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We present a general description of topological insulators from the point of view of Dirac equations. The Z_{2} index for the Dirac equation is always zero, and thus the Dirac equation is topologically trivial. After the quadratic B term in momentum is introduced to correct the mass term m or the band gap of the Dirac equation, the Z_{2} index is modified as 1 for mB>0 and 0 for mB<0. For a fixed B there exists a topological quantum phase transition from a topologically trivial system to a non-trivial one system when the sign of mass m changes. A series of solutions near the boundary in the modified Dirac equation are obtained, which is characteristic of topological insulator. From the solutions of the bound states and the Z_{2} index we establish a relation between the Dirac equation and topological insulators.
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Plasmons are the quantized collective oscillations of electrons in metals and doped semiconductors. The plasmons of ordinary, massive electrons are since a long time basic ingredients of research in plasmonics and in optical metamaterials. Plasmons of massless Dirac electrons were instead recently observed in a purely two-dimensional electron system (2DEG)like graphene, and their properties are promising for new tunable plasmonic metamaterials in the terahertz and the mid-infrared frequency range. Dirac quasi-particles are known to exist also in the two-dimensional electron gas which forms at the surface of topological insulators due to a strong spin-orbit interaction. Therefore,one may look for their collective excitations by using infrared spectroscopy. Here we first report evidence of plasmonic excitations in a topological insulator (Bi2Se3), that was engineered in thin micro-ribbon arrays of different width W and period 2W to select suitable values of the plasmon wavevector k. Their lineshape was found to be extremely robust vs. temperature between 6 and 300 K, as one may expect for the excitations of topological carriers. Moreover, by changing W and measuring in the terahertz range the plasmonic frequency vP vs. k we could show, without using any fitting parameter, that the dispersion curve is in quantitative agreement with that predicted for Dirac plasmons.
We investigate a quantum well that consists of a thin topological insulator sandwiched between two trivial insulators. More specifically, we consider smooth interfaces between these different types of materials such that the interfaces host not only the chiral interface states, whose existence is dictated by the bulk-edge correspondence, but also massive Volkov-Pankratov states. We investigate possible hybridization between these interface states as a function of the width of the topological material and of the characteristic interface size. Most saliently, we find a strong qualitative difference between an extremely weak effect on the chiral interface states and a more common hybridization of the massive Volkov-Pankratov states that can be easily understood in terms of quantum tunneling in the framework of the model of a (Dirac) quantum well we introduce here.
The possible realization of dissipationless chiral edge current in a topological insulator / magnetic insulator heterostructure is based on the condition that the magnetic proximity exchange coupling at the interface is dominated by the Dirac surface states of the topological insulator. Here we report a polarized neutron reflectometry observation of Dirac electrons mediated magnetic proximity effect in a bulk-insulating topological insulator (Bi$_{0.2}$Sb$_{0.8}$)$_{2}$Te$_{3}$ / magnetic insulator EuS heterostructure. We are able to maximize the proximity induced magnetism by applying an electrical back gate to tune the Fermi level of topological insulator to be close to the charge neutral point. A phenomenological model based on diamagnetic screening is developed to explain the suppressed proximity induced magnetism at high carrier density. Our work paves the way to utilize the magnetic proximity effect at the topological insulator/magnetic insulator hetero-interface for low-power spintronic applications.
The recent theoretical prediction and experimental realization of topological insulators (TI) has generated intense interest in this new state of quantum matter. The surface states of a three-dimensional (3D) TI such as Bi_2Te_3, Bi_2Se_3 and Sb_2Te_3 consist of a single massless Dirac cones. Crossing of the two surface state branches with opposite spins in the materials is fully protected by the time reversal (TR) symmetry at the Dirac points, which cannot be destroyed by any TR invariant perturbation. Recent advances in thin-film growth have permitted this unique two-dimensional electron system (2DES) to be probed by scanning tunneling microscopy (STM) and spectroscopy (STS). The intriguing TR symmetry protected topological states were revealed in STM experiments where the backscattering induced by non-magnetic impurities was forbidden. Here we report the Landau quantization of the topological surface states in Bi_2Se_3 in magnetic field by using STM/STS. The direct observation of the discrete Landau levels (LLs) strongly supports the 2D nature of the topological states and gives direct proof of the nondegenerate structure of LLs in TI. We demonstrate the linear dispersion of the massless Dirac fermions by the square-root dependence of LLs on magnetic field. The formation of LLs implies the high mobility of the 2DES, which has been predicted to lead to topological magneto-electric effect of the TI.
The non-trivial topology of the three-dimensional (3D) topological insulator (TI) dictates the appearance of gapless Dirac surface states. Intriguingly, when a 3D TI is made into a nanowire, a gap opens at the Dirac point due to the quantum confinement, leading to a peculiar Dirac sub-band structure. This gap is useful for, e.g., future Majorana qubits based on TIs. Furthermore, these Dirac sub-bands can be manipulated by a magnetic flux and are an ideal platform for generating stable Majorana zero modes (MZMs), which play a key role in topological quantum computing. However, direct evidence for the Dirac sub-bands in TI nanowires has not been reported so far. Here we show that by growing very thin ($sim$40-nm diameter) nanowires of the bulk-insulating topological insulator (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ and by tuning its chemical potential across the Dirac point with gating, one can unambiguously identify the Dirac sub-band structure. Specifically, the resistance measured on gate-tunable four-terminal devices was found to present non-equidistant peaks as a function of the gate voltage, which we theoretically show to be the unique signature of the quantum-confined Dirac surface states. These TI nanowires open the way to address the topological mesoscopic physics, and eventually the Majorana physics when proximitised by an $s$-wave superconductor.
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