“Smart Design” of Quantum Wells and Double-Quantum Wells Structures

Igor Tralle, Klaudiusz Majchrowski

Faculty of Mathematics and Natural Sciences, University of Rzeszow, Rzeszow, Poland.

DOI: 10.4236/wjcmp.2014.41004                

Abstract

In the work, we propose an approach to “smart design” of heterostructures (quantum wells and superlattices) based on the combination of Inverse Scattering Problem Method and the direct solution of the eigenvalue problem for the Schr?dinger equation with reconstructed potentials. Potential shape reconstructed in this way can be substituted then by some approximation, so that the output spectrum obtained by solving the Schr?dinger equation with such approximated potential, differs only slightly from the input one. In our opinion, the approach can be used in many applications, for instance, for developing the new electronic devices such as tunable THz detectors.

Keywords

Inverse Scattering Problem Method; Quantum Wells and Superlattices; Predetermined Energy Spectrum; Bargmann Reflectionless Potentials

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	V. Milanovic and Z. Ikonic, “Optimization of Nonlinear Optical Rectification in Semiconductor Quantum Wells Using the Inverse Spectral Theory,” Solid State Community, Vol. 104, No. 8, 1997, pp. 445-450. http://dx.doi.org/10.1016/S0038-1098(97)00396-7
[2]	I. Tralle and G. Petrov, “On the Semiconductor Well Engineering,” Molecular Physics Reports, Vol. 23, 1999, pp. 199-202.
[3]	S. Tomic, V. Milanovic and Z. Ikonic, “Quantum Well Shape Tailoring via Inverse Spectral Theory: Optimizing Resonant Second-Harmonic Generation,” Journal of Physics: Condensed Matter, Vol. 10, No. 29, 1998, pp. 6523-6532. http://dx.doi.org/10.1088/0953-8984/10/29/012
[4]	D. Bessis and G. A. Mezincescu, “Design of Semiconductor Heterostructures with Preset Electron Reflectance by Inverse Scattering Techniques,” Microelectronics Journal, Vol. 30, No. 10, 1999, pp. 953-974. http://dx.doi.org/10.1016/S0026-2692(99)00059-2
[5]	A. A. Suzko and I. Tralle, “Reconstruction of Quantum Well Potentials via the Intertwining Operator Technique,” Acta Physica Polonica B, Vol. 39, No. 3, 2008, pp. 545-567.
[6]	S. A. Sofianos, G. J. Rampho, H. A. Donfack, I. E. Lagaris and H. Leeb, “Design of Quantum Filters with Pre-Determined Reflection and Transmission Properties,” Microelectronics Journal, Vol. 38, No. 2, 2007, pp. 235-244. http://dx.doi.org/10.1016/j.mejo.2006.11.006
[7]	T. Wojtowicz, G. Karczewski and J. Kossut, “Excitons in Novel Diluted Magnetic Semiconductor Quantum Structures,” Thin Solid Films, Vol. 306, No. 2, 1997, pp. 271-282. http://dx.doi.org/10.1016/S0040-6090(97)00284-8
[8]	K. Kowalik, A. Kudelski, J. A. Gaj, T. Wojtowicz, O. Krebs and P. Voisin, “In-Plane Optical Anisotropy of Parabolic and Half-Parabolic Cd1-xMnxTe Quantum Wells,” Solid State Community, Vol. 126, No. 8, 2003, pp. 467-471. http://dx.doi.org/10.1016/S0038-1098(03)00190-X
[9]	B. M. Levitan and M. G. Gasymov, “Determination of Differential Equation by Two of Its Spectra,” Russian Mathematical Surveys, Vol. 19, No. 2, 1964, pp. 3-63.
[10]	B. M. Levitan, “Inverse Sturm-Liouville Problems,” Nauka, Moscow, 1984.
[11]	H. B. Thacker, C. Quigg and J. L. Rosner, “Inverse Scattering Problem for Quarkonium Systems,” Physical Review D, Vol. 18, No. 1, 1978, pp. 274-287. http://dx.doi.org/10.1103/PhysRevD.18.274
[12]	J. F. Schonefeld, W. Kwong, J. L. Rosner, C. Quigg and H. B. Thacker, “On the Convergence of Reflectionless Approximations to Confining Potentials,” Annals of Physics, Vol. 128, No. 1, 1980, pp. 1-28. http://dx.doi.org/10.1016/0003-4916(80)90055-X
[13]	I. M. Gel’fand and B. M. Levitan, “On the Determination of a Differential Equation from Its Spectrum Function Izv. Akad. Nauk SSSR Ser. Mat.,” Vol. 15, No. 4, 1951, pp. 309-360.
[14]	V. A. Marchenko, “Spectral Theory of Sturm-Liouville Operators,” Nauk. Dumka, Kiev, 1972.
[15]	B. N. Zakhariev and A. A. Suzko, “Direct and Inverse Problems. Potentials in Quantum Scattering,” Springer-Verlag, Berlin, 1990.
[16]	Z. S. Agranovich and V. A. Marchenko, “Sturm-Liouville Operators and Their Applications,” Nauk. Dumka, Kiev, 1977.
[17]	J. Hadamard, “Le Probleme de Cauchy et les équations aux Derives Partiels Linéaires Hyperboliques,” Hermann, Paris, 1932.
[18]	A. N. Tikhonov and V. Ya. Arsenin, “Methods of Solution of Ill-Posed Problems,” Nauka, Moscow, 1979.
[19]	A. M. Tomlinson, C. C. Chang, R. J. Stone, R. J. Nicholas, A. M. Fox, M. A. Pate and C. T. Foxon, “Intersubband Transitions in GaAs Coupled-Quantum-Wells for Use as a Tunable Detector at THz Frequencies,” Applied Physics Letters, Vol. 76, No. 12, 2000, pp. 1579-1581. http://dx.doi.org/10.1063/1.126101