E-K Diagrams

As previously concluded, solids can be characterized based on energy band diagrams. A conductor has a valence and conduction bands that are very close or overlap. In addition a conductor will have a completely filled valence band and an almost full conduction band. The “forbidden region of the conductor is very small and little energy is required for an electron to move from conduction to valence band. In the presence of an external field, it is very easy for electrons to move from the valence band to the conduction band.

For semiconductors, at absolute zero the valence band is also completely full and the bandgap is typically about 1eV to 3eV, however even a bandgap of .1eV could be considered a semiconductor. Therefore, a semiconductor at 0K is an insulator. Semiconductors are very temperature sensitive. The subsequent figure illustrates the temperature dependence. The resistivity is very high at absolute zero, making the semiconductor behave like an insulator. However at higher temperatures the semiconductor can become quite conductive. At room temperature (300k), the semiconductor behaves more like a conductor.

temp_semi

With band diagrams, not much information is given therefore it is necessary to also analyze an E-K (Energy momentum) diagram. E is the energy require for an electron to traverse the bandgap. For example in Silicon with a bandgap of 1.1eV, it would take an energy level of 1.1eV for an electron to move from conduction to valence band. Energy is given as E = kT where T is a given temperature.

For intrinsic semiconductors like Silicon, the structure is crystalline and periodic. The wavefunction (which describes probability of finding an electron) should therefore be of periodic nature (sinusoidal). From the Schrodinger equation, it can be found that the Energy is periodic with k as well. For the diagrams, E is plotted against k.

ek

The borders of the first Brillouin zone are from -π/a to π/a. These are cells of the crystalline lattice. Since the wavefunction is periodic, we only care about one of the zones. The above figure can be considered the “reduced zone” figure. Sometimes the x axis is given as the moment or wavenumber, since these only differ by a factor of Planck’s constant. From this diagram: the bandgap energy is shown, the effective mass of electrons and holes are shown as well as the density of states. The effective mass is shown by the curvature of the bands. For example, a heavy hole band could be found by observing the diagram that is less curved. From the above diagram, it is also noticeable that the material is direction bandgap (such as GaAs). The basic energy gap diagram compares to the E-k diagram in that the maximums and minimums correspond. However, the original band gap diagram does not give any other characteristics. It is for this reason the E-k diagram is so useful.

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