DFT+U practical session
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1 DFT+U practical session Matteo Cococcioni GGA and GGA+U calculations in FeO Calculation of U for bulk Fe Calculation of U for NiO Exercise I: evaluating U for Cu 2 O Exercise II: evaluating U for FePO 4
2 Rhombohedral unit cell FeO-GGA
3 FeO-GGA AFM order 2 types of Fe atoms ATOMIC_SPECIES O 1. O.pbe-rrkjus.UPF Fe1 1. Fe.pbe-nd-rrkjus.UPF Fe2 1. Fe.pbe-nd-rrkjus.UPF starting_magnetization(2)= 0.5 starting_magnetization(3)=-0.5 Rhombohedral unit cell
4 FeO-GGA AFM order 2 types of Fe atoms ATOMIC_SPECIES O 1. O.pbe-rrkjus.UPF Fe1 1. Fe.pbe-nd-rrkjus.UPF Fe2 1. Fe.pbe-nd-rrkjus.UPF starting_magnetization(2)= 0.5 starting_magnetization(3)=-0.5 Rhombohedral unit cell: CELL_PARAMETERS Rhombohedral unit cell
5 FeO-GGA AFM order 2 types of Fe atoms ATOMIC_SPECIES O 1. O.pbe-rrkjus.UPF Fe1 1. Fe.pbe-nd-rrkjus.UPF Fe2 1. Fe.pbe-nd-rrkjus.UPF starting_magnetization(2)= 0.5 starting_magnetization(3)=-0.5 Rhombohedral unit cell: CELL_PARAMETERS Rhombohedral unit cell No Hubbard correction lda_plus_u =.true., Hubbard_U(1) = 1.d-10, Hubbard_U(2) = 1.d-10, Hubbard_U(3) = 1.d-10,
6 FeO-GGA Check the output file: feo.scf.out LDA+U parameters: U( 2) = alpha( 2) = U( 3) = Alpha( 3) = Hubbard correction on atom 2 (Fe1) Hubbard_alpha perturbation on atom 2 (Fe1) Total occupations: atom 1 Tr[ns(na)] (up, down, total) = atom 3 Tr[ns(na)] (up, down, total) = atom 4 Tr[ns(na)] (up, down, total) = Oxygen atoms non-magnetic, n up = n down Iron atoms ordered anti-ferromagnetically
7 FeO-GGA Eigenvalues and eigenvectors of occupation matrices (5x5 matrices): basis set: z 2, xz, yz, x 2 -y 2, xy spin 1 eigenvalues: eigenvectors: Caution: In previous versions of QE, the eigenvectors are printed as row vectors and the coefficients are not squared. c 1 c 2 c 3 c 4 c 5 occupations: x5 matrix
8 FeO-GGA Level splittings: atom 3 spin 1 eigenvalues: spin 2 eigenvalues: atom 4 spin 1 eigenvalues: spin 2 eigenvalues:
9 FeO-GGA Level splittings: atom 3 spin 1 eigenvalues: spin 2 eigenvalues: atom 4 spin 1 eigenvalues: spin 2 eigenvalues: Split by rhombohedral symmetry (magnetic order) Partial occupations of atomic orbitals metallic bands
10 FeO-GGA Level splittings: atom 3 spin 1 eigenvalues: spin 2 eigenvalues: atom 4 spin 1 eigenvalues: spin 2 eigenvalues: Split by rhombohedral symmetry (magnetic order) Partial occupations of atomic orbitals Eigenvector corresponding to A 1g : metallic bands
11 FeO-GGA density of states Density of states: Majority spin states are almost full, they are below E F Minority spin states are partially occupied at E F metallic behavior
12 FeO-GGA+U
13 FeO-GGA+U Hubbard U correction of Fe d-states: Hubbard_U(1)=1.d-10, Hubbard_U(2)=4.3, Hubbard_U(3)=4.3
14 FeO-GGA+U Hubbard U correction of Fe d-states: Total occupations: Hubbard_U(1)=1.d-10, Hubbard_U(2)=4.3, Hubbard_U(3)=4.3 atom 1 Tr[ns(na)] (up, down, total) = atom 3 Tr[ns(na)] (up, down, total) = atom 4 Tr[ns(na)] (up, down, total) =
15 FeO-GGA+U Hubbard U correction of Fe d-states: Total occupations: Hubbard_U(1)=1.d-10, Hubbard_U(2)=4.3, Hubbard_U(3)=4.3 atom 1 Tr[ns(na)] (up, down, total) = atom 3 Tr[ns(na)] (up, down, total) = atom 4 Tr[ns(na)] (up, down, total) = Check minority spin occupations of either Fe atom: atom 4 spin 1 eigenvalues: eigenvectors: Still partial occupations! METALLIC
16 FeO-GGA+U density of states We found a local minimum of DFT+U energy functional, which is metallic!
17 FeO-GGA+U
18 FeO-GGA+U Let us try to suggest a different possible electronic configuration to the system: Check minority spin occupations of either Fe atom after the 1st iteration: atom 4 spin 1 eigenvalues: eigenvectors:
19 FeO-GGA+U Let us try to suggest a different possible electronic configuration to the system: Check minority spin occupations of either Fe atom after the 1st iteration: atom 4 spin 1 eigenvalues: eigenvectors:
20 FeO-GGA+U Let us try to suggest a different possible electronic configuration to the system: Check minority spin occupations of either Fe atom after the 1st iteration: atom 4 spin 1 eigenvalues: eigenvectors: We want to drive the system to a configuration where the A1g state is fully occupied: This state is singlet filling it completely will lead to an insulator Physically, it points along the [111] direction, perpendicular to (111) AFM planes
21 FeO-GGA+U2
22 FeO-GGA+U2 Override the 3rd eigenvalue of the spin 2 of atom 2 and spin 1 of atom 3 with 1.d0: starting_ns_eigenvalue(3,2,2)=1.d0, starting_ns_eigenvalue(3,1,3)=1.d0 eigenvalue index spin index atom type index
23 FeO-GGA+U2 Override the 3rd eigenvalue of the spin 2 of atom 2 and spin 1 of atom 3 with 1.d0: starting_ns_eigenvalue(3,2,2)=1.d0, starting_ns_eigenvalue(3,1,3)=1.d0 eigenvalue index spin index atom type index Check the resulting minority spin occupations of either Fe atom: atom 4 spin 1 eigenvalues: eigenvectors: A1g state is fully occupied Other are almost empty.
24 FeO-GGA+U2 density of states The insulating state is found! A1g is completely full. No partial occupations.
25 FeO-GGA+U2 density of states The insulating state is found! A1g is completely full. No partial occupations. Which of these local minima has lower energy? DFT+U metallic DFT+U insulating total energy = Ry total energy = Ry
26 FeO-GGA+U2 What about other local energy minima? Are there other minima with lower energy? How to find them? Physical intuition (and breaking of symmetry)! Cococcioni and de Gironcoli, PRB 71, (2005)
27 Calculation of U (bulk Fe) If we use the unit cell, perturbing an atom will perturb all the atoms of the same type in the crystal. Perturbations can not be isolated Interactions between periodic images will interfere with each other.
28 Calculation of U (bulk Fe) If we use the unit cell, perturbing an atom will perturb all the atoms of the same type in the crystal. We need a super cell to compute U Perturbations can not be isolated Interactions between periodic images will interfere with each other.
29 Calculation of U (bulk Fe) If we use the unit cell, perturbing an atom will perturb all the atoms of the same type in the crystal. We need a super cell to compute U Perturbations can not be isolated Interactions between periodic images will interfere with each other.
30 Calculation of U (bulk Fe) If we use the unit cell, perturbing an atom will perturb all the atoms of the same type in the crystal. Perturbations can not be isolated Interactions between periodic images will interfere with each other. We need a super cell to compute U Calculated value of U will converge with increasing super cell size.
31 Convergence of U with cell size The U parameter should be converged with cell size
32 Convergence of U with cell size The U parameter should be converged with cell size Bulk Fe
33 Convergence of U with cell size The U parameter should be converged with cell size Bulk Fe
34 Calculation of U-NiO 4 atoms CELL_PARAMETERS atoms ATOMIC_POSITIONS {crystal} Ni1 0.d0 0.d0 0.d0 Ni2 0.5d0 0.5d0 0.d0 Ni2 0.5d0 0.d0 0.5d0 Ni2 0.d0 0.5d0 0.5d0 Ni3 0.5d0 0.d0 0.d0 Ni3 0.d0 0.5d0 0.d0 Ni3 0.d0 0.d0 0.5d0 Ni3 0.5d0 0.5d0 0.5d0 O1-0.25d0 0.25d0 0.25d0 O1 0.25d0 0.25d0 0.25d0 O1-0.25d0 0.25d0 0.75d0 O1-0.25d0 0.75d0 0.25d0 O1 0.25d0 0.75d0 0.25d0 O1 0.25d0 0.25d0 0.75d0 O1-0.25d0 0.75d0 0.75d0 O1 0.25d0 0.75d0 0.75d0 3 types of Ni atoms: 1. Ni1 perturbed spin up (always put the perturbed atom at the origin) 2. Ni2 unperturbed spin up 3. Ni3 unperturbed spin down
35 Calculation of U-NiO Steps: After perturbing all Hubbard atoms we can now construct response matrices
36 Calculation of U-NiO Steps: 1.Perform the unperturbed scf run nio_1.scf.out 2.Save the wavefunctions and potential in temp1 3.Reading the initial conditions from temp1, perturb the Ni-d orbitals by Hubbard_alpha ni out, ni.0.0.out, ni.0.15.out After perturbing all Hubbard atoms we can now construct response matrices
37 Calculation of U-NiO Steps: 1.Perform the unperturbed scf run nio_1.scf.out 2.Save the wavefunctions and potential in temp1 3.Reading the initial conditions from temp1, perturb the Ni-d orbitals by Hubbard_alpha ni out, ni.0.0.out, ni.0.15.out This step concludes the perturbation of Ni-d states. Now we perturb O-p states To do so, first shift the coordinates such that one O atom is at the origin After perturbing all Hubbard atoms we can now construct response matrices
38 Calculation of U-NiO Steps: 1.Perform the unperturbed scf run nio_1.scf.out 2.Save the wavefunctions and potential in temp1 3.Reading the initial conditions from temp1, perturb the Ni-d orbitals by Hubbard_alpha ni out, ni.0.0.out, ni.0.15.out This step concludes the perturbation of Ni-d states. Now we perturb O-p states To do so, first shift the coordinates such that one O atom is at the origin 1.Perform the unperturbed scf run nio_2.scf.out 2.Save the wavefunctions and potential in temp1 3.Reading the initial conditions from temp1, perturb the Ni-d orbitals by Hubbard_alpha o out, o.0.0.out, o.0.15.out After perturbing all Hubbard atoms we can now construct response matrices
39 Calculation of U-NiO Response matrices by numerical derivative: This step is performed for all the atoms by the program resp_mat.f90 in the folder Ucalc. This program reads the files dn.i.da.j.dat that contain the occupation of atom I as a function of the perturbation on atom J
40 Calculation of U-NiO Response matrices by numerical derivative: This step is performed for all the atoms by the program resp_mat.f90 in the folder Ucalc. This program reads the files dn.i.da.j.dat that contain the occupation of atom I as a function of the perturbation on atom J These quantities are extracted from the output files by the script grepnalfa_nio_r16 For example: grep 'Tr' ni.$con.out tail -16 head -$at tail -1 awk '{print $10}' extracts the scf response of the occupation of atom ``$at'' when a perturbation of intensity ``$con'' is applied to the Ni d-states. This quantity is used to construct
41 Calculation of U-NiO Response matrices by numerical derivative: This step is performed for all the atoms by the program resp_mat.f90 in the folder Ucalc. This program reads the files dn.i.da.j.dat that contain the occupation of atom I as a function of the perturbation on atom J These quantities are extracted from the output files by the script grepnalfa_nio_r16 For example: grep 'Tr' ni.$con.out tail -16 head -$at tail -1 awk '{print $10}' extracts the scf response of the occupation of atom ``$at'' when a perturbation of intensity ``$con'' is applied to the Ni d-states. This quantity is used to construct likewise; grep 'Tr' ni.$con.out head -32 tail -16 head -$at tail -1 awk '{print $10}' extracts the bare response of the occupation of atom ``$at'' when a perturbation of intensity ``$con'' is applied to the Ni d-states. This quantity is used to construct
42 Calculation of U-NiO Important checks: The response must be linear. Screened (scf converged) and Kohn-Sham (at fixed potential) responses must coincide at zero perturbation. e.g.
43 Calculation of U-NiO Description of the input format for resp_mat.f90 &input_mat ntyp = 2 # Number of types of atoms na(1) = 8 # Number of atoms of type 1 na(2) = 8 # Number of atoms of type 2 nalfa = 3 # Number of perturbations magn =.true. #.true. for magnetic systems,.false. otherwise. filepos = 'pos_nio_r16' # file containing atomic positions back = 'no' # to add neutralizing background, see PRB (05) filednda = 'file.nio.r16' # file containing the names of dn*dat n1 = 5 n2 = 5 # number of unit cells used for extrapolating n3 = 5 # response matrices to larger supercells. &end This input is automatically generated and the executed by the script ucalc_nio_r16.j
44 Calculation of U-NiO As we discussed earlier, the calculated values of U need to be converged with increasing super cell size. However, super cell calculations are expensive. One then extrapolates the obtained response to larger super cells; schematically: The response between the equivalent pairs of neighbors is rescaled by multiplicity. Further pairs, response is equated to 0 We compute explicitly: extrapolate This step is also performed by resp_mat.f90
45 Calculation of U-NiO Check Ur16.out: Number of atoms In the super cell Calculated U on Ni-d Calculated U on O-p
46 Exercise: Cu 2 O Cu 2 O is a semiconductor. It is the parent material for many transparent conductive oxides (with p-type conductivity). Non-magnetic E g ~ 2.2 ev ibrav=1, celldm(1)=8.21 nspin = 1 ATOMIC_POSITIONS {crystal} Cu Cu Cu Cu O O
47 Exercise: Cu 2 O Perform the GGA calculation and plot the DOS. What is the band gap? Compute U on Cu-d states using the linear response approach. Using the computed value of U, perform the GGA+U calculation. Investigate the changes in the DOS. How does the occupation matrices change? Which states increase their occupation?
48 Exercise: Cu 2 O 48 atoms super cell 24 atoms super cell
49 Exercise: Cu 2 O GGA GGA+U Why the gap is still smaller than the experiment? Will it increase with U? What states do you find at the edge of valence and conduction bands?
50 Exercise: FePO 4 unit cell: 4 Fe, 4 P, 16 O (of 3 different kinds ) scf and perturbed runs have already been completed: just compute U
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