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]0.08219341986225487Then 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: -0.0006182370306134658
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:40
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3879279559 0.07 1.335 3.439 6.9 4.71s
2 -362.9630178336 0.20 -0.10 0.223 3.873 2.6 10.5s
3 -363.1923001221 -0.64 -0.20 0.000 3.776 3.1 2.53s
4 -363.2396400704 -1.32 -0.29 0.000 3.781 2.2 2.81s
5 -363.3699126412 -0.89 -0.30 0.000 3.688 4.0 3.01s
6 -363.3864067568 -1.78 -0.48 -0.000 3.657 2.0 2.06s
7 -363.3967697109 -1.98 -1.13 -0.000 3.676 3.0 3.03s
8 -363.3937278788 + -2.52 -0.92 0.000 3.677 2.0 2.08s
9 -363.3967604770 -2.52 -1.08 0.000 3.656 1.0 1.67s
10 -363.3975296092 -3.11 -1.39 0.000 3.645 1.5 1.83s
11 -363.3976085118 -4.10 -1.47 0.000 3.643 1.0 2.32s
12 -363.3976322557 -4.62 -1.50 0.000 3.643 1.0 1.69s
13 -363.3976648119 -4.49 -1.55 0.000 3.641 1.0 1.67s
14 -363.3976069418 + -4.24 -2.02 0.000 3.654 1.0 1.67s
15 -363.3976552188 -4.32 -2.30 -0.000 3.654 1.0 2.33s
16 -363.3976270967 + -4.55 -2.19 -0.000 3.656 1.1 1.70s
17 -363.3976231615 + -5.41 -2.18 -0.000 3.656 1.0 1.68s
18 -363.3976314579 -5.08 -2.21 -0.000 3.656 1.0 1.68s
19 -363.3976769351 -4.34 -2.40 -0.000 3.654 1.0 2.39s
20 -363.3977021797 -4.60 -2.75 -0.000 3.651 1.0 1.73s
21 -363.3977074188 -5.28 -3.03 -0.000 3.650 1.1 1.68s
22 -363.3977099366 -5.60 -3.88 -0.000 3.649 2.0 1.99s
23 -363.3977100050 -7.16 -4.08 0.000 3.648 3.1 3.00s
24 -363.3977100102 -8.29 -4.18 0.000 3.648 1.2 1.73s
25 -363.3977100028 + -8.13 -4.09 0.000 3.648 2.0 1.93s
26 -363.3977100125 -8.02 -4.35 0.000 3.648 1.0 2.33s
27 -363.3977100111 + -8.87 -4.27 0.000 3.648 2.0 2.07s
28 -363.3977100159 -8.32 -4.61 0.000 3.648 1.0 1.73s
29 -363.3977100169 -9.00 -4.69 0.000 3.648 1.2 1.70s
30 -363.3977100173 -9.48 -4.67 0.000 3.648 1.2 2.38s
31 -363.3977100175 -9.62 -4.52 0.000 3.648 1.0 1.69s
32 -363.3977100177 -9.76 -4.80 0.000 3.648 1.0 1.70s
33 -363.3977100178 -10.00 -5.53 0.000 3.648 1.0 1.67s
34 -363.3977100178 -10.32 -5.72 0.000 3.648 2.5 2.81s
35 -363.3977100178 -10.87 -6.17 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]0.11667604876482168With 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)