[Paper Review] Computational quantum transport

Xavier Waintal, Michael Wimmer+6 more|arXiv (Cornell University)|Jul 23, 2024

Quantum and electron transport phenomena4 citations

TL;DR

This paper provides a comprehensive, pedagogical review of computational quantum transport, unifying the scattering and Green’s function formalisms for coherent electron transport in nanoscale systems. It derives both approaches from first principles, proves their equivalence, and systematically analyzes numerical algorithms—especially recursive Green’s functions—offering stable, efficient methods for calculating conductance, current, and noise in quantum devices.

ABSTRACT

This review is devoted to the different techniques that have been developed to compute the coherent transport properties of quantum nanoelectronic systems connected to electrodes. Beside a review of the different algorithms proposed in the literature, we provide a comprehensive and pedagogical derivation of the two formalisms on which these techniques are based: the scattering approach and the Green's function approach. We show that the scattering problem can be formulated as a system of linear equations and that different existing algorithms for solving this scattering problem amount to different sequences of Gaussian elimination. We explicitly prove the equivalence of the two formalisms. We discuss the stability and numerical complexity of the existing methods.

Motivation & Objective

• To provide a unified, pedagogical derivation of the scattering and Green’s function formalisms for quantum transport in nanosystems.
• To establish the mathematical equivalence between the scattering matrix and Green’s function approaches.
• To analyze the numerical stability and computational complexity of existing algorithms for solving the quantum transport problem.
• To systematize the formulation of lead modes, scattering problems, and self-energy embeddings in discrete lattice models.
• To enable accurate and efficient simulation of conductance, current, and noise in realistic nanoelectronic devices.

Proposed method

• Derives the scattering formalism by formulating the scattering problem as a system of linear equations, solved via Gaussian elimination sequences.
• Uses eigendecomposition of the lead Hamiltonian to separate propagating and evanescent modes, enabling stable treatment of non-invertible hopping matrices.
• Applies the Fisher–Lee relation to connect the scattering matrix to the retarded Green’s function of the scattering region.
• Develops a stable formulation of the lead problem using the generalized S-matrix and proper mode normalization, even for non-orthogonal basis sets.
• Introduces the recursive Green’s function (RGF) algorithm as a numerically stable method for computing the Green’s function of large systems.
• Applies the Keldysh formalism to compute non-equilibrium observables such as current, noise, and higher-order correlations.

Figure 1: Sketch of a quasi one-dimensional wire of width $\mathcal{W}$ . The shaded scattering region in the middle symbolizes an arbitrary potential that scatters the plane waves coming from the left into reflected ( $r$ ) and transmitted ( $t$ ) wave.

Experimental results

Research questions

• RQ1 How can the scattering formalism and the Green’s function formalism be rigorously derived and shown to be equivalent in quantum transport?
• RQ2 What are the numerical stability and computational complexity trade-offs of different algorithms for solving the quantum transport problem?
• RQ3 How can the presence of evanescent and propagating modes in leads be systematically handled in discrete models with non-invertible hopping matrices?
• RQ4 What is the role of discrete symmetries (e.g., time-reversal, chiral) in constructing proper lead modes and simplifying the transport problem?
• RQ5 How can the recursive Green’s function method be generalized and stabilized for use in realistic nanoscale devices with complex geometries and interactions?

Key findings

• The scattering problem is mathematically equivalent to a system of linear equations, and all existing algorithms correspond to different sequences of Gaussian elimination.
• The equivalence between the scattering matrix and the Green’s function formalism is rigorously proven via the Fisher–Lee relation and mode decomposition.
• The recursive Green’s function (RGF) algorithm is shown to be numerically stable and efficient for large systems, especially when combined with proper mode normalization.
• The paper provides a systematic method to handle non-orthogonal basis sets by absorbing velocity and current normalization factors into the mode vectors.
• The generalized scattering matrix $S_{ ext{tp}}$ includes both propagating and evanescent modes, enabling accurate treatment of tunneling and evanescent decay in leads.
• The formalism is extended to include many-body effects, thermoelectric responses, spin currents, and quantum noise, all within a unified framework.

Figure 2: Examples of conductance calculations in a two dimensional square lattice. For each example the inset shows the tight-binding system that has been used with the scattering region in black and the first two layers of semi-infinite electrodes in red. Upper left: conductance quantization in a