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Le Journal de Physique Colloques

A MULTI-DIMENSIONAL TUNNELING THEORY WITH APPLICATION TO SCANNING TUNNELING MICROSCOPE

Z.-H. Huang, Paul H. Cutler, T. E. Feuchtwang, Emil Kazes

Published Nov 1, 1989• 0 citations
Publisher

Abstract

We have developed a formal general approach for evaluating the WKB wave function in a forbidden region of a non-separable potential in a three-dimensional space. The wave function is described by two sets of orthogonal wave-fronts, the equi-phase and equi-amplitude surfaces, or equivalently by two sets of paths defined to be normal to these surfaces respectively. We have extended a Huygens type construction to the forbidden region to obtain the multi-dimensional wave function. The construction determines the wave-fronts and the paths. It is found that the equations for the paths obtained from the construction are coupled to each other and do not reduce to a set of ordinary differential equations. However, if the incident wave is normal to the turning surface, the equations satisfied by the paths can be de-coupled. These equations are found to be equivalent to Newton's equations of motion for the inverted potential and energy. This characteristics has been used to calculate the current density distribution for an STM. In this paper, we use the same approach to derive the expressions for the lateral resolutions and corrugations for a model STM.

DOI
10.1051/jphyscol:1989806

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We have developed a formal general approach for evaluating the WKB wave function in a forbidden region of a non-separable potential in a three-dimensional space. The wave function is described by two sets of orthogonal wave-fronts, the equi-phase and equi-amplitude surfaces, or equivalently by two sets of paths defined to be normal to these surfaces respectively. We have extended a Huygens type construction to the forbidden region to obtain the multi-dimensional wave function. The construction determines the wave-fronts and the paths. It is found that the equations for the paths obtained from the construction are coupled to each other and do not reduce to a set of ordinary differential equations. However, if the incident wave is normal to the turning surface, the equations satisfied by the paths can be de-coupled. These equations are found to be equivalent to Newton's equations of motion for the inverted potential and energy. This characteristics has been used to calculate the current density distribution for an STM. In this paper, we use the same approach to derive the expressions for the lateral resolutions and corrugations for a model STM.
The WKB method is commonly used in semiclassical approximations to the wave function in both the classically allowed and the forbidden regions of a one-dimensional potential. In a multidimensional space, the method can be adapted to construct wave functions in an ``allowed'' region from classical trajectories or wave normals. However, in the ``forbidden'' region the WKB wave function is in general specified by two sets of wave fronts, the equiphase and the equiamplitude surfaces or equivalently by two sets of paths defined to be normal to these surfaces, respectively. We present a Huygens-type construction for obtaining these wave fronts and paths, which reveals that for non-normal incidence the paths are coupled to each other. The analysis enables us to answer some of the basic questions concerning tunneling in multidimensional nonseparable potentials. A special and important case occurs when the incident wave is normal to the turning surface. We show that for normal incidence the path equations are decoupled and are equivalent to Newton's equations of motion for the inverted potential and energy.
A general Wentzel–Kramers–Brillouin (WKB) theory of multidimensional tunneling is formulated and an illuminating physical picture of the effects of multidimensionality is provided. Two basic problems are solved: (i) Maslov’s semiclassical wave function in the classically accessible region is connected to the wave function in the classically inaccessible region and (ii) the latter is propagated into the deep tunneling region. It is found that there exist two distinct types of tunneling: pure tunneling and mixed tunneling. The former is the usual one in which the tunneling path can be defined by a certain classical trajectory on the inverted potential and its associated action is pure imaginary. In the latter case, no tunneling path can be defined and the Huygens-type wave propagation should be carried out. In this case, tunneling is always accompanied by classical motion in the transversal direction and the associated action is complex. A general procedure is presented for the evaluation of energy splitting ΔE in the double well. Moreover, under the locally separable linear approximation, a simple and convenient formula for ΔE is derived and is confirmed to work well by comparison with the exact numerical calculations.
In the case of tunneling in chemical reactions [see figure1.1(c)], the total energy of reagents cannot be generally considered to be small; treating it as the first-order term of h¯ as was successfully done in Chapters 6 and 7 does not lead to any sensible semiclassical theory. The semiclassical mechanics requires (1) construction of Lagrange manifolds, (2) determination of amultidimensional caustic surface, and (3) connection of wave functions between classically allowed and forbidden regions [59]. Unfortunately, these problems have not yet been solved mathematically. As was discussed in Section 3.1, in the framework of the ordinary WKB approach we can construct Lagrange manifolds and caustics in the case of two dimensions and the connection problem can be treated relatively well by assuming the local separability. Extension of this method to higher dimensions, however, cannot be practical. The complexvalued (complex coordinate and/or momentum) WKB theory has been devised, as briefly explained there, but cannot be simply extended to multidimensions (see, for instance, [15]). This formidable task can hardly be accomplished mainly because of the complexity of the Lagrange manifold and the fact that analytical properties of any realistic potential energy surfaces extended into the complex-valued coordinate space cannot be guaranteed. The use of a complex trajectory was nicely materialized by the double-ended complex trajectory in the classical S-matrix theory [206], in which the quantized boundary conditions are satisfied at both (initial and final) ends. Unfortunately, this method turned out not to be practically useful, because finding such trajectories is also a formidable task in a multidimensional complex coordinate (and/or momentum) space. We can, however, make a break through the difficulty of detecting caustics in multidimensional space. An efficient way of doing that has been devised along classical trajectory [47]. This is not a trivial task at all, since it is not possible to find caustics in high-dimensional space by just shooting classical trajectories. Since the caustics can be found along each classical trajectory, chemical dynamics with tunneling effects incorporated can now be treated by the on-the-fly method. In this subsection, this method is explained.
General WKB theory of multidimensional tunneling between tori is formulated and illuminating physical picture on the effects of multidimensionality is provided. On the basis of Wilkinson's work, (1) Maslov's semiclassical wavefunction in classically accessible region is connected to the wavefunction in classically inaccessible region and (2) the latter is propagated into deep tunneling region. It is found that there exist two distinct regions of tunneling : a region where action is pure imaginary (I-region) and another region where action is complex (C-region). In I-region tunneling path can be defined by a certain classical trajectory on the inverted potential, while in C-region no tunneling path can be defined and the Huygens type wave propagation should be carried out. Moreover, wavefunction of a vibrationally excited state has nodal lines in C-region but not in I-region.
Multi-dimensional coupled-channel equation is treated by the WKB approximation for the main purpose of deriving the semi-classical quantization rule for it. A somewhat general type of the multi-dimensional coupled-channel Hamiltonian is presented which enables us to derive the semiclassical quantization rule for the energy spectra. As in the one-dimensional case, the WKB wave function has an additional phase besides the usual phase ∫Σpsdxs, and the quantization rule should include the contribution from this additional phase. The structure of the WKB wave function in the adiabatic representation is also discussed.
The path integral formulation of the traversal time in tunneling utilizes a restricted Feynman amplitude. We present a derivation based on the Fisher Lee formula to compute the average time and the integrated distribution for arbitrary potentials. The (complex) tunneling time can be expressed as a derivative with respect to a potential which surrounds the region over which the particle is “detected”, and a correspondence with other tunneling time scales which depend on energy derivatives of the transmission coefficient is found. A semiclassical treatment confirms that there are two time scales associated with this measurement procedure. The imaginary component is related to the multiple instanton bounces in the inverted potential. The issue of positive times is found to depend crucially on the behavior of the transmission coefficient for asymptotic limits of the measurement potential. The calculation of higher moments of the distribution allows to define a positive barrier interaction time, without appealing to any regularization procedure.