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-EF)/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/Ek)1/2. Since the kinetic energy Ek 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: pp = pp. The planar barrier acts on the momentum transversal to the surface, pt, the energy outside is reduced by the barrier, E = E I = (1/2m)(pp2 + pt2). The transverse momentum inside is pt = p cos(q)
Thus the electron is refracted by the surface. Only electrons which have sufficient transverse momentum can escape to vacuum:
pt2/2m = (p cos(q))2/2m = E cos2(q) > I
This defines an "escape cone" of angle:
qmax = cos-1 (I/E)½ = cos-1(I/(E+I))½
This means that when q < qmax 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(qmax) » (1-E'/I)½, or qmax»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.
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) T2 exp(-f/kT)
where A = 4pmk2e/h3 = 120 Amperes/cm2 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 LaB6 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 Poissons 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 Va3/2 (law of Child-Langmuir) and saturating at large Va to the value predicted by the Richardson-Dushman equation. This describes well the behavior of vacuum diodes.
A closer look at a the current-voltage characteristic of a real diode show that it doesnt really saturate, it has a small slope that depends on the electric field at the cathode (hence it depends on geometry, for a given Va.) 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) T2 exp[-(f-F1/2)/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 Vanode at different temperatures T. One then extrapolates the "saturation" region to Vanode = 0 to get rid of the Schottky effect, the extrapolated currents are Jo(T). One then does a plot of ln Jo(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
Copyright 2002, by Raúl Baragiola, University of Virginia. All rights reserved.