Basic issue for graphene sheet

  • #1
1
0
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 [tex] a_0 \sqrt{3}[/tex] , where [tex]a_0[/tex] 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" [Broken]] :


[tex] \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} )[/tex]

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

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

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

[tex]\Phi = C_A\phi_A+ C_B\phi_B = (C_A+C_B)\exp(i{\mathbf kr})[/tex]

Is above true ? Now can we can say that [tex]\Phi[/tex] is eigenvector of basis function [tex]\phi_A[/tex] and [tex]\phi_B[/tex] ?

Assume [tex]\phi_A[/tex] and [tex]\phi_B[/tex] are known as :

[tex]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})[/tex]​

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 [Broken] -- or go to http://arxiv.org/abs/0709.1163 and download pdf -- Eq(9) page 5]:

[tex]\theta= - \arctan \frac{q_y}{q_x}[/tex]​

Where [tex]q_y,q_x[/tex] are displacements in relation K. In conduction band taking for [tex]q_y=0[/tex] , [tex] \Phi[/tex] 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.
 
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Answers and Replies

  • #2
117
0
[tex]\phi_A, \phi_B[/tex] 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] [Broken].
 
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