[問題] 關於Weyl semimetal
以下是我對Weyl semimetal的想法，想請大家幫我看看哪裡有誤，謝謝大家。
Weyl semi-metal is a kind of quantum materials. There are many kinds of quantum materials, one of the classifications is topological materials. Moreover, topological materials can be classified into topological insulators and topological semi-metals which contain Weyl semi-metal and Dirac semi-metal. Interestingly, topological insulators and topological semi-metal all originate from Band inversion and Spin-Orbit Coupling.
Band inversion is a very special case in band structure (I am still curious about the origin of band inversion), conduction band and valance band overlap each other. Conduction band is lower than the Fermi level and valance band is higher than the Fermi level.
Spin-Orbit coupling is a relativistic effect. In a very usual case, an electron moves around a nucleus. It seems like the only thing acting on the electron is the electric field generated by nucleus. However, that is an objective view from us. We are not electron; we cannot know what actually happens on electron. To understand that, we need to observe the same situation as an electron. In this case, the electron is fixed, and the nucleus moves around the electron. Therefore, the circular motion of
nucleus generates a magnetic field on the electron. Furthermore, the electron has its own magnetic moment, which is spin, so the magnetic moment from spin interacts with the magnetic moment from the orbital motion. That is called Spin-Orbit Coupling.
Band inversion with the effect of SOC (Spin-Orbit Coupling) generates the TI (topological insulator), WSM (Weyl Semi-Metal), and DSM (Dirac Semi-Metal). It depends on the strength of SOC. If the effect of SOC is very strong, it opens a full gap between conduction band and valance band, then it is TI. If the SOC is not very strong, the conduction band still touches the valance band at some critical points. These points are called Weyl points in WSM and Dirac points in DSM.
The next problem is how to distinguish WSM and DSM. To know that, we need to know some symmetry in physics and in our samples. Those are time-reversal symmetry and inversion symmetry. Inversion symmetry is just mirror symmetry. For example, consider a function or a potential, and change the position variable from x to -x. If it is invariant, then the function follows the inversion symmetry. If some cases do not follow the inversion symmetry, then we call them inversion symmetry breaking. Similarly, in
the case of time (t to -t) is time-reversal symmetry.
In topological semimetals, WSM requires the breaking of either time-reversal symmetry or inversion symmetry. If time-reversal symmetry and inversion symmetry coexist, then it is DSM. In WSM, there is quantity, Berry curvature. Berry curvature is an effective magnetic field, is can viewed as a magnetic field in momentum space. Berry curvature starts from one Weyl point and end in the other Weyl point. Therefore, Weyl points must be a pair, one is the source of Berry curvature, the other one is the sink of
Berry curvature. Otherwise, the Berry curvature will diverge. If time-reversal symmetry breaking and inversion symmetry breaking coexist, then a pair of Weyl points become degenerated and two Dirac points.
There are two kinds of WSM which are type Ⅰ and type Ⅱ WSM. They depend on the shape of their band structure. If conduction band is higher than the Fermi level, valance band is lower than the Fermi level, and they touch each other at Weyl point, then it is type Ⅰ WSM. If some of the conduction band is lower than the Fermi level, some valance band is higher than the Fermi level, then it is type Ⅱ WSM. In type Ⅱ WSM, we call the part of conduction lower than the Fermi level is electron pockets, and
the part of valance higher than the Fermi level is hole pockets.
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