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A simple approximative method for determination of Auger 1 lifetime in degenerate narrow gap semiconductors
Infrared Phys. Vol. 34, No. 6, pp. 601~05, 1993 Printed in Great Britain. All rights reserved
0020-0891/93 $6.00+ 0.00 Copyright © 1993 Pergamon Press Ltd
A SIMPLE A P P R O X I M A T I V E M E T H O D F O R D E T E R M I N A T I O N OF A U G E R 1 L I F E T I M E IN D E G E N E R A T E N A R R O W GAP S E M I C O N D U C T O R S Z. DJURIC, Z . JAK~Ir, A . VUJANIC a n d M . SMILJANIC Belgrade University, Institute of Microelectronic Technologies and Single Crystals, Njegogeva 12, 11000 Belgrade, Serbia
(Received 3 July 1993) Almtraet--Using the well-known Beattie-Landsberg expression for Auger 1 lifetime and the "effective" Auger energy gap (E#), we obtained a simple approximative expression for determination of the Auger 1 lifetime. We also obtained a formula for the electron concentration above which the effects of degeneration are significant.
I. I N T R O D U C T I O N During the last few decades it has been established °) that the processes of Auger generation (avalanche multiplication) and recombination play an important role in various semiconductor devices, for example in semiconductor lasers, infrared photodetectors, etc. The Auger 1 process in narrow gap semiconductors is especially important, and particularly in various devices operating in so-called nonequilibrium mode. (2~) In Refs (2-4) it was concluded that under high electron concentrations the decrease of the Auger 1 lifetime becomes less significant, because the conduction zone fills quickly due to its low density of states and the energy necessary for an occurrence of the Auger process increases. To take this effect into account, White used, in Ref. (2), an empirical relation assuming a linear increase in the intrinsic Auger lifetime in dependence on electron concentration ,-, (1 + ct • n), where ct = 1.9 x 1017cm -3. We did not find any other experimental data regarding the behavior of the Auger 1 lifetime for the case of high electron concentrations. However, an accurate determination of the transition rate for Auger processes requires time-consuming computer programs performing multidimensional integration of transition probabilities over all initial and final states. (s) It is especially true for n-type narrow-gap semiconductors.
II.
THEORY
The Auger 1 recombination process takes place when an electron from the conduction band recombines with a hole from the heavy holes band, and the energy is transferred to a second electron which transits high into the conduction zone [Fig. l(a)]. The calculations of Beattie and Landsberg (l) show that for the case of an intrinsic semiconductor (nondegenerate statistics are applied) the following expression can be used for the Auger 1 lifetime:
• rc2Eo2h3x//~ 2 ,, ,I/mo'~fEg'~ 3/2 [ 2exp(l+2#Eg) T~I~ ,,(I+ #)'/2(I+ z#)~,~--j~,~--~) IFIF2 17+--# k--T]
(1)
where Er, static dielectric permitivity of the semiconductor material; E0, vacuum dielectric permitivity; Eg, energy gap; IF1F21, overlap integrals, with a value in the range (0.14).3). It is supposed that the overlap integrals are weakly dependent on carrier concentration. 601
602
Z. DJURi• et al.
The basic idea for our modification of expression (1), in order to apply it to a degenerate n-type semiconductor, was to introduce a new threshold energy for the avalanche multiplication. In expression (1) the threshold energy of the Auger 1 process is: E =
1 + 2/~ m. E~, # = - l+/t mp
(2)
Expression (2) is obtained making use of the energy and quasimomentum conservation laws. The procedure assumes that the threshold energy for the Auger 1 process equals the minimum energy of electron 2', the one which is able to start avalanche multiplication by ionizing an electron from the valence band for a given value of the quasimomentum. Bearing in mind that in equation (2) E describes electron kinetic energy calculated from the bottom of the conduction zone and supposing that in degenerate semiconductor all the levels under Er - 4kT ~ Eg - 4kT are occupied, we can assume that the process is equal to the case when the energy gap in a degenerate semiconductor becomes Eg + Er - 4kT instead of Eg. Therefore instead of equation (2), we can write for the threshold energy: 1 +2/L Ed = 1 + # (E~ + EF - 4kT)
(3)
where we suppose that the effective masses are constant. Expression (3) is valid when the Fermi level Er/> 4kT. Thus the new expression for z~t in a degenerate semiconductor becomes
z~l = *iA, ( E8 + EF -- 4kT~ 3/2 f l + 21, EF --4kT'~ " EgJ exp~ -1 5r-; kV J
(4)
It is known that in narrow gap semiconductors with a low effective mass the Fermi level enters the conduction zone when the electron concentration increases slightly above the intrinsic one. This phenomenon is manifested by the absorption edge shift (the Moss-Burstein effect).(6) In the range 4kT <~EF <~ 10 k T we used a series expansion (7) to determine the dependence EF = f ( n ) , while for the case of strong degeneracy (EF > 10kT) the Sommerfeld model is valid, and, according to the latter, the electron concentration in dependence on Fermi level is approximately given as (7)
n 4 ( E v E F + E , ~ 3/2 N ¢ - 3 x / ~ \~--~ -~g .j
(5)
where
N = 2 ( 3 E s k T ~ 3/2 c \ 8rip2 j , P = 9 x 10 -8 eV cm (Kane's matrix element). The expression for the Auger 1 lifetime in an n-type semiconductor is ~A~
2~klP n +p'
(6)
and for a degenerate n-type semiconductor it becomes z~l = 2z~.p n
(7)
III. N U M E R I C A L RESULTS We performed our calculations for a Hg~ _xCdxTe (mercury cadmium telluride) single crystal, where x denotes the cadmium molar fraction. The parameters necessary to calculate composition
Auger 1 lifetime in narrow gap semiconductors
603
Ec a)
2' l ~
b)
Ec
4kT
I
Eg
1
2
EgA
Eg
Ehh - ~
h
EIh
Es
Es
Fig. 1. (a) Auger 1 recombination process in a nondegenerate semiconductor; (b) threshold for the Auger I process in a degenerate semiconductor and the definition of EgA.
and temperature dependencies o f energy, intrinsic concentration and other parameters appearing in equations (1-7) were taken from Ref. (8). Figure 2 presents the dependence o f z~/Z~,l on electron concentration for various temperatures. The energy gap at the chosen temperature was always 0.1 eV. As can be seen from the figure, the limiting concentration above which the increase o f the Auger 1 lifetime becomes marked, depends on temperature. F o r example, this concentration at 77K ~ 8 × 1015cm -3, while at 200 K ~ 1.5 x l017 cm -3, which is in g o o d accordance with the empirical expression o f White32) Figure 3 shows the dependence o f z~,~ on the electron concentration for various temperatures. The dotted lines represent the case when the degeneration is not taken into account. The energy gap for all the curves is 0.1 eV. As can be seen f r o m the figure, after passing the m i n i m u m the Auger lifetime increases with d o p a n t concentration and therefore at high concentration levels other recombination mechanisms m a y prevail.
lo,2 1010 _
1 - 77K 2 - 150K 3 - 200K 4 - 250K 5 - 300K
10 s
106 I O4
1
2 34
102 10o 10-1
I 14
10
I 15
10
I 16
10
n, cm-3
I 17
10
18
10
19
10
Fig. 2. The ratio of intrinsic Auger I lifetime with and without degeneration vs electron concentration for various temperatures. The energy gap is 0.1 eV; the material is Hg t_xCdxTe, x = 0.2106, T = 77 K; x =0.196, T = 150K; x =0.185, T=200K; x =0.174, T = 250K; x =0.161, T = 300K. INF 3d/6---E
604
Z. DJURICet
al,
-2
I0
77K
150 200K/
-6 I0
i(~I°
-14
10
1014
1015
i
I
I
1016
1017
1018
n, c m
1019
-3
Fig. 3. Auger lifetime vs electron concentration for various temperatures, The energy gap is 0.1 eV; the material is Hg~_~CdxTe, and the compositions as in Fig. 2. The dotted line represents the nondegenerate case. Figures 4 and 5 show the dependence o f "i'd I o n inverse temperature for two electron concentrations, for x = 0.161 (Eg = 0.1 eV at 300 K) and x = 0.215 (0.1 eV at 77 K) respectively. The diagrams show that, starting from a threshold temperature, the Auger 1 lifetime increases with a temperature decrease, quite the opposite o f when degeneration is neglected. It can also be seen that the effects are more p r o n o u n c e d for lower compositions of Hgl_xCdxTe.
IV. C O N C L U S I O N A n approximate theory is given for estimation o f the Auger 1 lifetime dependence on concentration and temperature in narrow b a n d g a p semiconductors. The theory m a y be o f great interest in the design and optimization o f both photoconductive and photovoltaic I R detectors, and especially for the devices operating under nonequilibrium conditions, i.e. exclusion p h o t o c o n d u c tors and exclusion-extraction photodiodes. The theory m a y also be applied for semiconductors with wider bandgaps, i.e. GaAs, GaA1As, InAsP, etc. In this case it is necessary to include the dependence o f a nominal bandgap, Eg = Eg(n), on electron concentration. The narrowing o f Eg is a consequence o f two effects: electron interaction -6
10
-7 10 10.8 5.1016cm "3 10 -9
1.1017Cm'3 ...........
10 -lo 10
-11
10 "12
'
3
I
5
'
I
'
7
I
9
'
I
11
'
13
1000/T, K -1
Fig. 4. The dependence of ~'d I o n inverse temperature for electron concentrations of 5 x 10~6cm 3 and 1 x I017crn-3. The material is Hgl_xCdxTe, and the composition x = 0.161 (300 K, E8 = 0.1 eV). The dotted line shows the nondegenerate dependence of ZA~.
Auger 1 lifetime in narrow gap semiconductors
605
lo 10.7
lOl O .9
101 o -11 10 10-12
'
3
I
5
'
I
'
7
I
'
9
I
11
13
1000/T, K "1 Fig. 5. The dependence of z~m on inverse temperature for electron concentrations of 5 x 10t6 cm -~ and 1 x 1017 c m -3, The material is Hgl _xCdxTe, and the composition x = 0.2106 (77 K, Eg = 0.1 eV). The dotted line shows the nondegenerate dependence of TA~.
w i t h i o n i z e d d o p a n t s a n d i n t e r a c t i o n o f free c a r r i e r s (these effects a r e n o t i n c l u d e d in the t h e o r y o f n a r r o w b a n d g a p s e m i c o n d u c t o r s d u e to insufficient l i t e r a t u r e data). T h e t h e o r y g e n e r a l i z e d in s u c h a m a n n e r m a y be u s e d for c a l c u l a t i o n o f s e m i c o n d u c t o r lasers, w h i c h will be c o n s i d e r e d in t h e n e a r future. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
A. R. Beattie and P. T. Landsberg, Proc. R. Soc. 249, 16 (1959). A. M. White, Infrared Phys. 25, 729 (1985). A. M. White, J. Crystal Growth 86, 840 (1988). C. T. Elliott, Semicond. Sci. Technol. 5, $30 (1990). R. Gerhardts, R. Dornhaus and G. Nimtz, Solid-St. Electron. 21, 1467 (1978). H. C. Huang, S. Yee and M. Soma, J. appl. Phys. 67, 1497 (1990). V. Altschul and E. Finkman, Appl. Phys. Lett. 58, 942 (1991). J. Piotrowski, W. Galus and M. Grudzien, Infrared Phys. 31, 1 (1991).
0020-0891/93 $6.00+ 0.00 Copyright © 1993 Pergamon Press Ltd
A SIMPLE A P P R O X I M A T I V E M E T H O D F O R D E T E R M I N A T I O N OF A U G E R 1 L I F E T I M E IN D E G E N E R A T E N A R R O W GAP S E M I C O N D U C T O R S Z. DJURIC, Z . JAK~Ir, A . VUJANIC a n d M . SMILJANIC Belgrade University, Institute of Microelectronic Technologies and Single Crystals, Njegogeva 12, 11000 Belgrade, Serbia
(Received 3 July 1993) Almtraet--Using the well-known Beattie-Landsberg expression for Auger 1 lifetime and the "effective" Auger energy gap (E#), we obtained a simple approximative expression for determination of the Auger 1 lifetime. We also obtained a formula for the electron concentration above which the effects of degeneration are significant.
I. I N T R O D U C T I O N During the last few decades it has been established °) that the processes of Auger generation (avalanche multiplication) and recombination play an important role in various semiconductor devices, for example in semiconductor lasers, infrared photodetectors, etc. The Auger 1 process in narrow gap semiconductors is especially important, and particularly in various devices operating in so-called nonequilibrium mode. (2~) In Refs (2-4) it was concluded that under high electron concentrations the decrease of the Auger 1 lifetime becomes less significant, because the conduction zone fills quickly due to its low density of states and the energy necessary for an occurrence of the Auger process increases. To take this effect into account, White used, in Ref. (2), an empirical relation assuming a linear increase in the intrinsic Auger lifetime in dependence on electron concentration ,-, (1 + ct • n), where ct = 1.9 x 1017cm -3. We did not find any other experimental data regarding the behavior of the Auger 1 lifetime for the case of high electron concentrations. However, an accurate determination of the transition rate for Auger processes requires time-consuming computer programs performing multidimensional integration of transition probabilities over all initial and final states. (s) It is especially true for n-type narrow-gap semiconductors.
II.
THEORY
The Auger 1 recombination process takes place when an electron from the conduction band recombines with a hole from the heavy holes band, and the energy is transferred to a second electron which transits high into the conduction zone [Fig. l(a)]. The calculations of Beattie and Landsberg (l) show that for the case of an intrinsic semiconductor (nondegenerate statistics are applied) the following expression can be used for the Auger 1 lifetime:
• rc2Eo2h3x//~ 2 ,, ,I/mo'~fEg'~ 3/2 [ 2exp(l+2#Eg) T~I~ ,,(I+ #)'/2(I+ z#)~,~--j~,~--~) IFIF2 17+--# k--T]
(1)
where Er, static dielectric permitivity of the semiconductor material; E0, vacuum dielectric permitivity; Eg, energy gap; IF1F21, overlap integrals, with a value in the range (0.14).3). It is supposed that the overlap integrals are weakly dependent on carrier concentration. 601
602
Z. DJURi• et al.
The basic idea for our modification of expression (1), in order to apply it to a degenerate n-type semiconductor, was to introduce a new threshold energy for the avalanche multiplication. In expression (1) the threshold energy of the Auger 1 process is: E =
1 + 2/~ m. E~, # = - l+/t mp
(2)
Expression (2) is obtained making use of the energy and quasimomentum conservation laws. The procedure assumes that the threshold energy for the Auger 1 process equals the minimum energy of electron 2', the one which is able to start avalanche multiplication by ionizing an electron from the valence band for a given value of the quasimomentum. Bearing in mind that in equation (2) E describes electron kinetic energy calculated from the bottom of the conduction zone and supposing that in degenerate semiconductor all the levels under Er - 4kT ~ Eg - 4kT are occupied, we can assume that the process is equal to the case when the energy gap in a degenerate semiconductor becomes Eg + Er - 4kT instead of Eg. Therefore instead of equation (2), we can write for the threshold energy: 1 +2/L Ed = 1 + # (E~ + EF - 4kT)
(3)
where we suppose that the effective masses are constant. Expression (3) is valid when the Fermi level Er/> 4kT. Thus the new expression for z~t in a degenerate semiconductor becomes
z~l = *iA, ( E8 + EF -- 4kT~ 3/2 f l + 21, EF --4kT'~ " EgJ exp~ -1 5r-; kV J
(4)
It is known that in narrow gap semiconductors with a low effective mass the Fermi level enters the conduction zone when the electron concentration increases slightly above the intrinsic one. This phenomenon is manifested by the absorption edge shift (the Moss-Burstein effect).(6) In the range 4kT <~EF <~ 10 k T we used a series expansion (7) to determine the dependence EF = f ( n ) , while for the case of strong degeneracy (EF > 10kT) the Sommerfeld model is valid, and, according to the latter, the electron concentration in dependence on Fermi level is approximately given as (7)
n 4 ( E v E F + E , ~ 3/2 N ¢ - 3 x / ~ \~--~ -~g .j
(5)
where
N = 2 ( 3 E s k T ~ 3/2 c \ 8rip2 j , P = 9 x 10 -8 eV cm (Kane's matrix element). The expression for the Auger 1 lifetime in an n-type semiconductor is ~A~
2~klP n +p'
(6)
and for a degenerate n-type semiconductor it becomes z~l = 2z~.p n
(7)
III. N U M E R I C A L RESULTS We performed our calculations for a Hg~ _xCdxTe (mercury cadmium telluride) single crystal, where x denotes the cadmium molar fraction. The parameters necessary to calculate composition
Auger 1 lifetime in narrow gap semiconductors
603
Ec a)
2' l ~
b)
Ec
4kT
I
Eg
1
2
EgA
Eg
Ehh - ~
h
EIh
Es
Es
Fig. 1. (a) Auger 1 recombination process in a nondegenerate semiconductor; (b) threshold for the Auger I process in a degenerate semiconductor and the definition of EgA.
and temperature dependencies o f energy, intrinsic concentration and other parameters appearing in equations (1-7) were taken from Ref. (8). Figure 2 presents the dependence o f z~/Z~,l on electron concentration for various temperatures. The energy gap at the chosen temperature was always 0.1 eV. As can be seen from the figure, the limiting concentration above which the increase o f the Auger 1 lifetime becomes marked, depends on temperature. F o r example, this concentration at 77K ~ 8 × 1015cm -3, while at 200 K ~ 1.5 x l017 cm -3, which is in g o o d accordance with the empirical expression o f White32) Figure 3 shows the dependence o f z~,~ on the electron concentration for various temperatures. The dotted lines represent the case when the degeneration is not taken into account. The energy gap for all the curves is 0.1 eV. As can be seen f r o m the figure, after passing the m i n i m u m the Auger lifetime increases with d o p a n t concentration and therefore at high concentration levels other recombination mechanisms m a y prevail.
lo,2 1010 _
1 - 77K 2 - 150K 3 - 200K 4 - 250K 5 - 300K
10 s
106 I O4
1
2 34
102 10o 10-1
I 14
10
I 15
10
I 16
10
n, cm-3
I 17
10
18
10
19
10
Fig. 2. The ratio of intrinsic Auger I lifetime with and without degeneration vs electron concentration for various temperatures. The energy gap is 0.1 eV; the material is Hg t_xCdxTe, x = 0.2106, T = 77 K; x =0.196, T = 150K; x =0.185, T=200K; x =0.174, T = 250K; x =0.161, T = 300K. INF 3d/6---E
604
Z. DJURICet
al,
-2
I0
77K
150 200K/
-6 I0
i(~I°
-14
10
1014
1015
i
I
I
1016
1017
1018
n, c m
1019
-3
Fig. 3. Auger lifetime vs electron concentration for various temperatures, The energy gap is 0.1 eV; the material is Hg~_~CdxTe, and the compositions as in Fig. 2. The dotted line represents the nondegenerate case. Figures 4 and 5 show the dependence o f "i'd I o n inverse temperature for two electron concentrations, for x = 0.161 (Eg = 0.1 eV at 300 K) and x = 0.215 (0.1 eV at 77 K) respectively. The diagrams show that, starting from a threshold temperature, the Auger 1 lifetime increases with a temperature decrease, quite the opposite o f when degeneration is neglected. It can also be seen that the effects are more p r o n o u n c e d for lower compositions of Hgl_xCdxTe.
IV. C O N C L U S I O N A n approximate theory is given for estimation o f the Auger 1 lifetime dependence on concentration and temperature in narrow b a n d g a p semiconductors. The theory m a y be o f great interest in the design and optimization o f both photoconductive and photovoltaic I R detectors, and especially for the devices operating under nonequilibrium conditions, i.e. exclusion p h o t o c o n d u c tors and exclusion-extraction photodiodes. The theory m a y also be applied for semiconductors with wider bandgaps, i.e. GaAs, GaA1As, InAsP, etc. In this case it is necessary to include the dependence o f a nominal bandgap, Eg = Eg(n), on electron concentration. The narrowing o f Eg is a consequence o f two effects: electron interaction -6
10
-7 10 10.8 5.1016cm "3 10 -9
1.1017Cm'3 ...........
10 -lo 10
-11
10 "12
'
3
I
5
'
I
'
7
I
9
'
I
11
'
13
1000/T, K -1
Fig. 4. The dependence of ~'d I o n inverse temperature for electron concentrations of 5 x 10~6cm 3 and 1 x I017crn-3. The material is Hgl_xCdxTe, and the composition x = 0.161 (300 K, E8 = 0.1 eV). The dotted line shows the nondegenerate dependence of ZA~.
Auger 1 lifetime in narrow gap semiconductors
605
lo 10.7
lOl O .9
101 o -11 10 10-12
'
3
I
5
'
I
'
7
I
'
9
I
11
13
1000/T, K "1 Fig. 5. The dependence of z~m on inverse temperature for electron concentrations of 5 x 10t6 cm -~ and 1 x 1017 c m -3, The material is Hgl _xCdxTe, and the composition x = 0.2106 (77 K, Eg = 0.1 eV). The dotted line shows the nondegenerate dependence of TA~.
w i t h i o n i z e d d o p a n t s a n d i n t e r a c t i o n o f free c a r r i e r s (these effects a r e n o t i n c l u d e d in the t h e o r y o f n a r r o w b a n d g a p s e m i c o n d u c t o r s d u e to insufficient l i t e r a t u r e data). T h e t h e o r y g e n e r a l i z e d in s u c h a m a n n e r m a y be u s e d for c a l c u l a t i o n o f s e m i c o n d u c t o r lasers, w h i c h will be c o n s i d e r e d in t h e n e a r future. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
A. R. Beattie and P. T. Landsberg, Proc. R. Soc. 249, 16 (1959). A. M. White, Infrared Phys. 25, 729 (1985). A. M. White, J. Crystal Growth 86, 840 (1988). C. T. Elliott, Semicond. Sci. Technol. 5, $30 (1990). R. Gerhardts, R. Dornhaus and G. Nimtz, Solid-St. Electron. 21, 1467 (1978). H. C. Huang, S. Yee and M. Soma, J. appl. Phys. 67, 1497 (1990). V. Altschul and E. Finkman, Appl. Phys. Lett. 58, 942 (1991). J. Piotrowski, W. Galus and M. Grudzien, Infrared Phys. 31, 1 (1991).