Optimization of Resonant Second-and Third-Order Nonlinearities in ...

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 5, MAY 1998

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Optimization of Resonant Second- and Third-Order Nonlinearities in Step and Continuously Graded Semiconductor Quantum Wells Dragan Indjin, Zoran Ikoni´c, Vitomir Milanovi´c, and Jelena Radovanovi´c

Abstract—Methods for systematic optimization of step-graded and continuously graded ternary alloy based quantum wells (QW’s), in respect to second- or third-order intersubband nonlinear susceptibilities at resonance, are discussed. The use of these methods is examplified on the design of Alx Ga10x N and Alx Ga10x As-based QW’s intended for resonant second harmonic or third harmonic generation with h  ! = 116 meV or h ! = 240 meV pump photon energies, the objective being the largest susceptibility achievable with the chosen material. The obtained results exceed those previously reported in the literature. Index Terms— Nonlinear optics, quantum wells, secondharmonic generation.

I. INTRODUCTION

B

ANDGAP engineering of semiconductor quantum-well (QW) structures has been extensively employed in recent years, via sophisticated use of molecular beam epitaxy technique, for optimizing the performance of various QW-based devices [1]. In effect, the quantized states energies and wave functions may be tailored, by varying the structure profile, so to best suit a particular application. A good example of the field where the bandgap engineering can be fully exploited is that of intersubband transition based optical nonlinearities in QW’s. Advantages in using these for second-harmonic generation (SHG) or third-harmonic generation (THG) are large transitions matrix elements ( nm) and the possibility of achieving resonance conditions (subsequent levels spacing equal to the pump photon energy), which highly enhance the nonlinearity. The design goal would then be to find a QW with relevant matrix elements as large as possible, tuned to resonance for a chosen pump wave length. Taking the case of resonance SHG, for instance, one needs three equidistant is states and intersubband second-order susceptibility proportional to the cyclic product of the three dipole matrix elements. Symmetric structures are ruled out (one of matrix elements therein would be zero), and there remains to optimize the profile of an asymmetric structure to maximize resonant . Various asymmetric QW’s have been considered in the literature for this purpose, e.g., compositionally graded (in a stepwise-constant manner) step QW’s [2]–[4], electric field biased QW’s [5], [6], and asymmetric coupled QW’s [7]–[9]. -optimized structures of this type relied The design of Manuscript received July 17, 1997; revised January 27, 1998. The authors are with the Faculty of Electrical Engineering, University of Belgrade, 11000 Belgrade, Yugoslavia. Publisher Item Identifier S 0018-9197(98)03062-0.

mainly on physical intuition or on analytical consideration of idealized structures (with infinite barriers, or constant effective mass) and subsequent corrections for nonideality by trial-anderror method. Recently, some research effort has been put into devising systematic optimization procedures for finding the best potential shape of, usually quaternary alloy based, QW’s [10], [11]. Similar considerations apply to resonant THG in QW’s. While asymmetry of an appropriate structure is not really necessary, it does not seem to be harmful either. Indeed, the only known design of structures with four equispaced states for resonant THG [8], [12] comprise asymmetric coupled QW’s. It is not stated in [8] and [12] whether these have been optimized in any way. In this paper, we discuss some systematic methods of optimizing the potential shape (profile) of QW’s based on or . The first method, ternary alloys, in respect to handling step asymmetric QW’s, relies on finding the solution to a system of nonlinear equations containing a few free parameters. The second method uses a single-parameterdependent isospectral transform (based on supersymmetric quantum mechanics) of an initial potential, which generates a class of asymmetric potentials, with the resonance conditions achieved in the initial potential left unperturbed. Both methods are systematic in the sense that all potentials of a given class are explored, i.e., no potential better than that found as optimal may exist in that class. While the task of global optimization (finding any general potential shape, possibly subject to some mild constraints like the allowed extent of QW or similar) seems too difficult at present, the obtained numerical results suggest that they are probably not far from being globally optimal. II. THEORETICAL CONSIDERATIONS We consider an n-doped QW structure based on direct bandgap semiconductors and take the bandgap throughout it to be large enough that interband transitions may be neglected. The polarization response of the structure to the pump field is then mainly governed by interwith photon energy subband transitions between quantized conduction band states . Nonlinear polarization at twice the frequency of the pump field, acting as the source of second-harmonic field is described which, in the by the second-order susceptibility (i.e., double-resonance regime

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with strictly equispaced states) takes the maximal value [3] (1) are the transition dipole matrix where the length of the structure, the electron sheet elements, density in the ground state (assumed to be the only populated state), and the off-diagonal relaxation rates are taken to be (though this is not essential). Similarly, equal , describing THG, in case of the third-order susceptibility triple resonance takes the form [8], [12] (2) or one should clearly maxIn order to maximize imize the corresponding products of dipole matrix elements in numerators of (1) and (2), by appropriate tailoring of QW profile (and hence the wave functions) while preserving the , the presence of rules out level spacing. In case of symmetric QW’s, because of definite parity of wave functions, so one should consider asymmetric structures only. The case of , however, is less clear. All the matrix elements in (2) are generally nonzero, even in symmetric systems. Yet, the linear , or, if truncated, but harmonic oscillator has very small. In another simple symmetric case, infinitely deep is 10 times smaller than matrix rectangular well, elements. These numerical evidence indicate that asymmetric is to systems may well be better than symmetric ones if be maximized (though we are not aware of any general proof that it is so). To find the best potential shape, which will maximize, e.g., , i.e., the product of matrix elements , , and (related to it in ternary one should vary the potential , subjected to the constraint that alloys) effective mass states spacing should be as desired. Quantized electron states in a QW structure with position-dependent effective mass may be found by solving the envelope function Schr¨odinger equation of the form [13] (3) Effects of bulk dispersion nonparabolicity may be conveniently described by energy-dependent effective mass, according to the two-band Kane model [14]

(a)

(b)

Fig. 1. The potential (conduction band edge) in (a) single-step and (b) double-step QW’s. The structure design parameters, used in the main text, are all denoted.

A. Optimization of Step QW’s Consider an asymmetric step QW with stepwise constant potential and effective mass [Fig. 1(a)], which is frequently used in resonant SHG. Its optimization has been considered in [3]. It first assumed an idealized model with infinite barriers and constant effective mass, with nonparabolicity neglected, allowing for analytic solution. Having optimized the parameters of such a QW, it was then modified to take a finite barrier height into account. A single composition of the barrier material was assumed, regardless of the pump photon energy that QW was designed for, i.e., the barrier height was not considered as a parameter for optimization. Yet, data presented in [3] for infinite and finite barriers QW’s indicate that the in the latter exceeds the value matrix elements product but is always in the former by a factor which depends on may more then double due to finite very significant, e.g., barriers. The effect is ascribed to the fact that finite barriers allow for wave function penetration, and more extended wave functions lead to larger matrix elements. With enhancement of as large as that (i.e., such difference between optimized idealized QW, and modified but nonoptimized realistic QW), one may wonder whether the barrier height should also be considered as a free parameter, on equal grounds with others, and perform the full optimization of this system. This is the problem we consider below, taking also the nonparabolicity (neglected in [3]) into account. Equations (3)–(4) should be solved, with observing the and boundary conditions (continuity of at and . With the conventional exponential or plane-wave type of solutions in separate layers of the structure [Fig. 1(a)], we get a system of six homogeneous equations, the nontrivial solution of which requires that

(4) is the material composition- (and hence the where denotes the parposition-) dependent bandgap, and abolic (band edge) effective mass. These effects become increasingly important at higher energies. They are technically significant in QW’s designed for SHG of CO laser radiation, and even more so at higher energies.

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in the energy range , where and are the energy-dependent nonparabolic effective masses in the barrier, well, and the step layers, respectively, and the corresponding wave vectors are , and . In , we define the energy range above the step and (5) is modified by subnew stituting . This defines the function , zeros of which are the energies of quantized states in asymmetric single-step QW, with nonparabolicity included. The corresponding wave functions are then simply derived from the boundary conditions . and the normalization condition Having chosen the alloy system to work with (e.g., As), it is reasonable to take the well layer to Al Ga comprise a pure “well-type” semiconductor (GaAs in this roughly scaling as [3] instance), because, with there is no benefit from allowing the well layer to be made of is defined from the start, and in the step the alloy. Thus, and barrier layers, which are made of the alloy, with suitable and , the effective mass and potential are compositions and uniquely related to each other, i.e., . Therefore, is a nonlinear function and of four independent parameters, say the widths and potentials and . All possible QW shapes, i.e., the values of the four parameters, which result in three states , may be obtained from the system of spaced by three nonlinear equations (6) , the ground state energy measured from the well where bottom, is an additional free parameter. Equation (6) may then be solved for three parameters out of five, the remaining two being “input” parameters to be used for the QW shape variation, with values of all the five parameters subject either to obvious physical constraints or to limitations imposed by the chosen alloy system or by technological feasibility of the structure. By evaluation of the matrix elements (that can be done analytically, though via rather cumbersome expressions) for each individual solution, it is quite straightforward to search the entire two-dimensional (2-D) free-parameter space and find . the best of all single-step QW’s, which maximizes

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Analogous procedure may also be applied to optimize single-step QW’s in respect to third-order nonlinear susceptibility. In that case, an additional equation, , should be appended to (6), which would then yield the solution for four parameters in terms of one free parameter. Therefore, there is still some room for (one-dimensional) optimization. Alternatively, to introduce more freedom, one may put in another step layer in Fig. 1(a), and such two-step QW [Fig. 1(b)] would then be described by seven parameters, now three of which would be free. The expression for reads as (7), shown at the bottom of the page, with all the , parameters defined in Fig. 1(b), which holds in case and with substitutions analogous to those explained below (5) and . in cases The described procedure may be employed for optimization of (also frequently encountered) coupled wells. QW’s intended for other nonlinear processes which may not require equispaced states (off-resonant harmonic generation, parametric down-conversion, etc.) can also be optimized in the same fashion. On the practical side, it may be implemented with reasonable effort and computation time only for structures comprising not more than a few layers of different widths and compositions. Yet it is exactly such simple structures that are of the largest technical importance at present. B. Optimization of a Continuously Graded QW Optimization of QW’s with nominally continuously graded alloy composition, and smooth potential, is quite a different task. One can imagine that such a QW approximately com10 of thin layers with constant prises a large number composition (there is much justification for it, both from the technological viewpoint and due to the fact that composition variation within one lattice constant is meaningless). However, setting up the above procedure for such a system, and performing the (essentially global) optimization would be highly impractical. Not only that deriving a reasonably complicated , analogous to (5) or (7), is very difficult, expression for but also solving a large system of nonlinear equations is extremely time-consuming and potentially unsuccessful unless a good initial guess is provided. In dealing with continuously graded QW’s, one thus has to satisfy with more modest goals, i.e., to do optimization within some class of potentials. Keeping the levels’ spacing at required values throughout

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the potential shape variation is to be obtained by means other than finding the solution of a large system of nonlinear equations. A very convenient way to vary the potential shape in that manner has been devised within the supersymmetric quantum mechanics (SUSYQM) theory [15]. Starting from an initial (“original”) potential for which it was achieved, in whatever way, that its quantized states energies or their spacing are as required, this technique allows one to generate a family of parameter-dependent potentials which are all isospectral to the original. The SUSYQM theory [15] was mainly developed to handle Hamiltonians with variable potential but constant mass. In [10] that form, we have applied it to optimization of in constant mass, quaternary alloy based QW’s. The summary of working expressions are given below. The supersymmetric , isospectral to the original , partner potential (3), is given by (8) and normalized eigenfunctions corresponding to related to those of the original, (3), via

are

Fig. 2. The product of matrix elements 5(2) = M01 M12 M20 in an Alx Ga10x As single-step QW from Fig. 1(a) as it depends on the choice of the lower well width cW for various values of barrier height UB , calculated under double resonance condition  h! = 116 meV.

optimization procedure is not global, but rather within the class of potentials derived from the chosen original. The fact that remains constant upon SUSYQM transform makes not realizable by graded ternary alloys. To make it realizable, we have to slightly modify , as will be explained in Section III.

(9) III. NUMERICAL RESULTS is the eigenfunction of th state of original, where denotes any chosen solutions of (3), and (10) , i.e., th state eigenSpecifically, choosing function as the factorization state, all the transformed wave are given by (9), and the one functions for states by corresponding to (11) The free scalar parameter in (8)–(11) may take any value . In the special case of except those in range symmetric original potential it may be shown that , so all physically different may only. This procedure generates be obtained with positive a single-parameter dependent family of potentials and corresponding wave functions, while subsequent application of the SUSYQM transform would introduce more parameters, i.e., . The potential is thus varied through the variation of the parameter (or more parameters, if introduced) continuously, and the evaluation of wave functions and matrix elements will then readily deliver the best potential shape. Along with the continuously variable parameter(s) that control the shape of the partner potential, there is an additional discrete parameter—the factorization state index , which adds more freedom. It is important to note, however, that it is the original potential that determines what may be somewhat variability,” i.e., all possible vaguely called “the range of may take. Therefore, the SUSYQM-based shapes that

AND

DISCUSSION

To illustrate the described methods we have performed alloys based QW’s in the optimization of ternary or . Both the step-type and continuously respect to graded QW’s were considered, based on either conventional As, or on Al Ga N (which may be useful for Al Ga higher energy SHG) alloy. (e.g., 116 meV) the For not very large As-based QW should perform better than a nitrideAl Ga based QW, because of smaller effective mass (note that ). The first set of calculations was done for As single-step QW, taking 116 meV. an Al Ga The material parameters are taken as [13]: ( -free electron mass), 1.42 eV, 2.67 eV (direct gap), 750 meV (the conduction band offset), and Vegard’s law was used for effective masses and bandgaps in the alloy layers. 116 meV, we performed the single-step QW Choosing and barrier optimization by taking the width of the well as free parameters. Other parameters height were coming out from the solution of (6). Results given in Fig. 2 underline the importance of the optimal choice of the , barrier height, and Fig. 3 displays the best values of together with the optimized structure parameters, as they (i.e., the barrier compositions). The depend on choice of ˚ is obtained in QW with 3090 A largest value ˚ and ˚ ( 370 meV, 125 meV, 40 A, 57 A 86 meV). This result exceeds by 20%–30% the best values ˚ [3] or As QW’s: 2400 A published for Al Ga ˚ 2640 A [4]. To underline the importance of including the nonparabolicity in these calculations, we have also found the states energies for the above optimal QW within the parabolic model. With

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Fig. 3. The maximal values of (2) in an Alx Ga10x As single-step QW at double resonance, h! E 116 meV, achievable with various values of barrier height UB , together with other corresponding structural parameters.

 =1 =

120 meV and 134 meV, the resonance conditions were unacceptably perturbed. These may be restored to 116 meV, within the parabolic model, by changing the QW ˚ 370 meV, 145 meV, 40 A, parameters to ˚ i.e., significantly deviating 68 A, and from and those obtained with nonparabolicity included. A few more sample calculations indicate that the (non)parabolicity affects the results (energies and QW parameters) by approximately 15%, therefore including the nonparabolicity, which really exists, is essentially mandatory. Al Ga As may turn out to offer insufFor larger for three equispaced states, and one may then ficient turn to nitrides, in spite of larger effective masses. So we N QW’s, now consider the wurtzite semiconductor Al Ga both compounds having direct bandgaps, and use the following parameters [16]–[19]: 3.45 eV, and 6.28 eV. There is a between GaN and AlN in the dispersion of data on 2 eV, based on literature. We have used the value recent photoemission spectroscopy measurements [20], [21] and calculations [22]–[24] which all suggest the valence band offset is about 0.8 eV. 240 meV (this corresponds to 5.1Choosing initially m CO laser or approximately to frequency-doubled CO used as a pump for the next SHG), we performed the singleachievable step QW optimization. Maximum values of (dictated by technology-related with specified values of constraints, for instance), together with values of optimally designed QW parameters, may be read from Fig. 4. Results given in Fig. 4 again clearly show the importance of proper choice of the barrier height, along with other parameters, in designing the optimized QW. The largest value of ˚ with the QW achievable in this structure amounts to 247 A ˚ and 850 meV, 287 meV, 17 A, parameters ˚ ( 28 A 195 meV). Despite the larger bandgap of N, due to higher photon energies the importance of Al Ga As nonparabolicity here is found to be similar to the Al Ga QW’s. The procedure was then repeated for various values of pump 100–300 meV. The fully photon energy in the range and corresponding QW parameters, as optimized value , are presented in Fig. 5. they depend on

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Fig. 4. Same as Fig. 3 but for an Alx Ga10x N single-step QW at h! E 240 meV.

 =1 =

5

Fig. 5. Maximum achievable (2) in an Alx Ga10x N single-step QW [Fig. 1(a)], together with the structural parameters, given for a range of pump photon energies under double resonance conditions.

Consider now the single-step QW optimization in respect at triple resonance. As noted in Section II, it is to possible to achieve four equispaced states in such a structure and there remains one free parameter for optimization. In reality, however, there is a very limited range of values that , calculated for QW parameters may take. The value of As based QW at 116 meV, as it depends Al Ga , is given in Fig. 6, together with the on the well width corresponding values of other design parameters. The best ˚ (with ˚ 24 290 A 20.5 A, value of ˚ ˚ and rather small ˚ is 25.9 A, 26 A, 1.75 A) ˚ 414 meV, 137 meV, 73 A, obtained with ˚ 36 A. and Introducing another step in the QW [Fig. 1(b)] significantly improves the design flexibility. There are now three free ) and the remaining four parameters (say and are calculated from the levels spacing As requirements, as above. In calculations for the Al Ga 116 meV, the free parameters were based structure, with ˚ 0.55 eV, 26 A varied in the range 0.45 eV ˚ and 30 A ˚ ˚ The extracted optimized values, 46 A, 50 A. together with the corresponding design parameters, are given ˚ (with ˚ 47 120 A 16.2 A, in Fig. 7. The largest ˚ ˚ ˚ 23.7 A, 25.8 A, and 4.14 A) is obtained 500 meV, 133 meV, in the structure with ˚ and ˚ ( 198 meV, 33 A, 75.6 A 107 meV). is twice as large as that obtained in the In this case matrix element. single-step QW, mostly due to increased

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5 =

Fig. 6. Same as Fig. 3, but for (3) M01 M12 M23 M30 in an Alx Ga10x As single-step QW, calculated under triple resonance condition h! 116 meV.

 =

Fig. 7. Same as Fig. 6 but for a double-step QW [Fig. 1(b)].

Optimization procedure was also employed for continuously graded ternary alloy QW’s. We have restricted our considerations to the single-parameter-dependent family of potentials, obtained via the SUSYQM transform, from the original truncated parabolic potential with the constant mass . In optimization of the Al Ga N 240 meV pump based QW for resonant SHG of we have taken well depth of truncated parabolic potential 1 eV and (the lowest three states are equispaced to within 1 meV accuracy). Performing , we the SUSYQM transform (8)–(11), with find that individual matrix elements and their product depend on the free parameter as displayed in Fig. 8. The ˚ is obtained at 300 A 0.12, with maximum given in Fig. 9. At this the corresponding potential assumes the point we recall that the Hamiltonian with original, constant effective mass, and thus is not realizable by ternary alloys, and also the nonparabolicity is neglected. By changing the effective mass, i.e., enforcing it to follow in the way corresponding to ternary alloy

(12) and , the relevant states would clearly loose the equidistance property to some extent (the term in is introduced to shift the minimum of

with

5

Fig. 8. The individual matrix elements and their product (2) , as they depend on the free parameter  in SUSYQM generated potentials with the original truncated parabolic potential, under double resonance conditions at h! E 240 meV.

 =1 =

back to zero corresponding to pure GaN at this point). In this example, using (12) instead of constant and the presently optimized in the Schr¨odinger equation delivers the levels’ spacing 227 meV and 220 meV, significantly deviating from the desired 240 meV. To restore these spacings to 240 meV, we retailor the potential by coordinate scaling, i.e., define with , and substitute , together with the effective mass following it [as in (12)] into the Schr¨odinger equation. By and asymmetric solving it numerically, the symmetric scaling coefficients are determined so as to achieve 240 meV (this may be done by simple “numerical experimenting” or by formulating the problem as a pair of nonlinear equations in two unknowns). Since this last step is clearly directed toward restoring energies, not optimization of QW shape, it is desirable that the “amount of retailoring” is as small as possible, i.e., , otherwise the QW shape may significantly walk off optimum. Here we find and . Since the nonparabolicity increases the effective mass and hence lowers the quantization in energies, the well necessarily has to contract order to keep the states at desired energies. The main point is that is very close to zero, i.e., the QW shape is almost preserved, implying that this final potential , also given in Fig. 9, is (at least almost) optimal. Unlike , the final can be realized by composition grading of ternary alloy Al Ga N (Fig. 9). The only adverse outcome of ˚ . However, this is retailoring is that now drops to 260 A inevitable and stems from nonparabolicity-enhanced effective mass which necessitates the well contraction and hence lowers the matrix elements. As The same effect occurs when optimizing the Al Ga QW for SHG of 116 meV radiation. The constant mass model with gives ˚ , which drops to 3300 A ˚ upon including 3840 A the nonparabolicity and the position-dependent mass. In any case, the SUSYQM-based result is better than that for the step QW, obviously at the expense of a more complicated structure. The SUSYQM approach is no more difficult to apply for QW optimization in respect to third- (or even higher)

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Fig. 9. The optimized supersymmetric partner potential USS (z ) (dashed line), the original U (z ) (dotted line), and final retailored U n (z ). The mole fraction grading function x(z ) in Alx Ga10x N alloy necessary for realization of U n (z ), together with corresponding effective mass m3 (z ), are given in the inset.

Fig. 11. The optimized supersymmetric partner potential USS (z ) (dashed line), the original U (z ) (dotted line), and final retailored U n (z ), the last one to be realized by composition grading of Alx Ga10x As alloy, providing maximum third-order nonlinear susceptibility at  h! = 116 meV under triple resonance conditions.

Fig. 10. The individual matrix elements and their product 5(3) = M01 M12 M23 M30 relevant for THG, as they depend on the free parameter  in SUSYQM generated potentials, starting from the original truncated parabolic potential, under triple resonance conditions at  h! = 116 meV.

supersymmetric partner , and finally retailored potentials are given in Fig. 9, the last one being fully realizable As QW. The product of matrix elements in graded Al Ga ˚ (with 53 200 A 15.1 in it is calculated to be ˚ ˚ ˚ This ˚ 21.6 A, 31.4 A, and 5.2 A). A, As is, to our knowledge, the best result obtained in Al Ga based QW’s. It is also interesting to compare this result with the value ˚ , obtained in Al In As/Ga In As 57 300 A based structure of coupled QW type [8], [12]. The effective mass in the well-type material Ga In As is , i.e., % lower than in GaAs. Using the for individual matrix elements, i.e., scaling law we expect that SUSYQM optimization of in semiconductor with would give ˚ . Corrections due to nonparabolicity 190 000 A ˚ , as we should decrease this value to about 120 000 A , i.e., roughly estimate from which is still larger than the value from [8] and [12]. Realization of such a QW would rely on continuous grading of a quaternary alloy, instead of definitely simpler slab-by-slab growth of stepgraded QW’s. Yet, this is still an indication of the advantages of SUSYQM-based optimization if one is willing to put effort into graded structures realization (in reality these might be made by digitizing the smooth grading profile into some larger number of fixed-composition layers).

order nonlinearity. Consider, for example, optimizing a graded As QW for maximal value of , (2), at Al Ga 116 meV. Following the same steps as above, we find that , with 540 meV and gives four levels equispaced by 116-meV to within 1-meV accuracy. The SUSYQM parameter potential for ˚ with gives the largest value of 78 590 A ˚ ˚ ˚ ˚ 14.5 A, 26.3 A, 36.8 A and 5.6 A (Fig. 10). A point to note from Fig. 10 is a rather narrow , in contrast to a much broader one optimization peak of in Fig. 8 for SHG, indicating the need for quite precise design of QW’s for THG. Upon introducing the effective mass following the potential and the nonparabolicity, the levels’ spacings are calculated to deviate by up to 15 meV from the desired 116 meV. Retailoring the potential, via coordinate scaling, would here normally require three free parameters, in view of three goals to be achieved. However, we find that in this example the simple symmetric asymmetric linear scaling as used in case and , still manages of SHG, but now with 115.3 meV, to restore the level spacing to acceptable 115.2 meV, and 114 meV. The initial ,

IV. CONCLUSION Two systematic methods for optimization of ternary semiconductor alloys based QW’s in respect to nonlinear optical susceptibilities were discussed. One of them is applicable to step graded QW’s, like the asymmetric step QW, the coupled QW, and others. The other method is applicable to continuously graded QW’s. Both methods were applied N and Al Ga As based for optimized design of Al Ga QW’s intended for resonant SHG or THG. The obtained results, where comparison could be made, exceed those quoted

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in the literature. Furthermore, continuously graded structures were found to offer somewhat larger nonlinearities than step graded ones, but require more elaborate fabrication techniques. REFERENCES [1] F. Capasso, “Bandgap and interface engineering for advanced electronic and photonic devices,” MRS Bull., vol. 16, pp. 23–29, 1991. [2] P. Boucaud, F. H. Julien, D. D. Yang, J. M. Lourtiouz, E. Rosencher, P. Bois, and J. Nagle, “Detailed analysis of second harmonic generation near 10.6 m in GaAs/AlGaAs asymmetric quantum wells,” Appl. Phys. Lett., vol. 57, pp. 215–218, 1990. [3] E. Rosencher and P. Bois, “Model system for optical nonlinearities: Asymmetric quantum wells,” Phys. Rev., vol. 44, pp. 11315–11327, 1991. [4] S. J. B. Yoo, M. M. Fejer, and R. L Byer, “Second-order susceptibility in asymmetric quantum wells and its control by proton bombardment,” Appl. Phys. Lett., vol. 58, pp 1724–1726, 1991. [5] M. M. Fejer, S. J. Yoo, R. L. Byer, J. Harris, and J. S. Harris, Jr., “Observation of extremly large quadratic susceptibility at 9.6–10.8 m in electric biased AlGaAs/GaAs quantum wells,” Phys. Rev. Lett., vol. 62, pp. 1041–1044, 1989. [6] Z. Ikoni´c, V. Milanovi´c, and D. Tjapkin, “Resonant second harmonic generation by a semiconductor quantum wells in electric field,” IEEE J. Quantum Electron., vol. 25, pp. 54–60, 1989. [7] C. Sirtori, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett., vol. 60, pp. 151–153, 1992. [8] F. Capasso, C. Sirtori, and A. Y. Cho, “Coupled quantum well semiconductors with giant electric field tunable nonlinear optical properties in the infrared,” IEEE J. Quantum Electron., vol. 30, pp. 1313–1326, 1994. [9] C. Lien, Y. Huang, and T.-F. Lei, “The double resonant enhancement of optical second harmonic susceptibility in compositionally asymmetric coupled quantum well,” J. Appl. Phys., vol. 75, pp. 2177–2183, 1994. [10] V. Milanovi´c and Z. Ikoni´c, “On the optimization of resonant intersubband nonlinear optical susceptibilities in semiconductor quantum wells,” IEEE J. Quantum Electron., vol. 32, pp. 1316–1322, 1996. [11] V. Milanovi´c, Z. Ikoni´c, and D. Indjin, “Optimization of resonant intersubband nonlinear optical susceptibility in semiconductor quantum wells: The coordinate transform approach,” Phys. Rev. B, vol. 53, pp. 10877–10883, 1996. [12] C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Giant triply resonant (3) third order nonlinear susceptibility 3! in coupled quantum wells,” Phys. Rev. Lett., vol. 68, pp. 1010–1012, 1992. [13] G. Bastard, Wave Mechanics Applied to Semiconductor-Heterostructure. Les Ulis, France: Les Editions de Physique, 1990. [14] D. F. Nelson, R. C. Miller, and D. A. Kleinman, “Band nonparabolicity effects in semiconductor quantum wells,” Phys. Rev. B, vol. 35, pp. 7770–7773, 1987. [15] F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep., vol. 251, pp. 267–385, 1995. [16] H. Morkoc, S. Strite, G. B. Gao, M. E. Lin, B. Sverdlov, and M. Burns, “Large-band-gap SiC, III–V nitride, and II–IV ZnSe-based semiconductor device technologies,” J. Appl. Phys., vol. 76, pp. 1363–1398, 1994. [17] R. Davis, “III–V nitrides for electronic and optoelectronic applications,” Proc. IEEE, vol. 79, pp. 702–712, 1991. [18] S. N. Mohammad, A. A. Salvador, and H. Morkoc, “Emerging gallium nitride based devices,” Proc. IEEE, vol. 83, pp. 1306–1355, 1995.

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Dragan Indjin was born in Zemun, Yugoslavia, on January 11, 1963. He received the B.S., M.S., and Ph.D. degrees from the Faculty of Electrical Engineering, University of Belgrade, Belgrade, Yugoslavia, in 1988, 1993, and 1996, respectively. He currently holds the position of Assistant Professor in Physical Electronics at the University of Belgrade. His research interests are in the electronic structure of quantum semiconductor devices and nonlinear optical properties of quantum wells and superlattices.

Zoran Ikoni´c was born in Belgrade, Yugoslavia, on October 23, 1956. He received the Ph.D. degree from the Faculty of Electrical Engineering, University of Belgrade, Belgrade, Yugoslavia, in 1987. He currently is an Associate Professor at the University of Belgrade. His research interests are in the band structure calculations and nonlinear optics of quantum wells and superlattices.

Vitomir Milanovi´c was born in Svetozarevo, Yugoslavia, on July 12, 1947. He received the Ph.D. degree from the Faculty of Electrical Engineering, University of Belgrade, Belgrade, Yugoslavia, in 1983. He currently is an Associate Professor at the University of Belgrade. His research interests are in the band structure and optical properties of quantum wells and superlattices.

Jelena Radovanovi´c was born in Belgrade, Yugoslavia, on July 16, 1973. She received the B.S.E.E. degree from the Faculty of Electrical Engineering, University of Belgrade, Belgrade, Yugoslavia, in 1997. She joined the Faculty of Electrical Engineering, University of Belgrade, in 1997 and currently is a graduate student and Research Assistant. Her research interest is in the optimization of intersubband optical nonlinearities in semiconductor quantum structures.