A continuum limit for the Kronig–Penney model

Abstract

We investigate the transmission properties of a quantum one-dimensional periodic system of fixed length L, with N barriers of constant height V and width λ, and N wells of width δ. In particular, we study the behaviour of the transmission coefficient in the limit N → ∞, with L fixed. This is achieved by letting δ and λ both scale as 1/N , in such a way that their ratio γ = λ/δ is a fixed parameter characterizing the model. In this continuum limit the multi-barrier system behaves as it were constituted by a unique barrier of constant height E o = (γV )/(1 + γ). The analysis of the dispersion relation of the model shows the presence of forbidden energy bands at any finite N .

Transport in Quantum Multi-barrier Systems as Random Walks on a Lattice

“…, N + 1. As explained in the Appendix A, this is the case ccr-quant.tex -19 marzo 2019 in the γ, L-continuum limit [10] with D = 0, namely, for N large enough when all the other parameters are kept fixed. The values of the hopping probabilities p n for several different values of the energy are plotted in Fig.…”

“…The potential E 0 , which was there called zero point energy in Ref. [10], will be used also here to discriminate various transport regimes. We now write the stationary Schrödinger equation in units such that the constant 2 /2m (m the mass of the electron) equals 1, denoting by a prime the space derivatives: As in Ref.…”

“…We now write the stationary Schrödinger equation in units such that the constant 2 /2m (m the mass of the electron) equals 1, denoting by a prime the space derivatives: As in Ref. [10], the solution corresponding to positive energy, E > 0, can be written as…”

“…In this appendix we consider the Kronig-Penney model introduced in the Section 2 in the case D = 0 (see the comment below (2.6)). As mentioned below the equation (2.4), see also [10], the γ, L-continuum limit is realized by keeping fixed all the parameters of the Kronig-Penney model but the number of barriers N , which tends to infinity. Recalling the matrix M defined in (2.15), following [10], we let Φ = Re(M 11 ) = cos(kδ) cosh(zγδ) + z 2 − k 2 2kz sin(kδ) sinh(zγδ) .…”

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A quantum finite multi-barrier system, with a periodic potential, is considered and exact expressions for its plane wave amplitudes are obtained using the Transfer Matrix method [10]. This quantum model is then associated with a stochastic process of independent random walks on a lattice, by properly relating the wave amplitudes with the hopping probabilities of the particles moving on the lattice and with the injection rates from external particle reservoirs.

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