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 Zh. Vychisl. Mat. Mat. Fiz., 2013, Volume 53, Number 11, Pages 1835–1855 (Mi zvmmf9946)

Asymptotic and numerical study of resonant tunneling in two-dimensional quantum waveguides of variable cross section

L. M. Baskina, M.  Kabardova, P. Neittaanmäkib, B. A. Plamenevskiic, O. V. Sarafanovc

a St. Petersburg State University of Telecommunications, nab. Moiki 61, St.-Petersburg, 191186, Russia
b University of Jyväskylä, P.O. Box 35(Agora), FI-40014, Jyväskylä, Finland
c St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia

Abstract: A waveguide is considered that coincides with a strip having two narrows of width $\varepsilon$. The electron wave function satisfies the Helmholtz equation with Dirichlet boundary conditions. The part of the waveguide between the narrows plays the role of a resonator, and there arise conditions for electron resonant tunneling. This phenomenon means that, for an electron of energy $E$, the probability $T(E)$ of passing from one part of the waveguide to the other through the resonator has a sharp peak at $E=E_{\mathrm{res}}$, where $E_{\mathrm{res}}$ is a “resonant” energy. To analyze the operation of electronic devices based on resonant tunneling, it is important to know $E_{\mathrm{res}}$ and the behavior of $T(E)$ for $E$ close to $E_{\mathrm{res}}$. Asymptotic formulas for the resonance energy and the transition and reflection coefficients as $\varepsilon\to0$ are derived. These formulas depend on the limit shape of the narrows. The limit waveguide near each narrow is assumed to coincide with a pair of vertical angles. The asymptotic results are compared with numerical ones obtained by approximately computing the waveguide scattering matrix. Based on this comparison, the range of $\varepsilon$ is found in which the asymptotic approach agrees with the numerical results. The methods proposed are applicable to much more complicated models than that under consideration. Specifically, the same approach is suitable for an asymptotic and numerical analysis of tunneling in three-dimensional quantum waveguides of variable cross section.

Key words: two-dimensional quantum waveguides, Dirichlet problem for Helmholtz’s equation, asymptotic and numerical studies.

DOI: https://doi.org/10.7868/S004446691311001X

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English version:
Computational Mathematics and Mathematical Physics, 2013, 53:11, 1664–1683

Bibliographic databases:

UDC: 519.634

Citation: L. M. Baskin, M.  Kabardov, P. Neittaanmäki, B. A. Plamenevskii, O. V. Sarafanov, “Asymptotic and numerical study of resonant tunneling in two-dimensional quantum waveguides of variable cross section”, Zh. Vychisl. Mat. Mat. Fiz., 53:11 (2013), 1835–1855; Comput. Math. Math. Phys., 53:11 (2013), 1664–1683

Citation in format AMSBIB
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This publication is cited in the following articles:
1. D. Sokolovski, L. M. Baskin, “Statistics of resonance and nonresonance tunneling of fermionized cold atoms”, Phys. Rev. A, 90:2 (2014), 024101
2. L. M. Baskin, M. M. Kabardov, N. M. Sharkova, “Electron transport in a multi-resonator system formed by constrictions of a quantum waveguide”, 2016 Days on Diffraction (DD) (St.Petersburg, Russia), eds. O. Motygin, A. Kiselev, P. Kapitanova, L. Goray, A. Kazakov, A. Kirpichnikova, IEEE, 2016, 52–55
3. D. Sokolovski, L. M. Baskin, “Quantum statistical effects in multichannel wave-packet scattering of noninteracting identical particles”, Phys. Rev. A, 94:2 (2016), 022115
4. S. Kondej, “Straight quantum layer with impurities inducing resonances”, J. Phys. A-Math. Theor., 50:31 (2017), 315203
5. M. M. Kabardov, B. A. Plamenevskiy, O. V. Sarafanov, N. M. Sharkova, “Comparison of asymptotic and numerical approaches to the study of the resonant tunneling in a two-dimensional symmetric quantum waveguide of variable cross-section”, J. Math. Sci. (N. Y.), 238:5 (2019), 641–651
6. O. V. Sarafanov, “Asymptotics of the resonant tunneling of high-energy electrons in two-dimensional quantum waveguides of variable cross-section”, J. Math. Sci. (N. Y.), 238:5 (2019), 736–749
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