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]0.08219324825849406Then 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 -361.3825292218 0.07 1.334 3.443 6.8 11.3s
2 -362.9704717364 0.20 -0.10 0.224 3.873 2.5 5.24s
3 -363.1934288492 -0.65 -0.20 0.000 3.776 3.1 2.57s
4 -363.2414507990 -1.32 -0.29 0.000 3.781 2.1 1.82s
5 -363.3692236025 -0.89 -0.31 0.000 3.684 4.0 2.87s
6 -363.3861527914 -1.77 -0.48 -0.000 3.653 2.0 1.77s
7 -363.3969248971 -1.97 -1.12 -0.000 3.674 2.8 1.98s
8 -363.3933647617 + -2.45 -0.92 0.000 3.678 2.0 2.18s
9 -363.3967283718 -2.47 -1.08 0.000 3.657 1.0 1.46s
10 -363.3974655102 -3.13 -1.34 0.000 3.647 1.0 1.46s
11 -363.3975734191 -3.97 -1.43 0.000 3.644 1.0 1.87s
12 -363.3975924707 -4.72 -1.42 0.000 3.643 1.0 1.52s
13 -363.3976969628 -3.98 -2.06 0.000 3.643 1.0 1.47s
14 -363.3977049445 -5.10 -2.60 -0.000 3.645 1.0 1.87s
15 -363.3977027390 + -5.66 -2.54 -0.000 3.645 1.6 1.69s
16 -363.3977037321 -6.00 -2.60 -0.000 3.646 1.0 1.46s
17 -363.3977056403 -5.72 -2.75 0.000 3.646 1.0 1.52s
18 -363.3977082167 -5.59 -2.96 0.000 3.647 1.0 1.80s
19 -363.3977094897 -5.90 -3.20 -0.000 3.648 1.6 1.52s
20 -363.3977099897 -6.30 -4.05 0.000 3.648 2.0 1.70s
21 -363.3977099117 + -7.11 -3.60 -0.000 3.649 3.0 2.30s
22 -363.3977099387 -7.57 -3.72 0.000 3.649 1.0 1.51s
23 -363.3977100078 -7.16 -4.19 0.000 3.649 2.0 1.80s
24 -363.3977099993 + -8.07 -4.00 0.000 3.649 1.0 1.83s
25 -363.3977099987 + -9.23 -4.00 0.000 3.649 1.0 1.49s
26 -363.3977099988 -9.79 -3.92 0.000 3.649 1.0 1.47s
27 -363.3977100135 -7.83 -4.38 0.000 3.648 1.0 1.86s
28 -363.3977100175 -8.40 -4.73 0.000 3.648 2.0 1.69s
29 -363.3977100176 -9.91 -5.00 0.000 3.648 1.0 1.46s
30 -363.3977100178 -9.93 -5.22 0.000 3.648 2.0 2.05s
31 -363.3977100178 -10.61 -5.31 0.000 3.648 1.0 1.47s
32 -363.3977100178 -10.30 -5.50 0.000 3.648 1.2 1.50s
33 -363.3977100178 -10.96 -5.81 0.000 3.648 1.4 1.59s
34 -363.3977100178 -11.12 -5.83 0.000 3.648 1.8 1.92s
35 -363.3977100178 -11.42 -5.74 0.000 3.648 1.2 1.52s
36 -363.3977100179 -11.97 -5.35 0.000 3.648 1.1 1.89s
37 -363.3977100179 -11.62 -5.68 0.000 3.648 1.0 1.48s
38 -363.3977100179 -11.94 -5.93 0.000 3.648 1.0 1.46s
39 -363.3977100179 -12.34 -5.93 0.000 3.648 1.0 1.47s
40 -363.3977100179 -12.55 -5.97 0.000 3.648 1.0 1.86s
41 -363.3977100179 -12.47 -7.01 0.000 3.648 1.0 1.49s
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]0.11667612187501941With 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)