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]0.08219322471717938Then 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: -0.0006182370306135099
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:39
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3880114763 0.07 1.336 3.439 6.9 4.20s
2 -363.2385558602 0.27 -0.21 0.014 3.624 3.2 9.05s
3 -363.3507764470 -0.95 -0.58 0.000 3.727 3.2 2.95s
4 -363.3889998988 -1.42 -1.18 0.000 3.717 2.6 3.02s
5 -363.3959711901 -2.16 -1.67 0.000 3.681 2.0 2.25s
6 -363.3973177825 -2.87 -2.04 0.000 3.656 1.5 1.92s
7 -363.3976071213 -3.54 -2.28 0.000 3.647 2.1 2.71s
8 -363.3976927604 -4.07 -2.65 0.000 3.647 1.6 2.04s
9 -363.3977070567 -4.84 -3.00 0.000 3.649 2.1 2.80s
10 -363.3977063224 + -6.13 -2.93 -0.000 3.649 2.0 2.01s
11 -363.3977092077 -5.54 -3.18 0.000 3.648 2.0 2.12s
12 -363.3977090618 + -6.84 -3.23 0.000 3.648 1.4 2.49s
13 -363.3977090096 + -7.28 -3.15 -0.000 3.648 2.0 2.11s
14 -363.3977086594 + -6.46 -3.05 -0.000 3.649 1.0 1.74s
15 -363.3977089480 -6.54 -2.93 -0.000 3.649 1.0 1.81s
16 -363.3977090294 -7.09 -2.93 -0.000 3.649 1.0 2.25s
17 -363.3977089642 + -7.19 -2.89 -0.000 3.649 1.0 1.75s
18 -363.3977093613 -6.40 -2.98 -0.000 3.649 1.0 1.75s
19 -363.3977094967 -6.87 -3.01 -0.000 3.649 1.0 2.31s
20 -363.3977096388 -6.85 -3.02 -0.000 3.649 1.0 1.75s
21 -363.3977096666 -7.56 -3.01 -0.000 3.649 1.0 1.75s
22 -363.3977099038 -6.62 -3.12 0.000 3.649 1.0 2.34s
23 -363.3977099275 -7.63 -3.13 0.000 3.649 1.0 1.80s
24 -363.3977099512 -7.63 -3.19 0.000 3.649 1.0 1.75s
25 -363.3977100070 -7.25 -3.57 0.000 3.648 1.0 2.30s
26 -363.3977100070 + -11.41 -3.76 0.000 3.648 1.0 1.68s
27 -363.3977100044 + -8.58 -3.87 0.000 3.648 1.1 1.68s
28 -363.3977100102 -8.24 -3.83 0.000 3.648 1.0 2.22s
29 -363.3977100115 -8.88 -3.86 0.000 3.648 1.0 1.64s
30 -363.3977100133 -8.74 -3.97 0.000 3.648 1.0 1.66s
31 -363.3977100161 -8.56 -4.12 0.000 3.648 1.0 1.68s
32 -363.3977100176 -8.81 -5.28 0.000 3.648 1.0 2.19s
33 -363.3977100178 -9.70 -5.46 0.000 3.648 3.4 2.51s
34 -363.3977100178 -11.15 -5.44 0.000 3.648 1.0 1.66s
35 -363.3977100178 -10.72 -5.94 0.000 3.648 1.9 2.40s
36 -363.3977100178 -11.45 -6.41 0.000 3.648 1.9 1.86s
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]0.116676104046878With 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)