As a newly discovered phase of matter, the topological insulator (TI) has become one of the hottest topics in condensed-matter physics [see refs 1-3 for review]. The pioneering works on the first TI materials including HgTe [4-5] and Bi2Se3 etc. [6-10] have inspired the booming-up of this field with exciting potential applications such as a quantum computer based on Majorana fermions [11]. However, from material viewpoint there are many emerging problems (e.g. bulk conducting) that hinder us from utilizing the exotic topological states. Excellent topological materials and related-devices with controllable electronic properties are urgently required to realize proposed novel applications.

The state-of-art first-principles approaches have already played an important role in the discovery of Bi2Se3 type of topological materials. [6,7]. Considering the great success in materials design and property optimization and device fabrication, these computational materials methods will certainly be a powerful tool for topological insulator study, not only in realizing exotic proposals from the theoretical models, but also in contemporary predicting and understanding new experiments. Now we are at the cutting edge where computation materials researchers interact with the TI community and discuss about possible solutions to improve and tailor the materials properties based upon computational approaches.

A topological insulator (TI) has an energy gap in the bulk but host metallic edge (2D TI) or surface states (3D TI) on the boundary, as a result of topology. These topological states have Dirac-type energy dispersion, where spin and momentum are helically locked-up. They are protected from backscattering by the time-reversal-symmetry (TRS). The first observed 2D TI material (also called as quantum spin Hall effect) was first theoretically predicted in 2006 [4] and soon observed by experiments [5] in HgTe quantum wells. In 2009 both theoretical calculations [6] and ARPES experiments reported the most famous 3D TI materials, Bi2Se3 [7,8], Bi2Te3 [9] and Sb2Te3 [10]. These pioneering works inspired very much the subsequent developments. Since now, many new materials have been predicted particularly by density-functional calculations [see ref.12 for review] and some of them [13-16] have already been confirmed by experiments. As a new quantum state of matter, TIs attract tremendous attention and become one of the hottest fields in the condensed-matter physics, both for the fundamental interest and for the great application potential in spintronics, quantum computation by Majorana Fermions, thermoelectronics and devices utilizing various topological magnetoelectric effects.

The most distinguished feature of TIs is the existence of topological edge or surface states in the bulk energy gap. The realization of above application proposals are all based on the basic requirement that topological materials are insulating in the bulk and only conducts on the surface or at the edge. However, most known TI materials have much conducting bulk carriers. For example, the well-known material Bi2Se3 is heavily defect-doped as n-type semiconductor. Up to now, the bulk non-insulating problem remains unsolved despite extensive efforts involving chemical doping [8,9,17-19], nanostructuring [20], and electric gating [21]. More sophisticated material growth and processing methods and new high-quality TI materials are urgent to be explored.

One of the most exciting potential applications of TIs is the creation of Majorana fermions [11] that are proposed to make a fault-tolerant topological quantum computer. This relies the fabrication of an interface between a TI and a superconductor. The topological surface states of a TI have `` half ” freedom of normal 2D electron system. When a TI contacts with a superconductor, the surface states become superconducting and hence turn into half an ordinary superconductor. This is the required way to host Majorana fermions. From the material viewpoint, choosing proper TI and superconductor materials [22] and the properties of their interfaces are key factors to fabricate reliable devices for this proposal.

The topological states are protected by time-reversal symmetry. Once a TRS breaking perturbation is introduced, many important topological phenomena can be realized. The 2D TI can evolve into a quantum anomalous Hall (QAH) insulator [23,24]. The QAH insulator is a band insulator with quantized Hall conductance to realize dissipationless charge current but without external magnetic field. In a 3D TI, with TRS broken on the surface the effective electromagnetic response is described by a topological term [25]. This topological response supports many novel topological phenomena, such as the image magnetic monopole induced by a point charge, topological Faraday and Kerr effect, the realization of axion field in condensed-matter physics and other interesting topological magnetoelectric effects. Therefore magnetic doping is of particular interest. Though some achievements in Bi2Se3 and Sb2Te3 have been reported [19,26] recently, well-controlled magnetically doped TIs are still under developing.

Materials computations, especially the well-developed density-functional methods, have played an important role in the study of topological insulators. From the prediction of topological materials particularly the extensively studied Bi2Se3 family [6] to the characterization of their properties, computational materials methods are employed closely together with the phenomenological models and experiments [12]. Before these methods have already made great success in the study of semiconductor properties, such as p/n doping, defect, surface and interface properties, magnetic element doping and so on. And certainly they will contribute more to this newly booming-up field, since they are generically and hopefully related to solve many problems in topological materials. In current stage of topological insulators, on one hand, experimentalists need materials simulations to understand their results. On the other hand, phenomenological theorists expect new materials or explicit device design using known materials to realize their proposals. More advances and collaborations are expected from the materials science side to develop topological materials for both fundamental physics and device applications.