ar X iv : c on d - m at / 0 20 92 12 v 1 9 S ep 2 00 2 COULOMB SINGULARITY EFFECTS IN TUNNELLING SPECTROSCOPY OF INDIVIDUAL IMPURITIES

Non-equilibrium Coulomb effects in resonant tunnelling through deep impurity states are analyzed. It is shown that Coulomb vertex corrections to the tunnelling transfer amplitude lead to power law singularity in current-voltage characteristics. Localized states of individual impurity atoms and interacting impurity clusters can play the key role in tunnelling processes in small size junctions and often determine the behavior of tunnelling characteristics in STM/STS contacts. Now it is evident that in tunnelling junctions of nanometer scale there exists non-equilibrium distribution of tunnelling electrons which changes local density of states and tunnelling conductivity spectra. Some interesting effects, such as resonance structure of tunnelling conductivity inside semiconductor band gap, increased value of observed band gap and non-equilibrium interaction of neighboring impurity atoms have been recently investigated experimentally and theoretically analyzed [1, 2, 3, 4]. But all these effects are caused by local changes of the initial density of states in the contact area due to interactions of non-equilibrium particles. The modification of tunnelling amplitude by the Coulomb interaction of conduction electrons in metallic tip with non-equilibrium localized charges was ignored. It is shown in the present paper that corrections to the tunnelling vertex caused by the Coulomb potential can also result in nontrivial behavior of tunnelling characteristics and should be taken into account. One encounters with effects similar to the Mahan edge singu-larities in the problem of X-ray absorption spectra in metals [5]. The effect is well pronounced if tunnelling rate from a deep impurity level to metallic tip γ t is much larger than relaxation rate γ of non-equilibrium electron distribution at localized state. This condition can be realized experimentally for a deep impurity state in the semiconductor gap. Direct tunnelling from such states to semiconductor continuum states is strongly reduced due to wide barrier formed by surface band bending. Relaxation rate connected with electron-phonon interaction can be estimated to be of the order 10 8 − 10 10 1/s at low temperatures [6]. As to γ t — it is parameter, which can be varied in STM/STS experiments changing tip-sample separation. Since tip-sample separation is comparable with atomic scale, γ t often exceeds the relaxation rate for deep impurity states. Typical experimental value of tunnelling current 1 nA corresponds to γ t ≃ 10 11 − 10 12 1/s [3]. As it will be shown below for γ t ≫ γ the impurity level becomes nearly empty when the value …