Orbital Degeneracy Research Articles

Class of distorted Landau levels and Hall phases in a two-dimensional electron gas subject to an inhomogeneous magnetic field

An analytic closed form solution is derived for the bound states of a two-dimensional electron gas subject to a static, inhomogeneous ($1/r$ in plane decaying) magnetic field, including the Zeeman interaction. The solution provides access to many-body properties of a two-dimensional, noninteracting, electron gas in the thermodynamic limit. Radially distorted Landau levels can be identified as well as magnetic field induced density and current oscillations close to the magnetic impurity. These radially localized oscillations depend strongly on the coupling of the spin to the magnetic field, which gives rise to nontrivial spin currents. Moreover, the Zeeman interaction introduces a unique flat band, i.e., infinitely degenerate energy level in the ground state, assuming a spin ${g}_{s}$-factor of two. Surprisingly, the charge and current densities can be computed analytically for this fully filled flat band in the thermodynamic limit. Numerical calculations show that the total magnetic response of the electron gas remains diamagnetic (similar to Landau levels) independent of the Fermi energy. However, the contribution of certain, infinitely degenerate energy levels may become paramagnetic. Furthermore, numerical computations of the Hall conductivity reveal asymptotic properties of the electron gas, which are driven by the anisotropy of the vector potential instead of the magnetic field, i.e., become independent of spin. Eventually, the distorted Landau levels give rise to negative and positive Hall conductivity phases, with sharp transitions at specific Fermi energies. Overall, our work merges ``impurity'' with Landau-level physics, which provides novel physical insights, not only locally, but also in the asymptotic limit. This paves the way for a large number of future theoretical as well as experimental investigations, e.g., to include electronic correlations and to investigate two-dimensional systems such as graphene or transition metal dichalcogenides under the influence of inhomogeneous magnetic fields.

Analysis of Facial Features of Patients With Sagging Eye Syndrome and Intermittent Exotropia Compared to Controls

To compare the facial features of patients with sagging eye syndrome (SES) and other ophthalmic diseases, and to evaluate the diagnostic usefulness of facial features for SES. Retrospective cross-section study. We evaluated frontal facial photographs of patients >60 years of age with SES and intermittent exotropia (IXT), and control patients who visited the ophthalmology outpatient clinics of 2 institutions between June 2020 and December 2021. Three ophthalmologists evaluated each eye for sunken upper eyelid, blepharoptosis, and baggy lower eyelid, using a scoring scale. The average scores for each parameter among the 3 groups were analyzed. Patients with glaucoma, visual acuity <16/20, SES with a vertical strabismus angle of ≥6 Δ, IXT that could not be maintained in the phoria position during photography, a history of previous oculoplastic or ophthalmic surgery, and use prostaglandin analogs for cosmetic purposes were excluded. A total of 86 patients were included: 23 with SES, 28 with IXT, and 35 in the control group. All were Japanese. In all, 45 patients were male and 41 were female. The mean age was 72.7 ± 7.4 years. The sunken upper eyelid scores were significantly higher in the SES group than in the control and IXT groups (P < .001), whereas the baggy lower eyelid scores were significantly higher in the IXT group than in the control group (P < .05). Age-related orbital connective tissue degeneration may manifest as SES in the upper eyelid and as IXT in the lower eyelid.

Mean-Field Study of Antiferromagnetic and Antiferroquadrupolar Orderings in Tetragonal CeCoSi

We investigate the stability of the multipolar orderings in f-electron material CeCoSi based on a self-consistent mean-field calculation for the effective localized model. This material has two ordered phases in the temperature–pressure phase diagram: the antiferromagnetic phase and the nonmagnetic phase, the latter of which has been suggested to be an antiferroquadrupolar phase. Meanwhile, the origin of the antiferroquadrupolar phase has been unclear, since a quadrupole degree of freedom is present only between the ground-state level and highly separated excited-state level under a tetragonal crystalline electric field (CEF), whose energy scale is much larger than the transition temperature. To understand the sequence of the phase transition from the paramagnetic phase, antiferroquadrupolar phase, and antiferromagnetic phase when decreasing the temperature, we examine the important interaction parameters in the effective localized model. We clarify that the 3z2 − r2-type of the antiferroquadrupolar interactions can renormalize the CEF level splitting, which might assist a quadrupolar ordering even in a tetragonal system without orbital degeneracy. Moreover, the stability of the antiferroquadrupolar and antiferromagnetic states in a magnetic field and the behavior of the magnetic/quadrupolar susceptibility are also presented for the information to identify the unknown order parameter in the nonmagnetic ordered phase.

H-Bonding Room Temperature Phosphorescence Materials via Facile Preparation for Water-Stimulated Photoluminescent Ink.

Pure organic room-temperature phosphorescence (RTP) materials built upon noncovalent interactions have attracted much attention because of their high efficiency, long lifetime, and stimulus-responsive behavior. However, there are limited reports of noncovalent RTP materials because of the lack of specific design principles and clear mechanisms. Here, we report on a noncovalent material prepared via facile grinding that can emit fluorescence and RTP emission differing from their components’ photoluminescent behavior. Exciplex can be formed during the preparation process to act as the minimum emission unit. We found that H-bonds in the RTP system provide restriction to nonradiative transition but also enhance energy transformation and energy level degeneracy in the system. Moreover, water-stimulated photoluminescent ink is produced from the materials to achieve double-encryption application with good resolution.

Excitonic transverse and amplitude fluctuations in noncollinear and charge-ordered RbFe2+Fe3+F6

RbFe$^{2+}$Fe$^{3+}$F$_{6}$ is an example of an antiferromagnet with charge ordering of the octahedrally coordinated Fe$^{2+}$ and Fe$^{3+}$ ions. As well as different spin values, Fe$^{2+}$ ($S=2$) and Fe$^{3+}$ ($S={5\over2}$) possess differing orbital ground states with Fe$^{2+}$ having an orbital degeneracy with an effective orbital angular momentum of $l=1$. The resulting low temperature magnetic structure is non collinear with the spins aligned perpendicular to nearest neighbors (S. W. Kim \textit{et al.} Chem. Sci. {\bf{3}}, 741 (2012)). The combination of an orbital degeneracy and non collinear spin arrangements introduces the possibility for unusual types of excitations such as amplitude modes of the order parameter. In this paper we investigate this by applying a multi-level analysis to model neutron spectroscopy data (M. Songvilay \textit{et al.} Phys. Rev. Lett. {\bf{121}}, 087201 (2018)). In particular, we discuss the possible origins of the momentum and energy broadened continuum scattering observed in terms of amplitude fluctuations allowed through the presence of an orbital degree of freedom on the Fe$^{2+}$ site. We extend previous spin-orbit exciton models based on a collinear spin structure to understand the measured low-energy excitations and also to predict and discuss possible amplitude mode scattering in RbFe$^{2+}$Fe$^{3+}$F$_{6}$.

Hydrogenic impurity effect on optical properties of Wannier-Mott exciton confined in a spherical quantum dot with Kratzer potential under magnetic field

Confinement effects of Kratzer potential on a Wannier-Mott Exciton(W-M) are studied in a spherical quantum dot(QD) in the presence of a static magnetic field. Time independent Schr$\ddot{o}$dinger equation is solved numerically to obtain the energy states. The excitonic transitions so realized have been used to explore the non-linear optical properties that are important for optical characterization of materials such as the optical absorption coefficients (ACs) and refractive index changes (RICs). Impact of magnetic field, strength of the laser field and transition parameters using familiar compact density matrix approach are also analyzed. It has been observed that optical properties get radically modified under confinement effects. Also, the shift of degeneracy of different excitonic energy levels with the magnetic field in confinement potential has been reported for the first time for W-M exciton in the spherical quantum dot, the study that may have crucial input to the literature and myriad of practical implications.

Orbital dependent band degeneracy and edge states in single layer and AA bilayer honeycomb lattice systems with p orbital degeneracy

We have investigated band structure and edge states of single and AA bilayer honeycomb lattice systems with p orbital degeneracy which would be relevant for relatively heavy metals such as Pb and Bi. The p electrons, which are triply degenerate in the atomic limit, form quadruple degeneracy at K point in a single layer honeycomb lattice. The spin-orbit interaction splits the quartet at K point into two singlets and one doublet which are mixtures of p x , p y and p z orbitals. In the AA bilayer honeycomb lattice, the Dirac node at K point is deformed into circular line node around it. With the intermediate interlayer coupling, there is a degenerate point along Γ-K with p z character. A p x /p y type degenerate point exists under the strong interlayer coupling. This degenerate point is robust against the spin-orbit interaction and the antiferromagnetic spin arrangement only along the intralayer bonds. Without SOI, localized edge states with p x /p y or p z character are created at the zigzag edges both in the single and AA bilayer systems. In the AA bilayer system with SOI, only the p x /p y edge states exhibit the Dirac-like dispersion.

F+ center exchange mechanism and magnetocrystalline anisotropy in Ni-doped 3C-SiC

• F + center exchange mechanism has been invoked in order to explain the long range magnetic order in the system. • The Curie temperature is found to be above 350 K for all the doped samples obtained from the modified Bloch’s T 3/2 law. • The dynamic nature of the BMPs sizes have been studied extensively using Hc vs T data. • The spontaneous spin polarization has been induced in the system due to strong hybridization between (C) 2p and (Ni) 3d orbital. • The decrease in magneto crystalline anisotropy constant value with increase in temperature essentially indicates the weakening of the magnetic interaction in the system. • The incomplete quenching of orbital angular momentum and partial removal of orbital degeneracy resulted in reduction of line width and enhancement of integrated intensity as the temperature rises. Towards the development of a magnetic semiconductor suitable for spintronic device applications in extreme environments, we explored the possibility of inducing magnetic interaction in SiC by doping Nickel. The X-ray diffraction and Raman Spectroscopy studies confirm the incorporation of Ni into the host lattice. The magnetic measurements and electron spin resonance studies indicate the presence of room temperature ferromagnetic interaction in the system. The Curie temperature of 1, 3, and 5% Ni-doped samples have been found to be 420 K, 520 K, and 540 K respectively. Electron spin resonance study reveals that the valence state of Ni is 2 + , which implies the creation of vacancies at both Silicon (V Si ) and Carbon (V C ) sites as they are tetravalent. The change in magnetization of the system with an increase in dopant concentration is consistent with the variation in the number of vacancies and free holes. The analysis of magnetization data using the Law of approach to saturation shows that the anisotropic constant decreases with an increase in temperature. The long-range magnetic interaction in the system is explained using the F + center exchange mechanism.

Electronic structure of the frustrated diamond lattice magnet NiRh2O4

The $A$-site spinel ${\mathrm{NiRh}}_{2}{\mathrm{O}}_{4}$ is the only known realization of a spin-1 diamond lattice magnet and is predicted to host unconventional magnetic phenomena driven by frustrated nearest and next-nearest neighbor exchange as well as orbital degeneracy. Previous works found no sign of magnetic order but found a gapped dispersive magnetic excitation indicating a possible valence bond magnetic ground state. However, the presence of many competing low energy degrees of freedom and limited empirical microscopic constraints complicates further analysis. Here we carry out resonant inelastic x-ray scattering (RIXS), x-ray absorption spectroscopy (XAS), and inelastic neutron scattering (INS) to characterize the local electronic structure and lattice dynamics of ${\mathrm{NiRh}}_{2}{\mathrm{O}}_{4}$. The RIXS data can be partly described by a single-ion model for tetrahedrally coordinated ${\mathrm{Ni}}^{2+}$ and indicates a tetragonal distortion $\mathrm{\ensuremath{\Delta}}{t}_{2}=70$ meV that splits the ${t}_{2}$ orbitals into a high energy orbital singlet and lower energy orbital doublet. We identify features of the RIXS spectra that are consistent with a Rh-Ni two-site excitation indicating strong metal-metal hybridization mediated by oxygen in ${\mathrm{NiRh}}_{2}{\mathrm{O}}_{4}$. We also identify signatures of electron-phonon coupling through the appearance of phonon sidebands that dress crystal field excitations. These results establish the key energy scales relevant to the magnetism in ${\mathrm{NiRh}}_{2}{\mathrm{O}}_{4}$ and further demonstrate that covalency and lattice dynamics play essential roles in controlling the magnetic ground states of $A$-site spinels.

Fractional quantum oscillator and disorder in the vibrational spectra

We study the role of disorder in the vibration spectra of molecules and atoms in solids. This disorder may be described phenomenologically by a fractional generalization of ordinary quantum-mechanical oscillator problem. To be specific, this is accomplished by the introduction of a so-called fractional Laplacian (Riesz fractional derivative) to the Scrödinger equation with three-dimensional (3D) quadratic potential. To solve the obtained 3D spectral problem, we pass to the momentum space, where the problem simplifies greatly as fractional Laplacian becomes simply k^mu , k is a modulus of the momentum vector and mu is Lévy index, characterizing the degree of disorder. In this case, mu rightarrow 0 corresponds to the strongest disorder, while mu rightarrow 2 to the weakest so that the case mu =2 corresponds to “ordinary” (i.e. that without fractional derivatives) 3D quantum harmonic oscillator. Combining analytical (variational) and numerical methods, we have shown that in the fractional (disordered) 3D oscillator problem, the famous orbital momentum degeneracy is lifted so that its energy starts to depend on orbital quantum number l. These features can have a strong impact on the physical properties of many solids, ranging from multiferroics to oxide heterostructures, which, in turn, are usable in modern microelectronic devices.

Spin-related symmetry breaking induced by half-disordered hybridization in BixEr2-xRu2O7 pyrochlores for acidic oxygen evolution

While acidic oxygen evolution reaction plays a critical role in electrochemical energy conversion devices, the sluggish reaction kinetics and poor stability in acidic electrolyte challenges materials development. Unlike traditional nano-structuring approaches, this work focuses on the structural symmetry breaking to rearrange spin electron occupation and optimize spin-dependent orbital interaction to alter charge transfer between catalysts and reactants. Herein, we propose an atomic half-disordering strategy in multistage-hybridized BixEr2-xRu2O7 pyrochlores to reconfigure orbital degeneracy and spin-related electron occupation. This strategy involves controlling the bonding interaction of Bi-6s lone pair electrons, in which partial atom rearrangement makes the active sites transform into asymmetric high-spin states from symmetric low-spin states. As a result, the half-disordered BixEr2-xRu2O7 pyrochlores demonstrate an overpotential of ~0.18 V at 10 mA cm−2 accompanied with excellent stability of 100 h in acidic electrolyte. Our findings not only provide a strategy for designing atom-disorder-related catalysts, but also provides a deeper understanding of the spin-related acidic oxygen evolution reaction kinetics.

Modulation of electrical and thermal transports through lattice distortion in BaTi1–x Nb x O3 solid solutions

The electron and heat transports in solids are through the movement of carrier electrons and quantized lattice vibrations (phonons), which are sensitive to the lattice distortion and ionized impurities, and are essential aspects for the development of novel thermoelectric materials. In this study, we systematically investigated the modulations of electrical and thermal conductivities of BaTi1–x Nb x O3 solid solution (BTNO, 0 ≤ x ≤ 1) epitaxial films. At room temperature, BaTiO3 belongs to tetragonal perovskite and exhibits electron conduction through doubly degenerated Ti 3d-t2g orbitals upon doping, while BaNbO3 belongs to cubic perovskite and exhibits metallic electron conduction through partially filled triply degenerate Nb 4d-t2g orbitals. By controlling the Ti/Nb ratio, we found a dual modulation effect on both the lattice structures and conduction band, which affects the electrical and thermal conductivities. Similar to the SrTi1–x Nb x O3 solid solution (STNO, 0 ≤ x ≤ 1) system, a phase transition was detected at x ∼ 0.5, at which both the electron and heat transports exhibit abrupt changes. Unlike the transition in STNO, which was attributed to a polaronic phase transition, the transition in BTNO was due to contributions from both the lattice distortion and polaron effect. By controlling the lattice distortion, conduction band, and polaronic phase transitions, the electrical and thermal conductivity of BTNO epitaxial films are modulated within a much greater range than those of the STNO epitaxial films. Due to the double contribution of electron carriers and phonon to thermal conductivity (κ), the maximum κ modulation ratio of BTNO epitaxial films was ∼6.9. Our research provides an effective route to design electrical/thermal management materials.

Magneto-optic Kerr effect in Gd20Co80 alloy

The magneto-optical Kerr effect in Gd20Co80 alloy and cobalt thin films has been studied in a broad spectral range applying spectral ellipsometry experimental technique. The results of the experiments showed the complex nature of the complex Kerr angle dispersion curves. A quantum mechanical formalism for degenerate and non-degenerate Landau levels for quasi-free electrons in ferromagnetic material has been developed in order to analyze the experimental data. The equivalence of relations for off-diagonal dielectric tensor elements for non-degenerate Landau levels to the classical case of the motion of quasi-free electrons along circular trajectories in a magnetic field has been theoretically shown. The degenerate Landau levels in this approach are the result of motion of electrons in small confined volumes near rare-earth alloy atoms. Rotation of light polarization occurs in this case due to transitions between subbands having different magnetic quantum numbers. This theoretical approach allowed us to interpret in detail shapes and sign of the complex Kerr angle dispersion curves, which actually include the contributions of optical transitions between degenerate and non-degenerate energy levels. The complex Kerr angle sign is determined by the magnetization magnetic field direction for non-degenerate Landau levels and the Hund rule for degenerate Landau levels.

Trigonal symmetry breaking and its electronic effects in the two-dimensional dihalides MX2 and trihalides MX3

We study the consequences of the approximately trigonal (${D}_{3d}$) point symmetry of the transition metal ($M$) site in two-dimensional van der Waals ${MX}_{2}$ dihalides and ${MX}_{3}$ trihalides. The trigonal symmetry leads to a 2-2-1 orbital splitting of the transition metal $d$ shell, which is best represented by $d$ orbitals that describe the trigonal rather than ${O}_{h}$ symmetry. The ligand-ligand bond length differences (rather than metal-ligand) take the role of a Jahn-Teller-like mode and in combination with interlayer distances and dimensionality effects tune the crystal field splittings and bandwidths. These effects, in turn, are amplified by electronic correlation effects---leading to crystal field splittings of the order of 0.1--1 eV between the singlet and the lowest orbital doublet---as opposed to the often assumed degenerate $3\phantom{\rule{0.16em}{0ex}}{t}_{2g}$ states. Our calculations explain why most of the materials in this family are insulating, and why these considerations have to be taken into account in realistic models of them. Further, orbital order coupled to various lower symmetry lattice modes may lift the remaining orbital degeneracies, and we explain how these may support unique electronic states using ${\mathrm{ZrI}}_{2}$ and ${\mathrm{CuCl}}_{2}$ as examples, and offer a brief overview of possible electronic configurations in this class of materials. By building and analyzing Wannier models adapted to the appropriate symmetry we examine how the interplay among trigonal symmetry, electronic correlation effects, and $p\text{\ensuremath{-}}d$ orbital charge transfer leads to insulating, orbitally polarized magnetic and/or orbital-selective Mott states, and we provide a simple framework and numerical tool for others. Our paper establishes a rigorous framework to understand, control, and tune the electronic states in low-dimensional correlated halides. Our analysis shows that trigonal symmetry and its breaking is a key feature of the two-dimensional halides that needs to be accounted for in search of novel electronic states in materials ranging from ${\mathrm{CrI}}_{3}$ to $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$.

Encircling exceptional points in non-Hermitian systems with quasidegenerate energy levels

Dynamically encircling an exceptional point in non-Hermitian systems shows a chiral state switching behavior as a result of the nonadiabatic transition between nonorthogonal eigenstates. It has been revealed that the chiral dynamics is protected by the topological structure of eigenvalue sheets where only one lowest-loss sheet exists. This raises the question on what the dynamics would be in non-Hermitian systems possessing multiple degenerate lowest-loss energy levels. Here, we address this question by studying a photonic-waveguide-array non-Hermitian system where two exceptional points are encircled dynamically. The system possesses four quasidegenerate lowest-loss eigenvalue sheets and such topological structure results in an exotic nonchiral behavior for switching eigenstates such that the final state is always a superposition of the four lowest-loss eigenstates. We find that this intriguing phenomenon is robust to a certain perturbation of the system parameters. Our findings enrich the understanding of exceptional point-encirclement physics and may inspire potential non-Hermitian applications for manipulating waves.

Effects of zigzag edge states on the thermoelectric properties of finite graphene nanoribbons

Thermoelectric properties of finite graphene nanoribbons (GNRs) coupled to metallic electrodes are theoretically studied in the framework of tight-binding model and Green’s function approach. When the zigzag sides are coupled to the electrodes, the electron transport through the localized edge states can occur only if the channel length between electrodes is smaller than the decay length of these localized zigzag edge states. When the armchair edges are coupled to the electrodes, there is an interesting thermoelectric behavior associated with the mid-gap states when the GNR is in the semiconducting phase. Here we show that the thermoelectric behavior of zigzag edge states of GNRs with armchair sides connected to electrodes is similar to that of two parallel quantum dots with similar orbital degeneracy. Furthermore, it is demonstrated that the electrical conductance and power factor given by the zigzag edge states are quite robust against the defect scattering.

Interplay of the Jahn-Teller effect and spin-orbit coupling: The case of trigonal vibrations

We study an interplay between the orbital degeneracy and spin-orbit coupling giving rise to spin-orbital entangled states. As a specific example, we analyze the interaction of electrons occupying triply degenerate single-ion $t_{2g}$ levels with trigonal vibronic modes (the $t\otimes T$ problem). A more general problem of the electron-lattice interaction involving both tetragonal and trigonal vibrations is also considered. It is shown that the result of such interaction crucially depends on the occupation of $t_{2g}$ levels leading to either the suppression or enhancement of the Jahn-Teller effect by the spin-orbit coupling.

Driving quantum systems with periodic conditional measurements

We investigate the effect of conditional null measurements on a quantum system and find a rich variety of behaviors. Specifically, quantum dynamics with a time independent $H$ in a finite dimensional Hilbert space are considered with repeated strong null measurements of a specified state. We discuss four generic behaviors that emerge in these monitored systems. The first arises in systems without symmetry, along with their associated degeneracies in the energy spectrum, and hence in the absence of dark states as well. In this case, a unique final state can be found which is determined by the largest eigenvalue of the survival operator, the non-unitary operator encoding both the unitary evolution between measurements and the measurement itself. For a three-level system, this is similar to the well known shelving effect. Secondly, for systems with built-in symmetry and correspondingly a degenerate energy spectrum, the null measurements dynamically select the degenerate energy levels, while the non-degenerate levels are effectively wiped out. Thirdly, in the absence of dark states, and for specific choices of parameters, two or more eigenvalues of the survival operator match in magnitude, and this leads to an oscillatory behavior controlled by the measurement rate and not solely by the energy levels. Finally, when the control parameters are tuned, such that the eigenvalues of the survival operator all coalesce to zero, one has exceptional points that corresponds to situations that violate the null measurement condition, making the conditional measurement process impossible.

Thermally driven p–n–p multiple heteromorphic transformation facilitating high Seebeck coefficient for Ca0.8La0.2CeNbWO8+δ

Given that different types of carriers tend to gather at the cold end to offset part of the induced voltage, the high Seebeck coefficient obtained by a single type of carrier brings voltage difference to the automatic sensor. Nevertheless, due to the significant charge compensation effect, the effective mass and average energy of carriers can be improved, and a record high Seebeck coefficient (S &amp;gt; 10 mV K−1) is excavated through heteromorphic transformation. Herein, by introducing Ce4+/Ce3+ redox pairs into the p-type doped scheelite CaCeNbWO8 with excellent structural/thermal stability, we can realize various types of p–n–p conversion under thermal driving, and each transformation ushers in a sharp increase in the Seebeck coefficient and conductivity simultaneously. The heteromorphic transition activates solid solution diffusion, promotes transition from variable range hopping transport to nearest neighbor hopping transport, and induces energy-level degeneracy. In the process of p–n transition, the carrier mobility increases sharply and the Seebeck coefficient is ∼5.75 mV K−1 and the conductivity is ∼0.06 S m−1 at 881 K, while metallization occurs during n–p transition accompanying a surge in carrier concentration, resulting in a maximum Seebeck coefficient of ∼17.35 mV K−1 and a conductivity of ∼0.08 S m−1 at 1068 K. The important role of multiple heteromorphic transformation in improving the Seebeck coefficient is revealed, which is expected to be used in low-cost, low pollution, and low-power thermoelectric devices.

Three-Dimensional Fully π-Conjugated Macrocycles: When 3D-Aromatic and When 2D-Aromatic-in-3D?

Several fully π-conjugated macrocycles with puckered or cage-type structures were recently found to exhibit aromatic character according to both experiments and computations. We examine their electronic structures and put them in relation to 3D-aromatic molecules (e.g., closo-boranes) and to 2D-aromatic polycyclic aromatic hydrocarbons. Using qualitative theory combined with quantum chemical calculations, we find that the macrocycles explored hitherto should be described as 2D-aromatic with three-dimensional molecular structures (abbr. 2D-aromatic-in-3D) and not as truly 3D-aromatic. 3D-aromatic molecules have highly symmetric structures (or nearly so), leading to (at least) triply degenerate molecular orbitals, and for tetrahedral or octahedral molecules, an aromatic closed-shell electronic structure with 6n + 2 electrons. Conversely, 2D-aromatic-in-3D structures exhibit aromaticity that results from the fulfillment of Hückel’s 4n + 2 rule for each macrocyclic path, yet their π-electron counts are coincidentally 6n + 2 numbers for macrocycles with three tethers of equal lengths. It is notable that 2D-aromatic-in-3D macrocyclic cages can be aromatic with tethers of different lengths, i.e., with π-electron counts different from 6n + 2, and they are related to naphthalene. Finally, we identify tetrahedral and cubic π-conjugated molecules that fulfill the 6n + 2 rule and exhibit significant electron delocalization. Yet, their properties resemble those of analogous compounds with electron counts that differ from 6n + 2. Thus, despite the fact that these molecules show substantial π-electron delocalization, they cannot be classified as true 3D-aromatics.