Theory of charge and spin ordering in the nickelates - Physics ...
Theory of charge and spin ordering in the nickelates - Physics ... Theory of charge and spin ordering in the nickelates - Physics ...
Theory of charge and spin ordering in the nickelates Leon Balents, KITP Cologne, 9/2010 $$
- Page 2 and 3: People Collaborators Sungbin Lee Th
- Page 4 and 5: Outline Introduction to Mott transi
- Page 6 and 7: Nickelates But... we should underst
- Page 8 and 9: Nickelates Ideally RNiO3, with R 3+
- Page 10 and 11: Charge order Alonso et al, 1999:
- Page 12 and 13: Valence skipping Mazin et al, 2007:
- Page 14 and 15: Naive expectations Rock-salt orderi
- Page 16 and 17: Hubbard Model A minimal model to co
- Page 18 and 19: Phase Diagram Which is more appropr
- Page 20 and 21: Itinerant Limit Ho ?
- Page 22 and 23: 3 2 1 1 0.5 1.0 1.5 2.0 2.5 3.0 Com
- Page 24 and 25: Comparisons FERMI SURFACES, ELECTRO
- Page 26 and 27: Susceptibility Susceptibility is pe
- Page 28 and 29: Outline Introduction to Mott transi
- Page 30 and 31: Landau Theory SDW order parameter S
- Page 32 and 33: Cubic symmetry Landau free energy F
- Page 34 and 35: SDW states Look along axis θ=0
- Page 36 and 37: Magnetic structures Refinements giv
- Page 38 and 39: Orthorhombicity Orthorhomic distort
- Page 40 and 41: Unequal moments orthorhombicity
- Page 42 and 43: Outline Introduction to Mott transi
- Page 44 and 45: Phase Diagram Which is more appropr
- Page 46 and 47: Strong Coupling Approach Consider p
- Page 48 and 49: Ordering of pairs Half of sites sho
- Page 50 and 51: Magnetism The spin of each Ni 2+ el
<strong>Theory</strong> <strong>of</strong> <strong>charge</strong> <strong>and</strong><br />
<strong>sp<strong>in</strong></strong> <strong>order<strong>in</strong>g</strong> <strong>in</strong> <strong>the</strong><br />
<strong>nickelates</strong><br />
Leon Balents, KITP<br />
Cologne, 9/2010<br />
$$
People<br />
Collaborators<br />
Sungb<strong>in</strong> Lee<br />
Thanks to:<br />
Susanne Stemmer<br />
Junwoo Son<br />
Jim Allen<br />
Ru Chen<br />
Dan Ouellette<br />
Andy Millis
Outl<strong>in</strong>e<br />
Introduction to Mott transitions <strong>and</strong> <strong>charge</strong>/<strong>sp<strong>in</strong></strong> order <strong>in</strong> <strong>the</strong><br />
RNiO3 <strong>nickelates</strong><br />
Hubbard model<br />
it<strong>in</strong>erant limit: <strong>sp<strong>in</strong></strong>-density wave<br />
L<strong>and</strong>au analysis <strong>of</strong> <strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
role <strong>of</strong> orthorhombicity<br />
Localized limit<br />
<strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
O<strong>the</strong>r tests for it<strong>in</strong>erant versus localized physics
Outl<strong>in</strong>e<br />
Introduction to Mott transitions <strong>and</strong> <strong>charge</strong>/<strong>sp<strong>in</strong></strong> order <strong>in</strong><br />
<strong>the</strong> RNiO3 <strong>nickelates</strong><br />
Hubbard model<br />
it<strong>in</strong>erant limit: <strong>sp<strong>in</strong></strong>-density wave<br />
L<strong>and</strong>au analysis <strong>of</strong> <strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
role <strong>of</strong> orthorhombicity<br />
Localized limit<br />
<strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
O<strong>the</strong>r tests for it<strong>in</strong>erant versus localized physics
ates rema<strong>in</strong><br />
temperature<br />
ds, <strong>the</strong> key<br />
ort high T c<br />
y, <strong>sp<strong>in</strong></strong> oneg<br />
antiferroperties<br />
are<br />
ered cobalhe<br />
presence<br />
0 where <strong>the</strong><br />
–localizedatistics<br />
may<br />
The multic<br />
‘‘normal’’<br />
icles, pseurconduct<strong>in</strong>g<br />
y correlated<br />
as Ti 3 <strong>and</strong><br />
2g hole) <strong>and</strong><br />
at possess a<br />
hese comperties<br />
[1];<br />
from which<br />
eracy is ‘‘to<br />
etry <strong>of</strong> <strong>the</strong><br />
ck <strong>of</strong> both<br />
bital degenk<strong>in</strong>ematical<br />
tly, a fermid<br />
<strong>in</strong>sulator<strong>the</strong><br />
pseudoormation<br />
<strong>of</strong><br />
with<strong>in</strong> just<br />
correlations<br />
lectrons are<br />
ifferent or<strong>in</strong>teractions<br />
that result <strong>in</strong> a rich variety <strong>of</strong> magnetic states <strong>in</strong> S 1=2<br />
oxides such as RTiO 3 , Na x CoO 2 , Sr 2 CoO 4 , RNiO 3 ,<br />
NaNiO 2 . In contrast, <strong>sp<strong>in</strong></strong> correlations <strong>in</strong> s<strong>in</strong>gle-b<strong>and</strong> cuprates<br />
are <strong>of</strong> AF nature exclusively <strong>and</strong> hence strong.<br />
How to suppress<br />
Mott<br />
<strong>the</strong> orbital degeneracy <strong>and</strong><br />
<strong>in</strong>terfaces<br />
promote<br />
cupratelike physics <strong>in</strong> o<strong>the</strong>r S 1=2 oxides? In this<br />
Letter, we suggest <strong>and</strong> argue <strong>the</strong>oretically that this goal<br />
can be achieved <strong>in</strong> oxide superlattices. Specifically, we<br />
focus on Ni-based superlattices (see Fig. 1) which can be<br />
fabricated us<strong>in</strong>g recent advances <strong>in</strong> oxide heterostructure<br />
technology ([3–5] <strong>and</strong> references <strong>the</strong>re<strong>in</strong>). While <strong>the</strong> proposed<br />
compound has a pseudocubic ABO 3 structure, its<br />
Lots <strong>of</strong> <strong>in</strong>terest<strong>in</strong>g suggestions!<br />
low-energy electronic states are conf<strong>in</strong>ed to <strong>the</strong> NiO 2<br />
planes <strong>and</strong>, hence, are <strong>of</strong> a quasi-2D nature. A substrate<br />
<strong>in</strong>duced compression <strong>of</strong> <strong>the</strong> NiO 6 octahedra fur<strong>the</strong>r stabilizes<br />
<strong>the</strong> x 2 -y 2 orbital. Net effect is a strong enhancement <strong>of</strong><br />
(a)<br />
MO 2<br />
LaO<br />
NiO 2<br />
LaO<br />
MO 2<br />
a<br />
c<br />
b<br />
(b)<br />
(c)<br />
(d)<br />
substrate<br />
FIG. 1. (a) Superlattice La 2 NiMO 6 with alternat<strong>in</strong>g NiO 2 <strong>and</strong><br />
MO 2 planes. MO 2 layers suppress <strong>the</strong> c-axis hopp<strong>in</strong>g result<strong>in</strong>g <strong>in</strong><br />
Chaloupka +<br />
2D electronic structure. Arrows <strong>in</strong>dicate <strong>the</strong> c-axis compression<br />
<strong>of</strong> <strong>the</strong> NiO 6 octahedron Khaliull<strong>in</strong>, imposed by 2008 tensile epitaxial stra<strong>in</strong> <strong>and</strong><br />
supported by Jahn-Teller coupl<strong>in</strong>g. (b) ,(c), (d) Stra<strong>in</strong>-<strong>in</strong>duced<br />
stretch<strong>in</strong>g <strong>of</strong> <strong>the</strong> NiO 2 planes occurs when superlattices with<br />
M Al, Ga, Ti are grown on SrTiO 3 or LaGaO 3 substrates<br />
hav<strong>in</strong>g large lattice parameter compared to that <strong>of</strong> LaNiO 3 .<br />
Expected deformations are <strong>in</strong>dicated by arrows.<br />
Al<br />
Ni<br />
Ga<br />
Ni<br />
Ti<br />
Ni<br />
vary d to tune<br />
Mott transition<br />
d
Nickelates<br />
But... we should underst<strong>and</strong> <strong>the</strong> bulk first<br />
One <strong>of</strong> <strong>the</strong> classic perovskites for study<strong>in</strong>g<br />
<strong>the</strong> Mott transition c.f. Goodenough + Raccah, 1965<br />
Torrance et al, 1992
Bulk transitions<br />
Transitions are all first order, w/ hysteresis<br />
GARCÍA-MUÑOZ et al.<br />
Resistance (ohms)<br />
Nd<br />
10 8<br />
10 7<br />
10 6<br />
10 5<br />
10 4<br />
10 3<br />
10 2<br />
10 9 0 50 100 150 200 250 300<br />
Magnetization (x 10 5 , emu/g)<br />
3.0<br />
2.6<br />
2.2<br />
180 190 200 210 220 230 240<br />
T(K)<br />
TABLE I. a Ref<strong>in</strong><br />
<strong>in</strong> space group Pbnm. <br />
50 K <strong>in</strong> space group P<br />
Atom<br />
a a=5.38712<br />
Nd 4c 0.9958<br />
Ni 4b 1/2<br />
O1 4c 0.0692<br />
O2 8d 0.7165<br />
x<br />
Torrance et al, 1992<br />
10 1<br />
T (K)<br />
FIG. 1. Color onl<strong>in</strong>e Resistance <strong>of</strong> NdNiO 3 on heat<strong>in</strong>g <strong>and</strong><br />
cool<strong>in</strong>g show<strong>in</strong>g an augment <strong>of</strong> 3 orders <strong>of</strong> magnitude after formation<br />
<strong>of</strong> a <strong>charge</strong> Garcia-Munoz order phase. Inset: k<strong>in</strong>k <strong>in</strong>et <strong>the</strong>al, susceptibility 2009 related<br />
with <strong>the</strong> metal-<strong>in</strong>sulator transition <strong>and</strong> <strong>the</strong> onset <strong>of</strong> <strong>the</strong> antiferromagnetic<br />
order T N =T MI . Hysteresis on heat<strong>in</strong>g <strong>and</strong> cool<strong>in</strong>g can be<br />
appreciated.<br />
Polycrystall<strong>in</strong>e NdNiO 3 was syn<strong>the</strong>sized under high oxygen<br />
pressure 200 bar follow<strong>in</strong>g <strong>the</strong> procedure described <strong>in</strong><br />
Ref. 28. The sample was extensively characterized by laboratory<br />
x-ray powder diffraction, magnetic <strong>and</strong> electric measurements,<br />
<strong>and</strong> synchrotron diffraction, which confirmed its<br />
quality. High-resolution synchrotron x-ray powder-<br />
290 K 2<br />
Pbnm 4.53<br />
Atom<br />
b a=5.37783<br />
Nd 4c 0.99321<br />
Ni1 2d 1/2<br />
Ni2 2c 1/2<br />
O1 4e 0.07521<br />
O2a 4e 0.71432<br />
x
Nickelates<br />
Ideally<br />
RNiO3, with R 3+ , Ni 3+ = 3d 7<br />
e<br />
1 g<br />
t<br />
6 2g<br />
Local moment <strong>and</strong> orbital degeneracy
Magnetic structure<br />
Neutron scatter<strong>in</strong>g<br />
R=Pr, Nd, Ho, Eu, Sm, Y all show <strong>the</strong> same,<br />
unusual magnetic structure<br />
k=(1/2,0,1/2) <strong>in</strong> orthorhombic coord<strong>in</strong>ates,<br />
equivalent to k=(1/4,1/4,1/4) <strong>in</strong> cubic ones<br />
unusual pattern: along cubic axes<br />
...<br />
<strong>in</strong>itially <strong>in</strong>terpreted as evidence for orbital order
Charge order<br />
Alonso et al, 1999: “rock salt” <strong>charge</strong> order<br />
S<strong>in</strong>ce seen <strong>in</strong> R=Y, Nd, Ho, Y, Eu, Tm, Yb,<br />
Er, Lu, by alternat<strong>in</strong>g expansion/<br />
contraction <strong>of</strong> octahedra via neutrons/xrays<br />
negligible Jahn-Teller distortions seen<br />
Implication (observed <strong>in</strong> Ho, Y, Eu)<br />
... ...
Charge <strong>and</strong> <strong>sp<strong>in</strong></strong><br />
order<br />
CO<br />
CO+AF<br />
Torrance et al, 1992
Valence skipp<strong>in</strong>g<br />
Maz<strong>in</strong> et al, 2007: <strong>the</strong>oretically suggest mixed<br />
valence PRL 98, state 176406 to (2007) enhance exchange<br />
FIG. 1 (color onl<strong>in</strong>e). Schematic electronic level diagram <strong>of</strong> Ni<br />
ions <strong>in</strong> RNiO 3 <strong>in</strong> two cases: (a) two JT distorted Ni 3 ions<br />
(energy ga<strong>in</strong> E JT per site), <strong>and</strong> (b) <strong>charge</strong> disproportionation.<br />
<strong>and</strong> <strong>in</strong> <strong>the</strong> fully delocalized limit, <strong>the</strong> additional Coulomb<br />
energy due to CO is reduced to <strong>the</strong> Hartree energy, a very<br />
strong reduction. Fur<strong>the</strong>rmore, delocalization <strong>of</strong>ten leads<br />
PHYSICAL REVIEW LETTERS<br />
even <strong>the</strong> magnetic <strong>in</strong>st<br />
paramagnetic metal do<br />
disproportionation. Ex<br />
manifests itself <strong>in</strong> oxyg<br />
netic moments on <strong>the</strong> tw<br />
contrary to popular be<br />
Mott-Hubbard <strong>in</strong>sulato<br />
tional b<strong>and</strong> picture give<br />
it is capable <strong>of</strong> expla<br />
occurs <strong>in</strong> <strong>the</strong> crossover<br />
b<strong>and</strong> structure calculat<br />
zation <strong>of</strong> <strong>the</strong> nonmag<br />
The <strong>in</strong>sulator-metal<br />
structure <strong>in</strong> RNiO 3 are<br />
0:7 B for Ni 2 <strong>in</strong> YNi<br />
1:4 B <strong>and</strong> 0:6 B for H<br />
A close <strong>in</strong>spection<br />
erant states along <strong>the</strong>
Valence skipp<strong>in</strong>g<br />
Maz<strong>in</strong> et al, 2007: <strong>the</strong>oretically suggest mixed<br />
valence PRL 98, state 176406 to (2007) enhance exchange<br />
FIG. 1 (color onl<strong>in</strong>e). Schematic electronic level diagram <strong>of</strong> Ni<br />
But: Why <strong>the</strong> complex magnetic <strong>order<strong>in</strong>g</strong><br />
ions <strong>in</strong> RNiO 3 <strong>in</strong> two cases: (a) two JT distorted Ni<br />
pattern?<br />
3 ions<br />
(energy ga<strong>in</strong> E JT per site), <strong>and</strong> (b) <strong>charge</strong> disproportionation.<br />
c.f. Mizokawa, Khomskii, Sawatzky, 2000<br />
<strong>and</strong> <strong>in</strong> <strong>the</strong> fully delocalized limit, <strong>the</strong> additional Coulomb<br />
energy due to CO is reduced to <strong>the</strong> Hartree energy, a very<br />
strong reduction. Fur<strong>the</strong>rmore, delocalization <strong>of</strong>ten leads<br />
PHYSICAL REVIEW LETTERS<br />
even <strong>the</strong> magnetic <strong>in</strong>st<br />
paramagnetic metal do<br />
disproportionation. Ex<br />
manifests itself <strong>in</strong> oxyg<br />
netic moments on <strong>the</strong> tw<br />
contrary to popular be<br />
Mott-Hubbard <strong>in</strong>sulato<br />
The <strong>in</strong>sulator-metal<br />
structure <strong>in</strong> RNiO 3 are<br />
0:7 B for Ni 2 <strong>in</strong> YNi<br />
1:4 B <strong>and</strong> 0:6 B for H<br />
A close <strong>in</strong>spection<br />
tional b<strong>and</strong> picture give<br />
it is capable <strong>of</strong> expla<br />
occurs <strong>in</strong> <strong>the</strong> crossover<br />
erant states along <strong>the</strong><br />
b<strong>and</strong> structure calculat<br />
zation <strong>of</strong> <strong>the</strong> nonmag
Naive expectations<br />
Rock-salt <strong>order<strong>in</strong>g</strong> <strong>of</strong> Ni 2+ ions forms an fcc<br />
lattice<br />
FCC antiferromagnet is degenerate <strong>and</strong><br />
orders at (keep<strong>in</strong>g orig<strong>in</strong>al cubic conventions)<br />
Q=(1/2,0,q) - usually (1/2,0,0)<br />
“type I AF”<br />
But Q=(1/4,1/4,1/4) found experimentally
Outl<strong>in</strong>e<br />
Introduction to Mott transitions <strong>and</strong> <strong>charge</strong>/<strong>sp<strong>in</strong></strong> order <strong>in</strong> <strong>the</strong><br />
RNiO3 <strong>nickelates</strong><br />
Hubbard model<br />
it<strong>in</strong>erant limit: <strong>sp<strong>in</strong></strong>-density wave<br />
L<strong>and</strong>au analysis <strong>of</strong> <strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
role <strong>of</strong> orthorhombicity<br />
Localized limit<br />
<strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
O<strong>the</strong>r tests for it<strong>in</strong>erant versus localized physics
Hubbard Model<br />
A m<strong>in</strong>imal model to consider is<br />
H = ij;ab<br />
t ab<br />
ij c † iaα c jbα + i<br />
Un 2 i − J( S i ) 2<br />
B<strong>and</strong> energy<br />
(hopp<strong>in</strong>g)<br />
Coulomb<br />
Hund’s<br />
exchange
Hubbard Model<br />
A m<strong>in</strong>imal model to consider is<br />
H = ij;ab<br />
t ab<br />
ij c † iaα c jbα + i<br />
Un 2 i − J( S i ) 2<br />
B<strong>and</strong> energy<br />
(hopp<strong>in</strong>g)<br />
Coulomb<br />
Hund’s<br />
exchange<br />
vs
Phase Diagram<br />
Which is more appropriate start<strong>in</strong>g po<strong>in</strong>t?<br />
<strong>and</strong> how do we tell experimentally?<br />
J H /t<br />
(localized)<br />
strongly correlated <strong>in</strong>sulator<br />
<strong>in</strong>termediate correlation<br />
(it<strong>in</strong>erant)<br />
metal<br />
U/t
Phase Diagram<br />
Which is more appropriate start<strong>in</strong>g po<strong>in</strong>t?<br />
<strong>and</strong> how do we tell experimentally?<br />
J H /t<br />
(localized)<br />
strongly correlated <strong>in</strong>sulator<br />
<strong>in</strong>termediate correlation<br />
start here<br />
(it<strong>in</strong>erant)<br />
metal<br />
U/t
It<strong>in</strong>erant Limit<br />
Ho<br />
?
Tight-b<strong>in</strong>d<strong>in</strong>g model<br />
Keep just σ-bond<strong>in</strong>g for 2 eg b<strong>and</strong>s<br />
t’<br />
t<br />
Ex Hopp<strong>in</strong>g Ni d x<br />
2 O p x Ni<br />
t<br />
!"# $# !"#<br />
Ex Hopp<strong>in</strong>g Ni d x<br />
2 O px Ni d x<br />
2<br />
t<br />
x<br />
<br />
!"# $# !"#<br />
x
3<br />
2<br />
1<br />
1<br />
0.5 1.0 1.5 2.0 2.5 3.0<br />
Comparison w/ LDA<br />
N. Hamada, 1993<br />
NORIM<br />
HAMADA<br />
Best fit<br />
w<br />
0<br />
t=0.75 eV<br />
t’/t=0.07<br />
-2<br />
-4<br />
P<br />
X r X M r<br />
Figure 1: B<strong>and</strong> structure <strong>of</strong> a cubic LaNiOs with a lattice parameter <strong>of</strong> 3,88A. The orig<strong>in</strong> <strong>of</strong><br />
energy is taken at <strong>the</strong> Fermi energy.<br />
Fits reasonably well <strong>the</strong> b<strong>and</strong>s cross<strong>in</strong>g <strong>the</strong><br />
Fermi energy, but misses some (t2g) states<br />
below<br />
M<br />
V<br />
Figure 2: Fermi surfaces <strong>of</strong> a cubic LaNKIs: (a) a electron pocket at <strong>the</strong> I? po<strong>in</strong>t conta<strong>in</strong>~g 0.04<br />
electrons, <strong>and</strong> (b) a large hole Fermi surface around <strong>the</strong> R po<strong>in</strong>t conta<strong>in</strong><strong>in</strong>g 1.04 holes <strong>in</strong>side.
Fermi surfaces<br />
Variation with t’/t<br />
0.24 0.06<br />
t’/t<br />
t’/t = 0.25 t’/t = 0.15 t’/t = 0
Comparisons<br />
FERMI SURFACES, ELECTRON-HOLE ASYMMETRY, AND…<br />
PHYSICAL REVIEW B 79, 115122 2009<br />
Γ-X<br />
a<br />
R-M<br />
b<br />
X<br />
R<br />
Γ<br />
M<br />
X<br />
R<br />
B<strong>in</strong>d<strong>in</strong>g energy (eV)<br />
0.0<br />
0.5<br />
1.0 0.5 0.0 1.0 0.5 0.0<br />
B<strong>in</strong>d<strong>in</strong>g energy (eV)<br />
B<strong>in</strong>d<strong>in</strong>g energy (eV)<br />
k F 1 k F 2 k F 3 k F 4<br />
1.0<br />
X Γ X<br />
R M R<br />
s<strong>of</strong>t x-ray ARPES, R.<br />
FIG.<br />
Eguchi<br />
3. Color onl<strong>in</strong>e a <strong>and</strong><br />
et<br />
bal, EDCs along<br />
2009<br />
-X <strong>and</strong> R-M<br />
direction, respectively. c <strong>and</strong> d The <strong>in</strong>tensity maps <strong>of</strong> EDCs.<br />
k F1-k F4 <strong>in</strong>dicate <strong>the</strong> MDC peak positions at E F.<br />
we measured ARPES at a fixed photon energy <strong>of</strong> 630 eV<br />
k z 0 <strong>and</strong> 710 eV k z /c. The FS mapp<strong>in</strong>g <strong>in</strong> horizontal<br />
k x -k y planes “” <strong>and</strong> “” <strong>of</strong> Brillou<strong>in</strong> zone <strong>in</strong> Fig. 1a<br />
t’/t=0.15<br />
was obta<strong>in</strong>ed as shown <strong>in</strong> Figs. 2b <strong>and</strong> 2c. In “” plane,<br />
FIG. 2. Color onl<strong>in</strong>e Fermi-surface mapp<strong>in</strong>g <strong>in</strong> a a vertical<br />
<strong>the</strong> small electron FS around po<strong>in</strong>t is aga<strong>in</strong> observed. This<br />
k result <strong>in</strong>dicates that <strong>the</strong> 3D spherelike FS around po<strong>in</strong>t can<br />
z-k x plane “” <strong>and</strong> b horizontal k x-k y planes “” <strong>and</strong> c “”<br />
<strong>of</strong> Brillou<strong>in</strong> zone <strong>in</strong> Fig. 1a. Solid white l<strong>in</strong>es correspond to <strong>the</strong> be observed by ARPES experimentally. The squarelike <strong>in</strong>tense<br />
area around M po<strong>in</strong>ts obta<strong>in</strong>ed from raw data without<br />
cubic Brillou<strong>in</strong> zone <strong>and</strong> dotted white l<strong>in</strong>es correspond to <strong>the</strong> highsymmetry<br />
l<strong>in</strong>es. k F1-k F4 <strong>in</strong>dicate <strong>the</strong> MDC peak positions at E F <strong>in</strong> symmetrization corresponds to <strong>the</strong> projection <strong>of</strong> <strong>the</strong> hole FS<br />
Figs. 3c <strong>and</strong> 3d. Blue gray l<strong>in</strong>es show nest<strong>in</strong>g character hole centered at R po<strong>in</strong>t. In “” plane, no small FS is observed.<br />
LNO<br />
FSs. All <strong>the</strong> <strong>in</strong>tensity maps are measured data over <strong>the</strong> displayed Instead large FSs centered at R po<strong>in</strong>t were observed. From<br />
range <strong>and</strong> no symmetrization has been used to obta<strong>in</strong> <strong>the</strong> maps. <strong>the</strong> complete data set <strong>of</strong> energy <strong>and</strong> angle-dependent FS<br />
RH<br />
maps, <strong>the</strong> experimental FS obta<strong>in</strong>ed by s<strong>of</strong>t x-ray ARPES is<br />
to <strong>the</strong> Ni 3d6t 2g b<strong>and</strong>s. In addition, EDCs for 600–660 eV <strong>in</strong> overall agreement with that predicted by <strong>the</strong> b<strong>and</strong><br />
clearly show a dispersive b<strong>and</strong> cross<strong>in</strong>g at <strong>the</strong> E F arrows calculation. 24 However, <strong>the</strong> actual b<strong>and</strong> dispersions reveal an<br />
around <strong>the</strong> po<strong>in</strong>t. This dispersive b<strong>and</strong> orig<strong>in</strong>ates <strong>in</strong> Ni 3d important difference <strong>in</strong> electron <strong>and</strong> hole FSs as discussed <strong>in</strong><br />
e g states <strong>and</strong> forms a small electron pocket as predicted by <strong>the</strong> follow<strong>in</strong>g.<br />
<strong>the</strong> local-density 4 approximation LDA b<strong>and</strong> calculation. The In order to discuss <strong>the</strong> b<strong>and</strong> structures form<strong>in</strong>g <strong>the</strong>se FSs,<br />
b<strong>and</strong> disappears for energies below 600 eV <strong>and</strong> above 660 we measured <strong>the</strong> ARPES <strong>in</strong> <strong>the</strong> high-symmetry l<strong>in</strong>es with<br />
eV photon energy <strong>and</strong> an <strong>in</strong>crease <strong>in</strong> <strong>in</strong>tensity is observed detailed momentum steps <strong>and</strong> an energy resolution E<br />
close to <strong>the</strong> M po<strong>in</strong>t at h=570 <strong>and</strong> 700 eV.<br />
150 meV. The EDCs Figs. 3a <strong>and</strong> 3b <strong>and</strong> <strong>the</strong> <strong>in</strong>tensity<br />
plots Figs. 3c <strong>and</strong> 3d along -X <strong>and</strong> R-M directions<br />
2<br />
Figure 2a shows <strong>the</strong> FS mapp<strong>in</strong>g <strong>in</strong> a vertical k z -k x <br />
plane “” <strong>of</strong> Brillou<strong>in</strong> zone as shown <strong>in</strong> Fig. 1a, which is are shown <strong>in</strong> Fig. 3. The FS cross<strong>in</strong>g k F po<strong>in</strong>ts are labeled as<br />
obta<strong>in</strong>ed by a plot <strong>of</strong> <strong>the</strong> <strong>in</strong>tegrated <strong>in</strong>tensity from −0.05 to k F 1-k F 4. The <strong>in</strong>tensity plots <strong>in</strong> both directions Figs. 3c <strong>and</strong><br />
0.05 eV b<strong>in</strong>d<strong>in</strong>g energy <strong>in</strong> EDCs. A small circle centered at 3d show an <strong>in</strong>tense feature around 0.0–0.2 eV at <strong>and</strong> M<br />
po<strong>in</strong>t is observed, while no 0.2 <strong>in</strong>tensity is observed 0.1 around <strong>the</strong> 0.0po<strong>in</strong>ts, correspond<strong>in</strong>g 0.1 to <strong>the</strong> electron 0.2 t’/t<br />
b<strong>and</strong> <strong>and</strong> <strong>the</strong> hole b<strong>and</strong><br />
X po<strong>in</strong>t. The existence <strong>of</strong> a nearly spherical small FS centered<br />
at po<strong>in</strong>t, correspond<strong>in</strong>g to <strong>the</strong> electron FS, was pre-<br />
bottom at M po<strong>in</strong>t is around 0.25 eV, <strong>the</strong> b<strong>and</strong> bottom at <br />
derived from Ni 3d e g states, respectively. While <strong>the</strong> b<strong>and</strong><br />
dicted by2<br />
b<strong>and</strong> calculations. 24 From photon energy po<strong>in</strong>t is not at 0.25 eV. The <strong>in</strong>tensity rema<strong>in</strong>s <strong>in</strong> high b<strong>in</strong>d<strong>in</strong>genergy<br />
region about 0.5–1.0 eV at po<strong>in</strong>t Fig. 3c, <strong>in</strong>-<br />
k z -dependent ARPES measurements, we can decide <strong>the</strong><br />
photon energy trac<strong>in</strong>g <strong>the</strong> -X <strong>and</strong> X-M directions as shown dicat<strong>in</strong>g that <strong>the</strong> b<strong>and</strong>s extend to high b<strong>in</strong>d<strong>in</strong>g energies. Because<br />
<strong>the</strong> t 2g b<strong>and</strong>s also appear above 0.5 eV at X po<strong>in</strong>t, <strong>in</strong> Fig. 1b. 4 In order to observe <strong>the</strong> <strong>in</strong>-plane cuts <strong>of</strong> <strong>the</strong> FSs,<br />
<strong>the</strong><br />
6<br />
115122-3<br />
c<br />
d<br />
S/T<br />
3<br />
2<br />
1<br />
1<br />
2<br />
3<br />
optics, D. Ouellette et al, 2010<br />
LNO<br />
0.4 0.2 0.0 0.2 0.4<br />
t’/t<br />
“Hole like” Hall conductivity<br />
“Electron like” <strong>the</strong>rmopower
Nest<strong>in</strong>g<br />
Large fermi surface conta<strong>in</strong>s ra<strong>the</strong>r flat<br />
portions<br />
Approximate “nest<strong>in</strong>g” leads to a large<br />
susceptibility at some wavevectors, <strong>and</strong> a<br />
tendency to CDW or SDW order
Susceptibility<br />
Susceptibility is peaks for k=(k,k,k), with<br />
k≈0.4π (close to π/2), for <strong>the</strong> FS closest to<br />
ARPES<br />
Χk, Ω0<br />
1200<br />
1000<br />
800<br />
600<br />
400<br />
200<br />
t't0.1 0.05<br />
t't0.2 0.1<br />
t't0.3 0.15<br />
0<br />
000 100 110 111 000 101<br />
kΠ<br />
Due to JH, this drives a Sp<strong>in</strong> Density Wave<br />
<strong>in</strong>stability (<strong>in</strong> e.g. RPA)
Hartree-Fock<br />
Two SDW phases appear<br />
SDW<br />
Ɵ =0<br />
0 < Ɵ < π/4<br />
Ɵ = π/4<br />
PM+M : paramagnetic metal<br />
SM : semimetal+SDW<br />
SDW+CO+I : <strong>sp<strong>in</strong></strong> density wave +<br />
<strong>charge</strong> <strong>order<strong>in</strong>g</strong> + <strong>in</strong>sulator<br />
SDW+M : <strong>sp<strong>in</strong></strong> density wave + metal<br />
n.b. calculated for ideal cubic structure
Outl<strong>in</strong>e<br />
Introduction to Mott transitions <strong>and</strong> <strong>charge</strong>/<strong>sp<strong>in</strong></strong> order <strong>in</strong> <strong>the</strong><br />
RNiO3 <strong>nickelates</strong><br />
Hubbard model<br />
it<strong>in</strong>erant limit: <strong>sp<strong>in</strong></strong>-density wave<br />
L<strong>and</strong>au analysis <strong>of</strong> <strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
role <strong>of</strong> orthorhombicity<br />
Localized limit<br />
<strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
O<strong>the</strong>r tests for it<strong>in</strong>erant versus localized physics
L<strong>and</strong>au <strong>Theory</strong><br />
SDW order parameter<br />
<strong>in</strong> reality <strong>the</strong>re are 4 (2)<br />
S i = Re ψe iQ·r equivalent Q a by cubic<br />
i (orthorhombic)<br />
symmetry, but we expect<br />
<strong>the</strong> system to choose only<br />
one to order <strong>in</strong>to
L<strong>and</strong>au <strong>Theory</strong><br />
SDW order parameter<br />
S i = Re ψe iQ·r i <br />
Charge order parameter<br />
n i =(−1) x i+y i +z i<br />
Φ<br />
Symmetry allows <strong>the</strong> term<br />
F = λΦRe [ψ · ψ]<br />
Φ ∝ Re [ψ · ψ]
Cubic symmetry<br />
L<strong>and</strong>au free energy<br />
F = rψ ∗· ψ + u 1 (ψ ∗· ψ) 2<br />
+u 2 |ψ · ψ| 2 + u 3<br />
<br />
(ψ · ψ) 2 +h.c. <br />
<strong>Physics</strong> <strong>of</strong> u2:<br />
u20: spiral state
Cubic symmetry<br />
L<strong>and</strong>au free energy<br />
F = rψ ∗· ψ + u 1 (ψ ∗· ψ) 2<br />
+u 2 |ψ · ψ| 2 + u 3<br />
<br />
(ψ · ψ) 2 +h.c. <br />
<strong>Physics</strong> <strong>of</strong> u2:<br />
u20: spiral state
Cubic symmetry<br />
L<strong>and</strong>au free energy<br />
F = rψ ∗· ψ + u 1 (ψ ∗· ψ) 2<br />
+u 2 |ψ · ψ| 2 + u 3<br />
<br />
(ψ · ψ) 2 +h.c. <br />
<strong>Physics</strong> <strong>of</strong> u3:<br />
fixes phase <strong>of</strong> SDW<br />
ψ = |ψ|ˆn e iθ<br />
F = constant +u 3 |ψ| 4 cos 4θ<br />
f<strong>in</strong>d θ=0 or π/4
SDW states<br />
Look along axis<br />
θ=0<br />
“site centered”<br />
max Φ<br />
θ=π/4<br />
“bond centered”<br />
Φ=0
Cubic lattice<br />
“site centered”<br />
“bond centered”<br />
“MCO”<br />
c.f. Mizokawa et al, 2000<br />
“OCO”<br />
m 1 =m, m 2 =0 m 1 =m 2 =m
Magnetic structures<br />
Ref<strong>in</strong>ements give no materials with<br />
momentless Ni sites<br />
material m1(µB) m2(µB) Reference<br />
Ho 1.4 0.6 PRB 64, 144417 (2001)<br />
Y 1.4 0.7 PRL 82, 3871 (1999)<br />
Eu 1.2 1.2 PRB 57, 456(1998)<br />
Sm 0.9 0.9* PRB 57, 456(1998)<br />
Nd 0.9 0.9* PRB 50, 978 (1994)<br />
Pr 0.9 0.9* PRB 50, 978 (1994)<br />
* observed <strong>charge</strong> order is <strong>in</strong>consistent with equal moments
1918 J B Good<br />
Orthorhombic<br />
Distortion<br />
orthorhombicity<br />
Figure 2. Cooperative MX 6/2 rotations giv<strong>in</strong>g (a) tetragonal (projection on (001) <strong>of</strong> MX 3<br />
(b) rhombohedral <strong>and</strong> (c) orthorhombic (Pbnm axes) symmetry. Note: Pbnm axes a, b, c, b<br />
c, a, b, <strong>in</strong> Pnma.<br />
The equilibrium (A–X) <strong>and</strong> (M–X) bond lengths are calculated for ambient cond<br />
from <strong>the</strong> sums <strong>of</strong> <strong>the</strong> ionic radii available <strong>in</strong> tables [1]; <strong>the</strong>y were obta<strong>in</strong>ed from x-ray data
Orthorhombicity<br />
Orthorhomic distortion quadruples <strong>the</strong> unit cell<br />
<strong>and</strong> allows an additional term:<br />
In <strong>the</strong> coll<strong>in</strong>ear SDW this becomes<br />
Now <strong>the</strong> m<strong>in</strong>ima are at<br />
u30: θ=π/4 + δ, with δ ~ v/u
SDW states<br />
Look along axis<br />
θ=0<br />
θ=π/4<br />
+ δ<br />
Charge order grows with orthorhombicity!<br />
“site centered”<br />
max Φ<br />
zero moments<br />
due to I center<br />
Φ ∝ δ<br />
“<strong>of</strong>f center”<br />
develops unequal<br />
moments <strong>and</strong> CO
Unequal moments<br />
orthorhombicity
Hartree-Fock<br />
With orthorhombicity, <strong>charge</strong> order is<br />
everywhere<br />
θ=0<br />
↑0↓0<br />
θ=π/4±δ<br />
SDW+CO<br />
↑↑↓↓<br />
Ɵ =0<br />
0 < Ɵ < π/4<br />
Ɵ = π/4<br />
PM+M : paramagnetic metal<br />
SM : semimetal+SDW<br />
SDW+CO+I : <strong>sp<strong>in</strong></strong> density wave +<br />
<strong>charge</strong> <strong>order<strong>in</strong>g</strong> + <strong>in</strong>sulator<br />
SDW+M : <strong>sp<strong>in</strong></strong> density wave + metal<br />
But, <strong>the</strong>re are still 2 types <strong>of</strong> magnetic states
Outl<strong>in</strong>e<br />
Introduction to Mott transitions <strong>and</strong> <strong>charge</strong>/<strong>sp<strong>in</strong></strong> order <strong>in</strong> <strong>the</strong><br />
RNiO3 <strong>nickelates</strong><br />
Hubbard model<br />
it<strong>in</strong>erant limit: <strong>sp<strong>in</strong></strong>-density wave<br />
L<strong>and</strong>au analysis <strong>of</strong> <strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
role <strong>of</strong> orthorhombicity<br />
Localized limit<br />
<strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
O<strong>the</strong>r tests for it<strong>in</strong>erant versus localized physics
Back to <strong>the</strong> Mott<br />
Transition<br />
What about <strong>the</strong> separation <strong>of</strong> <strong>charge</strong> <strong>and</strong> <strong>sp<strong>in</strong></strong><br />
<strong>order<strong>in</strong>g</strong> for smaller rare earths?<br />
Ho<br />
?
Phase Diagram<br />
Which is more appropriate start<strong>in</strong>g po<strong>in</strong>t?<br />
<strong>and</strong> how do we tell experimentally?<br />
J H /t<br />
(localized)<br />
strongly correlated <strong>in</strong>sulator<br />
look here<br />
<strong>in</strong>termediate correlation<br />
(it<strong>in</strong>erant)<br />
metal<br />
U/t
Strong Coupl<strong>in</strong>g<br />
Approach<br />
Consider perturbation <strong>the</strong>ory <strong>in</strong> <strong>the</strong> hopp<strong>in</strong>g<br />
t=t’=0:<br />
E=2U-3J H /2<br />
E=4U-2J H
Strong Coupl<strong>in</strong>g<br />
Approach<br />
Consider perturbation <strong>the</strong>ory <strong>in</strong> <strong>the</strong> hopp<strong>in</strong>g<br />
t=t’=0:<br />
E=2U-3JH/2<br />
E=4U-2J H<br />
JH/t<br />
paired<br />
J H =4U<br />
electrons are bound<br />
<strong>in</strong>to S=1 bosonic pairs<br />
<strong>in</strong>termediate correlation<br />
(it<strong>in</strong>erant)<br />
metal<br />
U/t
Order<strong>in</strong>g <strong>of</strong> pairs<br />
Half <strong>of</strong> sites should be occupied (Ni 2+ )<br />
or<br />
or...
Order<strong>in</strong>g <strong>of</strong> pairs<br />
Half <strong>of</strong> sites should be occupied (Ni 2+ )<br />
or<br />
or...<br />
At second order <strong>in</strong> hopp<strong>in</strong>g, <strong>the</strong> <strong>order<strong>in</strong>g</strong> is<br />
determ<strong>in</strong>ed to be <strong>of</strong> rock salt (fcc) type<br />
virtual hopp<strong>in</strong>g processes favor Ni 2+<br />
neighbor<strong>in</strong>g Ni 4+<br />
TCO ~ t 2 /JH
Magnetism<br />
The <strong>sp<strong>in</strong></strong> <strong>of</strong> each Ni 2+ electron pair is totally free<br />
at O(t 2 /J H )<br />
exchange coupl<strong>in</strong>g sets TAF ~ t 4 /JH 3
Magnetism<br />
The <strong>sp<strong>in</strong></strong> <strong>of</strong> each Ni 2+ electron pair is totally free<br />
at O(t 2 /J H )<br />
exchange coupl<strong>in</strong>g sets TAF ~ t 4 /JH 3 |J1| (needs t’)<br />
But: <strong>the</strong> ↑0↓0 state is expected here
Put it toge<strong>the</strong>r<br />
Strong coupl<strong>in</strong>g picture agrees with HF<br />
↑0↓0<br />
θ=0<br />
↑0↓0<br />
θ=π/4±δ<br />
SDW+CO<br />
↑↑↓↓<br />
Ɵ =0<br />
0 < Ɵ < π/4<br />
Ɵ = π/4<br />
PM+M : paramagnetic metal<br />
SM : semimetal+SDW<br />
SDW+CO+I : <strong>sp<strong>in</strong></strong> density wave +<br />
<strong>charge</strong> <strong>order<strong>in</strong>g</strong> + <strong>in</strong>sulator<br />
SDW+M : <strong>sp<strong>in</strong></strong> density wave + metal
Put it toge<strong>the</strong>r<br />
Strong coupl<strong>in</strong>g picture agrees with HF<br />
↑0↓0<br />
Ɵ =0<br />
θ=0<br />
↑0↓0<br />
θ=π/4±δ<br />
SDW+CO<br />
↑↑↓↓<br />
0 < Ɵ < π/4<br />
Ɵ = π/4<br />
PM+M : paramagnetic metal<br />
SM : semimetal+SDW<br />
SDW+CO+I : <strong>sp<strong>in</strong></strong> density wave +<br />
<strong>charge</strong> <strong>order<strong>in</strong>g</strong> + <strong>in</strong>sulator<br />
SDW+M : <strong>sp<strong>in</strong></strong> density wave + metal<br />
Off-center SDW seems most compatible with expt
Outl<strong>in</strong>e<br />
Introduction to Mott transitions <strong>and</strong> <strong>charge</strong>/<strong>sp<strong>in</strong></strong> order <strong>in</strong> <strong>the</strong><br />
RNiO3 <strong>nickelates</strong><br />
Hubbard model<br />
it<strong>in</strong>erant limit: <strong>sp<strong>in</strong></strong>-density wave<br />
L<strong>and</strong>au analysis <strong>of</strong> <strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
role <strong>of</strong> orthorhombicity<br />
Localized limit<br />
<strong>sp<strong>in</strong></strong> <strong>and</strong> <strong>charge</strong> order<br />
O<strong>the</strong>r tests for it<strong>in</strong>erant versus localized physics
How to tell?<br />
It<strong>in</strong>erant vs.<br />
Localized?<br />
Magnetism: “<strong>of</strong>f center” SDW (it<strong>in</strong>erant) vs.<br />
“site centered” SDW (localized)<br />
Order <strong>of</strong> phase transitions (1st order)<br />
O<strong>the</strong>r ways?<br />
Electronic structure<br />
c.f. arXiv:1008.2373v1<br />
Heterostructures: response <strong>of</strong> <strong>charge</strong>+<strong>sp<strong>in</strong></strong><br />
order to conf<strong>in</strong>ement, stra<strong>in</strong>
Electronic<br />
structure<br />
Ano<strong>the</strong>r way to dist<strong>in</strong>guish two SDWs is by<br />
electronic properties<br />
c.f. DOS (cubic symmetry taken here)<br />
DOS<br />
0.08<br />
0.06<br />
DOS<br />
0.07<br />
0.06<br />
0.05<br />
0.04<br />
0.04<br />
0.03<br />
0.02<br />
1.0 0.5 0.0 0.5 1.0 1.5 2.0<br />
E<br />
gap<br />
Ɵ =0<br />
0 < Ɵ < π/4<br />
Ɵ = π/4<br />
0.02<br />
0.01<br />
1 0 1 2<br />
E<br />
l<strong>in</strong>ear DOS
semi-metal state<br />
In Hartree-Fock, one f<strong>in</strong>ds a semi-metallic<br />
solution <strong>in</strong> <strong>the</strong> cubic system<br />
This corresponds to <strong>the</strong> bond-centered SDW,<br />
<strong>and</strong> essentially electrons <strong>of</strong> one <strong>sp<strong>in</strong></strong><br />
polarization are conf<strong>in</strong>ed to a pair <strong>of</strong><br />
parallel planes<br />
This forms a honeycomb lattice<br />
looks familiar?
Hartree-Fock<br />
graphene-like b<strong>and</strong> structure!<br />
<br />
t’=0<br />
t’=0.15<br />
protected by topology <strong>and</strong> <strong>in</strong>version symmetry<br />
but: orthorhombic distortion will open up a gap
Hartree-Fock<br />
small gap opens with orthorhombicity<br />
<br />
before<br />
after
Anisotropy<br />
SDW wavevectors<br />
cubic unit cell 4 wavevectors top down view<br />
Transport along Q is <strong>the</strong> “hard” direction - expect large<br />
<strong>in</strong>-plane anisotropy for a s<strong>in</strong>gle-doma<strong>in</strong> film
Films, Interfaces<br />
L<strong>and</strong>au analysis can be readily adapted:<br />
<strong>in</strong>clude terms reflect<strong>in</strong>g lowered symmety<br />
Example: <strong>in</strong>terface<br />
z-translation symmetry is removed<br />
δF = λ e iα ψ 1 · ψ 3 +c.c. <br />
Q 1 = 1 4 (1¯1¯1)<br />
Q 2 = 1 4 (¯11¯1)<br />
leads to “two-q” state near <strong>in</strong>terface<br />
Similar effects can lead to switch<strong>in</strong>g <strong>of</strong> SDW<br />
wavevector <strong>and</strong> hence can turn <strong>charge</strong> order on/<strong>of</strong>f
Nest<strong>in</strong>g<br />
Conf<strong>in</strong>ement affects Fermi surfaces<br />
N=4 layers N=3 N=2 N=1
1<br />
0<br />
1<br />
0<br />
0<br />
0<br />
1<br />
0<br />
1<br />
1<br />
1<br />
2<br />
Sp<strong>in</strong> susceptibility<br />
0 1 2 3 3 2 1 0 1 2 3<br />
2<br />
3<br />
3<br />
3<br />
Sp<strong>in</strong> susceptibility<br />
3 2 1 0 1 2 3<br />
usceptibility<br />
70<br />
200<br />
60<br />
3 2 1 0 1 2 3<br />
1<br />
2<br />
2<br />
3<br />
3 2 1 0 1 2 3<br />
k z Π4<br />
t't0.05 t't0.05<br />
t't0.1 t't0.1<br />
2<br />
3<br />
200<br />
3 2 1 0 1 2 3<br />
t't0.15<br />
k z Π4<br />
k z Π2<br />
50<br />
150<br />
t't0.15<br />
t't0.15<br />
150<br />
k z 3Π4<br />
Χk, Ω0<br />
40<br />
30<br />
Χk, Ω0<br />
100<br />
Χk, Ω0<br />
100<br />
20<br />
50<br />
50<br />
10<br />
0<br />
0<br />
00 10 11 00<br />
00 10 11 00<br />
kΠ<br />
kΠ<br />
0<br />
00 10 11 00<br />
N=1 N=3<br />
kΠ<br />
S<strong>in</strong>gle layer is enhanced for broad range <strong>of</strong> t’/t<br />
N=3 layers is peaked at 2π(1/4,1/4,1/8)
Summary<br />
<strong>Theory</strong> <strong>of</strong> <strong>the</strong> <strong>nickelates</strong> must bridge <strong>the</strong> gap between it<strong>in</strong>erant<br />
<strong>and</strong> localized behavior<br />
There are sharp ways to def<strong>in</strong>e <strong>the</strong> two regimes, that can be<br />
experimentally dist<strong>in</strong>guished<br />
“site centered” versus “<strong>of</strong>f-center” magnetic order<br />
electronic structure<br />
behavior <strong>in</strong> heterostructures<br />
These methods can readily be applied to heterostructures <strong>and</strong><br />
<strong>in</strong>terfaces, <strong>and</strong> lead to testable predictions<br />
e.g. Expect conductivity anisotropy enhanced <strong>in</strong> SDW state<br />
Reference: arXiv:1008.2373v1