Modern Physics Unit SS Solid State Physics

Overview: This unit gives an introduction to solid physics. It begins with a discussion of conduction and the distinctions among metals, insulators, and semiconductors. It also introduces the Maxwell-Boltzmann distribution, the equipartition theorem as applied to specific heats, and the exclusion principle and complex atoms, all of which have application to the theory of solids. A second section treats the elementary quantum theory of solids.

Prerequisite: QM5

Section I: Introduction to Solids:

Kip, Chapt. 7 - The Physics of Conductivity (printout provided), Sec. 7.1-7.4; R. Eisberg and R. Resnick Quantum Mechanics of Atoms, Solids, Nuclei and Particles (2nd Ed.) , Chapt. 13 - Solids, Conductors, and Semiconductors, Sec. 13.4.

After completing this section you should understand::

1. How mobile conduction electrons can account for Ohm's law classically.
2. How bands of allowed energy states account for metals, insulators, and semiconductors.
3. That drift velocity is different from random velocity in the classical picture of electrical conductivity.
4. The difficulties concerning heat capacity of a metal and the temperature dependence of electrical conductivity.

Questions:

1. Classically, how does an electrical current heat a wire?
2. A 60 Hz alternating current of 1 A flows in a 1 mm diameter wire. On the average how far does an individual electron move.

Discussion:

1. The Maxwell-Boltzmann Distribution

In a classical collection of similar particles, not all of the particles have the same energy. In general there is a spread or distribution of energies. The Maxwell-Boltzmann distribution gives the relative number of particles in different small energy ranges as a function of the energy and temperature. In reality particles obey the laws of quantum mechanics, rather than classical mechanics, and we must think of the energy states that the particles occupy. The energy states are very close together in most of the macroscopic systems we will work with, and it is convenient to speak of the density of states. Then the number of particles that are in a small energy interval at a particular energy is the product of the density of states at that energy times the probability that a state of that energy will be occupied at the given temperature.

The Maxwell-Boltzmann distribution is a useful approximation to the quantum mechanical distributions under certain conditions, such as high temperature. This distribution is described in unit T5 and can be reviewed in Appendix C of Eisberg and Resnick.

2. The Equipartition Theorem and Classical Specific Heats

A simple but important theorem can be derived from the Maxwell-Boltzmann distribution. The equipartition theorem states that an average energy of (1/2)kB T is associated with each coordinate or momentum component appearing quadratically in the energy, where kB is Boltzmann's constant. The kinetic energy of an ideal gas molecule, for example, contains the momentum terms px2, py2 and pz2, and the equipartition theorem thus predicts the average energy to be (3/2)kB T. The total energy of a gas of N molecules is thus U = (3/2)N kB T, and the specific heat CV = (∂ U/∂ T)V is (3/2)N kB .

We can also apply the equipartition theorem to solids. A simple model of the atoms in a solid pictures them as vibrating in three dimensions as if interconnected by springs. The total energy of this ensemble of three dimensional oscillators consists of a kinetic energy term with the same squared momentum components as above, but now with an additional potential energy term containing the squares of the coordinates x2, y2 and z2. With six quadratic terms the average energy of one atom is now 6 (1/2)kB T = 3 T kB, giving U = 3 N kB T and CV = 3N kB. Classically, then, the solid is expected to have a constant specific heat of 3kBper particle (the Dulong-Petit law) and this is observed at high temperatures. At low temperatures, however, CV -> 0 and the heat capacity of metals and insulators is observed to be about the same. In the above classical picture we would expect that the free electrons in a metal would act as an ideal gas, adding (3/2) kB per free electron for the specific heat of a conductor. These contradictions can be resolved only by the quantum theory of solids.

The equipartition theorem is reviewed on page 14 of Eisberg and Resnick and its application to solids is discussed on pages 421 and 422.

3. The Exclusion Principle and Atoms

4. Solids are made up of atoms, each of which when isolated has definite discrete energy levels, certain of which are important to the collective behavior of the solid. An approximate quantum description of atoms is in terms of single electron wave functions and energy levels: each electron is assumed to move in the electrostatic potential of the nucleus and the other electrons. Since the charge of the other electrons is spread out over the atom this potential does not depend upon the radius r as 1/r. Although the states can be labelled by the same quantum numbers n, l, ml, ms as the levels of the hydrogen atom, the energy depends upon both n and l for complex atoms, increasing with both n and l. In the usual notation the letters s, p, d, f, g,... are associated with the angular momentum quantum numbers l = 0, 1, 2, 3, 4,... and the energy levels increase according to 1s, 2s, 2p, 3s, 3p, (4s,3d), 4p,... with the parenthesis indicating that the 4s and 3d levels are nearly the same. The energy levels of the ground state of atoms are strongly influenced by the Pauli exclusion principle, which forbids two electrons to be in the same quantum state. Thus, for example, the ground state of the neutral sodium atom, with 11 electrons, has the configuration 1s2, 2s2, 2p6, 3s1, and its chemical properties are determined mainly by the presence of a single weakly bound electron in the 3s level. More details about the structure of multielectron atoms and the periodic table can be found on pages 359 and 362 of Eisberg and Resnick.

Section II: Quantum Theory of Solids:

1. Read Eisberg and Resnick, Chapter 11, Sect. 1-4, and understand
1. The energy levels od the electrons in a solid.
2. How the electrons occupy the energy levels.
3. The Fermi energy.
2. Read Eisberg and Resnick, Chapter 11, Sect. 11, and Kip, Chapter 7, Sec. 7, and understand
1. How heat changes the energy distribution.
2. How an electric field changes the distribution, how current flows (Ohm's law) and how resistance arises.
3. The meaning of the work function.
3. Read Eisberg and Resnick, Chapter 13, Sect. 13, and understand
1. How bands of allowed energy levels arise.
2. How to count the number of states in a band.
3. How to determine the number of electrons in a band.
4. What determines whether a solid is an insulator, a conductor or a semiconductor
5. The role of impurities in semiconductors.

Questions:

How does current heat a wire from the point of view of quantum mechanics.

Eisberg and Resnick, Chapter 11: 3,7,20,23,34; Chapter 13: 2,10,11,12

Problems

Kip, Chapter7: 2C,2E,2F