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]0.08219336125082821Then 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 -361.3881022848 0.07 1.335 3.440 7.0 4.09s
2 -362.9627528835 0.20 -0.10 0.223 3.874 2.6 11.0s
3 -363.1920874219 -0.64 -0.20 0.000 3.776 3.2 3.16s
4 -363.2394722624 -1.32 -0.29 0.000 3.781 2.1 2.07s
5 -363.3700380838 -0.88 -0.31 0.000 3.688 4.1 3.00s
6 -363.3864478596 -1.78 -0.48 -0.000 3.657 2.1 2.72s
7 -363.3968049264 -1.98 -1.13 -0.000 3.676 2.6 2.25s
8 -363.3935418687 + -2.49 -0.91 0.000 3.678 2.0 2.04s
9 -363.3967485811 -2.49 -1.08 0.000 3.656 1.0 2.29s
10 -363.3975177725 -3.11 -1.38 0.000 3.645 1.5 1.85s
11 -363.3976014669 -4.08 -1.46 0.000 3.643 1.0 1.65s
12 -363.3976290025 -4.56 -1.49 0.000 3.643 1.0 1.73s
13 -363.3976568025 -4.56 -1.52 0.000 3.641 1.0 2.27s
14 -363.3976727215 -4.80 -2.27 -0.000 3.650 1.0 1.71s
15 -363.3976810295 -5.08 -2.42 -0.000 3.652 1.0 1.65s
16 -363.3976734185 + -5.12 -2.37 -0.000 3.653 1.2 2.33s
17 -363.3976714832 + -5.71 -2.36 -0.000 3.654 1.0 1.70s
18 -363.3976930113 -4.67 -2.55 -0.000 3.652 1.0 1.66s
19 -363.3977078640 -4.83 -2.99 -0.000 3.650 1.0 1.67s
20 -363.3977094550 -5.80 -3.28 -0.000 3.649 1.2 2.37s
21 -363.3977099304 -6.32 -3.53 0.000 3.648 2.0 2.10s
22 -363.3977100026 -7.14 -3.79 0.000 3.648 1.0 1.67s
23 -363.3977100080 -8.26 -3.78 0.000 3.648 1.1 2.33s
24 -363.3977100037 + -8.36 -3.95 0.000 3.648 1.0 1.70s
25 -363.3977100154 -7.93 -4.10 0.000 3.648 1.0 1.65s
26 -363.3977100137 + -8.77 -4.29 0.000 3.648 1.2 1.71s
27 -363.3977100168 -8.51 -4.66 0.000 3.648 1.0 2.30s
28 -363.3977100177 -9.07 -5.12 0.000 3.648 2.0 1.92s
29 -363.3977100177 -10.12 -5.17 0.000 3.648 1.8 1.82s
30 -363.3977100178 -10.56 -5.13 0.000 3.648 1.2 2.40s
31 -363.3977100178 + -11.25 -5.18 0.000 3.648 1.0 1.66s
32 -363.3977100178 -10.22 -4.97 0.000 3.648 1.0 1.66s
33 -363.3977100178 -10.64 -5.84 0.000 3.648 1.0 1.67s
34 -363.3977100178 -11.27 -5.79 0.000 3.648 2.9 2.69s
35 -363.3977100178 -11.47 -5.82 0.000 3.648 1.0 1.69s
36 -363.3977100179 -11.68 -6.00 0.000 3.648 1.0 1.67s
37 -363.3977100179 -11.99 -6.33 0.000 3.648 1.4 2.41s
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]0.11667597678322728With 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)