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]0.08219332118802647Then 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:39
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3869701744 0.07 1.335 3.439 7.0 4.64s
2 -363.2383754583 0.27 -0.21 0.014 3.625 3.1 8.52s
3 -363.3508355336 -0.95 -0.58 0.000 3.727 3.2 3.46s
4 -363.3890234932 -1.42 -1.18 0.000 3.717 2.6 2.43s
5 -363.3959596730 -2.16 -1.66 0.000 3.681 2.0 2.78s
6 -363.3973180064 -2.87 -2.04 0.000 3.656 1.6 1.94s
7 -363.3976048305 -3.54 -2.27 0.000 3.647 2.2 2.13s
8 -363.3976891400 -4.07 -2.63 0.000 3.647 1.5 2.47s
9 -363.3977063742 -4.76 -2.94 0.000 3.649 2.1 2.20s
10 -363.3977059690 + -6.39 -2.89 -0.000 3.649 2.0 1.94s
11 -363.3977092564 -5.48 -3.23 -0.000 3.649 1.6 2.45s
12 -363.3977093219 -7.18 -3.24 0.000 3.648 2.0 2.11s
13 -363.3977091571 + -6.78 -3.22 -0.000 3.649 2.0 2.07s
14 -363.3977090617 + -7.02 -3.16 -0.000 3.649 1.0 1.81s
15 -363.3977080487 + -5.99 -2.70 -0.000 3.649 1.2 2.25s
16 -363.3977095913 -5.81 -3.28 -0.000 3.649 1.2 1.78s
17 -363.3977098889 -6.53 -3.43 -0.000 3.649 1.0 1.68s
18 -363.3977099902 -6.99 -3.53 0.000 3.648 1.4 2.28s
19 -363.3977099929 -8.57 -3.47 0.000 3.648 1.0 1.69s
20 -363.3977100015 -8.07 -4.20 -0.000 3.648 1.0 1.67s
21 -363.3977100135 -7.92 -4.53 -0.000 3.648 2.1 2.55s
22 -363.3977100155 -8.69 -4.64 0.000 3.648 2.2 2.09s
23 -363.3977100171 -8.80 -5.23 0.000 3.648 1.9 1.85s
24 -363.3977100174 -9.47 -5.35 0.000 3.648 2.6 2.75s
25 -363.3977100176 -9.80 -5.01 0.000 3.648 1.4 1.77s
26 -363.3977100177 -10.11 -4.77 0.000 3.648 1.5 1.80s
27 -363.3977100177 -10.15 -4.74 0.000 3.648 1.9 1.91s
28 -363.3977100178 -10.51 -4.77 0.000 3.648 1.0 2.27s
29 -363.3977100178 -10.73 -4.96 0.000 3.648 1.0 1.66s
30 -363.3977100178 -10.48 -5.02 0.000 3.648 1.0 1.67s
31 -363.3977100178 -10.80 -5.33 0.000 3.648 1.0 2.25s
32 -363.3977100178 -11.20 -5.54 0.000 3.648 1.0 1.67s
33 -363.3977100178 -11.65 -5.44 0.000 3.648 1.0 1.66s
34 -363.3977100179 -12.13 -5.28 0.000 3.648 1.0 1.66s
35 -363.3977100179 -11.73 -5.51 0.000 3.648 1.0 2.22s
36 -363.3977100179 -12.01 -5.75 0.000 3.648 1.0 1.66s
37 -363.3977100179 -12.47 -5.70 0.000 3.648 1.0 1.66s
38 -363.3977100179 -12.64 -5.66 0.000 3.648 1.0 2.23s
39 -363.3977100179 + -Inf -5.63 0.000 3.648 1.0 1.67s
40 -363.3977100179 + -13.25 -5.58 0.000 3.648 1.0 1.66s
41 -363.3977100179 -12.77 -5.67 0.000 3.648 1.0 1.67s
42 -363.3977100179 -12.55 -5.93 0.000 3.648 1.0 2.23s
43 -363.3977100179 + -13.25 -6.32 0.000 3.648 1.0 1.68s
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]0.11667610740708073With 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)