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Lattice parameter and density of Ge–Si solid solutions
Renewable Energy 24 (2001) 347–351 www.elsevier.nl/locate/renene
Lattice parameter and density of Ge–Si solid solutions M.F.A. Alias, N.N. Rammo, M.N. Makadsi
*
Physics Department, College of Science, University of Baghdad, Jadiriya, PO Box 47162, Baghdad, Iraq
Abstract Solid solutions of the Ge–Si system have been prepared by reaction under vacuum and gravity of the constituent elements followed by different quenching procedures. Least-squares refinement to the lattice parameters of the Ge–Si phase has been established in the composition range 0.5–0.9 at% Si. Almost homogeneous alloys were obtained by rapid quenching in liquid nitrogen, whereas pseudo-homogeneous alloys occur as an intermediate stage when quenched in water. The refined lattice parameters, although computed with very high precision, did not predict the formation rate of the solid solution. However, careful analysis of the X-ray diffraction pattern of residual Ge and Si revealed that the lattice parameter of each component is modified in accordance with the atomic percentage of either element present in the lattice of the other. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Germanium–silicon alloys are being used widely in thermoelements [1]. Logan et al. [2], Johnson and Christian [3] and Bush and Vogt [4] have all pointed out that inhomogeneity exists in the prepared Ge–Si alloys irrespective of the method of preparation, whether slow cooling from the melt or zone levelling. On the other hand, Dismukes et al. [5] have reported homogeneous Ge–Si alloy ingots prepared by zone levelling, from which the lattice parameter and density have been measured throughout the entire alloy system. However, the large discrepancy between the results of previous investigations of these properties [3–5] corresponds to an uncertainty in composition for a definite value of the lattice parameter or density.
* Corresponding author. 0960-1481/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 1 4 8 1 ( 0 1 ) 0 0 0 1 5 - 5
348
M.F.A. Alias et al. / Renewable Energy 24 (2001) 347–351
Compared with relaxed Ge–Si [2] alloys, the structure of strained Ge–Si alloys [6] in terms of their lattice parameters was accounted for as additional stretching of the unit cell caused by strain. As different methods of preparation impart a significant level of inhomogeneity on the prepared Ge–Si ingots, it becomes of interest to visualize the behaviour of these alloys prepared by rather unique quenching from high temperature — the aim of the present work — and their subsequent thin film properties (to be presented elsewhere). 2. Experimental procedure Ge–Si alloy ingots in the composition range 0.5–0.9 at% Si were prepared from high-purity Ge and Si by reaction of the well-mixed powders in a sealed quartz capsule at temperatures of 1200–1500°C for 6–8 h, followed by rapid quenching in liquid nitrogen or dry ice. Densities were measured by two methods. The first was hydrostatic weighing [7] employing Archimedes’ principle, with weighing precision of 0.2 mg; the accuracy of density measurement was shown to be within 0.3%. The second method was by means of a multipycnometer. Samples for X-ray diffraction (XRD) analysis were ground to pass through a 75 µm mesh screen. No annealing was done to reduce residual stresses due to grinding. Diffraction scans were performed on a Philips PW1710 automated powder diffraction system, using monochromatic Cu Kα1 radiation at a scan rate of 0.02°/2 s, an automatic divergence slit for constant irradiated area and a receiving slit of 0.2°. The 111, 220, 311, 400 and 331 reflections of the diamond structure were used in the least-squares refinement of the lattice parameters by employment of the FIRSTAR program [8] and Robert’s extrapolation function for the diffractometer as (cos q cot q+cot2 q). The lattice parameters were determined with a precision of one part in 20,000 at room temperature. 3. Results and discussion Table 1 lists data on the density for different alloy compositions as measured by the two methods. Table 1 Density of Ge–Si alloys as measured by Archimides’ principle and the multipycnometer At% Si
50 60 70 80 90
Density (g/cm3) Archimides
Multipycnometer
3.925 3.639 3.329 3.010 2.678
3.820 3.350 3.005
M.F.A. Alias et al. / Renewable Energy 24 (2001) 347–351
Fig. 1.
349
XRD pattern of Ge0.3Si0.7 fired at 1450°C for 8 h and quenched in liquid nitrogen.
A typical XRD pattern of the Ge–Si alloy is shown in Fig. 1. Data on the refined lattice parameters of Ge–Si for different alloy compositions, together with the refined lattice parameter for individual Ge and Si phases found as residuals in the XRD spectra, are shown in Table 2. The results of lattice parameters in the range of compositions prepared seem to agree with relevant data in the literature [3–5], indicating that the method of alloy preparation has little significance on the lattice parameters. It is noted, however, that the lattice parameters of the residual phases of Ge and Si (see Table 2) are modified in accordance with the atomic percentage of either component present in the lattice of the other, suggesting that appreciable diffusion has occurred although not enough to complete the formation of a homogeneous alloy. Table 2 Lattice parameters of Ge–Si alloys, and of Ge and Si, for different compositionsa At% Si 0 50 60 70 80 90 100 a
aSi (nm)
aGe–Si (nm)
0.54604(22)
0.55317(18) 0.55201(25) 0.54887(21) 0.54691(15) 0.54494(13)
0.54421(24) 0.54365(12) 0.54309 Values in brackets are standard deviations.
aGe (nm) 0.56580 0.56549(10) 0.56397(23) 0.56215(25)
350
M.F.A. Alias et al. / Renewable Energy 24 (2001) 347–351
This may be due partly to the growth rate, which was not controlled as the melt composition changed.
4. Analytical calculation of lattice parameters For the lattice parameters of Ge–Si as a function of composition, Fischetti and Laux [9] have used the experimental results of Logan et al. [2] that have been approximated analytically as aGe–Si(x)⫽aGe x⫹aSi(1⫺x)⫺bx(1⫺x), with parameter b=1.88×10⫺3 nm. Results from this equation have been calculated in terms of the composition range considered in this work and are tabulated in Table 3 for comparative purpose. Our data on lattice parameters and the data of Refs. [5,9] are plotted in Fig. 2; there is full agreement between the two.
5. Prediction of lattice parameters from atomic radii The Ge–Si alloy system is an attractive one for comparing experimental deviations from Vegard’s law with theoretical calculations based on the atomic radius and braves lattice of individual components. A detailed analysis of theoretical work on the model that predicts lattice parameters as a function of composition is given by Abdulah [10]. The formula derived for the binary system is aGe–Si⫽C 2GeaGe⫹2CGeCSiaGeSi⫹C 2SiaSi. The values of aGe and aSi can be found from the atomic radius as 8r/√3 for the diamond structure. CGe and CSi are the atomic fractions of the components Ge and Si; aGe–Si is a simple function of the initial slope of the curve that represents the lattice parameter of the solid solution on the basis of solute concentration. The average of Table 3 Theoretical calculation of lattice parameters At% Si
50 60 70 80 90
Lattice parameter of Ge–Si (nm) Vegard’s law
Analytical via Ref. [9]
Predicted by atomic radius, Ref. [10]
Quoted from Ref. [5]
0.55410 0.55185 0.54959 0.54734 0.54509
0.55398 0.55173 0.54952 0.54734 0.54520
0.55360 0.55137 0.54917 0.54702 0.54491
0.55373 0.55149 0.54928 0.54722 0.54522
M.F.A. Alias et al. / Renewable Energy 24 (2001) 347–351
Fig. 2.
351
Lattice parameter of Ge–Si solid solution as a function of at% Si.
four atomic radii taken from different sources were used for both Ge and Si in the calculations, as rGe=0.122406 nm and rSi=0.117521 nm. Values of lattice parameters according to the prediction model are given in Table 3. Based on the atomic radius and a purely theoretical approach, the lattice parameters of the Ge–Si alloy seem to be better in comparison with the analytical calculation that had been taken from the experimental results, especially at mid composition where the discrepancy is most. Moreover, the prediction model gave better agreement with the experimental values of lattice parameters obtained in this work and by Dismukes et al. [5]. This favours the prediction model in the calculation that necessitates the use of lattice parameters of a binary system.
References [1] Pisharody RK, Garvey LP. In: Proceedings of the 13th Intersociety Energy Conversion Engineering Conference. New York: IEEE, 1978:1963. [2] Logan RA, Rowell JM, Trumbore FA. Phys Rev A 1964;136:1751. [3] Johnson ER, Christian SM. Phys Rev 1954;95:560. [4] Bush G, Vogt O. Helv Phys Acta 1960;33:437. [5] Dismukes JP, Ekstrom L, Paff RJ. J Phys Chem 1964;68:3021. [6] Van De Walle CG, Martin RM. Phys Rev B 1986;34:5621. [7] Smakula A, Sils V. Phys Rev 1955;99:1744. [8] Ferguson IF, Rogenson AH. Comput Phys Commun 1984;32:95. [9] Fischetti MV, Laux SE. J Appl Phys 1996;80:2234. [10] Abdulah OG. M.Sc. thesis. Baghdad: University of Baghdad, 1999.
Lattice parameter and density of Ge–Si solid solutions M.F.A. Alias, N.N. Rammo, M.N. Makadsi
*
Physics Department, College of Science, University of Baghdad, Jadiriya, PO Box 47162, Baghdad, Iraq
Abstract Solid solutions of the Ge–Si system have been prepared by reaction under vacuum and gravity of the constituent elements followed by different quenching procedures. Least-squares refinement to the lattice parameters of the Ge–Si phase has been established in the composition range 0.5–0.9 at% Si. Almost homogeneous alloys were obtained by rapid quenching in liquid nitrogen, whereas pseudo-homogeneous alloys occur as an intermediate stage when quenched in water. The refined lattice parameters, although computed with very high precision, did not predict the formation rate of the solid solution. However, careful analysis of the X-ray diffraction pattern of residual Ge and Si revealed that the lattice parameter of each component is modified in accordance with the atomic percentage of either element present in the lattice of the other. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Germanium–silicon alloys are being used widely in thermoelements [1]. Logan et al. [2], Johnson and Christian [3] and Bush and Vogt [4] have all pointed out that inhomogeneity exists in the prepared Ge–Si alloys irrespective of the method of preparation, whether slow cooling from the melt or zone levelling. On the other hand, Dismukes et al. [5] have reported homogeneous Ge–Si alloy ingots prepared by zone levelling, from which the lattice parameter and density have been measured throughout the entire alloy system. However, the large discrepancy between the results of previous investigations of these properties [3–5] corresponds to an uncertainty in composition for a definite value of the lattice parameter or density.
* Corresponding author. 0960-1481/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 1 4 8 1 ( 0 1 ) 0 0 0 1 5 - 5
348
M.F.A. Alias et al. / Renewable Energy 24 (2001) 347–351
Compared with relaxed Ge–Si [2] alloys, the structure of strained Ge–Si alloys [6] in terms of their lattice parameters was accounted for as additional stretching of the unit cell caused by strain. As different methods of preparation impart a significant level of inhomogeneity on the prepared Ge–Si ingots, it becomes of interest to visualize the behaviour of these alloys prepared by rather unique quenching from high temperature — the aim of the present work — and their subsequent thin film properties (to be presented elsewhere). 2. Experimental procedure Ge–Si alloy ingots in the composition range 0.5–0.9 at% Si were prepared from high-purity Ge and Si by reaction of the well-mixed powders in a sealed quartz capsule at temperatures of 1200–1500°C for 6–8 h, followed by rapid quenching in liquid nitrogen or dry ice. Densities were measured by two methods. The first was hydrostatic weighing [7] employing Archimedes’ principle, with weighing precision of 0.2 mg; the accuracy of density measurement was shown to be within 0.3%. The second method was by means of a multipycnometer. Samples for X-ray diffraction (XRD) analysis were ground to pass through a 75 µm mesh screen. No annealing was done to reduce residual stresses due to grinding. Diffraction scans were performed on a Philips PW1710 automated powder diffraction system, using monochromatic Cu Kα1 radiation at a scan rate of 0.02°/2 s, an automatic divergence slit for constant irradiated area and a receiving slit of 0.2°. The 111, 220, 311, 400 and 331 reflections of the diamond structure were used in the least-squares refinement of the lattice parameters by employment of the FIRSTAR program [8] and Robert’s extrapolation function for the diffractometer as (cos q cot q+cot2 q). The lattice parameters were determined with a precision of one part in 20,000 at room temperature. 3. Results and discussion Table 1 lists data on the density for different alloy compositions as measured by the two methods. Table 1 Density of Ge–Si alloys as measured by Archimides’ principle and the multipycnometer At% Si
50 60 70 80 90
Density (g/cm3) Archimides
Multipycnometer
3.925 3.639 3.329 3.010 2.678
3.820 3.350 3.005
M.F.A. Alias et al. / Renewable Energy 24 (2001) 347–351
Fig. 1.
349
XRD pattern of Ge0.3Si0.7 fired at 1450°C for 8 h and quenched in liquid nitrogen.
A typical XRD pattern of the Ge–Si alloy is shown in Fig. 1. Data on the refined lattice parameters of Ge–Si for different alloy compositions, together with the refined lattice parameter for individual Ge and Si phases found as residuals in the XRD spectra, are shown in Table 2. The results of lattice parameters in the range of compositions prepared seem to agree with relevant data in the literature [3–5], indicating that the method of alloy preparation has little significance on the lattice parameters. It is noted, however, that the lattice parameters of the residual phases of Ge and Si (see Table 2) are modified in accordance with the atomic percentage of either component present in the lattice of the other, suggesting that appreciable diffusion has occurred although not enough to complete the formation of a homogeneous alloy. Table 2 Lattice parameters of Ge–Si alloys, and of Ge and Si, for different compositionsa At% Si 0 50 60 70 80 90 100 a
aSi (nm)
aGe–Si (nm)
0.54604(22)
0.55317(18) 0.55201(25) 0.54887(21) 0.54691(15) 0.54494(13)
0.54421(24) 0.54365(12) 0.54309 Values in brackets are standard deviations.
aGe (nm) 0.56580 0.56549(10) 0.56397(23) 0.56215(25)
350
M.F.A. Alias et al. / Renewable Energy 24 (2001) 347–351
This may be due partly to the growth rate, which was not controlled as the melt composition changed.
4. Analytical calculation of lattice parameters For the lattice parameters of Ge–Si as a function of composition, Fischetti and Laux [9] have used the experimental results of Logan et al. [2] that have been approximated analytically as aGe–Si(x)⫽aGe x⫹aSi(1⫺x)⫺bx(1⫺x), with parameter b=1.88×10⫺3 nm. Results from this equation have been calculated in terms of the composition range considered in this work and are tabulated in Table 3 for comparative purpose. Our data on lattice parameters and the data of Refs. [5,9] are plotted in Fig. 2; there is full agreement between the two.
5. Prediction of lattice parameters from atomic radii The Ge–Si alloy system is an attractive one for comparing experimental deviations from Vegard’s law with theoretical calculations based on the atomic radius and braves lattice of individual components. A detailed analysis of theoretical work on the model that predicts lattice parameters as a function of composition is given by Abdulah [10]. The formula derived for the binary system is aGe–Si⫽C 2GeaGe⫹2CGeCSiaGeSi⫹C 2SiaSi. The values of aGe and aSi can be found from the atomic radius as 8r/√3 for the diamond structure. CGe and CSi are the atomic fractions of the components Ge and Si; aGe–Si is a simple function of the initial slope of the curve that represents the lattice parameter of the solid solution on the basis of solute concentration. The average of Table 3 Theoretical calculation of lattice parameters At% Si
50 60 70 80 90
Lattice parameter of Ge–Si (nm) Vegard’s law
Analytical via Ref. [9]
Predicted by atomic radius, Ref. [10]
Quoted from Ref. [5]
0.55410 0.55185 0.54959 0.54734 0.54509
0.55398 0.55173 0.54952 0.54734 0.54520
0.55360 0.55137 0.54917 0.54702 0.54491
0.55373 0.55149 0.54928 0.54722 0.54522
M.F.A. Alias et al. / Renewable Energy 24 (2001) 347–351
Fig. 2.
351
Lattice parameter of Ge–Si solid solution as a function of at% Si.
four atomic radii taken from different sources were used for both Ge and Si in the calculations, as rGe=0.122406 nm and rSi=0.117521 nm. Values of lattice parameters according to the prediction model are given in Table 3. Based on the atomic radius and a purely theoretical approach, the lattice parameters of the Ge–Si alloy seem to be better in comparison with the analytical calculation that had been taken from the experimental results, especially at mid composition where the discrepancy is most. Moreover, the prediction model gave better agreement with the experimental values of lattice parameters obtained in this work and by Dismukes et al. [5]. This favours the prediction model in the calculation that necessitates the use of lattice parameters of a binary system.
References [1] Pisharody RK, Garvey LP. In: Proceedings of the 13th Intersociety Energy Conversion Engineering Conference. New York: IEEE, 1978:1963. [2] Logan RA, Rowell JM, Trumbore FA. Phys Rev A 1964;136:1751. [3] Johnson ER, Christian SM. Phys Rev 1954;95:560. [4] Bush G, Vogt O. Helv Phys Acta 1960;33:437. [5] Dismukes JP, Ekstrom L, Paff RJ. J Phys Chem 1964;68:3021. [6] Van De Walle CG, Martin RM. Phys Rev B 1986;34:5621. [7] Smakula A, Sils V. Phys Rev 1955;99:1744. [8] Ferguson IF, Rogenson AH. Comput Phys Commun 1984;32:95. [9] Fischetti MV, Laux SE. J Appl Phys 1996;80:2234. [10] Abdulah OG. M.Sc. thesis. Baghdad: University of Baghdad, 1999.