Computational perspective on recent advances in quantum electronics: from electron quantum optics to nanoelectronic devices and systems

Section 2 provides a short overview of the key computational methods identified in the research reviewed in the later sections. A particular focus is on the methods which are applied the most, such as for calculating the electronic structure and the electron quantum transport. However, we also discuss other, alternative methods which see considerable use. We do not express a preference of one method over the other but merely reflect the extent of use in the research presented in this work.

Section 3 represents the first of the review sections and covers the previously highlighted area of electron quantum optics and the flying charge qubit systems. The section starts off by discussing the principal enabling technology: single-electron sources. In particular, mesoscopic capacitors, electron pumps, surface acoustic waves (SAWs), levitons, and locally modulated quantum Hall edge state sources are covered. This is followed by a discussion of essential waveguides (which enable one to build coherent quantum circuits) and of electron transport dynamics. Finally, interferometers, a core focus of electron quantum optics, and tomography, spectroscopy, and detection methods are highlighted.

Section 4 provides an overview of computational research into quantum dots and the predominant approach for realizing quantum information processing: spin qubits. Although spin transport itself is not at the focus of this review, a few selected works are presented where the electron wave nature is of particular relevance.

Section 5 highlights research in superconducting junctions, in particular normal-superconductor interfaces. These interfaces are particularly relevant for Josephson junctions and superconducting qubits.

Section 6 delves into the particularly strong focus area of 2D materials and topological materials and systems. In addition, light–matter interaction effects in materials are highlighted as are combinations of materials in the form of heterostructures; the latter enable one to further tune material properties. Finally, research into molecular junctions and systems is discussed, where typically a specific molecule (chain) is considered between contacts.

Section 7 provides an overview into research on next-generation FETs. The focus is of course on GAA designs, such as, nanoribbon FETs and 2D FETs. The latter further underlines the importance of 2D materials for nanoelectronic devices. The section is concluded by research on tunneling and other types of FETs, such as FinFETs and ultra-thin body FETs.

Section 8 discusses research on one of the core topics of quantum electronics, quantum cascade LASER (QCL) devices. To this day, research on QCL devices is a highly active field, particularly focusing on THz operation.

Finally, section 9 highlights research where electron transport is extended to include heat transfer into the electron transport modeling picture. Focus areas include illuminating fundamental transfer processes but also thermoelectric effects.

Simulations in quantum electronics can be broadly divided into simulations of electronic structures and quantum transport simulations. As indicated before, sometimes these simulations are combined, for instance, in an electronic structure simulation (e.g., using DFT) which is followed by a transport simulation (e.g., non-equilibrium Green's function (NEGF) approach). Many examples of such combinations are available and are discussed in the technological sections below.

Looking at the underlying simulation models, specification of models for quantum transport is complex, especially for realistic devices [22, 23]. The quantum mechanical states are coupled with the environment. The system is driven out of equilibrium by an applied bias. States get filled through the contact regions and through scattering interactions. Electrons interact with phonons over long distances. Some systems are highly inhomogeneous due to different materials used for active layers. Typically, parts of the nanosystem have reduced dimensionality. Proper open boundary conditions have to be applied and non-normalizable states have to be treated numerically. Because of the thus resulting modeling complexity, practical transport simulations often contain semi-empirical approximations. A typical system which is regularly considered in quantum electron transport consists in a nanosystem between electrodes.

Full inclusion of many-body interactions in the simulation is in general computationally intractable. To formulate computationally tractable models a closure problem has to be solved, which is discussed in the following after covering computational complexity.

2.2.1. Computational complexity

Models in computational quantum electronics are based in many-body quantum physics. We discuss computational costs using the example of a many-body model Hamiltonian describing electrons interacting through Coulomb force which is given in first quantization (in atomic units) as

$$\begin{equation}H=\sum\limits _{i=1}^{N}-\frac{1}{2}{\nabla }_{i}^{2}+\sum\limits _{i=1}^{N}\enspace \sum\limits _{j=i+1}^{N}\frac{1}{\vert {r}_{i}-{r}_{j}\vert }.\end{equation} \tag{ 1 }$$

Because the Coulomb interaction involves two electrons this problem cannot be reduced to a single-body problem. The electron wave-functions have domain ${\mathbb{R}}^{3N}$. If the problem is discretized using K grid points in ${\mathbb{R}}^{3}$ the number of unknowns scales as K^N . A direct solution becomes computationally intractable already for low numbers N ∼ 10. However, often the system of interest consists of a large number of electrons or even worse one is interested in a limit N → ∞. Many computational methods work by effectively reducing the problem in some way to the case of independent particles.

2.2.2. The closure problem

On the highest level, models for quantum electronics are formulated in second quantization (Fock space) where spin and antisymmetry of the wave function are naturally included. If the underlying one-particle Hilbert space is modelled by a finite-dimensional Hilbert space with dimension K then the corresponding fermionic Fock space has dimension 2^K . In realistic applications this is not numerically tractable and full knowledge of the many-body wave function is in general not possible.

A natural idea to extract information about the system is to solve for correlation functionals from which the observables of interest can be calculated. Physical observables are calculated as ensemble averages over the density operator $\hat{\rho }$

$$\begin{equation}O(t)=\mathrm{Tr}[\hat{\rho }{\hat{O}}_{\mathrm{H}}(t)]=\langle {\hat{O}}_{\mathrm{H}}(t)\rangle .\end{equation} \tag{ 2 }$$

Here the subscript H denotes the Heisenberg picture where the state is time-independent and the time dependence is carried by the operators. A striking example is charge density n(r) which is the basic quantity in DFT.

If two or more operators at (possibly) different times are involved the operator trace defines a correlation functional

$$\begin{equation}{C}_{AB}(t,{t}^{\prime })=\mathrm{Tr}[\hat{\rho }{\hat{A}}_{\mathrm{H}}(t){\hat{B}}_{\mathrm{H}}({t}^{\prime })]=\langle {\hat{A}}_{\mathrm{H}}(t){\hat{B}}_{\mathrm{H}}({t}^{\prime })\rangle .\end{equation} \tag{ 3 }$$

However, formulating equations in terms of a few selected correlation functionals leads to a closure problem. Due to the interaction between particles the equations for a given fixed finite set of functionals will in general contain other functionals outside the fixed set and the resulting system is underdetermined (not closed). Closure problems emerge in many places in theoretical physics. We mention the closure problem for the BBGKY-hierarchy and solution methods for the Boltzmann equation, which try to reduce the distribution to a few of its moments. Hence, a useful way to classify models used in computational quantum electronics is the set of basic quantities (functionals) in which the theory is formulated together with a method of closure (see discussion of NEGF below).

For the sake of completeness, we note that in selected cases, such as the jellium model, an essentially exact calculation (i.e., without a closure) of specific ground-state functionals may be possible using quantum Monte Carlo methods [24, 25]. However, these methods are computationally far too expensive to use in all but the simplest cases [26].

Green's functions are a general mathematical technique to derive explicit solutions to differential equations, see [27, 28] and especially [29]. Many-body Green's functions provide a powerful formalism as basically all models can be specified in terms of many-body Green's functions and second quantization.

Several of the calculation techniques in use for many-body Green's functions have been originally developed in quantum field theory [30, 31]. The classical text book by Kadanoff and Baym from 1962 [32] on Green's functions in and out of equilibrium has recently been republished in an annotated edition [33] which is highly recommended. Other early texts devoted especially to the application of Green's functions in solid state physics include [34, 35].

We want to mention that the full power of Green's functions is not always used, even if the method might be superficially denoted as NEGF. For example, the quantum transmitting boundary method (theoretically corresponding to a single-particle Schrödinger equation) [36] can be specified using a self-energy for the open boundary conditions in NEGF. Put differently, a simple model may still be specified in a powerful formalism.

2.3.1. One-particle Green's functions

A central role in second quantization is played by the field operators $\hat{\psi }(r)$, ${\hat{\psi }}^{{\dagger}}(r)$ which annihilate/create a particle at location r. The Green's functions in statistical quantum mechanics are a linear combination of a small set of basic correlation functions involving the field operators. The simplest example is the single particle lesser Green's function

$$\begin{equation}{G}^{< }(r,t;{r}^{\prime },{t}^{\prime })=\frac{i}{\hslash }\langle {\hat{\psi }}_{\mathrm{H}}^{{\dagger}}({r}^{\prime },{t}^{\prime }){\hat{\psi }}_{\mathrm{H}}(r,t)\rangle \end{equation} \tag{ 4 }$$

from which the electron density is calculated as [37, p 153]

$$\begin{equation}n(r,t)=-i\hslash {G}^{< }(r,t;r,t).\end{equation} \tag{ 5 }$$

Most expectation values of interest can be calculated from the knowledge of the basic Green's functions. Under steady-state conditions, the Green's functions depend only on the time difference τ = t − t'. The two-times Green's functions allow the study of time-dependent and time-nonlocal phenomena such as retarded interactions and memory.

2.3.2. Higher-order Green's functions

To illustrate the previously mentioned closure problem (section 2.2.2) in the case of Green's functions we discuss the equation of motion for the one-particle Green's function with Coulomb interaction v(r − r').

For the one-particle Green's function g(x, x') we get the equation [29, p 255]

$$\begin{align}\hfill & \left(i\hslash \frac{\partial }{\partial t}+\frac{{\hslash }^{2}}{2m}{\nabla }_{r}^{2}\right)g(x,{x}^{\prime })\hfill \\ \hfill & \hspace{25.0pt}=\delta (x-{x}^{\prime })-i\hslash \int {\mathrm{d}}^{4}{x}_{1}v(r-{r}_{1}){g}_{2}(x,{x}_{1};{x}^{\prime },{x}_{1}^{+}){\vert }_{{t}_{1}=t}\hfill \end{align} \tag{ 6 }$$

where we used the shorthand x = (r, t) etc. Here, ${x}_{1}^{+}$ denotes (r₁, t₁ + s) as s → 0⁺. The two-particle Green's function g₂ entering above is defined as

$$\begin{equation}{g}_{2}({x}_{1},{x}_{2};{x}_{1}^{\prime },{x}_{2}^{\prime })=-\frac{1}{{\hslash }^{2}}\langle T[\hat{\psi }({x}_{1})\hat{\psi }({x}_{2});{\hat{\psi }}^{{\dagger}}({x}_{2}^{\prime }){\hat{\psi }}^{{\dagger}}({x}_{1}^{\prime })]\rangle \end{equation} \tag{ 7 }$$

with T the (fermionic) time-ordering operator. Therefore, in the presence of a pairwise interaction the equation of motion for the one-particle Green's function g involves the two-particle Green's function g₂. The equation for the two-particle Green's function involves the three-particle Green's function and so on.

If an infinite number of higher-order Green's functions is included we see that the Green's function formalism reformulates the full problem. However, the attractiveness of the method is based on the fact that perturbative methods (Feynman diagrams [38]) can be used which allow the problem to be attacked in a systematic approximative way. We remark that the properties of the perturbation expansion are mostly unknown. The series might diverge or the convergence is only asymptotic [39].

2.3.3. Equilibrium Green's functions (Matsubara formalism)

At finite temperature under thermodynamic equilibrium the density operator $\hat{\rho }$ is given as

$$\begin{equation}\hat{\rho }=\frac{{\text{e}}^{-\beta \hat{K}}}{\mathrm{Tr}[{\text{e}}^{-\beta \hat{K}}]}\end{equation} \tag{ 8 }$$

where we defined $\hat{K}=\hat{H}-{E}_{\mathrm{F}}\hat{N}$ with the number operator $\hat{N}$ and β = 1/k_B T.

Time dependency is described by the operator ${\text{e}}^{-\text{i}\hat{H}t}$ while an operator ${\text{e}}^{-\beta \hat{H}}$ for real β is contained in the density operator. This observation motivates the study of the analytic continuation of Green's functions for imaginary time.

Important quantities, which can be calculated from the knowledge of the equilibrium one-particle Green's function, are the spectral density function, the density of states function, conductivity, and dielectricity (linear response functions). Peaks in the analytic continuation of the spectral function give rise to an interpretation as quasiparticles.

2.3.4. Nonequilibrium Green's functions (Keldysh formalism)

Electronic structure methods stem from the field of quantum chemistry and are used to calculate the ground-state properties of condensed systems, e.g., exact ground state density and energy. In the following, we provide a short overview of the key methods in this area. For a recent review of electronic structure methods see [10].

2.4.1. Density functional theory

DFT is a method to calculate the electronic structure of atoms, molecules, and solids. The method is highly popular as it appears to occupy a sweet spot in terms of trade off between quality of results and computational costs.

The basic unknown quantity in DFT is the one-particle observable n(r). Conceptually, DFT is a mean-field many-body theory of the ground state. The potential used in DFT is the sum of external potentials (determined by the elemental composition of the system) and an effective potential which represents the exchange–correlation energy (i.e., electron–electron interaction beyond mean-field; respecting the Pauli exclusion principle). Here, the closure problem (section 2.2.2) emerges in the form of the exchange–correlation potential V_XC whose exact form is unknown. The effective single-particle potential can be written as

$$\begin{equation}{V}_{\mathrm{s}}(r)={V}_{\mathrm{e}\mathrm{x}\mathrm{t}}(r)+\int {\mathrm{d}}^{3}{r}^{\prime }\frac{n({r}^{\prime })}{\vert r-{r}^{\prime }\vert }+{V}_{\mathrm{X}\mathrm{C}}[n(r)].\end{equation} \tag{ 9 }$$

The ground state of this system has to be calculated in a self-consistent way which results in a non-linear system of equations coupling the one-particle wave functions.

DFT is sometimes referred to as the standard model for periodic solids [55]. Considering time-dependent DFT, there DFT is used for simulating electronic excitation processes [56]. Due to the popularity of the method the literature on DFT is extensive and without a claim of completeness, the reader is referred to textbooks, e.g., [57–62], and to reviews, e.g., [10, 63, 64]. Some examples for DFT simulation tools are, e.g., VASP [65], QuantumATK [66], DFTB+ [67], CP2K [68], Atomistix [52].

2.4.2. GW method

2.4.3. Tight-binding

We consider here quantum transport methods which solve a transport equation to describe the electron transport within any kind of electronic device or system. Particularly challenging yet realistic scenarios include interactions with the environment via, for instance, contacts, phonons, or electromagnetic fields. In the following, we give an overview of the key full quantum transport methods which are capable of describing the wave nature of electrons. We thus omit semi-classical approaches. For in-depth discussions of quantum transport we refer the reader to standard text books, such as [3, 22, 23, 77–79] in addition to the text books mentioned in section 2.3.4. We remind the reader that the NEGF formalism is also used to describe electron transport. The corresponding discussion is provided in section 2.3 and thus NEGF is not further mentioned here.

2.5.1. Scattering matrix

2.5.2. Master equations and reduced density operators

2.5.3. Wigner function

Historically, the Wigner function has been introduced for the analysis of quantum corrections to classical stochastic mechanics [91] and thus has been derived from operator mechanics. Later the Wigner function formalism was established as an independent formulation of quantum mechanics based on the Moyal bracket and the star-product, which is denoted as quantum mechanics in phase space [92–95].

The Wigner function is a real quasi-distribution in phase space [96]. The formalism transfers many concepts and notions from classical to quantum mechanics, such as the aforementioned phase space but also the distribution function and the way of calculating the physical average $\bar{A}$

$$\begin{equation}\bar{A}=\frac{1}{2\pi \hslash }\iint {A}_{\mathrm{w}}(x,p){f}_{\mathrm{w}}(x,p)\mathrm{d}p\enspace \mathrm{d}x.\end{equation} \tag{ 10 }$$

Here A_w denotes the Weyl transform of the classical observable A(x, p) and the Wigner function f_w plays the role of the classical probability distribution f_cl(x, p).

In contrast to the classical Boltzmann distribution the Wigner function may take on negative values which is interpreted as an indication of quantum effects. As the formalism is close to a classical phase-space description it allows for mixed quantum/semi-classical modelling.

Many applications of the Wigner function (also outside of the electronic scope) are demonstrated in monographs [5, 48, 97] and in recent reviews [7, 95]. Using a Wigner function approach transport can be analysed in terms of signed particles and stochastic jump processes, hence Monte Carlo methods may be employed for simulation (see [79] for a recent monograph on stochastic approaches to electron transport in micro- and nanostructures). More recent advancements concern, for example, an extension of the theoretical framework for Monte Carlo, accessible descriptions of general electromagnetic fields [98], handling of boundary conditions [99], and an operator splitting spectral method for a Wigner–Poisson system [100]. A 1D/2D signed-particle Wigner transport simulator is available via ViennaWD [101, 102]. For non-trivial interaction mechanisms a closure problem exists in the Wigner formalism [78, p 98]. Other very recent advances of Wigner function methods for electronics (and photonics, but not further discussed here) can be found in a recent special issue [103] and cover, among others, handling of boundary conditions [104–106] and modelling and solution approaches [107–113].

2.5.4. Schrödinger–Poisson

Here we collectively mention a few other methods which are also used by the community. Among the methods is the so-called wave function approach [119–121]. This is yet another method to study time-resolved quantum electronics. Another method is the single-electron approach which was proposed as an extension of the Landauer approach for time-dependent drive [122, 123]. Tensor networks can be used as representations of quantum many-body states based on their local entanglement structure [124, 125]. Bayesian formalism can be applied to simulate measurement backaction in an electron detection setting [126]. A program for calculating maximally-localised Wannier functions from a set of Bloch states is available by Wannier90 [127]. Neural networks can be used to predict transport properties of a quasi-one-dimensional (1D) tight-binding model with disordered on-site energies [128]. Finally, Bohmian mechanics can be used to describe electron quantum transport [20, 129, 130].

As indicated before, a major milestone for coherent electronics was the development of coherent single-electron sources. In essence, three different approaches emerged: the first is based on the discrete level spectrum or Coulomb blockade effect in strongly confined systems. Realizations have been made via mesoscopic capacitors in the quantum Hall regime (e.g., [15]), superconducting turnstiles (e.g., [133]), electron pumps (e.g., [134]), and sound waves (e.g., [135]). The second uses Lorentzian-shaped voltage pulses to generate so-called levitons (e.g., [136]). And the third approach utilizes the local gate modulation of a quantum Hall edge state for noiseless single-particle emission on top of the Fermi sea (e.g., [137]). The review of Bäuerle et al provides an excellent overview of different single-electron source technologies [6]. In what follows, we highlight recent work in these areas, underlining the highly active field of research.

3.1.1. Mesoscopic capacitors

Mesoscopic capacitors have been around since many decades [138–140]. In particular, Fève et al applied the mesoscopic capacitor concept in 2007 to implement an on-demand coherent single-electron source by periodically driving a quantum dot [15]. In essence, the approach is based on driving an RC circuit by alternating-current (AC) voltage so as to limit the charge and discharge of the capacitor to a single elementary charge. The generated electron and hole is then emitted into a ballistic quantum Hall channel.

Selection of recent computational research: Yin used a scattering matrix approach to introduce a general method to extract the wave function of quasi-particles (i.e., electrons and/or holes in the Fermi sea) [141]. The excitation probability and the one-body wave function can be derived from the scattering matrix. The author used the approach to compare the characteristics of sources based on quantum dots and on an alternative approach based on quantum point contacts, showing clear differences in the differently generated particle features.

Wagner et al introduced a theoretical model of a single-particle emitter of charge pulses that uses a quantum dot coupled to a quantum Hall edge state by using a generalized master equation approach [142]. The authors used the model to highlight the Coulomb interactions between the dot and the chiral quantum Hall edge caused a destruction of precise charge quantization in the emitted wave packet. The research thus questions the viability of this particular setup as a single-electron source of quantized charge pulses.

Gurvitz used the single-electron approach to investigate the time-dependent electron current through a quantum dot under external drive, which is coupled to leads at zero bias (one lead as in the case of single-electron sources) [123]. A simplified formula describing the time-dependent current through the quantum dot is derived and investigated by using different driving pulse shapes. It was found that the current is indeed time-dependent at steady-state, but is zero if averaged over a time period for any external drive.

Particularly interesting work focuses on extending the work of single-electron sources to multi-electron sources. Such multi-electron sources would enable entirely new experiments and applications, such as quantum tomography (section 3.4) protocols. Based on previous work (e.g., [143]), in [144], Moskalets et al used Floquet scattering theory to describe composite two-particle sources composed of several single-particle emitters connected in series to a chiral waveguide in the (non-)adiabatic regime, see figure 1. The flexible design allows for more than two sources and for mixing source types, such as mixing a leviton source (section 3.1.4) with a mesoscopic capacitor. In follow up work, emission from orbitally degenerate quantum levels that arise in quantum dots with a nontrivial ring topology were discussed using an excess correlation function of electrons [145]. Various ring setups with one or two leads and with(out) a magnetic field were analyzed. It was shown that degeneracy provides additional flexibility for single- and two-electron emission.

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3.1.2. Electron pumps

Electron pumps have been originally proposed in 1991 by Kouwenhoven et al [146] and are in essence dynamic semiconductor quantum dots. Since the beginnings, the accuracy of electron pumps has been a challenge [147]. Over the years, the initial design has been extended. For instance, using one [148, 149] or two [134] voltage-controlled barrier(s).

Selection of recent computational research: Yamahata et al shed light on the internal electron dynamics in a silicon single-electron pump with picosecond resolution by numerically solving the time-dependent Schrödinger equation [150]. Together with experimental investigations the authors were able, among others, to show that a non-adiabatically excited electron spatially oscillates quantum coherently at 250 GHz inside the source at 4.2 K.

Restrepo et al studied electron pumping in the strong coupling and non-Markovian regime [151]. The authors investigated a single quantum dot with periodically modulated energy and tunneling amplitudes. For the high-frequency regime, Floquet theory has been used to accurately treat the time dependencies, ultimately obtaining a nonsecular master equation. A rectification of the pumped current in the high-frequency regime has been observed, see figure 2.

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Emary et al theoretically studied the relaxation of hot quantum-Hall edge-channel electrons under the emission of both acoustic and optical phonons using a master equation approach [152]. The authors consider the electron energy distributions and detector arrival times and are thus able to determine the emission rate of optical phonons.

3.1.3. Surface acoustic waves

In 1953 Parmenter predicted that SAWs can drive a current through a conductor [153]. The experimental realization followed four years later by Weinreich and White [154]. The fundamental property of electron transport in SAW minima lead to the interpretation of moving quantum dots for wavelengths $\leqslant \;1\enspace \mu$m. It took almost 60 years until SAWs have been successfully used as single-electron source [135] (at the same time other important advances were made in the context of SAW-based quantum dot charge transfer [155]). For excellent recent reviews on SAWs see [156, 157].

Selection of recent computational research: Takada et al demonstrated a SAW-driven single-electron source that is triggered by a picosecond voltage pulse in the context of electron transfer between distant quantum dots [158]. The authors outline the mechanisms to convey quantum information through a circuit of quantum logic gates and solve the time-dependent Schrödinger equation over the entire tunnel-coupled region using Dirichlet boundary conditions. Figure 3 shows time-dependent simulations of the SAW-driven electron propagation.

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de Oliveira Sales et al analyzed the dynamics of a single electron (modeled as a wave packet) in a Morse chain considering SAWs [159]. The considered systems contain two sources of disorder: on-site energies and mass distribution. The authors solve the time-dependent Schrödinger equation.

3.1.4. Levitons

In 1996, Levitov et al advanced the theory of quantum measurement of electric current [160] which together with follow-up work represents the basis for a different type of electron source based on voltage pulses, later named levitons [136]. The fundamental approach is based on applying a periodic train of quantized Lorentzian-shaped pulses to a quantum conductor. Remarkably, this particular pulse shape excites minimal single-electron excitations without generating electron–hole pairs, making leviton single-electron sources an attractive candidate for single-electron sources, in particular for flying qubit systems [6].

Selection of recent computational research: Glattli and Roulleau [161] introduced randomized trains of Lorentzian pulses by using Floquet scattering theory, highlighting the role of the Pauli exclusion principle in electronic quantum signals. These randomized levitons were recently used by Roussel et al to propose a general signal-processing algorithm to extract the elementary single-particle states (the authors refer to these as electronic atoms of signal), see figure 4 [162]. Again, Floquet scattering theory was applied as well as the Wigner function to visualize the single-electron coherence.

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Yue et al investigated the generation of electron–hole pairs using a scattering matrix approach [163]. Gaussian-shaped voltage pulses and transitions to Lorentzians have been investigated. Among others, it was shown that the electron–hole pairs can always be classified into normal and anomalous electron–hole pairs, whose excitation probabilities depend on the pulse flux.

3.1.5. Locally modulated quantum Hall edge state

Generating individual electrons reliably is one thing, another is where to inject them into and how to coherently transfer them via waveguides (a term stemming originally from the field of microwaves and which has later been also introduced to the field of optics). Ultimately, electron sources and waveguides enable one to realize quantum circuits [165] which can be used, for instance, for the previously hinted interference experiments (section 3.3). As discussed, among the most used approaches are quantum Hall edge channels. Historically, such channels have been implemented by using a 2D electron gas (2DEG) in a GaAsAl/GaAs heterojunction which is exposed to a magnetic field so as to realize a quantum Hall regime [166] with no spin degeneracy and long decoherence lengths resulting in a 1D transport channel [15, 167–170]. However, with advances in materials science, alternative materials have been suggested [171].

Selection of recent computational research: Roussely et al conducted time-of-flight measurements of ultrashort few-electron charge pulses injected into a quasi 1D quantum conductor [172]. The density profile in the conductor was calculated using the Schrödinger equation and KWANT [74].

Nedjalkov et al introduced a gauge-invariant Wigner theory in terms of the kinetic momentum which is conserved after a change of the gauge [98]. As a consequence the authors were able to derive an explicit form of the Wigner transport equation corresponding to general, inhomogeneous, and time-dependent electromagnetic conditions.

In follow-up work, Ballicchia et al applied parts of this theoretical framework (by utilizing the Wigner signed-particle formulation provided in the ViennaWD simulator [101, 102]) to investigate the effect of a uniform magnetic field on the electron state interference pattern which arises in a focusing double-well potential structure [173]. The authors analyzed the electron density and the negativity of the Wigner function and showed how the magnetic field controls the electron state but also destroys the coherence of the evolution dynamics.

In related work, Ballicchia et al presented an analysis of the quantum processes involved in the electron evolution around a barrier placed in a waveguide using ViennaWD [174]. The quantum transport results have been compared to classical results (an inherent ability of a Wigner transport approach) to better highlight quantum effects. The relation between quantum effects and their impact on the current has been highlighted. In follow-up work, the authors introduced an approach for modeling surface roughness in waveguides [175] and investigated the relation between quantum coherence, interference, and the negative parts of the Wigner quasi-distribution in a beam splitter structure using ViennaWD [176].

Weinbub et al investigated single-electron interference effects as a result of interactions between single-electrons and two potential wells symmetrically placed in a waveguide, again using ViennaWD [177]. The results bear resemblance to the well-known double-slit experiment. The tight dependence between the well configurations (e.g., size, position) and resulting interference pattern triggered follow-up research with respect to a potential application as logic devices [178], see figure 5.

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Rodriguez et al used a scattering matrix approach to investigate experimental findings concerning the relaxation and revival of electrons injected at finite energy into a quantum Hall edge channel [179].

Sanz et al theoretically investigated the use of two crossed graphene nanoribbons for use as a beam splitter or mirror [180]. The authors used tight-binding, DFT, and Green's function theory to calculate the electron transport properties.

Ito et al developed a coherent beam splitter for single electrons driven through two tunnel-coupled waveguides by SAWs [181]. The authors solved the Schrödinger equation to describe the SAW-driven electron transport. A relation between the gate voltages and the current oscillations in the wires is being shown and related to the coherent electron tunneling.

Moreau et al investigated limitations of realizing quantum Hall edge channels in graphene [182]. The authors particularly looked at the combined role of electrostatics and antidots at graphene edges in quantum Hall breakdown. Tight-binding simulations provided by KWANT [75] were used to tune the position and configuration of the quantum Hall edge channels by optimizing the utilized scanning gate microscopy's tip voltage and position (figure 6).

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Varnava et al evaluated quantum point junctions on the surface of an antiferromagnetic topological insulator [171]. Antiferromagnetic topological insulators have been predicted by Mong, Essin, and Moore in 2010 [183]. Since then several material candidates have been identified, such as MnBi₂Te₄, MnBi₄Te₇, EuIn₂As₂, and NpBi, promising high temperature operation (contrary to conventional 2DEG systems). It this recent work, the authors used dynamic wave-packet simulations to extract the scattering-matrix of the quantum point junction.

Rebora et al studied electron–electron interactions between two edge channels at filling factor two considering electromagnetic radiation generated by a quantum Hall device in a quantum point contact geometry [184]. The two interacting quantum Hall edge channels are modelled using the edge-magnetoplasmon scattering matrix formalism.

Welland and Ferry developed a novel numerical method for solving the Wigner transport equation, without relying on evaluating the Wigner kernel [185]. The authors evaluated their method by simulating and analyzing a cat state, entangled Gaussians, and excited states.

In follow-up work to [186], Jensen et al developed a novel numerical solution to the time evolution equation of the Wigner distribution function [187]. A particular focus was to correctly capture the intricate dynamics of an impinging wave packet (electron) on a barrier (focus on Gaussian and parabolic), exhibiting transmission and reflection behavior, i.e., transmission and reflection delay times. The newly developed solution approach relies on the finite difference method using second-order accurate schemes in both position and momentum.

Benam et al introduced a computational approach for investigating Coulomb interaction using Wigner–Poisson coupling, thereby providing an option to investigate dynamic electron–electron entanglement [188]. The authors apply an approximation based on replacing the Wigner potential of the electron–electron interaction by a local electrostatic field, which is introduced through the spectral decomposition of the potential.

Based on research into electron sources and additional work on ballistic conductors, it became possible to investigate the fundamental wave-based quantum properties of electrons by conducting experiments similar to the quantum optical domain, most dominantly, interference experiments, such as Fabry–Pérot [16], Mach–Zehnder [17], Hanbury Brown–Twiss [18], and Hong–Ou–Mandel [19] interferometers. The reader is directed to the review of Bäuerle et al concerning an overview of various electron-based interferometers [6].

Selection of recent computational research: Barbarino et al considered a Hong–Ou–Mandel interferometer and showed how to engineer statistical transmutation of identical quantum particles, in particular, to cause fermions to bunch [189]. To that end, the authors used a scattering matrix approach and suggested to entangle the quantum particles with an external degree of freedom.

Bordone et al reviewed an approach where the Schrödinger equation is solved with a split-step Fourier method for interferometers based on quantum Hall edge channels [190]. Using this modeling and simulation framework, in earlier work, Bellentani et al investigated a multichannel electronic Mach–Zehnder interferometer, based on magnetically driven noninteracting edge states [191]. Later, the antibunching of two interacting fermionic wave packets impinging on a quantum point contact considering a Hong–Ou–Mandel interferometer based on quantum Hall edge states was simulated [192]. In follow-up work, single and two-electron transport in a full-scale, 2D Hall nanodevice as a potential candidate for a two-qubit conditional phase shifter was investigated [193]. The authors considered two parallel Mach–Zehnder interferometers, see figure 7. In particular, the spatial shift induced by Coulomb repulsion in the final two-electron wave function for two indistinguishable electrons was shown and discussed.

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Acciai et al studied the interplay between electron excitations, considering levitons, and electron–electron interaction in an integer quantum Hall system [194]. The authors showed that the excitations fractionalize into oppositely charged wave packets and show the derivations of the used equations of motion as well as using the Wigner function for visualizing the excitations.

Clark et al studied the decoherence mechanisms of hot electrons in a Mach–Zehnder interferometer, particularly focusing on acoustic and optical phonon emissions [170]. The authors used a quantum master equation for the electron density matrix based on rates derived from the Fröhlich Hamiltonian modified to include emission within interferometer arms.

Rebora et al investigated a Hong–Ou–Mandel interferometer with levitons injected into a quantum Hall edge channel [195]. A particular focus was on the relation between noise and the geometry of the setup. The dynamics of the two interacting quantum Hall edge channels are modelled in terms of the edge-magnetoplasmon scattering matrix formalism.

Kotilahti et al investigated multi-particle effects in an electronic Mach–Zehnder interferometer driven by a series of voltage pulses [196]. The authors used the Floquet scattering formalism to evaluate the interference current and the visibility in the outputs of the interferometer.

With the ability to reliably generate and transport single-electrons the natural next step is to be able to characterize them. This gives rise to tomography, spectroscopy, and detection techniques. This section summarizes a few key computational research advances in this setting.

Selection of recent computational research: in general, quantum tomography refers to extracting and analyzing the wave function of single-electron excitations [197]. In 2019, Bisognin et al introduced a quantum tomography protocol for generating electron and hole wave functions and their emission probabilities from any electrical current, focusing in particular on levitons [198], see figure 8. The authors evaluated the electron/hole wave functions based on calculated Wigner distributions using Floquet scattering theory.

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Fletcher et al introduced a continuous-variable, back-projection-based tomography scheme of solitary electrons [199]. The authors used energy-time filtering and reconstruction of the mixed-state density matrix Wigner representation.

Locane et al investigated single-electron scattering by a ballistic constriction in a fully depleted quantum Hall system under spatially uniform but time-dependent electrostatic potential modulation [200]. The authors applied the Wigner function to describe electrons distributed in time-energy space.

In contrast, spectroscopy provides insights into the time-frequency dependencies of the quantum systems. In 2018 Rossignol et al proposed a spectroscopy approach of flying qubits, by measuring the direct-current (DC) response to a high-frequency AC voltage drive [201]. The authors used Floquet theory and conducted time-dependent simulations based on different models and predicted the power-frequency map of the multi-terminal device using KWANT [74].

Burset et al proposed time-domain spectroscopy of waveguides by applying Lorentzian-shaped voltage pulses (levitons) to an input contact [165]. Floquet scattering theory was used to calculate the electron waiting times, considering an electronic Fabry–Pérot cavity and a Mach–Zehnder interferometer. The authors showed how the distribution of waiting times between charge pulses relates to the characteristic timescales of the waveguide.

In turn, Zilberber et al studied electron detection considering coherent transfer via adiabatic passage in a triple quantum-dot system, where a quantum point contact senses the charge of the middle dot [126]. The authors apply a Bayesian formalism based simulation to study the measurement backaction.

As previously indicated, the high pace of research in electronic devices has been fueling massive research efforts into novel materials. With the advent of graphene in 2007, 2D materials entered the race and have since then evolved into a major field of research giving rise to a plethora of new very exciting electronic materials, e.g., MoS₂ and many others [8]. In the following we list several advancements in computational materials science where treating electron transport quantum mechanically is vital for the gathered findings. We first start with graphene followed by research into other 2D materials.

Selection of recent computational research: Mosallanejad et al investigated the electronic transport properties of curved graphene waveguides by using an NEGF approach [242]. In particular, the authors studied the dependence of the confined waveguide modes on the potential difference, waveguide width, and side barriers. In similar research, Wei et al used graphene waveguides to build and analyze a four-terminal gate defined waveguide and an Aharanov–Bohm interferometer [243]. The authors used an NEGF approach (Landauer–Büttiker for spin-dependent current) and highlighted and discussed the predicted conductance plateaus and relations to experimental results.

Espinosa-Torres et al used DFT, molecular dynamics, and NEGF to theoretically analyze the electronic structure and transport properties of several single and double walled carbon nanotube configurations to identify optimal configurations [244].

Santos et al used a graphene Hall bar to introduce a novel self-contained description of the wave-function matching method to calculate electronic quantum transport properties of nanostructures using the Landauer–Büttiker approach [245]. The method is based on partitioning the system-to-be-solved into a central conductor and an asymptotic region for the leads. The thus resulting two subsystems are linearly coupled and can be solved simultaneously using a sparse linear solver.

Yamaletdinov et al used stochastic reactive molecular dynamics simulations for developing a statistical method for generating fluorinated graphene structures with a desirable fluorine distribution [246]. In addition to molecular dynamics, the transport properties are investigated using a recursive NEGF method based on a tight-binding Hamiltonian.

Díaz-Bautista et al used the Wigner function for analyzing the electron dynamics of uniaxially strained graphene [247]. In particular, the authors studied the effect of strain on the Wigner function of electrons considering a uniform magnetic field, see figure 11. It was shown that strain impacts the shape of the Wigner function of Landau and coherent states.

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Kraft et al investigated graphene superlattice miniband fermions using interferometry in magnetotransport experiments and tight-binding and Green's function quantum transport simulations [248]. The results revealed unconventional cyclotron motion on the length scale of the cavity, which reflects the reshaped hexagonal Fermi surface.

Silva et al used time-dependent DFT to investigate negative differential conductance, current oscillations, and molecular sensing in isolated finite armchair single wall carbon nanotubes without end contacts [249]. The authors showed that (i) the conductivity depends on the nanotube length, (ii) the nanotube current has Bloch oscillations, and (iii) that the adsorbed molecule on the nanotube impacts this oscillatory pattern, decreasing the conductivity.

Solomon et al analyzed valleytronic phenomena in graphene using a Schrödinger equation approach [250]. In particular, it was shown that strong valley-dependent scattering emerges from bilayer graphene quantum dots. Dots of course allow for tuning as the gap size can be modulated using the interlayer potentials in dual-gated devices. The authors investigated the role of dot size, mass strength, and other potential terms.

Barletti et al derived a mathematical model describing a general graphene heterojunction device consisting of two classical regions (diffusive transport) and one middle active region (quantum transport) [251]. Where the first are modelled by a conventional drift–diffusion modeling approach, the latter is modelled by an interface where suitable transmission conditions are imposed, which take the quantum scattering process into account.

Aktor et al used a combination of quantum transport formalisms (Green's function, Kubo, and Landauer–Büttiker formalisms) to discuss bulk and valley-polarized currents in graphene-based devices which are driven by spatially varying regions of broken sublattice symmetry [252]. The findings represent alternatives for generating the valley Hall effect in graphene and for valley-dependent electron optics.

We now turn to other 2D materials beyond graphene. As indicated before, there is a plethora of predicted and investigated 2D materials providing various advantages but also disadvantages. Klinkert et al conducted a large-scale computational study to investigate the suitability of 100 different 2D materials for potential use in ultra-scaled FETs [8]. To that end, the authors applied DFT and quantum transport simulations using OMEN (Schrödinger–Poisson solver via an NEGF formalism) [253] to calculate current–voltage characteristics for each material. In another review, Wang et al provided an overview on the advancements concerning Schottky barriers within the context of FETs based on 2D materials [254]. The authors investigate both theoretical and experimental work and identify in particular first-principles simulations to be most crucial for advancing the research in this area.

Ferry et al investigated the use of the Wigner function for describing transport in transition metal dichacogenides (WS₂) with a dominant spin–orbit interaction [255].

Szafran et al studied electron transport, in particular electron wave packet dynamics, along the zero line of a symmetry-breaking potential of a vertical electric field in buckled silicene [256], see figure 12. The authors used an atomistic tight-binding approach and solved the Schrödinger equation on the atomic lattice; for the initial conditions an analytical solution of the Dirac equation was utilized.

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Qu et al studied the electronic properties of 2D Sb₂Te₂Se and its use in a double-gated MOSFET by coupled DFT–NEGF simulations [257]. Monolayer Sb₂Te₂Se offers some intriguing properties, such as high carrier mobility, and the authors show that in particular the n-MOSFET is attractive because of a much higher on-current.

Zhang et al introduced an implementation of the exact muffin tin orbital method in combination with Kohn–Sham DFT for electronic-structure and quantum transport simulations [258]. The authors considered a representative device-material system in equilibrium condition based on, e.g., MoS₂ and black phosphorus, as evaluation platform and used the recursive Green's function method to calculate the device region.

Hu et al used DFT coupled to NEGF to investigate the electronic properties and ballistic transport performance of 2D SbSiTe₃ [259]. This particular material has attractive properties and a layered structure, thus monolayers can be created via exfoliation. The authors used a double-gated MOSFET device structure as benchmark platform.

Topological materials have an electronic band structure that has a band topology and offer topologically protected surface states with unique electronic properties (for a recent review on topological nanomaterials see [260]). Research into topological materials focuses, among others, on 2D materials, topological (crystalline) insulators, topological semimetals, topological superconductors, fractional quantum Hall effect, quantum rings, and Majorana zeromodes [260, 261].

Selection of recent computational research: Gioia et al used a scattering-matrix approach to calculate the electronic two-terminal conductance (including interference effects) of a quantum ring, considering Dirac-like charge carriers as are typical for 2D materials [262]. The authors apply a k ⋅ p approach to describe the Dirac-electron rings.

Rodríguez-Mena et al investigated the transport of the anomalous Floquet-Anderson insulator which is a topological phase unique to driven systems [263]. The authors used Floquet scattering theory for the transport calculations.

Wu et al used an NEGF approach to study the transport properties of the edge modes of a Fe₃Sn Kagome nanoribbon [264]. Kagome lattice materials are layered 2D materials and can possess topologically non-trivial electronic bands and are an attractive candidate as an interconnect material.

Ronetti et al analyzed the transport properties of a single edge of a 2D topological insulator (with Rashba spin–orbit coupling) which is driven by a train of Lorentzian-shaped pulses (levitons) and connected to a superconductor [265]. The suggested setup represents an alternative realization of an electron quantum optics experiment where the beam splitter is realized by the superconductor rather than a quantum point contact. The authors calculated the charge noise and derived the equations of motion for their setup and used the scattering matrix approach to take the effect of the superconductor into account.

Okugawa et al studied disorder induced topological phase transitions in magnetically doped (Bi, Sb)₂Te₃ thin films [266]. To that end, the authors used Landauer–Büttiker theory to calculate the disorder averaged conductance and compare their exact disorder simulation results with a self-consistent Born approximation.

Huang et al derived the exact master equation and transient quantum transport for studying dissipative topological systems, focusing on non-interacting topological insulators and topological superconductors and including initial system-reservoir entanglement [261]. The authors show that the dissipative dynamics is fully encapsulated in the spectral density through NEGFs.

Cepellotti et al introduced a first-principles model for electronic transport using the Wigner distribution function and used it to solve the electrons' equations of motion in Bi₂Se₃ [267], see figure 13. It was shown that interband tunneling dominates the electron transport dynamics.

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Light–matter interactions are quantum electrodynamical processes and typically described by electron quantum transitions and cover emission, absorption, and scattering of electromagnetic field quanta [268]. Here, we introduce computational research covering solar cells, photovoltaic effect, and LASER based energy transfer.

Selection of recent computational research: Hathwar et al published a review on the role of ultrafast carrier dynamics in the performance of advanced concept solar cells [269]. Aside from theoretical work, the authors particularly highlighted computational results based on ensemble Monte Carlo transport simulations and focused on multi-quantum well systems and as well as III–V nanowires. Li et al reviewed research focusing on excited state dynamics in perovskite solar cells [270]. Many other reviews on this topic exist, see for example an extensive review by Moskalenko et al [271].

Michelini et al introduced a theoretical framework to describe time-dependent energy transport in quantum networks, in particular focusing on energy transfer from a femtosecond LASER pulse to electrons and subsequent transport in a molecular circuit [272]. The authors used an NEGF approach to simulate the energy current in the system.

Bajpai et al conducted a computational study of shift current (a central mechanism of the bulk photovoltaic effect) dynamics in response to a femtosecond light pulse [273]. To that end, the authors employed a time-dependent NEGF approach to simulate the current dynamics.

Beltako et al used an NEGF approach to calculate the charge transfer dynamics triggered by a femtosecond LASER pulse at the donor–acceptor interface of molecular junctions [274] (for molecular junctions, see also section 6.5). The authors include Coulombic interaction within the Hartree–Fock approximation.

Aeberhard conducted a computational study of hot carrier photocurrent generation in quantum well solar cells using an NEGF approach [275, 276]. The author included GW self-energy to treat electron–electron scattering together with photogeneration and phonon-mediated carrier relaxation.

Karlsson et al introduced an NEGF approach for describing electron-boson dynamics with a particular focus on computational efficiency (numerical effort scales linearly with propagation time) and upholding the conservation law [277]. Figure 14 depicts the simulated electron–phonon dynamics in the considered showcase system.

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Tang et al investigated the layer-dependent photoresponse properties and photovoltaic effects of 2D Bi₂O₂X (X = S, Se, and Te) by combining first-principles calculations and quantum transport [278]. In particular, the authors calculated the absorption coefficients via time-dependent DFT and the photocurrent via an NEGF method.

Heterostructures consist of several combined materials with the typical motivation to improve the intrinsic characteristics of an individual material. Heterostructures are a success story and there are countless examples, both historically and in modern times, in particular with the advent of 2D materials. Practically, all modern electronic devices are actually heterostructures so it is hard to draw a clear line here. However, here in this section, we present computational research where the focus is on the heterostructure itself rather than on a complete device. For a recent overview on this matter see [279].

Selection of recent computational research: Jech et al used ab initio methods to study the interactions of a Si–H bond in a 3D Si/a-SiO₂ interface [280]. In particular, the authors used well-tempered metadynamics and nudged-elastic-band calculations based on DFT (via the CP2K package [68]), which allowed them to shed light on the dissociation kinetics and electronic properties.

Dong et al introduced atomistic full-band self-consistent quantum transport simulations of the graphene/MoS₂ edge contact using a Wannier-function basis [281]. The authors used DFT to obtain the band structures and for the quantum transport solver they applied the NEGF formalism and maximally localized Wannier functions.

Xu et al presented a computational study about the ohmic contact at bilayer InSe–metal interfaces within the context of FETs [282]. The authors employ DFT and ab initio quantum transport simulations [52]; the latter is particularly useful for correctly describing the lateral Schottky barrier heights.

Szabó et al used ab initio quantum transport simulations to investigate the role of an oxide layer between a metallic contact and a MoS₂ monolayer [283]. The authors analyzed the interface related edge- or area-dependent electron transport processes. The computational methodology consisted of combining plane-wave DFT, maximally localized Wannier functions, and NEGF.

Yang et al looked at ballistic transport properties with a particular focus on a change of the Hamiltonian in a lead-semiconductor-lead structure (1D atom chain model) [284]. A NEGF approach was used to derive the transient current.

Ning et al used ab initio electronic calculations and quantum transport simulations to investigate the electronic and transport properties of InAs/graphene heterostructures [285]. More specifically, the authors used VASP [65] for the DFT part and interfaced it with NEGF based transport simulations [52].

Bhattacharya et al investigated the bridge-site asymmetry for methanethiol adsorbed on Au(111) with two different S–C bond orientations using dispersion-corrected DFT [286]. Again, VASP [65] was used and various configurations regarding pseudo-potentials, exchange functionals, and dispersion correction were considered for optimal modeling.

Ducry et al introduced a mode-space approximation for transforming the Hamiltonian of a metallic structure expressed in a nonorthogonal DFT basis into a low-dimensional space suitable for electron-transport simulations, in particular for NEGF [287]. The authors applied it locally to inhomogeneous material stacks including amorphous phases and interfaces. Figure 15 shows simulation results for a Cu/a-SiO₂/Cu conductive bridging random-access memory cell structure.

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M'foukh et al discussed a full-band quantum transport model based on the NEGF formalism and empirical pseudopotentials and the application to heterojunctions between lattice-matched semiconductors [288]. The authors also applied their method to an Esaki tunneling diode and n- and p-type heterojunction tunnel FETs.

Smorka et al investigated non-equilibrium steady-state charge transport through a heterostructure consisting of a finite 2D lattice (Falicov–Kimball model) which is sandwiched between two non-interacting semi-infinite leads [289]. A particular focus was on the influence of non-homogeneous charge orderings on the transport. The computational approach was based on a Monte Carlo-NEGF solver.

Kumar et al studied the low field 2DEG transport properties in $\beta -{({\mathrm{A}\mathrm{l}}_{x}{\mathrm{G}\mathrm{a}}_{1-x})}_{2}{\mathrm{O}}_{3}/{\mathrm{G}\mathrm{a}}_{2}{\mathrm{O}}_{3}$ heterostructures [290]. The authors used a self-consistent Poisson–Schrödinger solver to calculate the sub-band energies and the wave functions in the quantum well. In turn, DFT (and density functional perturbation theory) was used to determine the phonon dispersion.

Li et al used neural networks to predict transport properties of a quasi-1D tight-binding model with disordered on-site energies [128]. The authors consider a scattering region between two leads as the evaluation platform. Different disorders of the on-site energies in the scattering region are considered.

Wu et al introduced a machine learning approach to replace NEGF quantum transport simulations and used as a showcase a simulation of a graphene–ferroelectric–metal ferroelectric tunnel junction [291]. Their approach is based on learning a sparse representation of a quantum transport property, which allows to train a model to yield device properties.

Wang et al investigated van der Waals p–n heterojunctions considering 2D–2D (e.g., black phosphorus/MoS₂) and mixed dimensional systems [292]. The interface physics is investigated by means of quantum transport simulations (encompassing incoherent scattering and carrier recombination). The transport simulations were based on an NEGF method (NEMO5 [50]) in combination with maximally localized Wannier function basis sets created for structures relaxed within DFT models (VASP [65]).

Jin et al used a Schrödinger–Poisson approach to calculate the dynamical electron transmission in an ultra-short two-terminal piezoelectronic device (ballistic regime) [293]. A metal–piezoelectric ZnO–metal structure exposed to sinusoidal and rectangular external stresses was considered.

This section discusses computational research into the electronic characteristics and electron transport in molecular junctions and systems. The majority of research focuses on junctions, where a metal–molecule–metal configuration is considered. However, other molecular systems are considered as well, such as loop structures. Excellent recent reviews on computational methods for molecular junctions cover electron transport [294] and specifically the use of Green's function methods [295].

Selection of recent computational research: Avriller et al investigated time-dependent electronic transport properties and vibrational dynamics of a molecular junction based on an NEGF approach [296]. Among others, the authors identify oscillations and establish a similarity with the classical harmonic oscillator.

Moura-Moreira et al analyzed electron tunneling through two 1D molecular junctions using a coupled DFT and NEGF approach [297]. The first junction consists of left and right carbyne wire electrodes with a sodium atom in the middle. In contrast, the second junction is based on two single-wall carbon nanotube electrodes with again a sodium atom in the middle, the scattering region.

Pohl et al presented time-dependent electron transport simulations through a graphene nanojunction under time-dependent potential biases [298]. The authors used the driven Liouville–von Neumann approach to simulate the one-electron density matrix dynamics under non-equilibrium conditions and to analyze scattering and relaxation processes as well as the steady-state, see figure 16.

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Chiang et al discussed time-dependent electron transport simulation results of a molecular junction (fumaronitrile molecule) in the adiabatic limit using DFT and the molecular dynamics-driven Liouville von Neumann (MD-DLvN) approach [299]. The modeling was done within the Born–Oppenheimer approximation, i.e., no vibronic coupling, and ignoring nuclear quantum effects.

Phuc et al showed that the electron transfer dynamics in molecular loop structures can be manipulated by Floquet engineering (i.e., Hamiltonian of the system is temporally modulated in a periodic manner) via LASER pulses [300].

Liu et al introduced the generalized input–output method for studying charge transport in molecular junctions considering strong electron-vibration interactions and including electronic and phononic environments [301]. The authors derive a Langevin-type equation of motion for system operators and position their framework relative to NEGF and perturbative quantum master equation methods. In follow-up work, the method is applied to various molecular junctions [302].

Ramakrishnan et al studied transport selectivity across isomeric molecular junctions by conducting computational electron dynamics studies considering many-body coherence and electron correlation effects [303]. To that end, the authors employed the real-time, many-body, and time-dependent configuration interaction method to model and simulate the ultrafast electron dynamics [304] and compared the method to Green's function and density matrix approaches. Cyanobenzene and in m/p-linked benzonitrile thiolate molecules bonded to a gold atom (acceptor terminal) were considered. Interference effects in the electron density were observed and discussed.

Tharammal et al discussed a simulation based design for a molecular switch considering a 1.4-benzene dithiol molecule with gold, silver, platinum, and palladium metal electrodes [305]. The electronic properties have been calculated using DFT and the Hartree–Fock method.

Romero-Muñiz et al presented a theoretical study within the context of protein electronics [306]. More specifically, the authors investigated the coherent electron transport in metal–protein–metal junctions based on the blue-copper azurin from Pseudomonas aeruginosa. The simulation methodology was based on molecular dynamics (junction geometries), DFT (electronic structure), and Landauer formalism/NEGF (electron transport).

Tuovinen et al introduced a generalized Kadanoff–Baym ansatz scheme (on top of NEGF theory) which considers initial correlations in a partition-free setting [307]. The method was evaluated considering carbon-based molecular junctions based upon electron correlations in transient current signatures.

Arasu et al employed atomistic simulations to establish a general relationship between the electronic spectra of aromatic, antiaromatic, and quinoidal molecules and the impact on electron transport [308]. To that end, the authors utilized a DFT–NEGF approach for simulating and analyzing the electronic properties.

Cheng et al investigated non-equilibrium transport in conjugated polymer by a time-dependent NEGF approach [309]. Specifically, a polymer chain, using a Su–Schrieffer–Heeger model Hamiltonian, was sandwiched between two reservoirs.

With the advent of nanotubes came the idea of using them as transport channels in FETs. This further evolved over the years and yielded various forms and approaches resulting in several loosely-defined technological terms as indicated by this section's title, which are thus differently used throughout research and industry but in essence refer to similar device concepts. Overall, these highly confined FETs, in particular the GAA approaches, are strong candidates to come out as the successor of the currently prevailing FinFET technology [312, 313].

Selection of recent computational research: Kim et al used the discrete Wigner transport equation to simulate the electrical characteristics of GAA junctionless nanowire transistors [314]. The authors particularly focused on the accuracy balancing problem and mitigation.

Markov et al investigated the source-to-drain direct tunneling limit in GAA Si nanowire FETs (3–8 nm gate lengths) [315]. A density functional tight-binding Hamiltonian together with an NEGF approach was applied, including self-consistent solution via the Poisson equation.

Medina-Bailon et al studied the electron mobility of n-type silicon nanowire transistors with a particular focus on the nanowire cross-section's shape and scattering [316]. Based on the findings, the authors identified the elliptical shape to be superior over the circular shape. To minimize computational effort, the authors self-consistently solved several cross sections of the nanowire via a Poisson–Schrödinger solver to calculate the respective potential profiles and eigenfunctions. Based on this, various scattering rates and mobilities (using the Kubo–Greenwood formalism) were calculated.

Hsiao et al introduced a coupled Poisson-NEGF approach to self-consistently calculate 3D transport in GaAs GAA nanowire FETs [317]. The approach included electron–phonon scattering, ionized impurity scattering, and surface roughness scattering.

Hwang et al theoretically and experimentally studied graphene-nanoribbon tunneling FETs at room temperature [318]. A particular focus was on the negative-differential resistance observed in graphene nanoribbons. The authors used an NEGF approach with a p/d orbital tight-binding model to corroborate the experimental findings.

Van de Put et al introduced an alternative to the prevailing tight-binding approach to describe the atomic configuration by using a plane-wave approximation of the atomic (pseudo-)potentials [319]. The developed approach separates the intrinsic Hamiltonian concerning the atomic configuration and the extrinsic Hamiltonian dealing with the external potential. The performance was evaluated by simulating a graphene nanoribbon FET containing more than 2000 atoms. In follow-up work, Chen et al investigated the effect of defects on the charge-transport properties of GAA graphene nanoribbons FETs [320], see figure 17.

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Poljak et al conducted a performance study of 15 nm-gate-length MOSFETs based on silicene nanoribbons [321]. The width scaling, series resistance, and transport characteristics were investigated by using an NEGF approach and corresponding silicene nanoribbon Hamiltonians that are expressed in an atomistic tight-binding basis.

Zhang et al studied different designs of vertically stacked horizontal Si and Ge nanosheet GAA pMOSFETs (sub-5 nm node) using an NEGF approach [322]. The ON-state current and subthreshold swing were calculated and evaluated.

Lv et al used quantum dynamics simulations to analyze the electronic fluctuation issue of the so-called armchair graphene nanoribbon MOSFETs [323]. In addition to electron transport, the authors also considered the behavior of the C and H nuclei and used DFT as well as an NEGF transport simulator provided by NanoTCAD ViDES. ⁶

Tamersit proposed a new double-gate band-to-band tunneling junctionless graphene nanoribbon FET with a sub-10 nm gate length by using a self-consistent Poisson-NEGF simulation approach (in the ballistic limit) [324]. Several key electrical properties, such as the I_DS–V_GS transfer characteristics, were simulated and discussed. Other work of the author, including colleagues and using a similar simulation approach, focused on investigating a nanoscale coaxial-gate negative-capacitance carbon nanotube FET with a metal–ferroelectric–metal–insulator–semiconductor structure [325].

Abdi et al looked into an optimized design of a textured graphene nanoribbon phototransistor [326]. Here, the graphene layer acts not only as the channel but also as the light absorber. The authors used an NEGF mode-space approach.

Sun et al investigated the operation modes of a dual-gate reconfigurable (silicon-nanowire) FET [327]. Reconfigurable FETs use electrostatic doping for virtual n-/p-regions and can be tuned to operate as unipolar n-/p-transistors. The authors used a self-consistent NEGF approach and solve it one-dimensionally on a finite-difference grid using an effective mass approximation (symmetric conduction and valence band).

The staggering advancement of 2D materials (see section 6.1) also triggered substantial research into utilization as channel materials for FETs, collectively named: 2D FETs [328–331].

Selection of recent computational research: Ye et al studied the performance limits of monolayer bismuthane (BiH) transistors considering spin-orbital coupling using a coupled DFT–NEGF approach [332]. The authors particularly focused on the required device width to achieve suitable OFF-state currents.

Kim et al used a computational simulation workflow which also uses a DFT–NEGF approach to investigate the device performance of double-gated silicene/gallium phosphide heterobilayer FETs [333]. The authors could predict that these FETs have attractive characteristics, such as high ON-state current.

Han et al analyzed the photoresponse mechanisms (photoconductive and photogating effects) in MoSe₂ transistors [334]. Using a self-consistent simulation approach based on solving the Poisson's equation, an NEGF approach, and considering charge trapping the authors were able to, among others, identify photogating effects as the primary contributor to the overall photocurrent.

Liu et al simulated sub-10 nm double-gate monolayer Cs₂PbI₄ MOSFETs by using a coupled DFT–NEGF approach [335]. Good subthreshold swing values, ON-state current, and delay time were predicted.

Guo et al studied monolayer α-CS with a puckered structure for potential application as a channel material for sub-5 nm FETs [336]. The authors used a coupled DFT–NEGF approach to calculate the electrical characteristics which yielded attractive results, in particular regarding the ON-current.

Klinkert et al used a coupled DFT–NEGF approach to investigate the orientation misalignments and their impact on the performance of black phosphorus FETs [337]. N- and p-type configurations were investigated including six alignment angles. Ballistic transport as well as electron–phonon and charged impurity scattering was assumed.

Ding et al investigated the ballistic transport performance and gate control mechanism of sub-10 nm monolayer GeS metal-oxide-semiconductor FETs by using a coupled DFT–NEGF approach [338], see figure 18. A particular focus was on evaluating the ON-state current, subthreshold swing, delay time, and power consumption.

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Zhang et al simulated sub-5 nm gate-length double-gate monolayer MoS₂ MOSFETs using a coupled DFT–NEGF approach [339]. Based on the results the authors predicted that 1–5 nm gate lengths fail to meet certain expectations (e.g., ON-state current) but with the introduction of a negative capacitance dielectric layer some of them could be mitigated.

Cao et al looked at dissipative transport in antimonene and arsenene double-gate n-type FETs using a coupled DFT–NEGF simulation approach [340]. Among others, the authors highlighted that the ON-current can be highly impacted by optical intravalley and intervalley phonon scatterings and that simulations not properly considering these are seriously overestimating device performance.

Tunneling FETs (TFETs) differ from conventional FETs by using a different switching mechanism which is based on quantum tunneling through a barrier. For reviews on this topic see, for instance, [341, 342].

Selection of recent computational research: Gnani et al investigated the performance of an inverter based on an n- and p-type TFET considering InAs/Al_0.05Ga_0.95Sb and interface traps and localized strain [343]. A four-band k ⋅ p Hamiltonian is used in combination with an NEGF approach self-consistently coupled to a Poisson solver.

Ahn et al introduced an advanced NEGF simulation approach to reduce the computational burden of atomistic-level simulations of FETs [344]. The approach is based on using the R-matrix (for details on the R-matrix method see [345]) and recursive Green's function method and was evaluated considering a germanane/InSe vertical tunneling FET.

Brahma et al studied electron–phonon coupling limited transport in phosphorene MOSFETs and TFETs [346]. Among others, the local density of states and spectral current densities were calculated (see figure 19) using hybrid DFT whereas transport is solved by an NEGF approach using a DFT-calibrated two-band k ⋅ p Hamiltonian. Electron–phonon scattering was incorporated. Among others, the authors could show that optical phonon modes are primarily responsible for the degradation of the ON-current.

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Chen et al proposed an improved design of a triple heterojunction tunneling FET with a 12 nm body thickness based on doping profile engineering [347]. To that end, the authors used a mode-space quantum transport simulation approach, considering thermalization and scattering, to optimize the device design.

This section collects all remaining FET designs, ranging from the predominant FinFET to ultrahin-body FETs and other types of related devices.

Selection of recent computational research: Pala et al presented a new method for full-band quantum transport simulations based on an empirical pseudopotential Hamiltonian and an NEGF approach [348]. The method was showcased by considering planar, ultrathin-body FETs and nanowire FETs. The authors further discussed different crystal orientations as well as various strain/stress scenarios.

Vyas et al introduced a method based on the Pauli master equation supporting dissipative quantum transport, which keeps the transfer-wave-vector dependence of the scattering processes [349]. A Schrödinger–Poisson solver using the effective-mass approximation was used in the 2D plane of the device. The quantum transmitting boundary method was applied because of open boundary conditions. Finally, dissipative transport was modeled using the Pauli master equation. Ultra-thin body FETs were studied with respect to electron–phonon and surface roughness scattering to evaluate the method.

Kao et al studied the subthreshold swing of source-to-drain tunneling short-channel MOSFETs at cryogenic temperatures [350]. To that end, a coupled Poisson-NEGF simulation approach was employed. The authors discussed four features concerning subthreshold swing saturation dominated by source-to-drain tunneling.

Cam et al coupled Landau–Khalatnikov and Poisson equations self-consistently with the NEGF formalism and used the simulation approach to study ultra-scaled (i.e., to sub-10 nm gate lengths) metal–ferroelectric–insulator–semiconductor negative-capacitance FETs [351]. Based on the simulations, the authors could show that the considered device structure holds attractive properties, e.g., concerning subthreshold slope.

Kumar et al considered a split-gate 2D MoTe₂ FET to design an AND gate and studied the device by using an NEGF approach [352]. Back-gate parameter variations were studied with respect to their impact on device performance.

Khaliq et al used doping engineering to improve the electrical characteristics in double-gate pMOSFETs hosting a 5 nm gate length [353]. A NEGF approach was used in combination with a six-band k ⋅ p Hamiltonian.

Damodaran et al investigated a quantum well modulation-doped FET for use as a quantum dot memory device [354]. Their approach centered around a self-consistent Schrödinger–Poisson solver and an eight-band k ⋅ p model. Among others, the results showed that information (i.e., electrons/holes) can be stored longer with increased dot sizes.

Related to FETs are diodes, which contain some form of junction with which to realize the diode functionality. In what follows, a few computational studies are highlighted which explicitly focus on diodes.

Schulz et al introduced a numerical approximation technique for the Wigner transport equation (including the spatial variation of the effective mass) and applied it to simulating a resonant tunneling diode with different material systems using a GaAs/Al_x Ga_1−x As structure [355], see figure 20.

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Abedi et al suggested a time-dependent quantum transport model considering elastic scattering and based on the time-dependent Schrödinger equation [356]. Similar to the NEGF approach, an imaginary potential is added to the Schrödinger equation for absorbing electron waves [357]. Also, a source term is added to model the injection of electron waves. Again, a resonant tunneling diode was used to showcase the model.

Dey et al applied a coupled DFT–NEGF approach to study quantum ballistic transport properties of a doped guanine-nanosheet-based bio-Zener diode [358]. Coherent tunneling and incoherent hopping processes are covered by the considered Hamiltonian.

Zhao et al introduced a boron intratube p-i-n junction via gas storage for diode applications [359]. The authors used a coupled DFT–NEGF approach to describe and simulate the electronic structure and transport properties.