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Optical characteristics of a superlattice avalanche photodiode
Solid-State Electronics Vol. 30, No. 7, pp. 675479, 1987 Printed in Great Britain. All rights reserved
0038-I 101/87 $3.00 +O.OO
Copyright 0 1987 Pergamon Journals Ltd
OPTICAL CHARACTERISTICS OF A SUPERLATTICE AVALANCHE PHOTODIODE P. CHAKRABARTI and B. B. PAL Department of Electronics Engineering, Institute of Technology, Banaras Hindu University, Varanasi, India (Received
22 May 1986; in revised form 13 September
1986)
Abstract-Avalanche photodiodes using III-V materials are suitable for use in long distance fiber optic communication systems due to their faster speed of response and high gain. The superlattice APD is expected to be far more attractive than the conventional APD for their better noise performance. Theoretical studies have been carried out on the photoresponse characteristics of an AI,Ga, _&/GaAs superlatticep+-i-n + structure. It is observed that for a particular d.c. multiplication factor the normalised gain of the device remains constant with frequency and falls steadily after a certain frequency. The band width of the response curve increases with decrease in d.c. multiplication factor. Furthermore, the output current of the superlattice structure increases with the increase in optical power. The device also shows a good percentage of quantum efficiency.
INTRODUCTION
The superlattice structure is one of the novel heterojunction configurations having potential applications
in photonic devices. Avalanche photodiodes (APD) are useful as detectors in long distance fiber optic communication systems in I .3-1.55 pm wavelength range. For a material having nearly equal electron and hole ionisation rates (CXand B respectively), the device becomes noisy due to feedback effect. In order to have a better noise performance, it is desirable to have an APD with u much larger than /?. It has been found that the higher the a//l ratio is, the better the noise performance of the APD. This idea led device design engineers to fabricate and test several revolutionary APD structures[ 11: (a) Graded gap APD where the difference between the ionisation energies and quasistatic field for electrons and holes have been employed. (b) Superlattice and stair case APD which utilises the difference in ionisation energies for electrons and holes created by the asymmetry between conduction and valence band discontinuities in III-V heterojunctions. (c) Channeling APD where a special separation technique of electrons and holes in materials of different band gap via p-n-p-n structures are used. The superlattice is a multilayered heterojunction arrangement consisting of low doped alternate layers of dissimilar, but lattice matched, materials having different band gaps sandwiched between n + and p + regions. Al,Ga, _ &/GaAs is an example of such a typical material combination[2]. The difference in the energy band gap of the two compounds produces a periodic potential in the structure with a period equal to the sum of the thicknesses of two successive layers. Miller et a/.[31 have shown that for parabolic quanS.S.E. M/,-A
tum well structure, the energy band gap discontinuity and valence band wells. While the paper compares these values with the widely accepted values of AE, = 0.85 AE, and AE, = 0.15 AE, based on square well spectra, fails to provide any explanation for the discrepancy. Recently, Brennan et a/.(4] have shown that the conduction band edge discontinuity in the AlGaAs/GaAs system is roughly equal to 75%. A Monte-Carlo simulation technique has been employed[5] to establish that the ratio of the electron to hole impact ionisation rate is roughly one order of magnitude in AlGaAs/GaAs superlattice structure. The enhancement in the LX/Bratio may be accounted for by the fact that when a hot electron accelerating in the AI,Ga, _ 3s layer enters the GaAs layer, it gains an energy equal to the conduction band edge discontinuity, AE,. Electrons thus find an ionisation energy in GaAs less by AE, (Ei N 1.5 eV) than that in bulk (Ei N 2 eV). This results in a large increase in effective IX.This is not the case with /3, as the valence band edge discontinuity AE, is only 15% of AE,[4]. Brennan[S] has also shown that the situation is quite different when the conduction band and valence band are comparable in size. In that case /l is considerably suppressed only for small step lengths. This is due to the fact that the holes lose more kinetic energy to the potential steps which are now larger than before. Experimentally fabricated and tested superlattice APD with AI,Ga, _&/GaAs showed an a//? ratio equal to 8[1]. Chin ef a1.[2] have calculated theoretically the impact ionisation in a multilayered heterojunction structure where it is shown that effective ionisation rates of electrons and holes are very different even if they are nearly the same in the bulk material. This paper presents theoretical results of different photoresponse characteristics of an Al,Ga, _ ,As/ AE, is evenly divided between conduction
P.
676
CHAKRABARTI
where E, is the high field threshold energy, or, ~~and 6, are the threshold fields for the carriers to overcome the decelerating effects of thermal, optical phonon and ionisation scattering. In the structure under consideration the conduction band edge discontinuity is much larger than the valence band edge discontinuity and equation (1) can be used to find out the multiplication factors for electrons and holes in different layers by adjusting the band edge discontinuities with the ionisation threshold energies. The thickness of the layers has been assumed to be such that after an electron impact ionises in GaAs layer it can gain sufficient energy in the layer to get out of the
THEORY
The structure under conisderation is a reverse biased p +-i-n + superlattice structure and is shown in Fig. 1 along with the energy band diagram. When the front surface of the device is exposed to carrier modulated optical radiation, multiplication of the light generated carriers occurs. Starting from the basic ionisation equation in terms of ionisation threshold and considering the effect of band edge discontinuity at the heterojunction we calculate: (a) Electron and hole multiplication factors M, and Mp and the corresponding intrinsic response time z, and z,, in each layer. (b) The effective multiplication factors M,,, Mp and their corresponding effective intrinsic response times me, 7p for electrons and holes respectively for the entire structure. (c) The overall multiplication gain considering the effect of optically generated carriers and the corresponding response time. (d) The frequency dependent multiplication gain of the device.
/ hv 0
PAL
The impact ionisation rate which depends expontentially on the impact ionisation threshold and the energy from which the electron starts to be accelerated has been computed using the relation[6]:
GaAs superlattice APD. We have considered the effect of optically generated carriers in different layers. The incident photons are modulated at a carrier frequency o. The theory has been presented in the next section.
,,Alx
and B. B.
well.
Let us consider a single layer of the multilayer structure of the central i-region. If the avalanche region extends from 0 to W, the d.c. multiplication factor at any point x within the avalanche region is given by[7]: exp( -s:(a 1 -Jowa exp( -s:(a
dx”)
-/?)dr”)dx”
Go,_,As
,GaAs
/ “Y
/
/
I
-
n+
P’ 1
II ’
- 8)
M(x) =
‘1
1
II ‘2
‘3
II ‘4
/
1
‘5
t
0
/d
II/ ‘6
‘7
‘6
AE,’
/
n+
Fig. 1. The schematic diagram of an AI,Ga, _ .As/GaAs superlattice structure along with a typical energy band diagram.
(*)
Superlattice avalanche photodiode and the electron and hole multiplication given by:
factors are
M”=M(x)l,=W,
(3)
Mp = M(x) L=o,
(4)
The corrected intrinsic response time of the carriers at x is defined a#]:
(5) where v, and up are the saturated velocity of electrons and holes respectively. K, = N(a//?), N being a slowly varying number from l/3 to 2 as p/a changes from 1 to 0.001[8]. The intrinsic response times for electrons and holes are given by: r,=r(x)L=W,
(6)
677
is the total junction area of the device and the integration extends over the entire avalanche region (0 to L). It may be pointed out here that in this case the avalanche region comprises alternate layers of AlGaAs and GaAs. Therefore, the integration is carried out separately for each layer using appropriate parameters and inserting suitable boundary values. Here:
where 4 is incident photon flux density and a, is the photo absorption coefficient per unit length. The total injected current is given by: L 1, =
I, +
I/,*+
qA
M=f.
and rp = r(x) Ix=O.
g(x) dx,
(15)
s0 where again the integration is computed separately for each layer and the resultant value has been used. The d.c. multiplication gain of the superlattice APD is thus given by: s
(7)
Now, the optically generated carriers will also Using the above equations and inserting the approaffect the effective response times of the carriers. The priate values of the parameters we calculate the expression for the overall response time of the carrier multiplication factors (M, LAS, (Mph,, , Wn)NGaAa, is obtained from the relation: Vfphma~s and intrinsic response times (r,)oaAs, (rp LAS, CT,hIGaAsand bJAIGaAs. The net multiplication arising out of a single AlGaAs/GaAs heterojunction is given by: 7 Mni = (Mn )oa~s X (Ma )AIG~A~ 9 Mpi = ‘iMp)~a~sX (M~)AIG~A~. For n such heterojunction of AlGaAs/GaAs, effective multiplication will be given by:
(8)
Ldx) -dx.
=:+$+qA
s0
(9) the
M,, = M,,, x Mn2 x . . . x M,, ,
(10)
Mp<=MpixMp2x.‘.xMpn.
(11)
Thus the effective multiplication factor is dependent on the number of heterojunctions. However, the effective intrinsic response time of the overall structure will be a sum of the intrinsic response times of the individual AlGaAs/GaAs heterojunctions. Thus if rni and rpi correspond to the intrinsic response times of electrons and holes for a single AlGaAs/GaAs layer, then
(17)
7(x)
The last term has been calculated for different layers separately as before. The frequency dependence of the multiplication gain is represented by[A: M M(w)=
(1 +
W2j,,f272)1/2'
(18)
where M(w) is the frequency dependent multiplication gain, o is the frequency of operation. Quantum eficiency The quantum efficiency is defined as the number of electron-hole pairs generated per incident photon and can be expressed as[6]:
n
7°C =
c i=l
7ni9
(12)
7pc =
C
i=,
4/s
(19)
’ = P,,,/Ahv
n 7pi.
(13)
Considering the carriers generated in the avalanche region due to absorption of optical radiation, the current Z can be written as[7]:
where I,, is the photogenerated current due to absorption of incident radiation, Poptis the incident power at a wavelength 1, hv is the energy of the photons. Assuming reflection only at the entrance, the photon flux density 4 can be written as:
Z=M,,Z,+M,Z,+qA
Lg(x)M(x)dx, (14) s0 where g(x) = photoelectric volume generation rate, A
dJ I
po,tu-m Ahv
’
678
P.
CHAKRABARTI
where R is the reflection coefficient at the entrance given by the relation[6]: (21) Here A is the refractive index of the semiconductor where the light is incident and K is the absorption constant given by:
(22) NUMERICAL
CALCULATIONS
AND DISCUSSION
Numerical computations have been carried out for the Al,,,Gq,,As/GaAs p +-i-n + superlattice structure. Total length of the i-region is 2.5 pm[l]. This consists of 50 alternating layers of Al,.,Ga,,,As(550 A) and GaAs(450 A). The first layer in the i-region being Al,,,Ga,,,As, the electrons after traversing the first layer then enter the second layer (GaAs) and experience a reduction in ionisation threshold energy by an amount of band edge discontinuity AE,. Since the central region is an i-layer, the field has been assumed to be constant in this region. The material parameters for the two semiconductors used in the calculation are taken from Refs[2,6,9]. The ionisation rates a and j3 in AlGaAs and GaAs layers are computed from equation (1) inserting appropriate values of ionisation thresholds (considering band edge discontinuity) and other parameters. The ionisation rates of bulk GaAs calculated from equation (1) for electric fields 4.0 x 10’ V/m and 2.0 x 10’ V/m were found to be within 5-l% of the corresponding experimental values of Bulman et aZ.[lO]. This indicates that the values used in the analysis are reliable. Using the values of ionisation rates we compute first the electron-hole multiplication factors for a single AlGaAs/GaAs heterojunction and their corresponding intrinsic response times. Using equations (10) and (11) we then compute the effective multiplication factors M,, and Mpe. Similarly use of equations (12) and (13) yields the overall intrinsic response time of carriers r,, and 7pe.
and B. B.
PAL
The total current Z is computed using equation (14). The integration jtg(x) M(x) dx extends over the entire i-region and includes alternate layers of GaAs and AlGaAs. The integration is carried out with the help of the computer. The total injected current is similarly computed from equation (15). Using the values of Z and Z,, we compute the d.c. multiplication factor from equation (16). Finally, we compute the value of the intrinsic response time r of the device using equation (17). It is found that the d.c. multiplication factor is different for different applied voltages across the device. For each value of the applied electric field we study how the d.c. multiplication gain changes as the frequency is increased. Equation (18) is utilised for this purpose. Figure 2 shows the variation of the effective multiplication factors for electrons and holes (M, and M, respectively) with the inverse of applied electric field. It is seen from the variation that as the field increases the values of M, and M, increase and become very large near the breakdown field. The values are found to be in good agreement with those of Capasso[l 11. Furthermore, the effective a,//( ratio is also found to be in agreement with the theoretical results of Chin et af.[2].
\
%,+ (W/m’)
Fig. 3. Total output current vs incident optical power density for two different applied fields.
lOOO-
E g 5
3
loo-
B f
z 0) 2 E t J w
j_
.P 0.6=
‘0
0.25
0.45
0.35
l/F
I 0.55x10-7
(m/v)
Fig. 2. Effective multiplication factors M, (for electrons) and Mpy(for holes) against inverse of applied electric field.
g
0
z
Fig.
I
I
I
10’
108
109
Frequency 4.
I 10’0
1 10”
(Gtiz)
Normal&d multiplication factor vs frequency for different d.c. multiplication factors.
Superlattice avalanche photodiode Figure 3 shows the plot of the total current I with the input optical power density for different values of the applied field. It is evident that for a particular field, the total current I increases steadily with the incident optical power density. This can be accounted for the additional generation of carriers due to the increase in absorption of optical power. Figure 4 shows the variation of the frequency dependent normalised multiplication factor M(w)/M against modulated frequency for different d.c. multiplication factors. The curves show that in the lower frequency range the normalised gain is almost unity and falls after a certain frequency. It is also seen that the lower the d.c. multiplication factor, the higher is the bandwidth. The quantum efficiency of the device has been calculated at 1 = 0.75 pm, using equations (19)-(22). It has been found that the quantum efficiency at this wavelength is as high as 70%.
CONCLUSION
The above study makes it clear that the incident optical radiation on a superlattice structure increases the total output current of the device and the current increases with the increase in optical power density. Such a device can respond to light modulated at
679
microwave frequencies. Since in such a structure a,/& is considerably high, the noise is expected to be moderately low. Moreover, the device gives a good percentage of quantum efficiency which can be further improved by using antireflection coating on the front surface. REFERENCES
1. F. Capasso, Surface Sci. 142, 513 (1984). 2. R. Chin, N. Holonyak Jun., G. E. Stillman, J. Y. Tang and K. Hess. Elecbon. Let?. 16, 467 (1980). 3. R. C. Miller. A. C. Gossard. D. A. Klemman and 0. Monteaner, &Jx Rev. 29B, .3740 (1984). 4. K. Brennan, T. Wang and K. Hess, IEEE Electron. Dev. Lett. EDM, 199 (1985). 5. K. Brennan, IEEE Trans. Electron Dev. ED-32, 2197
(1985). 6. S. M. Sze, Physics of Semiconductor Devices, 2nd Edn. Wiley, New Delhi (1981). 7. J. R. Grierson and S. O’Hara, Solid-St. Electron. 18, 1003 (1975). 8. D. P. Schinke, R. G. Smith and A. R. Hartman, Photodetectors, Topics in Applied Physics (Edited by H. Kressel), Vol. 39. Springer, Berlin (1980). 9. M. S. Burroughs, K. Hess, N. Holonyak Jun., W. D. Laidig and G. E. Stillman, So/id-St. Electron. 25, 161 (1982).
10. G. E.‘Bulman, V. M. Robbins, K. F. Brennan, K. Hess and G. E. Stillman. IEEE Electron Deu. Lett. EDL-4. 181 (1983).
11. F. Capasso, W. T. Tsang, A. L. Hutchinson and G. F. Williams, Appl. Phys. Lett. 40, 38 (1982).
0038-I 101/87 $3.00 +O.OO
Copyright 0 1987 Pergamon Journals Ltd
OPTICAL CHARACTERISTICS OF A SUPERLATTICE AVALANCHE PHOTODIODE P. CHAKRABARTI and B. B. PAL Department of Electronics Engineering, Institute of Technology, Banaras Hindu University, Varanasi, India (Received
22 May 1986; in revised form 13 September
1986)
Abstract-Avalanche photodiodes using III-V materials are suitable for use in long distance fiber optic communication systems due to their faster speed of response and high gain. The superlattice APD is expected to be far more attractive than the conventional APD for their better noise performance. Theoretical studies have been carried out on the photoresponse characteristics of an AI,Ga, _&/GaAs superlatticep+-i-n + structure. It is observed that for a particular d.c. multiplication factor the normalised gain of the device remains constant with frequency and falls steadily after a certain frequency. The band width of the response curve increases with decrease in d.c. multiplication factor. Furthermore, the output current of the superlattice structure increases with the increase in optical power. The device also shows a good percentage of quantum efficiency.
INTRODUCTION
The superlattice structure is one of the novel heterojunction configurations having potential applications
in photonic devices. Avalanche photodiodes (APD) are useful as detectors in long distance fiber optic communication systems in I .3-1.55 pm wavelength range. For a material having nearly equal electron and hole ionisation rates (CXand B respectively), the device becomes noisy due to feedback effect. In order to have a better noise performance, it is desirable to have an APD with u much larger than /?. It has been found that the higher the a//l ratio is, the better the noise performance of the APD. This idea led device design engineers to fabricate and test several revolutionary APD structures[ 11: (a) Graded gap APD where the difference between the ionisation energies and quasistatic field for electrons and holes have been employed. (b) Superlattice and stair case APD which utilises the difference in ionisation energies for electrons and holes created by the asymmetry between conduction and valence band discontinuities in III-V heterojunctions. (c) Channeling APD where a special separation technique of electrons and holes in materials of different band gap via p-n-p-n structures are used. The superlattice is a multilayered heterojunction arrangement consisting of low doped alternate layers of dissimilar, but lattice matched, materials having different band gaps sandwiched between n + and p + regions. Al,Ga, _ &/GaAs is an example of such a typical material combination[2]. The difference in the energy band gap of the two compounds produces a periodic potential in the structure with a period equal to the sum of the thicknesses of two successive layers. Miller et a/.[31 have shown that for parabolic quanS.S.E. M/,-A
tum well structure, the energy band gap discontinuity and valence band wells. While the paper compares these values with the widely accepted values of AE, = 0.85 AE, and AE, = 0.15 AE, based on square well spectra, fails to provide any explanation for the discrepancy. Recently, Brennan et a/.(4] have shown that the conduction band edge discontinuity in the AlGaAs/GaAs system is roughly equal to 75%. A Monte-Carlo simulation technique has been employed[5] to establish that the ratio of the electron to hole impact ionisation rate is roughly one order of magnitude in AlGaAs/GaAs superlattice structure. The enhancement in the LX/Bratio may be accounted for by the fact that when a hot electron accelerating in the AI,Ga, _ 3s layer enters the GaAs layer, it gains an energy equal to the conduction band edge discontinuity, AE,. Electrons thus find an ionisation energy in GaAs less by AE, (Ei N 1.5 eV) than that in bulk (Ei N 2 eV). This results in a large increase in effective IX.This is not the case with /3, as the valence band edge discontinuity AE, is only 15% of AE,[4]. Brennan[S] has also shown that the situation is quite different when the conduction band and valence band are comparable in size. In that case /l is considerably suppressed only for small step lengths. This is due to the fact that the holes lose more kinetic energy to the potential steps which are now larger than before. Experimentally fabricated and tested superlattice APD with AI,Ga, _&/GaAs showed an a//? ratio equal to 8[1]. Chin ef a1.[2] have calculated theoretically the impact ionisation in a multilayered heterojunction structure where it is shown that effective ionisation rates of electrons and holes are very different even if they are nearly the same in the bulk material. This paper presents theoretical results of different photoresponse characteristics of an Al,Ga, _ ,As/ AE, is evenly divided between conduction
P.
676
CHAKRABARTI
where E, is the high field threshold energy, or, ~~and 6, are the threshold fields for the carriers to overcome the decelerating effects of thermal, optical phonon and ionisation scattering. In the structure under consideration the conduction band edge discontinuity is much larger than the valence band edge discontinuity and equation (1) can be used to find out the multiplication factors for electrons and holes in different layers by adjusting the band edge discontinuities with the ionisation threshold energies. The thickness of the layers has been assumed to be such that after an electron impact ionises in GaAs layer it can gain sufficient energy in the layer to get out of the
THEORY
The structure under conisderation is a reverse biased p +-i-n + superlattice structure and is shown in Fig. 1 along with the energy band diagram. When the front surface of the device is exposed to carrier modulated optical radiation, multiplication of the light generated carriers occurs. Starting from the basic ionisation equation in terms of ionisation threshold and considering the effect of band edge discontinuity at the heterojunction we calculate: (a) Electron and hole multiplication factors M, and Mp and the corresponding intrinsic response time z, and z,, in each layer. (b) The effective multiplication factors M,,, Mp and their corresponding effective intrinsic response times me, 7p for electrons and holes respectively for the entire structure. (c) The overall multiplication gain considering the effect of optically generated carriers and the corresponding response time. (d) The frequency dependent multiplication gain of the device.
/ hv 0
PAL
The impact ionisation rate which depends expontentially on the impact ionisation threshold and the energy from which the electron starts to be accelerated has been computed using the relation[6]:
GaAs superlattice APD. We have considered the effect of optically generated carriers in different layers. The incident photons are modulated at a carrier frequency o. The theory has been presented in the next section.
,,Alx
and B. B.
well.
Let us consider a single layer of the multilayer structure of the central i-region. If the avalanche region extends from 0 to W, the d.c. multiplication factor at any point x within the avalanche region is given by[7]: exp( -s:(a 1 -Jowa exp( -s:(a
dx”)
-/?)dr”)dx”
Go,_,As
,GaAs
/ “Y
/
/
I
-
n+
P’ 1
II ’
- 8)
M(x) =
‘1
1
II ‘2
‘3
II ‘4
/
1
‘5
t
0
/d
II/ ‘6
‘7
‘6
AE,’
/
n+
Fig. 1. The schematic diagram of an AI,Ga, _ .As/GaAs superlattice structure along with a typical energy band diagram.
(*)
Superlattice avalanche photodiode and the electron and hole multiplication given by:
factors are
M”=M(x)l,=W,
(3)
Mp = M(x) L=o,
(4)
The corrected intrinsic response time of the carriers at x is defined a#]:
(5) where v, and up are the saturated velocity of electrons and holes respectively. K, = N(a//?), N being a slowly varying number from l/3 to 2 as p/a changes from 1 to 0.001[8]. The intrinsic response times for electrons and holes are given by: r,=r(x)L=W,
(6)
677
is the total junction area of the device and the integration extends over the entire avalanche region (0 to L). It may be pointed out here that in this case the avalanche region comprises alternate layers of AlGaAs and GaAs. Therefore, the integration is carried out separately for each layer using appropriate parameters and inserting suitable boundary values. Here:
where 4 is incident photon flux density and a, is the photo absorption coefficient per unit length. The total injected current is given by: L 1, =
I, +
I/,*+
qA
M=f.
and rp = r(x) Ix=O.
g(x) dx,
(15)
s0 where again the integration is computed separately for each layer and the resultant value has been used. The d.c. multiplication gain of the superlattice APD is thus given by: s
(7)
Now, the optically generated carriers will also Using the above equations and inserting the approaffect the effective response times of the carriers. The priate values of the parameters we calculate the expression for the overall response time of the carrier multiplication factors (M, LAS, (Mph,, , Wn)NGaAa, is obtained from the relation: Vfphma~s and intrinsic response times (r,)oaAs, (rp LAS, CT,hIGaAsand bJAIGaAs. The net multiplication arising out of a single AlGaAs/GaAs heterojunction is given by: 7 Mni = (Mn )oa~s X (Ma )AIG~A~ 9 Mpi = ‘iMp)~a~sX (M~)AIG~A~. For n such heterojunction of AlGaAs/GaAs, effective multiplication will be given by:
(8)
Ldx) -dx.
=:+$+qA
s0
(9) the
M,, = M,,, x Mn2 x . . . x M,, ,
(10)
Mp<=MpixMp2x.‘.xMpn.
(11)
Thus the effective multiplication factor is dependent on the number of heterojunctions. However, the effective intrinsic response time of the overall structure will be a sum of the intrinsic response times of the individual AlGaAs/GaAs heterojunctions. Thus if rni and rpi correspond to the intrinsic response times of electrons and holes for a single AlGaAs/GaAs layer, then
(17)
7(x)
The last term has been calculated for different layers separately as before. The frequency dependence of the multiplication gain is represented by[A: M M(w)=
(1 +
W2j,,f272)1/2'
(18)
where M(w) is the frequency dependent multiplication gain, o is the frequency of operation. Quantum eficiency The quantum efficiency is defined as the number of electron-hole pairs generated per incident photon and can be expressed as[6]:
n
7°C =
c i=l
7ni9
(12)
7pc =
C
i=,
4/s
(19)
’ = P,,,/Ahv
n 7pi.
(13)
Considering the carriers generated in the avalanche region due to absorption of optical radiation, the current Z can be written as[7]:
where I,, is the photogenerated current due to absorption of incident radiation, Poptis the incident power at a wavelength 1, hv is the energy of the photons. Assuming reflection only at the entrance, the photon flux density 4 can be written as:
Z=M,,Z,+M,Z,+qA
Lg(x)M(x)dx, (14) s0 where g(x) = photoelectric volume generation rate, A
dJ I
po,tu-m Ahv
’
678
P.
CHAKRABARTI
where R is the reflection coefficient at the entrance given by the relation[6]: (21) Here A is the refractive index of the semiconductor where the light is incident and K is the absorption constant given by:
(22) NUMERICAL
CALCULATIONS
AND DISCUSSION
Numerical computations have been carried out for the Al,,,Gq,,As/GaAs p +-i-n + superlattice structure. Total length of the i-region is 2.5 pm[l]. This consists of 50 alternating layers of Al,.,Ga,,,As(550 A) and GaAs(450 A). The first layer in the i-region being Al,,,Ga,,,As, the electrons after traversing the first layer then enter the second layer (GaAs) and experience a reduction in ionisation threshold energy by an amount of band edge discontinuity AE,. Since the central region is an i-layer, the field has been assumed to be constant in this region. The material parameters for the two semiconductors used in the calculation are taken from Refs[2,6,9]. The ionisation rates a and j3 in AlGaAs and GaAs layers are computed from equation (1) inserting appropriate values of ionisation thresholds (considering band edge discontinuity) and other parameters. The ionisation rates of bulk GaAs calculated from equation (1) for electric fields 4.0 x 10’ V/m and 2.0 x 10’ V/m were found to be within 5-l% of the corresponding experimental values of Bulman et aZ.[lO]. This indicates that the values used in the analysis are reliable. Using the values of ionisation rates we compute first the electron-hole multiplication factors for a single AlGaAs/GaAs heterojunction and their corresponding intrinsic response times. Using equations (10) and (11) we then compute the effective multiplication factors M,, and Mpe. Similarly use of equations (12) and (13) yields the overall intrinsic response time of carriers r,, and 7pe.
and B. B.
PAL
The total current Z is computed using equation (14). The integration jtg(x) M(x) dx extends over the entire i-region and includes alternate layers of GaAs and AlGaAs. The integration is carried out with the help of the computer. The total injected current is similarly computed from equation (15). Using the values of Z and Z,, we compute the d.c. multiplication factor from equation (16). Finally, we compute the value of the intrinsic response time r of the device using equation (17). It is found that the d.c. multiplication factor is different for different applied voltages across the device. For each value of the applied electric field we study how the d.c. multiplication gain changes as the frequency is increased. Equation (18) is utilised for this purpose. Figure 2 shows the variation of the effective multiplication factors for electrons and holes (M, and M, respectively) with the inverse of applied electric field. It is seen from the variation that as the field increases the values of M, and M, increase and become very large near the breakdown field. The values are found to be in good agreement with those of Capasso[l 11. Furthermore, the effective a,//( ratio is also found to be in agreement with the theoretical results of Chin et af.[2].
\
%,+ (W/m’)
Fig. 3. Total output current vs incident optical power density for two different applied fields.
lOOO-
E g 5
3
loo-
B f
z 0) 2 E t J w
j_
.P 0.6=
‘0
0.25
0.45
0.35
l/F
I 0.55x10-7
(m/v)
Fig. 2. Effective multiplication factors M, (for electrons) and Mpy(for holes) against inverse of applied electric field.
g
0
z
Fig.
I
I
I
10’
108
109
Frequency 4.
I 10’0
1 10”
(Gtiz)
Normal&d multiplication factor vs frequency for different d.c. multiplication factors.
Superlattice avalanche photodiode Figure 3 shows the plot of the total current I with the input optical power density for different values of the applied field. It is evident that for a particular field, the total current I increases steadily with the incident optical power density. This can be accounted for the additional generation of carriers due to the increase in absorption of optical power. Figure 4 shows the variation of the frequency dependent normalised multiplication factor M(w)/M against modulated frequency for different d.c. multiplication factors. The curves show that in the lower frequency range the normalised gain is almost unity and falls after a certain frequency. It is also seen that the lower the d.c. multiplication factor, the higher is the bandwidth. The quantum efficiency of the device has been calculated at 1 = 0.75 pm, using equations (19)-(22). It has been found that the quantum efficiency at this wavelength is as high as 70%.
CONCLUSION
The above study makes it clear that the incident optical radiation on a superlattice structure increases the total output current of the device and the current increases with the increase in optical power density. Such a device can respond to light modulated at
679
microwave frequencies. Since in such a structure a,/& is considerably high, the noise is expected to be moderately low. Moreover, the device gives a good percentage of quantum efficiency which can be further improved by using antireflection coating on the front surface. REFERENCES
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