Magneto-Optics of Massive Dirac Fermions in BulkBi2Se3

Abstract

We report on magneto-optical studies of Bi2Se3, a representative member of the 3D topological insulator family. Its electronic states in bulk are shown to be well described by a simple Diractype Hamiltonian for massive particles with only two parameters: the fundamental bandgap and the band velocity. In a magnetic field, this model implies a unique property -spin splitting equal to twice the cyclotron energy: Es = 2Ec. This explains the extensive magneto-transport studies concluding a fortuitous degeneracy of the spin and orbital split Landau levels in this material. The Es = 2Ec match differentiates the massive Dirac electrons in bulk Bi2Se3 from those in quantum electrodynamics, for which Es = Ec always holds.PACS numbers: 71.70. Di, 76.40.+b, Inspiring analogies to relativistic systems have largely helped to elucidate the electronic properties of twodimensional graphene [1,2], surface states of topological insulators (TIs) [3][4][5][6], novel three-dimensional (3D) semimetals [7][8][9] as well as certain narrow gap semiconductors [10]. Here, we report on magneto-optical studies of bulk Bi 2 Se 3 , which imply the approximate applicability of the Dirac Hamiltonian for massive relativistic particles to approach the band structure of this popular representative of the TI family.The dispersion relations of genuine massive Dirac fermions in quantum electrodynamics are defined by two parameters: the energy gap 2∆ between particles and antiparticles and velocity parameter v D . At low energies, i.e., in the non-relativistic limit, these dispersions become parabolic and characterized by the same effective mass m D = ∆/v 2 D (rest Dirac mass). Such dispersions resemble the cartoon sketch of a direct gap semiconductor, which may be conventionally described using Schrödinger equation, completed by extra Pauli terms in order to include the spin degree of freedom. In contrast, no additional terms are needed when Dirac equation is employed, since it inherently accounts for spin-related effects. For instance, when the magnetic field B is applied, Dirac equation describes both cyclotron (E c ) as well as spin (E s ) splitting of the electronic states and implies that these two splitting energies are the same and linear with B in the non-relativistic approximation:For free electrons, this condition is equivalent to the effective g factor of 2 in E s = gµ B B (Bohr magneton µ B = e /2m 0 ) [11].In this Letter, we demonstrate experimentally that the conduction and valence bands of Bi 2 Se 3 are both, with a good precision, parabolic (perpendicular to the c-axis) and characterized by approximately the same effective mass. This crucial observation implies a great simplification of the multi-parameter Dirac Hamiltonian [5,12] commonly used to describe the bands of this material.The resulting simplified Dirac Hamiltonian differs from that of the genuine quantum electrodynamics system only by relevant (additional) diagonal dispersive terms, and importantly, it remains to be defined by two parameters only: by the bandgap energy 2∆ an...

Fanoq-reversal in topological insulator Bi2Se3

“…A linear fit (shown with a dashed line) yieldshω c /B = 0.77 meV/T, from which we extract the cyclotron effective mass m * /m e = 0.15. This value is in excellent agrement with the value (0.14) extracted from a recent magnet-transmission measurement on a thick film [16]. A fraction of this data are shown in Fig.…”

“…This is another testimony to the excellent sample quality, which when combined with a high magnetic field allowed for the first time [2,3,14] the observation of a cyclotron resonance in bulk Bi 2 Se 3 samples. The cyclotron resonance in Bi 2 Se 3 has so far been observed only in thin films [8,16]. [18], the most prominent feature in reflectance is the plasma edge located at approximately 180 cm −1 .…”

“…Far-infrared magneto-reflectance ratios R(ω,B)/R(ω, 0 T) of Bi 2 Se 3 were collected at the National High Magnetic Field Laboratory in magnetic fields as high as 18 T. Magneto-optical spectra of Bi 2 Se 3 have previously been reported only in magnetic field up to 8 T [2,3,14,15,8] and only recently in higher magnetic field [16]. The reflectance ratios R(ω,B)/R(ω, 0 T) were supplemented with the zero field data to obtain the absolute values of reflectance in magnetic field R(ω, B), as the product of the ratios with the absolute value of reflectance in zero field [17].…”

Nearly massless Dirac fermions and strong Zeeman splitting in the nodal-line semimetal ZrSiS probed by de Haas–van Alphen quantum oscillations

“…The spin-orbit coupling introduces an interaction energy of λL⋅S, leading to an effective g-factor of g eff = g (1±λL⋅S/Δ) where Δ is the crystal field splitting and λ is the spin-orbital coupling constant [84]. Spin-orbit coupling is predicted to play a more important role in Dirac materials [85][86][87]. When the spin-orbit coupling and non-relativistic approximation are taken into account, the cyclotron energy is found to be equal to the Zeeman splitting energy derived from the Dirac equation [85][86][87], from which the g-factor is determined to be 2m 0 /m D where m 0 and m D represent free electron mass and Dirac electron mass respectively.…”

Magnetoinfrared Spectroscopy of Landau Levels and Zeeman Splitting of Three-Dimensional Massless Dirac Fermions inZrTe5

“…If neglecting the dispersion along the magnetic field direction, then E n ∝ √ B. As a comparison, for free electron systems, the magnetic induced Landau level obeys E n = (n + 1 2 ) ω c , where ω c is the the cyclotron angular velocity and proportional to B instead of √ B. Additionally, the Landau level energy of massive 3D Dirac fermions in topological insulator Bi 2 Se 3 is reported to be in linear scale with B as well [28]. Therefore, the √ B dependence in Fig.…”