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Yopajoe
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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 :
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]:
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.
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.
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