The second harmonic generation (SHG) of GaAs/AlxGa1 − xAs a wurtzite coupled quantum wells (CQWs)
is studied theoretically for different widths of well, barrier and values Al concentration, taking into account
the strain-induced piezoelectric (PZ) effects. The analytical expression of the SHG susceptibility is deduced
by using the compact density matrix approach. The confined wave functions and energies of electrons
GaN/AlxGA1 – x N are calculated in the effective-mass approximation, solving the Schrödinger equation by
Numerov’s method using six order approximations for the derivatives. The calculated results also reveal
that by adjusting the widths of well, the barrier and Al concentration respectively, a set of optimal structural
parameters can be found for obtaining a strong SHG susceptibility.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/2777

Abouelaoualim, D.

Elkadadra, A.

Oueriagli, A.

Outzourhit, A.

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 4 No 2, 02002(3pp) (2012) Том 4 № 2, 02002(3cc) (2012)   2077-6772/2012/4(2)02002(3) 02002-1 Ó 2012 Sumy State University High Second-Order Nonlinear Susceptibility Induced in GaN/AlxGa1 – xN Coupled Quantum Well for Infrared Photodectors Application  A. Elkadadra, D. Abouelaoualim*, A. Oueriagli, A. Outzourhit  LPSCM, Department of Physics, Faculty of Sciences Semlalia  Cadi Ayaad University P.O. Box 2390, 40000 Marrakech, Morocco  (Received 22 January 2012; published online 07 May 2012)  The second harmonic generation (SHG) of GaAs/AlxGa1 − xAs a wurtzite coupled quantum wells (CQWs) is studied theoretically for different widths of well, barrier and values Al concentration, taking into account the strain-induced piezoelectric (PZ) effects. The analytical expression of the SHG susceptibility is deduced by using the compact density matrix approach. The confined wave functions and energies of electrons GaN/AlxGA1 – x N are calculated in the effective-mass approximation, solving the Schrödinger equation by Numerov’s method using six order approximations for the derivatives. The calculated results also reveal that by adjusting the widths of well, the barrier and Al concentration respectively, a set of optimal struc-tural parameters can be found for obtaining a strong SHG susceptibility.  Keywords: Second-harmonic generation, Wurtzite quantum well, Strain-induced piezoelectric, Density matrix approach, Numerov method.   PACS numbers: 42.65.Ky, 85.60 – q   ______________ * abouelaoualim_d@hotmail.com 1. INTRODUCTION  Recently, there has been a considerable interest in interband transitions (IBTs) in nitride semiconductor quantum well (QW) structures [1-5]. These heterostruc-tures are characterized by very large conduction-band offsets (up to about 1.75 eV) allowing for ISBT at record short wavelengths in the near-infrared spectral region. At the same time, they feature especially fast ISBT re-laxation lifetimes due to the highly polar nature of (Al)(Ga)N. The spontaneous and piezoelectric polariza-tions of these structures [6] can give rise to large polari-zation sheet charges on the interfaces or surfaces of the system, which, in turn, create a definite internal built-in electric field (BEF) inside the heterostructure. This property is ideally well suited to the development of ul-trafast all-optical modulators and switches operating at fiber-optic communication wavelengths. In this paper, we show that the built-in electric fields induced by the piezoelectricity and spontaneous polariza-tion in a wurtzite GaN/AlxGa1 – xN  QW  breaks  the  sym-metry of the confinement potential profile and leads to large second-order susceptibilities. In particular, g(2) was investigated as a function of Al content, well width, bar-rier width and pump photo energy. The organization of this paper is as follows. In Section 2, theory and model are presented. The model of calculation used in this paper is based on using the effective mass Schrödinger with the Numerov numerical technique which we provide. Simula-tion results of our models and discussions are illustrated in Section 3. In this section, our results have been ex-plained and show that the proposed structure is a better model for applications in microwave and optical switches. Finally, the paper contains a short conclusion.  2. THEORY AND MODEL  The second-order susceptibilities ( )22wc  can  be  ex-pressed by   ( ){ ( )( )( )( )( ) ( )( ) }322 2 ,01212 +1 1+1 ,ln nm nlm nnl nl ml mlnl nl ml mlml ml nl nlmn mnq n z z zi ii ii iiwc ew g w gw g w gw g w gw g=´- W - - W -+- W - W --- W - - W -´- W -åh   where ijz  is i jzj j , /i jE E- h , 1/ ijg  is the dephasing times, and q the electronic charge. It can be easily checked that this expression is nonzero only for asymmetric wave functions. In order to calculate the c2, the eigenenergies and envelope wave function should be obtained. One solves the effective-mass Schrödinger equation:    ( ) ( ) ( ) ( )2 22 ,2d z V z z E zm dzj j j*- + =h    where m* the the effective mass of the particle, and the conduction band nonparabolicity has been neglect-ed, and V(z) the conduction band offset given by   ( ) ,c zV z E eF z= D +   where DEc is the conduction band discontinuity be-tween the well and barrier layers in the absence of strain, and Fz is the sum of the spontaneous (zero strain) and piezoelectric (strain-induced) field, which is given by   ( )( )1 1110 2x X x Xx Xx XAl Ga N Al Ga N GaNbPESP SPAl Ga NAl Ga NGaNe w e bP P P LFL Le e e- ---+ -= -+   A. ELKADADRA, D. ABOUELAOUALIM, A. OUERIAGLI, A. OUTZOURHIT J. NANO- ELECTRON. PHYS. 4, 02002 (2012)   02002-2  ( )( )1 11022x X x Xx XAl Ga N Al Ga N GaNwPESP SPGaNAl Ga NGaNe w e bP P P LFL Le e e- --+ -=+   where xSPP  and xPEP  respectively denote the spon-taneous and the piezoelectric polarization of layer x (x = GaN, AlxGa1 – xN) ee is the dielectric constant of materiel x,  and  Lw  and  Lb  respect  the  widths  of  the  GaN well-layer and AlxGa1 – xN barrier layer. Under the effective mass and envelope wave-function approx-imation, the electronic eigen-function are expressed by    12 34 5exp( )( ) exp ( ) exp ( )cosh( ) sinh( )a z zz a i kz a i kza z a zrjr ráìï + -íï +î   where 2 2 2 ( )bbm V Er *= -h  and 2 2 2 wk m E*=h . The eig-en-energy En of the double quantum well in z-direction can be numerically solved by standarded boundary condition of electronic bound state.   3. NUMERICALS RESULTS AND DISCUSSIONS  In order to find the optimal parameters of size structure for obtaining strong SHG coefficient  in the wurtzite nitride DQW, the numerical calculations on a GaN AlxGa1 – xN DQW are performed in the present section. Material parameters and values of polarization and elastic constants used in the calculation are given in Table 1 [7].   GaN AlxGa1-XN a(A0) 3.189 3.122 x + 3.189(1 – x) er 10 8.5 x + 10(1 – x)  e31(C/m2) – 0.490 – 0.600 x – 0.490(1 – x) e33(C/m2) 0.730 – 1.460 x + 0.730(1 – x) C13(GPa) 103 108 x + 103(1 – x) C13(GPa) 373 405 x + 373(1 – x) Psp(C/m2) – 0.029 – 0.081 x – 0.029(1 – x)  Taking DEc = 2 eV as conduction band discontinuity and 12 23 0.006emVg g g= = = , 00.33GaNm m* = , GaNm* =  ( ) 00.33 0.22 1x x mé ù= + -ë û , is the mass of a free electron, the sheet density n = 2 × 1012 cm – 2. Using Numerov’s method for the numerical solution of the Schrödinger equation (Eq.(4)) in one dimension, we find both for the eigenvalue and the wavefunction. The efficiency of Numerov’s method lies in the fact that one obtains a local error of O(h6) with just one evalua-tion of V and E per step. This should be compared to the Runge–Kutta algorithm that needs six function evalua-tions per step to achieve a local error of O(h6). Fig. 1. shows the second harmonic generation (SHG) coeffi-cients function of the pump photon energy	ħw. The well-width Lw, the barrier-width Lb and doped concentration x are parameters in Fig. 1(a), (b) and (c), respectively. It can be seen from the figure that, the strength of the SHG coefficient in the CQW achieves the magnitude of 10 – 4 m/V, which is one to two orders higher than that in GaN/AlGaN single quantum well [8]. The strong SHG effect is primarily ascribed to the large dipole-matrix elements in nitride DQW. Moreover, it is observed from   a   b   c  Fig. 1 – SHG coefficients c(2) 2ω as a function of the photon energy ħw for different parameters of well-width Lw (Lb = 1 nm, x = 0.3 in (a)), the barrier-width Lb (Lw = 3 nm, x = 0.3 in (b)) and the doped-concentration x (Lw = 3 nm, Lb = 1 nm in (c))   HIGH SECOND-ORDER NONLINEAR SUSCEPTIBILITY INDUCED… J. NANO- ELECTRON. PHYS. 4, 02002 (2012)   02002-3 Fig. 1(a) that, with the increase of the well-width Lw, the SHG coefficients increases apparently, and the resonant photon frequencies have a obvious red-shift. For example, as Lw increases  from  2.5  to  3.5  nm,  the  SHG coefficients increase nearly three times, and the resonant photon frequencies  ħw decrease. Hence it is an effective method that the DQW with relatively large Lw should be chosen for obtaining large SHG. On the other hand, the no table red-shift of the pump photon frequencies should also be taken into account. When the barrier-width Lb is  over  2.0  nm (Fig.  1(b))  and the doped concentrate x = 0.2 (Fig. 1(c)), with the decreases of Lb or x, the SHG coefficient also increases remarka-bly. Moreover, the resonant photon frequencies ħw have obviously blue-shift. The strong strength characteris-tics of the SHG susceptibility can be attributed to the complicated competitive relationships between the ef-fect of strong built-in electric fields (F AlGaN and F GaN) and the quantum size effect of the DQWs (the well-width Lw and barrier width Lb) [9,10]. Indeed, with the increase of Lw, the size confinement of the CQW for the carrier becomes weaker and weaker. But the built-in electric field F GaN in GaN-barrier becomes larger [14, 15], which leads to stronger confined effect to the DQW systems. On the other hand, as the barrier-width Lb increases, the quantum-size confined effect for the car-rier in QW will decrease. But the built-in electric field F AlGaN in the AlGaN -well region becomes stronger with the creases of Lb, which enhances the confined effect to the carrier in the DQW. Hence these integrat-ed factors strongly influence the distributions of elec-tronic wave-functions and eigen-energy, which affects the magnitude of the dipole-matrix elements at last. This is quite beneficial to getting strong SHG coefficients in nitride wurtzite DQWs. In conclusion, the second-order nonlinear optical susceptibilities of GaN/AlxGa1 – xN DQW by taking into account influences of polarization field have been per-formed in this paper, and a Numerov numerical tech-nique to solve Schrödinger equation is used. A large value of c(2) (several orders than that of bulk GaN) has been obtained. In addition, c(2) as functions of well width, barrier width, Al content and pump photo ener-gy are also given,  which can provide the structure pa-rameters to optimize c(2) of DQW. Based on the present results, we hope that the important infrared photode-tectors, near infrared laser amplifiers and can be con-structed based on the group- III nitride QWs.  RERFERENCES   1.  S.C.P. Rodrigues, O.F.P. dos Santos, L.M.R. Scolfaro, G.M. Sipahi, E.F. da Silva, Appl. Surf. Sci. 254, 7790 (2008). 2. A. Alemu, A. Freundlich, Microelectr. J. 40, 421 (2009). 3. M. Gladysiewicz, R. Kudrawiec, J. Misiewicz, M. Siekacz, G. Cywinski C. Skierbiszewski, Phys. Status Solidi (C) 8, 2273 (2011). 4. T. Lermer, A. Gomez-Iglesias, M. Sabathil, J. Müller, S. Lutgen, U. Strauss, B. Pasenow, J. Hader, J.V. Moloney, S.W. Koch, W. Scheibenzuber, U.T. Schwarz, Appl. Phys. Lett. 98, 021115 (2011).   5. H. Jönen, U. Rossow, H. Bremers, L. Hoffmann, M. Brendel, A.D. Dräger, S. Schwaiger, F. Scholz, J. Thalmair, J. Zweck, A. Hangleiter, Appl. Phys. Lett. 99, 011901 (2011). 6. M. Thomsen, H. Jönen, U. Rossow, A. Hangleiter, J. Appl. Phys. 109, 123710 (2011). 7. F. Bernandini, V. Fiorentini, Phys. 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High Second-Order Nonlinear Susceptibility Induced in GaN/AlxGa1 – xN Coupled Quantum Well for Infrared Photodectors Application

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