- Email: [email protected]
Quantum well shape tailoring via inverse spectral theory: Optimization of nonlinear optical rectification
16 February
1998
PHYSICS
EIS3VIER
Physics Letters A 238 (1998)
LETTERS
A
385-389
Quantum well shape tailoring via inverse spectral theory: Optimization of nonlinear optical rectification Stank0 Tomb5 a~1, Vitomir Milanowc’ ’ b*2, Zoran IkoniC b a Laboratory h Faculty
of Physics (OIO), VINdA
of Electrical
Engineering,
Received
Institute of Nuclear
University
17 October
of Belgrade,
Sciences, P.0. Box 522, 11001 Belgrade, Bulevar
Revolucije
1997; accepted for publication Communicated by L.J. Sham
73. 11000 Belgrade.
18 November
Yugoslavia Yugoslavia
1997
Abstract A procedure for the design of quantum well structures optimized in respect to the intersubband resonant optical rectification generation is proposed. It relies on the inverse spectral theory, allowing one to generate a family of potentials with prescribed level energies, and additionally vary the potential shape in an isospectral manner, via a scalar parameter. @ 1998 Elsevier Science B.V. PACS:
73.20.D~;
78.66.-w;
03.65.-w
Intersubband transitions in semiconductor quantum wells (QW) have attracted considerable research attention. This is mainly due to large values of dipole transition matrix elements and the possibility of achieving the resonance conditions, resulting in rather intense linear and, even more so, nonlinear optical processes. The medium polarization induced by the external harmonic electromagnetic field E(t) = Eejwt + E* e-jwt, keeping terms up to second order, may be written as P(t)
+
= ( .50*(1)Eejwr +
cox:fJi2 e2jor)
/y(y)82,
+ c.c.
(1)
where EO is the vacuum permittivity, x(t) the linear susceptibility, xi: and xi*) are the second-order susceptibilities at double or zero frequency, the latter one ’ E-mail: [email protected]. * E-mail: [email protected]. 0375.9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PII SO375-9601(97)00909-2
(also known as optical rectification coefficient) describing the generation of a static electric field. The density matrix calculations [ 1,2] for a two-level system shows that xr) becomes largest at resonance (when the spacing between the two relevant levels of the system equals the photon energy, i.e. &J = E2 - El = fiw21), and then amounts to (2)
-2
Xomax -
e3Tl T2 --&
(Pll
-
P22)&42.
(2)
Here pii, i = 1,2, denote electron densities on the ith quantized state per unit well surface, ~~2 = (11~ 12) and St2 = (1 Iz 11) - (212 12) are the transition dipole moment and the difference of the permanent dipole moments of the two states. Furthermore, T, and T2 are the diagonal and off-diagonal relaxation times in the density matrix equation (the excited state lifetime and the dephasing time, respectively). Concerning the possibilities of increasing xb2’,
386
S. Torn2 et al. /Physics
the dephasing time T2. arising from various scattering mechanisms, depends only slightly on the QW shape [ 3 1. The same is true for the lifetime Tl. In fact, the effective Tl may be increased by introducing additional, optically inactive, metastable states into the system, but ,&*’ is then increased at the expense of making the response slower [ 41. The dipole moments ,ui2 and 612, on the other hand, very much depend on the QW shape, which may thus be employed for maximizing xc’. While varying the QW shape, it is essential to keep the level spacing as specified, to match the resonance condition. Clearly, only asymmetric structures are useful for optical rectification, and may be realized by asymmetric composition grading in a stepwise constant or continuous manner. Some considerations of optimizing the QW shape, within the class of simple step-graded QWs, and within a somewhat idealized model, were presented in Ref. [ 41. If the search for the best potential shape requires any amount of trial-and-error type calculations, most effort is spent in restoring the level spacing upon changing the potential shape, rather than in checking the values of the matrix elements. Within the class of continuously graded QWs this problem may be very serious. For these we have recently used [5] a method based on supersymmetric quantum mechanics (SUSYQM) [ 61, which starts with an initial potential, with its states positioned as required, for resonance SHG, and then generates a family of potentials isospectral to the initial one. Their shape is controlled by one or more scalar parameters, variation of which allows one to search for the best potential shape. Here we describe another method for optimizing the optical rectification coefficient, which is even more versatile than that based on SUSYQM. It starts with an almost completely arbitrary initial potential, and, using the inverse spectral theory (IST), e.g. Ref. [ 71, shifts the states to their desired positions, while, at the same time, it introduces free parameter(s) for varying the shape of this modified potential in an isospectral manner. The outline of IST, necessary for the purpose of this work, is briefly described below. Consider an electron moving in a potential web U(z), with a constant effective mass rn*. Its bound states energies and wave functions (obtained from the effective mass Schrodinger equation) will be denoted as Ei and Jri, i = 1,2,. . . The IST enables one to construct a modified potential
Letters A 238 (1998) 385-389
&ST(Z; Ek + l , V(z ) , W{rclc, $k}), eigenenergies of which have the property that EiIST = E; (i # k), while the kth state is shifted by some amount E, i.e. EfT = Ek + E. The Wronskian in &ST, i.e. W(t&,,&} = ~/~~(z)+b_l(z>- $:(z)$k(z) is calculated with the kth eigenfunction of the original Schrodinger equation and the function $-c which is any solution of this equation at energy Ek + E, not coinciding with any of its eigenenergies (hence $t is not its normalizable wave function) and a prime denotes d/dz. The shift e in IST may take any value from the interval (&.-I - &. &+i - Ek), i.e. the shifted level may not cross any other level. It is convenient to write lCItas a linear combination of the two fundamental solutions of the original Hamiltonian at EisT = & + E, that is the functions [t,~( z) satisfying the fundamental initial conditions l,(O) = [i( 0) = 1 and Si (0) = l2(0) =O.Thefunctioncj/,(z;a) =51(t)+q”2(2) thus depends on one free parameter ((Y), aside from an arbitrary multiplicative constant which is irrelevant and left out. The expression for VIST( z ) reads
The normalized wave functions for i # k, corresponding to it, are given by
(I/fSyZ;*) = (I x
&)-“*
2m*E #e(z;aY) fi* wrwd~~k~
(
h(z) - Z
X
s
‘ki(Z’>‘h(Z’)dz’
-m
(4)
>
while -l/2 q$ST(z;a)
=
-- 2;:r(F+
-F-) >
@k(Z) ’
(5)
w(cll,(+bk}’
where F’(z) = l/$:(z) and Fk = lim,,h, F(z). It is important to note that Q in Eqs. (3)-(5) is an “isospectral” free parameter: its variation does not affect any eigenvalue of VIST, but does change its shape,
S. TomiC et al. /Physics
Letters A 238 (1998) 385-389
and the eigenfunctions (and hence the dipole matrix elements). In addition, if the original potential U( z ) .is symmetric, and therefore not suitable for optical rectification, the parameter (Yenables the construction of the asymmetric Utsr: for this purpose the fundamental solution [( z ) having the same parity as that of the to-be-shifted original state has to be present in IJ~. In order to optimize the QW shape in respect to xh2’ > therefore, one first sets some initial potential .!/( z ), the states of which are likely not to be properly spaced. Then, a family of modified potentials (3) is generated, with the spacing between the two states relevant for xh2’ corrected by making an appropriate shift E. By varying CX,one evaluates the matrix elements Pi2
=
(~fST~z>~Z~~;ST~z))
and
62 = (4vT(z)IZI+:ST(Z)~ - (~:sTtz>lzl@T
1zI < A,
Izl>A,
(6)
where A&(N) = 100 + N x 0.5 [meV], and N is an integer incremented from 0 to 145. The effective mass in (6) is taken constant and equal to that in GaAs (m* = 0.066 in free electron mass units), and the potential is truncated at V = 190 meV or V = 200 meV in two sets of calculations (note that the well width A changes with incrementing N). Level energies will change as N (i.e. the well width) varies, so the
387
potentials (6) generally do not meet the resonance condition, and the shifting of one of the levels is first necessary, which we perform for the second level (k = 2 in the above expressions) to obtain EiST - EfST = fiw, with EisT = Ez + E and E t”’ = El. For each specific initial potential (i.e. the value of N in (6) ) we vary the parameter LYand evaluate the dipole matrix elements, from the wave functions (4), (5), in order to find the largest 17(O) = pT26tz corresponding to this N, and repeat this procedure for all N. It is worth noting that, due to the symmetry of the initial potentials, it is enough to scan only over positive values of (Y(because Z7(‘) ( -a) = -Z7(‘) (a) ), otherwise both positive and negative values should be explored. Furthermore. one has to take care that the Wronskians in Eqs. (3)-(5) do not cross zero, in order to avoid singularities and physically unacceptable solutions. Results of the search done for the truncation V = 200 meV are given in Fig. la. We find that the largest value of n(O) occurs for the values of parameters N = 101 (i.e. AE(lO1) = 150.5 meV), and LY,,,~= 0.48. Here we have fl,$ = 15677.5 A”. with individual dipole moments ~12 = 15.75 A and 6,? = 63.20 A. The initial potential had its levels at El = 73.01 meV and Ez = 192.65 meV, so a shift of E = -3.64 meV of the second state was performed to bring the spacing AE21 = 119.64 meV to the desired 116 meV. In another search (Fig. I b), with the truncation V = 190 meV, we find D&“,? = 18770.6 A3, with ~12 = 13.51 8, and 612 = 102.84 A, which is obtained for AE(97) = 148.5 meV and (~,,r~= 0.70. The corresponding initial potential in this case had El = 7 1.75 meV and E2 = 185.7 1 meV, and required a shift E = 2.04 meV of the second level to correct the spacing. The optimized potentials IJIST( ; ; aopr ) are given in Fig. lc. While the optimized potentials may be realized by grading the AlGaAs alloy, in real structures it may be necessary to include corrections due to the position (i.e. the alloy composition) dependent mass and the nonparabolicity. However, with rather low energies ot states in the above examples, these corrections are here expected to be very small indeed. Comparison of 17kti found here against the values obtained elsewhere [ 2,81 in step-graded QWs (also based on AlGaAs. within the constant mass approximation, to make a fair comparison), shows that our U,$ is better by up to 40%.
388
S. TomiC et al. /Physics
Letters A 238 (1998) 385-389
2.00
and the IST results in the V = 200 meV case might have been physically expected from the rather small value of the shift E which was necessary here to correct the Ievel spacing. Indeed, the SUSUQM is in fact contained in the IST, as we show below. Consider, within the IST, the case E 4 0, when the transformed potential is isospectral to the original, just as is the case in SUSYQM. Taking, for convenience, that one of the fundamental solutions (say li ) has the property lim,,o ct ( z > = (elk( z >, the Wronskian may be written as w{lt, $k} + LY~{~z, $k}, where the first term becomes
1.75 1.50 1.25
ts
~~~~~~~,I
0
20
40
1.2,
1
!
20
40
60N
80
100
120
!
1
1
1
60
80
100
120
140 ._
0,
0.8 8
0.6 0.4 0.2 0.0
0
N
while the second term becomes a constant UC, where C = lim,+c w{lz;, &}. If Ly is given a finite value, then W{+,, $k} = cd? is constant, which corresponds to the identity transform of the potential, Utsr (z ) = U( z ) , for any LY.However, taking (Y tends to zero so that (Y(E) C -t 2m*eA/h2, where A is an arbitrary constant, gives the Wronskian
140
2m*e
I -100
-50
0 z
50
I 100
[Al
Fig. I. (a) Values of the matrix elements product nro) = /LL:~S~~ obtainable with various values of the QW design parameters N and a. with the initial parabolic potential truncated at V = 200 meV, and V = 190 meV (b) ; (c) the optimized potentials &sT( z ) for N = 101 and (I OPl= 0.48 corresponding to V = 200 meV (dashed line) and N = 97 and clopI = 0.70 corresponding to V = 190 meV (solid line).
and also somewhat better than the value we have previously obtained using the SUSYQM based optimization ( Lrco) = 15322 A3) [ 51. In the latter we have started with the potential (6) truncated at V = 200 meV, its parameters already fitted to provide the correct level spacing, and then varied it isospectrally in order to find the best potential shape (however, the potential truncated at V = 190 meV could not be handled by this approach). The similarity between the SUSYQM
=
+%s(z) Utsr (z ) isospectral
and the potential U(z) is &ST(z)=~(z)
-
$&In
to the original
&S(Z)=&(z)
(9) where the subscript SS refers to the expressions derived in the SUSYQM [ 61. Therefore, we conclude that the optimized potential obtained via IST will never be inferior to that obtained via SUSYQM, provided one starts with the same class of initial potentials in both approaches (though the improvement in some examples may not be drastic). In conclusion, a procedure of QW shape optimization in respect to the optical rectification coefficient, based on IST, was proposed and discussed. This is shown to be potentially superior to another method based on SUSYQM, as was demonstrated on the ex-
S. TomiC et al. /Physics
ample design of QW for optical rectification I 16 meV radiation.
of F&J=
References [I I I>. Ahn. S.L. Chuang, 2196. 121 E. Rosencher.
IEEE J. Quant.
Electron.
Ph. Bois, Phys. Rev. B 44 (1991)
23 (1987) 315.
Letters A 238 (1998) 385-389
389
[31 A. Selmeier, H. J. Hiibner, G. Abstreiter, G. Weimann. W. Schlapp, Phys. Rev. Lett. 59 (1987) 1345. [4] E. Rosencher, P. Bois, B. Vinter. J. Nagle. D. Kaplan, Appl. Phys. Lett. 56 (1990) 1822. L.51S. TomiC. V. MilanoviC, Z. IkoniC, Opt. Commun.. in press. [61 F. Cooper, A. Khare, U. Sukhatme. Phys. Rep. 251 ( 1995) 267. [7
I V.M. Chabanov, B.N. Zakhariev. S. Brandt. H.D Dahmen, T.
Stroch, Phys. Rev. A 52 (1995) 3389. [81 M. Zaluzny, J. Appl. Phys. 74 (1993) 4716.
1998
PHYSICS
EIS3VIER
Physics Letters A 238 (1998)
LETTERS
A
385-389
Quantum well shape tailoring via inverse spectral theory: Optimization of nonlinear optical rectification Stank0 Tomb5 a~1, Vitomir Milanowc’ ’ b*2, Zoran IkoniC b a Laboratory h Faculty
of Physics (OIO), VINdA
of Electrical
Engineering,
Received
Institute of Nuclear
University
17 October
of Belgrade,
Sciences, P.0. Box 522, 11001 Belgrade, Bulevar
Revolucije
1997; accepted for publication Communicated by L.J. Sham
73. 11000 Belgrade.
18 November
Yugoslavia Yugoslavia
1997
Abstract A procedure for the design of quantum well structures optimized in respect to the intersubband resonant optical rectification generation is proposed. It relies on the inverse spectral theory, allowing one to generate a family of potentials with prescribed level energies, and additionally vary the potential shape in an isospectral manner, via a scalar parameter. @ 1998 Elsevier Science B.V. PACS:
73.20.D~;
78.66.-w;
03.65.-w
Intersubband transitions in semiconductor quantum wells (QW) have attracted considerable research attention. This is mainly due to large values of dipole transition matrix elements and the possibility of achieving the resonance conditions, resulting in rather intense linear and, even more so, nonlinear optical processes. The medium polarization induced by the external harmonic electromagnetic field E(t) = Eejwt + E* e-jwt, keeping terms up to second order, may be written as P(t)
+
= ( .50*(1)Eejwr +
cox:fJi2 e2jor)
/y(y)82,
+ c.c.
(1)
where EO is the vacuum permittivity, x(t) the linear susceptibility, xi: and xi*) are the second-order susceptibilities at double or zero frequency, the latter one ’ E-mail: [email protected]. * E-mail: [email protected]. 0375.9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PII SO375-9601(97)00909-2
(also known as optical rectification coefficient) describing the generation of a static electric field. The density matrix calculations [ 1,2] for a two-level system shows that xr) becomes largest at resonance (when the spacing between the two relevant levels of the system equals the photon energy, i.e. &J = E2 - El = fiw21), and then amounts to (2)
-2
Xomax -
e3Tl T2 --&
(Pll
-
P22)&42.
(2)
Here pii, i = 1,2, denote electron densities on the ith quantized state per unit well surface, ~~2 = (11~ 12) and St2 = (1 Iz 11) - (212 12) are the transition dipole moment and the difference of the permanent dipole moments of the two states. Furthermore, T, and T2 are the diagonal and off-diagonal relaxation times in the density matrix equation (the excited state lifetime and the dephasing time, respectively). Concerning the possibilities of increasing xb2’,
386
S. Torn2 et al. /Physics
the dephasing time T2. arising from various scattering mechanisms, depends only slightly on the QW shape [ 3 1. The same is true for the lifetime Tl. In fact, the effective Tl may be increased by introducing additional, optically inactive, metastable states into the system, but ,&*’ is then increased at the expense of making the response slower [ 41. The dipole moments ,ui2 and 612, on the other hand, very much depend on the QW shape, which may thus be employed for maximizing xc’. While varying the QW shape, it is essential to keep the level spacing as specified, to match the resonance condition. Clearly, only asymmetric structures are useful for optical rectification, and may be realized by asymmetric composition grading in a stepwise constant or continuous manner. Some considerations of optimizing the QW shape, within the class of simple step-graded QWs, and within a somewhat idealized model, were presented in Ref. [ 41. If the search for the best potential shape requires any amount of trial-and-error type calculations, most effort is spent in restoring the level spacing upon changing the potential shape, rather than in checking the values of the matrix elements. Within the class of continuously graded QWs this problem may be very serious. For these we have recently used [5] a method based on supersymmetric quantum mechanics (SUSYQM) [ 61, which starts with an initial potential, with its states positioned as required, for resonance SHG, and then generates a family of potentials isospectral to the initial one. Their shape is controlled by one or more scalar parameters, variation of which allows one to search for the best potential shape. Here we describe another method for optimizing the optical rectification coefficient, which is even more versatile than that based on SUSYQM. It starts with an almost completely arbitrary initial potential, and, using the inverse spectral theory (IST), e.g. Ref. [ 71, shifts the states to their desired positions, while, at the same time, it introduces free parameter(s) for varying the shape of this modified potential in an isospectral manner. The outline of IST, necessary for the purpose of this work, is briefly described below. Consider an electron moving in a potential web U(z), with a constant effective mass rn*. Its bound states energies and wave functions (obtained from the effective mass Schrodinger equation) will be denoted as Ei and Jri, i = 1,2,. . . The IST enables one to construct a modified potential
Letters A 238 (1998) 385-389
&ST(Z; Ek + l , V(z ) , W{rclc, $k}), eigenenergies of which have the property that EiIST = E; (i # k), while the kth state is shifted by some amount E, i.e. EfT = Ek + E. The Wronskian in &ST, i.e. W(t&,,&} = ~/~~(z)+b_l(z>- $:(z)$k(z) is calculated with the kth eigenfunction of the original Schrodinger equation and the function $-c which is any solution of this equation at energy Ek + E, not coinciding with any of its eigenenergies (hence $t is not its normalizable wave function) and a prime denotes d/dz. The shift e in IST may take any value from the interval (&.-I - &. &+i - Ek), i.e. the shifted level may not cross any other level. It is convenient to write lCItas a linear combination of the two fundamental solutions of the original Hamiltonian at EisT = & + E, that is the functions [t,~( z) satisfying the fundamental initial conditions l,(O) = [i( 0) = 1 and Si (0) = l2(0) =O.Thefunctioncj/,(z;a) =51(t)+q”2(2) thus depends on one free parameter ((Y), aside from an arbitrary multiplicative constant which is irrelevant and left out. The expression for VIST( z ) reads
The normalized wave functions for i # k, corresponding to it, are given by
(I/fSyZ;*) = (I x
&)-“*
2m*E #e(z;aY) fi* wrwd~~k~
(
h(z) - Z
X
s
‘ki(Z’>‘h(Z’)dz’
-m
(4)
>
while -l/2 q$ST(z;a)
=
-- 2;:r(F+
-F-) >
@k(Z) ’
(5)
w(cll,(+bk}’
where F’(z) = l/$:(z) and Fk = lim,,h, F(z). It is important to note that Q in Eqs. (3)-(5) is an “isospectral” free parameter: its variation does not affect any eigenvalue of VIST, but does change its shape,
S. TomiC et al. /Physics
Letters A 238 (1998) 385-389
and the eigenfunctions (and hence the dipole matrix elements). In addition, if the original potential U( z ) .is symmetric, and therefore not suitable for optical rectification, the parameter (Yenables the construction of the asymmetric Utsr: for this purpose the fundamental solution [( z ) having the same parity as that of the to-be-shifted original state has to be present in IJ~. In order to optimize the QW shape in respect to xh2’ > therefore, one first sets some initial potential .!/( z ), the states of which are likely not to be properly spaced. Then, a family of modified potentials (3) is generated, with the spacing between the two states relevant for xh2’ corrected by making an appropriate shift E. By varying CX,one evaluates the matrix elements Pi2
=
(~fST~z>~Z~~;ST~z))
and
62 = (4vT(z)IZI+:ST(Z)~ - (~:sTtz>lzl@T
1zI < A,
Izl>A,
(6)
where A&(N) = 100 + N x 0.5 [meV], and N is an integer incremented from 0 to 145. The effective mass in (6) is taken constant and equal to that in GaAs (m* = 0.066 in free electron mass units), and the potential is truncated at V = 190 meV or V = 200 meV in two sets of calculations (note that the well width A changes with incrementing N). Level energies will change as N (i.e. the well width) varies, so the
387
potentials (6) generally do not meet the resonance condition, and the shifting of one of the levels is first necessary, which we perform for the second level (k = 2 in the above expressions) to obtain EiST - EfST = fiw, with EisT = Ez + E and E t”’ = El. For each specific initial potential (i.e. the value of N in (6) ) we vary the parameter LYand evaluate the dipole matrix elements, from the wave functions (4), (5), in order to find the largest 17(O) = pT26tz corresponding to this N, and repeat this procedure for all N. It is worth noting that, due to the symmetry of the initial potentials, it is enough to scan only over positive values of (Y(because Z7(‘) ( -a) = -Z7(‘) (a) ), otherwise both positive and negative values should be explored. Furthermore. one has to take care that the Wronskians in Eqs. (3)-(5) do not cross zero, in order to avoid singularities and physically unacceptable solutions. Results of the search done for the truncation V = 200 meV are given in Fig. la. We find that the largest value of n(O) occurs for the values of parameters N = 101 (i.e. AE(lO1) = 150.5 meV), and LY,,,~= 0.48. Here we have fl,$ = 15677.5 A”. with individual dipole moments ~12 = 15.75 A and 6,? = 63.20 A. The initial potential had its levels at El = 73.01 meV and Ez = 192.65 meV, so a shift of E = -3.64 meV of the second state was performed to bring the spacing AE21 = 119.64 meV to the desired 116 meV. In another search (Fig. I b), with the truncation V = 190 meV, we find D&“,? = 18770.6 A3, with ~12 = 13.51 8, and 612 = 102.84 A, which is obtained for AE(97) = 148.5 meV and (~,,r~= 0.70. The corresponding initial potential in this case had El = 7 1.75 meV and E2 = 185.7 1 meV, and required a shift E = 2.04 meV of the second level to correct the spacing. The optimized potentials IJIST( ; ; aopr ) are given in Fig. lc. While the optimized potentials may be realized by grading the AlGaAs alloy, in real structures it may be necessary to include corrections due to the position (i.e. the alloy composition) dependent mass and the nonparabolicity. However, with rather low energies ot states in the above examples, these corrections are here expected to be very small indeed. Comparison of 17kti found here against the values obtained elsewhere [ 2,81 in step-graded QWs (also based on AlGaAs. within the constant mass approximation, to make a fair comparison), shows that our U,$ is better by up to 40%.
388
S. TomiC et al. /Physics
Letters A 238 (1998) 385-389
2.00
and the IST results in the V = 200 meV case might have been physically expected from the rather small value of the shift E which was necessary here to correct the Ievel spacing. Indeed, the SUSUQM is in fact contained in the IST, as we show below. Consider, within the IST, the case E 4 0, when the transformed potential is isospectral to the original, just as is the case in SUSYQM. Taking, for convenience, that one of the fundamental solutions (say li ) has the property lim,,o ct ( z > = (elk( z >, the Wronskian may be written as w{lt, $k} + LY~{~z, $k}, where the first term becomes
1.75 1.50 1.25
ts
~~~~~~~,I
0
20
40
1.2,
1
!
20
40
60N
80
100
120
!
1
1
1
60
80
100
120
140 ._
0,
0.8 8
0.6 0.4 0.2 0.0
0
N
while the second term becomes a constant UC, where C = lim,+c w{lz;, &}. If Ly is given a finite value, then W{+,, $k} = cd? is constant, which corresponds to the identity transform of the potential, Utsr (z ) = U( z ) , for any LY.However, taking (Y tends to zero so that (Y(E) C -t 2m*eA/h2, where A is an arbitrary constant, gives the Wronskian
140
2m*e
I -100
-50
0 z
50
I 100
[Al
Fig. I. (a) Values of the matrix elements product nro) = /LL:~S~~ obtainable with various values of the QW design parameters N and a. with the initial parabolic potential truncated at V = 200 meV, and V = 190 meV (b) ; (c) the optimized potentials &sT( z ) for N = 101 and (I OPl= 0.48 corresponding to V = 200 meV (dashed line) and N = 97 and clopI = 0.70 corresponding to V = 190 meV (solid line).
and also somewhat better than the value we have previously obtained using the SUSYQM based optimization ( Lrco) = 15322 A3) [ 51. In the latter we have started with the potential (6) truncated at V = 200 meV, its parameters already fitted to provide the correct level spacing, and then varied it isospectrally in order to find the best potential shape (however, the potential truncated at V = 190 meV could not be handled by this approach). The similarity between the SUSYQM
=
+%s(z) Utsr (z ) isospectral
and the potential U(z) is &ST(z)=~(z)
-
$&In
to the original
&S(Z)=&(z)
(9) where the subscript SS refers to the expressions derived in the SUSYQM [ 61. Therefore, we conclude that the optimized potential obtained via IST will never be inferior to that obtained via SUSYQM, provided one starts with the same class of initial potentials in both approaches (though the improvement in some examples may not be drastic). In conclusion, a procedure of QW shape optimization in respect to the optical rectification coefficient, based on IST, was proposed and discussed. This is shown to be potentially superior to another method based on SUSYQM, as was demonstrated on the ex-
S. TomiC et al. /Physics
ample design of QW for optical rectification I 16 meV radiation.
of F&J=
References [I I I>. Ahn. S.L. Chuang, 2196. 121 E. Rosencher.
IEEE J. Quant.
Electron.
Ph. Bois, Phys. Rev. B 44 (1991)
23 (1987) 315.
Letters A 238 (1998) 385-389
389
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