Cohen-Bergstresser model¶

This example considers the Cohen-Bergstresser model[^CB1966], reproducing the results of the original paper. This model is particularly simple since its linear nature allows one to get away without any self-consistent field calculation.

[^CB1966]: M. L. Cohen and T. K. Bergstresser Phys. Rev. 141, 789 (1966) DOI 10.1103/PhysRev.141.789

We build the lattice using the tabulated lattice constant from the original paper, stored in DFTK:

In [1]:

using DFTK

Si = ElementCohenBergstresser(:Si)
lattice = Si.lattice_constant / 2 .* [[0 1 1.]; [1 0 1.]; [1 1 0.]]
atoms = [Si => [ones(3)/8, -ones(3)/8]];

Next we build the rather simple model and discretise it with moderate Ecut:

In [2]:

model = Model(lattice; atoms=atoms, terms=[Kinetic(), AtomicLocal()])
basis = PlaneWaveBasis(model, Ecut=10.0, kgrid=(1, 1, 1));

We diagonalise at the Gamma point to find a Fermi level ...

In [3]:

ham = Hamiltonian(basis)
eigres = diagonalize_all_kblocks(DFTK.lobpcg_hyper, ham, 6)
εF = DFTK.fermi_level(basis, eigres.λ)

Out[3]:

0.3842999767139468

... and compute and plot 8 bands:

In [4]:

using Plots
using Unitful

p = plot_bandstructure(basis; n_bands=8, εF, kline_density=10, unit=u"eV")
ylims!(p, (-5, 6))

Computing bands along kpath:
       Γ -> X -> U  and  K -> Γ -> L -> W -> X
Diagonalising Hamiltonian kblocks: 100%|████████████████| Time: 0:00:01

Out[4]: