Fundamental principles of semiconductors

In this paper we shall develop a qualita- tive description of the electronic structure of crystalline solids, and use the model obtained to describe t...
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Edward F. Gurnee The Dow Chemical Company


Midlond, Michigon 48640

Flwdamental Principles of Semiconductors

In this paper we shall develop a qualitative description of the electronic structure of crystalline solids, and use the model obtained to describe the fundamental optical and electric properties of these materials. Consider first an isolated hydrogen atom. It is convenient to sketch the electronic states on a potential energy diagram as indicated in Figure 1. The curved lines represent the potential energy of the electron in the electric field produced by the positive nucleus. The energy levels in this attractive "well" can be obtained exactly by solving the Schriidinger wave equation. The results are discussed in Appendix I. A similar potential well can be drawn for the silicon nucleus, as indicated in Figure 2, where each energy level can contain at most two electrons (of opposite spin). This figure must be interpreted with care, however, since each electron is influenced in a complicated manner by the repulsion of all the other electrons.

Figure 1.

Energy levslr for a hydrogen atom.

Figure 2.

Energy levels for



silicon atom.

/ Jourml of Chemical Education

Thus, while the innermost electrons actually see an attractive potential from a charge of approximately +14, the outer electrons see an attractive potential corresponding more nearly to a charge of +l. This inexactness is of little concern to our discussion-the important point is simply that for isolated atoms the electrons are in distinct energy levels. These ideas must he extended to describe the energy levels in a silicon crystal. In this case we have a tremendous number of electrons, perhaps more than loz0even in a small crystal. The energy levels, however, are derived in essentially the same manner as before-the wave equation is solved using the appropriate attractive potential. These results are discussed in the next section. Band Structure of Crystals ( l,2, 3)

A single crystal of silicon can he considered as one giant molecule that has been prepared by placing silicon atoms in a symmetrical spatial array. The attractive potential for this system of nucleii is sketched in Figure 3. For a real crystal, the periodic potential would have to he drawn in three dimensions, with a more careful consideration to the detailed crystal structure. The salient feature of the diagram-the periodic potential-will not he altered by these refinements. We will now simply state, as a mathematical fact, that the solution of the wave equation for an electron in an attractive periodic potential field results in a series of permissible energy bands rather than energy levels as in the case of isolated atoms. These energy bands are indicated in Figure 3. The required number of electrons (14 for each atom in the crystal) are now added, filling up the lowest energy hands first. The highest energy hand which contains electrons is called the valence band; in this case the valence band is completely filled. The next higher energy band which contains no electrons is referred to as the conduction band; the energy separation between these two bands is the energy gap. It is instructive to consider the origin of the energy bauds in crystals from a diierent standpoint. Instead of fixing the nucleii and then adding the electrons, we

Figure 3. Energy bands in 0 silicon crysfd. the fllled bonds are indicded d the left

The atomic levels t h d form

will imagine the crystal to be prepared by bringing the atoms to their lattice positions from an infinite separation, and see how the energy bands are formed from the energy levels of the individual atoms.

structure. The energy gaps for some of the Group IV elements are listed in Table 1. If the energy gap is zero, the crystal is a metal (Pb); an electrical insulator has a large energy gap (diamond) ; the intermediate values of energy gap are classified as semiconductors. Electrical Conductivity

\ -





Figure 4. Splitting of energy levels for identical atoms. lo1 Two iroloted o e m s lbl two interacting atoms, and 1.1 lhree interorting atoms.

Let us represent two isolated atoms by the potential wells indicated in Figure 4a; only the two lowest energy levels are shown. As the atoms are brought closer together the interaction results in a splitting of the energy levels so that the combined system appears as indicated in Figure 4b. If three identical atoms are brought together, three levels would be formed in the combined system for each level in the isolated atom (Fig. 4c). These are quantum mechanical results that are not obvious or intuitive. By extending this result to a crystal composed of N identical atoms, we infer that each band would contain N closely spaced levels and could therefore hold a maximum of 2N electrons, by reason of the Pauli exclusion principle. It should be kept in mind that we are considering a fictitious one-dimensional crystal. I n a real crystal the periodicity is determined by the primitive cell which may contain more than one atom; we have essentially sssumed one atom per unit cell. To summarize, the electronic structure of a crystal can be represented by energy bands that extend throughout the entire crystal. The hands are widest where the interaction between the atoms is the strongest (outer electrons); the inner electrons are localized in the vicinity of the nucleus in what are essentially unperturbed atomic orbitals. The electrical and optical properties of the crystal are determined by the magnitude of the energy gap, in much the same manner as the absorption spectra of simple molecules are determined by the energy difference between the ground state and an excited state. The magnitude of the energy gap, in turn, depends on the atoms that make up the crystal as well as the crystal Table 1.

Now that we have established the general features of the band structure of crystals, it is possible to inquire into the nature of electronic conductivity. In the presence of an electric field an electron will be accelerated and consequently its energy is increased. Electronic conductivity, therefore, requires a band containing some readily available empty electronic levels. A completely filled band cannot conduct because all available energy levels are filled with two electrons-the maximum permissible. A completely empty band, of course, cannot conduct because it has no charge carriers to be accelerated. We are left with the important conclusion that only partia2lyfilled bands permit electronic conduction. It is now clear that good conductors of electricity (metals) have zero energy gap. It is also apparent that a large energy gap implies that the crystal is an insulator-the applied electric field cannot supply enough energy to raise electrons to the available empty levels of the conduction band. Let us investigate electronic conduction in metals in more detail. The valence band in a sodium crystal composed of N atoms will contain N closely spaced levels resulting from the interactions between the N identical 3s levels of the individual atoms. Each level in the band will contain two electrons, and since N electrons (one from each atom) must be accommodated, the valence band will he only half full. The alkali metals therefore exhibit metallic conduction; the halffull band is equivalent to a zero energy gap. Now consider the alkaline earth metals. Magnesium has two electrons in the 3s atomic level; a crystal of N atoms would have a valence band of N levels and this would be completely filled by the 2N valence electrons of the magnesium atoms. Since a filled band cannot lead to electrical conduction we would expect magnesium to he an insulator-a contradiction with experimental fact.

=-. F' al C



Enerav Gaws for Grouw IV Elements

Internuclear Separation Figure 5. Energy gap and band widths as o function of internuclear separation, showing overlap of bands for strong interoctionr between OtomL

Volume 46, Number 2, Februory 1969



Figure 6.

A representation of a silicon cryslai illustrating the mechanism of charge carrier pmdustion b y light.

The conductivity of magnesium results from an overlapping of the valence and conduction bands. Figure 5 is a sketch to indicate how the atomic levels combine to form bands as the internuclear distance is reduced from infinity (isolated atoms) to the equilibrium value in the crystal. Notice that as the interaction increases, the energy bands become wider, the energy gap decreases, and finally the bands may overlap to give metallic conduction. (A reasonable understanding of the overlapping of energy bands can only be obtained by a more careful consideration of the three-dimensional crystal structure. I n a real crystal we would expect different band structures for different directions in the crystal, since the periodicity of the crystal lattice is a function of the direction being considered (4)). The energy gaps listed in Table 1 for Group IV elements now become more reasonable. These atoms contain an even number of valence electrons and consequently the valence band is completely full. The energy gap becomes smaller as the atomic number increases until finally the conduction and valence bands overlap to give metallic conduction. There seems to be no justification for referring to the small energy gap crystals as semiconductors, for the previous discussion clearly implies that they are nonconductors. If the energy gap is sufficiently small, however, it is possible to have some thermal excitation of electrons from the valence band to the conduction band. Conductivity would then be possible because we would now have a partially filled band-actually two of them. A more important reason for considering the small energy gap crystals as semiconductors is the fact that their band structure can he considerably modified in local regions by the appropriate addition of impurities. This leads to the formation of p and n type semiconductors. Optical Properlies of Silicon (5)

Before proceeding with a detailed discussion of the properties of semiconductors formed by "doping" silicon, it is instructive to consider the optical properties of the pure crystal. It is convenient to represent a silicon crystal by the two-dimensional network indicated in Figure 6. A silicon atom is located a t each intersection; each line segment represents two valence electrons, one furnished by each neighboring atom, as shown in more detail in part of the figure. A photon of light of sufficient energy (1.1 eV) can raise an electron to the conduction band and leave a vacancy ("hole") in the valence band. The crystal now contains two partially filled bands, each of which will contribute to electronic conduction. The conduction due to electrons in the almost empty con82 / Journal of Chemicol Education

Figure 7. A model cornporing hole motion with an oir bubble in o gravitational fleld.

duction band is called n type conduction; that due to electrons in the almost full valence hand is referred to as p type conduction, since it is most conveniently described in terms of the motion of the positive "holes." It is important to remember that the positive charge of a hole is simply the result of a deficiency of electrons in the vicinity of a silicon nucleus. If an electric field is applied to the crystal, the electron and hole will move in opposite directions; the motion of the hole actually involves the concerted motions of many electrons. A useful analogy for hole conduction can be drawn from Figure 7. A long sealed tube is completely filled with liquid (representing electrons) except for one small air bubble near the bottom (representing a hole) which is slowly rising. In reality, however, the fluid is falling under the influence of gravity; but we can describe the motion in terms of the motion of the bubble. To do this, we would simply say that the bubble has a negative mass-it moves in the opposite direction to that of a positive mass in the presence of a gravitational field. In the same manner a hole can be described as an electron of negative mass, or as is more convenient, a positive charge of positive mass. Further development of this analogy has been discussed in the literature (6). Returning to the silicon crystal, it can be seen that silicon is a photoconductor-it becomes a conductor in the presence of light of the appropriate wavelength. It may be asked why the electron and hole don't simply recombine to give off energy either to the crystal lattice or as a photon. It turns out that the coulombic attraction between the hole and electron is quite small due to the high dielectric constant of the crystal, and consequently these charges are available for conduction. Let us investigate the coulombic attraction between the hole and the electron in more detail. The association of a hole and an electron is called an "exciton." It is the analog of a hydrogen a6om in which the hole replaces the proton; the energy states for an excitou can be calculated as indicated in Appendix I, and they are sketched in Figure 8. The energy difference be-


Figure 8.

Energy levels for an exciton.


rier. This is only true for small amounts of doping-the amount of phosphorous can not be so large that it grossly disturbs the crystal structure. Typically, the concentration of doping agent would he in the order of parts per million. Other pentavalent metals, such as arsenic, exhibit the same behavior as phos~horous. I t is now reasonable to inquire into the result of doping silicon with a trivalent metal, say aluminum (Fig. 10). Instead of an extra electron, as with phosphorous, there will be one electron missing from the valence band for each aluminum atom added, i.e., a "hole" is present. The result is a p type semiconduc-

7 Si '14 Figure 9.

Si +I4



Si +I4

Si +I4

Band structure for silicon doped with phosphorous In type remisonductorl.

Si Si + 1 4 +14

Figure 10.

Si +I4

I t is important to emphasize that the concept of a hole and p type conductivity is more than just a convenient representation; many semiconductorl. ex~eriments clearlv indicate the difference in sign of the carriers in n and p type semiconductors. Consider the "Hall experiment" indicated in Figure 11. The electric field E, produces a current density j in the same direction an the electric field. Because of the magnetic field B, a t right angles to this current, the charge carriers will be deflected toward the top surface of the crystal. This is true whether the carriers are positive or negative. A p type semiconductor will therefore produce a positive potential on the top surface (with respect to the bottom surface) while n type materials give a negative potential. I n addition to the sign of the carriers it is possible to determine their mobility as well as the number per unit volume. This is discussed in Appendix 11.


A1 +14 + 1 3

Band itructure for silicon doped with olurninurn lp type

tween the lowest state (n = 1) and the conduction band is so small that excitons are relatively easy to ionize (i.e., dissociate), and consequently they will exist only at relatively low temperatures. Their existence has been confirmed experimentally for some crystals from the absorption spectrum. .P.


n and p Type Semiconductors

It is possible, as indicated earlier, to modify the electrical properties of silicon by the addition of appropriate impurities into the crystal structure. For example, assume that an occasional silicon atom is replaced by a phosphorous atom, as indicated in Figure 9. Phosphorous has five valence electrons and only four are needed to fit phosphorous into the silicon crystal. The remaining electron will he associated with the phosphorous at low temperatures, hut is readily "ionized" a> room temperature to give a free electron in the crystal and a fixed positive phosphorous ion. The relatively small amount of energy needed to ionize the phosphorous is a result of the large dielectric constant of the crystal, and it is readily calculated from the hydrogen-like energy formula, as shown in Appendix I. A more detailed consideration would take into. account that the phosphorous actually perturbs the periodic crystal structure, and consequently the valence hand and conduction bands are distorted in the vicinity of the impurity atom. This is also shown in Figure 9; the net result is that localized "donor" levels are produced that contain the extra electron at low temperatures, but are sufficiently near the conduction hand so that thermal ionization is possible at room temperatures to give a mobile electron. Phosphorous doped silicon will then exhibit electrical conductivity because i t has a partially filled hand-the conduction band. This is referred to as n type silicon (negative charge carriers). The conductivity will be proportional to the amount of phosphorous added, since each phosphorous atom furnishes one charge car-

Si +14

Si +14

p n Junction

The most important use for semiconductors is probably the formation of a p-n junction. Consider a crystal of silicon (Fig. 12) in which one side has been doped with phosphorous and the other side with aluminum. This produces n and p type semiconductors, respectively. The n type region consists of a number of fixed

Sign of c a r r i e r s ........ Force from


H a l l field Figure 11.



Current density j Force f r o m

-1 @


Bz and






Illustration of the "Hall Effect."

Volume 46, Number 2, February 1969



Figure 12. j""~ti0".

A p-n junction and the variation of potential ocmrr the

positive phosphorous ions and an approximately equal number of mobile negative charges. The p type region contains fixed negative ionic sites and mobile holes. The mobile electrons and holes will have a tendency to diffuse throughout the entire crystal. As the diffusion takes place, holes and electrons will neutralize each other in the vicinity of the junction. This results in the establishment of a potential harrier in which the n type region is positive with respect to the p type region; this harrier opposes the tendency for the carriers to diffuse from their original regions. A steady state is soon reached in which the net flow across the junction is zero-the diffusion flux is balanced by the flux due to the electric field a t the barrier. An elementary mathematical discussion of the p-n junction is available in the literature (3,7). It is important to keep a clear distinction between potential and potential energy. The potential is the potential energy per unit positive charge. The potential curve indicated in Figure 12 thus expresses the change in potential energy of a hole; the potential energy of an electron would be represented by the negative of this curve. The operation of most of the semiconductor devices that are so common in our present technology is based on the properties of the p-n junction. Some of these devices will now he briefly described. Semiconductor Devices

Diode. A p-n junction behaves as a rectifier; that is, if an external potential difference is applied across the junction the current is much larger when the p type side is made positive than when the applied potential is reversed. This is readily understandable if we recall that the potential harrier in Figure 12 prevails when the net flow of charge carriers is zero. When the potential barrier is reduced by making the p side more positive, current will flow. If the harrier is increased ( p side more negative) the current remains essentially zero, since the barrier for zero applied potential is already sufficient for zero net current. The action of a diodwcan .be explained by a more 84


Journal o f Chemical Education

intuitive approach. If the p side is made positive and the n side negative, the holes and electrons are forced toward the junction where they combine. The supply of electrons and holes is replenished a t the electrodes. When the potential is reversed, holes and electrons are withdrawn from the region of the junction, and the current soon falls to zero since there is no means for replenishing the supply of carriers. Photovoltaic Cell. Referring again to Figure 12, imagine a photon to he absorbed in the vicinity of the harrier to form an exciton. The hole and electron thus produced are separated by the electric field a t the junction so that the p type region tends to become more positive while the n type region becomes more negative. Furthermore, the current produced can flow through an external circuit and do work; i.e., the junction is a photovoltaic cell. One should avoid confusing a photoconductor with a photovoltaic material. A photoconductor (illustrated by a semiconductor) permits current to flow when light is absorbed; a photovoltaic material (a p-n junction) produces a potential difference under the same conditions. Transistor. The simplest transistor is a combination of two p-n junctions as indicated in Figure 13. For use Emitter ' O s e 7

Figure 13.


A n-p-n transistor.

as an amplifier, the input side (emitter) is biased in the forward direction; the output side (collector) is hiased in the reverse direction. The p region is quite thin and is only slightly doped as compared to t h e n regions; that is, the concentration of charge carriers is much greater in the emitter and collector than in the base. Most of the current between the emitter and the base is therefore carried by electrons. Furthermore, because of the thinness of the base, as well as the positive potential on the collector, most of these electrons from the emitter will pass directly through the base to the collector and return through the load resistor. A more detailed study, involving consideration of both hole and electron currents, shows that under these conditions current amplification is obtained. Another factor that contributes to the amplification ability of the transistor is the fact that the input impedance is low and the output impedance high, due to the manner in which the junctions are hiased. Thus, even if there were no current amplification, voltage amplification would still occur. It is clear that transistors can also he prepared in which the n and p type regions are interchanged. An interesting short historical account of the development of the transistor has been presented by Garrett (8).

Organic "Semiconductors" (9, 10) We nhnll uow cousidcr the pmhlem of electrical conductivity iu orgnuic mnterinls. 1x1 us conaidcr only elcctrouic couduction-iouic conductivity iurolvcn n ninss trnnsfcr : u d sonic electrode rmctiou, nutl is themforc less suitcd for r i ~ n ~ i c o n d u cdevices. t~r Attempts to extend the hnud model to orgnnic conductors hnve only hecn pnrtinlly succesful. The origin of the clcctmnic hnnds is eswutinlly iu the perioclic structure of the crystnl nud the internetions betwecn the electrous. 111n typicnl orgnnic crystnl, nny nnthrnecne, the periodic stmcturc is prelieut, hut the internetions betwect~clectrous on neighboring molecules nrc rclntively wenk. The electrolls tend to be locnlized on the individunl anthrnecne molecules, nnd ns n result the electronic bnnds will be vcry nnrrow; this m u l t s in low mohility of the chnrge cnmcrs. The low mohility in most orgnnic crystnls is confirmed by n vcry smnll, if any, Hall effect. Elcctricnl couductivity will be pmportionnl to the number of chnrgc cnrricrs nnd their mohility. Since we hnve pmlicted the mohility to he low, we cntl form the coucludou thnt orgnnic cryatnls must hnve n lnrge number of cnrricrn in order to exhibit npprccinble conductivity. I n nddition to discullsing conductivity in terms of bnnd theory, n "hopping" model has been used for crystnls in which the electrons tend to bc locnlized. According to this model n chnrgc on one molecule hnn n fiuite prohnhility of trnusferriug to nn ntljneent m o l e culc. In the 11resenee of :ut cxtcrnnl electric field, this results io an electric c u m u t . If the prohnhility for chnrge trnuafer to nn ndjaceut molecule is lnrge, the chnrgc is no longer locnlized nud the h m d model is prohnbly mom nppmprintr. Either of thew two thcorics cnn nccount for couduetivity sfter chnrge cnrricrs nre produced; the pmductiou of the cnrricrs, however, in counitlernhly morc complicnted. Evcu it1 the cnsc of photoconductivity, where eonsidcrnhlc c ~ ~ e r gisysupplied by the photon, chnrge cnrricr productiou mny involve severnl steps thnt nrc probnhly not the snmc for nll molcculnr cry*tnlq. Theoricn for "dnrk currcnt" nrc eveu morc vngue. Ihrther rewnrch iuto the mechnnism of cnrricr protluction will ~tudouhtcdlyfuruish informntioa ou the fennihility of organic .wmiconductor cleviccn. Appendix I The xnlation of the wave eql~aliutlfor a hydroen-like atom r n ~ ~ li lll senergy levrlr given hy the expmint~

Table 2. Gmund Stata Energy and Charge Separation for Some Hydrogen Anologs --

- ~-


Hydmn atom &at energy level





n Type center in x i l i w n

.. -. Rfl(11.X)' E./(ll.xF 2 x ll.8ra 11.~.



Ihc maw ,sf nu

rlrrtru~~, Ihcclielerlrir


in n i l i w t


m,~~.tnttt knr r d w w I* 1 1 3 ,

It is mmve~tietblIII dincum the llsll experimettt i n MKS anits. Ftwthermore, we will mnxider nmgnitartrs cmly i n lhc follow it^ equntion-the xigw and clirrclic~~w have Im~lo h t n i n l from the qwditalive dirn~x~ions i l l the text. If n is the n u m l r r of rn~~cltlrti~~g particlea per twit volume and 0. their drift velw-it? l~t~clcr tho itlfluenre of the elertrir field E,, t h e n t he current drnsit!, is given by By equnting the form on the moving r h n m due to the llnll electric field (E,) ned the applied mngnelir field, we have rE. m,N, where If, in the mapetic induction. If", can he Fnnn thexe t w o equations t h e llnll mnxla~~t. defi~mla.


and the value of n is readily determined from experimental qaaatitia. The mobility, rr. is defined M the drift velc~ityper ttnit cleclrir field. The current density ean therefore he wriltrn rr* and the mohility ix readily ohlni~ned. C W ( I ) KITWL, C., "I~~tr~htrtinn lo %lid State Physin," (2nd 4.)John Wiley & %lux, Inc., New Yurk, 1956. (2) I ~ K K I :A.~ ,J., % l i d S t l'hysin," I'r~nlirrllnll, Inr., Ihglewcxd Clill., Ncw Jewry, 1957. (3) SUITII,If. A,, "Semirt,t~d~~rlon," Cnn,l,riclm LTnivpnity


I'rew 1959.


1luut.-lforer:tw, W., "Atomic Thenry for S t ~ ~ d e n of tr

Volvma 46, Number 2, Fsbrv0t-y 1969 / 8 5