- Email: [email protected]
On the deformation potentials in CuInSe2 ternary semiconductors
002213697/a s3.00 + 0.00 0 1988 Pcrpml PIcsr pit
1. P&s. Chew. So&& Vol. 49, No. 4, pp. 391-393, 1988 Primed in Great Britain.
ON THE DEFORMATION POTENTIALS IN CuInSe, TERNARY SEMICONDUCTORS
c.
bJC6N
Centro de Estudios de Semiconductores, Departamento de Fisica, Facultad de Ciencias, Universidad de Los Andes, M&da, Venezuela (Received 10 April 1987; accepted 6 August 1987) Ah&me-From the analysis of the variation of the energy gap with temperature and pressure in the ternary compound CuInSe, the valence and conduction band deformation potentials are estimated to be 7.78 and 9.48 eV per unit dilation, respectively. It is found that both the valence and conduction band extrema move to higher energies on compression. It is also suggested that electron-electron scattering combined with polar opticaf modes has an important influence on the mobility in n-type CuIr&+ Keywork
Deformation potentials, scattering mechanisms, carrier mobility, energy gap, CuInSe,.
~RODU~ON
electron-lattice
The ternary compound semiconductor CuInSe,, which crystallizes in the chalcopyrite structure, appears to be a promising candidate for many technological applications and therefore has been receiving much attention in recent years [l-3]. In order to obtain the magnitude of the predominant scattering mechanisms in this compound, analyses of the variation of the carrier mobility with temperature, by using the Mathiessen’s approximation, have been reported by several authors [4-n. In this model the total mobility is calculated by the equation
P-‘=FI1;5
THEORY
The shift of the energy gap with temperature in a semiconductor is caused mainly by two contributing effects [S, 91. The most important is the
which is given by [S]
(dE,/dT), = - (8/9n)(3/4U)“’ x ( K~V~31~2~v2)(m~~~ + m,Cz>, (2)
(1)
where the p,s represent the mobilities of the charge carriers as influenced by different scattering mechanisms present in the crystal. However, discrepancies exist in the literature as to the individual contribution of these terms to the total mobility. For example, depending on the scattering mechanisms included in &he calculation, the value of the condution band deformation potential, a parameter that determines the strength of the acoustic mode scattering, varies from 11 to 55 eV [7]. In the present work, by analyzing the variation of the energy gap with temperature and pressure in CuInSe,, an attempt is made to estimate the valence and conduction band defo~ation potentials of this compound. From the study it is concluded that an additional scattering mechanism of the charge carriers should be considered in the mobility calculation for n-type CuInSe,.
interaction
where h4 and V are the mass and volume of the unit cell respectively, and v is the velocity of sound in the crystal. C,, and C, are the valence and conduction band deformation potentials. These are the rate of change of the valence and conduction bands energy extrema when the lattice is uniformly expanded or compressed HO, 1I]. The second contribution to the shift in the energy gap is due to the thermal expansion of the lattice. This is given by 1121 (dE,/dT), = f (3aJK) dE/dP = 2aL( f G, f CA
(3)
where aL and rc are the linear expansion coefficient and the compressibility of the crystal respectively. The alternate signs allow for all possible cases of the bottom of the conduction band and the top of the valence band moving either up or down with dilation. Comparison of the data of dE,/dT with the sum of expressions (2) and (3) would show which signs are to be used in eqn (3) 18, IO]. ANALYSIS OF THR ~P~~~
DATA
Measurements of the temperature dependence of the energy gap in CuInSe, have been reported by several authors [13,14]. Recently, Nakanishi et al.[ls] conclude, in agreement with Ref. [14], that this dependence is linear between around 100 and 300 K, and is given by dE,/dT = - 1.1 x lo-” eV/K. The pressure dependence of the energy gap in CuInSe, has also been stud&l, and it is established that the band-gap increases linearly with pressure [15]. Nakanishi et al. report that 391
392
C. RINC~N
dE,/dP = 2.8 x lo-” eV/Pa between normal pressure and 4 GPa. This value is in good agreement with dE,/dP = 2.6 x lo-” eV/Pa up to 7 GPa, obtained by Gonzalez (personal communication). The average linear expansion coefficient, between 200 and 300 K, and the compressibility of CuInSe, are xL = 8 x 10m6/K [16] and K = 2.3 x IO-“/Pa [15]. By using these values, we obtain from eqn (3) the thermal expansion contribution to the energy gap as (dE,/dT), = - 3(a,/K) dE,/dP = - 2.7 x lo-‘eV/K. The minus sign, as suggested in Ref. [15], is used in this equation. The electron-lattice contribution, (dE,/dT), = - 8.3 x IO-‘eV/K, is obtained by subtracting the thermal expansion contribution from the total value of dE,/dT. Putting these values into eqns (3), and by using for CuInSe, (2) and u = 2.18 x lo5 cm/s [7], we obtain 0.73 Ci + 0.09 Ca = 52.3 eV* +C,*C,=
- 1.7eV.
(4) (5)
Expression (5) is in good agreement with f C, f C, = - 2.1 eV, obtained by a theoretical analysis which takes into account the effect of the electron-phonon interaction in CuInSe, with its 24 vibrational modes [17, 181. In order to obtain the deformation potentials, as C,, < C,, we assume that + C, and - C, must be used in eqn (5). These opposite signs indicate that the valence and conduction band extrema move in the same direction on compression in CuInSe,. Under these conditions, from eqns (4) and (5) we find 7.78 and 9.48eV/unit dilation for C,, and C, respectively. DISCUSSION It is pointed out that the decrease in the band-gap of CuInSe,, as compared with its binary analog, is caused mainly by the p-d hybridization which produces a repulsive interaction between the p-, and d-like rlJ valence band states. As a result, the top of the valence band is pushed to higher energies [3]. It is suggested that, due to the reduction of the lattice constants, pressure should also lead to a stronger overlap between the T,,(p) and T,,(d) valence band states. This should also shift upwards the top of the valence band [19,20]. However, as observed experimentally [151,the band-gap of CuInSe2 increases with pressure. This leads us to conclude that the conduction band moves to higher energies with pressure and its relative upward shift with pressure is greater than that of the valence band. This is consistent with our result that these bands move in the same direction on compression and C, > C,,. This also should explain the small band-gap pressure coefficient observed in CuInSe* relative to its binary analog. These results are also consistent with those of Jaffe and Zunger [3]. They have pointed out that the excitation across the band-gap in Cu-ternaries couples states on the same
sublattice and it constitutes therefore an intra-atomic transition. This is stronger than the analogous interatomic transition in binary semiconductors and leads to an anomalously small band-gap pressure coefficient in Cu-III-VI, chalcopyrites [3]. The value of C,, obtained here is in excellent agreement with 7.5 and 8.0eV obtained by Wasim under the assumption that the dominant scattering mechanisms in p-type CuInSe, are due to ionized impurities, acoustic and non-polar optical modes [7]. On the other hand, if the scattering of holes by polar optical modes is included in the calculations, a slightly lower value of C,,, 6.5 eV is obtained from the mobility data. Therefore, our result suggests that holes are not predominantly scattered by polar optical modes in p-type CuInSe,. The conduction band deformation potential obtained in the present work is smaller than all the values reported in the literature. In fact, analysis of the mobility data of n-type CuInSe,, taking into account the scattering by ionized impurities, acoustic and polar optical modes, yields values which vary between 19.5 and 55 eV [6,7]. However, if the scattering of electrons by non-polar optical modes is included in the calculation, a relatively smaller value of C,, 13 eV instead of 19.5 eV, can be obtained. This is still higher than our calculated C, value of 9.48 eV. This suggests that an additional scattering mechanism of the charge carriers should be considered in the mobility calculations for n-type CuInSe,. It is pointed out that for n-type InSb, the electronelectron (e-e) scattering mechanism combined with the polar mode scattering can have an important influence on the mobility of this compound [21,22]. Bate et al. find that e-e scattering reduces the mobility of InSb by a factor of up to about 0.7 in the range of temperature where the polar-optical mode scattering is predominant. Thus, we can assume that e-e scattering also has an important influence on the mobility of n-type CuInSe, in the region where polar-optical mode scattering is predominant. If we suppose that this additional scattering mechanism, not considered in the calculations, reduces the total mobility of n-CuInSe, by about lo%, it is necessary to compensate this effect by increasing the contribution of the acoustic mode (p,) in fitting the experimental mobility data. Considering sample CIE,-1 of Ref. [5], in the temperature range where polar-optical mode scattering is predominant, we find that pa=N 4~. In this case, with p’ac- l/C: [7,8], we find that C, must be reduced by a factor of about 0.75. Thus, the value of C, obtained by including the non-polar scattering will be reduced from 13 eV to about 9.8eV. This is in good agreement with the value obtained in the present calculation. It may be mentioned that sample CIE,- 1 studied by Irie et al. has a relatively low electron concentration (n N 10’5-10’6cm-3 at room temperature) and for samples with higher electron concentration than the CIE,-1, one should expect a higher reduction of the
On the deformation potentials in CuInSe, ternary semiconductors
mobility by e-e scattering. This could perhaps explain large values of C, obtained in samples of high electron con~nt~tion. It can also be concluded that in p-type CuInSe2 hole-hole scattering does not play an important role in determining the total mobility of this compound.
Acknowledgements-This work was supported by El Consejo de Desarrollo Cientifico y Humanistico (C.D.C.H.) de la Universidad de Los Andes. The author also wishes to thank Dr S. M. Wasim for valuable discussions and MS C. Kauman for typing the manuscript.
RJBERENCES 1. Kazmerski L. L., Nuuvo Cim. D2, 2013 (1983). 2. Rincon C. and Bellabarba C.. P&s. _ Reu. 833, 7160 (1986). 3. Jaffe J. E. and Zunger A., Phys. Reu. B28,5822 (1983); B29, 1882 (1984). 4. Neumann H., Van Nam N., Hiibler H. J. and Kuhn G., Solid St. Commun. 25, 899 (1978). 5. Irie T.. Endo S. and Kimura S., &m. ._I.armi. _. Phvs. _ 18. 1303 (i979). 6. Wasim S. M. and Noguera A., Phys. Status Solidi A, 82, 553 (1984).
PC.S. 49/4-a
393
7. Wasim S. M., Solar Cells 16, 289 (1986). 8. Fan H. Y., Phys. Rev. 82, 900 (1951). 9. Schlilter M., Martinez G. and Cohen Marvin L., Pkys. Reu. BlZ, 650 (1975). 10. Bardeen J. and Shockley W., Phys. Rev. So,72 (1950). 11. Ziman J. M., Efectrons and Phonons, p. 175. Oxford University Press, Oxford (1960). 12. Moss T. S., Optical Properties of Semiconductors, pp. 43-45. Butte~orths, London (19.59). 13. H&rig W., Neumann H., Hlibler H. J. and Kuhn, G., Phys: Status Solidi B, 80, K21 (1977). 14. Rincbn C.. Gonzalez J. and Sanchez Perez G.. Phvs. Status Solidi B, 117, K123 (1983). 15. Nakanishi H., Endo S., Irie T. and Chang B. H., Ternary and ~u~t~ffry Corncobs (Edited by S. K. Deb. and A. Zunger), p. 99. Materials Research Society, Pittsburgh (1987). 16. Deus P., Neumann H., Kuhn G. and Hinze B., Phys. Status Solidi A, 80, 205 (1983). 17. Rincon C., Wasim S. M. and Neumann H., Solid St. Commun. 54, 629 (198.5). 18. Rin&n C. and Gonzalez J., S&r Cells 16, 357 (1986). 19. Jarayaman A., Narayanamurti V., Kasper H. M., Chin M. A. and Maines R. G., Phys. Rev. 814, 3516 (1976). 20. Jarayaman A., Dernier P. D., Kasper H. M. and Maines R. G., High Temp. High Pressures 9, 97 (1977). 21. Bate R. T, Baxter R. D., Reid F. J. and I&r A. C., J. Phvs. Chem. Soi&& 26. I205 119651. 22. Mathur P. C., Kataria N. D.; Jain S. and Sharma V., J. Phys. C9, L89 (1976).
1. P&s. Chew. So&& Vol. 49, No. 4, pp. 391-393, 1988 Primed in Great Britain.
ON THE DEFORMATION POTENTIALS IN CuInSe, TERNARY SEMICONDUCTORS
c.
bJC6N
Centro de Estudios de Semiconductores, Departamento de Fisica, Facultad de Ciencias, Universidad de Los Andes, M&da, Venezuela (Received 10 April 1987; accepted 6 August 1987) Ah&me-From the analysis of the variation of the energy gap with temperature and pressure in the ternary compound CuInSe, the valence and conduction band deformation potentials are estimated to be 7.78 and 9.48 eV per unit dilation, respectively. It is found that both the valence and conduction band extrema move to higher energies on compression. It is also suggested that electron-electron scattering combined with polar opticaf modes has an important influence on the mobility in n-type CuIr&+ Keywork
Deformation potentials, scattering mechanisms, carrier mobility, energy gap, CuInSe,.
~RODU~ON
electron-lattice
The ternary compound semiconductor CuInSe,, which crystallizes in the chalcopyrite structure, appears to be a promising candidate for many technological applications and therefore has been receiving much attention in recent years [l-3]. In order to obtain the magnitude of the predominant scattering mechanisms in this compound, analyses of the variation of the carrier mobility with temperature, by using the Mathiessen’s approximation, have been reported by several authors [4-n. In this model the total mobility is calculated by the equation
P-‘=FI1;5
THEORY
The shift of the energy gap with temperature in a semiconductor is caused mainly by two contributing effects [S, 91. The most important is the
which is given by [S]
(dE,/dT), = - (8/9n)(3/4U)“’ x ( K~V~31~2~v2)(m~~~ + m,Cz>, (2)
(1)
where the p,s represent the mobilities of the charge carriers as influenced by different scattering mechanisms present in the crystal. However, discrepancies exist in the literature as to the individual contribution of these terms to the total mobility. For example, depending on the scattering mechanisms included in &he calculation, the value of the condution band deformation potential, a parameter that determines the strength of the acoustic mode scattering, varies from 11 to 55 eV [7]. In the present work, by analyzing the variation of the energy gap with temperature and pressure in CuInSe,, an attempt is made to estimate the valence and conduction band defo~ation potentials of this compound. From the study it is concluded that an additional scattering mechanism of the charge carriers should be considered in the mobility calculation for n-type CuInSe,.
interaction
where h4 and V are the mass and volume of the unit cell respectively, and v is the velocity of sound in the crystal. C,, and C, are the valence and conduction band deformation potentials. These are the rate of change of the valence and conduction bands energy extrema when the lattice is uniformly expanded or compressed HO, 1I]. The second contribution to the shift in the energy gap is due to the thermal expansion of the lattice. This is given by 1121 (dE,/dT), = f (3aJK) dE/dP = 2aL( f G, f CA
(3)
where aL and rc are the linear expansion coefficient and the compressibility of the crystal respectively. The alternate signs allow for all possible cases of the bottom of the conduction band and the top of the valence band moving either up or down with dilation. Comparison of the data of dE,/dT with the sum of expressions (2) and (3) would show which signs are to be used in eqn (3) 18, IO]. ANALYSIS OF THR ~P~~~
DATA
Measurements of the temperature dependence of the energy gap in CuInSe, have been reported by several authors [13,14]. Recently, Nakanishi et al.[ls] conclude, in agreement with Ref. [14], that this dependence is linear between around 100 and 300 K, and is given by dE,/dT = - 1.1 x lo-” eV/K. The pressure dependence of the energy gap in CuInSe, has also been stud&l, and it is established that the band-gap increases linearly with pressure [15]. Nakanishi et al. report that 391
392
C. RINC~N
dE,/dP = 2.8 x lo-” eV/Pa between normal pressure and 4 GPa. This value is in good agreement with dE,/dP = 2.6 x lo-” eV/Pa up to 7 GPa, obtained by Gonzalez (personal communication). The average linear expansion coefficient, between 200 and 300 K, and the compressibility of CuInSe, are xL = 8 x 10m6/K [16] and K = 2.3 x IO-“/Pa [15]. By using these values, we obtain from eqn (3) the thermal expansion contribution to the energy gap as (dE,/dT), = - 3(a,/K) dE,/dP = - 2.7 x lo-‘eV/K. The minus sign, as suggested in Ref. [15], is used in this equation. The electron-lattice contribution, (dE,/dT), = - 8.3 x IO-‘eV/K, is obtained by subtracting the thermal expansion contribution from the total value of dE,/dT. Putting these values into eqns (3), and by using for CuInSe, (2) and u = 2.18 x lo5 cm/s [7], we obtain 0.73 Ci + 0.09 Ca = 52.3 eV* +C,*C,=
- 1.7eV.
(4) (5)
Expression (5) is in good agreement with f C, f C, = - 2.1 eV, obtained by a theoretical analysis which takes into account the effect of the electron-phonon interaction in CuInSe, with its 24 vibrational modes [17, 181. In order to obtain the deformation potentials, as C,, < C,, we assume that + C, and - C, must be used in eqn (5). These opposite signs indicate that the valence and conduction band extrema move in the same direction on compression in CuInSe,. Under these conditions, from eqns (4) and (5) we find 7.78 and 9.48eV/unit dilation for C,, and C, respectively. DISCUSSION It is pointed out that the decrease in the band-gap of CuInSe,, as compared with its binary analog, is caused mainly by the p-d hybridization which produces a repulsive interaction between the p-, and d-like rlJ valence band states. As a result, the top of the valence band is pushed to higher energies [3]. It is suggested that, due to the reduction of the lattice constants, pressure should also lead to a stronger overlap between the T,,(p) and T,,(d) valence band states. This should also shift upwards the top of the valence band [19,20]. However, as observed experimentally [151,the band-gap of CuInSe2 increases with pressure. This leads us to conclude that the conduction band moves to higher energies with pressure and its relative upward shift with pressure is greater than that of the valence band. This is consistent with our result that these bands move in the same direction on compression and C, > C,,. This also should explain the small band-gap pressure coefficient observed in CuInSe* relative to its binary analog. These results are also consistent with those of Jaffe and Zunger [3]. They have pointed out that the excitation across the band-gap in Cu-ternaries couples states on the same
sublattice and it constitutes therefore an intra-atomic transition. This is stronger than the analogous interatomic transition in binary semiconductors and leads to an anomalously small band-gap pressure coefficient in Cu-III-VI, chalcopyrites [3]. The value of C,, obtained here is in excellent agreement with 7.5 and 8.0eV obtained by Wasim under the assumption that the dominant scattering mechanisms in p-type CuInSe, are due to ionized impurities, acoustic and non-polar optical modes [7]. On the other hand, if the scattering of holes by polar optical modes is included in the calculations, a slightly lower value of C,,, 6.5 eV is obtained from the mobility data. Therefore, our result suggests that holes are not predominantly scattered by polar optical modes in p-type CuInSe,. The conduction band deformation potential obtained in the present work is smaller than all the values reported in the literature. In fact, analysis of the mobility data of n-type CuInSe,, taking into account the scattering by ionized impurities, acoustic and polar optical modes, yields values which vary between 19.5 and 55 eV [6,7]. However, if the scattering of electrons by non-polar optical modes is included in the calculation, a relatively smaller value of C,, 13 eV instead of 19.5 eV, can be obtained. This is still higher than our calculated C, value of 9.48 eV. This suggests that an additional scattering mechanism of the charge carriers should be considered in the mobility calculations for n-type CuInSe,. It is pointed out that for n-type InSb, the electronelectron (e-e) scattering mechanism combined with the polar mode scattering can have an important influence on the mobility of this compound [21,22]. Bate et al. find that e-e scattering reduces the mobility of InSb by a factor of up to about 0.7 in the range of temperature where the polar-optical mode scattering is predominant. Thus, we can assume that e-e scattering also has an important influence on the mobility of n-type CuInSe, in the region where polar-optical mode scattering is predominant. If we suppose that this additional scattering mechanism, not considered in the calculations, reduces the total mobility of n-CuInSe, by about lo%, it is necessary to compensate this effect by increasing the contribution of the acoustic mode (p,) in fitting the experimental mobility data. Considering sample CIE,-1 of Ref. [5], in the temperature range where polar-optical mode scattering is predominant, we find that pa=N 4~. In this case, with p’ac- l/C: [7,8], we find that C, must be reduced by a factor of about 0.75. Thus, the value of C, obtained by including the non-polar scattering will be reduced from 13 eV to about 9.8eV. This is in good agreement with the value obtained in the present calculation. It may be mentioned that sample CIE,- 1 studied by Irie et al. has a relatively low electron concentration (n N 10’5-10’6cm-3 at room temperature) and for samples with higher electron concentration than the CIE,-1, one should expect a higher reduction of the
On the deformation potentials in CuInSe, ternary semiconductors
mobility by e-e scattering. This could perhaps explain large values of C, obtained in samples of high electron con~nt~tion. It can also be concluded that in p-type CuInSe2 hole-hole scattering does not play an important role in determining the total mobility of this compound.
Acknowledgements-This work was supported by El Consejo de Desarrollo Cientifico y Humanistico (C.D.C.H.) de la Universidad de Los Andes. The author also wishes to thank Dr S. M. Wasim for valuable discussions and MS C. Kauman for typing the manuscript.
RJBERENCES 1. Kazmerski L. L., Nuuvo Cim. D2, 2013 (1983). 2. Rincon C. and Bellabarba C.. P&s. _ Reu. 833, 7160 (1986). 3. Jaffe J. E. and Zunger A., Phys. Reu. B28,5822 (1983); B29, 1882 (1984). 4. Neumann H., Van Nam N., Hiibler H. J. and Kuhn G., Solid St. Commun. 25, 899 (1978). 5. Irie T.. Endo S. and Kimura S., &m. ._I.armi. _. Phvs. _ 18. 1303 (i979). 6. Wasim S. M. and Noguera A., Phys. Status Solidi A, 82, 553 (1984).
PC.S. 49/4-a
393
7. Wasim S. M., Solar Cells 16, 289 (1986). 8. Fan H. Y., Phys. Rev. 82, 900 (1951). 9. Schlilter M., Martinez G. and Cohen Marvin L., Pkys. Reu. BlZ, 650 (1975). 10. Bardeen J. and Shockley W., Phys. Rev. So,72 (1950). 11. Ziman J. M., Efectrons and Phonons, p. 175. Oxford University Press, Oxford (1960). 12. Moss T. S., Optical Properties of Semiconductors, pp. 43-45. Butte~orths, London (19.59). 13. H&rig W., Neumann H., Hlibler H. J. and Kuhn, G., Phys: Status Solidi B, 80, K21 (1977). 14. Rincbn C.. Gonzalez J. and Sanchez Perez G.. Phvs. Status Solidi B, 117, K123 (1983). 15. Nakanishi H., Endo S., Irie T. and Chang B. H., Ternary and ~u~t~ffry Corncobs (Edited by S. K. Deb. and A. Zunger), p. 99. Materials Research Society, Pittsburgh (1987). 16. Deus P., Neumann H., Kuhn G. and Hinze B., Phys. Status Solidi A, 80, 205 (1983). 17. Rincon C., Wasim S. M. and Neumann H., Solid St. Commun. 54, 629 (198.5). 18. Rin&n C. and Gonzalez J., S&r Cells 16, 357 (1986). 19. Jarayaman A., Narayanamurti V., Kasper H. M., Chin M. A. and Maines R. G., Phys. Rev. 814, 3516 (1976). 20. Jarayaman A., Dernier P. D., Kasper H. M. and Maines R. G., High Temp. High Pressures 9, 97 (1977). 21. Bate R. T, Baxter R. D., Reid F. J. and I&r A. C., J. Phvs. Chem. Soi&& 26. I205 119651. 22. Mathur P. C., Kataria N. D.; Jain S. and Sharma V., J. Phys. C9, L89 (1976).