Elsevier

Surface Science

Volumes 502–503, 10 April 2002, Pages 26-33
Surface Science

Theory of vibrational tunneling spectroscopy of adsorbates on metal surfaces

https://doi.org/10.1016/S0039-6028(01)01894-5Get rights and content

Abstract

A theoretical model for studying elementary processes in vibrational scanning tunneling microscopy/spectroscopy is presented. General formulae for the elastic and inelastic tunneling currents are derived on the basis of the Anderson Hamiltonian coupled to a vibrational degree of freedom. Our treatment includes the vibrational damping due to electron–hole pair excitations. It is found that the non-equilibrium situation plays a crucial role for opening of the inelastic channel and excitations of adsorbate phonons. Elementary processes for both elastic and inelastic tunneling currents are clarified in a qualitative manner. The distribution function of adsorbate phonons is derived and may be of great importance for the understanding of the recently observed current-induced dynamical motions of adsorbates.

Introduction

A variety of interesting experimental results has been reported recently in the field of scanning tunneling microscopy/spectroscopy (STM/STS) of adsorbates on metal surfaces. Particularly, inelastic tunneling spectroscopy provides a new tool to investigate vibrational properties of a single molecule adsorbed on metal surfaces with atomic resolution [1]. This technique is also capable of mapping spatial distributions of vibrational intensities [2], [3]. In addition to the identification of single adsorbed molecule through vibrational fingerprint, it was demonstrated that the inelastic current may induce dynamical motions of adsorbate including rotation, dissociation, desorption and diffusion [4], [5], [6], [7]. These current-induced dynamic processes may be important for a comprehensive understanding of the mechanism of a single atom manipulation [8].

Many theoretical attempts have been made to elucidate the elementary properties involved in the vibrational STS. Persson and Demuth [9] first discussed the inelastic tunneling on the scheme of dipole scattering theory using Bardeen's formula for electric current. Later, Persson and Baratoff [10] showed that the resonant tunneling via adsorbate states could be a dominant channel for inelastic current and the inelastic fraction of electric current was related to the vibrational damping rate due to electron–hole pair excitations. Electron–phonon interaction effects in tunneling junctions were also studied by Caroli et al. [11]. They derived analytical expressions for elastic and inelastic currents based on the non-equilibrium Green's function method and suggested a possible application to the internal vibrations of adsorbate molecules. The calculation of spatially resolved inelastic current was performed by Mingo and Makoshi using a linear combination of atomic orbitals approach [12] and Lorente and Persson using density functional treatment [13]. However effects of vibrational damping rate on the inelastic current were neglected in both studies.

Recently, we have proposed [14] a theoretical description of vibrational STS on the basis of the Anderson Hamiltonian supplemented with a phonon system using a non-equilibrium diagram technique. Along this line, we develop our theory to derive the elastic and inelastic currents and a relation between the inelastic fraction of tunneling current and the vibrational damping. Interpretation of each component included in the expression of current provides a deep insight into the vibrational STS. The deviation of distribution function of a vibrational state from its equilibrium is also discussed. It is emphasized that this deviation is not necessarily small as was assumed in the previous works even for the weak electron–phonon coupling.

We analyze, depending on the material parameters of the system, when the differential tunneling conductance d2j/dV2 either increases or decreases upon the opening of the inelastic tunneling channel. It is demonstrated under what condition the results obtained by Persson and Baratoff [10] are reproduced. The distribution function for phonons is also derived. It is found that the phonon distribution function contains an additional contribution arising from the opening of the inelastic current. This enables us to introduce an effective temperature of the phonon system. It is suggested that the overheating by the inelastic current and the bias voltage may play important roles in understanding of the current-induced dynamical motions of adsorbates.

Section snippets

Model and formalism

The total system consists of subsystems, a substrate, a tip, an adsorbate orbital and an adsorbate phonon, whose energies and annihilation operators are denoted by ϵp, ϵk, ϵa, Ω and cp, ck, ca, b, respectively. For simplicity, only a single adsorbate orbital and a single adsorbate phonon mode are taken into consideration,H0=∑kϵkckck+∑pϵpcpcpacaca+Ω(bb+1/2).Here after, we use the atomic unit where ℏ=1. The tip and substrate systems are assumed to be in thermal equilibrium at the same

Electric current and other physical quantities

The static total current jtot is given by the time derivative of the total electron number in the tip Nt(t) or the substrate Ns(t) according to a recipe for the electronic transport via a quantum dot [17], [18], [19];jtot=−eddtNt(t)=+eddtNs(t),jtot=2e∫dϵπΔsΔtGra(ϵ)Gaa(ϵ)(ns(ϵ)−nt(ϵ))+2e∫dϵπΔsΔtGra(ϵ)Gaa(ϵ)ns(ϵ)−nt(ϵ)Γad(ϵ)Δ.Here denotes the time-average. The total current includes an inelastic current as well as an elastic current. The damping rate of electrons in the adsorbate due to

Concluding remarks

A theoretical framework for studying elementary processes of the vibrational STS has been presented on the basis of Anderson model for adsorbates coupled to a phonon degree of freedom. This enabled us to clarify the microscopic processes of the tunneling in the presence and absence of vibrational excitations and deexcitations. The present treatment also encompasses the statistical properties of phonon distribution. Upon the opening of the inelastic tunneling channel, the effective temperature

Acknowledgements

The authors are grateful to Professor K. Makoshi for many stimulating discussions. S.T. thanks Russian Basic Research Foundation and Russian Ministry of Science program “Nanostructures” for partial support.

References (21)

  • B.N.J. Persson et al.

    Solid State Commun.

    (1986)
  • S. Tikhodeev et al.

    Surf. Sci.

    (2001)
  • B.N.J. Persson et al.

    Solid State Commun.

    (1980)
  • K. Makoshi et al.

    Surf. Sci.

    (1996)
  • T. Mii et al.

    Surf. Sci.

    (2001)
  • R.E. Walkup et al.

    J. Electron. Spectrosc. Relat. Phenom.

    (1993)
  • B.C. Stipe et al.

    Science

    (1998)
  • L.J. Lauhon et al.

    Phys. Rev. Lett.

    (1999)
  • J.I. Pascual et al.

    Phys. Rev. Lett.

    (2001)
  • B.C. Stipe et al.

    Phys. Rev. Lett.

    (1998)
There are more references available in the full text version of this article.

Cited by (0)

View full text