First-principles investigations on the Berry phase effect in spin–orbit coupling materials
Graphical abstract
Introduction
The Berry phase effect [1], which is defined as that the nondegenerate eigenstate definitely comes back to itself but with an extra phase difference when the system is adiabatically evolved along a closed path in the parameter space, has recently caused a great deal of interest in a variety of fields in physics. The Berry phase has proved its importance due to three key physical properties [2], [3]. First, the Berry phase is gauge invariant in the sense that multiplying the eigenstate by an overall phase factor does not change the Berry phase within multiples of 2π. This property makes the Berry curvature physically observable. Second, the Berry phase is geometrical because it can be expressed in terms of local geometrical quantities along a loop in the parameter space. This property makes the Berry phase computable in practice. Finally, the Berry phase is analogous to gauge field theory and differential geometry. This makes the Berry phase a beautiful and intuitive concept.
In crystalline solid, the band structure within the independent electron approximation provides a natural starting point to study the Berry phase effect. The band structure for a given system is determined by the following Hamiltonian:where is the periodic potential with the Bravais lattice vector. The -dependent Hamiltonian reads as and its eigenstate can be written as , which is in fact the periodic part of the Bloch function . Since the Brillouin zone (BZ) can be regarded as the parameter space and varies in the momentum space, the Berry phase for Bloch state can be expressed as,where the integrand is usually known as the Berry connection, defined asIt should be emphasized that the path must be a closed path in the BZ such that is a gauge-invariant quantity. In analogy to electrodynamics, it is useful to introduce the Berry curvature from the Berry connection,Then according to Stokes’s theorem, the Berry phase can also be written as a surface integralwhere is an arbitrary surface enclosed by the path . The Berry curvature is an intrinsic quantity, which provides a local description of the geometric properties in the parameter space. Now, we have recognized that the Berry curvature plays an essential role in the description of the dynamics of the Bloch electrons. Thus, the Berry curvature is a more fundamental quantity in a large class of applications of the Berry phase.
In solid materials, especially for those with large spin–orbit coupling (SOC), the Berry phase has manifested itself having deep relations with various emergent quantum phenomena [3], including anomalous Hall effect, spin Hall effect, valley Hall effect, anomalous thermoelectric effect, electronic polarization, orbital magnetization, magnetoresistance, magneto-optic effect, and three/two-dimensional (3D/2D) topological insulator. The first-principles calculation within the density functional theory played an important role to understand the Berry phase in vast situations because the quantitative comparison with experimental results becomes possible. In the past decade, our research group has paid much attention to these kinds of studies and produced very fruitful works.
In this paper, we review our first-principles investigations on the Berry phase effect in real materials with SOC. The rest of this paper is organized as follows. In Section 2, we review the first-principles calculations of the anomalous Hall conductivity, spin Hall conductivity, anomalous Nernst conductivity, Chern number, and topological invariants, which all have been implemented in our homemade package based on the full-potential linearized augmented plane-wave (FP-LAPW) formalism. In Section 3, we focus on the anomalous Hall effect, spin Hall effect, and anomalous Nernst effect, and show that the intrinsic mechanism dominates at certain cases. Then, 3D and 2D topological materials as very hot topics in recent years are discussed in Sections 4 Two-dimensional topological materials, 5 Three-dimensional topological materials, respectively. In Section 6, we also introduce briefly our studies on group-VI dichalcogenide monolayers. Finally, in Section 7, we give a brief summary of our works on the Berry phase effect in solid materials with prominent spin–orbit coupling and presents some outlooks in further studies.
Section snippets
Computational methods and formulas
In this section, we review the computational methods of anomalous Hall conductivity, spin Hall conductivity, anomalous Nernst conductivity, Chern number, and topological invariants, following a brief introduction of the FP-LAPW formalism. The central job is to calculate various physical quantities, such as the velocity, parity, and time-reversal operators. The physical pictures for these fascinating quantum phenomena are schematically shown in Fig. 1.
Anomalous Hall, spin Hall, and anomalous Nernst effects
In solid materials, the spin–orbit coupling is responsible for various phenomena of transverse transports, e.g., the anomalous Hall effect (AHE) [27], [28], spin Hall effect (SHE) [29], and anomalous Nernst effect (ANE) [30], which have long been concerned in the fundamentally physical interest and practical applications.
Two-dimensional topological materials
Two-dimensional (2D) layered materials, such as successfully fabricated graphene [49], [50], [51], silicene [52], [53], [54], germanene, stanene, X-hydride/halide (X = N–Bi) monolayers, Bi4Br4, and graphite like metal–organic framework [55], are a hotspot in the fields of condensed matter physics and material science due to their novel low-energy Dirac electronic behaviors and promising applications in electronics. In this section, we will review our group’s works on novel quantum states of
Three-dimensional topological materials
Three-dimensional topological insulators (3D TIs) in strongly spin–orbit coupling materials with time-reversal symmetry has garnered great interest in the fields of condensed matter physics and materials science [81], [82]. In this section, several classes of 3D TIs predicted by us using first-principles calculations are introduced, including half-Heusler [83], [84], chalcopyrite [85], β-Ag2Te [86], stained InSb [87], core-hole affected Ge and InSb [88], Bi2Te3/BiTeI heterostructure [89],
Other spin–orbit coupling materilas— monolayers
The monolayers of group-VI dichalcogenides are visible-frequency direct-gap semiconductors with X-M-X sandwich structure in trigonal prismatic coordination for the transition metal M atom. They exhibit fascinating electronic and optoelectronic properties, such as valley-dependent circularly polarized optical selection rules, strong valley-spin coupling, valley contrasting Berry curvature, valley Hall and spin Hall effects, etc. [96]. All these properties are intimately related to their
Summary and outlook
In recent years, our research group has made great efforts to the studies of the Berry phase effect in SOC materials. Firstly, we developed computational methods for the anomalous Hall, spin Hall, and anomalous Nernst conductivities, which have been used to exactly describe the transverse Hall transports. In the Fe, Co, Ni, CuCr2Se4−xBrx, and Mn5Ge3, we first demonstrated that the intrinsic mechanism is dominated in the AHE other than the extrinsic scattering ones at certain case. We first
Acknowledgments
This work was supported by the MOST Project of China (Grant Nos. 2014CB920903, 2013CB921903, 2013CB934500, and 2011CBA00100), the NSF of China (Grant Nos. 11174337, 11225418, 11374033, 11304014, and 11404022), the SRFDPHE of China (Grant Nos. 20121101110046 and 20131101120052), the Excellent young scholars Research Fund of Beijing Institute of Technology (Grant No. 2014CX04028), and the Basic Research Fund of Beijing Institute of Technology (Grant Nos. 20131842001, 20141842001, and 20141842004).
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