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]0.08219338844153429Then 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: -0.0006182370306135084
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:39
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3854379424 0.07 1.335 3.441 6.9 4.74s
2 -363.2378512637 0.27 -0.21 0.014 3.624 3.2 8.60s
3 -363.3511061948 -0.95 -0.58 0.000 3.727 3.4 3.54s
4 -363.3890498560 -1.42 -1.18 0.000 3.716 2.5 2.37s
5 -363.3959673362 -2.16 -1.67 0.000 3.681 2.0 2.24s
6 -363.3973233767 -2.87 -2.05 0.000 3.656 1.6 1.95s
7 -363.3976033623 -3.55 -2.27 0.000 3.647 2.0 2.70s
8 -363.3976896096 -4.06 -2.63 0.000 3.647 1.8 1.96s
9 -363.3977064251 -4.77 -2.94 0.000 3.649 2.2 2.23s
10 -363.3977061524 + -6.56 -2.91 -0.000 3.649 1.8 2.56s
11 -363.3977093368 -5.50 -3.23 0.000 3.648 1.9 2.01s
12 -363.3977094080 -7.15 -3.33 0.000 3.648 1.8 1.95s
13 -363.3977089643 + -6.35 -3.20 -0.000 3.648 2.1 2.06s
14 -363.3977089127 + -7.29 -3.16 -0.000 3.649 1.0 2.26s
15 -363.3977072942 + -5.79 -2.74 -0.000 3.649 2.0 1.85s
16 -363.3977097281 -5.61 -3.26 -0.000 3.649 2.2 2.00s
17 -363.3977099548 -6.64 -3.39 -0.000 3.649 1.1 1.70s
18 -363.3977099890 -7.47 -3.50 0.000 3.648 1.2 2.28s
19 -363.3977100011 -7.92 -3.48 0.000 3.648 1.0 1.64s
20 -363.3977100031 -8.70 -3.77 -0.000 3.648 1.0 1.65s
21 -363.3977099844 + -7.73 -3.93 -0.000 3.648 1.1 1.68s
22 -363.3977100093 -7.60 -4.25 -0.000 3.648 1.0 2.26s
23 -363.3977100165 -8.14 -4.76 0.000 3.648 1.9 1.94s
24 -363.3977100171 -9.22 -4.79 0.000 3.648 2.4 2.14s
25 -363.3977100173 -9.64 -4.90 0.000 3.648 1.0 1.66s
26 -363.3977100177 -9.40 -5.20 0.000 3.648 1.8 2.42s
27 -363.3977100177 -10.26 -5.51 0.000 3.648 2.0 1.90s
28 -363.3977100178 -10.37 -5.47 0.000 3.648 1.4 1.76s
29 -363.3977100178 -10.58 -5.23 0.000 3.648 1.1 2.33s
30 -363.3977100178 -10.98 -4.99 0.000 3.648 1.0 1.64s
31 -363.3977100178 + -12.25 -4.79 0.000 3.648 1.2 1.70s
32 -363.3977100178 -10.80 -5.03 0.000 3.648 1.0 1.65s
33 -363.3977100178 -11.41 -5.03 0.000 3.648 1.0 2.26s
34 -363.3977100178 + -12.29 -4.96 0.000 3.648 1.0 1.68s
35 -363.3977100178 + -12.40 -4.97 0.000 3.648 1.0 1.67s
36 -363.3977100178 -11.38 -5.26 0.000 3.648 1.0 1.66s
37 -363.3977100178 -11.92 -5.19 0.000 3.648 1.5 2.47s
38 -363.3977100179 -11.78 -5.39 0.000 3.648 1.0 1.64s
39 -363.3977100179 + -12.47 -5.20 0.000 3.648 1.0 1.66s
40 -363.3977100179 -11.85 -5.51 0.000 3.648 1.0 1.64s
41 -363.3977100179 -12.20 -5.60 0.000 3.648 1.0 2.27s
42 -363.3977100179 -12.47 -5.81 0.000 3.648 1.0 1.64s
43 -363.3977100179 -12.94 -6.46 0.000 3.648 1.0 1.67s
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]0.11667610409744333With 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)