E-K diagram in case of semiconductors

Question

I am currently studying solid state electronic devices and want to build my concept in this subject. Can anyone explain to me what is E-k Diagram and what is its significance?

Answer 1

An E-k diagram shows characteristics of a particular semiconductor material. It shows the relationship between the energy and momentum of available quantum mechanical states for electrons in the material.

First, consider a basic E-k band diagram like this one (the x-axis can be either momentum, \$p\$, or wavenumber, \$k\$, since \$p=\hbar k\$):

(image source Best Innovative Source)

In this diagram you can see a few things:

• The band gap (E_G), which is the difference in energy between the top of the valence band and the bottom of the conduction band.

• The effective mass of electrons and holes in the material. This is given by the curvature of each of the bands.

• This diagram indicates (diagramatically) how the actual electron states are equally spaced in k-space. Which means that the density of states in E (\$\rho(E)\$) depends on the slope of the E-k curve.

There is a more complex form of E-k diagram that shows the relationship for different directions of k relative to the crystal lattice:

(image source: University of Kiel)

Here, the greek letters (\$\Gamma\$, \$\Delta\$, K, etc.) on the x-axis indicate different directions of k relative to the crystal axes.

In addition to showing the effective mass at different band extrema, this also shows that the effective mass varies depending on the direction of conduction relative to the crystal orientation.

This type diagram also shows whether the material is a direct-gap or indirect-gap semiconductor. Direct gap is when the valence band maximum and conduction band minimum occur at the same location in k-space. This is important in optoelectronics because only direct gap materials (like GaAs, but not including silicon) have efficient radiative absorption and emission, which is what makes LEDs and laser diodes work.

Answer 2

The E-k diagram is used to understand how a harmonic wave corresponding to one of the fundamental particles behaves in a supporting environment that is periodic, such as a semiconductor crystal (Si, GaAs, etc). This means that you can construct such a diagram for electrons, for photons, and phonons, to name the most popular. You can imagine that, as soon as the wavelength of the wave approaches the distance among the particles, then the wave will start to behave differently in trying to pass through the crystal. Some choices of particle wavelength will have an easier time, since it matches the crystal "wavelength", and other choices will produce interference.

It is a little like trying to run through a crowd of people, which gets difficult unless your step length matches the average gap between two persons, in which case you get through faster.

Since the crystal extends infinitely (idealisation!), and the wave too, you can "fold up" the picture, since a specific crystal position is no longer unique. This is the viewpoint shown above, you only need to plot the geometry of the wave around the unit cell of the crystal, but you have to make sure that the plot in the neighbouring cells is represented in your restricted view.

Now for the abstraction. The k-space is the Fourier transform of the real space. It is the spatial waveform, expressed in the wavelength of the crystal. So there are nice features that you can make use of, because a wave with a particular wavelength of course corresponds to a particular energy of the wave. And so on.