### Summary of Topological insulators

Topological insulators (TI) are materials that behave as insulators in their interiors or bulks while permitting the movement of charges (metallic) on their surface.

Inside these materials, as conventional insulators, the energy band structures are shown large energy gap, as shown as Fig. 1 (a) red and blue bands, and Fermi level is falling between conduction and valence bands. At room temperature, T is about 300 K, the gap is so wide that electrons energy level cannot leap from Fermi level to conduction levels. However, this situation only happens for the topological insulatorâ€™s interior. For surface parts, their band structures are much different and behaves as metallic non-gap band structure, described by the green line in Fig. 1 (a).

Meanwhile, the surface bandstructure is linear, ~ and has Z2 symmetry. Topological insulators GrapheneDue to topological insulatorâ€™s linear structure, it is many similarities to grapheme, a real two-dimension material whose bandstructure is also shown linear relationship at low energy. Especially, the grapheme armchair bandsrtructure is extremely similar to topological insulators, but the only different is the energy gap is at semi-conductor range.

Although the their bandstructures are similar to each other, their electrical motions shows different. In armchair grapheme ribbons, electrons move at centre part while for topological insulators, electrons only transport on surface, which is easily recall the phenomena of Hall effects.In several years ago, integral quantum Hall effect and fractional quantum Hall effect were discovered and explained in theory, which are the important quantum phenomena we could observe quantum phenomena in normal. For quantum Hall effect, boundary is a crucial factor for creating quantum Hall effect. In conventional quantum Hall effect, the electrons move along the surface, shown in Fig. 2 (a). However, for quantum spin Hall effect, the electrons have separated motional paths for different spin up and spin down.

Differ from quantum Hall effect (QHE), the path number is 2=1+1, the quantum spin Hall effect (QSHE) mode number is 4=2+2. In other words, QSHE separates the spin electrons.Based on TI spin electrons transport properties, it is potential to make spin devices. Another important application is for quantum information and the information is stored nonlocally due to the difficult with making a quantum computer is preventing the system from accidentally measuring itself.If I have a chance to research on topological insulators, I will approach to some theoretical works about spintronics. Because the most interesting properties of TI is the separated spin electrical transport paths on the material surfaces.