Generalized expression for the tunneling current in scanning tunneling microscopy

We present an analytical method for treating the tunneling current between a tip and a sample in scanning tunneling microscopy (STM) that goes beyond the independent-electrode (Bardeen) approximation and is valid for smaller tip-to-surface separations. The extremity of the tip is represented by a single spherical potential well. This well is strongly coupled to neighboring tip atoms, as well as the sample electrode, both of which we leave in a general form. The wave function for the entire system is obtained by a matching procedure, from which the total current is determined. If the current is associated with s-derived tip orbitals, the result is comparable in simplicity with that of J. Tersoff and D. Hamann [Phys. Rev. B 31, 805 (1985)]. The low-bias tunnel conductance is proportional to the local density of states (LDOS) of the surface, but renormalized to include multiple reflections to all orders: σ∝${\mathrm{\rho }}_{\mathit{s}}$(${\mathbf{r}}_0$,${\mathit{E}}_{\mathit{F}}$)/D, where D depends on both the tip and sample electronic structures and on the tip position ${\mathbf{r}}_0$. This effect includes the modification of the surface LDOS due to the presence of the tip. A compact expression is also obtained for orbitals of higher angular momenta: p and d states. The current then depends on the gradients of the surface spectral density, and not on the LDOS, and also has a characteristic denominator. We discuss the significance of this effect, both in the interpretation of STM images and related spectroscopies.