- Email: [email protected]
GaAlAs quantum wells
Superlattices and Microstructures 84 (2015) 192–197
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
The non parabolicity and the enhancement of the intersubband absorption in GaAs/GaAlAs quantum wells A. Rajira, H. Akabli, A. Almaggousi, A. Abounadi ⇑ Groupe d’Étude des Matériaux Optoélectroniques (G.E.M.O.), F.S.T.G, Cadi Ayyad University, BP 549 Marrakech, Morocco
a r t i c l e
i n f o
Article history: Received 2 May 2015 Accepted 6 May 2015 Available online 14 May 2015 Keywords: Quantum well Intersubband transitions Non parabolicity Absorption enhancement
a b s t r a c t We have performed calculations of the transition energies in a GaAs/AlGaAs quantum well with 38% of Aluminum. Three energy levels could be confined below a certain well width leading to two allowed transitions. Calculations were made in both the parabolic and the non parabolic cases. While the E23 transition stands above the E12 one in the first case, there is a clear crossing between these two transitions in the second case for a specific geometry (Lw = 7.7 nm). This suggests the possible enhancement in the absorption if a radiation has the appropriate frequency such that hm ¼ E12 ¼ E23 . This interesting proposition was investigated after the comparison of the absorption coefficient calculations for two different geometries. Such structures may be used in two photons absorption devices because the E3 level is close to the continuum. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Unipolar devices, based on intersubband transitions, are widely used in several applications. They are used on the absorption, the detection, and the emission of radiations, especially in photonic filters [1,2], Quantum Well Infrared Photodetectors (QWIP) [3–5] and in Quantum Cascade Lasers (QCLs) [6,7].
⇑ Corresponding author. E-mail address: [email protected] (A. Abounadi). http://dx.doi.org/10.1016/j.spmi.2015.05.008 0749-6036/Ó 2015 Elsevier Ltd. All rights reserved.
A. Rajira et al. / Superlattices and Microstructures 84 (2015) 192–197
193
More recently, there is a growing interest on mid-infrared and terahertz radiation sources [8,9]. However, these are often weak emitters. Innovative concepts are then needed for detecting the resulting weak signals. Good sensitivity was demonstrated in a quadratic two photon detector showing optical non linearity for a radiation with power as low as 0.1 W/cm2 [10]. These quadratic detectors are mostly based on three levels quantum well structures and require transitions of equal energy. The third level has, also, to be resonant with the continuum. In such devices, the modeling of absorption properties is closely linked to the accurate determination of energy levels and wave functions of electron. This can help to design devices with improved absorption. In this optic we explore GaAs/AlGaAs systems having three confined levels and study the behavior of the resulting two distinct transitions. The Aluminum composition was taken as high as 38% because it insures intersubband transitions for mid-infrared applications. Our electronic states calculations, presented in Section 2, show that below a given well width, there exit at most two distinct transitions. However, and particularly when including bands non parabolicity, our calculations predict the existence of a specific geometry for which the two transitions coincide in energy. In such geometry and for this energy the optical absorption could increase significantly. This was investigated on Section 3 through the absorption coefficient calculations. 2. Electronic states and transition energies The conduction band (C.B.) electronic states are calculated in the GaAs/AlGaAs symmetrical quantum well using the envelop function approximation and taking an offset [11]
V b ¼ DEc ¼ 0:65DEg ¼ 308 meV The calculations are developed within the parabolic approximation on one hand, and by taking into account the non parabolicity effects on another hand. In the last case the effective mass energy dependence follows Nelson’s model [12]:
mw ðeÞ ¼ mw 1 þ
e Ew
;
Vb e mb ðeÞ ¼ mb 1 Eb
EwðbÞ : the well’s (the barrier’s) band gap. mwðbÞ : the C.B. electron effective mass in the parabolic approximation in the well (in the barrier). The numerical resolution is based on the finite difference method. It considers the structure as a succession of n + 1 layers of very small but equal thicknesses, Dz. In each layer the mass (mn), the potential (Un) and the envelop functions (vn ) are considered constants. The envelop functions in the different layers, within this model, are linked together by:
vnþ1 ¼ vn ðDzÞ2 k2n þ
2
kn ¼
2mn ðU n EÞ 2 h
mn mn þ1 þ v mnþ1 mm1 n1
ð1Þ
:
The confined eigenenergies and the corresponding envelop functions are determined by expressing the boundary conditions. The evolution versus the well width, Lw, of the three confined levels ðE1p ; E2p ; E3p Þ and ðE1np ; E2np ; E3np Þ respectively for the parabolic and the non parabolic cases is reported in Fig. 1. In the parabolic approximation the E3 level increases but slows down before joining the continuum of states (E3 is no more confined below Lw = 8 nm). However, the results show that including the bands non parabolicity has the effect of lowering the positions of the excited levels (E2 and E3) whom reach the continuum for lower well widths (when compared to the parabolic case). Moreover, the slowing down of E3, around, Vb, is more pronounced. E3 seems to be ‘‘pinned’’ before reaching the continuum. The E3 level quits the well around 6 nm.
194
A. Rajira et al. / Superlattices and Microstructures 84 (2015) 192–197
350 E1 E2 E3 E1np E2np E3np
Levels energies (meV)
300 250 200 150 100 50 0 4
6
8
10
12
14
Well thikness (nm) Fig. 1. Energy levels evolution as a function of the well thickness for both the parabolic (Ei) and the non parabolic (Einp) cases. Symbols are added to distinguish between the curves.
Intersubband energies (meV)
240
Ep 12 Ep 23
200 160 120 80 40 4
6
8
10
12
14
Well thikness (nm) Fig. 2. The intersubband transition energies as a function of the well thickness. The parabolic case.
This behavior considerably influences the evolution, as a function of Lw, of the intersubband transitions energies in our structures, as shown in Figs. 2 and 3. From these figures and for large Lw, the E23 energy transition remains higher than that of E12 transition. This behavior is maintained, in the parabolic case, for any well width though the E23 energy position approaches closely the E12 one when E3 is close to Vb. Including non parabolicity has the effect of decreasing, notably, the E23 transition energy when decreasing Lw from about 9 nm. But the most interesting result is the clear crossing between the E12 and E23 transitions happening at Lw = 7.7 nm. The origin of this neat crossing is due to the particular behavior of the E3 level in the non parabolic case as previously explained. This result suggests the possible increase in the absorption of a radiation having a frequency such that hm ¼ E12 ¼ E23 . To quantify this phenomenon we propose to calculate the absorption coefficient for two different geometries. The first, for Lw = 9 nm, where the two transitions have separate energies, offers the possibility of simultaneously absorbing the two corresponding frequencies. The second, for Lw = 7.7 nm, where the transition energies coincide, a better absorption may occur for a radiation with the appropriate frequency.
A. Rajira et al. / Superlattices and Microstructures 84 (2015) 192–197
Intersubband energies (meV)
200
195
Enp 12 Enp 23
160
120
80
40 4
6
8
10
12
14
Well thikness (nm) Fig. 3. The clear crossing between the E12 and E23 transitions for Lw = 7.7 nm in the non parabolic case.
3. Absorption coefficient calculations To study the optical absorption one has to start by calculating the absorption coefficient aðxÞ. Its expression for intersubband transitions is given by [13]:
aðxÞ ¼
X 2pe2 e2z f ij xij dðEj Ei hxÞ f ðEj Þ f ðEi Þ ncxe0 Xm i;j 2
ð2Þ
2
k? where Ei ðk? Þ ¼ ei þ h2m is the in-plane dispersion relation of the ith subband (ei is the ith confined level: Fig. 1) xfi ¼ Ef Ei ¼ ef ei : f(E) is the Fermi–Dirac function and the fif is the oscillator strength related h to the Efi transition:
f if ¼
E2 2x me2 2 e2 vi pz vf ¼ fi z l2if mxfi h z h
ð3Þ
E
lif ¼ vi zvf is the dipolar matrix element and ez is the growth axis component of the polarization vector of the electromagnetic radiation. To take account of the transition’s peak broadening the d function is replaced by a Lorentzian one of 2 hCij FWHM. By summing over the whole k? states the intersubband coefficient can be definitely written as [14]:
3
2
aðxÞ ¼
7 hCij 4pe2 e2z 6 7 6X 6 ðni nj Þl2ij x2ij 7 2 2 ncxe0 L 4i;j¼1;2;3 ðej ei hxÞ þ ðhCij Þ 5
ð4Þ
ðiþjÞodd ihj
where the summation runs only for the allowed transitions as the quantum well is squared. ni is the free electron density per unit surface, of the ith subband. These quantities, on which the aðxÞ depends strongly, are calculated after the determination of the Fermi level’s position (EF), at a given temperature and for a given total electron density, ns. Following the calculations steps detailed in Ref. [15], the evolution, as a function of temperature, of each carrier’s density (ni) is reported for the three considered subbands on Fig. 4 for ns = 5.1011 cm2. We notice that an appreciable amount of electrons, exceeding 1010 cm2, are transferred to the first excited level (E2) for temperatures above 300 K (RT). This justifies the increase of the absorption at the
196
A. Rajira et al. / Superlattices and Microstructures 84 (2015) 192–197
Carries densities (Cm-2)
5x1011 4x1011
n1 n2 n3
3x1011 2x1011 1x1011
0 0
100
200
300
400
500
Temperature (K) Fig. 4. Carriers densities dependence on temperature.
E23 transition energy which becomes comparable, at RT, to that of the E12 transition. This is shown in Fig. 5 for the geometry where the transitions are separate. In this case one part of electrons participate to the absorption process of the first transition while the other part to the second transition. In another hand, it is possible to make all carriers contribute to the absorption of a same frequency. Indeed, Fig. 3 predicts the crossing of E12 and E23 transitions for a specific geometry. Fig. 6 presents the results of the absorption coefficient calculations for the structure (Lw = 7.7 nm, Vb = 308 meV) where the two distinct transitions correspond to the same energy (about 127 meV, k ¼ 9:75 lm), in the mid-Infrared domain. An increase of the total absorption coefficient magnitude (atot ), when compared to that of each transition, is remarked. This particular structure could then lead to a better detection of radiations having weak intensities. Such structures may be potentially useful in devices like the quadratic two-photon detectors [8].
Absorption coefficient (a. u.)
4
←α12 T = 300 K n = 5.10
3
s
11
-2
cm
←α23 2
1
0 60
100
140
180
220
Energy (meV) Fig. 5. The absorption coefficient as a function of energy. For Lw = 9 nm the two transitions are separate.
197
A. Rajira et al. / Superlattices and Microstructures 84 (2015) 192–197
Absorption coefficient (a. u.)
10
αtot T = 300 K
8
ns = 5.10 11 cm-2 α12
6
4
α23 2
0 60
100
140
180
220
Energy (meV) Fig. 6. The enhancement of the absorption at the energy where the two transitions coincide for a specific geometry (Lw = 7.7 nm).
4. Conclusion We have calculated the electronic states in a 38% Al composition GaAs/AlGaAs quantum well in both the parabolic and the non parabolic cases. Below a certain well width three levels, at most are confined leading to two distinct allowed transitions. When compared to the parabolic case, the positions of the excited levels were decreased when bands non parabolicity was included into the calculations. Besides, in the last case, the slowing down of the E3 level, around Vb, was found to be more pronounced. This explains the clear crossing between E12 and E23 for a specific configuration (E12 ¼ E23 = 127 meV for Lw = 7.7 nm). This suggests a possible enhancement of the absorption of a radiation whose frequency fulfills hm ¼ E12 ¼ E23 . This interesting prediction was confirmed after the analysis of the absorption results. The absorption coefficient calculations were performed for two geometries. One, corresponding to Lw = 9 nm and where the two distinct transitions are separate in energy, could be used in a multidetection process involving a unique system. The other, where the two distinct transitions E12 and E23 coincide, corresponds to Lw = 7.7 nm. We believe that devices based on the later specific structure could be engineered for a better detection of signals having weak intensities. References [1] C.P. Liu, T.J. Fan, Opt. Laser Technol. 62 (2014) 82–88. [2] C.C. Liao, S.F. Tang, T.C. Chen, C.D. Chiang, S.T. Yang, W.K. Su, Proc. SPIE 6119, Semiconductor Photodetectors III, 611905, 2006. [3] Y. Arslan, T. Çolakoglu, G. Torunoglu, O. Aktas, C. Besikci, Infrared Phys. Technol. 59 (2013) 108–111. [4] E.A. DeCuir Jr, K.K. Choi, J. Sun, P.S. Wijewarnasuriya, Infrared Phys. Technol. (2014), http://dx.doi.org/10.1016/ j.infrared.2014.09.018. [5] A. Almaggoussi, A. Abounadi, H. Akabli, K. Zekentes, M. Androulaki, Eur. Phys. J. Appl. Phys. 45 (2009) 20301. [6] M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, H. Melchior, Science 295 (2002) 301–305. [7] R. Colombelli, J.M. Manceau, Phys. Rev. X 5 (2015) 011031. [8] H. Schneider, H.C. Liu, S. Winner, O. Drachenko, M. Helm, J. Faist, Appl. Phys. Lett. 93 (101114) (2008) 49–51. [9] C. Franke, M. Walter, M. Helm, H. Schneider, Infrared Phys. Technol. (2014), http://dx.doi.org/10.1016/ j.infrared.2014.08.012. [10] H. Schneider, T. Maier, H.C. Liu, M. Walther, P. Koidl, Opt. Lett. 30 (2005) 287. [11] B.F. Levine, J. Appl. Phys. 74 (1993) R1. [12] D.F. Nelson, R.C. Miller, D.A. Kleinman, Phys. Rev. B 35 (1987) 7770. [13] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, EDP Sciences, Paris, 1988. [14] E. Rosencher, B. Vinter, Optoélectronique, Dunod, Paris, 2002. [15] H. Akabli, A. Rajira, A. Almaggoussi, A. Abounadi, Eur. Phys. J. Appl. Phys. 52 (2010) 20302.
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
The non parabolicity and the enhancement of the intersubband absorption in GaAs/GaAlAs quantum wells A. Rajira, H. Akabli, A. Almaggousi, A. Abounadi ⇑ Groupe d’Étude des Matériaux Optoélectroniques (G.E.M.O.), F.S.T.G, Cadi Ayyad University, BP 549 Marrakech, Morocco
a r t i c l e
i n f o
Article history: Received 2 May 2015 Accepted 6 May 2015 Available online 14 May 2015 Keywords: Quantum well Intersubband transitions Non parabolicity Absorption enhancement
a b s t r a c t We have performed calculations of the transition energies in a GaAs/AlGaAs quantum well with 38% of Aluminum. Three energy levels could be confined below a certain well width leading to two allowed transitions. Calculations were made in both the parabolic and the non parabolic cases. While the E23 transition stands above the E12 one in the first case, there is a clear crossing between these two transitions in the second case for a specific geometry (Lw = 7.7 nm). This suggests the possible enhancement in the absorption if a radiation has the appropriate frequency such that hm ¼ E12 ¼ E23 . This interesting proposition was investigated after the comparison of the absorption coefficient calculations for two different geometries. Such structures may be used in two photons absorption devices because the E3 level is close to the continuum. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Unipolar devices, based on intersubband transitions, are widely used in several applications. They are used on the absorption, the detection, and the emission of radiations, especially in photonic filters [1,2], Quantum Well Infrared Photodetectors (QWIP) [3–5] and in Quantum Cascade Lasers (QCLs) [6,7].
⇑ Corresponding author. E-mail address: [email protected] (A. Abounadi). http://dx.doi.org/10.1016/j.spmi.2015.05.008 0749-6036/Ó 2015 Elsevier Ltd. All rights reserved.
A. Rajira et al. / Superlattices and Microstructures 84 (2015) 192–197
193
More recently, there is a growing interest on mid-infrared and terahertz radiation sources [8,9]. However, these are often weak emitters. Innovative concepts are then needed for detecting the resulting weak signals. Good sensitivity was demonstrated in a quadratic two photon detector showing optical non linearity for a radiation with power as low as 0.1 W/cm2 [10]. These quadratic detectors are mostly based on three levels quantum well structures and require transitions of equal energy. The third level has, also, to be resonant with the continuum. In such devices, the modeling of absorption properties is closely linked to the accurate determination of energy levels and wave functions of electron. This can help to design devices with improved absorption. In this optic we explore GaAs/AlGaAs systems having three confined levels and study the behavior of the resulting two distinct transitions. The Aluminum composition was taken as high as 38% because it insures intersubband transitions for mid-infrared applications. Our electronic states calculations, presented in Section 2, show that below a given well width, there exit at most two distinct transitions. However, and particularly when including bands non parabolicity, our calculations predict the existence of a specific geometry for which the two transitions coincide in energy. In such geometry and for this energy the optical absorption could increase significantly. This was investigated on Section 3 through the absorption coefficient calculations. 2. Electronic states and transition energies The conduction band (C.B.) electronic states are calculated in the GaAs/AlGaAs symmetrical quantum well using the envelop function approximation and taking an offset [11]
V b ¼ DEc ¼ 0:65DEg ¼ 308 meV The calculations are developed within the parabolic approximation on one hand, and by taking into account the non parabolicity effects on another hand. In the last case the effective mass energy dependence follows Nelson’s model [12]:
mw ðeÞ ¼ mw 1 þ
e Ew
;
Vb e mb ðeÞ ¼ mb 1 Eb
EwðbÞ : the well’s (the barrier’s) band gap. mwðbÞ : the C.B. electron effective mass in the parabolic approximation in the well (in the barrier). The numerical resolution is based on the finite difference method. It considers the structure as a succession of n + 1 layers of very small but equal thicknesses, Dz. In each layer the mass (mn), the potential (Un) and the envelop functions (vn ) are considered constants. The envelop functions in the different layers, within this model, are linked together by:
vnþ1 ¼ vn ðDzÞ2 k2n þ
2
kn ¼
2mn ðU n EÞ 2 h
mn mn þ1 þ v mnþ1 mm1 n1
ð1Þ
:
The confined eigenenergies and the corresponding envelop functions are determined by expressing the boundary conditions. The evolution versus the well width, Lw, of the three confined levels ðE1p ; E2p ; E3p Þ and ðE1np ; E2np ; E3np Þ respectively for the parabolic and the non parabolic cases is reported in Fig. 1. In the parabolic approximation the E3 level increases but slows down before joining the continuum of states (E3 is no more confined below Lw = 8 nm). However, the results show that including the bands non parabolicity has the effect of lowering the positions of the excited levels (E2 and E3) whom reach the continuum for lower well widths (when compared to the parabolic case). Moreover, the slowing down of E3, around, Vb, is more pronounced. E3 seems to be ‘‘pinned’’ before reaching the continuum. The E3 level quits the well around 6 nm.
194
A. Rajira et al. / Superlattices and Microstructures 84 (2015) 192–197
350 E1 E2 E3 E1np E2np E3np
Levels energies (meV)
300 250 200 150 100 50 0 4
6
8
10
12
14
Well thikness (nm) Fig. 1. Energy levels evolution as a function of the well thickness for both the parabolic (Ei) and the non parabolic (Einp) cases. Symbols are added to distinguish between the curves.
Intersubband energies (meV)
240
Ep 12 Ep 23
200 160 120 80 40 4
6
8
10
12
14
Well thikness (nm) Fig. 2. The intersubband transition energies as a function of the well thickness. The parabolic case.
This behavior considerably influences the evolution, as a function of Lw, of the intersubband transitions energies in our structures, as shown in Figs. 2 and 3. From these figures and for large Lw, the E23 energy transition remains higher than that of E12 transition. This behavior is maintained, in the parabolic case, for any well width though the E23 energy position approaches closely the E12 one when E3 is close to Vb. Including non parabolicity has the effect of decreasing, notably, the E23 transition energy when decreasing Lw from about 9 nm. But the most interesting result is the clear crossing between the E12 and E23 transitions happening at Lw = 7.7 nm. The origin of this neat crossing is due to the particular behavior of the E3 level in the non parabolic case as previously explained. This result suggests the possible increase in the absorption of a radiation having a frequency such that hm ¼ E12 ¼ E23 . To quantify this phenomenon we propose to calculate the absorption coefficient for two different geometries. The first, for Lw = 9 nm, where the two transitions have separate energies, offers the possibility of simultaneously absorbing the two corresponding frequencies. The second, for Lw = 7.7 nm, where the transition energies coincide, a better absorption may occur for a radiation with the appropriate frequency.
A. Rajira et al. / Superlattices and Microstructures 84 (2015) 192–197
Intersubband energies (meV)
200
195
Enp 12 Enp 23
160
120
80
40 4
6
8
10
12
14
Well thikness (nm) Fig. 3. The clear crossing between the E12 and E23 transitions for Lw = 7.7 nm in the non parabolic case.
3. Absorption coefficient calculations To study the optical absorption one has to start by calculating the absorption coefficient aðxÞ. Its expression for intersubband transitions is given by [13]:
aðxÞ ¼
X 2pe2 e2z f ij xij dðEj Ei hxÞ f ðEj Þ f ðEi Þ ncxe0 Xm i;j 2
ð2Þ
2
k? where Ei ðk? Þ ¼ ei þ h2m is the in-plane dispersion relation of the ith subband (ei is the ith confined level: Fig. 1) xfi ¼ Ef Ei ¼ ef ei : f(E) is the Fermi–Dirac function and the fif is the oscillator strength related h to the Efi transition:
f if ¼
E2 2x me2 2 e2 vi pz vf ¼ fi z l2if mxfi h z h
ð3Þ
E
lif ¼ vi zvf is the dipolar matrix element and ez is the growth axis component of the polarization vector of the electromagnetic radiation. To take account of the transition’s peak broadening the d function is replaced by a Lorentzian one of 2 hCij FWHM. By summing over the whole k? states the intersubband coefficient can be definitely written as [14]:
3
2
aðxÞ ¼
7 hCij 4pe2 e2z 6 7 6X 6 ðni nj Þl2ij x2ij 7 2 2 ncxe0 L 4i;j¼1;2;3 ðej ei hxÞ þ ðhCij Þ 5
ð4Þ
ðiþjÞodd ihj
where the summation runs only for the allowed transitions as the quantum well is squared. ni is the free electron density per unit surface, of the ith subband. These quantities, on which the aðxÞ depends strongly, are calculated after the determination of the Fermi level’s position (EF), at a given temperature and for a given total electron density, ns. Following the calculations steps detailed in Ref. [15], the evolution, as a function of temperature, of each carrier’s density (ni) is reported for the three considered subbands on Fig. 4 for ns = 5.1011 cm2. We notice that an appreciable amount of electrons, exceeding 1010 cm2, are transferred to the first excited level (E2) for temperatures above 300 K (RT). This justifies the increase of the absorption at the
196
A. Rajira et al. / Superlattices and Microstructures 84 (2015) 192–197
Carries densities (Cm-2)
5x1011 4x1011
n1 n2 n3
3x1011 2x1011 1x1011
0 0
100
200
300
400
500
Temperature (K) Fig. 4. Carriers densities dependence on temperature.
E23 transition energy which becomes comparable, at RT, to that of the E12 transition. This is shown in Fig. 5 for the geometry where the transitions are separate. In this case one part of electrons participate to the absorption process of the first transition while the other part to the second transition. In another hand, it is possible to make all carriers contribute to the absorption of a same frequency. Indeed, Fig. 3 predicts the crossing of E12 and E23 transitions for a specific geometry. Fig. 6 presents the results of the absorption coefficient calculations for the structure (Lw = 7.7 nm, Vb = 308 meV) where the two distinct transitions correspond to the same energy (about 127 meV, k ¼ 9:75 lm), in the mid-Infrared domain. An increase of the total absorption coefficient magnitude (atot ), when compared to that of each transition, is remarked. This particular structure could then lead to a better detection of radiations having weak intensities. Such structures may be potentially useful in devices like the quadratic two-photon detectors [8].
Absorption coefficient (a. u.)
4
←α12 T = 300 K n = 5.10
3
s
11
-2
cm
←α23 2
1
0 60
100
140
180
220
Energy (meV) Fig. 5. The absorption coefficient as a function of energy. For Lw = 9 nm the two transitions are separate.
197
A. Rajira et al. / Superlattices and Microstructures 84 (2015) 192–197
Absorption coefficient (a. u.)
10
αtot T = 300 K
8
ns = 5.10 11 cm-2 α12
6
4
α23 2
0 60
100
140
180
220
Energy (meV) Fig. 6. The enhancement of the absorption at the energy where the two transitions coincide for a specific geometry (Lw = 7.7 nm).
4. Conclusion We have calculated the electronic states in a 38% Al composition GaAs/AlGaAs quantum well in both the parabolic and the non parabolic cases. Below a certain well width three levels, at most are confined leading to two distinct allowed transitions. When compared to the parabolic case, the positions of the excited levels were decreased when bands non parabolicity was included into the calculations. Besides, in the last case, the slowing down of the E3 level, around Vb, was found to be more pronounced. This explains the clear crossing between E12 and E23 for a specific configuration (E12 ¼ E23 = 127 meV for Lw = 7.7 nm). This suggests a possible enhancement of the absorption of a radiation whose frequency fulfills hm ¼ E12 ¼ E23 . This interesting prediction was confirmed after the analysis of the absorption results. The absorption coefficient calculations were performed for two geometries. One, corresponding to Lw = 9 nm and where the two distinct transitions are separate in energy, could be used in a multidetection process involving a unique system. The other, where the two distinct transitions E12 and E23 coincide, corresponds to Lw = 7.7 nm. We believe that devices based on the later specific structure could be engineered for a better detection of signals having weak intensities. References [1] C.P. Liu, T.J. Fan, Opt. Laser Technol. 62 (2014) 82–88. [2] C.C. Liao, S.F. Tang, T.C. Chen, C.D. Chiang, S.T. Yang, W.K. Su, Proc. SPIE 6119, Semiconductor Photodetectors III, 611905, 2006. [3] Y. Arslan, T. Çolakoglu, G. Torunoglu, O. Aktas, C. Besikci, Infrared Phys. Technol. 59 (2013) 108–111. [4] E.A. DeCuir Jr, K.K. Choi, J. Sun, P.S. Wijewarnasuriya, Infrared Phys. Technol. (2014), http://dx.doi.org/10.1016/ j.infrared.2014.09.018. [5] A. Almaggoussi, A. Abounadi, H. Akabli, K. Zekentes, M. Androulaki, Eur. Phys. J. Appl. Phys. 45 (2009) 20301. [6] M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, H. Melchior, Science 295 (2002) 301–305. [7] R. Colombelli, J.M. Manceau, Phys. Rev. X 5 (2015) 011031. [8] H. Schneider, H.C. Liu, S. Winner, O. Drachenko, M. Helm, J. Faist, Appl. Phys. Lett. 93 (101114) (2008) 49–51. [9] C. Franke, M. Walter, M. Helm, H. Schneider, Infrared Phys. Technol. (2014), http://dx.doi.org/10.1016/ j.infrared.2014.08.012. [10] H. Schneider, T. Maier, H.C. Liu, M. Walther, P. Koidl, Opt. Lett. 30 (2005) 287. [11] B.F. Levine, J. Appl. Phys. 74 (1993) R1. [12] D.F. Nelson, R.C. Miller, D.A. Kleinman, Phys. Rev. B 35 (1987) 7770. [13] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, EDP Sciences, Paris, 1988. [14] E. Rosencher, B. Vinter, Optoélectronique, Dunod, Paris, 2002. [15] H. Akabli, A. Rajira, A. Almaggoussi, A. Abounadi, Eur. Phys. J. Appl. Phys. 52 (2010) 20302.