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]0.08219341057513468Then 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: -0.000618237030613482
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:40
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3865739555 0.07 1.336 3.439 6.9 4.19s
2 -362.9622947166 0.20 -0.10 0.223 3.872 2.5 10.5s
3 -363.1928439674 -0.64 -0.20 0.000 3.775 3.1 3.19s
4 -363.2395444010 -1.33 -0.29 0.000 3.781 2.2 2.08s
5 -363.3694591296 -0.89 -0.30 0.000 3.688 4.1 2.92s
6 -363.3864363875 -1.77 -0.48 -0.000 3.657 2.0 2.74s
7 -363.3967984935 -1.98 -1.13 -0.000 3.676 2.6 2.21s
8 -363.3934474035 + -2.47 -0.89 0.000 3.677 2.0 1.99s
9 -363.3967464326 -2.48 -1.08 0.000 3.656 1.0 2.34s
10 -363.3975274204 -3.11 -1.39 0.000 3.645 1.5 1.85s
11 -363.3976061409 -4.10 -1.47 0.000 3.643 1.0 1.65s
12 -363.3976318100 -4.59 -1.50 0.000 3.643 1.0 1.65s
13 -363.3976627167 -4.51 -1.54 0.000 3.641 1.0 2.37s
14 -363.3976497465 + -4.89 -2.14 -0.000 3.652 1.0 1.66s
15 -363.3976714816 -4.66 -2.37 -0.000 3.653 1.0 1.63s
16 -363.3976475368 + -4.62 -2.25 -0.000 3.655 1.2 1.66s
17 -363.3976493065 -5.75 -2.27 -0.000 3.655 1.0 1.71s
18 -363.3976613224 -4.92 -2.31 -0.000 3.654 1.0 2.30s
19 -363.3976956810 -4.46 -2.58 -0.000 3.652 1.0 1.65s
20 -363.3977036297 -5.10 -2.79 -0.000 3.651 1.0 1.61s
21 -363.3977084971 -5.31 -3.13 0.000 3.650 1.4 1.78s
22 -363.3977098746 -5.86 -3.76 0.000 3.649 2.1 2.53s
23 -363.3977100068 -6.88 -4.10 0.000 3.648 3.1 2.34s
24 -363.3977100092 -8.62 -4.15 0.000 3.648 1.2 1.68s
25 -363.3977100096 -9.35 -4.15 0.000 3.648 1.0 1.64s
26 -363.3977100098 -9.98 -4.16 0.000 3.648 1.0 2.40s
27 -363.3977100141 -8.36 -4.43 0.000 3.648 1.0 1.63s
28 -363.3977100140 + -10.45 -4.38 0.000 3.648 1.0 1.63s
29 -363.3977100158 -8.74 -4.22 0.000 3.648 1.2 1.74s
30 -363.3977100171 -8.90 -4.59 0.000 3.648 2.0 2.53s
31 -363.3977100175 -9.40 -4.68 0.000 3.648 1.0 1.62s
32 -363.3977100177 -9.63 -4.88 0.000 3.648 1.0 1.66s
33 -363.3977100177 + -10.12 -5.01 0.000 3.648 2.0 1.86s
34 -363.3977100178 -9.83 -5.42 0.000 3.648 1.0 2.40s
35 -363.3977100178 -10.59 -5.37 0.000 3.648 2.1 1.90s
36 -363.3977100178 -11.30 -5.34 0.000 3.648 1.0 1.62s
37 -363.3977100178 -11.23 -5.60 0.000 3.648 1.0 1.73s
38 -363.3977100179 -11.45 -6.07 0.000 3.648 1.1 2.39s
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]0.1166760025140916With 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)