CarrierDispersion

graphenemodeling.graphene.bilayer.CarrierDispersion(k, u, band, model='Common')

Energy of charge carrier in bilayer graphene.

Returns the energy (J) of an electron with wavevector k (rad/m) in first (band=1) or second (band=2) conduction band.

Parameters:
  • k (array-like) – Wavenumber
  • u (scalar) – Potential difference between layers
  • model (string) – 'Common',``’LowEnergy’, or ``'FullTightBinding'
Returns:

disp – Dispersion
Return type:

array-like

Examples

Plot the four band of model=Common. Replicates Fig. 11 in Ref. [1].

>>> from graphenemodeling.graphene import bilayer as blg
>>> from scipy.constants import elementary_charge as eV
>>> import matplotlib.pyplot as plt
>>> k = np.linspace(-1e9,1e9,num=200)
>>> u = 0.3 * eV
>>> conduction1 = blg.CarrierDispersion(k,u,1,model='Common')
>>> conduction2 = blg.CarrierDispersion(k,u,2,model='Common')
>>> valence1 = blg.CarrierDispersion(k,u,-1,model='Common')
>>> valence2 = blg.CarrierDispersion(k,u,-2,model='Common')
>>> fig, ax = plt.subplots()
>>> ax.plot(k*1e-10,conduction1/eV,label='Band 1')
>>> ax.plot(k*1e-10,conduction2/eV,label='Band 2')
>>> ax.plot(k*1e-10,valence1/eV,label='Band -1')
>>> ax.plot(k*1e-10,valence2/eV,label='Band -2')
>>> ax.set_xlabel('k ($\AA^{-1}$)')
>>> ax.set_ylabel('E (eV)')
>>> plt.legend()
>>> plt.show()

(Source code, png, hires.png, pdf)

_images/graphene-bilayer-CarrierDispersion-1.png

Notes

\[E_{\pm}^2 = \frac{\gamma_1^2}{2}+\frac{u^2}{4}+\hbar^2v_F^2k^2 \pm \sqrt{\frac{\gamma_1^4}{4}+\hbar^2v_F^2k^2(\gamma_1^2+u^2)}\]

References

[1] Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., and Geim, A.K. (2009). The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162. https://link.aps.org/doi/10.1103/RevModPhys.81.109.