ferromagnetism of lacoo3 · ferromagnetism of lacoo 3 andrii sotnikov1 & jan kune s2 1 akhiezer...
TRANSCRIPT
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)
Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 2 / 11
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
Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 3 / 11
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!
Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 4 / 11
{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...
Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 5 / 11
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
Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 6 / 11
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)
Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 7 / 11
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!
Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 8 / 11
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:
Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 9 / 11
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
Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 10 / 11
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!
Andrii Sotnikov (Kharkiv, Ukraine) Ferromagnetism of LaCoO3 Stat.Phys.2019 11 / 11