Wavefunctions at surfaces

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. 

Fig. 4.1. Three types of wavefunctions at surfaces. (Reproduced from Himpsel, 1983, with permission.)...

Fig. 4.10. Angle dependence of photoelectrons from different states. The polar-angle dependence of the photoelectrons reflects the orbital characteristics of the wavefunctions at surfaces, (a) <7. peak on M0S2. (b) / . peak on GaSe. (c) An. v like state on M0S2. (Reproduced from Cardona and Ley, 1978, with permission.)...

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). 

Using the properties of the Green s function (see Appendix B), the evaluation of the effect of distortion to transmission matrix elements can be greatly simplified. First, because of the continuity of the wavefunction and its derivative across the separation surface, only the multiplier of the wavefunctions at the separation surface is relevant. Second, in the first-order approximation, the effect of the distortion potential is additive [see Eq. (2.39)]. Thus, to evaluate the multiplier, a simpler undistorted Hamiltonian might be used instead of the accurate one. For example, the Green s function and the wavefunction of the vacuum can be used to evaluate the distortion multiplier. 

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... 

For example, for an s state, the angular distribution should be a constant for a Pi state, the dependence should like cos 0 and for a state, a dependence of (3 cos 0-1)2 should be observed. This model has been verified by experiments (Smith, 1978). Figure 4.10 shows the polar-angle dependence of certain peaks in the spectra, which match what is expected from the knowledge of the orbital characteristics of the wavefunctions at these surfaces. 

According to the derivative rule, the tunneling matrix element for surface wavefunction at F from a p, tip state is identical to that from a spherical tip state. However, for a surface wavefunction at K, the tunneling matrix element from a p, tip state is ... 

A straightforward calculation using the tunneling matrix elements listed in Table 3.2 shows that the state results in a large but inverted corrugation amplitude on metal surface, because the tunneling matrix element for the sample wavefunction at the F point vanishes. The role of this state and the state in the inverted corrugation will be discussed in Section 5.5. 

A widely used approximate method of describing atomic states is the Slater atomic wavefunctions (Zener, 1930 Slater, 1930). In this section, we show that regarding STM, the Slater wavefunction is a convenient tool for describing localized atomic states at surfaces. 

Atoms at solid surfaces have missing neighbors on one side. Driven by this asymmetry the topmost atoms often assume a structure different from the bulk. They might form dimers or more complex structures to saturate dangling bonds. In the case of a surface relaxation the lateral or in-plane spacing of the surface atoms remains unchanged but the distance between the topmost atomic layers is altered. In metals for example, we often find a reduced distance for the first layer (Table 8.1). The reason is the presence of a dipole layer at the metal surface that results from the distortion of the electron wavefunctions at the surface. 

The rotational coordinates are Q 2 and Q 5. The rotational motion can be visualized by mapping the trough onto the surface of a 2D sphere the rotation is governed by the usual polar coordinate definitions, 6 and . This is also shown in equation (7) which has the usual form for a rotator with spherical harmonic solutions Ylm. The solutions will be written in the form I i//lo, hn ). For the high spin states case, it was found that l must be odd in order to obey the Pauli s exclusion principle and preserve the antisymmetric nature of the total wavefunctions at any point on the trough under symmetric operations [26]. In the current case, similar arguments show that l must be even. This is because the electronic basis is even under inversion and the whole vibronic wavefunction must also be even under inversion. A general mathematical proof can be found in Ref. [23],... 

There are several reasons for an energy shift of a surface state to occur. In fact, any modification of the potential and matching conditions of the wavefunctions at the (surface-vacuum) interface will modify the energy position (and dispersion) of a surface state. This includes the physisorption... 

With increasing atomic volume, one approaches the free atom limit where Hund s first rule postulates maximum spin, so that the individual spins of the electrons in a shell are aligned parallel. More generally, Pauli s exclusion principle implies that electrons with parallel spins have different spatial wavefunctions, reduces the Coulomb repulsion and is seen as exchange interaction. When the atoms are squeezed into a solid, some of the electrons are forced into common spatial wavefunctions, with antiparallel spins and reduction of the overall magnetic moment. At surfaces and interfaces, the reduced coordination reverses this effect, and a part of the atomic moment is recovered. 

A firm result will be this if the value of the function at a given neighborhood to a point on the surface is zero, whatever you do there will never be a spectral response derived from a quantum-mechanical interaction at that neighborhood no imprint mediated by the quantum state. Another one is that any finite value different from zero of the quantum state function at a given neighborhood of a point opens a possibility for a response from a properly sensitized surface that would reflect the wavefunction at that region (Cf. Eq. (3)). 

This depends on the value of the wavefunction at the nucleus (r = 0) for electrons averaged over the Fermi surface, where II is the volume per electron. This shift of frequency in a metal was first observed in copper in 1949 by W.D. Knight and was subsequently termed the Knight shift (Townes, Herring and Knight 1950, Knight 1956), given by... 

A much more common procedure for reference selection is to accept references whose estimated importance is greater than some given threshold this can involve perturbative estimates of a function s energetic contribution or its coefficient in some preliminary wavefunction. These approaches are more successful at obtaining the best wavefunction at the lowest expense, but they sacrifice the simplicity of the excitation class selection and can become more difficult to implement and to use. One complication is that potential energy surfaces determined using such methods may not be smooth to alleviate this, one may need to determine the important references at each geometry and use the union of these sets at every point. 

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