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Ab Initio Calculation of the Elastic and Optical Properties of Al3Sc Compound
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Available online at www.sciencedirect.com --&0
Acta Metall. Sin. (Engl. Lett.) Vol. 20 No. 6 pp425-428 Dec. 2007
ACTA METALLURGICA SINICA (ENGLISH LETTERS) www.ams.org.cn
AB INITIO CALCULATION OF THE ELASTIC AND OPTICAL PROPERTIES OF A@c COMPOUND M. Song* and D.H. Xiao State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China Manuscript received 10 January 2007; in revised form 3 March 2007
The ah initio method has been performed to explore the elastic and optical properties of AlJc compound, based on a plane wave pseudopotential method. It can be seen that the calculated equilibrium lattice parameter and elastic constants are in reasonable agreement with the previous experimentaldata. The elastic constants satisfv the requirement for mechunical stubilih. in the cubic structure of the AISc compound. The optical p r o p e q calculations show that a strong absorptive peak exists from 0-15eV and a relative small absorptive peuk exists around 3OeV. The form is caused by the optical transitions between high s, p , und d bands, and the latter results ,$-omthe opticul trunsitions from high s, p , and d hands to the low 2p band. KEY WORDS intermetallic compound; AIJc; elastic property; optical propem; ah initio calculation
1. Introduction Most age hardening aluminum alloys are limited to be used in the elevated temperature range, because of the rapid coarsening of the strengthening precipitates at high temperature"]. However, the addition of trace element Sc to aluminum alloys can substantially improve both their strength and ductility at elevated temperatures because of the formation of very fine coherent A1,Sc compounds[']. Thus, a lot of work has been carried out to study the effects of Sc on the microstructures and properties of the aluminum allo^[^-'], as well as, the structure of the A1,Sc corn pound^[^^^^^''^. It has been shown that AllSc is a stable L1, 1 and distributes along intermetallic compound. This compound has a very low density ( 3 . 0 3 ~03kg.m-')f'1, { 100) planed2] with a melting temperature of 1320°C ["I and very low coarsening ratesr2]. The electronic the structure of A13Sc has been calculated by using the first principle method, in the previous stlLdie~~'~-'~]. In this article, the elastic and optical properties are explored based on the DFT (density-fimctional theory), aiming to hrther understand other properties of the A1,Sc compound.
2. Calculation Method In the calculation, DFT'l4*"1is utilized based on the plane-wave method. In this approach, the correction 'Corresponding author. Tel. : +86 731 8877880. E-mail ruh1res.s : msong@mail. csu .edu. cn ( M . Song)
. 426
.
of the electronic density is considered - using exchange-correlation potentials parameterized according to the PBE (Perdew-Burke-Emzerhof) form, in the nonlocal model's generalized gradient approximation (GGA), where the main idea is to simplify the density hnctional problem by considering only the valence electrons. The calculation makes use of the ultrasoft pseudopotentials and an expansion of the electronic wave functions in plane waves with a kinetic energy cutoff of 290eV. The Brillouin zone is sampled with a [6 x 6 x 61 k-point mesh. This set of parameters assures a total energy convergence of 0.0 1eV per atom.
3. ResulcsandDiscussion A1,Sc is a binary stable L I Z intermetallic compound. The equilibrium lattice parameter a. is 0.4053393nm after ultrasoft pseudopotentials calcualtion, similar to the previous published data (0.4101nm by calculation and 0.4106nm by experiment)["'. The value of the equilibrium lattice parameter 0.4053393nm is very close to the lattice parameter of pure Al, with a value of 0.40496nm at 298KL"].Thus the A1 matrix can easily accommodate the elastic strain energy during the nucleation and growth of the A1,Sc compound, which exhibits a stronger coherent effect on the strength, compared to other A1J compounds (X= Zr, Cr, Er, etc.). To obtain the elastic constants of the A1,Sc compound with a cubic structure, the first principle calculation has been used by computing the stress tensor 6 for small strains (about 1/10 of the lattice parameter ao). It is well known that a cubic crystal has only three independent elastic constants (C,,, CI2,and C ), or only three independent compliance constants (SII,SI2,and S,). The bulk modulus (B,) and Poisson's ratio (u) can be determined through equations as follows B,=(C,,+2C,,)/3
(1)
The calculated elastic constants and previous experimental data[''' have been shown in Table 1. It can be seen that the calculated values are in reasonable agreement with the experimental data, except for ,C , where the result is smaller than the experimental counterpart. The mechanical stability in this cubic structure leads to the following restrictions on the elastic constants, CII - Clz> 0, C, > 0, CI1+ CI2> 0. From Table I , it can be seen that the calculated elastic constants obey these stability conditions, including the fact that C12must be smaller than C,,. The calculated elastic constants also obey the cubic stability condition that CI2
N=n+ik
where n is the refractive index and k is the absorption coefficient (or extinction coefficient). Table 1 Elastic constants calculated in this study with previous experimental data Data
C I I ,GPa
Cl2, GPa
C44rGPa
Bo, GPa
v
This work
181.00667
37.28050
38.86243
85.18922
0.1708
Experimental"'I
189
43
66
99.2
427 Fig. 1 shows the calculated refraction as a fbnction of energy (frequency) of A1,Sc compounds. It can be seen that the real part (refractive index) of the refraction has a strong peak at the position of low energy. This peak is accompanied by another strong peak at the imaginary part (absorption) of the refraction. As the intraband transitions in the far infrared (IR) region are not considered at all in this study, the strong peak is caused by the direct optical transitions between different bands. Detailed information about the optical transitions can be illustrated by checking the partial density of the states (PDOS). Fig.2 illustrates the calculated PDOS of the A1,Sc compound. It can be seen that the energy difference is about 20V between low s and p bands, whereas, it is more than 18eV between low p band and high s, p, and d bands. Thus, the strong peak is caused by the optical transitions between high s, p, and d bands. It can also be seen that a small absorption peak exists around 30eV. This small peak is caused by the optical transitions from high s, p, and d bands to the low 2p band. One should be careful of the calculated refraction, as the intraband transitions are not considered in this calculation. Normally, the intraband transitions are observed in metallic materials instead of insulators because electrons can only be excited into empty states because of the Pauli principle. It is common to evaluate the complex dielectric constant when calculating the optical properties. ) is,(w), where E is dielectric conThe frequency dependent complex dielectric constant, E ( W ) = E ~ ( W + stant, o is the angular frequency, e l is the real part and e2is the imaginary part, is known to describe the optical response of the medium at all phonon energies, E = tzw, E is energy and A is the Planck constant, using the formalism of Ehrenreich and C ~ h e n " ~The ]. 7, real and imaginary parts can be described as follows
6!l 5
e2(w)=2nk
(5)
Fig.3 shows the calculated dielectric function of the A13Sccompound. It can be seen that the curves in the dielectric function are similar to those in refraction. The imaginary (absorptive) part of the dielectric function has a strong peak at low energy and a small peak around 30eV. As described earlier, the strong 12 1
0
10
20
30
40
50
1
Energy, eV
Fig. 1 Calculated refraction of A1,Sc compound.
I
40
I
Small peak
-60
-50
-40
-30
-20
-10
0
10
Energy, eV
Fig.2 Calculated PDOS of AI$c compound.
20
0
10
20
30
40
50
60
Energy, eV
Fig.3 Calculated dielectric function of AI& compound.
. 428
.
peak is caused by the optical transitions between high s, p, and d bands. The small peak is because of the transition from high s, p, and d bands to low 2p band.
4. Conclusions Elastic and optical properties of the AI,Sc compound have been studied based on DFT. The calculation makes use of the ultrasoft pseudopotentials, and an expansion of electronic wavehnctions in plane waves. It has been shown that the elastic constants of A1,Sc compound, obtained by the calculations, satisfy the requirement of the mechanical stability in a cubic structure. The equilibrium lattice parameter of the A1,Sc compound is very close to the lattice parameter of pure Al, thus the A1,Sc compound exhibits a stronger coherent effect on the strength. It has also been shown that two absorptive peaks exist for the A1,Sc compound. The strong peak is caused by the optical transitions between high s, p, and d bands, whereas, the relative small peak is caused by the optical transitions from high s, p, and d bands to the low 2p band. Acknowledgements-This work wus supported by the Hunan Provinciul Natural Science Foundutinn of Chinu (No. 075531 17) and Chinese Postdoctoral Scientific Foundation (No. ,2OO7O410303).
REFERENCES 1 N. Blake and M.A. Hopkins, J. Mnter. Sci. 20 (1985) 2861. 2 R.W. Hyland, Melull. Trms. A 23 (1992) 1947. 3 C.B. Fuller, D.N. Seidman, and D.C. Dunand, Scr. Mater. 40 (1999) 691.
4 Z. Yin, Q. Lin, Y. Zhang, and F. Jiang, Mnter. Sci. Eng. A 280 (2000) 15 I . 5 R. Sawtell and C.L. Jensen, M e t d l . Trims. A 21 (1990) 42 I . 6 L.I. Kaygorodova and V.P. Domashnikov, Phys. Met. Metdl. 68 (1989) 160. 7 J. Hyung-Ho and F. Ahin-Itiroh, Mater. Sci. Eng. A 171 (1993) 15 I . 8 L.S. Kramer, W.T. Tack, and M.T. Femandes, A i h . Maier. Process. 152 (1997) 23. 9 Y. Harada and D.C. Dunand, A ctu Miuer. 48 (2000) 3477. 10 H. Okamoto, J Phme Equilibrin. 12 (1991) 612. 1 1 G. Bester and M. Fahnle, J. Phys.: Condens. M d t r r . 13 (2001) 11551. 12 J.H. Xu and A.J. Freeman, Phys. Rev. R 4 1 (1990) 12553. 13 M. Asta and V. Ozolins, Phys. Reu. B 64 (2001) 094104. 14 P. Hohenberg and W. Kohn, Phys. Rev. B 136 (1964) 864. 15 W. Kohn and L.J. Sham, Phys. Kei:. A 140(1965) 1133. 16 R.W. Jr. Hyland, M. Asta, S.M. Foiles, and C.L. Rohrer, A cta Mow. 46 ( 1 998) 3667. 17 Z.T. Wang and R.Z. Tian, Hwirlbook ofAluminum Alloys a d Their Fihriccltion (Changsha: Central South University of Technology Press, 1988) p. 152 (in Chinese). 18 R.W. Hyland and R.C. Stiffler, Scr. Metdl. Mater. 25 (1991) 473. 19 H. Ehrenreich and M.L. Cohen, Phys. Reo. 115 (1959) 786.
Available online at www.sciencedirect.com --&0
Acta Metall. Sin. (Engl. Lett.) Vol. 20 No. 6 pp425-428 Dec. 2007
ACTA METALLURGICA SINICA (ENGLISH LETTERS) www.ams.org.cn
AB INITIO CALCULATION OF THE ELASTIC AND OPTICAL PROPERTIES OF A@c COMPOUND M. Song* and D.H. Xiao State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China Manuscript received 10 January 2007; in revised form 3 March 2007
The ah initio method has been performed to explore the elastic and optical properties of AlJc compound, based on a plane wave pseudopotential method. It can be seen that the calculated equilibrium lattice parameter and elastic constants are in reasonable agreement with the previous experimentaldata. The elastic constants satisfv the requirement for mechunical stubilih. in the cubic structure of the AISc compound. The optical p r o p e q calculations show that a strong absorptive peak exists from 0-15eV and a relative small absorptive peuk exists around 3OeV. The form is caused by the optical transitions between high s, p , und d bands, and the latter results ,$-omthe opticul trunsitions from high s, p , and d hands to the low 2p band. KEY WORDS intermetallic compound; AIJc; elastic property; optical propem; ah initio calculation
1. Introduction Most age hardening aluminum alloys are limited to be used in the elevated temperature range, because of the rapid coarsening of the strengthening precipitates at high temperature"]. However, the addition of trace element Sc to aluminum alloys can substantially improve both their strength and ductility at elevated temperatures because of the formation of very fine coherent A1,Sc compounds[']. Thus, a lot of work has been carried out to study the effects of Sc on the microstructures and properties of the aluminum allo^[^-'], as well as, the structure of the A1,Sc corn pound^[^^^^^''^. It has been shown that AllSc is a stable L1, 1 and distributes along intermetallic compound. This compound has a very low density ( 3 . 0 3 ~03kg.m-')f'1, { 100) planed2] with a melting temperature of 1320°C ["I and very low coarsening ratesr2]. The electronic the structure of A13Sc has been calculated by using the first principle method, in the previous stlLdie~~'~-'~]. In this article, the elastic and optical properties are explored based on the DFT (density-fimctional theory), aiming to hrther understand other properties of the A1,Sc compound.
2. Calculation Method In the calculation, DFT'l4*"1is utilized based on the plane-wave method. In this approach, the correction 'Corresponding author. Tel. : +86 731 8877880. E-mail ruh1res.s : msong@mail. csu .edu. cn ( M . Song)
. 426
.
of the electronic density is considered - using exchange-correlation potentials parameterized according to the PBE (Perdew-Burke-Emzerhof) form, in the nonlocal model's generalized gradient approximation (GGA), where the main idea is to simplify the density hnctional problem by considering only the valence electrons. The calculation makes use of the ultrasoft pseudopotentials and an expansion of the electronic wave functions in plane waves with a kinetic energy cutoff of 290eV. The Brillouin zone is sampled with a [6 x 6 x 61 k-point mesh. This set of parameters assures a total energy convergence of 0.0 1eV per atom.
3. ResulcsandDiscussion A1,Sc is a binary stable L I Z intermetallic compound. The equilibrium lattice parameter a. is 0.4053393nm after ultrasoft pseudopotentials calcualtion, similar to the previous published data (0.4101nm by calculation and 0.4106nm by experiment)["'. The value of the equilibrium lattice parameter 0.4053393nm is very close to the lattice parameter of pure Al, with a value of 0.40496nm at 298KL"].Thus the A1 matrix can easily accommodate the elastic strain energy during the nucleation and growth of the A1,Sc compound, which exhibits a stronger coherent effect on the strength, compared to other A1J compounds (X= Zr, Cr, Er, etc.). To obtain the elastic constants of the A1,Sc compound with a cubic structure, the first principle calculation has been used by computing the stress tensor 6 for small strains (about 1/10 of the lattice parameter ao). It is well known that a cubic crystal has only three independent elastic constants (C,,, CI2,and C ), or only three independent compliance constants (SII,SI2,and S,). The bulk modulus (B,) and Poisson's ratio (u) can be determined through equations as follows B,=(C,,+2C,,)/3
(1)
The calculated elastic constants and previous experimental data[''' have been shown in Table 1. It can be seen that the calculated values are in reasonable agreement with the experimental data, except for ,C , where the result is smaller than the experimental counterpart. The mechanical stability in this cubic structure leads to the following restrictions on the elastic constants, CII - Clz> 0, C, > 0, CI1+ CI2> 0. From Table I , it can be seen that the calculated elastic constants obey these stability conditions, including the fact that C12must be smaller than C,,. The calculated elastic constants also obey the cubic stability condition that CI2
N=n+ik
where n is the refractive index and k is the absorption coefficient (or extinction coefficient). Table 1 Elastic constants calculated in this study with previous experimental data Data
C I I ,GPa
Cl2, GPa
C44rGPa
Bo, GPa
v
This work
181.00667
37.28050
38.86243
85.18922
0.1708
Experimental"'I
189
43
66
99.2
427 Fig. 1 shows the calculated refraction as a fbnction of energy (frequency) of A1,Sc compounds. It can be seen that the real part (refractive index) of the refraction has a strong peak at the position of low energy. This peak is accompanied by another strong peak at the imaginary part (absorption) of the refraction. As the intraband transitions in the far infrared (IR) region are not considered at all in this study, the strong peak is caused by the direct optical transitions between different bands. Detailed information about the optical transitions can be illustrated by checking the partial density of the states (PDOS). Fig.2 illustrates the calculated PDOS of the A1,Sc compound. It can be seen that the energy difference is about 20V between low s and p bands, whereas, it is more than 18eV between low p band and high s, p, and d bands. Thus, the strong peak is caused by the optical transitions between high s, p, and d bands. It can also be seen that a small absorption peak exists around 30eV. This small peak is caused by the optical transitions from high s, p, and d bands to the low 2p band. One should be careful of the calculated refraction, as the intraband transitions are not considered in this calculation. Normally, the intraband transitions are observed in metallic materials instead of insulators because electrons can only be excited into empty states because of the Pauli principle. It is common to evaluate the complex dielectric constant when calculating the optical properties. ) is,(w), where E is dielectric conThe frequency dependent complex dielectric constant, E ( W ) = E ~ ( W + stant, o is the angular frequency, e l is the real part and e2is the imaginary part, is known to describe the optical response of the medium at all phonon energies, E = tzw, E is energy and A is the Planck constant, using the formalism of Ehrenreich and C ~ h e n " ~The ]. 7, real and imaginary parts can be described as follows
6!l 5
e2(w)=2nk
(5)
Fig.3 shows the calculated dielectric function of the A13Sccompound. It can be seen that the curves in the dielectric function are similar to those in refraction. The imaginary (absorptive) part of the dielectric function has a strong peak at low energy and a small peak around 30eV. As described earlier, the strong 12 1
0
10
20
30
40
50
1
Energy, eV
Fig. 1 Calculated refraction of A1,Sc compound.
I
40
I
Small peak
-60
-50
-40
-30
-20
-10
0
10
Energy, eV
Fig.2 Calculated PDOS of AI$c compound.
20
0
10
20
30
40
50
60
Energy, eV
Fig.3 Calculated dielectric function of AI& compound.
. 428
.
peak is caused by the optical transitions between high s, p, and d bands. The small peak is because of the transition from high s, p, and d bands to low 2p band.
4. Conclusions Elastic and optical properties of the AI,Sc compound have been studied based on DFT. The calculation makes use of the ultrasoft pseudopotentials, and an expansion of electronic wavehnctions in plane waves. It has been shown that the elastic constants of A1,Sc compound, obtained by the calculations, satisfy the requirement of the mechanical stability in a cubic structure. The equilibrium lattice parameter of the A1,Sc compound is very close to the lattice parameter of pure Al, thus the A1,Sc compound exhibits a stronger coherent effect on the strength. It has also been shown that two absorptive peaks exist for the A1,Sc compound. The strong peak is caused by the optical transitions between high s, p, and d bands, whereas, the relative small peak is caused by the optical transitions from high s, p, and d bands to the low 2p band. Acknowledgements-This work wus supported by the Hunan Provinciul Natural Science Foundutinn of Chinu (No. 075531 17) and Chinese Postdoctoral Scientific Foundation (No. ,2OO7O410303).
REFERENCES 1 N. Blake and M.A. Hopkins, J. Mnter. Sci. 20 (1985) 2861. 2 R.W. Hyland, Melull. Trms. A 23 (1992) 1947. 3 C.B. Fuller, D.N. Seidman, and D.C. Dunand, Scr. Mater. 40 (1999) 691.
4 Z. Yin, Q. Lin, Y. Zhang, and F. Jiang, Mnter. Sci. Eng. A 280 (2000) 15 I . 5 R. Sawtell and C.L. Jensen, M e t d l . Trims. A 21 (1990) 42 I . 6 L.I. Kaygorodova and V.P. Domashnikov, Phys. Met. Metdl. 68 (1989) 160. 7 J. Hyung-Ho and F. Ahin-Itiroh, Mater. Sci. Eng. A 171 (1993) 15 I . 8 L.S. Kramer, W.T. Tack, and M.T. Femandes, A i h . Maier. Process. 152 (1997) 23. 9 Y. Harada and D.C. Dunand, A ctu Miuer. 48 (2000) 3477. 10 H. Okamoto, J Phme Equilibrin. 12 (1991) 612. 1 1 G. Bester and M. Fahnle, J. Phys.: Condens. M d t r r . 13 (2001) 11551. 12 J.H. Xu and A.J. Freeman, Phys. Rev. R 4 1 (1990) 12553. 13 M. Asta and V. Ozolins, Phys. Reu. B 64 (2001) 094104. 14 P. Hohenberg and W. Kohn, Phys. Rev. B 136 (1964) 864. 15 W. Kohn and L.J. Sham, Phys. Kei:. A 140(1965) 1133. 16 R.W. Jr. Hyland, M. Asta, S.M. Foiles, and C.L. Rohrer, A cta Mow. 46 ( 1 998) 3667. 17 Z.T. Wang and R.Z. Tian, Hwirlbook ofAluminum Alloys a d Their Fihriccltion (Changsha: Central South University of Technology Press, 1988) p. 152 (in Chinese). 18 R.W. Hyland and R.C. Stiffler, Scr. Metdl. Mater. 25 (1991) 473. 19 H. Ehrenreich and M.L. Cohen, Phys. Reo. 115 (1959) 786.