INTRODUCTION TO ELECTRON EMISSION PROBLEMS
blue problems are optional, meaning you don’t have to do them (and probably won’t)
Green problems are ones you should do, but only if you finish the others.
Black problems are ones you should do, and do first. They may use results from Blue or Green, however
(due: two weeks from 4/6/2005)
Problem 1: Electron Statistics
The equation for electron density [#/cm3] in terms of the Fermi Dirac Distribution
function fFD(E) is
 ,T  2 2 
3
 1 exp    E  d k
1


3
(1.1)
 
A. Show that the density at T = 0 K is  o ,0  kF3 / 3 2 , where kF is the Fermi
  / 2m .
momentum, where  o  hk F
2
B. Show that for T > 0, to leading order,
2

1   
(1.2)
(T )  o 1  
 
 12  o  
C. Assuming 1 free electron per atom, calculate (T) for: Gold, Silver, Cesium, Copper,
Magnesium, and Tungsten at room temperature. Compare to commonly accepted
values (e.g., Kittel, Solid State Physics, Chapter 6). Why are there differences?
Problem 2: Emission barrier (general)
A. Derive the Schottky factor   4QF from the Image Charge Potential. Evaluate
it for fields characteristic of thermionic emission, where the extraction grid is held at
5 kV and is 0.5 mm away from the surface of the cathode. How far away from the
surface of the cathode is the barrier maximum if the work function is that of a
dispenser cathode (2 eV)?
B. Evaluate it for fields characteristic of field emission from a hyperbolic gated field
emitter, where the gate voltage is 200 V, the gate radius is 1 micron, the apex radius
is 75 Angstroms, and the cone angle is 20 degrees. What is the width of the tunneling
barrier for an electron at the Fermi momentum of Copper (work function is 4.6 eV).
Problem 3: Tunneling Barrier
A. Under low fields, the emission barrier looks triangular. However, under high fields,
the barrier looks quadratic, or V (x)  Vo 1 (x / L)2 . Calculate the transmission


coefficient for the quadratic barrier. Assume the Fermi energy is 7 eV, and the work
function is 4.6 eV (Copper). Let L be the width of the tunneling barrier as calculated
1
in Problem 2. Plot the log of the transmission coefficient from E = 0 to the barrier
maximum.
B. The transmission probability at E = 0 should be zero. Why isn’t it in part A?
Problem 4: Thermionic emission
A. Based on the Richardson Equation, what is the change in current density as the grid
voltage in Problem 2B swings from 5 kV to 0 kV for an operating temperature of
1300 Kelvin for a dispenser cathode in which the work function is 2 eV? Ignore
space charge effects.
B. For the dispenser cathode in part A, how much does the temperature have to increase
for the current density to double (to 3 significant digits accuracy)?
C. The data from { J. A. Becker, Phys. Rev. 28, 341 (1926)} for clean and cesiated
tungsten is given below. From it, determine the work function of clean tungsten and
thoriated tungsten (hint: graphs are in the lecture notes). The work function you will
estimate for thoriated tungsten is larger than commonly accepted values. What might
have caused that?
T
Clean W [A/cm2]
T
1500.0
1550.0
1600.0
1650.0
1700.0
1750.0
1800.0
1850.0
1900.0
1950.0
2000.0
2050.0
2100.0
2150.0
2200.0
2250.0
2300.0
2350.0
2400.0
7.9885e-08
2.9197e-07
9.7494e-07
3.0014e-06
8.5858e-06
2.2980e-05
5.7899e-05
0.00013806
0.00031303
0.00067777
0.0014066
0.0028076
0.0054058
0.010068
0.018183
0.031912
0.054538
0.090929
0.14814
950.00
1000.0
1050.0
1100.0
1150.0
1200.0
1250.0
1300.0
1350.0
1400.0
1450.0
1500.0
1550.0
1600.0
Thoriated W
[A/cm2]
1.0292e-08
8.7295e-08
6.0085e-07
3.4554e-06
1.7007e-05
7.3071e-05
0.00027869
0.00095680
0.0029926
0.0086139
0.023019
0.057543
0.13545
0.30194
Problem 5: Field Emission
The data from {J. P. Barbour, W. W. Dolan, et al., Phys. Rev. 92, 45 (1953)} for clean
and coated tungsten needle cathodes is given below. In class, work function values were
estimated for the coated surfaces using a clean work function of 4.6 eV. Redo the
calculations with the assumption that the clean work function value is that of
polycrystalline tungsten (4.55 eV). Use only the “Direct Current” data values for the
calculation, which are given below. Hint: (a) use W [1] to infer the field enhancement
2
term and proceed from there; (b) the FN equation is for current density, but the values in
the table are for current – is this a problem (if so, why; if not, why not?)?
V [Volts]
W [1]
V [Volts] W+Ba [2] V [Volts]
6636.7
7.1020e-09
4012.1
4.7021e-09
3287.7
6815.4
1.3183e-08
4212.7
1.7354e-08
3390.5
7003.9
2.4471e-08
4421.7
3.9591e-08
3505.3
7069.1
3.0075e-08
4609.8
1.1100e-07
3593.6
7237.4
5.2118e-08
4814.7
2.9053e-07
3698.6
7413.8
9.6743e-08
5038.7
7.0995e-07
3803.5
7599.0
1.7958e-07
3907.8
7793.7
3.3334e-07
4003.9
8041.5
5.3930e-07
note: all currents are in Amps, all V’s are in Volts.
W+Ba [3]
7.7880e-09
1.5414e-08
2.9137e-08
6.0420e-08
1.0918e-07
2.2636e-07
3.9089e-07
7.0683e-07
Problem 6: Field Emitter Arrays
Data from MIT-Lincoln Labs on their very small field emitter arrays (the work function
for nickel-coated molybdenum was taken to be 3.7 eV) is given below. The gate
diameter is 0.1 microns and the cone angle is 23 degrees. The low gate voltage values
were noisy, and the high gate voltage values were affected by space charge (see K. L.
Jensen, J. Applied Physics 82, 845 (1997), Figure 8c, but you should be cautious of using
values therein as other effects are considered that we are ignoring). Pick the region of the
data that you find free of complications due to noise or space charge. From it, make an
estimation of the field enhancement factor, the apex radius and the area factor for a gate
voltage of 73 V. Under the assumption that the emitting tips were identical, what
percentage of the tips are emitting? Caution: this appears to be a “plug-into-theemission-formula” – but it is not. Do a consistency check: are the numbers you get
reasonable (e.g., if you get an emission area larger than your emitter tip, that is a
problem).
Gate Voltage [V]
50.000
51.990
53.980
55.970
57.950
59.910
61.870
63.800
65.710
67.590
69.400
Total Current [A]
4.0500e-06
9.1000e-06
1.8150e-05
3.0200e-05
5.0200e-05
8.2250e-05
0.00012635
0.00019045
0.00028065
0.00040090
0.00058120
Gate Voltage [V]
71.180
72.870
74.610
76.160
77.670
79.130
80.480
81.760
83.000
84.020
85.140
Total Current [A]
0.00080160
0.0011020
0.0013625
0.0018032
0.0022560
0.0027130
0.0031320
0.0034650
0.0038000
0.0040600
0.0042200
Problem 7: A Trick Question
The general current density formula predicts a monotonic increase in current density as
the temperature increases (assuming the chemical potential is constant). Using the
Thermal-field Fowler Nordheim Equation
3
 b fn   c fn /  
J (T , F )  a fn exp    
(1.3)

 F   sin(c fn /  ) 
and the approximations v(y) = 0.93685 – y2 and t(y) = 1.0566, calculate the field emission
current for a field of 3 GV/m and a work function of 4.5 eV for representative
temperatures from 400 K to 1500 K. What current would you anticipate for T =
1515.8743? Is your estimate born out by calculation? Why or why not?
Problem 8: Laser heating of a photoemitter surface
It was argued in class that a slab that absorbed an amount of energy E uniformly over a
length l would experience a change of temperature given by
E
(1.4)
T 
 To  3NkB l
where E  I o t and  = Ce(T)/T (and is generally constant for metals for temperatures
we consider). We took l as the laser penetration depth, and consequently predicted large
values of T for long laser pulses. Clearly, holding l so small was a bad idea. Develop a
better approximation to l and recalculate Eq. (1.4) for the times t = 0.001, 0.05, 1, and 6
ns for Copper (the example considered in class). Hint 1: solutions to the diffusion
equation t y  D2x y are
y(x,t)  yo 4 Dt  exp  x 2 / 4Dt 
(1.5)
Hint 2: an initial distribution of temperature can be thought of as a sum of Dirac Delta
pulses. Compare answers to numerical simulations, for which T = 0.498 K, 2.075 K,
10.010 K, and 24.806 K for t = 0.001, 0.05, 1, and 6 ns, respectively.
1/2
Problem 9: Impact of Space Charge on Thermal Emission
A. Show that a 1-D sheet of charge density q2 produces a field of F  q2 / (2 o ) .
B. Using Gyftopolous-Levine theory for a Barium dispenser cathode (assume f = 1; you
may wish to consult www.webelements.com for the radii of atoms) and the
Richardson equation, estimate the emission current density for T = 1300 Kelvin and
an applied field of 10 MV/m for when the coverage 60% (take the work function for a
monolayer to be 2.1 eV, the chemical potential for Tungsten is 18.08 eV).
C. How much does the answer of part B change if the applied field is decremented by
the space charge field produced by a top-hat pulse? Assume that the pulse is 1 ns in
duration.
Problem 10: Impact of Space Charge on Photoemission
A. Repeat Problem 9B for photoemission conditions but for a pulse of 10 ps. You will
need to calculate the scattering factor and the quantum efficiency using the “sort of”
Fowler-Dubridge formula:
4
U   h   
QE  f 1  R  
(1.6)
U  
Assumptions: the (normal incident) laser intensity is 10 MW/cm2 for a wave length of
355 nm; the temperature rise is that as calculated in Problem 8, but for tungsten
parameters and using your approximation for l; the initial temperature of the cathode
is 300 K; the reflectivity and penetration depth for tungsten @ 355 nm are 48.7% and
10.47 nm, respectively.
B. Repeat problem 9C but for a pulse of 10 ps.
note: this is a complex problem with many layers, requiring numerous factors. Solving Problem 9 first is
strongly suggested. If stymied in the determination of some of the needed parameters, develop reasonable
approximations or guestimates to their magnitude and show how they were obtained.
Problem 11: Thermal Emission, Field Enhancement, and Dark Current
In experiments at Los Alamos [i], photocathodes were observed to give dark current
when the laser was off. Using their observation that a 20% reduction in field caused a
factor of 5 decrease in background current, make an estimate of the beta factor, assuming
the macroscopic field is 26 MV/m. For sake of argument, assume that the work function
of the coated surface was 3 eV (it wasn’t – this is just a convenient number). How tall
would a needle-like protrusion have to be to give that field enhancement?
Alex H. Lumpkin, “Observations on Field-Emission electrons from the Lost Alamos FEL Photoinjector,”
Particle Accelerator Conference 1991.
i
5