[Paper Review] Generalized multi-terminal decoherent transport: Recursive algorithms and applications to SASER and giant magnetoresistance

Carlos J. Cattena, Lucas J. Fernández-Alcázar+3 more|Repositorio Digital de la UNC (National University of Cordoba)|Nov 9, 2013

Quantum and electron transport phenomena4 citations

TL;DR

This paper generalizes the D'Amato-Pastawski (DP) model for decoherent quantum transport to multi-terminal systems using recursive algorithms based on decimation procedures. It enables efficient computation of Green's functions, conductance, and voltage profiles in banded Hamiltonians, successfully reproducing key features of SASER efficiency degradation due to decoherence and the classical limit of giant magnetoresistance via a single decoherence rate parameter.

ABSTRACT

Decoherent transport in mesoscopic and nanoscopic systems can be formulated in terms of the D'Amato-Pastawski (DP) model. This generalizes the Landauer-Büttiker picture by considering a distribution of local decoherent processes. However, its generalization for multi-terminal setups is lacking. We first review the original two-terminal DP model for decoherent transport. Then, we extend it to a matrix formulation capable of dealing with multi-terminal problems. We also introduce recursive algorithms to evaluate the Green's functions for general banded Hamiltonians as well as local density of states, effective conductances and voltage profiles. We finally illustrate the method by analyzing two problems of current relevance. 1) Assessing the role of decoherence in a model for phonon lasers (SASER). 2) Obtaining the classical limit of Giant Magnetoresistance from a spin-dependent Hamiltonian. The presented methods should pave the way for computationally demanding calculations of transport through nanodevices, bridging the gap between fully coherent quantum schemes and semiclassical ones.

Motivation & Objective

• To extend the two-terminal D'Amato-Pastawski (DP) model for decoherent transport to multi-terminal configurations.
• To develop recursive algorithms for computing Green’s functions of general banded Hamiltonians, enabling efficient evaluation of local density of states, effective conductances, and voltage profiles.
• To provide a computationally efficient framework bridging fully coherent quantum transport and semiclassical descriptions in nanodevices.
• To demonstrate the method on two relevant problems: decoherence effects in phonon lasers (SASER) and the classical limit of giant magnetoresistance (GMR).

Proposed method

• Generalizes the DP model using a matrix formulation to handle multi-terminal setups, introducing a unified notation for transmittance and chemical potential relations.
• Develops recursive decimation procedures for banded Hamiltonians, enabling computation of the full Green’s function matrix via matrix continued fractions.
• Derives a compact matrix equation for effective transmittance $\tilde{\mathbb{T}}$ in the generalized multi-terminal scheme, leveraging symmetry and block structure.
• Applies the method to compute local chemical potentials, current distributions, and voltage profiles using self-consistent solutions of the generalized Landauer-Büttiker equations.
• Uses block tridiagonal structure of Hamiltonians to optimize computation, relying on recursive relations for non-diagonal blocks of the Green’s function.
• Introduces a single parameter, the local decoherence rate $\Gamma_{\phi i}$, to interpolate between quantum coherent and classical incoherent transport regimes.

Experimental results

Research questions

• RQ1 How can the D'Amato-Pastawski model for decoherent transport be generalized to multi-terminal systems?
• RQ2 What recursive algorithmic approach enables efficient computation of Green’s functions and transport properties in banded Hamiltonians?
• RQ3 How does decoherence affect the efficiency of a phonon laser (SASER) in a quantum dot model?
• RQ4 Can the classical limit of giant magnetoresistance (GMR) be reproduced using a single-parameter decoherence model within a Hamiltonian framework?
• RQ5 To what extent do quantum interference effects persist in large-scale molecular electronic systems under decoherence?

Key findings

• Decoherence suppresses antiresonances in the I-V curve of a SASER model, leading to degradation of the contrast between the valley and satellite peak, which limits device efficiency.
• The multi-terminal DP model successfully reproduces the spin-dependent chemical potential drop and current inversion on the length scale $L_{sd}$, consistent with established GMR behavior.
• The model captures the transition from coherent tunneling to incoherent hopping in large molecular systems by tuning the local decoherence rate $\Gamma_{\phi i}$.
• The recursive algorithm efficiently computes the full Green’s function matrix by reducing the problem to matrix continued fractions for diagonal blocks and recursive relations for off-diagonal blocks.
• The method enables accurate computation of effective conductances and voltage profiles in multi-terminal, spin-dependent, and decoherent nanoscale systems with reduced computational cost.
• The unified matrix formulation of the generalized Landauer-Büttiker equations allows for self-consistent solutions of chemical potentials and currents in complex multi-channel transport problems.