Ferromagnetism of LaCoO3 · Ferromagnetism of LaCoO 3 Andrii Sotnikov1 & Jan Kune s2 1 Akhiezer Institute for Theoretical Physics, NSC KIPT, Kharkiv, Ukraine 2 Institute for Solid

Stat.Phys.2019: July 3–6, Lviv, Ukraine

Ferromagnetism of LaCoO3

Andrii Sotnikov 1 & Jan Kunes 2

1 Akhiezer Institute for Theoretical Physics, NSC KIPT, Kharkiv, Ukraine

2 Institute for Solid State Physics, TU Wien, Vienna, Austria

Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 1 / 11

Intro: low-T physics of LaCoO3 (cubic structure)

The compound is studied since 1960’s, both experimentally and theoretically

La Co O

Ene

rgy 3+Co

Free ion

Orbital structure Relevant spin states

"LS"(low-spin)

"IS"(intermed.)

"HS"(high-spin)

LaCoO3Tanabe-Sugano diagram

En

erg

y (

eV

)

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

J (eV) 0.5 0.6 0.7 0.8 0.9 1

LS (1)

IS, T1g (9)

HS (15)

Tmetal (PM)

0K

~100K

~500K

ins.(PM)

ins.(NM)

Important facts:

X at low T : insulator, sizeable charge gap

X nonmagnetic −→ paramagnetic

X higher relevance of HS/IS among low-T excitations is still under debate

X “unusual” behavior at high magnetic fields (∼ 100 T)

X proximity to [A] HS–LS spin-state ordering, [B] BEC of IS (S = 1) excitons

X short-range ferromagnetic correlations (no long-range order)


Intro: low-T physics of LaCoO3 (stretched structure)

Relevant spin states

"LS"(low-spin)

"IS"(intermed.)

"HS"(high-spin)

Ene

rgy 3+Co

Free ion

LaCoO3film

La Co O

tensile strain

Orbital structure Magnetic ordering

T0K 94K

FMinsulator

PMinsulator

Important facts:

X insulator

X long-range ferromagnetic (FM) ordering with TC = 94 K

X magnetic structure with a propagation vector (1/4, -1/4, 1/4)

X g factor ≈ 0.7µB per Co atom

X controversial predictions from DFT+U approaches:

candidate ..–HS–LS–HS–LS–.. with magnetic moments on HS sitesAFM/FM alignment depends on U; the FM couplings are too small

?? 1/4 structure, TC , g factor and the physics behind


Intro: highly-dispersive IS, immobile HS (cubic structure)

LS state ≡ vacuumIS state ≡ exciton (electron+hole)

te

tt

ϵISϵHS

ϵLS

WIS

0

EWHS

LS IS

experiment + theory[R.-P. Wang, A. Hariki, et al., PRB 98, 035149 (2018)] IS(T1g)

IS(T2g)

HS

(a) RIXS (b) dens. of excit.

IS(T1g ) exciton:

+ =ISxzdx2-z2dxz

HS(T2g ) bi-exciton:

ISxz ISyz HSz

+ =X EHS < EIS ⇒ effective

on-site attraction of IS

X account forformation/decay of HSlowers its energy!


{IS,HS}-boson picture of the film LaCoO3

(d)

ISxz

ISyz

HSz

(c)

(e)

x

(f) (g)

ISxzISyzHSz

xy

z

0-1

-2-3

12

3-1

-1

0

0

1

1

-2

2

-3

3

ISxzISyzHSzLS

(a)

(b)

Q.: How we extract the hopping and interaction amplitudes for the real crystal?A.: Series of well-established and logically-consistent approximations...


Methodology

(iv)

(iii)

+O-2p

HS

+LS, IS

+3d5,3d7(ii)

(i)

px dx2-z2

pz dxz

pz

dyz

py

dy2-z2

(i) density functional theory (DFT)[wien2k]

(ii) Wannier projection to d-onlyFermi-Hubbard model, addinginteraction in the Slater form[wien2wannier, wannier90]

(iii) strong-coupling (t2/U)expansion in the d6 manifold[ED, 2nd order pert. theory]

⇒ effective Bose-Hubbard modeltreatment: exact diagonalization

————optional, but useful step:

(iv) reduction to the HS-only(spin-1/2) Ising model


level (iii): “isolated” HS states

ISxz

ISyz

HSz

(c)one IS state per line (length ∼ 10 sites),

known IS hopping amplitude t,

local attraction U is determined by EHS, EIS

(taken as tuneable parameter)

U/t0 2 4 6 8 10

HS:IS:

0nnnnn

den

sity

of

exci

tati

on

s

0

0.2

0.4

0.6

0.8

1

U/t0 2 4 6 8 10

E/t

−10

−8

−6

−4

−2

0

2D

Eat

EL

E1

3D

realistic regime: U/t ≈ 8

energy of HS is lowered by nnfluctuations

HS is surrounded by the IS “cloud”⇒ “dressed” state

Q.: What if we add excitations?A.: Rather not on the same line:(hard-core constraint → repulsion)

x

(g)


plaquettes with two “dressed” HS & their FM exchange

(e)

0

5

10

15

20

25

4 5 6 7 8 9 10

−70

−60

−50

−40

4 5 6 7 8 9 10

−80

−90

effective FM coupling on the diagonal: 4Jex ≈ −7 meV at U/t = 8(Jex < 0 results from the spin-resolved analysis)

diagonal ↑-↑ attraction V = 4|Jex| is comparable to Tc ≈ 94 K

the amplitude for the pair flip tp is of the same order

diagonal attraction & nn (nnn) repulsion along x , y ⇒ stripe formation!


decorating lattice with the “dressed” HS

HS

filli

ng

HS-AFM

SSO (HS-LS)

plaq.

0

0.5

1.0

0

pla

ne 1

pla

ne 2

pla

ne 3

pla

ne 4

3D:


supporting remarks: role of SOC; g -factor estimates

Spin-orbit coupling is analyzed in the atomic limit

Wei

gh

t

0

0.2

0.4

0.6

0.8

States

HSyHSx

HSz0

+1 -1 +1 -1

Wei

gh

t

0

0.2

0.4

0.6

0.8

0

+2

+1

0

+2

Wei

gh

t

0

0.2

0.4

0.6

0.8

0

-2

0

-2

-1

States

Wei

gh

t

0

0.2

0.4

0.6

0.8

-2

0

+2

+1 +1

States0

0.2

0.4

0.6

0.8

Wei

gh

t

+2

-2

-1 -1

En

erg

y (m

eV)

HSz-20

0

40

SOC strength0 1

HSx

HSy

20

main effects of SOC:

filters 3 out of 5 lowest spin states (j = 0,±1);

acts as a single-ion anisotropy term in spin space

adds a portion of HSx,y in the HSz set

g factors in thesubspace of HS withj = 0,±1:

Mr = µB(2sr + lr )

= µBgr jr

g‖ ≈ 2.73g⊥ ≈ 2.64(per HS atom)

Experiment:gCo ≈ 0.7± 0.1

gCo ≈1

4gHS


Summary / Acknowledgements

Brief∑

slides

FM in the insulating film LaCoO3 originates from the highly-fluctuativecharacter of excitations;

the lattice is filled by 1/4 of HS; these form diagonal stripes(agrees with exp.-observed propagation vector)

FM couplings on diagonals are of the same order as exp. TC

at nHS = 1/4, g factors fall close to exp. values

the low-energy physics of film LaCoO3 ⇔ the (extended) Ising model

Special thanks to:

X Jan Kunes (TU Wien);

X Kyo-Hoon Ahn (TU Wien);

X Juan F. Afonso (TU Wien);

X Atsushi Hariki (TU Wien);

X Ru-Pan Wang (Utrecht Uni);

X Frank de Groot (Utrecht Uni)

***

Thank you for your attention!


Ferromagnetism of LaCoO3 · Ferromagnetism of LaCoO 3 Andrii Sotnikov1 & Jan Kune s2 1 Akhiezer Institute for Theoretical Physics, NSC KIPT, Kharkiv, Ukraine 2 Institute for Solid

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