Tip wavefunctions

1990 Sacks and Noguera, 1991). All those authors used a spherical-harmonic expansion to represent tip wavefunctions in the gap region, which is a natural choice. The spherical-harmonic expansion is used extensively in solid-state physics as well as in quantum chemistry for describing and classifying electronic states. In problems without a magnetic field, the real spherical harmonics are preferred, as described in Appendix A. 

In this Chapter, we present step-by-step derivations of the explicit expressions for matrix elements based on the spherical-harmonic expansion of the tip wavefunction in the gap region. The result — derivative rule is extremely simple and intuitively understandable. Two independent proofs are presented. The mathematical tool for the derivation is the spherical modified Bessel functions, which are probably the simplest of all Bessel functions. A concise summary about them is included in Appendix C. 

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. 

The tip wavefunctions can be expanded into the spherical-harmonic components, T/, (0, ( )), with the nucleus of the apex atom as the origin. Each component is characterized by quantum numbers I and m. In other words, we are looking for solutions of Eq. (3.2) in the form 

The standard linear-independent solutions for Eq. (3.4) are the spherical modified Bessel functions, )(m) and ki(u) (Arfken, 1968). A brief introduction to them is provided in Appendix C. These so-called special functions are actually elementary functions.  

---

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. 1.25. The s-wave-tip model. The tip was modeled as a spherical potential well of radius R. The distance of nearest approach is d. The center of curvature of tip is To, at a distance (R + d) from the sample surface. Only the 5-wave solution of the spherical-potential-well problem is taken as the tip wavefunction. In the interpretation of the images of the reconstructions on Au(llO), the parameters used are R = 9 A, d = 6 A. The center of curvature of the tip is 15 A from the Au surface. (After Tersoff and Hamann, 1983.)...

Note that the expansion is done in terms of the tip wavefunctions, which form a complete and orthogonal set of states. Substituting Eq. (2.28) for Eq. (2.24), we obtain the exact equation for Cv(r) ... 

Oppenheimer (1928) estimated the error as follows. Since , are always several eV below vacuum level, the tip wavefunctions with energy levels close to that of the sample decay quickly outside the tip body. Thus, the matrix elements (Xp, Us Xx) for Ex < are much smaller than those with x 0. On the other hand, while x > 0, the tip wavefunction oscillates quickly, and the matrix elements become small again. Therefore, the numerator must have a sharp maximum somewhere at Ex E. By replacing Ex by E in the denominator of Eq. (2.37), we obtain the Oppenheimer error estimation term ... 

For z > zo, the distorted wavefunction has the same exponential dependence on z as the free wavefunction. Thus, d f dz gains the same factor. The tip wavefunction x gains a similar factor. Therefore, the transmission probability becomes... 

Obviously, the functions /,( ) diverge at large u, which are not appropriate to represent tip wavefunctions. The functions kiu) are regular at large u, which satisfies the desired boundary condition. Therefore, a component of tip wavefunction with quantum numbers I and m has the general form... 

In the absence of magnetic field, it is convenient to write those tip wavefunctions in real form, as listed in Table 3.1. The coefficients C are determined by comparing with first-principles calculations of actual tip states. 

Therefore, the. r-wave tip wavefunction is equal to the Green s function up to a constant, with the center of the apex atom taken as ro,... 

Taking the derivative with respect to zo on both sides of Eq. (3.25), and noticing that zo is a parameter in the integral (which does not involve the process of evaluating the integration), and the expression of the tip wavefunction, Eq. (3.14), we find... 

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 results can be summarized as the derivative rule Write the angle dependence of the tip wavefunction in terms of x, y, and z. Replace them with the simple rule. 

In this section, we present an alternative proof of the derivative rule, which provides an expression for the transmission matrix element from an arbitrary tip state expanded in terms of spherical harmonics. In the previous sections, we have expanded the tip wavefunction on the separation surface in terms of spherical harmonics. In general, the expansion is... 

The coefficients (3 , are determined by fitting the tip wavefunction on and beyond the separation surface. Inside the tip body, the actual tip wavefunction does not satisfy Eq. (3.2), and the expansion (Eq. (3.30)) does... 

Fig. 3.2. Derivation of the derivative rule general case. In the shaded region, the tip wavefunction does not satisfy the Schrodinger equation in the vacuum. However, the expansion in Eq. (3.30) satisfies the Schrodinger equation in the vacuum except at the nucleus of the apex atom. Thus the surface on which the Bardeen integral is evaluated can be deformed to be any surface that encloses the nucleus of the apex atom.

As we have shown, except at the nucleus of the apex atom, or the origin of the spherical harmonics, the expansion form of the tip wavefunction x Eq. (3.30), and the sample wavefunction ) satisfy the same Schrddinger equation, Eq. (3.8). Therefore, we can take any surface enclosing the nueleus of the apex atom to evaluate the transmission matrix element, Eq. (3.1), especially, a sphere of arbitrary radius ro centered at the nucleus of the apex atom. Substituting Equations (3.30) and (3.31) into Eq. (3.1),... 

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. 

We start our derivation by writing down the explicit form of the vacuum asymptote of a tip wavefunction (as well as its vacuum continuation in the tip body). As we have explained in Section 5.3, for the simplicity of relevant mathematics, the rather complicated normalization constants of the spherical harmonics are absorbed in the expression of the sample wavefunction. Up to 1=2, we define the coefficients of the expansion by the following expression ... 

This effect can be illustrated by Fig. 14.2. The effective range of local modification of the sample states is determined by the effective lateral dimension 4ff of the tip wavefunction, which also determines the lateral resolution. In analogy with the analytic result for the hydrogen molecular ion problem, the local modification makes the amplitude of the sample wavefunction increase by a factor exp( — Vi) 1.213, which is equivalent to inducing a localized state of radius r 4tf/2 superimposed on the unperturbed state of the solid surface. The local density of that state is about (4/e — 1) 0.47 times the local electron density of the original stale in the middle of the gap. This superimposed local state cannot be formed by Bloch states with the same energy eigenvalue. Because of dispersion (that is, the finite value of dEldk and... 

Tip treatment 281—293, 301 annealing 286 annealing with a field 288 atomic metallic ion emission 289 controlled collision 293 controlled deposition 288 field evaporation 287 for scanning tunneling spectroscopy 301 high-field treatment 291 Tip wavefunctions 76—81 explicit forms 77 Green s functions, and 78 Tip-state characterization 306, 308 ex situ 306 in situ 308... 

From the zero order Green s function the evaluation of expressions is straightforward. The exponential decay of surface and tip wavefunctions allows the conversion of the fourfold volume integration contained in the trace of Eq. (2) into surface integrals. An evaluation of the trace as well as the energy integral then leads to a modification of the standard Bardeen expression ... 

The most commonly used model in interpreting STM data is the Tersoff-Hamann model, in which the analysis is carried a step further. It is assumed that the tip wavefunction is an s-wave, and decays into the vacuum like... 

---