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Bulk modulus and its pressure derivatives of cuprous halides
Solid State Communications 134 (2005) 637–640 www.elsevier.com/locate/ssc
Bulk modulus and its pressure derivatives of cuprous halides G. Misra*, S.C. Goyal Department of Physics, Agra College, Agra 282002, India Received 18 November 2004; accepted 7 February 2005 by F. Peeters Available online 11 March 2005
Abstract The ab initio pseudopotential approach to the total crystal energy is presented using local DF formalism. The expressions for bulk modulus, its first and second pressure derivatives for group I–VII semiconductor binary compounds are derived. The expression for the second pressure derivative of the bulk modulus for four-fold crystal structures is derived for the first time within the pseudopotential framework. The computed results of the bulk modulus for cuprous halides are very close to the available experimental data. q 2005 Elsevier Ltd. All rights reserved. PACS: 62.20.Dc; 71.15.Ap; 71.15.Hx; 71.15.Mb; 71.15.Nc Keywords: A. Semiconductors; C. Four-fold crystal structure; D. Elasticity-bulk modulus; D. Electronic band structure
1. Introduction Various physical properties [1,2], thermal properties [3, 4] and the equation of states (EOS) [5–8] of solids sensitively depend upon the isothermal bulk modulus and its pressure derivatives. The reason being that the bulk modulus is defined as the derivative of volume, therefore, it is more sensitive to the variation in EOS than the volume itself. This provides a basis for studying the Earth’s deep interior [9]. In the past few years, it has become possible [10] to compute lattice constants, bulk moduli, cohesive energies, phonon spectra and other static properties knowing only the atomic numbers and masses of the atoms composing the materials. A simple formula for the bulk moduli was obtained by Cohen [11] using scaling arguments for the relevant energy and volume. Later on, Lam et al. [12] have explored the microscopic origin of the above simple relation. However, in both these studies [11,12] the authors have not included the cuprous halides though they
* Corresponding author. E-mail address: [email protected] (G. Misra).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.02.041
crystallize in the same four-fold crystal structure [13–15] and play an important role in various physical properties. The purpose of the present study is two fold (i) to extend the empirical relation [11] and (ii) to derive the expressions for bulk modulus, its first and second pressure derivatives of group I–VII cuprous halides within the pseudopotential frame work. The results based on the above derived relations for cuprous halides are very close to the available data.
2. Empirical method According to Cohen [11] the expression for the bulk modulus of semiconductor binary compounds is given by: B0 Z ð212:6 K 23:7lÞdK3:5 0
(1)
where B0 is the bulk modulus in GPa, d0 is the equilibrium nearest neighbour distance in a.u. and l is an empirical parameter which accounts for the effect of ionicity; lZ0, 1, 2 for group IV–IV, III–V, II–VI binary compounds, respectively. The value of l for group I–VII binary compounds is not included in the study [11]. More over the value of the
638
G. Misra, S.C. Goyal / Solid State Communications 134 (2005) 637–640
empirical parameter l such chosen has no physical significance. Now we propose the following empirical relation for the bulk modulus B0 of four-fold type semiconductor binary compounds in terms of equilibrium Seitz–Wigner radius R0 and DZ as: B0 Z
5696 K 636DZ R3:5 0
(2)
where B0 is in GPa, R0 is in a.u. and for a binary compound XY, DZZ(ZYKZX)/2, Z being the valency (the number of valence electrons). With this extension, it is possible to investigate the group I–VII cuprous halides. This empirical relation gives fairly good results for all four-fold type semiconductor binary compounds of group I–VII, II–VI, III–V and IV–IV.
X0 1 X0 GÞ C Ua VH ðGÞrð ½3XC ðGÞ EBS Z 30 K Ua 2
3. Pseudopotential method Within the pseudopotential framework [16], following the conventional density functional (DF) formalism [17,18] the total crystal energy (E); defined as the total energy difference between the solid and isolated cores or the negative of the sum of ionization potentials of the valence electrons plus the cohesive energy; under Frozen–Core approximation (FCA) is given by: E Z Te C Ve – e C Ve – ion C Vion – ion C EXC
(3)
The individual contributions are interpreted as the kinetic energy of electrons, the Coulomb energy due to electron– electron interaction, the energy due to electron–ion interaction, the Coulomb energy due to the ion–ion interaction and the electronic exchange-correlation energy. Since, the effect of core electrons is included in the pseudopotentials, the term ‘electrons’ used in this paper refers to the valence electrons only. The Schro¨dinger equation used in the plane-wave method can be easily derived variationally from the expression for the total crystal energy. Using the resulting eigenvalues 3i’s (the index i denote both the wave-vector ki and the band index n and runs over all occupied valence states) the total crystal energy per atom can be written in momentum space as: C UPS ðG Z 0Þ Z C gEwald E Z 3 C mXC ðG Z 0Þ Ua X0 1 X0 GÞ C Ua K Ua VH ðGÞrð ½3XC ðGÞ 2 G
energy density, Z is the valency of the ion, Ua is the atomic 0Þ are respectively the Ewald volume, gEwald and UPS ðGZ energy [19] and the pseudopotential term which measures only the integrated effects of the pseudopotential, higher The order effects are implicitly contained in 3i and rðGÞ. 0 prime on the summation in Eq. (4) denotes that the GZ term is excluded from the summation. In order to obtain the total crystal energy (E) as a function of the Seitz–Wigner radius (R), it is useful to write 3 Z 30 C 30 for systems where the band structure is not too far from the free-electron dispersion, with 30 being the average eigenvalue from the free-electron dispersion and 30 is the correction term. Then, Eq. (4) can be written as a sum of the uniform density term EUD and the band structure term EBS with: C UPS ðG Z 0Þ Z C gEwald (5) EUD Z 30 C mXC ðG Z 0Þ Ua
G
K mXC ðGÞrð GÞ (4) P where 3 Z NK1 i 3i is the mean eigen value averaged over is the Hartree potential, rðGÞ all the occupied states, VH ðGÞ is the exchangeis the (pseudo) charge density, mXC ðGÞ is the exchange-correlation correlation potential, 3XC ðGÞ
G
K mXC ðGÞrð GÞ
G
(6)
EUD can be shown to have the following explicit dependence on R: EUD Z
B A C 0:88Z K C K R2 R R3 R=Z 1=3 C 7:8
(7)
where AZ ð3=2pÞð9p=4Þ1=3 Z 4=3 C FS Z 2 z0:9163Z 4=3 C FS Z 2 with FS being the structure dependent Ewald constant and for diamond structure FSZ1.671, BZ ð3=5Þð9p=4Þ2=3 Z 5=3 z2:2099Z 5=3 , CZ ð3=4pÞZUPS ðGZ z0:2387ZUPS ðGZ 0Þ and the last term is the correlation 0Þ term. The first and second derivatives of this correlation term w.r.t. R are very small compared to the derivatives of the other terms. The contribution to the equilibrium volume and the bulk modulus from this correlation term is, therefore, neglected. Thus, Eq. (7) reduces to: EUD Z
B A C K C R2 R R3
(8)
The EBS will be investigated using perturbation theory following Heine and Weaire [20]. Up to second order in perturbation within the One-G model, the EBS is given by: (9) EBS Z KDR2 2 is a positive number, with VðGÞ being the where Df VðGÞ screened pseudopotential form factor. It should be pointed out that the R dependence of EBS is valid only for small deviations from the equilibrium volume; otherwise, the energy approaches negative infinity at large R. Furthermore, a weak pseudopotential is assumed such that second order perturbation theory is adequate. Now combining Eqs. (8) and (9), the final expression for total crystal energy becomes: EZ
B A C K C K DR2 R2 R R3
(10)
G. Misra, S.C. Goyal / Solid State Communications 134 (2005) 637–640
4. General relations for bulk modulus and its pressure derivatives On using Eq. (10) for the total crystal energy along with the equilibrium relation: AR20 K 2BR0 K 3C K 2DR50 Z 0
(11)
the bulk modulus B0 can readily be calculated using the following relation: 1 8B 3A 15C K C (12) B0 Z Ua;0 E 00 Z 9Ua;0 R20 R0 R30 where R0 and Ua,0 are the equilibrium Seitz–Wigner radius and equilibrium atomic volume, respectively. Here, the prime denotes differentiation w.r.t. atomic volume. The equilibrium condition (11) is used either to eliminate the dependence on C or D. Eq. (12) is in Ry/(a.u.)3Z 14700 GPa. The first and second pressure derivatives B0 0 and B0 00 , respectively, are then given by: dB0 E 000 ¼ K1 K Ua;0 00 dP E 7 2 A B B ¼ þ K 2 K 2 3 9Ua;0 B0 R0 R0 3R0 7 2 A B z þ K 3 9Ua;0 B0 R0 R20
B00 ¼
(13)
because for most materials 5DR20[B/3R20 d2 B0 E 000 E 00 00 ðE 000 Þ2 ¼ þ U K U a;0 a;0 dP2 ðE 00 Þ2 ðE 00 Þ2 ðE 00 Þ3 1 1116 ¼ K ð1 þ B00 Þð2 þ B00 Þ B0 81 1 120A 168B K 2 þ 81Ua;0 B0 R0 R0
aB000 ¼
(14)
5. Group I–VII semiconductor binary compounds The constants A and B in above Eqs. (12)–(14) are universal functions of the valence charge, i.e. Z. The two material parameters of this model are the average pseudopotential term C and the band structure term D, which are derivable from the pseudopotential. In principle, given a pseudopotential, one can obtain C or D and hence calculate B0, B0 0 and B0 00 . Empirically, for a given valency, B0 depends only on R0. This implies that C and D can be expressed as functions of R0. This turns out to be so and it can be illustrated using the Ashcroft empty core pseudopotential [21] that CZ3Z2r2c , where rc is the pseudopotential core radius. Also, the Seitz–Wigner radius varies linearly with pseudopotential core radius [22]. For group IV–IV binary compounds, ZZ4 (Z being the
639
average valency of atoms X and Y in the binary compound XY), so, A IV z32.6, B IV z22.3 and CIV Z 48rc2 z11ðR0 K 1:14Þ2 . Now to extend the calculation to other four-fold crystal structures, it is advantageous to adopt Phillip’s view that carbon and silicon are the purest covalent systems since there are no ionic and little d-like or metallic contributions to the bonding. For others the effect of ionicity must be included explicitly. To do so, it is useful to write: AZAIVCA 0 , BZBIVCB 0 , CZCIVCC 0 and DZ DIVCD 0 ; where AIV, BIV, CIV and DIV are the parameters for the covalent group IV material and the prime quantities are ionic parameters. The difference between A and A 0 comes from the Ewald energy of the group IV plus Madelung energy. Thus A 0 Z 1.1734(DZ)2 for four-fold structures. Now B being proportional to the average eigenvalue from the free-electron dispersion, 30 , suggests that B 0 Z0. To evaluate C 0 , recall 0Þ. An atom with a high valency that CZ ð3=4pÞZUPS ðGZ 0Þ, i.e. a high valency atom is more has a small UPS ðGZ electronegative. For the Ashcroft pseudopotential [21], 0ÞZ UPS ðGZ 4pZrc2 . As the pseudopotential core radius rc decreases with increasing Z, one would expect UPS ðGZ to scale as UPS ðGZ 0Þf Z h , where h is a negative 0Þ 0Þ about ZZ4, one obtains, number. Expanding UPS ðGZ C 0 ¼ fhðhK 1Þ=2gfDZ=Zg2 CIV . On substituting AZ AIV C A 0 z32:6C 1:1734ðDZÞ2 , BZBIVCB 0 z22.3 and C ¼ CIV þ C 0 z11ðR0 K 1:14Þ2 ! ½1 þ fhðhK 1Þ=2gfDZ=Zg2 in Eqs. (12)–(14), the bulk modulus and its pressure derivatives for group I–VII semiconductor binary compounds are simplified as: 1 178 f98 þ 3:52ðDZÞ2 g B0 ¼ K 9Ua;0 R20 R0 166ðR0 K 1:14Þ2 hðh K 1Þ DZ 2 14700 1þ C 2 Z R30 (15)
B00
7 2 f32:6 þ 1:1734ðDZÞ2 g 22:3 ¼ þ K 2 14700 3 9Ua;0 B0 R0 R0 (16)
aB000 ¼
1 1116 K ð1 þ B00 Þð2 þ B00 Þ B0 81 14700 f3912 þ 141ðDZÞ2 g 3746 K 2 C 81Ua;0 B0 R0 R0
(17)
where B0 is in GPa, B0 0 is dimensionless, B0 00 is in GPaK1 and R0 is in a.u.. It is found that the following choice for the scaling parameter hzK0.6 gives good results for I–VII cuprous halides (see Table 1).
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Table 1 Comparison of calculated and experimental values of bulk modulus B0, its first pressure derivative B0 0 and its second pressure derivative B0 00 Compounds
R0 in a.u.
Exp. [23]
GROUP I–VII CuCl CuBr CuI
3.66 3.84 4.08
B0 0 Calc.
B0 in GPa
39.4 38.7 35.5
Calc. (a)
(b)
40.4 34.1 27.6
41.1 37.5 33.0
B0 00 in GPaK1 Calc.
Present
Others [24]
6.42 5.79 5.32
4.13 4.50 4.97
K0.589 K0.451 K0.386
The bulk moduli are calculated (a) using empirical relation Eq. (2) and (b) using Eq. (15) with hZK0.6 for group I–VII cuprous halides. The first pressure derivatives are calculated with Eq. (16) using the experimental value for B0. The second pressure derivatives are calculated with Eq. (17) using the experimental value for B0 and calculated value for B0 0 .
6. Conclusion The empirical relation for bulk modulus of group I–VII, II–VI, III–V and IV–IV semiconductor binary compounds in diamond and zinc blende structures is proposed (Eq. (2)). This relation (Eq. (2)) is more general as compared to the relation proposed by Cohen [11] and works well for all fourfold type semiconductor binary compounds under study. Analytic expressions for the bulk modulus, its first and second pressure derivatives are derived from the ab initio pseudopotential total energy formalism. Using perturbation theory and a One-G model these expressions are written explicitly as a function of equilibrium Seitz–Wigner radius R0 alone. The expressions so derived are further simplified for group I–VII compounds. The agreement (Table 1) between the numerical and experimental values is very good. The effect of ionicity enters as DZ, being linear in empirical relation but quadratic in ab initio pseudopotential formalism.
Acknowledgements Authors are thankful to Dr A.V. Singh, Principal, Agra College, Agra for providing the facilities. One of the authors S.C.G. is thankful to the UGC for providing the financial assistance.
References
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[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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S.C. Goyal, M.M. Layas, Solid State Commun. 36 (1980) 335. Q. He, Z.T. Yan, Phys. Status Solidi 223 (2001) 767. P.P. Singh, M. Kumar, Physica B 344 (2004) 41. Z.-H. Fang, L.-R. Chen, Phys. Status Solidi B 180 (1993) K5. O.L. Anderson, Equation of State of Solid for Geophysics and Ceramic Science, Oxford University Press, Oxford, UK, 1995. M. Kumar, High Temp.–High Pressure 31 (1999) 221. S. Meenakshi, B.S. Sharma, Indian J. Phys. A73 (2001) 663. A.P. Jephcoat, S.P. Besedin, in: M.H. Manghnani, T. Yagi (Eds.), Properties of Earth and Planetary Materials at High Pressure and Temperature, AGU, Washington, DC, 1998. M.L. Cohen, Phys. Scr. T1 (1982) 5. M.L. Cohen, Phys. Rev. B 32 (1985) 7988. P.K. Lam, M.L. Cohen, G. Martinez, Phys. Rev. B 35 (1987) 9190. J.C. Phillips, Rev. Mod. Phys. 42 (1970) 317. L. Pauling, The Nature of Chemical Bond, Cornell University Press, Ithaca, New York, 1960. S.C. Goyal, Solid State Commun. 20 (1976) 269. W.A. Harrison, Pseudopotential in the Theory of Metals, Benjamin, New York, 1966. P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. R.A. Coldwell-Horsfall, A.A. Maradudin, J. Math. Phys. 1 (1960) 395. V. Heine, D. Weaire, in: H. Ehrenreich, F. Seitz, D. Turnbul (Eds.), Solid State Physics vol. 24, Academic, New York, 1970. N.W. Ashcroft, Phys. Lett. 23 (1966) 48. A. Zunger, Phys. Rev. B 22 (1980) 5839. K. Kunc, M. Balkanshi, M.A. Nusimov, Phys. Status Solidi B 71 (1975) 341. S.D. Chaturvedi, R.C. Dixit, A.A.S. Sangachin, Phys. Status Solidi B 147 (1988) 115.
Bulk modulus and its pressure derivatives of cuprous halides G. Misra*, S.C. Goyal Department of Physics, Agra College, Agra 282002, India Received 18 November 2004; accepted 7 February 2005 by F. Peeters Available online 11 March 2005
Abstract The ab initio pseudopotential approach to the total crystal energy is presented using local DF formalism. The expressions for bulk modulus, its first and second pressure derivatives for group I–VII semiconductor binary compounds are derived. The expression for the second pressure derivative of the bulk modulus for four-fold crystal structures is derived for the first time within the pseudopotential framework. The computed results of the bulk modulus for cuprous halides are very close to the available experimental data. q 2005 Elsevier Ltd. All rights reserved. PACS: 62.20.Dc; 71.15.Ap; 71.15.Hx; 71.15.Mb; 71.15.Nc Keywords: A. Semiconductors; C. Four-fold crystal structure; D. Elasticity-bulk modulus; D. Electronic band structure
1. Introduction Various physical properties [1,2], thermal properties [3, 4] and the equation of states (EOS) [5–8] of solids sensitively depend upon the isothermal bulk modulus and its pressure derivatives. The reason being that the bulk modulus is defined as the derivative of volume, therefore, it is more sensitive to the variation in EOS than the volume itself. This provides a basis for studying the Earth’s deep interior [9]. In the past few years, it has become possible [10] to compute lattice constants, bulk moduli, cohesive energies, phonon spectra and other static properties knowing only the atomic numbers and masses of the atoms composing the materials. A simple formula for the bulk moduli was obtained by Cohen [11] using scaling arguments for the relevant energy and volume. Later on, Lam et al. [12] have explored the microscopic origin of the above simple relation. However, in both these studies [11,12] the authors have not included the cuprous halides though they
* Corresponding author. E-mail address: [email protected] (G. Misra).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.02.041
crystallize in the same four-fold crystal structure [13–15] and play an important role in various physical properties. The purpose of the present study is two fold (i) to extend the empirical relation [11] and (ii) to derive the expressions for bulk modulus, its first and second pressure derivatives of group I–VII cuprous halides within the pseudopotential frame work. The results based on the above derived relations for cuprous halides are very close to the available data.
2. Empirical method According to Cohen [11] the expression for the bulk modulus of semiconductor binary compounds is given by: B0 Z ð212:6 K 23:7lÞdK3:5 0
(1)
where B0 is the bulk modulus in GPa, d0 is the equilibrium nearest neighbour distance in a.u. and l is an empirical parameter which accounts for the effect of ionicity; lZ0, 1, 2 for group IV–IV, III–V, II–VI binary compounds, respectively. The value of l for group I–VII binary compounds is not included in the study [11]. More over the value of the
638
G. Misra, S.C. Goyal / Solid State Communications 134 (2005) 637–640
empirical parameter l such chosen has no physical significance. Now we propose the following empirical relation for the bulk modulus B0 of four-fold type semiconductor binary compounds in terms of equilibrium Seitz–Wigner radius R0 and DZ as: B0 Z
5696 K 636DZ R3:5 0
(2)
where B0 is in GPa, R0 is in a.u. and for a binary compound XY, DZZ(ZYKZX)/2, Z being the valency (the number of valence electrons). With this extension, it is possible to investigate the group I–VII cuprous halides. This empirical relation gives fairly good results for all four-fold type semiconductor binary compounds of group I–VII, II–VI, III–V and IV–IV.
X0 1 X0 GÞ C Ua VH ðGÞrð ½3XC ðGÞ EBS Z 30 K Ua 2
3. Pseudopotential method Within the pseudopotential framework [16], following the conventional density functional (DF) formalism [17,18] the total crystal energy (E); defined as the total energy difference between the solid and isolated cores or the negative of the sum of ionization potentials of the valence electrons plus the cohesive energy; under Frozen–Core approximation (FCA) is given by: E Z Te C Ve – e C Ve – ion C Vion – ion C EXC
(3)
The individual contributions are interpreted as the kinetic energy of electrons, the Coulomb energy due to electron– electron interaction, the energy due to electron–ion interaction, the Coulomb energy due to the ion–ion interaction and the electronic exchange-correlation energy. Since, the effect of core electrons is included in the pseudopotentials, the term ‘electrons’ used in this paper refers to the valence electrons only. The Schro¨dinger equation used in the plane-wave method can be easily derived variationally from the expression for the total crystal energy. Using the resulting eigenvalues 3i’s (the index i denote both the wave-vector ki and the band index n and runs over all occupied valence states) the total crystal energy per atom can be written in momentum space as: C UPS ðG Z 0Þ Z C gEwald E Z 3 C mXC ðG Z 0Þ Ua X0 1 X0 GÞ C Ua K Ua VH ðGÞrð ½3XC ðGÞ 2 G
energy density, Z is the valency of the ion, Ua is the atomic 0Þ are respectively the Ewald volume, gEwald and UPS ðGZ energy [19] and the pseudopotential term which measures only the integrated effects of the pseudopotential, higher The order effects are implicitly contained in 3i and rðGÞ. 0 prime on the summation in Eq. (4) denotes that the GZ term is excluded from the summation. In order to obtain the total crystal energy (E) as a function of the Seitz–Wigner radius (R), it is useful to write 3 Z 30 C 30 for systems where the band structure is not too far from the free-electron dispersion, with 30 being the average eigenvalue from the free-electron dispersion and 30 is the correction term. Then, Eq. (4) can be written as a sum of the uniform density term EUD and the band structure term EBS with: C UPS ðG Z 0Þ Z C gEwald (5) EUD Z 30 C mXC ðG Z 0Þ Ua
G
K mXC ðGÞrð GÞ (4) P where 3 Z NK1 i 3i is the mean eigen value averaged over is the Hartree potential, rðGÞ all the occupied states, VH ðGÞ is the exchangeis the (pseudo) charge density, mXC ðGÞ is the exchange-correlation correlation potential, 3XC ðGÞ
G
K mXC ðGÞrð GÞ
G
(6)
EUD can be shown to have the following explicit dependence on R: EUD Z
B A C 0:88Z K C K R2 R R3 R=Z 1=3 C 7:8
(7)
where AZ ð3=2pÞð9p=4Þ1=3 Z 4=3 C FS Z 2 z0:9163Z 4=3 C FS Z 2 with FS being the structure dependent Ewald constant and for diamond structure FSZ1.671, BZ ð3=5Þð9p=4Þ2=3 Z 5=3 z2:2099Z 5=3 , CZ ð3=4pÞZUPS ðGZ z0:2387ZUPS ðGZ 0Þ and the last term is the correlation 0Þ term. The first and second derivatives of this correlation term w.r.t. R are very small compared to the derivatives of the other terms. The contribution to the equilibrium volume and the bulk modulus from this correlation term is, therefore, neglected. Thus, Eq. (7) reduces to: EUD Z
B A C K C R2 R R3
(8)
The EBS will be investigated using perturbation theory following Heine and Weaire [20]. Up to second order in perturbation within the One-G model, the EBS is given by: (9) EBS Z KDR2 2 is a positive number, with VðGÞ being the where Df VðGÞ screened pseudopotential form factor. It should be pointed out that the R dependence of EBS is valid only for small deviations from the equilibrium volume; otherwise, the energy approaches negative infinity at large R. Furthermore, a weak pseudopotential is assumed such that second order perturbation theory is adequate. Now combining Eqs. (8) and (9), the final expression for total crystal energy becomes: EZ
B A C K C K DR2 R2 R R3
(10)
G. Misra, S.C. Goyal / Solid State Communications 134 (2005) 637–640
4. General relations for bulk modulus and its pressure derivatives On using Eq. (10) for the total crystal energy along with the equilibrium relation: AR20 K 2BR0 K 3C K 2DR50 Z 0
(11)
the bulk modulus B0 can readily be calculated using the following relation: 1 8B 3A 15C K C (12) B0 Z Ua;0 E 00 Z 9Ua;0 R20 R0 R30 where R0 and Ua,0 are the equilibrium Seitz–Wigner radius and equilibrium atomic volume, respectively. Here, the prime denotes differentiation w.r.t. atomic volume. The equilibrium condition (11) is used either to eliminate the dependence on C or D. Eq. (12) is in Ry/(a.u.)3Z 14700 GPa. The first and second pressure derivatives B0 0 and B0 00 , respectively, are then given by: dB0 E 000 ¼ K1 K Ua;0 00 dP E 7 2 A B B ¼ þ K 2 K 2 3 9Ua;0 B0 R0 R0 3R0 7 2 A B z þ K 3 9Ua;0 B0 R0 R20
B00 ¼
(13)
because for most materials 5DR20[B/3R20 d2 B0 E 000 E 00 00 ðE 000 Þ2 ¼ þ U K U a;0 a;0 dP2 ðE 00 Þ2 ðE 00 Þ2 ðE 00 Þ3 1 1116 ¼ K ð1 þ B00 Þð2 þ B00 Þ B0 81 1 120A 168B K 2 þ 81Ua;0 B0 R0 R0
aB000 ¼
(14)
5. Group I–VII semiconductor binary compounds The constants A and B in above Eqs. (12)–(14) are universal functions of the valence charge, i.e. Z. The two material parameters of this model are the average pseudopotential term C and the band structure term D, which are derivable from the pseudopotential. In principle, given a pseudopotential, one can obtain C or D and hence calculate B0, B0 0 and B0 00 . Empirically, for a given valency, B0 depends only on R0. This implies that C and D can be expressed as functions of R0. This turns out to be so and it can be illustrated using the Ashcroft empty core pseudopotential [21] that CZ3Z2r2c , where rc is the pseudopotential core radius. Also, the Seitz–Wigner radius varies linearly with pseudopotential core radius [22]. For group IV–IV binary compounds, ZZ4 (Z being the
639
average valency of atoms X and Y in the binary compound XY), so, A IV z32.6, B IV z22.3 and CIV Z 48rc2 z11ðR0 K 1:14Þ2 . Now to extend the calculation to other four-fold crystal structures, it is advantageous to adopt Phillip’s view that carbon and silicon are the purest covalent systems since there are no ionic and little d-like or metallic contributions to the bonding. For others the effect of ionicity must be included explicitly. To do so, it is useful to write: AZAIVCA 0 , BZBIVCB 0 , CZCIVCC 0 and DZ DIVCD 0 ; where AIV, BIV, CIV and DIV are the parameters for the covalent group IV material and the prime quantities are ionic parameters. The difference between A and A 0 comes from the Ewald energy of the group IV plus Madelung energy. Thus A 0 Z 1.1734(DZ)2 for four-fold structures. Now B being proportional to the average eigenvalue from the free-electron dispersion, 30 , suggests that B 0 Z0. To evaluate C 0 , recall 0Þ. An atom with a high valency that CZ ð3=4pÞZUPS ðGZ 0Þ, i.e. a high valency atom is more has a small UPS ðGZ electronegative. For the Ashcroft pseudopotential [21], 0ÞZ UPS ðGZ 4pZrc2 . As the pseudopotential core radius rc decreases with increasing Z, one would expect UPS ðGZ to scale as UPS ðGZ 0Þf Z h , where h is a negative 0Þ 0Þ about ZZ4, one obtains, number. Expanding UPS ðGZ C 0 ¼ fhðhK 1Þ=2gfDZ=Zg2 CIV . On substituting AZ AIV C A 0 z32:6C 1:1734ðDZÞ2 , BZBIVCB 0 z22.3 and C ¼ CIV þ C 0 z11ðR0 K 1:14Þ2 ! ½1 þ fhðhK 1Þ=2gfDZ=Zg2 in Eqs. (12)–(14), the bulk modulus and its pressure derivatives for group I–VII semiconductor binary compounds are simplified as: 1 178 f98 þ 3:52ðDZÞ2 g B0 ¼ K 9Ua;0 R20 R0 166ðR0 K 1:14Þ2 hðh K 1Þ DZ 2 14700 1þ C 2 Z R30 (15)
B00
7 2 f32:6 þ 1:1734ðDZÞ2 g 22:3 ¼ þ K 2 14700 3 9Ua;0 B0 R0 R0 (16)
aB000 ¼
1 1116 K ð1 þ B00 Þð2 þ B00 Þ B0 81 14700 f3912 þ 141ðDZÞ2 g 3746 K 2 C 81Ua;0 B0 R0 R0
(17)
where B0 is in GPa, B0 0 is dimensionless, B0 00 is in GPaK1 and R0 is in a.u.. It is found that the following choice for the scaling parameter hzK0.6 gives good results for I–VII cuprous halides (see Table 1).
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G. Misra, S.C. Goyal / Solid State Communications 134 (2005) 637–640
Table 1 Comparison of calculated and experimental values of bulk modulus B0, its first pressure derivative B0 0 and its second pressure derivative B0 00 Compounds
R0 in a.u.
Exp. [23]
GROUP I–VII CuCl CuBr CuI
3.66 3.84 4.08
B0 0 Calc.
B0 in GPa
39.4 38.7 35.5
Calc. (a)
(b)
40.4 34.1 27.6
41.1 37.5 33.0
B0 00 in GPaK1 Calc.
Present
Others [24]
6.42 5.79 5.32
4.13 4.50 4.97
K0.589 K0.451 K0.386
The bulk moduli are calculated (a) using empirical relation Eq. (2) and (b) using Eq. (15) with hZK0.6 for group I–VII cuprous halides. The first pressure derivatives are calculated with Eq. (16) using the experimental value for B0. The second pressure derivatives are calculated with Eq. (17) using the experimental value for B0 and calculated value for B0 0 .
6. Conclusion The empirical relation for bulk modulus of group I–VII, II–VI, III–V and IV–IV semiconductor binary compounds in diamond and zinc blende structures is proposed (Eq. (2)). This relation (Eq. (2)) is more general as compared to the relation proposed by Cohen [11] and works well for all fourfold type semiconductor binary compounds under study. Analytic expressions for the bulk modulus, its first and second pressure derivatives are derived from the ab initio pseudopotential total energy formalism. Using perturbation theory and a One-G model these expressions are written explicitly as a function of equilibrium Seitz–Wigner radius R0 alone. The expressions so derived are further simplified for group I–VII compounds. The agreement (Table 1) between the numerical and experimental values is very good. The effect of ionicity enters as DZ, being linear in empirical relation but quadratic in ab initio pseudopotential formalism.
Acknowledgements Authors are thankful to Dr A.V. Singh, Principal, Agra College, Agra for providing the facilities. One of the authors S.C.G. is thankful to the UGC for providing the financial assistance.
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