Tunneling Current in Esaki Diodes

The integral giving the net tunneling current flowing across the junction in an Esaki diode, $I=A\int {E_c}^{E_v}\left\{f_c\mathrm{}\right(E)-f_v\mathrm{}(E\left)\right\}Z{\rho }_c\mathrm{}\left(E\right)\mathrm{}{\rho }_v\mathrm{}\left(E\right)\mathrm{dE}$ is evaluated under the normal assumptions that (${\xi }_c-E_c$) and ($E_v-{\xi }_v$) are of the order of $2kT$. The resulting expression is $I=-A^{\prime \prime }\frac{{(E_v-E_c)}^2(1-e^{\frac{\mathrm{qV}}{\mathrm{kT}}})}{\mathrm{}(m+n)\mathrm{}{e}^{\frac{a}{2}}+(1+e^{\frac{\mathrm{qV}}{\mathrm{kT}}})},$ where $A^{\prime \prime }$ is an arbitrary constant and $m$, $n$, and $a$ are functions of the Fermi levels on both sides of the junction, the location of the band edges and the absolute temperature. This expression is plotted as a function of the applied voltage for temperatures of 200°K, 300°K, and 350°K for donor and acceptor concentrations of ${10}^{19}$ ${\mathrm{cm}}^{-3\mathrm{}}$ and 1.6×${10}^{19}$ ${\mathrm{cm}}^{-3\mathrm{}}$, respectively. The resulting curves compare quite favorably with those of Esaki's.