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Linear and nonlinear optical properties in a double inverse parabolic quantum well under applied electric and magnetic fields
Superlattices and Microstructures 66 (2014) 129–135
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Linear and nonlinear optical properties in a double inverse parabolic quantum well under applied electric and magnetic fields F. Ungan a,b,⇑, M.E. Mora-Ramos b,c, C.A. Duque b, E. Kasapoglu a, H. Sari a, I. Sökmen d a
Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey Grupo de Materia Condensade-UdeA, Instituto de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medellín, Colombia c Facultad de Ciencias, Universidad Autonoma del Estado de Morelos, Ave. Universidad 1001, CP 62209, Cuernavaca, Morelos, Mexico d _ Turkey Department of Physics, Dokuz Eylül University, Buca, Izmir, b
a r t i c l e
i n f o
Article history: Received 3 October 2013 Accepted 10 December 2013 Available online 20 December 2013 Keywords: Double inverse parabolic quantum well Nonlinear optical property Magnetic field Electric field
a b s t r a c t In the present work, the effects of electric and magnetic fields on the optical absorption coefficient and refractive index changes associated with intersubband transitions in a GaAs/AlxGa1xAs double inverse parabolic quantum well are theoretically calculated within the effective-mass approximation. The expressions for the linear and third-order nonlinear absorption coefficients and refractive index changes are those obtained by using the compact density-matrix approach and iterative method. The results are presented as functions of the incident photon energy for different values of the applied electric and magnetic fields. It is found that the optical absorption coefficient and refractive index changes are strongly affected by the applied electric and magnetic fields. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In the last years, the linear and nonlinear optical properties related to intersubband transitions (ISBTs) in low-dimensional semiconductor systems in which the carriers are confined into one, two, and three dimensions such as quantum wells (QWs), quantum well wires (QWWs), and quantum dots (QDs) have been extensively studied from the viewpoints of both physical interests and novel ⇑ Corresponding author at: Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey. E-mail address: [email protected] (F. Ungan). 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2013.12.006
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F. Ungan et al. / Superlattices and Microstructures 66 (2014) 129–135
optoelectronic device applications. The nonlinear effects in these low-dimensional quantum systems are much stronger than in the bulk materials due to the existence of a strong quantum-confinement effect. Furthermore, these nonlinear properties have the potential for device application in semiconductor lasers [1], single-electron transistors [2], quantum computing [3], optical memories [4], farinfrared laser amplifiers [5], photo-detectors [6,7], and high-speed electro-optical modulators [8]. The development of material growth techniques, such as molecular beam epitaxy (MBE) and metal–organic chemical vapor deposition (MOCVD), makes possible to manufacture high-quality semiconductor QWs with desired profiles of the confinement potential. It is well known that the shape of the QW confining potential significantly affects the nonlinear optical properties. Therefore, the linear and nonlinear optical properties of these structures, with different confining potential functions have been intensively studied both theoretically and experimentally [9–23]. West and Eglash [9] and Levine et al. [10] reported their experimental results for the linear ISB optical absorption within the conduction band of a GaAs QW. The effects of applied electric fields on ISB transitions in modulation-doped single QW structures are studied experimentally by Mathur et al. [11]. Zhang and Xie [12] studied the electric field effect on the second-order nonlinear optical properties of parabolic and semiparabolic QWs. They found that the second-harmonic generation susceptibility sensitively depends on the relaxation rate of the systems. The THz nonlinear absorption of asymmetric double QWs is calculated by Bedoya and Camacho [13]. Yildirim and Tomak [14] studied the effects of the asymmetry and the electric field on the nonlinear ISB optical absorption in GaAs QWs. Chen et al. [15] discussed the linear and nonlinear ISB optical absorption in triangular QWs. Karabulut and Duque [16] calculated the nonlinear optical rectification and optical absorption in GaAs/AlxGa1xAs double QWs under applied electric and magnetic fields. The polaron effects on the optical absorption coefficient and refractive index changes in a square QW were calculated by Li et al. [17]. In addition, Keshavarz et al. [18– 20] have investigated the linear and nonlinear ISB optical absorption in symmetric and asymmetric double semi-parabolic QWs, with and without applied electric field. Baskoutas et al. [21] reported the linear and nonlinear optical absorption coefficients in inverse parabolic QWs under static external electric field. Unal et al. [22] theoretically examined the electric field effect on the refractive index changes in a Modified-Pöschl–Teller QW. Recently, the effects of geometry, hydrostatic pressure, and aluminum concentration on the donor-impurity-related linear and nonlinear optical absorptions in GaAs/AlxGa1xAs concentric quantum rings were calculated by Baghramyan et al. [23]. In previous works, we have studied the features of ISB optical transitions in single QWs [24–28] and double QWs [29,30] in the presence of the intense laser field [24–27], hydrostatic pressure [26,27], and the electric field [28,29]. We found that the application of one, or several, of these external probes has important influences on the optical properties of QWs. In the present article we investigate the linear and nonlinear optical absorptions associated with ISBTs within the conduction band of GaAs/AlxGa1xAs double inverse parabolic quantum well (DIPQW) under applied electric and magnetic fields. The paper is organized as follows: In Section 2, details of the calculations are presented. The numerical results are presented and discussed in Section 3. Finally, the conclusions are given in Section 4. 2. Theory Within the framework of the effective-mass approximation, the Hamiltonian for an electron in DIPQW in the presence of magnetic field B, applied perpendicular to the growth direction, and electric field F applied along the z-direction, can be written as
H¼
i2 1 h e P þ AðrÞ þ VðzÞ þ eFz; 2m c
ð1Þ
where z represents the growth direction, m is the electron effective mass, e is the elementary electron 1 charge, c is the speed of light in the free space, A ¼ 2m ðB rÞ the vector potential of magnetic field B [we choose a vector potential A ¼ ð0; Bz; 0Þ and magnetic field B ¼ ðB; 0; 0Þ], and VðzÞ is the finite confinement potentials in the z-direction. The functional form of the symmetric confinement potential is given by the expression
F. Ungan et al. / Superlattices and Microstructures 66 (2014) 129–135
8 jzj > Lw þ Lb =2 > < V 0; 4V VðzÞ ¼ rLw02 ðLb =2 jzjÞðLw þ Lb =2 jzjÞ; Lb =2 6 jzj 6 Lw þ Lb =2 > : jzj < Lb =2 V 0;
131
ð2Þ
with r ¼ xmax =xc (xmax ¼ 0:3 is the Al concentration at the barriers, xc ¼ 0:2 is the Al concentration at the well center), V0 (=228 meV) is the band discontinuity for xmax ¼ 0:3, V 0 =r is the maximum value of the potential at the center of the DIPQW, Lw is the well width, and Lb is the barrier thickness of DIPQW. Using the envelope wave-function approximation, the electron energy levels E and their corresponding wave functions wðzÞ in a DIPQW can be obtained by solving the Schrödinger equation. The one-dimensional Schrödinger equation is given as:
Hz wðzÞ ¼ EwðzÞ;
ð3Þ
where Hz is the z component of the whole Hamiltonian H [31]:
Hz ¼
h2 @ 2 e2 B2 2 þ z þ VðzÞ þ eFz: 2 2m @z 2m c2
ð4Þ
After the set of energies and their corresponding wave functions are obtained, by using the compact-density matrix method and the iterative procedure, the expression for the linear and nonlinear absorption coefficients, and refractive index changes can be clearly as [32–34]:
rffiffiffiffiffi
l jM21 j2 rV hC12 ; eR ðDE hxÞ2 þ ðhC12 Þ2 rffiffiffiffiffi C12 l I jM 21 j4 rV h að3Þ ðx; IÞ ¼ 2x Þ ð eR e0 nr c ½ðDE hxÞ2 þ ðhC12 Þ2 2 ! 2 2 jM22 M 11 j2 ðDE hxÞ ðhC12 Þ þ 2ðDEÞðDE hxÞ að1Þ ðxÞ ¼ x
1
j2M 21 j2
2
ðDEÞ2 þ ðhC12 Þ
Dnð1Þ ðxÞ rV jM 21 j2 DE hx ¼ nr 2n2r e0 ðDE hxÞ2 þ ðhC12 Þ2
ð5Þ ð6Þ
;
ð7Þ ð8Þ
Dnð3Þ ðx; IÞ lcjM21 j2 rV I ¼ nr 4n3r e0 ½ðDE hxÞ2 þ ðhC Þ2 2 12 ( " #) 2 2 jM 22 M 11 j ðhC12 Þ ð2DE hxÞ 2 2 : 4jM 21 j DEðDE hxÞ ðhC12 Þ 2 DE hx ðDEÞ2 þ ðhC12 Þ ð9Þ In these equations, rV is the electron density, nr is the refractive index, e0 is the permittivity of free space, l is the permeability of the system. On the other hand, C12 is the relaxation rate for states 1 and 2, I is the optical intensity of the incident electromagnetic wave (with the angular frequency x), which excites the structure and leads to the ISB optical transitions. Besides, DE ¼ E2 E1 (where E1 (E2 ) is the initial (final) energy state), eR is the real part of the permittivity, and M ij ¼ jhwi jezjwj ij (i; j ¼ 1; 2) is the electric dipole moment matrix element. Therefore, the total absorption coefficient is given by
aðx; IÞ ¼ að1Þ ðxÞ þ að3Þ ðx; IÞ:
ð10Þ
and the total refractive index change can be written as
Dnðx; IÞ Dnð1Þ ðxÞ Dnð3Þ ðx; IÞ ¼ þ : nr nr nr
ð11Þ
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3. Results and discussion In this section, we will discuss the linear and nonlinear optical properties of the DIPQW, which is shown schematically in Fig. 1, under the influence of external electric and magnetic fields. The following parameters are used in the calculations [35]: m ¼ 0:067m0 , where m0 is the free electron mass, l ¼ 4p 107 H m1 , rv ¼ 3 1016 cm3 , nr ¼ 3:2, C12 ¼ 1=T 12 where T 12 ¼ 0:14ps, I ¼ 0:2 MW=cm2 , Lw1 = Lw2 = 100 Å, and Lb = 25 Å. We plot the total changes in the refractive index as a function of the incident photon energy for three different electric fields values in Fig. 2. When the strength of the applied electric field increases, the asymmetry of the DIPQW becomes stronger, so the energy difference between the lowest two subbands of the system increases, and also the corresponding peak position shifts toward higher energies. In accordance, the total refractive index change becomes significantly reduced with the increasing applied electric field. These features reflect through the presence of two prominent features in the problem: the blue-shift of the relative maxima and minima, and a reduction in their amplitudes, as longer the electric field intensity grows. The first one relates with the enhancement in the energy distance between the ground and first excited levels. The ground state confines within the left-hand side potential well (see Fig. 1), which becomes shifted down by the field, whereas the first excited level confines mostly within the right-hand side quantum well, and is displaced upwards due to the corresponding shifting of the well bottom. On the other hand, the decrease in the amplitude of the total refractive index change has to do with the lost in symmetry reduced by the electric field effect. In our case, this reflects in the enhancement of the E2 E1 difference, as mentioned, and also in the fall in the values of the off-diagonal dipole moment matrix element, M 21 . This is associated with the spatial separation of the two involved wave functions, and with the superposition of the contributions with opposite sign in its evaluation. In Fig. 3, the variation of the total refractive index as a function of the incident photon energy is shown for three different magnetic field strength values. As can be seen from this figure, as long as the magnetic field increases, the change in the magnitude of the total refractive index decreases and also shifts toward higher energies. The main reason for this behavior is that the quantum confinement changes with the increasing magnetic field strength. This change is responsible for the increment of the electron energy difference between the first excited state and the ground state of the system, where an optical transitions occurs. The rise in the field intensity causes (as analogous to the case of a one-dimensional harmonic oscillator) that the energy separation between quantum levels has an increase as long as B augments. As a consequence, there will be a blue shift of the maxima and minima for the total refractive index change. However, the application of a magnetic field, in the
Fig. 1. Schematic diagram of a DIPQW without external fields (black line), with applied electric field (red line) and magnetic field (blue line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
F. Ungan et al. / Superlattices and Microstructures 66 (2014) 129–135
133
.
.
.
.
.
.
.
.
.
.
.
Fig. 2. The variation of total refractive index changes with the incident photon energy for three different electric field values. .
.
.
.
. .
.
.
.
.
.
.
Fig. 3. The variation of total refractive index changes with the incident photon energy for three different magnetic field values.
absence of an externally applied electric field, does not modify the confining potential symmetry. Therefore, diagonal dipole moment matrix elements are zero. At the same time, localization of ground and first excited state is now occurring in such a way that, although showing a reduction, the decrease of the off-diagonal element M 21 is not as pronounced as to induce a strong fail of the relative refractive index change amplitude. Fig. 4 shows the total absorption coefficient as a function of the incident photon energy for three different electric field values. It can be seen that as the electric field increases, the total absorption coefficient shifts toward higher energies. The main reason for this resonance shift is the increment of the energy difference between the ground state and the first excited state, DE ¼ E2 E1 , due to the increase of the electric field intensity. Therefore, the optical absorption coefficients are very sensitive to the applied electric field. These results are in good agreement with the results reported by Zhang and Xie [12]. Here, we better realize about the effects of an applied electric field on the optical responses under study. The behavior observed for the amplitudes of the resonant peaks comes from the combination of the growth in x21 ¼ ðE2 E1 Þ= h and the decrease in M 21 already discussed. It can be seen that the latter element becomes dominant, given the overall reduction in the resonant peak amplitudes. Finally, in Fig. 5, we display the total changes in the absorption coefficient as a function of the incident photon energy for three different magnetic field values. It is clear that as the magnetic field
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F. Ungan et al. / Superlattices and Microstructures 66 (2014) 129–135
.
.
.
.
.
.
.
Fig. 4. The total absorption coefficient versus the photon energy for three different values of the electric field.
.
.
.
.
.
.
.
Fig. 5. The total absorption coefficient versus the photon energy for three different values of the magnetic field.
increases, the total absorption coefficients increase and also shift toward higher energies. As known, increasing the magnetic field leads to a stronger confinement of the carriers and this causes the blue shift of the absorption spectrum. Furthermore, as the result of the increasing magnetic field, the magnitude of the absorption coefficient increases. Now, the increase of the resonant frequency dominates over the decrease of the off-diagonal electric dipole matrix element M 21 (see Eqs. (5) and (6)). In consequence, the resonant peak amplitude behaves as a growing function of B.
4. Conclusions In this work, we have studied the effect of applied electric and magnetic fields on the optical properties of a GaAs/AlxGa1xAs double inverse parabolic quantum well. The electron energy levels and their wave functions are calculated within the effective mass approximation. The total changes in the absorption and refractive index changes are investigated as a function of the incident photon energy for the different values of the applied electric and magnetic fields. The obtained numerical results show that (i) as the electric field increases, the total absorption coefficient (the refractive index) changes shift toward higher energies, and the magnitude of the total absorption coefficient (the refractive index) decreases. (ii) As the magnetic field increases, the total
F. Ungan et al. / Superlattices and Microstructures 66 (2014) 129–135
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absorption coefficient (the refractive index) changes shift toward higher energies, and the magnitude of the total absorption coefficient (the refractive index) increases (decreases). In summary, it is deduced that the electric and magnetic field can play an important role in the optical properties of semiconductor quantum wells. Thus, the modulation of the absorption coefficients and refractive index changes, which are suitable for good performance optical modulators and various infrared optical device applications, can be easily obtained by tuning the magnetic and electric field strength. Acknowledgments This research was partially supported by Colombian Agencies: CODI-Universidad de Antioquia (Estrategia de Sostenibilidad 2013–2014 de la Universidad de Antioquia), Facultad de Ciencias Exactas y Naturales–Universidad de Antioquia (CAD-exclusive dedication Project 2013–2014), and by EL PATRIMONIO AUTÓNOMO FONDO NACIONAL DE FINANCIAMIENTO PARA LA CIENCIA, LA TECNOLOGÍA Y LA INNOVACIÓN, FRANCISCO JOSÉ DE CALDAS. The Escuela de Ingeniería de Antioquia co-supported EIA-UdeA project: Efectos de láser intenso sobre las propiedades ópticas de nanoestrucuras semiconductoras de InGaAsN/GaAs y GaAlAs/GaAs. MEMR acknowledges support from Mexican CONACYT through Grant No. 101777. The authors are grateful to The Scientific Research Project Fund of Cumhuriyet University under the Project Number F-399. The work was done with the help of CENAPAD-SP, Brazil. References [1] N. Kristaedter, O.G. Schmidt, N.N. Ledentsov, D. Bimberg, V.M. Untinov, A.Y. Egorov, A.E. Zhukov, M.V. Maximov, P.S. Kop’ev, Z.I. Alferor, Appl. Phys. Lett. 69 (1996) 1226. [2] E. Leobandung, L. Guo, S. Chou, Appl. Phys. Lett. 67 (1995) 2338. [3] D. Loss, D.P. Divicenzo, Phys. Rev. A 57 (1998) 120. [4] K. Imamura, Y. Sugiyama, Y. Nakata, S. Muto, N. Yokoyama, Jpn. J. Appl. Phys. 34 (1995) L1445. [5] R.F. Kazarinov, R.A. Suris, Sov. Phys. Semicond. 5 (1971) 707. [6] D.A.B. Miller, Int. J. High Speed Electron. 1 (1991) 19. [7] X. Jiang, S.S. Li, M.Z. Tidrow, Physica E 5 (1999) 27. [8] S.Y. Yuen, Appl. Phys. Lett. 43 (1983) 813. [9] L.C. West, J.J. Eglash, Appl. Phys. Lett. 46 (1985) 1156. [10] B.F. Levine, R.J. Malik, J. Walker, K.K. Choi, C.G. Bethea, D.A. Kleinman, J.M. Vandenberg, Appl. Phys. Lett. 50 (1987) 273. [11] A. Mathur, Y. Ohno, F. Matsukura, K. Ohtani, N. Akiba, T. Kuroiwa, H. Nakajima, H. Ohno, Appl. Surf. Sci. 113/114 (1997) 90. [12] L. Zhang, H.J. Xie, Phys. Rev. B 68 (2003) 235315. [13] M. Bedoya, A.S. Camacho, Phys. Rev. B 72 (2005) 155318. [14] H. Yildirim, M. Tomak, Eur. Phys. J. B 50 (2006) 559. [15] B. Chen, K.X. Guo, R.Z. Wang, Z.H. Zhang, Z.L. Liu, Solid State Commun. 149 (2009) 310. [16] I. Karabulut, C.A. Duque, Physica E 43 (2011) 1405. [17] N. Li, K.X. Guo, S. Shao, Superlattices Microstruct. 50 (2011) 461. [18] A. Keshavarz, M.J. Karimi, Phys. Lett. A 374 (2010) 2675. [19] M.J. Karimi, A. Keshavarz, Superlattices Microstruct. 50 (2011) 572. [20] M.J. Karimi, A. Keshavarz, A. Poostforush, Superlattices Microstruct. 49 (2011) 441. [21] S. Baskoutas, C. Garoufalis, A.F. Terzis, Eur. Phys. J. B 84 (2011) 241. [22] V.U. Unal, E. Aksahin, O. Aytekin, Physica E 47 (2013) 103. [23] H.M. Baghramyan, M.G. Barseghyan, A.A. Kirakosyan, R.L. Restrepo, M.E. Mora-Ramos, C.A. Duque, J. Lumin. 145 (2014) 676. [24] F. Ungan, U. Yesilgul, S. Sakiroglu, E. Kasapoglu, H. Sari, I. Sokmen, Phys. Lett. A 374 (2010) 2980. [25] F. Ungan, E. Kasapoglu, C.A. Duque, H. Sari, I. Sokmen, Physica E 25 (2011) 2451. [26] F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. Sokmen, Opt. Commun. 285 (2012) 373. [27] F. Ungan, U. Yesilgul, S. Sakiroglu, M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, H. Sari, I. Sokmen, Opt. Commun. 309 (2013) 158. [28] U. Yesilgul, F. Ungan, S. Sakiroglu, M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, H. Sari, I. Sokmen, J. Lumin. 145 (2014) 379. [29] M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, H. Sari, I. Sokmen, J. Lumin. 135 (2013) 301. [30] F. Ungan, E. Kasapoglu, I. Sokmen, Solid State Commun. 44 (2011) 515. [31] E. Kasapoglu, I. Sokmen, Physica E 27 (2005) 198. [32] D. Ahn, S.L. Chuang, IEEE J. Quantum Electron. QE-23 (1987) 2196. [33] E.M. Goldys, J.J. Shi, Phys. Status Solidi (b) 210 (1998) 237. [34] S. Unlu, I. Karabulut, H. Safak, Physica E 33 (2006) 319. [35] S.L. Chuang, Physics of Optoelectronic Devices, Wiley, New York, 1995, p. 709.
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Linear and nonlinear optical properties in a double inverse parabolic quantum well under applied electric and magnetic fields F. Ungan a,b,⇑, M.E. Mora-Ramos b,c, C.A. Duque b, E. Kasapoglu a, H. Sari a, I. Sökmen d a
Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey Grupo de Materia Condensade-UdeA, Instituto de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medellín, Colombia c Facultad de Ciencias, Universidad Autonoma del Estado de Morelos, Ave. Universidad 1001, CP 62209, Cuernavaca, Morelos, Mexico d _ Turkey Department of Physics, Dokuz Eylül University, Buca, Izmir, b
a r t i c l e
i n f o
Article history: Received 3 October 2013 Accepted 10 December 2013 Available online 20 December 2013 Keywords: Double inverse parabolic quantum well Nonlinear optical property Magnetic field Electric field
a b s t r a c t In the present work, the effects of electric and magnetic fields on the optical absorption coefficient and refractive index changes associated with intersubband transitions in a GaAs/AlxGa1xAs double inverse parabolic quantum well are theoretically calculated within the effective-mass approximation. The expressions for the linear and third-order nonlinear absorption coefficients and refractive index changes are those obtained by using the compact density-matrix approach and iterative method. The results are presented as functions of the incident photon energy for different values of the applied electric and magnetic fields. It is found that the optical absorption coefficient and refractive index changes are strongly affected by the applied electric and magnetic fields. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In the last years, the linear and nonlinear optical properties related to intersubband transitions (ISBTs) in low-dimensional semiconductor systems in which the carriers are confined into one, two, and three dimensions such as quantum wells (QWs), quantum well wires (QWWs), and quantum dots (QDs) have been extensively studied from the viewpoints of both physical interests and novel ⇑ Corresponding author at: Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey. E-mail address: [email protected] (F. Ungan). 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2013.12.006
130
F. Ungan et al. / Superlattices and Microstructures 66 (2014) 129–135
optoelectronic device applications. The nonlinear effects in these low-dimensional quantum systems are much stronger than in the bulk materials due to the existence of a strong quantum-confinement effect. Furthermore, these nonlinear properties have the potential for device application in semiconductor lasers [1], single-electron transistors [2], quantum computing [3], optical memories [4], farinfrared laser amplifiers [5], photo-detectors [6,7], and high-speed electro-optical modulators [8]. The development of material growth techniques, such as molecular beam epitaxy (MBE) and metal–organic chemical vapor deposition (MOCVD), makes possible to manufacture high-quality semiconductor QWs with desired profiles of the confinement potential. It is well known that the shape of the QW confining potential significantly affects the nonlinear optical properties. Therefore, the linear and nonlinear optical properties of these structures, with different confining potential functions have been intensively studied both theoretically and experimentally [9–23]. West and Eglash [9] and Levine et al. [10] reported their experimental results for the linear ISB optical absorption within the conduction band of a GaAs QW. The effects of applied electric fields on ISB transitions in modulation-doped single QW structures are studied experimentally by Mathur et al. [11]. Zhang and Xie [12] studied the electric field effect on the second-order nonlinear optical properties of parabolic and semiparabolic QWs. They found that the second-harmonic generation susceptibility sensitively depends on the relaxation rate of the systems. The THz nonlinear absorption of asymmetric double QWs is calculated by Bedoya and Camacho [13]. Yildirim and Tomak [14] studied the effects of the asymmetry and the electric field on the nonlinear ISB optical absorption in GaAs QWs. Chen et al. [15] discussed the linear and nonlinear ISB optical absorption in triangular QWs. Karabulut and Duque [16] calculated the nonlinear optical rectification and optical absorption in GaAs/AlxGa1xAs double QWs under applied electric and magnetic fields. The polaron effects on the optical absorption coefficient and refractive index changes in a square QW were calculated by Li et al. [17]. In addition, Keshavarz et al. [18– 20] have investigated the linear and nonlinear ISB optical absorption in symmetric and asymmetric double semi-parabolic QWs, with and without applied electric field. Baskoutas et al. [21] reported the linear and nonlinear optical absorption coefficients in inverse parabolic QWs under static external electric field. Unal et al. [22] theoretically examined the electric field effect on the refractive index changes in a Modified-Pöschl–Teller QW. Recently, the effects of geometry, hydrostatic pressure, and aluminum concentration on the donor-impurity-related linear and nonlinear optical absorptions in GaAs/AlxGa1xAs concentric quantum rings were calculated by Baghramyan et al. [23]. In previous works, we have studied the features of ISB optical transitions in single QWs [24–28] and double QWs [29,30] in the presence of the intense laser field [24–27], hydrostatic pressure [26,27], and the electric field [28,29]. We found that the application of one, or several, of these external probes has important influences on the optical properties of QWs. In the present article we investigate the linear and nonlinear optical absorptions associated with ISBTs within the conduction band of GaAs/AlxGa1xAs double inverse parabolic quantum well (DIPQW) under applied electric and magnetic fields. The paper is organized as follows: In Section 2, details of the calculations are presented. The numerical results are presented and discussed in Section 3. Finally, the conclusions are given in Section 4. 2. Theory Within the framework of the effective-mass approximation, the Hamiltonian for an electron in DIPQW in the presence of magnetic field B, applied perpendicular to the growth direction, and electric field F applied along the z-direction, can be written as
H¼
i2 1 h e P þ AðrÞ þ VðzÞ þ eFz; 2m c
ð1Þ
where z represents the growth direction, m is the electron effective mass, e is the elementary electron 1 charge, c is the speed of light in the free space, A ¼ 2m ðB rÞ the vector potential of magnetic field B [we choose a vector potential A ¼ ð0; Bz; 0Þ and magnetic field B ¼ ðB; 0; 0Þ], and VðzÞ is the finite confinement potentials in the z-direction. The functional form of the symmetric confinement potential is given by the expression
F. Ungan et al. / Superlattices and Microstructures 66 (2014) 129–135
8 jzj > Lw þ Lb =2 > < V 0; 4V VðzÞ ¼ rLw02 ðLb =2 jzjÞðLw þ Lb =2 jzjÞ; Lb =2 6 jzj 6 Lw þ Lb =2 > : jzj < Lb =2 V 0;
131
ð2Þ
with r ¼ xmax =xc (xmax ¼ 0:3 is the Al concentration at the barriers, xc ¼ 0:2 is the Al concentration at the well center), V0 (=228 meV) is the band discontinuity for xmax ¼ 0:3, V 0 =r is the maximum value of the potential at the center of the DIPQW, Lw is the well width, and Lb is the barrier thickness of DIPQW. Using the envelope wave-function approximation, the electron energy levels E and their corresponding wave functions wðzÞ in a DIPQW can be obtained by solving the Schrödinger equation. The one-dimensional Schrödinger equation is given as:
Hz wðzÞ ¼ EwðzÞ;
ð3Þ
where Hz is the z component of the whole Hamiltonian H [31]:
Hz ¼
h2 @ 2 e2 B2 2 þ z þ VðzÞ þ eFz: 2 2m @z 2m c2
ð4Þ
After the set of energies and their corresponding wave functions are obtained, by using the compact-density matrix method and the iterative procedure, the expression for the linear and nonlinear absorption coefficients, and refractive index changes can be clearly as [32–34]:
rffiffiffiffiffi
l jM21 j2 rV hC12 ; eR ðDE hxÞ2 þ ðhC12 Þ2 rffiffiffiffiffi C12 l I jM 21 j4 rV h að3Þ ðx; IÞ ¼ 2x Þ ð eR e0 nr c ½ðDE hxÞ2 þ ðhC12 Þ2 2 ! 2 2 jM22 M 11 j2 ðDE hxÞ ðhC12 Þ þ 2ðDEÞðDE hxÞ að1Þ ðxÞ ¼ x
1
j2M 21 j2
2
ðDEÞ2 þ ðhC12 Þ
Dnð1Þ ðxÞ rV jM 21 j2 DE hx ¼ nr 2n2r e0 ðDE hxÞ2 þ ðhC12 Þ2
ð5Þ ð6Þ
;
ð7Þ ð8Þ
Dnð3Þ ðx; IÞ lcjM21 j2 rV I ¼ nr 4n3r e0 ½ðDE hxÞ2 þ ðhC Þ2 2 12 ( " #) 2 2 jM 22 M 11 j ðhC12 Þ ð2DE hxÞ 2 2 : 4jM 21 j DEðDE hxÞ ðhC12 Þ 2 DE hx ðDEÞ2 þ ðhC12 Þ ð9Þ In these equations, rV is the electron density, nr is the refractive index, e0 is the permittivity of free space, l is the permeability of the system. On the other hand, C12 is the relaxation rate for states 1 and 2, I is the optical intensity of the incident electromagnetic wave (with the angular frequency x), which excites the structure and leads to the ISB optical transitions. Besides, DE ¼ E2 E1 (where E1 (E2 ) is the initial (final) energy state), eR is the real part of the permittivity, and M ij ¼ jhwi jezjwj ij (i; j ¼ 1; 2) is the electric dipole moment matrix element. Therefore, the total absorption coefficient is given by
aðx; IÞ ¼ að1Þ ðxÞ þ að3Þ ðx; IÞ:
ð10Þ
and the total refractive index change can be written as
Dnðx; IÞ Dnð1Þ ðxÞ Dnð3Þ ðx; IÞ ¼ þ : nr nr nr
ð11Þ
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3. Results and discussion In this section, we will discuss the linear and nonlinear optical properties of the DIPQW, which is shown schematically in Fig. 1, under the influence of external electric and magnetic fields. The following parameters are used in the calculations [35]: m ¼ 0:067m0 , where m0 is the free electron mass, l ¼ 4p 107 H m1 , rv ¼ 3 1016 cm3 , nr ¼ 3:2, C12 ¼ 1=T 12 where T 12 ¼ 0:14ps, I ¼ 0:2 MW=cm2 , Lw1 = Lw2 = 100 Å, and Lb = 25 Å. We plot the total changes in the refractive index as a function of the incident photon energy for three different electric fields values in Fig. 2. When the strength of the applied electric field increases, the asymmetry of the DIPQW becomes stronger, so the energy difference between the lowest two subbands of the system increases, and also the corresponding peak position shifts toward higher energies. In accordance, the total refractive index change becomes significantly reduced with the increasing applied electric field. These features reflect through the presence of two prominent features in the problem: the blue-shift of the relative maxima and minima, and a reduction in their amplitudes, as longer the electric field intensity grows. The first one relates with the enhancement in the energy distance between the ground and first excited levels. The ground state confines within the left-hand side potential well (see Fig. 1), which becomes shifted down by the field, whereas the first excited level confines mostly within the right-hand side quantum well, and is displaced upwards due to the corresponding shifting of the well bottom. On the other hand, the decrease in the amplitude of the total refractive index change has to do with the lost in symmetry reduced by the electric field effect. In our case, this reflects in the enhancement of the E2 E1 difference, as mentioned, and also in the fall in the values of the off-diagonal dipole moment matrix element, M 21 . This is associated with the spatial separation of the two involved wave functions, and with the superposition of the contributions with opposite sign in its evaluation. In Fig. 3, the variation of the total refractive index as a function of the incident photon energy is shown for three different magnetic field strength values. As can be seen from this figure, as long as the magnetic field increases, the change in the magnitude of the total refractive index decreases and also shifts toward higher energies. The main reason for this behavior is that the quantum confinement changes with the increasing magnetic field strength. This change is responsible for the increment of the electron energy difference between the first excited state and the ground state of the system, where an optical transitions occurs. The rise in the field intensity causes (as analogous to the case of a one-dimensional harmonic oscillator) that the energy separation between quantum levels has an increase as long as B augments. As a consequence, there will be a blue shift of the maxima and minima for the total refractive index change. However, the application of a magnetic field, in the
Fig. 1. Schematic diagram of a DIPQW without external fields (black line), with applied electric field (red line) and magnetic field (blue line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. The variation of total refractive index changes with the incident photon energy for three different electric field values. .
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Fig. 3. The variation of total refractive index changes with the incident photon energy for three different magnetic field values.
absence of an externally applied electric field, does not modify the confining potential symmetry. Therefore, diagonal dipole moment matrix elements are zero. At the same time, localization of ground and first excited state is now occurring in such a way that, although showing a reduction, the decrease of the off-diagonal element M 21 is not as pronounced as to induce a strong fail of the relative refractive index change amplitude. Fig. 4 shows the total absorption coefficient as a function of the incident photon energy for three different electric field values. It can be seen that as the electric field increases, the total absorption coefficient shifts toward higher energies. The main reason for this resonance shift is the increment of the energy difference between the ground state and the first excited state, DE ¼ E2 E1 , due to the increase of the electric field intensity. Therefore, the optical absorption coefficients are very sensitive to the applied electric field. These results are in good agreement with the results reported by Zhang and Xie [12]. Here, we better realize about the effects of an applied electric field on the optical responses under study. The behavior observed for the amplitudes of the resonant peaks comes from the combination of the growth in x21 ¼ ðE2 E1 Þ= h and the decrease in M 21 already discussed. It can be seen that the latter element becomes dominant, given the overall reduction in the resonant peak amplitudes. Finally, in Fig. 5, we display the total changes in the absorption coefficient as a function of the incident photon energy for three different magnetic field values. It is clear that as the magnetic field
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Fig. 4. The total absorption coefficient versus the photon energy for three different values of the electric field.
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Fig. 5. The total absorption coefficient versus the photon energy for three different values of the magnetic field.
increases, the total absorption coefficients increase and also shift toward higher energies. As known, increasing the magnetic field leads to a stronger confinement of the carriers and this causes the blue shift of the absorption spectrum. Furthermore, as the result of the increasing magnetic field, the magnitude of the absorption coefficient increases. Now, the increase of the resonant frequency dominates over the decrease of the off-diagonal electric dipole matrix element M 21 (see Eqs. (5) and (6)). In consequence, the resonant peak amplitude behaves as a growing function of B.
4. Conclusions In this work, we have studied the effect of applied electric and magnetic fields on the optical properties of a GaAs/AlxGa1xAs double inverse parabolic quantum well. The electron energy levels and their wave functions are calculated within the effective mass approximation. The total changes in the absorption and refractive index changes are investigated as a function of the incident photon energy for the different values of the applied electric and magnetic fields. The obtained numerical results show that (i) as the electric field increases, the total absorption coefficient (the refractive index) changes shift toward higher energies, and the magnitude of the total absorption coefficient (the refractive index) decreases. (ii) As the magnetic field increases, the total
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absorption coefficient (the refractive index) changes shift toward higher energies, and the magnitude of the total absorption coefficient (the refractive index) increases (decreases). In summary, it is deduced that the electric and magnetic field can play an important role in the optical properties of semiconductor quantum wells. Thus, the modulation of the absorption coefficients and refractive index changes, which are suitable for good performance optical modulators and various infrared optical device applications, can be easily obtained by tuning the magnetic and electric field strength. Acknowledgments This research was partially supported by Colombian Agencies: CODI-Universidad de Antioquia (Estrategia de Sostenibilidad 2013–2014 de la Universidad de Antioquia), Facultad de Ciencias Exactas y Naturales–Universidad de Antioquia (CAD-exclusive dedication Project 2013–2014), and by EL PATRIMONIO AUTÓNOMO FONDO NACIONAL DE FINANCIAMIENTO PARA LA CIENCIA, LA TECNOLOGÍA Y LA INNOVACIÓN, FRANCISCO JOSÉ DE CALDAS. The Escuela de Ingeniería de Antioquia co-supported EIA-UdeA project: Efectos de láser intenso sobre las propiedades ópticas de nanoestrucuras semiconductoras de InGaAsN/GaAs y GaAlAs/GaAs. MEMR acknowledges support from Mexican CONACYT through Grant No. 101777. The authors are grateful to The Scientific Research Project Fund of Cumhuriyet University under the Project Number F-399. The work was done with the help of CENAPAD-SP, Brazil. References [1] N. Kristaedter, O.G. Schmidt, N.N. Ledentsov, D. Bimberg, V.M. Untinov, A.Y. Egorov, A.E. Zhukov, M.V. Maximov, P.S. Kop’ev, Z.I. Alferor, Appl. Phys. Lett. 69 (1996) 1226. [2] E. Leobandung, L. Guo, S. Chou, Appl. Phys. Lett. 67 (1995) 2338. [3] D. Loss, D.P. Divicenzo, Phys. Rev. A 57 (1998) 120. [4] K. Imamura, Y. Sugiyama, Y. Nakata, S. Muto, N. Yokoyama, Jpn. J. Appl. Phys. 34 (1995) L1445. [5] R.F. Kazarinov, R.A. Suris, Sov. Phys. Semicond. 5 (1971) 707. [6] D.A.B. Miller, Int. J. High Speed Electron. 1 (1991) 19. [7] X. Jiang, S.S. Li, M.Z. Tidrow, Physica E 5 (1999) 27. [8] S.Y. Yuen, Appl. Phys. Lett. 43 (1983) 813. [9] L.C. West, J.J. Eglash, Appl. Phys. Lett. 46 (1985) 1156. [10] B.F. Levine, R.J. Malik, J. Walker, K.K. Choi, C.G. Bethea, D.A. Kleinman, J.M. Vandenberg, Appl. Phys. Lett. 50 (1987) 273. [11] A. Mathur, Y. Ohno, F. Matsukura, K. Ohtani, N. Akiba, T. Kuroiwa, H. Nakajima, H. Ohno, Appl. Surf. Sci. 113/114 (1997) 90. [12] L. Zhang, H.J. Xie, Phys. Rev. B 68 (2003) 235315. [13] M. Bedoya, A.S. Camacho, Phys. Rev. B 72 (2005) 155318. [14] H. Yildirim, M. Tomak, Eur. Phys. J. B 50 (2006) 559. [15] B. Chen, K.X. Guo, R.Z. Wang, Z.H. Zhang, Z.L. Liu, Solid State Commun. 149 (2009) 310. [16] I. Karabulut, C.A. Duque, Physica E 43 (2011) 1405. [17] N. Li, K.X. Guo, S. Shao, Superlattices Microstruct. 50 (2011) 461. [18] A. Keshavarz, M.J. Karimi, Phys. Lett. A 374 (2010) 2675. [19] M.J. Karimi, A. Keshavarz, Superlattices Microstruct. 50 (2011) 572. [20] M.J. Karimi, A. Keshavarz, A. Poostforush, Superlattices Microstruct. 49 (2011) 441. [21] S. Baskoutas, C. Garoufalis, A.F. Terzis, Eur. Phys. J. B 84 (2011) 241. [22] V.U. Unal, E. Aksahin, O. Aytekin, Physica E 47 (2013) 103. [23] H.M. Baghramyan, M.G. Barseghyan, A.A. Kirakosyan, R.L. Restrepo, M.E. Mora-Ramos, C.A. Duque, J. Lumin. 145 (2014) 676. [24] F. Ungan, U. Yesilgul, S. Sakiroglu, E. Kasapoglu, H. Sari, I. Sokmen, Phys. Lett. A 374 (2010) 2980. [25] F. Ungan, E. Kasapoglu, C.A. Duque, H. Sari, I. Sokmen, Physica E 25 (2011) 2451. [26] F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. Sokmen, Opt. Commun. 285 (2012) 373. [27] F. Ungan, U. Yesilgul, S. Sakiroglu, M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, H. Sari, I. Sokmen, Opt. Commun. 309 (2013) 158. [28] U. Yesilgul, F. Ungan, S. Sakiroglu, M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, H. Sari, I. Sokmen, J. Lumin. 145 (2014) 379. [29] M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, H. Sari, I. Sokmen, J. Lumin. 135 (2013) 301. [30] F. Ungan, E. Kasapoglu, I. Sokmen, Solid State Commun. 44 (2011) 515. [31] E. Kasapoglu, I. Sokmen, Physica E 27 (2005) 198. [32] D. Ahn, S.L. Chuang, IEEE J. Quantum Electron. QE-23 (1987) 2196. [33] E.M. Goldys, J.J. Shi, Phys. Status Solidi (b) 210 (1998) 237. [34] S. Unlu, I. Karabulut, H. Safak, Physica E 33 (2006) 319. [35] S.L. Chuang, Physics of Optoelectronic Devices, Wiley, New York, 1995, p. 709.