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

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

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

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

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

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

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

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

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

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

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

It means, only the energy levels below the Fermi level are occupied.Meme it Energy states of the conduction band that are above the Fermi level are empty, so noMeme it electrons occupy these states.Meme it However, if the temperature goes up, the probability of occupation of higher energy levels increases.Meme it The more the temperature rises, the probability of occupation gets higher and more electronsMeme it can fill these states.Meme it

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

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

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

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

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

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

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

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

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

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

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

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

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

PV1x_2017_2.2.2_Carrier_concentrations-video.mp4