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Unified carrier density approximation for non-parabolic and highly degenerate HgCdTe semiconductors covering SWIR, MWIR and LWIR bands
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Infrared Physics & Technology 51 (2008) 259–262 www.elsevier.com/locate/infrared
Unified carrier density approximation for non-parabolic and highly degenerate HgCdTe semiconductors covering SWIR, MWIR and LWIR bands Sudha Gupta *, R.K. Bhan, V. Dhar Solid State Physics Laboratory, Lucknow Road, Delhi 110054, India Received 24 April 2007 Available online 27 July 2007
Abstract We propose a unified approximate analytical expression for the carrier density of HgCdTe semiconductors that have non-parabolic energy bands and are highly degenerate. The proposed expression is without any adjustable parameter. It can be applied to HgCdTe for the case where electron densities are very high and the material is strongly degenerate, e.g., a highly accumulated surface due to passivant-induced negative fixed charge density in an n-HgCdTe photoconductor device. The proposed expression is simultaneously valid for SWIR, MWIR and LWIR bands. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Carrier density; Narrow gap semiconductor; Degenerate; Non-parabolic energy bands; HgCdTe
1. Introduction The traditional parabolic Boltzmann’s approximation of carrier statistics is not applicable to the majority of narrow gap semiconductors (NGS), for example HgCdTe, due to a non-quadratic band structure. Boltzmann’s approximation also fails when the carriers are degenerate i.e. (Ef EC)/ kT > 0. To estimate electron density, n in these NGS one has to use the Fermi Dirac integral valid for non-parabolic bands. The detailed numerical tabulation of this integral was done by Bebb and Ratliff [1] and simple analytical approximations for a variety of special cases were given by them. However, in the case of strong degeneracy i.e. the normalized Fermi energy, / > 10 was not treated by them because it is difficult to handle. It was stated by the authors that this case is not of much use [1]. However, it is well known that infrared (IR) photoconductive detectors based on HgCdTe material use anodic oxide as a passivant, which yields positive fixed charges (Qf) of the order of *
Corresponding author. E-mail address: [email protected] (S. Gupta).
1350-4495/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.infrared.2007.07.008
1012 cm2 at the surface of n-HgCdTe [2]. This makes the surface highly n+ with carrier density of the order of 1017–1018 cm3. As a result, the normalized Fermi energy, /, at 77 K is very high at around 40–50 [3,4]. Hence, it is desirable to have an approximation of the Fermi integral for / > 10 for the case of passivated HgCdTe detectors. Bhan and Dhar presented a detailed model for the calculation of Qf including degeneracy and non-parabolicity involving the use of very high values of / up to 50 [5]. However, the error in the approximate expressions of n used by these authors was very high (100% near / 30). Therefore, more general and accurate expressions of n are desirable for such calculations. Recently, detailed analysis of various approximations available in the literature were reviewed by Bhan and Dhar [6]. Additionally, in Ref. [6], the authors presented two simple approximations for carrier density, n, applicable to MWIR and LWIR bands separately. In the present paper we propose a single analytical approximation applicable to SWIR, MWIR and LWIR bands simultaneously. It retains the basic advantages as mentioned in Ref. [5], besides being applicable to a larger range of wavelengths.
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This will be useful for modeling the moderate and highly degenerate surfaces in n-HgCdTe semiconductor devices. 2. Results and discussion The carrier density, n, including band non-parabolicity, valid for both NGS and wide band semiconductors is given by [1,7,8]: Z 2N C 1 e1=2 ð1 þ aeÞ1=2 ð1 þ 2aeÞde ð1Þ n ¼ pffiffiffi 1 þ expðe /Þ p 0 where e = (E EC)/kT is the normalized energy. The above integral is not analytically solvable. Its various approximations for some specific situations were given in Ref. [1]. In the above relation, a is the coefficient of nonparabolicity that can be calculated for the conduction band from the k Æ p model [8] as follows: a¼
1 2 ð1 m Þ eg
ð2Þ
where eg = (EC EV)/kT is the normalized band gap and m* = me/m0, me being the electron effective mass and m0 the free electron rest mass. There are only two basic material parameters in Eqs. (1) and (2) namely, eg and me. In the present analysis, me is calculated following Weiler’s relation (Eq. (5) of Ref. [9]) taking band gap dependence into consideration as follows: me 2 1 ¼ 1 þ 2F þ 0:33Ep þ ð3Þ EG ðEG þ DÞ m0
Eq. (1) for 10 < / < 50. The advantage of Eq. (4) is that it is in the form useful for efficient device modeling because numerical integration has been avoided without the addition of any adjustable parameters. Further, if one wants to calculate the Fermi energy for a given normalized carrier density, it can be used directly without any iterations. As will be shown, this equation is valid for kc in the range of 1–17 lm. Fig. 1 shows the comparison for the plot of n/NC vs. / using Eqs. (1) and (4) for SWIR, MWIR and LWIR bands. From Fig. 1 it is seen that agreement between our proposed approximation and Bebb’s numerical solution is very good for all the cases. The quantitative error will be presented later. Furthermore, it can be seen from this figure that our approximation shows excellent agreement for / > 10 i.e. for moderately and strongly degenerate cases. Compared to approximations given in Refs. [5,6] the proposed new relation (Eq. (4)) tries to cover larger range of wavelengths with additional advantages of being a single unified relation. Fig. 2 shows the plot of n/NC vs. kc for Eqs. (1) and (4) for / taking values of 10, 20, 30, 40, and 50. It can be seen from this figure that comparison is reasonable for / > 10 in
where F = 0.8, Ep = 19 eV, EG is the band gap in eV and D = 1 eV. The temperature T has been assumed to be 77 K. 2.1. Single approximation for SWIR, MWIR and LWIR case Following Eqs. (4) and (7) given in Ref. [6], in this section we propose a new modified version of the Fermi energy, /, for applications in SWIR (1–2.5 lm), MWIR (3–5 lm) and LWIR (8–14 lm). This has been obtained by further tuning the power of the exponents, as suggested in Ref. [6] and completely eliminating the additive constant. The new modified relation is given below: "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 0:0045ð1m Þ2 6:143a þ 0:01ð1 m Þ2 0:29 a / ¼ log ðn=N C Þ 2 ð1 m Þ B0 0:0002ð1m Þ2 0:45þ a þ 0:86B1 ð150n=N C Þ
Fig. 1. Comparison of the present approximation with the Bebb’s solution of the Fermi integral for typical SWIR, MWIR and LWIR cases.
ð4Þ 2
Additionally, for a < 0.5, B0 = 1 + 3.75a + 3.218a 2.461a3 and the factor B1 = (1/B30 )(0.3245 + 5.115a + 3.5166a2 2.3243 a3) [8]. The value of n (or n/NC) from Eq. (4) is estimated by using iterative procedure and the values calculated are compared with those from Bebb’s numerical solution of
Fig. 2. Comparison of the present approximation with the Bebb’s solution of the Fermi integral as a function of cut-off wavelength (range 1–17 lm) for / taking values between 10 and 50.
S. Gupta et al. / Infrared Physics & Technology 51 (2008) 259–262
the range 1 lm < k < 17 lm. However, at lower kc’s there is increasing disagreement between the Bebb’s numerical solution and our relation, particularly for very low (<10) and very high values of / (>50). This is because we have tuned our empirical fitting about the centre of the SWIR, MWIR and LWIR range (1–17 lm). This is the range often used by IR detectors based on HgCdTe material and covers all the practical cases. In Fig. 3 we show the percentage error of our approximation to the Bebb’s numerical solution as a function of cut-off wavelength. This figure shows that for / = 10, 20, 30, 40, and 50, the respective average errors are 10.3%, 6.3%, 3.0%, 6.5% and 11% in the cut-off wavelength region of 1–17 lm. This error is much lower than in Ref. [5] and its validity is for a larger range than in Ref. [6]. For the case of weak degeneracy (/ 6 10), there are many approximations that give reasonably accurate values as discussed in Ref. [6]. It may be stated here that there is a scarcity of approximations for strongly degenerate cases (/ > 30) [6]. We have tried to tune our approximations specifically for this range of / with increased wavelength range. In this regard, it may be seen from this figure that error for / > 30 shows oscillatory behavior and decreases beyond the range of 1–17 lm. Hence our approximation for this case is valid even for wider range of wavelengths. In general, the average percentage error for our approximation varies between 3% and 11% for the 1–17 lm wavelength range. The corresponding standard deviation of the percentage error varies from 2.6% to 6.3%. At this point, it may be pointed out that for the majority of numerical solution methods, when the upper limit of integration is infinity (as in the present case, Eq. (1)), then it is replaced by a suitable finite number and the accuracy of the solution depends upon the adjustment and optimization of this number. This number changes with the value of kc in the present case. For illustration, we have chosen the most widely used Gauss’s 15 point quadrature method to evaluate the integral (Eq. (1)).
261
Fig. 4. The comparison of Gauss quadrature (15 points) method with Bebb’s results for two different values of upper limit of integration being equal to 5eg and 3eg.
The results are depicted in Fig. 4. It can be seen that for higher kc (>8 lm) an upper integration limit of 5eg gives a good accuracy whereas the same value is not adequate for lower kc (<8 lm). The error w.r.t. Bebb’s calculations is as high as 46% for / = 10 at 2 lm (see symbol n in the figure). However, if one wants to improve the fitting in the lower cut-off wavelength region, then the upper integration limit should be <3eg. However, the results in that case degrade at higher values of / (>30) (see symbol in the figure), particularly for higher kc. The maximum error being 68% for / = 50 at 17 lm. In short there is no unique upper integration limit valid for all wavelengths. Thus, for modeling where evaluation of the Fermi integral is only a small part of the calculation, this is impractical. On the other hand, our proposed Eq. (4) may be used as such. In the case of PC detectors, our approximation could be used to calculate the areal surface electron density in cm2 used in Eq. (15) of Ref. [10] (Qf in that work). Furthermore, our approximation can be used to model the capacitance–voltage characteristics of a metal-semiconductor–insulator devices in the case of large gate voltages that involve the use of very high values of / and to calculate space charge density as a function of the Fermi energy (see Fig. 3 of Ref. [7]). In the case of PV detectors, the approximation can be used to estimate the carrier density in the heavily doped donor region of pn+ diodes. 3. Conclusions
Fig. 3. Percentage error vs. cut-off wavelength in calculating n/NC using our proposed approximation.
In summary, we have proposed an analytical approximation of the Fermi integral (valid for 10 < / < 50) to estimate the carrier density in an NGS, including the nonparabolicity of the energy bands and the degeneracy of the carriers. This approximation is useful for efficient modeling of IR detectors based on n-HgCdTe material in the SWIR, MWIR and LWIR range.
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Acknowledgement The authors are thankful to the Director, SSPL for his kind permission to publish this work. References [1] H.B. Bebb, C.R. Ratliff, J. Appl. Phys. 42 (1971) 3189. [2] P.C. Catagnus, C.T. Baker, US Patent No. 3, 977018, 1976; Y. Nemirovsky, I. Kidron, Solid State Electron. 22 (1979) 831; B.K. Janousek, M.J. Danghorty, R.B. Schoolar, in: G. Lucovsky, S.T. Pantellides, F.L. Gleener (Eds.), The Physics of MOS Insulators, Pergamon, New York, 1980, p. 217.
[3] P.E. Peterson, in: R.K. Willardson, A.C. Beer (Eds.), Semiconductors and Semimetals, vol. 18, Academic Press, 1981, p. 145. [4] J.F. Siliquini, L. Faraone, Semicond. Sci. Technol. 12 (1997) 1010. [5] R.K. Bhan, V. Dhar, Semicond. Sci. Technol. 18 (2003) 1043. [6] R.K. Bhan, V. Dhar, Semicond. Sci. Technol. 19 (2004) 413. [7] V. Ariel-Altschul, E. Finkman, G. Bahir, IEEE. Trans. Electron. Dev. ED-39 (1992) 1312. [8] V. Altschul, E. Finkman, Appl. Phys. Lett. 58 (1991) 942. [9] M.H. Wieler, in: R.K. Willardson, A.C. Beer (Eds.), Semiconductors and Semimetals, vol. 16, Academic Press, New York, 1981, pp. 119– 191. [10] I. Bloom, Y. Nemirovsky, Solid State Electron. 31 (1998) 17.
Infrared Physics & Technology 51 (2008) 259–262 www.elsevier.com/locate/infrared
Unified carrier density approximation for non-parabolic and highly degenerate HgCdTe semiconductors covering SWIR, MWIR and LWIR bands Sudha Gupta *, R.K. Bhan, V. Dhar Solid State Physics Laboratory, Lucknow Road, Delhi 110054, India Received 24 April 2007 Available online 27 July 2007
Abstract We propose a unified approximate analytical expression for the carrier density of HgCdTe semiconductors that have non-parabolic energy bands and are highly degenerate. The proposed expression is without any adjustable parameter. It can be applied to HgCdTe for the case where electron densities are very high and the material is strongly degenerate, e.g., a highly accumulated surface due to passivant-induced negative fixed charge density in an n-HgCdTe photoconductor device. The proposed expression is simultaneously valid for SWIR, MWIR and LWIR bands. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Carrier density; Narrow gap semiconductor; Degenerate; Non-parabolic energy bands; HgCdTe
1. Introduction The traditional parabolic Boltzmann’s approximation of carrier statistics is not applicable to the majority of narrow gap semiconductors (NGS), for example HgCdTe, due to a non-quadratic band structure. Boltzmann’s approximation also fails when the carriers are degenerate i.e. (Ef EC)/ kT > 0. To estimate electron density, n in these NGS one has to use the Fermi Dirac integral valid for non-parabolic bands. The detailed numerical tabulation of this integral was done by Bebb and Ratliff [1] and simple analytical approximations for a variety of special cases were given by them. However, in the case of strong degeneracy i.e. the normalized Fermi energy, / > 10 was not treated by them because it is difficult to handle. It was stated by the authors that this case is not of much use [1]. However, it is well known that infrared (IR) photoconductive detectors based on HgCdTe material use anodic oxide as a passivant, which yields positive fixed charges (Qf) of the order of *
Corresponding author. E-mail address: [email protected] (S. Gupta).
1350-4495/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.infrared.2007.07.008
1012 cm2 at the surface of n-HgCdTe [2]. This makes the surface highly n+ with carrier density of the order of 1017–1018 cm3. As a result, the normalized Fermi energy, /, at 77 K is very high at around 40–50 [3,4]. Hence, it is desirable to have an approximation of the Fermi integral for / > 10 for the case of passivated HgCdTe detectors. Bhan and Dhar presented a detailed model for the calculation of Qf including degeneracy and non-parabolicity involving the use of very high values of / up to 50 [5]. However, the error in the approximate expressions of n used by these authors was very high (100% near / 30). Therefore, more general and accurate expressions of n are desirable for such calculations. Recently, detailed analysis of various approximations available in the literature were reviewed by Bhan and Dhar [6]. Additionally, in Ref. [6], the authors presented two simple approximations for carrier density, n, applicable to MWIR and LWIR bands separately. In the present paper we propose a single analytical approximation applicable to SWIR, MWIR and LWIR bands simultaneously. It retains the basic advantages as mentioned in Ref. [5], besides being applicable to a larger range of wavelengths.
260
S. Gupta et al. / Infrared Physics & Technology 51 (2008) 259–262
This will be useful for modeling the moderate and highly degenerate surfaces in n-HgCdTe semiconductor devices. 2. Results and discussion The carrier density, n, including band non-parabolicity, valid for both NGS and wide band semiconductors is given by [1,7,8]: Z 2N C 1 e1=2 ð1 þ aeÞ1=2 ð1 þ 2aeÞde ð1Þ n ¼ pffiffiffi 1 þ expðe /Þ p 0 where e = (E EC)/kT is the normalized energy. The above integral is not analytically solvable. Its various approximations for some specific situations were given in Ref. [1]. In the above relation, a is the coefficient of nonparabolicity that can be calculated for the conduction band from the k Æ p model [8] as follows: a¼
1 2 ð1 m Þ eg
ð2Þ
where eg = (EC EV)/kT is the normalized band gap and m* = me/m0, me being the electron effective mass and m0 the free electron rest mass. There are only two basic material parameters in Eqs. (1) and (2) namely, eg and me. In the present analysis, me is calculated following Weiler’s relation (Eq. (5) of Ref. [9]) taking band gap dependence into consideration as follows: me 2 1 ¼ 1 þ 2F þ 0:33Ep þ ð3Þ EG ðEG þ DÞ m0
Eq. (1) for 10 < / < 50. The advantage of Eq. (4) is that it is in the form useful for efficient device modeling because numerical integration has been avoided without the addition of any adjustable parameters. Further, if one wants to calculate the Fermi energy for a given normalized carrier density, it can be used directly without any iterations. As will be shown, this equation is valid for kc in the range of 1–17 lm. Fig. 1 shows the comparison for the plot of n/NC vs. / using Eqs. (1) and (4) for SWIR, MWIR and LWIR bands. From Fig. 1 it is seen that agreement between our proposed approximation and Bebb’s numerical solution is very good for all the cases. The quantitative error will be presented later. Furthermore, it can be seen from this figure that our approximation shows excellent agreement for / > 10 i.e. for moderately and strongly degenerate cases. Compared to approximations given in Refs. [5,6] the proposed new relation (Eq. (4)) tries to cover larger range of wavelengths with additional advantages of being a single unified relation. Fig. 2 shows the plot of n/NC vs. kc for Eqs. (1) and (4) for / taking values of 10, 20, 30, 40, and 50. It can be seen from this figure that comparison is reasonable for / > 10 in
where F = 0.8, Ep = 19 eV, EG is the band gap in eV and D = 1 eV. The temperature T has been assumed to be 77 K. 2.1. Single approximation for SWIR, MWIR and LWIR case Following Eqs. (4) and (7) given in Ref. [6], in this section we propose a new modified version of the Fermi energy, /, for applications in SWIR (1–2.5 lm), MWIR (3–5 lm) and LWIR (8–14 lm). This has been obtained by further tuning the power of the exponents, as suggested in Ref. [6] and completely eliminating the additive constant. The new modified relation is given below: "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 0:0045ð1m Þ2 6:143a þ 0:01ð1 m Þ2 0:29 a / ¼ log ðn=N C Þ 2 ð1 m Þ B0 0:0002ð1m Þ2 0:45þ a þ 0:86B1 ð150n=N C Þ
Fig. 1. Comparison of the present approximation with the Bebb’s solution of the Fermi integral for typical SWIR, MWIR and LWIR cases.
ð4Þ 2
Additionally, for a < 0.5, B0 = 1 + 3.75a + 3.218a 2.461a3 and the factor B1 = (1/B30 )(0.3245 + 5.115a + 3.5166a2 2.3243 a3) [8]. The value of n (or n/NC) from Eq. (4) is estimated by using iterative procedure and the values calculated are compared with those from Bebb’s numerical solution of
Fig. 2. Comparison of the present approximation with the Bebb’s solution of the Fermi integral as a function of cut-off wavelength (range 1–17 lm) for / taking values between 10 and 50.
S. Gupta et al. / Infrared Physics & Technology 51 (2008) 259–262
the range 1 lm < k < 17 lm. However, at lower kc’s there is increasing disagreement between the Bebb’s numerical solution and our relation, particularly for very low (<10) and very high values of / (>50). This is because we have tuned our empirical fitting about the centre of the SWIR, MWIR and LWIR range (1–17 lm). This is the range often used by IR detectors based on HgCdTe material and covers all the practical cases. In Fig. 3 we show the percentage error of our approximation to the Bebb’s numerical solution as a function of cut-off wavelength. This figure shows that for / = 10, 20, 30, 40, and 50, the respective average errors are 10.3%, 6.3%, 3.0%, 6.5% and 11% in the cut-off wavelength region of 1–17 lm. This error is much lower than in Ref. [5] and its validity is for a larger range than in Ref. [6]. For the case of weak degeneracy (/ 6 10), there are many approximations that give reasonably accurate values as discussed in Ref. [6]. It may be stated here that there is a scarcity of approximations for strongly degenerate cases (/ > 30) [6]. We have tried to tune our approximations specifically for this range of / with increased wavelength range. In this regard, it may be seen from this figure that error for / > 30 shows oscillatory behavior and decreases beyond the range of 1–17 lm. Hence our approximation for this case is valid even for wider range of wavelengths. In general, the average percentage error for our approximation varies between 3% and 11% for the 1–17 lm wavelength range. The corresponding standard deviation of the percentage error varies from 2.6% to 6.3%. At this point, it may be pointed out that for the majority of numerical solution methods, when the upper limit of integration is infinity (as in the present case, Eq. (1)), then it is replaced by a suitable finite number and the accuracy of the solution depends upon the adjustment and optimization of this number. This number changes with the value of kc in the present case. For illustration, we have chosen the most widely used Gauss’s 15 point quadrature method to evaluate the integral (Eq. (1)).
261
Fig. 4. The comparison of Gauss quadrature (15 points) method with Bebb’s results for two different values of upper limit of integration being equal to 5eg and 3eg.
The results are depicted in Fig. 4. It can be seen that for higher kc (>8 lm) an upper integration limit of 5eg gives a good accuracy whereas the same value is not adequate for lower kc (<8 lm). The error w.r.t. Bebb’s calculations is as high as 46% for / = 10 at 2 lm (see symbol n in the figure). However, if one wants to improve the fitting in the lower cut-off wavelength region, then the upper integration limit should be <3eg. However, the results in that case degrade at higher values of / (>30) (see symbol in the figure), particularly for higher kc. The maximum error being 68% for / = 50 at 17 lm. In short there is no unique upper integration limit valid for all wavelengths. Thus, for modeling where evaluation of the Fermi integral is only a small part of the calculation, this is impractical. On the other hand, our proposed Eq. (4) may be used as such. In the case of PC detectors, our approximation could be used to calculate the areal surface electron density in cm2 used in Eq. (15) of Ref. [10] (Qf in that work). Furthermore, our approximation can be used to model the capacitance–voltage characteristics of a metal-semiconductor–insulator devices in the case of large gate voltages that involve the use of very high values of / and to calculate space charge density as a function of the Fermi energy (see Fig. 3 of Ref. [7]). In the case of PV detectors, the approximation can be used to estimate the carrier density in the heavily doped donor region of pn+ diodes. 3. Conclusions
Fig. 3. Percentage error vs. cut-off wavelength in calculating n/NC using our proposed approximation.
In summary, we have proposed an analytical approximation of the Fermi integral (valid for 10 < / < 50) to estimate the carrier density in an NGS, including the nonparabolicity of the energy bands and the degeneracy of the carriers. This approximation is useful for efficient modeling of IR detectors based on n-HgCdTe material in the SWIR, MWIR and LWIR range.
262
S. Gupta et al. / Infrared Physics & Technology 51 (2008) 259–262
Acknowledgement The authors are thankful to the Director, SSPL for his kind permission to publish this work. References [1] H.B. Bebb, C.R. Ratliff, J. Appl. Phys. 42 (1971) 3189. [2] P.C. Catagnus, C.T. Baker, US Patent No. 3, 977018, 1976; Y. Nemirovsky, I. Kidron, Solid State Electron. 22 (1979) 831; B.K. Janousek, M.J. Danghorty, R.B. Schoolar, in: G. Lucovsky, S.T. Pantellides, F.L. Gleener (Eds.), The Physics of MOS Insulators, Pergamon, New York, 1980, p. 217.
[3] P.E. Peterson, in: R.K. Willardson, A.C. Beer (Eds.), Semiconductors and Semimetals, vol. 18, Academic Press, 1981, p. 145. [4] J.F. Siliquini, L. Faraone, Semicond. Sci. Technol. 12 (1997) 1010. [5] R.K. Bhan, V. Dhar, Semicond. Sci. Technol. 18 (2003) 1043. [6] R.K. Bhan, V. Dhar, Semicond. Sci. Technol. 19 (2004) 413. [7] V. Ariel-Altschul, E. Finkman, G. Bahir, IEEE. Trans. Electron. Dev. ED-39 (1992) 1312. [8] V. Altschul, E. Finkman, Appl. Phys. Lett. 58 (1991) 942. [9] M.H. Wieler, in: R.K. Willardson, A.C. Beer (Eds.), Semiconductors and Semimetals, vol. 16, Academic Press, New York, 1981, pp. 119– 191. [10] I. Bloom, Y. Nemirovsky, Solid State Electron. 31 (1998) 17.