6.15 Thermionic Emission

The valence electrons in a block of metal have tremendous kinetic energy, of the order of electron volts. These electrons would like to escape the confines of the block, but attractive forces exerted by the nuclei hold them back. However, if the temperature is high enough, typically 1,000 to 2,500 K, a few electrons can pick up enough thermal energy to get away. The metal then emits a current of electrons. This is called “thermionic emission.” It is important for appli­cations such as electron tubes and fluorescent lamps.

The amount of thermionic emission depends not just on temperature, but also on how much energy electrons inside the metal need to escape. Now the energies of the most energetic electrons inside the metal are best expressed in terms of the Fermi energy level. Therefore, the energy required to escape is conven­tionally expressed relative to that level. In particular, the additional energy that a Fermi-level electron needs to escape is tradi­tionally written in the form $e\varphi_{\rm {w}}$ where $e$ is the electron charge and $\varphi_{\rm {w}}$ is called the “work function.” The magnitude of the work function is typically on the order of volts. That makes the energy needed for a Fermi-level electron to escape on the order of electron volts, comparable to atomic ionization energies.

The thermionic emission equation gives the current density of electrons as, {D.29},

$$j = A T^2 e^{-e \varphi_{\rm {w}}/k_{\rm B}T} %$$ (6.22)

where $T$ is the absolute temperature and $k_{\rm B}$ is the Boltzmann constant. The constant $A$ is typically one quarter to one half of its theoretical value

$$A_{\rm theory} = \frac{m_{\rm e}e k_{\rm B}}{2\pi^2 \hbar^3} \approx 1.2\;10^6 \mbox{ amp/m$^2$-K$^2$}$$ (6.23)

Note that thermionic emission depends exponentially on the temperature; unless the temperature is high enough, extremely little emission will occur. You see the Maxwell-Boltzmann distribution at work here. This distribution is applicable since the number of electrons per state is very small for the energies at which the electrons can escape.

Despite the applica­bility of the Maxwell-Boltzmann distribution, classical physics cannot explain thermionic emission. That is seen from the fact that the constant $A_{\rm {theory}}$ depends non­trivially, and strongly, on $\hbar$. The dependence on quantum theory comes in through the density of states for the electrons that have enough energy to escape, {D.29}.

Thermionic emission can be helped along by applying an additional electric field ${\cal E}_{\rm {ext}}$ that drives the electrons away from the surface of the solid. That is known as the “Schottky effect.” The electric field has the approximate effect of lowering the work function value by an amount, {D.29},

$$\sqrt{\frac{e{\cal E}_{\rm ext}}{4\pi\epsilon_0}}$$ (6.24)

For high-enough electric fields, significant numbers of electrons may also “tunnel” out due to their quantum uncertainty in position. That is called “field emission.” It depends exponentially on the field strength, which must be very high as the quantum uncertainty in position is small.

It may be noted that the term “thermionic emission” may be used more generally to indicate the flow of charge carriers, either electrons or ions, over a potential barrier. Even for standard thermionic emission, it should be cautioned that the work function depends criti­cally on surface conditions. For example, surface pollution can dramati­cally change it.

Key Points

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Some electrons can escape from solids if the temperature is sufficiently high. That is called thermionic emission.

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The work function is the minimum energy required to take a Fermi-level electron out of a solid, per unit charge.

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An additional electric field can help the process along, in more ways than one.