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Electrons in the lattice

Crystals are very regular structures of atoms or molecules. Depending on particle size and their preferred bonding, they form certain crystal classes. The most important structure for semiconductors is the face centered cubic (fcc) crystal which is the building block for silicon. This crystal structure allows to accomodate the atoms in a configuration with four nearest neighbours corresponding to the four valence electrons of silicon. An other very promenent member of this crystal structure is the diamomd modification of carbon which also gave the structure its name.

In the previous section we have seen how the simple parabolic relationship between k and the energy E is discretised into individual (k, E(k)) points but located on the same parabola by simply fitting the wave function into the finite dimensions of the material. The construction of an inverse space, k-space, allowed to calculate the electron energy distribution. Now we will add information on the crystal structure to our model. Crystallography defines the primitive cell which can be used to build the whole crystal, simply by stacking identical copies beside and above each other. Vice versa, all the information about the whole crystal is contained in one primitive cell. The underlying mathematical description is by means of a translational vectors which lead from one primitive cell to an adjacent one.

Also, we have to abandon the assumption of zero interactions between the electrons. While we cannot easily specify the nature of such interactions, it is reasonable to assume that they are described by a potential of some sort, and that this potential shows the periodicity of the crystal. Then, a theorem by Bloch states that the electron wavefunctions also shows the periodicity of the lattice. In different primitive cells they assume the same value except for a phase factor.

Just like real spaces, also the inverse lattice of k-space can be built from primitive cells. Here, we usually use a particular construction for the primitive cell, the so called Wiegner Seitz cell which is also called the first Brillouin zone. For the energy diagram of the previous section there is no longer the need to draw it for arbitrarily large k-values, rather, if k exceeds the first Brillouin zone we can make use of the periodicity and translate it back. This is called the reduced scheme of the energy diagram. Below is Wigner Seitz cell of the fcc crystal lattice, a truncated octahedron.

Due to multiple possibilities of translation the reduced scheme becomes more complicated. The figure below shows the band diagram of the fcc lattice in directions of high symmetry.

Band gaps

The figure below illustrates the translation procedure for the simpler two dimensioal case. Around the origin the simple parabolic relation between E and k is observed. If we move from the origin along the diagonal we hit the corner at W. All higher k-values along the diagonal are simply folded back (left part of the figure).

When moving from Γ to X, the boundary is met for a smaller energy value and we should also translate it back. This is not the case on the shown diagram. Rather, it illustrates a result of the interactions between the electrons. Remember that the electrons are described by wavefunctions now. Then, scattering of the waves at lattice planes violates the assumption of free electron movement (travelling waves). Rather, we deal with standing waves and the electron wavefunctions resemble more those bound states. As a result, in a direction of high symmetry like X there is a forbidden region where no travelling waves are allowed. Only if we provide more energy to the to the electron it can overcome the partial binding and states with k=X are allowed. This is exactly what we term an energy gap.

Depending on the particular interaction potential, the same sort of gap is also possible at the W point. If the two gaps are large enough to overlap there is no allowed state for this energy in the whole k-space. Then, the first prerequisite for a semiconducting material is met. Usually we call the total of all energy states on the parabolas below and above the gap bands. This is where the names band gap and band diagram come from. More precicely, below the gap we have the conduction band, above the gap is the valence band.

From the illustration it is immediately clear that a square Brillouin zone is not very suitable. The distances Γ-X and Γ-W differ by a factor of sqrt(2), for the parabolic case the energy at X would be only half of that at the point W. For an overlapping energy gap we would need big gaps at both points. Obviously, in three dimensions this is even more pronounced, remember the diagonal in a cube is sqrt(3) times the side. In a more favourable geomtery the boundary in the diagonal direcion towards W would be met for lower energies; this is possible if we truncate the corners of the primitve cell, for example like a hexagon or an octagon. We conclude that the first Brillouin zone of semiconductors should resemble as close as possiblea to a sphere (or a circle in the 2D illustration). Indeed, the primitive cell of the fcc diamond structure is already quite close, in the truncated octahedron the distances from the center to the faces (Γ-X and Γ-L), to the edges (Γ-K and Γ-U), and to the corners (Γ-W) are fairly similar.

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