Atomic Force Microscopy

This chapter discusses atomic force microscopy (AFM), focusing on the methods for atomic force detection. Although the force detection always requires a cantilever, there are two types of modes: the static mode and the dynamic mode. The general design and the typical method of manufacturing of the cantilevers are discussed. Two popular methods of static force detection are presented. The popular dynamic-force detection method, the tapping mode is described, especially the methods in liquids. The non-contact AFM, which has achieved atomic resolution in the weak attractive force regime, is discussed in detail. An elementary and transparent analysis of the principles, including the frequency shift, the second harmonics, and the average tunneling current, is presented. It requires only Newton’s equation and Fourier analysis, and the final results are analyzed over the entire range of vibrational amplitude. The implementation is briefly discussed.

Nanomechanical Effects

This chapter discusses the effect of force and deformation of the tip apex and the sample surface in the operation and imaging mechanism of STM and AFM. Because the contact area is of atomic dimension, a very small force and deformation would generate a large measurable effect. Three effects are discussed. First is the stability of the STM junction, which depends on the rigidity of the material. For soft materials, hysterisis is more likely. For rigid materials, the approaching and retraction cycles are continuous and reproducible. Second is the effect of force and deformation to the STM imaging mechanism. For soft material such as graphite, force and deformation can amplify the observed corrugation. For hard materials as most metals, force and deformation can decrease the observed corrugation. Finally, the effect of force and deformation on tunneling barrier height measurements is discussed.

Imaging Wavefunctions

The concept of wavefunction was introduced in the first 1926 paper by Erwin Schrödinger as the central object of the atomic world and the cornerstone of quantum mechanics. It is a mathematical representation of de Broglie’s postulate that the electron is a material wave. It was defined as everywhere real, single-valued, finite, and continuously differentiable up to the second order. Nevertheless, for many decades, wavefunction has not been characterized as an observable. First, it is too small. The typical size is a small fraction of a nanometer. Second, it is too fragile. The typical bonding energy of a wavefunction is a few electron volts. The advancement of STM and AFM has made wavefunctions observable. The accuracy of position measurement is in picometers. Both STM and AFM measurements are non-destructive, which leaves the wavefunctions under observation undisturbed. Finally, the meaning of direct experimental7 observation and mapping of wavefunctions is discussed.

Piezoelectric Scanner

This chapter discusses the physical principle, design, and characterization of piezoelectric scanners, which is the heart of STM and AFM. The concept of piezoelectricity is introduced at the elementary level. Two major piezoelectric materials used in STM and AFM, quartz and lead zirconate titanate ceramics (PZT), are described. After a brief discussion of the tripod scanner and the bimorph, much emphasis is on the most important scanner in STM and AFM: the tube scanner. A step-by-step derivation of the deflection formula is presented. The in-situ testing and calibration method based on pure electrical measurements is described. The formulas of the resonance frequencies are also presented. To compensate the non-linear behavior of the tube scanner, an improved design, the S-scanner, is described. Finally, a step-by-step procedure to repole a depoled piezo is presented.

Scanning Tunneling Spectroscopy

This chapter discusses various aspects of scanning tunneling spectroscopy (STS). It is an extension of the classical tunneling spectroscopy experiment to nanometer-scale or atomic-scale features on the sample surface. First, the electronics for STS is presented. The nature of STS as a convolution of tip DOS and sample DOS is discussed. Special tip treatment for the STS experiment, often different from the atomic-resolution STM, is described. The purpose is to produce tips with flat DOS, instead of special tip orbitals. Experimental methods to determine tip DOS is discussed. A detailed account of the inelastic scanning tunneling spectroscopy, or STM-IETS, is then discussed. It includes the principles, the electronics, and the instrumental broadening of the features. This chapter concludes with the STS study of superconductors, especially High-Tc supercondoctors.

Atomic Forces

This chapter discusses the physics and properties of four types of atomic forces occurring in STM and AFM: the van der Waals force, the hard core repulsion, the ionic bond, and the covalent bond. The general mathematical form of the van der Waals force between a tip and a flat sample is derived. The focus of this chapter is the covalent-bond force, which is a key in the understanding of STM and AFM. The concept of covalent bond is illustrated by the hydrogen molecular ion, the prototypical molecule used by Pauling to illustrate Heisenberg’s concept of resonance. The Herring-Landau perturbation theory of the covalent bond, an analytical incarnation of the concept of resonance, is presented in great detail. It is then applied to molecules built from many-electron atoms, to show that the perturbation theory can be applied to practical systems to produce simple analytic results for measurable physical quantities with decent accuracy.

Tip Treatment

This chapter discusses various methods for tip treatment. First, a general discussion about the experimental facts of STM and AFM tips is presented, which points to the subtleties and significance of the last few atoms at the tip apex. The standard method of making an STM tip is the electrochemical etching of a tungsten wire. The experimental procedure is described in detail. The study of the tip using field-ion microscopy is outlined. The tungsten tips freshly made from electrochemical etching often do not provide atomic resolution. Ex-situ and in-situ tip treatments are necessary. Several ex-situ tip treatment methods are described, inducing annealing, field evaporation, and annealing with a field. In-situ tip treatment method such as high-field treatment and controlled collision are described. Then, tip treatment for electrochemical STM is described. Tip treatment methods for spin-polarized STM are described. Finally, tip functionalization, especially with Xe atom and CO molecule, is described.

Overview

This chapter presents the basic designs and working principles of STM and AFM, as well as an elementary theory of tunneling and the imaging mechanism of atomic resolution. Three elementary theories of tunneling are presented: the one-dimensional Schrödinger’s equation in vacuum, the semi-classical approximation, and the Landauer formalism. The relation between the decay constant and the work function, and a general expression of tunneling conductance versus tip-sample distance are derived. A brief summary of experimental facts on the mechanism of atomic resolution STM and AFM is presented, which leads to a picture of interplay between the atomic states of the tip and the sample, as well as the role of partial covalent bonds formed between those electronic states. Four illustrative applications are presented, including imaging self-assembed molecules on solid-liquid interfaces, electrochemical STM, catalysis research, and atom manipulation.

Nanometer-Scale Imaging

This chapter discusses the imaging mechanism of STM at the nanometer scale, where the features of interest are of about one nanometer and up. Using an s-wave tip model, using the Bardeen tunneling theory, Tersoff and Hamann showed that the STM image in this case is tip-independent: it is determined by the local density of states of the bare sample surface at Fermi level, taken at the center of curvature of the tip. The Tersoff-Hamann model has found numerous applications in interpreting the STM images, from the superstructure of surface reconstruction to the confined or scattered waves of the surface states. However, as shown by Tersoff and Hamann in their original papers, for features much smaller than one nanometer, such as at the atomic features of 0.3 nm, the non-spherical electronic states of the tip could play a significant role and thus cannot be overlooked.

Atomic-Scale Imaging

This chapter discusses the imaging mechanism of STM and AFM at the atomic scale. Experimental facts show that at atomic resolution, tip electronic states play a key role. Analytic theoretical treatments provide quantitative explanation of the effect of the tip electronic states. On transition-metal tips, first-principle studies unanimously show that d-type tip electronic states dominate the Fermi-level DOS. First-principle studies of the combined tip-sample systems show that for both STM and AFM, the p- and d-type tip electronic states are the keys to understanding the atomic-scale images. The case of spin-polarized STM and the chemical identification of surface atoms are also discussed in terms of tip electronic structure. The chapter concludes with discussions of experimental verifications of the reciprocity principle: at atomic resolution, the role of tip electronic states and the sample electronic states are interchangeable.