Quantum electronic transport: Linear and nonlinear conductance from the Keldysh approach

This is a review of the derivation of the Landauer conductance using the Keldysh non-equilibrium Green's function (NEGF) formalism and the equations-of-motion (EOM) method. We consider the elastic quantum electronic transport through a multi-lead device and treat the conductor in the mean-field approximation. This is suitable for open quantum dots as well as for several molecular systems where charging effects are negligible. The focus of the presentation is to unveil the technical issues involved in the formalism. We show how the Landauer conductance emerges as a linear term in the current-voltage I-V characteristics and indicate how to go beyond this regime. We address the connection of the NEGF approach to recent developments in molecular transport and discuss the problems that arise when one tries to include interaction effects beyond the mean field.     

SymGF: a symbolic tool for quantum transport analysis and its application to a double quantum dot system

We report the development and an application of a symbolic tool, called SymGF, for analytical derivations of quantum transport properties using the Keldysh nonequilibrium Green's function (NEGF) formalism. The inputs to SymGF are the device Hamiltonian in the second quantized form, the commutation relation of the operators and the truncation rules of the correlators. The outputs of SymGF are the desired NEGF that appear in the transport formula, in terms of the unperturbed Green's function of the device scattering region and its coupling to the device electrodes. For complicated transport analysis involving strong interactions and correlations, SymGF provides significant assistance in analytical derivations. Using this tool, we investigate coherent quantum transport in a double quantum dot system where strong on-site interaction exists in the side-coupled quantum dot. Results obtained by the higher-order approximation and Hartree-Fock approximation are compared. The higher-order approximation reveals Kondo resonance features in the density of states and conductances. Results are compared both qualitatively and quantitatively to the experimental data reported in the literature.     

Nanoscale device modeling: the Green’s function method

The non-equilibrium Green’s function (NEGF) formalism provides a sound conceptual basis for the devlopment of atomic-level quantum mechanical simulators that will be needed for nanoscale devices of the future. However, this formalism is based on concepts that are unfamiliar to most device physicists and chemists and as such remains relatively obscure. In this paper we try to achieve two objectives: (1) explain the central concepts that define the ‘language’ of quantum transport, and (2) illustrate the NEGF formalism with simple examples that interested readers can easily duplicate on their PCs. These examples all involve a short n ^+  + – n ⁺ – n ^+  + resistor whose physics is easily understood. However, the basic formulation is quite general and can even be applied to something as different as a nanotube or a molecular wire, once a suitable Hamiltonian has been identified. These examples also underscore the importance of performing self-consistent calculations that include the Poisson equation. The I–V characteristics of nanoscale structures is determined by an interesting interplay between twentieth century physics (quantum transport) and nineteenth century physics (electrostatics) and there is a tendency to emphasize one or the other depending on one’s background. However, it is important to do justice to both aspects in order to derive real insights.     

Simulation of the electrical characteristics of a one-dimensional quantum dot array

A quantum dot array, consisting of Au dots, was prepared by the linear aggregation technique and assembled between two electrodes. We study the voltage–current characteristic of the quantum dot array, using a Non-Equilibrium Green’s Function (NEGF) model based on the Keldysh formalism. The results of our simulation and experimental data are compared. The simulated voltage–current curve is a reasonable fit with the measured data. It shows that the present model can be used to study quantum dot arrays. Furthermore, our results indicate that the electrical characteristics of an Au dot array are directly related to the coupling parameters.     

Disorder scattering in magnetic tunnel junctions: theory of nonequilibrium vertex correction

We report a first principles formalism and its numerical implementation for treating quantum transport properties of nanoelectronic devices with atomistic disorder. We develop a nonequilibrium vertex correction (NVC) theory to handle the configurational average of random disorder at the density matrix level so that disorder effects to nonlinear and nonequilibrium quantum transport can be calculated from atomic first principles in a self-consistent and efficient manner. We implement the NVC into a Keldysh nonequilibrium Green's function (NEGF) -based density functional theory (DFT) and apply the NEGF-DFT-NVC formalism to Fe/vacuum/Fe magnetic tunnel junctions with interface roughness disorder. Our results show that disorder has dramatic effects on the nonlinear spin injection and tunnel magnetoresistance ratio.     

Multisymplectic Geometry and Its Appiications for the Schrodinger Equation in Quantum Mechanics

Multisymplectic geometry for the Schrodinger equation in quantum mechanics is presented. This formalism of multisymplectic geometry provides a concise and complete introduction to the Schrodinger equation. The Schrodinger equation, its associated energy and momentum evolution equations, and the multisymplectic form are derived directly from the variational principle. Some applications are also explored.     

Numerical simulations of time-resolved quantum electronics

Numerical simulation has become a major tool in quantum electronics both for fundamental and applied purposes. While for a long time those simulations focused on stationary properties (e.g. DC currents), the recent experimental trend toward GHz frequencies and beyond has triggered a new interest for handling time-dependent perturbations. As the experimental frequencies get higher, it becomes possible to conceive experiments which are both time-resolved and fast enough to probe the internal quantum dynamics of the system. This paper discusses the technical aspects–mathematical and numerical–associated with the numerical simulations of such a setup in the time domain (i.e. beyond the single-frequency AC limit). After a short review of the state of the art, we develop a theoretical framework for the calculation of time-resolved observables in a general multiterminal system subject to an arbitrary time-dependent perturbation (oscillating electrostatic gates, voltage pulses, time-varying magnetic fields, etc.) The approach is mathematically equivalent to (i) the time-dependent scattering formalism, (ii) the time-resolved non-equilibrium Green’s function (NEGF) formalism and (iii) the partition-free approach. The central object of our theory is a wave function that obeys a simple Schrödinger equation with an additional source term that accounts for the electrons injected from the electrodes. The time-resolved observables (current, density, etc.) and the (inelastic) scattering matrix are simply expressed in terms of this wave function. We use our approach to develop a numerical technique for simulating time-resolved quantum transport. We find that the use of this wave function is advantageous for numerical simulations resulting in a speed up of many orders of magnitude with respect to the direct integration of NEGF equations. Our technique allows one to simulate realistic situations beyond simple models, a subject that was until now beyond the simulation capabilities of available approaches.     

Quantum thermal transport in nanostructures

In this colloquia review we discuss methods for thermal transport calculations for nanojunctions connected to two semi-infinite leads served as heat-baths. Our emphases are on fundamental quantum theory and atomistic models. We begin with an introduction of the Landauer formula for ballistic thermal transport and give its derivation from scattering wave point of view. Several methods (scattering boundary condition, mode-matching, Piccard and Caroli formulas) of calculating the phonon transmission coefficients are given. The nonequilibrium Green's function (NEGF) method is reviewed and the Caroli formula is derived. We also give iterative methods and an algorithm based on a generalized eigenvalue problem for the calculation of surface Green's functions, which are starting point for an NEGF calculation. A systematic exposition for the NEGF method is presented, starting from the fundamental definitions of the Green's functions, and ending with equations of motion for the contour ordered Green's functions and Feynman diagrammatic expansion. In the later part, we discuss the treatments of nonlinear effects in heat conduction, including a phenomenological expression for the transmission, NEGF for phonon-phonon interactions, molecular dynamics (generalized Langevin) with quantum heat-baths, and electron-phonon interactions. Some new results are also shown. We briefly review the experimental status of the thermal transport measurements in nanostructures.     

Bose-Einstein condensates beyond mean field theory: quantum backreaction as decoherence

We propose an experiment to measure the slow log(N) convergence to mean field theory (MFT) around a dynamical instability. Using a density matrix formalism instead of the standard macroscopic wave function approach, we derive equations of motion which go beyond MFT and provide accurate predictions for the quantum break time. The leading quantum corrections appear as decoherence of the reduced single-particle quantum state.     

Investigation of Schottky-Barrier carbon nanotube field-effect transistor by an efficient semi-classical numerical modeling

An efficient semi-classical numerical modeling approach has been developed to simulate the coaxial Schottky-barrier carbon nanotube field-effect transistor (SB-CNTFET). In the modeling, the electrostatic potential of the CNT is obtained by self-consistently solving the analytic expression of CNT carrier distribution and the cylindrical Poisson equation, which significantly enhances the computational efficiency and simultaneously present a result in good agreement to that obtained from the non-equilibrium Green's function (NEGF) formalism based on the first principle. With this method, the effects of the CNT diameter, power supply voltage, thickness and dielectric constant of gate insulator on the device performance are investigated.     

A two-dimensional domain decomposition technique for the simulation of quantum-scale devices

The simulation of realistically sized devices under the Non-Equilibrium Greens Function (NEGF) formalism typically requires prohibitive amounts of memory and computation time. In order to meet the rising computational challenges associated with quantum-scale device simulation we offer a 2-D domain decomposition technique. This technique is applicable to a large class of atomistic and spatial simulation problems. Considering a decomposition along both the cross section and length of the device, the framework presented in this work ensures efficient distribution of both memory and computation based upon the underlying device structure. As an illustration we stably generate the density of states and transmission, under the NEGF formalism, for the atomistic-based simulation of square 5 nm cross section silicon nanowires consisting of over one million atomic orbitals.     

Spin transport properties in a non-uniform quantum wire modulated by both Rashba and Dresselhaus spin–orbit couplings

Spin transport properties in a non-uniform quantum wire (QW) in the presence of both the Rashba and Dresselhaus spin–orbit couplings (SOCs) is investigated by using the non-equilibrium Green's function (NEGF) method combined with the Landauer Büttiker formalism. It is found that such a non-uniform quantum wire exhibits considerable spin polarization in its conductance in the influence of both the Rashba and Dresselhaus SOCs, and that the two SOCs' strengths strongly affect both the magnitude and sign of the electron spin polarization. Interestingly, the Rashba and Dresselhaus SOCs play the same modulating role in the electron spin polarization. The proposed nanostructure can potentially be utilized to devise an all-electrical spintronic device.     

Spacetime path formalism for massive particles of any spin

Earlier work presented spacetime path formalism for relativistic quantum mechanics arising naturally from the fundamental principles of the Born probability rule, superposition, and spacetime translation invariance. The resulting formalism can be seen as a foundation for a number of previous parametrized approaches to relativistic quantum mechanics in the literature. Because time is treated similarly to the three-space coordinates, rather than as an evolution parameter, such approaches have proved particularly useful in the study of quantum gravity and cosmology. The present paper extends the foundational spacetime path formalism to include massive, non-scalar particles of any (integer or half-integer) spin. This is done by generalizing the principle of translational invariance used in the scalar case to the principle of full Poincaré invariance, leading to a formulation for the non-scalar propagator in terms of a path integral over the Poincaré group. Once the difficulty of the non-compactness of the component Lorentz group is dealt with, the subsequent development is remarkably parallel to the scalar case. This allows the formalism to retain a clear probabilistic interpretation throughout, with a natural reduction to non-relativistic quantum mechanics closely related to the well-known generalized Foldy-Wouthuysen transformation.     

低维纳米材料量子热输运与自旋热电性质 ——非平衡格林函数方法的应用

低维材料不断涌现的新奇性质吸引着科学研究者的目光. 除了电子的量子输运行为之外, 人们也陆续发现和确认了热输运中显著的量子行为, 如 热导低温量子化、声子子带、尺寸效应、瓶颈效应等. 这些小尺度体系的热输运性质可以很好地用非平衡格林函数来描述. 本文首先介绍了量子热输运的特性、声子非平衡格林函数方法及其在低维纳米材料中的研究进展; 其次回顾了近年来在 一系列低维材料中发现的热-自旋输运现象. 这些自旋热学现象展现了全新的热电转换机制, 有助于设计新型的热电转换器件, 同时也给出了用热产生自旋流的新途径; 最后介绍了线性响应理论以及在此理论框架下结合声子、电子非平衡格林函数方法进行的一些有益的探索. 量子热输运的研究对热效应基础研究以及声子学器件、能量转换器件的发展有着不可替代的重要作用.     

Orientational Order Parameters for Arbitrary Quantum Systems

The concept of quantum-mechanical nematic order, which is important in systems such as superconductors, is based on an analogy to classical liquid crystals, where order parameters are obtained through orientational expansions. This method is generalized to quantum mechanics based on an expansion of Wigner functions. This provides a unified framework applicable to arbitrary quantum systems. The formalism recovers the standard definitions for spin systems. For Fermi liquids, the formalism reveals the nonequivalence of various definitions of the order parameter used in the literature. Moreover, new order parameters for quantum molecular systems with low symmetry are derived, which cannot be properly described with the usual nematic tensors.     

Applied Bohmian mechanics

Bohmian mechanics provides an explanation of quantum phenomena in terms of point-like particles guided by wave functions. This review focuses on the use of nonrelativistic Bohmian mechanics to address practical problems, rather than on its interpretation. Although the Bohmian and standard quantum theories have different formalisms, both give exactly the same predictions for all phenomena. Fifteen years ago, the quantum chemistry community began to study the practical usefulness of Bohmian mechanics. Since then, the scientific community has mainly applied it to study the (unitary) evolution of single-particle wave functions, either by developing efficient quantum trajectory algorithms or by providing a trajectory-based explanation of complicated quantum phenomena. Here we present a large list of examples showing how the Bohmian formalism provides a useful solution in different forefront research fields for this kind of problems (where the Bohmian and the quantum hydrodynamic formalisms coincide). In addition, this work also emphasizes that the Bohmian formalism can be a useful tool in other types of (nonunitary and nonlinear) quantum problems where the influence of the environment or the nonsimulated degrees of freedom are relevant. This review contains also examples on the use of the Bohmian formalism for the many-body problem, decoherence and measurement processes. The ability of the Bohmian formalism to analyze this last type of problems for (open) quantum systems remains mainly unexplored by the scientific community. The authors of this review are convinced that the final status of the Bohmian theory among the scientific community will be greatly influenced by its potential success in those types of problems that present nonunitary and/or nonlinear quantum evolutions. A brief introduction of the Bohmian formalism and some of its extensions are presented in the last part of this review.     

Star products and geometric algebra

The formalism of geometric algebra can be described as deformed super analysis. The deformation is done with a fermionic star product, that arises from deformation quantization of pseudoclassical mechanics. If one then extends the deformation to the bosonic coefficients of superanalysis one obtains quantum mechanics for systems with spin. This approach clarifies on the one hand the relation between Grassmann and Clifford structures in geometric algebra and on the other hand the relation between classical mechanics and quantum mechanics. Moreover it gives a formalism that allows to handle classical and quantum mechanics in a consistent manner.     

Nonequilibrium Green’s function method for quantum thermal transport

This review deals with the nonequilibrium Green’s function (NEGF) method applied to the problems of energy transport due to atomic vibrations (phonons), primarily for small junction systems. We present a pedagogical introduction to the subject, deriving some of the well-known results such as the Laudauer-like formula for heat current in ballistic systems. The main aim of the review is to build the machinery of the method so that it can be applied to other situations, which are not directly treated here. In addition to the above, we consider a number of applications of NEGF, not in routine model system calculations, but in a few new aspects showing the power and usefulness of the formalism. In particular, we discuss the problems of multiple leads, coupled left-right-lead system, and system without a center. We also apply the method to the problem of full counting statistics. In the case of nonlinear systems, we make general comments on the thermal expansion effect, phonon relaxation time, and a certain class of mean-field approximations. Lastly, we examine the relationship between NEGF, reduced density matrix, and master equation approaches to thermal transport.