In this video, we will define density of states. The ultimate goal that we have at this part of the course is to be able to predict the electrical current in a semiconductor. Now, any kind of motion, movement of charge carriers will lead to electrical current. In a semiconductor, we know there are two type of charge carriers, electrons and holes. Or to be more specific, electrons in the conduction band and the holes in the valence band. Then what caused them to move? There are two possible sources that could cause carrier movement. One is an application of external electric field. Now under electric field, the charge carriers will be pushed one way or the other depending on the sign of their charge. And this phenomenon is called the drift. And therefore the current induced by electric field is called the drift current. Carrier movement can also be caused by non-uniform carrier concentration distribution. In this case, carriers will diffuse from high concentration region to low concentration region. This phenomenon is called a diffusion. And the current induced by this diffusion process is naturally called the diffusion current. Now, let's be more quantitative and try to express mathematically the current and how it's related to these carrier properties. By definition, the electrical current is defined by the ratio of the total transfer charge per time. Now, if the carriers are moving at an average speed of v, then the time it takes for a carrier to move is simply given by the length of your material divided by the velocity at which the carriers move. Now physicists like to use the current density instead of current itself because current density is the quantity that is independent of the geometry of the material that you have. By definition, current density is the total current divided by the cross sectional area. Now let's plug in this expression that we just derived for I here. And then that leads to this expression here. Now in the denominator, we have a product of A and L. A is a cross sectional area, L is length. So the product of the two gives you a volume. So total charge divided by volume will give you charge density and multiplied by the average velocity at which your carrier has moved. Now if the current that we're describing here is a current due to electrons in the conduction band, the charged density here is simply given by the electronic charge -q, the charge that each electron carries, times the density of the electron. Density of the electron in the conduction band or sometimes we call it a concentration of electrons in the conduction band. So what do we have now? We can calculate current density if we know the carrier density or carrier concentration. And the velocity at which the carriers move. You'll see the key discovery here. So how do we know how much current flows in a semiconductor? Well, we need to know two things. How many carriers are there in the conduction band? How many flows are there in the valence band. Also we need to know their average velocity, how fast are they moving? Now in order to answer these two questions, we need to post a subsequent question. So we're going to tackle the first question first in order to answer the first question that is how many carriers are there in the conduction band and valence band? In order to answer this question, we need to note two information. One, how many available states are there in those bands, in the conduction band and the valence band? And what is the average probability of finding an electron in a given state. The answer to the first question is given by the density of states. The answer to the second question here is given by Fermi-Derac probability function. Once we know these two, then we can calculate the concentration of the carriers by simply multiplying these two. Now, let's look at the density of states first. What is an energy band? Energy band, we defined as a range of energy that is allowed. We treat it as if it is a continuum. But in reality it is composed of a very very large number of energy levels that are very very closely spaced. So for all practical purpose, we can treat it as a continuum, but in reality it is composed of many many energy levels. So it is logical to ask how many energy levels are there at a given energy? And that is the density of state. So density of state is defined as the number of state per unit volume per energy. Or, alternatively, you can define it as the g(E), density of state, times dE, the small energy interval, and call that the number of states per unit volume within an infinitesimal energy range between E and E + dE. Now, how do we calculate density of states then? Well, we need to know something about the energy band. And here, we adopt something called a parabolic energy band approximation. Now what is a parabolic energy band? Energy band that is a parabola in the e versus k energy band relationship. So typical parabolic energy band is written like this, E energy is proportional to k squared, and the curvature is inversely proportional to the effective mass. Now this is an exact expression for free space electron. And it is also a good approximation in semiconductor near the band edge, that is, at the bottom of the conduction band. At the top of the valence band, the energy band very much should look like a parabola. And we can use this approximation by appropriately choosing the value of the effective mass for each band. Now, once you adopt this parabolic energy band structure, then you can define a sphere with a radius given by this. What is this? This is the k value corresponding to energy E. So for a given energy E here, if you plot all the k values that corresponds to the same energy E, this equation will tell you that that will form a sphere. And the sphere radius is given by this. And this is pictorially shown here, okay? So there is a sphere, section one-eighth of a sphere in the first octet. In this sphere, the surface of this sphere defines all the k values corresponding to energy E. Now, then we can look at the volume occupied by this sphere, one eighth of a sphere. And that sphere represents the k-space volume corresponding to the energy range from zero to E. And you can easily calculate the volume. The volume of a sphere is 4pi over 3 times r cubed. And we're taking only the positive values of k here, so we take one-eighth of it. This is the volume in k-space corresponding to energy range from zero to E. Now I want to know what is the volume occupied by one k value? To look at that, we consider the simple case of rectangular solid. And electron in this rectangular solid, Can only survive if the electron's wave function form a standing wave. What that means is that the wavelength of the electron's wave must fit these rectangular solid perfectly. Otherwise, the wave reflected by this boundary and wave reflected by that boundary will interfere each other destructively and disappear. Only the standing wave can survive. So that condition is given by this here, half wavelengths into their multiple of half wavelengths must be equal to the length of the solid. And this from the definition of k which is 2pi over the wavelengths. We get the equation that k is equal to integer n times pi over L. So it's an integer multiple of pi over L, and therefore the interval between two adjacent allowed k values is simply given by pi over L. Now extend this argument to three dimensional solid. What is the volume occupied by one allowed value of k vector, a triplet of kx, ky, and kz? Well, that is simply given by the cubic power of this pi over L, because individual direction kx and ky, kz. You have an interval of pi over L, or pi over L sub x, pi over L sub y, and pi over L sub z if these x,y,z dimensions are different. And in any case these triple product LxLyLz simply gives you the volume of the solid. So the case space volume occupied by one allowed k vector is given by pi cubed over V. Now this is a very simple argument that derived this. To be more rigorous, you have to assume a periodic boundary condition. And that really is beyond the scope of this course. And I refer you to interested readers to a solid state physics textbook. But you end up getting the same results. The one allowed value of k occupies the volume, k-space volume of pi cubed over V, the real space volume of the solvent. Now, we can then divide the Total Volume of k-space corresponding to energy range from zero to E divided by the volume occupied by the one k. That will give you the total number of k factors contained in that volume, contained in that energy range from zero to E. And that simply give you by this. So this is the one-eight of the sphere that we calculated earlier. And pi cubed over V is the volume occupied by one k vector. And that gives you this. This is, again, the number of states, the number of different k values in the energy range from zero to E. Now we have multiplied by 2 here for spin degeneracy. In each energy level, you may have two electrons, spin up and spin down. Now, we can then calculate the number of states per unit energy per volume by taking a differentiation of this relative to E and divide it by 1 over V, volume. And that leads to simple calculus will give you this equation here. Now to apply this result to conduction and valence band, all you have to do is to substitute the effective mass for conduction band electron here. And effected mass for valence band hole here, excuse me >> Effective mass for electron here, effective mass for hole here. And shift the origin of energy to the bottom of the conduction band and the top of the valence band. So these are the density of states for conduction band here, and density of states for valence band here. The key part is that, in this case, the density of state goes as a square root of energy. So it is 0 at the bottom of the conduction band. It increases as square of E as you go up into higher energy. Likewise, the density of states for hole is zero at the top of the valence band. And as you go down in energy, deeper into the valence band, the density of state increases as a square root of the energy. Now, the square root of E dependence is the result of the three-dimensional space that we have considered here. What if we have a low dimensional structure? What if the electronic motion is restricted to a plane? Well, in that case, your parabolic energy band here is simply given by this two-dimensional expression, okay? So instead of kx squared plus ky squared plus kz squared, we have simply kx squared plus ky squared. And instead of the volume of the sphere that we consider, we will be considering the area, two-dimensional area, in k-space. We instead of the volume occupied by one k point, we will be talking about an area occupied by one k point using this, going through the exactly the same procedure. You will see an energy, you will see a density of state that is independent of energy. So in two dimensional structure, your density of state does not depend on energy. Now you can do the same for one-dimensional structure and zero-dimensional structure. And the summary is shown here. So in the three dimensional case, you have a density of state that is proportional to square root of E. In a two-dimensional structure, you have a density of states that is flat independent of the energy. And every time you introduce a new energy state, there's a step. In the one-dimensional case, you have a density of state that is proportional to 1 over square root of E. And in a zero-dimensional structure, every time there is allowed energy level, there is a spike in density of states. So density of states are given by a series of delta functions.
