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]0.08219335337954581Then 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));┌ Warning: Negative ρcore detected: -0.0006182370306134874
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:40
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3881700565 0.07 1.333 3.441 6.9 4.02s
2 -362.9627346210 0.20 -0.10 0.223 3.872 2.6 10.3s
3 -363.1925009043 -0.64 -0.20 0.000 3.776 3.1 2.58s
4 -363.2399002742 -1.32 -0.29 0.000 3.781 2.1 2.66s
5 -363.3691253564 -0.89 -0.30 0.000 3.687 4.1 2.99s
6 -363.3863097321 -1.76 -0.48 -0.000 3.656 2.1 2.11s
7 -363.3967933981 -1.98 -1.13 -0.000 3.676 2.8 2.93s
8 -363.3934195208 + -2.47 -0.89 0.000 3.677 2.0 2.05s
9 -363.3967790139 -2.47 -1.09 0.000 3.656 1.0 1.66s
10 -363.3975201214 -3.13 -1.39 0.000 3.646 1.5 2.47s
11 -363.3976036620 -4.08 -1.47 0.000 3.643 1.0 1.66s
12 -363.3976310919 -4.56 -1.50 0.000 3.643 1.0 1.67s
13 -363.3976524948 -4.67 -1.51 0.000 3.641 1.0 1.68s
14 -363.3976838089 -4.50 -2.39 -0.000 3.649 1.0 2.35s
15 -363.3976859122 -5.68 -2.47 -0.000 3.651 1.0 1.72s
16 -363.3976893727 -5.46 -2.51 -0.000 3.652 1.0 1.66s
17 -363.3976987054 -5.03 -2.65 -0.000 3.651 1.0 1.68s
18 -363.3977096261 -4.96 -3.28 -0.000 3.649 1.9 2.57s
19 -363.3977077813 + -5.73 -2.96 -0.000 3.648 2.4 2.23s
20 -363.3977089422 -5.94 -3.14 -0.000 3.648 1.1 1.69s
21 -363.3977095095 -6.25 -3.30 -0.000 3.648 1.0 2.35s
22 -363.3977098172 -6.51 -3.54 0.000 3.648 1.2 1.69s
23 -363.3977099954 -6.75 -4.11 0.000 3.648 2.0 1.97s
24 -363.3977100156 -7.69 -4.59 0.000 3.648 2.2 2.09s
25 -363.3977100166 -8.99 -4.94 0.000 3.648 1.8 2.57s
26 -363.3977100169 -9.48 -4.89 0.000 3.648 1.9 1.91s
27 -363.3977100174 -9.35 -5.21 0.000 3.648 1.0 1.67s
28 -363.3977100176 -9.60 -5.64 0.000 3.648 2.2 2.66s
29 -363.3977100177 -9.99 -5.47 0.000 3.648 2.5 2.10s
30 -363.3977100178 -10.25 -5.17 0.000 3.648 1.4 1.74s
31 -363.3977100178 -10.49 -4.95 0.000 3.648 1.5 1.81s
32 -363.3977100178 -10.81 -4.83 0.000 3.648 1.5 2.45s
33 -363.3977100178 -11.08 -4.76 0.000 3.648 1.2 1.75s
34 -363.3977100178 -11.75 -4.70 0.000 3.648 1.2 1.73s
35 -363.3977100178 -12.34 -4.66 0.000 3.648 1.1 2.39s
36 -363.3977100178 -12.29 -4.65 0.000 3.648 1.0 1.64s
37 -363.3977100178 -11.45 -4.67 0.000 3.648 1.0 1.66s
38 -363.3977100178 -11.75 -4.68 0.000 3.648 1.0 1.66s
39 -363.3977100178 + -12.20 -4.67 0.000 3.648 1.0 2.32s
40 -363.3977100178 + -11.69 -4.66 0.000 3.648 1.0 1.69s
41 -363.3977100178 -11.43 -4.69 0.000 3.648 1.0 1.66s
42 -363.3977100178 -11.35 -4.74 0.000 3.648 1.0 1.66s
43 -363.3977100178 -11.02 -4.91 0.000 3.648 1.0 2.32s
44 -363.3977100179 -11.02 -5.80 0.000 3.648 2.0 2.16s
45 -363.3977100179 -12.34 -5.81 0.000 3.648 1.0 1.66s
46 -363.3977100179 -13.25 -6.25 0.000 3.648 1.0 1.66s
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]0.11667610544332269With 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)