Anisotropic magnetic interactions in Iridium oxides from LDA+U - PowerPoint PPT Presentation

Anisotropic magnetic interactions in Iridium oxides from LDA+U calculations Alexander Yaresko Max Planck Institute for Solid State Research, Stuttgart, Germany What about U? Effects of Hubbard Interactions and Hunds Coupling in Solids ICTP, Trieste, October 17-21, 2016

outline motivations: j eff =1/2 beauty 1 anisotropic exchange in honeycomb Na 2 IrO 3 2 noncollinear ground state in Sr 2 IrO 4 and Sr 3 Ir 2 O 7 3 A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 2 / 29

anisotropic exchange interaction i � = j S T for general bi-linear pair interaction between spins: H = � i J ij S j J = J + + J − where J αβ is a real 3 × 3 matrix: J + = ( J + J T ) / 2 J − = ( J − J T ) / 2 with symmetric and antisymmetric αβ = 0 , J − no spin-orbit coupling (SOC) ( J + αα = J , J + αβ = 0 )   J 0 0  ; J = 0 J 0 H = J S i · S j  0 0 J isotropic Heisenberg weak SOC, no inversion ( J − xy = D � = 0 )   J D 0  ; J = − D J 0 H = J S i · S j + D ij · [ S i × S j ]  0 0 J antisymmetric Dzyaloshinsky-Moriya zz = J + K , J + strong SOC (e.g., J + xx = J + yy = J , J + αβ � = 0 ,. . . )   J 0 0  ; H = J S i · S j + KS z i S z J = 0 J 0  j 0 0 J + K A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 3 / 29

cubic crystal field + spin-orbit coupling (SOC) split t 2 g states into a Γ 8 ( j eff =3 / 2 ; d 3 / 2 + d 5 / 2 ) quartet:  � � � 1 d yz χ ± 1 2 ± id zx χ ± 1  3z 2 -1 2 χ Γ 8 = 2 Γ 8 � � � x 2 -y 2 j=5/2 1 2 d xy χ ∓ 1 2 ± d yz χ ± 1 2 + id zx χ ± 1  6 2 Γ 6 j eff =1/2 and a Γ 6 ( j eff = 1 / 2 ; pure d 5 / 2 ) doublet: � � � 1 χ Γ 6 = d xy χ ∓ 1 2 ∓ d yz χ ± 1 2 + id zx χ ± 1 xz, yz 3 xy 2 j=3/2 Γ 8 j eff =3/2 d 3 / 2 and d 5 / 2 = + isospin up spin up, l z =0 spin down, l z =1 j eff =1/2 half-filled Ir 4+ 5 d 5 ion in octahedral environment: j eff =3/2 completely filled Mott insulator already for moderate U j eff = ± 1 / 2 splitting is caused by U instead of J H A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 4 / 29

j eff =1/2 magnetism week ending P H Y S I C A L R E V I E W L E T T E R S PRL 102, 017205 (2009) 9 JANUARY 2009 Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models G. Jackeli 1, * and G. Khaliullin 1 1 Max-Planck-Institut fu ¨r Festko ¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany (Received 21 August 2008; published 6 January 2009) We study the magnetic interactions in Mott-Hubbard systems with partially filled t 2 g levels and with strong spin-orbit coupling. The latter entangles the spin and orbital spaces, and leads to a rich variety of the low energy Hamiltonians that extrapolate from the Heisenberg to a quantum compass model depending on the lattice geometry. This gives way to ‘‘engineer’’ in such Mott insulators an exactly solvable spin model by Kitaev relevant for quantum computation. We, finally, explain ‘‘weak’’ ferro- magnetism, with an anomalously large ferromagnetic moment, in Sr 2 IrO 4 . corner-sharing octahedra (a): (a) (b) Heisenberg + weak pseudo-dipolar interaction 180 o o 90 H = J 1 S i · S j + J 2 ( S i · r ij )( r ij · S j ) p z additional anisotropic terms if φ � = 180 ◦ yz xz xy p y xy H = J 1 S i · S j + J z S z i S z j + D · [ S i × S j ] xz yz are responsible for weak FM in Sr 2 IrO 4 xz p z xz p z A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 5 / 29

j eff =1/2 magnetism week ending P H Y S I C A L R E V I E W L E T T E R S PRL 102, 017205 (2009) 9 JANUARY 2009 Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models G. Jackeli 1, * and G. Khaliullin 1 1 Max-Planck-Institut fu ¨r Festko ¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany (Received 21 August 2008; published 6 January 2009) We study the magnetic interactions in Mott-Hubbard systems with partially filled t 2 g levels and with strong spin-orbit coupling. The latter entangles the spin and orbital spaces, and leads to a rich variety of the low energy Hamiltonians that extrapolate from the Heisenberg to a quantum compass model depending on the lattice geometry. This gives way to ‘‘engineer’’ in such Mott insulators an exactly solvable spin model by Kitaev relevant for quantum computation. We, finally, explain ‘‘weak’’ ferro- magnetism, with an anomalously large ferromagnetic moment, in Sr 2 IrO 4 . edge-sharing octahedra (b): j eff =1/2 hoppings via O 1 p z and O 2 p z cancel out (a) (b) 180 o o 90 isotropic superexchange J is suppressed p z strongly anisotropic interaction K αβ yz xz xy p y xy Kitaev-Heisenberg model: H HK = K αβ S γ i S γ xz yz j + J S i · S j xz p z xz p z with exotic spin-liquid ground state Can one get correct ground state and estimate J from LDA + U calculations? A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 5 / 29

back to U 1234 Ir d occupation matrix n σσ ′ mm ′ has off-diagonal in spin terms ( n σ − σ mm ′ � = 0 ): Coulomb energy: 1 � E U � n σσ m 1 m 2 ( � m 1 m 3 | V ee | m 2 m 4 � − � m 1 m 3 | V ee | m 4 m 2 � ) n σσ = m 3 m 4 2 σ, { m } + n σσ m 1 m 2 � m 1 m 3 | V ee | m 2 m 4 � n − σ − σ m 3 m 4 − n σ − σ m 1 m 2 � m 1 m 3 | V ee | m 4 m 2 � n − σσ � m 3 m 4 σ, m -dependent potential: mm ′ = ∂ ( E U − E dc ) E dc = 1 2 UN ( N − 1) − 1 V σσ ′ � , 2 J N σσ ( N σσ − 1) ∂n σσ ′ mm ′ σ A. Liechtenstein, et al PRB 52 , R5467 (1995), AY, et al PRB 67 , 155103 (2003), . . . rotationally invariant LDA+ U +SOC split j eff = 1 / 2 states but does not change their wavefunction A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 6 / 29

how to estimate J ? to calculate the total (band) energy as a function of angle between spins 1 using spin-spiral calculations and/or constraining magnetization direction to map ε ( { φ } ) onto an appropriate Heisenberg+Kitaev+. . . model 2 spin-spiral calculations do not work with SOC tricky to impose constraints on magnetization direction in LDA+ U calculations but we can do: calculations for magnetic configurations constrained by symmetry − limited number of magnetic configurations − not all exchange parameters can be determined simultaneously A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 7 / 29

motivations: j eff =1/2 beauty 1 anisotropic exchange in honeycomb Na 2 IrO 3 2 noncollinear ground state in Sr 2 IrO 4 and Sr 3 Ir 2 O 7 3 A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 8 / 29

crystal structure Ir Na 1 Na 2,3 O 1 Z O 2 Y X Y X monoclinic C/ 2 m space group honeycomb Ir layers separated by triangular Na layers trigonally distorted IrO 6 octahedra; Ir 4+ d 5 with half-filled j eff =1/2 states S.K. Choi, et al PRL 108 , 127204 (2012), F. Ye, et al PRB 85 , 180403 (2012) A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 9 / 29

experimental magnetic structure zigzag order (c) ordered Ir moment 0.22 µ B F. Ye, et al PRB 85 , 180403 (2012) explanations: isotropic Heisenberg model with long-ranged interactions Kitaev-Heisenberg model Kitaev-Heisenberg model + additional anisotropic exchanges Γ , Γ ′ A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 10 / 29

diffuse magnetic x-ray scattering results LETTERS PUBLISHED ONLINE: 11 MAY 2015 | DOI: 10.1038/NPHYS3322 Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na 2 IrO 3 Sae Hwan Chun 1 , Jong-Woo Kim 2 , Jungho Kim 2 , H. Zheng 1 , Constantinos C. Stoumpos 1 , C. D. Malliakas 1 , J. F. Mitchell 1 , Kavita Mehlawat 3 , Yogesh Singh 3 , Y. Choi 2 , T. Gog 2 , A. Al-Zein 4 , M. Moretti Sala 4 , M. Krisch 4 , J. Chaloupka 5 , G. Jackeli 6,7 , G. Khaliullin 6 and B. J. Kim 6 * a Na 2 lrO 3 Ir O zig-zag magnetic order Ir moments lie in ac plane form angle Θ = 44 . 3 ◦ with a axis y -bond scattering intensities above T N are y z explained by strongly anisotropic z -bond Θ x exchange interactions b c a x -bond A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 11 / 29

relativistic bands for Na 2 IrO 3 0.5 j eff =1/2 0.0 Energy (eV) -0.5 -1.0 -1.5 Ir d 3/2 Ir d 5/2 -2.0 Γ X S Y Γ Z 0 2 4 j eff =3/2 2.0 j eff =3/2 Ir t 2 g bands are split into sub-bands due to j eff =1/2 DOS (1/eV/atom) formation of quasi-molecular orbitals (MO) eg 1.5 eg I. Mazin, et al PRL 109 , 197201 (2012) 1.0 highest MO are coupled by SOC 0.5 dominant contribution of Ir d 5 / 2 states to 0.0 bands crossing E F ⇒ ∼ j eff = 1 / 2 states -2 -1 0 1 2 3 4 Energy (eV) A. Yaresko (MPI FKF) anisotropic magnetic interactions in iridates ICTP, 18.10.16 12 / 29