Magnetism and doping effects in spin-orbit coupled Mott insulators

Max Planck Institute for Solid State Research, Stuttgart

Giniyat Khaliullin

Witczak-Krempa, Chen, Y-B.Kim, Balents (Annu. Rev. 2014)

Multiorbital Hubbard: U vs λ phase diagram

spin-orbit coupling λ

corr

elat

ion

U

spin-orbit coupling

S p i n – o r b i t a l a l g e b r a

Fe r m i o n i c s t a t i s t i c s

next 40 min: stay in this area

corr

elat

ion

weak

strong

Spin-orbit coupled Mott insulators: „unconventional“ magnetism and doping effects

spin-orbit coupling

corr

elat

ion

Spin-orbit coupled Mott insulators: from 3d-Co to 5d-Ir

Ti Co Ru Rh Os Ir

Sr2IrO4

CoO2

Ca2RuO4

KOs2O6

3d 4d 5d

La2CuO4

spin-orbit quenched

spin-orbit coupling

corr

elat

ion Sr2IrO4

CoO2

Ca2RuO4

KOs2O6

D o

p i

n g

D o

p i

n g

D o

p i

n g

Fe r m i o n i c s t a t i s t i c s

„unconventional“ magnetism „unconventional“ metal & SC?

D O P I N G:

La2CuO4

Jahn-Teller Spin-orbit

Lifting the orbital degeneracy

moment J=S+L

trigonal field

real orbital L=0

complex L=1

|xy + yz + zx

λ

SL

quadrupole spin

Spin ½ algebra pseudospin spin

t2g t2g

g= 2 g= -2

reality is often „in-between“

coplanar, single-Q

H = (SS)

Magnetism Jahn-Teller

S=1/2 Spin-orbit

J=1/2

GKh (PTPS, 2005)

large magnetic unit cell non-coplanar, multi-Q

hosts spin vortex

H= (SS) + Ising(x,y,z)

yy

xx zz

coplanar, single-Q

H = (SS)

Magnetism Jahn-Teller

S=1/2 Spin-orbit

J=1/2

GKh (PTPS, 2005)

large magnetic unit cell non-coplanar, multi-Q

hosts spin vortex

H= (SS) + Ising(x,y,z)

yy

xx zz

JT vs SO incommensurate orderings

SC pairing Jahn-Teller Spin-orbit

GKh, Koshibae, Maekawa (PRL 2004) GKh (PTPS, 2005)

Spin-singlet d-wave SC Baskaran (2003) Kumar, Shastry (2003) Ogata (2003) P. Lee et al.(2004)

Pseudospin-triplet p-wave SC

J=1/2 S=1/2

nondegenerate

-

Knight-shift finite in all directions

SC pairing Jahn-Teller Spin-orbit

Nax(Co,Rh,Ir)O2

GKh, Koshibae, Maekawa (PRL 2004) GKh (PTPS, 2005)

Spin-singlet d-wave SC Baskaran (2003) Kumar, Shastry (2003) Ogata (2003) P. Lee et al.(2004)

Pseudospin-triplet p-wave SC

J=1/2 S=1/2

Orbital moment L has a „shape“:

The origin of „unconventionality“: ORBITAL magnetism

Lz=1 z(x+iy)

c

a

Lx=1 x(y+iz)

b

Ly=1 y(z+ix)

Hopping amplitude:

nontrivial band topology Shitade et al. (2009)

A B complex

(no inversion)

A B real

(inversion)

Orbital moment L interactions:

ORBITAL MAGNETISM

c

a b

1) non-Heisenberg 2) bond-dependent

unconventional models: bond-dependent Ising, biquadratic…

L-moment shape links magnetism and chemical bonding

Orbital moment L interactions:

ORBITAL MAGNETISM

c

a b

1) non-Heisenberg 2) bond-dependent

„orbital frustration“

H

GKh & Okamoto (PRL 2002)

Simple cubic lattice: Qx

Qy Qz

non-coplanar multi-Q no LRO at finite T

?

Pseudospin J=S+L inherits bond-dependent and frustrated

nature of orbitals

„unconventional“ magnetism

L S J

spin-nematics, multipolar order, spin-liquids…

Spin-orbit multiplels of TM-ions

GS-degeneracy:

Co,Rh,Ir

Nb,Mo,Re Ru,Re,Os

Mo,Re,Os

„pseudospin“

Kramers: dipole, octupole,… non-Kramers: quadrupole,…

large J multipoles: spin nematic, „hidden“ order quantum spin „nonmagnetic“

Abragam & Bleaney (1970)

„The old“ pseudospin J=1/2

Two branches split by spin-orbit: magnon & exciton NaxCoO2 ?

GKh et al. (2004,2005)

„Heisenberg“ magnon

exciton

3d Co-fluoride, neutron scattering

M a g n o n

E x c i t o n 600 meV

40 meV 3d

5d

× 10

K2CoF3

from 3d Co to 5d Ir B.J.Kim, Takagi,…

Holden et al. (1971) neutron scatteting

Kim et al. (2012) RIXS data

Sr2IrO4

spin one-half 2D Mott systems similar to cuprates ?

d1 d5 d7 d9

Ti ...

t2g electron eg electron

Ni …

eg hole

Cu

t2g hole

Co …

JAF=130 meV La2CuO4

d1 d5 d7 d9

t2g electron eg electron eg hole t2g hole

NaxCoO2 triangular lattice small J, low Tc

JAF=130 meV La2CuO4

spin one-half 2D Mott systems similar to cuprates ?

superlattices …/NiO2/…

?

Cu Ni Co

spin one-half 2D Mott systems similar to cuprates

d1 d5 d7 d9

t2g electron eg electron eg hole

Cu

t2g hole

Co Ir

Sr2IrO4

Cuprate-like magnetism and SC?

TN~240 K

(Cava, Cao,…) TN~320 K

JAF=130 meV La2CuO4

Layered “213” iridate Na2IrO3

honeycomb lattice planes

edge-shared octahedra 90° Ir-O-Ir bonds

…similar to NaxCoO2

spin one-half 2D Mott systems similar to cuprates

d1 d5 d7 d9

t2g electron eg electron eg hole

Cu

t2g hole

Ir

La2CuO4 Sr2IrO4 Na2IrO3

t2g eg „orbital chemistry“ spin-orbit coupling

Jackeli, GKh (2009) Chaloupka, Jackeli, GKh (2010, 2013)

J=1/2 magnetism in iridates: theory

Two-parameter Hamiltonian = Ising(x,y,z) + Heisenberg:

dominant in 180-bonding (Sr2IrO4 perovskite)

dominant in 90-bonding (honeycomb Na2IrO3)

= „cuprate“ model high-Tc SC ?

= Kitaev model (2006) „Majorana world“ ?

Jackeli, GKh (2009) Chaloupka, Jackeli, GKh (2010, 2013)

J=1/2 magnetism in iridates: theory

Two-parameter Hamiltonian = Ising(x,y,z) + Heisenberg:

dominant in 180-bonding (Sr2IrO4 perovskite)

dominant in 90-bonding (honeycomb Na2IrO3)

= „cuprate“ model high-Tc SC ?

= Kitaev model (2006) „Majorana world“ ?

Kim et al.(2012)

Sr2IrO4 TN~240 K

Coldea et al.(2001)

La2CuO4 TN~320 K

?

?

-theory predicts „nearly“ Heisenberg AF

Sr2IrO4

Pseudospin ½ in perovskites:

Fermiology of electron doped Sr2IrO4

B.J. Kim et al. (Science 2014)

„Fermi-arcs“ at low doping

Potassium K overlayer

„normal“ FS

B.J. Kim et al. (Science 2014)

„Fermi-arcs“ at low doping

Pseudogap opens at low T

…and closes at 110 K

„normal“ FS

T-dependent „pseudogap“ in Sr2IrO4

many-body effect !

Sr2IrO4 magnetism, fermiology & lattice: same as in

superconductivity?

SUPER! GREAT!

„…find 10 differences…“ Sr2IrO4 La2CuO4

La2CuO4

Current status: no definite answer

Jackeli, GKh (2009) Chaloupka, Jackeli, GKh (2010, 2013)

J=1/2 magnetism in iridates: theory

Two-parameter Hamiltonian = Ising(x,y,z) + Heisenberg:

dominant in 180-bonding (Sr2IrO4 perovskite)

dominant in 90-bonding (honeycomb Na2IrO3)

= „cuprate“ model high-Tc SC ?

= Kitaev model (2006) „Majorana world“ ?

The Kitaev model

Exactly solvable

Low-energy excitations: free Majorana fermions

Short-range RVB, large spin gap

Dirac cones EF

SxSx SySy

SzSz

Sα = c fα

Itinerant free band (uncharged)

local & high-energy

Honeycomb lattice theory: K should be dominant but J is finite

Na2IrO3

What is in between ? -If „some liquid“ is still left ?

Kitaev model Heisenberg model

order liquid

Chaloupka, Jackeli, GKh (2010)

Spins, magnons Majorana land

Quantum phase transition: spin fractionalization

spin-orbit coupling

J K

YES!

pseudospin spin

Na2IrO3

AM order TN~15K

(1) Mag. bandwidth: 40 meV~30 TN (Gretarrson et al.)

(3) SW gap is small ~1 meV (Coldea et al.)

(2) Intense q=0 scattering (Gretarrson et al.)

Exp.data

(2) non-Heisenberg (3) C3 symmetric

(1) strongly frustrated

Interactions:

Kitaev-Heisenberg K-J model with large K > J makes all three points „for free“

- Kitaev term seems to be dominant - Other terms have yet to be sorted out

(current status) Data collected so far suggests that

Na2IrO3

2014 (H.-Y. Kee et al. van den Brink et al. Imada et al.)

measure and fit, measure and fit…

most wanted: single-crystal S(q,w) data

-Did you lose your Majoranas here?

Na2IrO3

The streetlight effect

-No, but the light is much better over here !

New candidate: RuCl3 Plumb et al, 2014

honeycomb plane

„dreamland“

J=1/2 d5 Co, Rh, Ir

„nonmagnetic“ Mott insulators

…farther away from the streetlight

L S

L + S = 0

d4 Ru, Os,..

competing

singlets magnetic LRO

QCP

spin-orbit driven magnetic QCP J=0 physics:

singlet

triplet Jex λ λ

d4 d4

(iii) Condensation of spin-orbit exciton

magnetic LRO

d4 ion: Van-Vleck magnetism

(i) There are no „pre-existing“ moments

L S

M=2S-L AFM exci

ton

gap

exchange interaction

J=0

J=1 λ (ii) J=0 to J=1 transition: off-diagonal magnetic moment M=2S-L

(„spin-orbit exciton“) 0

Singlet-triplet examples

QCP

para magnetic

J

(C) eg orbital FeSc2S4 (Chen, Balents, Schnyder, 2009)

(D) Spin-state-crossover in Fe-based SC (Chaloupka & GKh, 2013)

(A) Weakly coupled dimers

(B) 4f Pr compounds (broad literature since 1970‘s)

Bilayer AF

(Nat.Phys. 2009)

birthplace for „unconventional“ physics

d4 : Intra-ionic „dimer“ made of S and L

λ S L

1) energetic (~100 meV) 2) generic (any lattice)

QUANTUM CRITICALITY

GKh (2013)

Sachdev, Keimer, Phys.Today, 2011

Inter-ionic dimers

-small energies -special geometry

LRO moment:

cond.density

distance from QCP

M

AFM (triplon gas) PM

J QCP

spin-orbit exchange

J

S=1 boson

d4 Mott insulator: singlet-triplet model (180°) (perovskites)

GKh (2013)

Excitations: 1. The amplitude mode changing the angle between S & L

2. The phase modes in-phase rotation of S & L

S L Goldstone

„Higgs“ S L

1 2

k

ω

Bond-dependent „xy-model“:

c-bond exchange:

hopping pair-generation

(triangular, honeycomb, kagome…)

x and y type bosons only involved

GKh (2013)

„Heisenberg“ „Kitaev“

NB: T is hard-core boson, not spin!

BOND-SELECTIVE boson interactions:

Honeycomb lattice: dimensionality reduction

Each flavor Tx, Ty, Tz has its own zigzag to move along 1D dispersion:

J=Jcrit :

VBS? zigzag order? spin nematic? …

zero-energy lines

unusual singlet-triplet model, yet to be solved…

Spin-orbit driven magnetic QCP

Ruthenates Ca2RuO4? Li2RuO3? or elsewhere?

Van Vleck-type d4 Mott insulators:

d4 d5 „triplon gas“ or AF S=1/2 AF

J=0 singlet

J=1 triplet

J=1/2 doublet

Ru4+ Os4+ Ir 5+

Ru3+ Os3+ Ir 4+

Doping of „nonmagnetic“ d4 Mott insulators

doping

electron doping

λ

d4 d5 PM or AF S=1/2 AF

singlet

triplet

doublet

doping doping

λ

doping

J

AF

doping

J

AF Para

Naive expectation

Doping of „nonmagnetic“ d4 Mott insulators Chaloupka & GKh (unpublished)

Doping of „nonmagnetic“ Mott insulators

d4 S=0

S=1

d5

S=1/2

Quantum mixture of S=0, S=1/2, S=1 objects

Hopping generates S=1 states („vacuum polarization“): t f+ifj (T+-T) - mixes triplons with Stoner continuum Stoner triplon

- activates double-exchange FM

Triplon mixes-up with Stoner continuum, q=0 paramagnon emerges

Paramagnon softens, its intensity increases with hopping t

„bare“ Stoner

paramagnon Van-Vleck exciton

Phase diagram: doping x vs exchange J

AF

Para

FM FM

Para

1.  Doping supresses AF (expected)

2.  Doping induces FM (new)

3.  Larger t wider FM area

AF

„triplon gas“

Phase diagram: doping x vs hopping t

AF

FM

„nonmagnetic“ metal

„nearly-FM“ metal

AF --- nearly AF liquid ---- triplon gas ---- nearly FM --- FM

…emerging out of the „nonmagnetic J=0“ state

„Paramagnon-glue“: triplet or singlet SC-pairing

AF

FM

...undecided…

FM-paramagnons

p-SC

Doping of „nonmagnetic“ d4 Mott insulators

t2g4 ions:

Mott insulator to start with, better 2D Spin-orbit comparable with hopping t

AF

FM

FM-paramagnons Ca2RuO4 ? p-SC

Doping of „nonmagnetic“ d4 Mott insulators

Os

Ir

Re

Ru

Two hopping paths: positive quantum interference

Path A: t/3 Path B: t/3

A + B = (2/3) t Net hopping: large, spin-isotropic

J (SiSj) Heisenberg coupling as in cuprates

Corner-shared octahedra (orbital-diagonal hopping)

teff = A + B = 0 Net hopping amplitude is zero!

Edge-shared octahedra (orbital non-diagonal hopping)

A = iσ t/3 B = - iσ t/3

Usual Heisenberg interaction is suppressed

GKh, Koshibae, Maekawa (2004) GKh (2005) Jackeli & GKh (2009) Shitade et al. (2009)

Two hopping paths: negative interference

…and look for J=1/2 K-J model on other lattices

Pseudospin AND geometrical frustration:

- amplify the chances to find exotic states !

GKh (PTPS, 2005)

spin vortex multi-Q order

(K=-J>0)

Rousochatzakis et al. (condmat, 2012)

Kimchi & Vishwanath (PRB, 2014)

„hidden“ Goldstone

Map H(S,L) onto singlet-triplet model

Spin-orbital exchange Hamiltonian

interchange (x y)j

t2g orbital degeneracy: unquenched L=1

S=1, no orbital degeneracy

J=0

J=1 λ

J=2

2λ

nonmagnetic, J=S+L=0 „Van Vleck-type ion“

Δ

Jahn-Teller Spin-orbit (SL)

d4 : Jahn-Teller vs spin-orbit

xy

xz yz

LDA+U result for Ca2RuO4