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Lattice dynamics of the chalcopyrite and defect stannite phases in the Cu–(In, Ga)–Se system
Journal of Crystal Growth 237–239 (2002) 2014–2018
Lattice dynamics of the chalcopyrite and defect stannite phases in the Cu–(In, Ga)–Se system Shigetaka Nomura*, Saburo Endo Department of Electrical Engineering, Faculty of Engineering, Science University of Tokyo, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
Abstract Raman spectra observed for bulk crystals of defect stannite phases in the Cu–(In, Ga)–Se system are analyzed mainly by the dispersion curve calculations with a superposed lattice model, assuming four fundamental structures. The calculations are performed on the basis of the Keating model, solving dynamical matrix of 24 24. Vacancy induced modifications on force interactions between nearest atoms are also discussed with effective bond overlap population and effective charges determined by the DV-Xa calculations for unit cell clusters. Apparently, invariable Raman profiles with respect to the In to Ga molar ratios through the alloy systems are explained by the superposed lattice model. r 2002 Elsevier Science B.V. All rights reserved. PACS: 63.20.Dj; 78.30.j Keywords: A1. Computer simulation; A1. Crystal structure; A1. Defects; A1. Dispersion curve calculation; A1. DV-Xa calculation; A1. Raman spectra; B1. Alloys; B2. Semiconducting ternary compounds
1. Introduction The defect stannite phases in the Cu–(In, Ga)–Se system have been known to have a non-stoichiometric structure that cannot be represented by a simple unit cell. Only the averaged structure was known to belong to a space group I4% 2m from the convergent electron beam diffraction study [1], categorized to the ‘defect stannite’. The authors have investigated these materials on the structural properties in view of the lattice dynamics [2,3]. In this work, Raman spectra
observed on these materials are analyzed by the dispersion curve calculations assuming superposed lattice model where the defect stannite structure consists of four kinds of fundamental structures in atomic occupation. The bond properties are also discussed with the effective charges and effective bond overlap populations estimated by the molecular orbital calculations using the DV-Xa method for a unit cell cluster.
2. Experimental procedure *Corresponding author. Tel.: +81-3-3260-4272x3397; fax: +81-3-5261-4805. E-mail address: [email protected] (S. Nomura).
Bulk crystals in the alloy systems of the Cu(In1xGax)3Se5 were prepared by the normal
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 2 2 4 6 - 1
S. Nomura, S. Endo / Journal of Crystal Growth 237–239 (2002) 2014–2018
freezing method. Single phase formations of these crystals were confirmed by the X-ray diffraction analysis, and the compositions were analyzed by the energy dispersive X-ray microanalysis (EDX). Raman spectra measurements were performed for these crystals in air at room temperature with the Ar+ ion laser as an excitation light source.
3. Calculations Dispersion curve calculations of the chalcopyrite and the defect stannite phases in the Cu– (In, Ga)–Se system were made by solving the respective dynamical matrices of 24 24 ranks which consist of force constants, atomic masses, tetragonal distortion parameter c/a and anion displacement parameters. Phonon branches were calculated between the main symmetrical points and assigned in reference to the group theory. From the factor group analysis, the 24 eigen modes are expressed as 1G1 þ 2G2 þ 3G3 þ 4G4 þ 7G5 for the chalcopyrite type and 2G1 þ G2 þ 2G3 þ 5G4 þ 7G5 for the stannite one, where only the G2 mode is silent, or both Raman and infrared inactive, and only the G5 is doubly degenerated. Effective charges and effective bond overlap populations for the constituent elements were estimated by the molecular orbital calculations with the DV-Xa method using SCAT [4] on a unit
2015
cell cluster for each crystal structure concerned here. Defect induced modifications on the bond properties are discussed.
4. Results and discussion Fig. 1 illustrates unit cells of the tetragonal chalcopyrite (I4% 2d) and the possible four types of (defect) stannite structures (I4% 2m) in the Cu– (In, Ga)–Se system. For the stannite structure, unoccupied site is limited to the 2b (or 2a, both of them are equivalent) site, as structural types (a) or (b) in Fig. 1. For example, the combination of 60%, 20% and 20% of the structural types of (a), (b) and (d), respectively, satisfies the composition of Cu(In, Ga)3Se5. Fig. 2 shows a lineup of X-ray diffraction profiles for bulk crystals of the Cu(In1xGax)3Se5 system. All these profiles satisfy the tetragonal symmetry, showing characteristic peaks of the defect stannite structure such as (1 1 0), (2 0 2) and (1 1 4). Lattice constants are very close to the corresponding chalcopyrite structure: a ¼ 0:5757; c ¼ 1:1534 nm for CuIn3Se5 with a ¼ 0:5782; c ¼ 1:1620 nm for CuInSe2, and a ¼ 0:5503; c ¼ 1:0974 nm for CuGa3Se5 with a ¼ 0:5596; c ¼ 1:1003 nm for CuGaSe2. The composition rates of CuIn3Se5 and CuGa3Se5 were Cu:In:Se=15.4:29.7:55.3 and Cu:Ga:Se=13.5:33.4:53.1, respectively. There was a
Fig. 1. Unit cells of the tetragonal chalcopyrite (I4% 2d) and the possible four types (different in atomic occupation at 2a and 2b sites) of stannite structures (I4% 2m) in the Cu–(In, Ga)–Se system, where V denotes vacancies.
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S. Nomura, S. Endo / Journal of Crystal Growth 237–239 (2002) 2014–2018
Fig. 2. A lineup of X-ray diffraction profiles of the Cu(In1xGax)3Se5 system.
Fig. 3. Non-polarized Raman spectra of the Cu(In1xGax)3Se5 system.
definite trend of Cu-rich deviation in composition from the starting ratios. Fig. 3 shows the typical non-polarized Raman spectra for some crystals of the Cu(In1xGax)3Se5 system. Raman profiles are very close to each other without peak shift at the most intense one around 160 cm1 through the system. This peak was assigned as the G1 mode from the polarized Raman spectra measurement for CuIn3Se5 [2]. However, detailed assignment for all irreducible modes has not been attained experimentally because of the structural complexity, or breaking rules observed. Regarding calculations, at first, we evaluated force constants a (bond stretching) and b (bond bending) for the chalcopyrite CuInSe2 and CuGaSe2 by fitting experimental Raman data [5,6] with our dynamical matrix. Table 1 lists the estimated force constants. For both compounds, assuming the first nearest Se–Se interaction in addition to the nearest Cu–Se and In(Ga)–Se ones improves the fitness especially for the G1 frequencies. Second, we calculated dispersion curves for the chalcopyrite and the stannite four structures in the
Table 1 Evaluated force constants for the chalcopyrite CuInSe2 and CuGaSe2 Compound
Bond
Force const. (N/m)
CuInSe2
Cu–Se
a b a b a
25.28 2.30 41.91 1.32 1.25
a b a b a
26.51 0.38 45.71 3.60 1.48
In–Se Se–Se CuGaSe2
Cu–Se Ga–Se Se–Se
Cu–(In, Ga)–Se system using force constants evaluated above. Application of the bond force constants of the chalcopyrite to the stannite is an approximation. In order to check the difference of bond properties between them, we also estimated effective charges of the constituent elements and effective bond overlap populations for these materials. Table 2 lists the results. The parameter
S. Nomura, S. Endo / Journal of Crystal Growth 237–239 (2002) 2014–2018
2017
Table 2 Effective bond overlap population ecov ; effective charges ecat and eSe ; and Coulomb effect on force constant kC for the Cu–Se, In–Se and Ga–Se bonds in the respective lattices Bond
Lattice
ecov
ecat
eSe
KC (N/m)
Cu–Se
CuInSe2 Cu–In–Se(a) Cu–In–Se(c) Cu–In–Se(d) CuGaSe2 Cu–Ga–Se(a) Cu–Ga–Se(c) Cu–Ga–Se(d)
0.40 0.41 0.42 0.39 0.41 0.42 0.40 0.42
0.25 0.21 0.34 0.39 0.36 0.28 0.34 0.43
0.06 0.01 0.07 0.13 0.06 0.05 0.07 0.02
0.46 0.06 0.86 1.61 0.66 0.43 0.84 0.33
In–Se
CuInSe2 Cu–In-Se(a) Cu–In–Se(b) Cu–In–Se(c) Cu–In–Se(d)
0.38 0.38 0.53 0.29 0.38
0.18 0.18 0.12 0.07 0.20
0.06 0.01 0.08 0.07 0.13
0.27 0.04 0.46 0.14 0.66
Ga–Se
CuGaSe2 Cu–Ga–Se(a) Cu–Ga–Se(b) Cu–Ga–Se(c) Cu–Ga–Se(d)
0.33 0.38 0.54 0.32 0.34
0.01 0.38 0.31 0.11 0.07
0.06 0.05 0.22 0.07 0.02
0.01 0.58 2.99 0.27 0.05
Fig. 4. Calculated dispersion curves of the chalcopyrite (Ch.) and the stannite (St.) phases for the Cu–In–Se and Cu–Ga–Se systems. For the stannite phases, dispersion curves calculated assuming possible four structural types are superposed.
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kC is defined by dfC qq kC ¼ ¼ c a3 ; 2p r dr
S. Nomura, S. Endo / Journal of Crystal Growth 237–239 (2002) 2014–2018
ð1Þ
where fC is Coulomb force between the nearest cation and anion with the effective charges qc and qa ; respectively, and r the distance. Calculated results reveal that the Coulomb effects on force constants are trivial in this material system except for the Cu–Se bond in the Cu–In–Se(d) and the Ga–Se bond in the Cu–Ga–Se(b). The effective bond overlap population ecov is a parameter expressing covalency of bonds. One can see that the values of ecov for the Cu–Se bonds are almost constant around 0.41 in all cases, and those for In(Ga)–Se bonds shift a little except for the type (b) without Cu atoms in lattices. Our assumption of the force constants of In(Ga)–Se bonds for the type (b) must be too small. However, the direct relation between force constants and ecov has not been clarified at present. Although the In(Ga)–Se bonds are tighter than the Cu–Se bond (see Table 1), those ecov are rather small. Force constants are considered to be determined not only by ecov but also by the total valence electrons of bonds, i.e. seven for the Cu–Se and nine for the In– Se, accordingly. Fig. 4 shows the calculated dispersion curves between the main symmetry points (notated after Ref. [7]) of the chalcopyrite (Ch.) and the stannite (St.) phases for the Cu–In–Se and Cu–Ga–Se systems. For the stannite phases, calculations on the fundamental four structures (see Fig. 1) are superposed here. The branch formations are very similar to each other between the chalcopyrite and the stannite structures. However, the branches
form the bands for the stannite. In the higher frequency range, the band is rather dispersive for the Cu–Ga–Se in comparison with the Cu–In–Se. On the other hand, the middle frequency bands resemble each other between the two systems especially at the G point. This is the reason for the resemblance in their Raman spectra. The possible irreducible modes in the middle frequency range are limited to G5 ½X5u þ G1 ½X1 þ G1 ½W1 þ G5 ½W3 þ G4 ½X3 ; where the silent G2 ½X1 mode for the chalcopyrite transforms to the detectable G1 ½X1 for the stannite.
5. Conclusions The Similarity in Raman profiles through the defect stannite Cu–(In, Ga)–Se system was explained by the dispersion curve calculations assuming a superposed structural model.
References [1] T. Hanada, A. Yamana, Y. Nakamura, O. Nittono, T. Wada, Technical Digest International PVSEC-9, Miyazaki, Japan, 1996, p. 595. [2] S. Nomura, S. O’uchi, S. Endo, Jpn. Appl. Phys. 36 (1997) L1075. [3] S. Nomura, S. O’uchi, S. Endo, Inst. Phys. Conf. Ser. 152 (1998) 649. [4] H. Adachi, M. Tsukada, C. Satoko, J. Phys. Soc. Jpn. 45 (1978) 875. [5] H. Tanino, T. Maeda, H. Fujikake, H. Nakanishi, S. Endo, T. Irie, Phys. Rev. B 45 (23) (1992) 13323. [6] C. Rincon, F.J. Ramirez, J. Appl. Phys. 72 (9) (1992) 4321. [7] S. Limpijumnong, W.R.L. Lambrecht, B. Segall, Phys. Rev. B 60 (11) (1999) 8087.
Lattice dynamics of the chalcopyrite and defect stannite phases in the Cu–(In, Ga)–Se system Shigetaka Nomura*, Saburo Endo Department of Electrical Engineering, Faculty of Engineering, Science University of Tokyo, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
Abstract Raman spectra observed for bulk crystals of defect stannite phases in the Cu–(In, Ga)–Se system are analyzed mainly by the dispersion curve calculations with a superposed lattice model, assuming four fundamental structures. The calculations are performed on the basis of the Keating model, solving dynamical matrix of 24 24. Vacancy induced modifications on force interactions between nearest atoms are also discussed with effective bond overlap population and effective charges determined by the DV-Xa calculations for unit cell clusters. Apparently, invariable Raman profiles with respect to the In to Ga molar ratios through the alloy systems are explained by the superposed lattice model. r 2002 Elsevier Science B.V. All rights reserved. PACS: 63.20.Dj; 78.30.j Keywords: A1. Computer simulation; A1. Crystal structure; A1. Defects; A1. Dispersion curve calculation; A1. DV-Xa calculation; A1. Raman spectra; B1. Alloys; B2. Semiconducting ternary compounds
1. Introduction The defect stannite phases in the Cu–(In, Ga)–Se system have been known to have a non-stoichiometric structure that cannot be represented by a simple unit cell. Only the averaged structure was known to belong to a space group I4% 2m from the convergent electron beam diffraction study [1], categorized to the ‘defect stannite’. The authors have investigated these materials on the structural properties in view of the lattice dynamics [2,3]. In this work, Raman spectra
observed on these materials are analyzed by the dispersion curve calculations assuming superposed lattice model where the defect stannite structure consists of four kinds of fundamental structures in atomic occupation. The bond properties are also discussed with the effective charges and effective bond overlap populations estimated by the molecular orbital calculations using the DV-Xa method for a unit cell cluster.
2. Experimental procedure *Corresponding author. Tel.: +81-3-3260-4272x3397; fax: +81-3-5261-4805. E-mail address: [email protected] (S. Nomura).
Bulk crystals in the alloy systems of the Cu(In1xGax)3Se5 were prepared by the normal
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 2 2 4 6 - 1
S. Nomura, S. Endo / Journal of Crystal Growth 237–239 (2002) 2014–2018
freezing method. Single phase formations of these crystals were confirmed by the X-ray diffraction analysis, and the compositions were analyzed by the energy dispersive X-ray microanalysis (EDX). Raman spectra measurements were performed for these crystals in air at room temperature with the Ar+ ion laser as an excitation light source.
3. Calculations Dispersion curve calculations of the chalcopyrite and the defect stannite phases in the Cu– (In, Ga)–Se system were made by solving the respective dynamical matrices of 24 24 ranks which consist of force constants, atomic masses, tetragonal distortion parameter c/a and anion displacement parameters. Phonon branches were calculated between the main symmetrical points and assigned in reference to the group theory. From the factor group analysis, the 24 eigen modes are expressed as 1G1 þ 2G2 þ 3G3 þ 4G4 þ 7G5 for the chalcopyrite type and 2G1 þ G2 þ 2G3 þ 5G4 þ 7G5 for the stannite one, where only the G2 mode is silent, or both Raman and infrared inactive, and only the G5 is doubly degenerated. Effective charges and effective bond overlap populations for the constituent elements were estimated by the molecular orbital calculations with the DV-Xa method using SCAT [4] on a unit
2015
cell cluster for each crystal structure concerned here. Defect induced modifications on the bond properties are discussed.
4. Results and discussion Fig. 1 illustrates unit cells of the tetragonal chalcopyrite (I4% 2d) and the possible four types of (defect) stannite structures (I4% 2m) in the Cu– (In, Ga)–Se system. For the stannite structure, unoccupied site is limited to the 2b (or 2a, both of them are equivalent) site, as structural types (a) or (b) in Fig. 1. For example, the combination of 60%, 20% and 20% of the structural types of (a), (b) and (d), respectively, satisfies the composition of Cu(In, Ga)3Se5. Fig. 2 shows a lineup of X-ray diffraction profiles for bulk crystals of the Cu(In1xGax)3Se5 system. All these profiles satisfy the tetragonal symmetry, showing characteristic peaks of the defect stannite structure such as (1 1 0), (2 0 2) and (1 1 4). Lattice constants are very close to the corresponding chalcopyrite structure: a ¼ 0:5757; c ¼ 1:1534 nm for CuIn3Se5 with a ¼ 0:5782; c ¼ 1:1620 nm for CuInSe2, and a ¼ 0:5503; c ¼ 1:0974 nm for CuGa3Se5 with a ¼ 0:5596; c ¼ 1:1003 nm for CuGaSe2. The composition rates of CuIn3Se5 and CuGa3Se5 were Cu:In:Se=15.4:29.7:55.3 and Cu:Ga:Se=13.5:33.4:53.1, respectively. There was a
Fig. 1. Unit cells of the tetragonal chalcopyrite (I4% 2d) and the possible four types (different in atomic occupation at 2a and 2b sites) of stannite structures (I4% 2m) in the Cu–(In, Ga)–Se system, where V denotes vacancies.
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S. Nomura, S. Endo / Journal of Crystal Growth 237–239 (2002) 2014–2018
Fig. 2. A lineup of X-ray diffraction profiles of the Cu(In1xGax)3Se5 system.
Fig. 3. Non-polarized Raman spectra of the Cu(In1xGax)3Se5 system.
definite trend of Cu-rich deviation in composition from the starting ratios. Fig. 3 shows the typical non-polarized Raman spectra for some crystals of the Cu(In1xGax)3Se5 system. Raman profiles are very close to each other without peak shift at the most intense one around 160 cm1 through the system. This peak was assigned as the G1 mode from the polarized Raman spectra measurement for CuIn3Se5 [2]. However, detailed assignment for all irreducible modes has not been attained experimentally because of the structural complexity, or breaking rules observed. Regarding calculations, at first, we evaluated force constants a (bond stretching) and b (bond bending) for the chalcopyrite CuInSe2 and CuGaSe2 by fitting experimental Raman data [5,6] with our dynamical matrix. Table 1 lists the estimated force constants. For both compounds, assuming the first nearest Se–Se interaction in addition to the nearest Cu–Se and In(Ga)–Se ones improves the fitness especially for the G1 frequencies. Second, we calculated dispersion curves for the chalcopyrite and the stannite four structures in the
Table 1 Evaluated force constants for the chalcopyrite CuInSe2 and CuGaSe2 Compound
Bond
Force const. (N/m)
CuInSe2
Cu–Se
a b a b a
25.28 2.30 41.91 1.32 1.25
a b a b a
26.51 0.38 45.71 3.60 1.48
In–Se Se–Se CuGaSe2
Cu–Se Ga–Se Se–Se
Cu–(In, Ga)–Se system using force constants evaluated above. Application of the bond force constants of the chalcopyrite to the stannite is an approximation. In order to check the difference of bond properties between them, we also estimated effective charges of the constituent elements and effective bond overlap populations for these materials. Table 2 lists the results. The parameter
S. Nomura, S. Endo / Journal of Crystal Growth 237–239 (2002) 2014–2018
2017
Table 2 Effective bond overlap population ecov ; effective charges ecat and eSe ; and Coulomb effect on force constant kC for the Cu–Se, In–Se and Ga–Se bonds in the respective lattices Bond
Lattice
ecov
ecat
eSe
KC (N/m)
Cu–Se
CuInSe2 Cu–In–Se(a) Cu–In–Se(c) Cu–In–Se(d) CuGaSe2 Cu–Ga–Se(a) Cu–Ga–Se(c) Cu–Ga–Se(d)
0.40 0.41 0.42 0.39 0.41 0.42 0.40 0.42
0.25 0.21 0.34 0.39 0.36 0.28 0.34 0.43
0.06 0.01 0.07 0.13 0.06 0.05 0.07 0.02
0.46 0.06 0.86 1.61 0.66 0.43 0.84 0.33
In–Se
CuInSe2 Cu–In-Se(a) Cu–In–Se(b) Cu–In–Se(c) Cu–In–Se(d)
0.38 0.38 0.53 0.29 0.38
0.18 0.18 0.12 0.07 0.20
0.06 0.01 0.08 0.07 0.13
0.27 0.04 0.46 0.14 0.66
Ga–Se
CuGaSe2 Cu–Ga–Se(a) Cu–Ga–Se(b) Cu–Ga–Se(c) Cu–Ga–Se(d)
0.33 0.38 0.54 0.32 0.34
0.01 0.38 0.31 0.11 0.07
0.06 0.05 0.22 0.07 0.02
0.01 0.58 2.99 0.27 0.05
Fig. 4. Calculated dispersion curves of the chalcopyrite (Ch.) and the stannite (St.) phases for the Cu–In–Se and Cu–Ga–Se systems. For the stannite phases, dispersion curves calculated assuming possible four structural types are superposed.
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kC is defined by dfC qq kC ¼ ¼ c a3 ; 2p r dr
S. Nomura, S. Endo / Journal of Crystal Growth 237–239 (2002) 2014–2018
ð1Þ
where fC is Coulomb force between the nearest cation and anion with the effective charges qc and qa ; respectively, and r the distance. Calculated results reveal that the Coulomb effects on force constants are trivial in this material system except for the Cu–Se bond in the Cu–In–Se(d) and the Ga–Se bond in the Cu–Ga–Se(b). The effective bond overlap population ecov is a parameter expressing covalency of bonds. One can see that the values of ecov for the Cu–Se bonds are almost constant around 0.41 in all cases, and those for In(Ga)–Se bonds shift a little except for the type (b) without Cu atoms in lattices. Our assumption of the force constants of In(Ga)–Se bonds for the type (b) must be too small. However, the direct relation between force constants and ecov has not been clarified at present. Although the In(Ga)–Se bonds are tighter than the Cu–Se bond (see Table 1), those ecov are rather small. Force constants are considered to be determined not only by ecov but also by the total valence electrons of bonds, i.e. seven for the Cu–Se and nine for the In– Se, accordingly. Fig. 4 shows the calculated dispersion curves between the main symmetry points (notated after Ref. [7]) of the chalcopyrite (Ch.) and the stannite (St.) phases for the Cu–In–Se and Cu–Ga–Se systems. For the stannite phases, calculations on the fundamental four structures (see Fig. 1) are superposed here. The branch formations are very similar to each other between the chalcopyrite and the stannite structures. However, the branches
form the bands for the stannite. In the higher frequency range, the band is rather dispersive for the Cu–Ga–Se in comparison with the Cu–In–Se. On the other hand, the middle frequency bands resemble each other between the two systems especially at the G point. This is the reason for the resemblance in their Raman spectra. The possible irreducible modes in the middle frequency range are limited to G5 ½X5u þ G1 ½X1 þ G1 ½W1 þ G5 ½W3 þ G4 ½X3 ; where the silent G2 ½X1 mode for the chalcopyrite transforms to the detectable G1 ½X1 for the stannite.
5. Conclusions The Similarity in Raman profiles through the defect stannite Cu–(In, Ga)–Se system was explained by the dispersion curve calculations assuming a superposed structural model.
References [1] T. Hanada, A. Yamana, Y. Nakamura, O. Nittono, T. Wada, Technical Digest International PVSEC-9, Miyazaki, Japan, 1996, p. 595. [2] S. Nomura, S. O’uchi, S. Endo, Jpn. Appl. Phys. 36 (1997) L1075. [3] S. Nomura, S. O’uchi, S. Endo, Inst. Phys. Conf. Ser. 152 (1998) 649. [4] H. Adachi, M. Tsukada, C. Satoko, J. Phys. Soc. Jpn. 45 (1978) 875. [5] H. Tanino, T. Maeda, H. Fujikake, H. Nakanishi, S. Endo, T. Irie, Phys. Rev. B 45 (23) (1992) 13323. [6] C. Rincon, F.J. Ramirez, J. Appl. Phys. 72 (9) (1992) 4321. [7] S. Limpijumnong, W.R.L. Lambrecht, B. Segall, Phys. Rev. B 60 (11) (1999) 8087.