Hamiltonian¶
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graphenemodeling.graphene.bilayer.Hamiltonian(k, u)¶ Returns the full tight-binding Hamiltonian of bilayer graphene.
Parameters: - k (array-like, rad/m) – Wavenumber
- u (scalar, J) – Interlayer potential energy difference
Returns: H – Tight-binding Hamiltonian of bilayer graphene.
Return type: array-like
Notes
The
FullTightBindingHamiltonian is (Eqn. 16 Ref. [1])\[ \begin{align}\begin{aligned}H=\left(\begin{matrix} -u/2 & -\gamma_0f(k) & \gamma_4f(k) & -\gamma_3f^*(k)\\ -\gamma_0 f^*(k) & -u/2 & \gamma_1 & \gamma_4f(k)\\ \gamma_4f^*(k) & \gamma_1 & u/2 & -\gamma_0f(k)\\ -\gamma_3f(k) & \gamma_4f(k) & -\gamma_0f^*(k) & u/2 \end{matrix}\right)\end{aligned}\end{align} \]The
Commontight-binding Hamiltonian is given by (Eqn. 30 Ref. [1])\[ \begin{align}\begin{aligned}H=\left(\begin{matrix} -u/2 & \hbar v_F k & -\sqrt{3/4}k a\gamma_4 & -\sqrt{3/4}k^* a\gamma_3\\ \hbar v_Fk^* & -u/2 & \gamma_1 & -\sqrt{3/4}ka\gamma_4\\ -\sqrt{3/4}k^*a\gamma_4 & \gamma_1 & u/2 & \hbar v_F k\\ -\sqrt{3/4}k a\gamma_3 & -\sqrt{3/4}k^*a\gamma_4 & \hbar v_F k^* & u/2 \end{matrix}\right)\end{aligned}\end{align} \]The
LowEnergyHamiltonian is\[ \begin{align}\begin{aligned}H=\left(\begin{matrix} u/2 & p^2 / 2m\\ p^2/2m & -u/2 \end{matrix}\right)\end{aligned}\end{align} \]References
[1] McCann, E., and Koshino, M. (2013). The electronic properties of bilayer graphene. Reports on Progress in Physics 76, 056503. https://arxiv.org/abs/1205.6953.