P
W
4
h California Association of Chemistry Teachers
Norman J. Juster' University o f California LOS Angeles
Conductiom and Semiconduction
Empirical observations of the electrical properties of matter antedate the time of Volta, and the semiconducting behavior of organic materials has been studied since about 1906 by Pocchettino and others (3, 4, 5) who discovered and worked with the photoconductivity of anthracene. Despite this, the theoretical ground-work of solid-state chemistry mas laid only a bit over 30 yeals aao and - by . Wagner Schottky (11). Semiconductors are usually crystalline solids (althouch liauid and amor~hous-solidsemiconductors are known) having, in general, an eiectrical resistivity which is intermediate between that of typical metals and that of typical insulators, and which decreases, usually exponentially, with temperature. Their low conductivity is highly sensitive to light and temperature changes (and to the presence of crystal defects and impurities). Because of these properties, the study of semiconductors has yielded new theories in solid state physics and chemistry and has provided a device to test established ones. I n industry, semiconducting devices are beginning to dominate the entire field of electronics, a phenomenon that is remarkable in view of the fact that the invention of the transistor occurred within the last 15years (1,6). The behavior of electrons in solids is not readily visualized by one accustomed to dealing with isolated atoms and molecules. Although other theoretical representations have been fruitfully applied (9, lo), the band-model theory for metals, devised from quantummechanical considerations, is the most widely referredto explanation for (their) electrical conductivity, and this was extended to cover both insulators and semiconductors. The starting point of the band theory is a collection of nuclei arrayed in space a t their final internuclear separations, a (see Fig. 1). The total number of available electrons is poured into the resultant field of force, a regularly periodic field. Consider
in Figure 1 the simplified model of a one-dimensional structure, the nuclei being those of sodium, for example, with a charge of +11. Due to the large electrostatic attraction, the position of each nucleus represents a deep potential energy well for the electrons. The electrons would all be a t fixed levels if these wells were infinitely deep, giving rise to ls2, 2s2, Zp', 3s' configurations typical of isolated sodium atoms in the ground state, as represented in Figure la. But the wells are
- .
Presented in part at Fourth Summer CACT Conference, Adomar California, August, 1962. The 6mt paper on this topic (Electronic Interpretations of Physical Properties) appeared in rnrs J ~ ~ N A30, L 596 , (1962); the third paper will appear in the October issue. Visiting Proiessor, 1963, on leave fromPasadenaCity College.
Figure 1. Energy levels in sodium; left, (a) isololed atoms; right, section of d i d matrix sodium.
(bl
not infinitely deep and the potential energy barriers separating the electrons on different nuclei are not infinitely high; this is shown in Figure l b . Hence there can be a quantum mechanical leakage of electrons through the barriers, or, in other words, a delocalization (resonance) of electron waves to cover a large number of identical positions. Thus, we must consider not the energy levels of single sodium atoms but those of the crystal as a whole. Since the Pauli principle states that at most two electron waves (of opposite sign spin quantum number) can occupy exactly the same energy level (same n, 1, m quantum numbers) and since the electron waves can now be considered as spread through the entire structure, it is clear that each energy level in an individual sodium atom is broadened in crystalline sodium into a band of closely packed energy levels (see Fig. lb). Each atomic orbital contributes one Volume 40, Number 9, September 1963
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level to a band, the levels being spaced at about 10WZ2 ev (so the hand is of the order of 1 ev wide). Thus, for N atoms, each having j electronic energy levels, there result N j levels for the crystal matrix as a whole (7). I n the lower bands (Is, 29, 2p, here) there are therefore just enough levels to accommodate the number of available electrons (i.e., the bands are completely filled), and these electrons cannot move under the influence of an external electric field. Such acceleration by the field would necessitate their moving into somewhat higher energy levels within the bands, and any levels ahove them are all filled. Of course, an electron may gain such a large amount of energy as to be shifted from its own hand to a higher unoccupied band, hut this is a rare event. I n the uppermost hand (3s here), which is only partially (half) filled, an electron toward the top of the band may be accelerated and move up into unfilled levels within the hand. I n Figure l b it can be seen that the uppermost band has actually broadened enough to overtop the potential energy barriers, so that these electrons can move freely through the crystal structure (or be delocalized); when an electric field is applied, current thus flows through a t this conduction band level. Ideally, if the nuclei were always arranged at all points of a periodic lattice, there should be no resistance to current flow; resistance arises from deviations from perfect periodicity. Since such deviations can he caused by thermal vibrations of lattice nuclei, sterically destroying perfect resonance between the electronic energy levels, it is easy to understand why resistance increases with temperature. Another such cause of deviations may be the introduction of foreign atoms or of imperfections in the crystal structure.
Figure 2.
Bond models of solids
I n the case of a metal such as magnesium, with two 3s electrons (zero-point ground state) and apparently completely filled 3s bands, conduction occurs because the 3p band is low enough to overlap the top of the 3s hand, providing a large number of available empty levels. Thus, conductors are characterized either by partially-filled hands or by overlapping of the topmost bands. Insulators have completely-filled lower bands with a wide energy gap between the topmost filled band and the lowest empty band (see Fig. 2). Energy bands in solids can be studied experimentally by the methods of X-ray emission spectroscopy. For example, if an electron is driven out of the 1s level in sodium metal (Figure lb), the K , X-ray emission occurs when an electron from the 3s band falls into the hole in the 1s level. Since the 3s electron can come from anywhere within the band of energy levels, the 490
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X-rays emitted will have a spread of energies (and hence frequencies) exactly corresponding with the spread of allowed energies in the 3s hand. The following widths (in ev) are found for the conduction bands in a few of the metals investigated: Li, 4.1; Na, 3.4; Be, 14.8; Mg, 7.6; and Al, 13.2. Note that within a given group of the periodic table, the smaller the nuclear charge the larger the conduction-hand width (as would be expected). Band models for semiconductors are also included in Figure 2. These models usually possess, in addition to normal bands, narrow impurity bands--either unfilled levels closely above a filled band or filled levels closely below an empty band. The extra levels are the result of either foreign atoms dissolved in the structure or a departure from the ideal stoichiometric composition (g). Thus, zinc oxide normally contains an excess of zinc, whereas copper(1) oxide normally contains an excess of oxygen. Both these compounds behave as typical semiconductors. Their conductivities increase approximately exponentially with temperature, because the number of conduction electrons depends on excitation of electrons into or out of the impurity levels, and excitation is governed by an eCAEglRT Boltzmann factor (3, 4), where AE, is the energy gap between the filled and unfilled bands. This is discussed below. If this energy gap between the filled valence band and the empty conduction band is sufficiently small so that electrons are excited to the conduction band by thermal energy, a crystal may be a semiconductor even in the absence of effects due to impurities. Germanium, with AE, = 0.72 ev, and grey tin, with 0.10 ev, are examples of such intrinsic semiconductors. I t should be noted that the removal of an electron from a filled band creates a positive "hole" (an empty energy-level) which allows conduction to occur in the valence band. Thus electrons (in the conduction band) and "holes" are the basic electrical carriers in semiconductors. The most striking difference between metals and semiconductors is that, in the former, the number of carriers is large and constant, whereas in the latter the number is smaller and variable and may be controlled. This control may be effected by control of impurity content, but the carrier density may also be varied for a material of fixed impurity content. One of the noteworthy characteristics of semiconductors is that they show marked photo-conductive effects (5, 8). This is readily understood in terms of the hand model. When the semiconductor is struck by light of frequency high enough so that a quantum absorbed by a valence electron has sufficientenergy to raise it from the top of the full hand to the conduction band, extra carriers are thereby created in the semiconductor, and these lead to increased conductivity. Photo-conduction due to excitation of electrons from impurity levels has also been observed; but since these electrons are much less numerous than the valence electrons and the levels are filled only a t low temperatures, this effect is much less marked. It also appears that electrons and holes may be injected into semiconductors from metallic contacts and by other means, and also that the carriers may be extracted under suitable conditions by an electric field. This ability to control the carrier density in semiconductors is the main reason for their great technological import.ance.
Generally, one type of impurity level, either donor or acceptor, predominates to a large extent in semiconductors used for practical purposes. When donors predominate and greatly exceed the numher of intrinsic carriers, then the electron concentration, n, greatly exceeds the hole concentration, p, and the material is called '%typev; the electrons are referred to as the majority carriers and the holes as the minority carriers. The situation is reversed for "p-type" material, in which acceptors predominate. Control of majority carriers is usually the mechanism employed in semiconductor applications, hut control of minority carriers is naturally less difficult (e.g., by light injection) and has many important examples in the technology. The electrical and optical properties of semiconductors can be explained simply on the basis of the energy band spacing of Figures 1 and 2. Figure 2 also shows the upper filled band, i.e., the valence band which contains the outer valence electrons, and the conduction band, separated by a region of width AEg normally forbidden to electrons by the principles of quantum mechanics. AE, is usually expressed in electron-volts. In an intrinsic semiconductor there are just sufficient electrons to fill all the bauds up to the valence band and a t absolute zero the conduction band would be completely empty. At a higher temperature, T, collision processes give rise to a distribution of thermal energies of the electrons around kT such that an electron in the valence band has a finite chance of being excited into the conduction band even though kT is less than A At temperature T ° K , the equilibrium concentration, n, of electrons excited thermally into the conduction band is given by
where Q is a factor characteristic of the material. Other things being equal, n is thus primarily dependent on hE,. For example, in germanium a t room temperature n = 2.4 X loL3per cc and in InSb a t room temperature n = 2 X 10" per cc; n is also equal to the concentration of positive holes in the valence hand of intrinsic semiconductors, p. The electrical conductivity of a pure semiconductor depends on the value of n and is given by
where e is the electronic charge and p, and p, are the mobilities of electrons and of holes respectively, i.e., their velocities under unit applied electric field. In some semiconductors, but not all, the carrier mobilities are proportional to T-"/'. Thus, from the expressions for n and o given above, the conductivity varies exponentically with
and a plot of log a versus T-' should he a straight line. A typical result obtained from experimental data shows that the exponential law is followed, but only a t elevated temperatures. The value of AE, can be calculated from the slope of the straight line region, but this will be accurate only if mobilities follow the T-"/' law; otherwise the exact nature of the temperature-dependency must be known. AE,itself, naturally, varies slightly with temperature. The deviation of the
log o curve from the straight line a t lower temperatures is often due to the presence of impurities or to defects in structure or composition of the crystal. These can give rise to additional energy levels in the normally forbidden region as shown in Figure 2c. The upper represents the possible situation when impurity atoms have surplus electrons which can he detached easily and excited into the conduction band. These are ntype or donor-impurities. For example, phosphorus in germanium has an average of one excess valence electron per impurity atom over the numher of valence electrons of the germanium substrate, and tellurium in InSb has an average excess of two such electrons. The conductivity a t low temperatures is almost entirely due to these donor electrons, and the material is said to he n-type. I n the lower case of Figure 2c the impurity atoms have a deficiency of electrons and can easily capture electrons from the valence band, e.g., aluminum in germanium or zinc in InSb. Here the impurities are p-type or "acceptors" and the material is p-type. When one type of carrier predominates, the conductivity is given by o = nep. The particular temperature a t which the log o curve deviates from a straight line depends on the impurity concentration and the positions of the impurity levels in the forbidden region. The temperature region where impurity carriers predominate is called "extrinsic" and the remainder "intrinsic." At sufficiently low temperatures the numher of carriers, impurity and intrinsic, n, can be negligibly smail (unless the impurity concentration is very high), so that the semiconductor becomes an insulator. This is the reverse of metals where, a t low temperatures, thermal vibrations of the periodic lattice are smoothed ont and the resistance decreases to negligible values. The Hall effect is a most useful effect for determining the numher and type of carriers in a semiconductor, and it finds its greatest application in semiconductor measurements. It occurs when a current flows in a slice of conducting material and a magnetic field is applied at right angles to the direction of current flow. The carriers are deflected to one side or other, depending on the relative directions of magnetic field and current and on the sign of the carriers. A transverse electric field is built up across the specimen, since more charge is carried along one side than the other, until a balance occurs; and this field can be measured with a suitable voltmeter, usually a potentiometer. The field is given by E = RJH, where J is the current density and H the magnetic field. R is a coefficient of proportionality, the Hall coefficient, which depends on the material. I n terms of measured quantities V, H, and I , where V is the Hall voltage in volts, I the total current in amps, and w and t the width and thickaess of the specimen in cm, the above equation becomes
The direction of V shows whether the current in the specimen is carried by electrons or positive holes. The magnitude of V and thus R depends on the carrier concentration, and it can be shown that in a specimen where one type of carrier predominates, e.g., in an impurity region Volume 40, Number 9, September 7963
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(In some circumstances the factor 3 r / 8 is replaced by 1.) Thus, R can give a measure of the concentration of "effective impurities," i.e., of those contrihuting carriers, and the nature of the contrihuting impurity can be inferred from the sign of V. When carriers of both types contribute to conduction, R is given by
Electrons can he excited into the conduction bands by absorbing energy from incident radiation, hut only if the quantum energy of the radiat,ion is greater than AEg. If this energy is insufficient, the electrons cannot jump the gap and the radiation cannot be absorbed by the process. A typical curve of absorption against wavelength for a semiconductor is shown in Figure 3.
\
Absorption Edge
harrier and only the highest-energy ones can cross the barrier. Thus, the metal contact at this junction is being depleted of its most energetic electrons and at the other junction; where no such harrier exists, electrons of excessive thermal energy are added to the metal. For p-type material, holes are the carriers and must enter against a similar harrier; thus, the effect is the same hut the sign of the effect is reversed. The same type of reasoning applies to the Seebeck effect. The mobile charge carriers tend to diffuse from the hot to the cold junction so that the latter acquires a potential of the same sign as the carrier, and a voltage difference results. The Seebeck effect has been used to generate power in underdeveloped areas of the world; the Peltier effect has been used to all-weather air condition homes (pump heat outside in summer, the reverse direction in the winter) in more developed nations. The measurements of the Hall effect and of th~rmoelectricpower have been widely used in semiconductor studies. The interpretation of the Hall effect is more straightforward, and it gives more precise results when it can be measured. On low mobility materials, however, it is sometimes difficult to measure, and thermoelectric power measurements can often be made in such cases. The qualitative measurement of thermoelectric power is so simple that it is also very useful for determining the conductivity type of a semiconductor from the sign of the effect. The p a Junction
Figure 3.
Spectral absorption in a remicondudor.
The wavelength a t which the absorption falls sharply, the "absorption edge," occurs when the quantum energy hv or hc/X becomes equal to AE,. Thus, at this wavelength AEq = hc/A = 1.24/A ev
This is a recognized method of determining AEg. Absorption beyond the edge is due to residual free carriers in the semiconductor and can give an indication of purity. Another important method for. studying semiconductors is the measurement of the Seebeck effect, or thermoelectric power, as it is often called. A temperature differenceacross a semiconductive wafer, AT, gives rise to an emf of QAT millivolts. This defines Q, the thermoelectric power (in millivolts/degree). The reverse of the Seebeck Effect is the Peltier effect. When a current is passed through a semiconductor, heat is absorbed a t one semiconductor-metal junction and it is liberated at the other. The Peltier coe5cient, T , is expressed as joules of heat ahsorhed or liherated reversibly per coulomb of charge passing through the junction (i.e., units of volts). Thus, QT = r, where Q is volts per degree. When a field is applied to the junction for n-type senliconductors, the electrons trying to flow into the semiconductor face an energy 492
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Journal o f Chemical Education
If, within the same single crystal, there are adjacent regions of n- and p-type semiconductors, the resulting boundary is called a "p-n junction." A n n-type material contains mobile negative charges (conduction electrons) and an equal concentration of lixed positive charges (the ionized donors); p-type material contains mobile positive charges (holes) and fixed negative charges (ionized acceptors). With the two regions in contact, the mobile electrons and holes might be expected to flow out of the n- and ptype regions, respectively, across the junction, because of the concentration gradients for these species. On the other hand, this flow leaves the lzrtype region with a net positive charge and the p-type region with a net negative charge, thus establishing a field in a direction which opposes further flow. At equilibrium, this field just balances the effect of the concentration gradient. The net charges appear in the regions immediately adjacent to the p-n junction and the field appears in the space-charge region. Such a junction will usually act as a rectifier. In practice p-n junctions are built-in during crystal growth or are formed by diffusion of suitable impurities into a crystal or by alloying. The latter processes are called "doping." Properties of the p-n Junction
In a semiconductor specimen a t thermal equilibrium, the product of the conduction-electron concentration and the hole concentration in the vicinity of any point equals a constant. For germanium the value of this constant is about 5.8 X 1020/cme,and for silicon, 2.3 X 1020/cm8. The constancy of this "pn product" is a very useful principle to invoke when describing junctions a t equilibrium because it applies in all
regions-within and near junctions as well as in uniformly-doped regions. We sometimes write where ni represents the equal hole and electron concentrations found in a region which is intrinsic (i.e., which has zero net doping). In Fignre 4a, x is an axis passing normally through a n abrupt (or step) junction. A p-region is on the left of x = 0, and an n region is on the right. Even though the net doping changes abruptly at x = 0, the carrier concentrations make their transition through a region of appreciable thickness. This is because a carrier concentration gradient is inevitably accompanied by a diffusion current, and therefore the gradient must relax to a low enough value so that an equilibrium will exist between diffusion and the counter-mechanism-driftas described earlier. The logarithms of the carrier concentrations p and n, are plotted versus x in Figure 4a.
approaches the junction; hence, there is a region enclosing the' junctions which is practically devoid of carriers. It is sometimes termed a "depletion layer'' because of this scarcity of carriers. Further, it can be accurately treated as a region with sharp boundaries, as represented by Figure 4c. Here, circled charges represent impurity ions locked in the lattice and uncircled charges represent mobile carriers. The absence of carriers in the depletion layer "exposes" the ions, which then constitute a space charge, as shown in Figure 4d, with the same quantity of charge on each side. Accompanying the space charge is a field directed from right to left and it is this field which gives rise to the carrier drift mentioned above. Holes diffuse to the right and drift to the left in precisely compensating fashion, while electrons do just the opposite. Thus a t equilibrium there are four distinct components of current crossing the junction but zero net current. The "built-in field" of a junction is responsible for the built in potential difference, which is simply related to carrier ratios across the junction as follows:
where 6+ is the height of the potential barrier in energy units and the other symbols have their customary meanings. Hence, when a voltage is applied to a junction, the pn product within the depletion layer is altered to
where 86 is the applied voltage. For modest forward biases (64 positive) the increase in the product a t the depletion layer boundaries is borne solely by an increase in minority carrier concentration, neglecting large biases. Thus
a t the right-hand boundary, and
Figure 4. ( 0 ) Logarithmic plot of carrier ancentrations through jundion; (bl linwr plot of carrier ancentrotions through junction; ( 4 ionic (circled symbols) space charge in junction; ond (dl space charge distribution in step
Their sum must be the same everywhere, by the pn product rule. Note that as a consequence, the height of the step in the logarithm of electron concentration is precisely equal to that for holes. Hence, carrier ratios must be equal across the junction. That is, P, ---n. P. nd
where the subscript indicates the region intended. Although useful, this plot conveys a distorted impression of carrier distributions within the region of changing concentration. Figure 4b shows carrier concentrations on a linear scale versus x. A break is indicated in the vertical direction because the majority carrier concentrations in this example are taken to differ by several orders of magnitude. Note that the majority carrier concentrations of both types drop sharply as one
where the diffusion length, Lp, is the characteristic constant of minority carrier decay in the n region. At the x-origin boundary, the current density, J, is given by the concentration gradient times the diffusion constant (for holes in this case) times the charge on each particle
where (Ap/Lp) can he found from the slope of the previous curve a t x = 0. Now
This last equation is known as the rectifier equation. A comparable equation can be written for the electron diffusion current into the p region, but in the example chosen this current is negligible because n, is several orders of magnitude less than p,. Thus, we have here a heavily doped p-region which is an effective emitter of holes into the more lightly doped n-region. The Volume 40, Number 9, September 1963
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condition just described is that of forward bias or easy conduction in a rectifier. An efficientemitter junction is necessary to a practical rectifier. When the injected hole density becomes comparable to the majority electron density, then additional electrons are pulled into the region adjacent to the junction to maintain charge neutrality. This increase in both carrier population, conductivity modulation, diminishes the voltage drop in the forward-biased rectifier below the level which would otherwise be expected-clearly a desirable effect. For reverse bias (64 negative) the exponential term in the parentheses of the rectifier equation given above quickly becomes negligible compared to unity (e.g., a t 64 = 0.1 volt, it equals 0.02). Thus, under these conditions the rectifier equation predicts that the current should approach a low saturation value,
The model here is one of depressed pn product a t the boundary (and through the depletion layer) and hence a depressed minority carrier concentration a t the boundary. Therefore minority carriers (holes in this example) diffuse to the boundary to be swept through by the field. Reverse voltages beyond about 0.1 volt serve to heighten the field and thicken the depletion layer but have a negligible effect on the boundary condition; at 0.1 volt p is already a mere 2% of p,, and hence the profile which gives rise to diffusion changes but little when reverse voltage is increased further. Electron current on the other side of the junction is again neglected for the same reasons as before.
signal is often "smeared out" in transit by diffusion. An important consequence of effects such as theseemitter inefficiency, recombination in the base region and signal attenuation-is that current through the collector junction and out the collector lead is less than current entering the emitter. If we apply a lowfrequency, small-amplitude signal a t the emitter, these factors will lead to a smaller value of signal current at the collector than at the emitter and the current gain,
where i is an ac signal and the bracket indicates that collector voltage is held constant (at ac ground) for purposes of this definition. What causes ac transverse majority carrier current? Anything which causes the curreut gain to be less than unity, as discussed above, will do it. Figure 6 shows an equivalent circuit useful for describing a transistor's ac properties. The ac resistance of the emitter, r., follows from the rectifier equation, above. If the dc emitter bias is 1 ma, then T. = 26 ohms. Thus, with freedom to increase emitter bias without limit, we can make r. as small as we like. The equivalent circuit also includes ohmic base resistance, r,', capacitance of the reverse-biased collector junction, C,, and a current generator ai., where a is a factor differing from w. In operation, the transistor develops a
Transistors
A simple transistor consists of three elements of semiconducting material: an emitter, a base and a collector, as shown schematically for the pnptype in Figure 5 in the common-base connection, with biases of the correct polarity applied. Since the emitter is forward-biased, minority carriers (holes once again) flood into the base. But the reverse-biased collector
Figvre 6.
Transistor equivolentsirsuil,
voltage across a load impedance. A portion of this voltage is fed back to the input by the voltage divider formed by r,' and C,. The fraction of output voltage which will be fed back is 16'
~b'
Figvre 5.
Boric honrirtor structure.
junction acts as a perfect sink for these holes, like the reverse-biased rectifier junctions just discussed. Further, since the diffusion length for holes in the base is usually substantially larger than the base thickness, the hole distribution is nearly a straight line going from a high value near the emitter to nearly zero a t the collector depletion layer edge. By diffusing "down" this gradient the holes move through the base region. If the emitter were 100% efficient, all of the current crossing it would take the form of hole current. I n practice, efficiencies between 99 and 100yo are common. Also some of the holes in transit recombine and, too, the 494
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+ (l/jwCr)
To minimize this feedback 7', and C, must be made small. Such feedback spoils the isolation between input and output. The importance of rb' and C, in determining amplifier gaiu and output can be expressed as follows. Recall that the transistor, operated as described so far, has a current gain less than unity. All of its gain is impedance gain-a current flowing in a low impedance circuit controls a current in a high impedance circuit. But r,' represents an irreducible minimum on input impedance, and C, imposes a limit on output impedance, a t least a t higher frequencies. Thus, these two parameters are directly related to the ratio of output to input impedance, or impedance gain. The signal currents at the three terminals are shown in Figure 7. Note that the current gain with the commonemitter connection (Fig. 7b) is The impedance gain is less in Figure 7b than in Figure 7a-though the power gaiu is about the same in
either; and this is better for some situations, such as working into a low-impedance load. I n the commonemitter connection, once again r,' represents the minimum obtainable input impedance. Using three of the above parameters we can construct a "figure of merit" which gives approximately the highest frequency of oscillation: 25
n' C.'
where fa is the alpha cut-off frequency, for which a has declined 3 db below the low-frequency value, aa [i.e., f, = frequency for which (a/ao) is 0.7071. Let us plot
Figure 7.
Transislor sign01 currents for (m) common bore connection and
Ibl common miner connection.
this number on the frequency axis of a diagram of frequency versus power, and plot a point on the other axis corresponding to the maximum allowable power dissipation for the same transistor. It is then possible to draw a smooth curve of some sort between these points as shown in Figure 8, which can be interpreted as follows: p represents the useful ac output power which can be taken from the transistor a t the moderate frequency, f, for oscillator operation. The expression for f, given above can be reinterpreted as a gain-band figure of merit in case one wants to know what the unit will do as an amplifier rather than as an oscillator. The expression is
These have the following disadvantages-low power, temperature limitations, upper frequency limitations, instability, noise, and poor resistance to radiation damage. Certain compounds of two or more elements, like GaAs, InP, and Sic have been developed for use as semiconductors with improved properties in one or more of the above areas. The most important of these new compounds for semiconductor purposes are those of the 111-V and IV-IV groups. This suggests that certain organic compounds might do as well. Keeded are those with a large band gap-preferably 1 ev-2 ev (for high-temperature use)-and high carrier (electron and hole) mobilities (for high frequency response), preferably 1000 cm2/volt sec, but at least 100 cm2/volt sec. The quantum theory of semiconductors and conductors has some serious major flaws, although it has been used with great success in solid-state physics and chemistry. The results it gives, while qualitatively true, are often not quantitatively correct. The band theory is based on the periodic nature of lattices, which causes the wave functions to be periodic and hence determines the laws of motion of electrons and holes. The same band structures are found in amorphous and liquid semiconductors, which lack long-range order; the main properties of these substances are determined by nearest-neighbor interactions. The spacings and arrays formed by the atoms are more responsible for the properties than is the periodic character of the lattice. The free-motion of an electron in a periodic field of any spacing, no matter what the height of the barriers between the atoms, is in practice limited by the necessity for the wave functions to overlap and by the time required for the transitions. Band theory makes various approximations, such as neglecting direct interactions of an injected electron or hole with others. Thus,
HIGHEST FREQUENCY O F O S C .,=,.f
f
L 1
Figure 8.
where r,, is the low-frequency power gain in the common emitter connection and B is the frequency a t which the power gain is down 3 db. In addition to the junction rectifier and transistor, other solid state electronic devices employing these principles are in common use. Some examples include the junction capacitor, the current limiter, the integrated circuit variable resistance, the gyrator, the inductor and the zener diode. Present electronic semiconductor devices, such as transistors, diodes, and switches nearly always utilize single-crystal wafers of ultra-pure germanium or silicon.
J-
.
-
FREQUENCY
Orcillotor power versus frequency for
0
particular transistor.
the predicted variation of the mobility, p, with T is found to agree roughly with theory only for bodies with purely homopolar bonds. Theoretical predictions of how thermoelectric power varies with T agree with experiment in only a few cases for limited temperature ranges; no acceptable reason has yet been suggested for this. We must thus recognize that our theoretical and practical studies of semiconductors, although numerous, are not yet adequate. There are many unexplained contradictions and frequently no certain correlations Volume 40, Number 9, September 1963
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between theory and experiment. The modern quantum theory is nevertheless useful, even if inadequate to give a full and correct understanding of the enormous variety of effects found in semiconductors and of technological use. Although there are many problems which have not been cleared up, the theory can explain most of the more important effects qualitatively, if not quantitatively. It is due to the qualitative encouragement derived from the theory that the field of organic semiconductors was entered seriously by investigators from various physics and chemistry disciplines. This field is an area of investigation of the author's and a general review of it will be the subject of a subsequent paper.2 THISJOURNAL, in press.
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Literature Cited (1) FORCAAEIMER, 0. I,., AND XEWKIRK, T. F., J . CHEM. Eouc., 38,183(1961). (2) GATOS, H. C., ed., "The Surface Chemistry of Metals and
Semiconductors," John Wiley and Sons, Inc., New York, 1960. (3)
(4) (5) (6) (7)
(8) (9) (10) (11)
HANNAY, N. R., ed.,
Semiconductors," Reinhold Publishing Corp., New Yark, 1959. JOFFE,A. F., ''Phyaic~of Semiconductors," Academic Press, Inc., New York, 1960. POCCHETTINO, A,, Aead. L i r ~Rendiconti, 15 (I), 355 (1906). SCOTT, A. B., J . CHEM.EDUC., 38, 224-250 (1961). SIATER,J. C., "Quantum Theory of Matter," McCrawHill Book Co., Ine., Xew Yark, 1951. SMITH, R. A., "Semiconductors," Cambridge University Press, Cambridge, 1959. SOMMERFELD, A,, Z. Phys., 47, 1 (1928). VOLMER, M., A m . Physik., 40, i75 (1913). WAGNER, C., AND SCHOTTKI,W.,Z. Ph&. Chen~ie,B 11, 163 (1930).
