- Email: [email protected]
Computation of the nonlinear optical properties of n-type asymmetric triple δ-doped GaAs quantum well
Superlattices and Microstructures 130 (2019) 76–86
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Computation of the nonlinear optical properties of n-type asymmetric triple δ-doped GaAs quantum well F. Ungan a, *, S. Pal b, M.K. Bahar c, M.E. Mora-Ramos d a
Department of Optical Engineering, Faculty of Technology, Sivas Cumhuriyet University, 58140, Sivas, Turkey Department of Chemistry, Hetampur Raj High School, Hetampur, Birbhum, 731124, West Bengal, India Department of Energy Systems Engineering, Faculty of Engineering, Karamano�glu Mehmetbey University, 70100, Karaman, Turkey d Centro de Investigaci� on en Ciencias, Instituto de Investigaci� on en Ciencias B� asicas y Aplicadas, Universidad Aut� onoma del Estado de Morelos, Av. Universidad 1001, CP, 62209, Cuernavaca, Morelos, Mexico b c
A R T I C L E I N F O
A B S T R A C T
Keywords: δ-doped quantum well Nonlinear optical response Intense laser field
In the present study, we present numerical results for the influence of non-resonant intense THz laser radiation as well as the structure parameters, such as well width and central doping con centrations, on the total optical absorption coefficients (TOACs) and relative refractive index changes (RRICs) in n-type asymmetric triple δ-doped GaAs quantum well (QW) by using the compact-density-matrix formalism via iterative method. In order to obtain these numerical re sults, we have first obtained the subband energy spectrum and the electronic wave functions of the structure by using the effective-mass and parabolic band approximation. We have then calculated the nonlinear optical (NLO) properties of the system, by employing these energy ei genvalues and eigenfunctions. The numerical results for these calculations show that; (i) the magnitude of the resonant peak of TOACs decreases and the peak position shifts towards the lower energies when the central doping concentration and the well width are increased; (ii) due to the intense laser field effect, the magnitude of the TOACs resonant peak initially increases and its position shifts towards higher energies and then it decreases, shifting towards lower energies; (iii) by increasing the central doping concentration (well width), the magnitude of the resonant peak of RRICs increases (decreases), and the resonant peak position shifts towards the lower energies; and finally (iv) because of the increase in the intensity of the laser field applied to the structure, the magnitude of the resonant peak of RRICs increases, and the resonant peak position initially shifts towards the higher energies and then shifting towards the lower energies.
1. Introduction With the advent of technological development on semiconductor nano-structures during the last two decades, the linear and nonlinear optical (NLO) responses of heterostructures with reduced dimensionality such as quantum wells (QWs), quantum wires, quantum dots and superlattices have attracted a great deal of interest in designing of optoelectronic devices. Space quantized energy levels originating in confinement of charge carriers in these structures make them potent candidates to express vivid nonlinear effects [1–3]. It is possible to create semiconductors with different level of confinement of charge carriers in one, two or three dimensions by the expeditious advancement of several growth techniques such as molecular beam epitaxy (MBE) and metal organic chemical vapor * Corresponding author.. E-mail address: [email protected] (F. Ungan). https://doi.org/10.1016/j.spmi.2019.04.023 Received 22 February 2019; Received in revised form 28 March 2019; Accepted 11 April 2019 Available online 16 April 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
deposition (MOCVD) etc [4]. The subband (SB) energy states of the confined charge carriers in the semiconductor QWs give rise aesthetic optical spectrum which can be monitored by tailoring structural parameters as well as by application of various external perturbations such as hydrostatic pressure, electric, magnetic, and high-frequency intense laser fields (ILF). For examples, the effects of the external electric and magnetic fields as well as hydrostatic pressure on the photoluminescence energy transitions in GaAs/GaAlAs coupled double quantum wells have been considered by using a variational method within the parabolic band approximation and the effective mass [5]. The effect of the laser field on the nonlinear optical properties of a square quantum well subject to the external electric field have been examined through the density matrix formalism and the effective mass approach [6]. The most appreciable outcome of such works is determination of the functionalities of various external perturbations such as hydrostatic pressure, electric, magnetic, and ILF on electronic and optical features of QWs. The triple-delta doped quantum wells are taken into consideration due to many advantages of the multiple quantum wells in applications. One of the most common application areas recently is Quantum Cascade Laser [7]. As the quantum efficiency is substantially restricted through optical absorption by electrons and holes, this effi ciency can be enhanced by replication quantum well. In this regard, the multiple quantum wells can be employed in order to be able to cancel out reducing in active layer thickness. In a word, the multiple quantum wells are more functional compared to the single quantum wells to fabricate higher efficiency lasers [8]. Gaggero-Sager et al. carried out an investigation on the influence of tem perature on transport properties in Si-δ-doped GaAs QWs [9]. Grimalsky and his co-workers studied theoretically the impact of temperature on electronic spectrum of δ-doped QWs through the Thomas-Fermi (TF) method [10]. Oubram et al. lucidly probed the influence of hydrostatic pressure on on intersubband (ISB) optical absorption of n-type δ-doped GaAs QW [11]. Ungan et al. nicely explained modulation of electronic states of triple δ-doped QW in presence and absence of an external monochromatic ILF [12]. Strong quantum confinement of the charge carriers in semiconducting heterostructures leads to large energy separation between bound states. Such large energy separation in turn diminishes dipole matrix elements enhancing the possibility of obtaining the resonance condi tions. Bound energy states of charge carriers in semiconductor QWs can be finely tuned by applying the non-resonant ILF without disturbing center of symmetry of confinement potential. Bound energy states and wave functions of charge carriers confined in QWs can be modified by non-resonant ILF. Therefore, the effect of non-resonant ILF has earned a great deal of attraction to the researchers as it can dress the bound energy states of charge carriers. Some of remarkable works on ILF investigations can be exemplified by; the ILF effects on the electronic properties of V-shaped and inverse V-shaped QWs generated by GaAs/GaAlAs heterostructure have been examined by using the transfer matrix method [13]. The ILF effects on the subband structure and polarizabilities of shallow donors in V-shaped QW generated by GaAs/GaAlAs heterostructure with different Al concentrations have been studied through the effective mass approach [14]. The introduction of dopant as highly localized impurity sheets within specially designed QWs has earned much attention to the noble researchers in modulating of ISB energy states of QWs. In this respect, δ-doped systems become very special because they are fabricated with two dimensional electron or hole gases having high density in semiconductor heterostructures [15–20]. Many startling experiments based on fabrication techniques have been being driven to fabricate these novel structures doped with -dopant. MBE is visible experimental method to manufacture delta-doped QWs. Liu et al. have been worked delta doping in QW structures by consisting a strained InGaAs sandwiched between two GaAs layers, analyzing various doses for structures and QW thicknesses [21]. Lee et al. have been studied the magnetic field effects on delocalization in multiple QWs with center-doped generated by GaAs/GaAlAs het erostructure [22]. In this regard, magneto-transport measurements and scaling analysis on a series of center-doped GaAs QWs have been demonstrated by considering the doping concentration in QWs. It was made possible to fabricate δ-doped QWs by doping a positive δ-function potential at the middle of the QWs [23]. By the help of growth-interruption technique in which donor or acceptor impurities could be grown randomly, the epitaxial growth of the δ-potential was done [24]. Such donor or acceptor impurities act as potent source of charge carriers. It is also achieved by depositing atoms of impurity on suspended epitaxial layers of matrix of QWs [25]. Restrepo et al. investigated the effects of an external electric field and impurity on NLO properties in δ-doped quantum rings. Effects of externally applied magnetic field and electric field on nonlinear optical absorption coefficients (OACs) was probed by R. L. Restrepo and his co-workers in a δ-doped AlGaAs=GaAs triple-quantum well (δ-TQW) [26]. Tang et al. examined effects of location and density of δ dopant on ISB transitions in AlGaN=GaN step QWs induced for terahertz (THz) applications [27]. In a word, δ-doped QWs have been studied comprehensively for its potentiality to offer addition quantum confinement to the charge carries and to carry out appreciable radiative recombination between the two dimensional electron gas and the photo-generated holes [28]. By considering remarkable contribution of asymmetric nature into the symmetric energy profile of confinement potential of QWs by means of various external perturbations as well as by presence of dopant, there have been occurred so many novel researches on several linear and nonlinear optical phenomena exhibited by these heterostructures [29–37]. Among various linear and NLO prop erties of semiconductor QWs, TOACs, and RRICs dependent on ISB transition of confined charge carriers have attracted special attention of these researchers. H. Dakhlaoui nicely investigated ISB transition of charge carriers between the ground state and first excited state along with the TOACs in the Si-δ-doped step AlGaN=GaN QW theoretically [38]. Zeiri et al. numerically demonstrated influence of well depth and barrier width on the TOACs, and RRICs in asymmetrically fabricated QWs [39]. Karimi and his co-workers theoretically investigated effects of structural parameters on the TOACs and RRICs of asymmetrically tuned double semi-parabolic QWs [40]. Transition of charge carriers from single to double QW by inducing ILF in semiconductor QWs have been examined by Lima et al. [41]. Kasapoglu and her co-workers elucidated influence of ILF on the TOACs, and RRICs in n-type double δ-doped GaAs QWs [42]. From their experiment it was revealed that, ground state and first excited energy state of double δ-doped GaAs QWs were energetically more separated by enhanced laser field intensity leading to occurrence of optical blue-shift in the ISB transitions. It can be concluded from their experiment that significant tunability in ISB transitions of charge carriers can be achieved by modulating in tensity of applied ILF. The motivation of our present work is to explore the effect of central doping concentration (CDC), and width of QW on the optical 77
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
properties such as TOACs, and RRICs in the n-type asymmetric triple δ-doped (ATDD) GaAs QW induced by ILF. The paper is orchestrated as follows: Section 2 contains theoretical background, Section 3 comprises obtained results and discussions, and finally, the main conclusions of present work are presented in Section 4. 2. Theory As mentioned in the introduction section, in the presence of non-resonant intense THz laser radiation which is applied in the growth direction, we will consider an electron confined in the n-type ATDD GaAs QW grown along the z-axis. Under this condition, taking advantage of the envelope wave function and effective-mass approximations, the Hamiltonian of the confined electron in this structure can be written as following � � �� ℏ2 d 2 H¼ þ V ðz; α Þ ; (1) ATDD 0 2m�e dz2 where m�e is the electron effective mass, ℏ is the reduced Planck constant, α0 ¼ eA0 =m� ϖ is allowed as the parameter that specifies the ILF strength, calling as the laser-dressing intensity parameter, e is the elementary charge, A0 is the laser field strength, and ϖ designates the angular frequency of the laser field, and hVATDD ðz; α0 Þi is the laser-dressed confinement potential given by Ref. [41]. *
+
VATDD ðz; α0 Þ
¼
ϖ 2π
Z
2π =ϖ
(2)
VATDD ðz þ α0 sinϖtÞdt: 0
The mathematical expression used in this study for the potential term was obtained using the TF approximation within the framework of the one-dimensional local density functional theory, as suggested by Ioriatti [43]. This theory was found to be highly successful in the theoretical modeling of the electronic structure in the δ-doped structures [44]. Moreover, with this theory, we do not need to make a self-consistent calculation with a long and laborious process. Thus, the mathematical expression of the confinement potential of n-type ATDD QW can be written as VATDD ðzÞ ¼
ζ2
ζ2 4
Lw=2j þ zL Þ
ðζjz
ζ2 4
ðζjzj þ zM Þ
ðζjz
Lw=2j þ zR Þ4
:
(3)
Here Lw is the effective width between the left and right-hand side δ-doped QWs, ζ ¼ ðe2 ð2m�e Þ3=2 Þ=ð15πεr ℏ3 Þ, zL ¼ 1=5
1=5
1=5
ð2εr ζ3 =ðπ e2 NL2d ÞÞ , zM ¼ ð2εr ζ3 =ðπe2 NM , zR ¼ ð2εr ζ3 =ðπe2 NR2d ÞÞ are the delta potential parameters, εr is the dielectric constant 2d ÞÞ L M R of GaAs, N2d , N2d and N2d represent the corresponding two-dimensional impurity density in the left, central and right-hand side δ-doped layers, respectively. The effect of non-resonant intense THz laser radiation on the electronic wave functions and SB energy spectrum of the structure is obtained from the solution of Eq. (1) using diagonalization method. On the other hand, applying a linearly polarized electromagnetic wave with ω angular frequency on the n-type ATDD QW is considered, and this time-dependent electromagnetic wave is given by Ref. [45]. ~ iωt þ Ee ~ EðtÞ ¼ E0 cosðωtÞ ¼ Ee
iωt
(4)
:
However, the electronic polarization PðtÞ and susceptibility χ ðtÞ are presented via the dipole operator M and the density matrix ρ: being V and ε0 are the volume of system, the dielectric constant of vacuum, respectively ~ PðtÞ ¼ ε0 χ ðωÞEe
iωt
~ iωt ¼ 1 TrðρMÞ: þ ε0 χ ð ωÞEe V
(5)
The linear and nonlinear susceptibilities are determined within the framework of same compact-density-matrix formalism, their analytical expressions are given as follows, � ρ10 �M10 j2 ε0 χ ð1Þ ðωÞ ¼ ; (6) E10 ℏω iℏΓ10 �
ε0 χ ð3Þ ðωÞ ¼
ðE10
"
�
Ej2 ρ10 �M10 j2 �~ � E10 ℏω iℏΓ10 ðE10
ðM11 M00 Þ2 iℏΓ10 ÞðE10 ℏω
� 4�M10 j2 ℏωÞ2 þ ðℏΓ10 Þ2
(7)
# iℏΓ10 Þ
;
where ρ10 ¼ m�e E10 =ðπℏ2 LwÞ is the electron density in the system, E10 is the energy difference between the two lowest electronic states, � � � � � � � � Mij ¼ �hψ i �ez�ψ j i� ði; j ¼ 0; 1Þ is the product of matrix elements and also means geometric factor, Γ10 ¼ 1=τ10 is the phenomenological relaxation rate.
78
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
Then, the RRICs change is presented as � � ΔnðωÞ χ ðωÞ ¼ Re 2 nr 2nr
(8)
where nr is the refractive index. When considering Eqs. (6)–(8), the linear and the third-order nonlinear RRICs are obtained by Ref. [45]. " # � 2 Δnð1Þ ðωÞ ρ10 �M10 j E10 ℏω ; (9) ¼ nr 2n2r ε0 ðE10 ℏωÞ2 þ ðℏΓ10 Þ2 �
μc�M10 j2 4n3r ε0 ½ðE 10
Δnð3Þ ðω; IÞ ¼ nr
ρ10 I
i2 ℏωÞ2 þ ðℏΓ10 Þ2 � ( " � ðM11 M00 Þ2 � 2 ðE10 ℏωÞ � 4ðE10 ℏωÞ�M10 j � ðE10 Þ2 þ ðℏΓ10 Þ2 i oi ðℏΓ10 Þ2 ð2ðE10 Þ ℏωÞ ; �½ðE10 ÞðE10 ℏωÞ ðℏΓ10 Þ2
(10)
where μ is the magnetic permeability, c is the speed of light in free space, I is the intensity of electromagnetic field, defining as � rffiffiffiffi � εR �� 2nr �� I¼2 EðωÞj2 ¼ (11) EðωÞj2 : � μ μc � where εR the real part of the permittivity. The RRICs change is figure out by considering the linear and nonlinear contributions [45]: Δnðω; IÞ Δnð1Þ ðωÞ Δnð3Þ ðω; IÞ ¼ þ : nr nr nr
(12)
However, the OACs change is stated as � �1=2
ηðωÞ ¼ ω
μ εR
(13)
Imðε0 χ ðωÞÞ;
χ ðωÞ is the Fourier component of χ ðtÞ. The linear and the third-order nonlinear OACs are obtained by �
rffiffiffiffi
�M10 j2 ρ ℏΓ10 μ 10 η ðωÞ ¼ ω ; εr ðE10 ℏωÞ2 þ ðℏΓ10 Þ2
(14)
ð1Þ
rffiffiffiffi μ I ð Þ� εR ε0 nr c
� �M10 j4 ρ ℏΓ10 10
ηð3Þ ðω; IÞ ¼ 2ω
ðE10 �
ℏωÞ2
½ðE10
ℏωÞ2 þ ðℏΓ10
ðℏΓ10 Þ2 þ 2ðE10 ÞðE10 ðE10 Þ2 þ ðℏΓ10 Þ2
�2 i2 ð1
� �M11 M00 j2 � �2M10 j2
(15)
ℏωÞ Þ
The TOACs change is given by summing up the linear and nonlinear contributions [45]: (16)
ηðω; IÞ ¼ ηð1Þ ðωÞ þ ηð3Þ ðω; IÞ:
Finally, it should be pointed out that all numerical calculations were performed using the following physical parameters that are suitable for GaAs material [12]: m�e ¼ 0:067m0 (m0 the single electron bare mass), εr ¼ 12:58, nr ¼ 3:2, μ ¼ 4π � 10 7 Hm 1 , NL2d ¼ 3:0 � 1012 cm 2 , NR2d ¼ 9:0 � 1012 cm 2 , I ¼ 0:05 MWcm 2 , and τ10 ¼ 0:14ps. 3. Results and discussion Fig. 1 schematically represents modulation of the confinement potential profile and the probability densities of lowest two energy levels as a function of growth direction (z) in the n-type ATDD QWs for two different values of CDC, width of QW, and laser-dressing parameter. Red line, and blue line indicate probability densities of ground state and first excited state respectively whereas black line indicates confinement potential profile. 12 2 12 2 From Fig. 1(a), it is evident that as CDC increases from NM to NM in absence of ILF, confinement 2d ¼ 4 � 10 cm 2d ¼ 6 � 10 cm potential energy of central QW also decreases without affecting depths of right and left QW. Increase in the depth and width of central QW with the increase of CDC signifies greater amount of confinement of charge carriers with increasing concentration of dopant. As we assume that, concentration of left side dopant (NL2d ¼ 3:0 � 1012 cm 2 ) is lesser than that of right side dopant (NR2d ¼ 9:0 � 1012 cm 2 ), 79
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
the depth of right side QW is larger than the depth of left side QW. The phenomenon clearly announces dependence of depth of QW with concentration of dopant in direct proportion. Therefore, escalation in the depth of central QW with increase in CDC goes in good conformity with the above fact. Fig. 1(a) also reveals that, probability density of first excited state diminishes more than that of ground state with enhancement in CDC. Probability density of first excited state goes through two maxima where, first maximum is more prominent with greater height than the second one. Although energy of first excited state decreases with increase in CDC but its two maxima become more sharp without making any positional shift. Probability density of ground state goes through only one maximum which becomes less prominent with increase in CDC. The above facts clearly suggest that first excited state suffers higher influence of dopant concentration than ground state. As the ground state is more localized to the base of QW it remains in greater confinement than that of first excited state which is more labile. Therefore, CDC plays more impactive role on modulating first excited energy state than modulating ground energy state.
From Fig. 1(b) it can be observed that as effective width between the left and right-hand side δ-doped QWs enhances from Lw ¼ 12 2 100� A to Lw ¼ 200� A in absence of ILF with a fixed value of CDC at NM 2d ¼ 6 � 10 cm , confinement potential energy also increases. It also reveals that, depth of three QWs specially left QW become more prominent with increase in Lw. Right quantum well enjoys blueshift whereas left QW experiences red-shift with increasing Lw. When effective width is small then the charge carrier is effectively confined in left and central QWs. At lower values of Lw triple QWs becomes double QWs. As Lw increases electron tunneling towards left QW also increases. On the other hand, square of wave functions of lowest two energy levels are also augmented with increasing effective width with a considerable blue-shift in the peak positions of probability densities. As Lw increases distance between dopant and charge carriers confined in QWs also increases which effectively increases spatial extension of wave functions of charge carriers. Therefore probability densities are enhanced with increasing Lw. The enhancement of probability densities with increasing Lw is more visible in case of ground energy state because at lower value of Lw ground sate is in more confined state than first excited state which is already in relaxed mood. Hence first excited state enjoys less relaxation than ground state by increasing Lw. Therefore in contrast with Fig. 1(a) it can be argued that ground energy state becomes more affected with increasing Lw whereas from Fig. 1(a) it has been established that first excited state was more deeply perturbed with increasing CDC. Increase in effective width means enhancement in separation between the three QWs which is evident from Fig. 1(b), which in turn offers lesser confinement to the charge carriers. Such lesser confinement plays important role in intensifying of spatial extension of squared ground and first excited state envelope wave functions. Fig. 1(c) clearly announces that, with increasing laser dressing parameter (α0 ) the charge carriers are practically going to be confined in central QW. Enhancement of ILF means enhancement of confinement potential of the charge carriers. As α0 increases from 0 to 50� A with a fixed value of Lw and NM at 200� A and 6 � 1012 cm 2 respectively left and right QWs become very faint. At higher ILF 2d
the confinement potential moves from a triple delta configuration towards a single triangle-like QW. This phenomenon is actually opposite to that reported by Lima and co-workers in the case of rectangular QWs under ILF [41]. They revealed the transition from single to a double potential well conduction band profile. With the increasing ILF parameter the squared wave functions corresponding to the ground and first excited state are also proliferated. As geometrical confinement increases with amplified ILF the wave functions become more localized in QW. It is observed from Fig. 1(c) that, the original region of spatial confinement of electron becomes wider in a stronger way in case of first excited state than that of ground state because first excited state is more labile to enjoy ILF than ground state which is more stable due to its lower energy. Besides, n-type asymmetric triple δ-doped QW becomes triangle-like QW due to enhancement of spatial extension of the electronic wave functions with increasing ILF parameter. Bottom of confinement potential shifts towards higher energy region for stronger ILF, which can be explained in the light of higher repulsion the electron in QWs experiences by n-type dopants that are brought closer to the electron by stronger ILF. Fig. 2(a) selectively represents intersubband energy difference between first excited state and ground state (ΔE ¼ E1 E0 ) with CDC in absence of ILF maintaining Lw at a constant value of 200� A. ΔE diminishes with increasing CDC. The fact can also be established
from Fig. 1(a) which clearly announces that squared ground and first excited state envelope wave functions are decreased with increasing CDC. Such a decrease is more prominent in case of first excited state as a result energy separation between these lowest two states are declined with increasing CDC. Increase in CDC actually offers greater amount of confinement which is responsible for decreasing the separation between first excited state and ground state. Thus escalation in dopant concentration allows the wave functions of two lowest energy states to overlap with each other in greater extent which is clearly reflected by increasing character of jM10 j with CDC. As a result transition energy for electronic transition from ground state to first excited state is also decreased with increasing CDC. The difference between geometric factor of first excited state and ground state (jM11 M00 j) gradually diminishes with increase in CDC which is quite obvious because as CDC increases, confinement of charge carriers also increases, which makes the wave functions more localized. Variation of main intersubband energy difference (ΔE) and electric dipole moment matrix elements as functions of effective width 12 2 between the left and right-hand side δ-doped QWs (Lw) in absence of ILF at a fixed CDC (NM 2d ¼ 6 � 10 cm ) is exhibited in Fig. 2(b). From this figure it is quite evident that ΔE gradually falls off with escalation of Lw which is supported by Fig. 1(b). In Fig. 1(b) it has been established that the probability densities of two lowest energy states are enhanced by enhancement of Lw, but the enhancements have been occurred in different extent for these two states. As a result energy difference between these two states are also reduced with increase in Lw. On the other hand, dipole matrix elements between first excited state and ground state jM10 j is also stepped up with increasing Lw signifying easy transition of electron from ground state to first excited state which is a direct outcome of decrease in ΔE. Besides, jM11 M00 j is sharply increased by increasing Lw which can explained by the fact that wave functions of ground state and first excited state is endowed with greater spatial extension as effective distance between dopant and charge carrier is increased. Similarly Fig. 2(c) evinces modulation of (ΔE) and electric dipole moment matrix elements as functions of laser dressing parameter 80
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
Fig. 1. The variation of the n-type asymmetric triple δ-doped QWs confinement potential profile and the squared ground and first excited state envelope wave functions versus the z-coordinate for two different values of (a) central doping concentration, (b) quantum well width, and (c) laser dressing parameter.
81
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
Fig. 2. Main intersubband energy difference (ΔE ¼ E1 E0 ) and electric dipole moment matrix elements as functions of (a) central doping con centration, (b) quantum well width, and (c) laser dressing parameter. 12 � (α0 ) keeping Lw and NM 2d at constant values of 200A and 6 � 10 cm
2 respectively. The modulation of (ΔE) with α0 is not very straightforward. Initially ΔE rises with increasing α0 attending a maximum nearly at 60� A then it starts to fall slowly. Increase in ILF parameter brings about magnification of energy of two lowest energy states by offering extended confinement of charge carriers. This magnification of energy is not occurred in equal extent for these two states. From Fig. 1(c), it can be concluded that first excited state is offered greater perturbation than ground state by increasing ILF, therefore ΔE after passing through a maximum starts to decrease with increasing ILF. At higher values of α0 energy of first excited state of confined electron do not enhance more readily than ground energy
82
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
state as in lower ILF region because at higher confinement energy of ground state also starts to increase rapidly as a result of higher repulsive interaction between the dopant and the electron. As a result ΔE starts to fall at higher α0 region. Matrix elements of dipole moment between ground state and first excited state (jM10 j) does not get affected so much with increasing α0 because undulatory nature of energy separation between ground state and first excited state. At very lower α0 region it delineates a feeble rise with increasing ILF signifying greater amount of overlap between the wave functions of the two lowest energy states. At some intermediate ILF jM10 j exhibits a faint declination around α0 � 60� A, this is the region where ΔE exhibits its maximum. The two results run in
Fig. 3. The variation of the TOACs as a function of the incident photon energy for different values of (a) central doping concentration, (b) quantum well width, and (c) laser dressing parameter. 83
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
Fig. 4. The variation of the RRICs as a function of the incident photon energy for different values of (a) central doping concentration, (b) quantum well width, and (c) laser dressing parameter.
84
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
accordance with each other. Increase in energy gap means decrease in extent of overlap of the wave functions which in turn diminishes transition of electron from ground state to first excited state. Again at higher ILF region jM10 j begins to rise with ILF as at this region ΔE starts to decrease allowing significant amount of overlap of wave functions of the electronic states. Besides, jM11 M00 j is gradually diminished with increasing ILF only showing a faint rise at the region of α0 � 60� A but after this region it again falls off with increasing
ILF. The wave functions of electronic states become more localized with increasing confinement offered by ILF, as a result jM11 M00 j diminishes with increasing ILF. The feeble increase in jM11 M00 j at the region mentioned above is due to exert of greater repulsion given by dopant to confined electrons which in turn increases spatial extension of wave functions. But with increasing ILF this repulsive interaction is overcome by extended confinement potential brought about through ILF which makes wave functions highly localized. 12 2 Variation of TOACs and RRICs as a function of the incident photon energy for five different values of CDC (NM 2d ¼ 3 � 10 cm ;4 � 12 2 12 2 12 2 12 2 � 10 cm ; 5 � 10 cm ; 6 � 10 cm ;7 � 10 cm ) in absence of ILF with a fixed value of Lw (200A) are represented by Figs. 3(a)
and 4(a) respectively. TOACs exhibit red-shift of absorption peak positions with increasing CDC but peak height are steadily decreased with increasing CDC. From Fig. 2(a) it has been established that jM10 j increases with CDC facilitating transition of electron from ground electronic state to first excited electronic state. Thus, the red-shift of peak positions due to enhancement of electron transition is a direct consequence of extended overlap of electronic wave functions with increasing CDC. At resonant condition (E10 ¼ ℏω), the linear and nonlinear absorption coefficients become rffiffiffiffi �� 2 μ M10 j ρ10 ηð1Þ ðωÞ ¼ ω ; (17) εr ℏΓ10 � � rffiffiffiffi �M10 j4 ρ �M11 M00 j2 μ I 10 � ð1 þ ð Þ� �2M10 j2 εR ε0 nr c ðℏΓ10 Þ3
ηð3Þ ðω; IÞ ¼ 2ω
(18)
ðℏΓ10 Þ2
� Þ ðE10 Þ2 þ ðℏΓ10 Þ2 From equations (17) and (18) it is evident that both linear and nonlinear AC increase with increase in M10 , but such an enhancement is more visible in case of nonlinear absorption coefficient. As a result nonlinear contribution in TOACs is magnified with CDC. Thus resonant peak height of TOACs are gradually petered out with CDC. Similarly resonant peak positions of RRICs experience a red-shift with increasing CDC as demonstrated by Fig. 4(a). The resonant peak heights of RRICs also enjoy enlargement with enhancement of CDC. Figs. 3(b) and 4(b) delineate regulation of TOACs and RRICs respectively as function of incident photon energy for five different values of Lw (100� A; 125� A; 150� A; 175� A; and200� A) in absence of ILF with CDC fixed at NM ¼ 6 � 1012 cm 2 . Both TOACs and RRICs 2d
experience red-shift of resonant peak positions with increasing Lw. From Fig. 2(b) it has been made clear that M10 increases with Lw helping electronic transition from ground state to first excited state. There resonant peak positions of TOACs and RRICs also suffer redshift with increasing Lw. Besides, resonant peak height gradually decreases with increasing Lw for both the optical responses. From Fig. 2(b) it has been also argued that both jM10 j and jM11 M00 j are intensified with enhancement in Lw imparting greater contribution of nonlinear parts of TOACs as well as of RRICs as evident from equation (18). As a result peak height of TOACs and RRICs are also diminished with increasing Lw. Figs. 3(c) and 4(c) deliciously describe variation of TOACs and RRICs respectively as a function of incident photon energy for five different values of ILF dressing parameter (α0 ¼ 0� A;25� A;50� A;75� A;100� A) with constant values of Lw ¼ 200� A, and NM ¼ 6 � 1012 cm 2 . 2d
From these two figures it can be ferreted out that resonant peak positions of TOACs and RRICs go through a blue-shift with increasing α0 from 0� A to 50� A , then they experience a red-shift on going from α0 ¼ 75� A to α0 ¼ 100� A. From Fig. 2(c) it can easily be observed that
A then it begins to fall with increasing α0 . energy separation between ground state and first excited state (ΔE) enhances up to α0 � 60� Similarly jM10 j exhibits a dim fall nearly at α0 � 60� A then it starts to rise again with further increase in α0 . Therefore, the blue-shift and
subsequent red-shift of resonant peak positions of these two optical responses can be explained in the light of undulatory nature of ΔE and jM10 j with ILF. Figs. 3(c) and 4(c) also delineate that resonant peak height of TOACs and RRICs at first increases with increase in α0 from 0� A to 50� A, then it starts to decline on going from α0 ¼ 75� A to α0 ¼ 100� A. Actually in the range of α0 � 60� A both jM10 j and jM11
M00 j manifest declination with ILF. As a result, nonlinear contribution to TOACs and RRICs becomes lesser leading to increase in A both the matrix elements begin to rise which impart larger contribution of resonant peak height of TOACs and RRICs. Beyond α0 � 60� nonlinear part to TOACs and RRICs leading to decrease in resonant peak height of these two optical responses. 4. Conclusions In this present investigation we furnish a detailed study of TOACs and RRICs in an n-type asymmetric triple δ-doped GaAs quantum well (QW) induced by non-resonant intense THz laser radiation (ILF). Applying ILF nicely reorganizes the confinement potential of the electron confined in QWs. Numerical calculations are done by using the effective-mass and parabolic band approximation to obtain intersubband energy spectrum and the electronic wave functions of the structure. The investigations show that, the energy difference between the ground state and first excited state steadily decreases with increasing CDC and width of QWs. As a result, resonant peak positions of TOACs and RRICs suffer red-shift with increase in CDC (NM 2d ) and effective width (Lw) of doped QW. Besides, resonant peak heights of TOACs and RRICs exhibit steady declination with increasing CDC and Lw. As ILF increases confinement of electron trapped 85
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
inside QWs also increases, which completely destroyed the energy barrier between the δ-doped layers and makes the triple δ-doped QWs into a shallow single confinement potential. The separation between lowest two energy states in ATDD GaAs QW increases with initial increase in ILF, which causes to an optical blue-shift in resonant peak positions of TOACs and RRICs. But with further increase in ILF such energy separation starts to decrease which in turn leads to an optical red-shift in ISB transitions. In a word, optical transitions in ATDD QW can be accomplished with eloquent tunability by modulating different parameters like, central doping concentration, width of QW and intensity of ILF. We hope that the work can motivate further theoretical investigations on the nonlinear optical properties in asymmetric n-type triple δ-doped QW, which leads to immense development of optoelectronic devices. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
S. Sakiroglu, F. Ungan, U. Yesilgul, M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, H. Sari, I. S€ okmen, Phys. Lett. A 376 (2012) 1875–1880. I. Karabulut, C.A. Duque, Physica E 43 (2011) 1405–1410. J.C. Martínez-Orozco, M.E. Mora-Ramos, C.A. Duque, J. Lumin. 132 (2012) 449–456. M.K. Gurnick, T.A. De Temple, IEEE J. Quantum Electron. 19 (1983) 791–794. S.Y. L� opez, M.E. Mora-Ramos, C.A. Duque, Physica B 404 (2009) 5181–5184. I. Karabulut, Appl. Sur. Sci. 256 (2010) 7570–7574. M. Razeghi, Technology of Quantum Devices, Springer, Evanston- USA, 2010, pp. 271–321. Q. Dai, et al., Appl. Phys. Lett. 94 (2009) 111109–111112. L.M. Gaggero-Sager, G.G. Naumis, M.A. Mun€ oz-Hernandez, V. Montiel-Palma, Physica B 405 (2010) 4267–4270. V. Grimalsky, L.M. Gaggero-Sager, S. Koshevaya, Physica B 406 (2011) 2218–2223. O. Oubrama, O. Navarro, L.M. Gaggero-Sager, J.C. Martínez-Orozco, I. Rodríguez-Vargas, Solid State Sci. 14 (2012) 440–444. F. Ungan, M.E. Mora-Ramos, E. Kasapoglu, H. Sari, I. S€ okmen, Philos. Mag. 99 (2019) 644–658. E.C. Liculescu, A. Radu, M. Stafe, Superlattice. Microst. 46 (2009) 443–450. L.M. Burileanu, E.C. Niculescu, N. Eseanu, A. Radu, Phys. E 41 (2009) 856–860. G.H. Dohler, Surf. Sci. 73 (1978) 97–105. C.E.C. Wood, G. Metze, J. Berry, L.F. Eastman, J. Appl. Phys. 51 (1980) 383–387. M. Zachau, F. Koch, K. Ploog, P. Roentegen, H. Beneking, Solid State Commun. 59 (1986) 591–594. M.E. Lazzouni, L.J. Sham, Phys. Rev. B 48 (1993) 8948–8958. R.V. de Walle, R.L.V. Meirhaeghe, W.H. Laflere, F. Condon, J. Appl. Phys. 74 (1993) 1885–1889. K.A. Rodriguez-Magdaleno, J.C. Martínez-Orozco, I. Rodriguez-Vargas, M.E. Mora-Ramos, C.A. Duque, J. Lumin. 147 (2014) 77–84. D.G. Liu, C.P. Lee, K.H. Chang, D.C. Liou, Appl. Phys. Lett. 57 (1990) 1887–1888. C.H. Lee, Y.H. Chang, Y.W. Suen, H.H. Lin, Phys. Rev. B 58 (1998) 10629–10633. L.N. Pandey, T.F. George, M.L. Rustgi, J. Appl. Phys. 68 (1990) 1933–1936. E.F. Schubert, J. Vac. Sci. Technol. (1990) 2980–2986. L.N. Pandey, T.F. George, Appl. Phys. Lett. 61 (1992) 1081–1083. R.L. Restrepo, L.F. Castan€o-Vanegas, J.C. Martínez-Orozco, A.L. Morales, C.A. Duque, Appl. Phys. A 125 (2019) 31–36. C. Tang, J. Shi, Physica E 69 (2015) 96–100. Y.C. Shih, B.G. Streetman, Appl. Phys. Lett. 59 (1991) 1344–1346. I. Karabulut, H. Safak, M. Tomak, J. Appl. Phys. 103 (2008) 103116–103123. I. Karabulut, U. Atav, H. Safak, M. Tomak, Eur. Phys. J. B 55 (2007) 283–288. I. Karabulut, H. Safak, Phys. B 368 (2005) 82–87. W.F. Xie, Phys. Stat. Sol. B 246 (2009) 2257–2262. R.H. Wei, W.F. Xie, Curr. Appl. Phys. 10 (2010) 757–760. S. Baskoutas, E. Paspalakis, A.F. Terzis, J. Phys. Condens. Matter 19 (2007) 395024–395033. S. Baskoutas, C. Garoufalis, A.F. Terzis, Eur. Phys. J. B 84 (2011) 241–247. S. Baskoutas, A.F. Terzis, Eur. Phys. J. B 69 (2009) 237–244. S. Baskoutas, A.F. Terzis, Phys. E 40 (2008) 1367–1370. H. Dakhlaoui, Superlattice. Microst. 97 (2016) 439–447. N. Zeiri, N. Sfina, S. Abdi-Ben Nasrallah, M. Said, Optik 124 (2013) 7044–7048. M.J. Karimi, A. Keshavarz, A. Poostforush, Superlattice. Microst. 49 (2011) 441–452. F.M.S. Lima, M.A. Amato, O.A.C. Nunes, A.L.A. Fonseca, B.G. Enders, E.F. da Silva Jr., J. Appl. Phys. 105 (2009) 123111–123118. E. Kasapoglu, U. Yesilgul, F. Ungan, I. S€ okmen, H. Sari, Opt. Mater. 64 (2017) 82–87. L. Ioriatti, Phys. Rev. B 41 (1990) 8340–8344. L.M. Gaggero-Sager, R. P�erez-Alvarez, J. Appl. Phys. 78 (1995) 4566–4569. R.W. Boyd, Nonlinear Optics, third ed., 2007. Rochester, New York.
86
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Computation of the nonlinear optical properties of n-type asymmetric triple δ-doped GaAs quantum well F. Ungan a, *, S. Pal b, M.K. Bahar c, M.E. Mora-Ramos d a
Department of Optical Engineering, Faculty of Technology, Sivas Cumhuriyet University, 58140, Sivas, Turkey Department of Chemistry, Hetampur Raj High School, Hetampur, Birbhum, 731124, West Bengal, India Department of Energy Systems Engineering, Faculty of Engineering, Karamano�glu Mehmetbey University, 70100, Karaman, Turkey d Centro de Investigaci� on en Ciencias, Instituto de Investigaci� on en Ciencias B� asicas y Aplicadas, Universidad Aut� onoma del Estado de Morelos, Av. Universidad 1001, CP, 62209, Cuernavaca, Morelos, Mexico b c
A R T I C L E I N F O
A B S T R A C T
Keywords: δ-doped quantum well Nonlinear optical response Intense laser field
In the present study, we present numerical results for the influence of non-resonant intense THz laser radiation as well as the structure parameters, such as well width and central doping con centrations, on the total optical absorption coefficients (TOACs) and relative refractive index changes (RRICs) in n-type asymmetric triple δ-doped GaAs quantum well (QW) by using the compact-density-matrix formalism via iterative method. In order to obtain these numerical re sults, we have first obtained the subband energy spectrum and the electronic wave functions of the structure by using the effective-mass and parabolic band approximation. We have then calculated the nonlinear optical (NLO) properties of the system, by employing these energy ei genvalues and eigenfunctions. The numerical results for these calculations show that; (i) the magnitude of the resonant peak of TOACs decreases and the peak position shifts towards the lower energies when the central doping concentration and the well width are increased; (ii) due to the intense laser field effect, the magnitude of the TOACs resonant peak initially increases and its position shifts towards higher energies and then it decreases, shifting towards lower energies; (iii) by increasing the central doping concentration (well width), the magnitude of the resonant peak of RRICs increases (decreases), and the resonant peak position shifts towards the lower energies; and finally (iv) because of the increase in the intensity of the laser field applied to the structure, the magnitude of the resonant peak of RRICs increases, and the resonant peak position initially shifts towards the higher energies and then shifting towards the lower energies.
1. Introduction With the advent of technological development on semiconductor nano-structures during the last two decades, the linear and nonlinear optical (NLO) responses of heterostructures with reduced dimensionality such as quantum wells (QWs), quantum wires, quantum dots and superlattices have attracted a great deal of interest in designing of optoelectronic devices. Space quantized energy levels originating in confinement of charge carriers in these structures make them potent candidates to express vivid nonlinear effects [1–3]. It is possible to create semiconductors with different level of confinement of charge carriers in one, two or three dimensions by the expeditious advancement of several growth techniques such as molecular beam epitaxy (MBE) and metal organic chemical vapor * Corresponding author.. E-mail address: [email protected] (F. Ungan). https://doi.org/10.1016/j.spmi.2019.04.023 Received 22 February 2019; Received in revised form 28 March 2019; Accepted 11 April 2019 Available online 16 April 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
deposition (MOCVD) etc [4]. The subband (SB) energy states of the confined charge carriers in the semiconductor QWs give rise aesthetic optical spectrum which can be monitored by tailoring structural parameters as well as by application of various external perturbations such as hydrostatic pressure, electric, magnetic, and high-frequency intense laser fields (ILF). For examples, the effects of the external electric and magnetic fields as well as hydrostatic pressure on the photoluminescence energy transitions in GaAs/GaAlAs coupled double quantum wells have been considered by using a variational method within the parabolic band approximation and the effective mass [5]. The effect of the laser field on the nonlinear optical properties of a square quantum well subject to the external electric field have been examined through the density matrix formalism and the effective mass approach [6]. The most appreciable outcome of such works is determination of the functionalities of various external perturbations such as hydrostatic pressure, electric, magnetic, and ILF on electronic and optical features of QWs. The triple-delta doped quantum wells are taken into consideration due to many advantages of the multiple quantum wells in applications. One of the most common application areas recently is Quantum Cascade Laser [7]. As the quantum efficiency is substantially restricted through optical absorption by electrons and holes, this effi ciency can be enhanced by replication quantum well. In this regard, the multiple quantum wells can be employed in order to be able to cancel out reducing in active layer thickness. In a word, the multiple quantum wells are more functional compared to the single quantum wells to fabricate higher efficiency lasers [8]. Gaggero-Sager et al. carried out an investigation on the influence of tem perature on transport properties in Si-δ-doped GaAs QWs [9]. Grimalsky and his co-workers studied theoretically the impact of temperature on electronic spectrum of δ-doped QWs through the Thomas-Fermi (TF) method [10]. Oubram et al. lucidly probed the influence of hydrostatic pressure on on intersubband (ISB) optical absorption of n-type δ-doped GaAs QW [11]. Ungan et al. nicely explained modulation of electronic states of triple δ-doped QW in presence and absence of an external monochromatic ILF [12]. Strong quantum confinement of the charge carriers in semiconducting heterostructures leads to large energy separation between bound states. Such large energy separation in turn diminishes dipole matrix elements enhancing the possibility of obtaining the resonance condi tions. Bound energy states of charge carriers in semiconductor QWs can be finely tuned by applying the non-resonant ILF without disturbing center of symmetry of confinement potential. Bound energy states and wave functions of charge carriers confined in QWs can be modified by non-resonant ILF. Therefore, the effect of non-resonant ILF has earned a great deal of attraction to the researchers as it can dress the bound energy states of charge carriers. Some of remarkable works on ILF investigations can be exemplified by; the ILF effects on the electronic properties of V-shaped and inverse V-shaped QWs generated by GaAs/GaAlAs heterostructure have been examined by using the transfer matrix method [13]. The ILF effects on the subband structure and polarizabilities of shallow donors in V-shaped QW generated by GaAs/GaAlAs heterostructure with different Al concentrations have been studied through the effective mass approach [14]. The introduction of dopant as highly localized impurity sheets within specially designed QWs has earned much attention to the noble researchers in modulating of ISB energy states of QWs. In this respect, δ-doped systems become very special because they are fabricated with two dimensional electron or hole gases having high density in semiconductor heterostructures [15–20]. Many startling experiments based on fabrication techniques have been being driven to fabricate these novel structures doped with -dopant. MBE is visible experimental method to manufacture delta-doped QWs. Liu et al. have been worked delta doping in QW structures by consisting a strained InGaAs sandwiched between two GaAs layers, analyzing various doses for structures and QW thicknesses [21]. Lee et al. have been studied the magnetic field effects on delocalization in multiple QWs with center-doped generated by GaAs/GaAlAs het erostructure [22]. In this regard, magneto-transport measurements and scaling analysis on a series of center-doped GaAs QWs have been demonstrated by considering the doping concentration in QWs. It was made possible to fabricate δ-doped QWs by doping a positive δ-function potential at the middle of the QWs [23]. By the help of growth-interruption technique in which donor or acceptor impurities could be grown randomly, the epitaxial growth of the δ-potential was done [24]. Such donor or acceptor impurities act as potent source of charge carriers. It is also achieved by depositing atoms of impurity on suspended epitaxial layers of matrix of QWs [25]. Restrepo et al. investigated the effects of an external electric field and impurity on NLO properties in δ-doped quantum rings. Effects of externally applied magnetic field and electric field on nonlinear optical absorption coefficients (OACs) was probed by R. L. Restrepo and his co-workers in a δ-doped AlGaAs=GaAs triple-quantum well (δ-TQW) [26]. Tang et al. examined effects of location and density of δ dopant on ISB transitions in AlGaN=GaN step QWs induced for terahertz (THz) applications [27]. In a word, δ-doped QWs have been studied comprehensively for its potentiality to offer addition quantum confinement to the charge carries and to carry out appreciable radiative recombination between the two dimensional electron gas and the photo-generated holes [28]. By considering remarkable contribution of asymmetric nature into the symmetric energy profile of confinement potential of QWs by means of various external perturbations as well as by presence of dopant, there have been occurred so many novel researches on several linear and nonlinear optical phenomena exhibited by these heterostructures [29–37]. Among various linear and NLO prop erties of semiconductor QWs, TOACs, and RRICs dependent on ISB transition of confined charge carriers have attracted special attention of these researchers. H. Dakhlaoui nicely investigated ISB transition of charge carriers between the ground state and first excited state along with the TOACs in the Si-δ-doped step AlGaN=GaN QW theoretically [38]. Zeiri et al. numerically demonstrated influence of well depth and barrier width on the TOACs, and RRICs in asymmetrically fabricated QWs [39]. Karimi and his co-workers theoretically investigated effects of structural parameters on the TOACs and RRICs of asymmetrically tuned double semi-parabolic QWs [40]. Transition of charge carriers from single to double QW by inducing ILF in semiconductor QWs have been examined by Lima et al. [41]. Kasapoglu and her co-workers elucidated influence of ILF on the TOACs, and RRICs in n-type double δ-doped GaAs QWs [42]. From their experiment it was revealed that, ground state and first excited energy state of double δ-doped GaAs QWs were energetically more separated by enhanced laser field intensity leading to occurrence of optical blue-shift in the ISB transitions. It can be concluded from their experiment that significant tunability in ISB transitions of charge carriers can be achieved by modulating in tensity of applied ILF. The motivation of our present work is to explore the effect of central doping concentration (CDC), and width of QW on the optical 77
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
properties such as TOACs, and RRICs in the n-type asymmetric triple δ-doped (ATDD) GaAs QW induced by ILF. The paper is orchestrated as follows: Section 2 contains theoretical background, Section 3 comprises obtained results and discussions, and finally, the main conclusions of present work are presented in Section 4. 2. Theory As mentioned in the introduction section, in the presence of non-resonant intense THz laser radiation which is applied in the growth direction, we will consider an electron confined in the n-type ATDD GaAs QW grown along the z-axis. Under this condition, taking advantage of the envelope wave function and effective-mass approximations, the Hamiltonian of the confined electron in this structure can be written as following � � �� ℏ2 d 2 H¼ þ V ðz; α Þ ; (1) ATDD 0 2m�e dz2 where m�e is the electron effective mass, ℏ is the reduced Planck constant, α0 ¼ eA0 =m� ϖ is allowed as the parameter that specifies the ILF strength, calling as the laser-dressing intensity parameter, e is the elementary charge, A0 is the laser field strength, and ϖ designates the angular frequency of the laser field, and hVATDD ðz; α0 Þi is the laser-dressed confinement potential given by Ref. [41]. *
+
VATDD ðz; α0 Þ
¼
ϖ 2π
Z
2π =ϖ
(2)
VATDD ðz þ α0 sinϖtÞdt: 0
The mathematical expression used in this study for the potential term was obtained using the TF approximation within the framework of the one-dimensional local density functional theory, as suggested by Ioriatti [43]. This theory was found to be highly successful in the theoretical modeling of the electronic structure in the δ-doped structures [44]. Moreover, with this theory, we do not need to make a self-consistent calculation with a long and laborious process. Thus, the mathematical expression of the confinement potential of n-type ATDD QW can be written as VATDD ðzÞ ¼
ζ2
ζ2 4
Lw=2j þ zL Þ
ðζjz
ζ2 4
ðζjzj þ zM Þ
ðζjz
Lw=2j þ zR Þ4
:
(3)
Here Lw is the effective width between the left and right-hand side δ-doped QWs, ζ ¼ ðe2 ð2m�e Þ3=2 Þ=ð15πεr ℏ3 Þ, zL ¼ 1=5
1=5
1=5
ð2εr ζ3 =ðπ e2 NL2d ÞÞ , zM ¼ ð2εr ζ3 =ðπe2 NM , zR ¼ ð2εr ζ3 =ðπe2 NR2d ÞÞ are the delta potential parameters, εr is the dielectric constant 2d ÞÞ L M R of GaAs, N2d , N2d and N2d represent the corresponding two-dimensional impurity density in the left, central and right-hand side δ-doped layers, respectively. The effect of non-resonant intense THz laser radiation on the electronic wave functions and SB energy spectrum of the structure is obtained from the solution of Eq. (1) using diagonalization method. On the other hand, applying a linearly polarized electromagnetic wave with ω angular frequency on the n-type ATDD QW is considered, and this time-dependent electromagnetic wave is given by Ref. [45]. ~ iωt þ Ee ~ EðtÞ ¼ E0 cosðωtÞ ¼ Ee
iωt
(4)
:
However, the electronic polarization PðtÞ and susceptibility χ ðtÞ are presented via the dipole operator M and the density matrix ρ: being V and ε0 are the volume of system, the dielectric constant of vacuum, respectively ~ PðtÞ ¼ ε0 χ ðωÞEe
iωt
~ iωt ¼ 1 TrðρMÞ: þ ε0 χ ð ωÞEe V
(5)
The linear and nonlinear susceptibilities are determined within the framework of same compact-density-matrix formalism, their analytical expressions are given as follows, � ρ10 �M10 j2 ε0 χ ð1Þ ðωÞ ¼ ; (6) E10 ℏω iℏΓ10 �
ε0 χ ð3Þ ðωÞ ¼
ðE10
"
�
Ej2 ρ10 �M10 j2 �~ � E10 ℏω iℏΓ10 ðE10
ðM11 M00 Þ2 iℏΓ10 ÞðE10 ℏω
� 4�M10 j2 ℏωÞ2 þ ðℏΓ10 Þ2
(7)
# iℏΓ10 Þ
;
where ρ10 ¼ m�e E10 =ðπℏ2 LwÞ is the electron density in the system, E10 is the energy difference between the two lowest electronic states, � � � � � � � � Mij ¼ �hψ i �ez�ψ j i� ði; j ¼ 0; 1Þ is the product of matrix elements and also means geometric factor, Γ10 ¼ 1=τ10 is the phenomenological relaxation rate.
78
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
Then, the RRICs change is presented as � � ΔnðωÞ χ ðωÞ ¼ Re 2 nr 2nr
(8)
where nr is the refractive index. When considering Eqs. (6)–(8), the linear and the third-order nonlinear RRICs are obtained by Ref. [45]. " # � 2 Δnð1Þ ðωÞ ρ10 �M10 j E10 ℏω ; (9) ¼ nr 2n2r ε0 ðE10 ℏωÞ2 þ ðℏΓ10 Þ2 �
μc�M10 j2 4n3r ε0 ½ðE 10
Δnð3Þ ðω; IÞ ¼ nr
ρ10 I
i2 ℏωÞ2 þ ðℏΓ10 Þ2 � ( " � ðM11 M00 Þ2 � 2 ðE10 ℏωÞ � 4ðE10 ℏωÞ�M10 j � ðE10 Þ2 þ ðℏΓ10 Þ2 i oi ðℏΓ10 Þ2 ð2ðE10 Þ ℏωÞ ; �½ðE10 ÞðE10 ℏωÞ ðℏΓ10 Þ2
(10)
where μ is the magnetic permeability, c is the speed of light in free space, I is the intensity of electromagnetic field, defining as � rffiffiffiffi � εR �� 2nr �� I¼2 EðωÞj2 ¼ (11) EðωÞj2 : � μ μc � where εR the real part of the permittivity. The RRICs change is figure out by considering the linear and nonlinear contributions [45]: Δnðω; IÞ Δnð1Þ ðωÞ Δnð3Þ ðω; IÞ ¼ þ : nr nr nr
(12)
However, the OACs change is stated as � �1=2
ηðωÞ ¼ ω
μ εR
(13)
Imðε0 χ ðωÞÞ;
χ ðωÞ is the Fourier component of χ ðtÞ. The linear and the third-order nonlinear OACs are obtained by �
rffiffiffiffi
�M10 j2 ρ ℏΓ10 μ 10 η ðωÞ ¼ ω ; εr ðE10 ℏωÞ2 þ ðℏΓ10 Þ2
(14)
ð1Þ
rffiffiffiffi μ I ð Þ� εR ε0 nr c
� �M10 j4 ρ ℏΓ10 10
ηð3Þ ðω; IÞ ¼ 2ω
ðE10 �
ℏωÞ2
½ðE10
ℏωÞ2 þ ðℏΓ10
ðℏΓ10 Þ2 þ 2ðE10 ÞðE10 ðE10 Þ2 þ ðℏΓ10 Þ2
�2 i2 ð1
� �M11 M00 j2 � �2M10 j2
(15)
ℏωÞ Þ
The TOACs change is given by summing up the linear and nonlinear contributions [45]: (16)
ηðω; IÞ ¼ ηð1Þ ðωÞ þ ηð3Þ ðω; IÞ:
Finally, it should be pointed out that all numerical calculations were performed using the following physical parameters that are suitable for GaAs material [12]: m�e ¼ 0:067m0 (m0 the single electron bare mass), εr ¼ 12:58, nr ¼ 3:2, μ ¼ 4π � 10 7 Hm 1 , NL2d ¼ 3:0 � 1012 cm 2 , NR2d ¼ 9:0 � 1012 cm 2 , I ¼ 0:05 MWcm 2 , and τ10 ¼ 0:14ps. 3. Results and discussion Fig. 1 schematically represents modulation of the confinement potential profile and the probability densities of lowest two energy levels as a function of growth direction (z) in the n-type ATDD QWs for two different values of CDC, width of QW, and laser-dressing parameter. Red line, and blue line indicate probability densities of ground state and first excited state respectively whereas black line indicates confinement potential profile. 12 2 12 2 From Fig. 1(a), it is evident that as CDC increases from NM to NM in absence of ILF, confinement 2d ¼ 4 � 10 cm 2d ¼ 6 � 10 cm potential energy of central QW also decreases without affecting depths of right and left QW. Increase in the depth and width of central QW with the increase of CDC signifies greater amount of confinement of charge carriers with increasing concentration of dopant. As we assume that, concentration of left side dopant (NL2d ¼ 3:0 � 1012 cm 2 ) is lesser than that of right side dopant (NR2d ¼ 9:0 � 1012 cm 2 ), 79
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
the depth of right side QW is larger than the depth of left side QW. The phenomenon clearly announces dependence of depth of QW with concentration of dopant in direct proportion. Therefore, escalation in the depth of central QW with increase in CDC goes in good conformity with the above fact. Fig. 1(a) also reveals that, probability density of first excited state diminishes more than that of ground state with enhancement in CDC. Probability density of first excited state goes through two maxima where, first maximum is more prominent with greater height than the second one. Although energy of first excited state decreases with increase in CDC but its two maxima become more sharp without making any positional shift. Probability density of ground state goes through only one maximum which becomes less prominent with increase in CDC. The above facts clearly suggest that first excited state suffers higher influence of dopant concentration than ground state. As the ground state is more localized to the base of QW it remains in greater confinement than that of first excited state which is more labile. Therefore, CDC plays more impactive role on modulating first excited energy state than modulating ground energy state.
From Fig. 1(b) it can be observed that as effective width between the left and right-hand side δ-doped QWs enhances from Lw ¼ 12 2 100� A to Lw ¼ 200� A in absence of ILF with a fixed value of CDC at NM 2d ¼ 6 � 10 cm , confinement potential energy also increases. It also reveals that, depth of three QWs specially left QW become more prominent with increase in Lw. Right quantum well enjoys blueshift whereas left QW experiences red-shift with increasing Lw. When effective width is small then the charge carrier is effectively confined in left and central QWs. At lower values of Lw triple QWs becomes double QWs. As Lw increases electron tunneling towards left QW also increases. On the other hand, square of wave functions of lowest two energy levels are also augmented with increasing effective width with a considerable blue-shift in the peak positions of probability densities. As Lw increases distance between dopant and charge carriers confined in QWs also increases which effectively increases spatial extension of wave functions of charge carriers. Therefore probability densities are enhanced with increasing Lw. The enhancement of probability densities with increasing Lw is more visible in case of ground energy state because at lower value of Lw ground sate is in more confined state than first excited state which is already in relaxed mood. Hence first excited state enjoys less relaxation than ground state by increasing Lw. Therefore in contrast with Fig. 1(a) it can be argued that ground energy state becomes more affected with increasing Lw whereas from Fig. 1(a) it has been established that first excited state was more deeply perturbed with increasing CDC. Increase in effective width means enhancement in separation between the three QWs which is evident from Fig. 1(b), which in turn offers lesser confinement to the charge carriers. Such lesser confinement plays important role in intensifying of spatial extension of squared ground and first excited state envelope wave functions. Fig. 1(c) clearly announces that, with increasing laser dressing parameter (α0 ) the charge carriers are practically going to be confined in central QW. Enhancement of ILF means enhancement of confinement potential of the charge carriers. As α0 increases from 0 to 50� A with a fixed value of Lw and NM at 200� A and 6 � 1012 cm 2 respectively left and right QWs become very faint. At higher ILF 2d
the confinement potential moves from a triple delta configuration towards a single triangle-like QW. This phenomenon is actually opposite to that reported by Lima and co-workers in the case of rectangular QWs under ILF [41]. They revealed the transition from single to a double potential well conduction band profile. With the increasing ILF parameter the squared wave functions corresponding to the ground and first excited state are also proliferated. As geometrical confinement increases with amplified ILF the wave functions become more localized in QW. It is observed from Fig. 1(c) that, the original region of spatial confinement of electron becomes wider in a stronger way in case of first excited state than that of ground state because first excited state is more labile to enjoy ILF than ground state which is more stable due to its lower energy. Besides, n-type asymmetric triple δ-doped QW becomes triangle-like QW due to enhancement of spatial extension of the electronic wave functions with increasing ILF parameter. Bottom of confinement potential shifts towards higher energy region for stronger ILF, which can be explained in the light of higher repulsion the electron in QWs experiences by n-type dopants that are brought closer to the electron by stronger ILF. Fig. 2(a) selectively represents intersubband energy difference between first excited state and ground state (ΔE ¼ E1 E0 ) with CDC in absence of ILF maintaining Lw at a constant value of 200� A. ΔE diminishes with increasing CDC. The fact can also be established
from Fig. 1(a) which clearly announces that squared ground and first excited state envelope wave functions are decreased with increasing CDC. Such a decrease is more prominent in case of first excited state as a result energy separation between these lowest two states are declined with increasing CDC. Increase in CDC actually offers greater amount of confinement which is responsible for decreasing the separation between first excited state and ground state. Thus escalation in dopant concentration allows the wave functions of two lowest energy states to overlap with each other in greater extent which is clearly reflected by increasing character of jM10 j with CDC. As a result transition energy for electronic transition from ground state to first excited state is also decreased with increasing CDC. The difference between geometric factor of first excited state and ground state (jM11 M00 j) gradually diminishes with increase in CDC which is quite obvious because as CDC increases, confinement of charge carriers also increases, which makes the wave functions more localized. Variation of main intersubband energy difference (ΔE) and electric dipole moment matrix elements as functions of effective width 12 2 between the left and right-hand side δ-doped QWs (Lw) in absence of ILF at a fixed CDC (NM 2d ¼ 6 � 10 cm ) is exhibited in Fig. 2(b). From this figure it is quite evident that ΔE gradually falls off with escalation of Lw which is supported by Fig. 1(b). In Fig. 1(b) it has been established that the probability densities of two lowest energy states are enhanced by enhancement of Lw, but the enhancements have been occurred in different extent for these two states. As a result energy difference between these two states are also reduced with increase in Lw. On the other hand, dipole matrix elements between first excited state and ground state jM10 j is also stepped up with increasing Lw signifying easy transition of electron from ground state to first excited state which is a direct outcome of decrease in ΔE. Besides, jM11 M00 j is sharply increased by increasing Lw which can explained by the fact that wave functions of ground state and first excited state is endowed with greater spatial extension as effective distance between dopant and charge carrier is increased. Similarly Fig. 2(c) evinces modulation of (ΔE) and electric dipole moment matrix elements as functions of laser dressing parameter 80
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
Fig. 1. The variation of the n-type asymmetric triple δ-doped QWs confinement potential profile and the squared ground and first excited state envelope wave functions versus the z-coordinate for two different values of (a) central doping concentration, (b) quantum well width, and (c) laser dressing parameter.
81
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
Fig. 2. Main intersubband energy difference (ΔE ¼ E1 E0 ) and electric dipole moment matrix elements as functions of (a) central doping con centration, (b) quantum well width, and (c) laser dressing parameter. 12 � (α0 ) keeping Lw and NM 2d at constant values of 200A and 6 � 10 cm
2 respectively. The modulation of (ΔE) with α0 is not very straightforward. Initially ΔE rises with increasing α0 attending a maximum nearly at 60� A then it starts to fall slowly. Increase in ILF parameter brings about magnification of energy of two lowest energy states by offering extended confinement of charge carriers. This magnification of energy is not occurred in equal extent for these two states. From Fig. 1(c), it can be concluded that first excited state is offered greater perturbation than ground state by increasing ILF, therefore ΔE after passing through a maximum starts to decrease with increasing ILF. At higher values of α0 energy of first excited state of confined electron do not enhance more readily than ground energy
82
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
state as in lower ILF region because at higher confinement energy of ground state also starts to increase rapidly as a result of higher repulsive interaction between the dopant and the electron. As a result ΔE starts to fall at higher α0 region. Matrix elements of dipole moment between ground state and first excited state (jM10 j) does not get affected so much with increasing α0 because undulatory nature of energy separation between ground state and first excited state. At very lower α0 region it delineates a feeble rise with increasing ILF signifying greater amount of overlap between the wave functions of the two lowest energy states. At some intermediate ILF jM10 j exhibits a faint declination around α0 � 60� A, this is the region where ΔE exhibits its maximum. The two results run in
Fig. 3. The variation of the TOACs as a function of the incident photon energy for different values of (a) central doping concentration, (b) quantum well width, and (c) laser dressing parameter. 83
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
Fig. 4. The variation of the RRICs as a function of the incident photon energy for different values of (a) central doping concentration, (b) quantum well width, and (c) laser dressing parameter.
84
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
accordance with each other. Increase in energy gap means decrease in extent of overlap of the wave functions which in turn diminishes transition of electron from ground state to first excited state. Again at higher ILF region jM10 j begins to rise with ILF as at this region ΔE starts to decrease allowing significant amount of overlap of wave functions of the electronic states. Besides, jM11 M00 j is gradually diminished with increasing ILF only showing a faint rise at the region of α0 � 60� A but after this region it again falls off with increasing
ILF. The wave functions of electronic states become more localized with increasing confinement offered by ILF, as a result jM11 M00 j diminishes with increasing ILF. The feeble increase in jM11 M00 j at the region mentioned above is due to exert of greater repulsion given by dopant to confined electrons which in turn increases spatial extension of wave functions. But with increasing ILF this repulsive interaction is overcome by extended confinement potential brought about through ILF which makes wave functions highly localized. 12 2 Variation of TOACs and RRICs as a function of the incident photon energy for five different values of CDC (NM 2d ¼ 3 � 10 cm ;4 � 12 2 12 2 12 2 12 2 � 10 cm ; 5 � 10 cm ; 6 � 10 cm ;7 � 10 cm ) in absence of ILF with a fixed value of Lw (200A) are represented by Figs. 3(a)
and 4(a) respectively. TOACs exhibit red-shift of absorption peak positions with increasing CDC but peak height are steadily decreased with increasing CDC. From Fig. 2(a) it has been established that jM10 j increases with CDC facilitating transition of electron from ground electronic state to first excited electronic state. Thus, the red-shift of peak positions due to enhancement of electron transition is a direct consequence of extended overlap of electronic wave functions with increasing CDC. At resonant condition (E10 ¼ ℏω), the linear and nonlinear absorption coefficients become rffiffiffiffi �� 2 μ M10 j ρ10 ηð1Þ ðωÞ ¼ ω ; (17) εr ℏΓ10 � � rffiffiffiffi �M10 j4 ρ �M11 M00 j2 μ I 10 � ð1 þ ð Þ� �2M10 j2 εR ε0 nr c ðℏΓ10 Þ3
ηð3Þ ðω; IÞ ¼ 2ω
(18)
ðℏΓ10 Þ2
� Þ ðE10 Þ2 þ ðℏΓ10 Þ2 From equations (17) and (18) it is evident that both linear and nonlinear AC increase with increase in M10 , but such an enhancement is more visible in case of nonlinear absorption coefficient. As a result nonlinear contribution in TOACs is magnified with CDC. Thus resonant peak height of TOACs are gradually petered out with CDC. Similarly resonant peak positions of RRICs experience a red-shift with increasing CDC as demonstrated by Fig. 4(a). The resonant peak heights of RRICs also enjoy enlargement with enhancement of CDC. Figs. 3(b) and 4(b) delineate regulation of TOACs and RRICs respectively as function of incident photon energy for five different values of Lw (100� A; 125� A; 150� A; 175� A; and200� A) in absence of ILF with CDC fixed at NM ¼ 6 � 1012 cm 2 . Both TOACs and RRICs 2d
experience red-shift of resonant peak positions with increasing Lw. From Fig. 2(b) it has been made clear that M10 increases with Lw helping electronic transition from ground state to first excited state. There resonant peak positions of TOACs and RRICs also suffer redshift with increasing Lw. Besides, resonant peak height gradually decreases with increasing Lw for both the optical responses. From Fig. 2(b) it has been also argued that both jM10 j and jM11 M00 j are intensified with enhancement in Lw imparting greater contribution of nonlinear parts of TOACs as well as of RRICs as evident from equation (18). As a result peak height of TOACs and RRICs are also diminished with increasing Lw. Figs. 3(c) and 4(c) deliciously describe variation of TOACs and RRICs respectively as a function of incident photon energy for five different values of ILF dressing parameter (α0 ¼ 0� A;25� A;50� A;75� A;100� A) with constant values of Lw ¼ 200� A, and NM ¼ 6 � 1012 cm 2 . 2d
From these two figures it can be ferreted out that resonant peak positions of TOACs and RRICs go through a blue-shift with increasing α0 from 0� A to 50� A , then they experience a red-shift on going from α0 ¼ 75� A to α0 ¼ 100� A. From Fig. 2(c) it can easily be observed that
A then it begins to fall with increasing α0 . energy separation between ground state and first excited state (ΔE) enhances up to α0 � 60� Similarly jM10 j exhibits a dim fall nearly at α0 � 60� A then it starts to rise again with further increase in α0 . Therefore, the blue-shift and
subsequent red-shift of resonant peak positions of these two optical responses can be explained in the light of undulatory nature of ΔE and jM10 j with ILF. Figs. 3(c) and 4(c) also delineate that resonant peak height of TOACs and RRICs at first increases with increase in α0 from 0� A to 50� A, then it starts to decline on going from α0 ¼ 75� A to α0 ¼ 100� A. Actually in the range of α0 � 60� A both jM10 j and jM11
M00 j manifest declination with ILF. As a result, nonlinear contribution to TOACs and RRICs becomes lesser leading to increase in A both the matrix elements begin to rise which impart larger contribution of resonant peak height of TOACs and RRICs. Beyond α0 � 60� nonlinear part to TOACs and RRICs leading to decrease in resonant peak height of these two optical responses. 4. Conclusions In this present investigation we furnish a detailed study of TOACs and RRICs in an n-type asymmetric triple δ-doped GaAs quantum well (QW) induced by non-resonant intense THz laser radiation (ILF). Applying ILF nicely reorganizes the confinement potential of the electron confined in QWs. Numerical calculations are done by using the effective-mass and parabolic band approximation to obtain intersubband energy spectrum and the electronic wave functions of the structure. The investigations show that, the energy difference between the ground state and first excited state steadily decreases with increasing CDC and width of QWs. As a result, resonant peak positions of TOACs and RRICs suffer red-shift with increase in CDC (NM 2d ) and effective width (Lw) of doped QW. Besides, resonant peak heights of TOACs and RRICs exhibit steady declination with increasing CDC and Lw. As ILF increases confinement of electron trapped 85
Superlattices and Microstructures 130 (2019) 76–86
F. Ungan et al.
inside QWs also increases, which completely destroyed the energy barrier between the δ-doped layers and makes the triple δ-doped QWs into a shallow single confinement potential. The separation between lowest two energy states in ATDD GaAs QW increases with initial increase in ILF, which causes to an optical blue-shift in resonant peak positions of TOACs and RRICs. But with further increase in ILF such energy separation starts to decrease which in turn leads to an optical red-shift in ISB transitions. In a word, optical transitions in ATDD QW can be accomplished with eloquent tunability by modulating different parameters like, central doping concentration, width of QW and intensity of ILF. We hope that the work can motivate further theoretical investigations on the nonlinear optical properties in asymmetric n-type triple δ-doped QW, which leads to immense development of optoelectronic devices. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
S. Sakiroglu, F. Ungan, U. Yesilgul, M.E. Mora-Ramos, C.A. Duque, E. Kasapoglu, H. Sari, I. S€ okmen, Phys. Lett. A 376 (2012) 1875–1880. I. Karabulut, C.A. Duque, Physica E 43 (2011) 1405–1410. J.C. Martínez-Orozco, M.E. Mora-Ramos, C.A. Duque, J. Lumin. 132 (2012) 449–456. M.K. Gurnick, T.A. De Temple, IEEE J. Quantum Electron. 19 (1983) 791–794. S.Y. L� opez, M.E. Mora-Ramos, C.A. Duque, Physica B 404 (2009) 5181–5184. I. Karabulut, Appl. Sur. Sci. 256 (2010) 7570–7574. M. Razeghi, Technology of Quantum Devices, Springer, Evanston- USA, 2010, pp. 271–321. Q. Dai, et al., Appl. Phys. Lett. 94 (2009) 111109–111112. L.M. Gaggero-Sager, G.G. Naumis, M.A. Mun€ oz-Hernandez, V. Montiel-Palma, Physica B 405 (2010) 4267–4270. V. Grimalsky, L.M. Gaggero-Sager, S. Koshevaya, Physica B 406 (2011) 2218–2223. O. Oubrama, O. Navarro, L.M. Gaggero-Sager, J.C. Martínez-Orozco, I. Rodríguez-Vargas, Solid State Sci. 14 (2012) 440–444. F. Ungan, M.E. Mora-Ramos, E. Kasapoglu, H. Sari, I. S€ okmen, Philos. Mag. 99 (2019) 644–658. E.C. Liculescu, A. Radu, M. Stafe, Superlattice. Microst. 46 (2009) 443–450. L.M. Burileanu, E.C. Niculescu, N. Eseanu, A. Radu, Phys. E 41 (2009) 856–860. G.H. Dohler, Surf. Sci. 73 (1978) 97–105. C.E.C. Wood, G. Metze, J. Berry, L.F. Eastman, J. Appl. Phys. 51 (1980) 383–387. M. Zachau, F. Koch, K. Ploog, P. Roentegen, H. Beneking, Solid State Commun. 59 (1986) 591–594. M.E. Lazzouni, L.J. Sham, Phys. Rev. B 48 (1993) 8948–8958. R.V. de Walle, R.L.V. Meirhaeghe, W.H. Laflere, F. Condon, J. Appl. Phys. 74 (1993) 1885–1889. K.A. Rodriguez-Magdaleno, J.C. Martínez-Orozco, I. Rodriguez-Vargas, M.E. Mora-Ramos, C.A. Duque, J. Lumin. 147 (2014) 77–84. D.G. Liu, C.P. Lee, K.H. Chang, D.C. Liou, Appl. Phys. Lett. 57 (1990) 1887–1888. C.H. Lee, Y.H. Chang, Y.W. Suen, H.H. Lin, Phys. Rev. B 58 (1998) 10629–10633. L.N. Pandey, T.F. George, M.L. Rustgi, J. Appl. Phys. 68 (1990) 1933–1936. E.F. Schubert, J. Vac. Sci. Technol. (1990) 2980–2986. L.N. Pandey, T.F. George, Appl. Phys. Lett. 61 (1992) 1081–1083. R.L. Restrepo, L.F. Castan€o-Vanegas, J.C. Martínez-Orozco, A.L. Morales, C.A. Duque, Appl. Phys. A 125 (2019) 31–36. C. Tang, J. Shi, Physica E 69 (2015) 96–100. Y.C. Shih, B.G. Streetman, Appl. Phys. Lett. 59 (1991) 1344–1346. I. Karabulut, H. Safak, M. Tomak, J. Appl. Phys. 103 (2008) 103116–103123. I. Karabulut, U. Atav, H. Safak, M. Tomak, Eur. Phys. J. B 55 (2007) 283–288. I. Karabulut, H. Safak, Phys. B 368 (2005) 82–87. W.F. Xie, Phys. Stat. Sol. B 246 (2009) 2257–2262. R.H. Wei, W.F. Xie, Curr. Appl. Phys. 10 (2010) 757–760. S. Baskoutas, E. Paspalakis, A.F. Terzis, J. Phys. Condens. Matter 19 (2007) 395024–395033. S. Baskoutas, C. Garoufalis, A.F. Terzis, Eur. Phys. J. B 84 (2011) 241–247. S. Baskoutas, A.F. Terzis, Eur. Phys. J. B 69 (2009) 237–244. S. Baskoutas, A.F. Terzis, Phys. E 40 (2008) 1367–1370. H. Dakhlaoui, Superlattice. Microst. 97 (2016) 439–447. N. Zeiri, N. Sfina, S. Abdi-Ben Nasrallah, M. Said, Optik 124 (2013) 7044–7048. M.J. Karimi, A. Keshavarz, A. Poostforush, Superlattice. Microst. 49 (2011) 441–452. F.M.S. Lima, M.A. Amato, O.A.C. Nunes, A.L.A. Fonseca, B.G. Enders, E.F. da Silva Jr., J. Appl. Phys. 105 (2009) 123111–123118. E. Kasapoglu, U. Yesilgul, F. Ungan, I. S€ okmen, H. Sari, Opt. Mater. 64 (2017) 82–87. L. Ioriatti, Phys. Rev. B 41 (1990) 8340–8344. L.M. Gaggero-Sager, R. P�erez-Alvarez, J. Appl. Phys. 78 (1995) 4566–4569. R.W. Boyd, Nonlinear Optics, third ed., 2007. Rochester, New York.
86