Energy bands

Energy bands consisting of a large number of closely spaced energy levels exist in crystalline materials. The bands can be thought of as the collection of the individual energy levels of electrons surrounding each atom. The wavefunctions of the individual electrons, however, overlap with those of electrons confined to neighboring atoms. The Pauli exclusion principle does not allow the electron energy levels to be the same so that one obtains a set of closely spaced energy levels, forming an energy band. The energy band model is crucial to any detailed treatment of semiconductor devices. It provides the framework needed to understand the concept of an energy bandgap and that of conduction in an almost filled band as described by the empty states.

In this section, we present the free electron model and the Kronig-Penney model. Then we discuss the energy bands of semiconductors and present a simplified band diagram. We also introduce the concept of holes and the effective mass.

Free electron model
The free electron model of metals has been used to explain the photo-electric effect (see section 1.2.2). This model assumes that electrons are free to move within the metal but are confined to the metal by potential barriers as illustrated by Figure 2.3.1. The minimum energy needed to extract an electron from the metal equals qFM, where FM is the workfunction. This model is frequently used when analyzing metals. However, this model does not work well for semiconductors since the effect of the periodic potential due to the atoms in the crystal has been ignored.

Figure 2.3.1.: The free electron model of a metal.

Periodic potentials
The analysis of periodic potentials is required to find the energy levels in a semiconductor. This requires the use of periodic wave functions, called Bloch functions
which are beyond the scope of this text. The result of this analysis is that the energy levels are grouped in bands, separated by energy band gaps. The behavior of
electrons at the bottom of such a band is similar to that of a free electron. However, the electrons are affected by the presence of the periodic potential. The
combined effect of the periodic potential is included by adjusting the value of the electron mass. This mass will be referred to as the effective mass.

The effect of a periodic arrangement on the electron energy levels is illustrated by Figure 2.3.2. Shown are the energy levels of electrons in a carbon crystal with
the atoms arranged in a diamond lattice. These energy levels are plotted as a function of the lattice constant, a.

Figure 2.3.2. : Energy bands for diamond versus lattice constant.

Isolated carbon atoms contain six electrons, which occupy the 1s, 2s and 2p orbital in pairs. The energy of an electron occupying the 2s and 2p orbital is indicated on the figure. The energy of the 1s orbital is not shown. As the lattice constant is reduced, there is an overlap of the electron wavefunctions occupying adjacent atoms. This leads to a splitting of the energy levels consistent with the Pauli exclusion principle. The splitting results in an energy band containing 2N states in the 2s band and 6N states in the 2p band, where N is the number of atoms in the crystal. A further reduction of the lattice constant causes the 2s and 2p energy bands to merge and split again into two bands containing 4N states each. At zero Kelvin, the lower band is completely filled with electrons and labeled as the valence band. The upper band is empty and labeled as the conduction band.

The Kronig-Penney model
The Kronig-Penney model demonstrates that a simple one-dimensional periodic potential yields energy bands as well as energy band gaps. While it is an
oversimplification of the three-dimensional potential and bandstructure in an actual semiconductor crystal, it is an instructive tool to demonstrate how the band
structure can be calculated for a periodic potential, and how allowed and forbidden energies are obtained when solving the corresponding Schrödinger equation.

The potential assumed in the model is shown in the Figure 2.3.3:

Figure 2.3.3 : The periodic potential assumed in the Kronig-Penney model. The potential barriers (region I) with width, b, are spaced by a distance (region II), a-b,
and repeated with a period, a.

The Kronig-Penney consists of an infinite series of rectangular barriers with potential height, V0, and width, b, separated by a distance, a-b, resulting in a
periodic potential with period, a. The analysis requires the use of Bloch functions, traveling wave solutions multiplied with a periodic function, which has the
same periodicity as the potential. The actual derivation can be found in section 2.3.8.

Solutions for k and E are obtained when the following equation is satisfied:

Energy bands of semiconductors

Complete energy band diagrams of semiconductors are very complex. However, most have features similar to that of the diamond crystal discussed in section 2.3.2. In
this section, we first take a closer look at the energy band diagrams of common semiconductors. We then present a simple diagram containing only the most important
features and discuss the temperature and doping dependence of the energy bandgap.

Energy band diagrams of common semiconductors

The energy band diagrams of semiconductors are rather complex. The detailed energy band diagrams of germanium, silicon and gallium arsenide are shown in Figure 2.3.3.
The energy is plotted as a function of the wavenumber, k, along the main crystallographic directions in the crystal, since the band diagram depends on the direction
in the crystal. The energy band diagrams contain multiple completely-filled and completely-empty bands. In addition, there are multiple partially-filled band.