We theoretically investigate the absorption and emission of light carrying orbital angular momentum (twisted-light) by quasi-two-dimensional (disc-shaped) quantum dots in the presence of a static magnetic field. We calculate the transition matrix element for the light-matter interaction and use it to explore different scenarios, depending on the initial and final state of the electron undergoing the optically-induced transition. We make explicit the selection rule for the conservation of the z-projection of the orbital angular momentum. For a realistic set of parameters (quantum dots size, beam waist, photon energy, etc.) the strength of the transition induced by twisted light is 10% of that induced by plane-waves. Finally, our analysis indicates that it may be possible to select precisely the electronic level one wishes to populate using the appropriate combination of light-beam parameters suggesting technological applications to the quantum control of electronic states in quantum dots.
It has been theoretically predicted that light carrying orbital angular momentum, or twisted light, can be tuned to have a strong magnetic-field component at optical frequencies. We here consider the interaction of these peculiar fields with a semiconductor quantum dot and show that the magnetic interaction results in new types of optical transitions. In particular, a single pulse of such twisted light can drive light-hole-to-conduction band transitions that are cumbersome to produce using conventional Gaussian beams or even twisted light with dominant electric fields.
We investigate the Aharonov-Bohm (AB) effect for the double quantum dots in the Kondo regime using the slave-boson mean-field approximation. In contrast to the non-interacting case, where the AB oscillation generally has the period of 4$π$ when the two-subring structure is formed via the interdot tunneling $t_c$, we find that the AB oscillation has the period of 2$π$ in the Kondo regime. Such effects appear for the double quantum dots close to the T-shaped geometry even in the charge-fluctuation regime. These results follow from the fact that the Kondo resonance is always fixed to the Fermi le…
Correlation effects on electron transport through a system of T-shaped double-dots are investigated, for which only one of the dots is directly connected to the leads. We evaluate the local density of states and the conductance by means of the non-crossing approximation at finite temperatures as well as the slave-boson mean field approximation at zero temperature. It is found that the dot which is not directly connected to the leads considerably influences the conductance, making its behavior quite different from the case of a single-dot system. In particular, we find a novel phenomenon in the…
We calculate the conductance through strongly correlated T-shaped molecular or quantum dot systems under the influence of phonons. The system is modelled by the extended Anderson-Holstein Hamiltonian. The finite-U mean-field slave boson approach is used to study many-body effects. Phonons influence both interference and correlations. Depending on the dot unperturbed energy and the strength of electron-phonon interaction the system is occupied by a different number of electrons which effectively interact with each other repulsively or attractively. This leads together with the interference effe…
Phonons strongly influence the optical control of semiconductor quantum dots. When modeling the electron-phonon interaction in several theoretical approaches the quantum dot geometry is approximated by a spherical structure, though typical self-assembled quantum dots are strongly lens-shaped. By explicitly comparing simulations of a spherical and a lens-shaped dot using a well-established correlation expansion approach we show that indeed lens-shaped dots can be exactly mapped to a spherical geometry when studying the phonon influence on the electronic system. We also give a recipe to reproduc…
We developed a set of equations to calculate the electronic Green's functions in a T-shaped multi-quantum dot system using the equation of motion method. We model the system using a generalized Anderson Hamiltonian which accounts for {\em finite} intradot on-site Coulomb interaction in all component dots as well as for the interdot electron tunneling between adjacent quantum dots. Our results are obtained within and beyond the Hartree-Fock approximation and provide a path to evaluate all the electronic correlations in the multi-quantum dot system in the Coulomb blockade regime. Both approximat…
We study the spin-resolved transport properties of T-shaped double quantum dots coupled to ferromagnetic leads. Using the numerical renormalization group method, we calculate the linear conductance and the spin polarization of the current for various model parameters and at different temperatures. We show that an effective exchange field due to the presence of ferromagnets results in different conditions for Fano destructive interference in each spin channel. This spin dependence of the Fano effect leads to perfect spin polarization, the sign of which can be changed by tuning the dots' levels.…
We study the effects of intraband relaxation processes on optical manipulation protocols for sp biexcitons hosted by CdTe disk-shaped quantum dots. The many-body states are calculated within the configuration interaction method starting from single-particle states given by the kp theory. The time-dependent occupations of relevant many-body states are extracted from the von Neumann-Lindblad equation for the density operator. We mainly investigate the generation of sp biexcitons with two pulses of different polarizations sigma_{+} and sigma_{-}. The fast hole relaxation processes prevent a high-…
We study Rabi rotations arising in intensity-dependent photon echoes from an ensemble of self-assembled InGaAs quantum dots. To achieve a uniform distribution of intensities within the excited ensemble, we introduce flattop intensity profiles of picosecond laser pulses. This allows us to overcome the damping of Rabi rotations imposed by the spatial inhomogeneity of Rabi frequencies by a Gaussian laser profile. Using photon echo polarimetry, we distinguish between the coherent optical responses from exciton and trion ensembles. Here, we demonstrate that a photo-induced charging of the quantum d…
We examine the influence of the superconducting proximity effect on the transport properties of a T-shaped double quantum dot strongly coupled to two normal, nonmagnetic or ferromagnetic leads. We show that the two-stage Kondo screening may be suppressed or enhanced by the presence of pairing correlations, depending on the specific geometric arrangement of the device. We explain our results by invoking effective decrease of Coulomb interactions by proximity effect and find qualitatively correct description in many cases, although spin-filtering effect stemming from spin-dependent Fano-Kondo in…
We theoretically investigate the thermoelectric and spin thermoelectric properties of a T-shaped double quantum dot strongly coupled to two ferromagnetic leads, focusing on transport regime where the system exhibits the two-stage Kondo effect. We study the dependence of the (spin) Seebeck coefficient, the corresponding power factor and the figure of merit on temperature, leads' spin polarization and dot level position. We show that the thermal conductance fulfills a modified Wiedemann-Franz law. We also demonstrate that the spin thermopower is enhanced at temperatures corresponding to the seco…
In this work, the effect of size and wetting layer on subband electronic envelop functions, eigenenergies, linear and nonlinear absorption coefficients and refractive indices of a dome-shaped InAs/GaAs quantum dot were investigated. In our model, a dome of InAs quantum dot with its wetting layer embedded in a GaAs matrix was considered. A finite height barrier potential at the InAs/GaAs interface was assumed. To calculate envelop functions and eigenenergies, the effective one electronic band Hamiltonian and electron effective mass approximation were used. The linear and nonlinear optical prope…
In this paper, optical and electronic properties of diamond shaped graphene quantum dots (DQDs) have been studied by employing large-scale electron-correlated calculations. The computations have been performed using the π-electron Pariser-Parr-Pople model Hamiltonian, which incorporates long-range Coulomb interactions. The influence of electron-correlation effects on the ground and excited states has been included by means of the configuration-interaction approach, used at various levels. Our calculations have revealed that the absorption spectra are red-shifted with the increasing sizes of qu…
Experimental realization of graphene-based stadium-shaped quantum dots (QDs) have been few and incompatible with scanned probe microscopy. Yet, direct visualization of electronic states within these QDs is crucial for determining the existence of quantum chaos in these systems. We report the fabrication and characterization of electrostatically defined stadium-shaped QDs in heterostructure devices composed of monolayer graphene (MLG) and bilayer graphene (BLG). To realize a stadium-shaped QD, we utilized the tip of a scanning tunneling microscope to charge defects in a supporting hexagonal bor…
We study the influence of a magnetic flux tube on the possibility to electrostatically confine electrons in a graphene quantum dot. Without magnetic flux tube, the graphene pseudospin is responsible for a quantization of the total angular momentum to half-integer values. On the other hand, with a flux tube containing half a flux quantum, the Aharonov-Bohm phase and Berry phase precisely cancel, and we find a state at zero angular momentum that cannot be confined electrostatically. In this case, true bound states only exist in regular geometries for which states without zero-angular-momentum co…
Electronic properties of quantum dots (QDs) depend sensitively on their parent materials. Therefore, confined electronic states in graphene QDs (GQDs) of monolayer and Bernal-stacked bilayer graphene are quite different. Twisted bilayer graphene (TBG) is distinct from monolayer and Bernal-stacked bilayer graphene because of the new degree of freedom: twist angle. In the past few years, numerous efforts have been made to realize the GQDs of monolayer and Bernal-stacked bilayer graphene and achieved great success. Thus far, however, strategies for realizing GQDs of TBG have been elusive. Here, w…
Finding out how symmetry encodes optical polarization information into the selection rule in molecules and materials is important for their optoelectronic applications including spectroscopic analysis, display technology and quantum computation. Here, we extend the polarization-dependent selection rules from atoms to solid systems with point group descriptions via rotational operator for circular polarization and $2$-fold rotational operator (or reflection operator) for linear polarization. As a variant of graphene quantum dot (GQD), twisted bilayer graphene quantum dot (TBGQD) certainly inher…
Moiré superlattices in twistronic heterostructures are a powerful tool for materials engineering. In marginally twisted (small misalignment angle, $θ$) bilayers of nearly lattice-matched two-dimensional (2D) crystals moiré patterns take the form of domains of commensurate stacking, separated by a network of domain walls (NoDW) with strain hot spots at the NoDW nodes. Here, we show that, for type-II transition metal dichalcogenide bilayers MoX$_2$/WX$_2$ (X=S, Se), the hydrostatic strain component in these hot spots creates quantum dots for electrons and holes. We investigate the electron/hole …
We derive a material-realistic real-space many-body Hamiltonian for twisted bilayer graphene from first principles, including both single-particle hopping terms for $p_z$ electrons and long-range Coulomb interactions. By disentangling low- and high-energy subspaces of the electronic dispersion, we are able to utilize state-of-the-art constrained Random Phase Approximation calculations to reliably describe the non-local background screening from the high-energy $s$, $p_x$, and $p_y$ electron states for arbitrary twist angles. The twist-dependent low-energy screening from $p_z$ states is subsequ…
Free standing beams containing a two-dimensional electron system are shaped from a GaAs/AlGaAs heterostructure. Quantum point contacts and (double) quantum dots are laterally defined using metal top gates. We investigate the electronic properties of these nanostructures by transport spectroscopy. Tunable localized electron states in freely suspended nanostructures are a promising tool to investigate the electron-phonon-interaction.
We analyze the zero temperature conductance of a parallel T-shaped double quantum dot system. We present an analytical expression for the conductance of the system in terms of the total number of electrons in both quantum dots. Our results confirm that the system's conductance is strongly influenced by the dot which is not directly connected to the leads. We discuss our results in connection with similar results reported in the literature.
This letter is devoted to the optimization of the Cooper pair splitting efficiency in a Y-shaped junction. The latter consists of two quantum dots, one superconducting and two normal leads. We tailor the bias in the two normal leads such that the Cooper pairs leaving the superconductor are split up resulting in entangled electrons, one on each quantum dot. We are able to achieve a splitting efficiency of more than 99% which is significantly better than the efficiencies obtained in experiments so far.
In quasi-1D $π$-conjugated polymers such as \emph{trans}-polyacetylene and polyenes, electron correlation effects determine the "reversed" excited state ordering in which the lowest two-photon $2A_{g}$ state lies below the lowest one-photon $1B_{u}$ state. In this work, we present conclusive theoretical evidence of reversed excited state ordering in fairly 2D $π$-conjugated systems, namely, diamond-shaped graphene quantum dots (DQDs). Our electron correlated calculations show that DQDs begin to exhibit reversed excited ordering with increasing size, in disagreement with independent-particle pi…
We engineer a system of two strongly confined quantum dots to gain reproducible electrostatic control of the spin at zero magnetic field. Coupling the dots in a tight ring-shaped potential with two tunnel barriers, we demonstrate that an electric field can switch the electron ground state between a singlet and a triplet configuration. Comparing our experimental co-tunneling spectroscopy data to a full many-body treatment of interacting electrons in a double-barrier quantum ring, we find excellent agreement in the evolution of many-body states with electric and magnetic fields. The calculations…
We systematically investigate electron transport through double quantum dots with particular emphasis on interference induced via multiple paths of electron propagation. By means of the slave-boson mean-field approximation, we calculate the conductance, the local density of states, the transmission probability in the Kondo regime at zero temperature. It is clarified how the Kondo-assisted transport changes its properties when the system is continuously changed among the serial, parallel and T-shaped double dots. The obtained results for the conductance are explained in terms of the Kondo reson…
The transport behavior of a double quantum dot side-attached to a topological superconducting wire hosting Majorana zero-energy modes is studied theoretically in the strong correlation regime. It is shown that Majorana modes can leak to the whole nanostructure, giving rise to a subtle interplay between the two-stage Kondo screening and the half-fermionic nature of Majorana quasiparticles. In particular, the coupling to the topological wire is found to reduce the effective exchange interaction between the two quantum dots in the absence of normal leads. Interestingly, it also results in an enha…
We present a theoretical investigation of the optical properties of polaronic excitons in stacked self-assembled quantum dots, which is based on the non-adiabatic approach. A parallelepiped-shaped quantum dot is considered as a model for a self-assembled quantum dot in a stack. The exciton-phonon interaction is taken into account for all phonon modes specific for these quantum dots (bulk-like, half-space and interface phonons). We show that the coupling between stacked quantum dots can lead to a strong enhancement of the optical absorption in the spectral ranges characteristic for phonon satel…
Computational studies based on 6 band k$\cdot$p theory are employed on lens-shaped III-nitride quantum dots (QDs) with focus on the polarization properties of the optical interband transitions. The results predict pronounced linear polarization of the ground-state related transitions for asymmetric QDs of a material with small split-off energy. It is demonstrated that a moderate externally applied electric field can be used to induce a linear polarization and to control its direction. InN is found to be the most efficient choice for dynamic polarization switching controlled by an electric fiel…
The spin-polarized transport through a coherent strongly coupled double quantum dot (DQD) system is analyzed theoretically in the sequential and cotunneling regimes. Using the real-time diagrammatic technique, we analyze the current, differential conductance, shot noise and tunnel magnetoresistance (TMR) as a function of both the bias and gate voltages for double quantum dots coupled in series, in parallel as well as for T-shaped systems. For DQDs coupled in series, we find a strong dependence of the TMR on the number of electrons occupying the double dot, and super-Poissonian shot noise in th…
We theoretically analyze the effect of electron-electron interactions on Aharonov-Bohm (AB) current oscillations in ring-shaped systems with metallic quantum dots pierced by external magnetic field. We demonstrate that electron-electron interactions suppress the amplitude of AB oscillations $I_{AB}$ at all temperatures down to T=0 and formulate quantitative predictions which can be verified in future experiments. We argue that the main physical reason for such interaction-induced suppression of $I_{AB}$ is electron dephasing while Coulomb blockade effects remain insignificant in the case of me…
Using the tight-binding approximation we calculated the magnetic susceptibility of graphene quantum dots (GQD) of different geometrical shapes and sizes, smaller than the magnetic length, when the magnetic properties are governed by the electron edge states. Two types of edge states can be discerned: the zero-energy states (ZES) located exactly at the zero-energy Dirac point, and the dispersed edge states (DES) with the energy close, but not exactly equal to zero. DES are responsible for the temperature independent diamagnetic response, while ZES provide the temperature dependent spin Curie pa…
We theoretically study the optical properties and the electronic structure of highly elongated quantum dots (quantum dashes) and show how carrier trapping due to geometrical fluctuations of the confining potential affects the excitonic spectrum of the system. We focus on the study of the optical properties of a single exciton confined in the structure. The dependence of the absorption and emission intensities on the geometrical properties (depth and position) of the trapping center of a quantum dash is analyzed and the dependence of the degree of linear polarization on these geometrical parame…
Electronic transport properties of two quantum dots side-coupled to a quantum wire are studied by means of the two impurity Anderson Hamiltonian. The conductance is found to be a superposition of a Fano and a Breit-Wigner resonances as a function of the Fermi energy, when the gate voltages of the quantum dots are slightly different. Under this condition, we analyze the time evolution of a Gaussian-shaped superposition of plane waves incoming from the source lead, and found that the wave packet can be splitted into three packets at the drain lead. This spatial pattern manifests in a direct way …
The Fano-Kondo effect in zero-bias conductance is investigated based on a theoretical model for the T-shaped quantum dot. The conductance as a function of the gate voltage is generally characterized by a Fano asymmetric parameter q. With varying temperature the conductance shows a crossover between the high and low temperature regions compared with the Kondo temperature T_K: two Fano asymmetric peaks at high temperatures and the Fano-Kondo plateau inside a Fano peak at low temperatures. Temperature dependence of conductance is calculated numerically by the Finite temperature density matrix ren…
We explore the quantum transmission through open oval shaped quantum dots. The transmission spectra show periodic resonances and, depending on the geometry parameter, a strong suppression of the transmission for low energies. Applying a weak perpendicular magnetic field changes this situation drastically and introduces a large conductance. We identify the underlying mechanisms being partially due to the specific shape of the oval that causes a systematic decoupling of a substantial number of states from the leads. Importantly a pairwise destructive interference of the transmitting states is en…
We investigate the magnetic interface states of graphene quantum dots that contain p-n junctions. Within a tight-binding approach, we consider rectangular quantum dots in the presence of a perpendicular magnetic field containing p-n, as well as p-n-p and n-p-n junctions. The results show the interplay between the edge states associated with the zigzag terminations of the sample and the snake states that arise at the p-n junction, due to the overlap between electron and hole states at the potential interface. Remarkable localized states are found at the crossing of the p-n junction with the zig…
In this work the intersubband electronic properties of two laterally coupled dome-shaped InAs/GaAs quantum dots were investigated. The envelope functions and eigenenergies were calculated as function of distance between the dots. The coupling between the dots was studied using transition lifetime between the dots. The results showed that in close distances (smaller than 3 nm) the quantum dots are coupled and by increasing the distance transition lifetime fall down drastically and the dots become uncoupled.
We investigate theoretically the electron spin states in disk-shaped HgTe topological insulator quantum dots (TIQDs) containing a single magnetic $Mn^{2+}$ ion. We show that the energy spectrum and the electron density distribution of the topological edge states in HgTe TIQD can be modulated significantly by the position of the magnetic $Mn^{2+}$ ion. The numerical results further demonstrate that the electric fields not only tune the spin splittings of edge states via the $\emph{sp-d}$ exchange interaction between the electron (hole) and the magnetic $Mn^{2+}$ ion but also give rise to the br…
A resonance behaviour of the far infrared absorption probability at a frequency \sim N^{1/4} is predicted for clusters of N electron-hole pairs (2\le N\le 110) confined in disk-shaped quantum dots. For radially symmetric dots, the absorption is dominated by a Giant Dipole Resonance, which accounts for more than 98 % of the energy-weighted photoabsorption sum rule.
We report temperature-dependent photoluminescence (PL) studies and PL relaxation dynamics for uncoated and silver-coated CdS quantum dots (QDs). Room-temperature photoluminescence studies indicate that the presence of silver coating on CdS QD samples enhances the blue luminescence corresponding to the band-to-band transitions by a factor of ten compared to the uncoated samples. Furthermore, the decay time measurements using Time Correlated Single Photon Counting technique demonstrate Förster resonance energy transfer (FRET) between silver-shell and CdS-core in silver-coated CdS QD samples. A d…
We propose phonon spectroscopy by electric measurements of the low-temperature conductance of coupled-quantum dots, specifically employing dephasing of the quantum electronic transport by the phonons. The setup we consider consists of a T-shaped double-quantum-dot (DQD) system in which only one of the dots (dot 1) is connected to external leads and the other (dot 2) is coupled solely to the first one. For noninteracting electrons, the differential conductance of such a system vanishes at a voltage located in-between the energies of the bonding and the anti-bonding states, due to destructive in…
We show that problem of interacting electrons in a quantum dot with chaotic boundary conditions is solvable in the large-g limit, where g is the dimensionless conductance of the dot. The critical point of the $g=\infty$ theory (whose location and exponent are known exactly) that separates strong and weak-coupling phases also controls a wider fan-shaped region in the coupling-1/g plane, just as a quantum critical point controls the fan in at T>0. The weak-coupling phase is governed by the Universal Hamiltonian and the strong-coupling phase is a disordered version of the Pomeranchuk transition i…
The embedding method for the calculation of the conductance through interacting systems connected to single channel leads is generalized to obtain the full complex transmission amplitude that completely characterizes the effective scattering matrix of the system at the Fermi energy. We calculate the transmission amplitude as a function of the gate potential for simple diamond-shaped lattice models of quantum dots with nearest neighbor interactions. In our simple models we do not generally observe an interaction dependent change in the number of zeroes or phase lapses that depend only on the sy…
We investigate the thermoelectric properties of a T-shaped double quantum dot system described by a generalized Anderson Hamiltonian. The system's electrical conduction (G) and the fundamental thermoelectric parameters such as the Seebeck coefficient ($S$) and the thermal conductivity ($κ$), along with the system's thermoelectric figure of merit (ZT) are numerically estimated based on a Green's function formalism that includes contributions up to the Hartree-Fock level. Our results account for finite onsite Coulomb interaction terms in both component quantum dots and discuss various ways leadi…
Spin-orbit effects on electron-electron interaction are studied theoretically. The corrections to the Coulomb interaction of quantum well electrons induced by the spin-orbit coupling are derived. The developed theory is applied to calculate the energy spectrum fine structure of an electron pair triplet states localized in small lateral disk-shaped quantum dots. We show that the spin degeneracy of a triplet state is completely lifted in anisotropic quantum dots. Isotropic quantum dots also demonstrate peculiar fine structure of triplet states caused by the spin-orbit interaction.
We theoretically investigate the effect of inhomogeneous light beams with (twisted light) and without (plane-wave light) orbital angular momentum on semiconductor-based nanostructures, when the symmetry axes of the beam and the nanostructure are displaced parallel to each other. Exact analytical results are obtained by expanding the off-centered light field in terms of the appropriate light modes centered around the nanostructure. We demonstrate how electronic transitions involving the transfer of different amounts of orbital angular momentum are switched on and off as a function of the separa…
We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of "slowing down" the twisting in the mean gives rise to a non-empty discrete spectrum.
The main challenge in the development of non-classical light sources remains their brightness that limits the data transmission and processing rates as well as the realization of practical devices operating in the telecommunication range. To overcome this issue, we propose to utilize universal and flexible in-situ electron-beam lithography and hereby, we demonstrate a successful technology transfer to telecom wavelengths. As an example, we fabricate and characterize especially designed photonic structures with strain-engineered single InGaAs/GaAs quantum dots that are deterministically integra…
The recent discovery of two-dimensional (2D) Van der Waals (VdW) magnets is a crucial turning point in the quantum magnet research field, since quantum fluctuations and experimental difficulties often elude stable magnetic orders in 2D. This opens new doors to delve for novel quantum and topological spin configurations, which may or may not have direct analogs in bulk counterparts. Here we study a twisted bilayer geometry of 2D magnets in which long-range spin-spin interactions naturally commence along the inter-layer Heisenberg ($J_{\perp}$) and dipole-dipole ($J_{\rm D}$) channels. The $J_{\…
ZnSe nanowire heterostructures were grown by molecular beam epitaxy in the vapour-liquid-solid growth mode assisted by gold catalysts. Size, shape and crystal structure are found to strongly depend on the growth conditions. Both, zinc-blende and wurtzite crystal structures are observed using transmission electron microscopy. At low growth temperature, cone-shaped nano-needles are formed. For higher growth temperature, the nanowires are uniform and have a high aspect ratio with sizes of 1-2 $μ$m in length and 20-50 nm in width as observed by scanning electron microscopy. Growing a nanowire on t…
Electronic and magnetic properties of a system of two charged vacancies in hexagonal shaped graphene quantum dots are investigated using a mean-field Hubbard model as a function of the Coulomb potential strength $β$ of the charge impurities and the distance R between them. For $β=0$, the magnetic properties of the vacancies are dictated by Lieb's rules where the opposite (same) sub-lattice vacancies are coupled antiferromagnetically (ferromagnetically) and exhibit Fermi oscillations. Here, we demonstrate the emergence of a non-magnetic regime within the subcritical region: as the Coulomb poten…
Employing oval shaped quantum billiards connected by quantum wires as the building blocks of a linear quantum dot array, we calculate the ballistic magnetoconductance in the linear response regime. Optimizing the geometry of the billiards, we aim at a maximal finite- over zero-field ratio of the magnetoconductance. This switching effect arises from a relative phase change of scattering states in the oval quantum dot through the applied magnetic field, which lifts a suppression of the transmission characteristic for a certain range of geometry parameters. It is shown that a sustainable switchin…
The emergence of alternating twist multilayer graphene (ATMG) as a generalization of twisted bilayer graphene (TBG) raises the question - in what important ways do these systems differ? Here, we utilize ab-initio relaxation and single-particle theory, analytical strong coupling analysis, and Hartree-Fock to contrast ATMG with $n=3,4,5,\ldots$ layers and TBG. $\textbf{I}$: We show how external fields enter in the decomposition of ATMG into TBG and graphene subsystems. The parallel magnetic field has little effect for $n$ odd due to mirror symmetry, but surprisingly also for any $n > 2$ if we ar…
