Basic issue for graphene sheet

[Yopajoe]

Hello ,

Maybe this topic has basic understandings of quantum mechanics and involves some mathematic theory that I haven’t learned yet, but it’s related on graphene sheet and Klein tunneling .


Consider that our observation is so much larger in scale then sub-lattice constant $$a_0 \sqrt{3}$$ , where $$a_0$$ is distance between two adjacent atoms , which form two different sub-lattice A and B. Bloch function for each sub-lattice A , B by tight binding model is [https://wiki.physics.udel.edu/phys824/Band_structure_of_graphene,_massless_Dirac_fermions_as_low-energy_quasiparticles,_Berry_phase,_and_all_that" ] : $$\phi_{A} = \frac{1}{\sqrt{N}} \sum \exp{i{\mathbf kR}} \varphi_{2pz} ( {\mathbf r} - { \mathbf R_n} )\\\ , \phi_{B} = \frac{1}{\sqrt{N}} \sum \exp{ik({\mathbf R}+{\mathbf \tau})} \varphi_{2pz} ({\mathbf r} -{ \mathbf R_n}-{\mathbf \tau} )$$

where $$\varphi_{2pz}$$ is eigenfunction 2pz orbital, and sum goes in term of all atoms in sub-lattice (n = 1…N ) , and N is number of atoms . $${\mathbf \tau}$$ is shift between two sub-lattice and its amount obviously is $$a_0$$.

(1)As mentioned before because of our point of view , we could regard $$\phi_A$$ and $$\phi_B$$ as plane wave function exp ( ikr ) . Is this right ? Or I don’t understand it well. Do $$\phi_A$$ and $$\phi_B$$ have some phase shift ?

(2) If (1) is true , there is general solution for electron in graphene

$$\Phi = C_A\phi_A+ C_B\phi_B = (C_A+C_B)\exp(i{\mathbf kr})$$

Is above true ? Now can we can say that $$\Phi$$ is eigenvector of basis function $$\phi_A$$ and $$\phi_B$$ ?

Assume $$\phi_A$$ and $$\phi_B$$ are known as :

$$C_A=\frac{1}{\sqrt{2}} \exp (i\frac{\theta(k)}{2}) \ ,\ C_B= \pm \frac{1}{\sqrt{2}} \exp (-i\frac{\theta(k)}{2})$$​

At site around K point in the first Brilluein zone with approximation near Dirac point , we suppose [ http://www.sciencetimes.com.cn/upload/blog/file/2009/8/200987222258483302.pdf -- or go to http://arxiv.org/abs/0709.1163 and download pdf -- Eq(9) page 5]:

$$\theta= - \arctan \frac{q_y}{q_x}$$​

Where $$q_y,q_x$$ are displacements in relation K. In conduction band taking for $$q_y=0$$ , $$\Phi$$ is obtained zero . What is wrong?

Any directions and clarifications would be welcome, also put any related links and sorry for grammar mistakes .

Thanks in advance for taking time to help me; I really appreciate your effort.

 

[PRB147]

$$\phi_A, \phi_B$$ constitute the two component of the Graphene spinor, can not simply added as scalar wave. they are isospin state.
Wavefunction is determined by the tight-binding Hamiltonian (5)[http://arxiv.org/abs/0709.1163] .