Topological insulators doped with transition metals have recently been found to host a strong ferromagnetic state with perpendicular to plane anisotropy as well as support a quantum Hall state with edge channel transport, even in the absence of an external magnetic field. It remains unclear however why a robust magnetic state should emerge in materials of relatively low crystalline quality and dilute magnetic doping. Indeed, recent experiments suggest that the ferromagnetism exhibits at least some superparamagnetic character. We report on transport measurements in a sample that shows perfect quantum anomalous Hall quantization, while at the same time exhibits traits in its transport data which suggest inhomogeneities. We speculate that this may be evidence that the percolation path interpretation used to explain the transport during the magnetic reversal may actually have relevance over the entire field range. 73.43.Fj, 75.45.+j, 75.50.Pp The recent report on the experimental observation of a quantum anomalous Hall effect (QAHE) in Cr-doped (Bi,Sb) 2 Te 3 [1] generated significant interest in this material system for its potential as a magnetic topological insulator and as a test bed for the study of the Quantum Hall effect without the need for an external magnetic field [2][3][4][5][6]. This original report showed that the anomalous Hall contribution [1] appeared to saturate to a value of one conduction quantum as the sample was cooled to mK temperatures, but did not yet provide evidence that the transport takes place in edge states. In order to convincingly verify that the transport takes place in quantum Hall like edge states, non-local geometries are required. Such measurements were first reported in [3], albeit in configurations where the signals were very small, and where their interpretation required invoking some loss mechanism in the edge channels, and subsequently in [4], where convincing evidence of edge state transport was reported. This last paper also observed some unusual temperature and sweep rate related features in their data, which were at least in part interpreted as additional cooling through adiabatic demagnetization mechanisms. Shortly after the first reports on Cr-doped layers, it was discovered by the Moodera group [5-7] that using V instead of Cr appears to lead to more reproducible samples with a more robust magnetic and quantum anomalous Hall state. Using this material system, the authors were able to reproduce both precise quantization of the Quantum Hall state [5], and unequivocal evidence of edge state transport [6]. While the described quantum anomalous Hall phenomenology is now well established, its microscopic origin remains much less clear. The proposed mechanism for the QAHE is the breaking of time reversal symmetry by a perpendicular to plane internal magnetic field which leads to the reversal of the band inversion of one of the two spin species in a ferromagnetic two dimensional topological insulator [1,8]. The origin of the ferromagnetic state in the original paper [1] w...

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...

The microstructure of Bi 2 Se 3 topological-insulator thin films grown by molecular beam epitaxy on InP(111)A and InP(111)B substrates that have different surface roughnesses has been studied in detail using X-ray diffraction, X-ray reflectivity, atomic force microscopy and probe-corrected scanning transmission electron microscopy. The use of a rough Fe-doped InP(111)B substrate results in complete suppression of twin formation in the Bi 2 Se 3 thin films and a perfect interface between the films and their substrates. The only type of structural defects that persist in the "twinfree" films is an antiphase domain boundary, which is associated with variations in substrate height.It is also shown that the substrate surface termination determines which family of twin domains dominates.

We report on the scaling behavior of V-doped (Bi,Sb)_{2}Te_{3} samples in the quantum anomalous Hall regime for samples of various thickness. While previous quantum anomalous Hall measurements showed the same scaling as expected from a two-dimensional integer quantum Hall state, we observe a dimensional crossover to three spatial dimensions as a function of layer thickness. In the limit of a sufficiently thick layer, we find scaling behavior matching the flow diagram of two parallel conducting topological surface states of a three-dimensional topological insulator each featuring a fractional shift of 1/2e^{2}/h in the flow diagram Hall conductivity, while we recover the expected integer quantum Hall behavior for thinner layers. This constitutes the observation of a distinct type of quantum anomalous Hall effect, resulting from 1/2e^{2}/h Hall conductance quantization of three-dimensional topological insulator surface states, in an experiment which does not require decomposition of the signal to separate the contribution of two surfaces. This provides a possible experimental link between quantum Hall physics and axion electrodynamics.

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