##

## Thermionic Emission

Inside the solid, there is a small tail in the energy distribution of electrons that
extends to energies above the Fermi level. The energy distribution inside the solid is
N(E)P(E) where N(E) is the density of states (dn/dE) and P(E) is the Fermi-Dirac
distribution. At high E, the energy distribution is dominated by the strongly varying tail
of P(E),

P(E) ~ exp-(E-E_{F})/kT].

This tail is determined by fluctuations or unusual collisions between electrons.

*
**Electron escape from the solid*

Not all electrons that have energy above the vacuum level can escape. They also need to
be moving in the right direction. As we have seen, the surface barrier is corrugated, but
not within the range of the electron wavelength l. Recall that l(Å)=(150 eV/E_{k})^{1/2}. Since the kinetic energy
E_{k} decreases when going across the surface, the wavelength increases. Close to
the energy of the vacuum level, the wavelength is very large outside the solid, and of the
order of l(Å)=(150/11)^{1/2 }~ 3.7 Å inside
(neglecting band structure effects), where *I* = 11 eV is a typical value for the
inner potential. One can then consider the surface barrier to be approximately planar.

In a planar barrier, the lattice cannot take up momentum (except in
the Bragg condition where a reciprocal lattice vector is exchanged). The momentum of the
electron parallel to the surface is conserved: p_{p }= p_{p’. }The
planar barrier acts on the momentum transversal to the surface, p_{t}, the energy
outside is reduced by the barrier, E’ = E – *I *= (1/2m)(p_{p}^{2}
+ p_{t}’^{2}). The transverse momentum inside is p_{t
}= p_{ }cos(q)

Thus the electron is
refracted by the surface. Only electrons which have sufficient
transverse momentum can escape to vacuum:

p_{t}^{2}/2m = (p cos(q))^{2}/2m
= E cos^{2}(q) > I

This defines an "escape cone" of angle:

q_{max} = cos^{-1}
(I/E)^{½}
= cos^{-1}(I/(E’+I))^{½}

This means that when q_{ }< q_{max}
the electrons suffer total internal reflection.

In thermionic emission, electrons have energies barely above the vacuum level
(internally, E very close to I). This means cos(q_{max})
»
(1-E'/I)^{½}, or q_{max}»2E’/I,
very small. That means that not only
the electrons must be in the tail of the energy distribution to be emitted, but they also
must be traveling nearly perpendicular to the surface. In this case, one also needs to
consider a quantum mechanical effect: an electron that is above the barrier might be
reflected because of the uncertainty in the momentum caused by the width d of the barrier (Dp = ®/d). The reflection
coefficient, *r*, decreases with energy above the vacuum level and for gradual
barriers.

**
****Thermoionic current**

If an anode (metal electrode) with a positive potential with respect to the electron
emitter (cathode) is placed nearby, it will collect the emitted electron current. The
current density at the cathode is:

J = A (1-r) T^{2} exp(-f/kT)

where A = 4pmk^{2}e/h^{3} = 120 Amperes/cm^{2}
K^{-2 }is a fundamental constant, r is the reflection coefficient of the electrons
mentioned above, T is the temperature and f the work function
of the surface. This is called the Richardson-Dushman equation.

The equation is dominated by the exponential factor which is a very small number, since
a typical work function is much larger than thermal energies. Large thermionic currents
require high T. For a surface with f=4.5 eV at room
temperature, exp(-f/kT) ~ 10^{-78}, and 1.5x10^{-8}
at 3000 K. Increasing T does not ensure observable electron emission since materials
evaporate, melt or decompose when heated. Only refractory materials like W are useful as
thermionic sources of electrons. Most low work function materials are very volatile, an
exception is LaB_{6} with f = 2.5 eV, which is used in
the electron emitting filaments of electron microscopes. Other materials are thoria-coated
iridium, used in ionization vacuum gage tubes.

If the anode voltage is small, electrons will travel slowly in vacuum. Therefore, their
density, n = J/(ev) will be large (v is the electron velocity and e its charge.) This
density, called the "space charge", induces an electrostatic potential by
Poisson’s law that is negative in front of the cathode and, therefore, presents a
barrier. In other words, an electron that exits the solid with see a cloud of electrons in
front of the surface which may return it to the cathode. The space charge is responsible
for J increasing initially as V_{a}^{3/2 }(law of Child-Langmuir) and
saturating at large V_{a }to the value predicted by the Richardson-Dushman
equation. This describes well the behavior of vacuum diodes.

#### Schottky Effect

A closer look at a the current-voltage characteristic of a real
diode show that it doesn’t really saturate, it has a small slope that depends on the
electric field at the cathode (hence it depends on geometry, for a given V_{a}.)
The reason is that the electric field curves the surface potential (as seen when
discussing field emission). The electric field in a normal diode is not large enough for
"cold" field-emission, the barrier is too broad. However, the barrier is
decreased slightly, allowing more electrons to come out. The decrease of the barrier is
proportional to the square root of the field F. The modified equation is:

J = A (1-r) T^{2} exp[-(f-F^{1/2})/kT]

where a is a constant. The lowering of the barrier due to the electric field is called
the Schottky effect.

The equation can be used to determine the work function of the
surface. One measures J(T) as a function of V_{anode} at different temperatures T.
One then extrapolates the "saturation" region to V_{anode} = 0 to get
rid of the Schottky effect, the extrapolated currents are J_{o}(T). One then does
a plot of ln J_{o}(T)/T vs. 1/kT, which should be a straight line, and obtains f from the slope. If the temperature range is wide, one finds a
deviation from the straight line. This is because the work function changes slightly with
temperature because the solid expands, and because of changes in surface composition and
structure.

*updated September 28, 2000*