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Solid state physics is a vast area of quantum physics in which we are concerned with understanding the mechanical, thermal, electrical, magnetic, and optical properties of solid matter. Some aspects have been discussed in earlier chapters, such as the lattice and electronic contributions to the specific heats of solids, radiation from a blackbody, thermionic emission, and contact potentials. Here we shall focus on the origin of the forces that hold atoms together in a solid and on the allowed energy levels of the electrons in the solid. This will lead us to the band theory of solids. That theory will then be applied to phenomena of much practical and theoretical interest, including semiconductors and semiconductor devices. Many electrical, thermal, and optical properties of solids will thereby become more clearly understood. In the next chapter we extend the theory to the phenomenon of superconductivity and consider magnetic properties of solids as well.
TYPES OF SOLIDS
In the gaseous state the average distance between molecules is large compared to the
size of a molecule, so the molecules may be regarded as isolated from one another.
Many substances, however, are in the solid state at ordinary temperatures and
pressures. In that state molecules (or atoms) can no longer be regarded as isolated.
Their separation is comparable to the molecular size, and the strength of the forces
holding them together is of the same order of magnitude as the forces binding the
atoms into a molecule. Hence, the properties of a molecule are altered by the presence
of neighboring molecules. Characteristic of crystalline solids is the regular arrangement
of atoms, a recurrent or periodic pattern called a crystal lattice. The solid can
be regarded as a large molecule, the forces between atoms being due to interaction
between atomic electrons, and the structure of the solid being determined as that
arrangement of nuclei and electrons which yields a quantum mechanically stable
system. Although the number of atoms involved is very large, they are arranged in a
regular pattern. In noncrystalline solids, such as concrete and plastic, the perfectly
regular pattern does not hold over long distances, but there is an orderly pattern in
the neighborhood of any one atom. We shall discuss only crystalline solids in this
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book. Such solids are classified according to the predominant type of binding, the principal types being molecular, ionic, covalent, and metallic. Molecular solids consist of molecules which are so stable that they retain much of their individuality when brought in close proximity. The electrons in the molecule are all paired so that atoms in different molecules cannot form covalent bonds with one another. The intermolecular binding force is the weak van der Waals attraction that is present between such molecules in the gaseous phase. The physical mechanism involved in the van der Waals attraction is an interaction between electric dipoles. Because of the fluctuating quantum mechanical behavior of the electrons in a molecule, all molecules have a fluctuating electric dipole moment, even though for many of them symmetry considerations require that it fluctuate about an average value of zero. At a time when a molecule has a certain instantaneous electric dipole moment, the external electric field that it produces will induce in the charge distribution of a nearby molecule a dipole moment. By drawing rudimentary sketches of the charges and field in various cases, the student can immediately convince himself that the force exerted between the inducing and the induced electric dipole is always attractive. The interaction energy is proportional to the mean square of the inducing electric dipole moment. The resulting attraction is weak, the binding energies being of the order of 10 -2 eV and the force varying with the inverse seventh power of the intermolecular separation. In the solid, successive molecules have electric dipole moments which alternate in orientation so as to produce successive attractions. Many organic comci. pounds, inert gases, and ordinary gases such as oxygen, nitrogen, and hydrogen form molecular solids in the solid state. Because the binding is weak, solidification takes place only at very low temperatures where the disruptive effects of thermal agitation are very small. (The melting point of solid hydrogen is 14°K, for example.) The weak binding makes molecular solids easy to deform and compress, and the absence of free electrons makes them very poor conductors of heat or electricity. Ionic solids, such as sodium chloride, consist of a close regular three-dimensional array of alternating positive and negative ions having a lower energy than the separated ions. The structure is stable because the binding energy due to the net electrostatic attraction exceeds the energy spent in transferring electrons to create the isolated ions from neutral atoms, just as for ionic binding in molecules. Ionic binding in solids is not directional because spherically symmetrical closed shell ions are involved. Hence the ions are arranged like close-packed spheres. The actual crystal geometry depends on which arrangement minimizes the energy, and this in turn depends principally on the relative sizes of the ions involved. Because there are no free electrons to carry energy or charge from one part of the solid to another, such solids are poor conductors of heat or electricity. Because ctf the strong electrostatic forces between the ions, ionic solids are usually hard and have high melting points. Lattice vibrations can be excited by energies corresponding to radiation in the far infrared, so that ionic solids show strong optical absorption properties in that region. But optical absorption by excitation of electrons requires energies in the ultraviolet, so that ionic crystals are transparent to visible radiation. Covalent solids contain atoms that are bound by shared valence electrons, as in covalent binding of molecules. The bonds are directional and determine the geometrical arrangement of atoms in the crystal structure. The rigidity of their electronic structure makes covalent solids hard and difficult to deform, and it accounts for their high melting points. Because there are no free electrons, covalent solids are not good heat or electrical conductors. Sometimes, as for silicon and germanium, they are semiconductors. At room temperature some covalent solids, such as diamond, are transparent; the energy required to excite their electronic states exceeds that of photons in the visible region of the spectrum so that such photons are not absorbed. But most covalent solids absorb in the visible and are therefore opaque.
BAND THEORY OF SOLIDS
To understand the effect of putting a great many atoms close together in a solid,
consider first two atoms only that are initially far apart. All of the energy levels of this
two-atom system have a twofold exchange degeneracy. That is, for the combined
system the space part of the eigenfunction for the electrons can contain either a
combination of the individual atom space eigenfunctions which is symmetric in an
exchange of pairs of electron labels, or which is antisymmetric in such a label exchange.
(The total eigenfunction of the system of electrons is, of course, antisymmetric,
since the symmetric space eigenfunction is associated with an antisymmetric
spin eigenfunction, and vice versa.) When the atoms are widely separated, the two
different types of eigenfunctions lead to the same energy, and so each of the energy
levels is said to have a twofold exchange degeneracy. But when the atoms are brought
together, the exchange degeneracy is removed. Because the electron charge density in
the important region between the atoms depends on whether the space eigenfunction
is symmetric or antisymmetric, when the atoms are close enough together that the
wave functions of the individual atoms overlap, the energy of the system depends
on the symmetry of the space eigenfunction. Hence, a given energy level of the system
is split into two distinct energy levels as overlap commences, and the splitting increases
as the separation of the atoms decreases. Of course a famous example of this
phenomenon is found in the ground state energy level of the system containing two
hydrogen atoms, as we saw in Section 12-3. Figure 12-4 shows this splitting for the
ground state level only, but each of the higher levels of the system splits in the
same way, and for the same reason, as the atoms are brought together.
If we had started with three isolated atoms, we would have had a threefold exchange
degeneracy of the energy levels. When the atoms are brought together in a
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uniform linear lattice, each of the levels splits into three distinct levels. Figure 13-1 illustrates this schematically for a typical energy level of a system of six atoms. The splitting commences when the center-to-center atomic separation R becomes small enough for the atoms to begin overlapping. As R decreases from this value there is a decrease in the energy of the levels for which the symmetry of the space eigenfunction leads to a favorable electron charge distribution (i.e., which puts electron charge where the ions exert the strongest binding), and an increase in the energy of the levels associated with space eigenfunctions whose symmetry leads to an unfavorable charge distribution. The more favorable, or unfavorable, the charge distribution is, the greater is the decrease, or increase, in the energy. So the levels are spread, by the quantum mechanical requirements of indistinguishability, about an average energy equal to the energy the system would have at a given R if there were no such requirements. Note that this average energy begins to increase rapidly for sufficiently small R. This is due to the Coulomb repulsion that the ions exert on each other.
ELECTRICAL CONDUCTION IN METALS
Some useful results concerning conduction electrons in metals can be obtained from classical ideas. In the absence of an applied electric field, the directions in which these electrons move are random. The reason is that the electrons frequently collide with imperfections in the crystal lattice of the metal, which arise from thermal motion of the ions about their equilibrium positions in the lattice or from the presence of impurity ions in the lattice. In colliding with these imperfections, the electrons suffer changes in speed and direction, and this makes their motion random. As in the case of molecular collisions in a classical gas, we can describe the frequency of electronlattice imperfection collisions by a mean free path 2, where 2 is the average distance that an electron travels between collisions. When an electric field is applied to a metal, the electrons modify their random motion in such a way that, on the average, they drift slowly in the direction opposite to that of the field, because their charge is negative, with a drift speed v d. This drift speed is very much less than the effective instantaneous speed v of the random motion. In copper vd is of the order of 10 -2 cm/sec, whereas v is of the order of 10 8 cm/sec. The drift speed can be calculated in terms of the applied electric field E and of v and 2. When a field is aplied to an electron in the metal, it wil experience a force of magnitude eE which will give it an acceleration of magnitude a given by a = eE/m. Consider now an electron that has just collided with a lattice imperfection. In general, the collision will momentarily destroy the tendency to drift and the electron will move in a truly random direction after the collision. Just before its next collision the electron will have changed its velocity, on the average, by a2/v where 2/v is the mean time between collisions. We call this the drift speed v d , so that
Semiconductors are of much interest because their behavior is the basis for many practical electronic devices, such as transistors. Also, they are excellent illustrations of the ideas discussed in previous sections. Semiconductors are covalent solids that may be regarded as "insulators" because the valence band is completely full and the conduction band is completely empty at the absolute zero of temperature, but they have an energy gap between the valence and conduction bands of no more than about 2 eV. For silicon the energy gap is 1.14 eV and for germanium the gap is 0.67 eV. Although the value of the Fermi distribution function governing the relative population of an energy state in the conduction band to an energy state in the valence band is small, since kT 0.025 eV at room temperature, the number of available states in the conduction band is high. Hence the thermal excitation from the valence band into the conduction band occurs for a significant number of electrons, this number being the product of the number of electrons per quantum state and the number of quantum states per energy interval. Furthermore, the conductivity of a semiconductor increases rapidly with rising temperature, the number of excited electrons in silicon, for example, increasing by a factor of about one billion with a doubling of temperature from 300°K to 600°K. Since the valence band is filled at low temperature, with the four valence electrons of silicon or germanium forming covalent bonds, each electronic excitation into the conduction band leaves a hole in the valence band. These holes, acting as positive charge carriers, also contribute to the conductivity. In Figure 13-14 we illustrate the semiconductor band scheme.
Fermi energyFermi energy level is a term that is used to represent the top of a collection of energy states at absolute zero temparature. It is the thermodynamic work required to add one electron to a body. The electrical conductivity or potential is actually dependent on this Fermi energy. Sometimes it is said that electric currents are driven by the electrostatic potential (Galvanic potential). But this is not the case. There exists electric potential between two semiconductor material when joined together. Yet without any accompanying net current, if any voltmeter is attached to these junctions voltmeter simply reads zero.
This concept is rooted in Fermi-Dirac statistics. Electrons occupy specific energy states in solids. Fermi and Dirac calculated that the probability of elctron occupying some states can be represented with a function which is known as Fermi-Dirac statistics. The equation has this simple looking form :
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