# Cohen-Bergstresser model¶

This example considers the Cohen-Bergstresser model1, 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.

1. 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]:
Ecut = 10.0
model = Model(lattice; atoms=atoms, terms=[Kinetic(), AtomicLocal()])
basis = PlaneWaveBasis(model, Ecut, 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]:
0.3842999767139182

... and compute and plot 8 bands:

In [4]:
using Plots

n_bands = 8
ρ0 = guess_density(basis)  # Just dummy, has no meaning in this model
ρspin0 = nothing
p = plot_bandstructure(basis, ρ0, ρspin0, n_bands, εF=εF, kline_density=10)
ylims!(p, (-5, 6))

Computing bands along kpath:
Γ -> X -> W -> K -> Γ -> L -> U -> W -> L -> K  and  U -> X
Diagonalising Hamiltonian kblocks: 100%|████████████████| Time: 0:00:02

Out[4]: