DEVIATIONS FROM STOICHIOMETRY AND LATTICE DEFECTS IN IV-VI COMPOUNDS AND THEIR ALLOYS. GENERAL FEATURES AND EXPERIMENTAL METHODS

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DEVIATIONS FROM STOICHIOMETRY AND LATTICE DEFECTS IN IV-VI COMPOUNDS AND

THEIR ALLOYS. GENERAL FEATURES AND EXPERIMENTAL METHODS

A. Strauss, R.F. Brebrick

To cite this version:

A. Strauss, R.F. Brebrick. DEVIATIONS FROM STOICHIOMETRY AND LATTICE DE- FECTS IN IV-VI COMPOUNDS AND THEIR ALLOYS. GENERAL FEATURES AND EX- PERIMENTAL METHODS. Journal de Physique Colloques, 1968, 29 (C4), pp.C4-21-C4-33.

�10.1051/jphyscol:1968403�. �jpa-00213607�

JOURNAL DE PHYSIQUE

Colloque C 4, suppiment au no 11-12, Tome 29, Novembre-Dkcembre 1968, page C 4 - 21

DEVIATIONS FROM STOICHIOMETRY AND LATTICE DEFECT S IN IV-V1 COMPOUNDS AND THEIR ALLOYS.

GENERAL FEATURE S AND EXPERIMENTAL METHOD S

A. J. STRAUSS and R. F. BREBRICK

Lincoln Laboratory (*), Massachusetts Institute of Technology Lexington, Massachusetts 02173

RBsumB. - On dispose de beaucoup plus de renseignements sur les kcarts A la stoichiomktrie dans les composks IV-V1 que dans n'importe quel autre groupe de composes semiconducteurs. Nous dB- crivons ici la remarquable combinaison des propriktks des composksIV-V1 qui en font des matkriaux particulikrement adaptks a l'ktude expkrimentale des karts a la stoichiom6trie. La prksentation la plus pratique des rbsultats de ces etudes est sous la forme de projections du diagramme de phase tridimensionnel : tempkrature-composition, pression-composition et pression-tempkrature. Ce sont les caractkres gknkraux de ces projections pour les composks IV-V1 qui sont dkcrits, ainsi que les principales mkthodes expkrimentales employkes pour leur dktermination. En conclusion, nous ras- semblons les diffkrents types de rksultats qui ont kte rapportks pour chaque compose particulier.

Abstract. - Considerably more information concerning deviations from stoichiometry is avail- able for the IV-V1 compounds than for any other group of semiconducting compounds. This paper first describes the unusual combination of properties exhibited by the IV-V1 compounds which makes them especially suitable materials for experimental studies of stoichiometric deviations.

The results of these studies are most conveniently presented in the form of temperature-composi- tion, pressure-composition, and pressure-temperature projections of the three-dimensional phase diagram. The general features of these projections for the IV-V1 compounds are described, together with the principal experimental methods which have been employed to determine them. The paper concludes by summarizing the types of data which have been reported for theindividual compounds.

Introduction. - For convenience, we designate solid chemical compounds by stoichiometric formulas, corresponding to ideal crystal structures, which state that the relative numbers of different atoms are given by the ratio of small integers. However, it is a well- established principle of statistical mechanics that at any finite temperature a compound in the solid state will in general be stable over a range of compositions, or homogeneity range, which may or may not include the stoichiometric composition specified by the for- mula.

The deviations from stoichiometry in the IV-V1 compounds and their alloys have been the subject of a large number of investigations conducted over the past fifteen years. As a result of these investigations, considerably more information concerning stoichio- metric deviations is available for the IV-V1 compounds than for any other group of semiconducting com-

(*)

Operated with support from the U. S. Air Force.

pounds. It is the purpose of this paper to give a general review of this information and of the methods used to obtain it. Before doing so, however, we shall give a general description of the unusual combination of physico-chemical properties exhibited by the IV-V1 compounds which have occasioned special interest in their deviations from stoichiometry and have made it relatively easy to obtain experimental data concern- ing these deviations.

The interest in deviations from stoichiometry in the IV-V1 compounds arises principally because of the marked and often dominant effect of these deviations on the electrical properties of the compounds. For almost every sample which is not intentionally doped, the deviations are large enough compared t o the impurity content and intrinsic carrier concentration that, at room temperature and below, the type and concentration of charge carriers are determined pri- marily by the nature and magnitude of the deviations.

For this reason, although the deviations are usually

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1968403

( 2 4 - 2 2 A. J.

STRAUSS AND R. F. BREBRICK

too small to be determined chemically, their relative to be equilibrated with the vapor phase in convenient magnitudes in samples of a particular compound times (not more than a few weeks).

can be found from the carrier concentrations, which are readily obtained by Hall coefficient measurements.

Furthermore, for many of the compounds the stoi- chiometric deviations can be varied over an appreciable range by growing or annealing samples at controlled temperatures and partial pressures. Therefore by using carrier concentration data for such samples it is possible to establish functional relationships between solid-phase composition, vapor pressure, and tempe- rature. Comparing these empirical relationships with theoretical models derived from thermodynamics and statistical mechanics has been one of the principal methods used to characterize the lattice defects responsible for the stoichiometric deviations.

Fundamental investigations on the physical chemis- try of the IV-V1 compounds are not the only source of interest in their deviations from stoichiometry.

An equally important source is the practical problem of preparing materials with controlled electrical properties for other scientific studies and for device applications, since in most cases this problem must be solved by control of composition rather than by control of impurity content, and in others control of both composition and impurity content is necessary.

It should be noted that with few exceptions the mini- mum attainable carrier concentration is determined by the minimum attainable deviation from stoichiom- etry, and that the natural and artificial p - ⁿ junc- tions used in devices are produced by changes in composition rather than in impurity content.

So far we have described the properties of the IV- V1 compounds which are primarily responsible for the continued interest in their deviations from stoi- chiometry. Now we shall list a number of other properties which, although not individually critical, in combination have greatly encouraged the study of the deviations because they make it relatively easy to obtain and interpret pertinent experimental data.

(In considering these properties, it should not be assumed that all of them are equally characteristic of all the compounds in the group.)

l) Large singIe crystals can be grown from the melt because the melting points of the compounds and their vapor pressures at the melting points are not excessive. The highest melting point is l l l l OC, the value for PbS. Single crystals of a number of compounds can also be grown from the vapor phase.

2) Diffusion rates in the solid phase are sufficiently high at elevated temperatures to permit samples of convenient thickness (1-2 mm) for Hall measurements

3) With decreasing temperature, the diffusion rates decrease sufficiently to permit the stoichiometric deviations established at elevated temperatures to be quenched in by rapid cooling and then retained in a metastable state for long periods at room temperature.

(In some cases discussed later, the ability to quench in deviations is limited by the rate of internal precip- itation rather than by the rate of equilibration with the vapor phase.)

4) Carrier mobility is sufficiently high and contact resistance sufficiently low to permit accurate Hall coefficient measurements even for samples with carrier concentrations exceeding 10'' cm-3.

5) In most cases, the electrical conductivity is predominantly due to one type of charge carrier, so that the carrier concentration can be obtained from the measured Hall coefficient by using the simple one-carrier formula. In some samples of p-type PbTe, SnTe, and possibly GeTe, however, the one-carrier formula is inapplicable to room temperature data because of the presence of two type of holes.

6 ) The lattice defects responsible for deviations from stoichiometry are completely ionized at all temperatures, so that each defect supplies an integral number of charge carriers.

From the list of favorable properties just given, it is apparent that the IV-V1 compounds are especially suitable materials for experimental studies of devia- tions from stoichiometry. The results of these studies are most conveniently presented in the form of phase diagrams which show the composition of the solid phase as a function of temperature, the partial pressures of the elements in the CO-existing vapor phase, and the composition of any CO-existing liquid phase.

The general features of these diagrams are similar for all the IV-V1 compounds and also for their alloys.

We shall describe these general features and also the

principal types of experiments used to obtain phase

diagram data, as a preliminary to presenting in a

subsequent paper the detailed results which have been

reported for individual compounds and alloys. We

shall not attempt to give an extensive theoretical

interpretation of the phase diagrams, since a compre-

hensive treatment in terms of mass action laws is

given by Kroger [l] in his book on imperfect crystals,

and Brebrick [2] has recently used a somewhat different

approach based on statistical mechanics to deal with

non-stoichiometry in binary semiconductorcompounds.

DEVIATIONS FROM STOICHIOMETRY AND LATTICE DEFECTS C 4 - 2 3 Phase diagrams for IV-V1 compounds. - The

phase relations for, a binary system of elements M and N can be completely specified by a three-dimen- sional pressure-temperature-composition or P - T - x diagram, where x represents the atom fraction of one of the elements. The information is generally presented in the form of the two:dimensional T - X, P - x, and P - T projections, and we shall discuss each of these in turn.

T - x PROJECTION.

Figure 1 shows a schematic T

x projection for a IV-V1 system in which there is a single congruently melting compound whose ideal

by volatilization. Liquidus points in the Ge-Te system have also been found by means of partial pressure measurements which will be described when we discuss the P - T projection.

In order to consider the deviations from stoichiom- etry exhibited by the compound MN, we shall examine in detail the portion of the T -

projection in the vicinity of 50 at. %. Figure 2 shows schematic

I

FIG. 2.

Schematic T

projections, with expanded abscissa, for the composition region near

FIG.

- Schematic T-

projection for a binary system o f ' elements M and

crystal structure corresponds to the formula MN, where M is the Group IV element and N is the Group V1 element. In this diagram x is the atom fraction of N. The compound exhibits such small deviations from stoichiometry that on the scale shown it is a line phase with x

0.5. This type of diagram is charac- teristic of the Pb-S, Pb-Se, Pb-Te, and Sn-Te systems.

The Ge-Te system also contains only one compound, GeTe. The diagrams for the Sn-S, Sn-Se, Ge-S, and Ge-Se systems include at least one additional com- pound, ,,but in the following discussion we shall consider only the MN compound. The liquidus curve, which gives the temperature at which a liquid of given composition is in equilibrium with a solid phase (either MN, M, or N), is generally found by thermal analysis experiments. Since at elevated temperatures IV-V1 materials have appreciable vapor pressures and also react with oxygen, these experiments are performed with the sample in a sealed quartz ampoule.

The gas space is made small enough that the composi- tion of the condensed phase is not appreciably changed

diagrams of this portion, with the abscissa scale greatly expanded, which illustrate the three types of homogeneity range observed for the IV-V1 compounds.

The stoichiometric composition does not lie within the homogeneity range at all temperatures in any of the diagrams. In the left-hand and center diagrams, the homogeneity range includes the stoichiometric compo- sition, but only at temperatures below a maximum value, and in the right-hand diagram it does not include the stoichiometric composition at all. In no case, therefore, does the maximum melting point, where the solid and CO-existing liquid have the same composition, occur at the stoichiometric composition.

In the left-hand diagram the composition at the maximum melting point is more than 50 at. % .M

(X

< 0.5). In the other two diagrams, the composition at the maximum melting point is more than 50 at. %

N (X > 0.5). In the left-hand and center diagrams, at sufficiently low temperatures, the solubility of the elements in the compound decreases with decreasing temperature. In this region the elements are said to exhibit retrograde solubility.

We shall now describe the principal methods which have been used to determine the solidus curves which form the boundaries of the homogeneity range of MN.

(To simplify the discussion, we shall restrict it to the

temperature region above the M - MN and N - MN

eutectics, since in most cases the data are limited to

this region. Similar principles would be applicable

below the eutectic temperatures.) Points on the solidus

curves give the compositions of samples which are in

equilibrium with M-rich and N-rich liquids. We

C 4 - 2 4

A. J. STRAUSS AND R. F. BREBRICK shall designate these samples as M-saturated and

N-saturated, respectively, even though it can be seen from figure 2 that some M-saturated samples contain more than 50 at. % N, and vice versa. The solidus curves are generally found by preparing samples which are M-saturated or N-saturated and then determining their compositions. We shall first discuss the problem of determining composition.

The compounds SnTe and GeTe exhibit such large deviations from stoichiometry that it has been possible to prepare solid samples with specified compositions inside the homogeneity range by weighing out the elements in appropriate proportions. Quantitative data concerning the composition of almost all other IV-V1 samples have been obtained by carrier concen- tration measurements, as discussed in the Introduction.

The carrier concentrations have generally been found by Hall coefficient measurements at room temperature or below, although in some cases measurements of resistivity (at room temperature) or thermoelectric power (at or above room temperature) have been used. From experimental results discussed below, we can conclude that - except perhaps for GeTe

in the absence of impurities, samples which are n- type contain more than 50 at. % M and those which are p-type contain more than 50 at. % N. Furthermore, the carrier concentrations increase monotonically as the deviations from stoichiometry increase. To obtain absolute composition values it is obviously necessary to know the quantitative relationship between the concentrations of carriers and lattice defects.

We shall now consider the methods which have been used to prepare saturated M N crystals. One such method is the solidification of crystals by direc- tional freezing of a melt under quasi-equilibrium conditions, as in the Bridgman method of crystal growth. The temperature of solidification is either found from the initial composition of the melt and the known iliquidus curve for the system, or fixed by quenching. The corresponding solidus compositions are those of the solids which are first-to-freeze and last-to-freeze before quenching, respectively. This method has not been used extensively to establish solidus curves for the IV-V1 compounds. However, directional solidification has been used several times in attempts to determine a particular solidus composi- tion, the one at the maximum melting point. In some experiments, carrier concentration measurements were made on ingots frozen from melts of various compo- sitions close to stoichiometric. The ingot which was most uniform in concentration along its length was assumed to have the congruently melting composition.

In other experiments, the congruently melting compo- sition was taken to be the first-to-freeze composition of ingots directionally frozen from stoichiometric melts. This is a good approximation for any melt composition close to the congruently melting composi- tion, due to the flatness of the liquidus curve near the maximum melting point, as shown in figure 2. However, this method is inapplicable to many of the IV-V1 compounds because of internal precipitation effects resulting from retrograde solubility, as discussed below.

In a second method of preparing saturated samples, equilibrium between a sample and an M-rich or N- rich liquid is established via the vapor phase. This method has been used in determining solidus curves for several IV-V1 compounds and their alloys. An MN crystal of any initial composition, usually 1-2 mm thick, is annealed at an elevated temperature in a sealed fused silica ampoule containing an ingot which at the annealing temperature is a mixture of solid MN and either M-rich or N-rich liquid. Generally the ingot is separated from the sample by some type of fused silica spacer, so that it does not adhere to the sample when the run is over. Since three phases (including the vapor) are present in the ampoule, the system has only one degree of freedom, and fixing the temperature fixes the equilibrium composition of each phase. Except for annealing temperatures close to the maximum melting point, the exact composition and amount of the ingot are not critical, since it is only necessary for the composition of the ingot and sample together to lie between the liquidus and solidus curves at the annealing temperature. If this condition is satisfied, the composition of the sample must change by solid state diffusion until at equilibrium it reaches the M-saturated or N-saturated value, depend- ing on whether the ingot is M-rich or N-rich. (If an M-rich sample is annealed in the presence of an N-rich ingot, or vice versa, a p

n junction can be formed by interrupting the diffusion process before equilibrium is attained. This method has been used to prepare p - n junctions for infrared detectors and lasers.)

The time required to reach equilibrium depends on

the rate of diffusion, which decreases exponentially

with decreasing temperature. The equilibrium time

therefore increases rapidly as the annealing temperature

decreases. It should be noted that the relevant diffusion

coefficients, the inter-diffusion coefficients, are orders

of magnitude larger than the self-diffusion coefficients

for M or N in MN. At the conclusion of the saturation

run, the ampoule is quenched as rapidly as possible

DEVIATIONS FROM STOICHIOMETRY AND LATTICE DEFECTS

in order to retain the solidus composition characteristic of the annealing temperature. The purpose of rapid quenching is not only to minimize interaction with the vapor phase during cooling but also to minimize internal precipitation due to retrograde solubility.

For most of the IV-V1 compounds, internal precip- itation is the greater problem.

The reason for internal precipitation is illustrated by figure 3, in which the left-hand T - x projection

LIQUID

FIG. 3.

Schematic T-

projection near

0.5, illustrating

reason for

precipitation.

from figure 2 is reproduced on a larger scale. Consider the cooling of an MN sample which has been N- saturated at the temperature and composition given by point A on the solidus curve. If its composition djd not change during cooling, its cooling curve would be the dashed line AB. Since this line lies outside the homogeneity range of MN, the sample would be supersaturated in N. The excess N can be removed from the lattice not only by loss to the vapor phase but also by the formation of microprecipitates of N (or more precisely, an N-rich phase) at sites dispersed throughout the crystal. The precipitation process can occur much faster, since it generally requires diffusion over a distance much shorter than the distance to the surface of the sample. Therefore a cooling rate may be fast enough to prevent loss of N to the vapor phase but not to prevent internal precipitation. In this case, the cooling curve for the lattice will coincide with the solidus curve, as indicated by the arrows in figure 3, down to a temperature at which the diffusion rate has been reduced sufficiently to prevent further precipitation. Below this effective quenching temper- ature, which is indicated by point C, the composition of the lattice remains constant, and the cooling curve

is given by CD. An exactly similar description applies to cooling on the M-rich side of the homogeneity range, where a typical cooling curve is given by A' C' D'. In general, for a given experimental proced- ure, the effective quenching temperatures C and C' are not the same since the diffusion rates and therefore precipitation rates are different for M-saturated and N-saturated MN.

The microprecipitates of M or N form a dispersed second phase which does not contribute charge carriers to the MN crystal. Therefore the measured carrier concentration is determined by the amount of M or N in excess of the stoichiometric composition which remains dissolved in the lattice. Consequently the carrier concentration measured at the completion of a saturation experiment is equal to the value corre- sponding to the solidus composition at the effective quenching temperature. For this reason, if the carrier concentration measured in a series of saturation experiments initially increases and then becomes constant as the annealing temperature is increased, it is likely that the upper limit on concentration is observed because of internal precipitation, not because the composition along the solidus curve becomes independent of temperature. Up to a certain point the effective quenching temperature can be increased by changing the experimental procedure to increase the quenching rate, but for every compound there is an intrinsic upper limit to the quenching rate which is determined by the heat capacity and thermal con- ductivity of the material.

It should be noted that internal precipitation will occur when crystals grown from the melt are cooled to room temperature, if the excess M or N incorporated during solidification is greater than the solubility at the effective quenching temperature determined by the cooling rate. This will be the case for many IV- V1 crystals which are grown by the Bridgman tech- nique, since these crystals are usually furnace cooled rather than quenched. The precipitation process is reversible, since heating a sample containing micro- precipitates will cause them to redissolve to the extent determined by the solidus curve at the annealing temperature. The dissolution rate increases strongly with increasing temperature, because of the increase in diffusion coefficients, and equilibrium times of the order of minutes are observed even at temperatures well below the maximum melting point.

Internal precipitation in IV-V1 compounds exhibit-

ing retrograde solubility is utilized as the basis of a

third method of preparing saturated samples. An M-

rich or N-rich sample is annealed in a sealed ampoule

C 4

- 26 A.

S T W S S AND R. F. BREBRICK at a temperature where the solubility of M or N,

respectively, is smaller than the total excess of that element present. When equilibrium is reached, the amount of excess M or N which remains dissolved in the lattice is equal to the solubility at the annealing temperature, with the remainder present in the form of microprecipitates. Equilibrium can be attained either by formation of precipitates or by dissolution of precipitates initially present. The sample is then quenched to room temperature at a rate which is sufficient to prevent further precipitation. The, mea- sured carxier concqtration c~rresponds to the compo- sition on the solidus curve at the annealing tempera- ture. (In one series of experipents, this procedure was modified by using Seebeck coefficient measurements made in situ at the annealing temperature to determine the carrier concentration.)

It has been found that the solidus' points obtained by precipitation experiments are in good agreement with those found by the method of equilibrating with a solid-liquid ingot. The principal advantage of the precipitation technique is that the equilibration rate at a given annealing temperature is many times faster than that for the ingot-anneal method. Therefore the minimum temperatures for making solidus deter- mination~ - the temperatures below which equili- bration times become impractically long - are considerably lower for precipitation experiments than for ingot-anneal experiments.

P

PROJECTION.

According to Gibbs' phase rule, fixing the temperature of an M - N system fixes the composition of the solid MN phase only when another condensed phase (either liquid or solid) is present in addition to the vapor phase.

Therefore the only solid-phase compositions appearing in T

projections down to the eutectic temperatures are those along the solidus curves. To specify compo- sitions inside the homogeneity range, it is necessary to specify another intensive variable in addition to the temperature. The partial pressure of one species in the vapor phase is a convenient choice.

The vapor phase in equilibrium with a IV-V1 compound is composed of MN molecules, M atoms, and one or more species containing N atoms. The relative proportions of the latter are given by the equilibrium constants characteristic of pure sulfur, selenium, or tellurium. For most temperatures and pressures of interest here, there is no appreciable concentration of N atoms, and the diatomic N, molecule is the predominant N, species. It has been found that the partial pressure of MN molecules does not change appreciably across the homogeneity

range of solid MN, whereas the partial pressures of the elementary species .generally change by orders of magnitude. (These properties are characteristic of a highly ordered solid phase with a narrow homogeneity range.) Therefore the relationship between solid and vapor-phase at a fixed temperature is best represented by plotting the partial pressure of either M atoms or NZ molecules as a function of the composition in a P -

projection. This relationship is of fundamental interest because the partial pressure of a vapor species is related in a known way to its chemical potential in the vapor phase and therefore in the solid phase in equilibrium with the vapor. In addition, solid-vapor relationships are of practical importance in materials preparation, since samples of IV-V1 compounds with compositions inside the homogeneity range are almost always obtained by equilibration with an elementary vapor of controlled pa;tial pressure.

Figure 4 shows a schematic P -

projection for an M - N system. The equilibrium partial pressures of M and NZ at constant temperature are plotted as functions of X, the average atom fraction of N for the condensed phase or phases. The quantitative features of the diagram are not significant. (For exam- ple, the abscissa scale has been distorted so that the

FIG.

Schematic P

projection for the M

N system in the vicinity of MN(c).

diagram can show the partial pressures over M-rich

and N-rich liquids as well as those in equilibrium

with solid MN, the relative pressures of M and NZ

are arbitrary, and all P -

curves are shown as

straight lines.) The qualitative features are a conse-

quence of the thermodynamic criterion for a stable

phase - the chemical potential of a component

increases monotonically with an increase in the atom

DEVIATIONS FROM STOICHIOMETRY AND LATTICE DEFECTS C 4 - 27 fraction of that component. It follows that the partial

pressures of ther components change with average composition in regions where there is only one con- densed phase and remain constant in two-phase regions, since in the latter the composition of each phase is independent of the average composition. Therefore, on crossing the P -

diagram from left to right, we see that the partial pressure of N2 increases with increasing

in the region containing only M-rich liquid, becomes constant when both M-rich liquid and solid MN are present, increases across the homoge- neity range of MN, becomes constant when both MN and N-rich liquid are present, and finally increases again when only N-rich liquid is present.

Since the atom fraction of M, (1 - X), decreases as

X

increases, the partial pressure of M decreases as that of N2 increases. When solid MN is present, the two partial pressures are related quantitatively by the equation

where P, and P,, are the partial pressures of M and NZ, respectively, R is the gas constant, Tis the absolute temperature, and AG, is the free energy of formation for the reaction 112 M(g, 1 atm) + 114 N2(g, 1 atm) + 112 MN(c). In deriving this equation, the small variation of AG, across the homogeneity range is neglected, and it is assumed that the vapor is ideal.

Whether or not the compound MN has any con- gruently subliming composition

a composition which is in equilibrium with a vapor of the same average composition - is determined by the relative pressures of M and N, for values of x within the homogeneity range of MN. In general, the congruently subliming composition, if one exists, will not be exactly stoi- chiometric. In the stoichiometric vapor, the partial pressure of N, is equal to one-half the partial pressure of M, if these are the only species other than MN molecules present in the vapor phase. Regardless of the species present, at constant temperature the total pressure of the vapor phase is lower for the congruently subliming composition than for any other.

When a sample of compound MN with composition inside the homogeneity range is heated in an initially evacuated, sealed ampoule under isothermal conditions, its composition will generally change because the vapor produced by sublimation has a different compo- sition from the solid. Whatever the initial composition, the effect of sublimation will be to shift it toward the congruently subliming composition, if the compound has one. Sufficient repetition of the process will cause

the sample to attain the congruently subliming compo- sition, after which there will be no further change.

The same result can be obtained by allowing the sample to sublime in a dynamic vacuum or in a stream of inert gas, provided that the rate of sublimation is slow enough to permit homogenization of the sample by solid state diffusion. If the compound has no congruently subliming composition, however, the vapor phase is always richer in one of the elements (say, N) than any solid within the homogeneity range. In this case, any of the sublimation procedures mentioned will cause the sample composition to continue changing until it reaches the limit of stability (for this example, the M-rich solidus). Still further sublimation will cause the formation of a liquid phase and eventually complete melting of the sample.

The type of projection shown in figure 4, but with coordinates of P'/' and (X - 1/2), has been used to present data for the partial pressures of Tez(g) in equilibrium with samples of SnTe and GeTe lying within the homogeneity range, which is wide enough in each case to permit preparation of samples with known values of x by weighing out the elements in desired proportions. The partial pressures were determined by an optical absorption method described below.

A different type of P -

projection has been used to present the experimental results for PbS, PbSe, and PbTe, for which the narrowness of the homoge- neity range makes it necessary to obtain composition data by means of carrier concentration measurements.

A projection of this kind is shown schematically in figure 5, where carrier concentration measured at room temperature on samples annealed at an elevated temperature and then quenched is plotted on a log-log

log P,,, +

2

FIG.

-Schematic plot of carrier concentration as a

function of

partial pressure for MN(c) at constant tem-

perature.

C 4 - 2 8

A. J. STRAUSS AND R. F. BREBRICK scale as a function of N, partial pressure at the anneal-

ing temperature. At the lowest partial pressures, the material is n-type. As the partial pressure increases, the electron concentration decreases monotonically, at first slowly with a constant slope, then more rapidly.

The material then becomes p-type, after which the hole concentration increases monotonically, at first rapidly, then more slowly with a constant slope.

The plot thus consists of two parts which are roughly symmetric.

As pointed out previously, from the thermodynamic criterion for a stable phase it folIows that

the atom fraction of element N in solid MN, increases mono- tonically as the partial pressure of NZ increases.

Therefore the plot in figure 5 shows that as

increases there is a monotonic decrease in electron concentration in n-type samples and a monotonic increase in hole concentration in p-type samples. This property justifies the basic model which has been adopted for the devia- tions from stoichiometry in the IV-V1 compounds (with the possible exception of GeTe), according to which the lattice defects associated with excess M and N are donors and acceptors, respectively.

Consequently the transition from n-type to p-type occurs at the stoichiometric composition in samples not containing electrically active impurities.

(It should be noted that in general the concentration of lattice defects associated with a deviation from stoichiometry determines not the electron concen- tration, n, or the hole concentration, p, but the differ- ence between them. In almost all samples of the IV-V1 compounds, however, at room temperature this difference is so large compared to the intrinsic carrier concentration that the concentration of minority carriers is negligible compared to that of majority carriers. Therefore in n-type samples

and in p-type samples (p - n) -- ^p. Consequently, the carrier concentrations measured at room temper- ature are directly related to the deviations from stoichiometry .)

Two different methods have been used to obtain carrier concentration versus partial pressure data for plots of the type shown in figure 5. Almost all these data have been obtained by annealing a sample of MN in a sealed, fused silica tube which contains a con- densed phase of pure M or N maintained at a temper- ature lower than that of the sample. The sample is placed at one end of the tube and a small quantity of the element at the other end. The tube is evacuated, sealed, and placed in a furnace with two zones, which

are then heated to the desired temperatures. (A low pressure of inert gas is frequently placed in the tube to reduce sublimation of the sample.) The vapor pressure of the solid or liquid element at the reservoir temperature determines its partial pressure in contact with the sample. Fixing this pressure and the sample temperature fixes the equilibrium composition of solid MN, and the sample composition changes by solid state diffusion until it reaches this equilibrium value. As in the saturation experiments described previously, the time required to attain equilibrium increases rapidly as the sample temperature decreases.

At the conclusion of the run, the sample is quenched as rapidly as possible in order to minimize interaction with the vapor phase during cooling. (Internal precip- itation is not a problem, except for equilibrium compositions close to the solidus curves.) The carrier concentration is then determined by measuring the Hall coefficient of the quenched sample.

Because of the exponential dependence of vapor pressure on temperature, it is necessary to control the reservoir temperature very accurately in order to obtain accurate partial pressure data by the two- temperature method. It is also necessary to prevent contamination of the reservoir by vapor of the other element produced by sublimation of the sample.

This generally limits the applicability of the method to equilibrium composi'tions not too close to the congru- ently subliming value, if there is one.

The other method of obtaining carrier concentration versus partial pressure data has been used only once, in an investigation of PbS. A mixture of gaseous H,S and H, was passed slowly through a heated quartz vessel containing the PbS sample. After annealing, the sample was cooled to room temperature and the carrier concentration was determined by Hall coeffi- cient measurements. The partial pressure of S, was calculated from the annealing temperature and the published equilibrium constant for the HZS-H, system.

The experimental methods just described have

been used to obtain data not only for undoped samples,

but also for samples intentionally doped with elec-

trically active impurities. The effects of impurities are

illustrated by the three schematic plots of carrier

concentration versus partial pressure shown in

figure 6. The upper, middle, and lower plots give

results for samples containing donor impurities, no

electrically active impurities, and acceptor impurities,

respectively. The middle plot is the same as the one

shown in figure 5. These plots show that the partial

pressure of NZ at which the material changes from

DEVIATIONS FROM STOICHIOMETRY AND LATTICE DEFECTS C 4 - 2 9 n-type to p-type is increased by donor impurities and

decreased by acceptor impurities. For each type of doping, there is a region in which the carrier concen- tration is almost independent of partial pressure.

In this region, which occurs in n-type material for donor doping and in p-type material for acceptor doping, the carrier concentration is nearly equal to the

impurity concentration because the latter is large l

compared to the concentration of lattice defects

due to deviations from stoichiometry. -

ACCEPTOR-DOPED

log

PN-

2

FIG.

- Schematic plots of carrier concentration

N2 partial pressure for donor-doped, undoped, and acceptor- doped MN(c).

P

T PROJECTION.

The third and last type of projection used to present phase diagram data for the IV-V1 compounds is the P

T projection. Such a projection for compound MN is shown schematically in figure 7, where the logarithm of partial pressure is plotted against the reciprocal of absolute temperature.

These are the cabrdinates-generally adopted, because in many cases log P and 1/T are linearly related.

The loop shown gives the partial pressures of N, along the solidus lines for MN. This loop is generally referred to as the three-phase line, since solid, liquid, and vapor coexist for points which lie along it. For points inside the loop, only solid and vapor coexist ; for points outside it, only liquid and vapor. The upper leg of the three-phase line gives pressures of N, for solid MN in equilibrium with N-rich liquid, while the lower leg gives pressures for solid M N in equilibrium with N-rich liquid. The two legs meet at the maximum

LIQUID

+

FIG. 7. - Schematic P

T projection for the M

N system in the vicinity of MN(c).

melting point of MN. At temperatures sufficiently far below this point, the pressures of NZ along the two legs generally differ by orders of magnitude, as previously noted.

The solid straight line in figure 7 gives the partial pressure of MN molecules in equilibrium with solid MN. As stated above, this pressure is essentially constant over the homogeneity range of MN. The slope of the line is related to the heat of sublimation by the Clausius-Clapeyron equation. In figure 7, at most temperatures the MN partial pressure lies between the maximum and minimum pressures of N,.

Although this is the case for PbTe and SnTe, there is no a priori reason for it to be true in general.

The two dashed straight lines in figure 7 set upper and lower limits on the partial pressures of NZ in equilibrium with solid MN. The upper line gives the partial pressure of N2 in equilibrium with pure N liquid (or solid, at low enough temperatures). This is the vapor pressure curve for N, if its vapor consists entirely of N, molecules. At sufficiently low temper- atures? where M-saturated MN is in equilibrium with almost pure N, the upper leg of the three-phase line for MN is practically the same as the line for pure N.

With increasing temperature, as the N-rich liquid in equilibrium with M N becomes richer in M, the three- phase line falls further and further below the line for pure N.

The lower dashed line in figure 7 gives the hypothet-

ical partial pressures of N, which would be in equi-

librium with solid MN if the latter were in equilibrium

with pure M liquid (or solid). They are calculated

from the vapor pressure curve for pure M by using

C 4 - 3 0

A. J. STRAUSS AND R. F. BREBRICK the equilibrium constant K ( T )

P,(PN,)1/2 which

relates the partial pressures of the elementary species in the presence of solid MN. At sufficiently low tem- peratures, the three-phase line for MN coincides with this hypothetical line for pure M.

If the partial pressures of M in equilibrium with solid MN are plotted on a log P - l/Tplot, the three- phase line will also be a loop. In this case, the upper and lower legs will give the M pressures along the M-rich and N-rich solidus lines, respectively. Figure 8 shows

the partial pressure of NZ is determined by the reser- voir temperature. (In principle, a reservoir of pure M could be used, but the Group V1 elements are more convenient because of their higher vapor pressures.) After the melt is equilibrated with the vapor, it is cooled and the liquidus temperature measured. This is the temperature on the three-phase line which corresponds to the partial pressure of NZ used.

An alternative method for determining the liquidus temperature for a fixed partial pressure of N, was used in an investigation bf P ~ S , A solid sample of

-

PbS was annealed in a sealed ampoule containing a reservoir of pure S at a lower temperature. The sample was then quenched to room temperature and examined for evidence of melting. If no melting had occurred, the sample was re-annealed at a temperature about 5 O C higher but with the reservoir at the same temper- ature as before. By repeating this process until melting was detected, the liquidus temperature was determined to within about 5 OC for the partial pressure of S, fixed by the reservoir temperature.

In two cases, the liquidus temperature for a given partial pressure was found by visually observing the melting and freezing of a sample sealed in an FIG.

P - T

on 'the left represents a compound without a congruently subliming composition, ,the one

on

schematic log P - 1/T plots which give both the M and NZ partial pressures over two different types of MN compounds. The right-hand plot, in which the areas within the three-phase lines overlap, is charac- teristic of a compound which sublimes congruently over a certain temperature range. For a compound represented by the left-hand plot, in which the partial pressure of N, is always greater than that of M, there is no congruently subliming composition at any temperature. There is no IV-V1 compound for which the three-phase loop for M lies above that for N,, since the vapor pressure of any Group V1 element is greater than the vapor pressure of any Group IV element.

Several methods have been used to determine partial pressures along the three-phase lines. One method employs thermal analysis to determine the liquidus temperature for an M - N melt which is in equilibrium with a known partial pressure of NZ. The melt is heated in a sealed, fused silica ampoule which also contains a reservoir of pure N at a lower temperature.

As in the solid-vapor experiments described previously,

ampoule containing a reservoir of pure N at a lower temperature. In an investigation of tin sulfide the partial pressure of S, was kept constant, and the sample temperature was varied. 'In an investigation of lead telluride, the sample temperature was fixed, and the partial pressure of Te2 was varied.

Points on the three-phase lines have also been obtained by three methods which do not require a direct determination of the conditions under which a sample melts and/or freezes. Instead, they involve a measurement of the partial pressure of N, in equilib- rium with an ingot which at the experimental tem- perature is a mixture of solid MN and either M-rich or N-rich liquid. Two of these methods have been used to investigate tin sulfide. The partial pressure of S, was found in one case by weighing the sulfur transported by a stream of argon gas passed over the ingot, and in the other by measurements of H2S and H, pressures.

The third method employs an optical absorption technique to determine the NZ partial pressure.

The experiment is performed with . a fused silica

ampoule which consists of an optical cell with parallel

windows and a sidearm reservoir at right angles to

the cell. After the ingot is placed at the end of the side

arm, the ampoule is evacuated, sealed, and positioned

in a furnace so that the optical cell intercepts the

sample beam of a double beam spectrophotometer.

DEVIATIONS FROM STOICHIOMETRY AND LATTICE DEFECTS

The optical cell is heated to a constant temperature, and the optical absorption of the vapor in the cell is then measured as a function of the temperature of the reservoir, which is the coldest part of the ampoule.

The partial pressure of NZ is found from the optical density of the vapor by using calibration data obtained by measuring the optical density of NZ vapor in equilibrium with pure liquid or solid N, together with the published curve for the vapor pressure of pure N as a function of temperature. It is assumed that the optical density due to N, is the same function of N, partial pressure in the mixed vapor over the compound as in the pure vapor. The calibration exper- iments show that the optical density is directly propor- tional to the partial pressure.

The optical absorption method has also been used to determine the partial pressures of Te, as a function of temperature for compositions inside the homo- geneity range of SnTe and GeTe. The procedure is the same, except that the reservoir of the optical ampoule contains a solid sample, whose composition is estab- lished by weighing out the elements in appropriate proportions, rather than a solid-liquid ingot. In these experiments, it is necessary to make the sample large enough compared to the vapor space to keep the sample composition from changing appreciably due to preferential vaporization of Te, and to pre-anneal sufficiently to obtain a homogeneous soIid phase.

On a log P - 1/T plot, as shown schematically in figure 9, the data for the various solid compositions

l / T -

FIG. 9.

Schematic P

T projection for MN(c), showing lines for samples of constant composition.

are represented by straight lines lying inside the three- phase loop. The plot shown gives the results for a

compound whose entire homogeneity range lies on the N-rich side of the stoichiometric composition.

As stated in our earlier discussion, at constant temper- ature the partial pressure of N, increases monotoni- cally as the atom fraction n increases.

For a given composition, the change of NZ partial pressure with increasing temperature is shown by the lines marked ABCD in figure 9. The partial pressure first increases until the straight line beginning at A reaches the three-phase line at B. The intersection point occurs at the solidus temperature. As the temperature is further increased, the sample begins to melt, and the partial pressure then follows the three-phase line, in the direction indicated by the arrows, until the liquidus temperature is reached at point C. Depending on composition, the partial pressure may either increase or decrease between B and C. At still higher temperatures, the sample is completely melted, and the partial pressure increases along a straight line CD lying outside the three-phase loop. It is seen that the change of partial pressure with composition is much greater when the samples are solid than when they are liquid. Partial pressure data obtained by the optical absorption method have been used to determine solidus and liquidus temperatures by finding the intersection points B and C, respectively.

A log P - 1/T plot is also used to present partial pressure data within the homogeneity range of compounds for which composition data are obtained by carrier concentration measurements.

Figure 10 shows a schematic plot of this kind for a compound whose homogeneity range includes the stoichiometric composition. The solid lines give the relationships between temperature and N, partial pressure for selected carrier concentrations, measured at room temperature, in n-type and p-type material.

These lines are based on points interpolated from isotherms of carrier concentration versus partial pressure like the one shown schematically in figure 5.

Partial pressures of NZ for the stoichiometric compo- sition are indicated by the dashed line marked n = p, which divides the area inside the three-phase loop into n-type (M-rich) and p-type (N-rich) regions.

In general the line is nof placed s y ~ m e t r i c a l y within

this area. With increasing temperature the lines for

constant carrier concentration approach the stoichio-

metric line, and those for sufficiently low concentra-

tions eventually merge with .it on the scale shown in

figure 10. Consequently, for samples equilibrated

with a controlled NZ partial pressure, the minimum

carrier concentration which can be achieved decreases

as the annealing temperature decreases, if the relative

C 4 - 3 2 A.

AND R. F. BREBRICK

l / T

FIG. 10.

Schematic P

T projection for MN(c), showing lines for samples

constant carrier concentration.

accuracy of pressure control is constant. However, the temperature cannot be decreased indefinitely because the time required for equilibrium eventually becomes too long. For this reason, few samples with carrier concentrations below 1017 cm-3 have been reported for any of the IV-V1 compounds.

The dashed line marked P,,, in figure 10 is an apl?roximation to the partial pressures of N, in

equilibrium with congruently subliming samples.

This line is calculated from the expression

which is derived from the condition for a stoichiometric vapor, P,,

112 P,, by using the equilibrium constant K(T)

P,(PN,)112. In general, the congruently subli- ming composition may be either a- or p-type. In the plot shown in figure 10, the congruently subliming samples are n-type, and the electron concentration increases with increasing annealing temperature.

Phase Diagram Data Reported for the IV-V1 Compounds.

No attempt will be made here to give a detailed review of the quantitative phase diagram data which have been reported for the individual IV-V1 compounds. Such a review will be presented in a later paper, together with a discussion of the nature of the lattice defects associated with the deviations from stoichiometry in these compounds. To conclude the present paper, we merely list in table I the types of quantitative data (other than liquidus curves) which have been obtained. Although more than thirty separate papers are represented by the entries in the table, there are still very considerable gaps in the data, and no results at all are available for GeS or GeSe.

Types of phase diagram data reported for IV-V1 compounds Compound

PbS PbSe PbTe SnS SnSe SnTe GeTe

- - - -

-

T - x projection (solidus curves)

Directional freezing ... [3 1 141 L51 Equilibrium with two-phase ingot ....

Internal precipitation [l01 [l l ] L61 L71 L81

C81 [l21 1131 [l41 1151 U61 [l71 [l71 [l81 [91

. . .

Partial pressure data (optical absorption)

Approach to melting at constant P. . . . _[21] ^{[l93 1201}

Lattice parameter ... ¹²²¹

P -

projection (inside homogeneity range) Interpolation from P

T curves for constant

(optical absorption). ....

Equilibrium with elementary vapor

(carrier concentration data). ... [23] [24] [25] [26]

Analysis of vapor increments. ...

P - T projection (along three-phase lines)

Melting-freezing at constant P. ... [21]

Melting-freezing at constant T. ... ^L251

Optical absorption ...

Analysis of vapor ...

P

T projection (inside homogeneity range) Optical absorption ...

Interpolation from isotherms of carrier

concentration vs pressure. ... ^{[24] 1341 [25] 1271}

DEVIATIONS FROM STOICHIOMETRY AND LATTICE DEFECTS C 4 - 3 3 References

[l] KROGER (F. A.), The Chemistry of Imperfect Crystals (North-Holland Publishing Co., Amsterdam, 1964).

[2] BREBRICK (R. F.), in Progress in Solid State Chemistry, edited b y H. Reiss (Pergamon Press, Oxford, 1966), p. 213.

[3] GOLDBERG (A. E.) and MITCHELL (G. R.), J. Chem.

Phys., 1954,22,220.

[4] MILLER (E.), KOMAREK (K.) and CADOFF (I.), Trans.

Met. Soc. AZME, 1959,215, 882.

[S] ABBOTT (F. C.), Ph. D. Thesis, University of Dela- ware, 1965.

[6] BREBRICK (R. F.) and GUBNER (E.), J. Chem. Phys., 1962, 36, 170.

[7] BREBRICK (R. F.) and ALLGAIER (R. S.), J. Chem.

Phys., 1960, 32, 1826.

[8] BREBRICK (R. F.) and GUBNER (E.), J. Chem. Phys., 1962,36,1283.

[9] BREBRICK (R. F.), J. Phys. Chem. Solids, 1963,24, 27.

1101 NENSBERG (Y. D.) and PETROV (A. V.), Znorg. Mater., 1965, 1, 1365.

[l11 STRAUSS (A. J.), Trans. Met. Soc. AZME, 1967, 239, 791.

[l21 KOVAL'CHIK (T. L.) and MASLAKOVETS (Y. P.), SOV.

Phys. Tech. Phys., 1956,1,2337.

[l31 FRITTS (R. W.), in Thermoelectric Materials and Devices, edited by I . Cadoff and E. Miller (Rein- hold Publishing Corp., New York, 1960), p. 143.

[l41 MILLER (E.), KOMAREK (K.) and CADOFF (I.), J.

Appl. Phys., 1961,32,2457.

[l51 SCANLON (W. W.), Phys. Rev., 1962,126,509.

[l61 ALBERS (W.), HAAS (C.), VINK (H. J.) and WASSCHER ( J . D.), J . Appl. Phys., 1961,32, 2220.

[l71 ALBERS (W.), HAAS (C.) and VINK (H. J.), Philips Res.

Repts, 1963,18,372.

[l81 ASANABE (S.), J. Phys. SOC. Japan, 1959,14,281.

[l91 BREBRICK (R. F.) and STRAUSS ( A . J.), 1. Chem. Phys., 1964, 41, 197.

L201 BREBRICK (R. F.), J. Phys. Chem. Solids, 1966, 27, 1495.

[21] BLOEM (J.) and KROGER (F. A.),

Physik. Chem., 1956, 7, 1.

1221 MAZELSKY (R.) and LUBELL (M. S.), Adv. in Chem., 1963. 39. 210.

[23] SCANLOG ^(W. W.) and BREBRICK (R. F.), Physica, 1954, 20, 1090.

[24] BLOEM (J.), Philips Res. Repts, 1956, 11, 273.

[25] OHASHI (N.) and IGAKI (K.), Trans. Japan. Inst.

Metals, 1964,5,94.

[26] NOVOSELOVA (A. V.), ZLOMANOV (V. P.) and MAT-

VEYEV

(0. V.), IZV. Neorg. Mater., 1967, 3, 1323.

[27] FUJIMOTO (M.) and SATO (Y.), Japan. J. Appl. Phys., 1966,5,128.

[28] RAU (H.), J. Phys. Chem. Solids, 1966,27,761.

[29] ALBERS (W.) and SCHOL (K.), Philips Res. Repts., 1961, 16. 329.

[30] BIS (R.'F.), J. Phys. Chem. Solids, 1963,24,579.

[31] BREBRICK (R. F.) and STRAUSS (A. J.), J. Chem. Phys., 1964, 40, 3230.

[32] BREBRICK (R. F.), J. Chem. Phys., 1964,41,1140.

[33] RAU (H.), Ber. Bunsenges. Phys. Chem., 1965, 69, 731.

[34] SCANLON (W. W.), in Properties of Elemental and Compound Semiconductors, edited by H . C. Gatos (Interscience Publishers, New York, 1960), p. 185.

DISCUSSION

CROCKER, A. J. - pM(pN2)'/'

exp[2 AGJRT]

Why the 2 ? Shouldn't it be absent ?

STRAUSS, A. J. - The 2 is present because AG, is the free energy of formation for one gram-atom, according to the equation :

+ ^{M(g) + $ N,(g)}

+

rather than for one mole.

ZEMEL, J. N. - In the congruently subliming case, one has a condition equivalent t o that in a molecular beam source commonly used in film preparation.

Assuming quasi-equilibrium conditions would hold in sublimation from such a source, the data on vapor composition should be valid. However, sublimation directly into the vacuum will be a vastly different problem since composition changes at the solid sur- face can have dramatic effects on the sublimation a t the various components. I t can be easily argued that under these conditions, the sublimation will be limited by solid state diffusion of the volatile components to the surface.

ESAKI, L. - Could you make any comment o n the application of ion implantation techniques which could extend the impurity concentration range ?

STRAUSS A. J.

Ion implantation might be a very useful technique for obtaining metastable samples, but I d o not know of anyone who is currently studying ion implantation of the IV-V1 compounds.

RODOT, M.

Have you any indication whether, during the achievement of equilibrium, the slower step is diffusion inside the solid or passage through the surface ?

STRAUSS, A. J.

I suspect that the diffusion in the solid is the slower step, but I d o not know of any data concerning this relation. Because both processes probably depend exponentially on temperature, it is possible that one dominates over a certain temperature range and the other dominates at other temperatures.

PAPARODITIS, C.

VOUS avez trait6 des conditions

?I I'tquilibre. Dans le cas dynamique de la sublimation stationnaire, que peut-on dire des especes chimiques B

1'6tat gazeux ?

STRAUSS, A. J.

I d o not know of any evidence

concerning the species under dynamic conditions

other than the mass spectrographic data of Porter on

PbTe. His results do not suggest any radical difference

between the static and dynamic cases.