- Email: [email protected]
Optimization of nonlinear optical rectification in semiconductor quantum wells using the inverse spectral theory
Solid State Communications,
Vol. 104, No. 8, pp. 445-450, 1997 @ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 $17.00 + .OO
Pergamon
PII:
SOO38-1098(97)00396-7
OPTIMIZATION OF NONLINEAR OPTICAL RECTIFICATION IN SEMICONDUCTOR QUANTUM WELLS USING THE INVERSE SPECTRAL THEORY V Milanovic and Z. Ikonic Faculty of Electrical Engineering, University of Belgrade, Bulevar Revolucije 73, 11000 Belgrade, Yugoslavia (Received 16 June 1997; accepted 26 August 1997 by R. Phillips)
Using the inverse spectral theory we devise a systematic procedure to generate quantum well profiles optimized in respect to the optical rectification coefficient. Cases that, in course of design, start with an infinite barrier rectangular or with parabolic quantum well allow for a large part of the procedure to be done analytically, and are considered in more detail. Calculations for GaAs based wells indicate somewhat larger values of the rectification coefficient may be obtained than are reported in the current literature. The design procedure may also be extended to optimization of other optical nonlinearities. @ 1997 Elsevier Science Ltd Keywords: A. nanostructures,
A. quantum wells, D. optical properties.
1. INTRODUCTION Intersubband transitions in semiconductor quantum wells (QW) have attracted considerable research attention. This is due to generally large values of dipole transition matrix elements and the possibility of achieving the resonance conditions. Therefore both the linear and, even more so, nonlinear optical processes in these structures are very intense. Large dipole matrix elements are associated to small effective mass m” of electrons, scaling approximately as m” -‘I2 (Ref. [l]). Within a given material (i.e. m”), however, a lot can be done to enhance those matrix elements relevant for a particular type of nonlinearity by proper shaping of the QW. Variation of the QW potential affects the quantized states wave functions, and hence the matrix elements, and has to be performed under the constraint that levels spacing should be kept as specified. The second-order nonlinear coefficients, describing, for instance, second harmonic generation and optical rectification, scale as third power of dipole matrix elements, and thus can be significantly enhanced by suitable QW design. Clearly, only asymmetric structures are useful for this purpose, and may be realized by either electric field biasing [2] or asymmetric composition grading [3]. Finding the best potential shape may be accomplished by semiquanti-
tative considerations plus trial-and-error method, and most effort in such an ap,proach is spent in restoring the levels spacing upon changing the potential shape in a realistic structure, rather than in checking the values of matrix elements. We have recently described an alternative optimization procedure [4] which completely eleviates the levels spacing problem. It relies on supersymmetric quantum mechanics (SUSYQM) which generates a family of asymmetric potentials isospectral to an original potential, their shape being varied via a scalar parameter (or a discrete set of scalar parameters). This approach was employed in Ref. [4] to maximize the nonlinear susceptibility relevant for second harmonic generation. Here we describe yet another technique for finding the best potential shape of QWs, based on the inverse spectral theory (IST) and examplify its use for optimizing the optical rectification coefficient.
2. THEORETICAL
CONSIDERATIONS
The electronic polarization P(t) of the QW slab due to an electric field E(t) = E cos(wt) can be written as (including terms up to second order in E)
445
p(t)
=
(,,x”‘E@” w
+ ~~x~~)E~e-‘~) w
446
OPTIMIZATION
OF NONLINEAR
OPTICAL RECTIFICATION
+ cc + E,,x;~‘E~
U(z) = 6(z) - $$[ln
where EOis the vacuuum dielectric permittivity, and x”) denotes the i-th order susceptibility at frequency given by its subscript. Optical rectification is thus described by xh2’. Density matrix considerations [I] of a two-level system in the presence of close-to-resonance light (i.e. under conditions when just two levels are important) give the expression for ~a), which takes the largest value at exact resonance, Aw = E2 - El = hum, and then amounts to (2) XO~X
-
e3fiG(N 2Eoh2
-
Vol. 104, No. 8
N2),42&2
W(z)].
The Wronskian W(z) in equation (3) is given by [5] W(Z;Ej
+ E, 0)
=
WE(Z)7
d@j(Z)
-
@j(Z)
!y
(4)
where cy, (z) is any solution of the original Schrodinger equation with 6(z) at an energy Ej + E, and the shift E may be given any value in the interval (Ej-i - Ej, Ej+i - Ej), i.e. the level which is moved should not cross any other level. It suffices to find the function I&(Z) (which, certainly, cannot be normalized) up to a multiplicative constant. It may be written as a linear combination of particular solutions ctri, and CUT,,i.e. &(z) = (111&-)+ CW.J~,(Z)where o( is an arbitrary constant. Therefore a family of scalar parameter dependent isospectral potentials U(z, 00 is in fact generated, sharing the same energy spectrum (Ei) but with different wave functions Cyi
where e is the electron charge, Nt and N2 are the electron surface densities of the lower 11 > and the upper 12 > state, ~12 = I < 11212> I is the transition dipole moment and 612 =< l/z]1 > - < 21212 > is the difference of permanent dipole moments, i.e. the mean electron displacement of the transition. Furthermore, T2 is the off-diagonal relaxation time in the Liouville equation, related to the linewidth, and 7’1is the diagonal relaxation time, i.e. the excited state lifetime. i*j The value of xh2A, clearly increases with increasing Tl and T2. The former may be significantly enhanced (5) by introducing a third, optically inactive metastable state, loosely coupled to the ground state due to spatial and for i = j displacement of their wave functions, but the operating speed of a potential device would then be sacrificed. (6) The latter (Tz) depends on the structure quality and on various scattering mechanisms, and is eventually It should be noted that the wave functions given by limited from above by electron-phonon scattering, i.e. equation (5) are normalized, i.e. < +Yif vi > = 1, while cannot be significantly enhanced by “band structure wj(z) as given by equation (6) has to be normalized engineering”. Thus, restricting our considerations to numerically, except in special cases. The requirement strictly two-level systems, and assuming that Tl and T2 for the wave functions to be finite everywhere implies are not much dependent on the QW profile, there rethat the Wronskian W(z) should not cross zero at any mains the product of matrix elements II = pf2 6 $2to be z. This restricts the range of values that the parameter optimized. This type of optimization, within the class (x may really take. We will elaborate on this point when of asymmetric step QWs has been described in Ref. [l], dealing with specific examples. using an analytic model which assumes infinitely high In case of position dependent effective mass m” = barriers of the structure and neglects the nonparabolm*(z), as exists in composition graded QWs, the icity and the effective mass position dependence (corSchrijdinger equation takes the Ben Daniel-Duke rections to account for these approximations have to form [6], and we find that all the above expressions be done numerically afterwards). Here we consider this remain valid upon substitution m* - m*(z), except problem by employing IST, the essentials of which we equation (3) which then reads describe next. Consider an electron moving in a potential 6(z), J&-$iln W(z)) (7) with a constant effective mass m" . Bound states ener- u(z) = Ok.1 - J&-$ [ gies of the system are denoted as Ei (i = 1,2,. . .), and the corresponding wave functions as @i. The IST al- One should note that m*(z) does not change in the lows one to construct a modified potential V(z) such transform 6(z) - U(z). Finally, within the constant mass model we consider that its eigenenergies Ei satisfy Ei = $ (i # j), while a very interesting case when E - 0, i.e. when the new Ej = &j + E, provided U(z) takes the form [5]
1
O~IMIZATION
Vol. 104, No. 8
OF NONLINEAR
potential is fully isospectral to the original one, just the same as in SUSYQM [7]. For further analysis it is convenient to take ~1 E(~, E) such that v~r&, E 4 0) = $J~(z). The Wronskian may then be written as W[qtE, @j] + rwW[+9ze,@J, where the first term 2em*
J 2
mYi,, sijl - -jp-
i@dz’
(8)
--m tends to zero in the limit E - 0, and the setond term becomes a constant orC, where C = lim,,O W[@ze, @j]. If o( is given a finite value then W(z) = 0rC is constant, whe~from we get the identity transform, U (2) = I?(z), for any P( rf 0. However, if cu is also taken to approach zero in such a way that o((E)C - 2cm*A/h2, where A is an arbitrary constant, the Wronskian becomes
=
2&m* -$--
WSUSY
f.d
and the potential U(z) isospectral to the original 6(z) is U(z) =
O(z) - $-$@I
= ususY(~)
~SVSY~Z))
(10)
where the subscript SUSY denotes the expressions known from SUSYQM theory [7]. Therefore SUSYQM is actually a special case of IST, the case when E - 0 and LXN E - 0. One may thus expect that IST applied to optimization of QW nonlinearity will give even better results than does the SUSYQM method 141. The IST based method of designing the QW optimized for, say, resonant optical rectification would include the following steps. First one sets an arbitrary potential and finds its levels 81 and J$ (and higher for other nonlinear processes). This potential may be input as strictly fixed, or, even better if possible, given in terms of some number of parameters that specify its shape or range of extension. The levels spacing &-El needs not coincide with the photon energy (more precisely should do so only very roughly), so there is a great deal of freedom in the choice of the initial potential 0 (z). Second, this potential is reshaped via IST to bring the levels spacing to the desired value. If & - & differs by E from the desired value this can be done either by shifting level 1 by E or level 2 by -6. The modified potential thus depends on the initial one, the type of shift, and also on the free parameter o[ which
OPTICAL RECTIFICATION
447
describes an isospectral family of potentials, as noted above. By varying o( and evaluating the matrix elements product II one finds the best potential shape U(z, ccopr), within the class derived from the initial 8 (z), which maximizes II. The resonance condition is automatically preserved throughout the variation of o(. As compared to the SUSYQM method [4], the one based on IST has an advantage in that even the initial potential needs not be taylored to satisfy the resonance condition(s), i.e. it may be choosen with much more freedom. Since SUSYQM is naturally included in IST, as noted above, the best results of SUSYQM should also be derivable by IST, and even better results may be expected from the latter. To demonstrate the applicability of the IST to optimization of optical rectification coefficient in QWs we take two examples in which the initial potential is either rectangular with infinite barriers, or parabolic (linear harmonic oscillator). Advantage of these two cases is that most of the job may be done analytically. It is highly probable that some other choices might result in better values of II, but the procedure would have to be done nume~~lly throughput. i) The case of infinite barriers rectangular initial QW. If the second level is to be shifted by E to get a proper spacing of Rw while Et (and all others except &) are kept fixed, it will be possible to proceed if this E belongs, to the interval (Er - &, $ - &). With the effective mass m* assumed constant and known in advance, levels energies are given by & = (A2rr2/2m*d2) *n2and the well width d can be considered as a parameter at disposal. Unnormalized electron wave functions of the first two states are @r(z) = sin(ktz),
@z(z) = sin(2k2z)
(11)
where kl = n/d, k2 = 2k3 and the solution corresponding to an energy & + e is &(z) = sin(k,z) + occos(k,z) - sh(k,z + Cp) (12) where the “wave number” kc is written k,mwith’ e = 2m”bd2 7SA2
as kE =
(13)
and rf, is a constant. It turns out that only the values of (b from the range 6, E [O, rr/21 result in physically different potentials after the IST transform. Having in mind that kc -c 3kl, because E2 cannot be shifted above the third level, it follows that B < 8, implying in turn that the well width d must not exotherwise resonance at ceed dmrrx = 2rrp1/ y/x, Aw cannot be achieved and optimizing the QW shape would be pointless.
448
OPTIMIZATION
OF NONLINEAR
OPTICAL RECTIFICATION
The Wronskian is now given by W(z) = 2kr cos(krz) sin&z +
(14)
and, as noted above, it must not cross zero in the range z E [-d/2, d/2]. This will be fulfilled if 4 E (--TT + y,
;I,
k,d E (2rr, 37-r)
a, E (TT- y,
;),
rc,d E (TT,27~)
Vol. 104, No. 8
which coincides with the corresponding SUSYQM expressions [7] for V~I,J(Z)upon introducing a new constant A = 2A*/d - l/2, ii) The case of par~olic initial QW (linear harmonic oscillator). Given the values of the constant effective mass m* and the levels spacing AE the Schrodinger equation reads
(15)
---
@=EW
(20)
or (16)
where the first (second) of these relations holds for E > 0 (E < 0). In the intervals given by equations (1516) W(z) is positive for all z. The normalized wave functions are given by
4r?z*e x
cos(klz) +
3fi2w(z)kI
sin(k,z + +) cos3 (krz)
1
(17) and q&z) =
a Fke
sin(%+ + 4) sin(%$ - 4) sin(k,d)
x sin(2krz) W(z) .
(18)
The matrix elements ~12 and 612 relevant for optical rectification may now be found, and this is the only part in this example which requires numerical integration. Another choice is to keep the second level fixed, determined by the well width d, and shift the first one to get the resonance condition E2 - El = Aw. Here we have 8 = m*(Alo)d2/2n2A2 and may take values from the interval [O,l], which corresponds to 81 r 0 and 81 -I-E < &. The constant + may take values from (0, k,d/2) if k,d -CIT, or from (0, rr - k,d/2) if k,d > n. The normalized wave function yr (z) is given by equation (18) in which sin(2kiz) - cos(klz), and (CIZ(Z)by equation (17) with cos(klz) -) sin(2krz) in its first term only. The SUSYQM situation is obtained by setting 4p 0 or # - rr/2 in the above two cases, respectively. Indeed, choosing dj = $$$A* (where A* is a constant) and taking the E - 0 limit, it can be shown that 2m*e W(z) - --jp-
A* +; -
sin(2kzz) 4k
2
where zi = ~2~m~AE. To recast it into the common form of Weber differential equation [S] we introduce a new variable t = 42~1 zo and equation (19) then becomes
d=Y --
(21)
dt2
with a = -EfAE. Within this case we will explore in more detail only the choice of keeping the first level fixed while the second is shifted by E until the resonance condition Ez - El = I?2 f E - El = fzw is achieved. The allowed range of E is here (& - &, $ - $), implying that AE E [&m/2, + co), or the parameter a in equation (21) should satisfy -512
(22)
where LXis an arbitrary constant. The wave function $2 corresponding to eigenenergy & is of the well known form $2 - t exp (- t2 /4) and the Wronskian reads W(t) = e-g x (1 - $m 1
+ aV(t)l
- t tu’tt)
+
cfV’(r)l . 1 (23)
Analysis of this expression shows that there are four ranges of of for which W (t ) has no zeros. These are: lo o( E (--s,O) for a E (-l/2, -I), 2“ o( E (-ce, 0) u (s, +a) for a E (-1, -3/2), 30 lx E (-00, -s) u (0, -t-m) for a E (-3/2, -2), 4O ci E (0,~) for a f (-2, -S/2), where s = r-r1sin(rru)l-llI% + ;)I-‘. Signs of o( and W(t) are always the same. The SUSYQM case is obtained by letting a - -312 and o( - 0 simultaneously. The normalized wave function of the ground state (El) can be written explicitely
1 w4
Vol. 104, No. 8
OPTIMIZATION
OF NONLINEAR
OPTICAL RECTIFICATION
449
while the wave function at E2 cannot be normalized analytically, and the unnormalized form reads 12 y2(t)
=
te_S
(25)
W(t)
(note that the real space coordinate z = zot/d). The matrix elements 1112and 612 may then be found numerically.
3. NUMERICAL EXAMPLES DISCUSSION
20
40
60
80
IO0 120 140 160 180 200
AND
Numerical calculations were performed with m* = 0.067, in free electron mass units, as corresponds to GaAs based QW, and photon (resonance) energy hco = 116 meV (CO2 laser radiation). The matrix elements product II = ~14~612for the class of potentials derived from rectangular QW depend on its width d (the only external parameter for this class) and the (isospectral) parameter ar. The II dependence is generally asymmetric bell shaped, the value and the position mopt of the maximum both depending on d. In Fig. 1 we give this, optimized-in-oc product II as it varies with d, for either fixed-i?r-shifted-$ or the opposite case. As noted above, there is a maximum value of the initial well width that allows for tuning to the desired resonance, and here it amounts to d,,,ax = 197 zfi in the fixed-&-shifted-& case, and to d,,, = 139 A in the fixed-&-shifted-& case. The largest values of II in the two cases are 12270 A3 and 12408 k3, respectively. The SUSYQM case occurs for d = 120.7 A (resonance already achieved), the largest value of Il now being 10570 A3. For the second choice, the parabolic initial potential, other conditions being unchanged, in Fig. 2 is given the optimized-in-a product II as it varies with the parameter a. The maximum of II = 12680 A3 occurs at a = - 1.80, but, as above, the maximum is comfortably broad. The corresponding best potential shape is given in Fig. 3. The SUSYQM situation, occuring at a = - 1.50, again gives a somewhat lower Il = 12680 A3. The optimized values of II obtained here compare favourably to the values reported elsewhere [l, 31 for the step asymmetric type of QWs, designed via approximate analytic considerations plus trial-and-error method. The values obtained here are in fact better by a few percent. Yet, we note that the two initial potentials we used were chosen only for purpose of carrying out most of the work analytically. They are
Fig. 1. The optimized-in-or values of II = ~17~612, calculated for the fixed-&-shifted-$ (solid line) and fixed-&-shifted-& (dashed line) cases as they depend on the width d of the starting infinite barrier rectangular potential, for m * = 0.067 and Rw = 116 meV. The SUSYQM case is also denoted.
-2.5
-2
-1.5 a
-1
-0.5
Fig. 2. Same as in Fig. 1, but for the parabolic starting potential described with the parameter a, the fixed-& shifted-& case only. rather idealized, and any realistic structure that would be made of graded Al,Gat-,As alloy would necessarily have finite barriers. The wave function penetration into the finite barriers, however, enhances the matrix elements and thus the nonlinear susceptibilities [l]. For instance, our calculations based on SUSYQM approach, that start with the parabolic potential truncated at 300 meV, indicate that II amounts to 15300 k3, as compared to 13445 a3 found above with strictly parabolic initial potential. Without doing explicit numerical calculations, we may estimate by simple scaling that using the IST method to design QWs with finite barriers would result in Il = 16200 A3. This would be
450
OPTIMIZATION
OF NONLINEAR
OPTICAL RECTIFICATION
o*61 0.5
SUSYQM based QW opt~i~tion in respect to second harmonic generation [4] indicates that these have a rather mild influence on the best potential shape (slightly squeezing the output of idealized calculations in order to compensate for the nonparabolicityincreased effective mass). The important conclusion from the above calculations for somewhat idealized cases is that the IST approach delivers some, though not drastic, improvement over the QW designs obtained by the SUSYQM approach.
”
0.4 z
0.3 -
5s 3 0.2 0.1 O-0.1 -150
Vol. 104, No. 8
-100
-50
0
50
100
150
REFERENCES
z PI 1.
Fig. 3. The optimized potential corresponding value a = -1.80 from Fig. 2.
to the
- 25% larger than what was found in Refs. [l] and [3].
2. 3. 4.
4. CONCLUSION The IST based method of finding the best potential shape of QWs in respect to resonant optical rectification was proposed and its use demonstrated in somewhat idealized situations. In the shape optimization of realistic QWs one would also have to take account of the effective mass variation in graded alloys, and of nonparabolicity, especially if higher photon energies are involved. These effects can at present be included only numerically. Our previous experience with
5.
6.
7. 8.
Rosencher, E. and Bois, Ph., Phys. Rev. B, 44 1991, 315. Ikonik, Z., Milanovik, V. and Tjapkin, D., IEEE J: Qaantam E~e~tro~~24, 1989, 54. Rosencher, E., Bois, P, Nagle, J., Costard, E. and Delaitre, S., App~ Phys. Lett., 55, 1989, 1589. Milanovi~, V. and IkoniC, Z., IEEE J; Q~ntam Electron., 32, 1996, 1316. Chabanov, V.M., Zakhariev, B.N., Brandt, S., Dahmen, H.D. and Stroch, T., Phys. Rev. A, 52, 19953389. Bastard, G., Wave Mechanics Applied to Semiconductor Heterostructures. Les Editions de Physique, CNRS, Paris, 1988. Cooper, F., Khare, A. and Sukhatme, U., Phys. Reports, 251, 1995,267. Handbook of Mathematical Functions, (Edited by M. Abramowitz and I. A. Stegun). Dover, New York, 1972.
Vol. 104, No. 8, pp. 445-450, 1997 @ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 $17.00 + .OO
Pergamon
PII:
SOO38-1098(97)00396-7
OPTIMIZATION OF NONLINEAR OPTICAL RECTIFICATION IN SEMICONDUCTOR QUANTUM WELLS USING THE INVERSE SPECTRAL THEORY V Milanovic and Z. Ikonic Faculty of Electrical Engineering, University of Belgrade, Bulevar Revolucije 73, 11000 Belgrade, Yugoslavia (Received 16 June 1997; accepted 26 August 1997 by R. Phillips)
Using the inverse spectral theory we devise a systematic procedure to generate quantum well profiles optimized in respect to the optical rectification coefficient. Cases that, in course of design, start with an infinite barrier rectangular or with parabolic quantum well allow for a large part of the procedure to be done analytically, and are considered in more detail. Calculations for GaAs based wells indicate somewhat larger values of the rectification coefficient may be obtained than are reported in the current literature. The design procedure may also be extended to optimization of other optical nonlinearities. @ 1997 Elsevier Science Ltd Keywords: A. nanostructures,
A. quantum wells, D. optical properties.
1. INTRODUCTION Intersubband transitions in semiconductor quantum wells (QW) have attracted considerable research attention. This is due to generally large values of dipole transition matrix elements and the possibility of achieving the resonance conditions. Therefore both the linear and, even more so, nonlinear optical processes in these structures are very intense. Large dipole matrix elements are associated to small effective mass m” of electrons, scaling approximately as m” -‘I2 (Ref. [l]). Within a given material (i.e. m”), however, a lot can be done to enhance those matrix elements relevant for a particular type of nonlinearity by proper shaping of the QW. Variation of the QW potential affects the quantized states wave functions, and hence the matrix elements, and has to be performed under the constraint that levels spacing should be kept as specified. The second-order nonlinear coefficients, describing, for instance, second harmonic generation and optical rectification, scale as third power of dipole matrix elements, and thus can be significantly enhanced by suitable QW design. Clearly, only asymmetric structures are useful for this purpose, and may be realized by either electric field biasing [2] or asymmetric composition grading [3]. Finding the best potential shape may be accomplished by semiquanti-
tative considerations plus trial-and-error method, and most effort in such an ap,proach is spent in restoring the levels spacing upon changing the potential shape in a realistic structure, rather than in checking the values of matrix elements. We have recently described an alternative optimization procedure [4] which completely eleviates the levels spacing problem. It relies on supersymmetric quantum mechanics (SUSYQM) which generates a family of asymmetric potentials isospectral to an original potential, their shape being varied via a scalar parameter (or a discrete set of scalar parameters). This approach was employed in Ref. [4] to maximize the nonlinear susceptibility relevant for second harmonic generation. Here we describe yet another technique for finding the best potential shape of QWs, based on the inverse spectral theory (IST) and examplify its use for optimizing the optical rectification coefficient.
2. THEORETICAL
CONSIDERATIONS
The electronic polarization P(t) of the QW slab due to an electric field E(t) = E cos(wt) can be written as (including terms up to second order in E)
445
p(t)
=
(,,x”‘E@” w
+ ~~x~~)E~e-‘~) w
446
OPTIMIZATION
OF NONLINEAR
OPTICAL RECTIFICATION
+ cc + E,,x;~‘E~
U(z) = 6(z) - $$[ln
where EOis the vacuuum dielectric permittivity, and x”) denotes the i-th order susceptibility at frequency given by its subscript. Optical rectification is thus described by xh2’. Density matrix considerations [I] of a two-level system in the presence of close-to-resonance light (i.e. under conditions when just two levels are important) give the expression for ~a), which takes the largest value at exact resonance, Aw = E2 - El = hum, and then amounts to (2) XO~X
-
e3fiG(N 2Eoh2
-
Vol. 104, No. 8
N2),42&2
W(z)].
The Wronskian W(z) in equation (3) is given by [5] W(Z;Ej
+ E, 0)
=
WE(Z)7
d@j(Z)
-
@j(Z)
!y
(4)
where cy, (z) is any solution of the original Schrodinger equation with 6(z) at an energy Ej + E, and the shift E may be given any value in the interval (Ej-i - Ej, Ej+i - Ej), i.e. the level which is moved should not cross any other level. It suffices to find the function I&(Z) (which, certainly, cannot be normalized) up to a multiplicative constant. It may be written as a linear combination of particular solutions ctri, and CUT,,i.e. &(z) = (111&-)+ CW.J~,(Z)where o( is an arbitrary constant. Therefore a family of scalar parameter dependent isospectral potentials U(z, 00 is in fact generated, sharing the same energy spectrum (Ei) but with different wave functions Cyi
where e is the electron charge, Nt and N2 are the electron surface densities of the lower 11 > and the upper 12 > state, ~12 = I < 11212> I is the transition dipole moment and 612 =< l/z]1 > - < 21212 > is the difference of permanent dipole moments, i.e. the mean electron displacement of the transition. Furthermore, T2 is the off-diagonal relaxation time in the Liouville equation, related to the linewidth, and 7’1is the diagonal relaxation time, i.e. the excited state lifetime. i*j The value of xh2A, clearly increases with increasing Tl and T2. The former may be significantly enhanced (5) by introducing a third, optically inactive metastable state, loosely coupled to the ground state due to spatial and for i = j displacement of their wave functions, but the operating speed of a potential device would then be sacrificed. (6) The latter (Tz) depends on the structure quality and on various scattering mechanisms, and is eventually It should be noted that the wave functions given by limited from above by electron-phonon scattering, i.e. equation (5) are normalized, i.e. < +Yif vi > = 1, while cannot be significantly enhanced by “band structure wj(z) as given by equation (6) has to be normalized engineering”. Thus, restricting our considerations to numerically, except in special cases. The requirement strictly two-level systems, and assuming that Tl and T2 for the wave functions to be finite everywhere implies are not much dependent on the QW profile, there rethat the Wronskian W(z) should not cross zero at any mains the product of matrix elements II = pf2 6 $2to be z. This restricts the range of values that the parameter optimized. This type of optimization, within the class (x may really take. We will elaborate on this point when of asymmetric step QWs has been described in Ref. [l], dealing with specific examples. using an analytic model which assumes infinitely high In case of position dependent effective mass m” = barriers of the structure and neglects the nonparabolm*(z), as exists in composition graded QWs, the icity and the effective mass position dependence (corSchrijdinger equation takes the Ben Daniel-Duke rections to account for these approximations have to form [6], and we find that all the above expressions be done numerically afterwards). Here we consider this remain valid upon substitution m* - m*(z), except problem by employing IST, the essentials of which we equation (3) which then reads describe next. Consider an electron moving in a potential 6(z), J&-$iln W(z)) (7) with a constant effective mass m" . Bound states ener- u(z) = Ok.1 - J&-$ [ gies of the system are denoted as Ei (i = 1,2,. . .), and the corresponding wave functions as @i. The IST al- One should note that m*(z) does not change in the lows one to construct a modified potential V(z) such transform 6(z) - U(z). Finally, within the constant mass model we consider that its eigenenergies Ei satisfy Ei = $ (i # j), while a very interesting case when E - 0, i.e. when the new Ej = &j + E, provided U(z) takes the form [5]
1
O~IMIZATION
Vol. 104, No. 8
OF NONLINEAR
potential is fully isospectral to the original one, just the same as in SUSYQM [7]. For further analysis it is convenient to take ~1 E(~, E) such that v~r&, E 4 0) = $J~(z). The Wronskian may then be written as W[qtE, @j] + rwW[+9ze,@J, where the first term 2em*
J 2
mYi,, sijl - -jp-
i@dz’
(8)
--m tends to zero in the limit E - 0, and the setond term becomes a constant orC, where C = lim,,O W[@ze, @j]. If o( is given a finite value then W(z) = 0rC is constant, whe~from we get the identity transform, U (2) = I?(z), for any P( rf 0. However, if cu is also taken to approach zero in such a way that o((E)C - 2cm*A/h2, where A is an arbitrary constant, the Wronskian becomes
=
2&m* -$--
WSUSY
f.d
and the potential U(z) isospectral to the original 6(z) is U(z) =
O(z) - $-$@I
= ususY(~)
~SVSY~Z))
(10)
where the subscript SUSY denotes the expressions known from SUSYQM theory [7]. Therefore SUSYQM is actually a special case of IST, the case when E - 0 and LXN E - 0. One may thus expect that IST applied to optimization of QW nonlinearity will give even better results than does the SUSYQM method 141. The IST based method of designing the QW optimized for, say, resonant optical rectification would include the following steps. First one sets an arbitrary potential and finds its levels 81 and J$ (and higher for other nonlinear processes). This potential may be input as strictly fixed, or, even better if possible, given in terms of some number of parameters that specify its shape or range of extension. The levels spacing &-El needs not coincide with the photon energy (more precisely should do so only very roughly), so there is a great deal of freedom in the choice of the initial potential 0 (z). Second, this potential is reshaped via IST to bring the levels spacing to the desired value. If & - & differs by E from the desired value this can be done either by shifting level 1 by E or level 2 by -6. The modified potential thus depends on the initial one, the type of shift, and also on the free parameter o[ which
OPTICAL RECTIFICATION
447
describes an isospectral family of potentials, as noted above. By varying o( and evaluating the matrix elements product II one finds the best potential shape U(z, ccopr), within the class derived from the initial 8 (z), which maximizes II. The resonance condition is automatically preserved throughout the variation of o(. As compared to the SUSYQM method [4], the one based on IST has an advantage in that even the initial potential needs not be taylored to satisfy the resonance condition(s), i.e. it may be choosen with much more freedom. Since SUSYQM is naturally included in IST, as noted above, the best results of SUSYQM should also be derivable by IST, and even better results may be expected from the latter. To demonstrate the applicability of the IST to optimization of optical rectification coefficient in QWs we take two examples in which the initial potential is either rectangular with infinite barriers, or parabolic (linear harmonic oscillator). Advantage of these two cases is that most of the job may be done analytically. It is highly probable that some other choices might result in better values of II, but the procedure would have to be done nume~~lly throughput. i) The case of infinite barriers rectangular initial QW. If the second level is to be shifted by E to get a proper spacing of Rw while Et (and all others except &) are kept fixed, it will be possible to proceed if this E belongs, to the interval (Er - &, $ - &). With the effective mass m* assumed constant and known in advance, levels energies are given by & = (A2rr2/2m*d2) *n2and the well width d can be considered as a parameter at disposal. Unnormalized electron wave functions of the first two states are @r(z) = sin(ktz),
@z(z) = sin(2k2z)
(11)
where kl = n/d, k2 = 2k3 and the solution corresponding to an energy & + e is &(z) = sin(k,z) + occos(k,z) - sh(k,z + Cp) (12) where the “wave number” kc is written k,mwith’ e = 2m”bd2 7SA2
as kE =
(13)
and rf, is a constant. It turns out that only the values of (b from the range 6, E [O, rr/21 result in physically different potentials after the IST transform. Having in mind that kc -c 3kl, because E2 cannot be shifted above the third level, it follows that B < 8, implying in turn that the well width d must not exotherwise resonance at ceed dmrrx = 2rrp1/ y/x, Aw cannot be achieved and optimizing the QW shape would be pointless.
448
OPTIMIZATION
OF NONLINEAR
OPTICAL RECTIFICATION
The Wronskian is now given by W(z) = 2kr cos(krz) sin&z +
(14)
and, as noted above, it must not cross zero in the range z E [-d/2, d/2]. This will be fulfilled if 4 E (--TT + y,
;I,
k,d E (2rr, 37-r)
a, E (TT- y,
;),
rc,d E (TT,27~)
Vol. 104, No. 8
which coincides with the corresponding SUSYQM expressions [7] for V~I,J(Z)upon introducing a new constant A = 2A*/d - l/2, ii) The case of par~olic initial QW (linear harmonic oscillator). Given the values of the constant effective mass m* and the levels spacing AE the Schrodinger equation reads
(15)
---
@=EW
(20)
or (16)
where the first (second) of these relations holds for E > 0 (E < 0). In the intervals given by equations (1516) W(z) is positive for all z. The normalized wave functions are given by
4r?z*e x
cos(klz) +
3fi2w(z)kI
sin(k,z + +) cos3 (krz)
1
(17) and q&z) =
a Fke
sin(%+ + 4) sin(%$ - 4) sin(k,d)
x sin(2krz) W(z) .
(18)
The matrix elements ~12 and 612 relevant for optical rectification may now be found, and this is the only part in this example which requires numerical integration. Another choice is to keep the second level fixed, determined by the well width d, and shift the first one to get the resonance condition E2 - El = Aw. Here we have 8 = m*(Alo)d2/2n2A2 and may take values from the interval [O,l], which corresponds to 81 r 0 and 81 -I-E < &. The constant + may take values from (0, k,d/2) if k,d -CIT, or from (0, rr - k,d/2) if k,d > n. The normalized wave function yr (z) is given by equation (18) in which sin(2kiz) - cos(klz), and (CIZ(Z)by equation (17) with cos(klz) -) sin(2krz) in its first term only. The SUSYQM situation is obtained by setting 4p 0 or # - rr/2 in the above two cases, respectively. Indeed, choosing dj = $$$A* (where A* is a constant) and taking the E - 0 limit, it can be shown that 2m*e W(z) - --jp-
A* +; -
sin(2kzz) 4k
2
where zi = ~2~m~AE. To recast it into the common form of Weber differential equation [S] we introduce a new variable t = 42~1 zo and equation (19) then becomes
d=Y --
(21)
dt2
with a = -EfAE. Within this case we will explore in more detail only the choice of keeping the first level fixed while the second is shifted by E until the resonance condition Ez - El = I?2 f E - El = fzw is achieved. The allowed range of E is here (& - &, $ - $), implying that AE E [&m/2, + co), or the parameter a in equation (21) should satisfy -512
(22)
where LXis an arbitrary constant. The wave function $2 corresponding to eigenenergy & is of the well known form $2 - t exp (- t2 /4) and the Wronskian reads W(t) = e-g x (1 - $m 1
+ aV(t)l
- t tu’tt)
+
cfV’(r)l . 1 (23)
Analysis of this expression shows that there are four ranges of of for which W (t ) has no zeros. These are: lo o( E (--s,O) for a E (-l/2, -I), 2“ o( E (-ce, 0) u (s, +a) for a E (-1, -3/2), 30 lx E (-00, -s) u (0, -t-m) for a E (-3/2, -2), 4O ci E (0,~) for a f (-2, -S/2), where s = r-r1sin(rru)l-llI% + ;)I-‘. Signs of o( and W(t) are always the same. The SUSYQM case is obtained by letting a - -312 and o( - 0 simultaneously. The normalized wave function of the ground state (El) can be written explicitely
1 w4
Vol. 104, No. 8
OPTIMIZATION
OF NONLINEAR
OPTICAL RECTIFICATION
449
while the wave function at E2 cannot be normalized analytically, and the unnormalized form reads 12 y2(t)
=
te_S
(25)
W(t)
(note that the real space coordinate z = zot/d). The matrix elements 1112and 612 may then be found numerically.
3. NUMERICAL EXAMPLES DISCUSSION
20
40
60
80
IO0 120 140 160 180 200
AND
Numerical calculations were performed with m* = 0.067, in free electron mass units, as corresponds to GaAs based QW, and photon (resonance) energy hco = 116 meV (CO2 laser radiation). The matrix elements product II = ~14~612for the class of potentials derived from rectangular QW depend on its width d (the only external parameter for this class) and the (isospectral) parameter ar. The II dependence is generally asymmetric bell shaped, the value and the position mopt of the maximum both depending on d. In Fig. 1 we give this, optimized-in-oc product II as it varies with d, for either fixed-i?r-shifted-$ or the opposite case. As noted above, there is a maximum value of the initial well width that allows for tuning to the desired resonance, and here it amounts to d,,,ax = 197 zfi in the fixed-&-shifted-& case, and to d,,, = 139 A in the fixed-&-shifted-& case. The largest values of II in the two cases are 12270 A3 and 12408 k3, respectively. The SUSYQM case occurs for d = 120.7 A (resonance already achieved), the largest value of Il now being 10570 A3. For the second choice, the parabolic initial potential, other conditions being unchanged, in Fig. 2 is given the optimized-in-a product II as it varies with the parameter a. The maximum of II = 12680 A3 occurs at a = - 1.80, but, as above, the maximum is comfortably broad. The corresponding best potential shape is given in Fig. 3. The SUSYQM situation, occuring at a = - 1.50, again gives a somewhat lower Il = 12680 A3. The optimized values of II obtained here compare favourably to the values reported elsewhere [l, 31 for the step asymmetric type of QWs, designed via approximate analytic considerations plus trial-and-error method. The values obtained here are in fact better by a few percent. Yet, we note that the two initial potentials we used were chosen only for purpose of carrying out most of the work analytically. They are
Fig. 1. The optimized-in-or values of II = ~17~612, calculated for the fixed-&-shifted-$ (solid line) and fixed-&-shifted-& (dashed line) cases as they depend on the width d of the starting infinite barrier rectangular potential, for m * = 0.067 and Rw = 116 meV. The SUSYQM case is also denoted.
-2.5
-2
-1.5 a
-1
-0.5
Fig. 2. Same as in Fig. 1, but for the parabolic starting potential described with the parameter a, the fixed-& shifted-& case only. rather idealized, and any realistic structure that would be made of graded Al,Gat-,As alloy would necessarily have finite barriers. The wave function penetration into the finite barriers, however, enhances the matrix elements and thus the nonlinear susceptibilities [l]. For instance, our calculations based on SUSYQM approach, that start with the parabolic potential truncated at 300 meV, indicate that II amounts to 15300 k3, as compared to 13445 a3 found above with strictly parabolic initial potential. Without doing explicit numerical calculations, we may estimate by simple scaling that using the IST method to design QWs with finite barriers would result in Il = 16200 A3. This would be
450
OPTIMIZATION
OF NONLINEAR
OPTICAL RECTIFICATION
o*61 0.5
SUSYQM based QW opt~i~tion in respect to second harmonic generation [4] indicates that these have a rather mild influence on the best potential shape (slightly squeezing the output of idealized calculations in order to compensate for the nonparabolicityincreased effective mass). The important conclusion from the above calculations for somewhat idealized cases is that the IST approach delivers some, though not drastic, improvement over the QW designs obtained by the SUSYQM approach.
”
0.4 z
0.3 -
5s 3 0.2 0.1 O-0.1 -150
Vol. 104, No. 8
-100
-50
0
50
100
150
REFERENCES
z PI 1.
Fig. 3. The optimized potential corresponding value a = -1.80 from Fig. 2.
to the
- 25% larger than what was found in Refs. [l] and [3].
2. 3. 4.
4. CONCLUSION The IST based method of finding the best potential shape of QWs in respect to resonant optical rectification was proposed and its use demonstrated in somewhat idealized situations. In the shape optimization of realistic QWs one would also have to take account of the effective mass variation in graded alloys, and of nonparabolicity, especially if higher photon energies are involved. These effects can at present be included only numerically. Our previous experience with
5.
6.
7. 8.
Rosencher, E. and Bois, Ph., Phys. Rev. B, 44 1991, 315. Ikonik, Z., Milanovik, V. and Tjapkin, D., IEEE J: Qaantam E~e~tro~~24, 1989, 54. Rosencher, E., Bois, P, Nagle, J., Costard, E. and Delaitre, S., App~ Phys. Lett., 55, 1989, 1589. Milanovi~, V. and IkoniC, Z., IEEE J; Q~ntam Electron., 32, 1996, 1316. Chabanov, V.M., Zakhariev, B.N., Brandt, S., Dahmen, H.D. and Stroch, T., Phys. Rev. A, 52, 19953389. Bastard, G., Wave Mechanics Applied to Semiconductor Heterostructures. Les Editions de Physique, CNRS, Paris, 1988. Cooper, F., Khare, A. and Sukhatme, U., Phys. Reports, 251, 1995,267. Handbook of Mathematical Functions, (Edited by M. Abramowitz and I. A. Stegun). Dover, New York, 1972.