Phonons

Type of physical science: Condensed matter physics

Field of study: Solids

As a result of the close packing of atoms, the displacement of a single atom from its equilibrium position in a crystalline lattice causes the neighboring atoms to move as well. This disturbance travels as a wave through the crystal and is called a phonon. These waves are responsible for the flow of heat in insulating materials, for electrical resistance in ordinary metals, and for superconductivity.

Overview

A phonon is a wave of displaced atoms that travels through a solid. Each atom moves in a repetitive fashion, or oscillates, about its equilibrium position, where it experiences no net force. The atoms in a solid are usually found at a series of very regular positions, called lattice sites. This array of atoms, known as a crystal, is very similar to a group of soldiers marching in formation. Each atom is at a particular position, which is a certain distance from its neighboring atoms. Every atom experiences a force from each of its neighbors. At the lattice sites, these forces are balanced so that the atom is at a position of stable equilibrium. If the atom moves away from the lattice site by a small amount, it will experience a net force that will pull the atom back toward its lattice site. This pull arises because the atoms in the solid are bound to one another. In general, the bonds between an atom and its nearest neighbors are the strongest bonds in the solid, just as a soldier interacts most effectively with the soldier next to him. The bonds between an atom and its second nearest neighbors are usually weaker. The bonds between an atom and its third nearest neighbors might still be significant, but bonds in crystals tend not to extend much farther. A similar situation arises for the soldiers in marching formation; a soldier finds it quite difficult to talk directly with a soldier who is very far away. The soldier can still talk with any other soldier by having each intermediate soldier pass the message along. This same mechanism is found in solids; there is order throughout the crystal as a result of the fact that each atom interacts with its immediate neighbors in an identical manner.

If an atom is displaced, the atom will push its neighboring atoms in the same direction before returning to its equilibrium position. These atoms then push their neighbors in the same direction, and so on. This effect is similar to what would happen if one soldier would fall, knocking over the next soldier who, in turn, would knock over a third soldier. In this manner, a wave of falling soldiers would travel through the formation. The same phenomenon happens in crystals: A wave of successive displacements of atoms travels through the crystal.

The bonds between neighboring atoms in a crystal can be accurately described by thinking of the atoms as being connected by small springs. An atom that is displaced from its equilibrium position experiences a restoring force when these springs push the atom back to its equilibrium position. Thus, the atom can be considered as an oscillator. The atom will execute simple harmonic motion about the equilibrium position. In a solid, there is not a single oscillator but approximately 10²³ oscillators, all of which are connected to one another.

The motion of these coupled atoms can be analyzed mathematically to yield a set of uncoupled equations describing the normal modes of oscillation. Any motion of the atoms in the solid can be described as a combination of these normal modes. The application of quantum mechanics to this analysis shows that the normal modes are quantized as phonons.

Striking a crystal will cause a wave of displaced atoms to travel through the crystal in the form of a sound wave, just as sound travels through air. The same principles are involved in both cases. When a person speaks, a pattern of disturbance is created of the molecules in the air.

In some regions, the air molecules have been moved closer together, whereas in other regions, the air molecules have been moved farther apart, forming a wave in the air. Such sound waves (or phonons) can also travel through liquids and solids.

An external action, such as someone speaking or a hammer striking a crystal, is not necessary to create these waves. Phonons are always present in any crystal as a result of thermal excitation. The connection between temperature and atomic motion is most easily understood for a boiling liquid. In this case, the heat causes the liquid to become a gas and form a bubble. The bubble travels to the top of the liquid and escapes into the air. The gaseous molecules move with a high velocity because of their high temperature. The average velocity of an atom in a liquid or solid is also related to the temperature. The greatest difference between atoms in a gas and atoms in a solid is that the atoms in a gas can travel anywhere, whereas the atoms in a solid cannot move very far from their equilibrium positions. The atom moves at its highest speed through its equilibrium position (where the spring is not stretched) and slows down as it moves away from equilibrium. Eventually, the spring brings the atom to rest and pulls it back toward the equilibrium position. As the atom moves away from its equilibrium position, it pushes its neighboring atoms, which then push their neighboring atoms. In this manner, phonons are always present in a solid, whether or not an external disturbance has been applied. At lower temperatures, the average velocity of each atom will be decreased and the atom will have a smaller displacement from the equilibrium position. Consequently, the atom will not be able to push the next atom so vigorously. The number of phonons present in the sample is a function of temperature: The lower the temperature, the fewer the phonons.

One important parameter in describing a phonon is the speed at which it propagates through the crystal. The speed of the phonon is related to the strength of the bonds connecting neighboring atoms. The bonds connecting atoms in a gas, such as air, are very tenuous; as a result, the speed of sound in air is about 330 meters per second. In contrast, very strong bonds hold together the atoms in a solid; consequently, the speed of sound in a solid is much greater, typically about 4,000 meters per second.

This discussion has assumed that the solid is composed of only one type of atom. In reality, a solid may be composed of more than one atom, such as in table salt (NaCl). In such solids, there is an array of atoms at each lattice site called the unit cell. Table salt has two atoms in its unit cell: one sodium ion and one chloride ion. A phonon in which both the sodium and the chloride ions move in the same direction is called an acoustic phonon (or a sound wave). A different kind of phonon, called an optic phonon, is also possible. In an optic phonon, the atoms of the unit cell move in opposite directions. Optic phonons involve significantly more stretching of the bonds connecting the atoms within a unit cell than acoustic phonons, since the atoms of the unit cell in an optic phonon are moving in opposite directions. As a result of the increased stretching of bonds, optic phonons carry more energy than acoustic phonons.

Applications

Phonons transport heat in any solid, since the atoms in a hot region have large displacements and create phonons that propagate into cold regions. In materials with mobile electrons such as metals or semiconductors, the mobile electrons carry the vast majority of the heat. In an insulator, however, there are no mobile electrons, and the only way for heat to flow from one part of the crystal to another is via the phonons. One important application of phonons is in integrated circuits. In these miniature devices, many electrical components are manufactured in a very small space. During normal operation, all electrical devices consume power in order to perform their functions. Since no physical system works with perfect efficiency, some of the electrical power is lost to heat. This heat must be transported away from the devices to avoid overheating. In order to conduct the heat away without disturbing the electrical function of the integrated circuit, an insulating material is frequently used. In hot regions, the atoms of the insulator are heated to high temperatures, which causes the atoms to move fairly large distances away from their equilibrium points. This big displacement causes the neighboring atoms to be pushed a large amount as well. Thus, the phonons generated in hot regions will move into cool regions. In the cool regions, this excess heat is lost to the surroundings. The most efficient way to remove excess heat with an insulator is with phonons that have a very high speed of propagation.

The speed of propagation is related to the strength of bonds in the solid. The stronger the bonds, the higher the speed of sound, just as a wave travels faster down a taut rope than down a relaxed rope. The insulator with the strongest bonds is diamond.

Another example of the use of an insulator to transport heat are the windows of an intense laser. As the laser beam travels through the window, a certain amount of the laser beam is absorbed by the window. The amount of energy absorbed is usually a very small fraction of the energy present in the laser beam. If the laser beam is extremely intense, then the amount of energy absorbed by the window can result in significant heating of the window, causing the center of the window to heat up. The differences in temperature between the center and edge cause stresses in the window. If these stresses are large enough, the window will break.

Consequently, windows in the most powerful lasers are usually made of the best conductor of heat, diamond.

Phonons also influence electrical resistance in ordinary conductors. In metals, electrons travel through the crystalline lattice. An individual electron will travel only a finite distance before it is scattered--before its direction of travel is changed. When considering electron motion in a crystalline solid, one could conclude that an electron can travel without being scattered if it could travel through the empty spaces of the solid. Quantum mechanics shows that the electrons do not need to travel through the empty spaces between the atoms; an electron can travel through a perfect lattice without being scattered. In practice, imperfections always exist. Impurities are present that break the perfect repetition. Even if a perfect lattice could be grown, the repetition of the lattice is broken by the presence of phonons, which move the atoms away from their regular lattice sites. This loss of perfect regularity of the lattice causes the electrons to be scattered. In this manner, phonons give rise to electrical resistance.

Phonons play an integral role in normal superconductivity. The electrical resistance of a superconductor is zero; in other words, electrons travel through a superconductor as if there is no opposition to their motion. In a superconductor, the individual electrons form pairs, known as Cooper pairs, in which the electrons are held together by the effect of a phonon. At first glance, it seems odd that two electrons could form a pair, since the electrons have the same charge and should repel each other. A phonon is a disturbance of the atoms from their regular lattice positions. In materials that become superconductors, the atoms at the lattice sites are actually positively charged ions. The mechanism of Cooper pairing is subtle. One of the two electrons moves through the lattice. As this electron passes the ions, the positive ions feel a net attraction to the negatively charged electron. This attraction causes the lattice of ions to distort, creating a phonon. The ions typically have a mass twenty thousand times greater than the mass of the electron. Consequently, the ions move more sluggishly than the electron. Long after the first electron has left, the lattice will still have a net distortion inward. For the second electron, this net distortion of the lattice appears as an effective positive charge, which is attractive to the second electron. The second electron is drawn toward the first electron by this distortion of the lattice. These two loosely bound electrons form a Cooper pair, which gives rise to superconductivity.

Context

The seventeenth century scientist Christiaan Huygens was among the first to suggest that a crystalline solid is composed of atoms packed in the regular pattern of a lattice. The development of the theory of the motion of atoms in such a lattice began with the three laws of motion established by Sir Isaac Newton. The basis of the oscillatory motion of the atoms in a lattice can be understood in terms of Newton's second law of motion.

In 1900, Max Planck initiated the revolution of quantum mechanics by suggesting that the energy of electromagnetic radiation exists in discrete amounts known as quanta. In 1907, Albert Einstein first applied the quantum hypothesis to crystalline solids. Combining the concept that energy comes in discrete "packets" with the prediction of waves in a solid resulting from Newton's second law, Einstein produced the first explanation of the low-temperature thermal properties of solids. Einstein's work provided the first description of a phonon. In addition, this work on solids was the first application of quantum principles to mechanical systems. By successfully describing both electromagnetic and mechanical phenomena, quantum theory was seen to have wide applicability in the physical world. It has since been shown that quantum principles govern all physical phenomena when viewed on a sufficiently small scale. In 1912, Peter Debye refined Einstein's work and showed that the concept of phonons is necessary to understand the observed differences in the abilities of various solids to absorb heat energy at low temperatures. His work formed the basis for the theory of lattice dynamics, which is the modern theory of the motion of atoms in a solid.

In 1928, Sir Chandrasekhara Venkata Raman discovered that light was changed by passing it through a transparent sample. He found that the energy of a small amount of light was changed by either absorbing or creating a phonon. In order to conserve energy, the light had either to gain or lose a small amount of energy (the energy of the phonon). This effect, known as Raman scattering, provided direct experimental evidence of the existence of phonons. Raman was awarded the Nobel Prize in Physics in 1930 for this discovery. Careful comparison of the theory of lattice dynamics with experimental techniques such as Raman scattering makes it possible to evaluate the microscopic origin of the thermal and mechanical properties of solids.

Principal terms

ATOM: the fundamental building block of all matter, composed of a nucleus of protons and neutrons orbited by electrons

CRYSTALLINE LATTICE: a repeating three-dimensional array of positions

DISPLACEMENT: the distance of an atom from its equilibrium position

EQUILIBRIUM POSITION: the position of an atom in a unit cell at which the atom experiences no net force; the natural position of the atom

OSCILLATOR: a system that undergoes a repetitive motion

RESTORING FORCE: a force that drives the atoms back toward their equilibrium positions

UNIT CELL: the basic grouping of atoms that occurs at each lattice position in a crystalline solid

WAVE: a repetitive motion traveling through a medium

Bibliography

Feynman, Richard P., Robert B. Leighton, and Matthew Sands. THE FEYNMAN LECTURES ON PHYSICS. Vol. 1. Reading, Mass.: Addison-Wesley, 1963. These famous lectures are based on a course of lectures in introductory physics given by Feynman at the California Institute of Technology. Covers all the basic topics necessary to understand the concept of the normal modes of oscillations.

Frautschi, S. C., R. P. Olenick, T. M. Apostol, and D. L. Goodstein. THE MECHANICAL UNIVERSE: MECHANICS AND HEAT. New York: Cambridge University Press, 1986. This book is the written version of the public television series THE MECHANICAL UNIVERSE. A clear presentation of the important topics, with many useful diagrams.

French, A. P. VIBRATIONS AND WAVES. New York: W. W. Norton, 1971. Part of the introductory physics course taught at the Massachusetts Institute of Technology. Fully develops all the concepts and provides the necessary mathematics to understand coupled modes of oscillations, including a discussion of phonons.

Gough, W., J. P. G. Richards, and R. P. Williams. VIBRATIONS AND WAVES. Chichester, England: Ellis Horwood, 1983. A well-written discourse on a number of different physical systems in which vibrations and waves are important. Intended as a textbook and includes a development of the mathematical derivations for the more advanced reader.

Hewitt, P. G. CONCEPTUAL PHYSICS: A NEW INTRODUCTION TO YOUR ENVIRONMENT. Boston: Little, Brown, 1971. Intended for use in an introductory physics course for those with little background in science. Every subject is discussed thoroughly with many helpful illustrations. Includes an excellent discussion of vibrations and waves, along with a chapter on the properties of solids.

Kittel, Charles. INTRODUCTION TO SOLID STATE PHYSICS. 6th ed. New York: John Wiley & Sons, 1986. This textbook has been used to introduce an entire generation of scientists to the concepts of solid-state physics. Crystal structure, phonons, and the effects of phonons are discussed in the first five chapters with a minimum of mathematics.

Reissland, J. A. THE PHYSICS OF PHONONS. New York: John Wiley & Sons, 1973. One of the few textbooks devoted exclusively to phonon physics. The first several chapters present a clear, accessible discussion of phonons, though later chapters are intended for those familiar with the more advanced concepts of physics.

Thermal Properties of Solids

Sounds Waves