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Tunneling escape rate in dc-biased periodic multibarrier semiconductor heterostructures
ARTICLE IN PRESS Physica B 405 (2010) 3409–3411
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Tunneling escape rate in dc-biased periodic multibarrier semiconductor heterostructures R. Biswas a, C. Sinha b, P. Panchadhyayee a,n, P.K. Mahapatra c a
Department of Physics, Prabhat Kumar College, Contai, P.O. Contai, Dist. Purba Medinipur, West Bengal 721401, India Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India c Department of Physics and Technophysics, Vidyasagar University, Midnapore, West Bengal 721102, India b
a r t i c l e in fo
abstract
Article history: Received 31 August 2009 Received in revised form 24 February 2010 Accepted 7 May 2010
The tunneling escape rate is calculated on the basis of ballistic miniband transport of electron in the presence of external electric field. The variation of the miniband escape rate is reported as functions of barrier and well widths as well as the externally applied bias. Interesting features with regard to the electric field modulation are observed in the semiconductor heterostructures with small barrier and well widths. For the system with constant periodic length, the maximum confinement of carriers is found for the well width almost equal to half of the barrier width. & 2010 Elsevier B.V. All rights reserved.
Keywords: Tunneling escape rate Ballistic transport Transmission coefficient Electric field
1. Introduction The electric-field-induced carrier escape mechanism through the semiconductor multibarrier heterostructures has been the subject of intense investigation with immense practical importance as well as basic physical interest since 1990s [1–3]. In the context of device applications the tunneling escape rate (ER) plays a pivotal role because the speed of carrier escape controls the span of charging and discharging times. Fast carrier escape leads to high saturation intensities of quantum well modulators [4] and thus improves the switching speed of self-electro-optic effect devices (SEED) [5]. Moreover, the tuning of tunneling current [6,7], a vital issue for device modeling, is achieved by the change in carrier escape dynamics whose key information comes from the dependence of tunneling lifetime on the electric field, the barrier width, and the temperature of the system [8]. It has also been noted that the maximum available power from high frequency resonant tunneling oscillation depends on the maximum escape through the tunneling structures [9]. In most of the previous works [10,11] the inverse of the tunneling lifetime is considered as the escape rate. Actually the rate at which the number of carriers tunnel out through the system should not only depend on the tunneling time [10,11] but also on the probability of transmission. The question of dependence on transmission coefficient (Tc) is immaterial in field-free condition as the n
Corresponding author. Tel.: + 91 3220255956. E-mail address: [email protected] (P. Panchadhyayee).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.05.014
resonant tunneling corresponds to unit transmission across the unbiased structure. But it finds relevance in the case of tunneling through biased systems. To our feeling the ER may be considered as the inverse of the quasi-resonant tunneling lifetime multiplied by the corresponding Tc. The motivation of the present work is to study the dependence of escape rate (in its present form) on the variation of applied bias as well as the width of the well and barrier materials forming the heterostructures.
2. Model and parameters Using the standard transfer matrix formalism and exact Airy function approach the transmission spectrum has been obtained for electrons in the first miniband of the periodic ¨ N-barrier heterostructure by solving the Schrodinger equation using effective mass-dependent boundary conditions [12]. In the present study, as an approximation, we have incorporated the two different dielectric constants (er) for the well (er ¼ 12.9) and barrier (er ¼ 12.048) regions instead of a single value used in our previous calculations [13,14]. The half-width at half-maximum (Dem) obtained numerically from the transmission spectra is deployed to compute the quasi-resonant tunneling lifetime using the energy–time uncertainty relation [13,15]
t¼
_ 2Dem
ð1Þ
Finally, the rate of escape corresponding to a quasiresonant energy level (em) having transmission coefficient (Tc)
ARTICLE IN PRESS R. Biswas et al. / Physica B 405 (2010) 3409–3411
25 6 x1014 ; nb = 2 ; E = 10 V/m
20
-1
and lifetime (t) is given by Tc/t. The sum of the ER for all the quasi-levels in a miniband gives the miniband escape rate (MER) across the structure. In the present model, the barrier width (b) and the well width (w) are expressed in units of corresponding number of cells in barrier (nb) and well (nw) regions. The variation is studied in three different ways: (i) keeping nb constant and changing nw, (ii) changing nb and nw symmetrically, (iii) changing the nb and nw simultaneously keeping the total number of cells constant over a period. For the numerical computation of electric-field-induced ER here we consider the triple-barrier periodic GaAs–Al0.3Ga0.7As system. It is worth mentioning that only one allowed miniband containing a maximum of two quasi-levels occurs in belowbarrier condition (0o e rV0, V0 being the conduction band discontinuity) for all the cases. The system parameters are chosen to be the same as in Ref. [14].
MER (s )
3410
7 x1014 ; nb = 2 ; E = 10 V/m 6 13 x10 ; nb = 4 ; E = 10 V/m
15
7 x1013 ; nb = 4 ; E = 10 V/m
10
5
0 2
4
6
8
10
nw 3. Results and discussion
Fig. 2. Variation of MER as a function of number of well cells (nw) for the system as in Fig. 1.
100
4
10
5
1
6 7
0.1
0.01 100 105
8 106 E (Vm-1)
107
Fig. 3. Variation of MER with applied field for different symmetric structures (nw ¼ nb) for the system as in Fig. 1.
40
32
ER ( x1012 s-1)
nb = nw = 3
MER ( x1012 s-1)
To analyze our results we have plotted the escape rate (Tc/t) and also 1/t with respect to the applied field for the two tunneling states in the first miniband corresponding to two different well– barrier combinations. It can be seen from Fig. 1 that the curves corresponding to 1/t and Tc/t are coincident for fields up to E 106 V m 1 and depart for higher fields. The nature of variation of 1/t and Tc/t shown in Fig. 1 is reproducible for other values of parameters also. For E Z106 V m 1, the difference between the two graphs corresponding to 1/t and Tc/t increases for the situation in which the widths of barrier and well differ. For the higher (solid symbols) of the two levels both the ER and 1/t reproduce the usual nature of variation (decrease in magnitude) with the applied field. However, for the lower level (open symbols), though 1/t increases sharply with field, the decrease in transmission probability lowers the term Tc/t with the increase in field. The lowering of ER in the field regime E 106–107 V m 1 is quite natural as a consequence of partial Stark localization [16]. It may be noted from Fig. 2 that for well width much larger than that of the barrier, the MER decreases with w almost in the
24
16
8
0 100 105
106
E
107
(Vm-1)
Fig. 1. Escape rate (ER) in the forms, 1/t and Tc/t, is plotted as a function of electric field (E) applied on a three-barrier GaAs–Al0.3Ga0.7As system. Here, solid triangle and open inverted triangle correspond to 1/t whereas solid and open circles correspond to Tc/t. Solid lines for nb ¼nw ¼ 5; dashed lines for nb ¼4 and nw ¼ 6. Solid (open) symbols correspond to the higher (lower) tunneling states in the miniband.
same manner for all the cases. This is due to the increase in the length of the structure with increasing nw. As is expected in both the cases of applied field the MER is lower for higher barrier width. The ER for the two fields crosses each other at a particular w–b combination. The point of cross-over shifts towards smaller nw with increasing b (for nb ¼6 the cross-over occurs almost near nw ¼2, not shown in figure). Thus, we conclude that the optimum choice of the electric field depends on the proper well–barrier combination for better performance of device so far as MER is concerned. Fig. 3 depicts the variation of MER with applied field for different symmetric cases of well and barrier widths. It is clear from the figure that when nb and nw are small the internal coupling is too strong to be modulated by the external field, the MER remains almost independent of the external field even up to E¼10 MV m 1. In fact, higher the value of nw (or nb) lower is the value of the field at which the MER gets affected by the field. In this situation the interaction between the barrier potential and the external field becomes much stronger than the internal coupling, thereby producing the crest in the MER characteristics. The position of the crest shifts towards lower fields at higher b (¼w) as expected.
ARTICLE IN PRESS R. Biswas et al. / Physica B 405 (2010) 3409–3411
3411
4. Conclusions
60 MER (x 1012 s-1)
101
MER ( x 1012 s-1)
45
10-1 10-3 10-5
30
1 2 3 4 5 6 7 8 9 nb
15
0 1
2
3
4
5 nb
6
7
8
9
Fig. 4. Variation of MER as a function of number of barrier cells (nb) for constant periodic length structure (nw + nb ¼10) for the system same as in Fig. 1. Inset: Three-barrier GaN–Al0.3Ga0.7N system. Solid line with open square symbol for E ¼106 V m 1 and dashed line with open circle symbol for E ¼107 V m 1.
Finally, Fig. 4 shows the variation of MER with respect to change in nb, keeping the periodic length almost constant. For low nb, the nature of variation of MER is different at the two different fields, although it remains almost similar at higher nb showing a minimum for the well width almost equal to half of the barrier width. This may be attributed to the increase in tunneling lifetime as a result of destructive interference among the reflected and the transmitted part of electron waves within the well under the condition mentioned above. Since the condition for obtaining the minimum remains invariant even at zero-bias condition (not shown), we conclude that the localization of electron waves (other than Stark localization) depends on barrier–well combination and the enhancement in localization becomes maximum when nw ¼nb/2. To apply the model for another III–V system we also present the results for GaN–Al0.3Ga0.7N system [14] in the inset of Fig. 4. As is noted from the figure that the magnitude of MER for the nitride system is much lower than the arsenide one and for the former case the change in MER with nb is much more sensitive compared to the later.
In this article, we have studied numerically the effect of electric field and barrier width on the miniband escape rate of a semiconductor triple-barrier system. So long as the variation of ER is concerned, it is noted that the range of applied field for better efficiency depends strictly on the choice of well and barrier widths. For nearly equal well and barrier widths the carrier transport is mainly controlled by the internal coupling when nw (or nb) is low and by the resultant effect of the internal coupling and the applied field when nw (or nb) is high. For multibarrier systems with constant periodic length, the escape rate becomes a minimum when the width of the barrier is almost double to that of the well, a condition independent of the external bias. The present work might provide a guideline to the experimentalists to study the confinement of carriers choosing the suitable parameters of the structure at which the MER is a minimum.
References [1] A.M. Fox, D.A.B. Miller, G. Livescu, J.E. Cunningham, W.Y. Jan, J. Quantum Electron. 27 (1991) 2281. [2] A. Miller, C.B. Park, K.W.P. Li, Appl. Phys. Lett. 60 (1992) 97. [3] G. Mahgerefteh, C.M. Yang, L. Chen, K. Hu, W. Chen, E. Garmire, A. Madhukar, Appl. Phys. Lett. 61 (1992) 2592. ¨ . Gobel, [4] G. von Plessen, J. Feldmann, K.W. Goossen, B. Schlichtherle, E.O D.A.B. Miller, J.E. Cunningham, Semicond. Sci. Technol. 9 (1994) 523. [5] D.A.B. Miller, Opt. Quantum Electron. 22 (1990) 61. [6] P. Vasilopoulos, F.M. Peeters, D. Aitelhabti, Phys. Rev. B 41 (1990) 10021. [7] W.B. Su, S.M. Lu, C.L. Jiang, H.T. Shih, C.S. Chang, T.T. Tsong, Phys. Rev. B 74 (2006) 155330. [8] L.L. Chang, E.E. Mendez, C. Tejedor, in: Resonant Tunneling in Semiconductors: Physics and Applications, Plenum, New York, 1991, p. 333. [9] S.D. Ganichev, E. Ziemann, Th. Gleim, W. Prettl, I.N. Yassievich, V.I. Perel, I. Wilke, E.E. Haller, Phys. Rev. Lett. 80 (1998) 2409. [10] N. Zou, J. Rammer, K.A. Chao, Phys. Rev. B 46 (1992) 15912. [11] K.R. Lefebvre, A.F.M. Anwar, IEEE J. Quantum Electron. 33 (1997) 187. [12] G. Bastard, Phys. Rev. B 24 (1981) 5693. [13] P. Panchadhyayee, R. Biswas, A. Khan, P.K. Mahapatra, J. Appl. Phys. 104 (2008) 084517. [14] P.K. Mahapatra, P. Panchadhyayee, S.P. Bhattacharya, A. Khan, Physica B 403 (2008) 2780. [15] P.J. Price, Phys. Rev. B 38 (1988) 1994. [16] S.M.A. Nimour, N. Zekri, R. Ouasti, Phys. Lett. A 226 (1997) 393.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Tunneling escape rate in dc-biased periodic multibarrier semiconductor heterostructures R. Biswas a, C. Sinha b, P. Panchadhyayee a,n, P.K. Mahapatra c a
Department of Physics, Prabhat Kumar College, Contai, P.O. Contai, Dist. Purba Medinipur, West Bengal 721401, India Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India c Department of Physics and Technophysics, Vidyasagar University, Midnapore, West Bengal 721102, India b
a r t i c l e in fo
abstract
Article history: Received 31 August 2009 Received in revised form 24 February 2010 Accepted 7 May 2010
The tunneling escape rate is calculated on the basis of ballistic miniband transport of electron in the presence of external electric field. The variation of the miniband escape rate is reported as functions of barrier and well widths as well as the externally applied bias. Interesting features with regard to the electric field modulation are observed in the semiconductor heterostructures with small barrier and well widths. For the system with constant periodic length, the maximum confinement of carriers is found for the well width almost equal to half of the barrier width. & 2010 Elsevier B.V. All rights reserved.
Keywords: Tunneling escape rate Ballistic transport Transmission coefficient Electric field
1. Introduction The electric-field-induced carrier escape mechanism through the semiconductor multibarrier heterostructures has been the subject of intense investigation with immense practical importance as well as basic physical interest since 1990s [1–3]. In the context of device applications the tunneling escape rate (ER) plays a pivotal role because the speed of carrier escape controls the span of charging and discharging times. Fast carrier escape leads to high saturation intensities of quantum well modulators [4] and thus improves the switching speed of self-electro-optic effect devices (SEED) [5]. Moreover, the tuning of tunneling current [6,7], a vital issue for device modeling, is achieved by the change in carrier escape dynamics whose key information comes from the dependence of tunneling lifetime on the electric field, the barrier width, and the temperature of the system [8]. It has also been noted that the maximum available power from high frequency resonant tunneling oscillation depends on the maximum escape through the tunneling structures [9]. In most of the previous works [10,11] the inverse of the tunneling lifetime is considered as the escape rate. Actually the rate at which the number of carriers tunnel out through the system should not only depend on the tunneling time [10,11] but also on the probability of transmission. The question of dependence on transmission coefficient (Tc) is immaterial in field-free condition as the n
Corresponding author. Tel.: + 91 3220255956. E-mail address: [email protected] (P. Panchadhyayee).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.05.014
resonant tunneling corresponds to unit transmission across the unbiased structure. But it finds relevance in the case of tunneling through biased systems. To our feeling the ER may be considered as the inverse of the quasi-resonant tunneling lifetime multiplied by the corresponding Tc. The motivation of the present work is to study the dependence of escape rate (in its present form) on the variation of applied bias as well as the width of the well and barrier materials forming the heterostructures.
2. Model and parameters Using the standard transfer matrix formalism and exact Airy function approach the transmission spectrum has been obtained for electrons in the first miniband of the periodic ¨ N-barrier heterostructure by solving the Schrodinger equation using effective mass-dependent boundary conditions [12]. In the present study, as an approximation, we have incorporated the two different dielectric constants (er) for the well (er ¼ 12.9) and barrier (er ¼ 12.048) regions instead of a single value used in our previous calculations [13,14]. The half-width at half-maximum (Dem) obtained numerically from the transmission spectra is deployed to compute the quasi-resonant tunneling lifetime using the energy–time uncertainty relation [13,15]
t¼
_ 2Dem
ð1Þ
Finally, the rate of escape corresponding to a quasiresonant energy level (em) having transmission coefficient (Tc)
ARTICLE IN PRESS R. Biswas et al. / Physica B 405 (2010) 3409–3411
25 6 x1014 ; nb = 2 ; E = 10 V/m
20
-1
and lifetime (t) is given by Tc/t. The sum of the ER for all the quasi-levels in a miniband gives the miniband escape rate (MER) across the structure. In the present model, the barrier width (b) and the well width (w) are expressed in units of corresponding number of cells in barrier (nb) and well (nw) regions. The variation is studied in three different ways: (i) keeping nb constant and changing nw, (ii) changing nb and nw symmetrically, (iii) changing the nb and nw simultaneously keeping the total number of cells constant over a period. For the numerical computation of electric-field-induced ER here we consider the triple-barrier periodic GaAs–Al0.3Ga0.7As system. It is worth mentioning that only one allowed miniband containing a maximum of two quasi-levels occurs in belowbarrier condition (0o e rV0, V0 being the conduction band discontinuity) for all the cases. The system parameters are chosen to be the same as in Ref. [14].
MER (s )
3410
7 x1014 ; nb = 2 ; E = 10 V/m 6 13 x10 ; nb = 4 ; E = 10 V/m
15
7 x1013 ; nb = 4 ; E = 10 V/m
10
5
0 2
4
6
8
10
nw 3. Results and discussion
Fig. 2. Variation of MER as a function of number of well cells (nw) for the system as in Fig. 1.
100
4
10
5
1
6 7
0.1
0.01 100 105
8 106 E (Vm-1)
107
Fig. 3. Variation of MER with applied field for different symmetric structures (nw ¼ nb) for the system as in Fig. 1.
40
32
ER ( x1012 s-1)
nb = nw = 3
MER ( x1012 s-1)
To analyze our results we have plotted the escape rate (Tc/t) and also 1/t with respect to the applied field for the two tunneling states in the first miniband corresponding to two different well– barrier combinations. It can be seen from Fig. 1 that the curves corresponding to 1/t and Tc/t are coincident for fields up to E 106 V m 1 and depart for higher fields. The nature of variation of 1/t and Tc/t shown in Fig. 1 is reproducible for other values of parameters also. For E Z106 V m 1, the difference between the two graphs corresponding to 1/t and Tc/t increases for the situation in which the widths of barrier and well differ. For the higher (solid symbols) of the two levels both the ER and 1/t reproduce the usual nature of variation (decrease in magnitude) with the applied field. However, for the lower level (open symbols), though 1/t increases sharply with field, the decrease in transmission probability lowers the term Tc/t with the increase in field. The lowering of ER in the field regime E 106–107 V m 1 is quite natural as a consequence of partial Stark localization [16]. It may be noted from Fig. 2 that for well width much larger than that of the barrier, the MER decreases with w almost in the
24
16
8
0 100 105
106
E
107
(Vm-1)
Fig. 1. Escape rate (ER) in the forms, 1/t and Tc/t, is plotted as a function of electric field (E) applied on a three-barrier GaAs–Al0.3Ga0.7As system. Here, solid triangle and open inverted triangle correspond to 1/t whereas solid and open circles correspond to Tc/t. Solid lines for nb ¼nw ¼ 5; dashed lines for nb ¼4 and nw ¼ 6. Solid (open) symbols correspond to the higher (lower) tunneling states in the miniband.
same manner for all the cases. This is due to the increase in the length of the structure with increasing nw. As is expected in both the cases of applied field the MER is lower for higher barrier width. The ER for the two fields crosses each other at a particular w–b combination. The point of cross-over shifts towards smaller nw with increasing b (for nb ¼6 the cross-over occurs almost near nw ¼2, not shown in figure). Thus, we conclude that the optimum choice of the electric field depends on the proper well–barrier combination for better performance of device so far as MER is concerned. Fig. 3 depicts the variation of MER with applied field for different symmetric cases of well and barrier widths. It is clear from the figure that when nb and nw are small the internal coupling is too strong to be modulated by the external field, the MER remains almost independent of the external field even up to E¼10 MV m 1. In fact, higher the value of nw (or nb) lower is the value of the field at which the MER gets affected by the field. In this situation the interaction between the barrier potential and the external field becomes much stronger than the internal coupling, thereby producing the crest in the MER characteristics. The position of the crest shifts towards lower fields at higher b (¼w) as expected.
ARTICLE IN PRESS R. Biswas et al. / Physica B 405 (2010) 3409–3411
3411
4. Conclusions
60 MER (x 1012 s-1)
101
MER ( x 1012 s-1)
45
10-1 10-3 10-5
30
1 2 3 4 5 6 7 8 9 nb
15
0 1
2
3
4
5 nb
6
7
8
9
Fig. 4. Variation of MER as a function of number of barrier cells (nb) for constant periodic length structure (nw + nb ¼10) for the system same as in Fig. 1. Inset: Three-barrier GaN–Al0.3Ga0.7N system. Solid line with open square symbol for E ¼106 V m 1 and dashed line with open circle symbol for E ¼107 V m 1.
Finally, Fig. 4 shows the variation of MER with respect to change in nb, keeping the periodic length almost constant. For low nb, the nature of variation of MER is different at the two different fields, although it remains almost similar at higher nb showing a minimum for the well width almost equal to half of the barrier width. This may be attributed to the increase in tunneling lifetime as a result of destructive interference among the reflected and the transmitted part of electron waves within the well under the condition mentioned above. Since the condition for obtaining the minimum remains invariant even at zero-bias condition (not shown), we conclude that the localization of electron waves (other than Stark localization) depends on barrier–well combination and the enhancement in localization becomes maximum when nw ¼nb/2. To apply the model for another III–V system we also present the results for GaN–Al0.3Ga0.7N system [14] in the inset of Fig. 4. As is noted from the figure that the magnitude of MER for the nitride system is much lower than the arsenide one and for the former case the change in MER with nb is much more sensitive compared to the later.
In this article, we have studied numerically the effect of electric field and barrier width on the miniband escape rate of a semiconductor triple-barrier system. So long as the variation of ER is concerned, it is noted that the range of applied field for better efficiency depends strictly on the choice of well and barrier widths. For nearly equal well and barrier widths the carrier transport is mainly controlled by the internal coupling when nw (or nb) is low and by the resultant effect of the internal coupling and the applied field when nw (or nb) is high. For multibarrier systems with constant periodic length, the escape rate becomes a minimum when the width of the barrier is almost double to that of the well, a condition independent of the external bias. The present work might provide a guideline to the experimentalists to study the confinement of carriers choosing the suitable parameters of the structure at which the MER is a minimum.
References [1] A.M. Fox, D.A.B. Miller, G. Livescu, J.E. Cunningham, W.Y. Jan, J. Quantum Electron. 27 (1991) 2281. [2] A. Miller, C.B. Park, K.W.P. Li, Appl. Phys. Lett. 60 (1992) 97. [3] G. Mahgerefteh, C.M. Yang, L. Chen, K. Hu, W. Chen, E. Garmire, A. Madhukar, Appl. Phys. Lett. 61 (1992) 2592. ¨ . Gobel, [4] G. von Plessen, J. Feldmann, K.W. Goossen, B. Schlichtherle, E.O D.A.B. Miller, J.E. Cunningham, Semicond. Sci. Technol. 9 (1994) 523. [5] D.A.B. Miller, Opt. Quantum Electron. 22 (1990) 61. [6] P. Vasilopoulos, F.M. Peeters, D. Aitelhabti, Phys. Rev. B 41 (1990) 10021. [7] W.B. Su, S.M. Lu, C.L. Jiang, H.T. Shih, C.S. Chang, T.T. Tsong, Phys. Rev. B 74 (2006) 155330. [8] L.L. Chang, E.E. Mendez, C. Tejedor, in: Resonant Tunneling in Semiconductors: Physics and Applications, Plenum, New York, 1991, p. 333. [9] S.D. Ganichev, E. Ziemann, Th. Gleim, W. Prettl, I.N. Yassievich, V.I. Perel, I. Wilke, E.E. Haller, Phys. Rev. Lett. 80 (1998) 2409. [10] N. Zou, J. Rammer, K.A. Chao, Phys. Rev. B 46 (1992) 15912. [11] K.R. Lefebvre, A.F.M. Anwar, IEEE J. Quantum Electron. 33 (1997) 187. [12] G. Bastard, Phys. Rev. B 24 (1981) 5693. [13] P. Panchadhyayee, R. Biswas, A. Khan, P.K. Mahapatra, J. Appl. Phys. 104 (2008) 084517. [14] P.K. Mahapatra, P. Panchadhyayee, S.P. Bhattacharya, A. Khan, Physica B 403 (2008) 2780. [15] P.J. Price, Phys. Rev. B 38 (1988) 1994. [16] S.M.A. Nimour, N. Zekri, R. Ouasti, Phys. Lett. A 226 (1997) 393.