Creator:L. Ekstrom,J. P. Dismukes,R. J. Paff
Date Created:
Place Created:New Jersey
Keywords:LATTICE PARAMETER
Context:
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LATTICE PARAMETER AND DENSITY IN THE Ge-Si ALLOY SYSTEM*
J. P. Dismukes, L. Ekstrom, and R. J. Paff
RCA Laboratories Radio Corporation of America Princeton, New Jersey
ABSTRACT
Th.e lattice parameter and density of chemically analyzed samples of homogeneous Ge-Si alloy have been measured throughout the entire alloy system. The temperature dependence of the lattice parameter was measured between . Compo-
sitional dependences of the lattice parameter and density are accurate to about atomic per cent in alloy composition.
Lack of chemical analysis or sample inhomogeneity may explain the large discrepancies between previous investigations of these properties. The excess volume of mixing is given by
Deviations from Vegard's law are negative as predicted by models based on first order elasticity theory, but smaller in absolute magnitude. This discrepancy is about the size of the positive deviations calculated from second order elasticity theory.
This research has been supported by the U. S. Navy Bureau of Ships under contract No. NObs-88595.
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INTRODU'CTION
Composition in the Ge-Si alloy system can be accurately determined, from measurements either of density or of lattice parameter provided the dependences of lattice parameter and density on composition are known. However, the large discre- pancy between the results of previous investigations^of these 1-5
properties, corresponds to an uncertainty in composition for a definite value of lattice parameter or density of  throughout most of the system. This discrepancy can be attributed to the fact that no previous investigator has both evaluated sample homogeneity and determined composition by chemical analysis. Therefore the variation of lattice parameter and density at room temperature has been reinvestigated throughout the Ge-Si alloy system, using chemically homogeneous specimens. The temperature dependence of lattice parameter has also been measured in the range 25-800°C for several alloy compositions.
EXPERIMENTAL PROCEDURE
Homogeneous Ge-Si alloy ingots were prepared from high purity Ge and Si by zone leveling using the procedure described by Dismukes and Ekstrom.^ Typical mass speotrographic analyses of impurities in these materials are shown in Table II. The procedure for this study consisted of first measuring the density
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and then either measuring the lattice parameter or the chemical
7
composition of the material .
Densities were measured by the method of hydrostatic
Q
weighing, employing Archimedes' principle. The samples in the form of slices weighed 0.7-1.5 gm, and the weight loss in water was 0.1-0.3 gm. The samples were suspended from a 0.003 inch diameter tungsten wire, and were weighed to a precision of 0.02 mgm using a semimicrobalance. Water, to which a small amount of wetting agent had been added to reduce the surface tension, was used as the immersion liquid. The measurements were corrected for water temperature and for the air displaced by the sample, but not for effects due to the wetting agent, dissolved air, or atmospheric press-are. The absolute accuracy of these density measurements is shown to be within by comparison in Table III of measurements
made on Si and Ge corrected to 25°C with the values of Smakula
Q
and Sils.
Samples for lattice parameter measurements were ground to pass through a 325 mesh screen. The sieved powder was mixed with Duco cement and was then rolled into a fiber approximately 0.1 mm in diameter. The fiber was mounted in a 114.6 mm diameter camera using the asymmetric method of mounting the film. Powder patterns were obtained using Ni filtered Cu radiation, with an exposure time of about 24 hours and at room temperature of . The back reflection
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lines, both Ko^ and. Ko^* were measured to within -0.05mm. The lattice parameter for each reflection was calculated, and the final value was obtained by extrapolation using the Nelson-Riley function. The absolute accuracy of the lattice parameter measurements is shown to be within -0.0005A by comparison in
v
Table III of measurements made on Si and Ge with the values
Q
of Smakula and Kalnajs. The variation of lattice parameter with temperature in the range 25-800°C was determined by scanning the (531) and (620) diffraction peaks with a diffrac-tometer. Values of the lattice parameter at each temperature were obtained by averaging the lattice parameter values for the two peaks. Good agreement between the diffractometer method and the Debye-Scherrer method was obtained at room temperature.
Chemical composition was determined by analyzing the
7
material for its Ge content by the method of Cheng and Goydish using samples containing 150-300 mgn of Ge.
RESULTS
Data on measurements of density and lattice parameter at 25°C for different alloy compositions are listed in Table III. The variation of lattice parameter with density is shown in Pig. 1, and this is compared with published data. The curve is drawn through the data from the present study.
2
The scatter in the data of Johnson and Christian and of
4
Busch and Vogt, and the large systematic error in the latter
-4-
data, may be due to sample innomogeneity. The preparative procedure employed by Johnson and Christian, slow cooling from the melt, could lead to this condition, since the growth
rate was not controlled and the melt composition changed during
4
growth. The growth rates employed by Busch and Vogt appear to have been too large for preparing homogeneous material by zone leveling in the middle of the system,^ where the discrepancy is greatest. There is good agreement between our result;
3
and the data of Wang and Alexander, who have shown that their material was homogeneous.
The curve for the variation of density with chemical composition, shown in Pig. 2, was drawn through the points determined by chemical analysis. Uncertainty in the recovery factor for Ge, -0.3%, is probably the largest source of error
7
in the analysis. This effect contributes to the relatively large scatter at the Ge-rich end of the system. We also calculated from the curve in Pig. 1 the average atomic weight, A, using the relation,
where N is Avagadro's number,10 and from this the alloy composition. Points for density versus composition determined in this manner are also shown in Pig. 2.
The variation of lattice parameter with composition is shown in Pig. 3. The chemical composition was determined by (a) combining Pigs. 1 and 2, and (b) using Eq.. 1. The results
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p
of Johnson and Christian (Pig. 3) show some scatter, which could be due to sample inhomogeneity as discussed above, or
to error in the polarographic analytical method as was pointed
7	1
out by Cheng and Goydish. The results of Stohr and Klemm are
in better agreement with our data, but they show a large deviation
3
at low Ge concentration. The values of Wang and Alexander show a large disagreement when plotted against the given compositions. This suggests that their specimens, though chemically homogeneous, differed in composition from the intended values
by an average of about 7at%Ge. The lattice parameter data of 5
Busch and Vogt was not compared with the results of the present work, because of both the lack of chemical analysis and of the large deviation observed in Fig. 1.
In Table IV are listed values of density and lattice parameter at 25°C for composition intervals of 5at?oGe. These values are derived from Pigs. 1-3, and their absolute accuracy is probably within io.3at#Ge. Values of density were taken as the mean of those from the curve in Pig. 2 and of those calculated from Eq. (1). The departure in lattice parameter from Vegard's law,A , given by,
where c&e = atomic fraction Ge, is also listed in Table IV. This quantity is negative throughout the system and reaches a broad minimum in the middle of the system.
The excess volume of mixing, , calculated using the
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expression,
(3)
is also small and negative throughout the alloy system. Prom the plot of against given in Pig. 4 it is
seen that	can he expressed by the empirical relation,
(4)
where cgi is the atomic fraction of Si.
The temperature dependence of the lattice parameter of Ge, Si, and three Ge-Si alloys is shown in Pig. 5 together with the average values of the linear expansion coefficient, o(, between 25°C and 800°C. Por Ge and the Ge-Si alloys, o< is independent of temperature, but the data for Si indicates an increase in o< of about 50% from 25°C to 800°C. A larger
13
increase in c< with temperature has been reported by Dutta. Por the alloys containing 20.1at%Ge and 34.7at%Ge, the deviation from Vegard's law and the excess volume of mixing are constant with temperature within the experimental uncertainty of about - 15$ of these quantities. In the 51.7at%Ge alloy, these deviations decrease about 25% between 25°C and 800°C.
DISCUSSION
Volume of Mixing
xs
The observation that AV^ can be expressed by equation (4) suggests that the Ge-Si alloy system might show a simple type of
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thermodynamic behavior.1^ Since is not zero, the system
15
is not an ideal solution, in agreement with Thurmond*s conclusion from the nature of the phase diagram. The next simplest type of behavior is regular solution theory in which
The quasichemical approach to regular solution theory16 leads to the expression,
(5)
where-o_ is related to the bond energies H(je_(je» HSi-Si' ^^ HGe-Si
(6)
xs
Prom quasichemical theory one would expect that and
17
have the same sign. Rastogi and Nigam have calculated a value of +20kcal mole"1 for -n» from a quasichemical regular solution treatment of the Ge-Si alloy phase diagram. We have repeated their calculation considering a larger section of the phase diagram, and obtain the value +2.4kcal mole"1. Thus while the value of-A. is quite uncertain, the sign is probably correct. Since-/L and are not of the same sign, a more
refined model will be required to explain the thermodynamic properties of the Ge-Si alloy system.
Deviations from Vegard's Law
The Ge-Si alloy system is an attractive one for comparing experimental deviations from Vegard's law with theoretical
calculations, since there is only one crystal structure in the
«
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system, 110 relative valency effect, and only small differences
in size and electronegativity between the constituents. A
summary of the theoretical work on this topic was recently
18
given by Gschneidner and Vineyard. The deviations from
Vegard's law predicted by theories which require data only
on the elastic properties of the pure components are shown
19
in Pig. 6. Pines used an elastic sphere model to derive the equation,
where A refers to the solvent, B to the solute,/-c is the
20
shear modulus, and is the compressibility. Pournet considered the effects of nearest neighbor interactions to obtain the equation,
Both equations (7) and (8) predict negative deviations when the element with the larger atom is softer. Thus they give the correct sign of the deviation for the Ge-Si alloy system, but the predicted magnitude is about twice the experimental value..
2±
Priedel treated the elastic sphere model so as to obtain the equation,
-9-
where is Poisson's ratio. This equation is valid only for
dilute solutions, but within this limit it is in better agree-
20
ment with experiment than those of Pine3 and Pournet.
17
Gschneidner and Vineyard applied second order elasticity theory to obtain the equation,
where p is pressure and B is the bulk modulus. This equation is also valid only for dilute solutions. They suggest the approximation,
where V is the molar volume and Cy is the molar heat capacity. This equation predicts only positive deviations from Vegard's law. However, the magnitude of its effect is comparable to the amount by which the first order theories overestimate the negative deviations. This suggests that a theory combining both first and second order elasticity effects would be a considerable improvement over current theories for predicting deviations from Vegard's lav/.
ACKNOWLEDGMENTS
The authors wish to thank B. Goydish for performing the chemical analyses, H. H. Whitaker for performing the mass spec orographic analyses, and J. G. White for measurements of the temperature dependence of the lattice parameter of Ge and Si.
(9)
(10)
(11)
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REPERENCES
1. H. Stohr and W. Klemm, Z. anorg. Chem. 24I, 313-18 (1939).
2. E. R. Johnson and S. M. Christian, Phys. Rev. £5, 560-1 (1954).
3. C.C. Wang and B. H. Alexander, Pinal Technical Report on Investigation of Germanium-Silicon Alloys, Bureau of Ships Contract No. N0bsr-63130, February 17, 1955.
4. G. Busch and 0. Yogt, Helv. Phys. Acta 12* 437-58 (I960).
5. A. V. Sandulova, P. S. Bogoiavlenski, and M. I. Droniuk, Lokl. Akad. Nauk. SSSR 143, 610-12 (1962).
6. J. P. Dismukes and L. Ekstrom, to be published.
7. X. L. Cheng and B. L. Goydish, Anal. Chem. 21, 1273-5 (1963).
8. A. Smakula and V. Sils, Phys. Rev. 1744-6 (1955).
9. A. Smakula and J. Kalnajs, Phys. Rev. 1737-43 (1955).
10. The value of N on the universal C scale, 6.02311 x 10 5
(gm mole)"1, was obtained from the value of Cohen and Dumond11
-1 C
on the 0 scale using the conversion factor of Cameron and 12
Wichers. Spectroscopically determined atomic weights
on the C12 scale were 72.628 and 28.086 for Ge and Si, 12
respectively.
11. E. R. Cohen and J. W. M. DuMond, Phys. Rev. Letters 1, 291-2 (1958).
12. A. E. Cameron and E. Wichers, J. Am. Chem. Soc. 84, 4175-97 (1962).
13. B.'N. Dutta, Phys. Status Solidi 2, 984-7 (1962).
14. J. H. Hildebrand and R. L. Scott, "The Solubility of Non-
electrolytes", Third Edition, Reihhold, New York, 1950, p. 141.
15. C. D. Thurmond, J. Phys. Chem. 5J 827-30 (1953).
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16. R. A. Swalin, Thermodynamics of Solids, John Y/iley and Sons, 1962, New York, Ch. 9.
17. R. P. Rastogi and R. K. Nigam, Proc. Natl. Inst. Sci. India 26, 134-94 (I960).
18. K. A. Gschneidner, Jr. and G. H. Vineyard, J. Appl. Phys. 33, 3444-50 (1962).
19. B. J. Pines, J. Phys. (U.S.S.R.) 309-19 (1940).
20. G. Pournet, J. phys. radium 14, 374-80 (1953).
21. J. Friedel, Phil. Mag. 1^6,	(1955).
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LIST OP TABLES
Table I Previous Investigations of Lattice Parameter and Density in the Ge-Si Alloy System
Table II Mass Spectrographic Analysis for Impurities in Ge, Si, and Ge-Si Alloy
Table III Experimental Values of Density, Lattice Parameter, and Chemical Composition for Ge-Si Alloy Samples
Table IV Accurate Values of Density and Lattice Parameter for Ge-Si Alloy Derived from Pigs. 1-3
LIST OP PIGuRES
Pig. 1 Variation of Lattice Parameter with Density in the Ge-Si Alloy System
Pig. 2 Variation of Density with Chemical Composition in the Ge-Si Alloy System
Pig. 3 Variation of Lattice Parameter with Composition in the Ge-Si Alloy System
Pig. 4 Variation of Reduced Volume of Mixing, ,
with cGe at 25°C
Pig. 5 Variation of Lattice Parameter with Temperature for Ge, Si, and Some Ge-Si Alloys
Pig. 6 Deviations from Vegard's law in the Ge-Si Alloy System Predicted by Several Theories
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TABLE I
Authors	Method of Preparation	Lattice Parameter	Density	Method of Analysis
Stohr and Klemm'''	Sintering	D	ND	Nonea
2 Johnson and Christian	Slow Cooling from	D	D	Polarographic
	the Melt			
3 Wang and Alexander	Zone Leveling	D	D	None^
4 Busch and Vogt	Zone Leveling	D	D	None°
Sandulova, et al.^	Vapor Transport	D	ND	None^
D = Determined
ND = Not Determined
a = Composition assumed to be that of the mixed components before sintering.
b = Composition assumed to be that of the mixed components before zome leveling.
c = Composition determined from the results of Johnson and Christian.
d = Composition assumed to be that of the mixed components before vapor transport.

TABLE II
Materials		TYPICAL	IMPURITY	CONTENTS,	Atomic	ppm	
	Fe	Cu	A1	Mg	0	C	H
Ge -Si	1	.2	5	1	300	30	100
Ge	6	ND	.4	.4	10	100	30
:Si	1	.2	1	I	300	100	10
ND = Not detected
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TABLE III
d (gm cm	a (A)	C (at.% Ge)
2.6382	= «,	7.5
2.6319	---	8.0
2.6357	---	8.0
2.6330	5.4492	g-4
3.0047	----	18.9
3.0098	---	20.0
3.0710	---	21.4
3.0830	---	21.7
3.0120	5.4726	ZCol
3.0822	5.4772	23.3
3.0884	---	22.6
3.1118	---	23.5
3.2723	___	28.2
3.2844	5.4898	7
3.3634	---	30.5
3.3585	---	31.6 A O
3.4735	5.5032	3±8
3.4762	---	35.4
3.5181	----	35.5
3.5715	----	36.6
3.6312	5.5151	s%n
3.6313	---	38.8
3.6340		39.8
3.6874	---	40.3
3.6766	___	41.4
3.7750	5.5250	^±3 7
3.7827	---	44.5
3.7883	___	44.6
3.9915	5.5404	SL4-
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TABLE III (Cont.)
d (gm cm	a (A)	C (at.% Ge)
3.8944	„ _ _	48.0
3.9940	---	49.9
4.0037	---	50. 6
4.0749	5.5475	
4.0673		53.8
4.0897	----	54.8
4.1652	---	57.0
4.1923	5.5567	
4.5289	5.5841	
4.5170	---	68.0
4.4969	---	68.9
4.6817	5.5966	
4.6652	___	71.9
4.6735	---	72.3
4.8246	---	79.4
4.8513	---	79.4
4.8410	5.6112	
5.0752	5.6325	
5.0754	---	87.3
5.0728	---	87.9
5.0970	---	88.8
5.0410	------	89.4
5.1090	-----	90.8
5.1697	5.6419	
5.1669	---	94.8
5.3256	5.6575	Gea
5.32674	5.65754	Geb
2.3277	5.4310	Sia
2.32902	5.43072	Sib
a Present work
b Smakula and Sils^ and Smakula and Kalnajs^
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TABLE IV
C (at 7o Ge)	d (gm cm )	a (1)	a - a (A)
0	2.3277	5.4310	V
5	2.5100	5.4419	-.0004
10	2.6825	5.4522	-.0014
15	2.8490	5.4624	-.0026
20	3.0075	5.4722	-.0041
25	3.1660	5.4825	-.0051
30	3.3265	5.4928	-.0062
35	3.4340	5.5038	-.0065
40	3.6405	5.5149	-.0067
45	3.7950	5.5261	-.0068
50	3.9470	5.5373	-.0069
55	4.0990	5.5492	-.0063
60	4.2465	5.5609	-.0060
65	4.3905	5.5727	-.0055
70	4.5335	5.5842	-.0053
75	4.6730	5.5960	-.0048
80	4.8115	5.6085	-.0027
85	4.9445	5.6206	-.0019
90	5.0740	5.6325	-.0023
95	5.1990	5.6448	-.0013
100	5.3256	5.6575