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Factors limiting lifetime of charge carriers in semi-insulated CdTe under high radiation flux
Nuclear Instruments and Methods in Physics Research A 633 (2011) S95–S96
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Factors limiting lifetime of charge carriers in semi-insulated CdTe under high radiation flux ¨ J. Franc n, R. Grill, J. Kuba´t, V. Dˇedicˇ, E. Belas, P. Hoschl Faculty of Mathematics and Physics, Institute of Physics, Prague, CZ 121 16, Czech Republic
a r t i c l e in fo
abstract
Available online 18 June 2010
With the growing interest in CdTe and (CdZn)Te X-ray detectors operating at high fluxes of X-ray photons ( 1010 cm 2 s 1) the questions arises, whether the lifetime of charge carriers is limited by trapping and recombination at deep levels or by band-to-band recombination due to a strongly elevated concentrations of free electrons and holes compared to detectors working at low fluxes. A set of numerical simulations was done to resolve this question. The approach is based on the self-consistent steady state solution of electron and hole drift-diffusion equations using an iterative method. The effect of space charge on the electric field distribution and carrier transport through the sample is evaluated by solving the Poisson equation. The material simulation parameters were chosen to describe typical situations in state-of-the art highly compensated semi-insulated CdTe and CdZnTe ( 1010 Ocm) with deep near-midgap levels with concentrations in the range of 1011–1012 cm 3.The band-to-band recombination parameter was calculated using the Van Roosbroeck–Shockley relation and was estimated 3 10 10 cm3/s. The results show, that under all studied conditions the lifetime of carriers and thus the detector performance is limited by carrier trapping and recombination at deep levels. & 2010 Elsevier B.V. All rights reserved.
Keywords: CdTe Detectors Recombination
1. Introduction During the last decades, CdZnTe and CdTe room-temperature x- and g-ray semiconductor detectors have played an evergrowing role in various aspects of human life. [1–6] Already, the field of applications of these detectors extends from health and medicine to nuclear security and imaging of energetic ions, and likely will extend further. Detectors in imaging applications operate at high photon fluxes generating high count rates (20–200) 108 counts/(cm2s) [7]. At such a high count rates the concentrations of free electrons and holes in the detector volume are substantially enhanced in comparison to detectors working at low fluxes. These free carriers are in ideal case removed from the material by drift and collected at the electrodes generating signal pulses. In real situation, part of the carriers can recombine at deep levels by Shockley–Read process or by band-to-band recombination. The purpose of this paper is to find out, whether at real operating conditions in state-of-the art CdTe and CdZnTe detector-grade materials the lifetime is limited by band-to-band recombination or by the Shockley–Read recombination. Answering this question is important in order to find out, whether the currently achieved values of mobility-lifetime products can be further increased by progress in material development or whether
n
Corresponding author. Tel.: +420 221911335. E-mail address: [email protected] (J. Franc).
0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.06.134
the limit due to inherent band-to-band recombination was achieved.
2. Theory The band-to-band recombination Rbb is obtained with Rbb ¼
R ðnpn0 p0 Þ n0 p0
ð1Þ
where the recombination rate R is calculated with the Van Roosbroeck–Shockley relation [8,9] R¼
n2r akb TE2g
p2 c2 _3
eðEg =kb TÞ
ð2Þ
In Eq. (2) nr and a are effective refractive index and absorption coefficient near absorption edge, Eg is the gap energy and c is the light velocity. Combining (1) with well-known form for intrinsic carrier density n2i ¼n0p0, Rbb results as 2p_3 n2r a Eg 2 Rbb ¼ ðnpn0 p0 Þ ð3Þ c2 m30 ðme mh Þ3=2 kb T Here m0 is the electron rest mass and me and mh are relative electron and hole masses, respectively. In CdTe at the room temperature we take nr ¼3.5 [10], a ¼2 104 cm 1, Eg ¼1.5 eV,
S96
J. Franc et al. / Nuclear Instruments and Methods in Physics Research A 633 (2011) S95–S96
16
τe(bb), 10 cm s
1
τe(SR), 10 cm s
0
τe(bb), 10 cm s
10
16
2
-2 -1
15
10
3
10
-2 -1
2
10
10
-2 -1
1
10
0
10
lifetime (s)
lifetime (s)
-1
10
-2
10
-3
10
-4
10
-2
10
-3
10
-4
10
-5
10
-5
10
-6
10
-6
10
-7
10
-1
10
0.00
0.02
0.04
0.06
0.08
0.10
X (cm)
-7
10
0.00
0.02
0.04
0.06
0.08
0.10
x (cm)
Fig. 1. Comparison of Shockley–Read and band to band recombination lifetimes of electrons (a ¼ 50 cm 1, U¼ 100 V). The illuminated cathode is at location x¼ 0.
Fig. 2. The sensitivity of te(bb) on the band-to-band recombination parameter rbb.
me ¼0.096, and mh ¼0.83. Substituting these values, Rbb is given by
Calculations for the case of high absorption (a ¼40,000 cm 1) lead to the same conclusion, i.e. to domination of the Shockley– Read recombination process over the band-to-band mechanism.
Rbb ¼ rbb ðnpn0 p0 Þ
ð4Þ 10
3
1
cm s . where rbb ¼3.0 10 A set of numerical simulations was performed in order to estimate the relative influence of Shockley–Read and band-toband recombination in state-of-the art detector-grade semiinsulated CdTe. The approach was based on the self-consistent steady state solution of electron and hole drift-diffusion equations coupled with Poisson equation using an iterative method [11].
3. Results and discussion We work with a model including one deep level. Parameters of midgap level from Ref. [7], which represents situation in state-ofthe art detector-grade fully compensated semi-insulating CZT were used (N ¼2 1011 cm 3, se ¼5 10 13 cm2, sh ¼3 10 14 cm2). The simulations were performed for two values of absorption coefficient a ¼40000 and 50 cm 1 representing the cases of high and low attenuation of heavy charged particles and medium energy X-ray photons, respectively. The band-to-band recombination parameter rbb ¼3 10 10 cm 3 s 1 was used in the calculations. The illuminated contact is located on the left hand side of all pictures at position x ¼0. The illumination is expressed in the number of generated electron– hole pairs per square centimeter and second throughout the paper. The results of the simulation of electron lifetime for the case of small attenuation coefficient a ¼50 cm 1 are presented in Fig. 1. The applied voltage was 100 V. The electric field attracts electrons to the anode positioned at x ¼0.1 cm while holes drift to the illuminated cathode. The band-to-band recombination lifetime te(bb) is in the whole sample volume approximately four orders of magnitude higher than the Shockley–Read lifetime te(SR). The increase of te(bb) towards the anode is caused by a decreased concentration of free holes. The sensitivity of te(bb) on the band-to-band recombination parameter rbb is demonstrated in Fig. 2. The te(bb) and te(SR) would become comparable at rbb 10 5 cm 3 s 1, several orders of magnitude above the value evaluated from standard theory discussed in the previous paragraph.
4. Conclusion The band-to-band recombination parameter was calculated using the Van Roosbroeck–Shockley relation and was estimated 3 10 10 cm3/s. The results of numerical simulations show that at standard operating conditions the Shockley–Read recombination is dominant independent of the radiation energy. Therefore, further increase of lifetime of carriers and charge collection efficiency is possible by material improvement.
Acknowledgments This paper was financially supported by the Grant Agency of the Czech Republic under No. GACR 102/10/0148, the Grant Agency of Charles University (No. 48910/2010) and Alexander von Humboldt foundation. It is also a part of the research plan MSM 0021620834 that is financed by the Ministry of Education of the Czech Republic. References [1] C. Szeles, Phys. Status Solidi(c) 241 (2004) 783. [2] H. Chen, S. Awadalla, J. Mackenzie, R. Redden, G. Bindley, A.E. Bolotnikov, G.S. Camarda, IEEE Trans. Nucl. Sci. NS-54 (2007) 811. [3] H. Chen, S. Awadalla, R. Harris, P.H. Lu, R. Redden, G. Bindley, A. Copete, J. Hong, J. Gridlay, M. Amman, J.S. Lee, P. Luke, I. Kuwetli, C. Budtz-Jorgensen, IEEE Trans. Nucl. Sci. 54 (2008) 1567. [4] V. Carcelen, N. Viayan, E. Dieguez, A. Zapettini, M. Zha, L. Sylla, A. Fauler, M. Fiederle, J. Optoelectron. Adv. Mater. 10 (2008) 3135. [5] E. Saucedo, P. Rudolph, E. Dieguez, J. Cryst. Growth 310 (2008) 2067. [6] H. Chen, S. Awadalla, R. Harris, P.H. Lu, R. Redden, G. Bindley, A. Copete, J. Hong, J. Gridlay, M. Amman, J.S. Lee, P. Luke, I. Kuwetli, C. Budtz-Jorgensen, IEEE Trans. Nucl. Sci. NS-55 (2008) 1567. [7] D. Bale, C. Szeles, Phys. Rev. B. 77 (2008) 033205. [8] W. Roesbroeck, W. Shockley, Phys. Rev. 94 (1954) 1558. [9] B. Sapoval, C. Hermann, Physics of Semiconductors, Springer, New York, 1995. [10] S. Adachi, T. Kumura, Jpn. J. Appl. Phys. 32 (1993) 3866. ¨ [11] J. Kuba´t, H. Elhadidy, J. Franc, R. Grill, E. Belas, P. Hoschl, IEEE Trans. Nucl. Sci. NS-56 (2009) (1706).
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Factors limiting lifetime of charge carriers in semi-insulated CdTe under high radiation flux ¨ J. Franc n, R. Grill, J. Kuba´t, V. Dˇedicˇ, E. Belas, P. Hoschl Faculty of Mathematics and Physics, Institute of Physics, Prague, CZ 121 16, Czech Republic
a r t i c l e in fo
abstract
Available online 18 June 2010
With the growing interest in CdTe and (CdZn)Te X-ray detectors operating at high fluxes of X-ray photons ( 1010 cm 2 s 1) the questions arises, whether the lifetime of charge carriers is limited by trapping and recombination at deep levels or by band-to-band recombination due to a strongly elevated concentrations of free electrons and holes compared to detectors working at low fluxes. A set of numerical simulations was done to resolve this question. The approach is based on the self-consistent steady state solution of electron and hole drift-diffusion equations using an iterative method. The effect of space charge on the electric field distribution and carrier transport through the sample is evaluated by solving the Poisson equation. The material simulation parameters were chosen to describe typical situations in state-of-the art highly compensated semi-insulated CdTe and CdZnTe ( 1010 Ocm) with deep near-midgap levels with concentrations in the range of 1011–1012 cm 3.The band-to-band recombination parameter was calculated using the Van Roosbroeck–Shockley relation and was estimated 3 10 10 cm3/s. The results show, that under all studied conditions the lifetime of carriers and thus the detector performance is limited by carrier trapping and recombination at deep levels. & 2010 Elsevier B.V. All rights reserved.
Keywords: CdTe Detectors Recombination
1. Introduction During the last decades, CdZnTe and CdTe room-temperature x- and g-ray semiconductor detectors have played an evergrowing role in various aspects of human life. [1–6] Already, the field of applications of these detectors extends from health and medicine to nuclear security and imaging of energetic ions, and likely will extend further. Detectors in imaging applications operate at high photon fluxes generating high count rates (20–200) 108 counts/(cm2s) [7]. At such a high count rates the concentrations of free electrons and holes in the detector volume are substantially enhanced in comparison to detectors working at low fluxes. These free carriers are in ideal case removed from the material by drift and collected at the electrodes generating signal pulses. In real situation, part of the carriers can recombine at deep levels by Shockley–Read process or by band-to-band recombination. The purpose of this paper is to find out, whether at real operating conditions in state-of-the art CdTe and CdZnTe detector-grade materials the lifetime is limited by band-to-band recombination or by the Shockley–Read recombination. Answering this question is important in order to find out, whether the currently achieved values of mobility-lifetime products can be further increased by progress in material development or whether
n
Corresponding author. Tel.: +420 221911335. E-mail address: [email protected] (J. Franc).
0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.06.134
the limit due to inherent band-to-band recombination was achieved.
2. Theory The band-to-band recombination Rbb is obtained with Rbb ¼
R ðnpn0 p0 Þ n0 p0
ð1Þ
where the recombination rate R is calculated with the Van Roosbroeck–Shockley relation [8,9] R¼
n2r akb TE2g
p2 c2 _3
eðEg =kb TÞ
ð2Þ
In Eq. (2) nr and a are effective refractive index and absorption coefficient near absorption edge, Eg is the gap energy and c is the light velocity. Combining (1) with well-known form for intrinsic carrier density n2i ¼n0p0, Rbb results as 2p_3 n2r a Eg 2 Rbb ¼ ðnpn0 p0 Þ ð3Þ c2 m30 ðme mh Þ3=2 kb T Here m0 is the electron rest mass and me and mh are relative electron and hole masses, respectively. In CdTe at the room temperature we take nr ¼3.5 [10], a ¼2 104 cm 1, Eg ¼1.5 eV,
S96
J. Franc et al. / Nuclear Instruments and Methods in Physics Research A 633 (2011) S95–S96
16
τe(bb), 10 cm s
1
τe(SR), 10 cm s
0
τe(bb), 10 cm s
10
16
2
-2 -1
15
10
3
10
-2 -1
2
10
10
-2 -1
1
10
0
10
lifetime (s)
lifetime (s)
-1
10
-2
10
-3
10
-4
10
-2
10
-3
10
-4
10
-5
10
-5
10
-6
10
-6
10
-7
10
-1
10
0.00
0.02
0.04
0.06
0.08
0.10
X (cm)
-7
10
0.00
0.02
0.04
0.06
0.08
0.10
x (cm)
Fig. 1. Comparison of Shockley–Read and band to band recombination lifetimes of electrons (a ¼ 50 cm 1, U¼ 100 V). The illuminated cathode is at location x¼ 0.
Fig. 2. The sensitivity of te(bb) on the band-to-band recombination parameter rbb.
me ¼0.096, and mh ¼0.83. Substituting these values, Rbb is given by
Calculations for the case of high absorption (a ¼40,000 cm 1) lead to the same conclusion, i.e. to domination of the Shockley– Read recombination process over the band-to-band mechanism.
Rbb ¼ rbb ðnpn0 p0 Þ
ð4Þ 10
3
1
cm s . where rbb ¼3.0 10 A set of numerical simulations was performed in order to estimate the relative influence of Shockley–Read and band-toband recombination in state-of-the art detector-grade semiinsulated CdTe. The approach was based on the self-consistent steady state solution of electron and hole drift-diffusion equations coupled with Poisson equation using an iterative method [11].
3. Results and discussion We work with a model including one deep level. Parameters of midgap level from Ref. [7], which represents situation in state-ofthe art detector-grade fully compensated semi-insulating CZT were used (N ¼2 1011 cm 3, se ¼5 10 13 cm2, sh ¼3 10 14 cm2). The simulations were performed for two values of absorption coefficient a ¼40000 and 50 cm 1 representing the cases of high and low attenuation of heavy charged particles and medium energy X-ray photons, respectively. The band-to-band recombination parameter rbb ¼3 10 10 cm 3 s 1 was used in the calculations. The illuminated contact is located on the left hand side of all pictures at position x ¼0. The illumination is expressed in the number of generated electron– hole pairs per square centimeter and second throughout the paper. The results of the simulation of electron lifetime for the case of small attenuation coefficient a ¼50 cm 1 are presented in Fig. 1. The applied voltage was 100 V. The electric field attracts electrons to the anode positioned at x ¼0.1 cm while holes drift to the illuminated cathode. The band-to-band recombination lifetime te(bb) is in the whole sample volume approximately four orders of magnitude higher than the Shockley–Read lifetime te(SR). The increase of te(bb) towards the anode is caused by a decreased concentration of free holes. The sensitivity of te(bb) on the band-to-band recombination parameter rbb is demonstrated in Fig. 2. The te(bb) and te(SR) would become comparable at rbb 10 5 cm 3 s 1, several orders of magnitude above the value evaluated from standard theory discussed in the previous paragraph.
4. Conclusion The band-to-band recombination parameter was calculated using the Van Roosbroeck–Shockley relation and was estimated 3 10 10 cm3/s. The results of numerical simulations show that at standard operating conditions the Shockley–Read recombination is dominant independent of the radiation energy. Therefore, further increase of lifetime of carriers and charge collection efficiency is possible by material improvement.
Acknowledgments This paper was financially supported by the Grant Agency of the Czech Republic under No. GACR 102/10/0148, the Grant Agency of Charles University (No. 48910/2010) and Alexander von Humboldt foundation. It is also a part of the research plan MSM 0021620834 that is financed by the Ministry of Education of the Czech Republic. References [1] C. Szeles, Phys. Status Solidi(c) 241 (2004) 783. [2] H. Chen, S. Awadalla, J. Mackenzie, R. Redden, G. Bindley, A.E. Bolotnikov, G.S. Camarda, IEEE Trans. Nucl. Sci. NS-54 (2007) 811. [3] H. Chen, S. Awadalla, R. Harris, P.H. Lu, R. Redden, G. Bindley, A. Copete, J. Hong, J. Gridlay, M. Amman, J.S. Lee, P. Luke, I. Kuwetli, C. Budtz-Jorgensen, IEEE Trans. Nucl. Sci. 54 (2008) 1567. [4] V. Carcelen, N. Viayan, E. Dieguez, A. Zapettini, M. Zha, L. Sylla, A. Fauler, M. Fiederle, J. Optoelectron. Adv. Mater. 10 (2008) 3135. [5] E. Saucedo, P. Rudolph, E. Dieguez, J. Cryst. Growth 310 (2008) 2067. [6] H. Chen, S. Awadalla, R. Harris, P.H. Lu, R. Redden, G. Bindley, A. Copete, J. Hong, J. Gridlay, M. Amman, J.S. Lee, P. Luke, I. Kuwetli, C. Budtz-Jorgensen, IEEE Trans. Nucl. Sci. NS-55 (2008) 1567. [7] D. Bale, C. Szeles, Phys. Rev. B. 77 (2008) 033205. [8] W. Roesbroeck, W. Shockley, Phys. Rev. 94 (1954) 1558. [9] B. Sapoval, C. Hermann, Physics of Semiconductors, Springer, New York, 1995. [10] S. Adachi, T. Kumura, Jpn. J. Appl. Phys. 32 (1993) 3866. ¨ [11] J. Kuba´t, H. Elhadidy, J. Franc, R. Grill, E. Belas, P. Hoschl, IEEE Trans. Nucl. Sci. NS-56 (2009) (1706).