# ECE Purdue Semiconductor Fundamentals L3.6: Equilibrium Carrier Concentration – Recap

[L3.6 Slide 1] Well, it’s time to wrap up Unit 3. In Unit 3, we have been discussing how we compute equilibrium concentrations of electrons and holes in a semiconductor. By equilibrium, we mean the semiconductor
is just sitting there. No voltages have been applied, we’re not shining light on it. It’s not the condition
that we’re interested in when we think about semiconductor devices, but it’s a good starting point for understanding devices. So I’d just like to summarize now, in this lecture, some of the key points that we should take away from this unit. [Slide 2] We began the unit by
discussing the Fermi function. The Fermi function was
this simple equation that gives us the
probability that a state, if it exists at a particular energy E, is occupied by an electron. The two important parameters
in the Fermi function are the Fermi level E sub F and the temperature. [Slide 3] If we plot the Fermi function, the probability is between 1 and 0 depending on the energy of the state. If the energy is equal
to the Fermi energy, then the probability is 50% that the state will be occupied. Notice that if the Fermi
level is in the bandgap, in the forbidden gap, there are no states. So, there will be no electrons there even though the Fermi function gives us a probability of 1/2. Now there’s a transition
between 1 and 0 that depends on the temperature. The higher the temperature, the broader that transition. The lower the temperature, the narrower that transition. At T equals zero, we would get a step function here. All states below the Fermi level would be occupied with probability 1, all states above the Fermi level would have no probability
of being occupied. The width of this transition region is a few kT in width and energy. [Slide 4] We then used the density of states that we discussed in the previous unit, and the Fermi function that we introduced in this unit, to ask ourselves the question, how are the electrons distributed in the conduction band? Where are they? Well, we could answer that question simply by looking at the number of states in an energy range, the density of states times the width of that energy range, and multiplying by the probability that those states are occupied. And we saw that it peaks very near the bottom
of the conduction band, in fact, about 1/2 kT from the bottom of the conduction band. That’s a small number, at room temperature that would be 0.013 electron volts, so that confirms our intuition that most of the electrons should be should be occupying states that are very near the bottom of the conduction band. Same things goes for the
top of the valence band. So the holes are near the top of the valence band, the electrons are near the bottom of the conduction band. [Slide 5] We then asked ourselves how do we relate the
position of the Fermi level to the total number of electrons in the conduction band, total number of holes in the valence band? To do that, we just take the number of states at an energy range and we integrate them across all energies in the conduction band. The number of states is density of states times dE. The probability that they’re occupied is the Fermi direct integral. We can do that integral, the result turns out to be
a collection of constants, parameters, and a function that we call the Fermi direct integral of order 1/2, and it depends on where the Fermi level is located with respect to the bottom of the conduction band. This parameter N sub C, we call the effective density of states. And it depends on temperature and some fundamental constants and, very importantly, on the effective mass
of the semiconductor. [Slide 6] Now if we look a little more carefully at this Fermi direct integral and plot it as a function of its argument, Eta is the argument here, the Fermi energy with respect to the bottom of the conduction band in units of kT. Then when the energy is very far below the Fermi energy, the two are identical. The Fermi direct integral is equal to the exponential. As they go to higher energies, then the Fermi direct integral is always less than the
corresponding exponential. A nondegenerate semiconductor is one in which the Fermi level is well below the bottom
of the conduction band. The Fermi direct integral
reduces to an exponential. That makes life simple, makes it easy to calculate
various quantities that we’re interested in. In this class, we will be making use of the nondegenerate approximation extensively. [Slide 7] And just to remind you, a nondegenerate semiconductor is one that the Fermi level stays inside the bandgap and doesn’t get too close to the bottom of the conduction band, or too close to the top
of the valence band. [Slide 8] So, just to summarize what
you should remember, is how the electron density is related to the location of the Fermi level. It’s the effective density of states times the exponential of Fermi level minus bottom of the conduction band over kT. Fermi level goes first because the higher the Fermi
level the more electrons. The hole density is related to the effective density of states
for the valence band, which depends on the
valence band effective mass. And then top of the valence band minus Fermi energy, over kT Minus the Fermi energy because the lower the Fermi energy, the more probable it is that states in the valence band are empty and we have holes. [Slide 9] Then having developed these expressions for electron concentration
and hole concentration we’ve multiplied the two together to get the product, the np product. This is an important quantity that we make use of
frequently in semiconductors. What we found when we
did that multiplication is that it is independent of the location of the Fermi level, np is equal to ni squared, there’s an assumption
here that we’re dealing with a nondegenerate semiconductor again. When we evaluated this expression we got a formula for the intrinsic carrier concentration squared. It depends on the square root of the product of the
two densities of states, effective densities of states, so the effective masses come in there. But very importantly, it depends exponentially on the bandgap and on the temperature. [Slide 10] We then asked ourselves, if we have an intrinsic semiconductor but we have an equal number of electrons and an equal number of holes, where is the Fermi level? We expect it to be near the middle of the bandgap. We don’t expect it to be exactly in the middle of the bandgap because the electron effective mass and the hole effective mass are usually slightly different. That means the effective density of states for electrons and the
effective density of state for holes are slightly different. So we were able to develop an expression that tells us that the intrinsic Fermi level is
in the middle of the gap plus a correction that can be either positive or negative
depending on the ratio between the hole and the
electron effective masses. [Slide 11] So, in summary then, we developed these relations that tell us how the electron concentration and the hole concentration is related to the position of the Fermi level with respect to the band edges. We also showed that there is a mathematically equivalent way that is sometimes more convenient if we know the intrinsic carrier
concentration accurately, in silicon it’s easy to remember that it’s one times 10 to the 10th, then it may be easier to use
the equivalent expression which says that the electron concentration is the intrinsic carrier concentration times e to the Fermi level minus intrinsic level over kT. The higher the Fermi level is above the intrinsic level, the more electrons we have. The lower the Fermi level is below the intrinsic level, the more holes that we have. The constants in these expressions, the effective densities of states, are given by these two expressions. The intrinsic carrier concentration is given by this expression. And the product of the two is just this material parameter we call ni squared. [Slide 12] Now, we’ve been talking about
how carrier concentrations are related to the location
of the Fermi level. When we draw energy band diagrams and we indicate the Fermi level, it is easy to see from the position of the Fermi level then, whether we’re dealing with an n-type semiconductor or
a p-type semiconductor. A heavily doped, or an n-type conductor
with a lot of electrons or with a few electrons, we just look at where the Fermi level is. If it’s above the intrinsic level we have an n-type semiconductor. If it’s below the intrinsic level we have a p-type semiconductor. [Slide 13] We then turned out attention to calculating carrier concentrations for a given doping density. We changed the concentrations of electrons and holes in a semiconductor by introducing dopants. So the question is, when we introduce a known number of donors or acceptors, how many electrons and holes result? We began that calculation by saying that semiconductors like to be neutral, the dopants will become ionized, and when they’re ionized they will create free electrons and free holes. The mobile charges, the free electrons and the holes, will flow around to the immobile charges, the dopants, the ionized dopants, such that the positive charges and negative charges cancel each other out and we’ll be left with a neutral region by putting this expression for space charged neutrality together with the fact the n times p is equal to n i squared. We have two equations and two unknowns, the two carrier densities, we can solve those two equations. We get quadratic equations. If we have an n-type material we would express the solution this way. If we have a p-type solution we would express the solution this way. So, given the known doping densities and the known intrinsic
carrier concentration for the semicondunctor, we can compute the resulting electron and hole concentrations. [Slide 14] We then look at the different regions, temperature wise, there is a region we call
the extrinsic region. In this region, the carrier concentrations are dominated by the dopants that we have put into the silicon lattice. Not by the intrinsic carriers caused by breaking the covalent bonds. In semiconductors like silicon where all of the dopants are fully ionized at room temperature, or the bandgap is wide enough that the intrinsic carrier concentration is small compared to the doping densities, then the answer is simple. In an n-type material, the electron concentration is equal to the net n-type dopant. And then the hole concentration is just ni squared divided by that
electron concentration. In a p-type material, the hole concentration is just the net p-type dopant. And the minority electron concentration is then ni squared divided by that hole concentration. [Slide 15] So as long as we stay
in the intrinsic region things are very simple
and easy to compute. Now if we move to higher temperatures, we know that the intrinsic carrier density is going to increase exponentially. Then it cannot be ignored, in those quadratic equation expressions, we have to solve the quadratic equation to find the electron and hole densities. In this region the
dopants are fully ionized, so we can make that assumption, but we cannot assume that the intrinsic carrier
concentration is negligible. We need to include it in the calculations. [Slide 16] If we turn to the low temperature region there are no intrinsic
carriers to worry about, so we can eliminate that term from the quadratic equation. But at low temperatures, the dopants might not be fully ionized. Only some fraction of
them are fully ionized. At very low temperatures the fraction is an exponential function of the energy level of the donor with respect to the band edge. So, as we go to lower and lower temperatures, we get fewer and fewer carriers coming from those donor atoms until they eventually, when we’re at a low enough temperature, we have no carriers at all and the semiconductor is an insulator. [Slide 17] And then the final topic
we discussed is that knowing the carrier
density versus temperature, how can we determine what the Fermi level is versus temperature? Because these are two parameters we can go back and forth with. If we know the Fermi level, we can deduce the carrier concentration. If we know the carrier concentration, we can deduce the Fermi level. And we argue that it’s easy, qualitatively, to see that at low temperatures the Fermi level and an
n-type semiconductor has to be above the donor level because most of them are occupied. At high temperatures a semiconductor is intrinsic, the Fermi level has to be
at the intrinsic level. And in a p-type semiconductor, an analogous thing occurs. The Fermi level is down here below the acceptor level but at high temperatures, when the semiconductor becomes intrinsic, it also ends up at the intrinsic level. [Slide 18] Okay so, we can basically summarize everything that early on in the course we discussed qualitatively why the carrier density versus
temperature characteristic has this characteristic shape. And in this unit, we discussed quantitatively how we compute the electron and
the hole concentrations in each of these three regions. So, having done this we have a good solid understanding of carrier concentrations under equilibrium conditions. [Slide 19] Some of the terms that we’ve introduced in this unit are listed here. As you review the material in this unit you should make sure that you’re familiar with what each of these
various terms means. So as you review this
material and prepare, we now have the background that we need to move from the equilibrium condition to the non equilibrium condition, which is the condition of interest for semiconductor devices
with applied voltages, or with sunlight shining
on them for example. So we’ll begin that
discussion in Unit 4 next. Thank you.

1. kari Alem says: