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Optical transitions near the band edge in bulk CuInxGa1−xSe2 from ellipsometric measurements
Materials Chemistry and Physics 70 (2001) 300–304
Optical transitions near the band edge in bulk CuInx Ga1−x Se2 from ellipsometric measurements C.A. Durante Rincón a,∗ , E. Hernández a , M.I. Alonso b , M. Garriga b , S.M. Wasim c , C. Rincón c , M. León d a
d
Departamento de F´ısica, Facultad Experimental de Ciencias, Universidad del Zulia, OPT Galerias, Aptdo. Postal 15645, Maracaibo, Venezuela b Institut de Ciència de Materials de Barcelona, CSIC, Campus de la UAB, 08193 Bellaterra, Spain c Centro de Estudios de Semiconductores, Facultad de Ciencias, Universidad de Los Andes, Mérida 5251, Venezuela Departamento de F´ısica Aplicada, Facultad de Ciencias, C-XII, Universidad Autónoma de Madrid, 28049 Madrid, Spain Received in revised form 16 July 2000; accepted 9 November 2000
Abstract From the analysis of the variation of optical absorption coefficient α with incident photon energy between 0.8 and 2.6 eV, obtained from ellipsometric data, the energy EG of the fundamental absorption edge and EG0 of the forbidden direct transition for CuInx Ga1−x Se2 alloys are estimated. The change in EG and the spin-orbit splitting ∆SO = EG0 −EG with the composition x can be represented by parabolic expression of the form EG (x) = EG (0) + ax + bx2 and ∆SO (x) = ∆SO (0) + a 0 x + b0 x 2 , respectively. b and b0 are called “bowing parameters”. Theoretical fit gives a = 0.875 eV, b = 0.198 eV, a 0 = 0.341 eV and b0 = −0.431 eV. The positive sign of b and negative sign of b0 are in agreement with the theoretical prediction of Wei and Zunger [Phys. Rev. B 39 (1989) 6279]. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Copper ternary alloys; Ellipsometry; Bandgap; Forbidden direct transition; Solar cells
1. Introduction The ternary compound semiconductor CuInSe2 (CIS) of the I–III–VI2 family, that has a bandgap EG around 1 eV, has emerged as a leading candidate for application in solar cells devices [1]. The other member of this family is CuGaSe2 (CGS) that has EG of about 1.65 eV [2,3]. For this reason, in recent years alloys of CuInx Ga1−x Se2 (CIGS) having bandgap between 1.4 and 1.5 eV with suitable values of x are also used in energy conversion devices. In fact, solar cells based on CIGS with an efficiency close to 20% have been reported [4,5]. Information about spectral dependence of optical parameters such as dielectric constants, refractive index, reflectivity and absorption coefficient are essential in the characterization of materials that are used in the fabrication of opto-electronic devices and also in the optimization of the efficiency of thin film solar cells. One of the non-destructive techniques that is employed lately for the characterization is spectroscopic ellipsometry [6]. This technique measures ∗ Corresponding author. E-mail address: [email protected] (C.A. Durante Rinc´on).
the change in the polarization state of light reflected from the sample’s surface. From the ellipsometric data, the complex dielectric function εs = ε1 −iε2 of the sample, where ε1 and ε2 are the real and imaginary parts, is calculated [7,8] using the isotropic two-phase model that involves the medium and the sample. ε s is related to complex refractive index N through the relation εs = N 2 = (n − ik)2 . In this relation, n is the real part of the refractive index and k the extinction coefficient. These parameters can be determined by solving the following equations: ε1 = n2 − k 2
(1)
ε2 = 2nk
(2)
The optical absorption coefficient α and the reflectivity R can be calculated through the relations: α=
4π k λ
(3)
R=
(n − 1)2 + k 2 (n + 1)2 + k 2
(4)
The variation of optical constants with photon energy hν, estimated from the analysis of ellipsometric data, of both
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bulk and thin film samples of CuInSe2 of the Cu–III–VI2 family [9,10] has been published. The bandgap EG and the energy ∆SO of the spin-orbit splitting of this semiconducting compound are determined. In the present work, we report on the spectral dependence, in the energy range between 0.8 and 2.6 eV, of the absorption coefficient at room temperature of CuInx Ga1−x Se2 alloys. The data were obtained using the ellipsometry technique. Following the approach of [10], the variation of EG and ∆SO with the alloy composition x is established and compared with those obtained earlier from other method.
2. Experimental methods Ingots of CuInx Ga1−x Se2 alloys with x = 0, 0.2, 0.4, 0.6, 0.8 and 1.0 were prepared by direct fusion of stoichiometric mixture of constituent elements of at least 5 N purity in evacuated quartz ampoules. These were previously coated with carbon. Details related to the crystal growth have been reported earlier [2,11]. The composition of each alloy was determined by atomic emission spectroscopy technique with an ICP Thermo Jarrel Polyscan 61E using HNO3 acid solution. As shown in Table 1, this was close to the ideal theoretical value of the starting composition. All ingots, as observed by a thermal probe, showed p-type conductivity. The unit cell parameters a and c for different composition x, obtained from the analysis of X-ray powder diffraction data using the method described in [2,11], are also given in Table 1. For comparison, values published earlier by Tinoco et al. [12] are also included. These data further confirm the chalcopyrite-type structure of CuInx Ga1−x Se2 alloys for the whole range of x from 0 to 1. Ellipsometric measurements were made, using a spectral ellipsometer SOPRA ES4G equipped with rotating polarizer, on samples that had surface area of about 4 mm2 and thickness of around 1 mm. To minimize the contribution to the experimental data of thin films of oxide that are formed on the sample surface due to oxidation and also to reduce the effect of surface roughness to which ellipsometry is very sensitive, the samples were polished to attain optical quality just before acquiring ellipsometric data. The focused light spot on the sample surface allowed to locate the adequate point to achieve the best response.
Fig. 1. Optical absorption coefficient of CuInx Ga1−x Se2 (x = 1, 0.8, 0.6, 0.4, 0.2 and 0) as a function of incident photon energy hν between 0.8 and 2.6 eV obtained from ellipsometry data.
3. Results and discussion The optical absorption coefficient α of CuInx Ga1−x Se2 alloys with different values of x, calculated from the ellipsometry data with Eqs. (1)–(3), is shown in Fig. 1 as a function of photon energy. It can be observed that α for CuInSe2 (x = 1) is about 104 cm−1 at roughly 0.8 eV, increases sharply with incident energy and is above 2 × 105 cm−1 at 2.6 eV. Such a high value around the fundamental absorption edge agrees very well with that reported in Fig. 1 of [13] and reflects good quality of the samples used in the present work. Similar behavior but with slightly lower values of α can also be noted in Fig. 1 near the band edge for other members of CIGS that have smaller x. It is well established [11,12] that CuInx Ga1−x Se2 alloys system have direct bandgap. Under this condition the variation of α with hν near the fundamental absorption edge is expected to follow the relation: α=
Ad (hν − EG )1/2 hν
(5)
Table 1 Stoichiometric composition and unit cell lattice parameters of CuInx Ga1−x Se2 alloysa
CuInSe2 CuIn0.8 Ga0.2 Se2 CuIn0.6 Ga0.4 Se2 CuIn0.4 Ga0.6 Se2 CuIn0.2 Ga0.8 Se2 CuGaSe2 a
Cu
In
Ga
Se
a (Å)
0.994 0.990 0.990 0.976 0.998 1.005
1.002 0.797 0.597 0.392 0.203 –
– 0.198 0.389 0.599 0.798 0.991
2.001 2.001 1.977 2.000 1.998 2.013
5.78315 5.74841 5.71275 5.68233 5.64608 5.61783
Values in the parenthesis indicate standard deviation in the last digit.
(13) (11) (16) (14) (12) (8)
a (Å) [12]
c (Å)
5.777 5.751 5.694 5.674 5.631 5.615
11.62226 11.50706 11.40058 11.25070 11.13238 11.02425
c (Å) [12] (38) (31) (48) (42) (31) (22)
11.590 11.507 11.370 11.230 11.104 11.019
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Fig. 2. A plot of (αhν)2 vs. hν of representative samples of CuInSe2 (䊐) and CuGaSe2 (䊊) of CuInx Ga1−x Se2 alloys. The bandgap EG is estimated from the extrapolation of the fitted lines to (αhν)2 = 0.
Fig. 3. Variation of the energy gap EG as a function of composition x in CuInx Ga1−x Se2 . The symbols (䊐) represent the present data and (䊊) those of [12]. The continuous line shows a polynomial fit to the present data with the parameters given in the text.
where the constant Ad , related to the transition probability, depends on the effective mass and the refractive index and EG , the bandgap, is the characteristic energy of the allowed direct transition. Using the method described in [14], EG is estimated from a plot of (αhν)2 vs. hν. This is shown in Fig. 2 for the two end members with x = 0 and 1, respectively. The values of 0.982 and 1.663 for CuInSe2 and CuGaSe2 , respectively, agree quite well with 0.955 and 1.620 eV reported by Tinoco et al. [12]. The bandgap of CIGS obtained by this method for all values of x together with their corresponding Ad are given in Table 2. For a comparative analysis, these data and those reported earlier [12] are also plotted in Fig. 3. They are fitted to a polynomial of the form:
a = −0.840 eV and b = 0.175 eV reported by Tinoco et al. [12]. The theoretical absorption coefficient α d near the bandgap, calculated from Eq. (5) with the parameters Ad and EG of Table 2, is found to be smaller than that obtained from the ellipsometry data for the CIGS alloys. This is shown in Fig. 4 for CuInSe2 and CuGaSe2 . This suggests the existence of an additional absorption process above the fundamental edge. It is found, as in the case of CuInSe2 samples of [10], that the difference α 0 = α − αd above the fundamental absorption in the CIGS alloys can also be expressed by the relation:
EG (x) = EG (0) + ax + bx2
where the constant A0 , like Ad , also depends on the refractive index and effective masses of electron and holes [15]. In Fig. 5 (α 0 hν)2/3 is plotted against hν for representative CuInSe2 and CuGaSe2 , the two end members of CuInx Ga1−x Se2 . From a linear fit to the data around the
(6)
where EG (0) = 1.662 eV, a = −0.875 eV and b = 0.198 eV. The standard deviation σ of the fit is 0.011 eV. These agree very well with the values of EG (0) = 1.620 eV,
α0 =
A0 (hν − EG0 )3/2 hν
(7)
Table 2 Values of the allowed and forbidden direct transition bandgaps EG and EG0 , optical parameters Ad and A0 related to this transitions and the spin-orbit splitting energy ∆SO = EG0 − EG in CuInx Ga1−x Se2 alloys
CuInSe2 CuIn0.8 Ga0.2 Se2 CuIn0.6 Ga0.4 Se2 CuIn0.4 Ga0.6 Se2 CuIn0.2 Ga0.8 Se2 CuGaSe2
EG (eV)
Ad (105 cm−1 eV1/2 )
EG0 (eV)
A0 (105 cm−1 eV−1/2 )
∆SO (eV)
0.982 1.099 1.196 1.352 1.491 1.663
1.066 1.738 1.814 2.723 3.033 3.182
1.265 1.451 1.605 1.791 1.918 2.021
0.798 1.793 2.645 2.592 2.921 3.480
0.283 0.352 0.409 0.439 0.427 0.358
C.A. Durante Rinc´on et al. / Materials Chemistry and Physics 70 (2001) 300–304
Fig. 4. Variation of the experimental optical absorption coefficient α exp (earlier called α) and the calculated absorption coefficient α d with energy hν for representative samples CuInSe2 (x = 1) and CuGaSe2 (x = 0) of CuInx Ga1−x Se2 alloys. α d was calculated from Eq. (5) using EG and Ad of Table 2.
second transition defined by Eq. (7), which is also referred in the literature as the forbidden direct transition [15], EG0 is estimated from the extrapolation of (α 0 hν)2/3 to zero. Values of EG0 and A0 thus obtained for the CIGS alloys for different values of x are summarized in Table 2. A tendency is observed, where Ad and A0 increase with the increase of Ga content. It is also found that the reflectivity R and
Fig. 5. A plot of (α 0 hν)2/3 vs. hν for the two end members of CuInx Ga1−x Se2 alloys with α 0 = αexp −αd . The forbidden direct bandgap EG0 is estimated from the extrapolation of the fitted lines to (α 0 hν)2/3 = 0.
303
refractive index n (details to be published elsewhere) in this system change very little with x near the fundamental absorption edge. This implies that the increase in Ad and A0 with Ga content is related mainly to the increase in the effective masses of the charge carriers in CuInx Ga1−x Se2 alloys. This is confirmed for the end members where me ∗ = 0.09me and mh ∗ = 0.73me for CuInSe2 are smaller than me ∗ = 0.14me and mh ∗ = 0.84me for CuGaSe2 [16]. As pointed out by Neumann [17], this type of forbidden direct transition in CuInSe2 is allowed by group theory. It is further asserted that the variation of α 0 with hν that follows Eq. (7) can only be explained if the energy dispersion of the split-off band, ascribed to the spin-orbit splitting ∆SO = 0.233 eV, contains a term linear in K⊥ . The presence of such a transition in the band structure of selenides and tellurides of other Cu-ternaries of the I–III–VI2 family has also been reported [18,19]. The values of ∆SO (x) = EG0 (x) − EG (x), calculated from EG0 (x) and EG (x) for different composition of x, are also given in Table 2. ∆SO = 0.283 eV for CuInSe2 is in reasonable agreement with 0.237 and 0.240 eV reported by Chichibu et al. [20] and Hidalgo et al. [10], respectively. However, ∆SO = 0.358 eV for CuGaSe2 is about 50% higher as compared to that reported in [20] which was obtained from optical absorption study. These results indicate that the assertion made about the presence of a term linear in K⊥ in the energy dispersion relation of the split-off band is also true for CuInx Ga1−x Se2 alloys. The variation of ∆SO with x, shown in Fig. 6, can be represented by the following parabolic expression: ∆SO (x) = ∆SO (0) + a 0 x + b0 (∆SO )x 2
(8)
Fig. 6. Variation of the spin-orbit splitting energy ∆SO with the composition x in CuInx Ga1−x Se2 . The continuous line represents a fit to the data with Eq. (8).
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where b0 (∆SO ) is the bowing parameter for the spin-orbit splitting. The theoretical fit gives a 0 = 0.341 eV and b0 (∆SO ) = −0.431 eV. The standard deviation σ for the fit is 0.012 eV. The negative sign of b0 (∆SO ) is in agreement with the analysis of the measurements reported in bulk ternary and epitaxial lattice-matched quaternary alloys [21]. This indicates that at Γ -point of the valence-band maximum, these materials exhibit an “upward concave” bowing (b0 (∆SO ) < 0). The model that explains b0 (∆SO ) < 0 in tetrahedrally bonded Ax B1−x C semiconducting alloys is due to Wei and Zunger [22]. Based on self-consistent electronic structure calculations, they have shown that while the bowing b(EG ) of the bandgap in these alloys should be positive, calculation predicts that b0 (∆SO ) should be negative. Depending on the crystalline structure of the alloys, b0 (∆SO ) is expected to vary between −0.01 and −0.33 eV. In the present case of CuInx Ga1−x Se2 , b0 (∆SO ) = −0.431 eV, although slightly higher than the upper theoretical limit of −0.33 eV, is smaller than −0.585 eV reported [23] for ZnSe1−x Tex alloys. The origin of the “upward concave” bowing of the spin-orbit splitting is attributed by these authors as due to the coupling between p-like wave functions at the valence-band maximum.
4. Conclusion From the variation of optical absorption coefficient α with incident photon energy hν, obtained from ellipsometric data, the direct energy bandgap EG is calculated for CuInx Ga1−x Se2 alloys. The magnitude of EG and its parabolic dependence on the composition x, where the bowing parameter b(EG ) is positive, are in good agreement with earlier works [11,12]. Additional absorption process above the fundamental edge EG is observed. The energy EG0 corresponding to the forbidden direct transition is estimated from the analysis of the residual absorption. The energy ∆SO (x) of the spin-orbit splitting, calculated from the difference between EG0 (x) and EG (x) is also found to have a parabolic dependence where the bowing parameter b0 (∆SO ) is negative. b(EG ) > 0 and b0 (∆SO ) < 0 are in agreement with the theoretical prediction of Wei and Zunger.
Acknowledgements This work was supported by grants from CONICIT (Contract No. G-97000670) and CONDES (Contract No. 01450-99). References [1] A. Rockett, R.W. Birkmire, J. Appl. Phys. 20 (1991) R81. [2] S.M. Wasim, C. Durante, C. Rincón, Mater. Lett. 28 (1996) 231. [3] J.H. Schön, J. Oestreich, O. Schenker, H. Riazi-Nijad, M. Klenk, N. Fabre, E. Arushanov, E. Bucher, Appl. Phys. Lett. 75 (1999) 2969. [4] M.A. Contreras, B. Egaas, K. Ramanathan, J. Hiltner, A. Swartzlander, F. Hasoon, R. Noufi, Prog. Photovolt. Res. Appl. 7 (1999) 311. [5] U. Rau, H.W. Schock, Appl. Phys. A 69 (1999) 13. [6] J.H. Ho, C.L. Lee, C.W. Yen, T.F. Lei, Solid State Electron. 30 (1987) 973. [7] R.M.A. Azzam, N.M. Bashara, Ellipsometry and Polarized Light, North-Holland, Amsterdam, 1976. [8] D.E. Aspnes, Optical Properties of Solids — New Developments, North-Holland, Amsterdam, 1976 (Chapter 15). [9] F.A. Abou-Elfoutouh, G.S. Horner, T.J. Coutts, M.W. Warlass, Solar Cells 30 (1991) 473. [10] M.L. Hidalgo, M. Lachab, A. Zouaoui, M. Alhamed, C. Linares, J.P. Peyrade, J. Galibert, Phys. Stat. Sol. (b) 200 (1997) 297. [11] C.A. Durante, S.M. Wasim, E. Hernandez, Cryst. Res. Technol. 31 (1996) 749. [12] T. Tinoco, C. Rincón, M. Quintero, G. Sanchez Perez, Phys. Stat. Sol. (a) 124 (1991) 427. [13] J.E. Jaffe, A. Zunger, Phys. Rev. B 29 (1984) 1882. [14] G. Marin, S.M. Wasim, G. Sanchez Perez, P. Bocaranda, A.E. Mora, J. Electron. Mater. 27 (1998) 1351. [15] J.I. Pankove, Optical Processes in Semiconductors, Dover, New York, 1971, 36 pp. [16] R. Marquez, C. Rincón, Mater. Lett. 40 (1999) 66. [17] H. Neumann, Solar Cells 16 (1987) 317. [18] W. Hörig, H. Neumann, B. Schumann, G. Kühn, Phys. Stat. Sol. (b) 85 (1978) K57. [19] H. Neumann, W. Hörig, E. Reccius, H. Sobota, B. Schumann, G. Kühn, Thin Solid Films 61 (1979) 13. [20] S. Chichibu, T. Mizutani, K. Murakami, T. Shioda, T. Kurafuji, H. Nakanishi, S. Niki, J. Fons, A. Yamada, J. Appl. Phys. 83 (1998) 3678. [21] P. Parayananthal, F.H. Pollak, Phys. Rev. B 28 (1983) 3632. [22] S.-H. Wei, A. Zunger, Phys. Rev. B 39 (1989) 6279. [23] A. Ebino, Y. Sato, T. Takuhashi, Phys. Rev. Lett. 32 (1974) 1366.
Optical transitions near the band edge in bulk CuInx Ga1−x Se2 from ellipsometric measurements C.A. Durante Rincón a,∗ , E. Hernández a , M.I. Alonso b , M. Garriga b , S.M. Wasim c , C. Rincón c , M. León d a
d
Departamento de F´ısica, Facultad Experimental de Ciencias, Universidad del Zulia, OPT Galerias, Aptdo. Postal 15645, Maracaibo, Venezuela b Institut de Ciència de Materials de Barcelona, CSIC, Campus de la UAB, 08193 Bellaterra, Spain c Centro de Estudios de Semiconductores, Facultad de Ciencias, Universidad de Los Andes, Mérida 5251, Venezuela Departamento de F´ısica Aplicada, Facultad de Ciencias, C-XII, Universidad Autónoma de Madrid, 28049 Madrid, Spain Received in revised form 16 July 2000; accepted 9 November 2000
Abstract From the analysis of the variation of optical absorption coefficient α with incident photon energy between 0.8 and 2.6 eV, obtained from ellipsometric data, the energy EG of the fundamental absorption edge and EG0 of the forbidden direct transition for CuInx Ga1−x Se2 alloys are estimated. The change in EG and the spin-orbit splitting ∆SO = EG0 −EG with the composition x can be represented by parabolic expression of the form EG (x) = EG (0) + ax + bx2 and ∆SO (x) = ∆SO (0) + a 0 x + b0 x 2 , respectively. b and b0 are called “bowing parameters”. Theoretical fit gives a = 0.875 eV, b = 0.198 eV, a 0 = 0.341 eV and b0 = −0.431 eV. The positive sign of b and negative sign of b0 are in agreement with the theoretical prediction of Wei and Zunger [Phys. Rev. B 39 (1989) 6279]. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Copper ternary alloys; Ellipsometry; Bandgap; Forbidden direct transition; Solar cells
1. Introduction The ternary compound semiconductor CuInSe2 (CIS) of the I–III–VI2 family, that has a bandgap EG around 1 eV, has emerged as a leading candidate for application in solar cells devices [1]. The other member of this family is CuGaSe2 (CGS) that has EG of about 1.65 eV [2,3]. For this reason, in recent years alloys of CuInx Ga1−x Se2 (CIGS) having bandgap between 1.4 and 1.5 eV with suitable values of x are also used in energy conversion devices. In fact, solar cells based on CIGS with an efficiency close to 20% have been reported [4,5]. Information about spectral dependence of optical parameters such as dielectric constants, refractive index, reflectivity and absorption coefficient are essential in the characterization of materials that are used in the fabrication of opto-electronic devices and also in the optimization of the efficiency of thin film solar cells. One of the non-destructive techniques that is employed lately for the characterization is spectroscopic ellipsometry [6]. This technique measures ∗ Corresponding author. E-mail address: [email protected] (C.A. Durante Rinc´on).
the change in the polarization state of light reflected from the sample’s surface. From the ellipsometric data, the complex dielectric function εs = ε1 −iε2 of the sample, where ε1 and ε2 are the real and imaginary parts, is calculated [7,8] using the isotropic two-phase model that involves the medium and the sample. ε s is related to complex refractive index N through the relation εs = N 2 = (n − ik)2 . In this relation, n is the real part of the refractive index and k the extinction coefficient. These parameters can be determined by solving the following equations: ε1 = n2 − k 2
(1)
ε2 = 2nk
(2)
The optical absorption coefficient α and the reflectivity R can be calculated through the relations: α=
4π k λ
(3)
R=
(n − 1)2 + k 2 (n + 1)2 + k 2
(4)
The variation of optical constants with photon energy hν, estimated from the analysis of ellipsometric data, of both
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301
bulk and thin film samples of CuInSe2 of the Cu–III–VI2 family [9,10] has been published. The bandgap EG and the energy ∆SO of the spin-orbit splitting of this semiconducting compound are determined. In the present work, we report on the spectral dependence, in the energy range between 0.8 and 2.6 eV, of the absorption coefficient at room temperature of CuInx Ga1−x Se2 alloys. The data were obtained using the ellipsometry technique. Following the approach of [10], the variation of EG and ∆SO with the alloy composition x is established and compared with those obtained earlier from other method.
2. Experimental methods Ingots of CuInx Ga1−x Se2 alloys with x = 0, 0.2, 0.4, 0.6, 0.8 and 1.0 were prepared by direct fusion of stoichiometric mixture of constituent elements of at least 5 N purity in evacuated quartz ampoules. These were previously coated with carbon. Details related to the crystal growth have been reported earlier [2,11]. The composition of each alloy was determined by atomic emission spectroscopy technique with an ICP Thermo Jarrel Polyscan 61E using HNO3 acid solution. As shown in Table 1, this was close to the ideal theoretical value of the starting composition. All ingots, as observed by a thermal probe, showed p-type conductivity. The unit cell parameters a and c for different composition x, obtained from the analysis of X-ray powder diffraction data using the method described in [2,11], are also given in Table 1. For comparison, values published earlier by Tinoco et al. [12] are also included. These data further confirm the chalcopyrite-type structure of CuInx Ga1−x Se2 alloys for the whole range of x from 0 to 1. Ellipsometric measurements were made, using a spectral ellipsometer SOPRA ES4G equipped with rotating polarizer, on samples that had surface area of about 4 mm2 and thickness of around 1 mm. To minimize the contribution to the experimental data of thin films of oxide that are formed on the sample surface due to oxidation and also to reduce the effect of surface roughness to which ellipsometry is very sensitive, the samples were polished to attain optical quality just before acquiring ellipsometric data. The focused light spot on the sample surface allowed to locate the adequate point to achieve the best response.
Fig. 1. Optical absorption coefficient of CuInx Ga1−x Se2 (x = 1, 0.8, 0.6, 0.4, 0.2 and 0) as a function of incident photon energy hν between 0.8 and 2.6 eV obtained from ellipsometry data.
3. Results and discussion The optical absorption coefficient α of CuInx Ga1−x Se2 alloys with different values of x, calculated from the ellipsometry data with Eqs. (1)–(3), is shown in Fig. 1 as a function of photon energy. It can be observed that α for CuInSe2 (x = 1) is about 104 cm−1 at roughly 0.8 eV, increases sharply with incident energy and is above 2 × 105 cm−1 at 2.6 eV. Such a high value around the fundamental absorption edge agrees very well with that reported in Fig. 1 of [13] and reflects good quality of the samples used in the present work. Similar behavior but with slightly lower values of α can also be noted in Fig. 1 near the band edge for other members of CIGS that have smaller x. It is well established [11,12] that CuInx Ga1−x Se2 alloys system have direct bandgap. Under this condition the variation of α with hν near the fundamental absorption edge is expected to follow the relation: α=
Ad (hν − EG )1/2 hν
(5)
Table 1 Stoichiometric composition and unit cell lattice parameters of CuInx Ga1−x Se2 alloysa
CuInSe2 CuIn0.8 Ga0.2 Se2 CuIn0.6 Ga0.4 Se2 CuIn0.4 Ga0.6 Se2 CuIn0.2 Ga0.8 Se2 CuGaSe2 a
Cu
In
Ga
Se
a (Å)
0.994 0.990 0.990 0.976 0.998 1.005
1.002 0.797 0.597 0.392 0.203 –
– 0.198 0.389 0.599 0.798 0.991
2.001 2.001 1.977 2.000 1.998 2.013
5.78315 5.74841 5.71275 5.68233 5.64608 5.61783
Values in the parenthesis indicate standard deviation in the last digit.
(13) (11) (16) (14) (12) (8)
a (Å) [12]
c (Å)
5.777 5.751 5.694 5.674 5.631 5.615
11.62226 11.50706 11.40058 11.25070 11.13238 11.02425
c (Å) [12] (38) (31) (48) (42) (31) (22)
11.590 11.507 11.370 11.230 11.104 11.019
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Fig. 2. A plot of (αhν)2 vs. hν of representative samples of CuInSe2 (䊐) and CuGaSe2 (䊊) of CuInx Ga1−x Se2 alloys. The bandgap EG is estimated from the extrapolation of the fitted lines to (αhν)2 = 0.
Fig. 3. Variation of the energy gap EG as a function of composition x in CuInx Ga1−x Se2 . The symbols (䊐) represent the present data and (䊊) those of [12]. The continuous line shows a polynomial fit to the present data with the parameters given in the text.
where the constant Ad , related to the transition probability, depends on the effective mass and the refractive index and EG , the bandgap, is the characteristic energy of the allowed direct transition. Using the method described in [14], EG is estimated from a plot of (αhν)2 vs. hν. This is shown in Fig. 2 for the two end members with x = 0 and 1, respectively. The values of 0.982 and 1.663 for CuInSe2 and CuGaSe2 , respectively, agree quite well with 0.955 and 1.620 eV reported by Tinoco et al. [12]. The bandgap of CIGS obtained by this method for all values of x together with their corresponding Ad are given in Table 2. For a comparative analysis, these data and those reported earlier [12] are also plotted in Fig. 3. They are fitted to a polynomial of the form:
a = −0.840 eV and b = 0.175 eV reported by Tinoco et al. [12]. The theoretical absorption coefficient α d near the bandgap, calculated from Eq. (5) with the parameters Ad and EG of Table 2, is found to be smaller than that obtained from the ellipsometry data for the CIGS alloys. This is shown in Fig. 4 for CuInSe2 and CuGaSe2 . This suggests the existence of an additional absorption process above the fundamental edge. It is found, as in the case of CuInSe2 samples of [10], that the difference α 0 = α − αd above the fundamental absorption in the CIGS alloys can also be expressed by the relation:
EG (x) = EG (0) + ax + bx2
where the constant A0 , like Ad , also depends on the refractive index and effective masses of electron and holes [15]. In Fig. 5 (α 0 hν)2/3 is plotted against hν for representative CuInSe2 and CuGaSe2 , the two end members of CuInx Ga1−x Se2 . From a linear fit to the data around the
(6)
where EG (0) = 1.662 eV, a = −0.875 eV and b = 0.198 eV. The standard deviation σ of the fit is 0.011 eV. These agree very well with the values of EG (0) = 1.620 eV,
α0 =
A0 (hν − EG0 )3/2 hν
(7)
Table 2 Values of the allowed and forbidden direct transition bandgaps EG and EG0 , optical parameters Ad and A0 related to this transitions and the spin-orbit splitting energy ∆SO = EG0 − EG in CuInx Ga1−x Se2 alloys
CuInSe2 CuIn0.8 Ga0.2 Se2 CuIn0.6 Ga0.4 Se2 CuIn0.4 Ga0.6 Se2 CuIn0.2 Ga0.8 Se2 CuGaSe2
EG (eV)
Ad (105 cm−1 eV1/2 )
EG0 (eV)
A0 (105 cm−1 eV−1/2 )
∆SO (eV)
0.982 1.099 1.196 1.352 1.491 1.663
1.066 1.738 1.814 2.723 3.033 3.182
1.265 1.451 1.605 1.791 1.918 2.021
0.798 1.793 2.645 2.592 2.921 3.480
0.283 0.352 0.409 0.439 0.427 0.358
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Fig. 4. Variation of the experimental optical absorption coefficient α exp (earlier called α) and the calculated absorption coefficient α d with energy hν for representative samples CuInSe2 (x = 1) and CuGaSe2 (x = 0) of CuInx Ga1−x Se2 alloys. α d was calculated from Eq. (5) using EG and Ad of Table 2.
second transition defined by Eq. (7), which is also referred in the literature as the forbidden direct transition [15], EG0 is estimated from the extrapolation of (α 0 hν)2/3 to zero. Values of EG0 and A0 thus obtained for the CIGS alloys for different values of x are summarized in Table 2. A tendency is observed, where Ad and A0 increase with the increase of Ga content. It is also found that the reflectivity R and
Fig. 5. A plot of (α 0 hν)2/3 vs. hν for the two end members of CuInx Ga1−x Se2 alloys with α 0 = αexp −αd . The forbidden direct bandgap EG0 is estimated from the extrapolation of the fitted lines to (α 0 hν)2/3 = 0.
303
refractive index n (details to be published elsewhere) in this system change very little with x near the fundamental absorption edge. This implies that the increase in Ad and A0 with Ga content is related mainly to the increase in the effective masses of the charge carriers in CuInx Ga1−x Se2 alloys. This is confirmed for the end members where me ∗ = 0.09me and mh ∗ = 0.73me for CuInSe2 are smaller than me ∗ = 0.14me and mh ∗ = 0.84me for CuGaSe2 [16]. As pointed out by Neumann [17], this type of forbidden direct transition in CuInSe2 is allowed by group theory. It is further asserted that the variation of α 0 with hν that follows Eq. (7) can only be explained if the energy dispersion of the split-off band, ascribed to the spin-orbit splitting ∆SO = 0.233 eV, contains a term linear in K⊥ . The presence of such a transition in the band structure of selenides and tellurides of other Cu-ternaries of the I–III–VI2 family has also been reported [18,19]. The values of ∆SO (x) = EG0 (x) − EG (x), calculated from EG0 (x) and EG (x) for different composition of x, are also given in Table 2. ∆SO = 0.283 eV for CuInSe2 is in reasonable agreement with 0.237 and 0.240 eV reported by Chichibu et al. [20] and Hidalgo et al. [10], respectively. However, ∆SO = 0.358 eV for CuGaSe2 is about 50% higher as compared to that reported in [20] which was obtained from optical absorption study. These results indicate that the assertion made about the presence of a term linear in K⊥ in the energy dispersion relation of the split-off band is also true for CuInx Ga1−x Se2 alloys. The variation of ∆SO with x, shown in Fig. 6, can be represented by the following parabolic expression: ∆SO (x) = ∆SO (0) + a 0 x + b0 (∆SO )x 2
(8)
Fig. 6. Variation of the spin-orbit splitting energy ∆SO with the composition x in CuInx Ga1−x Se2 . The continuous line represents a fit to the data with Eq. (8).
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where b0 (∆SO ) is the bowing parameter for the spin-orbit splitting. The theoretical fit gives a 0 = 0.341 eV and b0 (∆SO ) = −0.431 eV. The standard deviation σ for the fit is 0.012 eV. The negative sign of b0 (∆SO ) is in agreement with the analysis of the measurements reported in bulk ternary and epitaxial lattice-matched quaternary alloys [21]. This indicates that at Γ -point of the valence-band maximum, these materials exhibit an “upward concave” bowing (b0 (∆SO ) < 0). The model that explains b0 (∆SO ) < 0 in tetrahedrally bonded Ax B1−x C semiconducting alloys is due to Wei and Zunger [22]. Based on self-consistent electronic structure calculations, they have shown that while the bowing b(EG ) of the bandgap in these alloys should be positive, calculation predicts that b0 (∆SO ) should be negative. Depending on the crystalline structure of the alloys, b0 (∆SO ) is expected to vary between −0.01 and −0.33 eV. In the present case of CuInx Ga1−x Se2 , b0 (∆SO ) = −0.431 eV, although slightly higher than the upper theoretical limit of −0.33 eV, is smaller than −0.585 eV reported [23] for ZnSe1−x Tex alloys. The origin of the “upward concave” bowing of the spin-orbit splitting is attributed by these authors as due to the coupling between p-like wave functions at the valence-band maximum.
4. Conclusion From the variation of optical absorption coefficient α with incident photon energy hν, obtained from ellipsometric data, the direct energy bandgap EG is calculated for CuInx Ga1−x Se2 alloys. The magnitude of EG and its parabolic dependence on the composition x, where the bowing parameter b(EG ) is positive, are in good agreement with earlier works [11,12]. Additional absorption process above the fundamental edge EG is observed. The energy EG0 corresponding to the forbidden direct transition is estimated from the analysis of the residual absorption. The energy ∆SO (x) of the spin-orbit splitting, calculated from the difference between EG0 (x) and EG (x) is also found to have a parabolic dependence where the bowing parameter b0 (∆SO ) is negative. b(EG ) > 0 and b0 (∆SO ) < 0 are in agreement with the theoretical prediction of Wei and Zunger.
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