Chapter 19. Electrical Properties
19.2 Ohms Law
When an electric potential V is applied across a material, a current of magnitude I flows. In most metals, at low values of V, the current is proportional to V, according to Ohm's law:
I = V/R
where R is the electrical resistance. R depends on the intrinsic resistivity r of the material and on the geometry (length l and area A through which the current passes).
R = rl/A
19.3 Electrical Conductivity
The electrical conductivity is the inverse of the resistivity: s = 1/r.
The electric field in the material is E=V/l, Ohm's law can then be expressed in terms of the current density j = I/A as:
j = s E
The conductivity is one of the properties of materials that varies most widely, from 107 (W-m) typical of metals to 10-20 (W-m) for good electrical insulators. Semiconductors have conductivities in the range 10-6 to 104 (W-m).
19.4 Electronic and Ionic Conduction
In metals, the current is carried by electrons, and hence the name electronic conduction. In ionic crystals, the charge carriers are ions, thus the name ionic conduction (see Sect. 19.15).
19.5 Energy Band Structures in Solids
When atoms come together to form a solid, their valence electrons interact due to Coulomb forces, and they also feel the electric field produced by their own nucleus and that of the other atoms. In addition, two specific quantum mechanical effects happen. First, by Heisenberg's uncertainty principle, constraining the electrons to a small volume raises their energy, this is called promotion. The second effect, due to the Pauli exclusion principle, limits the number of electrons that can have the same property (which include the energy). As a result of all these effects, the valence electrons of atoms form wide valence bands when they form a solid. The bands are separated by gaps, where electrons cannot exist. The precise location of the bands and band gaps depends on the type of atom (e.g., Si vs. Al), the distance between atoms in the solid, and the atomic arrangement (e.g., carbon vs. diamond).
In semiconductors and insulators, the valence band is filled, and no more electrons can be added, following Pauli's principle. Electrical conduction requires that electrons be able to gain energy in an electric field; this is not possible in these materials because that would imply that the electrons are promoted into the forbidden band gap.
In metals, the electrons occupy states up to the Fermi level. Conduction occurs by promoting electrons into the conduction band, that starts at the Fermi level, separated by the valence band by an infinitesimal amount.
19.6 Conduction in Terms of Band and Atomic Bonding Models
Conduction in metals is by electrons in the conduction band. Conduction in insulators is by electrons in the conduction band and by holes in the valence band. Holes are vacant states in the valence band that are created when an electron is removed.
In metals there are empty states just above the Fermi levels, where electrons can be promoted. The promotion energy is negligibly small so that at any temperature electrons can be found in the conduction band. The number of electrons participating in electrical conduction is extremely small.
In insulators, there is an energy gap between the valence and conduction bands, so energy is needed to promote an electron to the conduction band. This energy may come from heat, or from energetic radiation, like light of sufficiently small wavelength.
A working definition for the difference between semiconductors and insulators is that in semiconductors, electrons can reach the conduction band at ordinary temperatures, where in insulators they cannot. The probability that an electron reaches the conduction band is about exp(-Eg/2kT) where Eg is the band gap and kT has the usual meaning. If this probability is, say, < 10-24 one would not find a single electron in the conduction band in a solid of 1 cubic centimeter. This requires Eg/2kT > 55. At room temperature, 2kT = 0.05 eV; thus Eg > 2.8 eV can be used as the condition for an insulator.
Besides having relatively small Eg, semiconductors have covalent bond, whereas insulators usually are partially ionic bonded.
19.7 Electron Mobility
Electrons are accelerated in an electric field E, in the opposite direction to the field because of their negative charge. The force acting on the electron is -eE, where e is the electric charge. This force produces a constant acceleration so that, in the absence of obstacles (in vacuum, like inside a TV tube) the electron speeds up continuously in an electric field. In a solid, the situation is different. The electrons scatter by collisions with atoms and vacancies that change drastically their direction of motion. Thus electrons move randomly but with a net drift in the direction opposite to the electric field. The drift velocity is constant, equal to the electric field times a constant called the mobility m,
vd= me E
which means that there is a friction force proportional to velocity. This friction translates into energy that goes into the lattice as heat. This is the way that electric heaters work.
The electrical conductivity is:
s = n |e| me
where n is the concentration of electrons (n is used to indicate that the carriers of electricity are negative particles).
19.8 Electrical Resistivity of Metals
The resistivity then depends on collisions. Quantum mechanics tells us that electrons behave like waves. One of the effects of this is that electrons do not scatter from a perfect lattice. They scatter by defects, which can be:
One can express the total resistivity rtot by the Matthiessen rule, as a sum of resistivities due to thermal vibrations, impurities and dislocations. Fig. 19.8 illustrates how the resistivity increases with temperature, with deformation, and with alloying.
19.9 Electrical Characteristics of Commercial Alloys
The best material for electrical conduction (lower resistivity) is silver. Since it is very expensive, copper is preferred, at an only modest increase in r. To achieve low r it is necessary to remove gases occluded in the metal during fabrication. Copper is soft so, for applications where mechanical strength is important, the alloy CuBe is used, which has a nearly as good r. When weight is important one uses Al, which is half as good as Cu. Al is also more resistant to corrosion.
When high resistivity materials are needed, like in electrical heaters, especially those that operate at high temperature, nichrome (NiCr) or graphite are used.
19.10 Intrinsic Semiconduction
Semiconductors can be intrinsic or extrinsic. Intrinsic means that electrical conductivity does not depend on impurities, thus intrinsic means pure. In extrinsic semiconductors the conductivity depends on the concentration of impurities.
Conduction is by electrons and holes. In an electric field, electrons and holes move in opposite direction because they have opposite charges. The conductivity of an intrinsic semiconductor is:
s = n |e| me + p |e| mh
where p is the hole concentration and mh the hole mobility. One finds that electrons move much faster than holes:
me > mh
In an intrinsic semiconductor, a hole is produced by the promotion of each electron to the conduction band. Thus:
n = p
Thus, s = 2 n |e| (me + mh) (only for intrinsic semiconductors).
19.11 Extrinsic Semiconduction
Unlike intrinsic semiconductors, an extrinsic semiconductor may have different concentrations of holes and electrons. It is called p-type if p>n and n-type if n>p. They are made by doping, the addition of a very small concentration of impurity atoms. Two common methods of doping are diffusion and ion implantation.
Excess electron carriers are produced by substitutional impurities that have more valence electron per atom than the semiconductor matrix. For instance phosphorous, with 5 valence electrons, is an electron donor in Si since only 4 electrons are used to bond to the Si lattice when it substitutes for a Si atom. Thus, elements in columns V and VI of the periodic table are donors for semiconductors in the IV column, Si and Ge. The energy level of the donor state is close to the conduction band, so that the electron is promoted (ionized) easily at room temperature, leaving a hole (the ionized donor) behind. Since this hole is unlike a hole in the matrix, it does not move easily by capturing electrons from adjacent atoms. This means that the conduction occurs mainly by the donated electrons (thus n-type).
Excess holes are produced by substitutional impurities that have fewer valence electrons per atom than the matrix. This is the case of elements of group II and III in column IV semiconductors, like B in Si. The bond with the neighbors is incomplete and so they can capture or accept electrons from adjacent silicon atoms. They are called acceptors. The energy level of the acceptor is close to the valence band, so that an electron may easily hop from the valence band to complete the bond leaving a hole behind. This means that conduction occurs mainly by the holes (thus p-type).
19.12 The Temperature Variation of Conductivity and Carrier Concentration
Temperature causes electrons to be promoted to the conduction band and from donor levels, or holes to acceptor levels. The dependence of conductivity on temperature is like other thermally activated processes:
s = A exp(Eg/2kT)
where A is a constant (the mobility varies much more slowly with temperature). Plotting ln s vs. 1/T produces a straight line of slope Eg/2k from which the band gap energy can be determined. Extrinsic semiconductors have, in addition to this dependence, one due to the thermal promotion of electrons from donor levels or holes from acceptor levels. The dependence on temperature is also exponential but it eventually saturates at high temperatures where all the donors are emptied or all the acceptors are filled.
This means that at low temperatures, extrinsic semiconductors have larger conductivity than intrinsic semiconductors. At high temperatures, both the impurity levels and valence electrons are ionized, but since the impurities are very low in number and they are exhausted, eventually the behavior is dominated by the intrinsic type of conductivity.
19.13 The Hall Effect (not covered)
19.14 Semiconductor Devices
A semiconductor diode is made by the intimate junction of a p-type and an n-type semiconductor (an n-p junction). Unlike a metal, the intensity of the electrical current that passes through the material depends on the polarity of the applied voltage. If the positive side of a battery is connected to the p-side, a situation called forward bias, a large amount of current can flow since holes and electrons are pushed into the junction region, where they recombine (annihilate). If the polarity of the voltage is flipped, the diode operates under reverse bias. Holes and electrons are removed from the region of the junction, which therefore becomes depleted of carriers and behaves like an insulator. For this reason, the current is very small under reverse bias. The asymmetric current-voltage characteristics of diodes (Fig. 19.20) is used to convert alternating current into direct current. This is called rectification.
A p-n-p junction transistor contains two diodes back-to-back. The central region is very thin and is called the base. A small voltage applied to the base has a large effect on the current passing through the transistor, and this can be used to amplify electrical signals (Fig. 19.22). Another common device is the MOSFET transistor where a gate serves the function of the base in a junction transistor. Control of the current through the transistor is by means of the electric field induced by the gate, which is isolated electrically by an oxide layer.
19.15 Conduction in Ionic Materials
In ionic materials, the band gap is too large for thermal electron promotion. Cation vacancies allow ionic motion in the direction of an applied electric field, this is referred to as ionic conduction. High temperatures produce more vacancies and higher ionic conductivity.
At low temperatures, electrical conduction in insulators is usually along the surface, due to the deposition of moisture that contains impurity ions.
19.16 Electrical Properties of Polymers
Polymers are usually good insulators but can be made to conduct by doping. Teflon is an exceptionally good insulator.
A dielectric is an electrical insulator that can be made to exhibit an electric dipole structure (displace the negative and positive charge so that their center of gravity is different).
When two parallel plates of area A, separated by a small distance l, are charged by +Q, Q, an electric field develops between the plates
E = D/ee0
where D = Q/A. e0 is called the vacuum permittivity and e the relative permittivity, or dielectric constant (e = 1 for vacuum). In terms of the voltage between the plates, V = E l,
V = Dl/ee0 = Q l/Aee0 = Q / C
The constant C= Aee0/l is called the capacitance of the plates.
19.18 Field Vectors and Polarization
The dipole moment of a pair of positive and negative charges (+q and q) separated at a distance d is p = qd. If an electric field is applied, the dipole tends to align so that the positive charge points in the field direction. Dipoles between the plates of a capacitor will produce an electric field that opposes the applied field. For a given applied voltage V, there will be an increase in the charge in the plates by an amount Q' so that the total charge becomes Q = Q' + Q0, where Q0 is the charge of a vacuum capacitor with the same V. With Q' = PA, the charge density becomes D = D0 E + P, where the polarization P = e0 (e1) E .
19.19 Types of Polarization
Three types of polarization can be caused by an electric field:
19.20 Frequency Dependence of the Dielectric Constant
Electrons have much smaller mass than ions, so they respond more rapidly to a changing electric field. For electric field that oscillates at very high frequencies (such as light) only electronic polarization can occur. At smaller frequencies, the relative displacement of positive and negative ions can occur. Orientation of permanent dipoles, which require the rotation of a molecule can occur only if the oscillation is relatively slow (MHz range or slower). The time needed by the specific polarization to occur is called the relaxation time.
19.21 Dielectric Strength
Very high electric fields (>108 V/m) can free electrons from atoms, and accelerate them to such high energies that they can, in turn, free other electrons, in an avalanche process (or electrical discharge). This is called dielectric breakdown, and the field necessary to start the is called the dielectric strength or breakdown strength.
19.22 Dielectric Materials
Capacitors require dielectrics of high e that can function at high frequencies (small relaxation times). Many of the ceramics have these properties, like mica, glass, and porcelain). Polymers usually have lower e.
Ferroelectric materials are ceramics that exhibit permanent polarization in the absence of an electric field. This is due to the asymmetric location of positive and negative charges within the unit cell. Two possible arrangements of this asymmetry results in two distinct polarizations, which can be used to code "0" and "1" in ferroelectric memories. A typical ferroelectric is barium titanate, BaTiO3, where the Ti4+ is in the center of the unit cell and four O2- in the central plane can be displaced to one side or the other of this central ion (Fig. 19.33).
In a piezolectric material, like quartz, an applied mechanical stress causes electric polarization by the relative displacement of anions and cations.