R-matrix Formalism for Electron Scattering in Two Dimensions with Applications to Nanostructures with Quantum Dots

Abstract

We investigate the scattering phenomena in two dimensions produced by a general finite-range nonseparable potential. This situation can appear either in a Cartesian geometry or in a heterostructure with cylindrical symmetry. Increasing the dimensionality of the scattering problem, new processes as the scattering between conducting channels and the scattering from conducting to evanescent channels are allowed. For certain values of the energy called resonance energy, the transmission through the scattering region changes dramatically in comparison with an onedimensional problem. If the potential has an attractive character, even the evanescent channels can be seen as dips of the total transmission. The multi-channel current scattering matrix is determined using its representation in terms of the R-matrix. The resonant transmission peaks are characterized quantitatively through the poles of the current scattering matrix. Detailed maps of the localization probability density sustain the physical interpretation of the resonances. Our formalism is applied to a quantum dot in a two-dimensional electron gas and to a conical quantum dot embedded inside a cylindrical nanowire.

Computation of Nanowire Transport Properties

“…(5) to the matrix eigenproblem, (11) where E ⊥ = s 2 j 2 m k is the transversal eigenenergy corresponding to the mode m k r . 1, 5 Appropriate truncation of Eq.…”

“…The second step is to construct the scattering states which contain all the information necessary to calculate the reflection and transmission coefficients. 10,11 These two steps provide a natural description of all electron transport properties in nanodevices.…”

Parameter optimization of an electron ballistic switch in a quantum network model

“…where λ k m is dimensionless energy of k m th branch-channel (of mth channel in the branch Ω k ), ε is dimensionless energy of electron, C sw is extended current scattering matrix of the switch [22,23], S sw is extended scattering matrix of the switch [22, p. 155], F −1 (η) = 1/(1 + e −η ) is Fermi-Dirac integral of order −1, ε k F is dimensionless Fermi level in kth branch. Connection of the dimensionless quantities with dimensional ones is set by expressions…”

“…where λ k m is dimensionless energy of k m th branch-channel (of mth channel in the branch Ω k ), ε is dimensionless energy of electron, C sw is extended current scattering matrix of the switch [22,23], S sw is extended scattering matrix of the switch [22, p. 155…”

Calculation of Electron Transport in Branched Semiconductor Nanostructures Using Quantum Network Model

“…Any quantum network can be considered as one junction with a complex structure, and one of the mentioned above methods [15][16][17][18][19][20][21][22][23][24][25][26] can be used to calculate its scattering matrix. However, for calculations, it may be more convenient to find a scattering matrix of network using the rule of combining scattering matrices of its junctions [17,27].…”