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Supplemental Information for Nature of the Magnetic Interactions in Sr3niiro6 Supplemental Information for Nature of the Magnetic Interactions in Sr3NiIrO6 Turan Birol Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA and Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA Kristjan Haule and David Vanderbilt Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA (Dated: October 7, 2018) METHODS 3 First principles density functional theory (DFT) cal- culations were performed using Vienna Ab Initio Sim- 2 ulation Package (VASP), which uses the projector aug- mented wave method [1{4]. PBEsol exchange correlation Ni functional [5] is used in conjunction with the DFT+U as introduced by Dudarev et al. [6]. The value of U has been chosen as U = 2 eV and U = 4 eV for Ir and Ni i Ir Ni Ir respectively. This choice of U gives the magnetic (spin 2 + orbital) moments of µIr ∼ 0:6µB and µNi ∼ 1:8µB, which are in reasonable agreement with µIr ∼ 0:5µB and Ni µNi ∼ 1:5µB observed in experiment [7]. Small variations in the values of U's give rise to quantitative changes, but the magnetic ground state does not change. A plane- Ir i wave basis cutoff energy of 500 eV, and a 8 × 8 × 8 k-point grid for the 22 atom primitive unit cell, which corresponds to one k-point per ∼ 0:02 2π=A,˚ is found to provide good convergence of the electronic properties. Spin-orbit coupling is taken into account in all calcula- tions unless otherwise stated. FIG. 1. (Color Online) Crystal structure of Sr3NiIrO6 consists of one-dimensional chains along the c axis that consist of face- CRYSTAL AND MAGNETIC STRUCTURE sharing NiO6 and IrO6 polyhedra. There is a 3-fold rotational symmetry axis along each chain, as well as 2-fold axes passing through the Ni sites, and inversion centers on the Ir sites. Sr3NiIrO6 crystallizes in the K4CdCl6 structure, which consists of parallel one-dimensional chains of alternating, face sharing NiO6 and IrO6 polyhedra (Fig. 1a and Table ble I and Fig. 1. Both of them have their d orbitals ¯ I) [8, 9]. Its space group is trigonal R3c, and the 3-fold split into 2+2+1. This splitting can be considered the rotation axis passes through the cations, parallel to the result of coexisting cubic and trigonal crystal fields on chains, along the crystallographic c axis (Fig. 1b). There each ion. The ratio of the trigonal to cubic crystal field are three 2-fold rotation axes perpendicular to the 3-fold strengths is expected to be larger for the Ni ion, because one, passing through the Ni cations, and inversion cen- it is on a trigonal prismatic site, whereas Ir is on a (only ters on the Ir cations. Ir ions are surrounded by oxygen slightly distorted) octahedral site [11, 12]. Using a cubic octahedra whereas Ni ions are at the center of oxygen point group as reference, we can consider a higher energy trigonal prisms. The octahedra and prisms are face shar- eg-like doublet, and three lower energy t2g-like orbitals ing with each other, which enables strong direct cation - which are split into 2+1 [13]. (Henceforth, we omit the cation direct interactions [10]. `-like' suffix and simply refer to eg and t2g states.) The relative energies of the t2g singlet and the doublet is not determined by symmetry. The t2g doublet transforms ORBITAL CONFIGURATIONS OF Ir4+ AND Ni2+ as the same irrep as the eg doublet, and hence can mix with it. (In the following, we ignore this mixing, which is The site symmetries of the Ir and Ni cations in not important for the current discussion.) A convenient Sr3NiIrO6 are 3¯ and 32 respectively, as shown in Ta- choice of cartesian basis is one with thez ^ along the crys- 2 Space Group: R3c¯ direction, which is the crystallographic c axis. In the Space Group Number: 167 absence of the trigonal crystal field, γ = 1, and these ˚ Lattice constants: a = 9:578 A states have exactly 33:3%¯ jAi = j3z2 −r2i character with c = 11:132 A˚ spin opposite to the rest of the state. This is very close Ion Wyckoff Position Site Symmetry 2 2 Sr 18e 1¯ to the first principles result of 36% j3z − r i character Ni 6a 32 for the t2g hole on Ir, showing that the trigonal field is Ir 6b 3¯ not strong enough to significantly change the orbital and O 36f 1 spin characteristics of the Ir ion. TABLE I. Details of the crystal structure of Sr3NiIrO6. Crys- tal structure data taken from Nguyen et al. [9]. DENSITY OF STATES OF NI tallographic c axis (chain direction), and thex ^ andy ^ on the a-b plane (Fig. 1c). With this axis choice, the three 2 2 2 2 t2g orbitals can be written as 3z z xy + x y + xz + yz | − ³| | − | | ´ ® ® ® ® ® 2 2 2.0 jAi = j3z − r i (1) M // z 1.5 ) 1 − V e i i 1 1 ( + 2 2 jE i = −p jxyi+ p jxzi+ p jyzi− p jx −y i (2) S O 1.0 D 3 6 6 3 l a i t r a P 0.5 i i 1 1 jE−i = +p jxyi− p jxzi+ p jyzi− p jx2 −y2i (3) 3 6 6 3 0.0 4 3 2 1 0 1 2 3 4 in terms of the cubic harmonics. 1.5 M // z Under a cubic crystal field, the spin-orbit coupling is 1.0 known to mix and split the t2g cubic harmonic orbitals to ) 1 0.5 − give rise to the higher energy pair of so called Jeff = 1=2 V e · states. For an octahedron that has corners along the B 0.0 µ ( ® z cartesian axes, these jJ1=2; "i and jJ1=2; #i states are often S 0.5 ­ written as 1.0 1 1.5 jJ1=2; "i = p (−|xy; #i − ijxz; "i + jyz; "i) (4) 4 3 2 1 0 1 2 3 4 3 Energy (eV) and 1 FIG. 2. (Color Online) (a) Densities of states of the Ni ion jJ1=2; #i = p (+jxy; "i + ijxz; #i + jyz; #i) : (5) projected onto the d orbitals in the FiM state with magnetic 3 moments along the z axis (magnetic ground state). (b) Energy These states exhibit spin-orbital entanglement in the resolved expectation value of the z component of spin, hSzi sense that the jxyi part has an opposite spin to the rest. for the d orbitals of the Ni ion in the FiM state with magnetic moments along the z axis. Similarly, with our choice of axes for the Ir atom (where the corners of the octahedra are along the h111i) and the ~ ~ 2+ trigonal crystal field, diagonalizing the L · S operator in The Ni cation has 8 electrons, and its t2g-like orbitals the t2g subspace gives the Jeff = 1=2 states as are completely filled. The unoccupied DOS between 1.5 and 2.0 eV's is that of the two holes in the half-filled p 1 + e -like orbitals, which have no j3z2 − r2i character as jJ1=2; "i = p iγjA; #i + 2jE ; "i (6) g jγj2 + 2 expected. and p 1 − jJ1=2; #i = iγjA; "i + 2jE ; #i (7) MAGNON SPECTRUM pjγj2 + 2 where we ignored the mixing with the eg orbitals. Here, In this section, we calculate the magnon spectrum of the magnetic moments of either state are along thez ^ Sr3NiIrO6 using the model 3 X Ir Ni Ni Ir Ni Ni Ir Ni Ni E = Jk Mn;z(Mn;z + Mn+1;z) + J? Mn;x(Mn;x + Mn+1;x) + Mn;y(Mn;y + Mn+1;y) (8) n Ir ^z with parameters from first principles to show that a Mn;z = −Rn (12) model without single ion anisotropy is sufficient to ex- plain the large gap in the magnon spectrum. We introduce new operators, Q^ and R^, for the pseudo- 1 M Ir = − R^+ + R^− = −R^ (13) spins of the two ions: n;x 2 n n x Ni ^z Mn;z = Qn (9) Ir 1 ^+ ^− ^ Mn;y = Rn − Rn = Ry (14) 1 2i M Ni = Q^+ + Q^− = Q^ (10) n;x 2 n n x Both Q^ and R^ satisfy the usual commutation relations such as [Q^ ; Q^ ] = i " Q^ . 1 i j ~ ijk k M Ni = Q^+ − Q^− = Q^ (11) The magnetic hamiltonian takes the form: n;y 2i n n y X n h ^z ^z ^z i h ^x ^x ^x ^y ^y ^y io H = −Jk Qn(Rn + Rn+1) + J? Qn(Rn + Rn+1) − Qn(Rn + Rn+1) (15) n ^z ^∓ ^∓ We now make the approximations that QnRm = µNiRm write these two equations as and R^z Q^∓ = µ Q^∓ This approximation is valid in the n m Ir m @ Q^+ Q^+ ordered state for small amplitude of excitations [14, 15]. i q = M q (22) ^− ^− It gives @t Rq Rq 1 iqc [R^−; H] = −2J µ R^− − J µ (Q^+ + Q^+ ) (16) −2JkµIr −J?(1 + e )µNi n k Ni n ? Ir n n−1 M = −iqc (23) ~ J?(1 + e )µIr 2JkµNi We define the magnon creation/annihilation operators 1 ^+ ^+ ^− ^− for the two branches via a matrix K that diagonalizes [Qn ; H] = +2JkµIrQn + J?µNi(Rn + Rn+1) (17) ~ M: Q^+ α^ Introducing the Fourier transformed operators q = K q (24) ^− ^y Rq βq ^+ X −iqn ^+ Qq = e Qn (18) which leads to n y y −!q,αα^q −1 α^q ^ = K MK ^ (25) !q,ββq βq ^− X −iqn ^− h i Rq = e Rn (19) P y ^y ^ Now H = !q,αα^ α^q + !q,ββ βq .
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