Mott insulators with strong spin-orbit coupling

Max Planck Institute for Solid State Research, Stuttgart

Giniyat Khaliullin

motivated by:

Sr2IrO4 -s=1/2, perovskite 214-str.

-s=1/2, perovskite 214-str.

-s=1/2, honeycomb lattice

Sr2VO4

Na2IrO3

LS driven unusual ground states & excitations

spin one-half quasi 2D Mott systems

Kitaev model physics in (Li/Na)2IrO3 (?)

magnetically hidden order in Sr2VO4

cuprate-like AF & magnons in Sr2IrO4

Outline:

Mott Insulators with t2g orbital degeneracy

eg

t2g

dz2 dx

2-y

2

dxy dyz dxz 3x orbital degeneracy

MT

O2-

d x5

Sr2IrO4 Na2IrO3 Sr2VO4

d1, t2g electron, S=1/2

d5, t2g hole, S=1/2

Like 2D cuprates but: orbital angular momentum L=1

d-orbitals

“Orbital physics” in TMO

d

charge

orbital

spin

structural / magnetic transitions: orbital order, spin structure

metal / insul. transition, doping: orbit-selective MIT, orbital polarons

oxide heterostructures&interfaces: novel phases via orbital reconstruction

exotic quantum states in oxides: spin and orbital liquids

spin-state crossover (Co,Fe…): orbital repopulation, magnetic collapse

„multi-dimensional“ d-electron in oxides

Three different couplings in spin-orbital systems

Orbital-Lattice coupling

H = ECF + JSE + λso

Spin-Orbital superexchange

spin-orbit coupling

J Exchange interaction:

EJT

Spin-orbit coupling: λ Jahn-Teller coupling: EJT

Three different regimes in spin-orbital systems

Goodenough-Kanamori spin-exchange rules

AF Ferro

H=J(SiSj)

J Exchange interaction: Spin-orbit coupling: λ Jahn-Teller coupling: EJT

Three different regimes in spin-orbital systems

AF Ferro

permutation operator Pij =

P(spin) P(orb)

SU(4) spin-orbital fluctuations

J Exchange interaction: Spin-orbit coupling: λ Jahn-Teller coupling: EJT

Three different regimes in spin-orbital systems

?

„quantum orbital physics“

orbital frustration

higher D more frustration

J Exchange interaction: Spin-orbit coupling: λ Jahn-Teller coupling: EJT

Three different regimes in spin-orbital systems

bond directional nature of orbital interactions = frustration

Orbital anisotropy and frustration are directly translated into magnetic sector

…new route to exotic Hamiltonians & unusual phases

Relativistic spin-orbit coupling

L

S

orbital angular momentum

spin-orbit coupling: H= λ(LS)

3d

4d

5d

λ(Ir4+) = 0.4 eV

λ(Ti3+) = 0.02 eV

Strong SO coupling

Low-spin Ir4+

Single t2g hole: s=1/2, l=1

λ~ 0.4 eV, unquenched L moment

Quantum number J =L+S is formed

J=3/2

J=1/2

t2g

spin-orbit entangled d-electron

weak LS-coupling strong LS-coupling: L+S=Jeff

of „cubic“ shape protected from JT

complex wave-function

carriers both spin-directions coherently

phase factor / quantum interference

-collects phase factor (spin dependent) -quantum interference between A, B, … depending on hopping geometry

A

B

i j

…going from site-i to j :

nontrivial topology of bands & interactions

An example: consider two types of bonding geometry

H= J ( ) H= -J

Strong AF-Heisenberg Ferromagnetic Ising, z-axis: out-of-plane

perovskite lattices triangular, honeycomb, pyrochlore,..

x

y

G.Jackeli, G.Kh, PRL 2009

Sr2IrO4 (t2g analog of high-Tc perovskite La2CuO4)

Iridium oxides, 5d(t2g5)

Na2IrO3 (depleted ABO2 ; Ir ions on a honeycomb lattice)

180° bonding

90° bonding

Crystal structure of Sr2IrO4

Octahedra elongated along c-axis Ir-Oab=1.98A Ir-Oc=2.06A

Staggered rotation of octahedra around c-axis by α∼11ο

Magnetic properties of Sr2IrO4

Magnetization data: Cao et al., PRB ‘98

Anomalously large “weak” FM MFM =0.14µB [La2CuO4: 0.2 x10-2 µB]

φ α

AFM, large canting angle φ∼α Spins rigidly follow rotation of octahedra

1

2 Ferromagnetic?

Two options:

Exchange Hamiltonian: 1800-bonds

Active orbitals and their overlap

Isospin Hamiltonian:

Predominantly of Heisenberg form. Pseudo dipolar anisotropy: J2/J1~JH/U Anisotropy solely due to Hund’s coupling

Microscopic Hamiltonian of Sr2IrO4

Dominant interactions

X ~ Y ~

φ X ~

Y ~

Rotated basis: isotropic Heisenberg:

bond angle

Spins parallel to Ir-O bonds: strong spin-lattice coupling

spin angle

Canting angle vs tetragonal distortion

Magnetic Hamiltonian including Hund’s coupling

Phase diagram

Sr2IrO4

Г1 changes sign at large elongation of octahedra spin-flop transition

Energy scale in Sr2IrO4: TN=240 K J~ 50 meV

tetragonal orthorhombic

Sr2IrO4 resonant (elastic) x-ray scattering B.J.Kim et al., Science 2009

exper. confirmation

Formation of isospin 1/2 Kramers doublet (selection rules, L3-edge only observed) Magnetic structure: strongly canted AF

Theoretical predictions for Sr2IrO4

Spin-wave spectrum: Large out-of-plane gap of classical origin.

Small in-plane gap of quantum origin.

In-plane compression -> spin-flop transition

RIXS: spin-orbit J=1/2 to 3/2 peak about 0.6 eV

L. Ament, M. Daghofer, J. van den Brink, G.Kh. (PRB 2011; cond-mat 2011)

Calculated RIXS intensity (magnetic spectra)

J=1/2 magnons

J=3/2 sector

hard x-rays, Ir L3 edge: entire BZ is probed

J=3/2

J=1/2

RIXS spectra in Sr2IrO4 B.J.Kim et al. (cond-mat 2011)

Magnons (J=1/2 sector) measured by RIXS

RIXS operator:

J=1/2 to 3/2 exciton moves like a hole in t-J model

exciton magnon use SCBA

magnons (charge density unaffected )

spin-orbit exciton (both orbital shape & spin)

Isospin ½, 2D AF, broad magnon band, plus higher energy magnetic mode

Experimental challenge: -doping of spin-orbit Mott insulators: SC? -unusual proximity effects?

Sr2IrO4 summary (exp & theory):

Iridates with 900-exchange bonds

Two active orbitals/oxygen ions – two different paths

Isospin Hamiltonian

Quantum Compass Model destructive interference between two paths: Heisenberg term vanishes exactly

each bond has its own Ising easy-axis

Layered Iridates A2IrO3 (A=Li,Na) Honeycomb lattice planes

Ir x

z

y

Kramers doublets interact as in Kitaev Model A. Kitaev Ann. Phys’06

bond-dependent Ising axes: FRUSTRATION

90°-bonding

“Engineering” the Kitaev model

Kitaev model

yy xx zz

Topological degeneracy Relevant for Quantum computation

Solid state realization? Li2IrO3 , Na2IrO3

Na Ir

O

Cold atoms, optical lattices? (Demler et al.)

-Magn. order ~10 K -Intrinsic? -Impurity effect?

The Kitaev model

Exactly solvable

Emergent Majorana fermions

Short-range RVB spin liquid

GS degeneracy: depends on topology

yy xx

zz

The Kitaev’s solution

yy xx

zz

Free Majorana fermions Dirac spectra like in graphene

Introduce four Majorana fermions:

Spin:

where commute with H and are thus constants

Ground state:

EF

Full Hamiltonian including 2Δ charge-transfer

oxygen

spin disordered conventional AF

(i)

(ii)

Final result

oxygen

Kitaev (similar to U-process)

Heisenberg (also direct dd)

Heisenberg-Kitaev model

Heisenberg AM Kitaev SL

Honeycomb lattice

One more exact reference point:

Chaloupka/Jackeli/GKh, PRL 2010

4 sublattices, spin rotations

H rotated

For arbitrary

simple ferromagnet

Original spin basis: stripy AF (no zero-point fluctuations!)

Three phases for A2IrO3

FM in a rotated frame

0.8 0.4

Chaloupka/Jackeli/GKh, PRL 2010

24-site cluster (exact): spin correlations

NN

NNN NNNN

„Short-range RVB“

0.12…(exact)

spins, magnons Majorana land

Quantum phase transition: spin fractionalization

spin-orbit coupling

Doping of Kitaev-Heisenberg model

Mean-field RVB phase diagram (JH/JK=1/2)

T.Hyart, A.R.Wright, G.Kh, B.Rosenov (cond-mat 2011)

Spin-orbit insulators: d5 versus d1

d5 (hole) d1 (electron)

doublet,1/2 quartet,3/2

Co4+,Ir4+,… Ti3+,V4+,Nb4+…

H= α x Heisenberg + β x Kitaev H = ?

3λ/2

3/2 1/2

3λ/2

Magnetically Hidden Order in Sr2VO4

214 perovskite d1-electron analog of cuprates

Crystal structure of Sr2VO4

Elongated along c-axis V-Oab=1.91A V-Oc=1.94A

Zhou et al., PRL ‘07

V4+, 3d1 –an electron analog of La2CuO4 3d9

d1

xz/yz degeneracy: ideal

Phase transition: Isostructural, of first order

Zhou et al., PRL ‘07 tetragonal both below and above Ts

sharp increase of c/a

Phase transition in Sr2VO4

Zhou et al., PRL ‘07

Looks like (canted) AFM transition… However, no magnetic Bragg peaks have been detected

Theoretical proposals on Sr2VO4

(1) Imai et al., PRL ‘05

Orbital & spin order

(2) Jackeli & Ivanov, PRB’07

Spin-singlet VBS

Both states break translational symmetry (not observed) State 1: elastic magnetic Bragg peaks (not observed)

„All happy families are alike; each unhappy family is unhappy in its own way.“

-- Leo Tolstoy (Anna Karenina)

„All happy families are alike; each unhappy family is unhappy in its own way.“

-- Leo Tolstoy (Anna Karenina)

…no universal theory

…so are the oxide families; each has its own skeleton in the cupboard

d1

SO coupling

Tetragonal field: Low energy quadruplet remains active Spin-orbit: Quadruplet split into Kramers doublets

G.Jackeli & GKh, PRL 2009:

Ideal xz/yz degeneracy & spin-orbit coupling

Quadruplet: Two Kramers doublets

The ground state doublet:

First excited level:

,

nonmagnetic Mspin=0 Morbital=0

magnetic

SE-interaction between quadruplets

...obtained by projecting t2g spin-orbital model onto the quadruplet subspace

Isospins Inter-doublet transitions

s-o splitting between doublets J=t2/U

(about 10 meV in LaVO3)

The ground state

Low energy doublet is stabilized. Charge density of axial symmetry. Enhances c/a ratio.

Staggered order of isospins and of the chirality of wave-functions.

In-plane isospin order is selected by Hund´s coupling:

+ + GS(J) = GS(L-S) =

Octupolar order

x

y

y

x

Color map for real spin density distribution Sx=+ (blue) Sx=- (red)

Nature of the ground state ordering

Isospin order in terms of physical spin & orbital moments

Both spin and orbital moments are zero on every site. (No Bragg peak. Drop in magnetic susceptibility.) No quadrupole ordering: tetragonal symmetry respected

Ordering of magnetic octupoles

Elementary & Magnetic Excitations

Pseudoorbitals

Isospins („octupon“)

magnon

(Interdoublet My) Continuum, Mx

octupolar Bragg peak (x-rays)

Octupoles in TM-oxides?

Octupolar Bragg peaks and “octupons” in Sr2VO4 (resonant x-ray scattering) Unconventional magnetic excitation spectrum (neutron scattering) Unquenched spin-orbit coupling: large LS~0.5 (spin-resolved photoemission)

Summary

Mott insulators with strong spin-orbit coupling

unusual symmetries and orderings

Spin & Orbital subspaces entangled

Open problems: - Doping of spin-orbit Mott insulators - 3d electron octupolar orderings & dynamics - Heisenberg-Kitaev model: phase transitions