Splitting Of Landau Levels Research Articles

Exceptional electronic transport and quantum oscillations in thin bismuth crystals grown inside van der Waals materials.

Confining materials to two-dimensional forms changes the behaviour of the electrons and enables the creation of new devices. However, most materials are challenging to produce as uniform, thin crystals. Here we present a synthesis approach where thin crystals are grown in a nanoscale mould defined by atomically flat van der Waals (vdW) materials. By heating and compressing bismuth in a vdW mould made of hexagonal boron nitride, we grow ultraflat bismuth crystals less than 10 nm thick. Due to quantum confinement, the bismuth bulk states are gapped, isolating intrinsic Rashba surface states for transport studies. The vdW-moulded bismuth shows exceptional electronic transport, enabling the observation of Shubnikov-de Haas quantum oscillations originating from the (111) surface state Landau levels. By measuring the gate-dependent magnetoresistance, we observe multi-carrier quantum oscillations and Landau level splitting, with features originating from both the top and bottom surfaces. Our vdW mould growth technique establishes a platform for electronic studies and control of bismuth's Rashba surface states and topological boundary modes1-3. Beyond bismuth, the vdW-moulding approach provides a low-cost way to synthesize ultrathin crystals and directly integrate them into a vdW heterostructure.

Spin splitting and disorder of Landau levels in HgTe-based Dirac fermions

Spin splitting and disorder of Landau levels in HgTe-based Dirac fermions

Unique signatures of the Rashba effect in the magneto-optical properties of two-dimensional semiconductors

The authors theoretically demonstrate two simple magneto-optical measurements which can be used to confirm the existence of Rashba terms in two-dimensional semiconductors even in samples of suboptimal quality, where the transition linewidths are comparable to, or larger than, the Landau-level splitting. The first suggested technique is magnetic circular dichroism (MCD), where a giant MCD signal should be observed at the band edge in the presence of Rashba terms. The second one is the comparison of the diamagnetic shift, measured in absorption, with that in photoluminescence which results from the dispersion minimum of the Rashba exciton.

Theory of quantized photonic spin Hall effect in strained graphene under a sub-Tesla external magnetic field.

The quantized photonic spin Hall effect (PSHE) in the strained graphene-substrate system is predicted under a sub-Tesla external magnetic field, which is two orders of magnitude smaller than required to produce the quantized effect in the conventional graphene-substrate system. It is found that in-plane and transverse spin-dependent splittings in the PSHE, exhibit different quantized behaviors and are closely related to the reflection coefficients. Unlike the quantized PSHE in the conventional graphene-substrate system formed by the splitting of real Landau levels, the quantized PSHE in the strained graphene-substrate system is attributed to the splitting of pseudo-Landau levels caused by the pseudo-magnetic field and the lifting of valley degeneracy of the n ≠ 0 pseudo-Landau levels induced by the sub-Tesla external magnetic field. At the same time, the pseudo-Brewster angles of the system are also quantized with the change of Fermi energy. The sub-Tesla external magnetic field and the PSHE appear as quantized peak values near these angles. The giant quantized PSHE is expected to be used for direct optical measurements of the quantized conductivities and pseudo-Landau levels in the monolayer strained graphene.

Magnetoconductance oscillations in electron-hole hybridization gaps and valley splittings in tetralayer graphene

We investigate magnetotransport on Bernal-stacked tetralayer graphene whose band structure consists of two massive subbands with different effective masses. Under a finite displacement field, we observe the valley splitting of Landau levels (LLs) only in the light-mass subband, consistent with a tight-binding model. At low density, we find unexpected magnetoconductance oscillations in bulk gaps which originate from a series of hybridizations between electronlike and holelike LLs due to band inversion in tetralayer graphene. In contrast to a trivial LL quantization gap, these inverted hybridization gaps can lead to a change in the number of edge states which explains the observed oscillations.

Origin of Structure Inversion Asymmetry in Double HgTe Quantum Wells

The density dependences of the splitting values of the dominant magneto-absorption lines in undoped p-type double HgTe/CdHgTe quantum wells are studied. An analysis of the splitting values of the magneto-absorption lines on the basis of self-consistent calculations of the band structure has made it possible to clarify the origin of structure inversion asymmetry in double HgTe/CdHgTe quantum wells and to determine the contributions caused by the built-in electric field, the difference between the thicknesses of the quantum wells, and the order of their arrangement in the structure. The discrepancy between the experimental energy splitting of the zero-mode Landau levels, extracted from the analysis of two types of lines, indicates the influence of many-particle effects on the transition energies between Landau levels.

Unusual Magnetotransport from two-dimensional Dirac Fermions in Pd3Bi2Se2

Pd$_{3}$Bi$_{2}$Se$_{2}$ has been proposed to be topologically non-trivial in nature. However, evidence of its non-trivial behavior is still unexplored. We report the growth and magneto-transport study of Pd$_{3}$Bi$_{2}$Se$_{2}$ thin films, revealing for the first time the contribution of two-dimensional (2D) topological surface states. We observe exceptional non-saturated linear magnetoresistance which results from Dirac fermions inhabiting the lowest Landau level in the quantum limit. The transverse magnetoresistance changes from a semi-classical weak-field $B^{2}$ dependence to a high-field $B$ dependence at a critical field $B^{\star}$. It is found that $B^{\star} \propto T^2$, which is expected from the Landau level splitting of a linear energy dispersion. In addition, the magnetoconductivity shows signatures of 2D weak anti-localization (WAL). These novel magnetotransport signatures evince the presence of 2D Dirac fermions in Pd$_{3}$Bi$_{2}$Se$_{2}$ thin films.

Magnetic field dependent two-photon absorption properties in monolayer MoS2

We report our theoretical investigations on the magnetic field dependence of two-photon absorption (TPA) properties in ${\mathrm{MoS}}_{2}$. Based on the Landau levels obtained by the effective massive Dirac model, we developed TPA theories associated with both interband transitions and intraband transitions for transitional metal dichalcogenides (TMDCs) with a magnetic field under the second-order perturbation theory with respect to the electron-photon interaction. Our results reveal that the TPA coefficient is effectively enhanced by five orders of magnitude under the application of an external magnetic field, which results in the quantized horizontal Landau levels with the spin-orbit coupling (SOC) and valley Zeeman effects. The TPA peaks are located in the terahertz regime for intraband transitions and in the visible regime for interband transitions. The amplitude of TPA contributed by intraband transitions is much larger than that of interband transitions owing to the resonant transitions. Oscillatory behavior appeared in the interband transition-related TPA spectra. The separation of TPA for the transitions in the spin-up and spin-down states caused by the spin splitting of Landau levels resulting from the strong SOC is illustrated. We also analyzed the anisotropy of TPA associated with the direction of the incident light. Our results are of much importance for the application of TMDCs in optical devices.

Magneto-transport evidence for strong topological insulator phase in ZrTe5

The identification of a non-trivial band topology usually relies on directly probing the protected surface/edge states. But, it is difficult to achieve electronically in narrow-gap topological materials due to the small (meV) energy scales. Here, we demonstrate that band inversion, a crucial ingredient of the non-trivial band topology, can serve as an alternative, experimentally accessible indicator. We show that an inverted band can lead to a four-fold splitting of the non-zero Landau levels, contrasting the two-fold splitting (spin splitting only) in the normal band. We confirm our predictions in magneto-transport experiments on a narrow-gap strong topological insulator, zirconium pentatelluride (ZrTe5), with the observation of additional splittings in the quantum oscillations and also an anomalous peak in the extreme quantum limit. Our work establishes an effective strategy for identifying the band inversion as well as the associated topological phases for future topological materials research.

Influence of charged impurity screening and surface optical phonons on the n=0 Landau level splitting of graphene with polar substrate

The effect of charged Coulombic impurity screening on the n = 0 Landau level splitting of graphene with a polar substrate under a static magnetic field is investigated theoretically considering the electron-phonon Fröhlich interaction from surface optical phonons of the substrate. The screening is improved using a static dielectric function under the random-phase approximation. A method of linear combination operators is adopted to deal with the influence from a magnetic field and a weak electron-phonon coupling theory is used to discuss the effect from the surface optical phonons. It is found that the splitting due to phonons can be broadened by the charged impurity screening and the broadening increases as increasing of the magnetic field. Our results may provide a possible way to enlarge the energy gap of monolayer graphene. • The n = 0 Landau level splitting of graphene with a substrate under a magnetic field affected by Coulombic impurity screening is obtained. • The screening uses a static dielectric function under the random-phase approximation. • The splitting due to phonons can be broadened by the charged impurity screening. • The broadening increases as increasing of the magnetic field.

Temperature effect of tunable zero Landau Level splitting energy of monolayer graphene on substrates

The temperature effects of the tunable zero-Landau lever (LL) energy are studied using the Lee-Low-Pines (LLP) variational and quantum statistical theory in the presence of a magnetic field B on monolayer graphene of different polar substrates. This zero-LL energy was induced by the surface optical phonon polaron in monolayer graphene on polar substrates. We found that zero-LL splitting energy can be tuned by controlling the polarization strength η of the polar substrate, confinement strength parameter g, magnetic field B , and temperature T . The magnetic field has a negligible impact on the zero-LL splitting energy. And the temperature effect of zero-LL splitting energy of graphene substrates is further enhanced by Coulomb potential . The results obtained in the study provide new information about the fundamental properties of graphene on substrates at different temperatures T . • Temperature effect of tunable zero-LL splitting energy of monolayer graphene(MG) on substrates is studied. • The temperature has significant effect on zero-LL splitting without coulomb impurity for the MG on the substrate. • The magnetic field has a small impact on the zero-LL splitting energy of MG on the substrate. • The coulomb bound potential reinforces the temperature effect of zero-LL splitting energy. • Temperature effect supplies the theoretical interpretations of the experimental measurements.

Quantum geometric exciton drift velocity

We show that the dipole moment of an exciton is uniquely determined by the quantum geometry of its eigenstates, and demonstrate its intimate connection with a quantity we call the dipole curvature. The dipole curvature arises naturally in semiclassical dynamics of an exciton in an electric field, adding a term additional to the anomalous velocity coming from the Berry's curvature. In a uniform electric field this contributes a drift velocity akin to that expected for excitons in crossed electric and magnetic fields, even in the absence of a real magnetic field. We compute the quantities relevant to semiclassical exciton dynamics for several interesting examples of bilayer systems with weak interlayer tunneling and Fermi energy in a gap, where the exciton may be sensibly described as a two-body problem. These quantities include the exciton dispersion, its dipole curvature, and Berry's curvature. For two gapped-graphene layers in a vanishing magnetic field, we find the dipole curvature vanishes if the layers are identical, but may be non-zero when the layers differ. We further analyze examples in the presence of magnetic fields, allowing us to examine cases involving graphene, in which a gap is opened by Landau level splitting. Heterostructures involving TMDs are also considered. In each case the dipole and/or Berry's curvatures play out differently. In some cases, the lowest energy exciton state is found to reside at finite momentum, with interesting possibilities for Bose condensation. We also find situations in which the dipole curvature increases monotonically with exciton momentum, suggesting that the quantum geometry can be exploited to produce photocurrents from initially bound excitons with electric fields. We speculate on further possible effects of the semiclassical dynamics in geometries where the constituent layers are subject to the same or different electric fields.

Determination of Carrier Density and Dynamics via Magnetoelectroluminescence Spectroscopy in Resonant-Tunneling Diodes

We study the magneto-transport and magnetoelectroluminescence properties of purely $n$-doped ${\mathrm{Ga}\mathrm{As}/\mathrm{Al}}_{0.6}{\mathrm{Ga}}_{0.4}\mathrm{As}$ resonant-tunneling diodes with an ${\mathrm{In}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{As}$ quantum well and an emitter prewell. Before the resonant-current condition, magneto-transport measurements reveal charge-carrier densities comparable for diodes with and without the emitter prewell. Landau-level splitting is observed in the electroluminescence emission from the emitter prewell, enabling the determination of the charge-carrier buildup. Our findings show that magnetoelectroluminescence spectroscopy techniques provide useful insights into the charge-carrier dynamics in resonant-tunneling diodes and comprise a versatile tool to complement magneto-transport techniques. This approach might pave the way for the development of potentially more efficient optoelectronic resonant-tunneling devices by, e.g., monitoring voltage-dependent charge accumulation for the improvement of built-in fields and hence the maximization of the photodetector efficiency and/or the minimization of optical losses.

Light-tunable 2D subband population in a GaN/AlGaN heterostructure

2D electron density tuning from 5.3×1012 cm−2 to 8.8×1012 cm−2 by UV illumination has been demonstrated for an ultra-pure GaN/Al0.25Ga0.75N heterojunction. A single subband is occupied in the dark with an electron density of 5.3×1012 cm−2, while in the illuminated sample with an electron density of 8.8×1012 cm−2, characteristic beating patterns appear in magnetotransport data due to the filling of the second quantized subband. Simultaneously, this distribution is unambiguously confirmed by Landau level splitting and intensity oscillations in photoluminescence spectra. The electron densities and quantum lifetimes in each individual subband have been extracted independently, and the intersubband energy spacing amounts to 82 meV. Surprisingly, the quantum scattering time for electrons residing in the second subband is increased compared to its ground state counterpart.

Electron Spectrum and Interband Magneto-Absorption of Light by Two-Dimensional Systems with Antidots

The degeneracy of Landau levels in perforated two-dimensional electronic structures is lifted in a perpendicular magnetic field. The cylindrical symmetry of the problem with a circular antidot holds and the magnetic quantum number m = 0, ±1, ±2, … is a good quantum number. The splitting of Landau levels for the antidot smaller than the magnetic length, as well as the frequencies and intensities of interband transition lines for normal semiconductors and transition metal dichalcogenide monolayers, has been analytically found.

Fractional quantum Hall states for moiré superstructures in the Hofstadter regime

We study the transition of $\nu=1/3$ and $2/5$ fractional quantum Hall states of the honeycomb Hofstadter model as we tune to a two-orbital moir\'e superlattice Hamiltonian, motivated by the flat bands of twisted bilayer graphene in a perpendicular magnetic field. In doing so, we address the extent to which these states survive in moir\'e systems and analyze the nature of the transition. Through the use of a Peierls substitution, we determine the Landau-level splitting for the moir\'e Hamiltonian, and study the structure of the Chern bands for a range of magnetic flux per plaquette. We identify topological flat bands in the spectrum at low energies, with numerically tractable lattice geometries that can support the fractional quantum Hall effect. As we tune the model, we find that the orbital-polarized $\nu=1/3$ and $2/5$ states corresponding to the honeycomb Hofstadter model survive up to $\approx 30\%$ of typical moir\'e superlattice parameters, beyond which they transition into an insulating phase. We present evidence for this through density matrix renormalization group calculations on an infinite cylinder, by verifying the charge pumping, spectral flow, entanglement scaling, and conformal field theory edge counting. We conclude that fractional quantum Hall states from the Hofstadter model can persist up to hopping amplitudes of the same order as those typical for moir\'e superlattice Hamiltonians, which implies generally that fractional states for moir\'e superstructures can be discerned simply by analyzing the dominant terms in their effective Hamiltonians.

Spectroscopic characterization of Landau-level splitting and the intermediate v=0 phase in bilayer graphene

Despite various novel broken symmetry states have been revealed in bilayer graphene (BLG) experimentally, the atomic-scale spectroscopic investigation has been greatly limited. Here, we study high-resolution spectroscopic characteristics of high-quality BLG and observe rich broken-symmetry-induced Landau level (LL) splittings, including valley, spin and orbit, by using ultralow-temperature and high-magnetic-field scanning tunneling microscopy and spectroscopy (STM and STS). Our experiment demonstrates that both the spin and orbital splittings of the lowest n = (0,1) LL depend sensitively on its filling and exhibit an obvious enhancement at partial-filling states. More unexpectedly, the splitting of a fully-filled and valley-polarized LL is also enhanced by partial filling of the LL with the opposite valley. These results reveal significant many-body effects in this system. At half filling of the n = (0,1) LL (filling factor v = 0), a single-particle intermediate v = 0 phase, which is the transition state between canted antiferromagnetic and layer-polarized states in the BLG, is measured and directly visualized at the atomic scale. Our atomic-scale STS measurement gives direct evidence that this intermediate v = 0 state is the predicted orbital-polarized phase.

Valley Polarization and Inversion in Strained Graphene via Pseudo-Landau Levels, Valley Splitting of Real Landau Levels, and Confined States.

It is quite easy to control spin polarization and the spin direction of a system via magnetic fields. However, there is no such direct and efficient way to manipulate the valley pseudospin degree of freedom. Here, we demonstrate experimentally that it is possible to realize valley polarization and valley inversion in graphene by using both strain-induced pseudomagnetic fields and real magnetic fields. Pseudomagnetic fields, which are quite different from real magnetic fields, point in opposite directions at the two distinct valleys of graphene. Therefore, the coexistence of pseudomagnetic fields and real magnetic fields leads to imbalanced effective magnetic fields at two distinct valleys of graphene. This allows us to control the valley in graphene as conveniently as the electron spin. In this work, we report a consistent observation of valley polarization and inversion in strained graphene via pseudo-Landau levels, splitting of real Landau levels, and valley splitting of confined states using scanning tunneling spectroscopy. Our results highlight a pathway to valleytronics in strained graphene-based platforms.

Landau quantisation of photonic spin Hall effect in monolayer black phosphorus

Abstract The photonic spin Hall effect (PSHE) is a promising candidate for controlling the spin states of photons and exploiting next-generation photonic devices based on spinoptics. Herein, the influences of a perpendicular magnetic field on the PSHE appearing on the surface of monolayer black phosphorus (BP) are investigated. Results reveal that both the in-plane and transverse spin-dependent shifts are quantised and show an oscillating pattern due to the splitting of Landau levels (LLs) induced by the external magnetic field B. And the oscillation period of spin Hall shifts gradually increases with strengthening B because of the increase of LL spacings. By contrast, for a fixed magnetic field, as the LL spacings become smaller and smaller with increasing the LL index, the oscillation period of spin Hall shifts gradually decreases as the photonic energy increases. Moreover, it is possibly due to the synergistic role of intrinsic anisotropy, high crystallinity, and quantisation-incurred localised decreases in beating-like complex conductivities of the BP film, giant spin Hall shifts, hundreds of times of the incident wavelength, are obtained in both transverse and in-plane directions. These unambiguously confirm the strong impact of the external magnetic field on the PSHE and shed important insights into understanding the rich magneto-optical transport properties in anisotropic two-dimensional atomic crystals.

Anisotropic Landau level splitting and Lifshitz transition induced magnetoresistance enhancement in ZrTe5 crystals

Magneto-transport study has been performed in ZrTe5 single crystals. The observed Shubnikov–de Hass quantum oscillation at low temperature clearly demonstrates the existence of a nontrivial band with small effective mass in ZrTe5. Furthermore, we also revealed the highly anisotropic nature of high-field Landau level splitting in ZrTe5, suggesting the dominant role of orbital contribution to the splitting. Besides these, an abnormal large enhancement of magnetoresistance appears at high temperatures, which is believed to arise from the Lifshitz transition induced two-carrier transport in ZrTe5. Our study provides more understanding of the physical properties of ZrTe5 and sheds light on potential application of ZrTe5 in spintronics.