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]0.08219361531529101Then 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.
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: -0.0006182370306134938
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:39
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3876179091 0.07 1.335 3.440 6.9 3.80s
2 -363.2381556624 0.27 -0.21 0.014 3.624 3.4 4.93s
3 -363.3508797074 -0.95 -0.58 0.000 3.727 3.2 2.91s
4 -363.3890380546 -1.42 -1.18 0.000 3.717 2.6 2.43s
5 -363.3959732864 -2.16 -1.67 0.000 3.681 2.0 2.23s
6 -363.3973181975 -2.87 -2.04 0.000 3.656 1.5 1.91s
7 -363.3976084615 -3.54 -2.28 0.000 3.648 2.4 2.17s
8 -363.3976907985 -4.08 -2.63 0.000 3.647 1.5 1.96s
9 -363.3977067591 -4.80 -2.98 0.000 3.649 2.1 2.19s
10 -363.3977062128 + -6.26 -2.92 -0.000 3.649 2.4 2.04s
11 -363.3977093177 -5.51 -3.22 0.000 3.648 1.9 2.00s
12 -363.3977086846 + -6.20 -3.15 0.000 3.648 2.0 2.14s
13 -363.3977091963 -6.29 -3.18 -0.000 3.648 2.1 2.19s
14 -363.3977086636 + -6.27 -3.04 -0.000 3.649 1.1 1.78s
15 -363.3977089554 -6.53 -2.94 -0.000 3.649 1.0 1.76s
16 -363.3977090929 -6.86 -3.01 -0.000 3.649 1.0 1.76s
17 -363.3977093221 -6.64 -3.04 -0.000 3.649 1.0 1.76s
18 -363.3977092734 + -7.31 -3.01 -0.000 3.649 1.0 3.05s
19 -363.3977093820 -6.96 -3.01 -0.000 3.649 1.0 1.74s
20 -363.3977093842 -8.66 -2.99 -0.000 3.649 1.0 1.74s
21 -363.3977096040 -6.66 -3.04 -0.000 3.649 1.0 1.75s
22 -363.3977099670 -6.44 -3.83 0.000 3.648 1.1 1.77s
23 -363.3977099782 -7.95 -3.86 0.000 3.648 1.9 1.90s
24 -363.3977099880 -8.01 -3.59 0.000 3.648 2.0 2.04s
25 -363.3977099775 + -7.98 -3.72 0.000 3.648 1.1 1.69s
26 -363.3977100092 -7.50 -4.07 0.000 3.648 1.0 1.68s
27 -363.3977100174 -8.09 -4.46 0.000 3.648 1.9 1.96s
28 -363.3977100177 -9.44 -4.73 0.000 3.648 1.1 1.71s
29 -363.3977100178 -10.07 -5.08 0.000 3.648 1.4 1.75s
30 -363.3977100178 -10.56 -5.35 0.000 3.648 1.8 1.77s
31 -363.3977100178 -11.28 -5.85 0.000 3.648 1.1 1.70s
32 -363.3977100178 -11.85 -6.04 0.000 3.648 2.8 2.26sRun 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]0.11667629023808829With 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)