Wavefunctions tunneling

Analysis of the radial pair distribution function for the electron centroid and solvent center-of-mass computed at different densities reveals some very interesting features. At high densities, the essentially localized electron is surrounded by the solvent resembling the solvation of a classical anion such as Cr or Br. At low densities, however, the electron is sufficiently extended (delocalized) such that its wavefunction tunnels through several neighboring water or ammonia molecules (Figure 16-9). 

When the linewidth exhibits no oscillations, this suggests the occurrence of an inner crossing, but two cases exist where an outer crossing is shown to display no linewidth oscillation. The first example concerns the OD molecule. Below the energy of the curve crossing, the bound free vibrational overlap comes only from the tail of the discrete wavefunction (tunnelling). The nonradiative decay rate is very slow, but it increases smoothly with J [predissociation of the OD A2E+(v = 0-2) levels by the 4E state (Bergeman, et ai, 1981)]. 

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

The basis of the scanning tunnelling microscope, illustrated schematically in Figure 3.5, lies in the ability of electronic wavefunctions to penetrate a potential barrier which classically would be forbidden. Instead of ending abruptly at a... 

Fig. 1 Schematic drawings of a tunnel diode, an STM, and the electronic energy diagram appropriate for both. U is the height of the potential barrier, E is the energy of the incident electron, d is the thickness of the barrier, A is approximately 1.02 A/(eV)1/2 if U and E are in electron volts and d is in angstroms, /0 is the wavefunction of the incident electron, and /d is the wavefunction after transmission through the barrier. I is the measured tunneling current, V is the applied bias, and M and M are the electrode metals...

Fig. 5. Tunneling describes the quantum mechanical delocalization of a particle into the region where the total energy (dotted line) is less than the potential energy V(x). The tunneling barrier V(x) — E enters directly into the WKB formula for the wavefunction tail, whereas the zeroth order wavefunctions are typically expressed in a basis set expansion which may or may not be optimized for its tunneling characteristics...

Since the particle (electron) in one redox site can respond to the presence of a second deep well out in the distance only to the extent that this quantum tail is non-zero, the tunneling splitting A decays essentially exponentially just as the wavefunction tail in Eqs. 18-19 above. More precisely, we write ... 

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 s-wave-tip model (Tersoff and Hamann, 1983, 1985), the tip was also modeled as a protruded piece of Sommerfeld metal, with a radius of curvature R, see Fig. 1.25. The solutions of the Schrodinger equation for a spherical potential well of radius R were taken as tip wavefunctions. Among the numerous solutions of this macroscopic quantum-mechanical problem, Tersoff and Hamann assumed that only the s-wave solution was important. Under such assumptions, the tunneling current has an extremely simple form. At low bias, the tunneling current is proportional to the Fermi-level LDOS at the center of curvature of the tip Pq. 

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.

We choose the bisection plane in the barrier as the separation surface. The correction for tunneling current can be obtained from the correction for the wavefunction on the bisection plane. Using the Green s function method, following Eq. (2.42), the correction factor for the wavefunction at z = W/2 is... 

The wavefunction of another electrode gets the same factor. The tunneling current is proportional to the square of the matrix element. Thus,... 

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 ... 

From Eq. (3.1), the tunneling matrix elements are determined by the wavefunctions of the tip and the sample at the separation surface, which is located roughly in the middle of the vacuum gap, as shown in Fig. 3.1. For both tip states and sample states near the Fermi level, the wavefunctions on and beyond the separation surface satisfy Schrodinger s equation in the vacuum. 

Therefore, the tunneling matrix element for a p, tip state is proportional to the j derivative of the sample wavefunction at the center of the apex atom. 

The tunneling matrix elements from the rest of the nine tip wavefunctions can be derived using the relation between the tip and Green s functions established in the previous section. For example, for the d), tip state. 

The steps from Eq. (3.34) to Eq. (3.36) simply mean that for each component of the tip wavefunction with angular dependence characterized by landm, the tunneling matrix element is proportional to the corresponding component of the sample wavefunction with the same angular dependence. 

As we have shown in Chapters 2 and 3, under the normal operating conditions of STM, the tunneling current can be calculated from the wavefunctions a few A from the outermost nuclei of the tip and the sample. The wavefunctions at the surfaces of solids, rather than the wavefunctions in the bulk, contribute to the tunneling current. In this chapter, we will discuss the general properties of the wavefunctions at surfaces. This is to fill the gap between standard solid-state physics textbooks such as Kittel (1986) and Ashcroft and Mermin (1985), which have too little information, and monographs as well as journal articles, which are too much to read. For more details, the book of Zangwill (1988) is helpful. 

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. 

Surface states on d band metals and semiconductors are important examples of surface wavefunctions, which may dominate the tunneling current. On many metal surfaces, the tails of the bulk states dominate. For example, on the surfaces of Pt and Ir, the tails of the bulk states dominate -he wavefunctions at surfaces, and can be represented with reasonable accuracy as linear combinations of atomic states (LCAO). 

---