English Subtitles for PV1x_2017_2.2.1_Carrier_concentrations-video



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In the previous video, we have introduced band diagrams and you learned about the conduction

and valence energy band. Now we will start taking a deeper look at the charge carriers that can occupy energy states in those bands. The occupation of energy states in the valence and conduction bands of semiconductors determines the concentrations of charge carriers. Concentration of charge carriers is a very important property for understanding performance

of solar cells. Let’s start with the equilibrium condition. We define equilibrium of a system as a state in which the system is unperturbed. Therefore, no external forces are applied on this system. These forces could be: external voltage, magnetic field, illumination or mechanical stress. We can define the thermal equilibrium of a system as a condition in which its parameters

do not change with time. We use the word thermal because the equilibrium conditions will change depending on the temperature. Several steps are necessary to take to determine the concentration of carriers in a semiconductor. The first one is to determine the available energy states for electrons in the conduction and valence band. In the previous videos we have described the concept of energy levels and energy bands.

We saw that only few of the levels are effectively occupied by charge carriers. On the left you can see our familiar simplified band diagram, with the conduction band in yellow, the valence band in blue and the band-gap in white. The density of energy states, which we denote as ‘g’, is an important parameter that tell us the number of allowed energy states per unit volume as a function of energy. Here you can see how the density of states varies with the energy.

We have the density of states on the x-axis and the energy level on the y-axis. You can see that there are no allowed states in the band gap. Then, the further we move from the edges of the bands, the more energy states are available. ‘g_C’ represents the density of states in the conduction band. If you remember from the previous video, the conduction band represents energy states of mobile electrons.

‘g_V’ on the other hand, represents the density of states of holes in the valence band. Holes essentially occupy the states in the valence band of missing electrons. The electrons that have been excited to the conduction band. You will learn much more about holes in future videos. Let’s look into the equations that represent these densities.

Here you see our density of states function plotted again. Let’s look at the conduction band. We can express the equation for the density of states in the conduction band as follows. I won’t derive this equation, but you can learn more about it in the textbook. The equation is based on a few constants like the Plank’s constant and the effective mass of an electron.

However, please note, that the dependence on energy is a square root relationship. As the energy gets further away from the conduction band edge denoted by E_C, the amount of allowed energy states increases. Now let’s continue with the valence band. The equations are very similar to that of ‘g_C’ but now we have to use the effective mass of holes instead of electrons.

So now we know the density of ALLOWED energy states of mobile electrons and holes. In order to calculate the total charge-carrier concentrations we also need to know how many of energy states are really occupied. For this purpose, we introduce what is called the occupation function. This function is known as the Fermi-Dirac distribution function. The Fermi-Dirac distribution function expresses the probability that an available energy state

will be occupied at a certain temperature. This function depends on the difference between the energy level of interest and the so called Fermi level E_F. I will explain the definition of the Fermi level in a couple of slides, but let’s first take a look at the temperature dependence of this distribution function. At zero Kelvin, the Fermi-Dirac distribution function is a step function.

It means, only the energy levels below the Fermi level are occupied. Energy states of the conduction band that are above the Fermi level are empty, so no electrons occupy these states. However, if the temperature goes up, the probability of occupation of higher energy levels increases. The more the temperature rises, the probability of occupation gets higher and more electrons can fill these states.

This is the result of thermal excitation. The excited electrons are getting energy from the ambient heat. If the temperature is higher than zero Kelvin, electrons can occupy energy levels above the conduction band edge. Before we go any further, let me explain a very important concept: the Fermi Level. So, what is exactly the Fermi level?

In general it represents the total averaged energy of valence electrons of a material. This energy takes into account the electro-chemical energy of all the electrons in the conduction and valence band. From the Fermi-Dirac distribution function we can easily calculate that the probability that the energy level corresponding to the Fermi level is occupied is zero point five. The position of the Fermi level in an intrinsic semiconductor is close to the middle of the

band gap. Depending on the position of the Fermi level in the band gap we can simplify the Fermi-Dirac distribution function. If the Fermi level is within 3 times k_b T from both the conduction band edge and the valence band edge, so in the pink area of the band diagram we can use this simplified equation.

This is known as the Boltzmann Approximation. We have now defined all the parameters necessary to determine the charge carrier concentrations in thermal equilibrium. Let’s recap. Firstly we understood the energy band diagrams. Then we defined the density of states function to describe all allowed states in our band

diagram. The next step has been to look at the occupation of these states. For this, we introduced the Fermi-Dirac distribution function. Now the last step is to determine the charge carrier densities in the conduction and valence bands. This is how the profile of the occupied states look like.

On the top we can see the occupation of the conduction-band states and on the bottom we see the occupation of the valence-band states. So how did we get these profiles? Here we see our profiles of occupied conduction and valence energy states. Let’s calculate the concentration of charge carriers. We start with electrons in the conduction band.

This is calculated by multiplying the density of states by the occupation function. If we want to know the total amount of mobile electrons, we just have to integrate this product from the conduction band edge up through the band. A similar equation is used for the number of holes in the valence band. However, remember that holes are simply the missing electrons at given states in the valence band.

Therefore, we have to multiply the density of allowed states by one minus the occupation function, to find out which states are UNOCCUPIED. Again, we can integrate this product to obtain the total number of holes in the valence band. If we use the Boltzmann approximations, we can calculate these concentrations. For electrons in the conduction band we get this expression and for holes in the valence band we get this expression.

I won’t go into the full calculation as you can find it in the textbook. There are two new parameters in the equations for carrier concentrations, N_C and N_V. We call these parameters effective densities of conduction and valence band states, respectively. As you can notice they are different from each other, since the effective mass of electrons is different from the effective mass of holes. Now we get to an important property of semiconductors: the intrinsic carrier concentration or n_i.

n_i squared is equal to the product of n, the concentration of electrons occupying states in the conduction band, and p the concentration of holes occupying states in the valence band. This product can be calculated as follows. In the simplified expression we can see that n_i is just dependent on the bandgap, temperature and effective densities of states. In a fully intrinsic semiconductor, since n and p are equal, they are both equal to

n_i. For c-Si at 300K n_i is equal to 10 to the power of 10 per cubic centimeter. We will use this value in many equations to come, so make sure to note it down. This graph shows the temperature dependence of charge carrier concentrations in intrinsic silicon. It can be expected that, with increasing temperature, more electrons are excited from the valence

into the conduction band since more thermal energy is available. This explains the increasing n_i. With this lecture you have learned how to determine the concentration of charge carriers within a semiconductor. However, this method is only valid for intrinsic semiconductors in thermal equilibrium. In the following videos we will first see how the concentration of carriers can be manipulated

by doping the material. Afterwards we will move our analysis to a non-equilibrium situation.



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PV1x_2017_2.2.2_Carrier_concentrations-video.mp4