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Estimates of isothermal bulk moduli for group iva crystals with the zincblende structure
J. Phys.
Chem.
Solids
Pergamon
Press
I97 I. Vol.
32. pp. 653-655.
Printed
in Great
Britain.
ESTIMATES OF ISOTHERMAL BULK MODULI GROlJP IVA CRYSTALS WITH THE ZINCBLENDE STRUCTURE Chemistry
Department, (Received
D. H. YEAN University I July
1970;
and J. R. RITER, Jr. of Denver, Denver, Colo. in revisedform
20 August
FOR
802 10. U.S.A. I 970)
Abstract-By equating the differential changes in macroscopic work of compression and potential energy for group IVA crystals of the zincblende structure we have obtained an expression for the isothermal bulk modulus as a function of pressure. The model used for the potential energy for these covalent crystals is that of distinct nearest-neighbor bonds with force constants taken directly from the molecular series X,H,, and XYHs. We calculate this isothermal bulk modulus in the limit of zero external pressure to be 4070 (4420, 5450,5600), 1060 (970.8,988), 960 (7 12, 724.3,77 I .7), 720% 50 (?) and 2240 (?) kbar for diamond, Si, Ge, Sn (a. gray), and SIC @I, cubic) respectively, with experimental data in parentheses. It would seem that the value for a-Sn lies in the range 400-800 kbar, while that for p-Sic may not differ much from the calculated one. The treatment of a zincblende structure covalent crystal as a giant molecule for purposes of estimating low pressure isothermal bulk moduli is thus fairly satisfactory. For extending the moduli to higher pressures it suffers a serious defect as the (dimensionless) pressure derivative of the isothermal bulk modulus in the limit of zero pressure turns out to be I.0 for all covalent solids regardless of the number of bound neighboring atoms, compared to e.g. 4.16 and 4.35 for Si and Ge.
Table 1. HJ - XH3 and H3C - XH3 molecular force constants in the diatomicapproximation
1. INTRODUCTION
IS attracted by th:: simplicity of picturing a covalent crystal as a giant molecule with well-localized bonds similar to those in the corresponding molecules. We have pursued this idea by the computation from molecular force constants of a well-defined physical property of the crystal, the isothermal bulk modulus ONE
*=-VaP (av>T'. this is, of course, the reciprocal thermal compressibility. 2. MOLECULAR
FORCE
Compound
c H Si2,H:
G-9-b W-b CH,SiH, CH,GeH, CH,SnH,
(1)
k (millidynes/A)
Source
4.36 1.73 1.62 140~0~10 2.93 2.67 2.19
Ref. Ref. Ref. Ref. Ref. Ref. Ref.
estimatedonly;
[2] [3] [4] [5] [6] [7] [8]
of the iso3. ISOTHERMAL FUNCTION
The differential unit volume is
CONSTANTS
The values of the HaXH3 and H3C XH3 stretching force constants in the diatomic approximation were determined with one exception from i.r. and Raman studies and are listed in Table 1. Only the first five are used in this work as no others correspond to known crystals. A thorough search for the compound GeC was unsuccessful [ 11.
BULK MODULUS OF PRESSURE
AS A
work of compression
-+--pdp 0
per
(2)
with p = V/V,; we equate this to the differential of the potential energy, written as a product of three factors [9]. 653
D. H. YEAN
654
and J. R. RITER, Jr.
(3) where, on the left, the first term is the atomic density, the second the energy per bond, and the third the number of bonds per atom. Here p is the density, No Avogadro’s number, A the .average atomic weight, as 20.05 for Sic, k the force constant, and r the nearest neighbor bond length, with r,, and V,, corresponding to the values at zero external pressure. One solves for pressure as a function of p by equating the expressions in equations (2) and (3), and calling the positive first factor on the right hand side of (3) C, we have (4) We then find
=%#.
(5)
T
The zero pressure limits of B and (U/U), called B. and Bd, are C/3 and I.0 respectively. B,,’ is evaluated as
regardless of the number of covalently bound nearest neighbors- that is, B,,’ is independent OfC. 4. ANDERSON’S
EXPANSION B=B(P)
FOR
Anderson [ lo] has successfully extended low-pressure measurements of B by ultrasonic techniques to higher pressures for some 16 oxides, alkali halides, and bee, fee hexagonal, and diamond-structure metals by assuming that B = B,,+ B, ‘P
(7) without any higher powers of P. As can easily
be verified, and as has been pointed out before[lO], integration of this relation leads to the Murnaghan equation [ 1 l] (8)
This form is often used for presenting the results of solid compressibility studies [ 121; B, and B& are sometimes approximately evaluated by fitting the compressibility data to this relation. 5. CONCLUSIONS
From Table 2 it is seen that this treatment is qualitatively quite satisfactory and begins to be quantitatively useful for isothermal bulk moduli at low pressures. The international Critical Tables [15] give values of the isothermal bulk modulus for Sn at zero external pressure of 502 and 530 kbar by two different compressibility methods; presumably these refer to white tetragonal p-Sn however. The ultrasonic value of 1110 kbar for gray a-Sn measured [ 161 and quoted [14] seems suspiciously high. We have taken the temperature-dependent measurements [16] of both the rigidity and Young’s moduli and determined a value for the isothermal bulk modulus of 2600 kbar at 30°C a temperature at which the stable form is white tetragonal /3-Sn. A separate determination of the result for p-Sn from compressibility measurements, 541.5 kbar[l4], clusters well with the earlier values mentioned above and leads us to reject the value of 2600 for /3-Sn and of 1110 for ol-Sn by implication. Reference[lS] gives a value for Si of 3 120 kbar at pressures from O-1 to 0.5 kbar which is quite different from the ultrasonic value of 970*8[10]; support for the lower value comes from the compressibility-derived result of 988 kbar [14], and the 3120 result is also rejected. For Ge the compressibility-derived values of 771.7[14] and 712 kbar[l5] check the ultrasonic value of 724.3 [IO] quite well. Recent ultrasonic measurements on Sic,
ISOTHERMAL
Table
BULK
MODULI
655
2. Calculated and experimental values for B,-, for group IVA crystals with the zincblende structure
Crystal zi (diamond) Ge
Sn(a, gray)
Sic (p, cubic)
B. (exptl.)(kbar)
B. (Calc.) = C/3 (kbar)@t 4070 1060 960 720 k 50 2240
4420 ” 5450’c’ 970.8’:‘, 988”‘. 7 I 21fl. 724.3”‘. 600 I!Z 2W”‘, (I
5600’“’ i3 120’“)“’ 771.7”’ 1 lCP’)[”
ca. 224(YX1
‘“‘C = 2pN,krz/3A, see text. tblfrom ultrasonic measurements, Ref. [13]. “‘from compilation of measurements, mostly compressibility, Ref. [ 141. tdlfrom curve fitting to Murnaghan equation, Ref. [12]. [“from ultrasonic measurements, Ref. [lo]. ‘“from compressibility measurements, Ref. [15]. hIestimated only, see text. fh’from ultrasonic measurements, Ref. [16]. [“values in parentheses are rejected, see text.
giving values of 2250 [ 171 and 2234 kbar [ 181, were definitely made on specimens of o-Sic (hexagonal) [ 171, [ 181. Reasoning as a periodic-table chemist would lead one to expect the correct value for gray c&n to be a fair amount lower than the calculated one; perhaps 600 + 200 kbar. The same ‘method’ of error vs. position would indicate that the calculated value for p-Sic might not be too bad. The extension of this method to higher pressures, for example in geophysical work, still suffers a serious drawback even assuming one could somehow fit a harmonic force constant to the value of C or BO for the solid. This drawback is that the calculated Bd is exactly 1.0 for all covalent solids in this approximation, regardless of the number of bound neighbors, where as, for the previously mentioned tabulation [ lo] for oxides, alkali halides, and metals the values range from 3.59 to 6.77 with 4.16 for Si and 4.35 for Ge. REFERENCES 1. SCACE R. 1. and SLACK G. A., in Silicon Carbide, (Edited by J. R. O’Connor and J. Smiltens), p. 24. Pergamon Press, New York (1960).
2.
3. 4. 5. 6. 7. 8.
HERZBERG G., Infrared and Raman Spectra of Polyatomic Molecules, p. 344. D. Van Nostrand, Princeton, New Jersey (I 945). BETHKE G. W. and WILSON M. K., J. Chem. Phys. 26,1107 (1957). CRAWFORD V. A., RHEE K. and WILSON M. K., .I. Chem. Phys. 37,2377 (1962). RITER J. R. Jr., Spectrochim. Acta. (In press). WILDE R. E..J. Mol. Spectr., 8,427 (1962). GRIFFITHS J. E., J. Chem. Phys. 38.2879 (1963). DlLLARD C. R. and MAY L., J. Mol. Spectr. 14, 250 (1964).
9. IO.
11. 12.
13. 14.
15.
RITER J. R. Jr., J. Chem. Phys. 52.5008 (1970). ANDERSON D. L., J. Phys. Chem. Solids 27, 547 (1966). MURNAGHAN F. D., Finife Deformation of an Elastic Solid, Chapt. 4. John Wiley New York (1944). DRICKAMER H. G., LYNCH R. W., CLENDENEN R. L. and PEREZ-ALBUERNE E. A., in Solid State Physics, (Edited by F. Seitz and D. Turnbull) Vol. 19, p. 135. Academic Press, New York (1966). McSKlMlN H. J. and BOND W. L.,Phys. Rev. 105, 116(1957). GSCHNEIDNER, K. A. Jr., in Solid State Physics, (Edited by F. Seitz and D. Turnbull) Vol. 16, p. 308, Academic Press, New York (1964). International Critical Tables, Vol. 3, p. 46. McGrawHill,
New
York,
(1927).
REDDY P. J. and SUBRAHMANYAM S. V., Nature Lond. 185.29 (1960). 17. CARNAHAN R. D., J. Am. ceram. Sot. 51, 223
16.
(1968). 18.
SCHREIBER 49,342
(1966).
E. and SOGA N., J. Am. ceram. Sot.
Chem.
Solids
Pergamon
Press
I97 I. Vol.
32. pp. 653-655.
Printed
in Great
Britain.
ESTIMATES OF ISOTHERMAL BULK MODULI GROlJP IVA CRYSTALS WITH THE ZINCBLENDE STRUCTURE Chemistry
Department, (Received
D. H. YEAN University I July
1970;
and J. R. RITER, Jr. of Denver, Denver, Colo. in revisedform
20 August
FOR
802 10. U.S.A. I 970)
Abstract-By equating the differential changes in macroscopic work of compression and potential energy for group IVA crystals of the zincblende structure we have obtained an expression for the isothermal bulk modulus as a function of pressure. The model used for the potential energy for these covalent crystals is that of distinct nearest-neighbor bonds with force constants taken directly from the molecular series X,H,, and XYHs. We calculate this isothermal bulk modulus in the limit of zero external pressure to be 4070 (4420, 5450,5600), 1060 (970.8,988), 960 (7 12, 724.3,77 I .7), 720% 50 (?) and 2240 (?) kbar for diamond, Si, Ge, Sn (a. gray), and SIC @I, cubic) respectively, with experimental data in parentheses. It would seem that the value for a-Sn lies in the range 400-800 kbar, while that for p-Sic may not differ much from the calculated one. The treatment of a zincblende structure covalent crystal as a giant molecule for purposes of estimating low pressure isothermal bulk moduli is thus fairly satisfactory. For extending the moduli to higher pressures it suffers a serious defect as the (dimensionless) pressure derivative of the isothermal bulk modulus in the limit of zero pressure turns out to be I.0 for all covalent solids regardless of the number of bound neighboring atoms, compared to e.g. 4.16 and 4.35 for Si and Ge.
Table 1. HJ - XH3 and H3C - XH3 molecular force constants in the diatomicapproximation
1. INTRODUCTION
IS attracted by th:: simplicity of picturing a covalent crystal as a giant molecule with well-localized bonds similar to those in the corresponding molecules. We have pursued this idea by the computation from molecular force constants of a well-defined physical property of the crystal, the isothermal bulk modulus ONE
*=-VaP (av>T'. this is, of course, the reciprocal thermal compressibility. 2. MOLECULAR
FORCE
Compound
c H Si2,H:
G-9-b W-b CH,SiH, CH,GeH, CH,SnH,
(1)
k (millidynes/A)
Source
4.36 1.73 1.62 140~0~10 2.93 2.67 2.19
Ref. Ref. Ref. Ref. Ref. Ref. Ref.
estimatedonly;
[2] [3] [4] [5] [6] [7] [8]
of the iso3. ISOTHERMAL FUNCTION
The differential unit volume is
CONSTANTS
The values of the HaXH3 and H3C XH3 stretching force constants in the diatomic approximation were determined with one exception from i.r. and Raman studies and are listed in Table 1. Only the first five are used in this work as no others correspond to known crystals. A thorough search for the compound GeC was unsuccessful [ 11.
BULK MODULUS OF PRESSURE
AS A
work of compression
-+--pdp 0
per
(2)
with p = V/V,; we equate this to the differential of the potential energy, written as a product of three factors [9]. 653
D. H. YEAN
654
and J. R. RITER, Jr.
(3) where, on the left, the first term is the atomic density, the second the energy per bond, and the third the number of bonds per atom. Here p is the density, No Avogadro’s number, A the .average atomic weight, as 20.05 for Sic, k the force constant, and r the nearest neighbor bond length, with r,, and V,, corresponding to the values at zero external pressure. One solves for pressure as a function of p by equating the expressions in equations (2) and (3), and calling the positive first factor on the right hand side of (3) C, we have (4) We then find
=%#.
(5)
T
The zero pressure limits of B and (U/U), called B. and Bd, are C/3 and I.0 respectively. B,,’ is evaluated as
regardless of the number of covalently bound nearest neighbors- that is, B,,’ is independent OfC. 4. ANDERSON’S
EXPANSION B=B(P)
FOR
Anderson [ lo] has successfully extended low-pressure measurements of B by ultrasonic techniques to higher pressures for some 16 oxides, alkali halides, and bee, fee hexagonal, and diamond-structure metals by assuming that B = B,,+ B, ‘P
(7) without any higher powers of P. As can easily
be verified, and as has been pointed out before[lO], integration of this relation leads to the Murnaghan equation [ 1 l] (8)
This form is often used for presenting the results of solid compressibility studies [ 121; B, and B& are sometimes approximately evaluated by fitting the compressibility data to this relation. 5. CONCLUSIONS
From Table 2 it is seen that this treatment is qualitatively quite satisfactory and begins to be quantitatively useful for isothermal bulk moduli at low pressures. The international Critical Tables [15] give values of the isothermal bulk modulus for Sn at zero external pressure of 502 and 530 kbar by two different compressibility methods; presumably these refer to white tetragonal p-Sn however. The ultrasonic value of 1110 kbar for gray a-Sn measured [ 161 and quoted [14] seems suspiciously high. We have taken the temperature-dependent measurements [16] of both the rigidity and Young’s moduli and determined a value for the isothermal bulk modulus of 2600 kbar at 30°C a temperature at which the stable form is white tetragonal /3-Sn. A separate determination of the result for p-Sn from compressibility measurements, 541.5 kbar[l4], clusters well with the earlier values mentioned above and leads us to reject the value of 2600 for /3-Sn and of 1110 for ol-Sn by implication. Reference[lS] gives a value for Si of 3 120 kbar at pressures from O-1 to 0.5 kbar which is quite different from the ultrasonic value of 970*8[10]; support for the lower value comes from the compressibility-derived result of 988 kbar [14], and the 3120 result is also rejected. For Ge the compressibility-derived values of 771.7[14] and 712 kbar[l5] check the ultrasonic value of 724.3 [IO] quite well. Recent ultrasonic measurements on Sic,
ISOTHERMAL
Table
BULK
MODULI
655
2. Calculated and experimental values for B,-, for group IVA crystals with the zincblende structure
Crystal zi (diamond) Ge
Sn(a, gray)
Sic (p, cubic)
B. (exptl.)(kbar)
B. (Calc.) = C/3 (kbar)@t 4070 1060 960 720 k 50 2240
4420 ” 5450’c’ 970.8’:‘, 988”‘. 7 I 21fl. 724.3”‘. 600 I!Z 2W”‘, (I
5600’“’ i3 120’“)“’ 771.7”’ 1 lCP’)[”
ca. 224(YX1
‘“‘C = 2pN,krz/3A, see text. tblfrom ultrasonic measurements, Ref. [13]. “‘from compilation of measurements, mostly compressibility, Ref. [ 141. tdlfrom curve fitting to Murnaghan equation, Ref. [12]. [“from ultrasonic measurements, Ref. [lo]. ‘“from compressibility measurements, Ref. [15]. hIestimated only, see text. fh’from ultrasonic measurements, Ref. [16]. [“values in parentheses are rejected, see text.
giving values of 2250 [ 171 and 2234 kbar [ 181, were definitely made on specimens of o-Sic (hexagonal) [ 171, [ 181. Reasoning as a periodic-table chemist would lead one to expect the correct value for gray c&n to be a fair amount lower than the calculated one; perhaps 600 + 200 kbar. The same ‘method’ of error vs. position would indicate that the calculated value for p-Sic might not be too bad. The extension of this method to higher pressures, for example in geophysical work, still suffers a serious drawback even assuming one could somehow fit a harmonic force constant to the value of C or BO for the solid. This drawback is that the calculated Bd is exactly 1.0 for all covalent solids in this approximation, regardless of the number of bound neighbors, where as, for the previously mentioned tabulation [ lo] for oxides, alkali halides, and metals the values range from 3.59 to 6.77 with 4.16 for Si and 4.35 for Ge. REFERENCES 1. SCACE R. 1. and SLACK G. A., in Silicon Carbide, (Edited by J. R. O’Connor and J. Smiltens), p. 24. Pergamon Press, New York (1960).
2.
3. 4. 5. 6. 7. 8.
HERZBERG G., Infrared and Raman Spectra of Polyatomic Molecules, p. 344. D. Van Nostrand, Princeton, New Jersey (I 945). BETHKE G. W. and WILSON M. K., J. Chem. Phys. 26,1107 (1957). CRAWFORD V. A., RHEE K. and WILSON M. K., .I. Chem. Phys. 37,2377 (1962). RITER J. R. Jr., Spectrochim. Acta. (In press). WILDE R. E..J. Mol. Spectr., 8,427 (1962). GRIFFITHS J. E., J. Chem. Phys. 38.2879 (1963). DlLLARD C. R. and MAY L., J. Mol. Spectr. 14, 250 (1964).
9. IO.
11. 12.
13. 14.
15.
RITER J. R. Jr., J. Chem. Phys. 52.5008 (1970). ANDERSON D. L., J. Phys. Chem. Solids 27, 547 (1966). MURNAGHAN F. D., Finife Deformation of an Elastic Solid, Chapt. 4. John Wiley New York (1944). DRICKAMER H. G., LYNCH R. W., CLENDENEN R. L. and PEREZ-ALBUERNE E. A., in Solid State Physics, (Edited by F. Seitz and D. Turnbull) Vol. 19, p. 135. Academic Press, New York (1966). McSKlMlN H. J. and BOND W. L.,Phys. Rev. 105, 116(1957). GSCHNEIDNER, K. A. Jr., in Solid State Physics, (Edited by F. Seitz and D. Turnbull) Vol. 16, p. 308, Academic Press, New York (1964). International Critical Tables, Vol. 3, p. 46. McGrawHill,
New
York,
(1927).
REDDY P. J. and SUBRAHMANYAM S. V., Nature Lond. 185.29 (1960). 17. CARNAHAN R. D., J. Am. ceram. Sot. 51, 223
16.
(1968). 18.
SCHREIBER 49,342
(1966).
E. and SOGA N., J. Am. ceram. Sot.