Landau Level Spectrum Research Articles

Landau levels in Weyl semimetal under uniaxial strain

The external strain can lead to the similar effect to the external applied magnetic field. Such pseudo magnetic field can be larger than typical magnetic fields, which gives the opportunity to experimentally study the Landau levels. In this paper we study the effects of uniaxial strain on the Weyl nodes, using continuum and lattice model. In the continuum model we show that the uniaxial strain leads to magnetic field renormalization, which in practice corresponds to the shift of the Landau levels to higher energy. We also investigate type-I and type-II Weyl nodes using lattice model. In this case, the magnetic field is introduced by the Pierels substitution, while uniaxial strain by the direction dependence of hopping integrals. This allowed us to probe the Landau level and system spectrum which takes form of Hofstadter butterfly. We show that the renormalization of magnetic field, similar to this observed in the continuum model, emerges.

Electron–photon–phonon interactions in Dirac semimetals: Magneto-optical absorption and mobility analysis

We study the magneto-optical properties of Dirac semimetal (DSM) slabs with particular emphasis on Cd3As2 through electron–photon–phonon interactions, focusing on the magneto-optical absorption coefficient (MOAC) and full-width at half maximum (FWHM). Studying the Landau level (LL) energy of DSMs in the (xy) plane and the z-direction revealed a unique deviation from the square root dependence on the magnetic field, distinguishing DSMs from other semiconductors. At high magnetic fields, the electron–hole symmetry in the LL spectrum is broken, indicating a topological phase in DSMs. For undoped DSMs, MOAC is driven by interband transitions, with peaks from one-photon absorption being smaller and positioned to the left of two-photon ones. Increasing the magnetic field increases peak values. FWHM for one- and two-photon processes increases with the magnetic field and follows a T dependence on temperature. In doped DSMs, both intraband and interband transitions occur, with new interband peaks emerging at higher temperatures near the Fermi energy. Increased electron density shifts the peak position slightly toward higher energy. Peaks from optical phonon emission are consistently higher and located to the right of those from optical phonon absorption, indicating a stronger emission process. The FWHM data allow for the estimation of electron mobilities, and using a reasonable broadening parameter, our predicted mobility values agree with experimental results.

Probing the tunable multi-cone band structure in Bernal bilayer graphene

Bernal bilayer graphene (BLG) offers a highly flexible platform for tuning the band structure, featuring two distinct regimes. One is a tunable band gap induced by large displacement fields. Another is a gapless metallic band occurring at low fields, featuring rich fine structure consisting of four linearly dispersing Dirac cones and van Hove singularities. Even though BLG has been extensively studied experimentally, the evidence of this band structure is still elusive, likely due to insufficient energy resolution. Here, we use Landau levels as markers of the energy dispersion and analyze the Landau level spectrum in a regime where the cyclotron orbits of electrons or holes in momentum space are small enough to resolve the distinct mini Dirac cones. We identify the presence of four Dirac cones and map out topological transitions induced by displacement field. By clarifying the low-energy properties of BLG bands, these findings provide a valuable addition to the toolkit for graphene electronics.

Optical properties and Landau quantisations in twisted bilayer graphene

ABSTRACT The optical properties and Landau quantisations in twisted bilayer graphene are investigated using a generalised tight-binding model that takes into account all interlayer and intralayer atomic interactions in the Moire superlattice. The system is a zero-gap semiconductor with double-degenerate Dirac-cone structures, and saddle-point energy dispersions appear at low energies for small twisting angles. The magnetic quantisation phenomena in this system are rich and unique, with Landau-level subgroups specified owing to specific Moire zone folding through modulation of the stacking angles. The Landau-level spectrum shows hybridised characteristics associated with those in monolayer, as well as AA and AB stackings. The detailed analysis explores the complex relations among the different sublattices on the same and different graphene layers.

Stabilizing the Inverted Phase of a WSe2/BLG/WSe2 Heterostructure via Hydrostatic Pressure.

Bilayer graphene (BLG) was recently shown to host a band-inverted phase with unconventional topology emerging from the Ising-type spin-orbit interaction (SOI) induced by the proximity of transition metal dichalcogenides with large intrinsic SOI. Here, we report the stabilization of this band-inverted phase in BLG symmetrically encapsulated in tungsten diselenide (WSe2) via hydrostatic pressure. Our observations from low temperature transport measurements are consistent with a single particle model with induced Ising SOI of opposite sign on the two graphene layers. To confirm the strengthening of the inverted phase, we present thermal activation measurements and show that the SOI-induced band gap increases by more than 100% due to the applied pressure. Finally, the investigation of Landau level spectra reveals the dependence of the level-crossings on the applied magnetic field, which further confirms the enhancement of SOI with pressure.

Anomalous Landau quantization in intrinsic magnetic topological insulators

The intrinsic magnetic topological insulator, Mn(Bi1−xSbx)2Te4, has been identified as a Weyl semimetal with a single pair of Weyl nodes in its spin-aligned strong-field configuration. A direct consequence of the Weyl state is the layer dependent Chern number, C. Previous reports in MnBi2Te4 thin films have shown higher C states either by increasing the film thickness or controlling the chemical potential. A clear picture of the higher Chern states is still lacking as data interpretation is further complicated by the emergence of surface-band Landau levels under magnetic fields. Here, we report a tunable layer-dependent C = 1 state with Sb substitution by performing a detailed analysis of the quantization states in Mn(Bi1−xSbx)2Te4 dual-gated devices—consistent with calculations of the bulk Weyl point separation in the doped thin films. The observed Hall quantization plateaus for our thicker Mn(Bi1−xSbx)2Te4 films under strong magnetic fields can be interpreted by a theory of surface and bulk spin-polarised Landau level spectra in thin film magnetic topological insulators.

Landau level spectrum and magneto-optical conductivity in tilted Weyl semimetal

We present a systematic investigation of the magnetoresponses of the Weyl points (WPs) with a topological charge of n = 2, 3 and 4, and with both linear and quadratic energy tilt. The linear tilt always tends to squeeze the Landau levels (LLs) of both conduction and valence bands of all the WPs, and eventually leads to LL collapse in the type-II phase. However, the quadratic energy tilt has more complex influences on the LLs of the unconventional WPs. For charge-n (n = 2, 4) WP, the influence of the quadratic tilt on the LLs of conduction and valence bands are opposite, i.e. if the LLs of conduction (valence) bands are squeezed, then that of the valence (conduction) bands are broadened, and the squeezed LL spectrum will be collapsed in type-III phase. But, the LL collapse generally can not be found in the type-III charge-3 WP. Moreover, for charge-n (n = 2, 3) WP, the quadratic tilt breaks the degeneracy of the chiral LLs regardless of the direction of the magnetic field, leading to additional optical transitions and magneto-optical conductivity peaks at low frequencies. Interestingly, the four chiral LLs in charge-4 WP are always not degenerate. Hence, there inevitably exist magneto-optical conductivity peaks at low frequencies for charge-4 WP. Since the density of state of the LL spectrum is very large, one can expect that the low-frequency magneto-optical responses in unconventional WPs would be significant and may be used for developing efficient terahertz photodetectors.

Analytical model of the energy spectrum and Landau levels of a twisted double bilayer graphene

Motivated by recent experiments Adak et al. (2020) and Szentpéter et al. (2021), in this work we have considered a twisted double bilayer graphene system. Within the low energy effective theory, the bandwidth expression of the moire flatbands (with weak dispersion and diverging effective mass) and Landau level spectrum of the Bernal stacked twisted double bilayer graphene are obtained analytically. We analyze the dependence of the bandwidth, effective mass, and Landau level spectrum on twist angle, inter layer tunneling, and interlayer potential difference. Also, we have investigated the phase diagram of the effective mass as a function of the twist angle and interlayer tunneling strength to seek the parameter space with diverging effective mass. We find that the magic angle at which the flat bands exist is increasing with increasing inter layer tunneling strength. • In this work, in this work we have considered a twisted double bilayer graphene system. • Within the low energy effective theory, the bandwidth expression of the moiré flatbands (with weak dispersion and diverging effective mass) and Landau level spectrum of the Bernal stacked twisted double bilayer graphene are obtained analytically. • We analyze the dependence of the bandwidth, effective mass, and Landau level spectrum on twist angle, inter layer tunneling, and interlayer potential difference. • Also, we have investigated the phase diagram of the effective mass as a function of the twist angle and interlayer tunneling strength to seek the parameter space with diverging effective mass. • We find that the magic angle at which the flat bands exist is increasing with increasing inter layer tunneling strength. • The results presented in this work will be useful for the experimental investigation of the tDBG system.

Effective Low-Energy Hamiltonians and Unconventional Landau-Level Spectrum of Monolayer C3N.

We derive low-energy effective k·p Hamiltonians for monolayer C3N at the Γ and M points of the Brillouin zone, where the band edge in the conduction and valence band can be found. Our analysis of the electronic band symmetries helps to better understand several results of recent ab initio calculations for the optical properties of this material. We also calculate the Landau-level spectrum. We find that the Landau-level spectrum in the degenerate conduction bands at the Γ point acquires properties that are reminiscent of the corresponding results in bilayer graphene, but there are important differences as well. Moreover, because of the heavy effective mass, n-doped samples may host interesting electron-electron interaction effects.

About an (Im‐)Possible Anomaly in Band Structure Theory

Based on an envelope function approach pioneered by Wannier and Slater, it can be shown that under the influence of strong effective magnetic fields in spinor space the band structure of certain materials gives rise to the analogs of tardyons and luxons, and in principle also to the analogs of tachyons. Such a (hypothetical) tachyonic band structure would be metallic, but with a gap in k ‐space and not in the range of allowed band energies . Some of the conditions to observe such a tachyonic anomaly in certain types of electronic band structures, which do not seem to be implausible, are discussed. Similar anomalies should appear in the spectrum of Landau levels. Furthermore, some of the pathologies that would come with such exotic band structure anomalies are pointed out.

High-energy Landau levels in graphene beyond nearest-neighbor hopping processes: Corrections to the effective Dirac Hamiltonian

We study the Landau level spectrum of bulk graphene monolayers beyond the Dirac Hamiltonian with linear dispersion. We consider an effective Wannier-like tight-binding model obtained from ab initio calculations that includes long-range electronic hopping integral terms. We employ the Haydock-Heine-Kelly recursive method to numerically compute the Landau level spectrum of bulk graphene in the quantum Hall regime and demonstrate that this method is both accurate and computationally much faster than the standard numerical approaches used for this kind of study. The Landau level energies are also obtained analytically for an effective Hamiltonian that accounts for up to third-nearest-neighbor hopping processes. We find an excellent agreement between both approaches. We also study the effect of disorder on the electronic spectrum. Our analysis helps to elucidate the discrepancy between theory and experiment for the high-energy Landau level energies.

Anomalous broadening of Landau levels at the saddle point energy of two dimensional square lattice

We study the magnetic characteristics of the two dimensional (2D) square lattice with one valence electron per lattice site in weak rational magnetic fields accessible experimentally, by solving numerically the exact system of discrete equations, which fully describes the broadening of Landau levels. The 2D square lattice is used as a prototype electron system in which the Fermi level lies at the energy of the saddle point (Van Hove peak) of the Brillouin zone and the corresponding density of states N(EF) is formally infinite (N(EF)→+∞). According to the electron band treatment this could lead to a formally divergent paramagnetic susceptibility and an infinite electron contribution to the specific heat at zero temperature, but our accurate analysis shows that both values remain finite. The energy spectrum of the Landau level at the saddle point has turned out to be principally continuous, so that even in a very small magnetic field this Landau level is always broadened in a miniband, in contrast to other energies, where the spectrum of Landau levels is discrete. Taking into account the electron spin polarization and the numerical solution, we reproduce the temperature dependence of the induced magnetic moment proportional to the magnetic susceptibility, and the electron contribution to the specific heat. Both plots demonstrate unusual dependencies reflecting the “metal”-like or “insulator”-like structure of the Landau minibands in the neighborhood of the Fermi energy. At low temperatures all values display oscillatory behavior. We also prove rigorously that the fully occupied electron band has no contribution to the diamagnetic susceptibility and specific heat.

Landau levels in curved space realized in strained graphene

The quantum Hall effect in curved space has been the subject of many theoretical investigations in the past, but devising a physical system to observe this effect is hard. Many works have indicated that electronic excitations in strained graphene realize Dirac fermions in curved space in the presence of a background pseudo-gauge field, providing an ideal playground for this. However, the absence of a direct matching between a numerical, strained tight-binding calculation of an observable and the corresponding curved space prediction has hindered realistic predictions. In this work, we provide this matching by deriving the low-energy Hamiltonian from the tight-binding model analytically to second order in the strain and mapping it to the curved-space Dirac equation. Using a strain profile that produces a constant pseudo-magnetic field and a constant curvature, we compute the Landau level spectrum with real-space numerical tight-binding calculations and find excellent agreement with the prediction of the quantum Hall effect in curved space. We conclude discussing experimental schemes for measuring this effect.

Embedded topological semimetals

Topological semimetals, such as Dirac, Weyl, or line-node semimetals, are gapless states of matter characterized by their nodal band structures and surface states. In this work, we consider layered (topologically trivial) insulating systems in $D$ dimensions that are composed of coupled multi-layers of $d$-dimensional topological semimetals. Despite being nominal bulk insulators, we show that crystal defects having co-dimension $(D-d)$ can harbor robust lower dimensional topological semimetals embedded in a trivial insulating background. As an example we show that defect-bound topological semimetals can be localized on stacking faults and partial dislocations. Finally, we propose how an embedded topological Dirac semimetal can be identified in experiment by introducing a magnetic field and resolving the relativistic massless Dirac Landau level spectrum at low energies in an otherwise gapped system.

Magneto-thermoelectric transport of bilayer phosphorene: A generalized tight-binding model study

In this study, we investigate the thermoelectric transport properties of bilayer phosphorene under a composite magnetic and electric field (Ez and Bz). The complete Landau level spectrum and the real-space Landau wavefunctions are obtained using the generalized tight-binding model, which simultaneously considers the effects of geometrical structures, atomic interactions, and external fields. To our knowledge, this is the first study that uses spatially distributed subenvelope functions based on real sublattices to calculate the transition rate of Landau states in a truly two-dimensional system. The results show the nonmonotonic dependences of conductivity on the Fermi level, Ez strength, and direction, which enrich the thermopower and Nernst signal behaviors.

Valley-dependent time evolution of coherent electron states in tilted anisotropic Dirac materials

The effect of the Dirac cone tilt of anisotropic two-dimensional materials on the time evolution of coherent electron states in the presence of electric and magnetic fields is studied. We propose a canonical transformation that maps the anisotropic Dirac-Weyl Hamiltonian with tilted Dirac cones to an effective and isotropic Dirac Hamiltonian under these fields. In this way, the well-known Landau-level spectra and wave functions allow calculating the Wigner matrix representation of Landau and coherent states. We found a valley dependency in the behavior of the Wigner function for both Landau and coherent electron states. The time evolution shows that the interplay of the Dirac cone tilt and the electric field keeps the uncertainties of both position and momentum in one valley significantly lower than in the other valley. The increment of quantum noise correlates with the emergence of negative values in the Wigner function. These results may help us to understand the generation of coherent electron states under the interaction with electromagnetic fields. The reported valley-dependent signatures in the Wigner function of materials with tilted Dirac cones may be revealed by quantum tomography experiments, even in the absence of electric fields.

Magneto-optical properties in thin film topological insulators with quadratic momentum term

In this work we investigate the influence of quadratic in momentum term (Schrodinger term) on magneto-transport properties of thin film topological insulators. The Schrodinger term modifies the Dirac cones into an hourglass shape which results in inter and intraband Landau levels crossings. Breaking of the particle–hole symmetry in Landau level spectrum in the presence of k 2 term leads to asymmetrical density of states profile. We calculate collisional and Hall conductivity for mixed Dirac–Schrodinger system in linear response regime and show oscillatory behavior in collisional conductivity, while Zeeman and hybridization terms provide a doubly split peak structure in collisional conductivity for the case m/m e → ∞. We calculate Hall conductivity analytically and show that for mixed system filling factor is not symmetric about Fermi energy unlike symmetric plateaus for pure Dirac case.

Optical conductivity signatures of open Dirac nodal lines

We investigate the optical conductivity and far-infrared magneto-optical response of ${\mathrm{BaNiS}}_{2}$, a simple square-lattice semimetal characterized by Dirac nodal lines that disperse exclusively along the out-of-plane direction. With the magnetic field aligned along the nodal line the in-plane Landau level spectra show a nearly $\sqrt{B}$ behavior, the hallmark of a conical-band dispersion with a small spin-orbit coupling gap. The optical conductivity exhibits an unusual temperature-independent isosbestic line, ending at a Van Hove singularity. First-principles calculations unambiguously assign the isosbestic line to transitions across Dirac nodal states. Our work suggests a universal topology of the electronic structure of Dirac nodal lines.

Geometric characterization of anomalous Landau levels of isolated flat bands

According to the Onsager’s semiclassical quantization rule, the Landau levels of a band are bounded by its upper and lower band edges at zero magnetic field. However, there are two notable systems where the Landau level spectra violate this expectation, including topological bands and flat bands with singular band crossings, whose wave functions possess some singularities. Here, we introduce a distinct class of flat band systems where anomalous Landau level spreading (LLS) appears outside the zero-field energy bounds, although the relevant wave function is nonsingular. The anomalous LLS of isolated flat bands are governed by the cross-gap Berry connection that measures the wave-function geometry of multi bands. We also find that symmetry puts strong constraints on the LLS of flat bands. Our work demonstrates that an isolated flat band is an ideal system for studying the fundamental role of wave-function geometry in describing magnetic responses of solids.

Fermi level fluctuations, reduced effective masses and Zeeman effect during quantum oscillations in nodal line semimetals

We probe quantum oscillations in nodal line semimetals (NLSM) by considering an NLSM continuum model under strong magnetic field and report the characteristics of the Landau level (LL) spectra and the fluctuations in the Fermi level as the field in a direction perpendicular to the nodal plane is varied through. Based on the results on parallel magnetization, we demonstrate the growth of quantum oscillation with field strength as well as its constancy in period when plotted against 1/B. We find that the density of states (DOS) which show series of peaks in succession, witness bifurcation of those peaks due to Zeeman effect. For field normal to nodal plane, such bifurcations are discernible only if the electron effective mass is considerably smaller than its free value, which usually happens in these systems. Though a reduced effective mass m* causes the Zeeman splitting to become small compared to LL spacings, experimental results indicate a manifold increase in the Lande g factor which again amplifies the Zeeman contribution. We also consider magnetic field in the nodal plane for which the DOS peaks do not repeat periodically with energy anymore. The spectra become more spread out and the Zeeman splittings become less prominent. We find the low energy topological regime, that appears with such in-plane field set up, to shrink further with reduced m* values. However, such topological regime can be stretched out in case there are smaller Fermi velocities for electrons in the direction normal to the nodal plane.