Photovoltaics: Band Diagram and the Dispersal Diagram.

In this post we will look at the energy of electrons in a material with periodic atomic structure. Most semiconductor materials, such as silicon, have periodic atomic structure. We call it a crystalline structure. By following a simplified quantum-mechanical approach, we will show that the energy of electrons in semiconductors are grouped in bands. This post aims to understand the origins and the importance of electron energy-bands for semiconductors.

Specifically, we demonstrate how we visualize these energy levels in so-called energy band diagrams. Understanding band diagrams is of utmost importance for solar cell engineers. In the previous post it was described how atoms bond together to form a solid material. Now we will take a look on the energy of electrons in a regular atomic structure. This first part of the post will essentially serve as a derivation for energy-band and dispersion diagrams.

Since the quantum-mechanical approach to determine the energy of an electron in a real solid is rather complex, we will start by making some simplifying assumptions. Let’s consider a case in which the lattice structure is perfect and all the atoms are fixed in one position. Let’s then consider a one dimensional crystal lattice with lattice constant equal to “a” and composed of “N” atoms. In quantum mechanics, the Schrödinger equation relates the energy of a particle to its wave function. This differential equation is of the second order. As you can expect, solving the Schrödinger equation for a real 3-dimensional case is mathematically complex. Therefore we will use models and simplifications that will allow us to derive the energy of an electron in a simple 1-dimensional lattice structure.

This diagram shows the potential energy experienced by an electron crossing one dimensional lattice. When considering only coulombic forces as a result of atom core-electron interactions, the resulting potential-energy function, U(x), will be periodic. It’s important to remark that electron-electron interactions within atoms are neglected, since their average contribution to the potential energy is almost zero.

When dealing with a periodic function we can apply an additional approximation: the Bloch theorem. As a result, we can solve the Schrödinger equation focusing only on a single unit cell. We assume that the lattice is build up of a number of unit cells and consider the other unit cells to be equivalent. In this equation the wave function in a certain position of the lattice is dependent on the lattice constant “a” and a new parameter, “k”, which is called the wavenumber. It is important to deduct k in order to determine the allowed energy states for an electron ina periodic lattice. The assumption of a 1-dimensional periodic system enables us to arrive to the following conclusions.

It can be shown that for a 1-dimensional system there can be only two values of “k” for each allowed energy state. Besides, the values of “k” are periodic and they repeat every 2pi over a, the lattice constant. Moreover, if we apply periodic boundary conditions in a finite crystal lattice, “k” has a distinct number of values. This number equals the number of units or in our case atoms that form the lattice. Let’s assume the number of atoms is capital “N”. Since “k” is inversely proportional to the lattice constant, it can be concluded that the closer the atoms are to each other, the closer the k-values will be spaced. This can lead to a quasi-continuous range of allowed k values.

The periodic-potential energy can be further idealized according to the Kronig-Pennig model. The periodic-potential energy is represented by a barrier denoted as U and a well. The energy of the well is set to be zero. The barrier and the well that describe the periodic potential-energy function form a unit cell. We define the dimensions of the unit cell by “minus b” and “plus a” as boundaries in the x axis.If we apply the boundary conditions within such unit cell, we can solve the Schrödinger equation.

The resulting solution includes the parameters alpha and beta, which are related to the energies in specific regions of the potential-energy function. From this solution we can see that each value of k corresponds to a specific energy. Besides, to satisfy the relationship, the cosine of k times (a plus b) must assume a value in the range between minus 1 and plus 1. In order to visualize the results, let’s now define Xi as the ratio between the particle’s energy(E) and the maximum potential energy (U zero). The allowed energy values for the system are then those that satisfy the previous equation. If we adapt this equation using the new defined components of Xi and alfa-zero, we can show that the allowed solution lie in the range between minus 1 and plus 1.

If we plot the adapted equation as function of Xi, we can observe that it is an oscillatory function. The regions, where the function have values between “minus 1 and plus 1”, are enclosed in the blue boxes. These boxes represent sets of allowed-energy states that the system (our one-dimensional lattice) can have. These ranges of allowed energy states for electrons are what we call “energy bands”, while the spaces in between are forbidden-energy states for electrons. But, why are we interested in wavenumbers and energy bands? To understand it, we will now look at the energy-dispersion diagrams. An energy dispersion diagram, also referred to as “E-k” diagram, relates the electron-energy states to the wavenumber “k”.

Doing so, in a 1-dimensional system, we are able to know the energy of an electron for any specific value of “k”. But not only that. The wavenumber is also related to the particle momentum. In classical mechanics it is defined as the product of the mass of an object and its velocity. This resulting vector plays an important role in the mechanisms and energetics of energy transitions in a semi-conductor. We can extend our analysis to a 3-dimensional system. In this case, we refer to k as “wave vector”. It depends on the crystal orientation that has to be taken into account. However, if we do so, the resulting plot will be very complex and difficult to analyze. For this reason, generally only the regions that are most likely to be occupied by electrons are plotted.

Moreover, due to the symmetry of crystals, we can choose to plot only the positive values of“k”. It is important to realize that these energy levels vary, depending on the orientation that is taken into account. This means that, if we look at an electron in a crystal lattice, it will be able to propagate differently in different directions. We can use dispersion diagrams to extrapolate accurate information regarding the energy of an electron. However, due to their complexity, these diagrams are not commonly applied to describe the working principle of solar cells. Instead we used so called energy-band diagrams to represent energy behaviour of electrons in semiconductor materials and solar cells.

When discussing 3-dimensional dispersion diagrams, we said that only the bands that are most likely to be filled with electrons are represented. For device operation, valence electrons are of the utmost importance and therefore we focus on the energy of valence electrons. The allowed energy levels of valence electrons form the so called valence and conduction bands. The valence band comprise the energy levels of all valence electrons in a semiconductor material at absolute zero.

These electrons are the ones making the covalent bonds and when forming a bond they are immobile. So electrons occupying energy states in the valence band are fixed in the lattice and cannot move around. However, when supplied extra energy, for example in the form of heat, they can be excited and leave the bond. When this happens they gain energy and fill available energy levels in the conduction band.

In semiconductors, electrons with energies in the conduction band are mobile and can move throughout the material allowing the material to conduct electricity. In the case of solar cells, as we will see in following posts, light can excite valence electrons to the energy levels of conduction band. This is the first step to convert energy of light directly into electricity.

The difference between the conduction and valence bands is known as the bandgap. There are no energy states available for electrons between the top of the valence band and the bottom of the conduction band. So with these three definitions in mind, let’s take a look at what a band diagram looks like. 

There are numerous energy states in both bands, but we will mostly be concerned with the edges of these bands. These parts of the bands are the most important parts for understanding photovoltaic applications. We can usually draw the bands like this with a solid yellow box for the conduction band and a solid blue box for the valence band. We can also define the bandgap, which we denote as ‘E_G’ as the difference between the bottom edge of the conduction band, ‘E_C’ and the top edge of the valence band, which we denote ‘E_V’.