An Introduction to Principles Paul F. Weller
State University College Fredonia, New York 14063
I
of the Solid State Extrinsic sernicondudors
Several differences between pure materials that are classed as metals, insulators, and semiconductors were recently presented (I), using automobile travel on expressways and on local streets as an analogy for the conduction process in solids. The concepts previously described apply primarily to solids that exhibit intrinsic conduction, that is, electrical conductivity in which charge carriers arise from the chemical bonds of the material itself. Most common semiconductors that are important to technology, however, are not of the intrinsic type but are extrinsic semiconductors, in which charge carrier production and consequently electrical properties, are determined by lattice imperfections or by trace amounts of impurities. The presence of these defects-foreign impurities dissolved in the host material or imperfections in the orderly geometric array of the host crystal lattice-can produce a conducting material with negatively charged conduction electrons as the primary charge carriers, an n-type semiconductor, or a material in which the charge carriers are positively charged electron "holes," a p-type semiconductor. The analogy used previously (1) for intrinsic semiconductors is extended in this article to cover extrinsic semiconductors, both n-type and p-type. Included are discussions of the concepts of donors and acceptors, donor and acceptor activation energies and corresponding charge carrier production a t varying temperatures, and the effects of the presence of both donors and acceptors. Wherever possible, solid state principles and theories have been related to the analogies.
I I I -S-a-S-M-S-Cd-
I
Figure 1.
I
I
I
I
I
I
I
I
Point dofedr in CdS.
different methods of describing the charges that exist on the lattice atoms and any imperfections that are present. Chemists usually discuss "ionic-type" materials, such as CdF2, ZnS, In?, PbTe, etc., in terms of the ions that exist a t the lattice sites. In solid ZnS, for example, we often talk about Zn2+ and S2- ions. That's all right, a perfectly good approach, which physicists also often use. But notice that when a Na+ ion substitutes for a Zn2+ion in ZnS, a chemist might refer to a substitutional atom with a +1 charge while a physicist might refer to the same substitutional sodium atom as having a -1 charge. Actually it's the same defect, but there could also be a defect in the communication. Remember that the subst,itutional Na+ ion must be referred to the rest of the lattice. There is a +1 charge deficiency at the substitutional site, which appears to be a -1 charge to the surrounding lattice. There are many problems similar to this one; keep the two different approaches in mind (4). Effects of lmperfections: A n Analogy
Imperfections or Defects
The properties of extrinsic semiconductors are governed primarily by the imperfections or defects that are present in all materials. While imperfections fall into three broad categories: (1) point defects, (2) line defects, and (3) plane defects, point defects are considerably more important' to the effects and the concepts that we will discuss and will be the only type considered (2, 3). Point defects fall into two categories: those that are native to the material itself; and those that involve the presence of a foreign impurity. These defects are summarized in Figure 1. Defect Charges. There are two things that you should note about the above defects. (1) Charge balance must always be maintained. This can occur in different ways, many of which are often present in the same material (4). (2) We must be conscious of, and careful of, two
Semiconductors:
n-type
The analogy for n-type extrinsic semiconductors containing donor centers is diagrammed in Figure 2(b) and compared to the pure or intrinsic semiconductor case in Figure 2(a), described previously (1). There is one basic difference between the two cases: the location of isolated houses near the expressu7ayin the case of the extrinsic semiconductor. What would happen in these two cases if an electric field was applied to the two materials? Would there be a difference in electrical conductivity? Indeed there is; and we can see the difference using the analogy. Think of the application of an electric field as being analogous to the question, "Will I go to t,he city to shop?", assuming that expressway travel leads into the city while local shopping takes place in the groups of houses or communities. With the intrinsic Volume 48, Number 12, December 1971
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831
EXPRESSWAY
Automobiles 0-0-
&
,Cornmun~fl~es, Houses
(b.)
(a)
Figure 2. An fo*; lo) on intrinsic semiconductor in which the h o w e ~or communitier ore analogous to the valence band, the exprewway to the conduction band, and the narrow vertical rood to the forbidden energy gap; (bl on n-type extrinric remiconductor, in which the i d o t e d hovrer ore anologwr to donor centerr.
semiconductor, the answer to our shopping question will, most likely, be "No." But in the extrinsic case, the possibility of gett,ing people t,o use t,he express~ray is much greater since access is much easier from the isolated houses located near it. Since these isolated houses effect,ively increase the number of automobiles on the expressway, they are analogous to donor centers in n-type semieonduotor8, which increase t,he conductivity of t,he material. Semiconductors:
p-type
The analogy for p-type semiconductors is similar to that for the n-t,ype case, as is shown in Figure 3(a). EXPRESSWAY
001 0 0 0-0-0 0 0 05000 Car Mwsmeni
10.)
Hole Mwement Ibl
Id A p-type extrinirc mniconductor, in which the parking goroger are onolagour la acceptor centerr; (hi oppo=itite diredionr of movement of con and corvocancier IholeJ in troffic.
Figwe 3.
The difference between the n- and p-type cases is in the t,ype and location of t,he isolated "impurity." In the p-type analogy t,here are isolated parking garages a t various locations near the community shopping centers. These garages accept cars from the local streets, removing them from the t,rafficflow and leaving vacancies or holes in the lines of cars on the streets. These parking garages are analogous to acceptor centers in solids, which remove electrons from the valence band creating elect.ron vacancies or holes. Note that, a vacancy or hole in a traffic pattern has a high "attraction" for other automobiles. As soon as the hole appears a car moves into it, thereby creat,ing another vacancy. The new vacancy is immediately filled by another car, and so on until the hole has '%raveledn to the rear of the pack while the cars have all moved forward. The cars and the hole move in opposite direct,ions, as illustrated in Figure 3(b). A similar effect occurs in semiconductors containing acceptor centers. Holes are created in the valence band as electzronsenter the accept,ors. Conductivity
I t is evident that the numher of expressway cars or of local t beet holes (the mobile charge carriers of a semi832
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Journal o f Chemical Education
conductor) will depend on two things: (1) the number of isolated houses near the expressway (the donor concentration) or the number of isolated parking garages near the local shops (the acceptor concentration), and (2) the distance separating the donor houses from the expressway (the donor activation energy) or the distance separating the acceptor garages from the local streets (the acceptor act,ivat.ionenergy). We can make t,he following genefalining statements. (1) For n-type semiconductors, the larger the number of donor houses and t,he shorter the distance to the expressway, the higher u-ill be the number of expressway cars. That is, t,he larger the donor center concentration and the lower the donor act,ivat,ionenergy the more conduction band electrons produced-which increases the conductivity. (2) Similarly for p-type conductorsa large hole c~ncent~ration, corresponding to a higher cond~ctivit~y is the result of more acceptors and lower acceptor activation energies. Then the conductivity of an n- or a p-type extrinsic semiconductor should be much higher than the conductivity of the pure or intrinsic material, even t,bough the donor or acceptor concentrations are very low compared to the number of lattice site atoms. And at high temperatures most donor (and acceptor) levels are ionized yielding high conduction electron (or hole) concentrations and relatively high conductivities. At low temperatures, only t,hose donor (or acceptor) levels with small activation energies are ionized which yields low carrier concentrations and low conductivities. Finally we can get a very general idea about the relative mobilities of conduction band electrons and valence band holes using the analogies. In Figure 2(b) automobiles are shown using the expressway. There are many expressFay lanes and only a few cars. To pass one another cars simply change lanes and move easily through t.he traffic. Their progress is quite free. On the ot,her hand, the progress of holes in the valence band is somewhat more restricted. I n Figure 3(b) me see that movement of a hole depends on the collective movement,sof many cars,, each succeeding car moving into a hole and leaving another hole behind. We might, t,hen, predict that conduction electron mobilities should be higher than hole mobilities. This is, in fact, the generally observed case in semiconducting materials. Effects of imperfections:
Concepts a n d Theory
Extrinsic Semiconductors
The types and concentrations of defects and their effects on the properties of materials have been considered elsewhere (6-7). It is known, for example, that a donor level is formed by an anion vacancy in a semiconduct,or and that cation (or metal) vacancies produce acceptor centers. Donors can also be formed by small met,allic interstitials, such as Liin GaAs, and by substitutional imp~rit~ies.If a latt,ice at,om is replaced by an atom that has more valence electrons, the substituted atom is generally a donor. For example, the substitution of Ge at a Ga site or Se at an As site in GaAs will produce donor centers and n-type GaAs, as is diagrammed in Figure 4(a). The substitution of Zn at a Ga site in GaAs produces holes and p-type conduct,ivity since there is a deficiency
( E D J ,and the valence band (E,). Then n in the above equation is given by
If we assume that ED,,ED,,and E , are larger than kT (where k is Boltzman's constant), then la 1
lbl
Figvre 4. (a1 A Se impurity in GoAr provider on "extro" electron, relatively weakly bound to the Sei I b l a i n impurity in GoAr provider one too few electrons for bonding. Motion of this hole is from positon 1 to position 6, os successive electron "jumps" into the "dangling bond" are indicated b y the arrows labeled 1, 2, etc.
of one electron for complete bond formation, as shown in Figure 4(b). The presence of donors and acceptors in a particular semiconducting material is conveniently indicated using energy band diagrams as shown in Figure 5.
where nlo and nzoare the donor center concentrations and nebois the valence band state concentration. Since = =
+ +
( n ~ nb n,a)eo n,0eoe-EolIkT + n20epe-Eo2ikT+
If it is true that E , >> ED,,t,hen the number of conduction electrons from the valence hand will be negligible and a = A,e-EmIM
Figure 5. lo) An energy band diagram for an n-type remiconductor ED,; containing two donor centers with different activation energies, ED? lb) o wtype semiconductor with two different acceptor senten.
>
Donor and acceptor centers are located-in energy terms-near the conduction and valence bands. Ionization of these levels requires different activation energies ED,,El,, etc., depending on t,he type of donor or acceptor present (7). Note that this energy band picture corresponds to the analogy used above and to Figures 2 and 3. While we can qualitatively locate a donor (or acceptor) level somewhere within the band gap of the host material ( 5 ) , it is, in general, not possible to calculate the activation energy for a particular donor center in a given semiconductor. In a few cases, for example, Si or Ge containing As or Ga, the experimentally observed donor or acceptor activation energies correspond closely to those calculated using a hydrogenlike atom approach (8). Extrinsic Conducfivify
We have seen from the analogy above that we urould expect increasing conductivity for an extrinsic semiconductor as its temperature was increased. Indeed this is just what is observed experimentally. An extrinsic semiconductor's conductivity increases exponentially with temperature. This dramatic rise in conduction is caused by increased numbers of charge carriers and is, to a first approximation, unaffected by the mobility of the carriers. Recalling that the conductivity a is given by where n is the number of free charge carriers, e is the electron charge, and p is the carrier mobility and referring to Figure 5(a) for n-type mat,erials, we see that conduction electrons can be obtained from three different sources, donor level 1 E D donor level 2
n&oe-B.lkT
+ Ale-SdkT
where A1 = nPep and A2 = nloep and the temperature dependence of p (which is small, = l / T , compared to that for n) has been neglected. Then the conductivity of the material is governed by both donor centers (and a for this extrinsic semiconductor will be considerably higher than a in the intrinsic case). But t,be two donors become important at different temperatures, depending on the values of ED, and ED,. Since ED, < ED,, the a of the semiconductor will be det,ermiued predominantly by donor 1 at low temperatures. Conduction electrons are obtained only from donor 1 by t,hermal ionizat,ion. However, as the temperature is increased, donor 2 begins to yield significant numbers of conduction electrons (this depends on the relative values of ED, and kT) which is reflected by another increase in t,he conductivity. A typical (and idealized) plot of the temperature dependence of the conductivit,y of a semiconductor with two active donor centers is given in Figure 6. Note that the plot is of log a versus 1/T and t,hat there are two essentially straight-line portions present in the graph. From t,he above equation for a in the form
2,zkT +
log s = log A, - ---
log As
we can see that a log a versus 1/T plot can yield Figure 6, given the proper relative values of ED, and ED,. At low temperatures Em log a log A, - --------
-
2.303kT
-TI Figure 6. Temperature dependence of the conductivity of an extrinsic semiconductor containing two donor centerr.
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833
and ED, can be determined from the slope of the low temperature straight-line portion of Figure 6. In the intermediate temperature region, where u goes through a maximum, we can explain the shape of the curve in the following way. At low temperatures o is governed by ED, only. As the temperature is increased, electrons are excited into the conduction band out of the donor centers located a t ED, until all of these donors have been ionized. At this temperature ED,is still too large for any significant thermal electron excitation from it. Since the number of conduction electrons is then constant, the temperature dependence of o is governed by the variation of p with temperature. We saw previously (1) that p decreases approximately linearly with increasing temperature. This behavior is shown in Figure 6 by the low temperature straightline section reaching a maximum value and then decreasing. The region around this maximum and the ensuing decrease in o is often called the exhaustion. region since the donor centers at ED,have contributed all electrons that they possibly can to the conduction band. Finally, the conductivity begins to rise again in a second straight-line portion which is characteristic of, and whose slope yields, the second activation energy ED,(or possibly E , in certain cases) (9).
enough of these particular donor houses here. The area can't support any more; local shopping possibilities are lousy." But note what happens if local parking possibilities are improved greatly, i.e., more acoeptor parking garages are built. Now local shopping becomes much more attractive, and rather than traveling to the city via the expressway, many people from the donor houses use the local shops. This attractive aspect of the locality increases the desirability of living in the donor houses. Hence, more of these houses are built, and the donor house concentration increases. This, in turn, can stimulate further increases in parking garage constmction, which will create new donor house possibilities, and so on. The actual limits of donor house and acoeptor garage concentrations will, of course, depend on the special conditions present in any given locality, i.e., on the zoning ordinances (bonding type present), the absolute numbers of houses and garages present (imperfection concentration), the types of houses and garages being added (imperfection type), the room for expansion (lattice type), etc. But we can say that, in general, an increase in the number of acceptor parking garages will increase the possibility of having more donor houses, and vice versa.
Presence of Donors Plus Acceptors
The interactions of donor and acceptor centers in solids can be considered in several different ways, and in quantitative fashion. One can, for example, write down all appropriate reactions involving the donors and acceptors and solve the corresponding mass action expressions simultaneously. Or a kinetic approach can be used in which the rates of electron excitation into, and trapping out of, the conduction band are used. Both approaches afford quantitative solutions to the interaction characteristics of donors and acceptors, but considerable tedium is involved in d e h i tions and algebraic manipulations (4, 10). But we can get an insight into the methods involved, and some qualitative answers, using the following arguments. Consider an n-type semiconductor containing a donor D. Conduction band electrons are produced by excitation from the donor, yielding an ionized donor and the free charge carrier. This process is represented by the eqwtion
In the discussion above we treated n-type semiconductors as if donor centers were present by themselves and p-type cases as if only acceptors were present. That is, we assumed that donor concentrations, and consequently electron c h ~ r g ecarrier concentrations, were much greater than acceptor, or hole, concentrations in n-type materials (and vice versa for p-type conductors). In many cases this is a valid and useful assumption, but there are important effectsthat arise when both acceptor and donor centers are present in the same semiconductor. This is, of course, the usual situation since native or foreign defects which serve as donors and as acceptors are present in all real solids, even though we might try hard to eliminate them (4,6,10). Let's consider just two of the effects that are caused by varying relative concentrations of donors and acceptors in the same host material. One is concerned with changes in the solubility of a given impurity and the second with changes in the electrical conductivity of the material. It has been found experimentally, and subsequently explained theoretically, that, for example, the conductivity of an n-type semiconductor can be varied over a wide range (several orders of magnitude) if the acceptor concentration in the material can be selectively and accurately controlled. The case is similar for solubilities, say, of donors in a given solid. If the acceptor concentration is changed, the solubility of the donor is affected. We can get a qualitative idea of these effects from the proposed analogy. Solubility Effects:
The Analogy
If, as shown in Figure 2(b), donor houses are located near the expressway and no acceptor parking garages (or a t least a negligibly small number) are present near the local shopping area, there is a tendency for prospective donor house builders to say, "There are 834
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Journal of Chemical Education
Solubility Effects:
Equilibrium Treatments
D-D++e-
At a given temperature equilibrium concentrations of the three species are present. According to LeChatelier's principle, this equilibrium condition can be changed by applying a stress, such as the removal of the electrons from the conduction band, to the system. Conduction electrons can be readily "trapped" by acceptor centers, which can be represented by the equation e-+A-A-
Then if acceptors, A, are added to an n-type semiconductor containing donors, the unionized-ionized donor equilibrium is upset and new D and D + concentrations are established D-Dt+e-
+ A-A-
This entire process favors a higher solubility of D in the host semiconductor. We can develop a similar argument using appropriate equilibria and the corresponding mass action expressions. Silicon containing an As donor is an n-type semiconductor. Conduction electron production is governed by the donor center ionization and its corresponding equilibrium constant
At a given temperature, a certain [As] is established. If an acceptor such as boron is added to the Si host along with As, the solubility of As is increased. This is true since B accepts a valence band electron forming an ionized acceptor B- and an electron vacancy, or a hole h+, in the valence band
The equilibrium constant for the recombination of electrons and holes is very high (recall the analogy of heavy automobile traffic and the very high tendency of cars to be bumper-to-bumper, with very few openings or holes between cars). Consequently, when we cousider the simultaneous donor and the acceptor equilibria
we see that the conduction band electrons (essentially free charge carriers) and the valence hand holes (also free charge carriers) combine and effectively anihilate one another. That is, they are no longer charge carriers; the electron is now localized in a chemical bond between atoms (recall Fig. 4). This combination of electrons and holes (h+e- above) is very highly favored; hence, very few free e- and h + exist in the presence of one another. Then for the equilibrium constant expression [e-] [As+]/[As] to be equal to K",, the [As] must decrease (and the [As+] increase) to compensate for the decrease in [e-1. Thus the [As] is lowered below the solubility limit and more As can be dissolved in the Si sample containing the acceptor boron.
expressway will shop locally-parking their cars in the garages rather than driving them on the expressway. The end result of the parking garage additions is to reduce the number of expressway cars. This does, of course, decrease the number of people going to the city to shop, which is analogous to a decrease in the conductivity of an n-type semiconductor. The conduction electron concentration is lower; they have been "trapped" by the acceptor centers. From an equilibrium point of view, we can consider the case of the n-type CdTe semiconductor containing Li interstitials, which act as donors. Essentially free electron charge carriers in the CdTe conduction hand are produced by thermal ionization of the interstitial Li atoms At a given temperature and [Lit, the conduction electron concentration, [e-1, is controlled by the equilibrium constant for the reaction. Let's assume for simplicity that the Li ionization reaction goes essentially to completion, i.e., [e-] = [Li]. Then the conductivity value for the CdTe sample is determined by [L'i]. Now if acceptor centers, such as Sb atoms substituting for Te atoms, are also present in the CdTe samplewhich still contains interstitial Li a t the same concentration-the conductivity of the CdTe will be decreased. This occurs because the acceptors reduce the conduction electron concentration. The effect of an acceptor can be considered in the following way. You can simply think of an unionized acceptor, the substitutional Sb atom with its one incomplete bond, attractingstrongly-one of the free conduction electrons. This process completes the local bonding at the expense of a conduction band electron and forms an ionized acceptor, i.e., an Sb atom that has lost its hole and now is associated with one electron more than the number of protons in the Sb nucleus Then for each Sb acceptor center present in the CdTe one electron is removed from the conduction band. For the constant Li donor concentration and assumed complete ionization, the decrease in conductivity will depend directly on the Sb acceptor concentration Li
-
Li+
+ e-
+
Sb Compensotion Effecfs
It is apparent from what we have just said about the effect of acceptors on free electron charge carrier concentrations (or donors on [h+])that the conductivity of a semiconductor can be changed markedly by adjusting donor and acceptor concentratoins. These are called compensation effects since the presence of an acceptor effectivelynegates the effect of a donor by trapping its "free" electron. We shall illustrate this further using the automobile analogy and also an equilibrium-based argument. People living in local houses near the expressway often travel to the city to shop via the expressway. But if local area shopping is made more attractive by the addition of parking garages, the people near the
-
Sb-
Since u = nep and n = [e-1, u decreases as n decreases; and n decrnases as [Sb] increases. The Water Analogy
There is another analogy for semiconductors that is especially appropriate for chemists, water and its dissociation equilibrium. We will consider briefly here some of the solid state concepts covered above, using the water analogy ( 1 1 ) . Dissociation of pure water is analogous to the simultaneous formation of conduction band electrons and valence band holes, via excitation from the valence band, in intrinsic semiconductors. These two equilibria can he represented by Volume 48, Number 12, December 1971
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835
H,O Ft H +
+ OH-
and h+e-
$h+
+ e-
Note that both are weakly dissociated systems. Ionization of HzO molecules is difficult as is excitation of valence band electrons. It is true that [H+] = [OH-] and that these concentrations are low; it is true that [h+] = [e-] and that these concentrations are low. We know that the H + and OH- concentrations in aqueous solution can be drastically changed by the addition of acids or bases to the solution. Acids ionize to form H + ions and bases produce OH-ions. Similarly in semiconductors: the h+ and e- concentrations are affected greatly by the presence of impurities. Acceptors ionize to form h+, holes, and donors produce e-, conduction electrons. Extrinsic semiconductors are formed. These relationships are shown by the equilibria NaOH(.,,
-
Na+w
The solubility of NH40H and the donor D are increased by the addition of HC1 and the acceptor A. We also know that the [H+] is affected by the prek ence, or addition, of OH-
--
HCI Na0H
C1-
Na+
+ H+
+ + OH4
HIO
The [H+] is rapidly decreased when OH- is added to the solution. Similarly in the extrinsic semiconductors: hole concentrations, [h+],in a p-type semiconductor are lowered by the addition of a donor, which produces free electrons, eA-A-+h+
+ OH-(,)
D-DC+e-
+ 4
h+e-
and
The [h+] is rapidly decreased when electrons are added to the semiconductor via the donor D. Acknowledgment
where the subscripts imply aqueous, solid, conduction band, and valence band. The activation energy for an acceptor (or donor) in a semiconductor is analogous to the ease of (or %) dissociation of an acid or base
A, + h +
+ AI;
low En
but HCN
low%
AS s h +
+ CN-
H+
+ A,-,
higher EA,
The solubility of a donor is increased by the presence of an acceptor, and vice versa. In H20 the solubilit,y of NHs(,, can be increased by the addition of an acid to the solution NHm
+ H 2 0 * NHhOH a NH*+ + OH+ HCI
-
C1-
+ H+
L
H10
The analogous equilibria in a solid are
The author is grateful for generous support from the National Science Foundation, the Research Corporation, and the SUNY Research Foundation. Literature Cited (1) WELLER,P. F.. J. CHEM.EDOC.. 47, 501 (1970). (2) VAN BUREN,H. G.. "Imperfections i n Cryetala" (2nd Ed.) NorthHolland Puhiiehing Co.. Amsterdam, 1961. A detailed treatment of imperfections of all types is given. (3) MOTT,N. F., A N D GDRNEI. R. W., "Eleotronic P r o ~ e s s ei~n Ionia Crystals" (2nd Ed.) Dover Publiaations, Inc.. New York, 1964. Defect descriptions for, and effeots on, ionic crystals. (4) K n s o ~ n ,F. A., "The Chemistry of Imperfect Crystals:' NorthHolland Publishing Co., Amsterdam, 1964, paitioularly Chapter 7. A detailed amount of imperfections ~ n their d effects on orysts1a. (5) All of the concepts mentioned i n this section are covered in each of t h e three excellent book8, appropriate for chemista: H A N N A ~N., B. (Editor), "Semioonduotors," Reinhold Publishing Corp., New York. 1959; HANNAY,N. B., "Solid-State Chemistry," Prentiee-Hall, Ino., Englewood Cliffs, New Jersey, 1967: Moone. W. J.. "Seven Solid States," W. A. Benjamin. Ino.. New York, 1967. (6) Knnoen. F . A,. AND VINE, H. J.. in "Soiid State Phgaics" (Editors: Smrz, F. AND TURNBOLL, D.) Academic Press. New York, Yol. 3. 1956.
(7) Gunam, E. F., J. CHEM. EDUC.,46, 80 (1969): W m m n , P. F.. J. CHEM. EDUC.,44, 391 (1967). Short and nonmsthematioal descriptions of energy bands and t h e effects of imperfections. (8) See, for e r a m d e , LEVY. R. A,, "Principles of Solid State Physios." Academic Presa. New York, 1968, Chapter 9. (9) Literature dhth similar t o those in Figure 6 were found by, for example. DEBYE, P. P., A N D CONWELC, E. M.. P 1 w . Reu.. 93, 693 (1954). (10) S ~ A G I R N., A.. thermodynamic^ of Solids," John Wiiev & Sons, Ine.. New York. 1962. (11) Fn=w=. C. 8.. i n "Semiconductors" (EdiIor: HANNAY, N. B.). Reinhold Publish'in~Cow.. N e v York. 1959.
