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]0.0821937585895175Then 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=[:red, :red])
plot_pdos(bands; p, iatom=1, label="3D", colors=[:yellow, :orange], 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: -0.0006182370306135057
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:39
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3850392885 0.07 1.334 3.441 6.8 3.86s
2 -363.2387128559 0.27 -0.21 0.014 3.624 3.1 3.63s
3 -363.3508571751 -0.95 -0.58 0.000 3.727 3.4 3.00s
4 -363.3890030656 -1.42 -1.18 0.000 3.717 2.6 2.49s
5 -363.3959736737 -2.16 -1.67 0.000 3.681 2.0 3.63s
6 -363.3973144444 -2.87 -2.04 0.000 3.656 1.5 1.92s
7 -363.3976158347 -3.52 -2.29 0.000 3.648 2.2 2.20s
8 -363.3976923499 -4.12 -2.63 0.000 3.647 1.5 1.99s
9 -363.3977065879 -4.85 -2.96 0.000 3.649 2.1 2.26s
10 -363.3977062997 + -6.54 -2.92 -0.000 3.649 2.1 2.02s
11 -363.3977092748 -5.53 -3.20 0.000 3.648 1.8 2.03s
12 -363.3977088884 + -6.41 -3.18 0.000 3.648 2.0 2.17s
13 -363.3977091008 -6.67 -3.16 -0.000 3.648 2.0 2.17s
14 -363.3977086608 + -6.36 -3.00 -0.000 3.649 1.0 1.82s
15 -363.3977093017 -6.19 -3.04 -0.000 3.649 1.0 1.82s
16 -363.3977095574 -6.59 -3.16 -0.000 3.649 1.0 1.81s
17 -363.3977097202 -6.79 -3.16 -0.000 3.649 1.0 1.82s
18 -363.3977097818 -7.21 -3.16 0.000 3.649 1.0 1.81s
19 -363.3977098094 -7.56 -3.14 -0.000 3.649 1.0 1.82s
20 -363.3977097936 + -7.80 -3.26 -0.000 3.649 1.0 1.82s
21 -363.3977098258 -7.49 -3.38 -0.000 3.649 1.0 1.76s
22 -363.3977099940 -6.77 -3.85 0.000 3.648 1.0 1.71s
23 -363.3977100129 -7.72 -3.94 0.000 3.648 1.4 1.77s
24 -363.3977100166 -8.43 -3.99 0.000 3.648 1.5 1.78s
25 -363.3977100148 + -8.74 -4.26 0.000 3.648 1.0 1.72s
26 -363.3977100154 -9.24 -4.18 0.000 3.648 1.0 1.72s
27 -363.3977100163 -9.03 -4.05 0.000 3.648 1.0 1.72s
28 -363.3977100173 -8.99 -4.43 0.000 3.648 1.0 3.06s
29 -363.3977100177 -9.42 -5.25 0.000 3.648 1.0 1.69s
30 -363.3977100178 -10.71 -5.21 0.000 3.648 2.9 2.27s
31 -363.3977100178 -10.25 -5.31 0.000 3.648 1.0 1.69s
32 -363.3977100178 -10.48 -5.85 0.000 3.648 1.4 1.76s
33 -363.3977100178 -11.13 -5.79 0.000 3.648 2.6 2.07s
34 -363.3977100178 -11.48 -5.85 0.000 3.648 1.1 1.74s
35 -363.3977100179 -11.74 -6.12 0.000 3.648 1.0 1.72sRun 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]0.11667614482600874With 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=[:blue, :blue], εrange)
plot_pdos(bands_hub; p, iatom=1, label="3D", colors=[:green, :purple], εrange)