Abstract
The time-independent Schrödinger equation is a linear, second-order equation with a coefficient that depends on position. An analytical solution can be found in a limited number of cases, typically one-dimensional ones. This chapter illustrates some of these cases, starting from the step-like potential energy followed by the potential-energy barrier. In both of them, the coefficient of the Schrödinger equation is approximated with a piecewise-constant function. Despite their simplicity, the step and barrier potential profiles show that the quantum-mechanical treatment may lead to results that differ substantially from the classical ones: a finite probability of transmission may be found where the classical treatment would lead to a reflection only, or vice versa. The transmission and reflection coefficients are defined by considering a plane wave launched towards the step or barrier. It is shown that the definition of the two coefficients can be given also for a barrier of a general form, basing on the formal properties of the second-order linear equations in one dimension. Finally, the case of a finite well is tackled, showing that in the limit of an infinite depth of the well one recovers the results of the particle in a box illustrated in a preceding chapter.
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Notes
- 1.
- 2.
The reflection at the barrier for k 1 s ≠i π explains why the experimental value of the Richardson constant A is lower than the theoretical one. Such a constant appears in the expression J s = A T 2 exp[−E W ∕(k B  T)] of the vacuum-tube characteristics [31]. This is one of the cases where the effect is evidenced in macroscopic-scale experiments. Still considering the vacuum tubes, another experimental evidence of the tunnel effect is the lack of saturation of the forward current-voltage characteristic at increasing bias.
- 3.
The electron Volt (eV) is a unit of energy obtained by multiplying 1 J by a number equal to the modulus of the electron charge expressed in C (Table D.1).
References
M. Born, E. Wolf, Principles of Optics, 6th edn. (Pergamon Press, London, 1980)
C.R. Crowell, The Richardson constant for thermionic emission in Schottky barrier diodes. Solid-State Electron. 8(4), 395–399 (1965)
I.I. Gol’dman, V.D. Krivchenkov, Problems in Quantum Mechanics (Pergamon Press, London, 1961)
E.L. Ince, Ordinary Differential Equations (Dover, New York, 1956)
M. Muskat, E. Hutchisson, Symmetry of the transmission coefficients for the passage of particles through potential barriers. Proc. Natl. Acad. Sci. USA 23, 197–201 (1937)
M. Rudan, A. Gnudi, E. Gnani, S. Reggiani, G. Baccarani, Improving the accuracy of the Schrödinger-Poisson solution in CNWs and CNTs, in Simulation of Semiconductor Processes and Devices 2010 (SISPAD), pp. 307–310, ed. by G. Baccarani, M. Rudan (IEEE, Bologna, 2010)
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Rudan, M. (2018). Elementary Cases. In: Physics of Semiconductor Devices. Springer, Cham. https://doi.org/10.1007/978-3-319-63154-7_11
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DOI: https://doi.org/10.1007/978-3-319-63154-7_11
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