gftool.lattice.triangular.dos_mp

gftool.lattice.triangular.dos_mp(eps, half_bandwidth=1)

Multi-precision DOS of non-interacting 2D triangular lattice.

The DOS diverges at -4/9*half_bandwidth.

This function is particularity suited to calculate integrals of the form \(∫dϵ DOS(ϵ)f(ϵ)\). If you have problems with the convergence, consider using \(∫dϵ DOS(ϵ)[f(ϵ)-f(-4/9)] + f(-4/9)\) to avoid the singularity.

Parameters:

epsmpmath.mpf or mpf_like

DOS is evaluated at points eps.

half_bandwidthmpmath.mpf or mpf_like

Half-bandwidth of the DOS, DOS(eps < -2/3`half_bandwidth`) = 0, DOS(4/3`half_bandwidth` < eps) = 0. The half_bandwidth corresponds to the nearest neighbor hopping \(t=4D/9\).

Returns:

mpmath.mpf

The value of the DOS.

See also

gftool.lattice.triangular.dos

Vectorized version suitable for array evaluations.

References

[kogan2021]

Kogan, E. and Gumbs, G. (2021) Green’s Functions and DOS for Some 2D Lattices. Graphene, 10, 1-12. https://doi.org/10.4236/graphene.2021.101001.

Examples

Calculate integrals:

>>> from mpmath import mp
>>> mp.quad(gt.lattice.triangular.dos_mp, [-2/3, -4/9, 4/3])
mpf('1.0')

>>> eps = np.linspace(-2/3 - 0.1, 4/3 + 0.1, num=1000)
>>> dos_mp = [gt.lattice.triangular.dos_mp(ee, half_bandwidth=1) for ee in eps]
>>> dos_mp = np.array(dos_mp, dtype=np.float64)

>>> import matplotlib.pyplot as plt
>>> _ = plt.axvline(-4/9, color='black', linewidth=0.8)
>>> _ = plt.axvline(0, color='black', linewidth=0.8)
>>> _ = plt.plot(eps, dos_mp)
>>> _ = plt.xlabel(r"$\epsilon/D$")
>>> _ = plt.ylabel(r"DOS * $D$")
>>> _ = plt.ylim(bottom=0)
>>> _ = plt.xlim(left=eps.min(), right=eps.max())
>>> plt.show()

(png, pdf)