Magneto-optics of massive dirac fermions in bulk Bi2Se3

Anomalous Hall effect in Nd-doped Bi1.1Sb0.9STe2 topological insulator single crystals

Topological insulators are emerging materials with insulating bulk and symmetry protected nontrivial surface states. One of the most fascinating transport behaviors in a topological insulator is the quantum anomalous Hall effect, which has been observed in magnetic-topological-insulator-based devices. In this work, we report successful doping of rare-earth element Nd into Bi1.1Sb0.9STe2 bulk-insulating topological insulator single crystals, in which the Nd moments are ferromagnetically ordered at ∼100 K. Benefiting from the in-bulk-gap Fermi level, electronic transport behaviors dominated by the topological surface states are observed in the ferromagnetic region. At low temperatures, strong Shubnikov-de Haas oscillations with a nontrivial Berry phase are observed. The topological insulator with long range magnetic ordering in Nd-doped Bi1.1Sb0.9STe2 single crystals provides a good platform for quantum transport studies and spintronic applications.

Effects of pressure and temperature on topological electronic materials X2Y3 (X = As, Sb, Bi; Y = Se, Te) using first-principles

We systematically study the thermal and topological properties of X2Y3 (X = As, Sb, Bi; Y = Se, Te) and the effects of pressure and temperature on their electronic properties using first-principles. We find that the external pressure-induced electronic topological transition occurs at about 5 GPa for Bi2Se3, and the type of band gap tends to become indirect with the increase of pressure. We also investigate the lattice expansion with temperature in quasi-harmonic approximation and further explore the effect of temperature on the volume, band gap, and volumetric thermal expansion coefficient of the studied selenides and tellurides. Finally, we calculate the evolution of the Wannier charge center of X2Y3 to determine their topological invariants, and theoretically suggest that Bi2Se3 changes from a topological to an ordinary insulator when the pressure decreases to -8 GPa; As2Se3 is found to be an ordinary insulator, while all other four compounds are always strong topological insulators at any pressure or temperature.

Milliwatt terahertz harmonic generation from topological insulator metamaterials

Achieving efficient, high-power harmonic generation in the terahertz spectral domain has technological applications, for example, in sixth generation (6G) communication networks. Massless Dirac fermions possess extremely large terahertz nonlinear susceptibilities and harmonic conversion efficiencies. However, the observed maximum generated harmonic power is limited, because of saturation effects at increasing incident powers, as shown recently for graphene. Here, we demonstrate room-temperature terahertz harmonic generation in a Bi₂Se₃ topological insulator and topological-insulator-grating metamaterial structures with surface-selective terahertz field enhancement. We obtain a third-harmonic power approaching the milliwatt range for an incident power of 75 mW-an improvement by two orders of magnitude compared to a benchmarked graphene sample. We establish a framework in which this exceptional performance is the result of thermodynamic harmonic generation by the massless topological surface states, benefiting from ultrafast dissipation of electronic heat via surface-bulk Coulomb interactions. These results are an important step towards on-chip terahertz (opto)electronic applications.

The effect of mechanical strain on the Dirac surface states in the (0001) surface and the cohesive energy of the topological insulator Bi2Se3

The band gap (E _g) engineering and Dirac point tuning of the (0001) surface of 8 QLs (quintuple layers) thick Bi₂Se₃ slab are explored using the first-principles density functional theory calculations by varying the strain. The strain on the Bi₂Se₃ slab primarily varies the bandwidth, modifies the p _z - orbital population of Bi and moves the Dirac point of the (0001) surface of Bi₂Se₃. The Dirac cone feature of the (0001) surface of Bi₂Se₃ is preserved for the entire range of the biaxial strain. However, around 5% tensile uniaxial strain and even lower value of volume conservation strain annihilate the Dirac cone, which causes the loss of topological (0001) surface states of Bi₂Se₃. The biaxial strain provides ease in achieving the Dirac cone at the Fermi energy (E _F) than the uniaxial and volume conservation strains. Interestingly, the transition from direct E _g to indirect E _g state of the (0001) surface of Bi₂Se₃ is observed in the volume conservation strain-dependent E _g. The strain on Bi₂Se₃, significantly modifies the conduction band of Se2 atoms near E _F compared to Bi and Se1, and plays a vital role in the conduction of the (0001) surface of Bi₂Se₃. The atomic cohesive energy of the Bi₂Se₃ slab is very close to that of (0001) oriented nanocrystals extracted from the Raman spectra. The strain-dependent cohesive energy indicates that at a higher value of strain, the uniaxial and volume conservation strain provides better stability than that of the biaxial strain (0001) oriented growth of the Bi₂Se₃ nanocrystals. Our study establishes the relationship between the strained lattice and electronic structures of Bi₂Se₃, and more generally demonstrates the tuning of the Dirac point with the mechanical strain.

Unraveling the Topological Phase of ZrTe_{5} via Magnetoinfrared Spectroscopy

For materials near the phase boundary between weak and strong topological insulators (TIs), their band topology depends on the band alignment, with the inverted (normal) band corresponding to the strong (weak) TI phase. Here, taking the anisotropic transition-metal pentatelluride ZrTe_{5} as an example, we show that the band inversion manifests itself as a second extremum (band gap) in the layer stacking direction, which can be probed experimentally via magnetoinfrared spectroscopy. Specifically, we find that the band anisotropy of ZrTe_{5} features a slow dispersion in the layer stacking direction, along with an additional set of optical transitions from a band gap next to the Brillouin zone center. Our work identifies ZrTe_{5} as a strong TI at liquid helium temperature and provides a new perspective in determining band inversion in layered topological materials.

Electron-Hole Asymmetry of Surface States in Topological Insulator Sb2Te3 Thin Films Revealed by Magneto-Infrared Spectroscopy

When surface states (SSs) form in topological insulators (TIs), they inherit the properties of bulk bands, including the electron-hole (e-h) asymmetry but with much more profound impacts. Here via combining magneto-infrared spectroscopy with theoretical analysis, we show that e-h asymmetry significantly modifies the SS electronic structures when interplaying with the quantum confinement effect. Compared with the case without e-h asymmetry, the SSs now bear not only a band asymmetry, such as that in the bulk, but also a shift of the Dirac point relative to the bulk bands and a reduction of the hybridization gap of up to 70%. Our results signify the importance of e-h asymmetry in the band engineering of TIs in the thin-film limit.

Nonperturbative Matrix Mechanics Approach to Spin-Split Landau Levels and the g Factor in Spin-Orbit Coupled Solids

We propose a fully quantum approach to nonperturbatively calculate the spin-split Landau levels and g factor of various spin-orbit coupled solids based on the k·p theory in the matrix mechanics representation. The new method considers the detailed band structure and the multiband effect of spin-orbit coupling irrespective of the magnetic-field strength. We show an application of this method to PbTe, a typical Dirac electron system. Contrary to popular belief, we show that the spin-splitting parameter M, which is the ratio of the Zeeman to cyclotron energy, exhibits a remarkable magnetic-field dependence. This field dependence can rectify the existing discrepancy between experimental and theoretical results. We also show that M evaluated from the fan diagram plot is different from that determined as the ratio of the Zeeman to cyclotron energy, which also overturns common belief.

Revealing Optical Transitions and Carrier Recombination Dynamics within the Bulk Band Structure of Bi2Se3

Bismuth selenide (Bi₂Se₃) is a prototypical 3D topological insulator whose Dirac surface states have been extensively studied theoretically and experimentally. Surprisingly little, however, is known about the energetics and dynamics of electrons and holes within the bulk band structure of the semiconductor. We use mid-infrared femtosecond transient reflectance measurements on a single nanoflake to study the ultrafast thermalization and recombination dynamics of photoexcited electrons and holes within the extended bulk band structure over a wide energy range (0.3 to 1.2 eV). Theoretical modeling of the reflectivity spectral line shapes at 10 K demonstrates that the electrons and holes are photoexcited within a dense and cold electron gas with a Fermi level positioned well above the bottom of the lowest conduction band. Direct optical transitions from the first and the second spin-orbit split valence bands to the Fermi level above the lowest conduction band minimum are identified. The photoexcited carriers thermalize rapidly to the lattice temperature within a couple of picoseconds due to optical phonon emission and scattering with the cold electron gas. The minority carrier holes recombine with the dense electron gas within 150 ps at 10 K and 50 ps at 300 K. Such knowledge of interaction of electrons and holes within the bulk band structure provides a foundation for understanding how such states interact dynamically with the topologically protected Dirac surface states.

Chiral Landau levels in Weyl semimetal NbAs with multiple topological carriers

Recently, Weyl semimetals have been experimentally discovered in both inversion-symmetry-breaking and time-reversal-symmetry-breaking crystals. The non-trivial topology in Weyl semimetals can manifest itself with exotic phenomena, which have been extensively investigated by photoemission and transport measurements. Despite the numerous experimental efforts on Fermi arcs and chiral anomaly, the existence of unconventional zeroth Landau levels, as a unique hallmark of Weyl fermions, which is highly related to chiral anomaly, remains elusive owing to the stringent experimental requirements. Here, we report the magneto-optical study of Landau quantization in Weyl semimetal NbAs. High magnetic fields drive the system toward the quantum limit, which leads to the observation of zeroth chiral Landau levels in two inequivalent Weyl nodes. As compared to other Landau levels, the zeroth chiral Landau level exhibits a distinct linear dispersion in magnetic field direction and allows the optical transitions without the limitation of zero z momentum or [Formula: see text] magnetic field evolution. The magnetic field dependence of the zeroth Landau levels further verifies the predicted particle-hole asymmetry of the Weyl cones. Meanwhile, the optical transitions from the normal Landau levels exhibit the coexistence of multiple carriers including an unexpected massive Dirac fermion, pointing to a more complex topological nature in inversion-symmetry-breaking Weyl semimetals. Our results provide insights into the Landau quantization of Weyl fermions and demonstrate an effective tool for studying complex topological systems.

Engineering Dirac Materials: Metamorphic InAs1-xSbx/InAs1-ySby Superlattices with Ultralow Bandgap

Quasiparticles with Dirac-type dispersion can be observed in nearly gapless bulk semiconductors alloys in which the bandgap is controlled through the material composition. We demonstrate that the Dirac dispersion can be realized in short-period InAs_1-xSb_x/InAs_1-ySb_y metamorphic superlattices with the bandgap tuned to zero by adjusting the superlattice period and layer strain. The new material has anisotropic carrier dispersion: the carrier energy associated with the in-plane motion is proportional to the wave vector and characterized by the Fermi velocity v_F, and the dispersion corresponding to the motion in the growth direction is quadratic. Experimental estimate of the Fermi velocity gives v_F = 6.7 × 10⁵ m/s. Remarkably, the Fermi velocity in this system can be controlled by varying the overlap between electron and hole states in the superlattice. Extreme design flexibility makes the short-period metamorphic InAs_1-xSb_x/InAs_1-ySb_y superlattice a new prospective platform for studying the effects of charge-carrier chirality and topologically nontrivial states in structures with the inverted bandgaps.

Negative Longitudinal Magnetoresistance from the Anomalous N=0 Landau Level in Topological Materials

Negative longitudinal magnetoresistance (NLMR) is shown to occur in topological materials in the extreme quantum limit, when a magnetic field is applied parallel to the excitation current. We perform pulsed and dc field measurements on Pb_{1-x}Sn_{x}Se epilayers where the topological state can be chemically tuned. The NLMR is observed in the topological state, but is suppressed and becomes positive when the system becomes trivial. In a topological material, the lowest N=0 conduction Landau level disperses down in energy as a function of increasing magnetic field, while the N=0 valence Landau level disperses upwards. This anomalous behavior is shown to be responsible for the observed NLMR. Our work provides an explanation of the outstanding question of NLMR in topological insulators and establishes this effect as a possible hallmark of bulk conduction in topological matter.

Determination of the energy band gap of Bi2Se3

Despite intensive investigations of Bi₂Se₃ in past few years, the size and nature of the bulk energy band gap of this well-known 3D topological insulator still remain unclear. Here we report on a combined magneto-transport, photoluminescence and infrared transmission study of Bi₂Se₃, which unambiguously shows that the energy band gap of this material is direct and reaches E_g = (220 ± 5) meV at low temperatures.

Spectroscopic evidence for bulk-band inversion and three-dimensional massive Dirac fermions in ZrTe5

Three-dimensional topological insulators (3D TIs) represent states of quantum matters in which surface states are protected by time-reversal symmetry and an inversion occurs between bulk conduction and valence bands. However, the bulk-band inversion, which is intimately tied to the topologically nontrivial nature of 3D Tis, has rarely been investigated by experiments. Besides, 3D massive Dirac fermions with nearly linear band dispersions were seldom observed in TIs. Recently, a van der Waals crystal, ZrTe₅, was theoretically predicted to be a TI. Here, we report an infrared transmission study of a high-mobility [∼33,000 cm²/(V ⋅ s)] multilayer ZrTe₅ flake at magnetic fields (B) up to 35 T. Our observation of a linear relationship between the zero-magnetic-field optical absorption and the photon energy, a bandgap of ∼10 meV and a [Formula: see text] dependence of the Landau level (LL) transition energies at low magnetic fields demonstrates 3D massive Dirac fermions with nearly linear band dispersions in this system. More importantly, the reemergence of the intra-LL transitions at magnetic fields higher than 17 T reveals the energy cross between the two zeroth LLs, which reflects the inversion between the bulk conduction and valence bands. Our results not only provide spectroscopic evidence for the TI state in ZrTe₅ but also open up a new avenue for fundamental studies of Dirac fermions in van der Waals materials.

Temperature-driven massless Kane fermions in HgCdTe crystals

It has recently been shown that electronic states in bulk gapless HgCdTe offer another realization of pseudo-relativistic three-dimensional particles in condensed matter systems. These single valley relativistic states, massless Kane fermions, cannot be described by any other relativistic particles. Furthermore, the HgCdTe band structure can be continuously tailored by modifying cadmium content or temperature. At critical concentration or temperature, the bandgap collapses as the system undergoes a semimetal-to-semiconductor topological phase transition between the inverted and normal alignments. Here, using far-infrared magneto-spectroscopy we explore the continuous evolution of band structure of bulk HgCdTe as temperature is tuned across the topological phase transition. We demonstrate that the rest mass of Kane fermions changes sign at critical temperature, whereas their velocity remains constant. The velocity universal value of (1.07±0.05) × 10⁶ m s⁻¹ remains valid in a broad range of temperatures and Cd concentrations, indicating a striking universality of the pseudo-relativistic description of the Kane fermions in HgCdTe.

Kane fermions are predicted to be tunable with external parameters such as temperature. Here, Teppe et al. show a band structure evolution of bulk HgCdTe as temperature is tuned across topological phase transition, demonstrating that Kane fermions change sign in rest-mass and remain constant in velocity.

Crystalline spin-orbit interaction and the Zeeman splitting in Pb1-x Sn x Te

The ratio of the Zeeman splitting to the cyclotron energy ([Formula: see text]), which characterizes the relative strength of the spin-orbit interaction in crystals, is examined for the narrow gap IV-VI semiconductors PbTe, SnTe, and their alloy Pb1-x Sn x Te on the basis of the multiband [Formula: see text] theory. The inverse mass α, the g-factor g, and M are calculated numerically by employing the relativistic empirical tight-binding band calculation. On the other hand, a simple but exact formula of M is obtained for the six-band model based on the group theoretical analysis. It is shown that M  <  1 for PbTe and M  >  1 for SnTe, which are interpreted in terms of the relevance of the interband couplings due to the crystalline spin-orbit interaction. It is clarified both analytically and numerically that M is not a quantized value but a continuous one, and M  =  1 is obtained just at the band inversion point, where the transition from trivial to nontrivial topological crystalline insulator occurs. By using this property, one can detect the transition point only with the bulk measurements. It is also proposed that M is useful to evaluate quantitatively a degree of the Dirac electrons in solids.

Zero-bias photocurrent in ferromagnetic topological insulator

Magnetic interactions in topological insulators cause essential modifications in the originally mass-less surface states. They offer a mass gap at the Dirac point and/or largely deform the energy dispersion, providing a new path towards exotic physics and applications to realize dissipation-less electronics. The nonequilibrium electron dynamics at these modified Dirac states unveil additional functions, such as highly efficient photon to spin-current conversion. Here we demonstrate the generation of large zero-bias photocurrent in magnetic topological insulator thin films on mid-infrared photoexcitation, pointing to the controllable band asymmetry in the momentum space. The photocurrent spectra with a maximal response to the intra-Dirac-band excitations can be a sensitive measure for the correlation between Dirac electrons and magnetic moments.

By magnetic-doping, the electronic band structure of a topological insulator can be significantly altered to yield functional behaviour. Here, the authors demonstrate a large photocurrent response, and its control, under zero-bias in an optimally-designed magnetically-doped topological insulator thin film.

Fano q-reversal in topological insulator Bi2Se3

We studied the magneto-optical response of a canonical topological insulator Bi2Se3 with the goal of addressing a controversial issue of electron-phonon coupling. Magnetic-field induced modifications of reflectance are very pronounced in the infrared part of the spectrum, indicating strong electron-phonon coupling. This coupling causes an asymmetric line-shape of the 60 cm(-1) phonon mode, and is analyzed within the Fano formalism. The analysis reveals that the Fano asymmetry parameter (q) changes sign when the cyclotron resonance is degenerate with the phonon mode. To the best of our knowledge this is the first example of magnetic field driven q-reversal.

Theory of electron spin resonance in bulk topological insulators Bi2Se3, Bi2Te3 and Sb2Te3

We report a theoretical study of electron spin resonance in bulk topological insulators, such as Bi2Se3, Bi2Te3 and Sb2Te3. Using the effective four-band model, we find the electron energy spectrum in a static magnetic field and determine the response to electric and magnetic dipole perturbations, represented by oscillating electric and magnetic fields perpendicular to the static field. We determine the associated selection rules and calculate the absorption spectra. This enables us to separate the effective orbital and spin degrees of freedom and to determine the effective g factors for electrons and holes.

Massive and massless Dirac fermions in Pb1-xSnxTe topological crystalline insulator probed by magneto-optical absorption

Dirac fermions in condensed matter physics hold great promise for novel fundamental physics, quantum devices and data storage applications. IV-VI semiconductors, in the inverted regime, have been recently shown to exhibit massless topological surface Dirac fermions protected by crystalline symmetry, as well as massive bulk Dirac fermions. Under a strong magnetic field (B), both surface and bulk states are quantized into Landau levels that disperse as B^1/2, and are thus difficult to distinguish. In this work, magneto-optical absorption is used to probe the Landau levels of high mobility Bi-doped Pb_0.54Sn_0.46Te topological crystalline insulator (111)-oriented films. The high mobility achieved in these thin film structures allows us to probe and distinguish the Landau levels of both surface and bulk Dirac fermions and extract valuable quantitative information about their physical properties. This work paves the way for future magnetooptical and electronic transport experiments aimed at manipulating the band topology of such materials.

Strong interband Faraday rotation in 3D topological insulator Bi2Se3

The Faraday effect is a representative magneto-optical phenomenon, resulting from the transfer of angular momentum between interacting light and matter in which time-reversal symmetry has been broken by an externally applied magnetic field. Here we report on the Faraday rotation induced in the prominent 3D topological insulator Bi2Se3 due to bulk interband excitations. The origin of this non-resonant effect, extraordinarily strong among other non-magnetic materials, is traced back to the specific Dirac-type Hamiltonian for Bi2Se3, which implies that electrons and holes in this material closely resemble relativistic particles with a non-zero rest mass.

Origin of the Large Anisotropic g Factor of Holes in Bismuth

The ratio of the Zeeman splitting to the cyclotron energy (M=ΔE_{Z}/ℏω_{c}) for holelike carriers in bismuth has been quantified with great precision by many experiments performed during the past five decades. It exceeds 2 when the magnetic field is along the trigonal axis and vanishes in the perpendicular configuration. Theoretically, however, M is expected to be isotropic and equal to unity in a two-band Dirac model. We argue that a solution to this half-a-century-old puzzle can be found by extending the k·p theory to multiple bands. Our model not only gives a quantitative account of the magnitude and anisotropy of M for holelike carriers in bismuth, but also explains its contrasting evolution with antimony doping and pressure, both probed by new experiments reported here. The present results have important implications for the magnitude and anisotropy of M in other systems with strong spin-orbit coupling.

Magnetoinfrared Spectroscopy of Landau Levels and Zeeman Splitting of Three-Dimensional Massless Dirac Fermions in ZrTe(5)

We present a magnetoinfrared spectroscopy study on a newly identified three-dimensional (3D) Dirac semimetal ZrTe(5). We observe clear transitions between Landau levels and their further splitting under a magnetic field. Both the sequence of transitions and their field dependence follow quantitatively the relation expected for 3D massless Dirac fermions. The measurement also reveals an exceptionally low magnetic field needed to drive the compound into its quantum limit, demonstrating that ZrTe(5) is an extremely clean system and ideal platform for studying 3D Dirac fermions. The splitting of the Landau levels provides direct, bulk spectroscopic evidence that a relatively weak magnetic field can produce a sizable Zeeman effect on the 3D Dirac fermions, which lifts the spin degeneracy of Landau levels. Our analysis indicates that the compound evolves from a Dirac semimetal into a topological line-node semimetal under the current magnetic field configuration.