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

$$A_{\rm theory} = \frac{m_{\rm e}e k_{\rm B}}{2\pi^2 \hbar^3} \approx \mbox{1.2 10$\POW9,{6}$ amp/m$\POW9,{2}$K$\POW9,{2}$}$$ (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

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

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

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

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

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

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