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]0.08219258847129957Then 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. 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: -0.000618237030613506
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:39
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3873100308 0.07 1.336 3.439 6.8 5.05s
2 -363.2379566755 0.27 -0.21 0.014 3.624 3.1 3.51s
3 -363.3509585401 -0.95 -0.58 0.000 3.727 3.2 2.89s
4 -363.3890515773 -1.42 -1.18 0.000 3.717 2.6 2.41s
5 -363.3959760313 -2.16 -1.67 0.000 3.681 2.1 2.22s
6 -363.3973136019 -2.87 -2.04 0.000 3.656 1.5 1.89s
7 -363.3976108350 -3.53 -2.29 0.000 3.647 2.0 2.11s
8 -363.3976854700 -4.13 -2.55 0.000 3.646 1.4 1.90s
9 -363.3977064982 -4.68 -2.94 0.000 3.649 2.0 2.15s
10 -363.3977071331 -6.20 -2.96 -0.000 3.649 2.0 1.94s
11 -363.3977089175 -5.75 -3.08 -0.000 3.649 1.0 1.77s
12 -363.3977092250 -6.51 -3.03 0.000 3.648 1.0 1.77s
13 -363.3977097474 -6.28 -2.99 0.000 3.648 1.1 1.78s
14 -363.3977094071 + -6.47 -2.84 -0.000 3.648 1.0 1.76s
15 -363.3977089161 + -6.31 -2.72 0.000 3.648 1.0 1.77s
16 -363.3977081992 + -6.14 -2.59 -0.000 3.647 1.0 1.76s
17 -363.3977077356 + -6.33 -2.55 -0.000 3.647 1.0 1.77s
18 -363.3977086234 -6.05 -2.68 -0.000 3.647 1.0 1.76s
19 -363.3977094765 -6.07 -2.86 -0.000 3.648 1.0 1.76s
20 -363.3977094598 + -7.78 -2.83 -0.000 3.648 1.0 1.76s
21 -363.3977095985 -6.86 -2.84 0.000 3.648 1.0 1.77s
22 -363.3977098813 -6.55 -3.13 -0.000 3.648 1.0 1.76s
23 -363.3977099950 -6.94 -3.55 0.000 3.648 1.0 3.04s
24 -363.3977100085 -7.87 -3.64 0.000 3.648 1.0 1.69s
25 -363.3977100137 -8.28 -3.89 0.000 3.648 1.0 1.64s
26 -363.3977100162 -8.60 -4.11 0.000 3.648 1.2 1.69s
27 -363.3977100175 -8.89 -4.43 0.000 3.648 1.5 1.73s
28 -363.3977100177 -9.76 -4.84 0.000 3.648 1.6 1.86s
29 -363.3977100176 + -10.26 -4.83 0.000 3.648 2.0 1.95s
30 -363.3977100177 -9.97 -5.00 0.000 3.648 1.0 1.65s
31 -363.3977100178 -10.05 -5.32 0.000 3.648 1.0 1.66s
32 -363.3977100178 -10.70 -5.70 0.000 3.648 1.1 1.67s
33 -363.3977100178 + -11.18 -5.61 0.000 3.648 2.0 2.08s
34 -363.3977100178 -11.22 -5.75 0.000 3.648 1.0 1.66s
35 -363.3977100179 -11.32 -6.06 0.000 3.648 2.0 1.92sRun 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]0.11667586135327568With 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)