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]0.08219260026892422Then 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: -0.0006182370306135084
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:40
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3853020868 0.07 1.334 3.441 6.8 4.67s
2 -363.1985388655 0.26 -0.18 0.037 3.654 3.0 10.1s
3 -363.3349657563 -0.87 -0.50 0.000 3.738 3.4 2.77s
4 -363.3851549766 -1.30 -1.04 0.000 3.724 2.8 2.96s
5 -363.3961670029 -1.96 -1.69 0.000 3.683 2.0 2.07s
6 -363.3973006481 -2.95 -1.98 0.000 3.658 1.5 1.77s
7 -363.3976157853 -3.50 -2.25 0.000 3.648 2.1 2.14s
8 -363.3976862739 -4.15 -2.53 0.000 3.647 1.4 2.32s
9 -363.3977070763 -4.68 -2.96 0.000 3.648 2.1 1.93s
10 -363.3977081931 -5.95 -3.11 -0.000 3.649 2.2 2.08s
11 -363.3977090067 -6.09 -3.33 -0.000 3.649 1.0 1.70s
12 -363.3977094647 -6.34 -3.44 -0.000 3.648 2.0 2.69s
13 -363.3977097779 -6.50 -3.56 -0.000 3.648 1.0 1.73s
14 -363.3977098164 -7.41 -3.20 -0.000 3.648 1.0 1.70s
15 -363.3977099246 -6.97 -3.23 -0.000 3.648 1.2 1.74s
16 -363.3977099716 -7.33 -3.40 -0.000 3.648 1.0 2.32s
17 -363.3977099818 -7.99 -3.34 -0.000 3.648 1.0 1.72s
18 -363.3977099862 -8.36 -3.28 -0.000 3.648 1.0 1.69s
19 -363.3977099913 -8.29 -3.30 -0.000 3.648 1.0 1.70s
20 -363.3977100011 -8.01 -3.36 -0.000 3.648 1.0 2.32s
21 -363.3977100083 -8.14 -3.51 -0.000 3.648 1.0 1.74s
22 -363.3977100104 -8.66 -3.95 0.000 3.648 1.0 1.69s
23 -363.3977099997 + -7.97 -3.40 0.000 3.648 1.5 1.95s
24 -363.3977100155 -7.80 -4.31 0.000 3.648 1.0 2.31s
25 -363.3977100170 -8.82 -4.41 0.000 3.648 2.0 2.00s
26 -363.3977100172 -9.91 -4.83 0.000 3.648 1.0 1.69s
27 -363.3977100174 -9.60 -4.61 0.000 3.648 1.1 1.72s
28 -363.3977100177 -9.51 -5.25 0.000 3.648 1.0 2.34s
29 -363.3977100178 -10.16 -5.23 0.000 3.648 2.2 1.98s
30 -363.3977100178 -10.50 -5.43 0.000 3.648 1.4 1.78s
31 -363.3977100178 -10.90 -5.96 0.000 3.648 1.1 1.72s
32 -363.3977100178 -11.24 -5.79 0.000 3.648 2.6 2.75s
33 -363.3977100178 -11.45 -5.99 0.000 3.648 1.1 1.73s
34 -363.3977100179 -11.69 -5.93 0.000 3.648 1.4 1.80s
35 -363.3977100179 -12.34 -5.41 0.000 3.648 1.1 1.73s
36 -363.3977100179 -11.85 -5.62 0.000 3.648 1.0 2.32s
37 -363.3977100179 -12.17 -5.69 0.000 3.648 1.0 1.67s
38 -363.3977100179 -12.94 -5.77 0.000 3.648 1.0 1.68s
39 -363.3977100179 -12.47 -6.23 0.000 3.648 1.0 1.71s
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]0.11667611878594336With 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)