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]0.08219338925143688Then 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 -361.3875269478 0.07 1.333 3.441 6.9 4.67s
2 -362.9630721623 0.20 -0.10 0.222 3.873 2.5 10.5s
3 -363.1925191579 -0.64 -0.20 0.000 3.776 3.1 3.20s
4 -363.2398431809 -1.32 -0.29 0.000 3.781 2.2 2.17s
5 -363.3688762097 -0.89 -0.30 0.000 3.687 4.1 3.07s
6 -363.3861672499 -1.76 -0.48 -0.000 3.656 2.1 2.76s
7 -363.3967926553 -1.97 -1.13 -0.000 3.676 2.8 2.32s
8 -363.3933997989 + -2.47 -0.89 0.000 3.677 2.0 2.10s
9 -363.3967432508 -2.48 -1.08 0.000 3.656 1.0 1.77s
10 -363.3975160910 -3.11 -1.38 0.000 3.646 1.5 2.40s
11 -363.3975997351 -4.08 -1.46 0.000 3.643 1.0 1.70s
12 -363.3976301294 -4.52 -1.50 0.000 3.644 1.0 1.71s
13 -363.3976401897 -5.00 -1.49 0.000 3.643 1.0 2.36s
14 -363.3976942825 -4.27 -2.59 -0.000 3.649 1.0 1.70s
15 -363.3976970963 -5.55 -2.59 -0.000 3.651 2.0 2.08s
16 -363.3977018455 -5.32 -2.69 -0.000 3.650 1.0 2.33s
17 -363.3977074964 -5.25 -2.95 -0.000 3.650 1.0 1.68s
18 -363.3977090928 -5.80 -3.15 -0.000 3.649 1.4 1.85s
19 -363.3977098210 -6.14 -3.45 -0.000 3.649 1.9 1.94s
20 -363.3977099445 -6.91 -3.60 -0.000 3.649 1.4 2.43s
21 -363.3977099228 + -7.66 -3.59 -0.000 3.649 1.0 1.71s
22 -363.3977099798 -7.24 -3.81 -0.000 3.649 1.0 1.70s
23 -363.3977100092 -7.53 -4.15 0.000 3.648 1.0 1.77s
24 -363.3977100159 -8.17 -4.91 0.000 3.648 1.9 2.43s
25 -363.3977100166 -9.14 -4.68 0.000 3.648 2.9 2.34s
26 -363.3977100171 -9.34 -4.61 0.000 3.648 1.0 1.72s
27 -363.3977100173 -9.57 -4.56 0.000 3.648 1.0 2.33s
28 -363.3977100175 -9.87 -4.36 0.000 3.648 1.0 1.73s
29 -363.3977100177 -9.64 -4.58 0.000 3.648 1.1 1.75s
30 -363.3977100178 -10.09 -4.80 0.000 3.648 1.0 2.34s
31 -363.3977100178 -10.49 -4.83 0.000 3.648 1.0 1.69s
32 -363.3977100178 -10.67 -4.90 0.000 3.648 1.0 1.72s
33 -363.3977100178 -10.70 -6.08 0.000 3.648 1.0 1.71s
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]0.11667643897023472With 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)