## Introduction

Imaging the landscapes of solid surfaces ex-situ and in-situ is nowadays an established method for gaining information about their local structure. Reliable data about the structure of adsórbate overlayers on top of the substrate can also be obtained by this technique (see other papers in this volume), if the atoms are strongly adsorbed. What to do, however, with the sluggish adsorbed species, solvent, and any other molecules moving in the vicinity of or under the tip? They remain invisible in the conventional scanning probe methods, contributing to the noise. Noise in STM current is something that one usually tends to depress, get rid of or subtract as a factor obscuring the main signal. However, an alternative strategy may be followed, based on the theory of elastic tunneling including the current fluctuations due to motions of particles between the tip and the sample [1]. The latter suggests new ways for the study of local dynamics of weakly adsorbed species on metallic substrates. The ideas of the program, „Noise: from an enemy to a friend", are discussed in this paper.

The second question we tackle here is how to restore the true shape of the adsórbate nanometer-scale clusters from the images drawn by a tip that is not atomically sharp.

These „exotic" ideas for nanoprobe imaging are not pure speculations: some progress in their experimental realization is briefly discussed.

### 2 Adatom diffusion from STM noise

Adatoms moving along the substrate surface under an STM tip introduce an additional current noise [2, 3]. Random jumps of adatoms under an STM tip and the current dependence on time, I(t), caused by these jumps, are shown schematically in Fig. 1. The random function 7(f) found experimentally can be used to obtain information about the diffusion process. It is the task of the theory to connect 1(f) with the parameters of the system under investigation, in particular with the diffusion coefficients of adatoms and their density.

Gomer in his pioneering work [4] solved this problem (in the context of field emission spectroscopy) for the case of weak interaction between tunneling electrons and adsorbed atoms, i.e., when the current enhancement due to the appearance of an adsorbed atom under an STM tip is small. However, in experiments [2] one finds the opposite situation: the current peaks that appear when adsórbate is passing the region under an STM tip is several times larger than the stationary current. The solution of the indicated problem for the case of relatively strong electron-adatom interaction was obtained by Sumetskii et al. [5], where the strict asymptotic approach for finding the transparency of potential barriers with nonseparable variables [5,6] and the ideas of Sumetskii and Kornyshev [1] were combined. We briefly discuss the main ideas and results of this work.

The idea of noise spectroscopy is to determine for a measured 7(t) the average current, J = < 7(i) >, and also the temporary current correlation function,

BEYOND THE LANDSCAPES:IMAGING THE INVISIBLE K(t) = <l(t + r)l(r)>T-<I>2

Fig. 1. (a) Random jumps of adatoms (black dots) along the substrate surface under STM tip. (b) Sketch of the current / as a function of time t caused by adatom diffusion. The peaks correspond to individual adatoms passing near the STM tip. Steps in each peak correspond to the elementary jumps of the adatom.

In the course of the measurement, the tip is positioned at a place of interest above the sample. Having obtained enough i-points to perform the time averaging, one may move to another space point in order to end up with the maps of I (u) and K(t,u) [u=(x,y) is the lateral coordinate along the surface]. Another important characteristic would be the current noise spectral density S(co), the Fourier transform of K(t):

and its spatial map 5(o,u)

Which information, can be extracted from the measured </>, K(t), and S(m)l This is the question that the theory is required to solve. The answer can be given for a specified type of random process and some simplifying assumptions. The process of random diffusion was studied by Sumetskii et.al. [5] and assumptions were made that allowed expression of the introduced functions via the parameters of the tunneling gap and of the moving atoms. The central result was the analytical expression for the correlation function K(t), found for a relatively small adsórbate density.

### 2.1 Assumption and basic equations

Having ignored the lattice structure of the flat substrate surface, the barrier between the STM tip and the substrate was modeled by a bare potential Fo(r). The absolute value of electron momentum in the empty tunneling gap is then given by where Ef is the Fermi energy.

The adatom j, disposed in the barrier region at a point rj , is modeled by a potential well, U(r-rj), so that the potential barrier between the tip and the substrate, F(r), is the sum of the bare barrier and the atom potentials:

Different approaches were used by Sumetskii and Kornyshev [1] in order to solve the Schrodinger equation with potential (Eq. (4)) having a rectangular form for the strong and relatively weak electron-atom interaction, J7(r-rj), respectively. When applied to the problem of diffusion along the flat, surface, these approaches give similar results expressing <I(t)> and K(t) through the scattering cross-section a of the tunneling electron on an adsorbed atom. The results of Ref. [1] were generalized [5] by means of the semiclassical perturbation theory with respect to, £/(r-rj), to the case of an arbitrary semiclassical potential Fo(r). For this the scattering cross-section

<7= ji/u was introduced, where U(u,z) = U(r) and p(z)=p(0,0,z) (see Fig. 1). Usually, the scattering cross-section a has the order of atom "geometrical" cross-section nb2 = 10 A2.

The analytical expression for the current through the tunneling gap was obtained and then averaged over the positions of atoms assuming that they move independently. The latter is valid for relatively low adsórbate density. Similarly to [1], the average current is then:

where 7o is the tunneling current through the empty gap, A^d is the density of adatoms, and N3I¡o-= N^b2« 1. Thus, adatoms in the low coverage limit introduce only a small correction to the average current.

The basic quantity which determines surface diffusion is the adatom distribution function /(u0,u,,/), i.e., the density of probability that the atom which is initially at a point uo appears at time t at the point uj. For the assumed low adsórbate density the expression for the correlation function K(t) through /(u0, u,, t) reads [5]:

K(t)=N*i°22Q2 Zp Jdu0du, /(u0,u„f)exp[-Q(ul + M,2)], (7)

where Q is the parameter characteristic of the bare gap. There is a defined system of equations for its determination [5] which we do not reproduce here. In the simplest case of a rectangular barrier when p(r) =po = constant in the barrier region, it gives

where d is the width of the tunneling gap. For the general form of the potential barrier, one can use the following approximate expression for Q [6]:

-l where zm is the coordinate of the maximum of the potential barrier. This expression is good for estimates of Q but, for a fixed value of d, one would rather treat Q as the free parameter of the tip sample configuration. Equation (7) is the basis for further calculations.

2.2. Temporary current correlation function and low-frequency noise for adatom diffusion

Let us use the distribution function which corresponds to the (generally) anisotropic surface diffusion of adatoms:

where Dx and Dy are the diffusion coefficients along axes x and y of the surface, the atomic structure of which is ignored. The integral of Eq. (7) with distribution function in Eq.(10) gives:

exp exp

where the two characteristic correlation times, are the times needed for an adatom to cross the current tube of tunneling along the x or y direction. In the case of isotropic diffusion Dx= Dy= D, tjx> = tc(y) = tc,, and Eq. (11) reduces to:

In the opposite case of strongly anisotropic (one-dimensional) diffusion with negligible

Vh^F

Noise spectral density (Eq. (2)) can be expressed through special functions. Since there is no problem in numerical calculation of S(co) using Eqs. (11) and (2), we will not set it out here. We present, however, simple results for the low-frequency noise. In the case of comparable tj^ and tc(v)):

Equation (15) diverges logarithmically. In the case of one-dimensional diffusion, the divergence is stronger: for <<co~' <<i'y) it follows 1!-4a> law:

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