mott insulators with strong spin-orbi coupling -...
TRANSCRIPT
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
“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)
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
H rotated
For arbitrary
simple ferromagnet
Original spin basis: stripy AF (no zero-point fluctuations!)
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
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) =
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)