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]0.0821934455977405Then 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 -361.3890664018 0.07 1.334 3.439 7.0 4.16s
2 -362.9635183582 0.20 -0.10 0.223 3.872 2.5 11.4s
3 -363.1923353323 -0.64 -0.20 0.000 3.776 3.2 2.54s
4 -363.2398552781 -1.32 -0.29 0.000 3.781 2.2 2.19s
5 -363.3690499242 -0.89 -0.30 0.000 3.687 4.1 3.62s
6 -363.3862198270 -1.77 -0.48 -0.000 3.656 2.0 2.05s
7 -363.3968019720 -1.98 -1.13 -0.000 3.676 2.9 2.37s
8 -363.3934538752 + -2.48 -0.90 0.000 3.677 2.0 2.71s
9 -363.3967525282 -2.48 -1.09 0.000 3.656 1.0 1.65s
10 -363.3975190358 -3.12 -1.39 0.000 3.645 1.5 1.81s
11 -363.3976031570 -4.08 -1.47 0.000 3.643 1.0 1.67s
12 -363.3976315229 -4.55 -1.50 0.000 3.644 1.0 2.30s
13 -363.3976485764 -4.77 -1.50 0.000 3.642 1.0 1.66s
14 -363.3976875974 -4.41 -2.45 -0.000 3.649 1.0 1.66s
15 -363.3976875968 + -9.22 -2.48 -0.000 3.651 1.1 2.38s
16 -363.3976939647 -5.20 -2.56 -0.000 3.651 1.0 1.66s
17 -363.3977060792 -4.92 -2.91 -0.000 3.650 1.1 1.69s
18 -363.3977097914 -5.43 -3.45 -0.000 3.649 2.0 2.05s
19 -363.3977094431 + -6.46 -3.25 -0.000 3.648 2.1 2.67s
20 -363.3977097737 -6.48 -3.40 -0.000 3.648 1.0 1.64s
21 -363.3977099736 -6.70 -3.53 -0.000 3.648 1.1 1.67s
22 -363.3977095989 + -6.43 -3.26 -0.000 3.648 1.9 2.47s
23 -363.3977098742 -6.56 -3.44 -0.000 3.648 1.0 1.65s
24 -363.3977096985 + -6.76 -3.33 -0.000 3.648 1.0 1.66s
25 -363.3977096117 + -7.06 -3.31 -0.000 3.648 1.0 1.66s
26 -363.3977096895 -7.11 -3.34 -0.000 3.648 1.0 2.25s
27 -363.3977099019 -6.67 -3.49 -0.000 3.648 1.0 1.69s
28 -363.3977099771 -7.12 -3.66 -0.000 3.648 1.0 1.69s
29 -363.3977099978 -7.68 -3.95 0.000 3.648 1.0 1.66s
30 -363.3977100170 -7.72 -4.59 0.000 3.648 2.0 2.54s
31 -363.3977100173 -9.51 -4.77 0.000 3.648 1.9 1.91s
32 -363.3977100178 -9.29 -5.44 0.000 3.648 1.6 1.78s
33 -363.3977100178 -10.48 -5.44 0.000 3.648 2.8 2.23s
34 -363.3977100178 -10.64 -5.87 0.000 3.648 1.2 2.29s
35 -363.3977100178 -11.35 -5.89 0.000 3.648 2.1 1.94s
36 -363.3977100178 -11.25 -6.11 0.000 3.648 1.5 1.74s
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]0.11667608702862003With 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)