Hubbard correction (DFT+U)
In this example, we'll plot the DOS and projected DOS of Nickel Oxide with and without the Hubbard term correction.
using DFTK
using PseudoPotentialData
using Unitful
using UnitfulAtomic
using PlotsDefine the geometry and pseudopotential
a = 7.9 # Nickel Oxide lattice constant in Bohr
lattice = a * [[ 1.0 0.5 0.5];
[ 0.5 1.0 0.5];
[ 0.5 0.5 1.0]]
pseudopotentials = PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf")
Ni = ElementPsp(:Ni, pseudopotentials)
O = ElementPsp(:O, pseudopotentials)
atoms = [Ni, O, Ni, O]
positions = [zeros(3), ones(3) / 4, ones(3) / 2, ones(3) * 3 / 4]
magnetic_moments = [2, 0, -1, 0]4-element Vector{Int64}:
2
0
-1
0First, we run an SCF and band computation without the Hubbard term
model = model_DFT(lattice, atoms, positions; temperature=5e-3,
functionals=PBE(), magnetic_moments)
basis = PlaneWaveBasis(model; Ecut=20, kgrid=[2, 2, 2])
scfres = self_consistent_field(basis; tol=1e-6, ρ=guess_density(basis, magnetic_moments))
bands = compute_bands(scfres, MonkhorstPack(4, 4, 4))
lowest_unocc_band = findfirst(ε -> ε-bands.εF > 0, bands.eigenvalues[1])
band_gap = bands.eigenvalues[1][lowest_unocc_band] - bands.eigenvalues[1][lowest_unocc_band-1]0.08219334371654585Then we plot the DOS and the PDOS for the relevant 3D (pseudo)atomic projector
εF = bands.εF
width = 5.0u"eV"
εrange = (εF - austrip(width), εF + austrip(width))
p = plot_dos(bands; εrange, colors=[1, 1])
plot_pdos(bands; p, iatom=1, label="3D", colors=[3, 4], εrange)
To perform and Hubbard computation, we have to define the Hubbard manifold and associated constant.
In DFTK there are a few ways to construct the OrbitalManifold. Here, we will apply the Hubbard correction on the 3D orbital of all nickel atoms. To select all nickel atoms, we can:
- Pass the
Nielement directly. - Pass the
:Nisymbol. - Pass the list of atom indices, here
[1, 3].
To select the orbitals, it is recommended to use their label, such as "3D" for PseudoDojo pseudopotentials.
Note that "manifold" is the standard term used in the literature for the set of atomic orbitals used to compute the Hubbard correction, but it is not meant in the mathematical sense.
U = 10u"eV"
# Alternative:
# manifold = OrbitalManifold(:Ni, "3D")
# Alternative:
# manifold = OrbitalManifold([1, 3], "3D")
manifold = OrbitalManifold(Ni, "3D")OrbitalManifold(Ni, "3D")Run SCF with a DFT+U setup, notice the extra_terms keyword argument, setting up the Hubbard +U term. It is also possible to set up multiple manifolds with different U values by passing each pair as a separate entry in the Hubbard constructor (i.e. Hubbard(manifold1 => U1, manifold2 => U2, etc.)) or as two vectors (i.e. Hubbard([manifold1, manifold2, etc.], [U1, U2, etc.])).
model = model_DFT(lattice, atoms, positions; extra_terms=[Hubbard(manifold => U)],
functionals=PBE(), temperature=5e-3, magnetic_moments)
basis = PlaneWaveBasis(model; Ecut=20, kgrid=[2, 2, 2])
scfres = self_consistent_field(basis; tol=1e-6, ρ=guess_density(basis, magnetic_moments));n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3828037656 0.07 1.334 3.443 6.8 3.59s
2 -362.9705318360 0.20 -0.10 0.224 3.874 2.5 10.6s
3 -363.1934351686 -0.65 -0.20 0.000 3.776 3.1 2.43s
4 -363.2414700714 -1.32 -0.29 0.000 3.781 2.1 1.79s
5 -363.3692566780 -0.89 -0.31 0.000 3.684 4.0 2.74s
6 -363.3861533573 -1.77 -0.48 -0.000 3.653 2.0 1.75s
7 -363.3969240213 -1.97 -1.12 -0.000 3.674 2.8 2.30s
8 -363.3933834265 + -2.45 -0.92 0.000 3.678 2.0 1.74s
9 -363.3967303204 -2.48 -1.08 0.000 3.657 1.0 1.43s
10 -363.3974657133 -3.13 -1.34 0.000 3.647 1.0 1.77s
11 -363.3975735873 -3.97 -1.43 0.000 3.644 1.0 1.45s
12 -363.3975924359 -4.72 -1.42 0.000 3.642 1.0 1.49s
13 -363.3976969969 -3.98 -2.06 0.000 3.643 1.0 1.78s
14 -363.3977049886 -5.10 -2.61 -0.000 3.645 1.0 1.43s
15 -363.3977028422 + -5.67 -2.54 -0.000 3.646 1.6 1.66s
16 -363.3977038682 -5.99 -2.60 -0.000 3.646 1.0 1.80s
17 -363.3977057944 -5.72 -2.76 0.000 3.646 1.0 1.43s
18 -363.3977083430 -5.59 -2.98 0.000 3.647 1.0 1.78s
19 -363.3977095565 -5.92 -3.22 -0.000 3.648 1.6 1.49s
20 -363.3977099881 -6.36 -3.99 0.000 3.648 2.0 1.66s
21 -363.3977099060 + -7.09 -3.59 -0.000 3.649 2.9 2.18s
22 -363.3977099380 -7.50 -3.72 0.000 3.649 1.0 1.45s
23 -363.3977100045 -7.18 -4.11 0.000 3.649 2.0 1.68s
24 -363.3977099981 + -8.19 -3.93 0.000 3.649 1.0 1.83s
25 -363.3977100115 -7.87 -4.05 0.000 3.648 1.0 1.42s
26 -363.3977100127 -8.92 -4.05 0.000 3.648 1.0 1.43s
27 -363.3977100145 -8.75 -3.97 0.000 3.648 1.0 1.76s
28 -363.3977100172 -8.57 -4.68 0.000 3.648 1.0 1.43s
29 -363.3977100176 -9.39 -4.72 0.000 3.648 1.5 1.48s
30 -363.3977100176 -10.40 -4.81 0.000 3.648 1.2 1.80s
31 -363.3977100178 -9.78 -5.14 0.000 3.648 1.0 1.43s
32 -363.3977100178 -10.71 -5.59 0.000 3.648 1.1 1.44s
33 -363.3977100178 -11.00 -5.92 0.000 3.648 2.1 2.07s
34 -363.3977100178 -11.42 -6.04 0.000 3.648 1.8 1.55s
Run band computation
bands_hub = compute_bands(scfres, MonkhorstPack(4, 4, 4))
lowest_unocc_band = findfirst(ε -> ε-bands_hub.εF > 0, bands_hub.eigenvalues[1])
band_gap = bands_hub.eigenvalues[1][lowest_unocc_band] - bands_hub.eigenvalues[1][lowest_unocc_band-1]0.11667593301855295With the electron localization introduced by the Hubbard term, the band gap has now opened, reflecting the experimental insulating behaviour of Nickel Oxide.
εF = bands_hub.εF
εrange = (εF - austrip(width), εF + austrip(width))
p = plot_dos(bands_hub; p, colors=[2, 2], εrange)
plot_pdos(bands_hub; p, iatom=1, label="3D", colors=[3, 4], εrange)