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]0.0821933375805689Then 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: -0.0006182370306135084
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/terms/xc.jl:39
n Energy log10(ΔE) log10(Δρ) Magnet |Magn| Diag Δtime
--- --------------- --------- --------- ------ ------ ---- ------
1 -361.3876477182 0.07 1.336 3.441 6.9 4.83s
2 -363.2372728596 0.27 -0.21 0.014 3.625 3.4 8.99s
3 -363.3509998440 -0.94 -0.58 0.000 3.727 3.1 2.91s
4 -363.3890084297 -1.42 -1.18 0.000 3.717 2.6 2.46s
5 -363.3959693287 -2.16 -1.67 0.000 3.681 2.0 2.93s
6 -363.3973141439 -2.87 -2.04 0.000 3.656 1.5 1.93s
7 -363.3976161234 -3.52 -2.29 0.000 3.648 2.2 2.22s
8 -363.3976915821 -4.12 -2.62 0.000 3.647 1.5 2.01s
9 -363.3977066796 -4.82 -2.97 0.000 3.649 2.0 2.24s
10 -363.3977064310 + -6.60 -2.93 -0.000 3.649 2.1 2.69s
11 -363.3977092762 -5.55 -3.19 0.000 3.648 1.8 1.97s
12 -363.3977090114 + -6.58 -3.18 0.000 3.648 2.0 2.10s
13 -363.3977090833 -7.14 -3.13 -0.000 3.648 2.0 2.14s
14 -363.3977087133 + -6.43 -2.97 -0.000 3.648 1.0 1.78s
15 -363.3977093089 -6.23 -3.08 -0.000 3.649 1.0 1.78s
16 -363.3977094910 -6.74 -3.15 -0.000 3.649 1.0 2.37s
17 -363.3977096047 -6.94 -3.13 -0.000 3.649 1.0 1.78s
18 -363.3977096451 -7.39 -3.11 -0.000 3.649 1.0 1.77s
19 -363.3977097478 -6.99 -3.13 -0.000 3.649 1.0 1.77s
20 -363.3977099758 -6.64 -3.24 0.000 3.648 1.0 1.78s
21 -363.3977099947 -7.72 -3.32 0.000 3.648 1.0 1.75s
22 -363.3977099909 + -8.42 -3.28 0.000 3.648 1.0 2.29s
23 -363.3977099897 + -8.93 -3.22 0.000 3.648 1.0 1.67s
24 -363.3977099957 -8.22 -3.29 0.000 3.648 1.0 1.68s
25 -363.3977099985 -8.56 -3.55 0.000 3.648 1.0 1.68s
26 -363.3977100049 -8.19 -3.74 0.000 3.648 1.0 1.69s
27 -363.3977100134 -8.07 -3.98 0.000 3.648 1.0 1.68s
28 -363.3977100175 -8.38 -4.59 0.000 3.648 1.0 2.28s
29 -363.3977100177 -9.88 -5.04 0.000 3.648 1.9 1.96s
30 -363.3977100177 -10.73 -5.08 0.000 3.648 1.8 1.98s
31 -363.3977100178 -10.03 -5.27 0.000 3.648 1.0 1.67s
32 -363.3977100178 -10.76 -5.37 0.000 3.648 1.0 1.68s
33 -363.3977100178 -10.42 -5.69 0.000 3.648 2.0 2.01s
34 -363.3977100179 -11.39 -5.99 0.000 3.648 1.1 2.27s
35 -363.3977100179 -12.25 -6.02 0.000 3.648 1.2 1.78s
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]0.11667634982583269With 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)