Magnetism in transition metal oxides by post-dft methods

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Magnetism in transition metal oxides by post-dft methods Cesare Franchini Faculty of Physics & Center for Computational Materials Science University of Vienna, Austria Workshop on Magnetism in Complex Systems, 29

Outline 1 Methodology DFT, HF, DFTh, DFT+U VASP 2 Bulk (1): Transition metal monoxide CuO MnO 3 Bulk (2): Multivalent oxides Multivalent manganese oxides Mn x O y 4 Surfaces MnO(1) & MnO(11) surfaces 5 Thinlayers Mn x O y on Pd(1)

DFT, HF, DFTh, DFT+U DFT (1) Ĥ of N interacting electrons in an external potential Ĥ = ˆT + ˆV + Ŵ ˆT = N 2 i i=1 2 (kinetic operator) ˆV = N i=1 v(r i) (potential operator) Ŵ = 1 2 N i=1 N j=1,i j Hohemberg-Kohn theorem 1 (Coulomb interaction operator) r i r j 1 Many particle ground state unique functional of n(r) 2 Variational Principle: E = E v [n ] < E v [n] 3 Universal functional F[n]: E v [n] = F[n] + d 3 rv (r)n(r) F[n] = Ψ[n] ˆT + Ŵ Ψ[n], density dependence unknown

DFT, HF, DFTh, DFT+U DFT (2) The Kohn-Sham method: single particle scheme [ 2 2 + v s(r)] = ǫ i ϕ i (r), n(r) = occ i ϕ i (r) 2 Central idea: to construct the single-particle potential v s (r) in such a way that the density of the auxiliary non-interacting system equals the density of the interacting system of interest F[n] = T s [n] + W H [n] + E xc [n], W H [n] = 1 2 d 3 r d 3 r n(r)n(r ) r r

DFT, HF, DFTh, DFT+U DFT (3) Exchange-correlation energy E xc [n] = T[n] + W[n] W H [n] T s [n] = E x [n] + E c [n] v s [n](r) = v(r) + d 3 r n(r ) r r + v xc[n](r) v xc [n](r) = δexc[n] δn(r) Self consistent loop [ 2 Z v s[n](r) = v(r) + 2 + vs(r)] = ǫ iϕ i (r) (1) Xocc n(r) = ϕ i (r) 2 (2) i d 3 r n(r ) + vxc[n](r) (3) r r

DFT, HF, DFTh, DFT+U DFT (4) E c = E exact E KS Correlation energy Short range screening term to account for the approx. we make in assuming that an electron moves in the average field for all the others. g(x,y) = f 1 (x)f 2 (y) only if when x and y independent E x = KS Ŵ KS W H Exchange energy It is a correction to the Hartree term: self interaction (the Coulomb interaction of an electron with itself)

DFT, HF, DFTh, DFT+U DFT (5) F[n] = T s [n] + W H [n] + E xc [n], xc functionals?: DFT does not give any hint on how to construct E xc, it only holds the promise that E xc is a universal functional of the density LDA Exc LDA [n] = d 3 rn(r)exc unif (n(r)) Good for spatially slowly varying density GGA Exc GGA [n] = d 3 rf(n(r), n(r)) Non unique, many different forms. meta-gga Exc MGGA [n] = d 3 rf(n(r), n(r), τ(r)) Additional flexibility

DFT, HF, DFTh, DFT+U Hartree-Fock Theory (1) Single-particle problem Single-electron operators: Ĥ = N i=1 (ĥ(i) + ˆv (i) HF ) ĥ (i) = Kinetic + Potential ˆv (i) HF = e-e interaction ( j<i 1 r ij ) Slater Determinants Ψ = ψ i (x 1 )ψ j (x 2 )...ψ k (x N ) ψ(x) = φ(r)α(σ) or φ(r)β(σ) Self-Interaction Free (exact exchange) 1 N N 2 i=1 j=1 (ii ˆv jj) - (ij ˆv ji) If ψ i = ψ j the self interaction term (ii ˆv ii) is canceled by the exchange part

DFT, HF, DFTh, DFT+U Hartree-Fock Theory (2) No correlation Correlated movement of the electrons in HF only for like-spin electrons. Whenever two electrons occupy the same orbital the determinant will vanish (two rows are identical). HF considered correlation free. Notorious failure: metal state A pseudo-gap at the Fermi level due to the unscreened exchange interaction: δǫhf (k) δk, k = k F

DFT, HF, DFTh, DFT+U Summing up... DFT Approx. EXCHANGE HF Exact EXCHANGE Approx. CORRELATION No Correlation DFT+HF: Hybrid Functionals (DFTh) 1 MIX a fraction of exact HF exchange with GGA exchange: E HYB x a = mixing empirical parameter [n] = aex HF [n] + (1 a)ex GGA [1] Becke, A. D., J. Chem. Phys. 98, 1372 (1993).

DFT, HF, DFTh, DFT+U DFTh PBE 1,2 E PBE xc = 1 4 E HF x + 3 4 E PBE x + E PBE c HSE3 3 (PBE - long-ranged Fock exchange) E HSE3 xc = 1 4 E HF,sr,µ x + 3 4 E PBE,sr,µ x 1 r r = S µ( r r ) + L µ ( r r ) + E PBE,lr,µ x + E PBE c [1] M. Ernzerhof, and G.E. Scuseria, J. Chem. Phys., 11, 529 (1999). [2] J.P. Perdew, M. Ernzerhof, K. Burke, J. Chem. Phys. 15, 9982 (1996). [3] A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. 125, 22416 (26).

DFT, HF, DFTh, DFT+U DFT+U 1 DFT (LDA or GGA) De-localized electrons (s,p) + Hubbard Hamiltonian Localized electrons (d,f) LDA+U E = E LDA UN(N 1)/2 + 1 2 U i j n in j N = n i, (n i are d/f-orbital occupancies) ǫ i = δe/δn i = ǫ LDA + U( 1 2 n i) LDA+U shifts the LDA orbital energy by -U/2 for occupied orbitals (n i = 1) and by +U/2 for unoccupied orbitals (n i = ) [1] V. I. Anisimov, J. Zaanen and O. K. ANdersen, Phys. Rev. B 44, 943 (1991).

DFT, HF, DFTh, DFT+U Applicability: Single Slater Determinant LDA HSE LDA+U HF DOS Energy Energy Energy weak exchange Jellium (UEG) intermediate strong exchange Mott insulator Multi Slater determinant systems (highly correlated) DMFT

VASP Vienna ab initio Simulation Package (VASP) Plane Wave basis set: Pseudopotentials φ n,k (r) = G C n,k+ge i(k+g)r Projector augmented wave method (PAW)

CuO Rocksalt CuO? T N (K) 1 9 8 7 6 5 4 3 2 Rock salt structure Rock salt CuO? Monoclinic 1 MnO FeO CoO NiO CuO Transition metal compound

CuO Monoclinic CuO Cu 2+ (d 9 ): Jahn-Teller distortion Unlike other members of the TM oxide series CuO is unstable upon a Jahn-Teller distortion, which splits apart d x 2 y 2 and d2 z and the monoclinic crystal field polarizes the d 2 z states. LDA/GGA Metal - No splitting HSE Insulating, gap = 2.1 ev

CuO Rocksalt-Tetragonal CuO [1] Siemons et al., cond-mat FM q=(.5,.5, ) AFM1 q=(.5,.5, ) AFM2 q=(.5,.5,.5) AFM3 q=(,,.5) AFM4 TET1 TET2 HSE HSE Expt.[1] a=b (Å) 4.4 3.98 3.95 c (Å) 3.87 5.381 5.3 c/a.82 1.377 1.357 E g (ev) 1.57 2.7 - m Cu (µ B).71.69 - AFM5 q=(.5,, -.5)

CuO TET1: (d x 2 y 2)2 (d yz, d xz ) 4 (d xy ) 2 (d z 2) 1 TET2: (d z 2) 2 (d yz, d xz ) 4 (d xy ) 2 (d x 2 y 2)1 Intensity (arb.) Total Expt. Cu d O p -8-6 -4-2 Relative Binding Energy (ev)

CuO Magnetic interactions & Neél Temperature (T N ) Heisenberg H H = i j J ijs i S j Mol. Field theory T N = 2S(S+1) 3k B i j J ij Results (mev, K) J xy J xz J x J z T N TET1-7. 4. -51.6-197.7 17 TET2 63. 32.3-159.4 6.5 865 C2/c[1] -14 2.4-38.4 17 [1] Filippetti et al. PRL95, 8645 (25) T N (K) 1 9 8 7 6 5 4 3 2 elongated-rocksalt CuO Rock salt structure Monoclinic 1 MnO FeO CoO NiO CuO Transition metal compound

MnO Rhombohedrally distorted rocksalt structure (9 +α) PBE PBE+U PBE HF Expt. 1.44 2.3 4.2 12.6 3.6-4.2 m 4.31 4.69 4.52 4.7 4.58, 4.79 v 1.3 3. 1.7 3.9 1.8, 1.9 a 4.37 4.48 4.4 4.38 4.43, 4.4448 α 1.66.56.88.62 Energy(eV) PBE PBE+U PBE 8 6 4 2-2 -4-6 -8 X Γ Z X Γ Z X Γ

MnO DOS (States/eV - atom - spin) 4 2 2 Mn1, d 1-1 -2 2 1-1 -2 Total Mn2, d.8 O, p PBE PBE+U.4 PBE -1-8 -6-4 -2 2 4 6 8 1 Energy (ev) t 2g e g t 2g e g PBE PBE+U PBE HF Expt. 1.44 2.3 4.2 12.6 3.6-4.2 m 4.31 4.69 4.52 4.7 4.58, 4.79 v 1.3 3. 1.7 3.9 1.8, 1.9 a 4.37 4.48 4.4 4.38 4.43, 4.4448 α 1.66.56.88.62 Intensity (arb.) Expt Theory e g t 2g e g -1-8 -6-4 -2 2 4 6 8 1 Relative Binding Energy (ev) t 2g e g

MnO Origin of the Rhombohedral distortion: Magnetism Heisenberg Hamiltonian H = P NN J 1S i S j + P NNN J 2S i S j. H B1 = 3J 2 S 2 H dist = 3J 1 S2 + 3J 1 S2 3J 2 S 2 H dist H B1 = (J 1 J 1 )3S2 < which means J 1 < J 1 PBE PBE+U PBE HF Semiempirical 1,2 J 1 17.6 8.2 11.5 2.7 J 2 27.9 4.3 13.7 4.3 1.3 9.6 J 1 2.7 8.7 12.8 1. 9.9 J 1 14.3 7.7 1.6 7.9 7.5 J 2 / J 1 1.6.5 1.2 1.6 1.2 1.1 J 3.2.5 1.1 1.1 1.2 [1] M. Kohgi et al., SSC 11, 391 (1972). [2] G. Pepy et al., JPC 35, 433 (1974).

Multivalent manganese oxides Mn xo y Mn x O y : Crystal structures MnO, 2+, 5% Mn Mn 3 O 4, 2+ 3+, 43% Mn Mn 2 O 3, +3, 4% Mn MnO 2, 4+, 33% Mn

Multivalent manganese oxides Mn xo y Mn x O y : Results Calc. volume (Å 3 ) 11.6 11.2 1.8 1.4 +2,+3 Mn 1 3 O 4 9.6 9.2 +4 +3 MnO 2 Mn 2 O 3 MnO +2 PBE PBEU4 PBE HSE Intensity (a.u.) MnO a) Mn 3 O b) 4 Intensity (a.u.) 8.8 8.8 9.2 9.6 1 1.4 1.8 11.2 11.6 Expt. volume (Å 3 ) -15-1 -5 Relative Binding Energy (ev) -15-1 -5 Relative Binding Energy (ev) Calc. enthalpy of formation (ev) -2-4 -6-8 -1-12 -14-16 +2, +3 Mn 3 O 4 +3 Mn 2 O 3 +4 MnO 2 MnO +2 PBE PBEU4 PBE HSE -16-14 -12-1 -8-6 -4-2 Expt. enthalpy of formation (ev) Intensity (a.u.) Mn 2 O c) 3 MnO d) 2-15 -1-5 Relative Binding Energy (ev) Intensity (a.u.) -15-1 -5 Relative Binding Energy (ev) MARE PBEU6 PBEU4 PBEU3 PBE PBE HSE Volume 7.7 5.4 4.1 1. -.6 Enthalpy 14 7 8 22 9 9 Hybrid functionals provide the most balanced and consistent description

MnO(1) & MnO(11) surfaces Magnetism at surfaces Surface effects Surfaces effects (changes of the coordination, structural and electronic reconstructions, etc.) can strongly influence the magnetic structure: magnetic moments on O atoms modifications of the superexchange interactions reduced magnetic moment of the metal atom formation of incommensurable magnetic structures

(1) (11) (11) MR Ordering Spin Configuration E Spin Configuration E Spin Configuration E AFM II FM + + + + + + + 264 + + + + + + + + + 313 + + + + + + + + + + + 375 AFM 1 + + + + + + 178 + + + + + + + + 199 + + + + + + + + + + 262 AFM 2 + + + + + 13 + + + + + + + 166 + + + + + + + + + 15 AFM 3 + + + + + 94 + + + + + 41 + + + + + 68 AFM 4 + + + + + 157 + + + + + + + 162 + + + + + + + + + 29 AFM 5 + + + 157 + + + + + 163 + + + + + + + 25 AFM 6 + + + + 37 + + + + + 15 + + + + + + 31 AFM 7 + + + + 72 + + + + + + 44 + + + + + + + 38 AFM 8 ± + + + + + ± 191 + + + + + 215 AFM 9 + + + + + 171 + + + + + 132 MnO(1) & MnO(11) surfaces Magnetic interactions MnO(1) MnO(11)

MnO(1) & MnO(11) surfaces Missing row (MR) reconstruction γ PBE PBE+U ev/å 2 ev/å 2 (1).45.54 (11).82.18 (11) MR.58.78 DOS (bulk gap = 2.3 ev) DOS (states/ev atom spin) 2 1 2 1 2 1.8.4.8.4.8.4 Mn, d Mn, d Mn, d O, p MnO (11) O, p S O, p gap =.5 ev PBE+U PBE S-1-8 -6-4 -2 2 4 6 8 Energy (ev) S C S-1 C DOS (states/ev atom spin) 2 1 2 1 2 1.8.4.8.4.8.4 Mn, d Mn, d Mn, d O, p MnO (11) MR C gap = 1.66 ev PBE+U O, p PBE C S-1 O, p S-1-8 -6-4 -2 2 4 6 8 Energy (ev) S S J s (bulk: J 1 =.71, J 2 =.36) (1) (11) (11) MR J1 S.86.9 1.8 J S 1 1.7.62.67 J S 2 1.72.76.83 J S 3 1.7.72 J2 S.85.78 1.51 J S 1 2.57.22.73 J S 2 2.51.2 J S 3 2.48 J S 1.53 J S 2.4

Mn xo y on Pd(1) Two-dimensional Mn x O y layers on Pd(1) Phase diagram Structural model: Mn 3 O 4 -c(4 2)

Mn xo y on Pd(1) Two-dimensional c(4 2)-Mn 3 O 4 on Pd(1) Structural model: Mn 3 O 4 -c(4 2) Relative stability PBE vs. HSE FM AFM1 AFM2 PBE RH1 41 97 RH2 28 273 171 RH3 23 31 18 HSE RH1 45 22

Mn xo y on Pd(1) Two-dimensional c(4 2)-Mn 3 O 4 on Pd(1) FM AFM2 δ Mn1 Mn1 Mn2 δ O δ Mn2 δ O δ Mn2 b O o =.33 A

Mn xo y on Pd(1) Two-dimensional c(4 2)-Mn 3 O 4 on Pd(1) FM Mn1 AFM2 Mn2

Mn xo y on Pd(1) Two-dimensional c(4 2)-Mn 3 O 4 on Pd(1) Dipole active mode PBE HSE Expt. FM 36.4 43.3 43.5 AFM2 35.4 39.6

Summary Magnetic effects in bulk, surfaces and 2D thinfilms 1 CuO: Submitted (29) 2 MnO: PRB72, 45132 (25) 3 Mn x O y : PRB75, 195128 (27) 4 MnO surfaces: PRB73, 15542 (26) PRB75, 3544 (27) 5 Mn x O y on Pd(1): JCP13, 12477 (29) PRB79, 3542 (29) JPCM21, 1348 (29)

Acknowledgments Acknowledgments Methodology: J. Paier, M. Marsmann & G. Kresse, University of Vienna CuO: X.-Q. Chen & C. L. Fu, Oak Ridge National Laboratory Mn x O y : V. Bayer, G. Kresse & R. Podloucky, University of Vienna Surfaces and Thinfilms: V. Bayer & R. Podloucky, University of Vienna F. Allegretti, F.Li, G. Parteder, S. Surnev & F.P. Netzer, University Graz

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