D.30 Num­ber of con­duc­tion band elec­trons

This note finds the num­ber of elec­trons in the con­duc­tion band of a semi­con­duc­tor, and the num­ber of holes in the va­lence band.

By de­f­i­n­i­tion, the den­sity of states ${\cal D}$ is the num­ber of sin­gle-par­ti­cle states per unit en­ergy range and unit vol­ume. The frac­tion of elec­trons in those states is given by $\iota_{\rm {e}}$. There­fore the num­ber of elec­trons in the con­duc­tion band per unit vol­ume is given by

$$i_{\rm e} = \int_{{\vphantom' E}^{\rm p}_{\rm c}}^{{\vphan... ...m top}} {\cal D}\iota_{\rm e} {\,\rm d}{\vphantom' E}^{\rm p}\$$

where ${\vphantom' E}^{\rm p}_{\rm {c}}$ is the en­ergy at the bot­tom of the con­duc­tion band and ${\vphantom' E}^{\rm p}_{\rm {top}}$ that at the top of the band.

To com­pute this in­te­gral, for $\iota_{\rm {e}}$ the Maxwell-Boltz­mann ex­pres­sion (6.33) can be used, since the num­ber of elec­trons per state is in­vari­ably small. And for the den­sity of states the ex­pres­sion (6.6) for the free-elec­tron gas can be used if you sub­sti­tute in a suit­able ef­fec­tive mass of the elec­trons and re­place $\sqrt{{\vphantom' E}^{\rm p}}$ by $\sqrt{{\vphantom' E}^{\rm p}-{\vphantom' E}^{\rm p}_{\rm {c}}}$.

Also, be­cause $\iota_{\rm {e}}$ de­creases ex­tremely rapidly with en­ergy, only a very thin layer at the bot­tom of the con­duc­tion band makes a con­tri­bu­tion to the num­ber of elec­trons. The in­te­grand of the in­te­gral for $i_{\rm {e}}$ is es­sen­tially zero above this layer. There­fore you can re­place the up­per limit of in­te­gra­tion with in­fin­ity with­out chang­ing the value of $i_{\rm {e}}$. Now use a change of in­te­gra­tion vari­able to $u$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{({\vphantom' E}^{\rm p}-{\vphantom' E}^{\rm p}_{\rm {c}})/{k_{\rm B}}T}$ and an in­te­gra­tion by parts to re­duce the in­te­gral to the one found un­der ! in the no­ta­tions sec­tion. The re­sult is as stated in the text.

For holes, the de­riva­tion goes the same way if you use $\iota_{\rm {h}}$ from (6.34) and in­te­grate over the va­lence band en­er­gies.