The Bloch Function

1.5.1 The Bloch Function

Before discussing the Bloch function, it is important to write a few lines on the ‘Electron Kinet- ics’ in solids :

A convenient approach to the electronic properties of a solid consists of using the ‘one-electron’ motion, where core electrons are treated almost completely ‘localised’, and the energy eigenvalue problem for the valence electron system reduces to → solving the Schrodinger wave equation for each valence electron → as it moves in the spatially-dependent potential energy [V(r)].

Electron Kinetics.

As we deal with a large number of electrons in a solid, their interactions are important to describe by various electronic transport phenomena in solids. Although the ‘nuclear motion’ is not stationary, we still assume it to be at rest → so we need ‘approximation’, like Born-Oppenheimer approximation. In such an approximation, the total wave function for the system is given by a combination of wave func- tions. Each of these involves the ‘coordinates’ of only one electron → which gives rise to “one-electron approximation’’.

Within this framework → → → → → there are two different approaches :

1. Heitler – London or Valence Bond Scheme. when the atoms are far apart from each other i.e., the electrons are in ‘localised’ states.

2. Bloch Approach. where the electron is considered to be belonging to the three-dimensional crystal as a whole rather than to a particular atom.

The problem here involves an electron in a potential with the periodicity of the crystal lattice, which leads to a “natural distinction” between metals, insulators and semiconductors → in terms of band structure of these solids. However, first of all, we have to consider the 'anlogy' between the elec- tronic motion and elastic waves.

The Analogy

(a) The propagation of ‘elastic waves’ in a continuum or in a periodic structure, and (b) Electronic motion in a constant or a periodic potential.

(a) Elastic Waves For elastic waves in a continuous medium, the frequency ( ν) is inversely proportional to the

wavelength ( λ) with a ‘linear’ relation between ν and the ‘wave number’ or the ‘wave vector’ → which implies that the “velocity of propagation” is independent of λ, with no upper limit for the ‘frequency of

NANO MATERIALS

vibrational modes’ in a continuous medium. However, when we consider the ‘modes of vibration’ in a “lattice” of discrete points → which form a periodic structure → two characteristic features appear :

There exists ALLOWED frequency bands, which are separated by FORBIDDEN regions of frequency or energy.

The frequency is no longer proportional to the “wavenumber”, but a periodic function of the “lattice”.

(b) Electron Motion In a constant potential, i.e., the ‘free’ electron theory, the energy of the electron as a function of

‘wave vector’ k is given by : E = p , where k = =

D 22 k

D , 2 λ = wavelength, p = momentum of the m λ electron [V(x) = 0]. In this case, there is no upper limit to the energy → i.e., the energy is ‘quasi-

continuous’. However, if we consider the motion of an electron in a ‘periodic potential’ → we arrive at the

following results : There again exists ‘allowed’ energy bands, which are separated by ‘forbidden’ regions of energy,

and the energy function E(k) are periodic in k. In (a) above, we deal with the “elastic waves”, and in (b) we deal with the ‘waves’ associated

with the electrons → and hence this analogy is not surprising → both waves are moving in a periodic structure of a given lattice. It should be remembered that the ‘discrete nature of solids’ creating separate ‘allowed’ and ‘forbidden’ energy regions manifest in the observation of sharp resonance-like structures in the optical spectra of solids. Now, let us get into the most important theorem in formulating the ‘band structure’ in solids [7, 8].