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]0.08219326323962373Then 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 -361.3821629479 0.07 1.334 3.443 6.6 2.18s
2 -362.9705471296 0.20 -0.10 0.224 3.873 2.5 7.05s
3 -363.1935140528 -0.65 -0.20 0.000 3.776 3.1 1.54s
4 -363.2414425180 -1.32 -0.29 0.000 3.781 2.1 1.01s
5 -363.3690067031 -0.89 -0.31 0.000 3.683 4.0 1.40s
6 -363.3861093762 -1.77 -0.48 -0.000 3.652 2.0 1.36s
7 -363.3969362530 -1.97 -1.11 -0.000 3.674 2.8 1.11s
8 -363.3932922193 + -2.44 -0.91 0.000 3.678 2.0 1.00s
9 -363.3967164128 -2.47 -1.08 0.000 3.657 1.0 1.19s
10 -363.3974631242 -3.13 -1.34 0.000 3.647 1.0 816ms
11 -363.3975717257 -3.96 -1.43 0.000 3.644 1.0 818ms
12 -363.3975918643 -4.70 -1.42 0.000 3.643 1.0 1.21s
13 -363.3976970097 -3.98 -2.04 0.000 3.643 1.0 826ms
14 -363.3977049171 -5.10 -2.60 -0.000 3.645 1.0 820ms
15 -363.3977023544 + -5.59 -2.53 -0.000 3.645 1.8 946ms
16 -363.3977034448 -5.96 -2.60 -0.000 3.646 1.0 1.18s
17 -363.3977055331 -5.68 -2.77 0.000 3.646 1.0 826ms
18 -363.3977078043 -5.64 -2.90 0.000 3.647 1.0 821ms
19 -363.3977092246 -5.85 -3.11 0.000 3.648 1.0 829ms
20 -363.3977098366 -6.21 -3.44 0.000 3.648 1.9 1.23s
21 -363.3977100079 -6.77 -4.23 0.000 3.648 2.0 1.00s
22 -363.3977099910 + -7.77 -3.92 0.000 3.648 2.5 1.04s
23 -363.3977100079 -7.77 -4.22 0.000 3.648 1.4 1.24s
24 -363.3977100119 -8.40 -4.22 0.000 3.648 1.2 860ms
25 -363.3977100073 + -8.34 -4.07 0.000 3.648 1.0 817ms
26 -363.3977100105 -8.49 -4.09 0.000 3.648 1.0 830ms
27 -363.3977100136 -8.50 -4.12 0.000 3.648 1.0 1.21s
28 -363.3977100126 + -9.00 -4.10 0.000 3.648 1.0 807ms
29 -363.3977100118 + -9.07 -4.07 0.000 3.648 1.0 820ms
30 -363.3977100153 -8.46 -4.33 0.000 3.648 1.0 827ms
31 -363.3977100177 -8.61 -4.86 0.000 3.648 1.1 1.19s
32 -363.3977100178 -10.21 -5.11 0.000 3.648 1.5 859ms
33 -363.3977100178 -10.43 -5.69 0.000 3.648 1.5 855ms
34 -363.3977100178 -10.98 -5.84 0.000 3.648 2.1 1.38s
35 -363.3977100178 -11.18 -5.87 0.000 3.648 1.0 816ms
36 -363.3977100178 -11.28 -6.06 0.000 3.648 1.8 889ms
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]0.11667605339165604With 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)