Bloch wavefunctions

There is a condition of momentum conservation for photons and electrons which must also be satisfied in the photoemission process. For band electrons, for which the Bloch wavefunctions are characterized by the wavenumber k (proportional to the momentum p of the electron), the momentum conservation condition is important to determine the angular distribution of the photoemitted electrons. Angular J esolved FhotoEmission spectroscopy (ARPES), schematized in Fig. 2, is potentially able to provide, and has been used to obtain, the E(fc) dispersion curves for solids. 

The cross-sections for itinerant electrons, as, e.g., electrons in broad bands, are evaluated by taking into account that the electrons in the initial as well as in the final state may be represented by Bloch-wavefunctions P = u,t(/ ) exp(i R) (see Chap. A). In these wavefunctions atomic information is contained in the amplitude factor Uj (i ), whereas the wave part exp (i R) is characterized by the wavenumber k of the propagating wave (proportional to the momentum of the electron). 

Even though in similarity to free particle wavefunctions the Bloch wavefunctions are characterized by the wavevector k, and even though Eq. (4.80) is reminiscent of free particle behavior, the functions V nkfr) are not eigenfunctions of the momentum operator. Indeed for the Bloch function (Eqs (4.78) and (4.79)) we have... 

The unit cell group description of the normal modes of vibration within a unit cell, many of which are degenerate, given above is adequate for the interpretation of IR or Raman spectra. The complete interpretation of vibronic spectra or neutron inelastic scattering data requires a more generalized type of analysis that can handle 30N (N=number of unit cells) normal modes of the crystal. The vibrations, resulting from interactions between different unit cells, correspond to running lattice waves, in which the motions of the elementary unit cells may not be in phase, if ky O. Vibrational wavefunctions of the crystal at vector position (r+t ) are described by Bloch wavefunctions of the form [102]... 

The scope of the method was then extended to a periodic system via the following Bloch wavefunction, constructed in terms of atomic orbitals... 

More general expressions use a two-dimensional (2D) wavefunction and more than one band, i.e., for a (2/i -f-1) band model with Bloch wavefunctions in... 

The Hamiltonian of our system comprises the conduction and valence band energies, e. v, the exchange interaction, Kij the spin-orbit coupling, A and the crystal field splitting, The matrix elements of this Hamiltonian in the basis of the Bloch wavefunctions a,b,c and 5, Equations (6.16), whichcomprise the spin-symmetry-adapted configuration state functions, have been previously... 

The periodic nature of crystalline matter can be utilized to construct wavefunctions which reflect the translational synnnetry. Wavefiinctions so constructed are called Bloch functions [1]. These fiinctions greatly simplify the electronic structure problem and are applicable to any periodic system. 

We wish to construct linear combinations of the atomic orbitals such that the overall wavefunction meets the Bloch requirement. Suppose the s orbitals in our lattice are labelled X , where the wth orbital is located at position x = na. An acceptable linear combination of these orbitals that satisfies the Bloch requirements is ... 

Appendix 3.1 Alternative Expression for a Wavefunction Satisfying Bloch s Function... 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

The translational periodicity of the potential is the necessary and sufficient condition for describing the wavefunction as a linear combination of Bloch functions... 

When an external electric field is applied along the periodicity axis of the polymer, the potential becomes non periodic (Fig. 2), Bloch s theorem is no longer applicable and the monoelectronic wavefunctions can not be represented under the form of crystalline orbitals. In the simple case of the free electron in a one-dimensional box with an external electric field, the solutions of the Schrddinger equation are given as combinations of the first- and second-species Airy functions and do not show any periodicity [12-16],... 

At crystalline surfaces, there are three types of wavefunctions as shown in Fig. 4.1. (1) The Bloch states are terminated by the surface, which become evanescent into the vacuum but remain periodic inside the bulk. (2) New states created at the surfaces in the energy gaps of bulk states, which decay both into the vacuum and into the bulk, the so-called surface states. (3) Bloch states in the bulk can combine with surface states to form surface re.sonances, which have a large amplitude near the surface and a small amplitude in the bulk as a Bloch wave. 

The local modification of sample wavefunctions due to the proximity of the tip, and consequently the involvement of the Bloch functions outside the energy window Er eV in the tunneling process, has an effect on the limit of the energy resolution of scanning tunneling spectroscopy. This effect is discussed in detail by Ivanchenko and Riseborough (1991). First, if the tunneling current is determined by the bare wavefunctions of the sample and the tip, the process is linear, and there is no effect of quantum uncertainty. The effect of quantum uncertainty is due to the modification or distortion of the sample wavefunction due to the existence of the tip. Here, we present a simple treatment of this problem in terms of the MBA. 

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... 

Equation (8.4.2) suggests that a wavefunction uk(r) needs to be found by standard quantum-chemical means for only the atoms or molecules in the one direct-lattice primitive unit cell. For each of the Avogadro s number s worth of fermions in a solid, the factor exp(ik R) in Eq. (8.4.2) provides a new quantum "number," the wavevector k, that guarantees the fermion requirement of a unique set of quantum numbers. The Bloch waves were conceived to explain the behavior of conduction electrons in a metal. 

Simply put, the Bloch theorem guarantees that, if the correct wavefunction is found for the zeroth cell, the wavefunctions outside the cell are a repetition of that wavefunction, multiplied by the factor exp (ik R). Among many choices, the wavefunctions uk(r) can be Wannier45 functions, which are defined to be mutually orthogonal, (uk(r) uk (r)) — kk while for atomic or molecular wavefunctions this orthogonality does not necessarily hold. 

The introduction of the concept of one-electron crystal orbitals (CO s) considerably reduces difficulties associated with the many-electron nature of the crystal electronic structure problem. The Hartree-Fock (HF) solution represents the best possible description of a many-electron system with a one-determinantal wavefunction built from symmetry-adapted one-electron CO s (Bloch functions). The HF approach is, of course, only a first approximation to the many-particle problem, but it has many advantages both from practical and theoretical points of view ... 

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