A two-dimensional honeycomb lattice harbours a pair of inequivalent valleys in the k-space electronic structure, in the vicinities of the vertices of a hexagonal Brillouin zone, K±. It is particularly appealing to exploit this emergent degree of freedom of charge carriers, in what is termed 'valleytronics'. The physics of valleys mimics that of spin, and will make possible devices, analogous to spintronics, such as valley filter and valve, and optoelectronic Hall devices, all very promising for next-generation electronics. The key challenge lies with achieving valley polarization, of which a convincing demonstration in a two-dimensional honeycomb structure remains evasive. Here we show, using first principles calculations, that monolayer molybdenum disulphide is an ideal material for valleytronics, for which valley polarization is achievable via valley-selective circular dichroism arising from its unique symmetry. We also provide experimental evidence by measuring the circularly polarized photoluminescence on monolayer molybdenum disulphide, which shows up to 50% polarization.

A magnetoconductivity formula is presented for the surface states of a magnetically doped topological insulator. It reveals a competing effect of weak localization and weak antilocalization in quantum transport when an energy gap is opened at the Dirac point by magnetic doping. It is found that, while random magnetic scattering always drives the system from the symplectic to the unitary class, the gap could induce a crossover from weak antilocalization to weak localization, tunable by the Fermi energy or the gap. This crossover presents a unique feature characterizing the surface states of a topological insulator with the gap opened at the Dirac point in the quantum diffusion regime.PACS numbers: 73.25.+i, 03.65.Vf, Topological surface states, composed of an odd number of massless Dirac cones, are peculiar to threedimensional (3D) topological insulators [1][2][3]. Electrons in these states have a helical spin structure in momentum space, and acquire a π Berry's phase after completing a closed trajectory adiabatically around the Fermi surface. The π Berry phase could lead to the absence of backscattering[4], weak antilocalization [5], and the absence of Anderson localization [6,7]. In the quantum diffusion regime (mean free path ≪ system size ∼ phase coherent length), an electron maintains its phase coherence after being scattered by static centers for many times. As a result, the destructive interference due to the π Berry phase can give a quantum enhancement to the classical electronic conductivity, leading to weak antilocalization (WAL) [8,9]. Applying a magnetic field tends to break the destructive interference, giving rise to negative magnetoconductivity (MC), a key signature of WAL. WAL is expected in systems with symplectic symmetry. Much effort has been devoted to observing WAL in graphene [5,[10][11][12][13]. However, graphene has two valleys of gapless Dirac cones with opposite chiralities, and the intervalley scattering will inevitably suppress WAL [5,[10][11][12][13]. In contrast, the surface states of recently discovered topological insulators Bi 2 Te 3 and Bi 2 Se 3 have only one helical Dirac cone [14][15][16], and WAL is intrinsic to them. Many observations of WAL in Bi 2 Te 3 and Bi 2 Se 3 have been reported recently [17][18][19][20][21][22]. In particular, there is great interest in the effect of magnetic doping, which is considered to be an efficient way to open an energy gap in the Dirac cone by breaking time reversal symmetry (TRS) [23][24][25]. This gap is expected to give rise to many interesting phenomena, such as Majorana fermion [26], topological magnetoelectric effect [27] and quantized anomalous Hall effect [28]. These developments call for a thorough theoretical investigation on WAL in topological insulators, in particular, in the presence of magnetic doping.

We report that Bi₂Se₃ thin films can be epitaxially grown on SrTiO₃ substrates, which allow for very large tunablity in carrier density with a back gate. The observed low field magnetoconductivity due to weak antilocalization (WAL) has a very weak gate-voltage dependence unless the electron density is reduced to very low values. Such a transition in WAL is correlated with unusual changes in longitudinal and Hall resistivities. Our results suggest a much suppressed bulk conductivity at large negative gate voltages and a possible role of surface states in the WAL phenomena.

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Liouville's theorem on the conservation of phase space volume is violated by Berry phase in the semiclassical dynamics of Bloch electrons. This leads to a modification of the phase space density of states, whose significance is discussed in a number of examples: field modification of the Fermi-sea volume, connection to the anomalous Hall effect, and a general formula for orbital magnetization. The effective quantum mechanics of Bloch electrons is also sketched, where the modified density of states plays an essential role.PACS numbers: 73.43.-f, 72.15.-v, 75.20.-g Semiclassical dynamics of Bloch electrons in external fields has provided a powerful theoretical framework to account for various properties of metals, semiconductors and insulators [1]. In recent years, it has become increasingly clear that essential modification of the semiclassical dynamics is necessary for a proper understanding of a number of phenomena. It was known earlier that global geometric phase effects [2,3] on Bloch states are very important for insulators in our understanding of the quantum Hall effect [4], quantized adiabatic pumps [5], and electric polarization [6,7]. It was shown [8,9] later that geometric phase also modifies the local dynamics of Bloch electrons and thus affects the transport properties of metals and semiconductors. Recently these ideas have been successfully applied to the anomalous Hall effect in ferromagnetic semiconductors and metals [10,11,12,13], as well as spin transport [14,15].In this Letter, we reveal a general property of the Berry phase modified semiclassical dynamics which has been overlooked so far: the violation of Liouville's theorem for the conservation of phase space volume. Liouville's theorem was originally established for standard classical Hamiltonian dynamics, and its importance cannot be over emphasized as it serves as a foundation for classical statistical physics. The Berry phase makes, in general, the equations of motion non-canonical [8,9,16,17,18], rendering the violation of Liouville's theorem. Nevertheless, we are able to remedy the situation by modifying the density of states in the phase space.This modified phase-space density of states enters naturally in the semiclassical expression for the expectation value of physical quantities, and has profound effects on equilibrium as well as transport properties. We demonstrate this with several examples. First, we consider a Fermi sea of electrons in a weak magnetic field, and show that the Fermi sea volume can be changed linearly by the field. Second, we show how the Berry phase formula for the intrinsic anomalous Hall conductivity may be derived from equilibrium thermodynamics using the Středa formula [19]. Third, we provide a general derivation of an orbital-magnetization formula which is convenient for first-principles calculations.In addition, we present an effective quantum mechanics for Bloch electrons in solids by quantizing the semiclassical dynamics with the geometric phase. The density of states enters in a nontrivial manner into...

Conventional electronics are based invariably on the intrinsic degrees of freedom of an electron, namely its charge and spin. The exploration of novel electronic degrees of freedom has important implications in both basic quantum physics and advanced information technology. Valley, as a new electronic degree of freedom, has received considerable attention in recent years. In this paper, we develop the theory of spin and valley physics of an antiferromagnetic honeycomb lattice. We show that by coupling the valley degree of freedom to antiferromagnetic order, there is an emergent electronic degree of freedom characterized by the product of spin and valley indices, which leads to spinvalley-dependent optical selection rule and Berry curvatureinduced topological quantum transport. These properties will enable optical polarization in the spin-valley space, and electrical detection/manipulation through the induced spin, valley, and charge fluxes. The domain walls of an antiferromagnetic honeycomb lattice harbors valley-protected edge states that support spin-dependent transport. Finally, we use first-principles calculations to show that the proposed optoelectronic properties may be realized in antiferromagnetic manganese chalcogenophosphates (MnPX 3 , X = S, Se) in monolayer form.antiferromagnetism | valleytronics

Based on standard perturbation theory, we present a full quantum derivation of the formula for the orbital magnetization in periodic systems. The derivation is generally valid for insulators with or without a Chern number, for metals at zero or finite temperatures, and at weak as well as strong magnetic fields. The formula is shown to be valid in the presence of electron-electron interaction, provided the one-electron energies and wave functions are calculated self-consistently within the framework of the exact current and spin-density functional theory. DOI: 10.1103/PhysRevLett.99.197202 PACS numbers: 75.10.Lp, 73.20.At Magnetism is one of the most important properties of materials. Both spin and orbital motion of electrons can contribute to the total magnetization. While the spin magnetization can already be calculated from first principles with high accuracy by state-of-art methods such as the spin-density functional theory (SDFT), the study of orbital magnetization is still in a comparatively primitive stage.A first difficulty arises from the fact that there is still no theoretically well-established formula for calculating the orbital magnetization of a crystalline solid. The nonlocality of the orbital magnetization operatorM ÿe=2r is the major obstacle to obtaining a closed formula for an extended periodic system. Recently, Xiao et al. [1] and, independently, Thonhauser et al. [2,3], obtained an orbital magnetization formula which avoids the nonlocality problem and looks very promising for applications. However, up to date, there exists no general quantum mechanical derivation of this formula. The derivation presented in Ref.[1] relies on the semiclassical wave-packet dynamics of Bloch electrons [1,4,5], and its validity in the quantum context is not completely clear. On the other hand, the derivation presented in Ref.[2] is quantum mechanical but relies on the existence of localized Wannier functions, and it cannot be easily generalized to metals or insulators with nonzero Chern number. In addition, both derivations are limited to noninteracting systems. The shortcomings of these approaches call for a full quantum mechanical and many-body theory of the orbital magnetization.A second difficulty is that a first principle calculation of the orbital magnetization (taking into account many-body effects) should be based on the spin current density functional theory (SCDFT) [6] rather than the conventional SDFT. Unfortunately, SCDFT has been hindered so far by the lack of reliable expressions for the magnetizationdependent effective potentials. This may partly explain why the orbital moments of ferromagnetic transition metals such as Fe, Co, and Ni calculated in SCDFT were found to be significantly smaller than the experimentally determined values [7]. How problematic these calculations are is well explained in the review article by Richter [8]. The situation, however, has been rapidly changing in recent years. The advent of optimized effective potentials [9,10] which treat exchange exactly and may systemati...

We obtain a set of general formulas for determining magnetizations, including the usual electromagnetic magnetization as well as the gravitomagnetic energy magnetization. The magnetization corrections to the thermal transport coefficients are explicitly demonstrated. Our theory provides a systematic approach for properly evaluating the thermal transport coefficients of magnetic systems, eliminating the unphysical divergence from the direct application of the Kubo formula. For a noninteracting anomalous Hall system, the corrected thermal Hall conductivity obeys the Wiedemann-Franz law.