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]0.08219384513398592Then 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 -361.3862797287 0.07 1.335 3.439 7.1 4.01s
2 -363.2386981010 0.27 -0.21 0.014 3.625 3.1 10.5s
3 -363.3505940871 -0.95 -0.58 0.000 3.727 3.4 2.71s
4 -363.3889261846 -1.42 -1.17 0.000 3.717 2.5 2.19s
5 -363.3959816520 -2.15 -1.67 0.000 3.681 2.0 2.73s
6 -363.3973118053 -2.88 -2.04 0.000 3.656 1.5 1.73s
7 -363.3976115463 -3.52 -2.29 0.000 3.647 2.2 1.96s
8 -363.3976931850 -4.09 -2.65 0.000 3.647 1.8 1.84s
9 -363.3977067101 -4.87 -2.97 0.000 3.649 2.1 2.69s
10 -363.3977065453 + -6.78 -2.94 -0.000 3.649 1.8 1.76s
11 -363.3977093596 -5.55 -3.24 0.000 3.648 1.9 1.81s
12 -363.3977094418 -7.08 -3.35 0.000 3.648 1.8 1.78s
13 -363.3977088670 + -6.24 -3.18 -0.000 3.648 2.1 2.60s
14 -363.3977085600 + -6.51 -3.10 -0.000 3.649 1.0 1.51s
15 -363.3977088031 -6.61 -2.86 -0.000 3.648 1.0 1.53s
16 -363.3977095383 -6.13 -3.01 -0.000 3.649 1.0 1.52s
17 -363.3977099774 -6.36 -3.36 -0.000 3.648 2.0 2.40s
18 -363.3977094283 + -6.26 -3.33 0.000 3.648 1.9 1.70s
19 -363.3977099644 -6.27 -3.66 -0.000 3.648 1.8 1.68s
20 -363.3977100089 -7.35 -3.75 -0.000 3.648 2.0 1.74s
21 -363.3977099578 + -7.29 -3.70 -0.000 3.648 1.6 1.68s
22 -363.3977099883 -7.51 -3.45 -0.000 3.648 2.0 2.46s
23 -363.3977099514 + -7.43 -3.56 0.000 3.648 1.0 1.52s
24 -363.3977099948 -7.36 -3.94 -0.000 3.648 1.0 1.56s
25 -363.3977100158 -7.68 -4.83 0.000 3.648 1.5 1.64s
26 -363.3977100176 -8.75 -5.19 0.000 3.648 3.0 2.88s
27 -363.3977100177 -9.99 -5.14 0.000 3.648 1.8 1.58s
28 -363.3977100177 + -10.60 -5.21 0.000 3.648 2.0 1.87s
29 -363.3977100178 -10.19 -5.32 0.000 3.648 1.0 1.51s
30 -363.3977100178 -10.39 -5.35 0.000 3.648 1.0 1.53s
31 -363.3977100178 -10.89 -5.26 0.000 3.648 1.0 2.21s
32 -363.3977100178 -10.59 -5.15 0.000 3.648 1.4 1.54s
33 -363.3977100178 -11.16 -5.01 0.000 3.648 1.0 1.50s
34 -363.3977100178 + -11.99 -4.87 0.000 3.648 1.0 1.51s
35 -363.3977100178 + -10.82 -4.84 0.000 3.648 1.5 1.58s
36 -363.3977100178 -10.88 -4.91 0.000 3.648 1.0 2.25s
37 -363.3977100178 -11.65 -5.03 0.000 3.648 1.0 1.51s
38 -363.3977100178 + -11.13 -4.83 0.000 3.648 1.0 1.52s
39 -363.3977100178 -10.89 -5.13 0.000 3.648 1.0 1.51s
40 -363.3977100179 -11.29 -5.50 0.000 3.648 1.0 1.52s
41 -363.3977100179 -11.99 -5.82 0.000 3.648 1.0 2.22s
42 -363.3977100179 -12.64 -6.02 0.000 3.648 1.0 1.49s
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]0.1166761077678708With 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)