Field emission as a probe of the surface density of states

Field-emission measurements of the total-energy distribution from a clean metal surface are shown to provide information about the density of states near the surface. Specifically, we find the field-emitted current per unit energy at energy $\omega$ to be given approximately by $j\left(\omega \right)\simeq \left(\frac{2\hslash }{m}\right)S{\lambda }^{-2\mathrm{}}\left(\omega \right)\Sigma m^{}{D}_0^2\left(E_{\perp }^m\right)\times {\left|{\psi }_m\right(x_m\left)\right|}^2\delta (\omega -{\epsilon }_m)$, where $D_0^2$ is the usual barrier-penetration probability with image potential corrections, $E_{\perp }^m=\omega -\frac{{\hslash }^2{k}_{\parallel }^2}{2m}$, where $k_{\parallel }$ is the electron momentum parallel to the surface, $\left|{\psi }_m\right(x_m\left)\right|$ is the amplitude of the metal electron at the classical turning point ($x_m\sim 1-2$ ÅA), $\lambda \left(\omega \right)$ is a slowly varying function of $\omega$, and $S$ is the metal surface area. The $D_0^2$ factor in $j\left(\omega \right)$ strongly weights electron states with small $k_{\parallel }$ and consequently $j\left(\omega \right)$ measures the density of states at $x_m$ arising from the component of the bulk band structure normal to the surface. Measurements of $j\left(\omega \right)$ for several single-crystal planes of tungsten will be presented and compared to the relevant photoemission data.