Bardeen tunneling

Fig. 2.10. Different approximate methods for the square harrier problem. (Parameters used W= 2 A = 4 eV Utf= 16 eV.) The original Bardeen theory breaks down when the barrier top comes close to the energy level. The modified Bardeen tunneling theory is accurate with separation surfaces either centered L = lV/2) or off-centered L = VV73). By approximating the distortion of wavefunctions using Green s functions, the error in the entire region is only a few percent.

As we have discussed in Chapter 2, a direct consequence of the Bardeen tunneling theory (or the extension of it) is the reciprocity principle If the electronic state of the tip and the sample state under observation are interchanged, the image should be the same. An alternative wording of the same... 

J. Bardeen, Tunneling from a many-particle point of view, Phys. Rev. Lett. 6, 57 (1961). 

This section recalls very briefly the basic concepts necessary for the discussion. Electron tunneling has been established for several decades in vacuum [23] as well as in solid-state structures [24]. Quantum mechanics predicts that electrons can flow between two conductors separated by a distance of the order of 20 A. The energy diagram of a tunneling junction is sketched in Fig. 1 a. From the Bardeen tunneling... 

The most extensively used theoretical method for the understanding of the MIM tunneling junction is the time-dependent perturbation approach developed by Bardeen (1960). It is sufficiently simple for treating many realistic cases, and has been successfully used for describing a wide variety of effects (Duke, 1969 Kirtley, 1982). 

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

Fig. 1.20. The Bardeen approach to tunneling theory. Instead of solving the Schrddinger equation for the coupled system, a, Bardeen (1960) makes clever use of perturbation theory. Starting with two free subsystems, b and c, the tunneling current is calculated through the overlap of the wavefunctions of free systems using the Fermi golden rule.

In the interpretation of the experiment of Giaever (1960), Bardeen (1960) further assumed that the magnitude of the tunneling matrix element M does not change appreciably in the interval of interest. Then, the tunneling current is determined by the convolution of the DOS of two electrodes ... 

A systematic method of obtaining local tunneling spectra with STM was developed by Feenstra, Thompson, and Fein (1986). The details of this method will be described in Chapter 14. As expected from the Bardeen formula, the tip DOS plays an equal role as the sample DOS in determining the tunneling spectra. Because the tip is made of transition metals or semiconductors, the tip DOS is usually highly structured. In order to obtain reproducible tunneling spectra of the sample, special tip treatment procedures have to be conducted... 

The polarization, or the van der Waals interaction, can be accounted for by a stationary-state perturbation theory, effectively and accurately. The exchange interaction or tunneling can be treated by time-dependent perturbation theory, following the method of Oppenheimer (1928) and Bardeen (1960). In this regime, the polarization interaction is still in effect. Therefore, to make an accurate description of the tunneling effect, both perturbations must be considered simultaneously. This is the essence of the MBA. 

Fig. 2.6. Quantum transmission through a thin potential harrier. From the semi-classical point of view, the transmission through a high barrier, tunneling, is qualitatively different from that of a low barrier, ballistic transport. Nevertheless, for a thin barrier, here W = 3 A, the logarithm of the exact quantum mechanical transmission coefficient (solid curve) is nearly linear to the barrier height from 4 eV above the energy level to 2 eV below the energy level. As long as the barrier is thin, there is no qualitative difference between tunneling and ballistic transport. Also shown (dashed and dotted curves) is how both the semiclassical method (WKB) and Bardeen s tunneling theory become inaccurate for low barriers.

In this subsection, we show that using Schrbdinger s equations, the tunneling matrix element can be converted to a surface integral similar to Bardeen s. Using Schrbdinger s equation for the tip states, Eq. (2.25), the matrix element is converted into... 

As the semiclassical tunneling theory and Bardeen s original approach become inaccurate for potential barriers close to or lower than the energy level, the validity range of the MBA is much wider. In this subsection, the accuracy of the MBA is tested against an exactly soluble case, that is, the one-dimensional transmission through a. square barrier of thickness W=2 A (see Fig. 2.9). 

In Chapter 2, we showed that the tunneling current can be determined with a perturbation approach. The central problem is to calculate the matrix elements. Those are determined by the modified Bardeen surface integral, evaluated from the wavefunctions of the tip and the sample (with proper corrections) on a separation surface between them, as shown in Fig. 3.1 ... 

The surface states observed by field-emission spectroscopy have a direct relation to the process in STM. As we have discussed in the Introduction, field emission is a tunneling phenomenon. The Bardeen theory of tunneling (1960) is also applicable (Penn and Plummer, 1974). Because the outgoing wave is a structureless plane wave, as a direct consequence of the Bardeen theory, the tunneling current is proportional to the density of states near the emitter surface. The observed enhancement factor on W(IOO), W(110), and Mo(IOO) over the free-electron Fermi-gas behavior implies that at those surfaces, near the Fermi level, the LDOS at the surface is dominated by surface states. In other words, most of the surface densities of states are from the surface states rather than from the bulk wavefunctions. This point is further verified by photoemission experiments and first-principles calculations of the electronic structure of these surfaces. 

The following method, based on the modified Bardeen approach, allows the electronic states of the tip to be characterized. If the energy scale of feature in the DOS is larger than knT, the tunneling current is... 

Metal-vacuum-metal tunneling 49—50 Method of Harris and Liebsch 110, 123 form of corrugation function 111 leading-Bloch-waves approximation 123 Microphone effect 256 Modified Bardeen approach 65—72 derivation 65 error estimation 69 modified Helmholtz equation 348 Modulus of elasticity in shear 367 deflection 367 Mo(lOO) 101, 118 Na-atom-tip model 157—159 and STM experiments 157 NaCl 322 NbSej 332 NionAu(lll) 331 Nucleation 331... 

In the perturbative "transfer Hamiltonian approach developed by Bardeen 58), the tip and sample are treated as two non-interacting subsystems. Instead of trying to solve the problem of the combined system, each separate component is described by its wave function, i tip and i/zj, respectively. The tunneling current is then calculated by considering the overlap of these in the tunnel junction. This approach has the advantage that the solutions can be found, for many practical systems, at least approximately, by solution of the stationary Schrodinger equation. 

The purpose of doing STM is to learn about surface structures, and the tip as such is regarded as an uninteresting probe. In this sense, it is a problem that the electronic structure of the tip is contained in the formula for the tunnel current in the original work by Bardeen 58). Tersoff and Hamann 59,60), however, extended Bardeen s formalism and showed by simple, yet relevant approximations that the impact of the unwanted electronic structure of the tip is in many cases less pronounced for typical tunneling parameters. Fortunately, the Tersoff-Hamann model provides a simple conceptual framework for interpreting STM images, and therefore it is still the most widely used model. 

Despite the fact that both states belong to the same pes, an approach has been developed that enables this type of indirect photodissociation to be described as a quantum transition (33). The method is analogous to Bardeen s theory of tunneling (34)... 

By contrast the approach here is based on the theory of quantum transitions and is similar in approach to Bardeen s theory of tunneling (34). Further, in the present development, terms corresponding to higher-order transitions contain products of FC factors for different virtual transitions which results in additional orders of smallness in a perturbative sense. (This is additional justification for limiting consideration here to eq. 46.) This is in contrast to the theory of Schatz and Ross (36) and C. Villa et al. (37,38), which leads to a single FC factor. 

Usually tunneling through a potential barrier is considered on the basis of the stationary Schroedinger equation with the use of matching conditions. A different approach has been developed by Bardeen (34). Bardeen s method enables one to describe tunneling as a quantum transition and to use the Golden Rule in order to evaluate the probability of penetration through the barrier. A similar method has been used in Section III to describe vibrational predissociation. This section contains a short description of Bardeen s method (see refs. 39,82-84). 

In parallel, the related activity was in the field of single-electron shuttles and quantum shuttles [143-153]. Finally, based on the Bardeen s tunneling Hamiltonian method [154-158] and Tersoff-Hamann approach [159,160], the theory of inelastic electron tunneling spectroscopy (IETS) was developed [113-116,161-163],... 

Bardeen showed that to derive the tunneling matrix element, which represents the amplitude of electron transfer between the sample and tip, explicit expressions for the wavefunctions of the tip and sample were... 

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