To describe the unitary evolution of the density matrix, consider that the dynamics of a closed quantum system is specified by the solutions of the time-dependent Schrödinger equation. i ℏ ∂ t Ψ = H ^ Ψ , (1) where Ψ denotes the wave function, H ^ denotes the Hamiltonian, and ℏ is Planck’s constant. The time-dependent Schrödinger equation is a first order partial differential equation where both Ψ and H are functions of space and time. For computational purposes, it is useful to expand Eq. 9 in a set of basis functions that span the Hilbert space of the Hamiltonian, and to write it in matrix and vector form i ℏ ∂ t Ψ = H Ψ   , where Ψ is the state vector and H is the matrix representation of the Hamiltonian. This matrix equation can, in principle, be solved by diagonalizing the Hamiltonian matrix H. In practice, however, it is difficult to perform this diagonalization unless the size of the Hilbert space (i.e., the dimension of the matrix H) is small.

While the evolution of the state vector in a closed quantum system is deterministic, open quantum systems are stochastic in nature. The effect of an environment on the system of interest is to induce stochastic transitions between energy levels, and to introduce uncertainty in the phase difference between states of the system. The state of an open quantum system is therefore described in terms of ensemble averaged states using the density matrix formalism. The density matrix ρ describes a probability distribution of quantum states Ψ n , in a matrix representation ρ = ∑ n p n Ψ n   Ψ n , (2) where p n denotes the classical probability that the system is in the quantum state Ψ n . Calculation of the time evolution of the density matrix ρ ˙ t o t a l of the tunnel-coupled quantum dot system is the objective of the mesoscopic transport calculation discussed herein. Thus, at the level of mesoscopic transport calculation, we are solving the time-evolution of the density matrix at different charging potentials of the quantum dot and applied magnetic field.

The quantum statistical mechanics approach for deriving equations of motion of an open quantum system that is interacting with its environment is to expand the scope of the system to include the environment [44, 45]. Since the combined quantum dot and metallic lead system is a closed quantum system, its evolution is governed by the von Neumann equation ρ ˙ t o t a l = ∂ t   ρ ^ t o t a l = − i ℏ − 1 H ^ t o t a l , ρ ^ t o t a l (3) Equation 11 is the equivalent Schrödinger equation expressed in density matrix formalism in Eq. 1. Here, the total Hamiltonian, H_total = H_sys + H_env + H_int, includes the original system Hamiltonian H_sys, the Hamiltonian for the environment H_env, and an additional term H_int that represents the interaction between the system and its environment. This corresponds to the systems Hamiltonian defined in Section 2.3. Since the interest is only in the dynamics of the system, it is possible to perform a partial trace over the environmental degrees of freedom in Eq. 3 to obtain a master equation for the motion of the original system density matrix.

The generalized form of the completely positive trace-preserving time evolution of the density matrix is the Lindblad master equation for the reduced density matrix ρ = Tr_env [ρ _total]. This is expressed in terms of the time-dependent Hamiltonian H ^ t and dissipation terms D ρ ^ as [46], ρ ˙ t = ∂ t ρ ^ = − i ℏ − 1 H ^ t , ρ ^ t + D ρ ^ , (4) where D ρ ^ = ∑ 1 2 2 C n ρ t C n † − ρ t C n † C n − C n † C n ρ t . The term C n = γ n A n denotes collapse operators (or the Lindbladian) while A n denotes the operators through which the environment couples to the system, and γ n are the corresponding collapse rates. Equation (4) is the primary equation that is solved in the transport model. The formal derivation of the Lindblad master equation is presented in several standard sources, including Refs [47–53], and will not be repeated here. Nonetheless, if one retains only the first term on the right-hand side of Eq. 4, then the Liouville-von Neumann equation results. The unitary evolution of the density operator of the QD is denoted by the Liouvillian. The second term on the right-hand side of Eq. 4 is the Lindbladian, which describes the non-unitary evolution of the density operator. It emerges only when one takes the partial trace of the interacting degrees of freedom between the electron in the QD and the bath of conduction electrons in the metallic leads. By adopting the interaction H_int of the form, H int = ℏ ρ t C n † − ρ t C n † C n − C n † C n ρ t , (5)

the physical meaning of the Lindblad operators is understood as representation of the contribution of the QD charging energy to the interaction with conduction electrons. This is crucial since Eq. (4) is derived from the Liouville-von Neumann equation by tracing the degrees of freedom of the bath.

Analytically, it is a formidable task to calculate the dynamics of a QD system that has more than two states using the von Neumann equation (i.e., Eq. (3)). This becomes even more complicated when, in addition, one must consider dissipations due to the interaction of the QD with the surrounding environment using the Lindblad master equation (i.e., Eq. (4)). It is therefore necessary to resort to numerical approximations for these computational tasks to be performed in realistic situations. To this extent, the role of efficient and openly accessible computational tools, such as QuTip [54] and Kwant [55], in facilitating the numerical study of the dynamics of open quantum system calculations cannot be overstated. Herein, exact diagonalization calculations were performed to solve Eqs (3), (4) using different approximate master equations (i.e., Pauli, von Neumann, Redfield and Lindblad) as implemented in the Quantum master equation for Quantum (QmeQ) package [56]. QmeQ is an opensource PYTHON utility for simulating the stationary-state mesoscopic transport in QDs. The computational cost for solving the Lindblad equation numerically grows as additional complexity is introduced into the QD model. Hence, better numerical approximations are required for efficient numerical implementation of the solution. For derivation of the different approximate master equations that are implemented in the QmeQ package, and their pros and cons, see Appendices A–F in Ref; [56]. Key features of the approximate master equations used in the numerical simulations are listed below.

Second order von Neumann (2vN): This approach bridges the gap between rate equations and the transmission formalism which neglect level broadening and cotunneling which is essentially based on the single-particle picture thereby treating many-particle interactions on an approximate level. The 2vN is an approximation that is based on decoupling of the equations of motion for the density matrix. Only terms that involve up to two excitations n = 2 are retained in the Runge-Kutta algorithm for numerically solving the coupled ordinary differential equations. A Markov approximation is made to the two-excitation term. Electrons in the leads are assumed to be thermally distributed according to the Fermi-Dirac distribution, f. The 2vN resembles the resonant tunneling approximation, which is based on a suitable infinite re-summation of the perturbation series for the generalized master equation. (2vN) yields exact currents for non-interacting systems with H _Coulomb = 0. When interactions are introduced during charging (i.e., H_Coulomb ≠ 0) the results are only reliable at weak to moderate coupling strengths, wherein the Kondo temperature T_K is larger than (or equal to) the tunnelling rate Γ during the coupling. Also, the simulation temperature T must be larger than any Kondo temperature T_K in the system [54].

First order von Neumann (1vN): The 1vN approximation assumes that only the terms that involve one excitation n = 1 are retained in the Runge-Kutta algorithm for solving the coupled ordinary differential equations. A Markov approximation is made to the one-excitation term.

First-order Redfield: Retains all the assumptions, advantages, and disadvantages of the 1vN approximation. However, it implements an important exception wherein the Markov approximation to the one-excitation term is set to the steady-state solution of the Schrodinger equation, corresponding to Eq. 1, subject to the normalization condition. Thus, one can go to the Hamiltonian by transforming the first order Redfield master equation (also known as the second order in perturbation approximation) to the Lindblad master equation way.

Pauli master equation: Only the terms that involve one excitation n = 1 are retained, and all higher terms in the Runge-Kutta algorithm solving the coupled ordinary differential equations are discarded. The Markov approximation is made to the one-excitation term. The coherences of the reduced density matrix of the QD system are neglected. The Pauli master equation is obtainable from the 1vN or the Redfield approaches by neglecting the coherence.

Lindblad master equation: An approximate form of Eq. 4 is solved for the steady-state solution of the Schrodinger equation, corresponding to Eq. 1, subject to the normalization condition. This Lindblad approximation is the first order in rates and can describe the sequential tunneling in the presence of coherences. It preserves the positivity of the reduced density matrix as in the 2vN approach.

It is important to first characterize atomic network structure of the monolayer materials considered to appreciate their effects on electronic transport. Significant progress has been made in understanding the roles of defect in a covalently bonded solid. For instance, it is now well-known from our previous studies of tetrahedrally-bonded materials that the presence of an open-volume defect introduces localized electron states into the mid-gap region due to their dangling bonds [61, 62]. The density of the mid-gap defect states in the hydrogen-free amorphous silicon (a-Si) reduces significantly when it is passivated with hydrogen [63]. In this sense, the passivating hydrogen is an impurity in the disordered silicon network. In sp²-bonded crystalline materials, such as the hBN bilayer [64] and monolayer [65, 66], we also found that introduction of a point defect also introduces localized mid-gap electronic states. A single carbon substitutional defect introduced on the boron or nitrogen site creates s-resonant electronic states at the Fermi-level with finite magnetic moments. By contrast, the introduction of a few-carbon or on both boron and nitrogen sites either at near-neighbor positions or over an extended region of the monolayer also induces finite magnetic moments [67].

Although hydrogen-related point defects and vacancies are not considered in the QDs discussed in this work, it is demonstrated in the next section that an embedded carbon nanoline also introduces mid-gap defect states in the hBN monolayer leading to flat bands in the electronic structure. This leads to formation of the Fermi level resonant state when the number of embedded carbons is odd. By contrast, a significantly narrowed band gap is obtained when the number of embedded carbons is even. Such monolayer materials have been studied extensively for about 2 decades and graphene and monolayer boron nitride have both emerged as ideal Dirac material candidates for designing HPMs. More recently, our work has showcased different technology platforms that can utilize the ballistic transport capability of carriers in graphene, monolayer hBN or their derivatives in diverse technological applications like spintronics [68], skyrmionics [69], catalysis [70] and plasmonics [71]. Insights from these previous studies suggest that similar computational approaches can be used to gain unique insights into the efficacy of HPMs in energy conversion applications.

Figures 2A–F show the local atomic networks in the carbon nanoline-embedded hexagonal boron nitride monolayers. The short line of carbon (SLC), carbon dimer chain (CDC), hexagonal line of carbon (HLC), long line of carbon (LLC), and hexagonal ring carbon (HRC) represent unique nanoline geometries, respectively. The embedded SLC acts as a carrier conducting channel in the semiconducting layer. Its role in the hBN monolayer is analogous to the spin nanoroad observed in the spin-resolved charge density of the metal-insulator-metal heterostructure as a 1D conducting channel [71]. The nanoroad denotes the graphene conducting channels embedded within the hexagonal boron nitride layer [72]. As such, it is suggested that monolayer hBN QDs derived from carbon nanoline induced narrow bandgap semiconducting nanowires can serve as the building blocks for spin-based quantum information processing, and as HPMs for energy applications. QDs made with narrow bandgap semiconducting monolayers are advantageous because of their high electron mobility and small effective masses making it possible to manipulate the spin transport state.

Figure 3 shows the electronic band structure of nanoline-embedded monolayer hBN calculated with the GLLBSC functional. Firstly, the GLLBSC functional gives a significantly improved band gap of the pristine hBN monolayer relative to the experimental values. The band structure of the PML is characterized by a wide Kohn-Sham band gap of 5.827 eV. Figure 3A shows that the position of the highest occupied valence band state in the PML is localized at over 4 eV below the Fermi level. With a derivative discontinuity of 2.08 eV from the GLLBSC functional of the particle exchange-correlation hole, the fundamental quasiparticle band gap of 7.87 eV is obtained as expected. The calculations reveal that the electron states in the PML systems are time-reversal symmetric. Electrons from the two spin channels are not sensitive to spin polarization. As such, changes in direction of spin quantization does not affect the electron states. This causes the position of the corresponding highest occupied valence state to be localized at an energy level that is located at over 4 eV below the Fermi level. This leads to invariant direct interband transition from the valence to conduction states at Г-point of the Brillouin zone.

Using the nanoline geometry to embed carbon species in the monolayer introduces impurity band states in the gap region. Embedding the nanolines reduce the wide band gap in pristine hBN monolayer. Generally, Figure 3 shows that the carbon impurity states cause the position of the corresponding highest occupied valence band to shift towards the Fermi level. Except in the case of an embedded hexagonal line carbon, the lowest unoccupied conduction band states are shifted in energy to a position above 2 eV in the conduction band. These culminate in significant reductions in the size of the wide band gap obtained in the PML. As such, the band structure in each embedded system is only shown here in the energy interval of ±4 eV around the Fermi level (see Figures 3B–F). Figure 3B shows that the highest occupied valence band state lies directly at the Fermi level. This leads to metallic transport property in the SLC system. The other embedded systems yield a semiconducting band structure. Other embedded structures yield significantly reduced band gaps with a strong sensitivity to the local structure of the nanoline conducting channel.

There is a nearly flat band resonating with the Fermi level in Figure 3B. This band exists at the mid-gap region. It is due primarily from the carbon atoms at the zigzag edge nanoline of the SLC geometry. The behavior of this flat band state resembles a non-degenerate defect state in the pristine zigzag-edged graphene ribbon [70, 73]. Our previous studies [65–67] showed that the threshold for reconstruction of the local structure in the layer is sensitive to the presence of a substitutional carbon impurity. This implies that the layers are stable against puckering. The present study considers the embedded nanoline as a line defect and not as point defects. However, the origin of an associated magnetic moments behaves like the magnetic moments obtained with the substitutional carbon defects in the hBN monolayer. These are different from the magnetic moments obtained from d-electron states in dilute magnetic allots. As such, the associated Kondo resonance effects cannot be ascribed to a dispersive defect state. The Kondo effect associated with spin-dependent transport of carriers in a non-dispersive defect state is different to those in materials where magnetism arises from the d-band electron states. In the latter, d-electron states in transition metals lead to ferromagnetism but there are no d-band states in the monolayers considered herein.

Table 1 also shows the renormalized quasiparticle band gaps obtained in the GLLBSC approach. This description of electronic states in terms of the momentum space dispersion of energy leads to the non-magnetic ground state. Thus, the band structures shown in Figures 3C–F each has a finite gap, and the net magnetic moment is zero in each case. The ferromagnetism of the SLC-embedded system is due to the Fermi-level resonant state formed due to strong s-p hybridization. This gives rise to the dispersionless (or flat) band seen along the M-K direction of the Brillouin zone in Figure 3B. This flat band character is seen also along the Γ-M and the K-Γ directions. This is attributable to the combined effects of quantum confinement and the low dimensionality of the layer.

These have a strong influence on the interaction of electrons with local magnetic moments. The bandwidth of this flat band is ∼.05 eV. The magnitude of this band width is far smaller than the error of .2 eV expected from electronic energies obtained from DFT calculations with GPAW. This suggests that the SLC model of the flat bands will strongly contribute to the density of states (DOS) for occupied electrons and can be modelled within the SSQD approximation due to its unpaired electron spin. Presence of such a high DOS in the valence channel around the Fermi level position has crucial implications for carrier transport. This ferromagnetic moment give rise to interesting Kondo phenomena in QDs in ways that are different from the Kondo effect observed in DMAs in 3D materials.

Table 1 summarizes the defect induced local fields at ground state as the number of carbon changes. This shows the electric and magnetic dipole moments, changes in the GGA and GLLBSC band gaps for different nanolines, and derived position of the flat bands. This shows that the distribution of the energy level position of the most dominant flat band states in the nanoline embedded systems is sensitive to the number of carbons embedded in the layer. It is found that the position of the flat conduction band state decreases steadily towards the Fermi level while the flat valence band position is shifted deeper into the valence band as the number of carbons is increased in the layer. Similarly, the magnitude of the electric dipole moment also increases continuously with increasing number of carbons. The electronic structure of the SLC yields a distinctly metallic transport character while all the other embedded systems have a semiconducting character with reduced band gaps. It is important to also note that at the ground state of the SLC-embedded layer, the spin-split eigenvalues for up (↑) and down (↓) spin channel is −20.73912912029137 states/eV.cell and −20.477150085834335, respectively. This culminates in a total magnetic moment of 1.00 μB/cell. This is 3 orders of magnitude higher than the net magnetic moments in other embedded structure. The GLLBSC functional yields a total magnetic moment of 1.00 μB/cell for the SLC system. The energy-level positions of the flat bands are obtained in the vicinity of the Fermi level. In the SLC-embedded system, this was set to the Fermi level (i.e., 0 eV) since there is no band gap.

The positive net magnetic moment in the metallic system is suggestive of a ferromagnetically ordered local moment because the direction of spin quantization is along the z-axis of the monolayer. Thus, the GLLBSC band structure accurately describes the magnetic exchange interactions expected in the induced flat band ferromagnetism. This is important because when electronic spin in important in the tunnelling process making it convenient to view the spin tunneling as a magnetic exchange coupling. Several previous studies of the Hubbard model of ferromagnetism have shown that when the one-electron band structure contains flat bands [74–76], the magnitude of the repulsive electron–electron interaction will guarantee the emergence of a fully spin-polarized ground state whenever the flat band is half-filled. The above prediction of flat band ferromagnetism of the dominate carriers in the polarized spin-state was also confirmed in our previous calculations [69, 77]. This led to the conclusion that only small changes are expected in electron momentum due to deviations from perfect flat band dispersion in the free-standing monolayers compared to stacked multilayers [78]. This suggests that the signatures of Kondo physics in hBN are more likely to be easily delineated in the monolayer systems considered herein than in the multilayers.

The size of the narrowed band gaps of semiconducting sheets in Table 1 shows a strong sensitivity to the number of carbons in the embedded nanoline. With increasing number of carbons in the embedded layer, the overall electric dipole moment gets enhanced. However, the GLLBSC band gap decreases as number of carbons increases and gives rise to a progressively decreasing position of the flat bands for up and down spin channel. In the SLC, the odd number of carbons in the layer means there is an unpaired electron state in the system. This unpaired electron state forms an electronic resonance at the Fermi level leading to zero band gap—as would be expected. All the other embedded systems have a finite but reduced band gap. Only the SLC geometry is characterized by an odd number of carbons. Hence, the unpaired electron is the only physically plausible cause of the band (or itinerant) ferromagnetism observed in the carbon nanolines embedded monolayer. This is quite unlike the more localized Ising-type spins observed in DMAs due to the presence of localized d-band states in transition metal ions. Thus, our recent justification for using the Heisenberg exchange to model the interaction between localized spin moment in stacked monolayer systems is equally applicable in this case [79].

Figure 4 shows the effects of spin orbit coupling (SOC) on the electronic band structure of pristine monolayer hBN in the absence of carbon species. The color bar shows the evolution of the charge pseudospin degree of freedom under the effect of time reversal symmetry. It is crucial to emphasize that the presence of carbon in the hexagonal lattice of the hexagonal boron nitride (hBN) monolayer induces ferromagnetism. The phenomenon of carbon induced magnetism in monolayer hBN has been confirmed both experimentally [80–83] and theoretically [84–86]. Thus, consider too that the carbon-induced ferromagnetism and the intrinsically weak SOC of the nanoline-embedded monolayer enforces a broken time-reversal symmetry on the electron state of the hybrid monolayer. Since the presence of the carbon nanoline also breaks the local structural inversion symmetry of the material, it is the pseudospin ¹/₂ degree of freedom of the electron that gives rise to observable the Kondo effect. Irrespective of whether the QD has a magnetic or non-magnetic ground state, the pseudospin charge density contribution from each valley can be screened by being shifting spontaneously within the monolayer to reveal Kondo-like behavior because of Coulombic charging when the dot is tunnel-coupled to metallic leads.

To gain additional insights for understanding of the effect of the different flavors of the intrinsic broken structural inversion symmetry on the electron state caused by embedding of the carbon nanolines in hBN; Figure 4 shows the band structure of pristine hBN monolayer obtained from PBE (Figure 4A) and from GLLBSC (Figure 4B). The ground state electronic structure shows an intrinsically reversed magnitude of the “charge” pseudospin-½ degree of freedom at the K- and K ’-points of the Brillouin zone. These denote the Brillouin zone points at which the electron state should remain invariant under time-reversal symmetry when the dynamics of the electron is observed in imaginary time under the effect of a symmetry-breaking field such as the SOC. The phenomenon of charge pseudospin reversal also occurs in pristine graphene—even though the two inequivalent sublattice sites are occupied by carbon species only quite unlike boron and nitrogen in monolayer hBN. The top panels of Figure 4 show the reversal of the charge pseudospin from +1.0 (at point K) to −1.0 (at point K’) in the absence of carbon. Hence, introduction of carbon nanoline in the QD influences the pseudospin as well as the valley degrees of freedom for carrier transport [87]. The calculated electronic DOS (bottom panels) in the absence of carbon species shows a consistently wide bang gap—as expected. These indicate that neither the spin nor charge is significant in the physics of the broken symmetry state expected in the Kondo phenomenon reported herein. It is instead the charge pseudospin in the carbon nanoline-embedded layers that needs to be screened by the cloud of valence electrons in the metallic leads of the dots that gives rise to the Kondo resonance effects. This is crucial because the pseudospin of carriers in Dirac-like materials is known to play the central role in determining the generic transport properties in both fully spin-polarized and channel-symmetric two-channel Kondo models [88].

From the experimental perspective, it is now common knowledge nanoelectronics that a new paradigm has emerged more recently wherein a large QD is coupled at quantum point contacts to source and drain leads to realize the charge-Kondo QDs. This perspective allows for realization of more complex QI models to be experimentally by controlling the microscopic interactions without any explicit reference to the localized ferromagnetic spin or spinless fermion. This is encapsulated by the fact that when the original two-channel charge-Kondo model [34] is combined with the pseudogap Kondo physics of graphene [89], one realizes a novel two-channel pseudogap charge-Kondo effect akin to the phenomenon investigated herein. As discussed in Refs. [34, 89, 90], it is the frustrated Kondo screening of the single “impurity-like” state that is caused by the two distinct channels of metallic conduction electrons that displays all the characteristic features of the Kondo physics. This single “impurity-like” state is akin to a screened charge-pseudospin-1/2 state in the QDs considered herein. Thus, even when a gated “spinless” dot is connected to two metallic leads, the frustrated screening of the QD charge pseudospin is the mechanism that realizes the Kondo physics.

The impurity-like state arises from the intrinsic pseudospin SU(4) symmetry of the honeycomb lattice. When there is a coupling of the dot charge pseudospin to two channels of conduction electrons (each with the characteristic linear pseudogap DOS of graphene), the pseudospin screening breaks the particle-hole symmetry in the spin and valley pseudospin degrees of freedom. This physics is investigated here using the SDQD model of the HRC cluster to unravel the Kondo resonance effects. Thus, the results of the first principles calculations shows that insertion of different local geometries of the carbon nanoline conducting channel in the semiconducting hBN monolayer leads to two distinct (i.e., magnetic, and non-magnetic) ground states at 0-K depending on whether the number of embedded carbons is odd or even. This interpretation is consistent with the two distinct ground states correspond to an electronic structure with either a “finite” or “zero” magnetic moment. Based on the obtained ground states, a quantum dot can be modelled with either a “finite” or “zero” magnetic moment. Quantum dots are designed within these two categories of ground states and described using either the spinful single-orbital quantum dot (SSQD) model or spinless double quantum dot (SDQD) model via a tunable coupling to metallic leads—depending on a tunable Coulomb interaction that is scaled by the ratio, V/U, and by using the gate voltage, V=V_g.

Figures 5A,B shows the dependence of the magnitude of the tunneling current on gate voltage and strength of the orbital hybridization in the SDQD model at 0 K. The current is determined using the Pauli, Redfield, first-order von Neumann and Lindblad approaches using two different values of the itype variable in solution method implemented in QmeQ. The results are obtained using Γ= .5, U = 5, Ω = 2 as the simulation parameters. SDQD model data are obtained using energy position of 1.50 eV determined from the spin-split level of the two Fermi-level resonant flat bands at the Brillouin zone point K in the SLC nanoline model (see Figure 2B). Figure 4 shows the performance of different quantum master equation in the SDQD model for itype = 1. The colourmap denotes the distribution of the current at different charging potential. Note that setting itype = 0 yields a zero-tunnelling current irrespective of the simulation temperature. Hence, only the non-vanishing currents obtained with Pauli and Lindblad master equation for itype = 1 are displayed in Figures 4A,B.

Nevertheless, it is also important to emphasize that the results presented here for the SDQD are not sensitive to values of the itype parameter. The Pauli and Lindblad master equations yield a similar non-vanishing current contour while the first order von Neumann (1vN) and Redfield master equations give rise to zero current. However, the Pauli method is known to accurately reproduce the current profile of the and Lindblad approaches only within the high-charging regime of the SDQD model. In Ref. [54] for instance, the current predicted with the Pauli master equation for the SDQD model is anomalously high in the limit of minimal (or low) charging of the QD wherein Ω/Г→ 0 irrespective of the value of the itype variable. The non-zero current obtained when Ω/Г < 0 suggests a finite current flow at negative charging. This trend is not seen obtainable with other master equations. Secondly, the tunnelling current behaves quantum quantum-mechanically. In this sense, changes in the current contour shows an asymptotic jump between maximal values and a minimal value in both Pauli and Lindblad methods. This is denoted by the sequential change in the color of the current contour from white at the dotted spots through the red line to the blue region at both values of 0 and 1 for the itype parameter. This characterizes the discrete quantal behavior of current flowing in the QD at 0 K. This shows that only a discrete number of electrons are transported across the quantum dot.

Figures 6A–D show contours of the current profile at 1 μK derived from the four quantum master equations. At extremely low but non-zero temperatures important differences are observed in each case. It is observed that the Redfield and first order von Neumann master equations only give a finite (i.e., nonzero) current at extremely low (but nonzero) temperatures, in each case. This shows that the Redfield and first order von Neumann currents match the Pauli and Lindblad currents exactly at the high values of ratio Ω/Г. When the Pauli and Lindblad currents are compared to the 0 K (see Figures 5A,B), it is observed that the linearly distributed white dots on blue the background has disappeared. This leaves only the localized white spots seen on the red contour line. When the strength of the electronic coupling between the flat band states is set to Ω/Г ≈ 1.5, two distinct localized regions of high tunnelling current are observed around the region where the Coulomb potential (denoted by V_g/U) is −1 and 5. These two regions are denoted by the two regions of smoothly graded white halos in Figures 6B,C. Otherwise, all the four forms of the quantum master equation yield the same Kondo current. The results show that the Kondo current survives up to 10 K before it disappears. The other signatures of the Kondo resonance are studied using the Lindblad formalism.

It is well-known in the Kondo physics of a dilute magnetic alloy that at a characteristic temperature, which is also known as the Kondo temperature, the resistivity (or conductance) increases (or decreases) abruptly at a low temperature starting near the Kondo temperature. This phenomenon depends on the energy position of the impurity band state and on the nature of the metallic host in DMAs. However, it has recently been associated with the impurity-induced resonance peaks at the Hubbard band edges due to an isolated magnetic atom deposited onto monolayer 1T-TaSe₂ [94, 95]. Experimental measurements show that the observed Kondo resonance peaks disappear when the spatial overlap at the band edges is reduced or when the magnetic impurities are replaced with non-magnetic impurities. These effects are ascribed to the presence of spinons. A spinon is an exotic charge-neutral quasiparticle present only in the quantum spin liquid phase. As such, we have also studied the Kondo resonance effect associated with the effect of temperature on the conductance under zero and finite magnetic field. This includes the carrier stability diagram and the associated effects of temperature and magnetic field. Evidence of the tunneling transport of a well-defined number of electrons through the dot is obtainable from the charge stability diagram.

Figure 7 shows the charge stability diagram of a QD at four different but evenly separated low temperatures under zero magnetic field. Here, the Coulomb diamonds are labelled to show the quantized energy levels or microstates that are accessible to the electron. Firstly, there is considerable asymmetry in the size of the diamonds, i.e., the available energy levels. The size of the diamonds at N = 0 and N = 1 is not equal. This asymmetry is attributable to the two-fold degeneracy expected when the available energy levels in the HPM are filled with electrons at the valleys located at K and K’ in the Brillouin zone. Clearly, temperature has a non-trivial effect on the boundaries of the diamonds but not on their size. Secondly, negative charging yields an empty QD because the diamond labelled N = 0 does not close at all temperatures. At the center, the diamond closes at all temperatures with a relatively low energy penalty for adding the first electron. Secondly, excited states are visible outside the Coulomb diamonds. These appear near the diamonds as off-resonance conduction. This Kondo resonance effect is low at 0 K. However, the signature can be seen clearly up to .1 K (see Figure 7B). The edges of the Coulomb diamonds are no longer sharp at 1 K and the effects becomes more pronounced at 10 K. The smeared coulomb diamond edges are suggestive of the thermal broadening of accessible energy levels; Figures 6B–D illustrates the broadening of excited states in the interval between .1 and 10 K. The broadened off-resonance conduction characterizes the effect of temperature on quantized dc conductance at B = 0.

The color bar in Figure 7 depicts the magnitude of the conductance. It is observed that the dc conductivity saturates at the peak value of 5 S/m at 0 K. As temperature increases to 1 K the conductance drops to 14 S/m and this decreases further to 025 S/m at 10 K. The results show that the dc conductance survives up to 10 K before it disappears. The above trend in the results agrees temperature dependence of the Kondo resonance effect of a Co atom observed experimentally by Otte, et al; [96]. Their experiments demonstrated that a single Co atom that is bound to the Cu₂N surface shows a sharp zero-voltage peak in its conductance spectrum. This peak was attributed to increased density of electronic states near the Fermi level. The expected high DOS around the Fermi level was ascribed to the effect of the Kondo screening of the Co impurity. In our case, the absence of a magnetic impurity means that the high DOS originated explicitly from the small bandwidth of the flat bands.

To gain experimentally useful insights into the transport calculations, it is important to recall that when the QD model (either SSQD or SDQD) is connected as shown in the measurement geometry of Figure 1, it will gain access to an effective electron base temperature (T_bas), which cannot be lower than the Kondo temperature, T_K. With experimental QD devices made from the GaAs/AlGaAs gated structure, a T_base of more than 45 mK was measured using a variable dc bias and/or an ac bias of 1 mV at 13.5 Hz for a lock-in detection [57]. This was experimentally measured from the widths of the conduction band peaks in the weak tunneling regime. The expected pseudospin coupling interactions that give rise to localized single electron states (shown in Figure 7A) are accessible experimentally even though the Kondo temperature T_K of the system is well-above 0 K. This is crucial because the effective tunnelling rate Г of the coupled QD system is a function of the Coulomb energies U, as well as the tunnelling rates of the right ( Г R ) and left ( Г L ) leads and these combine to determine the T_K using [97], T K = U Г L Г R Г L + Г R   exp π ( Г L + Г R ) ( E − E F 2 Г L Г R   (17) where the Coulomb energy U and the Fermi energy E_F are both determined from the DFT calculations. Thus, in the QD charging limit where interactions are introduced (i.e., H_Coulomb ≠ 0), parameters ( Г R ) and left ( Г L ) are tunable to the required weak to moderate coupling strengths when different materials are used as leads ( Г R ≠ Г L ) by setting the gate so that the conduction band peaks overlap. Alternatively, the same material can be used as leads ( Г R = Г L ) to access the same physics. For instance, these present calculations show that setting V_gate = −U/2 and B = .375U localizes a single electron state to the experimentally accessible particle-hole symmetric point. This is because during charge coupling, the pseudospin contribution is only significant when T_K is larger than (or equal to) the combined tunnelling rate, Γ = Г L Г R / Г L + Г R . The predicted Kondo temperature of the two-quantum dot (i.e., SSQD and SDQD) models is 10 K and 50 K suggesting that T_K is structure tunable. These are consistent with the experimental T_K of ∼4 K evaluated from the width of zero-bias conductance peak of graphene QDs obtained by electron-beam lithography and reactive-ion etching [98], as well as the tunable T_K of up to ∼90 K in defective graphene QDs [99].

Figure 8A shows the summary of the changes in dc conductance with temperature when the magnetic field is set to B = .375U in comparison to B = 0. Figure 8B shows the magnetic field independent conductance of the quantum dot at 40 K. The applied magnetic field has minimal effects on the single electron conductance. However, as more electrons are added to the accessible energy levels, it is expected that magnetic field is capable of form Landau levels under strong magnetic fields. It is noted too that raising the temperature up to 40 K at a fixed value of B = .375 U only reduces the conductance to 009 S/m (see Figure 8B). The observed decrease in the dc conductance peak with increasing temperature is a systematic trendline. It does not depend on the applied magnetic field. This agrees with the arguments of thermally broadened Fano lineshape introduced to explain the temperature dependence of Kondo resonance effects of a single Kondo impurity [100].

Recently, Lu, et al. [101] have reviewed the methods for transforming energy using QDs. The dependence of particle-hole symmetry on gate voltage has crucial implications in typical energy conversion applications. Consider that the Hamiltonian of the Hubbard model is symmetric under the exchange of particles and holes [102]. The Hamiltonian (see Eq. (1)) has a fascinating particle-hole symmetry (PHS) that allows to establish a direct relationship between its emergent physical properties at different values of the Hamiltonian tuning parameters. In this case, particle-hole symmetry denotes the anti-unitary operation which anti-commutes with the Hamiltonian. Since the local network structures of the QD systems considered herein preserve to the local symmetry of the Brillouin zone valleys at KK’, the symmetry property of the electronic structure enforces the constraint ε_k = ε _−k on the electronic structure as the necessary condition for particle-hole symmetry. Thus, electrons and holes feel the same Coulombic interactions in the material without changing their effective mass, conductivity, and response to bosonic fields. This only happens at the particle-hole symmetry point when carrier transport preserves time reversal symmetry.

Figure 9 shows the bias trace at the particle–hole symmetric point V_gate = −U/2 for B = .375 U for the SLC (Figure 9A) and HRC (Figure 9B) models. These parameters are chosen to simulate the response of the single electron that occupies the n = 1 C diamond in both QD models (Figure 7A). Results are also shown from calculation done with the Pauli and second order von Neumann (2vN) methods, in each case. In Figure 9A, the dc conductance rises exponentially. There is no peak conductance since does not saturate. This signature is characteristic of the absence of the global symmetry that would have been expected on time reversal. This can also be attributed to the presence of the broken symmetry states in the transport phase. By contrast; Figure 9B shows conductance peaks at V/U = ∼1.3, both Pauli and 2vN methods give a well-defined peak at the particle-hole symmetry point. This peak characterizes the global symmetry of the transport state wherein electrons and holes feel the same interactions without changing. Interpretation of these results overlaps with our recent proposal for the dynamic bosonization of the spin-½ transport state in interaction-driven lattices [103]. In this sense, artificial gauge field coupling to the transport state of carriers create quasiparticle states during the ultrafast process of energy interconversion. Although the proposal for dynamic bosonization has been implemented within cavity quantum electrodynamics, compelling evidence for the formation of quasiparticle-like transport state [i.e., the charge (or spin) density wave state] was reported recently in an electromagnetically driven Dirac materials-based metamaterial [73].

Figure 8A shows the effect of Zeeman spin splitting magnetic field on the “broad-spectrum” of the tunneling conductance. By contrast, Figure 9B shows the effect of same Zeeman splitting magnetic field on the electronic structure of the localized electron state at the specific pair of points (K and K’) in the Brillouin zone of the HRC-model of the QD (see Figure 4) where particle-hole symmetry would have been expected to hold. The clear lack of particle-hole symmetry in Figure 8B means that the local structure of the QD model is capable of hosting quasiparticle states. Such states dominate the second-order scattering processes that mediate energy interconversion using quantum dots.

To understand the non-equilibrium physics of the energy interconversion process, consider that Hoffmann [104] has recently calculated the electromagnetic response of carriers in the half-filled Landau level. The calculations used the effective field theory proposed for composite fermion by Son [105]. The calculated electromagnetic response shows a cancellation or suppression of spurious interband transitions and collective modes that are present in the unmodified theory when the dipole terms from composite fermions is included in the calculation. It is crucial to note that cancelled (or suppressed) collective modes in the electromagnetic response are quasiparticles. The quasiparticles denote the bosonic field configurations of broken symmetry states in the Chern-Simons field theory [106, 107]. Within the framework for bosonization of the carrier transport state, the appearance of a collective transport state also applies to a QD when there is particle-hole asymmetry, i.e., εk ≠ ε−k. This is realizable in practice because gauge fields that preserve the U (1) gauge and translational symmetries of the lattice can be obtained by modifying the local atomic geometry of the HPM with a nanoline or by modulating the Bloch state of carriers by coupling them to bosonic fields.

Thus, to describe the fast processes of energy interconversion within the Hubbard model, an interaction term must be introduced in the Hamiltonian for coupling the carriers to artificial gauge fields in the QD. This reduces the Hubbard model to a many-body problem. Figure 8A suggests that the SLC-based QD system has comparably higher efficacy when integrated in energy applications. The above analyses allow for unique insights in to the non-equilibrium transport of electrons with a spin polarization in 2D materials. It is advantageous especially for understanding the effects of including the non-equilibrium magnetic moments in the energy transport mechanism. This is because the observed Kondo resonance effects are driven mainly by differences in chemical potentials of the QD model and by differences in temperature when a QD is tunnel coupled to metallic leads. Zirnbauer [108] has clarified the related particle-hole conjugation as an algebra automorphism that swaps single-fermion creation and annihilation operators. This was used to construct its invariant lift to the Fock space. This permits PHS to emerge as the concatenation of particle-hole conjugation with one (or another) involution that reverses the sign of the first quantized Hamiltonian.

The presence of broken PHS in the transport state is observed with the HRC-based QD model. This underscores the fact that it is experimentally feasible to form quasiparticles in the ultrafast energy conversion process using a suitably designed QD local structure. The above framework allows the PHS to be understood as a function of QD tunning parameters. Thus, the local network structure of the carbon nanoline based QD has a non-trivial effect on the particle–hole symmetric point. At DFT-level, the point is the time-reversal symmetric points of the Brillouin zone. The pseudospin state at points KK ’ (see Figure 4) indicated that this is the point in which the macroscopic “charge” pseudospin ½ symmetry of the single electron state is broken dynamically due to the charging effect. The broken particle-hole symmetry state is observed in the bias trace of the HRC (Figure 9B). This therefore supports the expected formation of the quasiparticle states through second-order scattering processes expected when the QDs are used in devices to interconvert energy. Since the transported energy current is a sum of kinetic and mass (i.e., excitation) energy but not potential energy [109]. The above interpretation of the role of quasiparticle states is consistent with recent experimental evidence for a non-vanishing quasiparticle population during spin dependent transport of energy in a superconductor [110]. Thus, the presence of such broken symmetry state has crucial effects on energy interconversion with HPM-based QDs. This is because the energy current is obtainable as a sum of the Poynting vector, the kinetic energy flow and mass energy flow of the particles. It is physically plausible that the Coulombic many-body interactions in the QD renormalize the single electron state in the presence of quasiparticles leading to a dynamic bosonization of the single electron state.

Some experimental methods of energy interconversion with graphene quantum dots (GQDs) are derivable from their physical and chemical properties leading to broad prospects in energy storage and conversion. For instance, in addition to the excellent properties of graphene, GQDs also have quantum confinement effects and edge effects. The size of GQDs and types of their edges determine their properties [111]. Carbon-based QDs can generate electron-hole pairs, and well as act like an electron sink to suppress electron-holes pairs recombination. Due to their high carrier conductivity, the graphene-based QDs can effectively transport photo-generated electrons through its conductive networks in solar cells [112, 113]. Non-etheless, the main arguments advanced herein for the experimental realization of energy interconversion with QDs is from the perspective of light-matter interaction in cavity quantum electrodynamics [114]. When the quasiparticle (e.g., a photon field) couples coherently to the tunnelling ‘single’ electron from the QD (see Figure 7A), the electron will be confined within a single mode cavity. The reduced dimensionality of the QD enables the occurrence of few-level systems with strong non-linear features akin to those that occur in the process of energy conversion. This facile process can lead to the formation of polarons, excitons or plasmons, etc as the quasiparticles that play major roles in the energy conversion using a photovoltaic [115, 116], optoelectronic [117, 118], excitonic [119, 120] or photonic [121, 122] device. These analyses suggest that a suitable device can be designed to implement the weak or moderate couplings of the QD to specific bosonic (i.e., quasiparticle) modes, which is realizable in practice with the single photon coupling mode at minimized dissipation [103].