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]0.0821938680535208Then 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 -361.3851476016 0.07 1.335 3.441 6.9 4.66s
2 -363.2383216840 0.27 -0.21 0.014 3.624 3.4 8.45s
3 -363.3510672971 -0.95 -0.58 0.000 3.727 3.2 3.43s
4 -363.3890363699 -1.42 -1.18 0.000 3.717 2.6 2.47s
5 -363.3959742861 -2.16 -1.67 0.000 3.681 2.0 2.22s
6 -363.3973233102 -2.87 -2.05 0.000 3.656 1.6 2.51s
7 -363.3976120911 -3.54 -2.29 0.000 3.648 2.2 2.15s
8 -363.3976925991 -4.09 -2.65 0.000 3.647 1.6 2.03s
9 -363.3977067788 -4.85 -2.97 0.000 3.649 2.2 2.73s
10 -363.3977060399 + -6.13 -2.91 -0.000 3.649 1.6 1.88s
11 -363.3977091371 -5.51 -3.15 0.000 3.648 2.0 2.69s
12 -363.3977090081 + -6.89 -3.21 0.000 3.648 1.4 1.85s
13 -363.3977088857 + -6.91 -3.09 -0.000 3.649 2.0 2.09s
14 -363.3977082874 + -6.22 -2.96 -0.000 3.649 1.0 1.82s
15 -363.3977092034 -6.04 -3.01 -0.000 3.649 1.0 2.22s
16 -363.3977095586 -6.45 -3.13 -0.000 3.649 1.0 1.74s
17 -363.3977096869 -6.89 -3.14 -0.000 3.649 1.0 1.75s
18 -363.3977097949 -6.97 -3.16 0.000 3.649 1.0 2.29s
19 -363.3977098974 -6.99 -3.14 0.000 3.649 1.0 1.76s
20 -363.3977098357 + -7.21 -3.29 -0.000 3.649 1.0 1.74s
21 -363.3977098202 + -7.81 -3.34 -0.000 3.649 1.0 2.21s
22 -363.3977099362 -6.94 -3.49 0.000 3.649 1.0 1.65s
23 -363.3977099662 -7.52 -3.73 0.000 3.648 1.0 1.65s
24 -363.3977099724 -8.21 -3.61 0.000 3.648 1.0 1.66s
25 -363.3977100111 -7.41 -4.07 0.000 3.648 1.0 2.20s
26 -363.3977100148 -8.43 -4.07 0.000 3.648 1.2 1.68s
27 -363.3977100173 -8.61 -4.36 0.000 3.648 1.6 1.74s
28 -363.3977100175 -9.65 -4.68 0.000 3.648 1.2 1.76s
29 -363.3977100177 -9.59 -4.96 0.000 3.648 1.4 2.19s
30 -363.3977100178 -10.74 -5.17 0.000 3.648 1.2 1.69s
31 -363.3977100178 -10.46 -5.34 0.000 3.648 1.6 1.74s
32 -363.3977100178 -10.43 -5.73 0.000 3.648 2.0 2.55s
33 -363.3977100178 + -11.92 -5.61 0.000 3.648 2.1 1.94s
34 -363.3977100178 -10.91 -5.95 0.000 3.648 1.8 1.86s
35 -363.3977100179 -11.74 -6.41 0.000 3.648 1.1 2.24s
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]0.11667610527019273With 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)