Effects of strain on orbital ordering and magnetism at perovskite oxide interfaces:
LaMnO
3
Õ SrMnO
3
B. R. K. Nanda and Sashi Satpathy
Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
共Received 31 March 2008; revised manuscript received 8 July 2008; published 19 August 2008
兲
We study how strain affects orbital ordering and magnetism at the interface between SrMnO
3
and LaMnO
3
from density-functional calculations and interpret the basic results in terms of a three-site Mn-O-Mn model.
Magnetic interaction between the Mn atoms is governed by a competition between the antiferromagnetic
superexchange of the Mn t
2g
core spins and the ferromagnetic double exchange of the itinerant e
g
electrons.
While the core electrons are relatively unaffected by the strain, the orbital character of the itinerant electron is
strongly affected, which in turn causes a large change in the strength of the ferromagnetic double exchange.
The epitaxial strain produces the tetragonal distortion of the MnO
6
octahedron, splitting the Mn e
g
states into
x
2
−y
2
and 3z
2
−1 states, with the former being lower in energy, if the strain is tensile in the plane and opposite
if the strain is compressive. For the case of the tensile strain, the resulting higher occupancy of the x
2
−y
2
orbital enhances the in-plane ferromagnetic double exchange owing to the larger electron hopping in the plane,
causing at the same time a reduction in the out-of-plane double exchange. This reduction is large enough to be
overcome by antiferromagnetic superexchange, which wins to produce a net antiferromagnetic interaction
between the out-of-plane Mn atoms. For the case of the in-plane compressive strain, the reverse happens, viz.,
that the higher occupancy of the 3z
2
−1 orbital results in the out-of-plane ferromagnetic interaction, while the
in-plane magnetic interaction remains antiferromagnetic. Concrete density-functional results are presented for
the 共LaMnO
3
兲
1
/ 共SrMnO
3
兲
1
and 共LaMnO
3
兲
1
/ 共SrMnO
3
兲
3
superlattices for various strain conditions.
DOI: 10.1103/PhysRevB.78.054427 PACS number共s兲: 75.70.Cn, 71.20.⫺b, 73.20.⫺r, 71.70.⫺d
I. INTRODUCTION
Recent advances in successfully designing atomically
sharp interfaces between dissimilar transition-metal oxides
have revealed the formation of new electronic and magnetic
phases at the vicinity of the interface, which are qualitatively
different from the parent compounds. The interfacial phases
show diverse magnetic properties due to the coupling be-
tween charge, orbital, and spin degrees of freedom. For ex-
ample, the magnetic ordering at the interface between the
two antiferromagnetic 共AFM兲 insulators SrMnO
3
共SMO兲共G
type兲 and LaMnO
3
共LMO兲共A type兲, schematically shown in
Fig. 1, could be ferromagnetic 共FM兲 along all directions,
ferromagnetic in the xy plane and antiferromagnetic normal
to the plane or antiferromagnetic in the plane and ferromag-
netic normal to the plane depending on the composition of
the parent compounds and epitaxial strain on the interface.
1–6
The epitaxial strain, arising due to lattice mismatch be-
tween the constituent compounds of the superlattice and the
substrate, induces anisotropic hopping between orbitals to
cause orbital ordering at the interface. By varying the strain
condition the orbital ordering changes which in turn changes
the magnetic ordering at the interface. In this paper we ex-
amine the magnetic properties at the interface of SrMnO
3
and LaMnO
3
for different epitaxial strain conditions through
first-principles electronic structure calculations.
Experimental studies show that if the substrate induces
tensile strain at the interface of the LMO/SMO superlattice,
where the in-plane lattice parameter a is greater than the
out-of-plane lattice parameter c, as in the case of
共LMO兲
3
/ 共SMO兲
2
superlattice grown on SrTiO
3
共STO兲 sub-
strate, the magnetic ordering of the interfacial Mn atoms is
A type with in-plane 共MnO
2
plane兲 FM ordering and out-of-
plane 共between MnO
2
planes兲 AFM ordering.
1
Quite interestingly, when the 共LMO兲
3
/ 共SMO兲
2
superlat-
tice is grown on La
0.3
Sr
0.7
Al
0.65
Ta
0.35
O
3
共LSAT兲 substrate,
which induces no strain 共a ⬃c兲, the interface shows a three-
MnO
2
MnO
2
MnO
2
x
2
−y
2
d
x
2
−y
2
d
3z
2
−1
d
+
3z
2
−1
d
Sr
La
Mn
J
J
’
J
SrO
LaO
Tensile Lattice Matched
C
ompressive
AFC
1 1
1 1
1
1 1
1 1
1
1
1 1
1 1
1
1 1
1 1
1
1 1
1 1
1
1
1 1
1 1
1
1
1
1
1
1
1
1
1 1
1 1
1
1 1
1 1
1
1
1 1
1 1
1
1
1 1
1
1
1
1
1
1 1
1
1
1
1
1
1 1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1 1
1 1
1 1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1 1
1 1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
FIG. 1. Different magnetic configurations considered in this pa-
per for the 共LMO兲
1
/ 共SMO兲
1
superlattice. A stands for the structure
with ferromagnetic ordering in the MnO
2
plane and antiferromag-
netic ordering between the planes, while “F” stands for ferromag-
netic ordering in all directions and “C” stands for antiferromagnetic
ordering in the MnO
2
plane and ferromagnetic ordering between the
planes. The schematic orbital ordering shown in the figure was
found from our density-functional results presented below and was
also inferred from the experiments 共Ref. 1兲. The symbols J and J
⬘
denote, respectively, the out-of-plane and in-plane exchange inter-
actions between the Mn atoms. The strain condition under which
each structure is stabilized has been indicated in the figure. The
oxygen atoms which occur at the midpoint between two neighbor-
ing Mn atoms have not been shown.
PHYSICAL REVIEW B 78, 054427 共2008兲
1098-0121/2008/78共5兲/054427共12兲 ©2008 The American Physical Society054427-1
dimensional FM ordering 共F type兲.
1
If the interface experi-
ences a compressive strain 共a⬍c兲, as in the case of LMO/
SMO superlattice grown on LaAlO
3
共LAO兲 substrate, the
magnetic ordering is C type with in-plane AFM ordering and
out-of-plane FM ordering.
1
Substrates are instrumental in inducing epitaxial strain
and thereby enforce tetragonal distortion to the superlattice.
As a consequence, in case of LMO/SMO superlattice, the
substrate distorts the MnO
6
octahedron and splits the degen-
erate Mn e
g
states into x
2
−y
2
and 3z
2
−1 states. Varied tetrag-
onal distortion changes the on-site energy and hence the oc-
cupancy of these two nondegenerate e
g
states 共Fig. 2兲. Since
the electronic configuration of Mn atoms away from the in-
terface is the same as in the bulk compounds, 共Mn
4+
,t
2g
3
e
g
0
兲
for SMO and 共Mn
3+
,t
2g
3
e
g
1
兲 for LMO, strain is not expected to
affect the magnetic configuration of the inner MnO
2
layers to
a large extent. However, at the interface, where we see the
valence state of the Mn atoms lies between 3+ and 4+ be-
cause of charge reconstruction,
2,7
the varied occupancy of the
nondegenerate e
g
orbitals imposes different orbital orderings
for different strain conditions 共Fig. 2兲 and influences the in-
terface magnetism considerably.
In this paper, we have studied in detail the interfacial
magnetic properties of LMO/SMO superlattices for different
strain conditions by performing electronic structure calcula-
tions based on the density-functional theory 共DFT兲. To illus-
trate the strain effect on magnetism, we have proposed a
simple three-site model to calculate the interfacial Mn-O-Mn
magnetic exchange both in the MnO
2
plane and between the
planes for different strain conditions. From the model we see
that the on-site energy difference between x
2
−y
2
and
3z
2
−1 orbitals 共Fig. 2兲 is instrumental in switching the
ferromagnetic and antiferromagnetic interactions. When the
3z
2
−1 orbital is sufficiently lower in energy than the x
2
−y
2
orbital 共compressive strain兲, the Mn-O-Mn exchange is anti-
ferromagnetic in the plane and ferromagnetic between the
planes and opposite when x
2
−y
2
orbital is sufficiently lower
in energy 共tensile strain兲. If the energy levels of both the e
g
orbitals are close enough 共lattice-matched interface兲, then the
Mn-O-Mn exchange is ferromagnetic in all directions.
The rest of the paper is organized as follows. In Sec. II we
describe the structural and computational details. A detailed
analysis of the electronic structure of the 共LMO兲
1
/ 共SMO兲
1
superlattice at different strain conditions, obtained from the
density-functional calculations, is carried out in Sec. III. In
Sec. IV, we illustrate the effect of epitaxial strain on the
magnetic ordering, with the aid of a proposed three-site 共Mn-
O-Mn兲 model. Electronic and magnetic properties of the
共LMO兲
1
/ 共SMO兲
3
superlattice at different strain conditions
are discussed in Sec. V. Finally in Sec. VI we present the
summary.
II. STRUCTURAL AND COMPUTATIONAL DETAILS
We have taken the equivalent cubic perovskite structure
of LMO and SMO in order to study the electronic and mag-
netic properties at the interface of these two compounds with
the aid of first-principles electronic structure calculations.
The effect of epitaxial strain, which arises due to lattice mis-
match between the substrate and the LMO/SMO superlattice,
is taken into account by applying tetragonal distortion to the
superlattice.
The tetragonal distortion is quantified by the c / a ratio
which differs from one. Here a is the in-plane 共xy-plane兲
lattice parameter which coincides with the lattice parameter
of the substrate and c is the average out-of-plane lattice pa-
rameter 共along z axis兲. The c / a ratio is determined from the
linear relation: c −a
0
=−4
共a −a
0
兲, where a
0
is the in-plane
lattice parameter of the superlattice when there is no strain
共c / a =1兲 and coefficient
is the Poisson ratio which is ap-
proximately 0.3 for perovskite manganites.
1,8
Experimentally
it is found that for LMO/SMO superlattices, a
0
matches
with the weighed average of the lattice constants of bulk
LMO 共3.936 Å兲 and bulk SMO 共3.806 Å兲.
1
For example,
for 共LMO兲
1
/ 共SMO兲
1
superlattice, the value of a
0
is
1
2
共3.936+3.806兲 Å.
In this paper, we have considered two superlattices, viz.,
共LMO兲
1
/ 共SMO兲
1
and 共LMO兲
1
/ 共SMO兲
3
, to study the elec-
tronic and magnetic properties at different strain conditions.
As is well known, the strength of the Jahn-Teller 共JT兲 distor-
tion is less in the mixed compounds 共La,Sr兲MnO
3
as com-
pared to that of LaMnO
3
, we have considered a small Jahn-
Teller distortion 共Q
2
⬇0.05 Å兲 in the basal plane for the
interfacial MnO
2
layers. However, test calculations showed
that a small variation of Q
2
does not change the electronic
and magnetic properties of the superlattice qualitatively.
All electronic structure calculations reported in this work
have been performed using the self-consistent tight-binding
linearized muffin-tin orbital 共TB-LMTO兲 method with the
atomic sphere approximation 共ASA兲.
9
Self-consistent cal-
culations are done within the framework of generalized
gradient approximation including Coulomb correction
共GGA+U兲. All results are obtained with U=5 eV and
J=1 eV unless otherwise stated.
III. ELECTRONIC STRUCTURE OF THE
(LaMnO
3
)
1
Õ (SrMnO
3
)
1
SUPERLATTICE
In this section, we describe the effect of strain on the
electronic structure at the interface from ab initio DFT cal-
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
x
2
−y
2
x
2
−y
2
3z
2
−1
∆
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3z
2
−1
O
Mn
x
2
−y
2
3
z
2
−1
∆
Mn
O
TensileCompressive
FIG. 2. Energy splitting of the Mn共e
g
兲 orbitals at the LMO/SMO
interface for compressive and tensile strain conditions. The param-
eter ⌬ is the difference between the energies of the Mn d
x
2
−y
2 and
the Mn d
3z
2
−1
orbitals. Compressive strain makes the 3z
2
−1 orbital
lower in energy, while tensile strain makes it higher.
B. R. K. NANDA AND SASHI SATPATHY PHYSICAL REVIEW B 78, 054427 共2008兲
054427-2
culations. We focus on the 共LMO兲
1
/ 共SMO兲
1
superlattice and
our results suggest that many of the interfacial electronic and
magnetic properties shown by this superlattice should also be
valid for the more general 共LMO兲
n
/ 共SMO兲
m
superlattices.
We briefly summarize the electronic structure and magne-
tism for the bulk SMO and LMO compounds. In bulk SMO,
the Mn atoms that are in 4+ charge state have three d elec-
trons which are occupied in the triply degenerate t
2g
states.
The doubly degenerate e
g
states, which are higher in energy
with respect to t
2g
states because of a MnO
6
octahedral
crystal-field split, remain unoccupied. The t
2g
3
spin-majority
states mediate an antiferromagnetic superexchange to stabi-
lize the G-type antiferromagnetic ordering in the bulk SMO
compound, where spin of each Mn atom is opposite to that of
the nearest-neighbor Mn atoms.
10,11
In bulk LMO the Mn atoms that are in 3+ charge state
have four d electrons. Three electrons are occupied in the
localized t
2g
states and the remaining one electron is occu-
pied in the e
g
state. The JT distortion to the MnO
6
octahe-
dron further splits the e
g
states into two nondegenerate states:
e
g
1
which is lower in energy and e
g
2
which is higher in
energy.
12
The one e
g
electron is occupied in the e
g
1
state
whose lobes are pointed toward the longest Mn-O bond. The
JT distortion stabilizes the A-type antiferromagnetic structure
in the LMO compound.
13
At the LMO/SMO interface the Mn atoms do not satisfy
the 4+ charge state or the 3+ charge state to support the bulk
magnetism of SMO or LMO. The mixed-valence nature of
the Mn atoms as well as the effect of epitaxial strain create
diverse magnetic phases at the interface, which will be ana-
lyzed in this section.
Epitaxial strain, arising due to the substrate on which the
interface is grown, induces tetragonal distortion to the cubic
interface which is quantified by the c / a ratio that differs
from one. Experimental studies show different magnetic be-
haviors at the interface for different c/ a ratios.
1
To obtain the
dependence of the magnetic ground state on the strain con-
dition, we have performed total-energy calculations in the
range 0.95ⱕ c / a ⱕ 1.05 for three possible magnetic configu-
rations 共A, F, and C兲共Fig. 1兲. Magnetic configuration A rep-
resents the FM ordering in the MnO
2
plane and AFM order-
ing between the planes. Magnetic configuration C represents
the AFM ordering in the MnO
2
plane and FM ordering be-
tween the planes and F represents the FM ordering in all
directions. The energetics is shown in Fig. 3 共top兲.
From the figure we see that for a strong compressive
strain 共e.g., c / a =0.95兲, “A” is the most stable magnetic con-
figuration. For the lattice-matched structure 共c/ a =1, no
strain兲, the interface stabilizes with magnetic configuration F
and in case of a strong compressive strain 共e.g., c / a =1.05兲 it
stabilizes with magnetic configuration C. The results are in
accordance with the experimental observations which show
that when the substrates are STO 共c/ a =0.98兲 , LSAT 共c / a
=1.01兲, and LAO 共c/ a =1.05兲, the respective magnetic con-
figurations at the LMO/SMO interface are A, F, and C.
1
We see that as strain changes, the occupancy of the e
g
orbitals, which controls the magnetic interaction at the inter-
face, also changes. This is shown in Fig. 3 共bottom兲. For the
tensile strain condition 共c / a ⬍ 1兲 the occupancy of the
x
2
−y
2
orbital is greater than the occupancy of the 3z
2
−1
orbital and opposite if the strain is compressive 共c / a ⬎ 1兲.
For the lattice-matched structure 共c / a=1兲 both the e
g
orbitals
are more or less equally occupied. Figure 3 also shows that
for any value of c/ a, the nondegenerate e
g
states combinedly
occupy 0.5 electrons which along with three t
2g
core elec-
trons make the average valence of the interface Mn atoms to
be +3.5 as expected.
Magnetic interaction between the Mn atoms is determined
by the competition between ferromagnetic double
exchange
14–16
via the itinerant Mn e
g
electrons and antiferro-
magnetic superexchange between the localized Mn t
2g
core
spins. When x
2
−y
2
is more occupied and 3z
2
−1 orbital is
less occupied 共or unoccupied兲, the strong double exchange in
the MnO
2
plane strengthens the ferromagnetic ordering
while superexchange stabilizes the antiferromagnetic order-
ing between the planes. The magnetic ordering is opposite to
the above when the occupancies of the two e
g
orbitals are
reversed. If both the e
g
orbitals are more or less equally
occupied, the double exchange stabilizes the ferromagnetic
ordering both in the plane and between the planes. As de-
scribed in Secs. III A–III C, a detailed analysis of the
density-functional electronic structure of the LMO/SMO in-
terface under different strain conditions gives us a better un-
derstanding on the strain induced orbital ordering and its
effect on magnetic properties at the interface.
A. c Õ a =0.95, tensile strain
Tensile strain reduces the out-of-plane lattice parameter c
and enhances the in-plane lattice parameter a. In other words
it decreases the Mn-O bond length between the MnO
2
planes
and increases it in the plane. In such a scenario, the total-
x
−y
2
2
3z
2
−1
(
LM
O)
1
/
(
S
M
O
)
1
c/a
Tensile Compressive
Lattice
Matched
0.96 1.00 1.04
0
0.2
0.4
O
rbital
O
ccupancy
0.96 1.00 1.04
Energy
(
eV
)
GGA+U
0
−0.05
0.05
0.10
0.15
0.20
E
F
E
A
E
C
FIG. 3. Total energies for the magnetic configurations, A and C,
relative to the energy for configuration F, as a function of the te-
tragonal distortion c/ a 共top兲. The magnetic configurations A, F, and
C are shown in Fig. 1. Bottom figure shows the occupancies of the
x
2
−y
2
and the 3z
2
−1 orbitals per Mn atom as a function of the
tetragonal distortion.
EFFECTS OF STRAIN ON ORBITAL ORDERING AND… PHYSICAL REVIEW B 78, 054427 共2008兲
054427-3
energy calculation 共Fig. 3兲 suggests a stable A-type magnetic
configuration 共Fig. 1兲 when tetragonal distortion 共c / a兲 is
close to 0.95. In Fig. 4, we have shown the total and partial
densities of states 共DOSs兲 for the 共LMO兲
1
/ 共SMO兲
1
superlat-
tice 共 c / a =0.95兲 in the A-type structure obtained from the
GGA+U calculations.
The characteristic features of the electronic structure un-
der tensile strain as seen from Fig. 4 are as follows. The
localized Mn t
2g
states lie far below the Fermi level 共E
F
兲
because of the octahedral crystal field and strong Coulomb
repulsion. The O p states occur in the energy range of −6 to
−1 eV. The x
2
−y
2
and 3z
2
−1 orbitals are predominant at
E
F
. Since the intraplane 共on the xy-plane兲 Mn-O bond is
longer than the interplane 共along the z-axis兲 one 共Fig. 2兲, this
lowers the energy of the x
2
−y
2
orbital making it more occu-
pied and raises the energy of the 3z
2
−1 orbital, which be-
comes less occupied.
The origin behind the stability of A-type magnetic con-
figuration for the tensile interface will be explained below. In
the bulk LMO, Mn 共3+兲 atom has the electronic configura-
tion t
2g
3
e
g
1
and in the bulk SMO, Mn 共4+兲 atom has the elec-
tronic configuration t
2g
3
e
g
0
. Since at the interface, the MnO
2
layers are surrounded by 共SrO兲
0
layer and 共LaO兲
1+
layer, the
interface Mn atoms are left with the average valence state of
+3.5. In such a scenario, the t
2g
orbitals will occupy three
electrons in the spin-majority states and the e
g
orbitals will
occupy the remaining 0.5 electrons.
Without any occupancy of the e
g
states, the only contri-
bution to the energy comes from the superexchange interac-
tion between the localized t
2g
states to stabilize the G-type
AFM phase as in the case of SMO. However, the itinerant e
g
states, if partially occupied, can mediate the Anderson-
Hasegawa double exchange
14
to stabilize the FM phase.
The strength of the FM ordering in the plane or out of the
plane depends on the occupancy of the individual x
2
−y
2
and
3z
2
−1 orbitals. From our calculations 共Fig. 3兲 we find that
for tensile strain condition 共c / a =0.95兲, the occupancy of
x
2
−y
2
orbital is close to 0.45, while for 3z
2
−1 orbital it is
less than 0.1. This is also reflected in the Mn e
g
band disper-
sion for the A-type magnetic configuration shown in Fig. 5.
Since, the unit cell for the magnetic structure is doubled
along the xy plane 关i.e., 2⫻共LMO兲
1
/ 共SMO兲
1
兴, for the AFM
magnetic configuration we have two Mn↑ and two Mn↓ at-
oms. Hence, for the local spin-majority channel, we have
two x
2
−y
2
orbitals and two 3z
2
−1 orbitals. From the figure
we see that the 3z
2
−1 orbitals are mostly in the conduction
band and only one x
2
−y
2
orbital of two crosses the Fermi
level and lies mostly in the valence band. This implies that
almost one electron per two Mn atoms in the x
2
−y
2
states is
occupied which is consistent with the orbital occupancy cal-
culation.
In such a case the x
2
−y
2
orbitals will mediate the double
exchange mechanism in the MnO
2
plane to stabilize a FM
ordering in the plane. The gain in kinetic energy due to the
planar orbital order, induced by the anisotropic hopping, is
more than the loss of superexchange energy. Since the
3z
2
−1 orbitals are only marginally occupied, superexchange
between the localized t
2g
electrons stabilizes the AFM order-
ing between the MnO
2
planes. The net result is an A-type
AFM ordering at the interface.
The valence electron charge-density contours for states in
the vicinity of the Fermi level 共E
F
兲, shown in Fig. 6, provide
a visualization of the above analysis. The charge contours
show that the orbital ordering is predominantly Mn x
2
−y
2
,
t
2g
O (p)
x
2
−y
2
3z
2
−1
t
2g
e
g
1
/(S
M
O)
1
(
LM
O)
La (d)
Sr (d)
La (f)
0
10
20
-8 -4 0 4 8
GGA+U E
F
Ener
gy (
eV
)
D
OS
(states
/
eV cell)
Tensile c
/
a = 0.95
FIG. 4. Total and partial DOSs for the 共LMO兲
1
/ 共SMO兲
1
super-
lattice 共c / a =0.95兲 in the A-type magnetic configuration. The sym-
bols ↑ and ↓ represent the local spin of the atoms. The Mn e
g
↑ state
at the Fermi level 共E
F
兲 splits into x
2
−y
2
and 3z
2
−1 states. The
orbital character of the e
g
states at E
F
is shown in Fig. 5.
(LM
O
)
1
/(SMO)
1
3z
2
−1
3z
2
−1
x
2
−y
2
x
2
−y
2
x
2
−y
2
3z
2
−1
3z
2
−1
x
2
−y
2
x
2
−y
2
Energy
(
eV
)
−2
−1
2
0
0
2
4
−2
0
2
Γ XZRΓ XZRΓ XZ
R
(
GG
A+U)
CAF
FIG. 5. 共Color online兲 Orbital character of the electron bands
near E
F
for the three magnetic structures A, F, and C. The bands are
plotted along the high-symmetry points ⌫共0,0,0兲, X共
2a
,−
2a
,0兲,
Z共0,0,−
2c
兲, and R共
2a
,−
2a
,
2c
兲. The unit cell for the magnetic
structures is doubled along the xy plane with the f.u.
2⫻共LMO兲
1
/ 共SMO兲
1
. For the AFM configurations 共A and C兲,we
have two Mn↑ and two Mn↓ atoms. Only spin-up bands are shown
for the FM configuration F.
(LMO)
1
/(SMO)
1
MnO
2
MnO
2
GGA+U
c/a = 0.95
Mn
Mn Mn
O LaO
SrO
SrO
Mn
x
z
FIG. 6. 共Color online兲 Valence electron charge-density contours
plotted on the xz plane in the energy range E
F
−0.15 eV to E
F
to
indicate the orbital ordering for the A-type magnetic configuration.
Contour values are
n
=
0
⫻10
n
␦
e/ Å
3
, where
0
=3.7⫻10
−3
,
␦
=0.4, and n labels the contours. The charge contours on the yz
plane 共not shown兲 are identical to that of xz plane. The orbital
ordering is mainly x
2
−y
2
.
B. R. K. NANDA AND SASHI SATPATHY PHYSICAL REVIEW B 78, 054427 共2008兲
054427-4
O p
x
, and p
y
, while the occupancies of the 3z
2
−1 and p
z
orbitals are small. As a result we see a strong coupling be-
tween the Mn e
g
and O p orbitals in the plane while it is
rather weak between the planes. Therefore, the in-plane mag-
netic exchange interaction J
⬘
is ferromagnetic, while the out-
of-plane J is antiferromagnetic 共Fig. 1兲. Our results are con-
sistent with the experimental results that the magnetic
ordering at the interface for 共LMO兲
3
/ 共SMO兲
2
superlattice
grown on STO substrate 共c/ a =0.98兲 is A type.
1
B. c Õ a =1.0, lattice-matched structure
Lattice-matched interfaces are without any tetragonal dis-
tortion and hence the in-plane and out-of-plane Mn-O bond
lengths are identical. Total-energy calculation 共Fig. 3兲 in this
case favors a three-dimensional FM ordering 共F type兲.To
gain insight into the origin behind the FM ground state, we
analyze the electronic structure for the lattice-matched inter-
face. In Fig. 7, we have shown the total spin-up and spin-
down DOSs for the F-type magnetic configuration.
General features of the electronic structure of the lattice-
matched interface are similar to that of the tensile interface.
However, now on either side of the Fermi level, both
x
2
−y
2
and 3z
2
−1 orbitals are predominant in the spin-up
channel and they have nearly equal on-site energies. It is due
to the fact that the Mn-O bond lengths are same both in plane
and out of plane, making the e
g
states nearly degenerate in
energy. This is also substantiated from the dispersion of the
spin-up Mn e
g
bands for the F-type structure shown in Fig. 5.
Since the f.u. is doubled along the xy plane, there are four
Mn atoms and all are in the same spin orientation. Hence, in
the spin-up channel, we have eight e
g
bands of which almost
six lie in the conduction band. Of the remaining two bands,
which are part of the valence bands, one is predominantly of
3z
2
−1 character, while the other is predominantly x
2
−y
2
.
Hence the occupancy of each of these orbitals is close to a
quarter electron per Mn atom which is also seen from the
orbital occupancy results of Fig. 3. The valence charge-
density contours of Fig. 8 indicate the orbital occupancy of
the two Mn e
g
orbitals as well as their hybridization with the
O p orbitals.
The partially occupied x
2
−y
2
and 3z
2
−1 orbitals mediate
a ferromagnetic double exchange, strong enough to over-
come the antiferromagnetic superexchange both in the plane
and out of the plane to stabilize a three-dimensional FM
ordering. We have shown earlier
17
that in the case of
CaMnO
3
/ CaRuO
3
interface, a leaking of 0.2 electrons from
the metallic CaRuO
3
side to the Mn e
g
states near the inter-
face, which were otherwise unoccupied, is sufficient to sta-
bilize the FM ordering of the Mn spins. In the present case,
both the e
g
orbitals being occupied substantially 共more than
0.2 electrons in each orbital兲, a strong ferromagnetic double
exchange coupling along all directions is expected. This is
consistent with the experimental observation of ferromag-
netism in the LMO/SMO interface structures grown on the
LSAT substrate 共c / a =1.01兲.
1
The other prominent feature in
the electronic structure of the lattice-matched interface is the
opening of a gap at the Fermi level in the spin-down channel
which makes the system half-metallic 共Fig. 7兲.
C. c Õ a =1.05, compressive strain
When the strain is compressive, the Mn-O bond length
reduces in the MnO
2
plane while it increases between the
planes. As a result, the 3z
2
−1 orbital is lower in energy and
is more occupied, while the x
2
−y
2
orbital is higher in energy
and is less occupied which is seen from the densities of states
共Fig. 9兲 as well as from the band structure 共Fig. 5, right
panel兲.
As in the case of A-type magnetic configuration in the
tensile strain condition discussed earlier, here also we have
two Mn↑ and two Mn↓ atoms. So for the local spin-majority
channel, we have two x
2
−y
2
orbitals and two 3z
2
−1 orbitals.
From Fig. 5 we see that the x
2
−y
2
orbitals lie in the conduc-
tion band and only one of the two 3z
2
−1 orbitals lies in the
valence band. This shows that the occupancy of the 3z
2
−1
orbital per Mn atom is close to 0.5 and x
2
−y
2
orbitals are
basically unoccupied. This is seen from the orbital occu-
pancy 共Fig. 3兲 as well as from the charge-density contour
t
2g
t
2g
e
g
x
2
−y
2
3z
2
−1
-20
-10
0
10
20
-8 -4 0 4
8
E
F
D
OS
(states
/
eV cell)
Energy (eV)
O (p)
La (f)
La (d)
Sr (d)
Spin−Down
Spin−Up
(S
M
O)
1
/(
LM
O)
1
Lattice matched, c
/
a = 1.0
GGA+U
FIG. 7. Total spin-up and spin-down DOSs for
共LMO兲
1
/ 共SMO兲
1
superlattice in the F-type magnetic configuration.
Both x
2
−y
2
and 3z
2
−1 orbitals are more or less equally occupied.
The orbital character of the e
g
states at E
F
in the spin-up channel is
shown in Fig. 5. The superlattice shows the half-metallic behavior.
MnO
2
MnO
2
(
LM
O)
1
/
(
S
M
O
)
1
x
z
Mn Mn
Mn
O
Mn
LaO
SrO
SrO
c
/
a = 1.0
FIG. 8. 共Color online兲 Valence electron charge-density contours
plotted on the xz plane in the energy range E
F
−0.15 eV to E
F
to
indicate the orbital ordering for the F-type magnetic configuration.
Contour values are:
n
=
0
⫻10
n
␦
e/ Å
3
, where
0
=3.7⫻10
−3
,
␦
=0.4, and n labels the contours. The charge contours on the yz
plane 共not shown兲 are identical to that of xz plane. The orbital state
of each Mn is a mixture of x
2
−y
2
and 3z
2
−1.
EFFECTS OF STRAIN ON ORBITAL ORDERING AND… PHYSICAL REVIEW B 78, 054427 共2008兲
054427-5
