Band theory of conduction

FIGURE 22.21 The valence orbitals of the silicon atom combine in crystalline silicon to give two bands of closely spaced levels. The valence band Is almost completely filled, and the conduction band Is almost empty. 

Calculate the longest wavelength of light that can excite electrons from the valence to the conduction band in silicon. In what region of the spectrum does this wavelength fall  

The energy carried by a photon is hv = hdX, where h is Planck s constant, v the photon frequency, c the speed of light, and A the photon wavelength. For a photon to just excite an electron across the band gap, this energy must be equal to that band gap energy. Eg = 1.94 X 10 J, where 

This wavelength falls in the infrared region of the spectrum. Photons with shorter wavelengths (for example, visible light) carry more than enough energy to excite electrons to the conduction band in silicon. 

An important feature of the band system is that electrons are delocalised or spread over the lattice. Some delocalisation is naturally expected when an atomic orbital of any atom overlaps appreciably with those of more than one of its neighbours, but delocalisation reaches an extreme form in the case of a regular, 3-dimensional lattice. We can understand this best if we choose to think of the wave nature of electrons, and from that point of view we can formulate band theory as follows. 

A free electron propagates through space as a wave1 characterised by a wave vector k (k=2n/X, where X is the wavelength), whose magnitude is related to the momentum p of the electron by the fundamental relation 

Analytical solutions for the band structure are possible under simplifying approximations. In the adiabatic approximation the electronic states are calculated for a static crystal lattice and the problem of the vibrations of the lattice is treated independently. Finally, the interaction between the electrons and the vibrations is introduced to combine these results. In discussing the electronic states we will assume a static lattice. Limiting the problem to a linear 

Substitution of the wavefunction exp( ikx) leads to the solution for the kinetic energy of the electron  

This is the parabolic curve shown in Fig. 4.4(a). A linear atomic lattice will provide a periodic rather than a constant potential, i.e. V(x) = V[x + a), where a is the repeat distance of the array. Bloch s theorem, also known as Floquet s theorem, states that possible solutions of the Schrodinger equation with a periodic potential are  

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In Section 6.3, we saw that the ability of metals to conduct heat and electricity can be explained by considering the metal to be an array of positive ions immersed in a sea of highly mobile delocalized valence electrons. In this section, we will use our knowledge of quantum mechanics and molecular orbital theory to develop a more detailed model for the conductivity of metals. The model we will use to study metallic bonding is the band theory of conductivity, so called because it states that delocalized electrons move freely through "bands formed by overlapping molecular orbitals. We will also apply band theory to understand the properties of semiconductors and insulators. 

The conductivity of metals generally decreases with increasing temperature. Give an explanation of this in terms of the band theory of conductivity. 

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