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) ψ

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

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

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) Ψ E electron energy X V o the potential barrier Uncertainty of electron energies ΔE = pΔp/m, ≈ (ΔP) 2 /m (or position)

Electron Optics Two essential components: 1)Electron source (gun) 2)Focusing system (lenses) Add scanning apparatus for imaging Electron gun Cathode Anode Alignment coils Lenses condensers objective Objective aperture assembly sample

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 x10 18 electrons 1 amp = 1 coulomb/sec SEMs typically operate in the picoamp ( A) to nanoamp (10 -9 A) range (final beam current at sample) so at 1nA ~ 9X10 9 electrons/sec

Current and Voltage Cameca SX50Carl Zeiss EVO50 Beam current At sample Gun emission current Filament heating current Beam current Control (condensers) Beam voltage (Read) (HV set)

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) LaB 6 / CeB 6 cathodes 4) Field emission and Schottky sources

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… MetalVacuum Interface EwEw EfEf E E x 2-5 Volts in metals Thermionic emission Work function of metal: Energy required to elevate an electron from the metal to vacuum

Thermionic emission Work function of metal: Energy required to elevate an electron from the metal to vacuum E f = Fermi level highest energy state in conduction band in this case E = Work necessary to remove electron to infinity from lowest state in metal E w = Work function E w = E – E f Heat electrons to overcome work function MetalVacuum Interface EwEw EfEf E E x

Self – biased electron gun 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

Cathode current density (emission current density) Richardson Law: J c = A c T 2 exp(-E w /kT)in A/cm 2 A c = material dependant constant T = emission temperature k = Botzmann’s constant For W: T = 2700KE w = 4.5ev J c = 3.4 A/cm 2 Improve current density?Use cathode material of lower E w Emitted electrons repelled by Wehnelt Column lenses produce demagnified image of the gun crossover to give the final beam spot at the sample

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

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 High bias negative field gradient strong Focusing action strong Emitted e- see only - field from Wehnelt = return to filament Emission current → 0 Cathode tip Down column toward anode

An optimum bias setting exists in conjunction with the filament – Wehnelt distance for maximum brightness Bias and distance are adjustable parameters on most instruments Emission Current (  A) Bias Voltage (V) Emission current Brightness Optimum bias voltage

Cameca SX50Carl Zeiss EVO50 Gun emission current Filament heating current Saturation

Want a well regulated beam current Increase i f – heat filament to overcome E w of cathode = emission Proper bias = i b does not vary as i f increased above critical value = saturation plateau As i f increases, bias increases also negative field increases and limits the rise in i b Emission Current (  A) Filament Current (A) Operating filament current

Saturation Emission Current (  A) Filament Current (A) Operating filament current

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 (E 0 ) Increase current density by lowering work function (E w )

Cathode types: Tungsten LaB 6 – CeB 6 Field Emission cold thermal Schottky Tungsten cathode Wire filament ~ 100μm diameter hairpin – V shaped operating temperature = 2700K J c = 1.75 A/cm 2 E w = 4.5ev electrons leave from emission area ~ 100x150 μm

Could theoretically increase brightness by increasing temperature D 0 = 100μm α = 3x10 -3 rad At 2700K and 25kV J c = 1.75 A/cm 2 β = 6x10 4 A/(cm 2 sr) brightness = measure of radiant intensity Filament life ~ 320/J c (hrs) hrs Increase temperature to 3000K J c = 14.2 A/cm 2 β = 4.4x10 5 A/(cm 2 sr) ~23 hrs

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

From Richardson equation: J c = A c T 2 exp(-E w /kT)in A/cm 2 A c = material dependant constant T = emission temperature k = Botzmann’s constant So current density (and brightness) increase by lowering work function (E w ) LaB 6 – CeB 6 cathodes At ~ 2700K, each 0.1eV reduction in E w → increase in J c by 1.5X REE hexaborides have much lower E w compared to W

Principle: Use LaB 6 or CeB 6 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) Can equal W current density at 1500K J c then nearly 100A/cm 2 at 2000K Mini Vogel Mount Mo-Re supports Graphite blocks 5000 psi Crystal made by electric arc melting of REEB 6 powder stick in inert atmosphere

2 results: 1)Low evaporation rate at low temperature → long lifetime 2)From Langmuir relation: β = 11,600J c E 0 /(πT) two sources of same current density and E 0 one at 1500K, one at 3000K low T source = twice as bright Advantages: Long lifetime Small d 0 = high resolution

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 E w depends on crystal orientation As crystallites evaporate, emission can change Best orientation = E w less than 2.0eV Better processing has improved performance lowest E w better stability Mechanical failure eventually…

LaB 6 vs. CeB 6 CeB 6 has generally lower evaporation rate and is less sensitive to C contamination

Principle: Cathode = tungsten rod, very sharp point (<100nm) Apply 3-5kV potential relative to first anode (very strong field at tip, >10 7 V/cm) Electrons can escape cathode without application of thermal energy Very high vacuum ( torr or better) Use second anode for accelerating electrons Field Emission (Fowler-Nordheim Tunneling) First Anode Second Anode Field Emission Tip V1V1 V0V0 Etched carbide tip (AP Tech)

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

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 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 Tunneling: Ralph Fowler Lothar Nordheim

Werner Heisenberg Ralph Fowler

Tunneling: If ∆x is on the order of the barrier width, there will be a finite probability of finding an electron on either side Thermionic Field emission E f for ZrO2/W E f for W Cathode Vacuum nm EwEw E w (SE)

Field emission = very high current density ~10 5 A/cm 2 (recall ~3 A/cm 2 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 torr) Current instabilities prevent practical application to microanalysis Expensive Limited current output Disadvantages: Easily poisoned Requires very high vacuum (better than torr) Current instabilities prevent practical application to microanalysis Expensive Limited current output

Schottky emitters: Thin layer of ZrO x 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.

SchottkyCold FieldLaB 6 Tungsten Source Size (nm) >10 4 Energy Spread (ev) Brightness (A/cm 2 SR) 5x Short-term beam Current stability (%RMS) <14-6<1 Typical service life >1yr hrs Now down to 0.15ev with monochromator

36 Monochromator gun concept Extend two mode approach to make a monochromator: 1)use an off-axis extractor aperture and 2)a strong C0-lens setting to create dispersion: C0 on >20 nA C0 on beam off-axis gun tip extractor C0 lens (Segmented electrode gun lens) off-axis aperture on-axis aperture

37 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 extractor C0 lens 2 nd gun deflector off-axial axial beam tip beam aperture slit

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