Splitting Of Landau Levels Research Articles

Magnetoelectronic and optical properties of a <span class="aps-inline-formula"><math><msub><mrow><mi>MoS</mi></mrow><mn>2</mn></msub></math></span> monolayer

A generalized tight-binding model is utilized to study Landau-level spectra of monolayer molybdenum disulfide. The intrinsic spin-orbit coupling effectively gives rise to multiple splitting of Landau levels. With a close inspection of wave-function characteristics, these levels can be classified into specific groups in terms of their orbital, spin, and valley signatures. In the calculation of magnetoabsorption spectra, the physical origin of the optical selection rules is clearly resolved. In particular, crossings of absorption lines are available and serve as a unique feature of the spin-orbit coupling. Our numerical results clearly demonstrate the magnetic control of spin and valley charge carriers and provide a basis for future experiments.

Landau level splitting in rotationally faulted multilayer graphene

We show that valley degeneracy in rotationally faulted multilayer graphene may be broken in the presence of a magnetic field and interlayer commensurations. This happens due to a simultaneous breaking of both time-reversal and inversion symmetries leading to a splitting of Landau levels linear in the field. Our theoretical work is motivated by an experiment [Y. J. Song et al., Nature (London) 467, 185 (2010)] on epitaxially grown multilayer graphene where such linear splitting of Landau levels was observed at moderate fields. We consider both bilayer and trilayer configurations and, although a linear splitting occurs in both cases, we show that the latter produces a splitting that is in quantitative agreement with the experiment.

Observation of three-dimensional massless Kane fermions in a zinc-blende crystal

Solid state physics and quantum electrodynamics with its ultra-relativistic (massless) particles meet, to their mutual beneit, in the electronic properties of one-dimensional carbon nanotubes as well as two-dimensional graphene or surfaces of topological insulators. However, clear experimental evidence for electronic states with conical dispersion relations in all three dimensions, conceivable in certain bulk materials, is still missing. In the present work, we fabricate and study a zinc-blend crystal, HgCdTe, at the point of the semiconductor-to-semimetal topological transition. Three-dimensional massless electrons with a velocity of about 10$^6$ m/s are observed in this material, as testifed by: (i) the dynamical conductivity which increases linearly with the photon frequency, (ii) in a magnetic field $B$, by a $\sqrt{B}$ dependence of dipole-active inter-Landau-level resonances and (iii) the spin splitting of Landau levels, which follows a $\sqrt{B}$ dependence, typical of ultra-relativistic particles but not really seen in any other electronic system so far.

Analysis of the spin splitting of maxima of quantum oscillations of the resistivity of n-Bi-Sb semiconductor alloys in a magnetic field parallel to the bisector axis

In samples of semiconductor alloys n-Bi0.93Sb0.07 with different electron concentrations (n1 = 8 × 1015 cm−3, n2 = 1.2 × 1017 cm−3, and n3 = 1.9 × 1018 cm−3), dependences of the electrical resistivity on magnetic fields up to 45 T parallel to the current and the bisector axis (H ‖ C1 ‖ j) have been measured at temperatures of 1.5, 4.5, and 10 K. The obtained dependences ρ22(H) demonstrate quantum oscillations of the resistivity (Shubnikov-de Haas effect), and, in high magnetic fields, there is a resistivity maximum far away from other maxima. On assumption that this maximum is related to the spin-split Landau level N = 0− for electrons of the main ellipsoid, the spin-splitting parameters are calculated for electrons of the main ellipsoid: γ1 = 0.87, γ2 = 0.8, and γ3 = 0.73. Using these values, the oscillation maxima can be reliably related to the numbers of split Landau levels for electrons of the main and secondary ellipsoids. The dependences of the resistivity ρ11 and the Hall coefficient R31.2 on magnetic field have been measured in a transverse magnetic field at H ‖ C1 and j ‖ C2 on the sample with the electron concentration n4 = 1.4 × 1017 cm−3. Using similar analysis, the spin-splitting parameter is found to be γ4 = 0.85, which is close to the value of γ2 = 0.8 obtained for the sample with close electron concentration (n2 = 1.2 × 1017 cm−3) during the measurements in a longitudinal magnetic field. The quantum oscillation maxima of Hall coefficient R31.2 are shifted to the range of high magnetic fields as compared to the quantum oscillation maxima of resistivity ρ11.

Splitting of the Zero-Energy Landau Level and Universal Dissipative Conductivity at Critical Points in Disordered Graphene

We report on robust features of the longitudinal conductivity (σ(xx)) of the graphene zero-energy Landau level in the presence of disorder and varying magnetic fields. By mixing an Anderson disorder potential with a low density of sublattice impurities, the transition from metallic to insulating states is theoretically explored as a function of Landau-level splitting, using highly efficient real-space methods to compute the Kubo conductivities (both σ(xx) and Hall σ(xy)). As long as valley degeneracy is maintained, the obtained critical conductivity σ(xx) =/~ 1.4e(2)/h is robust upon an increase in disorder (by almost 1 order of magnitude) and magnetic fields ranging from about 2 to 200 T. When the sublattice symmetry is broken, σ(xx) eventually vanishes at the Dirac point owing to localization effects, whereas the critical conductivities of pseudospin-split states (dictating the width of a σ(xy) = 0 plateau) change to σ(xx) =/~ e(2)/h, regardless of the splitting strength, superimposed disorder, or magnetic strength. These findings point towards the nondissipative nature of the quantum Hall effect in disordered graphene in the presence of Landau level splitting.

Quantum magnetotransport properties of ultrathin topological insulator films

We study the quantum magnetotransport in ultrathin topological insulator films in an external magnetic field considering hybridization between the upper and lower surfaces of the film. We investigate the two possible mechanisms for splitting of Landau levels, Zeeman and hybridization effects, and show that their interplay leads to minima in the collisional and Hall conductivities with a metal-to-insulator phase transition at the charge neutrality point. Hall plateaus arise at unusual multiples of e2/h. Evidence of a quantum phase transition for the zeroth and splitting of the higher Landau levels is found from the temperature and magnetic field dependences of the transport.

Landau level splitting due to graphene superlattices

The Landau level spectrum of graphene superlattices is studied using a tight-binding approach. We consider noninteracting particles moving on a hexagonal lattice with an additional one-dimensional superlattice made up of periodic square potential barriers, which are oriented along the zigzag or along the armchair directions of graphene. In the presence of a perpendicular magnetic field, such systems can be described by a set of one-dimensional tight-binding equations, the Harper equations. The qualitative behavior of the energy spectrum with respect to the strength of the superlattice potential depends on the relation between the superlattice period and the magnetic length. When the potential barriers are oriented along the armchair direction of graphene, we find for strong magnetic fields that the zeroth Landau level of graphene splits into two well-separated sublevels, if the width of the barriers is smaller than the magnetic length. In this situation, which persists even in the presence of disorder, a plateau with zero Hall conductivity can be observed around the Dirac point. This Landau level splitting is a true lattice effect that cannot be obtained from the generally used continuum Dirac-fermion model.

Tight-binding description of Landau levels of graphite in tilted magnetic fields

The electronic structure of Bernal-stacked graphite subject to tilted magnetic fields is studied theoretically. The minimal nearest-neighbor tight-binding model with the Peierls substitution is employed to describe the structure of Landau levels. We show that, while the orbital effect of the in-plane component of the magnetic field is negligible for massive Dirac fermions in the vicinity of the K point of the graphite Brillouin zone, at the H point it leads to the experimentally observable splitting of Landau levels, which grows approximately linearly with the in-plane field intensity.

Dirac donor states controlled by magnetic field in gapless and gapped graphene

In this paper, the exact solutions of Dirac electronic states of graphene in Coulomb and magnetic fields are acquired. The Coulomb field not only causes the splitting of Landau levels in gapless graphene but also leads to the variation of the energy level ordering in gapped graphene. The dependence of the binding energies on the gap and the magnetic field is discussed. Furthermore, the valley degree of freedom and the valley splitting spacing can be controlled by the Coulomb and magnetic fields in gapped graphene. The intervalley mixing of graphene is estimated and calculated in the direct sum spaces of the two valleys. The results obtained help us to understand the behaviors of the planar Dirac electron in electromagnetic fields and can be applied to the controlling of the electron's behaviors in graphene.

Quantum Hall Effect and Semimetallic Behavior of Dual-Gated ABA-Stacked Trilayer Graphene

The electronic structure of multilayer graphenes depends strongly on the number of layers as well as the stacking order. Here we explore the electronic transport of purely ABA-stacked trilayer graphenes in a dual-gated field-effect device configuration. We find that both the zero-magnetic-field transport and the quantum Hall effect at high magnetic fields are distinctly different from the monolayer and bilayer graphenes, and that they show electron-hole asymmetries that are strongly suggestive of a semimetallic band overlap. When the ABA trilayers are subjected to an electric field perpendicular to the sheet, Landau level splittings due to a lifting of the valley degeneracy are clearly observed.

Quantum Hall effect in a singly and doubly connected three-dimensional topological insulator

The surface states of topological insulators, which behave as charged massless Dirac fermions, are studied in the presence of a quantizing uniform magnetic field. Using the method of D.H. Lee[1], analytical formula satisfied by the energy spectrum is found for a singly and doubly connected geometry. This is in turn used to argue that the way to measure the quantized Hall conductivity is to perform the Laughlin's flux ramping experiment and measure the charge transferred from the inner to the outer surface, analogous to the experiment in Ref.[2]. Unlike the Hall bar setup used currently, this has the advantage of being free of the contamination from the delocalized continuum of the surface edge states. In the presence of the Zeeman coupling, and/or interaction driven Quantum Hall ferromagnetism, which translate into the Dirac mass term, the quantized charge Hall conductivity sigma_{xy}=n e^2/h, with n=0,\pm 1,\pm 3,\pm 5... Backgating of one of the surfaces leads to additional Landau level splitting and in this case n can be any integer.

Quantum transport of two-dimensional Dirac fermions in SrMnBi2

We report two-dimensional quantum transport in SrMnBi${}_{2}$ single crystals. The linear energy dispersion leads to unusual nonsaturated linear magnetoresistance since all Dirac fermions occupy the lowest Landau level in the quantum limit. The transverse magnetoresistance exhibits a crossover at a critical field ${B}^{*}$ from semiclassical weak-field ${B}^{2}$ dependence to the high-field linear-field dependence. With an increase in temperature, the critical field ${B}^{*}$ increases and the temperature dependence of ${B}^{*}$ satisfies the quadratic behavior which is attributed to the Landau-level splitting of the linear energy dispersion. The effective magnetoresistant mobility ${\ensuremath{\mu}}_{\text{MR}}\ensuremath{\sim}3400$ cm${}^{2}$/V s is derived. Angular-dependent magnetoresistance and quantum oscillations suggest dominant two-dimensional (2D) Fermi surfaces. Our results illustrate the dominant 2D Dirac fermion states in SrMnBi${}_{2}$ and imply that bulk crystals with Bi square nets can be used to study low-dimensional electronic transport commonly found in 2D materials such as graphene.

Chiral orbital current and anomalous magnetic moment in gapped graphene

We present a low-energy effective-mass theory to describe chiral orbital current and anomalous magnetic moment in graphenes with band gap and related materials. We show that a Bloch electron generally contains an anomalous current density due to inter-band matrix elements, which describes a chiral current circulation associated with magnetic moment. In gapped graphenes, the chiral current is opposite between different valleys, and corresponding magnetic moment accounts for valley splitting of Landau levels. In gapped bilayer graphene, in particular, the valley-dependent magnetic moment is responsible for divergence of paramagnetic susceptibility at the band bottom, and full valley polarization is achieved in relatively small magnetic field range. The formulation also applies to the gapped surface states of three-dimensional topological insulator, where the anomalous current is related to the magneto-electric response in spatially-modulated potential.

Landau level spectra and the quantum Hall effect of multilayer graphene

The Landau level spectra and the quantum Hall effect of ABA-stacked multilayer graphenes are studied in the effective mass approximation. The low-energy effective mass Hamiltonian may be partially diagonalized into an approximate block-diagonal form, with each diagonal block contributing parabolic bands except, in a multilayer with an odd number of layers, for an additional block describing Dirac-like bands with a linear dispersion. We fully include the band parameters and, taking into account the symmetry of the lattice, we analyze their affect on the block-diagonal Hamiltonian. Next-nearest layer couplings are shown to be particularly important in determining the low-energy spectrum and the phase diagram of the quantum Hall conductivity, by causing energy shifts, level anti-crossings, and valley splitting of the low-lying Landau levels.

Quantum Hall states under conditions of vanishing Zeeman energy

We report on magneto-transport measurements of a two-dimensional electron gas confined in a Cd$_{0.997}$Mn$_{0.003}$Te quantum well structure under conditions of vanishing Zeeman energy. The electron Zeeman energy has been tuned via the $s-d$ exchange interaction in order to probe different quantum Hall states associated with metallic and insulating phases. We have observed that reducing Zeeman energy to zero does not necessary imply the disappearing of quantum Hall states, i.e. a closing of the spin gap. The spin gap value under vanishing Zeeman energy conditions is shown to be dependent on the filling factor. Numerical simulations support a qualitative description of the experimental data presented in terms of a crossing or an avoided-crossing of spin split Landau levels with same orbital quantum number $N$.

Effect of Magnetic Field on the Electronic Transport in Trilayer Graphene

The perpendicular magnetic field dependence of the longitudinal resistance in trilayer graphene at various temperatures has been systematically studied. For a fixed magnetic field, the trilayer graphene displays an intrinsic semiconductor behavior over the temperature range of 5-340 K. This is attributed to the parabolic band structure of trilayer graphene, where the Coulomb scattering is a strong function of temperature. The dependence of resistance on the magnetic field can be explained by the splitting of Landau levels (LLs). Our results reveal that the energy gap in the trilayer graphene is thermally activated and increases with √B.

Two-dimensional electron gas inδ-dopedSrTiO3

It is shown that Shubnikov-de Haas oscillations in ${\text{SrTiO}}_{3}$ $\ensuremath{\delta}$-doped with La can be understood as arising from a two-dimensional electron gas of one subband immersed in the space charge layer. Despite the inherent complexity of a subband that is derived from four $d$-band states near the conduction-band minimum of ${\text{SrTiO}}_{3}$, the quantum oscillations can be modeled quantitatively by recognizing that the magnetic field $(B)$ induced effective spin splitting and Landau-level splitting are comparable. The oscillations are not strictly periodic in $1/B$, which can be understood as caused by a weak dependence of the electron density on the magnetic field in the subband that produces the observed oscillations.

Spin edge states in two-dimensional electron systems in the presence of Zeeman effect

We study the spin edge states, induced by the combined effect of Bychkov-Rashba spinorbit and Zeeman interactions or of Dresselhaus spin-orbit and Zeeman interactions in a twodimensional electron system, exposed to a perpendicular quantizing magnetic field and restricted by a hard-wall confining potential. We derive an exact analytical formula for the dispersion relations of spin edge states and analyze their energy spectrum versus the momentum and the magnetic field. We calculate the average spin components and the average transverse position of electron. It is shown that by removing the spin degeneracy, spin-orbit interaction splits the spin edge states not only in the energy but also induces their spatial separation. Depending on the type of spin-orbit coupling and the principal quantum number, the Zeeman term in the combination with spin-orbit interaction increases or decreases essentially the splitting of bulk Landau levels while it has a weak influence on the spin edge states.

High-Field Electronic Properties of Graphene

We have measured the energy gaps in single-layer and bilayer graphene by means of temperature dependent transport experiments in high magnetic fields up to 33 T. They follow the expected Landau level splitting when a finite level width is taken into account. The quantum Hall effect, hitherto only observed up to 30 K, remains visible up to 200 K in bilayers and even up to room temperature in single-layer graphene. Our experiments in single-layer graphene show that the lowest Landau level, shared equally between electrons and holes at zero energy, becomes extremely narrow in high magnetic fields. It is this narrowing, together with the large Landau level splitting in graphene that leads to an extremely robust localization and makes the quantum Hall effect visible up to room temperature. In high magnetic fields (B>20 T) we observe a strongly increasing resistance with decreasing temperature. These results are explained with field dependent splitting of the lowest Landau level of the order of a few Kelvin, as extracted from activated transport measurements.

Collapse of ρ xx Ringlike Structures in 2DEGs Under Tilted Magnetic Fields

In the quantum Hall regime, the longitudinal resistivity $\rho_{xx}$ plotted as a density--magnetic-field ($n_{2D}-B$) diagram displays ringlike structures due to the crossings of two sets of spin split Landau levels from different subbands [e.g., Zhang \textit{et al.}, Phys. Rev. Lett. \textbf{95}, 216801 (2005)]. For tilted magnetic fields, some of these ringlike structures "shrink" as the tilt angle is increased and fully collapse at $\theta_c \approx 6^\circ$. Here we theoretically investigate the topology of these structures via a non-interacting model for the 2DEG. We account for the inter Landau-level coupling induced by the tilted magnetic field via perturbation theory. This coupling results in anti-crossings of Landau levels with parallel spins. With the new energy spectrum, we calculate the corresponding $n_{2D}-B$ diagram of the density of states (DOS) near the Fermi level. We argue that the DOS displays the same topology as $\rho_{xx}$ in the $n_{2D}-B$ diagram. For the ring with filling factor $\nu=4$, we find that the anti-crossings make it shrink for increasing tilt angles and collapse at a large enough angle. Using effective parameters to fit the $\theta = 0^\circ$ data, we find a collapsing angle $\theta_c \approx 3.6^\circ$. Despite this factor-of-two discrepancy with the experimental data, our model captures the essential mechanism underlying the ring collapse.