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]0.08219250763755787Then 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: -0.0006182370306134945
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:39
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3847312727 0.07 1.334 3.440 6.8 4.80s
2 -363.2380528481 0.27 -0.21 0.014 3.624 3.1 8.71s
3 -363.3509519808 -0.95 -0.58 0.000 3.727 3.2 2.93s
4 -363.3890512681 -1.42 -1.18 0.000 3.717 2.6 3.08s
5 -363.3959794771 -2.16 -1.67 0.000 3.681 2.0 2.21s
6 -363.3973149231 -2.87 -2.04 0.000 3.657 1.5 1.93s
7 -363.3976123014 -3.53 -2.29 0.000 3.648 2.4 2.83s
8 -363.3976916906 -4.10 -2.63 0.000 3.647 1.6 1.98s
9 -363.3977066883 -4.82 -2.98 0.000 3.649 2.0 2.19s
10 -363.3977063089 + -6.42 -2.93 -0.000 3.649 1.8 1.95s
11 -363.3977092950 -5.52 -3.21 0.000 3.648 2.0 2.66s
12 -363.3977090307 + -6.58 -3.22 0.000 3.648 2.0 2.07s
13 -363.3977089416 + -7.05 -3.13 -0.000 3.648 2.2 2.16s
14 -363.3977083093 + -6.20 -2.99 -0.000 3.649 1.1 1.81s
15 -363.3977091240 -6.09 -2.98 0.000 3.649 1.0 2.41s
16 -363.3977094408 -6.50 -3.11 -0.000 3.649 1.0 1.74s
17 -363.3977096538 -6.67 -3.13 0.000 3.649 1.0 1.76s
18 -363.3977097481 -7.03 -3.16 0.000 3.649 1.0 2.45s
19 -363.3977097896 -7.38 -3.13 0.000 3.649 1.0 1.73s
20 -363.3977098272 -7.43 -3.31 0.000 3.649 1.0 1.77s
21 -363.3977099044 -7.11 -3.50 0.000 3.649 1.0 1.67s
22 -363.3977099985 -7.03 -3.85 0.000 3.648 1.0 2.34s
23 -363.3977100115 -7.88 -3.97 0.000 3.648 1.6 1.73s
24 -363.3977100149 -8.46 -3.79 0.000 3.648 1.5 1.73s
25 -363.3977100163 -8.87 -4.17 0.000 3.648 1.2 1.70s
26 -363.3977100170 -9.14 -4.38 0.000 3.648 1.0 2.34s
27 -363.3977100175 -9.31 -4.48 0.000 3.648 1.0 1.64s
28 -363.3977100178 -9.61 -4.97 0.000 3.648 1.0 1.66s
29 -363.3977100178 -10.33 -5.65 0.000 3.648 2.2 1.87s
30 -363.3977100178 -10.72 -5.56 0.000 3.648 3.0 2.17s
31 -363.3977100178 -10.99 -5.78 0.000 3.648 1.0 2.27s
32 -363.3977100178 -11.31 -5.69 0.000 3.648 1.4 1.73s
33 -363.3977100178 -11.54 -5.99 0.000 3.648 1.0 1.66s
34 -363.3977100179 -11.77 -6.18 0.000 3.648 1.4 1.78s
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]0.11667613542010691With 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)