kevin l. jensen - psec.uchicago.edu · kevin l. jensen naval research laboratory, 4555 overlook...

Kevin L. JensenNaval Research Laboratory,

4555 Overlook Ave. SW, Washington DC 20375 USA

:

Funding: We gratefully acknowledge support provided by

Naval Research Laboratory, Office of Naval Research, & Joint Technology Office

Colleagues: We are indebted to too many to name, but in particular:

• D. Feldman, P. O’Shea, E. Montgomery, Z. Pan (UMD), J. Yater, J. Shaw (NRL), M. Kordesch (Ohio)

• N. Moody, D. Nguyen (LANL), J. Lewellen (NPS), D. Dowell, J. Schmerge (SLAC)

• J. Smedley (BNL), C. Brau (Vanderbilt), Y. Y. Lau (UM), J. Petillo (SAIC), S. Biedron (ANL)

http://psec.uchicago.edu/photocathodeConference/photocathodeIndex.html

1st Workshop on Photo-cathodes

We Shall Discuss Electron Emission (Principally

Photoemission) With A Focus On:

1. The Canonical Equations

2. Photoemission From Metals & Semiconductors: How Terms Are Calculated,

Typical Values And Estimates, Comparison Of Models

• Absorption & Reflection,

• Transport To Surface

• Emission

3. Complications: Emittance, Diffusion, Evaporation

4. Other Issues: Geometry, Dark Current, Space Charge

2


Equation Formula

3

• PhotoemissionFowler-DubridgeL.A. DuBridgePhysical Review 43, 0727 (1933).

• Thermal EmissionRichardson-Laue-DushmanC. Herring, And M. Nichols, Reviews Of Modern Physics 21, 185 (1949).

• Field EmissionFowler NordheimE.L. Murphy, And R.H. Good, Physical Review 102, 1464 (1956).

JFN F( ) = AFNF2 exp

B 3/2

F

JRLD T( ) = ARLDT2 exp

kBT

JFD F( ) ( )

2


Simple Restatement of Obvious:

• QE = # of electrons emitted (proportional to charge emitted Q)

# of photons absorbed (proportional to energy absorbed E)

• RULE OF THUMB:

4

QE =Q / q

E /

J / q

Io /

If Emission Is Prompt, Then Emitted Current Has Same Temporal Shape As Absorbed Laser Power:

Common Factors Of Pulse Duration ( t) Drop Out

Left Discussing CURRENT DENSITY And LASER INTENSITY

Electrons Transport To Surface Subject To Collisions

(Look At Scattering)

# That Escape Depends On Impact Of Surface Barrier If Present

(Look At Emission Probability)

# Of Photons Absorbed Depends On Reflectivity Of Surface

(Look At Reflectivity)

Photo-excitation Depth Depends On How Deeply Photon Penetrated

(Look At Dielectric Constant)

QE %[ ] =123.98J A/cm2

Io W/cm2 μm[ ]


• Spicer’s Model For Semiconductors (This Version Looks Different From Spicer, But Is Same)

• p: quasi-empirical, argued to be 3/2

• B: (Escape)x(Transport) = B exp(- x)

• g: absorption factor “over” + “under” barrier terms

• Vo: Band gap Eg + Electron Affinity Ea

5

QEB

1+ g E Vo( )p

0.0

0.050

0.10

0.15

0.20

0.25

1.5 2 2.5 3 3.5 4

Spicer

Fit (Spicer)

Taft

Fit (Taft)

Qu

an

tum

Eff

icie

ncy

hf [eV]

Taft Spicerp 1.5 1.5B 0.397 0.1863g 1.701 0.6992Vo 1.876 2.038

Originals: W.E. Spicer. "Photoemissive, Photoconductive, and Optical Absorption Studies of Alkali-antimony Compounds." Physical Review 112, 114 (1958).

E.A. Taft, and H.R. Philipp. "Structure in the Energy Distribution of Photoelectrons From K3Sb and Cs3Sb." Physical Review 115, 1583 (1959).

For Metals: C.N. Berglund, and W.E. Spicer. "I - Photoemission Studies of Copper and Silver: Theory." Physical Review 136, A1030 (1964).

Modern Usage: D.H. Dowell, F.K. King, R.E. Kirby, J.F. Schmerge, J.M. Smedley. "In Situ Cleaning of Metal Cathodes Using a Hydrogen Ion Beam." Physical Review Special Topics Accelerators and Beams 9, 063502 (2006).

QE =q1 R( )( )F ,( )PFD ( )

abso

rptio

n

scat

terin

g loss

estra

nsm

ission

• Fowler-Dubridge Model For Metals• L.A. DuBridge. "Theory of the Energy Distribution of Photoelectrons." Physical Review 43, 0727 (1933).

• K.L. Jensen, D.W. Feldman, N.A. Moody, and P.G. O'Shea. "A Photoemission Model for Low Work Function Coated Metal Surfaces and Its Experimental Validation." J. Appl. Phys. 99, 124905 (2006).

QE PFD ( ) ( )2+

kBT( )2

6

• (Modified)

T TemperatureWork Function


THREE STEP MODEL OF PHOTOEMISSIONABSORPTION of light in bulk material and photo-excitation of electrons

• reflectivity R( )

• laser penetration depth ( )

TRANSPORT of photo-excited electrons to surface subject to scattering f (cos ,E)

• electron energy

• scattering rates (relaxation times)

EMISSION probability D(E)

• Metal: Chemical Potential μ, Work Function (work function measured from Fermi level)

• Semiconductor: barrier height Ea, band gap Eg

(Electron affinity measured from conduction band minimum)

6

D E,Ea ,Eg ,F( )

Semiconductor

D E,μ, ,F( )

f cos ,E( )

k

( )

R( )Metal


10-1

100

101

102

0 2 4

n[W]k[W]n[Cu]k[Cu]n[Mg]k[Mg]

n,

k

Photon Energy [eV]

0

0.2

0.4

0.6

0.8

1

0 2 4

R(W)

R(Cu)

R(Mg)

R [

%]

For METALS, spline fitting of readily available n, k data works well

• k = extinction coefficient

• n = index of refraction

• Off-normal reflectivity related to normal values

7

0

= n ik( )2

R ( ) =n 1( )

2+ k2

n+1( )2+ k2

( ) =4 k

=c

2k

n2 k2 K + Ko K( )T2

T2 2( )

2T2( )2+ o T( )

2

2nk Ko K( ) o T3

2T2( )2+ o T( )

2

For SEMICONDUCTORS... A Drude-Lorentz model makes up for incomplete n,k data

• Ko, K = static & high freq. dielectric const

• o = damping term

• T = transverse optical phonon

• Some semiconductors may require multiple T

Cs3Sb

0

1

2

3

4

1 2 3 4

n Wallisk Wallisn Theok Theo

Photon Energy [eV]

n,

k

0

10

20

30

40

1 2 3 4

Wallis

Theory

multialkali

R [

%]


10-6

10-4

10-2

100

102

104

106

10-5

10-4

10-3

10-2

10-1

100

101

-3 -2 -1 0 1 2 3

(x)

x5 W

– (5,1

/x)

ln(x)

x/4

120x5

x5W

(x)

x3/4

x3 / 8

After Photon Absorption, Electrons Migrate And Scatter...

• ... Off Of Each Other (metals; Semiconductors If E > “magic Window” (Off Of Valence Electron If Final State Allowed)

• ... Off Of Lattice Vibrations (Phonons - Primarily Acoustic For Single Atom Material, Polar Optical If Multicomponent)

Metals: Primary Mechanism Is Electron-Electron, But Acoustic Phonon Can Contribute

8

ac

3 vs2

4m 2k kBT( )T

4

W 5,T

1

= 13.1 fs Cu,E = μ + , = 266nm,T = 300K( )

ee=

4 Ks

2 E( )2

fs

2 mc21+ E μ( )

2

2kF

qo

1

= 2.61 fs Cu,E = μ + , = 266nm,T = 300K( )

General

• = 1/kBT (To = RT)

• kB = Boltzmann’s constant

• ao = Bohr Radius

• fs = Fine structure constant

• m = e– mass (eff. or rest)

• E = Electron energy

Metal-Specific

• μ = Fermi level

• qo = Thomas Fermi Screening

• Ks = Dielectric constant

Semiconductor-Specific

• = Debye Temperature

• = Deformation Potential

• = Mass density

• vs = sound velocity

• hk = Momentum 2 (2mE)1/2

Fermi Level for Cu

• 1/ ee = AT2

• 1/ ac = AT

(x) =x3

4arctan x( ) +

x

1+ x2

arctan x 2+ x2( )2+ x2

; qo

2=

4kF

Ksa

o


0

0.08

0.16

0.24

0.32

10 100

Alpha SemiPhysical SemiCs

3Sb

Eg /

R

mo / m

AlP

ZnS

ZnO

CdSZnSe

ZnTe

CdSe

GaP

AlSbCdTe

InP

GaAs

GaSb

InAsInSb

R = 13.6057 eV

mc2 = 510999 eV

Cs3Sb

An Alpha - Semiconductor Model Can Provide Needed Parameters (e.g., Electron Effective Mass) If Such Quantities Are Unknown / Ill-defined... And Even If They’re Not...

Also, Gives Forms Of Polar Optical And Ionized Impurity Scattering That Are Related To, But Different Than, Small Electron Energy Representations Found In Transport / Scattering Tomes

9

Eg

R=m

mo

Alpha Semiconductor Model Restricts Upper Limit On Electron Velocity In Semiconductor To Half Of

Product Of Fine Structure Constant With Speed Of Light; Implies Relationship Between Band Gap

Energy And Electron Effective Mass Of:

1

pop q ,E( )= 2 q

1

Ko

1

K2n q( ) +1( )

E

Eg

u( ) =16u2 +18u + 3

3 1+ 2u( ) u u +1( )

1

ii E( )

4 2

fsm2c

Ni

Ko2

ln 2( )kBT

Eg2

E E + Eg( )

General

• = 1/kBT (To = Room T)

• kB = Boltzmann’s constant

• Ko,K = Static & High Freq. Dielectric Const.

• fs = Fine structure constant

• m = electron mass (eff.)

• E = Electron energy

• Ni = ionized impurity concentration

• n(x) = Bose-Einstein Distribution 1/[ex – 1]

Scattering for Cs3Sb: Typical Values at RT in fs

• Polar Optical on the order of 26.6

• Ionized Impurity on the order of 4395.0

• Acoustic Phonon on the order of 694.0

For u small, (u) 1/u1/2


In Polar Coordinates, Velocity of e- at angle to normalAssume Any Scattering Event Is Fatal To Emission

Matthiessen’s Rule:

Ratio of penetration depth to distance between events

Fraction Of Surviving Electrons

Weighted Scattering Fraction (e.g. MFD Eq.)(1/y Acts As Cosine Of Escape Cone Angle)

10

f cos ,E( )

k

p E( ) =( )

l E( )=

m ( )

k E( ) E( )

f cos , p( ) =

expx x

l E( )cosdx

0

expxdx

0

=cos

cos + p y( )

total1

= j1

j

F = xf x, p( )1/ y

1dx

= p2 lny 1+ p( )

1+ yp+1

2y21 y( ) 2yp y 1( )

Example: Cs3Sb-like

• = 27 nm

• v/c = 0.8%

• = 31 fsp 0.36

Example: Cu-like

• = 12.6 nm

• v/c = 0.675%

• = 2.6 fsp 2.38

Ea

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5

F(y

)

y

po = 0.1

po = 0.3

po = 0.5

po = 1.0

po = 2.0

po = 4.0

E Eay2

p po

• for semiconductors, measure E w.r.t. Ea

• IF scales as 1/k, then p is constant


“Moments” Method Of Calculating QE Is Via Solutions To Schrödinger’s Eq.

• CLASSICAL: f(x,k,t) Is Distribution Function Where x & k Are Conjugate Coordinates => Boltzmann’s Eq.

• Integration Of kn • f(x,k) = Moments: Continuity Eq. Relates 1st (Density) & 2nd (Current Density) Moment:

11

f x + dx, k + dk, t + dt( ) f (x, k, t)

dt t+

k

m x+

F

kf (x, k, t) = 0

tx, t( ) =

t

1

2f (x, k, t) dk =

x

1

2

k

mf (x, k, t) dk =

xJ x, t( )

velocity acceleration

• QUANTUM MECHANICS

tˆ(t) =

iH , ˆ(t) =

2m xk, ˆ(t){ } =

xj t( ) j(x,t) =

2mx ˆ(t), k{ } x =

2mi

†

x x

†{ }“pure state” form

Tsu-Esaki-like formula

J(F,T ) =q

2

k

mD k( ) f k( )

0

dk

• Velocity

• Transmission Probability

• Supply Function

ˆ(t) = f

FDE

k( ) k(t)

k(t) x( ) = 2( )

3

dk dk fFD

E(k)( ) kx( )

2

Mixed state form

f(x,k) Approximated By Product Of Supply Function f(k) X Probability Of Transmission D(k) Past Barrier

D k( )jtrans

k( )jinc

k( )

E k( )2

2mk 2

+ k 2( )

f k( ) = 2( )2

fFD

E k,k( )( ) dk

Transmission Probability = ratio of transmitted to incident

current density for given k

Commonly assumed that energy is parabolic in

momentum k

Supply Function


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

D(E

)

E [eV]

Surface Barrier Subject To Applied Field Does Not Entail That D(E) Is A Step Function

• Triangular Barrier (Fowler-Nordheim Potential)Reasonable If No Image Charge Modification

• Exact Solution: Airy Functions

• Approximation: JWKB Method (As Commonly Used)

D(E) Calculation Requires Auxiliary TermsObserve Definitions Work For E > Ea Too

12

E Ea +E Ea( )

2+4

25

2F2

2m

2/3 1/2

E < Ea( ) = 2 k(x)dx =2 2

3mF

2m2Ea E( )

3/2

D E( ) =4 E E Ea +

2 E E Ea ++ E Ea +

+ E( )e (E )

D E,μ, ,F( )

D E,Ea ,Eg ,F( )

F

Ea

E

•Even For Small Ea, D(E) Is NOT A Step Function

•To Extend To Metals ( Image Charge ), Append v(y) to (E)

Ea = 0.5 eVF = 100 MV/m

D> E( )4 E E Ea( )

E Ea( )1/2

+ E1/22

D< E( ) =4

Ea

E Ea E( )e (E )

JWKB “area under curve”


Common Forms Of Emission Equations Obtained From One Formulation Using Energy Slope Terms (field) F, (temperature) T & Expansion Point Eo1D Approach To Evaluation (E Is “forward” Energy)

13

JT ARLD kB T( )2 T

F

exp T Eo μ( )

JF ARLD kB F( )2 F

T

exp F Eo μ( )

QEMFD = 1 R( )( )F( )

2+ 2 (2) T

2+ F

2( )2 2μ( )

Limit (n = 0) Richardson (Thermal) Eq

Limit (n = ) Fowler-Nordheim (Field) Eq

Energy Augmented By Photon (s<0) Fowler-Dubridge Equation

ARLDT2 exp T[ ]

AFNF2 exp

BFNF

( )2

10-3

10-2

10-1

100

101

0 0.4 0.8 1.2 1.6

ExactAprox

Diffe

ren

ce

x

J(F,T ) =e

2

m

T2

ln 1+ exp T μ E( ){ }1+ exp F Eo E( ){ }0

dE = ARLDT2N T

F

, F Eo μ( )

• Numerator: Supply Function With T = 1/kBT

• Denominator: Kemble Form Of D(k) With

N n, s( ) = n21

ne s

+ n( ) e ns=1

2n2s2 + 2( ) n2 +1 N n, s( )

x( ) 1+ 1 21 2 j( ) 2 j( ) x2 j

j=1

When F & T Become Comparable, Can Be Large

(x) = Riemann zeta function

E( ) = F Eo E( )


DEFINE the “Moments” function Mn by generalizing distribution function approachmetals - final state may be occupied (blue)

semiconductors - final state unoccupied & in conduction band (creates “magic” window)

To Calculate Emittance, Swap Forward Momentum With Transverse:

14

Mn= 2( )

3 2m2

3/ 2

E1/ 2 dE sin d2m

2E cos2

n / 2

0

/ 2

0

D (E + )cos2{ } f cos , p ( ) fFD

(E) 1 fFD

(E + ){ }

QE = Ratio Of Emitted M1 To Possible M1 (3D: E Is Total Energy; Semiconductor Shown)

QE = 1 R( )( )E xf x,E( )D E +( ) x2 dx

Ea /E

1dE

Ea

Eg

2 E xdx0

1dE

0

Eg

Leading Order (FD) Approximation: Ignore cos Dependence In D (i.e., take x = 1)

This form will lead to the comparison with the Spicer Model form

QE0 =1 R( )( )

2 Eg( )2

ED E +( ) xf x,E( )dxEa /E

1dE

Ea

Eg

This Term Is Same As Scattering Factor In MFD

“magic” window


Most Adaptable Model For Particle-In-Cell (PIC) Codes Modeling Beams Is QE0

• How Does QE0 Compare To QE?

• How Does QE0 Compare To Spicer 3-Step Model?

15

lim0G y( )

1

po +1( ) 1+Ea

2

In The Limit Of Small (asymptote)

QE0 =1

21 R( )( )G 1+

Ea

PIC - Ready Formulation

Comparison To Spicer 3-Step Model

• Interpretation Of B, g Altered

• Power Of And Dependence Changed

QEspicer

B

1+ g 3/2

0

0.2

0.4

0.6

0 1 2 3

Exact

Spicer Fit

Asymptote

G(y

)

0

0.04

0.08

0.12

2 2.4 2.8 3.2 3.6

QE0

QE0 - QE

QE

Quantu

m E

ffic

iency

Photon E [eV]

Eg = 1.6 eV

Ea = 0.3 eV

T = 300 K

F x( ) = sf s,Eax2( )ds

1/ x

1G y( ) =

8

y4x3F x( )dx

1

yRecall: Define:

= Eg + Ea( )

Define:

Spicer Parametric Fit Is Good QE0 Works Well - Good For PIC


Theory Of Photoemission From Cesium Antimonide Using An Alpha-semiconductor Model

Kevin L. Jensen, Barbara L. Jensen, Eric J. Montgomery, Donald W. Feldman, Patrick G. O'Shea, and Nathan Moody

J. Appl. Phys. 104, 044907 (2008); DOI:10.1063/1.2967826

A model of photoemission from cesium antimonide (Cs3Sb) that does not rely on adjustable parameters is proposed and compared to the experimental data of Spicer [Phys. Rev. 112, 114 (1958)] and Taft and Philipp [Phys. Rev. 115, 1583 (1959)]. It relies on the following components for the evaluation of all relevant parameters: (i) a multidimensional evaluation of the escape probability from a step-function surface barrier, (ii) scattering rates determined using a recently developed alpha-semiconductor model, and (iii) evaluation of the complex refractive index using a harmonic oscillator model for the evaluation of reflectivity and extinction coefficient.

16

10-3

10-2

10-1

0 20 40 60 80 100

375

405

532

655

808

Quantu

m E

ffic

iency [

%]

Coverage %

QE in %

0

0.04

0.08

0.12

0.16

1.6 2 2.4 2.8 3.2 3.6 4

300 K

77 K

QE0

QE

QE0 + QE

x 1.15

Qu

an

tum

Eff

icie

ncy

Photon E [eV]

Exp Data:W.E. Spicer

PR112, 114 (1958)Figure 8

QE not in %

Photoemission From Metals And Cesiated Surfaces

Kevin L. Jensen, N. A. Moody, D. W. Feldman, E. J. Montgomery, and P. G. O'Shea

J. Appl. Phys. 102, 074902 (2007); DOI:10.1063/1.2786028

A model of photoemission from coated surfaces is significantly modified by first providing a better account of the electron scattering relaxation time that is used throughout the theory, and second by implementing a distribution function based approach (“Moments”) to the emission probability. The latter allows for the evaluation of the emittance and brightness of the electron beam at the photocathode surface. Differences with the Fowler-Dubridge model are discussed. The impact of the scattering model and the Moments approach on the estimation of quantum efficiency from metal surfaces, either bare or partially covered with cesium, are compared to experiment. The estimation of emittance and brightness is made for typical conditions, and the derivation of their asymptotic limits is given. The adaptation of the models for beam simulation codes is briefly discussed.


THERMAL EMISSION

No photons

Uniform emission

Richardson Approx.

No Scattering

Maxwell-Boltzmann f(x,k)

No “final state” issues

17

= 0

2 x2 =2

= c2

D k( ) = E(k) μ( )

f x, p( ) =1

D(k) f (k) exp T E(k) μ( ){ }

1 fFD E( ) 1

Mn= 2( )

3 2m2

3/2

E1/2 dE sin d2m

2E +( )sin2

n/2

0

/2

0D (E + )cos2{ } f cos , p( )

fFD

(E) 1 fFD

(E + )( )

+ E Eg( )

n,rms (thermal) = mcx2 kx

2

=mc

c

2

M 2

2M 0

1/2

=c

2

kBT

mc2

1/2

Transverse Moments (metal & semiconductors)

Explanation of addition of photon energy to E for kn term:

• Schrödinger’s Eq.: E of e- in vacuum measured wrt conduction band min decreased by barrier height BUT

• Continuity of & x means k is conserved*

* See also: D.H. Dowell, J.F. Schmerge, SLAC-PUB-13535 (2009) they show eq. (klj) didn’t include hf in k so small by [μ/(μ+hf)]1/2

kvacuum = ksemiconductor =2m2 E +( )

1/2

sin

THEREFORE:

PHOTO-EMISSION

Photons

Uniform emission

JWKB Approx.

Scattering

Schottky Lowering

> 0

2 x2 =2

= c2

D k( ) = DJWKB E,F( )

f x, p( ) = x / x + p E( )( )

= q2F / 4 0

n,rms (photo) = mcc

2

M 2

2M 0

1/2

c

2

( )3mc2

1/2

leading order (metal)

Note: for metals, p large & f cos /p: therefore, emittance indep. of p. Semiconductors larger due to p small, but D behavior also has impact

1st Workshop on Photo-cathodes 18

0 0.2 0.4 0.6 0.8 1

WangTaylor

Gyfto Lev

2

3

4

5

0 0.2 0.4 0.6 0.8 1

Longo-1Longo-2Haas

SchmidtGyfto Lev

Work

Function [

eV

]

Coverage

Cs on WBa on W

related to of surface via Gyftopoulos-Levine Theory Experiments at UMD to understand relation of QE to coating & rejuvenation methods

• Work function depends on crystal face, bulk material, and alkali (or alkali earth) coating

• Work function minimizes at sub-monolayer

• Coatings on metals simplest - increasingly more complex semiconductor surfaces under study

• With evap/diff, possibility of uniform coverage at desired factor (e.g., dispenser photocathode)

Evap of Cs on W shows power-law dependence on coverage of form c(T) x n

BUT c & n change depending on

• Yellow: c exp(-68.1 + 5.19/kBT) n = 18

• Blue: c exp(-39.6 + 3.04/kBT) n = 10

850K800K

750K700K

0.1

1

10

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Evap R

ate

x10

-12 [

ato

ms/c

m2-s

ec]

Coverage

Data: Becker

Dash: c(T) ^10

Solid: c'(T) ^18Cs on W

J.A. Becker, Physical Review 28, 0343 (1926).

1st Workshop on Photo-cathodes 19

10 nm Cs on 200 nm Sb on SILICON

365 nm Hg, Heated by lamp, T~320-330KE = vo s jj=1

s0j=1

s j

Energy of coating atoms:

• vo = depth of well

• = interaction energy

• = # of nearest neighbors

• sj = 0,1 occupation factor

• = osc. frequency

• = hopping distance

• ( ) = hopping prob.

• P(E>vo) = Prob. evap

What’s happening:

These are PEEM images of squares of Cs (25 μm on a side) laid down on Sb, & heated. QE images: frames taken about 10 seconds apart; intensity adjusted for image (not held fixed)

Pjump

sisj

vo

Ej

D ( ) = ( ) 2 1

1+ Do =

3

Ro

voM

1/2

exp vo( )

1= e vo

1+ e vo( )1+ e vo( )

( ) = 0( )1+ e vo( )

3

1+ e vo( )4

Diffusion and evaporation of coatings: atoms as harmonic oscillators

• Diffusion D( ) = product of oscillation freq , jump length 2, and jump probability Pjump

• Both Jump prob & evap rate proportional to exp(- vo)

• Question: If atom only sees four ( ) nearest neighbors (microscopic view), how does it “know” what local coverage is (macroscopic view)?

• Answer: can be shown value of vo is related to

coverage, and therefore affects evaporation (through c(T)) & diffusion (through Pjump)

t Do2


POINT CHARGE MODEL

• Goal: Impact of surface features on emittance

• Serendipity: Dark current model (field emission if enhancement is high,

low, or both) is analytically tractable

• 3D, get trajectories, estimate emittance

20

V , z( ) Foa0Vn a0,z

a0

Vn , z( ) z + 2+ z2( )

1/2

+ j2+ z z j( )

2

( )1/2

j=1

n2+ z + z j( )

2

( )1/2

Vn 0, zn+1( ) 0 boundary conditions

dipole term

monopole term

PCM is dimensionless, scalable analytical method to get tip radii, field enhancement, total current

n = # chargesr = radii scaling

n = 1 2 3 4 5 6 7r = 0.75

monodipl

Background field Fo

radius of first charge ao

zn = ajj=0

n= r j 1a0j=0

n

tip height factor z

n r( ) = zVn 0, z( )z=zn+1

an r( ) =Vn , z( )2Vn , z( )

=0,z=zn+1

Field Enhancement

Apex Radius

& an all that is needed for analytical model of current from an apex: well-tested against real conical emitters

JFN F( ) = AF2 exp B / F( )

=8Q

9

2m

A =q

16 2 t yo( )2

2e6

4Q

B =4

32m 3

I F( )1

2a2

s

1/2

Erf p F( )s F( ) Erf s F( ){ }exp s F( )2( )J F( )

p x( ) =5B 4 2( )x

2B 2 1( )x

s x( ) =B 1( )x

2x 3B 2 3( )x

F = field at apex of emission site

100

101

102

103

10-5

10-4

10-3

10-2

10-1

100

2 4 6 8 10 12

0.40.71.0

0.4

0.7

1.0

n

dipole

an

n

10-4

10-2

100

102

104

1 3 5 7

4.5 eV3.6 eV2.0 eV

Pro

trusio

n C

urr

ent

[μ]

Apex Field [eV/nm]

atip

= 10 nm


• Copper Photoemission from Boss

• Bulk Temp = 300 Kelvin

• Wavelength = 266 nm

• Field = 100 MV/m

• Laser Intens = 160 MW/cm2

• 45 μm length (cath-anode)

• 12 μm center to center spacing

21

Cs3Sb:

• Outer: Cs-depleted el. affinity

• Inner: Cs-monolayer el. affinity

• Field = 10 MV/m

Emitted Charge Changes Fields On Surface That Affects Subsequent Emissions - Oscillations Induced By A Sudden Influx Of Charge Can Persist. Demonstration For Metal

(Cu) And Semiconductor (Cs3Sb) Using Particle-in-Cell (PIC) Code MICHELLE

Flat metal copper surface without & with hemispherical bump

Flat Sb surface with an inner square coated with Cs monolayer & depleted


What We Attempted

• Treatment Of Photoemission

• Spicer Model, Fowler-Dubridge, Moments-based Approach, And Their Relation

• Absorption - Dielectric Constant And Drude-Lorentz Model

• Transport - Scattering Mechanisms (electron-electron, Acoustic, Polar Optical, Ionized Impurity)

• Emission - Transmission Probability Through The Fowler-Nordheim Barrier

• General Thermal-Field-Photoemission Equation

• Comparison Of Theory To Experiments (UMD & NRL)

• Related Matters

• Emittance

• Evaporation And Desorption

• Dark Current Via Point Charge Model

• Space-Charge Induced Current Oscillations

22

What We Did

kevin l. jensen - psec.uchicago.edu · kevin l. jensen naval research laboratory, 4555 overlook...

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