Relations between the Concentrations of Imperfections in Crystalline Solids

Relations between the Concentrations of Imperfections in Crystalline Solids F. A. KROGERAND H. J. VINK Philips Research Laboratories, N . V . Philips’ Gloeilampenfabrieken, Eindhoven-Netherlands

I. Pure Stoichiometric Compounds.. . . ............................. 310 1. Atomic Disorder.. . . . . . . . . . . . . ............................. 310 2. Electronic Disorder.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 a. Electronic Disorder Not Involving Atomic Imperfections. . . . . . . . . . 315 b. Electronic Disorder Involving Atomic Imperfections. . . . . . . . 3. Complete Equilibrium Involving Both Electronic and Atomic D . . . . . . . . . . . . . 324 a. Equilibrium a t High Temperatures. ........ b. Processes Taking Place during Cooling. . . . . . . . . . . . . . . . . . . . . . . . 325 4. Migration of Atomic Imperfections; Ionic Conductivity. . . . . . . . . . . . . . 326 5. Electronic Conductivity Arising from Lattice Defects. . . . . . . . . . . . . . . . 328 11. Nonstoichiometric Compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 6. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Reactions Involving Crystal and Vapor.. . . . . . . . . . . . 8. Complete Equilibrium Crystal-Vappr for a Simple Case with Frenkel Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. A Method for Obtaining an Approximate Solution of Multi-Equation Relations Such as Those of Section 8 . . . . . . . . . . 10. Complete Equilibrium between Crystal and Vapor for a Crystal Containing Completely Ionized Frenkel Defects.. .......................... 342 11. A More Complicated Case of Schottky-Wagner Disorder. . . 12. Equilibrium between Crystal and Vapor for a Crystal M X structure Disorder, . , ...................... . . . . . . . . . . . . . . . . 347 a. Pure Antistructure Disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 b. Combination of Antistructure and Schottky-Wagner Disorder. . . . . . 349 13. Cooling; the State of the Crystals at Low Temperatures 14. The Deviation from the Simple Stoichiometric Ratio as a Function of the Atmosphere ....................... ...................... 357 15. The Position of the Fermi Level.. . . . . . . . ...................... 360 ................................ 362 16. Practical Limitations. ....... ................................ 363 17. Comparison with Experiment 18. The Effect of Heating in a Stream of Inert Gas.. . . . . . . . . . . . . . . . . . . . 369 19. Equilibria Vapor-Liquid and Solid-Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 20. Surface Layers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 .............................. 373 111. Crystals Containing Foreign Atoms. 21. Introduction.. . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . 373 307

.

308

F. A. KROGER

AND H. J. VINK

22. The Sites Occupied by Foreign Atoms . . . . . . . . 374 23. Energy Levels Due to Foreign Atoms.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 24. General Remarks Concerning Equilibria Solid-Vapor Containing Foreign Atoms. .......................... 25. Equilibrium between Crystal and Vapor for Binary Containing Foreign Atoms with Deviating Valence a t Normal Lattice Sites ........................................................... 379 a. One Level per Imperfection.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 b. Several Levels per Imperfection.. . . . . . . . . . . . . . . . . . c. Crystals Containing On d. Practical Limitations.. . . . . . . . . . . . . . . . 391 26. The Influence of Foreign A rystals . . . . . . . . . 392 a. Ionic Conductivity, Diffusion, and Dielectric Loss. . . . . . . . . . . . . . . . . 392 b. Electronic Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c. Optical Properties.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d. Magnetic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . 405 e. Deviations from the Simple Stoichiometric Ratio. . . . . . . . . . . . . . . . . 406 27. Comparison of Compounds and Monatomic Crystals.. . . . . . . . . . . . . . . . 407 28. Crystals Containing Two Types of Foreign Atoms.. . . . . . . . 29. Controlling the Charge of Foreign Atoms.. . . . . . . . . . . . . . . . IV. Solubility Relations.. .. . . . . . . . . . . . . 416 30. Introduction.. ..... 31. Equilibrium between Crystal and Vapor.. . . . . . . . . . . . . . . . . . . . . . . . . . 418 419 a. CdS Containing Chlorine Alone.. . . . . . . . . . . . . . . . . . . . . . 422 b. CdS Containing Silver Alone.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c. CdS Containing Chlorine as Well as Silver . . . . . . . . . . . . 422 d. Practical Limitati Foreign Elements ..................................... 425 32. Comparison with Ex t . . . . .‘. . . . . . . . . . . . . . . . . . . 427 428 33. Equilibrium between Solid and Liquid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Comparison of Various Models Used in Describing a Crystal. . . . . . . . . . . . . 431 34. A Justification of the Atomic Notation; The Problem of Determining the Equilibrium Constants ............................................ 431 432 35. Comparison of the Atomic and Ionic Notation.. . . . . . . . . . . . . . . . . . . . .

Many properties of crystalline solids, such as the electronic or ionic conductivity, the color, the luminescence, and the magnetic susceptibility, are determined by the presence of imperfections. Generally, six types of primary imperfections are distinguished, namely phonons, electrons and holes, excitons, vacant lattice sites and interstitial atoms or ions, foreign atoms or ions in either interstitial or substitutional positions, and dislocations.’ I n addition atoms of the base crystal may be present at lattice sites normally occupied by other atoms. Five types of primary imperfections, namely electrons and holes, vacant lattice sites, interstitials, misplaced lattice atoms, and foreign 1

F. Seita, in “Imperfections in Nearly Perfect Crystals’’ (W. Shockley, ed.), p. 3. Wiley. New York, 1952.

IMPERFECTIONS IN CRYSTALLINE SOLIDS

309

atoms, will be dealt with in this paper. Phonons will not be mentioned explicitly, although their presence will be assumed in order to account both for thermal disorder and for the establishment of thermal equilibrium. The concentrations of the various imperfections are not independent of each other. It is the purpose of this paper to show the type of relations existing between the concentrations of the imperfections. Given an understanding of these relations, one may hope to regulate the concentration of various imperfections in a crystal. The principle of the method to be outlined below was contained in papers by Wagner and Although this principle was published more than twenty years ago, its application has been restricted so far to a comparatively limited field, and has not been used to its full potentialities. The reason may be that the theory was given in an analytical form which makes it difficult to appreciate all its implications. In this paper, a new treatment of these problems will be given, making use of a graphical representation. This treatment, together with the use of a band scheme for the electronic energy levels, greatly facilitates the application of the theory and the deduction of conclusions from it. This treatment was first applied in a paper on CdS;gJoa simple mathematical technique of handling the problem has been given by Brouwer.11 In the present paper, apart from a few exceptions, binary nonmetallic compounds of the formula M X will be considered almost exclusively. Here M indicates an element of a more electropositive character (metal) and X an element of a more electronegative character. In Section I pure compounds will be considered, whereas compounds having deviations from the simple stoichiometric ratio will be treated in Section 11. The influence of foreign atoms is discussed in Section 111. The consequences of the statistical interaction between imperfections in determining the solubility of foreign atoms is the subject of Section IV. Throughout the article, the statistical interaction between imperfections

*

C. Wagner and W. Schottky, 2. physik. Chem. B11, 163 (1931). C. Wagner, 2. physik. Chem. Bodenstein Festband, p. 177 (1931). C. Wagner, Z. physik. Chem. B22, 181 (1933). C. Wagner, 2. EZektrochem. 39, 543 (1933). * W. Schottky, Z. physik. Chem. B29, 335 (1935). W. Schottky, 2. EZektrochem. 46, 33 (1939). W. Schottky, i n “Halbleiterprobleme I” (W. Schottky, ed.), p. 139. Vieweg, Braunschweig, 1954. 0 F. A. Kroger, H. J. Vink, and J. van den Boomgaard, Z. physik. Chem. 203,l (1954); see also F. A. Kroger and H. J. Vink, i n “Halbleiterprobleme I ” (W. Schottky, ed.), p. 128. Vieweg, Braunschweig, 1954. 10 F. A. Xroger and H. J. Vink, Physica 20,950 (1954). 11 G. Brouwer, Philips Research Repts. 9, 366 (1954). a

’ ’

310

F. A . KROGER AND H. J. VINK

is expressed in the simple form known as the law of mass action. The justification for this procedure and the limits of its applicability are discussed in Section 2a. The general arguments given in the article are independent of the type of bonding. For this reason, all reactions and relations have been formulated in terms of atoms. Section V contains a justification of the use of this type of formulation; further, it shows how the atomic formulation may be translated into a formulation corresponding to a purely ionic model. The material of this article is presented in rather a dogmatic way. A more argumented presentation cannot be given because of the scarcity of suitable experimental data. The theory nevertheless deserves the atterition of those who are interested in the extrinsic properties of crystals, particularly of compound crystals, for one cannot hope that the intricate relations between the chemistry and the physics of such crystals can ever be fully understood without the aid of a theory of this type. In the few cases in which it has been possible thus far to compare the theory with experiment, the results indeed support this view. Acknowledgment

We wish to thank Dr. E. J. W. Verwey for his valuable criticisms and also our colleagues D. Polder, F. H. Stieltjes, J. H. v. Santen, H. J. G. Meyer, W. v. Cool, W. Hoogenstraaten, and J. Bloem for their stimulating discussions.

I. Pure Stoichiometric Compounds120

1. ATOMICDISORDER

The state of lowest energy of a crystal is that in which the constituents are arranged according to a purely periodic pattern. This state prevails a t low temperatures. At higher temperatures, states with a higher energy may occur if the effect of the increase in energy on the free energy is compensated by an increase in entropy. Therefore, at higher temperatures less ideal crystals are formed (atomic disorder). I n a pure compound M X , five primary types of atomic disorder can be distinguished :2*3*4 6s12b

(1) Interstitial sites occupied by atoms M ( = Mi), together with an equal concentration of vacancies, that is sites where an M atom has been removed (V,). (2) Interstitial sites occupied by atoms X ( = Xi), together with an equaI concentration of sites where an X atom is missing (Vx). In this paper the term “stoichiometric” is used to indicate that the constituents of a compound are present in a simple stoichiometric ratio. 1*b J. Frenkel, 2.Physik 36, 652 (1926). 12a

IMPERFECTIONS I N CRYSTALLINE SOLIDS

311

(3) Vacant M sites ( V M ) , together with a n equal concentration of vacant X sites ( V X ) . (4) Interstitial atoms Mi together with a n equal concentration of interstitial atoms Xi. (5) Part of the atoms M occupy X positions ( M x ) and a n equal number of X atoms occupy M positions (XM). Combinations of these five types are also possible. The first and second type, first mentioned b y Frenkel,lZbis generally called the Frenkel disorder. The third type is usually called the Xchottky-Wagner disorder. The fourth is thought t o be very rare and will not be considered further. The fifth type might be called antistructure disorder. I n each type of disorder, the two compensating imperfections may be distributed a t random or may be present a t adjacent sites forming associations. I n the following, only those associations which consist of a pair of imperfections ( V M M J , (VXXJ, (VMVx), and (MxXM) will be considered. More intricate cases can be treated along the same lines. The formation of the various types of disorder b y thermal motion can be described by means of the following quasi-chemical reactions. I n these equations, the atoms of the crystal occupying normal sites are denoted by M and X without suffix. Imperfections are designated by a symbol indicating the nature of the imperfection, that is whether it is a n atom or a vacancy, and by a suffix indicating its position. The quantity 6 is the molar concentration of the imperfection, which is assumed to be small compared t o one. Although the same symbol 6 has been used in all the equations, it is not intended to suggest that all these concentrations are the same. Equations of type (a) describe the formation of 6 associated imperfections, whereas equations of type (b) describe the formation of 26 isolated imperfections. For Frenkel type of disorder, one has

312

F. A . KROGER AND H. J. VINK

In the formulations (1.1)to (11.4) describing the occupation by an atom M or X of a site in the interlattice, the decrease of the concentration of unoccupied interstitial sites is not explicitly accounted for. Similarly the formulations (1.5) and (1.6) do not include a term representing the expansion of the crystal (or, in other words, the volume of “vacuum” dissolved in the crystal). The omission of these terms is justified by the fact that it does not influence the results obtained noticeably for the purposes in which they are used in this paper (assuming 6 << 1). From the equations given above, relations between the concentrations of the imperfections may be obtained in the following way. According to the laws of chemical t h e r r n o d y n a m i c ~ ,the ~ ~ sum ~ ~ ~ of ~ ~the thermodynamic potentials of the reaction partners on the left-hand side of the reaction equations equals that of the reaction partners a t the right-hand side under equilibrium conditions. Or, taking the terms on the right-hand side positive and those on the left negative,

2 nipi = 0. i

Here ni is the number of atoms i taking part in the reaction, whereas is the thermodynamic potential of the reaction partner i: pi =

wi - T s ~ .

pc

(1.10)

The partial enthalpy and the partial entropy of i are wi and si, respectively, and T is the temperature. The thermodynamic potential pa is a function of the concentration xi of the component i. On the basis of elementary statistics it can be shown that = ( d o

+ RT In xi

(1.11)

as long as the atoms are statistically independent. Here ( p i ) o is the thermodynamic potential of i under standard conditions (T = 25”C, xi = l), xi is the concentration expressed as a mole fraction, and R is the universal gas constant. It follows from (1.9) and (1.11)that Zni In xi = -zni(pJo/RT

or

nzini= exp (-Zni(pi)o/RT)

(1.12) (1.13)

in which IIxini st,ands for the product of terms xini with different values of i; ni is positive for the reaction partners on the right-hand side of the reaction equation, and negative for those a t the left. This e,quation 13 14

W. Schottky, H. Ulich, and C. Wagner, “Thermodynamik.” Springer, Berlin, 1929. H. Ulich, “ Chemische Thermodynamik.” Steinkopf, Dresden, 1930.

IMPERFECTIONS I N CRYSTALLINE SOLIDS

313

indicates that there is a relation between the concentrations of the reaction partners: IIxini = K (1.14) with (1.15) K = exp ( Zn&) o/RT).

-

The relation (1.14) is known as the law of mass action. Inserting (1.10) into (1.15), one obtains

K

= exp

(Zn,(si)o/R) exp ( - 2 n i ( ~ ~ ) ~ / R = TC) exp (-2ni(zoi)o/RT). (1.16)

I n dealing with reactions in the solid such as (1.1)-(1.8), the densities of imperfections are usually so low that the concentrations of the normal components of the crystal may be assumed to be constant (6 << 1). Therefore, only the concentrations of the imperfections appear in the law of mass action. The constant K contains of course a contribution arising from pi = (pi)o, the chemical potential of the components of the crystal. I n the following, the concentrations will be expressed in numbers per cm3instead of in mole fractions. This affects the value of the constants K , but leaves the form of the expression (1.14) unaltered. Equation (1.11) and therefore also (1.14) hold only a t low concentrations. At higher densities the concentrations have to be replaced by the corresponding activities, which may be calculated by methods16-18similar to that proposed by Debye and Hiickel for liquid solutions of electrolytes. Applying the law of mass action to the reaction equation (1.1) and taking 6 << 1, one gets for the equilibrium involving associated Frenkel defects

[ ( V M M ~=) ]K W M = ) CCVM) exp ( - W ( V M ~ R T ) (1.17) and similar formulas for (1.3), (1.5), and (1.7). For the equilibrium involving single Frenkel defects, one obtains from (1.2) [VM][Mi]= K V , M K p = C p exp ( - W p / R T ) (1.18) and for the equilibrium involving single Schottky-Wagner defects, one obtains similarly from (1.6) (1.19) [VM][Vx] = Kv,v = Kg = Cs exp ( - W s / R T ) . I n these formulas [ ] indicates the concentration. Different from what has been said above about x, from now on the concentration will be J. Teltow, Ann. Physik 6, 63, 71 (1949). B. Lidiard, Phys. Rev. 94, 29 (1954). A. B. Lidiard, Repts. Conf. on Dejects in Crystalline Solids, Univ. Bristol 1964,p. 283, 1955. I. Ebert and J. Teltow, Ann. Ph&k 16, 268 (1955).

16A. 1’

Is

314

AND H. J. VINK

F. A. KROGER

expressed in numbers per em3, the constants assuming values in accordance herewith. The W are the energies involved in the various reactions and the C are constants determined by the change in entropy (see 1.16). It is seen that the concentration is a constant for the associated pairs of imperfections at a given temperature, decreasing exponentially with inversed temperature. The product of the concentrations of the compensating imperfections is a constant at a fixed temperature for single imperfections and varies exponentially with temperature. The concentrations of associated pairs and of isolated imperfections are connected by a reaction relation of the type

* +

(1.20) (VMMi) V M Mi which is comparable to the dissociation of molecules in a gas or a liquid. Application of the law of mass action to this reaction leads to (1.21) being the dissociation energy of (1.20). Comparison with (1.17) and (1.18) shows that

W D

C D = CF/C(VM)

and

W D =

Wp -

W(VM).

(1.22)

Similar relations exist for V x , X i ; V x , V M ; and X M , M I . When W D > 0, the fraction of imperfections that is present in the dissociated form is the greater the higher the temperature, and the lower the concentration of imperfections. The concentrations of vacancies and interstitials have actually been found to be rather small, the highest values observed (near the melting point) being in the range from 1018-1019defects emw3 ( g 10-4-10-3 defect per mole) in compounds like AgBr, NaC1, and PbS. 19--23b For this reason associated imperfections will be neglected in the following when considering equilibria a t high temperatures. It is advisable, however, to be aware of the possibility of such association. The concentrations of the compensating imperfections cannot remain equal of course if the various types of disorder are present simultaneously. Thus, when Frenkel disorder involving V M and M i and Schottky disorder involving V M and V Xare present together, the concentration of V Mwill S. W. Kurnick, J . Chem. Phys. 20, 218 (1952). F. Seitz, Revs. Mod. Phys. 26, 7 (1954). 2 1 J. Bloem, Philips Research Repts. 11, 273 (1956). 2 2 C. Wagner and P. Hantelmann, J . Chem. Phys. 18, 72 (1950). 230 H. Kelting and H. Witt, 2. Physik 126, 697 (1949). 23b Strictly speaking these figures refer to charged imperfections, although uncharged imperfections are considered in this section. The order of magnitude of the concentrations of imperfections in cases in which they are predominantly neutral will however be the same. Charged imperfections are treated in the next section. 19

20

IMPERFECTIONS I N CRYSTALLINE SOLIDS

315

be greater than is compensated b y either V x or Mi alone. In fact, if the crystal is t o remain stoichiometric (ionization, as treated in Section 2b, will be neglected here) : (1.23) [VMI = [VXI [Mil.

+

The concentrations of V,, Mi, and V x can be found by solving Eqs. (1.18), (1.19), and (1.23). It is easily seen th a t [V,] is smaller than the sum of the values which we would obtain if either Frenkel or Schottky disorder were present alone. This is an elementary and primary example of the statistical interaction between various types of disorder. Similar effects occur when other types of disorder are present together. The energies of formation of the various combinations of imperfections corresponding t o the different primary types of disorder often vary widely. Since these energies determine the equilibrium constants, only one type of disorder usually is predominant. Thus Schottky-Wagner defects prevail a t all temperatures in the alkali halidesz0 and PbSzl. However, in AgBr19sz4J5and AgClZ6sz7 two types of disorder are present, the Frenkel type prevailing a t temperatures below 600"K, the Schottky-Wagner type becoming predominant a t higher temperatures. Antistructure disorder may be expected in cases in which the components of the crystal have nearly the same electronegativity and about the same atomic radius. There are indications th a t it occurs in compounds such as CdSb,z8,z9 ZnSb, MgzSn, and BizTes (see Section 17). 2. ELECTRONIC DISORDER

All electrons in an insulating crystal are in the state of lowest energy a t low temperatures. At higher temperatures when the entropy plays a role, a part of the electrons occupy higher states under equilibrium conditions. This may be considered a type of disorder parallel to atomic disorder and may be called electronic disorder. Two types of excited electronic states can be distinguished: (a) states not involving atomic imperfections; (b) states involving atomic imperfections.

a. Electronic Disorder N o t Involving A t o m i c Imperfections Let us first consider electronic disorder of type (a). The electrons in the ground state of a nonconductor are not free to move under the 0. Stasiw and J. Teltow, Z. Naturforsch. 6a, 363 (1951). 0. Stasiw, Z. Physik 130, 39 (1951). ZeF. Seitz, Revs. Mod. Phys. 23, 328 (1951). 27 K. Kobayashi, Phys. Rev. 86, 150 (1952). 28 E. Justi and G. Lautz, Z. Naturforsch. 7a, 191 (1952). 29 E. Justi and G. Lauta, Abhandl. braunschweig. wiss. Ges. 4, 107 (1952).

z4 26

-

316

F. A. KROGER

AND H. J. VINK

influence of an applied electric field. Although there also are low-lying excited states of the crystal in which the electrons are unable to move in an electric field, namely exciton states, most of the higher excited states describe situations in which some of the valence electrons can move in an applied field. The excitation process can usually be described in terms of the excitation of individual electrons. The excited valence electrons which may move in a field usually act as particles with the normal negative electronic charge. The unexcited valence electrons, which now may also move in a field to a degree determined by the number of the excited electrons, customarily behave like particles having a positive electronic charge. They are called hole^.^^,^^^,^^^ Free electrons and holes generated by thermal motion recombine continuously. The formation and the annihilation of the quasi-free charge carriers can be described by the following equilibrium equation : ground state $ free electron

+ free hole - Ei.

(2.1)

Applying the law of mass action to this reaction, one finds the following relation between the concentrations of the electrons (n)and the holes ( p ) :

-

n p

=

Ki

=

Ci exp ( - E J R T ) .

(2.2)

Here E; is the excitation energy, and Ci a constant determined by the change in entropy involved in the reaction (2.1). When discussing free electrons and holes and presenting Eq. (2.2), we have not had to make a detailed commitment concerning the mechanism by which the motion through the crystal actually takes place. I n most known cases., the free electrons and holes move through the crystal by means of quantum-mechanical tunneling, that is without the need for an activation energy. This description is usually called the itinerant electron picture. The energy states describing the motion of free electrons and holes through the crystal lie in bands. These bands are indicated in Fig. 1, the 30

F. Seitz, “The Modern Theory of Solids,” p. 271. McGraw-Hill, New York, 1940. W. Schockley, “Electrons and Holes in Semiconductors.” Van Nostrand, New

310

31b

York, 1950. Using the notions of free electrons and holes, it is possible t o say something more about the exciton states a t this point. Such states can be viewed as composed of associated electron-hole pairs. Thus, just as in the case of atomic imperfections (Section l),association-dissociationequilibria involving the electronic imperfections are also possible. The excitons are present in appreciable concentrations only a t high temperature; they disappear upon cooling. Since they influence the other equilibria in no way, they shall not be considered more in the following.

IMPERFECTIONS I N CRYSTALLINE SOLIDS

317

lower one representing the states of the holes, the upper one the states of the electrons. The gap between the bands corresponds to the lowest thermal energy required to excite an electron and thereby form one free electron and one hole. This energy is the Ei of formula (2.2). Often the upper band is called the conduction band and the lower the valence band. The value of Ci appearing in (2.2) can be calculated readily if it is assumed that the relation between the momentum k and the kinetic

FIG.1. Electronic energy band scheme of a crystal omitting imperfections.

-

energy Eklnfor the free electrons and holes has the same form as for an electron moving in the free space (&in k2).One 0 b t a i n s a ~ ~ 3 ~ ~ :

in which k is the Boltzmann constant, h is Planck’s constant, and me and mh are the effective masses of free electrons and holes. In more complicated band structures, as, for example, those for Ge and Si,3233somewhat different results are obtained. In principle, another, fundamentally different transport mechanism involving electrons may also exist. This would occur if simple tunneling between neighboring atoms in equilibrium positions is negligible. The motion of free electrons and holes then requires vibration of the nuclei, tunneling becoming possible if adjacent atoms approach each other closely. The motion would then involve an activation energy. This mechanism may be called the jumping electron mechanism. Equation (2.2) would still hold for it; however, (2.3) would no longer apply. 81

aa

F. Herman, Physica 20, 801 (1964). C. Kittel, Physica 20, 829 (1954).

318

F. A. KROGER AND H. J. VINK

It is possible that considerations of this type must be applied to account for the complicated behavior of Fe304.34-39 b. Electronic Disorder Involving Atomic Imperfections

Imperfections may make three kinds of additional excitation processes possible, namely :

(1) excitation within the center formed by the imperfection, giving to an excited imperfection; (2) excitation in which electrons associated with the imperfections brought into states in which they can move freely through crystal; and (3) excitation in which normal valence electrons of the crystal transferred to the imperfections.

rise are the are

In the following, the first kind of excitation will be omitted. Excitation 2 can be indicated in a band picture as shown in Fig. 1 by introducing local occupied levels. The excitation corresponds to transitions between the local level and the conduction band. Excitation 3 can be indicated similarly by introducing local empty levels. The excitation corresponds to transitions between the valence band and the local level; free holes are formed in the valence band. One may ask whether an imperfection gives rise to an occupied or an unoccupied level, or both, and where the levels lie. The answers to such questions depend on the specific character of the imperfection. Precise data concerning the position of the levels can only be obtained experimentally. In order to interpret the experimental results, however, it is necessary to have some notion concerning the expected position of the levels. I n the following, arguments about the position of levels due to the imperfections met in Section 1 will be given. Let us first consider the interstitial atoms of type M . An interstitial atom M ican be ionized once according to the relation

E. J. W. Verwey and P. W. Haayman, Physica 8, 979 (1941). E. J. W. Verwey and E. L. Heilmann, J . Chem. Phys. 16, 174 (1947). 36 E. J. W. Verwey, P. W. Haayman, and F. C. Romeyn, J . Chem. Phys. 16,181 (1947). 37 C. A. Domenicali, Phys. Rev. 78, 458 (1950). 38 D. 0. Smith, in “Progress Report XI,” p. 53. Laboratory for Insulation Research, Massachusetts Institute of Technology, Cambridge. O.N.R. Contracts N50ri-07801; N5ori-07858, May, 1952. 39 B. A. Calhoun, Phys. Rev. 94, 1577 (1954). 34 35

IMPERFECTIONS IN CRYSTALLINE SOLIDS

319

and still further in accordance with the relation

Mi’ + Mi’‘

+ 0 - Ez.

(2.5)

Here Q indicates a free electron, the heavy dots indicate positive charges remaining on the center, whereas El and Ez are the ionization energies. A reliable estimate of the order of magnitude of these ionization energies can often be obtained by the following considerations. Since the dielectric constant of the crystal is usually high, the electron that is dissociated in accordance with the relation (2.4) is situated a t a rather large distance from the positive core Mi‘. The dissociation process therefore can be considered to an approximation as the ionization of a hydrogen atom embedded in a medium having the dielectric constant E of the

13‘5e~.4~341*42 Similarly Ez may be crystal. In this way one obtains El = €2

found using a model resembling a helium ion. According t o this picture, Ez 3 4E1.It must be said, however, that the energies estimated in this way are very rough. For a slightly better approximation, see S i m p ~ o n , ~ ~ and L e h ~ v e c .It~ ~is conceivable that energy is gained by splitting off an electron. This can be expressed by taking El < 0. It is also possible, a t least in principle, that the atom Mi exhibits a degree of electro-affinity for free electrons according to the relation

Mi

+ Q+

Mi’

+ E,

(2.6)

in which the sign ‘ stands for a negative charge on the center, whereas E, is the binding energy of the electron. Figure 2 gives a possible scheme of the energy levels caused by the centers belonging to the Mi “family.” The energy levels are represented by solid lines if the energy level is occupied by an electron. Empty levels are represented by dashed lines. A level is designated by the symbol of the imperfection with which it is associated if the particular level is occupied by an electron. Although the relative sequence of the various levels in the vertical direction is as shown in the figure, stress must be placed on the fact that the absolute positions of the energy levels in Fig. 2 are quite arbitrary, both with respect to one another and with respect to the conduction band. It might occur, for instance, that it is more difficult t o generate free electrons from Mi’ centers than from the conF. Mott and R. W. Gurney, “Electronic Processes in Ionic Crystals,’zp. 80, 2nd ed. Oxford U.P., New York, 1950. 4 1 H. A. Bethe, Mass. Inst. Technol. Radiation Lab. Rept. No. 43-12, 1942. 42 H. M. James and K. Lark-Horowita, Z . physik. Chem. 198, 107 (1951). 43 J. H. Simpson, Proc. Roy. SOC. A197, 269 (1949). 44 S. Pekar, J . Phys. U.S.S.R. 10, 341 (1946). 4 5 s . Pekar, J . Exptl. Theoret. Phys. U.S.S.R. 18, 481 (1948). 46 K. Lehovec, Phys. Rev. 92, 253 (1953). 40N.

320

F. A. KROGER AND H. J. VINK

stituents of the base crystal. In this case, the M< level would lie under the top of the valence band.

FIQ.2. A possible electronic energy level scheme of a crystal of composition M X containing interstitial M atoms.

Let us now consider interstitial X atoms. Since they are the more electronegative component of the crystal, the tendency to attract an electron probably will be greater for Xi atoms than for those of type Mi. In fact, it may even happen that an Xi atom can bind two electrons. Such binding of electrons can be represented by the following reactions: and Of course, it is also possible that the X , atom can be ionized according to the relation 0 - Er,. X t + Xi' (2.9)

+

Figure 3 gives a possible scheme of energy levels associated with the centers belonging to the Xi family. Again each level is designated by the symbol of the imperfection which obtains if that particular level is occupied by an electron. As can be seen by comparing Figs. 2 and 3, the difference in electronegativity of M and X is indicated by a difference in the position of levels, all Mi levels lying higher than the corresponding Xi levels. In principle, the ionization of Mi and Xi can also be described by reactions in which holes, rather than free electrons take part. These reactions are (2.10) @ - Eg Xi-iX: X [ - + X [ ' + @ - Ee (2.11) X ; - t X { + @ - E, (2.12)

+

IMPERFECTIONS IN CRYSTALLINE SOLIDS

32 1

for the Xi levels. Here @ indicates a free hole. Of course, the relations E , Eg = Ed EG = Ea E, = Ei are satisfied. This kind of description has no particular advantages for the Mi levels. However, it is preferable for the low-lying Xi levels, for the energies Es and Escan sometimes be calculated roughly on the base of hydrogen and helium like models involving holes instead of electrons. The case Es < 0 cannot be excluded for reasons similar to those given in the discussion of El. Different arguments hold for the energy levels arising from vacanIt would lie outside the scope of this article, however, to go cies.42s47-61

+

+

+

.’////,

// A

Conduction band

A

/ / / / / / A

A

4 Ei

FIG.3. A possible electronic energy level scheme of a crystal of composition MAcontaining interstitial X atoms.

into these considerations in detail. Therefore, we shall summarize the situation by stating that a vacancy ( V M )created by the removal of an atom of the more electropositive constituent of the compound tends to bind rather than to donate electrons, whereas a vacancy (V X )created by the removal of an atom of the more electronegative constituent tends to lose electrons rather than bind them. In complete analogy with the reactions (2.10), (2.11), and (2.12), the freeing of the holes from the vacancy V M ,and the less probable process of electron dissociation can be formulated in terms of the following reaction equations : (2.13) VM+ V M ‘ + @ - E, VM’-+ VM” @ - Es (2.14) (2.15) V M .-+ V Mf @ - Ed.

+

K. Lark-Horowitz, in “Semi-conducting Materials” (H. K. Henisch, ed.), p. 47. Academic Press, New York, 1951. ** H. M. James and G.W. Lehman, in “Semi-conducting Materials” (H. K. Henisch, ed.), p. 74. Academic Press, New York, 1951. 49 F. Seitz, Revs. Mod. Phys. 23, 328 (1951). D. C. Cronemeyer and M. A. Gilleo, Phys. Rev. 82, 975 (1951). 61 W.Ehrenberg and J. Hirsch, PTOC. Phys. SOC.(London)B64, 700 (1951). 47

322

F. A. KROGER AND H. J. VINK

Figure 4 represents a possible energy scheme for the levels caused by the centers belonging to the V Mfamily.

FIG.4. A possible electronic energy level scheme of a crystal of composition M X containing M vacancies.

The freeing of electrons from the vacancy Vx is described by the reactions V X 4 Vx'+@ - E, (2.16) (2.17) Vx' + VX.' 0 - El&

+

The less probable process of binding of an electron by a V Xvacancy can be formulated in the manner

Vx

+ 0- Vx' + E,.

(2.18)

Figure 5 gives a possible scheme of energy levels associated with centers belonging to the V Xfamily.

L%.---

FIG.5. A possible electronic energy level scheme ofIaIcrysta1 of composition M X containing X vacancies.

Attention is drawn t o the fact that there is a certain resemblance between the energy scheme for crystals containing interstitial M atoms

IMPERFECTIONS I N CRYSTALLINE SOLIDS

323

(Fig. 2) and for crystals containing X vacancies (Fig. 5 ) . Similarly, the scheme for crystals containing interstitial X atoms (Fig. 3) resembles that of crystals containing M vacancies (Fig. 4). As a consequence, the energy scheme of a crystal containing both V x and V M , as when SchottkyWagner disorder prevails, is similar to that of a crystal in which V Xand X i (or V M and Mi)are present, as when Frenkel disorder prevails. The position of energy levels arising from atoms in antistructure positions, i.e., an M atom at an X site (Mx) or an X atom at an M site ( X M ) , depends on the number of valence electrons on the M and X atoms. Since M was assumed to be the more electropositive and X the more electronegative component, the number of valence electrons associated with the M atom will normally be smaller than that associated with the X atom. Under these conditions X M will form a center that tends to give off one or more electrons, whereas the tendency to bind electrons will be small. Therefore, the energy level diagram will be like that of M i(Fig. 2). On the other hand, M X forms a center which tends to bind rather than to contribute electrons, the energy levels being like those of X i (Fig. 3). The reactions (2.4)-(2.18) are balanced by reactions in the opposite direction under equilibrium conditions. Application of the law of mass action leads to relations between the concentrations of free electrons or holes, and the imperfections in their various states of ionization. Thus, for reaction (2.4), one obtains (2.19)

-

Similar relations are obtained for the other reactions. If the assumption Eki, k2, made concerning the structure of the conduction band, is also made for the valence band, the constants C1 etc. have the form (2.20) (see Section 2a). Here m* is the effective mass m, of free electrons in the case of relations involving the concentration of free electrons and is the effective mass mh of the holes for relations involving the concentration of holes. The factor a denotes the statistical weight of the level of the center.

3. COMPLETE EQUILIBRIUM INVOLVING BOTH ELECTRONIC AND ATOMIC DISORDER Relations between the concentrations of neutral imperfections, formed as the result of thermal disorder, were obtained in Section 1. Similar relations between the concentrations of neutral imperfections, of charged imperfections and of free electrons or the holes were obtained in Section 2.

,

324

F. A. KROGER AND H. J. VINK

Since the different relations have factors in common, the various equilibria cannot be independent of each other; the equilibrium for one subprocess must be influenced by that of another, and vice versa. The first example of a mutual influence of this kind was met in Section 2, namely the case in which Schottky-Wagner and Frenkel disorder are present simultaneously. The object of this section is to demonstrate the statistical interaction between atomic and electronic disorder. This will be done by considering a particular case, vix. a crystal with Schottky-Wagner disorder, in which the vacancies may be singly or doubly ionized. A corresponding energy level diagram is shown in Fig. 6. It will be easy to extend

1 . 1

El0

Ei

-v,’

FIG.6. A possible electronic energy level scheme of a crystal of composition M X with Schott.ky-Wagner atomic disorder.

the argument to Frenkel or antistructure disorder and to imperfections possessing either more or less than two energy levels. a. Equilibrium at High Temperatures The relations pertaining to our example are the following:

[VMI[VXI = Ks (3.1) = (1.19) (3.2) = (2.2) n p = Ki under equilibrium conditions. Application of the law of mass action to the reactions (2.13), (2.14), (2.16), and (2.17) gives

IMPERFECTIONS I N CRYSTALLINE SOLIDS

325

The following relation must hold as a consequence of the fact that the compound is assumed to be purely stoichiometric:

Maintenance of electro-neutrality is expressed by the condition

The eight unknown concentrations of 0, 69, V M ,VM',Vd', V X ,VX',and VX" can be computed from the eight equations (3.1)-(3.8) if the constants Ks, Ki,K7 . . . Klo are known. In general, a solution is obtained in which all the concentrations have finite values. Since the values of the constants K , which depend on the temperature and energies E involved in the reactions, may vary widely, the solutions obtained also may vary considerably. In some cases the imperfections will be present predominantly in a neutral state, in others mainly in an ionized one. b. Processes Taking Place during Cooling

The concentrations obtained above correspond to complete equilibrium a t a certain, usually rather high temperature. Often, however, one is not interested in the properties of the solid a t a high temperature such as one a t which it may have been prepared, but a t a much lower temperature such as room temperature or even lower. Thus in such cases one has to consider what happens in the crystal during the cooling from the temperature of preparation to room temperature. Various possibilities exist, depending on the rate of c ~ o l i n g If . ~ the sample is cooled infinitely slowly, all equilibria have time to become established. The final situation will then be the equilibrium state a t room temperature. If the sample is cooled very rapidly, atomic configurations may have no opportunity to change, but free electrons and holes will tend respectively to occupy lower and higher states, that is they will recombine or fill local levels, although not necessarily in accordance with the true electronic equilibrium a t room temperature. Thus, further changes may occur when the specimens are kept a t room temperature. (1) Electrons and holes will tend to rearrange themselves over the various centers. Since the mobilities of the electrons and holes can be considerable a t room temperature, the rate a t which this process proceeds usually will be determined by the rate of release of the electrons or holes from local levels into the conduction or the valence band. This, in turn, will depend mainly on the difference in energy

326

F. A. KROGER AND H. J. VINK

between the levels and the bands. If all the levels are sufficiently close to the bands, the equilibrium a t room temperature is reached rapidly. However, the system may need a very long time if levels far from the bands are present. In the latter case, the system remains in a metastable state, similar to that of an excited phosphor containing deep traps. Imperfections tend to agglomerate, leading to a decrease in concentrations. The time needed to reach equilibrium may be long since these processes require atomic migration, which usually involves a considerable activation energy and therefore may be very slow at room temperature. In principle, charged as well as uncharged imperfections may take part in these processes, forming charged or uncharged agglomerations. Thus, uncharged associated pairs ( V M V X ) may be formed from V M V X , from V M ' Vx', and from V M " VX". Similarly, charged associated pairs such as (VMVX)'may be formed from VM' V X and from VM" V X ' . The energies involved in forming a particular pair vary of course with the type of imperfections taking part in the association. The rate at which the association takes place depends on both the diffusion constants and the charges of the imperfections. If the diffusion constant were approximately the same for imperfections in various states of ionization, the rate of association would be greatest for imperfections having opposite effective charges, smaller for association between charged and uncharged imperfections and between uncharged imperfections, and smaller still for the association between imperfections which have an effective charge of the same sign. I n consequence of the electrostatic repulsion between the charges of the same sign, it is doubtful whether association would occur a t all in the latter case. The centers formed by association, namely associated pairs or larger agglomerations, may give rise to localized electronic levels. These levels will be involved in the rearrangement of electrons and holes, discussed in (1) above. Thus, the entire process of the rearrangement of electrons and holes and of the atomic imperfections is very complicated. This complication is enhanced in practice by the fact that the situation usually corresponds neither to the case of infinitely rapid nor to that of very slow cooling, but to something in between.

+

+

+

+

+

4. MIGRATION OF ATOMICIMPERFECTIONS ; IONIC CONDUCTIVITY

Atomic migration in a crystal is possible only in the presence of imperfections. 12b

IMPERFECTIONS I N CRYSTALLINE SOLIDS

327

Atoms or ions on interstitial sites may jump from one interstitial site to another or from an interstitial site to a normal one (interstitialcy migration). Moreover, an atom on a normal site may jump into a vacancy. In general, these processes may be described as diffusion of imperfections. Migration of this type has been encountered in the foregoing section. If the imperfections are charged relative to the surrounding crystal, they may move in a preferred direction under the influence of an applied electric field. The resulting migration is usually called ionic conductivity. Since charged imperfections may be present in any crystal, independent of the type of bonding,62 ionic conductivity may occur in materials such as purely covalent compounds which do not contain ions in the proper sense of the word. As a consequence, the occurrence of ionic conductivity is not necessarily a proof of the ionic character of the bonds. As we shall see in the following, a compound hardly ever has a composition corresponding to a simple stoichiometric ratio; moreover, a certain concentration of impurities is inevitably present. As a consequence, the properties of actual crystals always depend more or less on the deviations and impurities. In many cases, however, it is possible to choose the conditions of an experiment in such a manner that the concentrations of imperfections arising from thermal disorder exceed those of the imperfections associated with small deviations from stoichiometry or from impurities. This is generally the case at higher temperatures. Examples of compounds in which the ionic conductivity originating in thermal disorder has been observed are LiF, LiC1, NaF, NaI, KF, KBr, KI, RbC1, RbBr, AgC1, TlC1, TlBr,63AgBr,9s16J'3*64 KCl,63J'6,66,67 NaC1,63J'8 NaBr,33J8s69PbI2, and PbC12.60rE1 See Tubandt62for a discussion of the older literature. Self-diffusion arising from thermal disorder has been observed A striking example of this is provided by Ge, in which the occupied levels due to Gei and Gei' lie above the two empty levels due to V Q and V G ~Accordingly '. the state of lowest electronic energy is that in which Gei" and V G ~are " present in Ge possessing Frenkel disorder. J. W. Cleland, J. H. Crawford, Jr., and J. C. Pigg, Phys. Rev. 99, 1170 (1955). 6 3 W. Lehfeldt, Z. Physik 86, 717 (1933). 54 I. Pfeiffer, K. Hauffe, and W. Jaenicke, Z. Elektrochem. 66, 728 (1952). 65 H. Witt, Z.Physik 134, 117 (1953). 6 6 F . Kerkhoff, Z. Physik 130,449 (1951). 67 G. Ronge and C. Wagner, J. Chem. Phys. 18, 74 (1950). 68D. Mapother, H. N. Crooks, and R. Maurer, J. Chem. Phys. 18, 1231 (1950). 5 @ H. W. Schamp, Jr. and E. Katz, Phys. Rev. 94, 828 (1954). 60 T. E. Phipps, W. Lansing, and T. Cook, J. Am. Chem. SOC. 48, 112 (1926). 6 1 See reference 40, p. 53. 62 C. Tubandt, in "Handbuch der Experimental Physik," Vol. 12, Part 1, p. 383. Akademische Verlagsges., Leipzig, 1932. 52

328

F. A. KROGER AND H. J. VINK

in NaCl (Na) ,6 3 , 5 8 , 6 4 NaBr (Na) , 6 * 3 9 KC1(K),66 AgBr (Ag ;Br),66 66 67 aAgI(Ag),68 aAg2Hg14(Ag ;Hg),68 PbClz(Pb) ,69 PbIz(Pb) ,6 9 s70 and AgzS04(Ag).?I Examples of cases in which the self-diffusion probably has been influenced by a deviation from the stoichiometric composition will be given in Section 17. 8

9

5. ELECTRONIC CONDUCTIVITY ARISINGFROM LATTICEDEFECTS

As has been shown in Section 2, electronic conduction may be derived from electrons and holes produced when electrons are excited from the valence band into the conduction band (intrinsic process). It may also be derived from electrons thermally excited from local occupied levels into the conduction band, or from holes excited thermally from empty levels into the valence band. In general, the number of electrons (or holes) originating from centers may be neglected in comparison with the intrinsic carriers if the levels from which they are excited lie below (or above) the center of the forbidden region. If levels arising from two types of imperfection are present in equal concentrations in a compound M X , as is the case for Frenkel or SchottkyWagner disorder, and if the levels arising from these centers are situated symmetrically relative to the bands, the full levels are always below, and the empty levels always above the middle of the gap. As a consequence, they exert practically no inffuence on the conductivity. A different situation arises, however, when there is a marked asymmetry either in the position of the levels arising from the various imperfections or in the composition of the compound. This is the case, for example, if the compound has the form M,Xb in which a and b are simple integers, not equal to one another. In this event, it may happen that occupied levels lie above, or empty levels below, the center of the band gap and, as a consequence, contribute to the conductivity. It is possible that the p-type conductivity observed in stoichiometric Cr20372 can be explained in this way. Figure 7 gives a possible electron R. J. Maurer, Forschungen u. Fortschr. 26, 3rd Sonderheft, p. 4 (1950). A. Murin and B. Lure, Doklady Akad. N a u k S.S.S.R. 73, 933 (1950). 65 J. Teltow, Z. Elektrochem. 66, 767 (1952). 6 6 A. Murin and J. Tausch, Doklady Akad. N a u k S.S.S.R. 80, 579 (1951). 67 K. E. Zimen, in “Fundamental Mechanisms of Photographic Sensitivity” (J. W. Mitchell ed.), p. 53. Academic Press, New York, 1951. 68 K. E. Zimen, G. Johansson, and M. Hillert, J . Chern. SOC.p. 392 (1949). 69 G. von Hevesy and W. Seith, 2. Physik 66, 790 (1929). l o W. Seith, 2. Physik 67, 869 (1929). G. Johansson and R. Lindner, Acta Chem. Scand. 4, 782 (1952) I e K. Hauffe and J. Block, 2. physik. Chem. 198, 232 (1951). 63

64

IMPERFECTIONS I N CRYSTALLINE SOLIDS

329

energy diagram for this case. Such an-effect has also been observed with the monatomic solid Te.73,74 II. Nonstoichiometric Compounds

6. INTRODUCTION

The situation discussed in Section I, in which a pure compound has

a composition corresponding to a simple stoichiometric ratio, is an

interesting one t o consider as an ideal limit. It occurs in practice only under special conditions. Normally the compound contains a small excess or deficit of one of the components. Such compounds may be called nonstoichiometric compounds, or compounds having a deviation from

/

///////////

/ / / / /

conduction band

FIG.7. A possible electronic energy level scheme for CrlOs containing Frenkel disorder, showing the distribution of electrons over the various centers. This corresponds to equilibrium a t low temperature (T = 0).

the stoichiometric composition. It is the purpose of this part of the article to consider the factors which determine the magnitude of the deviation from stoichiometric composition and the chemical reactions leading to it. Variation of the composition of a crystal involves the transfer of atoms between the lattice and another phase, which may be either a vapor, a liquid, or another solid. Reactions involving crystal-vapor equilibrium are dealt with in Sections 7 to 17 whereas reactions involving gas-liquid and solid-liquid equilibrium are discussed in Section 19. Reactions between solid phases will not be considered. They can be treated along the same lines as the reactions liquid-solid.

7. REACTIONS INVOLVING CRYSTAL AND VAPOR A crystal M X which is a t a temperature above absolute zero tends to form a vapor that may consist of molecules M X , atoms M or X , and pos78

7'

H. Fritsche, Science 116, 571 (1952). K. Kronmiiller, Angew. Chem. 64, 364 (1953).

330

F. A. KROGER AND H. J. VINK

sibly agglomerates such as ( M X ) N ,M,, X,, M,X,, where the integers N , n,m, p , and q can have values greater than 1. The molecules or atoms actually present in a particular case are determined by reactions taking place in the gas phase. Examples are 1 1 - ( M X ) N ~M X e - M m N m M , em M

x, ~

+;1 X n

nx.

(7.1) (7.2) (7.3)

Reactions of the type (7.2) and (7.3) also regulate the composition of the vapor which is in equilibrium with the pure elements. The data concerning such equilibria are available for most elements. Thus it is known, for instance, that the vapor of most metals consists of separate atoms a t temperatures exceeding 800°C. On the other hand, the vapor of oxygen and sulfur consists of O2 and Sz molecules. In the following considerations, it will be assumed that, apart from the possible presence of molecules ( M X ) N , the electropositive component M is present in the vapor in monatomic form, and the electronegative component in the diatomic form. It must be emphasized, however, that the method used is not restricted to this case, and can be applied to other cases easily. Transfer of molecules of type ( M X ) N between the vapor and the crystal has no influence on the composition of the latter and therefore does not need further attention. However transfer of the separate constituents of the crystal between the crystal and the vapor may change the composition and therefore may have important consequences for the atomic and electronic disorder of the crystal. If a quantity 6 of atoms M are transferred from the vapor t o the crystal and the extra atoms occupy normal lattice sites, an equal number of vacancies V x are formed:

+

(1 - 6 ) M X 6Mg MXu-a,(Vx)a. The same process can also be formulated more simply -+

(7.4)

+

Mg-+ M M vx (7.5) in which the suffixes g and M denote respectively that the atom M is present in the gas and in the solid a t a normal M site. The reaction equation (7.5) is obtained from (7.4) by leaving out all the symbols that appear on both sides of the equation in equal concentration and dividing by 6.76 n Strictly speaking the formulation (7.4) should include a term describing the solution

of vacuum (vac) in the crystal: (1

- 6 ) M X + 6 M , + 6 (vac) + MXI-&(VX)&

Instead of (7.5) one then obtains

(7.4')

IMPERFECTIONS IN CRYSTALLINE SOLIDS

331

It may also happen that the atoms of type M which are added occupy interstitial sites in the crystal: MX

+ 6Mg+ M(Mi)*X

(7.6)

or, in simpler notation,

M , + Mi.

(7.7)

Similarly, incorporation of the atoms X may lead to the formation of vacancies V Mor to interstitial X atoms ( X i ) . In the simplified notation, or

8(X,),

8(X2),

-+

-+

xx

+ VIK

xi.

Whether interstitials or vacancies are formed depends on the factors which also determine thetype of thermal disorder, particularly the energy of formation of the various imperfections. Thus if Frenkel disorder, involving M iand V M , prevails over Schottky-Wagner disorder, involving V x and V M ,one may conclude that the formation of V X apparently requires more energy than the formation of Mi. Accordingly, extra M atoms will be incorporated a t interstitial sites rather than a t lattice sites, with the consequent formation of vacancies V X . The reactions (7.4)-(7.9) may of course also take place in the opposite direction; equilibrium is reached as a consequence of the opposing reactions. Application of the law of mass action leads to relations between the equilibrium concentrations of vacancies or interstitials in the crystal, and the concentration of the component being considered in the vapor. Expressing the concentration in the vapor as a partial pressure, for example in atmospheres, one obtains a relation between the partial pressure of a component and the concentration of this component, or the appropriate imperfection, in the solid. For (7.5)

Or, since [ M M ] is also constant (CM)for small deviations from the stoichiometric composition, [ V X l / P M = C d C M = Kr (7.11) or [Vxl = K r P M M,

+ (vac)

+

MM

+ VX

I

(7.5’)

in which (vac) is a volume of vacuum, equal to the expansion of the crystal. This complication is unnecessary for our purposes and therefore will be omitted.

332

F. A. KROGER

in which p

M

AND H. J. VINK

indicates the partial pressure of M in the vapor. Similarly (7.12) (7.13) (7.14)

The subscripts rand R, and ox and Ox are employed because the reactions (7.4) to (7.6), read from left to right, are, chemically speaking, identical with a reduction, whereas the reactions (7.8) and (7.9) are identical with an oxidation. The constants in Eqs. (7.11), (7.12), (7.13), and (7.14) are interrelated. Multiplying Eqs. (7.12) and (7.13), one obtains (7.15) According to Section 1, Hence,

[VMI[Mil

(7.16)

= KF.

constant = K

M X

=

(1.18) (7.17)

Similarly, multiplication of (7.11) with (7.14) leads to (7.18)

According to the relations discussed in Section 1, (7.19) (7.20) Comparison of (7.17) and (7.20) shows that (7.21) As a result of the existence of these relations, only one of the four equations (7.11) to (7.14) is needed to account for the interaction between the crystal and the atmosphere when calculating the complete equilibrium between the crystal and the vapor. Examples will be given in the following sections. Normally one chooses the relation best fitted to the problem under consideration, i.e. (7.11) or (7.13) in the case of Schottky-Wagner disorder, (7.12) or (7.13) in the case of Frenkel disorder involving M atoms, and (7.11) or (7.14) in the case of Frenkel disorder involving X atoms. It is also seen from (7.17) and (7.20) that, for a vapor in equilibrium

333

IMPERFECTIONS I N CRYSTALLINE SOLIDS

with a crystal, there is a relation between p M and p x , of such a type that p x , varies in the direction opposite to pM. This relation can also be derived directly by considering the reaction in which the solid evaporates with the formation of molecules M X , namely (MX), (MX), (7.22)

*

and the reaction corresponding t o the dissociation of ( M X ) , that is, Eq. (7.1). Application of the law of mass action to (7.22) and (7.1) gives p ~ =x constant = K ,

(7.23) (7.24)

and hence p ~ , , , "* ~ PX"''~

=

K. * K,j

KM~.

(7.25)

Equation (7.25) is identical with (7.17) and (7.20) for m = 1 and n = 2, as assumed in the foregoing treatment. All relations obtained in terms of pM can be transcribed in terms of px,l with the aid of (7.17) or (7.20) and vice versa. Thus, instead of (7.12), we may write [Mi] = K'pxL-* (7.26) in which K' = KR * K M ~ . The vacancies or interstitials formed in the solid as a consequence of the oxidation or reduction reactions (7.4)-(7.9) will take part in the intricate interplay between atomic and electronic imperfections similarly as discussed in Section I for stoichiometric compounds. Particular examples will be discussed in the following sections. 8. COMPLETE EQUILIBRIUM CRYSTAL-VAPOR FOR WITH

FRENKEL DISORDER

A

SIMPLECASE

The complete equilibrium of a crystal-vapor system will be discussed in this section for a crystal showing Frenkel disorder in the lattice of M atoms. I n order t o simplify the discussion, we shall first treat a case in which the energy level diagram is as simple as possible. More complicated cases will be discussed subsequently. The electronic energy diagram of the crystal is shown in Fig. 8. Only two of the levels which, according to the discussion of Section 2b (Figs. 2 and 4), might be caused by interstitial M and by M vacancies, are assumed to be present, namely . others are assumed not to lie in the those designated by M iand V M The forbidden gap. According to (7.12), there is a simple relation between the partial

334

F. A. KROGER AND H. J. VINK

pressure of M in the vapor and the concentration of interstitial M atoms under equilibrium conditions, namely

[Mi] = K R P M .

(8.1)

=

(7.12)

We have seen in Section I, however, that other relations involving [Mi] hold within the crystal. Thus, according to the Frenkel mechanism,

[Mi][v~ =]K F

(8.2)

=

(1.18)

if we assume the imperfections are distributed at random. Further there is a relation between [Mi] and the concentration of electrons, according

FIG. 8. The electronic energy scheme corresponding to a simple case of Frenkel disorder.

to (2.4), because of the fact that M i may donate an electron. Application of the law of mass action gives the equation

(8.3) = (2.19) Similarly, V Mmay lose holes in accordance with the equation (2.13). This leads to the equation (8.4)

=

(3.3)

Still further, the concentration of electrons and holes are interrelated in the manner n - p = Ki (8.5) = (2.2) as a result of the intrinsic process (2.1). Finally, the sum of all the negative charges must be equal to the sum of all the positive charges because the

IMPERFECTIONS I N CRYSTALLINE SOLIDS

crystal is neutral as a whole. Hence, n

+

[VM']

=p

+ [Mi'].

335

(8.6)

Thus, we have six independent relations between the six unknown conand Mi'. The concentracentrations of the quantities @, @, V M ,V M ' , Mi, tions can be calculated from these if the constants KR, KF, K1, K T ,Kd, and the partial vapor pressure p M of M are known. I n other words, the concentrations of the electrons, the holes, and all the centers can then be calculated as a function of p M . Since K R always appears with p~ as a coefficient, the calculation can be carried out using the product K R P M as a variable instead of p M alone. The results of the calculations will depend on the temperature, since the constants are functions of this variable. It must be emphasized that the procedure followed here is very similar to that discussed in Section 3a. I n both cases we terminate with a number of simultaneous equations from which the unknowns can be calculated. The principal difference between the two cases, apart from the fact that we dealt with Schottky-Wagner disorder in the earlier case and with Frenkel disorder in the present one, resides in the fact that relation (8.1) originates in the reaction between solid and vapor, whereas (3.7) depended on the condition that the compound was stoichiometric. 9. A METHODFOR OBTAINING AN APPROXIMATE SOLUTION OF MULTI-EQUATION RELATIONS SUCHAS THOSE01"SECTION8

One is faced with difficulties of two types in attempting to apply the method outlined in the previous section. In the first place, the precise calculation is rather tedious, although possible in principle. In the second place, the values of the constants are often not known precisely, if a t all, because one has only a very rough notiod of the energy scheme representing the crystal and its imperfections in most cases. One may attempt to proceed, being guided by experimental data. Particular values are assigned to the constants, and the corresponding set of equations is solved using them. Then, the properties that can be predicted on the basis of the results are compared with experimental data. Modifications are made if there are discrepancies. The whole cycle is then repeated until agreement is obtained. It is obvious that this process requires a method of calculation which can lead to the solution as quickly as possible, even if speed is achieved a t the cost of the precision. Brouwer" has proposed such a method. It is based on the following principle. If one takes the logarithm of the two sides of the equations entering in our problem, it is readily seen that all equations except the

336

F. A. KROGER

AND H. J. VINK

last lead to linear relations between the logarithms of the unknown concentrations. The relations are log “,.I = log KRPM 1% [Mil log [ V M I = log K F log [Mi’] - log [Mi] log n = log K 1 log [VM’] - log [VM] log p = log K ? log n log p = log Ki

+

+ +

+

(9.1) (9.2) (9.3) (9.4) (9.5)

for Eqs. (8.1) to (8.5). Only the last equation, corresponding to the condition (8.6) for neutrality, off&s difficulties, for it contains a sum instead of a product. It can be shown, however, that two terms in the neutrality condition are usually far larger than the others. As a consequence, the neutrality condition may be simplified by approximating it by an equality of the type A = B

so that the procedure of taking the logarithm of both sides yields log A = log B

(9.6)

which is also a linear equation. The approximate neutrality condition (9.6) and the five relations (9.1) to (9.5) can now be solved easily. The pair of terms in the neutrality condition which dominate depend both on the values of the constants K1, Kv, Ki, K F and on the value of KRpM. Thus they change when K R ~ M varies. At extremely small values of K=I)M,corresponding to strongly oxidizing conditions, the neutrality condition simplifies to p = [VM’]. (9.7) On the other hand, the relation becomes

n = [Mi’] (9.8) at extremely large values of K R p M ,corresponding to strongly reducing conditions. Simplified neutrality conditions can be found for the intermediate range with the aid of either analytical or graphical methods. The former has the advantage of being quite general. The latter requires that particular values of the constants be inserted, but has the advantage that it gives a plot of the concentrations of all the unknown quantities as a funcM Both methods will be demonstrated in the tion of K R ~ immediately. following. Let us start with the analytical method. Solving Eqs. (9.1) to (9.5), along with the logarithmic equation log p

=

log

[VM’]

(9.9)

IMPERFECTIONS IN CRYSTALLINE SOLIDS

337

corresponding to (9.7), one obtains the concentrations of a11 the unknowns, expressed in terms of KRp, and the constants. The ensuing equations are

Thus it is seen that n, [Mi],and [Mi’]increase with increasing K R ~ M , whereas p , [VM], and [VM’] decrease. Since the concentrations of the neutral centers Mi and V , do not appear in the neutrality condition, only n or [M,’], rather than p and [V,’], can increase sufficiently to become dominant in the neutrality condition. Since @ is negative and Mi’ positive, we readily see that only two alternative cases can occur. Either n takes the place of [V,’], and the new simplified neutrality condition, for , higher values of K R P Mbecomes log n

=

log p

(9.15)

or [Mi’]takes the place of p , and the condition becomes log [V,’]

=

log [Mi’].

(9.16)

The case which is actually realized depends on whether the value of K R P Ma t which n outgrows [V,’] lies above or below the value for which [Mi’]exceeds p . The point where n = [V,’] may be found from (9.10) and (9.11). It lies a t log K R ~ = Mlog K F K ~ log Ki. (9.17) Similarly, it is seen from (9.10) and (9.13) that the point where

Hence, the simplified neutrality condition, holding in the next range of K R P Mis, (9.15) if Ki2 > K F K ’ I K I (9.19) and is (9.16) in the opposite w e .

338

F. A. KROGER AND H. J. VINK

By solving Eqs. (9.1) to (9.5) with the condition (9.15), one obtains the following set of equations: log n = log p = 8 log Ki log [Mi] = log K R p M log [Mi’] = log K R p M log K1 - 4 log Ki log [V,] = log K p - log K R p , log [V,’] = - log K R p , log KpK7 - 4 log Ki.

+

+

(9.20) (9.21) (9.22) (9.23) (9.24)

Similarly one obtains log [V,’] = log [Mi’] = log KFK7K1 - .t- log Ki log [Mi] = log K R p M log [V,] = log KJi’- log K R p M log n = log K R p M 4 log KlKi - .t- log KFK7 log p = - log KQM 4 log K F K ~ K - ~ log K1

+

+

+

(9.25) (9.26) (9.27) (9.28) (9.29)

from Eqs. (9.1) to (9.5),in combination with (9.16). It is seen from (9.20) and (9.25) that the members dominating the neutrality condition are independent of K R P M in both cases. Of the concentrations of charged imperfections, only [Mi‘] increases with K R P M when n = p . Therefore, this concentration will finally exceed p . I n the case [ V d ]= [Mi’], on the other hand, n increases with p , ~ . Therefore, we attain the relation

n

=

[Mi’]

(that is the relation (9.8) already derived above) in both cases. The point where this occurs marks the upper end of the intermediate range of values of K R p , where (9.15) or (9.16) holds. The point where n = [Mi’]lies a t log K R P M when n

=

=

log Ki - log K1

(9.30)

p. On the other hand, the point lies a t

log K R P M

=

log KFK7 - log Ki

(9.31)

in the case in which [ V M ’ ] = [Mi’],i.e., we get precisely the relations (9.17) and (9.18) already found above for the limits of the range in which

p

= [VM’I.

Thus we see that the width of the intermediate range is equal to A log K R P M= llOg K F K ~ K-I 2 log Kil

(9.32)

IMPERFECTIONS IN CRYSTALLINE SOLIDS

339

when either (9.15) or (9.16) holds in the intermediate range. Finally, the set of equations belonging to the last range of KRPMis

+ +

(9.33) log = log [Mi'] = 4 log KRPM 4 log K1 (9.34) log p = -6 log KRPM- log K1 log Ki (9.35) log [Mil = log K R P M (9.36) log [ V M ]= - log K R P M log K F log [VM']= -4 log KRpM log K F K ~ log Ki 6 log K i . (9.37) Comparing the sets of equations belonging to the various simplified forms of the neutrality condition, it is seen that the relations referring to the uncharged centers V M and M iare the same in all cases. The reason for this is that the concentrations of these centers do not enter into the neutrality condition (8.6). Therefore, changes in this condition do not affect these quantities. Further, it is easily seen, for example from (9.10) and (9.13), that there is a similar relation between the concentrations of the ionized imperfections, namely

+

+

+

+

[VM'][Mi")= K F K ~ K , / K3( KF' (9.38) in addition to the normal Frenkel relation involving the concentrations of the uncharged imperfections, Thus, the product of these concentrations is also constant. We have designated the product KF'. Inserting this new Frenkel constant into the relation (9.19) determining the mechanism of charge compensation in the intermediate range, one gets Ki > KF'. (9.39) Similarly, one obtains (9.40) A log KRPM = llog KF' - log Kil for the width (9.32) of the intermediate range. We shall now proceed to demonstrate the graphical method. Since all the relations of the four groups of equations (9.10) to (9.14), (9.20) to (9.24), (9.25) to (9.29), and (9.33) to (9.37) are linear, on a log concentration versus log KRpM plot, they are represented by straight lines with slopes 0, k6, 1. The slopes are different in each range in which a particular, simplified form of the neutrality condition holds. Owing to the linearity, the width of the ranges can easily be obtained graphically, provided particular values of the constants are inserted. Let us consider a case having the energy levels shown in Fig. 8, selecting the following constants: K F = 1030cm-6 Ki = cm-6 K 1 = K 7 = lo1*~ m - ~ and thus KF' = cm-6 [see (9.38)].

340

F. A. KROGER AND H. J. VINK

Lines representing the concentrations of the various centers as functions of K R p M according to Eqs. (9.10) to (9.14) are shown at the left-hand side of Fig. 9a for the case p = [V,’]. It is seen that p and [ V M ’ ] gradually decrease with increasing K R p M , the lines having a slope -6; however,

P

FIG.9. (a) Concentrations of imperfections in a crystal of composition M X with Frenkel disorder and an energy level scheme similar to that,shown in Fig. 8. This corresponds to complete equilibrium of crystal and vapor a t a high temperature for a case KF‘> Ki. The thick lines give the approximate solution; the thin lines represent the exact solution. (b) The position of the Fermi level relative to the bands (see Section 15).

++.

[Mi’] and n increase, the lines having a slope Since [Mi’] > n, the concentrations of M i first grows sufficiently t o become equal t o the concentrations of @ and VM’.This happens when log K R p M = 14, which marks the limit of the range where p = [VM’].From here onward [VM’I

=

[Mi’]

IMPERFECTIONS I N CRYSTALLINE SOLIDS

341

and the variations of concentration follow the relations (9.25) t o (9.29). In this range n increases, and p decreases more rapidly than in the foregoing range, the lines having a slope k l . In contrast, [VM']and [Mi']are constant. The end of this range (marked 11) is set by the point where the line designated 0intersects the [Mi']line. This happens when

FIQ. 10. Concentrations of imperfections in a crystal of composition M X with Frenkel disorder and the energy level scheme shown in Fig. 8. This corresponds to complete equilibrium of crystal and vapor a t a high temperature for a case KR' < Ki.

log K R p =~ 16. At larger values of K R p ~the , simplified neutrality condition obviously is n = [Mi'] and the relations (9.33) to (9.37) are valid. It is obvious that near the boundary between two adjacent ranges the simplified neutrality conditions are no longer good approximations. For example, the neutrality condition should be [VM'I

=p

+ [Mi']

at the boundary between ranges I and I1 rather than the simple relations used in range I and 11. The error which is made in this way is not serious. To demonstrate this, the concentrations of the various imperfections corresponding t o the exact solution are also given in Fig. 9a (thin lines). It is seen that the main difference between the exact solution and the approximate one is that the sharp corners in the lines at the boundaries of the approximate solution are rounded off in the exact solution. Figure 10 shows the situation for a similar system having the con-

342

F. A. KROGER

AND H. J. VINK

stants K F = lo2*cm-6, Ki = lOS2 cm-6, K7 = 3.1016 cm-3. Accordingly, KF’ = 1030 cm-6. Thus, KF‘ < Ki, which leads to a middle range in which n = p. The complete symmetry of Fig. 9, which was the result of the equality of K , and K,, is no longer present in Fig. 10, for now K , f K7. I n the cases corresponding to both Fig. 9 and Fig. 10, the crystal shows predominantly n-type conductivity a t high, and predominantly . transition from n to p p-type conductivity a t low values of K R ~ MThe type differs, however, for the two cases. The transition is gradual in the case of Fig. 9, a crystal with intrinsic properties (n = p) appearing only at one particular value of K R p ~In. the case of Fig. 10 the crystal is intrinsic over the entire middle range where the simplified neutrality condition n = p is satisfied. 10. COMPLETE EQUILIBRIUM BETWEEN CRYSTAL AND VAPORFOR A CRYSTAL CONTAINING COMPLETELY IONIZED FRENKEL DEFECTS Many systems have an energy diagram of the type shown in Fig. 8, in which the values of El and E7 are so small that the levels nearly coincide with the bands. I n this case the centers M i and V M are practically completely ionized. Two ways of obtaining the equilibrium concentrations of the imperfections are available in this case. In the first place, it is possible to use the set of equations given above, inserting very large values of K 1 and K,, and very small values of KF. Since the concentrations of M iand V M are negligible, however, it is more sensible to recast the entire problem, leaving Mi and V Mout entirely. Thus it is preferable to write M,*Mi’+@ (10.1) in place of the reduction equation (7.7). Application of the law of mass action t o this reaction leads to the equilibrium relation

n [ M i ] = KR’pna

(10.2)

which now has to take the place of (8.1). Further, we have the modified Frenkel relation [VM’][M;]= KF’. (10.3) = (9.38) The ionization relations (8.3) and (8.4)may be dropped. The intrinsic relation (8.5) and the neutrality condition (8.6) remain unchanged. The solution of this problem is very similar to that derived in the previous section. There are also three ranges in this case, namely

n

=

[Mi’],

p

=

[VM’],

and

n =p

or

[V,’] = [Mi’]

IMPERFECTIONS I N CRYSTALLINE SOLIDS

343

Fro. 11. Concentrations of imperfections in a crystal of composition M X with completely ionized Frenkel defects (Fig. 8; El = El = 0). This corresponds to complete equilibrium of crystal and vapor a t a high temperature for the case in which KF' > Ki.

-

10 K; PM

FIG. 12. Concentrations of imperfections in a crystal of composition M X with completely ionized Frenkel defects (Fig. 8, El = El = 0). This corresponds to complete equilibrium between crystal and vapor at a high temperature for the case that

KF'

< Ki.

the choice between the latter two being dependent on (9.39). Figure 11 shows an example of this case when K F f > Ki, that is, when Ki = 1030 cm-6 and Kp' = cm-s. Figure 12 shows an alternative case for which Kg' < K;, the actual values being K i = cm+' and Kp' = 10aocm-6. Comparing Figs. 9 and 10 with Figs. 11and 12, we 0ee that the dependence

344

F. A. KROGER

AND H . J. VINK

of the concentrations of the charged imperfections on KRPMor KR'PMis similar in corresponding cases. 11. A MORECOMPLICATED CASE OF SCHOTTKY-WAGNER DISORDER

Systems may be more complicated than those just considered for two reasons. In the first place, the different kinds of imperfections may possess more energy levels (see Section 2b). Second, the composition of the compound may be less simple than that of the binary compound M X discussed thus far. The same methods may be applied nevertheless to obtain the concentrations of the various imperfections in the equilibrium

FIG.13. A possible electronic energy level diagram for a crystal of composition M 2 X with Schottky-Wagner disorder.

state of the crystal and vapor. As an example, we will discuss the case of a compound of the type MZX, corresponding, for instance, t o Cu20 or K2S. We shall assume it exhibits Schottky-Wagner type of disorder predominantly and has the electronic energy scheme shown in Fig. 13. The component X will be assumed to be present in the vapor phase as X2 molecules. If we assume that the imperfections formed are distributed a t random, the equilibrium relations for this case are (a) Oxidation-reduction : [ V M I = Koxpx,t (b) Schottky-Wagner disorder: [VX][VMI' = Ks.

(11.1) (11.2)

The square of the concentration of V Moriginates in the fact that 2vM centers are formed for every V X . (c) Ionization of the atomic imperfections:

IMPERFECTIONS IN CRYSTALLINE SOLIDS

n - p = Ki

(d) Intrinsic excitation: (e) Neutrality condition: n

345 (11.4)

+ [VM'] p + [VX']+ ~ [ V X " ] . =

(11.5)

The factor 2 arises from the double charge of VX**. The presence of the term involving [Vx"]in the neutrality condition implies that the ways of obtaining a simplified neutrality condition must be investigated anew. Other ranges will be possible. Furthermore, the presence of the factor 2, associated with the double charge on Vx",

FIQ.14.Concentrations of the imperfections in a crystal of composition M2X with Schottky-Wagner disorder (energy scheme of Fig. 13). This corresponds to complete equilibrium between crystal and vapor at a high temperature.

implies that the point a t which VX" predominates in the neutrality condition is determined by the condition that the concentration of VX" becomes half that of the single-charged center of opposite charge which is being compensated. This means that the line in the double-logarithmic plot referring to the concentration of VX" reaches a point lying a distance log 2 = 0.3 below that of the concentration of the other center. The result of the calculations is shown in Fig. 14 for a system having the constants

K S = 8.1048 ~ r n - ~ , K 7 = K e and

Ki

= 6.3 * 10l8 CM-~,

=

1.6 *

cm-6.

Klo = 4.1016 ~ m - ~

346

F. A. KROGER AND H. J. VINK

It is seen that there are four ranges in this case namely, n =

[Vx']; [V,']

=

[Vx']; [V,']

= 2[Vx"];

and

p

=

[VM'].

Comparing Fig. 14 with Figs. 9, 10, 11, and 12, which correspond to simpler cases, one is struck by the fact that the concentration of metal vacancies (V,') increases in range I1 of Fig. 14 when the X Z pressure is decreased, that is when the metal pressure is decreased, in distinction to what happens in the other cases. This is a consequence of the fact that [V,'][Vx*l

=

Ks'

(11.6)

in the cases associated with Figs. 9, 10, 11, and 12, whereas one finds from Eqs. (11.1) to (11.4) that (11.71

in the case of Fig. 14. Thus in this case the product of the concentrations of VM' and Vx' is no longer a constant, but increases with decreasing px,. Therefore, the concentration of VM' as well as that of VX' has to increase with decreasing px, in the range 11, where the simplified neutrality condition has the form [VM']

= [VX.].

The variation of n and p with pM is similar to the variation observed in the simple cases discussed earlier, that is, n increases and p decreases correspondingly with increasing P M . Under particular conditions, the concentration of M vacancies in a solid of composition M X may-increase with increasing values of p~ in the vapor. This may happen in the case of Schottky-Wagner disorder if the simplified neutrality condition 2[VM"] = [Vx']

(11.8)

is valid for a range. It may also occur in the case of Frenkel disorder if the condition 2[VM"] = [Mi'] (11.9)

is valid. Such ranges may occur if M vacancies give rise to two levels within the forbidden gap and not just one, as assumed in the calculations leading to Figs. 9 to 12 (see Fig. 8), for example, if levels of type VM'and V," are present. Similarly, the concentration of X vacancies, or interstitial M atoms, increases with increasing px, in compounds of type M X if there are ranges (1 1 .lo) 2[Vx"] = [V,']

347

IMPERFECTIONS I N CRYSTALLINE SOLIDS

or 2[Mi”]

=

[V,’].

(11.11)

This may occur if X vacancies, or interstitial M atoms provide two levels within the forbidden gap. In general, it can be said that the concentration of a particular type of imperfections present in a considerable concentration varies with the atmosphere in an anomalous way only if: (1) the imperfection appears in the simplified neutrality condition; (2) the two types of imperfections dominating the neutrality condition cannot be generated by a simple disorder process in the proportion required by this condition. BETWEEN CRYSTAL AND VAPORFOR 12. EQUILIBRIUM M X WITH ANTISTRUCTURE DISORDER

A

CRYSTAL

In the cases treated in the foregoing sections, in which Frenkel or Schottky-Wagner disorder predominate, the concentration of free electrons increases and the concentration of the holes correspondingly decreases with increasing KRpM. This occurs because both M i and V X , which are formed by incorporation of extra M and are therefore favored by large values of K R p M , give rise to occupied levels close to the conduction band. In contrast, V M which is formed by the incorporation of extra X and is therefore favored by low values of K R ~ Mgives , rise to empty levels close to the valence band. Quite a different behavior is to be expected when antistructure disorder occurs. In this case, extra M will be incorporated at X sites ( M x ) at large values of K R p n a , whereas extra X will occupy M sites ( X M )a t low values of KRpM. According to Section 2b, M X may produce empty levels close to the valence band, and XM occupied levels close to the conduction band. Accordingly, the concentration of free electrons will decrease and the concentration of holes increase with increasing KRPMin this case. We shall consider a simple case of pure antistructure disorder initially in the following. Following this, we shall consider a case in which both Schottky-Wagner and antistructure disorder are present.

a. Pure Antistructure Disorder Figure 15 shows a possible position of the electronic energy levels for a crystal M X with M atoms at X sites and X atoms a t M sites. Since neither vacancies nor interstitials are assumed to be present, the incorporation of extra atoms of type M from the gas phase must take place in accordance with the reaction 2 M , - Mjtg Mx. (12.1)

+

348

F. A. KROGER AND H. J. VINK

FIG.15. A possible electronic energy scheme for a crystal of composition M X with antistructure disorder.

Application of the law of mass action to this reaction, taking into account the fact t ha t the concentration of M a t M sites ( M M ) may be considered to be constant, leads t o the relation

[Mx] = KRPM’.

(12.2)

Similarly, the interchange of M and X in accordance with the equation

MM

leads t o the relation

+Xx+Mx +XM

(12.3)

[MX][XMI = KA. (12.4) We have, in addition, the conditions giving the intrinsic electronic excitation and ionization of the imperfections, namely

and, finally, the neutrality condition

n

+ [MA-’] P + [XM*I. =

(12.6)

~ M ~ Again, all the unknowns can be calculated as functions of K I ~ from ( 12.2), (12.4) , ( 12.5) and (12.6). Just as for the other types of disorder (see Section 9), it can be shown tha t [IMx’][x2\1’]= K~KiiKi2/KiE KA’ (12.7) )

and that different solutions are obtained depending on whether KA‘ is greater or less than Ki. Figure 16 shows a solution for the case

K i = 1030cm-e, K A = for which

loz6cm-6,

Kll

KA‘ =

cm-6

=

l O I 9 ~ m - ~ Klz , = 10l8 cmW3

> Ki.

349

IMPERFECTIONS I N CRYSTALLINE SOLIDS

'1

t

F I ~ 16. . Concentrations of imperfections in a crystal of composition M X with antistructure disorder, in equilibrium with its vapor a t a high temperature for the case that KA' > Ki. For the energy levels see Fig. 15.

As anticipated, the concentration of free electrons is seen to decrease and that of holes to increase with increasing values of K R p M 2 . The material is intrinsic (n = p ) a t log K R p M = 13.5. The approximation [Mx'l = [XM'I holds in the middle range. When K A ' < Ki, the middle range is dominated by the relation n = p exactly as was found for the other types of disorder (Section 9).

b. Combination of Antistructure and Schottky-Wagner Disorder The levels of a crystal of the type M X in which both antistructure disorder and Schottky-Wagner disorder occur together are shown in Fig. 17 for a simple case. In this example, it is possible to use the reduction equation in the normal form which leads to

M,+ M M -k v x [vx]= K P M .

(12.8)

350

F. A. KROGER AND H. J. VINK / / / / / /

conduction band / / / / / /

FIG. 17. Electronic energy scheme for a crystal of composition M X with both Schottky-Wagner and antistructure disorder.

According to the Schottky-Wagner mechanism of disorder]

[ V ~ l [ v x= l Ks.

(12.9)

Occupation of an X site by an M atom and of an M site by an X atom are not directly coupled in this case. Hence, we have two equations

+ VXSMX + xx + VM*XM+ vx. VM

M M

and

(12.10) (12.11)

These lead to the relations76a (12.12) In addition to the ionization relations (12.5), there are two more relations associated with ionization of the vacancies (12.13) Finally, the neutrality condition is

n

+ [VM'I + [Mx'l

=

P

+ Vx'l + [XM'I.

(12.14)

Figure 18 shows an approximate solution obtained from the relations Multiplying the two relations (12.12)] we see that the simple interchange equilibrium still holds with the relation K A = K q ~ l K.cz,.

-

IMPERFECTIONS I N CRYSTALLINE SOLIDS

35 1

(12.5), (12.8), (12.9), (12.12), (12.13), and (12.14) in the usual way by using the c0nstants7~~

K s = 1030cm-6, Ki = cm-6, Kucl, = Kucz,= 10’2 cm-3 K s = Klz = 1019 cm-6. K , = K1l = 1017 cm-3, The conditions which dominate the neutrality relation in the various ranges are marked a t the top of the figure. Apart from ranges already known from the normal Schottky-Wagner problem (e.g., ranges 11, 111,

FIG.18. High temperature equilibrium concentrations of the imperfections for a crystal MX with Schottky-Wagner and anti-structure disorder; the energy lev& are m shown in Fig. 17 but with E7 > En.

and IV), we now have new types of ranges in which both vacancies and misplaced atoms play a role (ranges I and V). The variation of n and p with K,pM is normal in the “normal” ranges, n increasingland p decreasing with increasing K , p M . In the other ranges, n and p show the opposite behavior. As a consequence, the concentrations of electrons and holes change in a remarkable way, increasing or decreasing in the sequence of ranges. The material changes from n to p type a t three points on the KrpMaxis. It passes through an intrinsic range during the transition from n to p type in the central part of the figure. These values of the constants correspond to a position of the energy levels slightly different from that shown in Fig. 17. Namely E7 > E I Z ,and not E , < E ~ as z assumed in Fig. 17.

70b

352

F. A. KROGER

AND H. J. VINK

It must be emphasized that the variation of the concentrations shown in Fig. 18 is only one of a great variety of possible variations. Changes in the values of the constants may increase or decrease the effect which the different types of disorder have in determining the dependence of concentration of the imperfections upon K,pM.It can easily be shown that the larger K S is relative t o the constants K A ,Kacl),and Kacz), the wider is the range in which the Schottky-Wagner disorder dominates the properties. On the other hand, large relative values of and K+, increase the range in which antistructure disorder determines the properties. The situation shown in Fig. 16 is a limiting case. Similar conclusions hold for cases in which both antistructure disorder and Frenkel disorder play a part. 13. COOLING; THE STATEOF

THE

CRYSTALS AT Low TEMPERATURES

The results obtained in the previous sections correspond t o complete equilibrium both within the crystal and between the crystal and the surrounding atmosphere. In practice this state is achieved only a t a rather high temperature, such as that at which the crystal is prepared. In many cases, one is interested not in the state existing at the temperature of preparation, but in that attained after cooling to room temperature. As mentioned in Section 3b there are various possible differences between the state reached by cooling the crystal infinitely slowly, that is, the true equilibrium crystal-vapor at room temperature, and the state achieved by cooling very quickly. In the latter case, and also in intermediate cases, processes involving an activation energy may be frozen in. The degree of freezing-in is determined not only by the energy of activation, but also by the number of steps involved. As an example let us consider the possibility of a variation of the concentration of atomic imperfections. Such a variation necessarily involves the migration of defects. In this case the freezing-in will be more effective the greater the distances over which migration must take place. It seems safe t o conclude that oxidation-reduction reactions, which involve the transport of atoms over relatively large distances between the inside of the crystal and the surrounding atmosphere, are among the first t o be frozen in. The process of association of imperfections within the crystal, which involves diffusion over relatively short distances, may occur a t much lower temperatures. In practice, one must make an intelligent guess, guided by experiment, of what is going on during the cooling in each particular case. The behavior of CuzO, which is still far from being fully u n d e r s t o ~ d , ~illustrates ~~-~~ just how intricate the situation may be. 770 77)

G. Blankenburg and K. Kassel, Ann. Physik 10,201 (1952).

G.Blankenburg, C. Fritsche, and G. Schubart, Ann. Physik 10, 217 (1952).

IMPERFECTIONS I N CRYSTALLINE SOLIDS

353

One of the simplest assumptions to make is that all processes involving the migration of atoms are frozen in, whereas all reactions involving electrons or holes take place freely and lead t o equilibrium. I n this case, the total concentration of atomic defects remains fixed, and the state a t room temperature is found by calculating the distribution of electrons or holes over the levels arising from the various centers and the lattice bands. This attitude has been adopted in discussing the equilibrium between crystal and vapor for CdS.g The same assumption will be made in the following. We should stress once more, however, th a t this is a relatively arbitrary choice, many other assumptions being possil$e. Let us consider the simple case corresponding t o Fig. 11 for which the equilibrium concentrations a t high temperature have been discussed. T o simplify the calculation, the state will be determined at T = 0 rather than that a t room temperature. I n discussing the redistribution of electrons over the local levels arising from the atomic imperfections with the aid of the approximate results given in Fig. 11, one must bear in mind the nature of the approximations th at have been made. Most particularly one must remember that the neutrality condition must hold exactly in spite of all the approximations. Thus in range I of Fig. 11, the simplified neutrality condition p = [V,'], used in obtaining the approximate solution of the high temperature equilibrium, must be refined to the form

the concentration of the electrons being neglected. It is clear from this more refined equation that all the holes are trapped in the VM' centers upon cooling, forming V Mcenters. However, since the concentration of holes is smaller than th at of V M ' by the amount [Mi'], not all the V M ' centers will be filled by the holes. A concentration [VM']= [Mi'] will remain in the form V M ' , The result is shown in Fig. 19 in which the final concentra,tions are given as functions of the values of KR'pM a t which the high temperature equilibrium has been established. Proceeding in a similar way in range 111,we end up with Md and small, equal amounts of Mi' and VM'. The equality [ V M ' ] = [Mi'],used in obtaining the approximate solution for high temperature equilibrium shown in Fig. 11, must be refined in the following way in the central range (11).I n the left half of the range, there is a slight excess of V M ' , corresponding t o the relation G . Blankenburg and 0. Bottger, Ann. Physik 10, 241 (1952). G. Blankenburg, Ann. Physilc 14, 290, 308 (1954). C. Fritsche, Ann. Physik 16, 178 (1955).

78b 79

354

F. A. KROGER AND H. J. VINK

The concentrations of electrons is negligible. The holes are trapped at the VMf upon cooling, and form a concentration of VM centers equal to the concentration of holes initially present in the range. The remainder of the V M ' which , is the great majority in view of the fact that [ V M E ~ ][.Mi']

+

-

i

19 K ~ P M

FIG.19. Concentrations of centers at T = 0 for a case in which Fig. 11 shows the equilibrium at a high temperature.

p, remains unchanged. Similarly, we see that a fraction of the Mi' equal to the concentration of electrons is transformed into M iin the right half of range I1 where [VMl] n E [Mi'].

+

A concentration of Mi' equal to that of VM' remains unchanged. In the middle of the range the concentrations of V Mand M idrop to zero suddenly (Fig. 19). Now V M centers produce empty levels close to the valence band, whereas Mi centers contribute occupied levels close to the conduction band. Thus holes and electrons will be liberated in the bands a t temperature T > 0 and give rise to p-type conductivity to the left of the point where [V,] and [Mi]become zero, and n-type conductivity at the right. The transition from n to p type takes place very sharply. A discussion of this effect will be given in Section 15 in terms of the Fermilevel. I n a crystal whose high temperature equilibrium is described by Fig. 12, cooling under similar conditions will give rise to the states depicted in Fig. 20. The concentrations of the uncharged Mi centers and V Mcenters are found to vary as in Fig. 19. As a consequence, the semiconducting behavior of such a crystal will show a great resemblance to that of a

355

IMPERFECTIONS I N CRYSTALLINE SOLIDS

crystal for which Fig. 19 is valid. It will be p type a t low values of KR'PM and n type a t high and will exhibit a sharp transition in between.80eThe concentrations of the charged imperfections are quite different, however. Therefore, properties such as optical absorption, fluorescence, and ionic conductivity, will be quite different in the two cases, insofar as they are determined by the imperfections.

7

FIQ.20. Concentrations of centers a t T = 0 for the case i n which Fig. 12 shows the equilibrium concentrations a t a high temperature.

Figure 21 shows the state, at O'K, of a crystal in which the imperfections are not completely ionized a t high temperature (Fig. 9). The state at room temperature, or, in fact, a t any temperature where the restrictions are valid, can also be obtained. This, however, involves a slightly more complicated calculation. A simple graphical method for carrying out such calculations, which makes use of the Fermi level, has been presented by Shockley;80bMooser81° has devised an apparatus for making calculations of this type. (See Section 15 also.) We saw that the concentration of donors (Mi) or acceptors ( V M )is a simple function of p M over a wide range of pressure in the cooled crystals and possesses a sharp transition in between (Figs. 19 and 20). This be804

SO*

The semiconducting properties of the two systems are not quite identical because of t h e presence of Mi' and VM'in the case associated with Fig. 19. The ionized donors and acceptors influence the ionization equilibrium by increasing the probability of trapping of the carriers. Thus they tend to decrease the conductivity. This is a quantitative effect, however, and does not alter the conclusions regarding the transition from n- to p-type conductivity. See reference 31@,p. 464. E. Mooser, Z.angew. Math. u. Phys. 4, 433 (1953).

356

F. A. KROGER AND H. J. VINK

havior has consequences which are important for the preparation of compound semiconductors. Crystals containing Mi or V Min concentrations which correspond to the ranges where the curves giving log [ V M and ] log [Mi]as functions of log KR'PMare linear with slopes 0.5 or 1 can be

/

t I

/ / / / /

I I I I

1-

conduction band

/ / / / / /

- -- - --

FIG.21. (a) Concentrations of imperfections at T = 0 for the case in which Fig. 9 shows the equilibrium concentrations at a high temperature. (b) Position of the Fermi level relative to the bands (see Section 15).

obtained by heating a specimen in an atmosphere having the required value of KR'PM.However, crystals which have concentrations lying in the range where the crystal changes from n to p type cannot be obtained in practice. In this range, a small variation in KR'PJI,or in the temperature, will change the crystal from n to p type. Inhomogeneities are bound to occur as a consequence. The lowest concentration of donors or acceptors a t which homogeneous crystals can still be obtained corresponds to the

IMPERFECTIONS I N CRYSTALLINE SOLIDS

357

values where [V,] and [Mi] drop t o zero. This is equal to the smaller of Ki’ or KF‘*in the present examples. Since these constants decrease with decreasing temperature, the lowest concentrations of acceptors or donors that can be established are lower, the lower the temperature a t which the crystals are prepared.81bThis behavior actually has been observed with PbSe21,82a,82b The effect of cooling on crystals made a t a high temperature has another aspect t hat deserves consideration, namely th a t connected with surface layers. I n the considerations given a t the beginning of this section, it was shown that atomic migration will be frozen in more readily the longer the distance the atoms must migrate. As a consequence, equilibrium between the atmosphere and the crystal will be frozen in most effectively when the section of the crystal we are interested in lies far from the surface. I n contrast, layers close to the surface tend to remain in equilibrium with the atmosphere when the specimen is cooled to a much lower temperature. Cases in which crystals have different properties in the bulk and near the surface as a consequence of this effect have been observed. Examples are provided by 2n0,830-83b C U Z O and , ~ ~ CdS.g For the consequences which this effect may have on the electrical properties of polycrystalline, sintered samples, see references 85u-85d. THE SIMPLESTOICHIOMETRIC RATIOAS A FUNCTION OF THE ATMOSPHERE I n this paper, a compound in which the constituents are present in a simple stoichiometric ratio is called a compound with the stoichiometric composition or a stoichiometric compound. The deviation Ax from stoichiometric composition can be given as a simple function of the Concentrations of the atomic imperfections. I n the

14. THE DEVIATION FROM

The lowest temperature that can be used in practice is the temperature where the equilibrium between crystal and vapor can be established in a reasonable time. 820 J. Bloem, F. A. Kroger, and H. J. Vink, Repts. Conj. on Dejects in Crystalline Solids, Univ. Bristol 1964,p. 273, 1955. 82b PbS has Schottky-Wagner disorder; however, the main argument remains the same. 8 3 O D. J. M. Bevan, J. P. Shelton, and J. S. Anderson, J. Chem. Soe. p. 1729 (1948). 8ab P. H. Miller, Jr., in “Semi-Conducting Materials” (H. K. Henisch, ed.), p. 172. Academic Press, New York, 1951. a4 C. Fritsche, Ann. Physik 14, 135 (1954). 860 E. J. W. Verwey, in “Semi-Conducting Materials” (H. K. Henisch, ed.), p. 151. Academic Press, New York, 1951. E L b J. Volger, in “Semi-Conducting Materials” (H. K. Henisch, ed.), p. 162. Academic Press, New York, 1951. 8 s F. ~ G. Brockman, P. H. Dowling, and W. G. Steneck, Phys. Rev. 77, 85 (1950). C. G. Koops, Phys. Rev. 83, 121 (1951).

81*

358

F. A. KROGER AND H . J . VINK

following, Ax will be taken positive for an excess of M and negative for an excess of X . Consider a compound M X with Schottky-Wagner, Frenkel, or antistructure disorder, or with a combination of two or more. The deviation Ax can be written AX =

Z[Mi]

+

Z[VX] 2Z[Mx] - Z [ X i ]-

Z [ v M ]

- 28[xM] (14.1)

in which the sums are to be taken over all the states of each imperfection, including ionized and unionized. The quantity Ax will be expressed in atoms per cma. Equation (14.1) simplifies to AX =

[Mi] -I- [Mi'] - [VM]- [ V M ' ]

(14.2)

for a crystal M X having Frenkel disorder and only one level per imperfection, which corresponds to the cases treated in theSections3,4,5,and 7. Ax is known as a function of the variables appearing in this relation since the concentrations of the various imperfections have been obtained as functions of K R p M or K R ' p M . The deviation from stoichiometric composition to be expected a t high temperature can be obtained from the high temperature figures. The results for cooled crystals depend of course on whether or not changes in composition take place during the cooling. The high temperature data also apply to cooled samples if it is assumed that such changes do not occur, as in the examples of Section 13. In obtaining Ax from the figures giving the approximate concentrations of imperfections a t high temperature, with the aid of (14.1) or (14.2), one must keep in mind the approximations which have been made. Just as in Section 13, subtraction of the concentrations of the major imperfections to obtain the concentrations of the imperfections present in lower concentrations often leaves only small quantities which were left out of the simplified form of the neutrality condition. When applied to the case of Fig. 9, arguments similar to those given in Section 13 lead to a dependence of Ax on K R P M . At high values of K R P M , Ax is positive, corresponding to an excess of M , and negative at low values of K R P M , corresponding to an excess of x. The exact stoichiometric point lies at an intermediate value of KRPM.The dependence of log Ax on log K R P M is shown in Fig. 22. In order to indicate the variation of sign of Ax, log IAxI is plotted above the axis for positive values of Ax and below the axis for negative values. The slope of the function log Ax versus log K R P M varies from 0.5 or 1in the various ranges, to a very large, even infinite value at the point where Ax passes suddenly from a finite positive to a finite negative value.

IMPERFECTIONS I N CRYSTALLINE SOLIDS

359

It is easy to see from this figure that specimens, in which the deviation from stoichiometry corresponds to the flat parts of the Ax curve, can be obtained by bringing the crystal in equilibrium with a vapor in which P M , and thus KRPM,has the appropriate value. Heating in a vapor for which KRPMhas a value associated with the region where Ax changes rapidly

FIG.22. The deviation from the stoichiometric composition (Ax) as a function of K R ~ for M the cases in which Fig. 9 and Fig. 21 apply.

from positive to negative values, however, cannot lead to a homogeneous product. One ends up almost certainly with crystals consisting of zones having either an excess of M or an excess of X . This behavior has the consequence that values of Ax below that for which the crossing occurs cannot be obtained. In the present example, . to this value is equal to the smaller of the quantities Kih or K F ’ ~Thus prepare crystals possessing a small deviation in stoichiometry, we must work under conditions in which these constants are small, that is at a low temperature. The discussion given here is essentially identical with that given in Section 13, as far as it relates to the preparation of crystals hav. reason for ing a low concentration of donors (Mi)or acceptors ( V M )The

360

F. A. KROGER AND H. J. VINK

this is t ha t there is a one to one relation between the concentration of donors or acceptors and the excess of M or X for the example under discussion. This in turn is a consequence of the complete symmetry of both the composition of the compound ( M :X = 1 :1) and the position of the energy levels. The quantity A x varies in a similar way in cases in which antistructure disorder plays a role. The excess of M decreases monotonically with K R P M until, a t a particular point, the system suddenly changes from one with an excess of M to one with an excess of X . The latter excess then increases monotonically with decreasing KRpM. Attention must be drawn t o the fact t ha t the oscillations in the concentrations of the free electrons and holes (Fig. 18) are not accompanied by similar variations in Ax. The formula for A x is more complicated for compounds M,Xa when a # b. I n the case of compounds of composition M2X possessing SchottkyWagner or Frenkel disorder or a mixture of the two, A x can be expressed in the form AX = Z[Mi] 2Z[Vx] - Z[V,[ - 2Z[XL] (14.3)

+

I n this equation, A x is equal to the density of defects divided by the valence of the atoms. When only Schottky-Wagner disorder is present, and the energy levels of the imperfections have the form assumed in the example discussed in Section 11, this simplifies to AX

=

+

+

~ [ V X ] ~ [ V X ' ]~ [ V X "-] [V,] - [V,'].

It may be seen that A x

=

(14.4)

0 somewhere in range I11 of Fig. 14 where 2[Vx*.]

[V,'].

= 34.25 where The exact point will be close to the point log KOx2pz,3

qvx.1 = [VMI. The p-n transition occurs a t the point where the lines for n and p cross, that is, when log Kox2pZ,t= 31.3. Hence, the two points do not coincide in this case. The points where A x = 0 and p = n differ because the purely stoichiometric crystal shows p-type conductivity arising from its imperfections (see Section 5). OF 15. THEPOSITION

FERMILEVEL It is common to describe the semiconducting properties of a solid with the aid of a parameter known as the Fermi level which is, in fact, ~~~,~~ the thermodynamical, or chemical potential of the e l e ~ t r o n s .The concentration of electrons in the conduction band is connected with the 86

THE

R. A. Hutner, E. S. Rittner, and F. K. du Pr6, Philips Research Repts. 6 , 188 (1950).

IMPERFECTIONS I N CRYSTALLINE SOLIDS

361

position of the Fermi level p, measured relative to the lower edge of the conduction band, by the relation n =

/*

4n(2mekT)$ h3

x4dx+ 1.

ex-{/kT

(15.1)

This reduces to (15.2) for low concentrations of electrons where Maxwell-Boltzmann instead of Fermi-Dirac statistics hold. Here me is the effective mass of the electrons in the conduction band. Similar relations hold for holes in the valence band, mh replacing me, and E, p replacing p. Relation (15.2) holds with an error of 0.7% or less for T = 1000"K, me E mo, and n < 3.1018~ m - ~ . The error is less than 10% even when n = 6.1019 cm-3. Thus the latter expression is usually sufficiently precise for our purpose. Since n is given as a function of the partial pressure of one of the components of the system in the figures referring to the high temperature equilibria, it is easy to derive the position of the Fermi level as a function of the same variables from the figures using (15.2). An example showing the dependence of the position of the Fermi level on log KRpMat the temperature at which the complete equilibrium (Fig. 9a) has been established, is given in Fig. 9b. According to Fig. 9a, the position of the Fermi level coincides with the position of the energy levels of the centers arising from the imperfections at pressures where this level is half-filled with electrons. Still further, it may be seen that the Fermi level rises gradually with increasing metal pressures. When T = O"K, the position of the Fermi level coincides with the highest level occupied by electrons, regardless of the degree to which the level is occupied. As an example, the position of the Fermi level a t 0°K is shown in Fig. 21b for a rapidly cooled crystal, corresponding to the specimen whose state a t high temperature is described by Fig. 9. The Fermi level is independent of KRpil.r over a wide range or pressure, and coincides with the position of a particular center level. It jumps suddenly from that level to the next higher a t the value of K R P Mwhere the lower lying level is just completely filled. This jump marks the transition from p- to n-type conduction in the present case. The variation of the Fermi level is intermediate between that depicted in Fig. 9b and Fig. 21b a t intermediate temperatures. A method for calculating the position of the Fermi level a t intermediate temperatures is described in references 80b and 81a.

+

362

F.

A. K R ~ ~ G E AND R H. J. VINK

16. PRACTICAL LIMITATIONS

The theory presented in the foregoing sections permits us to obtain the concentrations of the various imperfections as a function of p M or px,. Thus far no limitations on the partial pressures have been taken into account. Actually these pressures cannot increase indefinitely, but are limited by the occurrence of new phases. The highest pressure of M that can ever be applied obviously is the vapor pressure of pure M a t the temperature a t which the equilibrium is studied. The actual limit is even lower still because M will take up X when in equilibrium with M X or with an atmosphere containing X . It will form either solid or molten M containing some X , or a new compound M,X, for which n > m. The partial pressure p M in equilibrium with this phase lies below that of pure M in all cases. Thus if, starting with the compound M X , the partial pressure of M is increased, M X becomes richer in M until a new phase is formed a t a critical value of p ~ The . deviation from stoichiometry and the imperfections follow the relations described in the foregoing sections below this critical pressure. Similarly, phases richer in X are formed on the side of low p ~ corresponding , to high pressures px,. The range of partial pressures in which the solid compound M X is stable might well cover only one or a t most a few of the many ranges in which simplified neutrality conditions hold. I n the case of mixed Schottky-Wagner (or Frenkel) disorder and antistructure disorder (Section 12b), this limitation may have the consequence that only a few of the many ups and downs in the concentrations of free electrons and holes suggested by the theory (see Fig. 18) can be realized. Thus it may happen that the concentration of electrons or holes either increases or decreases monotonically or passes through a single maximum or a minimum when the total pressure of one of the components is increased. If the conditions are such that only one range of the simplified neutrality condition can be realized, the solid will obey one simple law over the entire stability range of the solid. Both PbS and Cu20 are substances in which the phase relations and the dependence of the concentrations of the various imperfections have been studied as functions of the atmosphere. Under the conditions in which PbS can reach equilibrium with an ambient atmosphere (i.e., a t a temperature T > 5OO0C), the range of solid PbS is limited by the points where lead- or sulfur-rich liquids are formed (i.e., by points of the threephase line (PbS),-liquid-ges) .21,820-87n Under the conditions in which equilibria CuzO-gas are usually studied, the limits are set by the formation of solid copper or solid CuO (three-phase equilibria (Cu2O),-(Cu),870

J. Bloem and

F. A. Kroger, 2. physik. Chem. (Frankfurt) 7, 1 (1956).

IMPERFECTIONS I N CRYSTALLINE SOLIDS

363

gas, and (C~2O)~-(CuO)~-gas) .77°,78b,87b,88-Q1 The statistics of the formation of new solid phases has been studied by AndersonQ2and more recently by Reesag3 The statistics of melt,ing has been treated by M a t y a ~ . ~ ~

17. COMPARISON WITH EXPERIMENT The theory presented in the foregoing predicts a variation in the concentration of the various imperfections with the composition of the atmosphere in which the material is prepared. The theory can be checked by comparing this predicted variation with the variations of the properties determined by the imperfections. Such properties are the electronic and ionic conductivity, optical absorption, luminescence, photoconductivity and magnetism. So far, the general theory has been tested experimentally only for CdSgslo and PbS.21s820 Good agreement was observed in both cases. The great majority of cases in which experimental data are available have been explained using the Schottky-Wagner theory in a simplified form. This, as we shall see, corresponds to one of the ranges in the general theory. For example, the dependence of the electronic conductivity of ZnO on the pressure of oxygen has been discussed on the assumption that an extra Zn atom gives rise to an interstitial Zni ion and one free electron :4,95-96

Application of the law of mass action to this reaction, taking 6 to n[Zn<] po2-*. If all the free electrons arise from the extra Zn, that is, if

-

then Since

<< 1, leads (17.2) (17.3) (17.4) (17.5)

H. Dunwald and C. Wagner, Z . physik. Chem. B17,467 (1932). J. Gundermann, I(.Hauffe, and C. Wagner, 2.physik. Chem. BS7, 148 (1937). s8 C. Wagner and H. Hammen, 2. physik. Chem. B40, 197 (1938). 0. BSttger, Ann. Physik 10, 232 (1952). 91 C. A. Hogarth, Z . physik. Chem. 198, 30 (1951). 92 J. S. Anderson, Proc. Roy. Soe. A186, 69 (1946). 93 A. L. G. Rees, Trans. Faraday Soc. 60, 335 (1954). 9 4 Z. Matyas, Czechoslou. J . Phys. 4, 14 (1954). 95 K. Hauffe, Ergeb. ezukt. Nuturw. 26, 193 (1951). K. Hauffe, “Reaktionen in und an festen Stoffen,” p. 127. Springer, Berlin, 1955. 87b 88

364

F. A. KROGER AND H. J. VINK

-

according t o relation (7.25), Eq. (17.4) may also be written in the form n

pz,s.

(17.6)

If this result is compared with the general theory presented in the foregoing sections, i t is seen that the condition (17.3) corresponds t o one of the various possible simplications of the neutrality condition. Accordingly, the relations (17.4) or (17.6) hold in the range of the general theory associated with the condition (17.3). The general theory shows th a t the condition (17.3) holds only over a limited range of pressures. Moreover, it provides values for the exact limits of this range. The following examples furnish other illustrations of the correspondence between a simplified form of the theory and a range in the general one. Wagner4 recognized th at the electronic conductivity may become independent of the partial pressures of the components of the ambient gas when the concentration of intrinsic electrons and holes is sufficiently great. This principle has been used to explain the properties of C U O . ~Wagner ~~ also noted that ionic conductivity may become independent of the partial pressures if the concentration of atomic imperfections arising from thermal disorder exceeds that originating in a deviation from stoichiometric composition. These two cases are clearly similar t o those associated with range I1 in Figs. 10 and 12 and Figs. 9a and 11, respectively. The general theory not only provides the conditions that must be fulfilled in order to achieve these cases (viz.,Ki KF‘ in the case of Frenkel disorder, and Ki 2 Ks’ in th at of Schottky-Wagner disorder), but also provides the width (9.40) and position of th e range on the log KRpM or log KR’pM axis associated with each case. As shown in Section 16, it may well happen that the partial pressures which may be realized fall within one range of the general theory. This explains why the simplified theory has been adequate to account for the experimental results in many cases. The following are examples of cases in which the electronic conductivity of crystals has been determined as a function of the partial pressures of the components, and in which the results have been explained in terms of a simplified theory:97bZn0,91~98-104 Cd0,91~100~106~106 H. H.von Baumbach, 13. Diinwald, and C. Wagner, 2. physik. Chem. B22, 226 (1933). 97b For a more detailed presentation see references 95 and 96. 98 See reference 30, p. 466. 99 K. Hauffe and J. Block, 2. physik. Chem. 196, 438 (1950). 100 H. H. von Baumbach and C. Wagner, 2. physik. Chem. B22, 199 (1933). 101 C. A. Hogarth, Phil. Mag. [7] 39, 260 (1948). 102 D. J. M. Bevan and J. S. Anderson, Discussions Faraday SOC. 8, 238 (1950). 103 E. Scharowsky, 2. Physik 136, 318 (1953). 970

IMPERFECTIONS IN CRYSTALLINE SOLIDS

365

CdSe,l07 Fe0,108 CU0,91,97a,109NiS 7 110 SnS9 111,112 NiO 9 99,106 MgO1 113,114 Ag2S,115,116 Cu2S,117v118 Ti02,119-122 FeS2,lZ3SnOz,120Bi2O3,lZ4 In203,125 PrOZ,lz6and Fe304.34Indications of the occurrence of two ranges have been observed in cu20,s7b~s8-91~127 Ba0,1283129 Bi2Sa,130 Ca0,131and possibly CuI.132-134 The concentration of free electrons or holes varies with the partial pressure of the components in the vapor phase in the way expected for crystals possessing Frenkel or Schottky-Wagner disorder in all these cases. Examples of a variation of the concentration of free electrons and holes with the vapor pressure of the type characteristic of simple antistructure disorder or of a combination of antistructure disorder and Schottky-Wagner (or Frenkel) disorder are not available. There are several cases, however, where such concentrations have been determined F. Stockmann, Z . Physik 127, 563 (1950). C. A. Hogarth and J. P. Andrews, Phil. Mag. 171 40, 273 (1949). lo8 C. A. Hogarth, Nature 161, 60 (1948);167, 521 (1951); Proc. Phys. Soe. (London) B64, 691 (1951). 10' K. Hauffe and H. G. Flint, Ann. Physik 16, 141 (1955). l o 8 K.Hauffe and H. Pfeiffer, Z . Metallkunde 44, 27 (1953). l o g J. Gundermann and C. Wagner, Z . physik. Chem. B37, 157 (1937). K. Hauffe and H. G. Flint, 2.physik. Chem. 200, 199 (1952). J. S. Anderson and M. C. Morton, Nature 166, 112 (1945). J. S. Anderson and M. C. Morton, Proc. Roy. Soc. A184, 83 (1945). 113 E. Yamaka and K. Sawamoto, Phys. Rev. 96, 848 (1954). 114 E. Yamaka and K. Sawamoto, J. Phys. Soe. Japan 10, 176 (1955). 116 C. Wagner, Z . physik. Chem. B21, 42 (1933). 116 C. Wagner, Z. physik. Chem. B22, 181 (1933). L. Eisenmann, Ann. Physik 10, 129 (1952). E. Hirahara, J. Phys. SOC.Japan 6, 428 (1951). 119 M. D. Earle, Phys. Rev. 61, 56 (1942). l2O M. Foex, Bull. SOC. chim. France [5]11, 6 (1944). lz1 R. G. Breckenridge and W. R. Hosler, Phys. Rev. 83, 227 (1951). 122 D.C. Cronemeyer, Phys. Rev. 87, 876 (1952). lZ3 J. Bittncr, Jena. Zeiss Jahrb. p. 177 (1950). l z 4 R. Mansfield, Proc. Phys. SOC. B62,476 (1949). 125 G. Rupprecht, Z . Physik 139, 504 (1954). lZ6 R. L. Martin, Nature 166, 202 (1950). lZ7 N. Feldmann, Phys. Rev. 64, 113 (1943). lZ8 Y. Ishikawa, T. Sato, K. Okumura, and T. Sasaki, Phys. Rev. 84, 371 (1951). l z e S. Narita, J. Phys. SOC.Japan 7, 221 (1952). 130 G. Galkin, G. Dolgich, and W. Jurkov, J. Tech. Phys. (U.S.S.R.) 22, 1533 (1952). 131 K.Hauffe and G. Tranckler, Z . Physik 136, 166 (1953). 132 K.Nagel and C. Wagner, 2. physik. Chem. B26, 71 (1934). 133R.J. Maurer, J. Chem. Phys. 13, 321 (1945). 134 B. H. Vine and R. J. Maurer, 2. physik. Chem. 198, 147 (1951). lo*

lo5

366

F. A. KROGER AND H. J. VINK

as functions of the deviation A x from stoichiometric composition. According to Section 14, A x varies monotonically with the pressure of the components of the system in the vapor. Therefore the variation of properties with A x is closely related to the variation of the same properties with the partial pressures. There are several cases in which the dependence of concentration of free electrons or holes on A x cannot be explained by assuming SchottkyWagner or Frenkel disorder alone. Instead, a contribution from antistructure disorder is indicated. Examples of such cases are provided by CdSb,28,29 ZnSb,136-137 and MgzSn.1380 These all show a minimum in the density of charge carriers near the stoichiometric point. The concentration increases with A x both for positive and negative values of the parameter, but a change from p t o n type is absent. Probably BizTea provides a case of pure antistructure disorder. It exhibits n-type conductivity for an excess of the electronegative component (Te) and p-type conductivity for an excess of the electropositive component (Bi).13sb-13sf The presence of antistructure disorder is not unlikely in these substances, since they are intermetallic compounds whose components exhibit a relatively small difference in electronegativity. The systems KBr K,139CuI I,133 CuzS 5,"' and CUZO Os9 provide examples of cases in which the deviation from stoichiometric composition Ax has been measured for crystals in equilibrium with a vapor. The result A x p~ for KBr can be explained by the formation of nonionized imperfections (range IIIb of Fig. 22). Further experiments with CuI and CuzO are needed to explain the experimental results. The influence of the deviation in stoichiometry on the density of KBr K,1404 FeS S,I4Ob NiS S1400has been observed. The presence of localized levels in the forbidden energy gap can be

+

-

+

+

+

+

+

+

E. D. Deviatkova, I. P. Maslakovits, and L. S. Stilbans, J . Tech. Phys. (U.S.S.R.) 22, 129 (1952). 136 A. Eucken and G. Gehlhoff, Verhandl. deut. physik. Ges. 14, 169 (1912). 137 A. Eucken and G. Gehlhoff, Z. Metallkunde 12, 194 (1920). 1380 W. D. Robertson and H. H. Uhlig, Trans. Am. Inst. Mining Met. Engrs. J . Metals 180, 345 (1949). 138b W. Haken, Ann. Physik 32, 319 (1910). 13& F. Korber and U. Hoshimoto, Z. anorg. u. allgem. Chem. 188, 114 (1930). 138d F. I. Vasenin, J . Tech. Phys. (U.S.S.R.) 26, 397 (1955). 138eR. M. Vlasova and L. S. Stilbans, J . Tech. Phys. (U.S.S.R.) 26, 569 (1955). 138f T.C. Harman, S. E. Miller, and H. L. Goering, Bull. Am. Phys. Soc. SO, (7), 35 (1955). 189 H. Rogener, Ann. Physik 29, 386 (1937). l4O0 H. Witt, Nachr. Akad. Wiss. GGttingen Physik. Kl. No. 4, 17 (1952). G. Hagg, Nature.131, 167 (1933);G. Hagg and I. Sucksdorff, 2. phy8ik. chem. B22,444 (1933). 14(b M. Laffitte, Compt. rend. 243, 58 (1956).

136

367

IMPERFECTIONS I N CRYSTALLINE SOLIDS

determined, at least in principle, by observing optical transitions between these levels and the valence or conduction band. In fact, such transitions can lead to luminescence, and photoconductivity, as well as absorption. A classical example of the relation between the strength of an absorption arising from an imperfection and the composition of the vapor in which the crystal has been prepared is provided by the system KC1 K. Here the concentration of color centers, which are usually called “E’ centers,” and which correspond t o V X in the present terminology, is found to be proportional to the partial pressure of potassium in the vapor139(range IIIb of Fig. 22). On the other hand, excess halide produces absorption due to “ V centers,” which are designated Vaa in our t e r m i n o l ~ g y . ~Other ~ , ~ ~cases . ~ ~ in ~ which absorption due to imperCd0,144p146 AgBr,146 fections has been observed are NiO,142 Zn0,109J43,144 AgI,14’-149b Mg0,160,161 Ba0,162-167 Ti02,60,122 Th02,168-160CUZO,“~ CeOz ce203,162-164 UO 2, 166 CaF2,’66 M o O ~ , and ~ ~ Proz ~ J ~ ~Pr~O3.l’~

+

+

+

L. M. Shamovski, Doklady Akad. Nauk S.S.S.R. 91,229 (1953). M. Le Blanc and H. Sachse, 2.Elektrochem. 32, 58 (1926). 143 E. Mollwo and F. Stockmann, Ann. Physik 3, 223 (1948). 144 W. A. Weyl and T. Forland, 0. N. R. Tech. Rept. Nr. 2, Contract Nr. N 60nr 269, Task Order Nr. 11 NR 032-265, March (1949). 146 J. Stuke, 2.Physik 137, 401 (1954). 146 W. Kaiser, 2. Physik 132, 497 (1952). 147B.A. Bartsjewski, J . Exptl. Theoret. Phys. (U.S.S.R.) 12, 225 (1942). 148K. W. Sjalimova and N. C. Mendokov, Doklady Akad. Nauk S.S.S.R. 82, 575 (1952). 149a K. W. Sjalimova, Dokludy Akad. Nauk S.S.S.R. 96, 487 (1954). 1496 P. N. Kokhanenko and L. V. Grigoruk, Soviet Physics 2, 530 (1956). 160 H. Weber, Naturwissenschaften 38, 140 (1951). 161 H. Weber, 2. Physik 130, 392 (1951). 162 M. Schriel, 2. anorg. u . allgem. Chem. 231, 313 (1937). l E 3 W. W. Tyler and R. L. Sproull, Phys. Rev. 83, 548 (1951). 1 5 4 E . 0. Kane, J . Appl. Phys. 22, 1214 (1951). lK6 W. C. Dash, Phys. Rev. 92, 68 (1953). 166 R. L. Sproull, R. S. Bever, and G. G. Libowita, Phys. Rev. 92, 77 (1953). 167 G. G. Libowita, J . Am. Chem. Soe. 76, 1501 (1953). 168 W. E. Danforth, Phys. Rev. 86, 416 (1952). 169 0. A. Weinreich and W. E. Danforth, Phys. Rev. 88, 953 (1952). 160 J. H. Bodine and F. B. Thiess, Bull. Am. Phys. SOC.30, (2), 9 (1955). K. Kassel, Ann. Physik 10,211 (1952). 162 G. Brauer and H. Holtschmidt, 2. anorg. u. allgem. Chem. 266, 105 (1951). 163 K. Hauffe and H. Peters, 2. anorg. u. allgem. Chem. 266, 345 (1951). G. Brauer and H. Gradinger, 2. anorg. u. allgem. Chem. 277, 89 (1954). 166 K. B. Alberman and J. S. Anderson, J . Chem. SOC. 2, 303 (1949). 166 E. Mollwo, Nachr. Akad. Wiss. GGttinger, Math-physik-chem. Abt. I l a , 1,79 (193t). 16’ 0. Glemser,Angeur. Chem. 63, 449 (1950). 168 L. Kihlborg and A. Magnhlli, Acta Chem. Scand. 9,471 (1955). 141

142

368

F. A. K R ~ ~ G E AND R H. J. VINK

Photoconductivity arising from absorption involving imperfections has been studied extensively in the alkali halides and has also been observed in Cuz0169-1sz and Ba0.163,166 Luminescence involving imperfections has been observed in Zn0,173,174 ZnS,173*176-177 CaO,Is8 Ba0,ls9 (Ba,Sr)0,180LiF,lB1NaCl,lE2and SiOz.18s Under special conditions, vacancies may give rise to ferrimagnetism. This behavior may be expected in antiferromagnetic substances when the vacancies are 0rdered.18~An example is provided by FeS.186-187 The same behavior may occur in CrS1881189 and Cr2S3.190 Paramagnetism arising from electrons for holes bound at vacancies has been observed in KC1,191-194 NaC1, KBr, and LiF.I94 As we saw in Section 4, the migration of atoms usually requires imperfections, particularly interstitial atoms or ions or vacancies. Examples of systems in which the migration (self-diffusion)of lattice constituents under the influence of a concentration gradient has been measured are : ZnO(Zn),196,196 R. W. Pohl, Physik. 2. 39, 36 (1938). F. Seitz, Revs. Mod. Phys. 18, 384 (1946). 171 B. Gudden, Physik. 2. 32, 831 (1931). 1 7 * B. Schonwald, Ann. Physik 16, 395 (1932). 173F. A. Kroger and H. J. Vink, J. Chem. Phys. 22, 250 (1954). 174 E. Mollwo, 2. Physik 138, 478 (1954). 176 R. H.Bube, Phys. Rev. 80,655 (1950). 176 R. H. Bube and S. Larach, J . Chem. Phys. 21, 5 (1953). 117 A. Addamiano, J . Chem. Phys. 23, 1541 (1955). 178 J. Ewles and N. Lee, J. Electrochem. SOC.100,402 (1953). 179 V. L. Stout, Phys. Rev. 89, 310 (1953). lSo R. E. Aitcheson, Nature 164, 1088 (1949). C . C. Klick, Phys. Rev. 79, 894 (1950). lS2 Th. P. J. Botden, C. *Z. van Doorn, and Y. Haven, Philips Research Repts. 9, 469 (1954). l S 3J. Ewles and R. F. Youell, Trans. Faraday SOC. 47, 1060 (1951). lS4L. Ne61, Revs. Mod. Phys. 26, 58 (1953). lS6 R. Juza and W. Biltz, 2. anorg. u. allgem. Chem. 206, 273 (1932). lS6F. Bertaut, Compt. rend. 234, 1295 (1952). lS7 F. K.Lotgering, 2. physik. Chem. (Frankfurt) 4, 238 (1955). ls8 H. Haraldsen and A. Neuber, Naturwissenschuften 24, 280 (1936). Is9 H. Haraldsen and E. Kowalski, 2. anorg. u. allgem. Chem. 234, 388 (1937). l o o F. Jellinek, Koninkl. Ned. Akad. Wetenschap. Proc. Ser. B68, 213 (1955). I g 1 A. B. Scott, H. J. Hrostowski, and L. P. Bupp, Phys. Rev. 79, 346 (1950). lg2C. A. Hutchinson, Phys. Rev. 87, 1125 (1952). lg3A. F. Kip, C. Kittel, R. A. Levy, and LA. M. Portis, Phys. Rev. 91, 1066 (1953). l g 4W.Kanzig, Phys. Rev. 99, 1890 (1955). lg6P. H. Miller, Jr., Phys. Rev. 60, 894 (1941). l g 8R. Lindner, A d a Chem. Scand. 6, 457 (1952). 170

IMPERFECTIONS I N CRYSTALLINE SOLIDS

369

PbO(Pb),lg7 PbS(Pb),198~199 BaO(Ba),156~z00 FeO(Fe),201 Fe304(Fe),201 Fez03(Fe),202 AgzS(Ag $3) ,203,204 CuzO(Cu) ,205~206 Cu Be(Cu), 2 0 7 BaTi03 209 ZnFezOl (Zn;Fe) ,196,202 Na,W03(Na) ,210a and (Ba) ,208 CaFez04(Ca;Fe)) ZnSb(Sb).210bFor a survey, see Hauffe211and Jost.212Although most of these measurements have been carried out without paying particular attention to the stoichiometry, it is likely that the results have been influenced by deviations from the stoichiometric composition. The system aAgzTe Te213provides an example of a case in which ionic conductivity is directly related to a deviation in stoichiometric composition.

+

18. THE EFFECT OF HEATINGIN A STREAM OF INERT GAS As has been shown in Section 7 atoms M and molecules X , will occur in the vapor which is a t equilibrium with a compound M X . The partial pressures p~ and px, satisfy the relation PM . PX"''~ = K MX.

(18.1)

Various combinations of pM and px, are possible at any given temperature. Each corresponds to a particular composition of the compound M X , that is, to a particular stoichiometric ratio and thus to a particular value of Ax. It can easily be shown that whenever K M Xis constant, the sum of the partial pressure of M and X

+

PM PX, = p passes through a minimum when pM = npx,, that is, when the vapor has the composition M X if the composition of the vapor is varied a t a fixed temperature. This is not quite correct, for K M Xis always a slowly varying R. Lindner, Arkiv Kemi 4, 385 (1952). J. S.Anderson and J. R. Richards, J . Chem. SOC.p. 537 (1946). 199 R. F. Brebricq and W. W. Scanlon, Phys. Rev. 96, 598 (1954). 2OO R. W. Redington, Phys. Rev. 87, 1066*(1952). 201 L. Himmel, R. F. Mehl, and C. E. Birchenall, Trans. Am. Inst. Mining Met. Engrs. J . Met& 6 , 827 (1953). 2ozR. Lindner, Arkiv Kemi 4, 381 (1952). 203 D . Peschanski, J . Chim. phys. 47, 433 (1950). 204 C. Wagner, J . Chem. Phys. 21, 1819 (1953). 206 W. J. Moore and B. Selikson, J . Chem. Phys. 19, 1539 (1951). 206 W. J. Moore and B. Selikson, J . Chem. Phys. 20, 927 (1952). 207 H. Reinhold and H. Moehring, 2.physik. Chem. B38, 221 (1937). 208 A. Garcia-Verduch and R. Lindner, Arkiv Kemi 6, 313 (1952). 209 J. A. Hedvall, C. Brisi, and R. Lindner, Arkiv Kemi 6, 377 (1952). 2100 J. F. Smith and G. C. Danielson, J . Chem. Phys. 22, 266 (1954). 210b B. I. Boltaks, Doklady Akad. Nauk S.S.S.R. 100, 901 (1954). 211 See reference 96, p. 350. 212 W. Jost, "Diffusion in Solids, Liquids, Gases," p. 201. Academic Press, New York, 1952. 213 J. Appel, Z. Naturforsch. 10a, 530 (1955). 197 19s

370

F. A. KROGER

AND H. J. VINK

function of Ax, and therefore of p~ and px,. As a consequence, the ratio of concentrations of M and X found in the vapor at the minimum of P will usually differ slightly from unity. On the other hand, it can be shown214that whenever a crystal is heated in such a way that its vapor is continuously removed under conditions which maintain the equilibrium, the solid tends to change its composition until crystal and vapor have the same composition. Furthermore, it is clear that this procedure leads to a system in which P has the minimum value. Thus the crystal which is in equilibrium with the vapor a t the minimum point also has a composition which deviates slightly from the simple stoichiometric ratio. The following procedure can be used to determine the precise composition of the vapor and the solid at this point: (a) Determine the values of p~ and px. a t the minimum from the relation (18.1), using the value of K M Xknown for the compound in which [ M ] : [ X= ] 1. (b) Determine Ax for a crystal which is at equilibrium with the vapor having the partial pressures P M and px, found in (a) by use of the methods outlined in Section 14.

If Ax is found to have a value <1020 that is, less than about gram atoms per mole, the error made in the calculation of pjf and px, with the use of (a) is negligible. We can improve the result for larger values of Ax, by going through the cycle again, using the relation E

p,Cl+G (px,)””

=

KMX

(18.2)

in (a), instead of (18.1). Here 6 is A x expressed in units of gram atoms per mole. These considerations can be checked experimentally by heating specimens in a slow stream of inert gas, or by heating them in a “bad” vacuum. This has been done for PbS214and satisfactory agreement between the theory and experiment has been found.

19. EQUILIBRIA VAPOR-LIQUID AND SOLID-LIQUID The equilibrium between a liquid and an ambient gas phase or between a liquid and solid may be treated by the methods used above for the equilibrium between solid and gas. In order t o apply the theory, it is necessary t o assume a model for the liquid which is equivalent to the normal crystalline model of the solid. Two types of models may be used. One may be called the “gas-like” and the other the “ crystal-like”mode1. 214 J. Bloem and F. A. Kroger, 2. physik. Chem. (Frankfurt) 7, 15 (1956).

I M P E R F E C T I ONS I N C R YS T AL LI N E SOLIDS

37 1

I n the gas-like model, the perfect liquid is assumed to consist of molecules M X . Dissociation products of the molecules, M and X , act as imperfections. Ionization may occur and lead to additional imperfections such as M ,X’, n, p. This model has been used by Schottky et al. in their classical treatise on thermodynamic^.'^ I n the crystal-like model, the liquid is regarded as a crystal with an exceptionally large concentration of imperfections of the SchottkyWagner or Frenkel type. Intrinsic electronic excitation and ionization are assumed to take place exactly as in a solid, with values of the constants corresponding to the conditions in the liquid. This model has recently been used by van den Boomgaard216to discuss zone melting of dissociating compounds. Both models are equivalent, as far as the statistics of imperfections is concerned, and therefore give the same results. The crystal-like model will be used in the following. The equilibria between liquid and vapor need no further comment, for all the considerations of the previous sections apply to them unchanged. New reactions are met in discussing the equilibria between liquid and solid. I n addition to the reactions governing the equilibria of the imperfections within each phase, there are reactions describing the transfer of atoms from one phase to the other, analogous to the evaporation-condensation reactions associated with the equilibrium between a condensed phase and a vapor. Transfer of electrons or ions need not be considered since both phases remain neutral. If such particles are transferred, electrons and ions are transferred in equal concentrations, so that the final reactions may be expressed in terms of the transfer of neutral atoms.216o As an example, let us consider a crystal and a liquid in both of which Schottky-Wagner disorder predominates. The transfer of atoms is then described by the equations

The suffixes S and L indicate whether an atom of type M or X or a vacancy V M or Vx is present in the solid or liquid. Since (M,)s and (XX)S are the normal constituents of the solid, the concentrations of these atoms are not markedly influenced by the reactions (19.1) and (19.2). The same holds for (AIM),and (XX),. Application of the law of mass action to 21K

J. van den Boomgaard, Philips Research Repts. 11, 27 (1956). This statement holds only as far as the bulk of the phases is concerned. Charges may accumulate a t the boundary. The consequences this may have for the properties of the phases near the boundary will be considered in Section 20.

216n

372

F. A. KROGER AND H. J. VINK

the foregoing reactions, under the assumption that the vacancies in both phases are distributed a t random, leads to the relation (19.3)

I n other words, the ratio of the concentrations of vacancies in the two condensed phases is constant, The constants K V M and Kv, are similar to the distribution coefficients regulating the distribution of foreign atoms between two phases. I n fact (19.3) describes the distribution of atoms present in excess of the stoichiometric composition between the two phases. The concentrations of all the imperfections, and thus the compositions of the two phases, can be calculated with the use of (19.3), the two sets of equations corresponding to the internal reactions in the two phases, and a neutrality condition for each of the two phases, provided the constants are known. Although gas-solid, gas-liquid, and liquid-solid equilibria can be calculated in this way, one should not infer that all the phases are stable. Coexistence of the three phases is possible only under particular conditions, viz. a t a giver) pressure a t a particular temperature. 20. SURFACE LAYERS

It was shown in Section 13 that the crystal may acquire a surface layer having a composition differing from that of the bulk material upon cooling. There are several reasons apart from this why the properties of the crystal near the surface may differ from those of the bulk. Thus surface states and adsorbed atoms may lead to a variation of the concentration of electrons and holes in both free and bound states. A survey of these phenomena is given by Engell.216 For similar effects see also r e f e r e n ~ e s . These ~ ~ ~ . effects ~ ~ ~ will not be treated here since they lie outside the scope of this paper. There is still another effect that needs our attention however. For various reasons there may be a potential difference between the bulk of the crystal and the vapor. Part of the potential difference may arise from the fact that the crystal has a net charge. The remainder may arise from two sources: (a) A dipole layer formed as a consequence of a shift in the H. J. Engell, in “Halbleiterprobleme I ” (W. Schottky, ed.), p. 249. Vieweg, Braunschweig, 1954. 217 C. G. B. Garrett and W. H. Brattain, Phys. Rev. 99, 376 (1955). 218 R. H. Kingston, Phys. Rev. 98, 1766 (1955); J. Appl. Phys. 27, 101 (1956). 216

IMPERFECTIONS I N CRYSTALLINE SOLIDS

373

position of the positive and negative ions and their electron clouds. Calculations of this effect have been made for pure ionic crystals by V e r ~ e y . I~nl ~ general, the shift involves only one or two atomic layers. (b) The formation of a diffuse space-charge layer involving the electrons or holes and the charged atomic imperfections. This layer extends much deeper into the crystal. The effect has been considered by LehovecZz0 for crystals such as NaCl containing only charged atomic imperfections. The concentrations of the charged imperfections vary with the depth below the surface in consequence of the second effect. The concentrations decrease or increase exponentially in such a way that the total charge is zero deep in the crystal, and corresponds to the solutions discussed in the foregoing sections. Ill. Crystals Containing Foreign Atomszz0"

21. INTRODUCTION

Foreign atoms which are incorporated in crystals are found to influence the properties in various ways. Thus the incorporation of cadmium or calcium in AgCl and AgBr enhances the ionic conductivity.1b.221 Incorporation of lithium in NiO under oxidizing conditions increases the concentration of free holes and, therewith, the p-type ~ o n d ~ ~ t i v i t y . ~ ~ ~ ~ Gallium incorporated in CdS under sulfurizing conditions changes the color of pure CdS from yellow to a bright red. I n contrast, the incorporation of gallium under reducing conditions leaves the color unchanged, however, it causes the electronic conductivity to increase sub~tantia1ly.g~~~ Small concentrations of manganese incorporated in AlZO3increase the magnetic susceptibility by an amount corresponding to a magnetic moment of 4.9 Bohr magnetons per Mn atom. This moment is associated with the Mna+ ion. In contrast, the increase corresponds to a magnetic moment of 3.84 per Mn atom, associated with the Mn4+ ion, Finally, divalent manganese when manganese is incorporated in Ti02.224 has been found present when manganese is incorporated in ZnS, the solid 219E. J. W. Verwey, Rec. trav. chim. 66, 521 (1946). 820 K. Lehovec, J . Chem. P h p . 21, 1123 (1953). 220s We refrain from using the term impurities. Nowadays foreign atoms are added on purpose and in concentrations that are regulated accurately. It seems advisable to speak of impurities only in cases where one is dealing with undesired contaminations. 221 W. Koch and C. Wagner, 2. physik. Chem. B38, 295 (1937). 222 E. J. W. Verwey, P. W. Haayman, and F. C. Romeyn, Chem. WeekbZad 44, 705 (1948). 2zaE. J. W. Verwey, P. W. Haayman, F. C. Romeyn, and G . W. van Oosterhout, Philips Research Repts. 6, 173 (1950). 2p4 P. W. Selwood, T. E. Moore, M. Ellis, and W Wethington. J . Am. Chem. SOC.71, 693 (1949).

374

F. A . KROGER AND H. J. VINK

solution showing a characteristic orange-yellow luminescence.226These few examples may suffice to show that the incorporation of foreign atoms may have a variety of consequences which depend on the base crystal, the foreign atom, and the composition of the adjacent phase. Part of the effects arising from incorporation of a foreign atom is associated with the foreign atom and its particular state; another part is caused by the electronic and atomic inperfections of the base crystal. In the following the latter will be designated as native imperfections. It will be shown that there is a relation between the states and concentrations of the foreign atom and the native imperfections. Moreover, it will be demonstrated that all depend on the composition of the adjacent phases in equilibrium with the crystal and thus can be regulated by altering this composition. The effects of fixed concentrations of foreign atoms will be discussed in this part of the article, both for the case in which only one type of foreign atom is present and for that where two are present. The effects of these relations on the solubility of the foreign atoms will be treated in Section IV. Just as in Section 11, equilibria between solid and vapor will be considered in the main. I n all of the following considerations, the concentrations of the foreign atoms and of the imperfections will be assumed to be small atoms per mole = 1020 atoms per cm3, or less).

22. THESITESOCCUPIED BY FOREIGN ATOMS I n a compound of the type M X , foreign atoms may be incorporated a t normal M‘ or X lattice sites, or at interstitial sites. The position occupied depends on the energy balance. Factors such as the chemical bonding, van der Waals’ forces, Born repulsion, etc., will play a role. As far as lattice positions are concerned, it can be said that a foreign atom tends to occupy the position of the atom to which it is closest on the electronegativity scale if there is a marked difference in electronegativity between M and X . Thus “metal” atoms usually occupy M positions, and “metalloid” atoms X positions. The influence of size may dominate if the difference in electronegativity is small, or if the electronegativity of the foreign atom is approximately intermediate between that of M and X . Thus it has been suggested226that lead atoms may occupy indium sites in InSb, but antimony sites in AlSb. Foreign atoms substituting for M or X atoms will be indicated by F M or F x , respectively. Interstitial foreign atoms are denoted by F;. Here again, it is not necessary to distinguish the 225 226

F. A. Kroger, Physica

6, 369 (1939).

H. Wellcer, 2. Naturforsch. 7% 744 (1952).

IMPERFECTIONS IN CRYSTALLINE SOLIDS

375

various possibilities for chemical bonding. The symbols F M or FX will stand for the substitution of the foreign atom F for an atom M or X, regardless of the actual type of chemical bonding.

23. ENERGY LEVELSDUE TO FOREIGN ATOMS

If an atom is incorporated in a crystal M X in one of the ways indicated in Section 22, it may yield or trap electrons and thus take part in reactions involving electrons or holes. The extent to which it will do so is determined by the energy needed for the reactions, that is by the position of the levels associated with the foreign atom relative to the energy bands of the base crystal. We noted in Section 2b, in connection with the discussion of imperfections induced by thermal disorder, that the actual positions of the electronic levels can only be determined on an experimental basis. In principle, the same is true of the levels associated with foreign atoms. Nevertheless, we must have a t least a general notion of the positions of the levels in order to proceed with our discussion. We shall attempt to formulate a few general rules. In the case of interstitial foreign atoms, we may expect to apply arguments similar to those used in discussing the positions of levels of interstitial atoms in the pure compound (Section 2b). Such atoms may produce levels of the type illustrated in Figs. 2 and 3, the first case being valid for electropositive and the second for electronegative atoms. Cases in which the energies E land ESare negative may also occur. The position of the levels associated with substitutional foreign atoms depends to a large extent on the relative number of valence electrons on the foreign atom and on the lattice constituent which normally occupies this position. If the foreign atom has more valence electrons than the atom which is replaced, the center may tend to give off one or more of the excess electrons to the conduction band. The tendency to bind additional electrons will be small. I n general, the position of the levels of such atoms is similar to that of interstitial M atoms (Fig. 2). If, on the other hand, the foreign atom contains fewer valence electrons than the normal lattice constituent, the center will tend to bind extra electrons rather than to release them. I n this case, the levels are similar to those of interstitial X atoms (Fig. 3). These rules hold regardless of whether the foreign atom occupies a n M or an X site. Thus, in CdS, Ga possessing three valence electrons, may replace Cd, having two valence electrons, and C1, with seven valence electrons, may replace S with six. Both give rise to occupied levels close to the conduction band. It is not always possible to see which of the atomic electrons may be

376

F. A. KROGER AND H. J. VINK

regarded as valence electrons. I n many cases a decision can be reached only by considering experimental data. I n most cases only one, or perhaps two electrons can be dissociated from atoms FM or FX or bound to them. The ionization energies of such electrons can sometimes be estimated roughly by using hydrogen- or helium-like models. For higher steps of ionization, the ionization energy almost certainly becomes larger than the band gap. Therefore, the levels corresponding to the higher ionization processes lie outside the gap and need not be considered. The convention applied to the native atomic imperfections will be used to designate the foreign atoms and their various states. For instance, FM' indicates a center obtained by replacing an M atom by a foreign atom F and removing one electron. Again, the levels are designated by the symbol of the center belonging to the state in which the level is occupied by an electron. If one compares the group of levels which arise from atoms which have more valence electrons than the atoms replaced in the lattice (Fig. 2) with the group of levels arising from atoms having fewer valence electrons (Fig. 3), it will be seen that the levels of the first group tend to lie above the corresponding levels of the second group. As a consequence, we must expect the levels arising from foreign atoms having the same number of valence electrons as the atoms which they replace to lie a t an intermediate position. If the substituting atom is very similar to the normal atom, no localized levels will be formed within the forbidden gap. Examples of this case GaAs have been found in the systems CdS ZnS,221Ge Si,228,229,230 Gap, and InAs InP.231 Since the criteria determining the position of the levels of atoms placed a t interstitial and normal lattice sites, respectively, are totally different, the levels associated with a particular atom in either position may be different. Thus the relatively electropositive copper atom will give rise to the levels shown in Fig. 2 when placed interstitially in ZnS. I n contrast, a neutral copper atom, possessing only one valence electron, will produce the levels in the positions shown in Fig. 3232when substituted for a zinc atom, possessing two valence electrons. On the other hand, the incorporation of the electropositive element aluminum, possessing three

+

+

+

+

F. A. Kroger, Physicu 7, 1 (1940). E. R. Johnson and S. N . Christian, Phys. Rev. 96, 560 (1954). 2 2 p A . Levitas, C. C. Wang, and B. H. Alexander, Phys. Rev. 96, 846 (1954). 230 A. Levitas, Phys. Rev. 99, 1810 (1955). 231 C. G . Folberth, Z. Nuturforsch. 10a, 502 (1955). 132 R. Bowers and N. T. Melamed, Phys. Rev. 99, 1781 (1955). 227

228

IMPERFECTIONS IN CRYSTALLINE SOLIDS

377

valence electrons, a t either site in ZnS will produce levels a t the positions shown in Fig. 2. The distribution of electrons in a center giving rise to a particular level may differ considerably from one case to another. If an extra electron, or hole, is bound in a small orbit, it may be considered to belong to the foreign atom, which therefore changes its charge and certain specific properties, such as those related to magnetism, optical absorption, etc. If, in contrast, the extra electron, or hole, is bound in a large orbit, it resides mainly a t neighboring atoms. Accordingly, the charge of the foreign atom is unchanged, whereas the charges on the neighboring lattice atoms are altered. The positions of the levels will differ markedly from one foreign atom to another if the atoms are of the first type. On the other hand the effective charge rather than the specific character of the foreign atom will determine the positions if the levels represent states in which electrons are bound in large orbits. The situation actually realized depends to a large extent on whether the atoms contain partly filled electronic shells in which electrons can easily be accomodated, or from which electrons can easily be extracted. This question can be answered only if one has precise knowledge of the distribution of electrons in the base crystal. If, for a given type of bonding, the introduction of a foreign atom with a particular charge would involve disruption of a completely filled electron shell, the particular situation will not occur in practice. The actual center will then consist of the atom, with its complete shell, and a hole which circles around it in a relatively large orbit. An example may be the center LiNi formed by replacing a nickel atom in NiO with a lithium a t ~ r n . ~ This ~ ~ Jmay * ~ be represented schematically in two ways for an ionic model: Ni2+02-Ni2+02-Li2+0 2-Ni2+0 2(a> Ni2+02-Ni2+02-Li+$02-Ni2+02-. (b) The possibility (a) must be excluded since the formation of an Liz+ ion involves disruption of a complete electron shell. Thus in this case the center will consist of the Li+ ion and a hole which moves in the neighborhood of the ion. Since the hole will probably be localized on the nickel ions, the center can also be represented in the manner Ni2+O2-Ni2+0 2-Li+O 2-Ni3+0 2-. (c>

If the formation of the Similar arguments hold for Cu and Ag in ZnS.233J34a foreign atom with a particular charge does not involve disruption of filled shells, the electron or hole may be accomodated within the foreign atom. 233 2340

F. A. Kroger and J. A. M. Dikhoff, J . Electrochem. SOC.99, 144 (1952). H.A. Klasens, J . Electrochem. SOC.100, 72 (1953).

378

F. A. KROGER

AND H. J. VINK

An example is manganese which is incorporated in A1203 in the form of Mn3+ ions on Al3+ sites.224

EQUILIBRIA SOLID-VAPOR 24. GENERALREMARKS CONCERNING FOR COMPOUNDS CONTAINING FOREIGN ATOMS I n a pure compound, complete equilibrium between solid and vapor involves both reactions in which atoms are transferred between the two phases and reactions in the solid which center about the formation of defects, association and dissociation and both ionization and recombination. All the reactions are governed by the necessity for maintaining electroneutrality. Foreign atoms, when present, may influence the equilibrium by combining with particular atomic imperfections (association) or by capturing or giving up electrons. They acquire a charge if they lose or gain electrons, and therefore must appear in the neutrality condition. The equilibrium between solid and vapor can be treated in terms of the following equations in the case of a solid containing foreign atoms: (1) all the equations which involve defects and electrons and holes in the pure crystal, except the neutrality condition. (2) equations which describe the association of the foreign atoms present with atomic imperfections, and (3) the equations which describe the ionization of the foreign atoms; (4) an equation which states that the sum of the concentrations of the ionized and un-ionized, associated and unassociated foreign atoms is equal to the total concentration of foreign atoms; (5) the complete neutrality condition, including the concentrations of foreign atoms which are charged relative to the base lattice.

This set of equations is sufficient to calculate all the unknown concentrations. According to Lidiard,16 who studied the NaCl CdC12 system, association between Cd and the sodium vacancies can be neglected a t temperatures above 400°K when the cadmium concentration is below gram-atoms per mole atoms ~ m - ~ )Kurnick19 . calculated that association of the same type may be neglected in the AgBr CdBrz system when T > 500°K and [Cd] < 10-3 gram-atom per mole. Association will not be taken into account in this part of the article since we shall restrict ourselves to low concentrations of both foreign atoms and other imperfections. As has been noted in Section 11, the explicit relations between the unknowns and the partial pressures of M or X are rather involved. An approximate solution can be obtained along the lines indicated by

+

+

IMPERFECTIONS I N CRYSTALLINE SOLIDS

379

Brouwer,'l however, just as in the case of the pure compounds. From the state of complete equilibrium calculated for the high temperature of preparation, the state a t T = 0 can again be obtained if it is assumed that some processes are frozen in during cooling, whereas others continue to take place. The arguments put forward in Sections 3b and 13 hold practically unchanged, the only difference being th a t now association processes involving the foreign atoms should also be taken into account. Yet in the following we shall again make the simple assumption that all atomic processes are frozen in, only electronic processes remaining possible. Particular examples will be treated in detail in the following sections. Antistructure disorder, of either the lattice conitituents or the foreign atom, will not be considered. The arguments of the next sections can easily be extended to include these possibilities if required. As long as we restrict our attention to the effects observed a t low concentrations of foreign atoms, the limiting pressures determined by the formation of new phases (see Section 16) are practically identical with those of the pure crystals. Another type of limit obviously is set by the solubility of the foreign atoms. Since we are dealing with a fixed concentration of foreign atoms in this part of the article, we need not be concerned with this question. By definition, we shall be limited to concentrations sufficiently low to fall within the range of solubility. 25. EQUILIBRIUM OF CRYSTAL AND VAPORFOR BINARY COMPOUNDS M X CONTAINING FOREIGN ATOMSWITH DEVIATING VALENCE AT NORMAL LATTICESITES a. One Level per Imperfection

Let us consider the effects of foreign atoms in a crystal of composition M X possessing disorder of the Frenkel type as discussed in Section 8. The level scheme now contains levels due to foreign atoms in addition to the levels of the imperfections shown in Fig. 8. It will be assumed that the foreign atoms F are present only a t M sites, and not a t X or at interstitial sites. If F has more valence electrons than M , each foreign atom F M produces a level close to the conduction band. An example might be AgCl containing Cd a t silver sites or BnS containing A1 a t zinc sites. 'Figure 23 shows the simple energy level diagram applying to this case. Applying the law of mass action to the various possible reactions, summarized in Section 24, one gets a system of relations from which the

380

F. A. KROGER AND H. J. VINK

14 FIG. 23. The electronic energy scheme of a crystal of composition M X with Frenkel disorder and a foreign atom F at an M site, for the case that F has more valence electrons than M .

concentration of the various centers can be calculated. First, relations involving the normal imperfections are (25.1) = (8.1) (25.2) = (2.2) (25.3) = (1.18) (25.4) (25.5) Second, relations involving the foreign atom are (25.6)

(25.7)

and Finally, the neutrality condition is n

+ [V,']

=p

+ [Mi'] +

tFM.1.

(25.8)

Taking the logarithms of the relations (25.1) t o (25.6), we obtain linear relations between the logarithms of the concentrations and log Knp,. The logarithm of (25.7) and (25.8) does not, however, lead to a linear relation between the logarithms of the concentrations. As indicated by Brouwer,ll approximate solutions can be obtained in this case by simplifying both (25.7) and (25.8), each solution holds over the range of KRPM in which the solution satisfies the simplifying assumptions made.234b See Fig. 27 and reference 11 for cases in which both the approximate and the exact solution have been obtained.

234b

38 1

IMPERFECTIONS IN CRYSTALLINE SOLIDS

Just as in the case of pure nonstoichiometric compounds (Sections 8 and 9), the limits of these ranges are easily determined graphically. The deviation from the simple stoichiometric ratio, in this case an excess of X , will always exceed the concentration of foreign atoms at sufficiently low values of K R p M . As a consequence, in this range the neutrality condition may be simplified to p

(25.9)

= [VM'I.

The Fermi level lies low under these conditions. Therefore, the F M centers are practically completely ionized. Hence, [FM'I

(25.10)

= [FMIs:*l.

The linear equations obtained by taking the logarithms of the relations (25.1) to (25.6), as well as of (25.9) and (25.10) can now be solved easily. We obtain expressions for all the unknown concentrations as functions of log K R p M . These relations can be represented in a double logarithmic plot. Let us assume for the constants the values Ki

=

cm-6;

KF

=

1030cm-s;

K 1 = K7

=

10l8cm-3

used for a pure compound in Section 9 (Fig. 9); let us further take K 1 3= lo1' ~ m - ~and

[ F M ]=~ 4~ 10l8 ~ ~ ~~ m - ~ .

The result is sh&wn in Fig. 24a (range I). In order to find the limit of the range in which the relations (25.9) and (25.10) hold, we must find which of the simplifying relations fails first. This occurs at the point

where p becomes smaller than [FM']. Therefore, the next range (11) is associated with the simplified relations (25.11) and (25.10) The equations can again be solved easily when these conditions are satisfied. The limit of this range, designated as I1 in Fig. 24a, lies a t

382

F. A. KROGER AND H. J. VINK 22

‘4

2r 20

I

19

IB

n a -

16 15

rc I3

I2

I

FIG.24. (a) High temperature equilibrium for a crystal of composition M X with Frenkel disorder containing a foreign atom F M having more valence electrons than M . For the energy diagram, see Fig. 23. (b) Concentration of centers a t T = 0.

IMPERFECTIONS I N CRYSTALLINE SOLIDS

383

because [ F M ] is no longer negligible compared to [FM']a t that point. In the next range (111), the simplified relations are (25.11) and

(25.12)

r

Beyond K R ~ M = KK 7 K F 1 , the neutrality condition can no longer be simplified by relation (25.11), for [VM'] becomes smaller than [FM']at this point. It is easy to see that the simplified form of the neutrality condition valid in the next range (IV) is (25.13) whereas This range comes to an end at KRpM =

(25.12) K13[FM1t0ta1 K1

because [M,'] becomes

important in the neutrality condition at that point. Therefore, the simplified relations are n = [M,'] (25.14) and [FM]= [i47~1totsl (25.12) in the last range (V). No other ranges are possible because range V also embodies the cases, corresponding to high values of p M , where the deviation from stoichiometry, that is the excess of M , exceeds the density of ionized foreign atoms. Figure 24b shows the state corresponding to T = 0 after the system has been cooled rapidly from equilibrium a t high temperature. The concentrations of the various centers are shown as functions of K R p ~This . parameter characterizes the atmosphere with which the crystal has been in equilibrium a t the previous high temperature. Figure 24b has been calculated from Fig. 24a under the assumption that the cooling is so rapid that all processes involving the migration of atoms are frozen in. On the other hand, all reactions involving electrons or holes are assumed to take place freely, so that they are in equilibrium. I n deriving Fig. 24b from the approximate solution shown in Fig. 24a, one must always keep in mind that the neutrality condition must be satisfied. A discussion of this point has already been given in Section 13. The deviations from stoichiometric comp.osition exceed the concentration of the foreign atoms by far in the extreme ranges of Figs. 24a and

384

F. A. K R ~ G E RAND H. J. VINK

24b, that is in the range of K R P M below 11.4 and K R p M beyond 18.6. As a consequence, the concentrations of the imperfections are not appreciably influenced by the presence of the foreign atoms in these ranges. In intermediate ranges, the foreign atoms play a dominant role in determining the concentrations of imperfections associated with thermal disorder. We can distinguish two regions.

< K 7 K 1 s K Fthe

foreign atoms are incorporated KilF~lt~t~i along with an equal concentration of M vacancies. Thus we are dealF X 2 . This corresponds to the ing a solid solution of the form M X CdC12 when M X = AgCl and F E Cd. The solid solution AgCl occurrence of FM' and V M ' rather than FM and V M is a consequence of the fact that in this case FM has been assumed to give rise to an occupied level lying above the empty level associated with V M . Hence, the electrons from the FM centers have been transferred to the V Mcenters.

At values of

K R p M

+

+

When KRPM> K7K13KF~ the foreign atoms simply replace M atoms, Ki[F~]tot*~ and form a solid solution M X F X 2 (e.g., AgCl CdC12).

+

+

+

The solid solution M X FX2 (region a) contains two X atoms for each F atom rather than one for each M atom. This is in agreement with the initial assumption that F should have more valence electrons than M. The composition FX2 of the foreign compound formally introduced by FX2 may correspond to that of the compound of F with X writing M X that is normally known, but need not necessarily do so. Thus in the example AgCl CdCl2 the compound CdCl2 that is incorporated has the composition normally known for the compound formed by Cd and C1. On the other hand if M X = ZnS and F = All F X 2 = AlS2. The sulfide of A1 normally has the formula A12S3.The reason that incorporation of Al2Ss does not take place is the following. Incorporation of AlzS3 would mean the formation of one M vacancy per two A1 atoms. Since the A1 center assumes one position charge (Aleno),the M vacancy must be able to carry a double negative charge ( VM").But if the V Mlevels are situated as shown in Fig. 23, VM"cannot be formed. Therefore, in assuming the energy levels as shown in Fig. 23, the possibility of incorporation X ~excluded right from the start. A case in which one of the of ( F M ) ~ is native atomic imperfections gives rise to two levels within the forbidden gap will be discussed in Section 25b. F X (region b), the foreign atom F is In the solid solutions M X incorporated with fewer atoms of X than in region a. Chemically speaking

+

+

+

385

IMPERFECTIONS IN CRYSTALLINE SOLIDS

this corresponds to chemical reduction, favored by low values of p x , , or high values of p M , as is shown in the figure. The effect of a foreign atom FM having fewer valence electrons than the atoms of type M and producing the levels shown in Fig. 25 can be obtained in a similar way. This case might correspond to ZnS containing Ag a t zinc sites. Using the same constants for the base crystal and taking K I =~ loL7~ m and - ~ [FM]total = 4.10'8 cm-3, as previously, the results are given in Fig. 26a (high temperature equilibrium) and Fig. 26b (the state for T = 0°K). It is seen that Figs. 26a and 26b are mirror images of Figs. 24a and 24b in some respects.

FIQ.25. The electronic energy scheme of a crystal of composition M X with Frenkel disorder and a foreign atom F at a n M site, for the case in which F has fewer valence electrons than M.

Just as for Fig. 24, two mechanisms of incorporation of F can be distinguished in Fig. 26. When KRPM< [

+

~ l t o t K a ~we are dealing 7K14

+

with asolid

solution of composition M X FX (e.g., ZnS AgS). At higher values of KRpM, the incorporation of F is accompanied by the formation of an equal concentration of Mi' and FM'. This corresponds to a solid solution F (e.g., ZnS Ag). The solid solution M X FX of composition M X represents a higher oxidation state than M X F ; as a consequence, MX F X appears for lower values of KRPMthan M X F does. It is noteworthy that there are wide ranges in which the foreign atoms are incorporated as F X or as F ; however, there is no distinguished range in which it is incorporated in the form F,X with n > 1 (e.g., ZnS Ag2S), such as would correspond to the initial assumption that F has a lower valence than M . The explanation of this effect is similar to that given above for ZnS AlzSa. This mode of incorporation may become possible if the atomic imperfections involved (viz., the interstitial

+

+

+

+

+

+

+

+

386

F. A. KROGER AND H. J. VINK

-

29 KRPM

FIG.26. (a) High temperature equilibrium between vapor and solid for a crystal of composition M X with Frenkel disorder and containing foreign atoms F M with fewer valence electrons than M . Energy levels as shown in Fig. 25. (b) The concentration of centers at T = 0.

IMPERFECTIONS I N CRYSTALLINE SOLIDS

387

M atoms) contribute more than one level within the forbidden energy gap. (See Section 25b.) Incorporation of F produces interstitial M atoms rather than X vacancies because Frenkel disorder has been assumed to be predominant for the M atoms. One might ask if we should allow the possibility of interstitial F , excluded in the assumptions made a t the beginning of this section. This question can be answered by considering the equilibrium between F atoms a t lattice sites and F atoms a t interstitial sites. If one makes the assumption that the energy involved in bringing an F atom from a lattice site to an interstitial site, in accordance with the relation

is of the same order as that involved in bringing an M atom into the interlattice, (25.16) M M-+M , V M

+

it is easily seen that the ratio of the concentrations of F, and M , is roughly equal to that between the concentrations of F M and MM. This means that the concentration of F , may be neglected,236for the total concentration of F M is much smaller than that of M M . Interstitial F will only occur if there is a marked preference of F for the interlattice, that is if the energy associated with the reaction (25.15) is much smaller than that of (25.16). The method of treating such a case will be given in Section 25c. Since the effect of a foreign atom is determined solely by the position of its energy level, results similar to those displayed in Figs. 24a and 24b or Figs. 26a and 26b can be obtained for foreign atoms having other valences and located in other positions, provided the energy levels are the same (see Section 23). Thus the replacement of X by a foreign atom having a lower valence has exactly the same effect as replacement of M by a foreign atom of a higher valence, and vice versa. Replacement of Cd in CdS by Ga or I n has the same effect as replacement of S by CL9 If the levels of the imperfections and foreign atoms lie so close to the band edges that the centers are completely ionized a t the temperature a t which the equilibrium is studied, that is if El = E7 = E l , % 0, in Fig. 23, the problem can be solved by omitting Eqs. (25.4), (25.5), and (25.6) dealing with the ionization Further relation (25.1) is replaced by n[M,'] = KR'PM,whereas the Frenkel relation (25.3) is replaced by (9.38) : [M,'][VM'] = Kp'. Equation (25.7) is simplified to [ F M ' ] = [FMItotal. 236

C. Wagner, J. Chem. Phys. 18, 62 (1950).

(25.10)

388

F. A. KROGER AND H. J. VINK

Accordingly, the total concentration may be introduced into the neutrality condition (25.8) in place of F M ' . Figure 27 shows both the approximate and the exact solutions for [ F M ] = 4.1018 ~ m with - ~

KF' = cm-6 Ki = cm-6. It is seen that the result is much simpler than that for the corresponding case with partly ionized centers (Fig. 24a). The two modes of incorporating F , corresponding to M X FX2 and M X F X , remain the

+

+

1' A ii; >FiH *,I,

FIG.27. High temperature equilibrium between solid and vapor for a crystal of composition M X containing foreign atoms F M with more valence electrons than M for the case t h a t the foreign atoms and the atomic imperfections are completely ionized. Energy levels as shown i n Fig. 23 with El = Er = Eta S 0. The thick lines give the approximate solution; the thin lines represent the exact solution.

same however. This type of simplification holds for many compound semi-conductors, for example PbS-Bi and PbS-Ag.21.825It may also happen that the positions of the energy levels of the imperfections are asymmetrical, some levels lying close to one band but the others lying farther from the other band (Fig. 23, El= Ell G 0, E , # 0). This is the case, for example, in CdS,g CdTe,236s237 and GaSb.2as

b. Several Levels per Imperfection When the imperfections produce more than one level in the forbidden gap, so that the centers may occur in more than two different states, A. Jenny and R. H. Bube, Phys. Rev. 96, 1190 (1954). F. A. Kroger and D. de Nobel, J . Electronics 1, 190 (1955). 238 H. N. Leifer and W. C. Dunlap, Jr., Phys. Rev. 96, 51 (1954).

236D. 237

IMPERFECTIONS I N CRYSTALLINE SOLIDS

389

additional ionization reactions have to be taken into account. Moreover, the neutrality condition contains terms associated with charges higher than one, much as in the discussion of Section 11. As an example, let us consider the case of a crystal M X possessing Schottky-Wagner disorder, and containing a foreign atom F at M sites which has fewer valence electrons than an M atom. The X vacancy is

I

14

-vx’

I ITv4 FIG.28. An electronic energy scheme for a compound of composition M X with Schottky-Wagner disorder and containing foreign atoms F M with fewer valence electrons than M .

assumed to possess two levels, one lying so close to the conduction band that complete ionization occurs, the other lying much lower (Fig. 28). This case might correspond to CdS containing silver.9 The equations for this case are ~ [ V X ’=] K,’p,w (25.17) np = K i (25.18) (25.19) (25.20) (25.21) (25.22) (25.23)

2380

There are reasons for choosing the values of K , and Kx4to be of the same order of magnitude. The fact that they have been assumed t o be equal has no essential ground.

390

F. A. KROGER AND H. J. V I N K

The foreign atoms may be incorporated in more than two different ways because of the more complicated form of the neutrality condition (25.24). Three regions can be distinguished. (1) When log K , l p ~< -32,; the: foreign atoms simply replace the M atoms, giving rise to a solid solution M X F X (e.g., CdS AgS)

+

+

FIG.29. High temperature equilibrium between crystal and vapor for a crystal with composition M X possessing Schottky-Wagner disorder and containing foreign atoms FM with fewer valence electrons than M . Energy levels as shown i n Fig. 28.

(2) When 4 3 2 < log K i p , < 37.8 the foreign atom is incorporated as FM' along with VX",in a concentration equal to ~ [ F M ' This ] . corresponds t o a solid solution M X F2X (e.g., CdS A g 8 ) (3) When log K,lpM > 37.8, the foreign atom is incorporated as FM' together with an equal concentration of V X ' .This corresponds to a solid solution M X F (e.g., CdS Ag).

+

+

c.

+

+

Crystals Containing One T y p e of Foreign Atoms in Various Positions

It has been assumed so far that the foreign atoms present in a crystal may occupy only one position. Examples have been given of atoms occupying lattice sites, but cases with foreign atoms occupying interstitial sites may be treated in a similar way. It may also happen, however, th at a foreign atom can occupy two types of position, viz. a normal lattice site M and a lattice site X , or a

IMPERFECTIONS I N CRYSTALLINE SOLIDS

391

lattice site M and an interstitial site. The latter situation may be realized in ZnS-Cu phosphors, in which copper ions at lattice sites (CUZ,’) give rise to a green luminescence, whereas pairs consisting of substitutional copper ions and interstitial copper atoms (CuZn’Cui)are believed to be responsible for a blue f l ~ o r e s c e n c eAnother . ~ ~ ~ ~ example ~~~ is provided by Ge alloyed with copper. The copper may be present at Ge sites as well as at interstitial site~.~4l In such cases the set of equations necessary for a complete description contains the following equations, in addition to those the pure crystal, (25.25) as a consequence of the reaction

Fi

+ VM

FM

(25.26)

describing the transfer of F from one site to another; ionization relations involving the foreign atoms in their various positions ; possible association-dissociation reactions between foreign atoms and vacancies and between foreign atoms a t various sites (FA$’<). equations

in which x and y indicate the effective charge. The summation has of course to include all F atoms, independent of whether they form isolated centers or whether they belong to associates of the type indicated under (c). Finally, the neutrality conditions must contain the charged forms of the foreign atom in each position. Again, the concentrations of all the imperfections corresponding to equilibrium can be calculated from these equations.

d. Practical Limitations

It will usually not be possible to realize all the different ranges discussed in the previous sections because of the fact that the vapor pressures that can be achieved in practice are limited by the occurrence F. A. Kroger, J. E. Hellingman, and N. W. Smit, Physica 16, 990 (1949). F. A. Kroger, J . Chem. Phys. 20, 345 (1952). z 4 1 F. van der Maessen and J. A. Brenkman, J. Electrochem. SOC. 102, 229 (1955). 238

24u

392

F. A.

KRGGER

AND H. J. VINK

of new phases (see Section 24). It may even occur that pressures corresponding to only one range can be realized. Examples of such cases will be given in the following sections. The solution of the equations governing equilibrium for a particular case depend entirely on the values of the constants which enter the relations. These constants, in turn, depend exponentially on the temperature. Thus it may happen that preparation of the crystal at one temperature may lead to a specimen in the state corresponding to one range, whereas preparation a t a different temperature will produce a crystal in a state corresponding to a different range. 26. THEINFLUENCE OF FOREIGN ATOMSON OF CRYSTALS

THE

PROPERTIES

As has been stated in the introduction to this part of the article, the properties of crystals depend both on the foreign atoms present and on the native imperfections. It has been shown in the foregoing sections that there is a correlation between the states of the foreign atoms and the concentrations of various imperfections. A particular feature of this correlation is the occurrence of ranges in which the state of the foreign at,om and the concentration of a particular imperfection are practically independent of the partial pressures of M and X z in the adjacent vapor phase. As a consequence, the properties determined by the foreign atom, in whatever state it may be, and by a particular native imperfection are also independent of the atmosphere. This result may be used in the preparation of solids having well-defined properties. Particular examples will be discussed in the following subsections.

a. Ionic Conductivity, Diffusion, and Dielectric Loss Ionic conductivity requires the presence of charged atomic imperfections (Section 4). We have seen in Figs. 24a, 24b, 26a, 26b, and 29 that ranges occur in which the foreign atom is incorporated along with a constant and equal concentration of a particular type of charged imperfection. In Fig. 24b, for instance, it is seen that when log KRPM< 14.4, the concentration of VM’is equal to that of the foreign atoms. Similar results are obtained by applying the theory to more intricate compounds, such as M a x bwith a # b, which have either Frenkel or Schottky-Wagner disorder. It is found, in general, that the incorporation of a foreign atom can occur with the formation of charged imperfections which are present in a concentration that is independent of the atmosphere, and bears a

393

IMPERFECTIONS IN CRYSTALLINE SOLIDS

simple quantitative relation to the concentration of the foreign atom. These ranges may be called ranges of controlled atomic imperfection. The first examples of materials in which the concentration of charged imperfections, and hence the ionic conductivity, has been regulated with CdClz or the aid of foreign atoms are provided by the systems AgCl CdBrz, PbBr2, or ZnBrz studied by Koch and WagPbClz and AgBr ner.221These systems were investigated later by many other^.'^^'^ SimiCdC12,242~243J44 lar effects have been observed in the systems NaCl LiF MgF2, LiCl MgClZ, KC1 CaC12, SrClz or BaC1z1230,22.245,246,56 MgBrz, LiI MgIz,247 and NaCl BaC12.248 LiBr In all these cases the atom M has one valence electron and F has two. The incorporation takes place in such a way that one M vacancy is formed for each atom of F , that is, incorporation occurs according to the composiFX2. The centers actually present could be either FM' and tion M X VM'or F M and V M . Since the ionic conductivity is found to be markedly increased by the incorporation of the foreign atoms in all cases, the first situation is actually realized in practice. The presence of vacant M sites can also be proved by density measurements. Such proof has been given for the systems KC1 CaC12, SrC1z,249a CdBrz.249b and AgBr The regulated formation of VX" (or VX')centers has been confirmed by conductivity measurements in the systems T h o z Yz03,260~261 ~ ~ ~ JY203,263~254 ~~ CeOz L a ~ 0 3ZrOz , ~ ~+Ca0,268 ~ T h o z L a ~ 0 3 , ZrOz and KCl KzO, NazS.246 The presence of suchvacancies, SrS Ce&33,267 in concentrations determined by the concentration of the foreign atoms,

+

+

+ +

+

+

+

+

+

+

+

+

+ +

+ +

+

+

0. Stasiw and J. Teltow, Ann. Physik 1, 261 (1947). E. Schone, 0. Stasiw, and J. Teltow, 2.physik. Chem. 197, 145 (1951). 244 H. W. Etzel and R. J. Maurer, J . Chem. Phys. 18, 1003 (1950). 2 4 5 G. Ronge and C. Wagner, J . Chem. Phys. 18, 74 (1950). 246 F. Kerkhoff, Z. Phyaik 130, 449 (1951). Z 4 7 Y. Haven, Rec. trav. chim. 69, 1259, 1471, 1505 (1949). 248 Yu. K. Delimarskil, L. N. Sheiko, and V. G. Feshchenko, J . Phys. Chem. (U.S.S.R.) 242

Z43

29, 1499 (1955).

H. Pick and H. Weber, Z. Physik 128, 409 (1950). H. Junghans and H. Staude, 2. Electrochem. 67, 391 (1953). 260 F. Hund and R. Mezger, Z. physik. Chem. 201, 268 (1952). 2 6 1 G. Brauer and H. Gradinger, Z. anorg. u. allgem. Chem. 276, 209 (1954). 262 F. Hund, Z. anorg. u. allgem. Chem. 274, 105 (1953). 268 C. Wagner, Nalurwissenschaften 31, 265 (1943). 264 F. Hund, Z. Electrochem. 66, 363 (1951). 266 U. Croatto and M. Bruno, Guzz. chim. ital. 78, 95 (1948). 2 6 6 F . Hund, Z. physik. Chem. 199, 142 (1952). 26' E. Banks and R. Ward, J . Electrochem. Soc. 96, 297 (1949). z49a

2496

394

F. A. KROGER AND H. J. VINK

has been deduced from density measurements and rontgenographic data, or both, in the following cases: Biz03 PbO, Bi203 Sr0,258~269 Cd0,260,261 CeOz Ndz03, CeOz Smz03, CeOz Gdz03, Biz03 Dyz03, CeOz Ybz03, T h o z Cez03, T h o z Ndz03, T h o z CeOz Smz03, ThOz Gdz03,z51ZrOz MgO, ZrOz Ca0262and MgClz LiC1.263 Controlled formation of Mi’ has been demonstrated in the system AgBr AgzS264,265 by conductivity measurements, whereas the presence of interstitial M ions has been deduced from density measurements in the case of ZrOz Mg0.266Finally, the regulated formation of X ! has been substantiated by ionic conductivity measurements in the system SrFz LazF3,267 and by density measurements in the systems pPbFz BiF3,268CaFz YF3,269and CaFz ThF,.269Atomic imperfections present in the range where the incorporation of the foreign atom can be represented in terms of solid solutions M X FXz, MX FzX, etc., need not necessarily have a charge, a t least a t low temperatures, and therefore need not always enhance ionic conductivity. Charged imperfections are formed only when (a) an empty level of the uncharged imperfection lies below the occupied level of the uncharged foreign atom F , or (b) when the occupied level of the uncharged imperfection lies above an empty level of the uncharged F . This actually happens to be the case in the examples treated in Sections 25a and 25b. Charged as well as uncharged native imperfections which are formed in the presence of foreign atoms which possess a valence different from that of the lattice constituents for which they substitute may influence the diffusion of the foreign atoms or ions. Foreign atoms which have the same valence as the lattice atoms they replace on the other hand do not influence the concentrations of the native imperfections markedly. Accordingly, the diffusion of the foreign atoms will be practically independent of the presence of such atoms. Similar conclusions hold for the influence of foreign atoms on the migration of normal lattice constituents (self-diffusion). 268 L. G. SillBn and B. Aurivillius, 2. Krist. 101, 483 (1939).

+ +

+ +

+

+

+

+

+

+ +

+

+

+

+

+

+

+

+

+

+

+

2t9L. G. Sillen and B. Aurivillius, Naturwissenschaften 27, 388 (1939). 260 L. G. SillBn and B. SillBn, 2. physik. Chem. B49, 27 (1941).

+

K. Hauffe and H. Peters, 2. physik. Chem. 201, 121 (1952). A. I. Augustinik and N. S. Antselevich, J. Phys. Chem. (U.S.S.R.) 27, 973 (1953). 263 G. Bruni and A. Ferrari, 2. Rrist. 89, 499 (1934). 264 J. Teltow, 2. physik. Chem. 196, 197 (1950). 26s 0. Stasiw, 2. Physik 127, 522 (1950). 2 6 6 F. Ebert and E. Cohn, 2. anorg. u. allgem. Chem. 213, 321 (1933). 267 U. Croatto and A. Mayer, Gazz. chim.ital. 73, 199 (1943). 268 U. Croatto, Gazz. chim.ital. 74, 20 (1944). 2 6 9 E. Zintl and A. Udgard, 2. anorg. u. a!’gem. Chem. 240, 150 (1939). 261

262

395

IMPERFECTIONS I N CRYSTALLINE SOLIDS

Experiments concerning the diffusion of foreign atoms possessing the same number of valence electrons as the atoms they replace have been carried out for the systems NaC1-Cs, K, Rb, Ag,270,271 N ~ C ~ - A U , ~ ~ ~ J NaC1-C~,273,274KC1-A~,272,273,275KC1-Cu,2Wq73 AgCl-Cu,276 AgCl-Na,276 A~BI--N~,~?~ AgBr-C1, Br, I,'j5 A g B r - c ~ ,A~ ~~ I~- C U , ~ ~ ~ AgI-Li,276 AgI-C1,277 C U B ~ - A C~ U , ~I -~ A~ ~ C , ~U~ ~~ S - A C ~U , ~~~S~- S ~ , ~ ~ ~ A ~ Z S - C UBizSe3-Sb,279 ,~~~)~~ and ~ BizTe3-Sb.279 Diffusion of foreign atoms possessing a valence different from that of the substituted lattice atoms has been studied in the following cases: KCl-Ni273 NaC1-xi,273,?80s281NaCl-P, 5 , 2 8 2 AgBr-S,66 AgBr_Cd,Pb,66,243,283 AgC1-Pb,2s3,284 Fez03-Zn,202BizSe3-Sn,279Bi2Te3-Sn,279 and ZnSb-Sn.210a The influence of the presence of foreign atoms possessing a different number of valence electrons on self-diffusion has been measured in the system KC1 SrC12(K).65For a survey, see references 211 and 212. Indications of the presence of associated vacancy-foreign atom pairs ~ - ~ 8 ~ loss. Such can be obtained from m e a s ~ r e m e n t s l ~ ~of~ ~dielectric measurements have been carried out with the systems KCl SrClz, CaC12, NaCl M n C 1 ~ , ~ AgBr 8 ~ , ~ ~ CdBrz ~ KC1 PbC12,2g0 NaCl and AgBr PbBrz.292Association has been found a t low temperatures in all these cases. See also reference 293.

+

+

+

+

+

+ +

M. Chemla, Compt. rend. 236, 484, 2397 (1953). M. Chemla, Compt. rend. 238, 82 (1954). 272 M. J. Bogomolova, Acta Physicochim U.R.S.S. 6, 161 (1936). 273 S. A. Arzybitchew, Physik. 2. Sowjetunion 11, 636 (1937). 2T4 S. A. Arzybitchew and N. B. Borissov, Physik. Z . Sowjetunion 10, 44, 56 (1936). 276S.A. Arzybitchew, Doklady Akad. Nauk S.S.S.R. 8, 157 (1935). 276 C. Tubandt, H. Reinhold, and W. Jost, Z . anorg. ZL. allgem. Chem. 172,253 (1928). 2 7 ' W . Jost, dissertation, Halle, 1926. 278 H. Braune and 0. Kahn, 2. physik. Chem. 112, 270 (1924). 279 B. Boltaks, J. Tech. Phys. (U.S.S.R.) 26, 767 (1955). 280 J. A. Parfianowitch and S. A. Schipirzyn, Actu Physicochim. U.R.S.S. 6,263 (1937). 281 S. Pogodaw, Khim. Referat. Zhur. 4, 12 (1941). 282 M. Chemla, Compt. rend. 232, 1553 (1951). 283 J. Teltow, Phil. Mag. 46, 1026 (1955). 284 C. Wagner, J. Chem. Phys. 18, 1227 (1950). 285 R. G. Breckenridge, J. Chem. Phys. 16, 959 (1948). 2 8 s T . B. Grimley, J. Chem. Phys. 17,496 (1949). 287 R. G. Breckenridge, J. Chem. Phys. 18, 913 (1950). 288 B. W. Henvis, J. W. Davisson, and E. Burstein, Phys. Rev. 82, 774 (1951). 2 8 9 Y. Haven, Repts. Conf. on Defects in Crystalline Solids, Univ. Bristol 1964, p. 261, 2To

271

1955.

E. Burstein, J. W. Davisson, and N. Sclar, Bull. Am. Phys. Soc. 29, (5), 8 (1954). 2 9 1 Y. Haven, J. Chem. Phys. 21, 171 (1953). 292 J. Teltow and G. Wilke, Naturwissenschuften 41, 423 (1954). 293 R. Freyman, Physica 20, 1115 (1954).

290

396

F. A. KROGER AND H. J. VINK

b. Electronic Conductivity Whenever the concentrations of electrons and holes in a compound M X containing a foreign atom F which possesses more valence electrons than M (Fig. 24a) are compared with the concentrations in a pure compound MX (Fig. 9a), it is seen that the presence of the foreign atom has increased the concentration of electrons and decreased that of the holes at intermediate values of KRp,. Comparison of Fig. 26a with Fig. 9a /

/ / / / /

/

4

t

t

I-

4

conduction band

t

(€12

/

1

8

I

9

I

10

/ / / / / /

I

11

I

12

I

13

1

14

I

H

1

16

I

17

I

I

18

19 -19

I

20

I

I

2f 22 KR PM

FIG.30. Position of the Fermi level as a function of KnpM for the high temperature equilibrium of a crystal M X , which is pure or contains foreign atoms F M having either more or fewer valence electrons than M. (a) pure M X (see Fig. 9). (b) valence of F > valence of M (see Fig. 24a). (c) valence of F < valence of M (see Fig. 26a).

shows that foreign atoms having fewer valence electrons than M have the opposite effect. These effects are also clearly demonstrated in Fig. 30 giving the positions of the Fermi level for these cases (seelsection 15). At very large and very small values of KRPM,the concentration of electrons or holes, and hence the position of the Fermi level, is not appreciably affected by the presence of the foreign atoms, because the concentrations of electrons and holes resulting from the deviation from the stoichiometric composition are far greater than those due to the foreign atoms in these ranges. Curves b and c in Fig. 30, for the system MX-FM, contain ranges in which the position of the Fermi level is independent of the atmosphere.

IMPERFECTIONS IN CRYSTALLINE SOLIDS

397

These ranges correspond to the ranges in Fig. 24a and 26a in which n and p are independent of the atmosphere and proportional t o the concentrations of F. The existence of such ranges opens the possibility of controlling the concentration of free electrons or holes, that is, of the electronic imperfections much more precisely than would be possible for the pure compound in which the concentration of electrons or holes varies with the These ranges may be called composition of the atmosphere.222*223~294,86a ranges of controlled electronic imperfection. Their width can be calculated easily. The width of range IV of Fig. 24a is (26.1) Using KF' namely,

=

[VM'J[M~'] instead of K F = [VM][MiJ, and applying (9.38), KFf = Kp * KiK7 ___ K13

this simplifies to A log KQM = log

[ F ~ I t o t 13. d

KF'

(26.2)

A similar expression involving K14 instead of KI3is obtained in the case of Fig. 26a. For cases in which the level of Fnflies close to the band edge, so that complete ionization occurs (e.g., Fig. 27), the width becomes greater (range 111): (26.3)

It is seen that the width of the range of controlled electronic imperfection increases with the concentration of the foreign atoms, and decreases with increasing KF' in all cases. The decrease can be explained as follows. I n the range of controlled electronic imperfection, the foreign atoms are incorporated without being accompanied by the formation of native atomic imperfections. One side of the limit of this range is set by the point a t which M vacancies (V,') would be incorporated. The other limit is determined by the occurrence of a large concentration of interstitial M(Mi'). It follows that the width will be large when either one or both of these two atomic imperfections are not easily formed, and this corresponds to a small value of Kp'. A similar conclusion holds in the case of Schottky-Wagner disorder. 2g4E,

J. W. Verwey, Bull.

SOC.

chim. France, p. D122 (1949).

398

F. A. KROGER A N D H. J. VINK

Thus far the discussion has been concerned with high temperature equilibrium. After cooling, crystals corresponding to the high temperature range of controlled electronic imperfection will also display a definite though smaller concentration of free electrons or holes proportional t o the concentration a t the temperature of preparation. Therefore, the principle that the electronic imperfections can be regulated by means of foreign atoms having a different number of valence electrons will also hold after the crystals are cooled. The possibility of regulating the concentration of free electrons or holes and thereby the electronic conductivity by incorporating foreign atoms having different numbers of valence electrons was first recognized They applied this principle to NiO-Li. Later this by Verwey et group of authors, and others, applied the ideas to many other systems. I n the papers in which Verwey et a1.222,223,294,86a described the regulation of conductivity with the aid of foreign atoms, the authors coined the term controlled ualence. This term is derived naturally from the model used by the authors in describing a semiconductor. Their model had the following characteristics. ~

1

.

~

~

~

3

~

~

~

(1) The bonding was assumed to be purely ionic. ( 2 ) Free electrons or holes were assumed to move predominantly via the ions of one of the components of the base crystal. (3) The electrons or holes which are bound a t foreign atoms are assumed not to occupy orbits within the electronic shell of the foreign atom, but t o move about the atom in a more or less wide orbit, visiting predominantly the ions on which they move when free. When additional electrons or holes are a t an ion, they change the charge of th a t ion, that is, its valence. Thus the valence of a corresponding concentration of ions of the base crystal is altered by the addition of a foreign atom in an appropriate concentration.

We shall consider the case of NiO-Li as an example. Lithium is assumed t o produce Li+ ions on Ni2+sites when incorporated in Ni2+02-. The deficiency of charge is compensated by the formation of a n equal concentration of Ni3+ ions. Ni3+ ions far from Li+ ions represent free holes, whereas Ni3+ ions close to Li+ ions represent bound holes. (See also Section 23.) The range of controlled valence corresponds to that in which foreign atoms are incorporated without the formation of native atomic imperfections. As is seen in Figs. 24a and 26a, it is part of the range of the solid solution M X F X , that is, the part in which the concentration of the foreign atoms exceeds the concentration of native imperfections. This is the range -15 < log R R p < ~ -18.5 in Fig. 24a. It lies at -11.5

+

IMPERFECTIONS I N CRYSTALLINE SOLIDS

399

< log K R P M < -15 in Fig. 26a. I n these cases, the range of controlled electronic imperfection lies within that of controlled valence. They obviously have the same width in the case of completely ionized centers (Fig. 27) where [ F M M ]=~ ~[FM'] ~ ~ = n. Another consequence of the addition of a foreign atom having a valence different from that of the atom it replaces is that the transition from n-type t o p-type conductivity lies a t a value of K R p M different from that of the pure compound. Let us consider Fig. 30 again. If we assume that the effective masses of the electrons and holes are the same, the transition from n- t o p-type conductivity occurs a t the point where the Fermi level lies in the middle of the forbidden energy gap. This happens at the point x (log KQM = 15) for the pure compound. It is seen that the addition of foreign atoms having more valence electrons than the replaced atoms shifts the p-n transition point to lower values of K R P M . This happens a t y (log K R p M = 14.4) for curve b. The addition of foreign atoms with fewer valence electrons on the other hand shifts the transition to higher values of K R p M (curve c, point x a t log K R P M = 16.6). The shift is equal to the width of the range of controlled electronic imperfection. It is given by (26.2) and is thus proportional to the concentration of the foreign atoms. If we start with the pure compound, the addition of foreign atoms influences the conductivity in a way which depends on the value of K R P M . Thus when log K R p M = 15 (point z on curve a of Fig. 30), the addition of foreign atoms with more valence electrons than the replaced constituent causes an increase in the n-type conductivity, whereas the addition of foreign atoms with fewer valence electrons causes an increase in the p-type conductivity. At a larger value of K R p M (e.g., point w on curve a a t log K R P M = 16.6), the addition of foreign atoms with more valence electrons again causes an increase in n-type conductivity. Foreign atoms with fewer valence electrons cause a decrease in the n-type conductivity until point z is reached, which happens for [ F M ] t , t , l = 4.1018 ~ m in - this ~ case. Additional F M causes p-type conductivity t o appear. A similar, albeit opposite, behavior is observed a t values of log K R P M< 15. An example in which all these effects occur is PbS-Ag or Bi.21v820Many systems will show only part of the effects, either because of limitations in the pressures (see Section 25d) or because the concentration of the foreign atom, needed t o produce a certain effect, lies above the limit of solubility for a given K R p M . Often only an increase or decrease of one type of conductivity is observed in these cases. Table I gives a summary of the solids in which the electronic con-

400

F. A . KROGER AND H. J. VINK

TABLEI. SYSTEMSIN WHICHTHE ELECTRONIC CONDUCTIVITY HAS BEEN INFLUENCED BY THE ADDITIONOF FOREIGN ATOMSOF A DIFFERENT VALENCE Main compound C (graphite) Ge *

Si * Sn (gray) CaO CdS* CdSe CdTe

coo

CUO MnS NiO NiO NiO NiO PbS* S i c* ZnO ZnO ZnTe AlSb AlSb AlSb AlSb GaAs GaP InAs InAs In P InSb CdSb ZnSb ZnSb AggTe cu z o SnOs Ti02 Ti02 Ti02

* Single crystals.

References

Foreign atom B P, As, Sb, Li; B, Al, Ga, In, Ni, Co, Fe, Zn, Cu, Au, Pt P, As, Sb, Au, Li, Al, Ga, I n Bi, As, Sb, Al, Ga, In, Zn, Cd, Au Li, Y, La Ga, In, P, Sb, C1, Cu T1 Gal In, P, Sb, I Li Li, Cr, Y, Zr Li Li Ag Cr c1 Ag, Bi All B, P, N, Fe, Cr, Ca, Mg Ga Li, Al, Cr Cu, In, A1 Pb, Se Te Pb As, Bi, Ge, Pb, Sn, Se, T e Se S Zn, S Cd, Se Zn Pb Ag, Pb, Ni Ag, Sn Ag, Sn, Pb, Te, Cd, Bi, In Sn, Sb, Ge c1 Sb Cr, Ta, Sb, W, Mo, Nb, V, P, Al, Fe, Ga, Y Nil Be, Cr, Ga, Al, W Ta

332 296 296 333 131 9,334 107 236 223 297 223 222, 223, 294, 85a, 298, 299, 300, 96, 301 298 302 299, 300 21, 82a 303, 304, 335 235 305a 305b 226 306 307 308 309 310 311, 312 313 309 226 28, 29 314 135 213 299, 315 223 298, 316

317 223, 298

401

IMPERFECTIONS I N CRYSTALLINE SOLIDS

TABLE I. SYSTEMS IN WHICHTHE ELECTRONIC CONDUCTIVITY HAS BEEN I N F L U E N C E D BY THE ADDITION O F FOREIGN ATOMSO F A DIFFERENT VALENCE(Continued) ~~

~

~

Main compound

~~

~~

Foreign atom ~~

~

Na, Li, Ca, K

NbOa Biz03 Crz03 Fez03 BizTer MgWO4 CaTiOI SrTiOa BaTiOI CaMnOt LaMn03 LaCoO3 LaFe03 LaMn~1,-,~Cr,Fe,0~ CoFezOl MgFezOl NiFez04 ZnFezO4

296

Li, Na, K, Ca, Sr, Ba Cd Nil Ti, W Ti, Sn, W Pb, I Cr La La La La Ca, Sr Sr

Sr

Ba Ti Ti Ti Ti

References ~~

144, 318, 319, 320, 321, 322, 323, 324, 325 326 261 72 222, 223, 327, 328a 3283 223 223 223 223 223, 329 223, 329, 330, 331 329 223, 331 331 223 223 223 223

E. J. W. Verwey, P. B. Braun, E. W. Gorter, F. C. Romeyn, and J. H. van Santen,

Z. physik. Chem. 198, 6 (1951). 296 J. A. Burton, Physica 20, 845 (1954).

K. Hauffe and H. Grunewald, Z. physik. Chem. 198, 248 (1951). G. H. Johnson and W. A. Weyl, J . Am. Cerarn. SOC.52, 398 (1949). 299 W. Capps, M.S. thesis, Pennsylvania State College, 1950. 30° W. Capps and W. A. Weyl, O.N.R. Tech. Rept. Nr. 26, Contract Nr.N6 onr 269, Task Order 8, NR 032-265, Jan. 1951. 301 G. Parravano, J . Chem. Phys. 23, 5 (1955). 302 K. Hauffe, Ann. Physik 8, 201 (1950). 303 J. A. Lely, Colloq. Intern. Union Pure and A p p l . Chem. Mcnster p. 21 (1954). 304 J. A. Lely, Ber. deut. keram. Ges. 32, 229 (1955). 3060 K. Hauffe and A. L. Vierk, Z. physik. Chem. 196, 160 (1950). 306b E. L. Lind and R. H. Bube, Bull. Am. Phys. SOC.(11) 1, 110 (1956). 308 R. F. Blunt, M. P. L. Frederikse, J. H. Becker, and N. R. Hosler, Phys. Rev. 96, 29'

578 (1954). E. Justi rlnd G. Lautz, Abhandl. braunschweig. wiss. Ges. 6, 36 (1953). so* A. R. Regel and M. S. Sominskii, J . Tech. Phys. (U.S.S.R.) 26, 708 (1955). 309 0. G. Folberth and H. Weiss, Z. Naturforsch. 10a, 615 (1955). 310 0. G. Folberth and F. Oswald, Z. Naturforsch. 9a, 1050 (1954). 3 ~ 0 G. . Folberth, R. Grimm, and H. Weiss, Z. Naturforsch. 8a, 826 (1953). a12 0. G. Folberth, 0. Madelung, and H. Weiss, 2. Naturforsch. Qa,954 (1954).

307

402

F. .4. KROGER

AND H. J. VINK

TABLE11. SYSTEMSI N WHICH THE ABSORPTION SPECTRUMHAS BEEN INFLUENCED BY THE ADDITIONOF FOREIGN ATOMS ~~

Foreign atom

References

S, Se Sr Ca Pb Pb Pb Cd Ca, Sr, Ba 0, s Eu EU Ga Co, Ni Li Eu Eu, Ce, Sm Mn U W, Al, Nd La Pr Pr, Ta Y Nb, Ta, Sb, W, Mo, F e Ce

25, 336, 337, 338, 339, 340, 341, 342 343, 344 345 346 346 346 347 348, 349, 350 351 352 352 9 353, 354 222, 223, 294, 85a 352 352, 369 371 355, 356, 357 144, 358 359 162 144, 358 250 144,298, 360 361 362 254 256 144, 358 363, 364 365 I44 144 144, 318, 366 367 144, 368 372

Main compound AgBr NaCl NaCl NaCl RbCl KC1 KCI KC1 KC1 Bas CaS CdS MgO NiO MgS SrS ZnS CeOz CeOz CeOz CeOz ThOz ThOz Ti02 UOZ

uoz

ZrOz

ZrOz ZrOz A1203 A1203

Gad33 M03

wo

3

Vz05

VzOs Zn d3iO4

Y Y

Ca Ta, Pr, V Cr Mn Cr H Na Na, Li, Ag K Mn

R. M. Talley and D. P. Enright, Phys. Rev. 96, 1092 (1954). M. Telkes, J . Appl. Phys. 18, 1116 (1947). 315 W. Capps and W. A. Weyl, O.N.R. Tech. Rept. Nr. 27, Contract Nr. N6 onr 269, Task Order 8, N R 032-265, Jan. 1951. 318 G. H. Johnson, J . Am. Ceram. SOC.36, 97 (1953). 31; K. Hauffe, H. Grunewald, and R. Tranckler-Greese, 2. Electrochem. 66, 937 (1952). 318 G. Hagg, 2. physik. Chem. B29, 192 (1935). 313

314

IMPERFECTIONS I N CRYSTALLINE SOLIDS

403

E. J. Huibregtse, D. B. Barker, and G. C. Danielson, Phys. Rev. 84, 152 (1951). 32oB. W. Brown and E. Banks, Phys. Rev. 84, 609 (1951). 321 H. J. Juretschke, Phys. Rev. 86, 124 (1952). 322 W. R. Gardner and G. C. Danielson, Phys. Rev. 93, 46 (1954). 323B. W. Brown, J . Am. Chem. SOC.76, 963 (1954). 3Z4 M. E. Straumanis and A. Dravnicks, J . Am. Chem. SOC. 71, 683 (1949). 325 M. E. Straumanis, S. C. Das Gupta, and C. H. Ma, 2. anorg. u. allgem. Chem. 266, 209 (1951). 3 2 6 E . I. Krylov and A. A. Sharnin, J . Gen. Chem. (U.S.S.R.) 26, 1680 (1955). 327 H. Grunewald, Ann. Physik 14, 129 (1954). 3280 F. J. Morin, Phys. Rev. 83, 1005 (1951). 3286 S. E. Miller, J. C. Harman, and H. L. Goering, Bull. Am. Phys. SOC.30 (7), 35 (1955). 329 G. H. Jonker and J. H. van Santen, Physica 16, 337 (1950). 330 J. H. van Santen and G. H. Jonker, Physica 16, 599 (1950). 331 G. H. Jonker, Physica 20, 1118 (1954). 332 R. 0. Grisdale, A. C. Pfister, and W. van Roosbroeck, Bell System Tech. J . 30, 271 (1951); R. L. Shepard, H. S. Pattin, R. D. Westbrook, Bull. Am. Phys. SOC.(11) 1, 119 (1956). 333 G. Busch and E. Mooser, Helv. Phys. Acta 26,611 (1953); G. Busch and J. Wieland, Helv. Phys. Acta 26, 697 (1953); A. W. Ewald, Bull. Am. Phys. SOC.29, 30 (1954). 334D. C. Reynolds, L. C. Greene, R. G. Wheeler, and R. S. Hogan, Bull. Am. Phys. SOC. (11) 1, 111 (1956). 335 L. I. Iwanov and V. I. Bruzhinina-Granovskaja, J . Tech. Phys. (U.S.S. R.) 26, 220 (1956). 336 G. Seifert, 0. Stasiw, and C. H. Volke, Naturwissenschaften 41, 58 (1954). 337 0. Stasiw, 2. Physik 138, 246 (1954). 338 0. Stasiw, Science et inds. phot. 22, 424 (1951). 33@0. Stasiw, Z. Physik 127, 522 (1950). 340 0. Stasiw, Ann. Physik 6, 151 (1949). 341 0. Stasiw, 2. Elektrochem. 66, 749 (1952). 342 J. W. Mitchell, Phil. Mag. [7], 40, 249 (1949). a43 P. Camagni, Phil. Mag. [7], 46, 225 (1954). 344 H. Pick, 2. Physik 114, 127 (1939). 345 H. W. Etzel, Phys. Rev. 87, 906 (1952). 346 E. Burstein, J. J. Oberly, B. W. Henvis, and J. W. Davisson, Phys. Rev. 81, 459 (1951). 347 L. M. Shamovski and M. T. Gosteva, J . Phys. Chem. (U.S.S.R.) 28, 1266 (1954). 348 G. Heiland and H. Kelting, 2. Physik 126, 689 (1949). 349 H. Pick, Ann. Physik 36, 73 (1939). 350 S. Kondo, J . Phys. SOC. Japan 6, 200 (1950). 361 S. Akpinar, Ann. Physik 37, 429 (1940). 352 P. Brauer, 2. Naturforsch. 6a, 560 (1951). 353 S. Holgersson and A. Karlsson, 2. anorg. u. allgem. Chem. 182, 255 (1929). 354 F. A. Kroger and H. J. Vink, Physica 18, 77 (1952). 365 W. Rudorff and G. Valet, 2. anorg. u. allgem. Chem. 271, 257 (1953). 356 G. Brauer and R. Tiessler, 2. anorg. u. allgem. Chem. 271, 273 (1953). 357 F. Hund, R. Wagner, and U. Peetz, 2. Elektrochem. 66, 61 (1952). 358 W. A. Weyl, in “Outline of Course ‘Ceramics 503,”’ p. 118. Pennsylvania State College, 1952. 31s

404

F. A. KROGER AND H. J. VINK

ductivity has been influenced by the addition of foreign atoms possessing a number of valence electrons different from the number on the atom replaced (see also references 95 and 96). The few cases in which the investigations dealt with single crystals have been marked by an asterisk. The great majority involve polycrystalline samples made either from a melt or by pressing and sintering powders. One should be very careful in giving a quantitative interpretation of the results in the second CaSe.85a,b,c,d;295 c. Optical Properties The imperfections and the foreign atoms present in the crystal in the various states determined by the equilibria may give rise to absorption of light in two ways. (1) The centers may have characteristic excited states. The center can be raised from its ground state to the excited state by absorption. (2) Electrons may be excited from an occupied localized level of the imperfection or foreign atom into the conduction band, or from the valence band into a level of the centers.

If the absorption bands produced in this way lies in the visible part of the spectrum, and in a range in which the base crystal itself does not absorb light, the extra absorption changes the color of the crystal. The electrons excited into the conduction band or the holes brought to the valence band may be detected by an increase in conductivity (photoconductivity). In both cases light may be emitted when the system returns to the ground state (luminescence). Examples of cases in which the E. Zintl and U. Croatto, 2. anorg. u. allgem. Chem. 242, 79 (1939). W. A. Weyl and T. Forland, I n d . Eng. Chem. 42, 257 (1950). 361 W. Rudorff and G. Valet, 2. Naturforsch. 7b, 57 (1952). 869

860

Hund, H. Peets, and G. Kottenhahn, 2. anorg. u. allgem. Chem. 278,184 (1955). B. N. Grechusknikov, Doklady Akad. N a u k S.S.S.R. 99, 707 (1954). 364 E. Thilo, J. Jander, H. Seemann, and R. Sauer, Naturwissenschaften 37,399 (1950). 366 F. A. Kroger, “Some Aspects of the Luminescence of Solids,” p. 84. Elsevier, New

36zF. 361

York, 1948. M. E. Straumsnis, J . Am. Chem. SOC.71, 679 (1949). 367 H. Flood, T. Krog, and H. Sorum, Tidsskr. K j e m i Berguesen Met. S, 55 (1943); 6, 32, 59 (1946). 368 R. P. Ozerow, Doklady Akad. N a u k S.S.S.R. 99, 93 (1954). 369 F. Urbach, H. Hemmendinger, and D. Pearlman, in “Preparation and Characteristics of Solid Luminescent Materials’’ (G. R. Fonda and F. Seits, eds.), p. 279. Wiley, New York, 1948. 870 See reference 365, p. 40. 3 7 1 F. A. Kroger, Physica 6, 369 (1939). 371 F. A. Kroger, Physica 6, 764 (1939). 366

405

IMPERFECTIONS I N CRYSTALLINE SOLIDS

incorporation of foreign atoms causes a marked change in the absorption spectrum are given in Table 11. In addition, absorption measurements have been carried out on a great number of phosphors containing foreign atoms as an acti~ator.37~33~4 In a few cases, it is possible to assign the absorption bands to particular electronic transitions. Thus in MgO NiO, the green color is caused by transitions between levels of the Ni2+ ion.364In SrS-Eu and SrS-Ce, on the other hand, the absorption is caused by a transition from a characteristic Eu or Ce level t o the conduction band (or to a nonspecific excited state close to this band369,370). The absorption in CdS-Ga is produced by a transition from a Vcdl level into the conduction band.g In many cases, additional measurements will be needed in order t o ascribe the absorption bands to particular excitation mechanisms. Luminescence associated with foreign atoms involves either characteristic transitions within the foreign atom or transition from levels caused by the foreign atom to one of the bands. For a survey, see references 373 and 374. In a few cases, levels associated with vacancies whose concentration is determined by that of the foreign atoms are responsible for the luminescence. 173, 176,176

+

d. Magnetic Properties

Just as for the optical properties, both the native imperfections and the foreign atoms may influence the magnetic properties of the base crystal. An example of a case in which vacancies change the magnetic properties is provided by the system AgBr Ag.376Cases in which the presence of the foreign atoms markedly influences the magnetism are given by the following systems: (1) A1203 Mn203,Ti02 Mn02,224 A1203 c1-203,~~~ TiOz Cr02,377MgO NiO, rAl2Oa Ni203,Ti02 Ni02,378A1203 Fe203,379 rAl2O3 Fe203, Ti02 Fe02;380a~3soa (2) and LaCo03 CaMnO3 LaMn03,329 LaMn03 SrMn03,329J30v331 S ~ C O O ~ . ~ ~ ~ In the first ten systems, the foreign atom is incorporated together with

+

+

+

+

+

+

+

+

+ +

+

+

+

+

See reference 365,Table 111,p. 262. F. A. Kroger, Ergeb. ezakt. Naturw. 20, 61 (1956),Table 11. 376 N. Perakis, Compt. rend. 236, 1474 (1953). 376 R.P. Eischens and P. W. Selwood, J . Am. Chem. Soc. 69, 1590 (1947). 377 P.W.Selwood and L. Lyon, Discussions Faraday SOC. 8, 222 (1950). 3 7 8 F. N. Hill and P. W. Selwood, J . Am. Chem. SOC.71,2522 (1949). 379 P. W.Selwood, L. Lyon, and M . Ellis, J . Am. Chem. SOC.73, 2310 (1951). 3 8 0 0 P.W.Selwood, M. Ellis, and K . Wethington, J . Am. Chem. SOC. 71,2181 (1949). 380b Most of these experiments have been carried out with adsorbed layers of the foreign compound, the matrix crystal serving as a base. In view of the temperatures used in the preparation, it ie likely that formation of solid solution has taken place. a73

174

406

F. A. KROGER AND H. J. VINK

the same amount of oxygen as the metal in the base crystal. The magnetic measurements show that the foreign atom is present as an ion having the same valence as the metal ions of the base crystal. This is remarkable since the pure oxides of the foreign atoms have a different composition, corresponding to a different valence of the foreign atom when prepared under the same conditions. This behavior prompted Selwoodagl to speak of induced valence. In the three systems mentioned under (2), the foreign atom is again incorporated without the formation of vacancies. This necessarily requires a change in valence of the constituents of the system. Magnetic measurements show, however, that, in contrast to what happens in the systems mentioned in (I), a corresponding number of the ions of the base crystal, viz. Mn or Co, and not the foreign atoms, change their valence. Apparently this is an example of what Verwey et al. called controlled valence (see Section 26b). In our (atomic) notation, in which no attention is paid to the exact position of the electrons or holes bound a t the foreign atom (see Section 23), both modes of incorporation are regarded as the incorporation of F M centers.

e. Deviations from the Simple Xtoichiometric Ratio In describing the deviation from stoichiometric composition in a crystal containing foreign atoms, we must first define the composition considered as the normal stoichiometric one. A stoichiometric solid solution may be defined in three ways. (1) The solid solution is stoichiometric if the proportions of the concentration of the constituents of the base crystal is the same in the solid solution and in the pure stoichiometric base compound. (2) The solid solution is stoichiometric if the replacement of the foreign atoms by the corresponding constituents of the base crystal leads to the pure stoichiometric base compound. (3) The solid solution is stoichiometric if it can be formulated as a mixed crystal of the stoichiometric base compound and a compound of the foreign atom in which the latter has its normal chemical valence. I

Thus, according to (l),a solid solution of gallium in CdS is considered to be stoichiometric if the solid solution may be written as CdS Ga. In this case, the deviation from stoichiometry is given by the formula

(Ax>,= (Ax), - [Galct,, a8lP.

W. Selwood, J . Am. Chem. SOC.70, 883 (1948).

+

(26.4)

407

IMPERFECTIONS I N CRYSTALLINE SOLIDS

in which (Ax), is given by formula (14.1). According to (2), a solid solution of gallium in CdS is stoichiometric if it can be written as CdS Gas, i.e., the gallium is incorporated in accordance with the mechanism associated with the ranges 111, IV, and V in Fig. 24a. The deviation from the stoichiometric composition in this case is given by the formula holding for the pure solid (relation (14.1)) :

+

(26.5)

(Ax), = (Ax)o.

According to the third definition, the solid solution is stoichiometric if it can be formulated in terms of the system CdS GazSJ.The deviation from the stoichiometric composition is now given by the relation

+

(Ax),

=

(Ax),

[Gal + 7-

(26.6)

The corresponding expression would be

+

for the system AgCl CdCL The definitions (1) and (2) have the advantage of being unambiguous. Definition (3) is probably more attractive to the chemist, but it has the disadvantage that it requires a definition of a “normal” composition of the compound of the foreign atom. This is not always unambiguous as the cases of transition elements and lead indicate. Once a choice has been made, the variation of the deviation from the stoichiometric composition can be determined from the equilibrium concentrations of the centers and the imperfections, as described for compounds without foreign atoms (Section 14). The deviation from stoichiometry has been measured as a function of the atmosphere for KC1 SrC12.349

+

27. COMPARISON OF COMPOUNDS AND MONATOMIC CRYSTALS

As we have seen in Section 25, foreign atoms can be incorporated in various ways in compounds. The type of incorporation depends on the reducing power of the atmosphere. The incorporation of foreign atoms in monatomic crystals (Ge, Si, C) may, in principle, also take place in various ways. However, we lack the opportunity of influencing the mechanism of incorporation via the atmosphere in such cases. If a certain type of incorporation predominates in one case, it does so under all circumstances. I n most monatomic semiconductors ( M ) ,foreign atoms ( F ) are incorporated substitutionally, without forming vacancies, and give rise to solid solutions M F. Depending on the number of valence electrons F possesses relative to M , the foreign atoms produce a high or a low Fermi level. Examples are provided by Ge or Si containing pentavalent atoms

+

408

F. A. KROQER AND H. J. VINK

like P, As, Sb, etc., which show n-type conductivity. The same substances exhibit p-type conductivity when they contain trivalent atoms like B and Al. The concentration of electrons or holes is equal to that of foreign atoms at room temperature.296It might also happen, however, that incorporation of such substitutional foreign atoms would be accompanied by the formation of vacancies. This would occur if the vacancies could trap electrons or holes, and if the free energy gained by the transfer of an electron or hole from the foreign atom to the vacancy would be greater than the free energy needed to form the vacancy. Although vacancies may trap electrons, the second condition apparently is not usually fulfilled, so that the first mechanism of incorporation is the predominant one. 28. CRYSTALS CONTAINING Two TYPESOF FOREIGN ATOMS

When various types of foreign atom are present simultaneously, their effects can be explained on exactly the same basis as those for a single foreign atom, described above. There is a relation of the type

(28.1) for each type of foreign atom. Here c indicates the effective charge. In the neutrality condition, the concentrations of the charged species of the atoms appear, each being multiplied by a factor equal to its charge. I n seeking the approximate solutions by Brouwer's" method, one finds that each range is governed by as many simplified conditions of type (28.1) as there are types of foreign atoms present and by one simplified neutrality condition. This means that, in seeking the limit of a particular range, the algebraic sum of the concentrations of the charged foreign atoms must be compared with the concentrations of the charged native imperfections. It has been shown in Section 26b that foreign atoms possessing more valence electrons than the constituent of the base crystal which they replace, iend to raise the position of the Fermi level and shift the point at which the material changes from n to p type to lower values of KRp,. Foreign atoms with fewer valence electrons have the opposite effect. It must be expected that when foreign atoms (F,) possessing more valence electrons than the atoms they replace and foreign atoms ( F z ) possessing less are present at the same time, the effect of the former is counteracted by that of the latter. Such eff ects have been observed inNi0223 and Ti02.a1sIt has also been observed in monatomic semiconductors where

IMPERFECTIONS IN CRYSTALLINE SOLIDS

409

it is designated “compensation.’Ja82-a~~ The same term may be used with compounds. At the point where the crystal changes from n to p type, the Fermi level lies approximately halfway between the bands. I n this case, the foreign atoms possessing more valence electrons are ionized, and hence are present as (FM’), centers. The levels of the foreign atoms possessing fewer valence electrons, on the other hand, are occupied by electrons, that is, are present as ( F M ’ ) l centers. I n this case, the total concentration of each type of foreign atom appears in the neutrality condition, one on the left side and the other on the right. As a consequence, only the difference in the total concentrations of F , and Fl influences the position of the p-n transition point. Further consequences of the incorporation of various types of foreign atoms will be discussed both in the next section and in Section IV. 29. CONTROLLING THE CHARGE OF FOREIGN ATOMS

It has been shown in Section 26b that foreign atoms may be used to keep the concentration of electrons or holes constant a t a certain level, that is, t o keep the Fermi level a t a certain position. If other foreign atoms are present, the levels associated with these atoms will be filled as long as they are below the Fermi level and they will remain empty when they lie above the Fermi level. Since the filling of levels is identical with the forming of centers possessing different charges, foreign atoms of one type can be used to stabilize foreign atoms of another type in a certain state of charge. Speaking in terms of ions, we would say t h at ions are stabilized in a particular valence state. This effect may appear with any combinationof foreign atoms. It is particularly striking, however, when we are dealing with two types of foreign atoms, one possessing only a single level between the bands of the crystal, and the other type possessing several. Or, expressed in chemical terms, if one atom has a fixed and the other a variable valence. I n this case, the atoms with fixed valence will regulate the valence of the other. This is of importance in the preparation of phosphors. An example is the system MgO-Mn, Li. Here manganese may form Mn, Mn’, and Mn” centers, corresponding t o Mn2+, Mn3+, and Mn4+ ions in the ionic model (see Section V), whereas lithium may form only Li’ and Li centers, corresponding t o Li+ and Liz+ ions. C. S. Hung and V. A. Johnson, Phys. Rev. 79, 535 (1950). M. B. Prince, Phys. Rev. 92, 681 (1953). 384 F. W.G. Rose and E. W. Timmins, PTOC. Phys. SOC.(London) B66, 984 (1953). P.Ransom and F. W. G. Rose, PTOC.Phys. Soc. (London) B67, 646 (1954). 386 V. Ozarow, Phys. Rev. 93, 371 (1954).

383

410

F. A. KROGER AND

a. J.

VINK

As has been found by P r e n e ~ , ~ manganese ~' usually forms Mn centers ( = Mn2+ions) in the absence of lithium. I n the presence of an excess of lithium, however, the fluorescence properties indicate that Mn" centers ( = Mn4+ions) are formed. Since Mn' centers ( = Mn3+ ions) may exist next to Mn and Mn" centers, one may expect that Mn' will be formed in the presence of a smaller concentration of lithium. In fact, if incorporation occurs according to the rules of simple chemical reactions, and if the system has the greatest stability for the lowest concentrations of native atomic imperfections, one would expect that one lithium atom would stabilize one manganese ion in the Mn' form, whereas two lithium atoms would be needed to stabilize it in the Mn" form. Only in these cases are atomic imperfections not involved in the incorporation:

(29.1)

K l a ~ e n has s ~ ~shown ~ that the stabilization of high valent Mn is not absolute. It is only stabilized against reducing influences. I n the case of Mn", this stabilization is the more pronounced the larger the excess of lithium. This can be seen easily with the aid of the method discussed in this article. Since the energy levels are not known, a definite level scheme and definite values of the constants will be adopted. Schottky-Wagner disorder will be assumed, and it will be supposed that each vacancy gives rise to only one energy level in the forbidden band. Lithium is assumed to produce one level close to the valence band and manganese to produce two levels in the positions shown in Fig. 31. The following relations hold a t equilibrium in an atmosphere characterized by a definite partial pressure of oxygen: (29.3) (29.4) (29.5)

41 1

IMPERFECTIONS I N CRYSTALLINE SOLIDS

The following values of the constants are used: Ks' = 1030 cm-6, K , = loz1cm-6, K g = l O I 9 ~ m - K ~ s, = K14= 1014~ m - K ~ I, 3= 2.10'' ~ m - ~ and KIE,= 4.10'O ~ m - ~The . equilibrium concentrations of the various imperfections and centers are shown in Fig. 32 as functions of K,,poC. Figure 32a refers to MgO possessing 10'9 atoms of Mn per 01113. Figure 32b, 32c, and 32d refer to MgO containing Mn and Li in the concentrations 1019 Mn l O I 9 Li, 1019Mn 2.10'9 Li, and 10'9 Mn lozoLi, respectively.

+

+

-Li

+

@a

FIQ. 31. A possible electronic energy scheme for MgO containing lithium and manganese.

It may be seen from Fig. 32a that Mn occurs as Mn, Mn', Mn" depending on the pressure of oxygen, high oxygen pressure favoring the highly charged form. Figure 32b shows that the addition of one atom of lithium per manganese atom widens the stability range of the Mn' form of the system a t the cost of the Mn form. Addition of more lithium, to bring the proportion to 2: 1, narrows the Mn' range again, but widens the Mn" range (Fig. 32c). Upon increasing the proportion of Li to Mn further, the stability limits Mn-Mn' and Mn'-Mn" are gradually shifted to lower values of the oxygen pressure (Fig. 32d). The variation of the positions and the widths of the stability ranges are shown separately in Fig. 33 as functions of oxygen pressure. It can easily be shown that the width of the Mn' range is equal to (A log K o x p o ! ) ~ ~=. log 387

388

K13 ~

Kl5

J. S. Prener, J . Chem. Phys. 21, 160 (1953). H. A. Klasens, Philips Research Repts 9, 377 (I 854).

(29.10)

22 21

I

I

MnMg

-

-0-

I

'

n-

I

1-

Mn" M9

I

Mnhg

1-

I I

614

-

5

-A -;

; ; i 1 ;.;

; 6

I0

:1

1;

1:

I:

1;

I6 7;

Itp

-

:2

2: 21'

;5

65216

7;

19 KQXPo2

FIG.32a. High temperature equilibrium concentrations of the imperfections and the centers formed by the foreign atoms in the system MgO-Mn.

FIG.32b. High temperature equilibrium concentrations of the imperfections and the centers formed by the foreign atoms in MgO containing 1019Mnand 10k9Liatoms per cc.

5

W

FIG.32c. High temperature equilibrium concentrations of the imperfections and the centers formed by the foreign atoms in MgO containing lO’9Mn and 2.lO19Li atoms per cc.

FIG.32d. High temperature equilibrium concentrations of the imperfections and the centers formed by the foreign atoms is MgO containing 101QMnand lOeOLi per cc.

416

F. A. K R ~ G E RAND H. J. VINK

a t both high and low values of [Li]/[Mn]. It is greater only near the proportion [Li]/[Mn] = 1. Another example of the influence of one foreign atom'on the mechanism by which another is incorporated is probably given by the effect of Na on Alz03-Mn.389Whereas manganese is incorporated as Mn203in A1203 containing manganese alone (the manganese assuming the valence of the aluminum), the manganese is incorporated with more oxygen in a state corresponding to valence 4 3 . 5 in the presence of sodium. This indicates that sodium tends to stabilize the manganese in a state of valence greater than 3.

Mn2+- ion),

IV. Solubility Relations 30. INTRODUCTION

The influence of a fixed concentration of foreign atoms on the concentrations of the native imperfections has been discussed in Section 111. Once such effects are established, one may ask about the influence of the imperfections on the foreign atoms. One aspect of this influence has already been considered, namely the influence upon the charge of the center formed by the incorporated foreign atom (Section 29). Another aspect, which is closely connected to the one just mentioned, is the influence upon the solubility of the foreign a t ~ mI n principle, ~ . ~ ~ 189

P. W. Selwood and L. Lyon,. J. Am. Chem. SOC.74, 1051 (1950). C.Wagner, J. Phys. Chem. 67, 738 (1953).

8900

~

~

IMPERFECTIONS IN CRYSTALLINE SOLIDS

417

when foreign atoms are present in the crystal, they will also be present in adjacent phases (e.g., vapor, liquid, or other solid phases). The distribution of foreign atoms over the various phases is determined by the free energy balance. If, as a consequence of oxidation-reduction reactions, the charge of foreign atoms in the solid is changed, their thermodynamic potential is changed. Accordingly, the distribution of foreign atoms over the two phases may be influenced. If the thermodynamic potential of the foreign atoms in the solid is increased relative to that of the foreign atoms in another phase, the foreign atoms will tend to leave the crystal and enter the other phase. I n doing so, the concentration in the crystal, and thereby the thermodynamic potential, is diminished. This process continues until equilibrium is re-established. In the opposite case, atoms from the other phase tend to enter the crystal. I n any event, the solubility of the foreign atom is changed.ag0b The interrelation of the native imperfections and the solubility of foreign atoms can be studied by considering the transfer of the foreign atoms and the lattice constituents between the crystal and the adjacent phases under the condition that all internal reactions, such as formation of defects, electronic excitation, association or dissociation, also take place. The equilibria between crystal and gas will be considered in the following sections. A similar treatment of the equilibrium crystal-liquid will be given in Section 33. The solubility of a foreign atom F in a crystal can be defined in various ways. (a) The solubility of the foreign atom in a crystal a t a constant thermodynamic potential. If the adjacent phase is a gas, the thermodynamic potential pF of the foreign atom is related to the partial pressure p F of the foreign atom in the following way:

+ RT In

PF = ( ~ F ) o

PF

(30.1)

in which ( p F ) o is a constant. Thus in this case a constant thermodynamic potential means a constant partial pressure p ~ . (b) The solubility of F in M X in contact with a pure compound of F, e.g. F X .

It can be easily seen that the solubility may be entirely different in the two cases. The partial pressure of F in equilibrium with a compound FX 390b

Looking at the problems from this view, the results of Section I11 can also be regarded as a variation of solubility, viz. of the solubility of components of the base crystal in excess of the stoichiometric composition, under the influence of foreign atoms (see Wagner286).

418

F. A. KROGER AND H. J. VINK

follows the relation (see Section 7) (30.2) regardless of whether F X is a solid, a liquid, or a gas. Thus, p a may vary widely under variable oxidation-reduction conditions, that is variable px,. The first definition of the solubility will be used in the following sections. The solubility will be studied as a function of p F and p ~ The . second concept of solubility will be met in the discussion of the factors limiting p F and p~ (Section 31d). 31. EQUILIBRIUM BETWEEN CRYSTAL AND VAPOR The relation between the solubility of foreign atoms in a crystal and the oxidizing power of the atmosphere will be demonstrated for a practical example, namely for the solubility of silver and chlorine in crystalline CdS. The vapor in contact with the crystal will be assumed to contain a definite partial pressure of Ag, Clz and Cd and thus also of Sz. The energy level diagram for the CdS-Ag system corresponds to that shown in Fig. 28. If chlorine is present at a sulfur site it gives rise t o a level close to the conduction band, a t a position similar to that of V X . If V , and ClS are assumed to be completely ionized, the relations between the concentrations of the imperfections within the crystal are

n .p

=

Kg

[VCd'][V€4.1

=

K3'

(31.1) (31.2)

The transfer of Cd atoms between the crystal and the atmosphere can be accounted for by the relation n[VS'] = Kr'pM

(31.4)

If the transfer of silver and chlorine between the crystal and the atmosphere takes place according to the reactions (31.5) (31.6)

419

IMPERFECTIONS IN CRYSTALLINE SOLIDS

instead of relations like (25.23), we are led to the equations

,

(31.7) (31.8) The concentrations of all centers can be calculated as a function of and pclz from the nine relations (31.1), (31.2), (31.3), (31.4),

pod, p~~

log

FIQ.34. Concentrations of centers in the system CdS-C1 as a function of Kcipci: for Kr‘pCd = 36.5.

(31.7), and (31.8). The values of the constants used in Section 25b are employed, viz. .-

K,

=

cm-6,

Ks’ = 2.1032cm-6, K, Klo = 3 loL7CM-~.

=

KI4 = 1019 ~ m - ~

a. CdS Containing Chlorine Alone Let us first consider the case of the system CdS-C1 without silver, i.e.,

pA. = 0. The best way to proceed is to assume certain values of Kr’pcd, and calculate all concentrations as functions of Kclpcl,t for each value of K r ’ p C d . A typical result for a large value of K r ’ p C d (log K,’pCd = 36.5),

corresponding to strongly reducing conditions, is shown in Fig. 34. It is

420

F. A. KROGER AND H. J. VINK

I

FIG.35. Concentrations of centers in the system CdS-C1 as a function of Kcipci,+ for log Kr’pcd = 29.5.

seen that for low values of K C l p c l 2 ~ “ClSltOtsl

=

[Cls’l

-

-

KCIPC,?.

(31.9)

At high value of Kcrpcl:*(uiz., above log K C I ~ C=I ?O ) , [ClSI,o*,l = [Cls’l

Kc,*pc,,t.

(31.10)

Figure 35 shows similar results for more oxidizing conditions (log K,’pcd

=

29.5).

Using results obtained in this way for various values of Kr’pod, contour diagrams of the concentrations of all centers can be constructed as functions of K,’pc, and Kclpcl?.Figure 36 shows such a diagram in which the contours of [Cl,’], n, and [V,,] are shown. It is seen that there is interdependence between the effect of chlorination and reduction over the entire diagram. The range a t upper right, where [Cl,’] = ?z and that a t the lower right where [Cls’l = [VCd’I are particularly interesting. The first range obviously is one of controlled electronic imperfection (see Section 26b). The second is a range of con-

FIG. 36. Contours of the concentrations of C!Is*, VCI'and 0 for the system CdS-C1 in equilibrium with an atmosphere having various partial pressures of Cd and CL. The curves are marked with numbers giving the logarithm of the concentrations.

422

F. A. KROGER AND H. J. VINK

trolled atomic imperfection (Section 26a). The ranges n p = [VC,'] have also been indicated.

=

[Vs*] and

b. CdS Containing Silver Alone The case of CdS containing silver exclusively (pel, = 0) can be treated in the same way. A diagram showing contours of [Aged'], [VS'], ~ V S ' *and ], p is given in Fig. 37. This diagram is more complicated than that of CdS-C1 (Fig. 36) in the sense that it contains three ranges in which the foreign atom plays a dominant part in the neutrality condition. The three ranges are (a) the one at upper right where

[Aged']

=

[Vs']

=

~[VS"]

(b) t ha t in the middle right where [Aged']

( c ) t ha t at the bottom right, where [Agca']

=

p.

Range (c) is one of controlled electronic imperfection in which silver is incorporated without the formation of vacancies (CdS AgS). I n ranges (a) and (b) silver is incorporated along with vacancies, (b) corresponding to the incorporation of Ag2S, (a) to the incorporation of silver atoms. These simple relations hold only for the charged silver centers (Agcd'). At low values of K,'pcd (i.e., oxidizing conditions), a large fraction of the silver is present in a n uncharged form (Aged). Therefore, the total silver concentration differs markedly from that of Aged' in this range. This behavior can also be seen at the left-hand side of Fig. 29. When interested in the properties of a crystal with a certain constant concentration of the foreign atoms F under varying oxidation-reduction conditions (see Section 111), one follows a path [FtOtal] = constant through figures like Fig. 36 and Fig. 37. As is seen from these figures, pF varies along such a path so tha t the thermodynamic potential of F is not constant.

+

c. CdS

Containing Chlorine as Well as Silver

Interesting situations occur when both silver and chlorine are present. Although a complete solution of the equations can easily be obtained for any combination of P C d , pAg,and pel,, the representation of the results requires a four-dimensional figure. This difficulty can be solved by considering solutions for particular values of K,'pcd. Contour diagrams of

IMPERFECTIONS I N CRYSTALLINE SOLIDS

423

424

F. A. KROGER AND H. J. VINK

the concentrations of Aged' and C13' for K,'p,, Fig. 38. This figure contains four regions:

=

3.10a6are shown in

(1) that at the lower left (low values of pA, and pCl2);the system behaves like pure CdS. (2) upper left (low values of pel, but higher values of p A g ) ;the system behaves like CdS-Ag. Silver is incorporated as AgzS with the formation of one sulfur vacancy for each two silver atoms (see also Fig. 37 of this section, and Fig. 29 in Section 111).

FIG.38. Contours of the concentrations of Aged' and CIS' in CdS a t equilibrium and pel,. The contours are with an atmosphere having various partial pressures marked with numbers giving the logarithms of the concentrations.

(3) lower right (low silver pressures and higher chlorine pressures) ; the system behaves like CdS-C1. Chlorine is incorporated as CdCI, and produces one free electron per chlorine atom. (See also Fig. 36.) (4) upper right (large values of pcl and pAg); silver and chlorine are incorporated in equal concentrations without the formation of vacancies or free change carriers. Here we are dealing with solid solutions of the form CdS AgC1.

+

The simplified neutrality conditions holding in the regions (l), (2), and (3) vary for other values of K,'pcd but that valid in region (4) remains the same.

IMPERFECTIONS IN CRYSTALLINE SOLIDS

425

It is seen from Fig. 38 that for a given pAgthe concentration of Ag which dissolves in the crystal is independent of pclt when Kclpcll)< 0, and increases with pa, when log Kclpcll)> 0. Similarly, the concentration of C1 which dissolves in the crystal for a given value of pCl9 is independent of pA, when log KASpAr< 4 and increases with PA,, for higher values of KA,pA,. When log KAgpAg > 4 and log Kclpcl9*> 0, silver and chlorine mutually increase each others solubility, the concentrations that are incorporated being equal. It should be noted that all these solubilities correspond to definition (a) given in the introduction (Section 30). Similar results are obtained for other foreign atoms having levels in positions similar to those of Ag or C1. For instance, monovalent electropositive atoms such as Cu and Au behave like Ag when replacing Cd. Trivalent electropositive atoms such as Ga, In, and Al replacing Cd behave like C1 when replacing S. The theory can also be applied to compounds like CdSe, ZnS, ZnSe, ZnTe, InSb, Sic. Of course the constants should have different values; therefore, the particular ranges which occur with CdS may be lacking for the other compounds. However, the results obtained will be very similar on the whole. d. Practical Limitations; Solubility in Contact with a Compound of the

Foreign Elements As has been remarked in Sections 16 and 25d, the partial pressures of the lattice constituents and the foreign atoms, namely P C d , ps,, pel,, and pA,, in the present case, cannot increase infinitely. Limits are set by the formation of other condensed phases. Thus when CdS is heated in an atmosphere containing Cd, SZand Clz, a liquid phase consisting mainly of Cd, but with some S and C1 dissolved in it may be formed on increasing the partial pressure of Cd. If the concentration of C1 dissolved in this phase is small, the partial pressure ( P C d ) L a t which the liquid phase is formed is independent of pel,. This limits the use of a plot like Fig. 36 to a sector below a horizontal line a t K,'(pCJL. On the other hand, solid or liquid CdClz may also be formed. The exact composition of this phase, including atomic and electronic imperfections, depends on the pressures P C d , pcl, and ps,, as just shown for the CdS phase. Following reasoning similar to that used in Section 7, it can easily be shown that the partial pressures of pCd and pcl, in the vapor which is in contact with condensed CdCL obey the relation P C d ' pCl*

=

KCdCll

as long as the concentration of S in the CdClz phase is small. Thus there is a value of pclpthat cannot be exceeded, for each value of PCd. In a plot like Fig. 36, this behavior gives rise to the boundary represented by a

426

F. A. KR6GER AND H. J. VINK

straight line with slope -2. Its position can be specified only when the A schematic values of K,' and Kc, are known, in addition to that of KCICI,. representation of the limits for the cases shown in Fig. 36 are indicated in Fig. 39. If other condensed phases were formed, for example S, SZCL, S2C12, etc., they would limit the useful zone of the figure in a similar way.

-

% ~gKczPcza

?

Fia. 39. Contours of the concentrations of Cls i n CdS in equilibrium with vapor containing Cd and Clz. The limits set by the formation of CdClz and a cadmium-rich melt (L) are shown.

In the case of Fig. 37, limits are set by the formation either of solid or liquid Ag, containing some Cd and S, or of liquid Cd, containing some Ag and S. AgzS may be formed at low Cd pressures, that is, a t high pressures of Sz. In the case of Fig. 38, the limits are set by the formation of the same condensed phase of Ag and by the formation of AgC1, the latter giving rise to a limiting line with slope - 1. It may occur that interesting regions cannot be realized a6 a consequence of these limitations. The solubility of Ag and Cl in CdS under conditions in which a specific compound such as AgzS, AgC1, or CdClz

IMPERFECTIONS I N CRYSTALLINE SOLIDS

427

is formed, can be obtained directly from diagrams such as Fig. 39. This solubility corresponds to that given in definition (b) of Section 21. 32. COMPARISON WITH EXPERIMENT The solubility of chlorine in crystals of CdS grown from a gas phase in a known partial pressure of chlorine has been determined.g There is satisfactory agreement on the whole between the experiment and the theory outlined above. Experimental conditions which lead to the incorporation of CIS' compensated by electrons, that is reducing conditions, can easily be attained in the system CdS-C1. Under more oxidizing conditions, incorporation in accordance with the relation [Cls'l

=

[VCd'I

occurs. The VCd'centers formed in this way are believed to be responsible for the extra absorption and the red fluorescence of CdS-C1.g Similarly Vz,l is believed responsible for the extra absorption and the blue fluorescence of ZnS-C1. 173 A striking difference between CdS and ZnS lies in the fact that the range of controlled conductivity has not been realized thus far in the second compound. Perhaps it falls outside the stability range (see Section 25d). The ability of Ag (or Cu and Au) and C1 (or Al, Ga, and In) to increase their mutual solubility has been observed Similarly, oxygen has been found to inin ZnS phosphors.239~391~392~3g3 crease the solubility of U in CaFz.394 In the papers dealing with such phosphors, ionic models have been used for the most part. The mutual increase in solubility of atoms having a higher or lower valence than the atoms they replace has been presented as an effect of charge compensation. Thus the decrease of positive charge accompanying the replacement of Cd2+or Zn2+by Cu+ was considered to be compensated either by the decrease of negative charge accompanying the replacement of S2- by C1- or by the extra positive charge accompanying the replacement of Cd2+ (or Zn2+)by A13+,Ga3+,etc. As we shall see in Section V, ionic and atomic models lead to exactly the same relationships; the simple explanations of the experimental results given in the papers just mentioned, are essentially identical with the theoretical results of the present section. I n studying the solubility of N in S i c which is at equilibrium with an atmosphere containing Nz a t 25OO0C, it was found that the nitrogen atoms substitute the C atoms and that their concentration varies as a F. A. Kroger and J. E. Hellingman, J. Electrochem. SOC.93, 156 (1948). F. A. Kroger and J. E. Hellingman, J. Electrochem. SOC.96, 68 (1949). 393 F. A. Kroger and J. A. M. Dikhoff, Physica 16, 297 (1950). 3g4 F. A. Kroger, Physica 14, 483 (1948). 391

392

428

F. A. KROGER AND H. J. VINK

function of pN, in accordance with the reIation [Nc] = pN23it.3033304 Since the Nc centers are undoubtedly completely ionized a t the high temperature of preparation, this relation might correspond to the situation represented in the left-hand part of Fig. 34. It may also be explained by the presence of a large concentration of intrinsic electrons and holes, for the concentration of the vacancies in S i c probably will be very small.

33. EQUILIBRIUM BETWEEN SOLIDAND LIQUID The distribution of foreign atoms between coexisting liquid and solid phases plays a part both in the preparation of pure crystals from a melt (segregation, zone refining)396*396and in the growing of crystals containing a definite concentration of foreign atoms (zone leveling,397preparation of phosphors). The influence oxidation-reduction reactions may have on the distribution of foreign atoms between the liquid and solid phases can be discussed along the lines used for the case of solid and vapor, provided an appropriate model is assumed for the liquid (see Section 19). A complete discussion of the equilibrium between solid and liquid has not been carried out thus far. In the few cases in which these problems have been, treated simplifying assumptions have been made, namely, (a) Ionization equilibria have been neglected; this means that a choice concerning the charge of the atoms of which the system is composed must be made. In most cases the presence of definite ionic forms has been assumed (b) The effect of the imperfections in the liquid on the distribution of foreign atoms between the two phases has been neglected. This means, in effect, that the concentration of vacancies in the liquid has been assumed to be larger than that of the foreign atoms.398 (c) A particular mechanism of incorporation of the foreign atom in the crystal has been assumed, corresponding to one of the various ranges discussed in Section 25. This means that the influence of oxidationreduction reactions on the distribution of the foreign atom has been neglected. Thus the distribution of Cd between molten and solid AgCl was disby assuming the atoms are present as ions of the type Ag+, Cd2+, and C1-, whereas the solid phase was regarded as the solid solution AgCl CdC12.

+

W. G. Pfann, J . Metals 4, 747, 861 (1952). W.G.Pfann, Bell Labs. Record 33, 201 (1955). 397 W.G.Pfann and I (.M. Olsen, Phys. Rev. 89, 322 (1953). 898 C. Wagner and P. Hantelmann, J . Phys. & Colloid Chem. 64,426 (1950). 395

398

IMPERFECTIONS I N CRYSTALLINE SOLIDS

429

The transfer of CdClz from the melt (L) to the crystal (S) is then described in terms of the reaction

Application of the law of mass action, under the assumption that the concentration of C1- in the liquid and solid is constant leads to the relation (33.2)

If the concentration of Cd2+Ag+ is large relative to the thermal disorder in the solid, one has [(CdZ+~,+)'s] = [(VA~+)'S] (33.3) and hence [(Cd2+Ag+)*8] = Kdistrf[(C~'+)L]* (33.4) or [(Cdz+)L]/[(CdZ+~g+)'s] = Kdi~tr[(Cd~+~g+)'~l* (33.5) The quotient [(Cd2+)L]/[(Cd2+Ag+)'s] is commonly known as the distribution coefficient. It increases with the concentration of Cd2+in the solid in this case. In the present example, the distribution coefficient is less than 1 for concentrations of (Cd2+Ag+)'s less than 10 mole per cent and greater than 1 for higher concentrations. This leads to a maximum for the melting point CdClz a t a concentration of (Cdz+An+)*B near of the solid phase AgCl 10 mole per cent. The distribution coefficient is then nearly equal to 1. The influence of the thermal disorder in the solid has been explained quantitatively in a slightly more general treatment. The result has been applied to the distribution of CdCl2, SrC12,and BaClz between solid and molten KCl.23a Transfer of CaClz from the melt to the crystal leads to a relation similar to (33.2), namely

+

(33.6)

Furthermore, one has the relation [(vK+)'Sl[(vCl-)'Sl = KS'

(33.7)

for the thermal disorder in the solid, whereas the concentrations of imperfections and foreign atom in the solid are subject to the following condition of electro-neutrality

+

= [(VCI-)~'] [(Caz+~+>~'l. [(VK+>S']

(33.8)

430

F. A. KROGER AND H. J. VINK

It follows from these three equations that (33.9)

in which C = Kdi8tr/(Ks’)+. Comparison of theory and experiment for SrClz leads to a value of K,’ equal to 1 . 4 . cm-6 a t the melting point and to a value of KdiJtr equal to 2.1017 cm-3. The value of Ks’ corresponds to a concentration of imperfections of the order of 1.2 10l8 cm-s at the melting point (see Section 1). It is also possible to explain the decrease of the solubility of CdClz in AgCl in the presence of PbC1z236 with the use of a similar simple theory. See reference 399 for a discussion of the factors influencing the solubility of Mn in NaCl and KC1. A discussion of the equilibrium distribution of a foreign atom between a monatomic solid, namely Si, and a liquid has been given, using a gaslike model of the The theory has been checked experimentally for the distribution of Li between solid Si doped with B and a molten alloy of Sn and Li.4°1~,401b~401c A case in which atoms having a higher or a lower valence than the atoms which they replace increase the mutual solubility of the pair, in a way similar to that encountered in Sections 21c and 23 for solids, has been found for liquids. The distribution of Sm3+ and Ce3+ions between molten SrCL and solid SrSe or SrS was found to be influenced by the presence of oxygen in a way such that the concentration in the melt was increased by addition of oxygen. The distribution of Eu2+ could not be influenced by oxygen4°2-~osin the same way. These effects can be explained on the basis of the considerations given in this article if a semicrystalline model is adopted for the melt, whereas further the concentration of vacancies normally present in the melt is supposed to be small relative to the concentration of Sm and 0.4p6If, in this case, Sm3+or the corresponding chloride SmCl3, is dissolved in the melt of SrC12, strontium P. Brauer, 2. Naturforsch. 7a, 741 (1952). H. Reiss, J. Chem. Phys. 21, 1209 (1953). 4010 H. Reiss and C. S. Fuller, Phys. Rev. 97, 559 (1955). 401b H. Reiss and C. S. Fuller, J. Metals 8, 276 (1956). 'Ole H. Reiss, C. S. Fuller, and F. J. Morin, Bell Syst. Tech. J . 36, 535 (1956). 402 R. Ward, J . Electrochem. SOC.93, 171 (1948). 403 R. W. Mason, C. F. Hiskey, and R. Ward, J. Am. Chem. SOC.71, 509 (1949). 404J.S. Prener, R. W. Mason, and R. Ward, J . Am. Chem. SOC.71, 1803 (1949). 406 A. Dreeben and R. Ward, J. Am. Chem. Soc. 73, 4679 (1951). 406 F. A. Kroger, Brit. J . A p p l . Phys. 6 , Suppl. 4, S58 (1954). 399

400

43 1

IMPERFECTIONS I N CRYSTALLINE SOLIDS

vacancies are formed in a concentration equal to half the Sm concentration. However, vacancies need not be formed if Sr2+ is replaced by Sm3+ and C1- is simultaneously replaced by 02-:SmOCl has the same number of electropositive and electronegative atoms as SrC12. It is easily seen that the replacement of S2- by 02in the solid phase will not increase the solubility of Sm3+. Thus, in the presence of oxygen, the concentration of Sm3+ (or Ce3+) in the liquid phase of SrClz will be favored relative t o the concentration of Sma+ in the solid SrS (or SrSe) phase, as found experimentally. On the other hand, the distribution of foreign ions such as Eu2+,which has the same valence as Sr2+,will not be influenced by oxygen. Another case in which the properties of a melt can be explained on the basis of a semicrystalline model occurs in the system CaF2 CaO. CaO increases the concentration of fluorine vacancies, and, therefore, the ionic c o n d u ~ t i v i t y 4when ~ ~ dissolved in molten CaF2.

+

V. Comparison of Various Models Used in Describing a Crystal

34. A JUSTIFICATION OF THE ATOMICNOTATION; THE PROBLEM OF DETERMINING THE EQUILIBRIUM CONSTANTS I n discussing the properties of crystals, models involving a definite assumption concerning the distribution of valence electrons are often used. Thus covalent bonding is assumed in discussing solids such as Ge, Si, and intermetallic compounds such as InSb, GaAs, etc. Similarly, an ionic model is usually used in discussing NaC1, NiO, etc. An objection can be made to such choices. Although the bonding may be mainly covalent in some substances and mainly ionic in others, a marked contribution of both types of bonding is present in most cases. Therefore, the choice of a particular model is always somewhat arbitrary, and never quite correct. As will be shown, this uncertainty is not important in applications of the theory presented in this paper. Various energy parameters play a role in the considerations of this paper. They may be divided into two major classes: (1) the energies of formation of the various atomic imperfections. They are contained in the constants Ks, K F ,K R ,Kox,etc. (2) the electronic energies, given by the position of electronic energy levels. They are contained in the constants K,, K1 . K14.

..

I n principle, there are two ways of obtaining the values of the energies, namely by calculation or from experiment. The calculation of values of these energies would require complete knowledge of the distribution of 407

T.BBllk, Acta Chem. Scand. 9, 1406 (1955).

432

F. A. KROGER AND H. J. VINK

valence electrons in the crystal, that is, of the type of bonding. Such calculations are very difficult and have been carried out only approximately even in the simplest cases. In practice, therefore, the values of the energies, or constants involving them, must be obtained from experiment. The constants mentioned under (1) are obtained by comparing the results of the theory outlined in this paper with experiment (see reference 9, for example). The electronic energies, on the other hand, may be obtained from optical experiments involving absorption, photoconductivity, and luminescence, and from measurements of the Hall coeffcients or the thermoelectric power as functions of temperature. It is necessary to have a rough notion of the energies to be expected for particular centers in attempting to interpret such measurements. Such a notion can, however, be obtained without assuming a particular type of bonding (see Sections 2b and 23). Consideration of this type have led us to use the atomic notation which does not refer to a special type of bonding. 35. COMPARISON OF

THE

ATOMICAND IONIC NOTATION

Although the type of bonding need not be known exactly, models in which a particular type is assumed may nevertheless be used and are in fact'employed often. A system described with the use of atomic notation can usually be described with the use of ionic notation as well. It is easy to see that corresponding centers in the two models have the same chemical composition and the same charge relative to the normal crystal, that is, the same effective charge. As an example, let us consider the center formed in CdS by replacing an atom of Cd by one of Al. This center is indicated by Alcd in the atomic notation. The center chemically equivalent to it in the ionic representation is also formed by replacing Cd with Al. Since CdS is written as Cd2+S2-in ionic nomenclature, the replacement of Cd by A1 produces an A12+ion at a Cd2+site (AlCdr+2+)or (A~w+~+@). The effective charge obviously is zero in both cases, for no charge has been added or removed. Similarly, the replacement of a cadmium atom by Al+ gives rise to a center which is APCdin the atomic notation and A13+Cd9+ in ionic nota\el. The effective charge tion. Both centers have an effective charge may be indicated by means of a dot in both systems of notation. This leads to the symbols (Al+cd)' and (Al*+Cd;)* respectively. As in the atomic notation always the added charge is equal to the effective charge, the symbol may be simplified by omitting either the

+

IMPERFECTIONS IN CRYSTALLINE SOLIDS

433

effective charge or the charge inside the brackets. I n the present article we have chosen the latter possibility. According to this convention the atomic notation of the center discussed is Alca’. Similar arguments hold for vacancies although particular attention must be paid in this case. In atomic terminology one logically speaks of atomic vacancies, whereas one usually speaks of ionic vacancies in ionic models. Obviously the situation which arises from removal of an atom should not be compared with that arising from removal of an ion since the latter consists of an atom and a charge. One should, however, compare centers which arise in the same chemical process, that is centers which come into existence by the removal of atoms and charges in either models. The equality of effective charge is then maintained just as in the case of foreign atoms For example, a cadmium vacancy, which is represented by the symbol Vc,,in atomic notation, corresponds to a cadmium ion vacancy that has lost two electrons, namely v 2 + C d n + , in the ionic representation. Similarly,

It is possible to make a list of comparable centers in both representations in this way. Table I11 shows such a list for the case of MgO possessing vacancies, interstitial Mg atoms, Al, Li, and Mn atoms at Mg sites, and C1 atoms at 0 sites. In order to simplify the symbols in the ionic notation, the ions of the base crystal which appear in the suffix may he replaced by C (cation) or A (anion). In the German literature, a notation due to Schottky408is frequently used. This notation is indicated in the third column of Table 111;it em. ploys squares to indicate vacancies, dots to indicate atoms a t lattice sites, and open circles to indicate atoms at interstitial sites. Yet another notation has been proposed by R e e ~This . ~ ~ notation ~ is demonstrated on the fourth column of Table 111.It makes use 01squares to indicate lattice sites and vacancies a t such sites (a plus sign indicating cation sites, a minus sign anion sites) ; triangles indicate interstitial positions, whereas e and p denote quasi-free electrons and holes. The “system” behind the various notations (and also an occasional lack of logic) can easily be found by comparing corresponding formulations in the various notations. This is not the place to discuss the merits of the different notations in detail. It goes without saying that the present authors favor the atomic W. Schottky and F. Stockmann, in “Halbleiterprobleme I” (W. Schottky, ed.), pp. 80, 135. Vieweg, Braunschweig, 19544 409 A. L. G. Rees, “Chemistry of the Defect Solid State.” Wiley, New York, 1954. 408

434

F. A. KROGER

AND H. J . VINK

TABLE111. COMPARISON OF VARIOUSNOTATIONS OF CENTERS FOR THE CASE OF MgO CONTAINING VACANCIES, INTERSTITIALS A N D FOREIQN ATOMS Notation of the present paper Atomic notation

Ionic notation

Schottky's notation

Rees's notation

Effective charge (in electron charges)

0 -1 -2

0

0 OX

0 0'

+1 +2

00" MgOX MgO' L i W (Mg) LiO'(Mg) GaOX(Mg) GaO' (Mg) ClOX(0)

0

+1

0 -1

0 +I 0 +1 0 +1 +2

Cl'. (0) Mnox(Mg) MnO' (Mg) MnO" (Mg)

notation used throughout this article; the reader may make his own choice. Reaction equations expressed in a notation belonging to one model can be translated into the notation associated with another model by means of a "dictionary" such as Table 111. Thus Eq. (29.1) which is written in the form

in the atomic notation, becomes 6 + -Li+z02+ 6Mn2+02-+ 46 2

(1 - 26)Mg2+02-

-+

(Mg")

(1-26)

0 2

(Li+MI*+)'a( Mn3+~gz+)*aOz-

in the ionic formulation. Similarly, the incorporation of lithium in NiO in the range of controlled valence is described by ( 1 - 6)NiO

+ 36 Li20 + 46

0 2

+ K~(I-*)(L~W)~O

IMPREFECTIONS I N CRYSTALLINE SOLIDS

435

in atomic notation, or by (1 - 6)Ni2+02-

or (1 - 6)Ni2+02-

+ -26 Li+202- + -46 o2-+ (Ni2+)~l-*)(Li+Ni~+~)ao2-

+ 2-6 Li+202- + -46 O2+ (Ni2+)~~-~a)(Li+~i*+Ni3+)~02

in ionic notation (see also Section 23). The incorporation of lithium in a range of controlled atomic imperfection can be formulated in the manner (1 - 26)NiO

+ 6LizO + Ni(l-za"ir;i')za(Vo'')a0(1--6)

in the atomic notation, or by

(1 - 26)Ni2+02in the ionic notation.

+ 6Li+202--+ (Ni2f)(~-~~)(Li+~in+)'za(V~~-)''602-(1-6)