cathode = emission Proper bias = ib does not vary as if increased above critical value = saturation plateau As if increases, bias increases also negative field increases and limits the rise in ib

18. Saturation 200 Operating filament current 100 Emission Current (mA) 50 0 2.0 4.0 Filament Current (A)

19. Improvements in beam performance: Increase current density (more potential signal in smaller beam spot) Can increase the current density at the gun crossover by increasing brightness Higher brightness = More current for same sized beam Smaller beam at same current Increase brightness by: Increase voltage (E0) Increase current density by lowering work function (Ew)

20. Cathode types: Tungsten LaB6 – CeB6 Field Emission cold thermal Schottky Tungsten cathode Wire filament ~ 100μm diameter hairpin – V shaped operating temperature = 2700K Jc = 1.75 A/cm2 Ew = 4.5ev electrons leave from emission area ~ 100x150 μm

21. Could theoretically increase brightness by increasing temperature D0 = 100μm α = 3x10-3 rad At 2700K and 25kV Jc = 1.75 A/cm2 β = 6x104 A/(cm2sr) brightness = measure of radiant intensity Filament life ~ 320/Jc (hrs) 180-200 hrs Increase temperature to 3000K Jc = 14.2 A/cm2 β = 4.4x105 A/(cm2sr) ~23 hrs

22. Brighter sources are attractive, but tungsten: reliable stable relatively inexpensive Failure due to W evaporation at high temperature in good vacuum Sputtering from ion bombardment in poor vacuum

23. LaB6 – CeB6 cathodes From Richardson equation: Jc = AcT2exp(-Ew/kT) in A/cm2 Ac = material dependant constant T = emission temperature k = Botzmann’s constant So current density (and brightness) increase by lowering work function (Ew) At ~ 2700K, each 0.1eV reduction in Ew→ increase in Jc by 1.5X REE hexaborides have much lower Ew compared to W

24. Principle: • Use LaB6 or CeB6 single crystal • La atoms are mobile in B lattice when heated • - Evaporate during thermionic emission • La (or Ce) replenished at tip by diffusion • Low work function relative to W ~2.4eV (~ 4.5eV for W) Crystal made by electric arc melting of REEB6 powder stick in inert atmosphere Graphite blocks Mo-Re supports Can equal W current density at 1500K Jc then nearly 100A/cm2 at 2000K 5000 psi Mini Vogel Mount

25. 2 results: • Low evaporation rate at low temperature → long lifetime • From Langmuir relation: • β = 11,600JcE0/(πT) • two sources of same current density and E0 • one at 1500K, one at 3000K • low T source = twice as bright • Advantages: • Long lifetime • Small d0 = high resolution

26. Disadvantages of REE hexaboride cathodes Very chemically reactive when hot (forms compounds with all elements except C – poisons cathode Requires exceptionally good vacuum (10-7 torr or better) Expensive Ew depends on crystal orientation As crystallites evaporate, emission can change Best orientation = Ew less than 2.0eV Better processing has improved performance lowest Ew better stability Mechanical failure eventually…

27. LaB6 vs. CeB6 CeB6 has generally lower evaporation rate and is less sensitive to C contamination

28. Field Emission (Fowler-Nordheim Tunneling) Principle: Cathode = tungsten rod, very sharp point (<100nm) Apply 3-5kV potential relative to first anode (very strong field at tip, >107 V/cm) Electrons can escape cathode without application of thermal energy Very high vacuum (10-10torr or better) Use second anode for accelerating electrons Field Emission Tip V1 V0 First Anode SecondAnode Etched carbide tip (AP Tech)

29. Werner Heisenberg and the uncertainty principle (1927, age 25) The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. --Heisenberg, uncertainty paper, 1927 Tunneling: Quantum effect by which electrons can “pass” through the potential barrier to overcome the work function The applied field deforms the potential barrier, and unexcited electrons “leak” through the barrier ∆p • ∆x≈ ħ/2 Heisenberg uncertainty implies an uncertainty in position ∆x

30. Tunneling: Electrons near the Fermi level… Uncertainty in momentum corresponds to the barrier height(φ, work function): (2mφ)1/2 If the uncertainty in position ∆x≈ ħ/2(2mφ)1/2 is approximately the barrier width x= φ/Fe (Fe = applied field) then electrons will have a high probability of existing on either side Ralph Fowler Lothar Nordheim

31. Ralph Fowler Werner Heisenberg

32. Tunneling: If ∆x is on the order of the barrier width, there will be a finite probability of finding an electron on either side Cathode Vacuum Thermionic Ew(SE) Ew Field emission Ef for ZrO2/W Ef for W nm 0 1 2 3 4 5

33. Field emission = very high current density ~105 A/cm2 (recall ~3 A/cm2 for W thermionic cathodes) Very small emission region (~ 10nm) So brightness = 100s of times greater than thermionic emission at the same voltage Advantages: Long lifetime Very high resolution High depth of field Disadvantages: Easily poisoned Requires very high vacuum (better than 10-10torr) Current instabilities prevent practical application to microanalysis Expensive Limited current output

34. Schottky emitters: Thin layer of ZrOx further lowers work function. Using both high tip potential and thermal activation (2073K) to enhance emission Suppressor cap eliminates unwanted emission away from the tip Results in larger and more stable current compared to cold field emission Resolution approaches that of cold field emission.

35. Now down to 0.15ev with monochromator

36. C0 on C0 on >20 nA beam off-axis Monochromator gun concept • Extend two mode approach to make a monochromator: • use an off-axis extractor aperture and • a strong C0-lens setting to create dispersion: gun tip extractor on-axis aperture off-axis aperture C0 lens (Segmented electrode gun lens)

37. tip off-axial axial beam beam extractor C0 lens 2nd gun aperture slit deflector UC gun optics design • UC = “UniColore”: monochromator gun • 2 extractor apertures: • 1 for on-axial beam: normal beam • 1 for off-axial beam: UC beam • C0-lens focuses off-axial beam: • select beam energies with aperture • dispersion is in 1 direction: use slit • ΔE ≈ 0.15 eV • Extra deflector below slit: • steers off-axis beam onto optical axis • Geometry fits into Elstar gun module

38. Schottky-FEG extractor,2 apertures segmentedgun lens aperture and slit deflector Magellan XHR SEM: three beam modes available Standard High current Monochromated (UC)