Paul F. Weller
University College Fredonia, New York 14063
An Introduction to Principles
State
of the Solid State
M a n y of the fundamental principles and concepts of solid state chemistry and physics are quite unfamilinr to most chemists and are essentially unknown to undergraduate chemistry students. There are, no doubt, many reasons for this lack of exposure t o the principles of solids, but one frequently cited reason is the need for a relatively sophisticated mathematical background. Such a foundation is, of course, helpful, just as it is in all fields of science, but a n understanding of many solid state principles can be acquired by using commonly underst,ood situations as models or analogies. In fact, these qualitat,ive models often make the mathematical approaches more easily understood and, in some eases, even reasonable. I n any case, the author believes that most practicing chemists approach chemical problems by developing general models or qualitative pictures about them. Illore quantitative ideas follow furt,herstudy. The analogies presented in this communication have proved helpful in introducing many ideas and concepts of the solid state to undergraduate chemistry students. One basic analogy, with variations, has been used t,o consider: electrical conductivity, variat,ions in conductivity between metals, semiconductors, and insulators, and conductivity temperature dependencies. Where appropriate, short descriptions of the corresponding solid state principles or theories are given. Electrical Conductivity:
the Free Electron Theory
Somc mat,erials conduct electricity quite well, copper, silver, gold (a11 metals), while other materials are very poor conductors, diamond, AI2O3, CaF2 (insulators). This difference in behavior between metals and insulators is quite interesting and will be considered in more detail later. First, let us look a t the problem of the conductivit.y of met.als. We know that electrical charge can he transported from, say, one end of a metal wire to the other through the movement of electrons. One question that. might be asked is, "What determines the conductivity of the metal, i.e., how easily is charge transported from one end of the wire t o the other"? A simple model or analogy for this process can be developed in the following way. Assume that a common aut,omobile with its driver is analogous to an electron with its negative charge (only one person per automobile is allowed). The ear, of course, t,ravels along streets and highways. Now suppose that you have the problem of transporting people (charge) from your suburban home into the center of the nearest city, say, Buffalo, New York. How can the number of people getting into Buffalo he maximized? One way is to use a large number of ears. But there is also a second im-
portant variable, the speed of the cars. The faster they go, the greater the number of people moved. Then there are two important variables involved in transporting people into t,he city: (1) the number of ears; more ears, more people transported, (2) the speed of the cars; the faster they can move, t,he more people transported. The number of people arriving in Buffalo is directly proportional to the number of automobiles involved and also directly proportional t,o the speed of the ears. Applying this proposed analogy to the conduction process in metals, we can conclude that the amount of electrical charge transported in a metal, or its conductivity, should be proportional to the number of electrons (ear plus driver) and to the speed with which the electrons move. The larger the number of mobile electrons the higher is the electrical conductivity; the higher the electron mobility, the greater is the conductivity. The Theory of Drude and Lorenfz
Thefvee eleclvon theory of metals was first proposed by several scientists shortly after the discovery of the electron. Among these were P. Drude in 1902 and H. A. Lorentz in 1909 (1). Very briefly, the ideas of Drude and Lorentz were as follows (Z). They assumed that a metal was composed of positive ion cores imbedded in a perfect gas of freely moving electrons. When an electric field was applied to the metal, the electrons were free to flow along the potential gradients giving rise to an electric current. After some mean time of motion, the electrons were envisioned as colliding with one another or with the ion cores which destroyed the component of their motion in the direction of the field. Then using classical equations of motion for a particle in a field F and defining the conductivity, u, as u = j/F where j is the current obtained, they were able to derive the relation where n is the number of free electrons, e is the electron charge and fi is the electron mobility (free electron velocity per unit field strength). This equation for the electrical eonductivit,y of a metal, developed from free electron theory, corresponds quite well to the qualitative relations derived from the proposed analogy. The conductivity depends directly on the number of electrons and on their mobility with the electric charge as a constant of proport,ionality. (Common units are: (ohm cm)-' for u, e-/cc for n , coulomb fore- and cm2/v-sec for fi). While the free electron theory of Drude and Lorentz succeeds in explaining some aspects of the electrical conductivity of metals, it fails when applied t o other Volume 47, Number 7, July 7970
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metallic properties, such as specific heats and the paramagnetic properties of the free electrons. Furthermore, it does not provide a very good working model for solids in general since it treats the valence electrons in metals essentially as gaseous particles. I t is evident, then, that there was a need to develop a better understanding of the interactions of electrons in solids. One of the requirements of a new theory was its application to many types of materials, insulators, and semiconductors as well as metals. Band Theory, Analogies An Analogy for Metals, Semiconductors, and Insulators
We can extend the analogy presented above to illustrate the differences between metals, semiconductors, and insulators. One of the major differences between these three types of materials is their ability to conduct an electric current. I n general, metals are good conductors and insulators very poor conductors, while semiconductors show intermediate conductivities. There is, however, another difference in the electrical properties of these three materials. The electrical conductivity of a metal decreases as its temperature is increased while the conductivities of semiconductors and insulators are higher a t increased temperatures. Our proposed analogy should account for these observed differences in the various materials.
/ l 7 EXPRESSWAY-
AIJTOMOBI,(ES
METAL 101
The case is similar in mctals. Electrons are free to move throughout the metal; they are not localized or bound to atomic sites. There is, therefore, essentially no activation energy required for the conduction process. A n Insulator. The analogy for an insulator is very similar to that for a metal. The differenceisin the location of the expressway in relation to the houses. Figure l(c) shows that the individual houses (or communities) are located a considerable distance from the expressway. Travel to the expressway is difficult and time consuming; the trip is made only under the most severe provocation. Almost all travel is local. When the question is asked "Will I go to the city to shop"?, the answer is almost alwaysl "No." I t is just too difficult and requires the expenditure of too much energy. Local shopping is much easier and the drivers do not generally travel over large distances. Then in insulating materials we can assume that elect,ron movements are localized. Electrons are not free to move throughout the bulk of the material but are bound to atomic sites. As seen above, the opposite situation occurs in metals; electrons are free to move throughout the bulk of the material. A Semiconductor. The analogy used for an insulator also applies to a semiconductor, wit,h one slight alteration. The distance between the expressway and the individual houses is shorter, as illustrated in Figure 1(b). Since the expressway is more easily reached in the case of the semiconductor (as compared to the case of the insulator), the answer to the question "Will I go to t'he city to shop"? is sometimes answered "Yes." But there is considerable hesitation to giving a positive answer since the trip to the expressway still requires the expenditure of significant amounts of energy. While the trip is not as difficult as it is in the case of insulators, an energy barrier to travel is present and must be overcome. Again this situation is opposite to that found for metals where electrons are essentially free to move throughout the material. I n semiconductors the number of freely moving electrons (say, for room temperature conditions) is far less than the number present in metals hut generally somewhat greater than the number present in insulators. This difference between metals, semiconductors, and insulators in the number of free electrons present can account for the fact that m e t a l are much better electrical conductors than either semiconductors or insulators, but another significant difference is indicated by the proposed analogies. There is an activation energy required for the production of highly mobile charge carriers in the cases of the semiconducting and insulating materials while free charge carriers are always available in metals, with essentially zero activation energy. This difference in act,ivation energies is responsible for the observed differencesin the temperature dependences of the electrical conductivities of the three types of materials.
SEMICONDUCTOR
ib)
INSULATOR (El
Figure 1. Analogies for electron movement in: lo] metols-automobiles hove easy occerr to e r p r e r w a y r and can travel over wide areas relato expreswoyr is limited since tively freely; (b) remiconductom-access communities ore located some dirtonce away; mo3t travel is conflned within and between communities on local roods; irl insulotom-o similar case to ibl with even less expressway use and greater locolired trovel since e i p r e r r w a y accerr is so difficult.
A Metal. We can envision a metal as shown in Figure l(a), again assuming that an electron is analogous to a car aud its driver. I n Figure l(a), a multilane expressway is diagrammed which contains cars and which runs directly by large numbers of houses, i s . , there is essentially immediate access to the expressway. Now we can think of the application of an electric field to a metal as being analogous to the question "Will I go to the city to shop"? When an electric field is applied to a metal, a current flows through the metal. When wc ask the above question of anyone living on the expressway, the answer is, "Sure, no trouble." With very little difficulty he can drive into the city using the expressway. Essentially no energy is expended in getting into the rapid traffic flow and away from the house. 502 / Journal o f Chemical Education
Conductivity:
Temperoture Dependence
Metals. Using the proposed analogy for conduction in metals, Figure l(a), we can consider the effect of changing temperature on the conductivity of a metal. To do this, let us contrast two somewhat extreme cases: (1) what happens to a car and driver on a hot day, say
40°C (104"F), during the summer and (2) what happens to a car and driver on a cold day, say -20°C (-4'F), during the winter. On the hot day as the driver travels along the expressway, he is constantly distracted by many inviting places to stop, a swimming pool, an air-conditioned theater, an ice cream store, a bar, or a shady rest area. Quite often he pulls into one of these places for refreshments or a cool rest. Sometimes he might even get sidetracked completely and spend the day a t the beach. All of these stops do, of course, slow the progress of the driver. It takes longer to get to his destination, the main shopping center. On a cold day, on the other hand, the desire to stop at these side attractions is considerably less. It is much better to remain in the nice warm car and to get into the shopping center buildings as quickly as possible. Hence, there are many fewer side trips during cold weather and the apparent speed of the car, i.e., how fast the driver gets from his home to the shopping center, is greater than on the hot day. Since the electrical conductivity of a material depends on the number of charge carriers and also on the speed of the carriers, our analogy predicts that the conductivity of a metal should be lower a t elevated temperatures. As the temperature goes up, the number of side trips or stops that the driver makes increases. A greater number of delays lowers the average speed of the car and increases transit times. Since the number of cars present at both the low and high temperatures is the same but their average speed a t elevated temperatures is lower, the conductivity of a metal should decrease as the temperature is increased. Experimentally the electrical conductivity, u, of metallic-type conductors is found to decrease linearly as the temperat,ure of the metal is increased. This decrease in u is caused by a decrease in the electron mobility, p, as is predicted by the analogy. The observed temperature dependence of the conductivity of semiconductors and insulators, on the other hand, is quite different-approximately.the reverse of that for metals. Semiconductow and Insulato~s. Since the number of free charge carriers, cars plus drivers on the expressway, is governed by an activation energy in the case of semiconductors and insulators, the change in their electrical conductivity with changing temperature is not entirely dependent on the mobility of the carriers. I n fact, the temperature variations in the conductivity of semiconductors and insulators depend much more strongly on the number of charge carriers than on the carrier mobilities. Again we can consider the two extreme cases of traveling to the city on a hot and a cold day, as was done above for metals. On the very cold day the answer to the question, "Will I go to the city to shop"?, is most certainly ''No" for both the semiconducting and insulating cases. Figures l(b) and l(c) show the relatively long distances to the expressway; the resolve required for the drivers to get to the expressway is just too great on such a cold day. Only the most venturesome go far from home during such cold weather; most shopping is local. Then at very low temperatures our analogy indicates that very few drivers will he taking the expressway into the city. Even though they can travel rapidly with very few side trips, the number of people arriving in the
city is small since the number of cars is so small. I n the case of insulators, the amount of expressway travel is close to zero since the trip to the expressway is so difficult. While expressway travel in semiconductors is also quite limited, it might be somewhat larger than in the case of insulators since the difficulty of getting to the expressway is less. On the hot day the situation can be considerably different in the case of semiconductors but, in general does not change greatly for insulators. When thc question is asked, "Will I go to t,he city to shop"? the driver in the case of the insulator still generally answers "No." Even the attractions of air-conditioned stores or, better yet, swimming pools and beaches are not sufficiently great to overcome t,he difficultyof traveling to the expressway. The energy required to make thc trip is just too great; local shopping is preferable. Consequently, expressway travel in insulators is very limited on hot days just as it is on cold days. For semiconductors, on the other hand, t,his need not be true; it depends on the ease of the trip to the expressway. I n some semiconducting cases the answer to the question, "Will I go to the city to shop"?, can be answered "Yes" quite often since the attractions of the city, or what can be found en route, are sufficient to overcome any reluctance to travel to the expressway. I n these semiconductors the number of cars on the expressway is very much greater on hot days than on cold days. This leads to many more people arriving in the city on hot days, even though the speed of the cars is somewhat less. We can make some general statements about the temperature dependence of the electrical conductivity of semiconductors and insulators from the analogies just presented. The conductivity of a semiconductor or an insulator should be greater a t higher temperatures because the number of charge carriers increases as the temperature increases. Whether this is a significant increase in conductivity depends on the amount of energy required to travel to the expressway. If this trip is not too difficult, then the conductivity of the semiconductor is dramatically increased by an increase in temperature. The analogy that we have used for semiconductors and insulators has included a community of houses and, a t some distance, an expressway. For any given semiconductor or insulator, the cars using the expressway were assumed to come only from one community, over one specific route. This is the case for intrinsic semiconductors in which charge carriers arise from the chemical bonds of the semiconductor itself. I n general, however, most important semiconductors are of a variety called extrinsic semiconductors in which the charge carriers are produced much more easily from, for example, impurities that are present in the semiconducting material. Extrinsic semiconductivity will be covered in a scparate communication. Band Theory, Formulations Energy Bond Formofion in Solids (3, 4 )
The analogies used above for metals, semiconductors, and insulators have pict,ured good electrical conductivity as arising from rapid travel on expressways, compared to slowly moving local traffic. The expressVolume 47, Number 7, July 1 9 7 0
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503
ways and local streets are analogous to energy bands t,liat are formed in solid materials. The formation of these rclat,ively broad electron energy levels in solids can be considered using two different approaches: (1) t,he interaction of localized electrons and their energy levels (often called the tight binding approximation) and (2) a collcctive or essentially free electron approximat,ion. Only a brief and rather qualitative discussion of t,heset,wo approaches to hand formation is given here. I t is hoped that this abbreviated presentat,ion will make the concept of enerm bands in solids plausible and that the analogies t,o energy bands used throughout will seem reasonable. Localized Electrons. When two hydrogen atoms interact to form a diatomic hydrogen molecule, their 1s wave functions overlap producing two energy levels (each of which can contain two electrons) appropriate to the H, molecule ( 5 ) ,as diagrammed in Figure 2. If
INTERATOMIC SPACING
-
Figure 2. Potential energy Venus internuclear seporotion for two hydrogen otorn. ( I S atomic orbitals) forming a hydrogen molecule.
a group of six atoms are placed in a row with equal separations between the atoms, a treatment similar to that for two hydrogen atoms yields a set of six allowed energy levels split from the is atomic levels in a fashion similar to that for the two levels in H2 (6). AS illustrated in Figure 3 the 2s wave functions overlap a t
larger atomic spacings, but the same general pattern of energy level splittings is preserved. Solids, of course, contain not six but about at,oms/cc. When these huge numbers of atomic wave functions combine, the maximum amount of splitting of the atomic energy levels is not changed appreciably, but the number of energy levels split from any given atomic level is vast. Consequently, there is a very small energy difference between the levels split from any one atomic level; these energy levels essentially form an energy continuum, i.e., an energy band. There remains an energy separation between the 1s and 2s (the 2s and 2p, 2 p and 3s, etc.) energy bands, but within a given band the discrete energy levels have an extremely small separation in energy (7). Free Electrons. A similar description of energy bands in solids is obtained by considering an assembly of essentially free electrons and the manner in which they interact with, and are affected by, the atoms in a crystallattice. An approximate solution to this problem first given by Sommerfeld in 1928 (a),is obtained by requiring that, the electrons behave according to the principles of quantum mechanics and that the potential energy well (created by the atoms a t the lattice sites) in which the electrons move has infinitely high walls, i.e., V = 0 within the solid (metal) hut V = m 'outside the solid so that electrons can move freely within but cannot escape the solid. This is the "particle in a box" problem (O), and in the one-dimensional case the Schrodinger equation is
Here E is the total energy, V is t,he potential energy of the electron of mass m, h is Planck's constant and $ is the wave function associated with the electron. Since V = 0 within the solid, the equation reduces to
The solutions for this equation are standing waves of the form
*
=
e%k=
where k is called a wave number and k
= (2rplh) = (2r/X)
where p and X are the electron momentum and wavelength, respectively. The corresponding energy levels are given by
INTERATOMIC SPACING
-
Figure 3. A potential energy diagram, sirnilor to Figure 2, for o row of six equally spaced atom. (offer Shockley (611.
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where n is the principal quantum number and a is the "box" dimension. This approximation, then, yields essentially one continuous band of energy levels for different values of k (or n). An improvement is needed, and a major flaw in the original set of assumptions is rather apparent. Since the atoms in a crystal lattice occur periodically, the potential energy of the electrons within a solid will not he zero but will also vary periodically, because of the charges a t the lattice sites. This periodicity must be included in the solutions to the wave equation and in the associat,ed expressions for t,he allowed energy levels.
The problem for a single electron moving in a linear array of square potential wells (corresponding to the atomic cores a t the lattice sites) was first solved by Krouig and Penney (10). The wave function for an electron in this case, is composed of the wave function of a free electron, +(x), essentially that given above, combined with a function which has the periodicity of the lattice, u(x), or
*.
= *(z)u(z)
Functions of this form are called Bloch functions and have solutions composed of running waves modulated by the periodic function and given by =
eCk%(z)
At this point one might be tempted to say "so what," but what appears to be only a slight modification in results actually turns out to be quite significant. The periodic function permits only specific values for electron energies within the solid; other energy values aTe not allowed. Just as a periodic array of lattice atoms reflects certain X-ray wavelengths while other wavelengths pass through a solid, this same atomic array permits only certain electron energies (or wavelengths) and disallows others. Instead of one (essentially) continuous band of energy levels, as in the case of the "electron in a box" given above, the allowed electron energy levels fall in a series of energy bands with the bands separated by regions of forbidden energy or energy gaps. The energy bands and gaps correspond to certain values of electron energies, E, (or wave numbers, lc or wavelengths, A, since k = (2s/X). An expression for E similar to that given above for the "electron in a box" can be used if a factor called the effective electron mass is substituted for the free electron mass
sidered (nor shown in a diagram such as the one in Figure 4) since they involve, primarily, core electrons and are even more localized to the individual atoms. The lowest lying energy band that is not completely occupied by electrons (it can be, and often is, empty) is generally called the conduction band. Electrons located in this band are free (approximately) to move throughout the solid with essentially zero activation energy. There are, of course, bands of higher energy; these are usually not shown in typical diagrams. There is a forbidden energy region between the valence band and the conduction band; this energy separation is generally called the energy yap or band gap (Eg in Fig. 4). Energy Bands and the Analogy. As is seen by comparing Figures 1and 4, the valence band in a solid corresponds to-local streets within a community (or between small communities) while the conduction band is analogous to the expressway. Travel on local streets is slow and generally limited, as is electron movement in the valence band since bonding electrons are involved. Travel in the expressway, on the other hand, is rapid and extends over large areas. Electron movement in the conduction band is essentially free and delocalized over the entire solid. The energy gap Ey (Fig. 4) in a semiconductor or insulator is analogous to the distance between the local community and the expressway (Fig. l(b) and l(c)). Distances and band gaps are large for insulators and are smaller in the case of semiconductors. Energy Bands: Mefals, Semiconducfors, and lnsulofors
Band Diagrams. The differences between metals, semiconductors, and insulators can be described in terms of energy band diagrams, as shown in Figure 5. I n CONDUCTION BAND
The efective mass m* depends on k (m* = fi2/(d2E/dk2)) and, consequently, can be used to account for the energy band structure; Energy Band Diagrams. A very common method of describing allowed electron energy bands and their corresponding forbidden energy gaps is shown in Figure 4. The band of highest energy that is completely filled by electrons is generally called the valence band, i.e., electrons associated with this band are involved in chemical bonding and are, consequently, rather localized and not free to move throughout the solid. Bands of lower energy are usually not con-
CONDUCTION BAND
I
ENERGY GAP
Figure 4. A typical energy band diagram for solids showing o conduction band containing no electrons separated b y a forbidden energy region, the energy gop Eg, from the valence bond which ha, energy stater completely occupied by electrons.
-
' 1 METAL
SEMICONDUCTOR
(bl
(0)
INSUUTOR (c)
Figwe 5. Typical bond diogroms illustrating the differences between: 1-1 metols-the conduction band it partially occupied b y electrons; Ibl semiconductors-the conduction bond contoinr eeentiolly no electron. if Eg kT; when Eg * kT, electrons ore excited ocmrr the band gap, Eg, from the valence b m d into the conduction band; lcl inrulotorr-the conduction bond i. errentiolly free of electrons at all norm01 temperatures since Eg is very large IEg kT1.
>
>>
metals, Figure 5(a), the conduction band is partially occupied by electrons at all temperatures including OoIZ. I n insulators, Figure 5(c), the conduction band is essentially empty at all temperatures. I n semiconductors, Figure 5(b), the conduction band is partially populated a t "higher" temperatures but is completely empty a t O0IZ (or, more generally, a t "low" temperatures). While metals differ from the other two types of materials in the occupancy of the conduction band states, the difference between semiconductors and insulators is much less definite. Conduction band elecVolume 47, Number 7, July 7 970
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trons in semiconductors (intrinsic) and insulators are produced by excitation of electrons from tlie valence band (out of the chemical bonds). Thermal excitation of these electrons, then, depends on the band gap, i.e., the energy difference between the valence and conduction bands. In insulators the band gap energy is large with respect to available thermal energies, a t temperatures below the melting point. I n semiconductors (intrinsic) thermal production of electrons, say a t or above room temperature, becomes possible since band gaps are smaller than they are in insulators. At lower temperatures less thermal energy is available and fewer electrons are excited across the forbidden energy gap; consequently, fewer conduction electrons are present a t low temperatures in semiconductors. Conductivity: Tempevature Dependence. The differences shown in Figure 5 and described above indicate the reasons for different conductivity variations with changing temperature between metals and semiconductors (or insulators). Using the free electron relationship for the specific conductivity, o = nep, we see that o depends on both n and p. Then the temperature dependence of u stems from the variations of n or p, or a combination of the two, with temperature. In the case of metals, the free electron concentration ?L (the number of conduction e- per cc) remains essentially constant a t all temperatures below the melting point (electrons occupying conduction band states in Fig. 5(a)). Then the temperature dependence of the conductivity a is caused by the variation in electron mobility p . An increase in the temperature increases the thermal vibrations of the metal ions a t the lattice sites. Conduction electrons interact more strongly with the more vigorously vibrating positive ions and are scattered more often from their straight-line motion and pathway. These scattering processes ( I I ) decrease the electron mobility and, consequently, the conductivity as the temperature is increased. At most temperatures (above a certain temperature, the 8 or Debye temperature, characteristic of each metal) the mobility aud conductivity are proportional to 1/T. Hence, a plot of a versus T is a straight line. For sen~iconductorsthe situation is quite different since n can change as the temperature is varied. I n fact, the productiou of charge carriers follows an exponential dependence on temperature. The number of conduction electrons n is given by
where no is the concentration of atoms a t the lattice sites, Ey is the band gap energy and k is Boltzman's constant (assuming Eg > k T ) . Then the conductivity relationship becomes
Since p varies approximately as 1 / T (a very weak temperature dependence), the major temperature variation in the conductivity of a semiconductor is caused by the deperldeuce of the carrier concentration n . Very often no, e , and p are combined into a preexponential constant, assumed to be temperature independent,, to give = Ae-PdXT
where 506
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Journol o f Chemkol Education
and log s = log A
- (Eg/2.303kT)
Then a plot of log o versus 1/T should give a straight line with a slope of -Eg/2.303k. The measurement of o as a function of temperature will, therefore, permit calculation (approximate) of the band gap energy Eg of a ~emiconductil;~ material. (A similar treatment can be used for insulators, but energy gaps are generally so large that thermal production of free carriers is negligible.) Energy Gaps and lonicity
From the analogies for semiconductors and insulators presented above we can illustrate a relationship that exists between the band gap energy Eg and the amount of ionic character present in the solid (or the electronegativity difference in a biuary compound). Consider the analogy for a semiconductor with a small activation energy for conduction, i.e., a small Eg. I n this case a driver might well take the expressway into the city to shop; his trip to the expressway is relatively easy. Travel is not restricted to the local area. On the other hand, a semiconductor with a long and difficult trip to the expressway, a very large band gap, has very restricted travel; drivers tend to stay in the local area. Then semiconductors with large band gaps have quite localized electron populations while semiconductors with small band gaps have, a t least partially, delocalized electrons. The more electrons that can be localized, the larger is the forbidden energy gap. Increased localization of electrons can be accomplished quite easily with our analogy. Simply close the expressway and open a new one a t a greater distance. This corresponds to a significant change in a solid material, such as, for example, changing from ZnTe to ZnSe. While these two materials have many similarities, they are also different in several respects. One of these differences is the band gap energy; the Eg for ZnSe is larger (see the table). Dependence of Eg on Per Cent Ionic Charocter
Substance
Approximate yoIonic Character6
NaCl NaI AlN AgCl AgBr ZnS ZnSe AIP Ad ZnTe A l As AlSb InSb GnN Gap GaAs GaSb ZnTe CdTe HeTe a Values for Eg are for 2Tr°C; 1 eV = 23.05 kcal/mole and corresponds to n temperature of 1.16 X 10' OK, st wave number of 8,066 cm-I and a wave length of 1.24 p . "&dated using Pauling electronegativity differences.
When the chemical constituents of a compound are changed, we can make qualitative predictions about the degree of localization of the bonding elfictrons involved using the concept of electronegativity or the degme of ibnicity of a chemical bond. As the per cent ionic character of a chemical bond increases, the electrons become more tightly bound to the cores of the atoms involved. Since there is a greater degree of localization of the possible charge carriers with strong ionic bonding, we would predict that highly ionic compounds would have large band gap encrgics. A t least qualitatively this trend is observed. A highly ionic suhstance, such as ATal", bas a band gap (at 25°C) of about 12 cV while a compourid with a low per cent ionic character, sucli as InSb, has a band gap of ahout 0.16 cV. I k d i e r exaniplcs are given in the tahle. Further Applications of the Analogy
Actually the electrical properties of most semiconductors depend, not 011 the intrinsic host properties as discussed above, but on the presence of various types of imperfections or impurities. This can be quickly seen from the table or by considering the elemental semiconductor Si (probably the most important semiconducting material technologically). At 2.5"C Si has a band gap energy of 1.11 eV. Note that this energy corresponcln to a temperature of about 12,9OO0Ii. Consequently, the thermal production of a significant number of conduction electrons from the valence band in Si is not possible a t ordinary temperatures (below -lOOOoIi). But Si can be made into a relatively good electrical conductor a t room temperature by the addition of the proper kinds of impurities. When this is done, Si bchnve as a n extrinsic semiconductor. The analogies used in this communication can be readily extended to extrinsic semiconductors a i d to related concepts such as donor and acceptor levels (producing nor p-type semiconductivity, respectively), variations in donor and acceptor solubilities and the lpermi level.
Acknowledgement
The author is grateful for generous support from the National Science Foundation, Grant No. GP-8801, the Research Corporation, and the S U S Y Research Foundation. Literature Cited
(1) DRWDE,P., Ann. d. Physik, 7, 687 (1902); L o n i m ~ z ,H. A,, "Theory of Electrons," Leipsig, 1909. (2) A short summary of the DrudeLorentz treatment is given by I f o w , N. F. A N D JONES,H., "The Theory of the Properties of Metals and Alloys," Dover, New Yolk, 1936, pp. 240-2. (3) All of tKe concepts mentioned in this section are covered in each of the three exrelleat books, appropriate fur rhemists: H.mNaY, N. B., (Ed.), "Semirondurtars," Reinhold , B., Publishing Carp.. New Yark, 1959; H . ~ N A I -K. "Solid-state Chemistry," Prentice-Hall, Inc., Englemood Cliffs, N.J., 1967; R l o n w , W.J., "Seven Solid States," U'. A. Benjamin, Inr., New York, 1967. 14) Far mare detailed and mare mathematical tre,ztmenls of bnnd theory: Bnolw, F. C., "The Physirs af Solids," W. A. Benjamin, Inc., New York, 1967; KIPTIX.,C., "Introduction to Solid State Phyaici;," (3rd. Ed.), J,ilrti Wiley & Sons, Ine., New York, 1966; I ~ K K I X A , J., "Solid State Physics," P~.entirc-Hall, Inr., Englewoml Cliffs, N.J., 1957. See far example, I ~ .11. LY C., ,.\No S l i ~ m s J., , "Theorelirnl Inorgmic Chemistry," (2nd. Ed.), Reinhold Book Gorp., New York. 1969.. nn.176-184:~ PACLIZIG. , L.., . A N D\VIJ.SOS. E. B., "Introdoction to Quantum illevhnnicc," 1lr.Ch.o.wHill Book Co., In?., New Ymk, 1935, pp. 310-.i. (6) SHOCKLEY, W., "Electlm~sand Holes in Semicorrductc,l.s," D. Van Nost~xndCa., Inc., Prinreton, N.J., 1950, pp. 120-34. (7) For a. detailed ireatment and further referenr:es w e , for example, D i . : ~ n m A. , J., reference ( 4 ) ,pp. 2i7-G". (8) S O M M E ~ P I ~ A,, LD Z., Physik, 47, 1 (1928). (9) See far example, C.\STICI.LAN, G. W.,"Ph\-sical Chemislt.y," Addison-Wesley Publishiag Co., I n c , Reading, llnss., 1964, pp. 406-12. (10) KRONIO,11. 1)li L., A N D PI:NNI.:Y,W. G., PIU(..I?"!/, SO?. (London), A130, 499 (1930). (11) For a discussion of sentt,ering processes in solids: PV,IW:Y, E. H., "The Hall Efl'ect and Related Phermnenn," Butterwarths, London, 1960,pp. 13SA3.
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