Hamiltonian¶
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graphenemodeling.graphene.monolayer.Hamiltonian(k, model, g0prime=0)¶ Tight-binding Hamiltonian in momentum space.
Parameters: - k (array-like, complex, rad/m) – Wavevector of carrier. Use complex
k=kx + 1j*kyfor 2D wavevectors. - model (string) –
'LowEnergy','FullTightBinding' - g0prime (scalar, J) – The particle-hole asymmetry parameter \(\gamma_0'\). Typically \(0.02\gamma_0\leq\gamma_0'\leq 0.2\gamma_0\).
Returns: H – Tight-binding Hamiltonian evaluated at k.
Return type: 2x2 complex ndarray
Raises: ValueError– if model is not ‘LowEnergy’ or ‘FullTightBinding’Notes
Let \(k=k_x+ik_y\). Then the
model=FullTightBindingexpression is given by\[ \begin{align}\begin{aligned}H = \left(\array{ -\gamma_0' & \gamma_0f(k)\\ \gamma_0f(k)^* & -\gamma_0' } \right)\end{aligned}\end{align} \]where \(f(k)= e^{ik_x a/2} + 2 e^{-i k_x a/ 2}\cos(k_y a \sqrt{3}/2)\)
The more common
model=LowEnergyapproximation is\[ \begin{align}\begin{aligned}H = \hbar v_F\left(\array{ 0 & k\\ k^* & 0\\ } \right)\end{aligned}\end{align} \]References
[1] Wallace, P.R. (1947). The Band Theory of Graphite. Phys. Rev. 71, 622–634. https://link.aps.org/doi/10.1103/PhysRev.71.622
[1] Slonczewski, J.C., and Weiss, P.R. (1958). Band Structure of Graphite. Phys. Rev. 109, 272–279. https://link.aps.org/doi/10.1103/PhysRev.109.272.
[2] Falkovsky, L.A., and Varlamov, A.A. (2007). Space-time dispersion of graphene conductivity. Eur. Phys. J. B 56, 281–284. https://link.springer.com/article/10.1140/epjb/e2007-00142-3.
- k (array-like, complex, rad/m) – Wavevector of carrier. Use complex