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]0.08219316224644707Then 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: -0.0006182370306135145
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:39
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3867778827 0.07 1.334 3.440 6.9 3.89s
2 -363.2380727440 0.27 -0.21 0.014 3.623 3.4 8.20s
3 -363.3510618283 -0.95 -0.58 0.000 3.727 3.2 2.85s
4 -363.3889971192 -1.42 -1.18 0.000 3.717 2.6 2.34s
5 -363.3959844887 -2.16 -1.67 0.000 3.681 2.0 2.86s
6 -363.3973183611 -2.87 -2.05 0.000 3.656 1.5 1.86s
7 -363.3976164498 -3.53 -2.29 0.000 3.648 2.2 2.11s
8 -363.3976945041 -4.11 -2.67 0.000 3.647 1.6 2.63s
9 -363.3977067678 -4.91 -2.97 0.000 3.649 2.1 2.18s
10 -363.3977064266 + -6.47 -2.94 -0.000 3.649 1.9 1.83s
11 -363.3977089761 -5.59 -3.14 0.000 3.648 2.0 2.08s
12 -363.3977093126 -6.47 -3.31 0.000 3.648 1.0 2.43s
13 -363.3977092223 + -7.04 -3.18 -0.000 3.649 2.0 1.98s
14 -363.3977087319 + -6.31 -3.03 -0.000 3.649 1.2 1.64s
15 -363.3977090909 -6.44 -2.98 -0.000 3.649 1.0 1.62s
16 -363.3977093582 -6.57 -3.04 -0.000 3.649 1.0 2.31s
17 -363.3977095871 -6.64 -3.09 -0.000 3.649 1.0 1.64s
18 -363.3977096906 -6.98 -3.16 0.000 3.649 1.0 1.63s
19 -363.3977098368 -6.83 -3.36 0.000 3.649 1.0 1.61s
20 -363.3977099180 -7.09 -3.43 0.000 3.649 1.0 1.72s
21 -363.3977099396 -7.66 -3.45 0.000 3.649 1.0 2.20s
22 -363.3977100022 -7.20 -4.12 0.000 3.648 1.0 1.64s
23 -363.3977100167 -7.84 -4.58 0.000 3.648 2.5 1.96s
24 -363.3977100175 -9.11 -5.23 0.000 3.648 2.1 1.92s
25 -363.3977100177 -9.83 -5.16 0.000 3.648 2.2 2.72s
26 -363.3977100177 -10.04 -5.23 0.000 3.648 1.2 1.65s
27 -363.3977100178 -10.34 -5.33 0.000 3.648 1.0 1.61s
28 -363.3977100178 -10.61 -5.52 0.000 3.648 1.1 1.66s
29 -363.3977100178 -10.86 -5.63 0.000 3.648 1.4 2.42s
30 -363.3977100178 -11.08 -5.94 0.000 3.648 1.4 1.70s
31 -363.3977100178 -11.30 -6.05 0.000 3.648 2.0 1.89s
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]0.11667622914561349With 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)