I. Introduction
All the molecules or atoms composing solids are locked up into given positions and are considered either crystalline or amorphous depending on the degree of regularity of their arrangements. The Coulomb forces which hold molecules in their places in the lattice of a solid are the same as the forces which keep electrons in their places around the nucleus in an atom. Lattices form when two or more atoms closely encounter each other redistribute their electrons into new configurations. There are several intrinsically different electron rearrangements which can result, which correspond to several different kinds of bonds. (Ohanian 309)
Ionic bonds result when in an interaction of two (or more) atoms, an electron in a high energy orbital which is not held tightly by one atom is completely removed by another atom which has a vacant spot in a lower energy state. The result is two ions which usually acquire spherical symmetry and are held together by the Coulomb interaction between them whose potential energy is given by U = -Qq/(4PiEr), where Q and q are the charges on the two ions and E is the electric permitivity constant. (Even though we are considering a two atom interaction in this example, the analysis holds in general.) However, as the ions get closer, a repulsive effect arises because of the Exclusions Principle--as the electron wavefunctions of each ion begin to overlap, distortion of the wavefunctions results in which electrons move to vacant higher energy states, so that two electrons do not occupy the same state. The repulsive effect due to the Exclusion Principle which atoms offer to resist interpenetration is called a repulsive core. (Ohanian 311-313)
When an neither both atoms have an affinity for electrons, an even stronger bond, called a covalent bond may result in which the two atoms "share" electrons, usually in pairs of opposite spin (one electron per pair from each atom). If more than one pair of electrons is shared, i.e. an atom forms more than one covalent bond with more than one other atom, the electrons from the different bonds repel each other and thus the bonds exhibit directional preferences--planar triangles, tetrahedrons, etc.
Often, atoms or molecules have to available electrons to give up or share. In such cases attractive forces result among neighboring atoms known as Van de Waals forces or London Dispersion forces. Those bonds, which are much weaker than ionic bonds result from random rearrangements of electrons within an atom which result in temporary, partial, aligned dipoles, which attract each other. Because of the generality of those forces, they are present in all interactions of atoms, however, they are only noteworthy when ionic and covalent bonds are absent. (Ohanian 214)
When a hydrogen loses or shares its only electron, the H+ ion is merely a proton. When places between two negatively charged ions, that proton forms a bond between them, which resembles a covalent bond. Bonds like that are called hydrogen bonds and are responsible for holding the water molecules together in ice crystals and the two strands of DNA into a double helix.
An interesting phenomenon is observed in solid metals: instead of pairs of atoms sharing electrons, each atoms shares one or more valence electrons with all the other atoms in the lattice which results in a gas of free electrons. The Coulomb interactions between the positive charges and the sea of electrons holds all ions in place. Since the binding energy per atom is smaller for metallic bonds than that for covalent or ionic bonds, metallic bonds are considered weaker. The electrons are free to move around in the lattice, however, the potential energy is not uniform at all places. Classically, each electron speeds up near an ion and slows down between ions. However, since this disturbance is averaged over one wavelength of the electron, for low energy electrons it becomes negligible. (Ohanian 315)
II. The classical approach
The free electrons in the metal lattice, called conduction electrons are responsible for the high electric conductivity of metals. Around 1900, Drude treated the electrons are classical particles with large velocities of random directions and thus arrived at an important calculation. Because of the random velocities, in a lattice no net transport of charge occurs. However, if the ends of the lattice are connected to the terminals of a battery, and electric field E is induced in the lattice causing all the electrons to accelerate toward the anode. The additional velocity caused by the electric field, called drift velocity is limited because of decelerations due to collisions between the electrons and the ions in the lattice. Since the drift speed of an electron (around 1 cm/s) is quite small compared to its random speed, the trajectory of an electron is described by numerous zigzags vaguely, but noticeably favoring the direction of -E. The collision rate for an electron is proportional to its speed and is designate by 1/r, where r is the average time between two collisions (also known are relaxation time and on the order of 1E-14 seconds). The drift speed which results in the current through the lattice can be calculated by considering the change in momentum of each electron. In a collision all of the electron's forward momentum is destroyed leaving it only with a random thermal velocity. Therefore, the rate at which an electron loses momentum is
dp/dt = mv/r,
where m is the mass of the electron and v is the drift speed. On the other hand, the rate at which an electron gains momentum due to the electric field is
dp/dt = -eE.
Thus in a steady state conditions
mv/r = -eE, so the drift velocity is
v = -eEr/m in the direction opposite the electric field.
Suppose the number of free electrons per unit volume in a lattice is n, which usually takes values of 10^28 to 10^29 /m^3. For a wire of cross sectional area A and length l, the number of free electrons is nAl, so the total charge is
Dq = -enAl (where Dq represents "delta q").
Within the time interval Dt = l/v all electrons emerge from one end of the wire, so the current in the wire is
I = Dq/Dt = enAl/(l/v) = (nre^2/m)*AE
Letting DV = El be the potential difference across the wire, we get
I = (nre^2/m)(A/l)DV,
which we recognize as Ohm's law--the current is proportional to the potential difference. The resistance of the wire is therefore
R = DV/I = m/(nre^2) * (l/A), and its conductivity c is
c = l/(RA) = nre^2/m. (Ohanian 317-318)
III. The quantum mechanical approach
Even though we arrived at a correct result, there are several problems with the classical approach: 1) Calculations of the relaxation time r both disagreed with experiment and showed no dependence of conductivity on temperature and were thus a failure; 2) The classical theory predicts that the electron cloud has its own heat capacity of about 3 cal/K mole in conflict with experiments showing that the heat capacity of metals is not noticeably different from that of other crystals. Thus it is necessary to treat the electrons using wave mechanics and to consider the Exclusion Principle, which plays a crucial role in the electron gas properties by forcing electrons into states of much higher kinetic energies than those determined by the classical thermal KE of 3/2kT. (Ohanian 319)
The electron cloud is a gas of quantum mechanical particles of half integral spin obeying the Exclusion Principle, and are thus considered a Fermi gas, whose properties are quite different from those of a classical gas. For example, at the absolute zero, all particles of a classical gas have zero kinetic energy, which contradicts the Exclusion Principle by putting them all into the ground state. Quantum mechanically, at T = 0K the electrons will just occupy the lowest energy states which are available, with two electrons with opposite spins in each state. Looking at the free electrons as particles in a cubical box with side L, the available states are described by standing waves with energies given by the Schroedinger's Equation:
E = (nx^2 + ny^2 + nz^2)(HPi)^2/(2mL^2), where H = h/2Pi.
Letting R^2 = x^2 + y^2 + z^2, we can find all the available states up to a given energy E. We have
E = R^2*Pi^2*H^2/(2mL^2), so that
R = ((2mL^2)/(Pi^2*H^2)*E)^1/2
The number of available states of energy < E is just the number of lattice points inside the sphere is radius R in the first octant (in 3d, of course), and as R gets big it approaches the volume of that eighth of a sphere, (1/8)(4Pi/3)R^3. Since there are two electrons per energy state, the number of available states, N, is
N = 2 * (1/8)(4Pi/3)(2mL^2/(Pi^2H^2))^(3/2) * E^(3/2) = 1/3 (2m)^(1/3) * V/(Pi^2H^3) * E^(3/2)
If we set the number of electrons equal to the number of available energy states with energy less than E, then E is the energy of the highest orbital states filled by the electrons at 0K, called Fermi level. We obtain
E (Fermi level) = (3Pi^2)^(2/3) * H/(2m) * (N/V)^(2/3),
which is the Fermi energy of the electron gas and is dependent only of the density (N/V) of the gas.
Calculating the speeds of the free electrons in a given metal we find out that they are quite large (on the order of 10^6 m/s). A classical gas containing such energetic particles would have to have tens of thousands of kelvins of temperature. Differentiating N we can get the number of available states per unit energy to be
dN/dE = 1/2 * (2m)^(3/2) * (V/(Pi^2H^3)) * E^(1/2).
Thus the total energy E (total) of the Fermi gas is the integral from 0 to E (Fermi level) of E dN, which comes out to be
E (total) = 3/5 N E (Fermi level)
and the average energy per electron at zero energy is 3/5 E (Fermi level).
Suppose the gas was heated to some non-zero temperature. The only electrons which make transitions are those which can obtain enough energy from a collision with an ion to jump into a vacant state. Since at temperature T the energy of the ions is around kT, that is the greatest energy an electron can obtain. Because of the Exclusion Principle, only electrons with energies within kT (which is around 0.03 eV at 293K) of the zero temperature Fermi level can make transitions. Thus, only a few electrons make transitions to nearby available states, while the majority of the electron cloud remain unchanged even at temperatures as high as the metal's boiling point. The fact that the electron gas absorbs so little of the thermal energy explains why its specific heat is so small compared to that of the metal. As the specific heat of a classical gas is 3/2 Nk and the fraction of electrons which absorb energy is roughly (kT/E (Fermi level)), the specific heat per mole of Fermi gas must be around
(kT/E (Fermi level)) * (3/2Nk)
Since E (Fermi level) for a metal is usually several eV, kT/E (Fermi level) comes out to be about 1 %.
In an acceleration due to an electric field, the Exclusion Principle plays only a marginal role. Though the electrons are making transitions to neighboring states of higher energy, the electric field acts on all of the electrons at the same rate, including those of the highest energies, thus providing available states which the lower state electrons can occupy. However, in the deceleration process the Exclusion Principle is crucial--the only allowed collisions are those which reverse the direction of motion of electrons with highest speeds. As they start moving against the current they are then once again accelerated by the electric field.
The wave mechanical result for conductivity has the same form as Drude's result. However, the expression for the collision rate 1/r is quite different and is proportional to the temperature, in harmony with experiment. The reason that the conductivity decreases with temperature is that the amplitudes of the oscillations of ions increase and a collision with a by-passing electron becomes more likely, i.e. the effective friction force opposing the motion of the electron increases. (Ohanian 322-324)
IV. The energy levels
Due to the fact that the electrons collide with the ions in the lattice, the energy levels in a solid occur in bands. Whether a solid is a conductor or an insulator depends on whether those bands are completely filled (thus restricting electron motion) or partially filled. Assuming that the crystal lattice is infinite, considering the periodic potentials due to the ions we can derive The wavefunction of a free electron is e^(ikx) or e^(-ikx), depending on the direction of motion and its energy (proportional to the square of the wave number) is thus
E = H^2/(2m) * k^2
We have to modify these expressions due to the interactions of the electron with the ions. For small values of k, the energy must be of the form
E = A + Bk^2
A linear term would introduce dependence of the energy on the sign of k, thus its coefficient must be 0 since the potential is symmetric. Writing energy in a form resembling the energy of a free electron, we get
E = Eo + H^2/(2m)*k^2
where m is a constant called the effective mass. The fact that usually m > m shows that due to interactions with ions the electron in the lattice has higher inertia.
Whenever k becomes comparable to the reciprocal of the distance between the ions in the lattice, our approximation of k starts to fail and completely falls apart when k = Pi/a, where a is the half distance between two ions. At integer multiples of this k, the energy is discontinuous: if all the portions of the wave reflected by the ions are in phase, the amplitude of the wave traveling backward equals to the magnitude of that traveling forward. This happens when the distance between ions equals a multiple of 1/2 wavelength:
2a = nY (where Y is the wavelength), or
k = nPi/a
If, for example, k = Pi/a, the superposition of the forward and the backward traveling waves yields
W(i)= e^(iPix/a) - e^(-iPix/a) = 2isin(Pix/a); W(ii) = e^(iPix/a) + e^(-iPix/a) = 2cos(Pix/a)
which is a standing wave. Since one of those waves places a greater probability near an ion than the other, the two waves have different energies even though their wavelengths are equal. Thus, at k = Pi/a and similarly at k = nPi/a (where n is an integer) the energy experiences discontinuities. The ranges of continuous E(k) are called Brillouin zones and are separated by the forbidden ranges of energies. If the crystal is infinite, then each permitted range of energies within a Brillouin zone, called an energy band has a continuous range of energies. Since crystals have boundaries, each energy band consists of a set of discrete energy levels whose number is equal to the number of atoms in the lattice.
At T = 0K, the lower energy levels are completely filled and the higher energy levels are either partially filled (in conductors) or also completely filled (in insulators). For example, copper's highest level is half filled, thus electrons in that energy band can absorb energy from an electric field and move to other states in the same band. In an insulator, all energy bands are fill, so if an electron is to move, it has to move to a higher energy band. However, to jump over the forbidden band, electrons need typically about 6eV of energy and are therefore unaffected by small electric fields. In semiconductors, all the energy bands are filled, however, since the gap between the highest energy filled band (valence band) and the next energy band (conduction band) is small, they can jump into the empty band and carry current. Since greater temperatures provide the electrons with more excitation energy, they can more easily jump into the conduction band. Even though just like in conductors the effective friction increases with temperature, this effect is overcompensated by the number of conduction electrons participating in the current. When electrons move to the conduction band of a semiconductor, they leave behind vacant energy states, called holes, which allow for the conduction of current in the valence shell. Since the holes are lack of negative charge, if they are regarded as positive charges, the current in the valence shell can be described as a flow of holes. If a semiconductors contains different kinds of atoms in the lattice, they are called impurity conductors. The impurity atoms can be electron donors (like arsenic) or electron acceptors (like gallium), which either increase the number of electrons or increase the number of hole by trapping electrons, and in either case increase the conductivity of the semiconductor.
As even small quantities of impurity atoms greatly increase the conductivity of semiconductors, impurity semiconductors find wide applications in technology. When two semiconductors one of which is contaminated by electron donors and the other by electron acceptors are joined together, flow of charge and electric potential appears near the interface, which is at the basis of many semiconductor devices, such as rectifiers (diodes), transistors, light emitting diodes, and solar cells. (Ohanian 329-330)
V. Superconductivity
A remarkable property of conductors is superconductivity. When metals are cooled to temperatures close to the absolute zero, they reach their transition temperatures, at which their resistivity drops to zero, and they go into a different thermodynamic state of decreased entropy, called the superconducting state. Some alloys of conductors start being superconductors at slightly higher temperatures, the highest known being 23.2 K for triniobium germinate.
A superconductor is a perfect conductor--a current flowing in a closed superconducting loop has been proven experimentally to decay in a time greater than 10^5 years experimentally and over 10^40,000,000 theoretically. The current in a superconducting loop, called persistent current, induces a magnetic field, thus such loops can be used for magnets which do not need to be powered by emf or other energy sources. The interaction of a superconductor with a magnetic field is quite interesting. A conductor in the shape of a long cylinder parallel to the magnetic field expels magnetic field lines going through it before it becomes superconductive, a phenomenon called the Meissner effect. A superconductor pushes the magnetic field lines around its volume when brought into a weak magnetic field. This and the Meissner effect makes a superconductor a perfect diamagnet. Stronger magnetic fields destroy superconductivity, which limits the intensity of the magnetic fields induced by superconducting loops. (Ohanian 334-337)
For more information on superconductors, see Superconductivity by Schuyler Corry.
Bibliography
Ohanian, Hans C., Modern Physics. Prentice-Hall Inc. Englewood Cliffs 1987.
Beiser, Arthur., Concepts of Modern Physics. McGraw-Hill Book Company. New York, 1981.