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Ga1-xAlxAs double inverse parabolic quantum well
Optical Materials 47 (2015) 1–6
Contents lists available at ScienceDirect
Optical Materials journal homepage: www.elsevier.com/locate/optmat
Effects of applied electric and magnetic fields on the nonlinear optical properties of asymmetric GaAs=Ga1x Alx As double inverse parabolic quantum well E.B. Al a, F. Ungan b,⇑, U. Yesilgul b, E. Kasapoglu a, H. Sari a, I. Sökmen c a b c
Faculty of Science, Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey Faculty of Technology, Department of Optical Engineering, Cumhuriyet University, 58140 Sivas, Turkey _ Turkey Faculty of Science, Department of Physics, Dokuz Eylül University, 35160 Buca, Izmir,
a r t i c l e
i n f o
Article history: Received 15 May 2015 Received in revised form 23 June 2015 Accepted 23 June 2015
Keywords: Double inverse parabolic quantum well Nonlinear optical properties Electric and magnetic fields
a b s t r a c t The combined effects of electric and magnetic fields on the optical absorption coefficients and refractive index changes related to the intersubband transitions within the conduction band of asymmetric GaAs=Ga1x Alx As double inverse parabolic quantum wells are studied using the effective-mass approximation and the compact density-matrix approach. The results are presented as a function of the incident photon energy for the different values of the electromagnetic fields and the structure parameters such as quantum well width and the Al concentration at the well center. It is found that the optical absorption coefficients and the refractive index changes are strongly affected not only by the magnitudes of the electric and magnetic fields but also by the structure parameters of the system. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Recently, there has been a remarkable interest in the study of the physical properties of the low-dimensional semiconductor heterostructures such as quantum wells (QWs), quantum well wires (QWWs), and quantum dots (QDs). The studies on these structures open a new field in solid state physics, and thus get a lot of potential applications for optoelectronic devices such as hig h-speed-electron-optical-modulators [1], field-effect transistors [2], infrared detectors [3] and semiconductor lasers [4]. Because of the vast variety of technological applications, single and multiple QWs have been extensively investigated in various circumstances such as hydrostatic pressure, temperature, electric, magnetic and laser fields, different doping processes and so on [5–9]. By applying these circumstances, the subband state energies and their related wave functions can be modified according to the request. Thus, with the confinement of the carriers in the QWs, discrete energy levels are formed within the well, and this results in major optical nonlinearity in the semiconductor structure by comparison with that in the bulk material [10,11]. In recent years, the inverse parabolic quantum wells (IPQWs) have attracted considerable attention owing to their unusual ⇑ Corresponding author. E-mail address: [email protected] (F. Ungan). http://dx.doi.org/10.1016/j.optmat.2015.06.048 0925-3467/Ó 2015 Elsevier B.V. All rights reserved.
electronic and optical properties and possible practical applications [12–17]. Chen et al. [12,13] growed IPQWs by molecular-beam epitaxy using digital and analog techniques and studied its quantum-confined Stark effect. Vlaev et al. [14] studied an IPQW in the framework of a technique for doping practical tight-binding calculations and compare the results with experimental data. The linear an nonlinear optical absorption coefficients in IPQWs under static external electric field was calculated by Baskoutas et al. [15]. Niculescu [16] investigated the combined effects of the intense laser radiation and applied magnetic field on the shallow donor binding energy in IPQWs by using a nonperturbative theory within the effective mass approximation. The electron states and related optical responses in asymmetric IPQWs are studied by Duque and Mora-Ramos [17]. The linear and nonlinear optical properties of low-dimensional semiconductor systems have been studied by many authors in recent years [18–24]. Radu [18] has discussed the laser-dressing of electronic quantum states in graded semiconductor nano-structures. Karimi and Vafaei [19] studied the optical rectification and the second harmonic generation coefficients in a strained InGaN/AlGaN quantum well by taking into account impacts of the spontaneous and piezoelectric polarization fields on the potential profile. Zeiri et al. [20] calculated intersubband (ISB) resonant enhancement of the nonlinear optical properties in asymmetric (CdS/ZnSe)/X-BeTe based QWs. Keshavarz and Karimi
2
E.B. Al et al. / Optical Materials 47 (2015) 1–6
[21] investigated linear and nonlinear ISB optical absorption in symmetric double semi-parabolic QWs. The conduction band electron states and the associated ISB-related linear and nonlinear optical absorption coefficient and relative refractive index change are calculated for an asymmetric double n-type d-doped QW in a GaAs-matrix by Rodriguez-Magdaleno et al. [22]. Safarpour et al. [23] developed a finite difference method as well as have used compact density-matrix approach to analyze the effect of laser radiation and geometrical anisotropy on the optical properties of a GaAs=Ga1x Alx As nano-wire super-lattice (NWSL). And Duque et al. [24] obtained the exciton states in their system within the effective mass theory and the band anti-crossing model. The main conclusions of these works can be summarized as not only the geometry of systems but also the external perturbations can significantly influence the nonlinear optical properties of semiconductor structure. In this study, we will focus our study on the effects of electric and magnetic fields on the linear and nonlinear optical properties in asymmetric GaAs=Ga1x Alx As double inverse parabolic QW (DIPQW). We organized our paper as follows: In Section 2, the theoretical knowledge is defined. Then in Section 3, numerical results and discussion are performed. And finally in Section 4, our conclusions are given. 2. Theory
where dn is;
( dn ¼
0;
n is odd
p;
n is ev en
2
ð4Þ
and the wave function described the system is formed from a complete set as [28]:
wðzÞ ¼
X cn /n
ð5Þ
n¼1
where cn is the coefficient of the expansion. After obtaining energy levels and wave functions corresponding to these energy levels, the first-order linear and the third-order nonlinear absorption coefficients (ACs) for the ISB transitions between two subbands can be calculated respectively as follows [29]:
bð1Þ ðxÞ ¼ x
rffiffiffiffi
rffiffiffiffi
ð6Þ
l I jM 21 j4 rv hC21 er e0 nr c ½ðDE hxÞ2 þ ðhC21 Þ2 2 ! 2 2 jM 22 M 11 j2 ðDE hxÞ ðhC21 Þ þ 2ðDEÞðDE hxÞ
bð3Þ ðxÞ ¼ 2x 1
l jM 21 j2 rv hC21 er ðDE hxÞ2 þ ðhC21 Þ2
j2M 21 j2
2
ðDEÞ2 þ ðhC21 Þ
ð7Þ
In this study, we consider the asymmetric GaAs=Ga1x Alx As DIPQW grown along the z-axis, which consists of two QWs of widths Lw1 and Lw2 , separated to each other by a central barrier of width Lb . Using the effective-mass approximation, we studied the effects of electric and magnetic fields on the linear and nonlinear optical properties of this system. Where the electric field is taken in the growth direction ~ F ¼ F ^z and the magnetic field is taken x. The Hamiltonian for perpendicular to the growth direction ~ B ¼ B^
where x is the angular frequency of the incident photon, l is the magnetic permeability, er is the real part of the permittivity, rv is h is the reduced Planck constant, C21 is the ISB the carrier density, relaxation time, DE ¼ E2 E1 is the energy difference between the two lowest levels, E1 and E2 are the quantized energy levels for the initial and final states, respectively, I ¼ IðwÞ is the optical intensity of the incident electromagnetic wave that excites the semiconductor structure and leads to the ISB optical transition, e0 is the permittivity of free space, and nr is the refractive index. Mfi is the dipole matrix element and defined by [30]:
the electron confined in such a model takes on the form of the following [25,26]:
Mfi ¼
1 h e i2 ~ A þ VðzÞ þ e~ F ~ pe þ ~ r H¼ 2me c
ð1Þ
where me is the electron effective mass, ~ pe is the electron momenr is tum operator, e is the electron charge, c is the velocity of light, ~ the electron coordinate, ~ rÞ is the magnetic vector potential A¼~ Að~ ~ ~ ^), which is determined from the condition ~ B¼r (~ A ¼ Bzy A. VðzÞ is the confinement potential and by choosing as the origin of the z-axis the middle of the central barrier, confinement potential is given by following form [27]: 8 V 0; z 6 L2b Lw1 ; z P L2b þ Lw2 and z 6 jLb j > > > L > L > < 4V 0 ð 2b jzjÞð 2b Lw1 jzjÞ ; Lw1 6 z 6 L2b Lw21 VðzÞ ¼ rL > > > ðLb jzjÞðLb þLw jzjÞ > Lb 2 > 2 2 0 : 4V ; 6 z 6 Lw2 rR 2 Lw2 2
ð2Þ where
rL;R ¼ xxmax (xmax ¼ 0:3 is Al concentration at central barrier L;R
and xL;R is Al concentration at the left and right well center). V 0 is band discontinuity and calculated by V 0 ¼ Q c ð1155x þ 370x2 Þ, where Q c ¼ 0:6 is the conduction band offset parameter for GaAs=Ga1x Alx As QW. To solve Eq. (1), we take as base the eigenfunctions of the infinite potential well of width L. These bases are given at following:
rffiffiffi npz 2 cos /n ðzÞ ¼ dn L L
ð3Þ
Z
/f ðzÞjejz/i ðzÞdz;
ði; f ¼ 1; 2Þ
ð8Þ
Thus the total ACs calculated by:
bðxÞ ¼ bð1Þ ðxÞ þ bð3Þ ðxÞ
ð9Þ
The linear and nonlinear relative refractive index changes (RICs) can be expressed respectively as [31]:
" # Dnð1Þ ðxÞ rv jM 21 j2 DE hx ¼ nr 2n2r e0 ðDE hxÞ2 þ ðhC21 Þ2
ð10Þ
Dnð3Þ ðxÞ lcjM21 j2 rv I ¼ nr 4n3r e0 ½ðDE hxÞ2 þ ðhC Þ2 2 21 2
½4ðDE hxÞjM21 j
ðM22 M11 Þ2 2
ðDEÞ2 þ ðhC21 Þ 2
fðDE hxÞ
2
½ðDEÞðDE hxÞ ðhC21 Þ ðhC21 Þ ð2ðDEÞ hxÞg
ð11Þ
and finally the total relative RIC can be written as:
DnðxÞ Dnð1Þ ðxÞ Dnð3Þ ðxÞ ¼ þ nr nr nr
ð12Þ
3. Results and discussion In this paper, we consider the two different asymmetric GaAs=Ga1x Alx As DIPQW models. Firstly, we take into account the asymmetric structure due to the difference in Al concentration in the center of the two QWs ðLw1 ¼ Lw2 ¼ 100 ÅÞ separated each other
E.B. Al et al. / Optical Materials 47 (2015) 1–6
by a barrier of width Lb ¼ 25 Å. Secondly, we take into account the asymmetric structure due to the different width of the two QWs ðLw1 ¼ 100 Å and Lw2 ¼ 50 ÅÞ separated each other by a barrier of width Lb ¼ 25 Å with the same Al concentration in the center of the QWs. In both models, we examined the effects of electric and magnetic fields on the linear and nonlinear optical properties of the system. The parameters used in this study are: me ¼ 0:067m0 (where m0 is the free electron mass), V 0 ¼ 228 meV;
l ¼ 4p 107 H m1 ; rv ¼ 1:0 1017 cm3 ; s12 ¼ 0:14 ps (where s21 ¼ C121 ), I ¼ 0:4 MW cm2 and nr ¼ 3:2. In Fig. 1(a) and (b), we show the changes of the confinement potential of the asymmetric structure due to the difference in Al concentration in the center of the QWs, the lowest two sub-band energy levels, and the square of the corresponding wave functions under the electric and the magnetic fields. As seen from Fig. 1(a), the electron is localized in the left well ðLw1 Þ in the ground state ðE1 Þ, while it is localized in the right well ðLw2 Þ in the first excited state ðE2 Þ in the absence of the electric field. With the increase of the electric field applied on the growth direction, the charge carriers are swept towards left-hand side of the structure, since the electric field leads to the creation of a deeper QW on left-hand side of the structure. Furthermore, the localization of the electron in the
(a)
3
first excited state decreases in Lw2 while it increases in Lw1 . So, E1 (E2 ) decreases (increases) with the increasing of the electric field value. As a result, the energy difference between the two lowest levels ðDEÞ increases. Fig. 1(b) shows that the effect of the magnetic field applied perpendicular to the growth direction brings an additional geometric confinement to the structure. Both E1 and E2 levels increase with this effect. Consequently, DE also increases. In Fig. 2(a) and (b), we plot the effects of the electric and the magnetic fields on the confinement potential of the asymmetric structure with the same Al concentration in the center of the QWs, the lowest two sub-band energy levels, and the square of the corresponding wave functions, respectively. As demonstrated in Fig. 2(a), E1 (E2 ) decreases (increases) with the effect of the electric field. As a result, DE increases. As seen in Fig. 2(b), the magnetic field creates an additional geometric confinement to the structure. So E1 and E2 increase. As a result, DE decreases in this case according to previous case. For different electric and magnetic field values, the total relative RIC as a function of the incident photon energy are presented in Fig. 3(a) and (b), respectively. These calculations are for rL ¼ 6; rR ¼ 3 Lw1 ¼ Lw2 ¼ 100 Å, and Lb ¼ 25 Å. As shown in Fig. 3(a), the total relative RIC depends strongly on the electric field. The total relative RIC shifts towards to higher photon
(b)
Fig. 1. For rL ¼ 6; rR ¼ 3; Lw1 ¼ Lw2 ¼ 100 Å and Lb ¼ 25 Å, the variations of the confinement potential profile, ground and first excited state energies and the squared wave functions for related energy levels for two different values: (a) electric and (b) magnetic fields.
(a)
(b)
Fig. 2. The variations of the confinement potential profile, ground and first excited state energies and the squared wave functions for related energy levels for two different values: (a) electric and (b) magnetic fields for rL ¼ rR ¼ 3; Lw1 ¼ 100 Å; Lw2 ¼ 50 Å and Lb ¼ 25 Å.
4
E.B. Al et al. / Optical Materials 47 (2015) 1–6
(a)
(b)
Fig. 3. The change in the total refractive index as a function of the incident photon energy for three different values: (a) electric and (b) magnetic fields for rL ¼ 6; rR ¼ 3; Lw1 ¼ Lw2 ¼ 100 Å and Lb ¼ 25 Å.
energies (blue-shift) with increasing electric field values. These results are in good qualitative agreement with some previous works [25,32,33]. The main reason for the blue-shift is the increment in DE with the effect of the electric field. This case is seen clearly in Fig. 1(a). Also, the magnitude of the total relative RIC first decreases, and then increases with the increase of electric field value. This is resulted from the matrix elements jM 11 j; jM 22 j and jM21 j in Eqs. (10) and (11). With the increase of electric field, jM11 j (jM 22 j) increases (decreases), and jM 21 j first decreases, and then increases. Where jM 21 j is the dominant term and the magnitude of the total relative RIC varies as jM 21 j under the electric field. As seen in Fig. 3(b), the total relative RIC depends weakly on the magnetic field. The resonance peak of the total relative RIC shifts to blue by applying the magnetic field. Similar results were obtained in studies previously carried out [25,31,34,35]. The main reason for this behavior is that the quantum confinement varies with increasing the magnetic field. The variation of the quantum confinement causes to an increment in DE as seen in Fig. 1(b). Furthermore, increasing the magnetic field causes an increase in the peak amplitude of the total relative RIC. This is resulted from the matrix jM 21 j, as we have mentioned above. With the increase of magnetic field, jM 11 j and jM 22 j decrease while jM 21 j increases.
(a)
In Fig. 4(a) and (b), for different electric and magnetic field values the changes in the total relative RIC as a function of the incident photon energy are shown, respectively. These results are for
rL ¼ rR ¼ 3; Lw1 ¼ 100 Å; Lw2 ¼ 50 Å and Lb ¼ 25 Å. It is seen in Fig. 4(a) that the total relative RIC depends strongly on the electric field. It is observed that the peak of the total relative RIC shifts to blue with increasing electric field value. The accuracy of these results is observed by the compatibility with earlier studies [36]. This is a result of increased in DE with increasing electric field shown in Fig. 2(a). At the same time, the magnitude of the peak first decreases, and then increases for the further electric field values. The reason lies in the variation of the dipole transition matrix element jM21 j shown in Fig. 4(a). The dominant term jM 21 j decreases and than increases with the electric field. Fig. 4(b) shows that the total relative RIC depends weakly on the magnetic field. In this case, the total relative RIC shifts towards to lower photon energy (red-shift) with increasing magnetic field contrary to the previous case. However, the magnetic field applied on the asymmetric structure which consists of the two different width of QWs with the same Al concentration in the center of the QWs has a negligible effect on the linear and nonlinear optical properties of the structure. These results are similar to previous studies
(b)
Fig. 4. The variation of total refractive index changes with the incident photon energy for three different values: (a) electric and (b) magnetic fields for rL ¼ rR ¼ 3; Lw1 ¼ 100 Å; Lw2 ¼ 50 Å and Lb ¼ 25 Å.
5
E.B. Al et al. / Optical Materials 47 (2015) 1–6
(a)
(b)
Fig. 5. The variation of total absorption coefficients as a function of the incident photon energy for three different values: (a) electric and (b) magnetic fields for rL ¼ 6; rR ¼ 3; Lw1 ¼ Lw2 ¼ 100 Å and Lb ¼ 25 Å.
(a)
Fig. 6. The total absorption coefficients versus rL ¼ rR ¼ 3; Lw1 ¼ 100 Å; Lw2 ¼ 50 Å and Lb ¼ 25 Å.
(b)
the
incident
photon
energy
[37]. The reason for this shift is decrease in DE by the effect of the magnetic field as seen in Fig. 2(b). Also, the magnitude of the total relative RIC decreases with the increase of magnetic field value due to the variation of jM 21 j. Fig. 5(a) and (b) show the total AC spectra as a function of the incident photon energy for different electric and magnetic field values, respectively. Where rL ¼ 6; rR ¼ 3; Lw1 ¼ Lw2 ¼ 100 Å and Lb ¼ 25 Å. These figures clearly show that the magnitude and peak position of total ACs are sensitive to electric and magnetic fields values. With the increase of the electric and magnetic field values, the total AC towards higher photon energies (blue-shift) with increasing magnitude significantly for B ¼ 0 and F ¼ 0. Since the jM21 j (jM 22 j) dipole matrix elements increases (decreases), while jM 11 j remains almost constant. When the electric and magnetic fields increase, total AC shifts to blue with increasing the magnitude, which is a reason of increment in energy difference between corresponding energy levels. Furthermore, Blue shift in the electric field is more prominent than that of the magnetic field. Since, the variation of DE under the magnetic field is almost constant. These cases are observed easily from Figs. 1–4. Finally, in Fig. 6(a) and (b), we display the variation of total AC as a function of the incident photon energy for three different values: (a) electric and (b) magnetic fields. These results are for rL ¼ rR ¼ 3; Lw1 ¼ 100 Å; Lw2 ¼ 50 Å and Lb ¼ 25 Å. When we
for
three
different
values:
(a)
electric
and
(b)
magnetic
fields
for
compare the results obtained for this figure with that of Fig. 5, we can see that resonant peak of total AC shifts to the higher photon energies and it’s magnitude increases with increasing electric and magnetic fields values due to physical reasons mentioned above. Additionally, as seen in Fig. 6(b), the magnitudes of the total AC decrease with increasing magnetic field. But increment in the magnitude of the jM 22 M 11 j dipole matrix elements is larger than that of the jM 21 j dipole matrix elements. Thus, the magnitude of the total AC decreases with magnetic fields. 4. Conclusions In summary, we have investigated the combined effects of electric and magnetic fields on the ACs and RICs of asymmetric GaAs=Ga1x Alx As DIPQW. Based on the compact density-matrix approach and the iterative procedure, the calculation is performed within the effective-mass approximation. The numerical results show that both total AC and RIC are sensitive to structure parameters and the effects of electric and magnetic fields. By changing the magnitudes of the external fields together with structure parameters, we can obtain a blue or red shift, without the need for the growth of many different samples. This also gives a new degree of freedom in various infrared optical device applications based on the ISB transition of electrons.
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E.B. Al et al. / Optical Materials 47 (2015) 1–6
We hope that the calculation of the nonlinear optical properties in asymmetric GaAs=Ga1x Alx As DIPQW would make a significant contribution to the theoretical studies and provides a new model for the potential application of optoelectronic devices such as electro-optical modulators, optical switches and infrared photo-detectors.
[15] [16] [17] [18] [19] [20] [21] [22]
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Contents lists available at ScienceDirect
Optical Materials journal homepage: www.elsevier.com/locate/optmat
Effects of applied electric and magnetic fields on the nonlinear optical properties of asymmetric GaAs=Ga1x Alx As double inverse parabolic quantum well E.B. Al a, F. Ungan b,⇑, U. Yesilgul b, E. Kasapoglu a, H. Sari a, I. Sökmen c a b c
Faculty of Science, Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey Faculty of Technology, Department of Optical Engineering, Cumhuriyet University, 58140 Sivas, Turkey _ Turkey Faculty of Science, Department of Physics, Dokuz Eylül University, 35160 Buca, Izmir,
a r t i c l e
i n f o
Article history: Received 15 May 2015 Received in revised form 23 June 2015 Accepted 23 June 2015
Keywords: Double inverse parabolic quantum well Nonlinear optical properties Electric and magnetic fields
a b s t r a c t The combined effects of electric and magnetic fields on the optical absorption coefficients and refractive index changes related to the intersubband transitions within the conduction band of asymmetric GaAs=Ga1x Alx As double inverse parabolic quantum wells are studied using the effective-mass approximation and the compact density-matrix approach. The results are presented as a function of the incident photon energy for the different values of the electromagnetic fields and the structure parameters such as quantum well width and the Al concentration at the well center. It is found that the optical absorption coefficients and the refractive index changes are strongly affected not only by the magnitudes of the electric and magnetic fields but also by the structure parameters of the system. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Recently, there has been a remarkable interest in the study of the physical properties of the low-dimensional semiconductor heterostructures such as quantum wells (QWs), quantum well wires (QWWs), and quantum dots (QDs). The studies on these structures open a new field in solid state physics, and thus get a lot of potential applications for optoelectronic devices such as hig h-speed-electron-optical-modulators [1], field-effect transistors [2], infrared detectors [3] and semiconductor lasers [4]. Because of the vast variety of technological applications, single and multiple QWs have been extensively investigated in various circumstances such as hydrostatic pressure, temperature, electric, magnetic and laser fields, different doping processes and so on [5–9]. By applying these circumstances, the subband state energies and their related wave functions can be modified according to the request. Thus, with the confinement of the carriers in the QWs, discrete energy levels are formed within the well, and this results in major optical nonlinearity in the semiconductor structure by comparison with that in the bulk material [10,11]. In recent years, the inverse parabolic quantum wells (IPQWs) have attracted considerable attention owing to their unusual ⇑ Corresponding author. E-mail address: [email protected] (F. Ungan). http://dx.doi.org/10.1016/j.optmat.2015.06.048 0925-3467/Ó 2015 Elsevier B.V. All rights reserved.
electronic and optical properties and possible practical applications [12–17]. Chen et al. [12,13] growed IPQWs by molecular-beam epitaxy using digital and analog techniques and studied its quantum-confined Stark effect. Vlaev et al. [14] studied an IPQW in the framework of a technique for doping practical tight-binding calculations and compare the results with experimental data. The linear an nonlinear optical absorption coefficients in IPQWs under static external electric field was calculated by Baskoutas et al. [15]. Niculescu [16] investigated the combined effects of the intense laser radiation and applied magnetic field on the shallow donor binding energy in IPQWs by using a nonperturbative theory within the effective mass approximation. The electron states and related optical responses in asymmetric IPQWs are studied by Duque and Mora-Ramos [17]. The linear and nonlinear optical properties of low-dimensional semiconductor systems have been studied by many authors in recent years [18–24]. Radu [18] has discussed the laser-dressing of electronic quantum states in graded semiconductor nano-structures. Karimi and Vafaei [19] studied the optical rectification and the second harmonic generation coefficients in a strained InGaN/AlGaN quantum well by taking into account impacts of the spontaneous and piezoelectric polarization fields on the potential profile. Zeiri et al. [20] calculated intersubband (ISB) resonant enhancement of the nonlinear optical properties in asymmetric (CdS/ZnSe)/X-BeTe based QWs. Keshavarz and Karimi
2
E.B. Al et al. / Optical Materials 47 (2015) 1–6
[21] investigated linear and nonlinear ISB optical absorption in symmetric double semi-parabolic QWs. The conduction band electron states and the associated ISB-related linear and nonlinear optical absorption coefficient and relative refractive index change are calculated for an asymmetric double n-type d-doped QW in a GaAs-matrix by Rodriguez-Magdaleno et al. [22]. Safarpour et al. [23] developed a finite difference method as well as have used compact density-matrix approach to analyze the effect of laser radiation and geometrical anisotropy on the optical properties of a GaAs=Ga1x Alx As nano-wire super-lattice (NWSL). And Duque et al. [24] obtained the exciton states in their system within the effective mass theory and the band anti-crossing model. The main conclusions of these works can be summarized as not only the geometry of systems but also the external perturbations can significantly influence the nonlinear optical properties of semiconductor structure. In this study, we will focus our study on the effects of electric and magnetic fields on the linear and nonlinear optical properties in asymmetric GaAs=Ga1x Alx As double inverse parabolic QW (DIPQW). We organized our paper as follows: In Section 2, the theoretical knowledge is defined. Then in Section 3, numerical results and discussion are performed. And finally in Section 4, our conclusions are given. 2. Theory
where dn is;
( dn ¼
0;
n is odd
p;
n is ev en
2
ð4Þ
and the wave function described the system is formed from a complete set as [28]:
wðzÞ ¼
X cn /n
ð5Þ
n¼1
where cn is the coefficient of the expansion. After obtaining energy levels and wave functions corresponding to these energy levels, the first-order linear and the third-order nonlinear absorption coefficients (ACs) for the ISB transitions between two subbands can be calculated respectively as follows [29]:
bð1Þ ðxÞ ¼ x
rffiffiffiffi
rffiffiffiffi
ð6Þ
l I jM 21 j4 rv hC21 er e0 nr c ½ðDE hxÞ2 þ ðhC21 Þ2 2 ! 2 2 jM 22 M 11 j2 ðDE hxÞ ðhC21 Þ þ 2ðDEÞðDE hxÞ
bð3Þ ðxÞ ¼ 2x 1
l jM 21 j2 rv hC21 er ðDE hxÞ2 þ ðhC21 Þ2
j2M 21 j2
2
ðDEÞ2 þ ðhC21 Þ
ð7Þ
In this study, we consider the asymmetric GaAs=Ga1x Alx As DIPQW grown along the z-axis, which consists of two QWs of widths Lw1 and Lw2 , separated to each other by a central barrier of width Lb . Using the effective-mass approximation, we studied the effects of electric and magnetic fields on the linear and nonlinear optical properties of this system. Where the electric field is taken in the growth direction ~ F ¼ F ^z and the magnetic field is taken x. The Hamiltonian for perpendicular to the growth direction ~ B ¼ B^
where x is the angular frequency of the incident photon, l is the magnetic permeability, er is the real part of the permittivity, rv is h is the reduced Planck constant, C21 is the ISB the carrier density, relaxation time, DE ¼ E2 E1 is the energy difference between the two lowest levels, E1 and E2 are the quantized energy levels for the initial and final states, respectively, I ¼ IðwÞ is the optical intensity of the incident electromagnetic wave that excites the semiconductor structure and leads to the ISB optical transition, e0 is the permittivity of free space, and nr is the refractive index. Mfi is the dipole matrix element and defined by [30]:
the electron confined in such a model takes on the form of the following [25,26]:
Mfi ¼
1 h e i2 ~ A þ VðzÞ þ e~ F ~ pe þ ~ r H¼ 2me c
ð1Þ
where me is the electron effective mass, ~ pe is the electron momenr is tum operator, e is the electron charge, c is the velocity of light, ~ the electron coordinate, ~ rÞ is the magnetic vector potential A¼~ Að~ ~ ~ ^), which is determined from the condition ~ B¼r (~ A ¼ Bzy A. VðzÞ is the confinement potential and by choosing as the origin of the z-axis the middle of the central barrier, confinement potential is given by following form [27]: 8 V 0; z 6 L2b Lw1 ; z P L2b þ Lw2 and z 6 jLb j > > > L > L > < 4V 0 ð 2b jzjÞð 2b Lw1 jzjÞ ; Lw1 6 z 6 L2b Lw21 VðzÞ ¼ rL > > > ðLb jzjÞðLb þLw jzjÞ > Lb 2 > 2 2 0 : 4V ; 6 z 6 Lw2 rR 2 Lw2 2
ð2Þ where
rL;R ¼ xxmax (xmax ¼ 0:3 is Al concentration at central barrier L;R
and xL;R is Al concentration at the left and right well center). V 0 is band discontinuity and calculated by V 0 ¼ Q c ð1155x þ 370x2 Þ, where Q c ¼ 0:6 is the conduction band offset parameter for GaAs=Ga1x Alx As QW. To solve Eq. (1), we take as base the eigenfunctions of the infinite potential well of width L. These bases are given at following:
rffiffiffi npz 2 cos /n ðzÞ ¼ dn L L
ð3Þ
Z
/f ðzÞjejz/i ðzÞdz;
ði; f ¼ 1; 2Þ
ð8Þ
Thus the total ACs calculated by:
bðxÞ ¼ bð1Þ ðxÞ þ bð3Þ ðxÞ
ð9Þ
The linear and nonlinear relative refractive index changes (RICs) can be expressed respectively as [31]:
" # Dnð1Þ ðxÞ rv jM 21 j2 DE hx ¼ nr 2n2r e0 ðDE hxÞ2 þ ðhC21 Þ2
ð10Þ
Dnð3Þ ðxÞ lcjM21 j2 rv I ¼ nr 4n3r e0 ½ðDE hxÞ2 þ ðhC Þ2 2 21 2
½4ðDE hxÞjM21 j
ðM22 M11 Þ2 2
ðDEÞ2 þ ðhC21 Þ 2
fðDE hxÞ
2
½ðDEÞðDE hxÞ ðhC21 Þ ðhC21 Þ ð2ðDEÞ hxÞg
ð11Þ
and finally the total relative RIC can be written as:
DnðxÞ Dnð1Þ ðxÞ Dnð3Þ ðxÞ ¼ þ nr nr nr
ð12Þ
3. Results and discussion In this paper, we consider the two different asymmetric GaAs=Ga1x Alx As DIPQW models. Firstly, we take into account the asymmetric structure due to the difference in Al concentration in the center of the two QWs ðLw1 ¼ Lw2 ¼ 100 ÅÞ separated each other
E.B. Al et al. / Optical Materials 47 (2015) 1–6
by a barrier of width Lb ¼ 25 Å. Secondly, we take into account the asymmetric structure due to the different width of the two QWs ðLw1 ¼ 100 Å and Lw2 ¼ 50 ÅÞ separated each other by a barrier of width Lb ¼ 25 Å with the same Al concentration in the center of the QWs. In both models, we examined the effects of electric and magnetic fields on the linear and nonlinear optical properties of the system. The parameters used in this study are: me ¼ 0:067m0 (where m0 is the free electron mass), V 0 ¼ 228 meV;
l ¼ 4p 107 H m1 ; rv ¼ 1:0 1017 cm3 ; s12 ¼ 0:14 ps (where s21 ¼ C121 ), I ¼ 0:4 MW cm2 and nr ¼ 3:2. In Fig. 1(a) and (b), we show the changes of the confinement potential of the asymmetric structure due to the difference in Al concentration in the center of the QWs, the lowest two sub-band energy levels, and the square of the corresponding wave functions under the electric and the magnetic fields. As seen from Fig. 1(a), the electron is localized in the left well ðLw1 Þ in the ground state ðE1 Þ, while it is localized in the right well ðLw2 Þ in the first excited state ðE2 Þ in the absence of the electric field. With the increase of the electric field applied on the growth direction, the charge carriers are swept towards left-hand side of the structure, since the electric field leads to the creation of a deeper QW on left-hand side of the structure. Furthermore, the localization of the electron in the
(a)
3
first excited state decreases in Lw2 while it increases in Lw1 . So, E1 (E2 ) decreases (increases) with the increasing of the electric field value. As a result, the energy difference between the two lowest levels ðDEÞ increases. Fig. 1(b) shows that the effect of the magnetic field applied perpendicular to the growth direction brings an additional geometric confinement to the structure. Both E1 and E2 levels increase with this effect. Consequently, DE also increases. In Fig. 2(a) and (b), we plot the effects of the electric and the magnetic fields on the confinement potential of the asymmetric structure with the same Al concentration in the center of the QWs, the lowest two sub-band energy levels, and the square of the corresponding wave functions, respectively. As demonstrated in Fig. 2(a), E1 (E2 ) decreases (increases) with the effect of the electric field. As a result, DE increases. As seen in Fig. 2(b), the magnetic field creates an additional geometric confinement to the structure. So E1 and E2 increase. As a result, DE decreases in this case according to previous case. For different electric and magnetic field values, the total relative RIC as a function of the incident photon energy are presented in Fig. 3(a) and (b), respectively. These calculations are for rL ¼ 6; rR ¼ 3 Lw1 ¼ Lw2 ¼ 100 Å, and Lb ¼ 25 Å. As shown in Fig. 3(a), the total relative RIC depends strongly on the electric field. The total relative RIC shifts towards to higher photon
(b)
Fig. 1. For rL ¼ 6; rR ¼ 3; Lw1 ¼ Lw2 ¼ 100 Å and Lb ¼ 25 Å, the variations of the confinement potential profile, ground and first excited state energies and the squared wave functions for related energy levels for two different values: (a) electric and (b) magnetic fields.
(a)
(b)
Fig. 2. The variations of the confinement potential profile, ground and first excited state energies and the squared wave functions for related energy levels for two different values: (a) electric and (b) magnetic fields for rL ¼ rR ¼ 3; Lw1 ¼ 100 Å; Lw2 ¼ 50 Å and Lb ¼ 25 Å.
4
E.B. Al et al. / Optical Materials 47 (2015) 1–6
(a)
(b)
Fig. 3. The change in the total refractive index as a function of the incident photon energy for three different values: (a) electric and (b) magnetic fields for rL ¼ 6; rR ¼ 3; Lw1 ¼ Lw2 ¼ 100 Å and Lb ¼ 25 Å.
energies (blue-shift) with increasing electric field values. These results are in good qualitative agreement with some previous works [25,32,33]. The main reason for the blue-shift is the increment in DE with the effect of the electric field. This case is seen clearly in Fig. 1(a). Also, the magnitude of the total relative RIC first decreases, and then increases with the increase of electric field value. This is resulted from the matrix elements jM 11 j; jM 22 j and jM21 j in Eqs. (10) and (11). With the increase of electric field, jM11 j (jM 22 j) increases (decreases), and jM 21 j first decreases, and then increases. Where jM 21 j is the dominant term and the magnitude of the total relative RIC varies as jM 21 j under the electric field. As seen in Fig. 3(b), the total relative RIC depends weakly on the magnetic field. The resonance peak of the total relative RIC shifts to blue by applying the magnetic field. Similar results were obtained in studies previously carried out [25,31,34,35]. The main reason for this behavior is that the quantum confinement varies with increasing the magnetic field. The variation of the quantum confinement causes to an increment in DE as seen in Fig. 1(b). Furthermore, increasing the magnetic field causes an increase in the peak amplitude of the total relative RIC. This is resulted from the matrix jM 21 j, as we have mentioned above. With the increase of magnetic field, jM 11 j and jM 22 j decrease while jM 21 j increases.
(a)
In Fig. 4(a) and (b), for different electric and magnetic field values the changes in the total relative RIC as a function of the incident photon energy are shown, respectively. These results are for
rL ¼ rR ¼ 3; Lw1 ¼ 100 Å; Lw2 ¼ 50 Å and Lb ¼ 25 Å. It is seen in Fig. 4(a) that the total relative RIC depends strongly on the electric field. It is observed that the peak of the total relative RIC shifts to blue with increasing electric field value. The accuracy of these results is observed by the compatibility with earlier studies [36]. This is a result of increased in DE with increasing electric field shown in Fig. 2(a). At the same time, the magnitude of the peak first decreases, and then increases for the further electric field values. The reason lies in the variation of the dipole transition matrix element jM21 j shown in Fig. 4(a). The dominant term jM 21 j decreases and than increases with the electric field. Fig. 4(b) shows that the total relative RIC depends weakly on the magnetic field. In this case, the total relative RIC shifts towards to lower photon energy (red-shift) with increasing magnetic field contrary to the previous case. However, the magnetic field applied on the asymmetric structure which consists of the two different width of QWs with the same Al concentration in the center of the QWs has a negligible effect on the linear and nonlinear optical properties of the structure. These results are similar to previous studies
(b)
Fig. 4. The variation of total refractive index changes with the incident photon energy for three different values: (a) electric and (b) magnetic fields for rL ¼ rR ¼ 3; Lw1 ¼ 100 Å; Lw2 ¼ 50 Å and Lb ¼ 25 Å.
5
E.B. Al et al. / Optical Materials 47 (2015) 1–6
(a)
(b)
Fig. 5. The variation of total absorption coefficients as a function of the incident photon energy for three different values: (a) electric and (b) magnetic fields for rL ¼ 6; rR ¼ 3; Lw1 ¼ Lw2 ¼ 100 Å and Lb ¼ 25 Å.
(a)
Fig. 6. The total absorption coefficients versus rL ¼ rR ¼ 3; Lw1 ¼ 100 Å; Lw2 ¼ 50 Å and Lb ¼ 25 Å.
(b)
the
incident
photon
energy
[37]. The reason for this shift is decrease in DE by the effect of the magnetic field as seen in Fig. 2(b). Also, the magnitude of the total relative RIC decreases with the increase of magnetic field value due to the variation of jM 21 j. Fig. 5(a) and (b) show the total AC spectra as a function of the incident photon energy for different electric and magnetic field values, respectively. Where rL ¼ 6; rR ¼ 3; Lw1 ¼ Lw2 ¼ 100 Å and Lb ¼ 25 Å. These figures clearly show that the magnitude and peak position of total ACs are sensitive to electric and magnetic fields values. With the increase of the electric and magnetic field values, the total AC towards higher photon energies (blue-shift) with increasing magnitude significantly for B ¼ 0 and F ¼ 0. Since the jM21 j (jM 22 j) dipole matrix elements increases (decreases), while jM 11 j remains almost constant. When the electric and magnetic fields increase, total AC shifts to blue with increasing the magnitude, which is a reason of increment in energy difference between corresponding energy levels. Furthermore, Blue shift in the electric field is more prominent than that of the magnetic field. Since, the variation of DE under the magnetic field is almost constant. These cases are observed easily from Figs. 1–4. Finally, in Fig. 6(a) and (b), we display the variation of total AC as a function of the incident photon energy for three different values: (a) electric and (b) magnetic fields. These results are for rL ¼ rR ¼ 3; Lw1 ¼ 100 Å; Lw2 ¼ 50 Å and Lb ¼ 25 Å. When we
for
three
different
values:
(a)
electric
and
(b)
magnetic
fields
for
compare the results obtained for this figure with that of Fig. 5, we can see that resonant peak of total AC shifts to the higher photon energies and it’s magnitude increases with increasing electric and magnetic fields values due to physical reasons mentioned above. Additionally, as seen in Fig. 6(b), the magnitudes of the total AC decrease with increasing magnetic field. But increment in the magnitude of the jM 22 M 11 j dipole matrix elements is larger than that of the jM 21 j dipole matrix elements. Thus, the magnitude of the total AC decreases with magnetic fields. 4. Conclusions In summary, we have investigated the combined effects of electric and magnetic fields on the ACs and RICs of asymmetric GaAs=Ga1x Alx As DIPQW. Based on the compact density-matrix approach and the iterative procedure, the calculation is performed within the effective-mass approximation. The numerical results show that both total AC and RIC are sensitive to structure parameters and the effects of electric and magnetic fields. By changing the magnitudes of the external fields together with structure parameters, we can obtain a blue or red shift, without the need for the growth of many different samples. This also gives a new degree of freedom in various infrared optical device applications based on the ISB transition of electrons.
6
E.B. Al et al. / Optical Materials 47 (2015) 1–6
We hope that the calculation of the nonlinear optical properties in asymmetric GaAs=Ga1x Alx As DIPQW would make a significant contribution to the theoretical studies and provides a new model for the potential application of optoelectronic devices such as electro-optical modulators, optical switches and infrared photo-detectors.
[15] [16] [17] [18] [19] [20] [21] [22]
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