3D topological insulator

If we stack quantum hall insulator on upon other then we will get current only in the edge but this does not give rise a edge current on to and bottom of this state. Now for 3D topological hall insulator what we want is edge state driven current through the 2D surface state of a 3D bulk. So what we want is following:

For 2D topological insulator, edge state formed a cross sign "X"by up and down spin, for 3D topological insulator we want them to form cone, the 3D vrsion of the "Cross X " . So i am talking is a Dirac cone.

Bi based materials:
Bi has strong spin orbit coupling. If we take Bi-n, Bi-As then because of Rashba coupling give a pair of Fermi surface with up down spin at the surface. These are pair of Rashba states by breaking the time reversal symmetry at surface. 
 Now for  Bi2Se3, Bi-Sb the situtaion is different, we have only one Fermi surface and in momentum space the  K+ ,K- has opposite spin. As shown in following figure z axis is energy and x,y axis are momentum. If we cut the circle formed by the top of cone, the spin momentum has always opposite sign.So the spin form a clockwise spin texture at the top of the cone. Now if go towards center point of the cone the this opposite nature of spin is maintained. It has half the degree of freedom  compare to Rashba system. So at he end the surface is gapless 

Now this type band dispersion is exist only in the surface not for the bulk. Because such dispersion of energy(E) moment(k_x, k_y) is studied by doing ARPES at the surface of Bi2Se3. In order to make sure that it exist only at the surface not in the bulk, a K_z dependent study is needed. Now a K_z study revealed that the surface edge states , lates call them now Dirac band are non dispersive as function of K_z where as the bulk band is dispersive.

So the surface state contain, K_+ has spin polarization has up and K_ - has spin down. So this surface state has half the degree of freedom than the Rashba system. Remember the whole system is nonmagnetic. Now the Dirac cone make sure that back scattering is not allowed. The surface electron has spin momentum lock. These are called helical Fermion.

Now what is the fundamental difference between the normal insulator and such surface states with metallic state and bulk being insulator.

In the above figure one of Selenium site has been substituted by lower Z value atom salphar to decrease spin orbit coupling. The extreme the bulk and surface is also insulator. So this is  spin orbit coupled Bloch band insulator. In the right side there is formation of surface state provided the bulk band has closed the gap at 0.6. So there is phase transition , only odd number of band inversion has happen in he bulk then only surface states appear.