Modern Theory of Solids Paper

7/30/2019 Modern Theory of Solids Paper

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Modern Theory of Solids. IFrederick Seitz and R. P. JohnsonCitation: J. Appl. Phys. 8, 84 (1937); doi: 10.1063/1.1710273View online: http://dx.doi.org/10.1063/1.1710273View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v8/i2Published by theAmerican Institute of Physics.Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/Journal Information: http://jap.aip.org/about/about_the_journalTop downloads: http://jap.aip.org/features/most_downloadedInformation for Authors: http://jap.aip.org/authors

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Dr. Johnson

Modern Theoryof Solids. 1*

FREDERICK SEITZUniversity o fRochester, Rochester, New Yorkan d

R. P. JOHNSONGeneral Electric Company, Schenectady, New York

Introduction

Dr. Seitz

T HIS series of articles is presented with adual purpose: to emphasize the necessityfor using quantum-mechanical concepts in discussing the properties of the solid state; and toreview the progress of

always govern the behavior of these, whetherthey are isolated, are associated in small groupsas atoms or molecules, or are piled together invast numbers to form a macroscopic solid. Atheory of the solid state based on this viewpoint

is certain, obviously,to be a unified theory.\Vhether it is success-

the modern theorybased on these con-cepts.

Until recently, thevanous theories ofdifferen t solids wereconspicuously lackingin unity. To interpretth e distinctive properties of the three solidscopper, diamond, androcksalt, for example,one had to begin withthree widely difterentpictures of their internal constitution.And the attempts tointerrelate these various pictures, or tofit them in with prop-erties of the isolatedatoms, had no verysatisfactory success.

This article is the first of a series ofnonmathematical papers describing themodern theory of solids and its applica-tion to large scale properties of matter.The editor is particularly glad to be ableto present these papers at this time be-cause of the widespread interest ofmetallurgists in the fundamental de-velopments which the physicist has beenable to make in the last few years withregard to the nature of the solid state.Some of the properties to be consideredin this series are cohesion, conductivity,photoelectric effect, thermionic emis-sion, adsorption, fluorescence, photo-gra phic sensitivity and breaking strength.

-Editor.

ful in explaining theproperties of actualsolids will depend onwhether the fundamental quantum lawsreally are sufficient,on whether they arethoroughly known,and on whether one isskillful in deducingtheir large-scale consequences. Agreementbetween predictionand observation is theonly decisive test. Oneof our aims, then, is toshow how the quantum theory has suc-ceeded in interpretingmany of the observedproperties of solids

In the present-day theory, atomic nuclei andelectrons are taken as the prime entities, and theassumption is made that the same quantum laws

which the classical pictures left unexplained.In addition we shall wish to survey, in non

mathematical fashion, the present status ofdevelopment of the theory, to see where progresshas been made and where large gaps in our understanding still exist.

* These articles are based on a talk given by one of us ina symposium on industria! physics at the Rochester meeting of the American Physical Society in June, 1936.84 JOURNAL OF ApPLIED PHYSICS

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About half of the series will be devoted to adiscussion of volume properties, such as cohesion,order of magnitude of electrical conductivity,etc., which are determined by the cooperativeaction of the principal atomic constituents. I t isin this field that the theory has thus far beenmost successful.Surface properties, which are involved in thephotoelectric effect, thermionic emission, adsorption, etc., will be discussed in a part of the

third article. This class of properties is notnearly so well understood as are the volumeproperties.Structure sensitive properties, determined bythe presence of impurities or small flaws in thelattice, have only begun to be the subject oftheoretical investigation. They have tremendoustechnical importance, however-we mention

fluorescence, photographic sensitivity and breaking strength of materials as fields where structuresensitive properties are primarily involved. Inthe final article we shall consider the current explanation of these properties.1. Developments before Quantum Theory

1. Classification of solidsAn acceptable general theory of solids has toexplain a large body of facts accumulated during

the past two hundred years by investigationsnecessarily empirical. These investigations haveled to various divisions of solids into groups.Classification has been based on chemicalconstitution, on crystallographic symmetry, andon tensorial properties such as conductivity,elasticity, and dielectric behavior. Consideringall these characteristics, one finds that solidscan be grouped rather naturally into fiveprominent divisions:

(1) metals,(2) ionic crystals,(3) valence crystals,(4) semiconductors,(5) molecular crystals.

The metals are distinguished by high thermaland electrical conductivity. Most electropositiveelements form solids in this group. Ionic crystalsare marked by good ionic conductivity at hightemperatures, strong infrared absorption spectra,and good cleavage. Practically all the salts, suchFEBRUARY, 1937

as NaCl, MgO, etc., fall into this group. Valencecrystals, such as diamond and carborundum,have poor electronic conductivity, great hardness, and poor cleavage. They are formed of thelighter elements in the middle columns of theperiodic table. Semiconducting crystals arechiefly marked by a feeble electronic conductivity which increases with temperature. CuO,CU02 and ZnO are examples. Semiconductorsusually resemble valence crystals in cleavage,hardness, and lattice structure, but do notalways obey valence rules in combining. Molecular crystals, finally, are the class to 'whichbelong most solid organic compounds. They havelow melting and boiling points, evaporate ingeneral in the form of stable molecules, andbehave, in short, as orderly aggregates of looselybound molecules.

p-FIG. 1. (a) Energy distribution at several temperatures(T1>T.>T.>OOK) of the free electron gas in a metal

(Lorentz theory, Maxwell-Boltzmann statistics). Theaverage energy is (3j2)kT, about 0.04 electron volts atroom temperature. (b) Dependence of electronic energy Eon momentum p.There are, of course, many solids which lieintermediate among these five types. We shalluse this classification, however, in arranging thediscussion.

2. The Lorentz theory of metalsBefore 1900, theoretical work on solids wasmainly phenomenological. Various physical prop

erties were interrelated by the use of thermodynamics and electromagnetic theory. This stageof development is summarized in the books ofDrude,! Voigt,2 and others. The few provisionalattempts to interpret physical properties on thebasis of a picture of the internal constitution ofa solida are interesting chiefly as historicallandmarks.

The first work we shall treat is the theory ofmetaIlic conduction proposed by Lorentz m85

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1905.4 He regarded the valence electrons inmetals as particles of a perfect gas, free to movethroughout the lattice, and distributed inenergy according to Maxwell's law. The numberversus-energy function, and the variation ofenergy with momentum, are shown in Fig. 1.The average energy is (3/2) kT, where k is Boltzmann's constant and T is the absolute temperature. All the electrons have zero energy andzero velocity at T=O. When a potential difference is applied across a metal, the electron gas,on Lorentz' view, moves through the latticeunder the force which the field exerts on theindividual particles, and an electric currentresults. Resistance is interpreted as arising fromelastic collisions of the electrons with the ions ofthe lattice. Lorentz showed that one couldaccount for the magnitude of the resistance andfor its temperature variation in the neighborhood of room temperature by assuming that allthe valence electrons are free and that the meandistance traveled between collisions is approximately one lattice spacing.

This picture of an internal electron gas alsofitted satisfactorily with the observations onelectron emission from heated metals. I f thepotential energy of an electron outside the metalis chosen as zero, the total energy of an internalelectron is E= W+p2/2m, where W is thenegative internal potential energy, and p isthe magnitude of the momentum. At absolutezero, p is zero, and W is the work required toremove the electron from the metal-that is, Wis the work function (Fig. 2). At higher temperatures, those electrons in the Maxwellian distribution which strike the surface, at x = 0, with mo-

(a)

}POSSII3LETHERMIONICELECTRONS

-----------------1- -----w

(6) x - -FIG. 2. Thermionic emission on th e Lorentz theory.Th e few electrons in the Maxwellian distribution (a)reaching the surface with energy E.(E. = px2/2m) greaterthan the work function W (shown in (b) can escape.

86

mentum components Px so large that W+Px2/2mis positive, can escape and be measured as athermionic current. Richardson's5 work showedthat this simple picture is able to describepractically all the main features of thermionicemISSIOn.

The theory faced a major difficulty, however,in assigning a heat capacity of (3/2) k to eachvalence electron. Experimentally, the specificheats of most metals, like other solids, are withina few percent of the value 3R per mole, associatedwith the thermal motion of the lattice ions. Sofar as specific heat is concerned, the valenceelectrons in a metal behave not as if they wereall free, but as if only about one in a hundredwere free. To avoid this difficulty, one mightassume arbitrarily that most of the valenceelectrons are bound to atoms while the remainingfew are free to conduct current, but then themean free path between collisions would haveto be increased from about one lattice spacingto about a hundred, to account for the observedconductivity. In this dilemma, it had to be admitted that the theory, while basically plausible,was, on the whole, unsatisfactory. No promisingnew attack on the problem of metals was madeuntil the advent of the quantum theory.3. The Madelung ionic model

The next important development began withthe attempt of Madelung6 to calculate the binding energy of ionic crystals by treating them aslattice systems of positive and negative pointcharges, and determining the electrostatic energyof such configurations. Born, von Karman, andmany others used this ionic model during a longand fruitful period of computations. 7 Theywere able to interrelate many of the properties ofionic crystals in a semiquantitative way. Thus,the infrared absorption frequencies were connected with the elastic constants by consideringthe dependence of both on the assumed latticeforces. The Madelung model was most successfulin handling the monovalent alkali halide crystals,and failed rather badly with divalent compoundssuch as MgO. The reason for this failure nowappears to be, that the ion-lattice picture is no tstrictly correct in an y case, and becomes lessand less correct as the atbms involved approachthe center of the periodic table.

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4. VJllence and van der Waals bindingThe development of electron valence theories

by Lewis,8 Langmuir,9 and others in the periodfollowing 1916 naturally aroused speculation onthe electronic structure of valence crystals. Thesevalence theories did not attempt a dynamicinterpretation of the electronic bonds, but triedrather to correlate all cases through certainelementary concepts-principally, the "tendencyof atoms to form closed groups"-which haveempirical rather than mechanical justification.How these "atomic tendencies" are related toother characteristics of the atom, such as itsspectrum, was not known. The quantum theoryoffers an insight into these tendencies, but it alsoindicates that the valence rules are not universal,and that each substance requires a special andunfortunately intricate consideration.

Cohesion in semiconductors was left unexplained before 1925. The Madelung model iscertainly inapplicable, and furthermore thesecrystals do not always obey the valence rules.

Binding in molecular crystals was also not toothoroughly understood. It was generally believed10 that the molecules are held together bythe so-called van der Waals forces, which hadoriginally been introduced to explain deviationsin the behavior of real gases from that of aperfect gas. These forces were considered as dueto the interactions of mutually-induced dipole orquadripole moments in neighboring molecules;no exact picture of the dynamical behavior of thevalence electrons was forthcoming.5. The problem of conductivity

I t is worth emphasizing, before we leave thishistorical summary, that in all the classicalpictures of nonconducting crystals the valenceelectrons were fixed to particular atoms ormolecules or particular regions of space betweenparticular neighboring atoms. Lack of conductivity was blamed on the inability of the electrons to move over long distances. On the otherhand, the Lorentz theory, in spite of its defects,strongly supported the view that in metalssome of the electrons are somehow free to roamthroughout the lattice. Table I shows, for contrast, the conductivity, at room temperature, ofseveral metals and of several typical nonconductors. I t is evident that the vast gap in con-FEBRUARY, 1937

TABLE I. Resistivity for some metals and other solids (fromInternational Critical Tables).Solid Toe p ohm-emAg 20 1.6X10 6Al 20 2.8XlO-6Be 20 10.1 X 10-6W 0 5 X 10-6

B 0 1.8 X 106C (diamond) 15 ",1014Si0 2 (crystal) 20 ",1016Mica 10 9 to 10"Paraffin 10 16 to 10 19

ductivity between metals and other solids mustbe due to some fundamental difference. I f thevalence electrons are free in metals and boundin nonconductors, it seems reasonable thatmetals should be strikingly different in otherrespects than in conductivity. Yet, in most oftheir other properties, conductors and nonconductors are very much alike. In particular,one finds both low and high melting points inall the various groups of solids (excepting molecular crystals), and one is thus led to supposethat the cohesive forces in conductors and innonconductors arise from similar sources. Theclarification of this problem of conductivity isone of the most notable advances of the moderntheory.II . The Pauli-Sommerfeld Theory of Metals ll

Very soon after the quantum laws of atomicsystems were discovered, they were applied tothe interpretation of the properties of solids.The first such application was by Pauli, who wasable to explain the feeble paramagnetism ofmetals, which was previously mysterious.

Pauli supposed, as Lorentz had done, that allthe valence electrons in metals are free. He assumed that the energy E of each valence electronis given by the classical relation

(1)where Vo is the potential energy, taken to beconstant throughout the metal, and p is themagnitude of the momentum. Then, in keepingwith the quantum rules for atomic systems, heassumed that not all values of vector momentump are allowed, but only those for which thecomponents px, Pu and P. satisfy the relations

87



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ENERGYSPECTRUM

FIG. 3. Energy distribution at T = 0, when thefree electrons are subjectto the exclusion principle.Th e filled levels extendover several electron volts.

Here n x, ny and nz arearbitrary in t e g e r s .Lx, Ly and Lz are thelengths of the edges ofthe crystal sample alongthe x, y and z directions, respectively, andh is Planck's constantdivided by 27r. Thismeans that the allowed energy values forelectrons in a finitecrystal are not continuous as in the Lorentztheory, but discrete.However. the energydifference between adjacent energy levels turnsout to be so extremelysmall for a crystal ofordinary size (1 cmon edge, say) that the

energy spectrum could be regarded as continuous,except for another quantum rule, the Pauliexclusion principle.

The exclusion principle says that the loweststate (nx=ny=nz=O) can contain at most twoelectrons, one with its spin vector in, say, thepositive z direction, and the other with its spinvector oppositely directed. The same statementis made for each of the levels corresponding toother values of n x, ny and n z; no more than twoelectrons in the entire crystal can have the samevector momentum and the same energy. Atabsolute zero of temperature, in the absence ofexternal fields, only the lowest levels are filled,and we have for a metal with one valence electron per atom the energy-level diagram of Fig. 3:a filled band of levels is immediately overlainwith an empty quasi-continuum of levels. Theaverage energy of the electrons at T=O is notzero, as it was in the Lorentz theory, but isseveral electron volts, and the average momentu m is COl respondingly large. In the filled band,there are two electrons for each set of n's. Thene t magnetic moment of the crystal is zero,since each electron with spin in one direction isbalanced by another with spin in the oppositedirection.

In the presence of an external magnetic field88

of intensity H, the energy of a free electron is nolonger given simply by (1), but is changed by aterm (eh/2mc)H, where eh/2mc is the magneticmoment of the spinning electron, and the signdepends on whether the moment is antiparallelor parallel to the field H. The energy level associated with a given momentum thus splits intotwo levels; the one, corresponding to a momentparallel to the field, decreases by (eh/2mc)H; andthe other, corresponding to antiparallel moment,increases by the same amount (Fig. 4). Thedistribution giving minimum total energy isthen no longer ~ that in which equal numbersof electrons have the two directions of spin.The number having moment parallel to thefield increases, the number with antiparallelmoment decreases, and the crystal as a wholeacquires a magnetic moment.

The susceptibility computed from this modelagrees in order of magnitude (10-6 c.g.s.) with themeasured susceptibilities of the simpler metals.The deviations can be attributed largely toelectron interactions which the picture neglects.

The success of this first application of quantumANT/-PARALLELMOMENr

eliHme

PARALLELMOMENT

FIG. 4. Displacement (greatly exaggerated) of the energystates of different spin in a magnetic field. In equilibriumboth bands are filled to the same height, so there are moreelectrons with magnetic moment parallel to th e field thananti-parallel. -JOURNAL OF ApPLIED PHYSICS



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I

N()IE

E -(a ) p -(/))

FIG. 5. (a) Energy distribution at several temperatures(T 2>T,>To=OOK) of a free electron gas subject to theexclusion principle (Pauli-Sommerfeld theory, FermiDirac statistics). The energy a is several electron volts.(b) Dependence of electronic energy on momentum. Thisis the same as in the Lorentz theory (see Fig. 1).theory led Sommerfeld to reconsider the Lorentztheory of metallic conduction, using the newconcepts. When the particles of a gas are subject to the exclusion principle, the distributionin-energy varies with temperature according tothe Fermi-Dirac function, rather than theMaxwellian. The difference between the two ismarked at low temperatures or when the densityof the gas is high, and the valence-electron gas ina metal is so dense (2.56Xl022 particles/cc forNa, for example) that the Maxwell law is quiteinapplicable. The more correct Fermi-Diracdistribution is shown in Fig. 5, for several temperatures. Increase of temperature affects theT=O distribution only in the neighborhood ofhigher energies, smoothing it out to an exponential decrease. Most of the free electrons haveenergies corresponding to the flat part of thecurves. Though these electrons are movingfreely throughout the lattice, they behave intwo important respects as if they were bound.

First, as is evident from Fig. 5, they do not contribute to the specific heat. When the temperature of the metal is raised, only those electronswhich are already near the top of the energydistribution can gain still more energy. By theexclusion principle, an electron in one of thelower levels cannot increase it s energy slightly,since the level to which it would have to move isalready occupied. Thus, the chief difficulty of!he Lorentz theory is automatically avoided.Second, if an e.m.f. is applied across the metal,the only electrons that can change their momenta-that can contribute to a current in thedirection of the field-are those in the upperfilled levels of the energy spectrum, near theFEBRUARY, 1937

vacant states. As Bloch showed, the probabilitythat an electron will change it s energy by morethan a small fraction of a volt is negligible,unless the field is extremely large. The electronsin the lower levels are unaffected by an externalfield, and carry no net current, since for everyone with momentum p there is another withmomentum -poSommerfeld's equation for conductivity resembles that of Lorentz very closely, with thefew electrons in states bordering the unoccupiedlevels playing the same role as the free electronsin the Lorentz theory.

In the Sommerfeld equation, as in that ofLorentz, appears a parameter having the significance of a mean free path. This distance, to fitthe observed conductivities, must be about ahundred lattice spacings, since only about onepercent of the free electrons assist in conduction. Just as in the Lorentz theory, such a longmean free path had to be assumed arbitrarily.However, when the wave properties of electrons,to be described presently, are taken into account,a detailed consideration shows that a free pathof this order of magnitude is to be expectednaturally. It turns out, at variance with Lorentz'idea, that the conducting electrons do not makecollisions with the lattice ions so long as the ionsare stationary. Bloch found that the electronssuffer inelastic collisions with the ions when theseundergo thermal vibrations. The free path between such collisions is not directly connectedwith the interatomic distance, and is stronglytemperature-dependent at low temperatures.The temperature dependence computed byBloch is shown in Fig. 6. It has the proper orderof magnitude at room temperature, though thelow temperature treatment is not entirelysatisfactory. Later calculations using the sameapproach have improved this aspect.

As for thermionic phenomena, the PauliSommerfeld picture fitted with the observationsjust as completely as did the Lorentz theory.The Fermi-Dirac distribution and the Maxwellian distribution both have the same exponentialshape at high energies, and it is only the highenergy electrons that are emitted thermionically.

The Pauli-Sommerfeld theory thus offered aself-consistent interpretation of most of thegeneral properties of metals. The major question,89



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100

FIG. 6. Temperature dependence of mean free path ofelectrons in a metal, as calculated by Bloch. The dottedline shows the dependence predicted by the Lorentztheory.what is the essential difference between metalsand nonconductors, remained approximately asobscure as it had been in 1925.

III. The Zone Theory of Solids 121. Electrons in a periodic field

During the past few years a general theory ofthe solid state has been developing, which offersafl explanation of the difference between conductors and nonconductors, and which is havingconsiderable success in interpreting many of theother properties of solids. This modern theorydiffers from the Pauli-Sommerfeld theory inincluding two salient facts: the wave nature ofthe electron, and the periodic potential distribution in the solid lattice.

The wave mechanics picture may be summarized in a few words as follows: With everyparticle moving with momentum p is associateda wave of wave-length X=h/p. The square of theamplitude of this wave at an y point in space isproportional to the probability of finding theparticle there. For an electron of mass m, moving90

with constant total energy E in a potentialfield Vex, y, z), the wave function 1/;(x, y, z) ISgiven by the Schrodinger equation

VZif;+(87r2m/hZ) (E- V)Jf=O.The reader is undoubtedly acquainted with thesuccess of the picture in interpreting atomicspectra-quantization rules, selection rules, andthe uncertainty principle fall naturally out of it-and with the electron-diffraction experiments ofDavisson and Germer, Thomson, and others,which further demonstrate its correctness.\Ve shall need to discuss, first, the spectrumof possible energies of wave electrons moving ina periodic potential field, and, second, how theseenergy levels are populated.

The solutions of the Schrodinger equation foran electron in a solid are most convenientlydescribed in terms of the wave number vector, u(in magnitude, u= l/X). This quantity has thedimensions of momentum divided by action. Ifthe electron moves with constant total energyE in a constant potential field Vo, such as theLorentz and Pauli-Sommerfeld theories picturedfor the interior of a metal, then

E - Vo=p2/2m = h2/2mX2 = h2(j2/2m,and p=hu.That is, the energy E varies parabolically with pand therefore parabolically with u (Fig. 7a). In

: \I\I\\\\\

F: 1L.ArTICE IJ

III

FIG. 7. Dependence of E on wave number vector u, for(a) an electron in a uniform potential field, and (b) anelectron in a simple one-dimensional periodic field.an actual solid lattice, however, the potentialfield in which a given electron moves is notuniform, but is a.:complicated function of position. I t depends no t only on where the atomicnuclei are located, but also on how all the otherelectrons move. The potential distribution, the

JOURNAL OF ApPLIED PHYSICS



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wave functions 1/1 of the individual electrons, andthe charge distribution must be calculated alltogether, by methods of approximation whichwe shall not go into. It is fairly obvious, however,that the average of the potential and of the

!II II II I(41:I II ,, I;\b) (a)I I, II I, I Ii raj I: : :1 1 1I I I, I, , ,

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-q;-q;-Od:

of the possible directions of 0' . Fortunately. thequestion can usually be answered by calculatingE(1) for prominent lattice directions, since theenergy function may be expected to have itsextreme behavior in these directions. Fig. 8

POSSIBLf':INSULATOR "\ i

II

// . / , /

I,I I"d, V

I ,1: i l,II,I I,I:,I I/ : II I/

shows the two possibilities. In Fig. 8a, theallowed ranges for thethree directions of 0'overlap, so that noenergy value is absolutely forbidden. InFig. 8b, on the otherhand, the allowed ranges do not completelyoverlap, and a forbidden range ~ ' { i s t s .

FIG. 8. (a) E(u) for three typical directions in a cubic lattice (the (111), (110) an d(100) directions, for example). Note that the gaps occur at different points ua, Ub,uc, and that th e allowed ranges overlap, leaving no completely forbidden range ofenergy. (b) The gaps here are so wide that complete overlapping does not occur, and acertain range of energy is left unallowed.

Fig. 9 shows someequal-energy contoursfor an electron movingin a simple potentialfield periodic in twodimensions. The discontinuities in theE(tT""u y) function occur atthe boundaries of regular polygons centered

electronic charge distribution must both beperiodic, with the symmetry of the lattice.Strutt, Morse, Peierls, Brillouin13 and othersfound that in such a periodic potential field theenergy E of an electron is not proportional tou2, as it is in a uniform field, but depends on theconfiguration of the field and on the direction of(1 as well as on its magnitude. For a given direction of (1 , E(u) generally has such a form as isshown in Fig. 7b. At certain values of u, theenergy E has discontinuities--certain ranges ofenergy are forbidden to the electron, if its wavenumber vector has that direction. For a differentdirection of the vector (1 , these discontinuitieswill, in general, leave different energy rangesforbidden. Unless a forbidden range for onedirection of (1 is completely covered by allowedranges for other directions, there will be a gap inthe energy spectrum, which no electron, what-ever its momentum, can enter. To find ou twhether such overlapping of allowed ranges doesoccur, it is necessary to examine E(1) for eachFEBRUARY, 1937

about U" ,= Fora three-dimensional periodic field the corresponding equal-energy contours are complicated surfaces in a (1 space, with discontinuitieswhich occur at boundaries of regular polyhedracentered about the origin.

FIG. 9. Equal energy contours (drawn light) for electronsin a simple two-dimensional periodic field. Th e discontinui.ties in energy occur at th e boundaries of regular polygons(two squares are shown). For a three-dimensional periodicfield th e discontinuities occur at th e surfaces of regularpolyhedra.91



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In=the Pauli-Sommerfeld theory, the energyspectrum was practically a continuum of discretelevels-discrete, because the momentum components px, py and pz were quantized. Inthe new picture, the discrete levels still appear in fact, the quantization now comes in automatically as a consequence of the wave nature ofthe electron. But it turns out that the energylevels are no longer so uniformly spaced. If wehave a crystal block containing N unit cells, the

-.3a - 2 0 -a 2a .30-0

FIG. 10. Illustrates how th e continuous ranges of theE(u) plot on the extended zone scheme CA) are displacedhorizontally to form th e more compact reduced zonescheme (B).energy spectrum of the electrons has to bedivided into sets of N levels. Within each suchset, the N levels are always so closely spaced that,for all practical purposes, they form a continuousband, as did the levels in the Pauli-Sommerfeldpicture. We shall refer to such a band of Nlevels as a zone. In the one-dimensional case(Fig. 7b), each continuous range of energy between the discontinuities in the E(u) functioncontains the N energy levels comprised in onezone, and a similar correspondence holds in threedimensions. Th e distribution-in-energy of thediscrete levels thus depends directly on the wayin which the energy varies with wave numbervector. When the forbidden ranges of energy, forone direction of ( J , are covered by allowed rangesfor other directions, the zones will overlap.When they are not so covered, the zones willbe separated by energy-gaps in which no allowed levels exist.

We next ask which levels in this zone type ofenergy spectrum are occupied by electrons.

Each level in a zone is doubly degenerate, inthe sense that it can contain two electrons, withspins oppositely directed. The exclusion principlestill restricts the population of a given level tono more than these two electrons, so the electrons are distributed among the levels accordingto the Fermi-Dirac function, as in the Sommerfeld model. At T = 0 the lowest levels are alldoubly occupied, and all the levels above theseare completely empty. At higher temperaturesa few of these higher levels can be occupied byelectrons thermally excited from levels immediately lower.

I t is to be noted that whenever the forbiddenranges in energy are narrow (Fig. Sa) the E(u)curve, except in the immediate vicinity of adiscontinuity, is closely parabolic, like that of afree electron (i.e., an electron in a uniformfield). We may consequently expect the electronto move largely as if the potential field wereuniform instead of periodic. Conversely, we mayexpect less similarity to free electron behaviorwhen the allowed zones are narrow and theunallowed are wide-the situation sketched inFig. Sb is an example.

There is another way of plotting the relationbetween E and (J which does not emphasize thedeviations from the free electron relation, buthas compensating advantages. We shall illustrateit by a one-dimensional example and then statethe result of generalization to the actual threedimensional lattice. The E(u) plot for a onedimensional periodic potential field is shown inFig. lOa. The discontinuities occur at u= a,2a, 3a, na. Now, if we move thecurve between a and 2a horizontally back to therange -a>u>O, move that between -2a and-a horizontally over to the range O>u>a, anddo the same for all the other continuous sectionsbetween discontinuities, we finally get the schemeof Fig. lOb. E is then a many valued function inthe range - a> u>a, and each branch is con-tinuous. Each branch corresponds to one zone,and we may number them consecutively aszone 1, zone 2, etc. We shall refer to this sort ofE(a) plot as the reduced zone scheme, in contrast tothe extended zone scheme discussed previously.

Exactly the same procedure may be carriedthrough in the two- and three-dimensional cases.In two dimensions, the discontinuities occur at

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th e boundaries of regular polygons centeredabout 0'=0. A simple case is shown in Fig. 9.The area between anyone of these polygonsand th e next largest is always equal to the areaof the first, smallest polygon, so that the energyfunction in each such region may be mapped inthis first polygon. E will then be a many valuedfunction in this first polygon, and a consistentplan of mapping gives continuous energy surfaces,like the continuous curves in the one-dimensionalcase. In three dimensions, the discontinuitiesare at boundaries of regular polyhedra centeredabout 0'=0; the volume between succeSSive

FIG. 11 (a) an d (b). Illustrates th e reduced zone plotscorresponding, respectively, tu the extended zone plots ofFig. 8 (a) an d (b). A larger energy range is included, toshow th e characteristic behavior of upper zones. In th elower example, the second band contains three zones.Two of these are degenerate in the a an d b directions, soonly two curves are shown. In th e c direction this degener-acy is no t present. This case is similar to that of LiF.FEBRUARY, 1937

polyhedra is equal to th e volume of the first; an dall the energy functions in these regions may bemapped in this first polyhedron as a se t ofcontinuous functions E1(1T), E 2(1T) , etc., with 0'extending over this polyhedron alone. Whenthis is done, curves of the type shown in Fig. 8may be replaced by the system of Fig. 11.

It turns out that two neighboring energyfunctions are generally not equal at the samevalue of u---that is, they generally do not cross.In particular degenerate cases, however, theymay be equal at 0'=0, or for all values of 0' in agiven direction, such as along a symmetry axisof the lattice. In nondegenerate cases they areoften equal at different values of 0'; it is thiskind of equality that corresponds to th e overlapping of zones in the energy spectrum.

~ n n n n n n ( V l __FIG. 12. Th e lower curve symbolizes th e periodic potential field in a lattice; th e upper curve, th e corresponding

periodic probability distribution of electronic charge.The reduced zone scheme has several ad

vantages, notably compactness. This advantagebecomes apparent if four or five zones are to beplotted. Also, the selection rules for absorption oflight are expressed quite simply in terms of thisscheme.

We emphasize here that we are viewing a solidas a large molecule (containing, for example,8.S X 1022 atoms if the sample is a copper crystal1 em in edge). Each valence electron is in adefinite energy state, just as are the electrons inan isolated single atom, and each valence elec-tron is free to wander throughout the solid justas the electrons in th e isolated atom are free tomove in the potential field of the nucleus andthe other electrons. The probability of finding anelectron at a given point in the lattice dependson th e configuration of the potential field, andhas the periodicity of this field (see Fig. 12)

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Bloch, in a discussion of metals, was the first touse this mode of description.

We have been thinking of each electron asmoving in the a'{;eraged potential field of thenuclei and all the other electrons, and the zonestructure of th e energy spectrum is based on thisnotion. A closer examination shows that th epicture is not strictly correct. Classically, eachelectron has a space around it in which anotherelectron is no t likely to be found, because of themutual electrostatic repulsion. The exclusionprinciple, also, forbids two electrons to approacheach other too closely in space; the magnitude ofthis virtual repulsion depends on how nearly th etwo electrons have the same momenta, and onwhether their spins are parallel or anti-parallel.Both these electronic interactions have analogs in the wave mechanical picture, and bothare neglected in the zone theory as we havedescribed it . It turns out, however, that formany purposes the zone picture is a fairly goodapproximation for all valence electrons both inmetals and in nonconductors (even, for example,for the eight valence electrons in NaCl). Weshall use it with its limitations in mind. Electronsin the closed atomic shells, which are concernedin x-ray spectra, are, on any picture of thesolid, actually bound to particular atoms.We shall generally leave these electrons out ofthe discussion.2. Conductivity in the zone theory

Since th e valence electrons are free to movethrough th e lattice, in nonconductors as well asin metals, we must look to the zone structure ofthe electronic energy levels for an explanationof the difference between conductors and nonconductors. \Ve shall examine several cases indetail.

ferred to the higher states lying closely above.the statistical equilibrium will be disturbed, anda current will flow. This is identically the casediscussed by Sommerfeld. It is thus evident thatthe solids of the monovalent alkalis, with oneatom in the unit cell, must be metals. .

Suppose, instead, that each unit cell of thesolid under consideration contains two valenceelectrons, th e zones still being widely separated.Then, the first zone will be completely filled(Fig. 13c). I f an electric field of ordinary magnitude is applied, th e lowest unfilled levels are nowso far above the uppermost filled levels thatpractically no electrons can be raised to them.The statistical equilibrium is practically undisturbed, and practically no current flows. Simplecomputations show that a field of about 106volts/cm would be required to cause a ~ m e a s u r -

,xi!.!.X'>.:E

('l.X'I.'I::

(a) (b) (c) (d) (e)FIG. 13. (a) Typical energy spectrum for valence elec

trons in a crystal. The shaded regions are allowed, theclear regions are prohibited. (b) The lowest band is halffilled, th e crystal having an odd number of valence electrons per unit cell. The crystal is a good conductor. (c) Th elowest band is completely filled (even number of valenceelectrons per unit cell) and is distant from th e next allowedband. The crystal is a good insulator. (d) The allowedbands overlap. The crystal is a good conductor, whether th eunit cell contains an even or an odd number of valenceelectrons. (e) The lowest band is filled. but the gap aboveit is narrow. Electrons thermally excited to the upper band,and th e "holes" they leave in th e lowest band. allow thecrystal a feeble electronic conductivity at ordinary temperatures.Suppose, first, that the zones do no t overlap, able current. Though th e valence electrons are

so that there is one lowest zone containing N free to move throughout th e lattice, the sublevels, which is separated by several volts from -1_stance is a good insulator. Actually, some subth e next lowest zone (Fig. 13a). Suppose that stances with two valence electrons per unit cell,th e solid under consideration contains one such as Ca, are good conductors. We infer thatvalence electron per unit cell (e.g., an alkali th e zones here are not separated but overlapmetal). The lowest zone will be half-filled (Fig. 13d). The computations show that the(Fig. 13b). Electrons will be moving in all zones actually do overlap in both the alkalinedirections through the lattice, and there will be earth metals and the alkali metals. I t is clearno net electric current. I f a field is applied, some that th e alkalis would be conductors even if theof the electrons of highest energy will be trans- zones were separated, while th e overlapping is94 JOURNAL OF ApPLIED PHYSICS



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essential to the conductivity of the alkali earthmetals. Where the filled and unfilled zones overlap, the substance will be a conductor, whetherthe unit cell contains an even or an odd numberof valence electrons.- All valence and ionic crystals, on the otherhand, have an even number of valence electronsin the unit cell. Since they are good insulators,we conclude that the occupied zones are separated from the unoccupied. To avoid overlapping, the allowed zones must be narrow; onefinds, in fact, that it is just this narrowness ofthe zones that characterizes the nonconductors,as contrasted with the metals.

I f the energy gap is small (Fig. 13e), electrons from a completely filled lower zonemay be thermally excited to the upper zone,so that the solid will acquire a measurableelectronic conductivity, which will increasewith temperature.3. Zones and atomic energy levels

Some interesting correspondences exist between the electronic states in the isolated atomsand the zones in the solid. In many of the simplersolids each zone can be associated with a particular atomic state by tracing the behavior ofthe zones as the interatomic spacing is continually increased while the symmetry of thelattice is preserved. The zones become narrowerand narrower and finally are reduced to theatomic levels.

In the isolated atoms, the energy levels arecompletely narrow, and particular valence electrons are certain to be found in "orbits" aboutthe particular nuclei. Analogously, when theenergy zones in the solid are narrow, and wheneach zone can be traced definitely back to aparent atomic level, any particular valenceelectron is very likely to be found in some sort ofquasiatomic orbit about one or another of thelattice nuclei. To this extent the modern theoryagrees, in certain cases, such as the alkalihalides, with the classical concept that the"atomic character" of the constituent atoms isJess disturbed in the formation of an insulatingsolid than in the formation of a metal. In nocase does it support the classical view thatparticular valence electrons in insulators arebound to particular atoms.FEBRUARY, 1937

I f the formation of the lattice has a pronouncedeffect on the electronic orbits, as it does inpractically all metals, the zones may becomevery broad and may overlap in intricate fashion:the association with atomic states becomes moresymbolic than actual. When the overlappingbrings filled and unfilled levels into adjoiningpositions, the solid is a conductor. Such overlapping becomes very probable when the atomicstates are very dense, as they are in almost allatoms of large atomic number.

In the alkali halides, it proves simplest totrace the zones back to energy states of theisolated ions. With this convention, it is possibleto refer unambiguously to the Cl- 3s and 3pzones, for example. The zones are narrow at theequilibrium spacing, and the atomic (ionic)

-/ 0 zp;;:-~ >. -20 Zs

-30

-4Q z r-3 4 5FIG. 14. Semi-quantitative representation of the zonescheme of diamond, varying with lattice spacing.character of the electronic orbits is largelypreserved.

When the atomic states are well separated,as in carbon, for example, it may happen that asthe lattice spacing is decreased the zones meet, Iintermingle, and then separate into groups(Fig. 14), the components of each group comingfrom various atomic levels. IS Such a solid cannotconduct if the filled levels extend to the edge of aforbidden region. We shall discuss cases such asthese more fully in the next article.

In building up molecular crystals we mayproceed from either of two beginnings. Observingthe electronic energy states of the constituentmolecules as the lattice spacing is decreased, weexpect to find that at the observed intermolecular spacing these have not broadened enough tooverlap. I f we start with the energy states of theatoms, we expect, especially if carbon is present,

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that the zones will have overlapped intricatelyand at the actual interatomic spacing will haveseparated into groups. The final distribution ofzones will, of course, be the same, whicheverstarting point we choose.4. Binding energy

The cohesive energy of a solid-the work required to dissociate one gram-molecule into itsconstituent atoms- is the algebraic sum of theenergies of the electrons in the occupied statesand the energy of nuclear repulsion. The electrons as a group have lower energy in the solidthan in the isolated atoms, and this term therefore favors binding. The nuclear repulsion becomes larger, of course, as the spacing betweenatoms is decreased, so this term opposes binding.The sum is small, compared with the two individual terms; one must therefore be careful incomputing these terms if the sum is to be at allreliable. The nuclear repulsion energy is relatively easy to calculate, as is the energy of theelectrons in the closed shells, which are notgreatly affected by the formation of the solid.I t is the valence electrons, moving in complicated orbits through the lattice, that givetrouble. I f we have the energy spectrum of these(the zone diagram) at the actual lattice spacing,then to a first approximation the total energy ofall the valence electrons is the sum of the energies of the individual electrons in the occupiedlevels. But this must always be corrected to takeinto account the detailed electronic interactionswhich, as we have pointed out, the zone pictureneglects. The correction terms which arise fromthe virtual repulsion introduced by the exclusionprinciple are usually called exchange terms.Those which arise from the fact that two electrons are unlikely to be close together, because oftheir mutual electrostatic repulsion, are calledcorrelation terms. Both terms generally favorbinding, and both are of the same order ofmagnitude as the net binding energy, so theyhave to be included to get a correct value. Whenthey are neglected, the computed cohesiveenergy usually has the wrong sign, so that thesolid appears to be highly unstable at the zero oftemperature-this in face of the fact that alii

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