Lecture 32

Hello friends, I already have discussed about the concept of hole. In a periodic potential, electrons sometimes behave in a way as if it has a positive charge, which is termed as a hole. In this lecture, I am going to introduce another important concept, concept of effective mass. Let me start with a brief summary of electrons and holes. Say the electric field is in this direction. Since electrons are negatively charged, electrons will move in a direction opposite to the direction of the electric field. On the other hand, since holes are positively charged, holes will move in a direction parallel to the electric field. However, the direction of current remains the same. Motion of an electron is determined by the semi-classical equation of motion. Let us consider only the electric field. Since the external electric field is in the negative direction, k is going to increase in the positive direction with time. There are two possible scenarios. First one is velocity increasing as k increases. In this case, acceleration dv dt is parallel to dk dt. This is exactly what you would expect for a negatively charged free particle accelerating in the direction of external force. This is the direction of external force on a negatively charged particle, which is the same as the direction of acceleration in this region. A second possible scenario is velocity decreasing with increasing k. In this case, the electron decelerates. In other words, acceleration is anti-parallel to dk dt. This is a rather strange situation where acceleration is opposite to the external force. This is the direction of external force on a negatively charged particle and direction of acceleration is opposite to the external force in this region. Let us try to understand the case of acceleration anti-parallel to dk dt. Instead of a negatively charged particle, if we consider a positively charged particle, this is what we would expect. The particle accelerating in the direction of the external electric field. As shown in this figure, this is the direction of external force on a positively charged particle and in this region, acceleration is parallel to the direction of external force for a positively charged particle. Thus, in this region, which is closer to the zone center, we can describe the motion in the electric field as the motion of a negatively charged particle or electron. And in this region, which is closer to the zone boundary, we can describe the motion in the electric field as the motion of a positively charged particle or hole. However, keep in mind that positively charged holes are fictitious particles. We have introduced the idea for our convenience. In reality, acceleration of electrons is anti-parallel to the external force because the lattice provides some additional force, which is related to the back reflection. Note that in this region, the net force on electron is parallel to the external force due to the electric field and the electron accelerates in this region. However, in this region, the net force on electron is anti-parallel to the external force due to the electric field and as a result, it decelerates. However, instead of considering the motion of a negatively charged particle, if we consider the motion of a positively charged particle in an electric field, then in this region, the net force on hole is parallel to the external force. Thus, instead of deceleration of electrons, we can equivalently represent this as acceleration of positively charged holes. Whether we prefer an electron description or hole description depends on our convenience. Let us discuss motion of electron near band maximum. Unoccupied levels lie near the top of the band. We do not know what is the exact form of Ek or energy dispersion. All we know is that Ek is maximum at some point k0. So, this is the point k0 and energy is maximum at this point. Now, we expand Ek about the maximum point using Taylor series expansion. So, this is energy as a function of k, which is approximately equal to Ek0. This is the energy maximum and as we go away from this k0, we just assume that the energy is decreasing quadratically. The linear term vanishes as we are expanding about the maximum point and the first derivative of Ek has to be equal to zero. Moreover, since Ek is maximum at k0, A has to be positive. We define a positive quantity, m star, having a dimension of mass and we write A equal to h cross square divided by 2m star. Thus, we can rewrite this equation as Ek is approximately equal to Ek0 minus h cross square by 2m star k minus k0 whole square. Now, we calculate the first derivative of E with respect to k to get the velocity. V of k is equal to 1 by h cross first derivative of energy with respect to k. And from this equation, we have minus h cross square k minus k0 divided by m star. And we can write acceleration A is equal to d dt of V, which is equal to minus h bar divided by m star, k0 is a constant. So, we just have dk dt. Thus, near the band maximum, acceleration is opposite to dk dt. This is the semi-classical equation of motion. We have found that near band maximum, the acceleration can be written as this. And we can rewrite this as h cross dk dt is equal to minus m star times A. Now, we put it in this equation and we get minus m star times A is equal to the force. Thus, near band maximum, electrons respond to external fields as if it has a negative mass minus m star and a negative charge of minus E. Now, we can cancel the negative sign from both sides of the equation and rewrite this as m star times A is equal to E times electric field plus 1 by c V cross d. This equation is an alternate way of describing the previous equation. However, we are describing the motion of a particle with positive charge or hole with a positive mass m star. Now, let us discuss the motion of electron near band minimum. Thus, previously, what we did was relevant for unoccupied levels, but now what we are going to do is relevant for occupied levels. We do not know what is the exact form of E k. All we know is that E k is minimum at some point k naught. So, this is the point k naught and E k naught is minimum at this point. As we did before, we expand E of k about the minimum point k naught using Taylor series expansion. So, we write E k as E k naught. So, this is the minimum energy and as we go away from the point k naught energy increases quadratically. The linear term vanishes as we are expanding about a minimum point and the first derivative has to vanish. Moreover, since E of k is minimum at k naught, A has to be positive. Now, we define a positive quantity m star, which has a dimension of mass and write A is equal to h cross square divided by 2 m star. Replacing this in this equation, we get E of k is E of k naught plus h cross square by 2 m k minus k naught whole square. Now, we calculate the first derivative of E with respect to k to get the velocity. Velocity V of k is 1 by h bar times the first derivative of E with respect to k, which is approximately equal to h cross k minus k naught divided by m star. Now, we derive the acceleration, which is equal to d dt of V and using this equation, we can write it as h bar by m star dk dt. Thus, near the band minimum acceleration of electron is parallel to dk dt. This is the semi-classical equation of motion. We have found that near band minimum acceleration can be written as h bar divided by m star times dk dt. And we just rewrite this as h bar dk dt is equals to m star times A. Replacing this in equation of motion, we get m star times A is equals to minus E times the electric field plus 1 by c V cross the magnetic field. So, near the band minimum electron responds to external field as if it has a positive mass m star and a negative charge minus E. Let me quickly summarize what I have done so far. In case of free electrons, if we apply some external field, the acceleration is parallel to dk dt. However, in case of block electrons, this is not necessarily true. If we look at a region near the band minimum, that is this region, then it is very similar to the free electron picture. The acceleration is parallel to dk dt and we can write A is equals to h bar by m star dk dt. In this case, the equation of motion is m star, which has a dimension of mass times acceleration is equals to external force. This looks like the motion of free particle with negative charge and positive mass. Now, if we focus in a region near the band maximum, that is in this region and in this region, in this case, the acceleration is opposite to dk dt. That is, we can write acceleration is equals to minus h bar divided by m star dk dt, where m star has the dimension of mass. In this case, the equation of motion looks like minus m star times acceleration is equals to external force. This looks like the motion of a free particle with negative charge as well as negative mass. Thus, near the band maximum motion of electron looks like motion of a free particle with negative charge and negative mass. However, it is more intuitive to write it as motion of a free particle with positive charge and positive mass. From this equation, we can just cancel the negative sign from both the sides and we can write it as mass times acceleration is equals to e electric field plus 1 by c velocity cross the magnetic field. This is the motion of a free particle with positive charge and positive mass. Note that in the electron picture, charge remains negative, but mass becomes negative. However, in the whole picture, mass remains positive, but charge becomes positive. Near the band maximum, we can use either electron picture or whole picture. However, it is more common to use whole description near the band maximum. Now, let me formally introduce the concept of effective mass. Let us start with the energy dispersion relation of free electrons, Ek is equals to h cross square k square divided by 2m. The curvature of Ek curve for free electron is equal to h cross square divided by m. The curvature is given by the second derivative of the Ek curve with respect to k. Thus, 1 by h bar square times the curvature has the dimension of inverse mass. We have seen that near the band maximum equation of motion of an electron looks like this. The right-hand side is the Lorentz force on a negatively charged particle. The left-hand side is minus m star times acceleration, where m star is a positive number and it has a dimension of mass. Near band maximum, it appears from the equation of motion that the mass of the electron is negative. Near the band minimum, the equation of motion looks like this. Right-hand side of the equation is the Lorentz force on a negatively charged particle. On the left-hand side, we have m star times acceleration, where m star is a positive number and it has a dimension of mass. In case of band minimum, m star of electron is positive. Now, let us look at the Ek curve of block electrons. Curvature is negative near the band maximum. Since curvature of Ek curve is related to mass, negative curvature corresponds to negative effective mass in case of states near the band maximum. This agrees very well with the equation of motion near band maximum. On the other hand, curvature of Ek curve is positive near band minimum, which corresponds to positive effective mass. This also agrees very well with the equation of motion near the band minimum. We already have seen the connection between curvature and m star of electron. m star has the dimension of mass and we call it the effective mass. Let me explain why do we call it effective mass. Electron in a periodic potential accelerates relative to the lattice due to the external field as if the mass of the electron is m star. Due to external force, acceleration of an electron in a lattice effectively depends on m star, not on the free electron mass m, which is a fundamental constant. This is why m star is known as effective mass. Velocity of electron is the first derivative of energy with respect to k. Now, let us calculate acceleration, which is dv dt, which is equal to 1 by h bar del 2 E del k del t, which is equal to 1 by h bar del 2 E del k square del k del t. Let me denote this as equation 1. Now, we know that h bar dk dt is equal to external force such that dk dt is equal to force divided by h bar. So, I denote it as equation 2. Now, replacing 2 in 1, we can write dv dt is equal to 1 by h bar square del 2 E del k square times the external force, which implies that the external force is equal to h bar square I del 2 E del k square times dv dt. Note that equation 3 is Newton's second law, if h bar square divided by del 2 E del k square has a dimension of mass. And this is what we call it as effective mass. So, effective mass is related to the curvature of EK curve. If the curvature is negative, as in case of block electrons near the band maximum, the effective mass of an electron can even be negative in a periodic potential. Using a simple EK diagram, let me summarize various concepts discussed so far. The solid line represents the EK curve for block electrons and the dashed line represents the EK curve for free electrons. The curves are drawn only in the first belowing zone. We can see that near the zone center, EK curves of free and block electrons are identical and the difference becomes prominent as we go away from the zone center. Same is true for velocity. Velocity of block electrons is same as free electron velocity near zone center. Velocity increases with increasing k, it reaches some maximum value and then it decreases and becomes 0 at the zone boundary. In the unshaded region of the first belowing zone, velocity increases with k, m star is positive and equal to the free electron mass near the zone center and acceleration is positive. Thus, in the unshaded region closer to the zone center, the behavior of block electrons is somewhat similar to free electrons. On the other hand, block electrons differ from free electrons significantly in the shaded region. Velocity decreases with k, m star is negative and acceleration is negative. That is, acceleration is opposite to the external force which is related to the Bragg reflection. In conclusion, the effect of periodic potential is manifested mainly near the zone boundary while free electron description remains valid near the zone center.