Electronic and magnetic structure of the (LaMnO
3
)
2n
Õ (SrMnO
3
)
n
superlattices
B. R. K. Nanda and S. Satpathy
Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
共Received 22 October 2008; published 24 February 2009
兲
We study the magnetic structure of the 共LaMnO
3
兲
2n
/ 共SrMnO
3
兲
n
superlattices from density-functional cal-
culations. In agreement with the experiments, we find that the magnetism changes with the layer thickness n.
The reason for the different magnetic structures is shown to be the varying potential barrier across the
interface, which controls the leakage of the Mn-e
g
electrons from the LaMnO
3
side to the SrMnO
3
side. This
in turn affects the interfacial magnetism via the carrier-mediated Zener double exchange. For the n =1 super-
lattice, the Mn-e
g
electrons are more or less spread over the entire lattice so that the magnetic behavior is
similar to the equivalent alloy compound La
2/3
Sr
1/3
MnO
3
. For larger n, the e
g
electron transfer occurs mostly
between the two layers adjacent to the interface, thus leaving the magnetism unchanged and bulklike away
from the interface region.
DOI: 10.1103/PhysRevB.79.054428 PACS number共s兲: 75.70.Cn, 71.20.⫺b, 73.20.⫺r
I. INTRODUCTION
Superlattices made up of strongly correlated transition-
metal oxides such as LaMnO
3
共LMO兲 and SrMnO
3
共SMO兲
are of current interest because of the diverse magnetic and
electronic phases they exhibit. For example recent experi-
mental results reveal that 共LMO兲
2n
/ 共SMO兲
n
superlattice is
uniformly ferromagnetic for the short-period structure 共n
=1兲, while the long-period superlattices 共nⱖ 3兲 show bulk
antiferromagnetic ordering away from the interface and fer-
romagnetic ordering at the interface.
1,2
In this paper, we report results of our electronic structure
calculations, based on the density-functional theory 共DFT兲,
performed to understand the change in the magnetic proper-
ties of the 共LMO兲
2n
/ 共SMO兲
n
superlattices as a function of
the layer thickness n. We show that there exists a potential
barrier for the electrons, in particular, for the Mn-e
g
elec-
trons, the strength of which differs with the layer thickness n.
This varying potential barrier, which controls the leakage of
the Mn-e
g
electrons from the LMO side to SMO side, in turn
determines the stable magnetic configurations in the
共LMO兲
2n
/ 共SMO兲
n
superlattices. In agreement with the ex-
periments, our calculations predict a uniform ferromagnetic
共FM兲 ordering in the short-period superlattice 共n =1兲 and the
co-existence of interface FM phase and inner bulk antiferro-
magnetic 共AFM兲 phases in the long-period superlattices 共n
ⱖ 3兲. The magnetism can be qualitatively understood in
terms of the two competing interactions, viz., the antiferro-
magnetic superexchange between the core spins and the Ze-
ner ferromagnetic double exchange mediated by the itinerant
e
g
electrons.
II. COMPUTATIONAL AND STRUCTURAL DETAILS
The results presented in this paper are obtained from the
DFT studies of three superlattices, namely, 共LMO兲
2
/ 共SMO兲
1
共schematically shown in Fig. 1兲, 共LMO兲
4
/ 共SMO兲
2
, and
共LMO兲
6
/ 共SMO兲
3
using the linear muffin-tin orbitals
共LMTO兲 method
3
with general gradient approximation
4
and
on-site Coulomb correction 共GGA+U兲.
5
The Coulomb 共U兲
and the exchange parameter 共J兲 are taken as 5 and 1 eV,
respectively. Each superlattice consists of twice the formula
unit because of the magnetic structures considered in the
paper.
The bulk lattice parameters of LMO and SMO are, re-
spectively, 3.935 and 3.802 Å. However, since most of the
experimental results reported in the literature are based on
the LMO/SMO superlattices grown on the SrTiO
3
共STO兲
substrate,
1,2,6,7
we have taken the in-plane lattice parameter
for the 共LMO兲
2n
/ 共SMO兲
n
superlattices as the bulk STO lat-
tice parameter 共3.905 Å兲. The out-of-plane lattice param-
eters are taken to be 3.99 共LMO兲 and 3.65 Å 共SMO兲 which
preserve the bulk volumes. A somewhat better estimate of
the out-of-plane lattice parameters may be obtained from the
linear relation containing the Poisson’s ratio,
8–10
which
would yield the values 3.95 共LMO兲 and 3.78 Å 共SMO兲.We
do not expect these differences to change the basic physics
discussed here. However, a substantial change in the strain
condition, obtained for example by growing the superlattice
on different substrates, can alter the orbital ordering and
through it the interfacial magnetic structure as discussed
MnO
2
MnO
2
MnO
2
MnO
2
J
2
J
3
J
3
J
1
J
2
J
0
J
0
SrO
LaO
LaO
La
Sr
O
Mn0
Mn0
Mn0
Mn1
Mn
S
M
O
LM
O
FIG. 1. Schematic unit cell of 共LMO兲
2
/ 共SMO兲
1
superlattice and
the magnetic structure as predicted from the DFT calculations.
Mn-0 represents the interfacial Mn atoms surrounded by both SrO
and LaO layers and Mn-1 represents the Mn atoms inside the LMO
part. Because the SMO part is small, there is no Mn atom sur-
rounded by two SrO layers in this structure. The nearest-neighbor
Mn-Mn exchange interactions are indicated by the J’s.
PHYSICAL REVIEW B 79, 054428 共2009兲
1098-0121/2009/79共5兲/054428共6兲 ©2009 The American Physical Society054428-1
elsewhere.
8,9
The basal Jahn-Teller 共JT兲 distortion 共Q
2
兲 for
the inner Mn layers in the LMO site is taken the same as the
bulk value 共0.15 Å兲. The value of Q
2
for the interface Mn
layers is taken as 0.07 Å in view of the fact that the JT
distortion is reduced in the mixed compound 共La,Sr兲MnO
3
,
and one expects the distortion to scale roughly linearly with
the number of e
g
electrons on the Mn atom, which is ap-
proximately half for the interfacial Mn atom.
III. ELECTRONIC STRUCTURE OF THE (LMO)
2
Õ (SMO)
1
SUPERLATTICE
Before discussing the electronic and magnetic properties
of the 共LMO兲
2n
/ 共SMO兲
n
superlattices, we summarize the
electronic structure and magnetism of the bulk SMO and
LMO compounds. In bulk SMO, the Mn atoms are in the 4
+ charged state so that they have three d electrons occupying
the triply-degenerate t
2g
states. The doubly-degenerate e
g
states, which are higher in energy with respect to the t
2g
states because of the MnO
6
octahedral crystal field, remain
unoccupied. The t
2g
3
core spins interact via an antiferromag-
netic superexchange so as to stabilize the G-type AFM or-
dering in the bulk SMO compound.
11,12
In bulk LMO, the Mn atoms are in the 3+ charged state
with four occupied d electrons. Three electrons are present in
the t
2g
states and the remaining one in the e
g
states. The
Jahn-Teller distortion of the MnO
6
octahedron 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.
13
The e
g
1
orbital,
occupied by the lone electron, has its lobes pointed toward
the longest Mn-O bond. The JT distortion stabilizes the
A-type AFM structure in the LMO compound due to a com-
bination of the superexchange and Zener double exchange.
14
The charge reconstruction at the LMO/SMO interface
15,16
is
expected to change the electronic and magnetic properties of
the 共LMO兲
2n
/ 共SMO兲
n
superlattices, which will be discussed
in the remaining part of the paper.
Out of a number of magnetic configurations that we con-
sidered, the DFT calculations predict a ferromagnetic ground
state for the 共LMO兲
2
/ 共SMO兲
1
superlattice. In Fig. 2, we have
shown the total and partial spin-resolved densities of states
共DOSs兲 for the ferromagnetic configuration of this superlat-
tice. The characteristic features of the electronic structure as
seen from the figure are as follows. The 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 Mn-e
g
states occur
around the Fermi level E
F
, while the Sr-d, La-d, and La-f
states lie far above it.
As Fig. 2 shows, the most important feature in the elec-
tronic structure of 共LMO兲
2
/ 共SMO兲
1
is that the delocalized e
g
states of both Mn-0 共Mn atoms at the interface兲 and Mn-1
共Mn atoms inside the LMO part兲 are partially occupied,
which is in agreement with the earlier electronic structure
calculations.
17
These partially occupied e
g
states will mediate
a strong Zener ferromagnetic double exchange
18–20
between
the Mn-t
2g
core spins, which wins over the antiferromagnetic
superexchange, so that a uniform ferromagnetic ordering
throughout the superlattice is stabilized. The calculation of
the Mn-Mn exchange interactions discussed below indicates
that the FM ordering is stable, quite similar to the equivalent
alloy compound La
2/3
Sr
1/3
MnO
3
.
1
IV. MAGNETIC EXCHANGE INTERACTION
In order to study the magnetic ground state for the
共LMO兲
2n
/ 共SMO兲
n
superlattices, we have calculated the
neighboring Mn-Mn exchange interaction energies 共J’s兲 for
various exchange interactions as shown in Figs. 1 and 3.In
these figures the symbol J
1
represents the out-of-plane ex-
change interactions across the SrO layer close to the inter-
face, while J
3
and J
4
represent the same across the LaO
layers close to the interface and away from the interface,
respectively. J
2
denotes the in-plane exchange interaction for
the interfacial MnO
2
layer, which is surrounded by LaO and
SrO layers, whereas J
0
and J
5
denote the same for the MnO
2
layer inside the LMO and SMO part of the superlattice, re-
spectively.
The exchange interaction J is defined as the energy dif-
ference between the ferromagnetic alignment and the antifer-
romagnetic alignment of two neighboring Mn spins 共J =E
↑↑
(
LM
O)
2
/
(
S
M
O
)
1
t
2g
t
2g
e
g
e
g
t
2g
t
2g
t
2g
e
g
E
F
e
g
t
2g
O−p
La−f
La−d
O−p
Mn0−d
Mn1−d
−8 −4
04
5
0
10
20
20
0
0
20
20
0
10
5
Ener
g
y (eV)
DOS (states/eV cell)
FIG. 2. Total 共upper panel兲 and partial spin-resolved DOS for
the ferromagnetic 共LMO兲
2
/ 共SMO兲
1
superlattice. The labeling of the
Mn atoms is as in Fig. 1. Upper and lower segments within each
panel correspond, respectively, to the majority 共↑兲 and minority 共↓兲
spin densities.
B. R. K. NANDA AND S. SATPATHY PHYSICAL REVIEW B 79, 054428 共2009兲
054428-2
−E
↑↓
兲. We computed them by performing a number of total-
energy calculations for various magnetic configurations for
each superlattice and fitting the energies with the results of a
nearest-neighbor Heisenberg model. The results are listed in
Table I. The in-plane magnetic interaction J
0
inside the LMO
part, which is strongly ferromagnetic, was not computed. For
the case of the 共LMO兲
6
/ 共SMO兲
3
superlattice, the values of
the exchange interaction for the Mn layers away from the
interface in the LMO part and SMO part are, respectively, 12
共J
4
兲 and 19 meV 共J
5
兲. These values are in good agreement
with the experimental results for the bulk LMO 共J
⬃9.7 meV兲 and bulk SMO 共J ⬃13.1 meV兲.
21–23
From Table I, we see that for 共LMO兲
2
/ 共SMO兲
1
, the in-
plane exchange interaction J
1
, as well as the out-of-plane
exchange interactions J
2
and J
3
, is strong and negative so as
to stabilize the FM ordering throughout the superlattice, con-
sistent with the experimental observations.
1,2
Turning now to
the 共LMO兲
4
/ 共SMO兲
2
superlattice, the in-plane interactions
共J
0
and J
2
兲 are FM as also are the out-of-plane interactions
共J
3
and J
4
兲 within the LMO part. In the SMO part, the out-
of-plane exchange interaction 共J
1
兲 is AFM, but this being
weaker as compared to the in-plane J
2
共FM兲 and J
5
共AFM兲,
the magnetic configuration within the SMO part is controlled
by the latter two exchange interactions as shown in Fig. 3.
Finally, for the 共LMO兲
6
/ 共SMO兲
3
superlattice, the values of
J’s are similar to those of the 共LMO兲
4
/ 共SMO兲
2
superlattice,
except that now the out-of-plane exchange interaction for the
inner MnO
2
layers in the LMO side 共J
4
兲 is positive so as to
establish an A-type AFM configuration as in the bulk LMO
compound.
We note from the above discussions that as we increase
the layer thickness n, the FM interactions between the Mn
spins occurring on the two sides of the LaO layers 共see J
3
and J
4
in Table I兲 gradually become weak, which eventually
makes the LMO part type-A AFM like in the bulk. This
already happens for n=3. The transition from the FM to
AFM ordering for the Mn layers away from the interface
with the increase in the layer thickness n is indicative of the
fact that the charge reconstruction is essentially confined to
the few interface layers for the long-period superlattices 共n
ⱖ 3兲.
The calculated magnetic moments of the Mn atoms
共within the muffin-tin sphere radius of 1.15 Å兲 are also
given in Table I. Our magnetic unit cells consisted of two Mn
atoms per layer, with slightly different magnetic moments,
which we have averaged over to obtain the results presented
in Table I. For the n =2 and 3 superlattices, where bulklike
SMO and LMO regions exist, the magnetic moments are
consistent with the t
2g
3
↑ occupancy of the Mn-1 共bulk SMO
like兲, t
2g
3
↑e
g
1
↑ occupancies of the Mn-1, Mn-2, Mn-3 共bulk
LMO like兲, and the t
2g
3
↑e
g
0.5
↑ occupancy of the Mn-0 共inter-
facial Mn兲, which gives rise to the nominal magnetic mo-
ments of 3
B
,4
B
, and 3.5
B
, respectively. For the n =1
superlattice, the e
g
electrons are spread more or less all over
the lattice, and this is reflected in the near equality of the
magnetic moments of the two Mn atoms: 3.57
B
for Mn-0
and 3.70
B
for Mn-1, as seen from Table I.
V. ELECTRIC POTENTIAL PROFILE AND CHARGE
RECONSTRUCTION AT THE INTERFACE
The potential seen by the electrons varies as one crosses
the interface from one side to the other. This for example
J
2
J
4
J
3
J
4
J
3
J
2
J
1
J
5
J
0
La
Sr
Mn−1
Mn0
Mn1
Mn2
Mn1
Mn0
LM
OS
M
O
FIG. 3. Schematic unit cell of 共LMO兲
4
/ 共SMO兲
2
superlattice and
the magnetic structure as obtained from the DFT calculations. Oxy-
gen atoms occur at the intersections of the checkered lines forming
the MnO
6
octahedron. Mn atoms of each MnO
2
layer are labeled as
shown in the figure. Definitions of the exchange interactions for the
共LMO兲
6
/ 共SMO兲
3
superlattice are identical to the ones shown here,
and they are also consistent with Fig. 1 for the 共LMO兲
4
/ 共SMO兲
2
superlattice.
TABLE I. Calculated magnetic moments radius 1.153 Å and the exchange interactions. A negative J
corresponds to an FM interaction and a positive J corresponds to an AFM interaction.
Superlattice
Magnetic moment
共
B
兲
Exchange interaction
共meV兲
Mn-1 Mn-0 Mn-1 Mn-2 J
1
J
2
J
3
J
4
J
5
共LMO兲
2
/ 共SMO兲
1
3.57 3.70 −11 −39 −26
共LM兲
4
/ 共SMO兲
2
2.99 3.52 3.77 3.80 10 −36 −18 −4 17
共LMO兲
6
/ 共SMO兲
3
2.98 3.51 3.75 3.77 14 −37 −6 12 19
ELECTRONIC AND MAGNETIC STRUCTURE OF THE… PHYSICAL REVIEW B 79, 054428 共2009兲
054428-3
leads to the well-known band offset in the semiconductors.
Our calculations show that for the present superlattices, there
is a potential barrier as one goes from the LMO to the SMO
side. This controls the leakage of the Mn-e
g
electrons across
the barrier, which in turn affects the magnetic exchange in-
teractions near the interface leading to diverse magnetic
phases.
In Figs. 4共a兲–4共c兲, we plot the calculated oxygen 1s core
energies, indicating the potential barrier across the interface.
However, the valence states experience a somewhat different
potential than the core states because of different energy
terms. Since the Mn-e
g
electrons are mainly the electrons
that are transferred across the interface, we now examine the
potential V共z兲 felt by these electrons. In order to obtain the
variation in this potential, we have studied the band structure
and the atomic characters of the wave functions in each su-
perlattice by examining the so-called “fat” bands in the
LMTO results, which indicate the relative contributions of
the various orbitals to the wave function making the band.
From the fat bands, the lowest Mn-e
g
state belonging to a
particular Mn layer can be identified, which is then indica-
tive of the potential experienced by the Mn-e
g
electrons in
the various layers.
These results are shown in Fig. 4共d兲. The variation in V共z兲
for the n =1 superlattice is quite similar to the variation in the
oxygen 1s core energies and hence is not shown in the figure.
For this superlattice, we have a weakly varying potential due
to the close proximity of the interfaces to one another, which
results in the overlap of the attractive Coulomb potential
formed by the positively charged interfacial LaO
+
layers. In
this case the Mn-e
g
electrons are more or less spread
throughout the superlattice as seen from the layer-projected
DOS 共Fig. 2兲, where all Mn atoms have partially filled e
g
states. These itinerant e
g
electrons mediate the Zener double
exchange stabilizing the FM ordering throughout the super-
lattice.
With the increase in the layer thickness n, the variation in
the potential becomes stronger, leading to the formation of a
potential barrier at the interface with the LMO side having a
lower potential than the SMO side. This results in restricting
the leakage of the Mn-e
g
electrons to the SMO side 共Fig. 4兲.
Thus, for example in the case of the n =3 superlattice, there
is very little e
g
electron on the Mn-1 atom belonging to the
SMO side 共Fig. 5, topmost panel兲. Since the Mn-e
g
states are
unoccupied in the SMO side, a G-type AFM structure is
stabilized as in the bulk SMO.
The case of the n=2 superlattice is intermediate between
the short-period and the long-period 共nⱖ 3兲 superlattices.
Here, on one hand, the leakage of electrons to the SMO side
is small enough that the G-type AFM is maintained there as
in the bulk. On the other hand the number of e
g
electrons
leaving the LMO side is large enough that the LMO part
behaves like a hole doped bulk 共La
1−x
Sr
x
MnO
3
兲, thereby sta-
bilizing the FM structure as in the short-period superlattice
共n =1兲. However, as the calculated ferromagnetic stabiliza-
tion energy is relatively small here as compared to the n=1
case, it is only weakly ferromagnetic 共J
4
=−4 meV, Table I兲.
In contrast to this, in the long-period superlattices 共n
ⱖ 3兲, a much stronger potential barrier prevents any signifi-
cant leakage of the electrons to the SMO side, except to the
very first interfacial layer. This leads to the bulk magnetic
behavior inside the LMO as well as the SMO parts. The only
layers affected by the electron leakage are just two layers at
the interface so that the magnetic structure as indicated in
Fig. 4共a兲 is of the type ...兩FGGF·FAAAF兩..., where the ver-
tical line indicates the interface. The calculated ground-state
magnetic structures for the three superlattices discussed in
this paper agree with those observed in the experiments.
1,2
VI. ELECTRONIC STRUCTURE OF THE (LMO)
6
Õ (SMO)
3
SUPERLATTICE
We now turn to the electronic structure of the
共LMO兲
6
/ 共SMO兲
3
superlattice, which would be typical of the
F
GF F FGF
F
Mn−1
Mn0
Mn1
Mn2
Mn1
Mn0
Mn0
Mn−1
Mn−1
Mn0
Mn1
Mn2
Mn3
Mn1
0
Mn0 Mn0
Mn0 Mn−1 Mn2
A
Mn0
Mn1
Mn0
Mn0
Mn1
Mn1
Mn0
FF FF FFF
0
1
0
1
2
1
2
3
FFF
Mn0
FGGFFA A
F
V
(
z
)(
eV
)
12345678
Layer No. (z)
9
0
1
3
2.0
O
xygen core energy E
(
z
)(
eV
)
(LMO)
2
/(SMO)
1
(LMO)
4
/(SMO)
2
(LMO)
6
/(SMO)
3
(LMO)
6
/(SMO)
3
(LMO)
4
/(SMO)
2
SMO LMO
SMO LMO SMO
LMO
SMO
SMO
LMO
Interface
(a
)
(b
)
(c)
(d
)
FIG. 4. 关共a兲–共c兲兴 Variations in the oxygen 1s core energy and 共d兲
the energy of the lowest Mn-e
g
state of each MnO
2
layer, obtained
from the layer-projected wave-function characters. Mn atoms of
each MnO
2
layer are labeled as shown in the figure. The interfacial
manganese atoms 共Mn-0兲, which are sandwiched by the LaO and
SrO layers, are shown by open circles with vertical dashed lines,
indicating the position of the interface. The magnetic ordering of
Mn spins for each layer as obtained from the DFT calculations is
shown with the symbols F 共FM兲, G 共G-AFM兲, and A 共A-AFM兲.A
potential barrier is clearly seen for the n=2 and n =3 superlattices.
B. R. K. NANDA AND S. SATPATHY PHYSICAL REVIEW B 79, 054428 共2009兲
054428-4
long-period superlattices 共n ⱖ 3兲. The spin-resolved layer-
projected Mn-d DOS for this case is shown in Fig. 5.Ina
solid with complex magnetic structure, it is convenient to
discuss the electron occupancy with a local spin-quantization
axis defined with respect to the local moment of a specific
magnetic atom. This was done in Fig. 5.
As seen from the figure, deep inside both the SMO and
LMO parts, the electron occupancies are more or less similar
to those of the respective bulk compounds. The bulk behav-
ior occurs already beyond just one Mn layer on either side of
the interface. In the SMO part 共Mn-1 densities, topmost
panel兲, the Mn-t
2g
spin-up states are filled while the e
g
states
are empty just like bulk SMO. In the LMO part 共Mn-2 den-
sities, bottommost panel兲, the e
g
states are Jahn-Teller split
into two bands, with the lower one occupied, again, as in the
bulk LMO.
13
As one approaches the interface from the LMO
side, the e
g
occupancy is reduced slightly from one due to the
leakage of the electron to the interfacial Mn-0 layer. The
transferred e
g
electron across the interface controls the mag-
netic behavior of the interfacial layers as already discussed.
SUMMARY
In summary, we have studied the change in the magnetic
properties of the 共LMO兲
2n
/ 共SMO兲
n
superlattices as a func-
tion of the layer thickness n and explained the observed mag-
netic structure in terms of the electron leakage across the
interface and a double exchange interaction between these
electrons and the Mn-t
2g
moments. For the short-period su-
perlattice 共n =1兲, we find a weak variation in the potential
leading to the spreading of the Mn-e
g
electrons throughout
the superlattice, resulting in a FM structure via the carrier-
mediated Zener double exchange, much like the alloy com-
pound La
2/3
Sr
1/3
MnO
3
. For higher n there is a potential bar-
rier restricting the electron leakage to the SMO side. For n
ⱖ 3, the charge leakage is restricted to just two layers at the
interface, beyond which a bulklike electronic and magnetic
structure results.
ACKNOWLEDGMENTS
This work was supported by the U.S. Department of
Energy under Grant No. DE-FG02–00ER45818. We thank
J. W. Freeland for stimulating this work and for valuable
discussions.
1
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(
LM
O)
6
/(S
M
O)
3
E
F
−8 −4 0 4
3
0
3
3
0
3
3
0
3
3
0
3
t
2g
e
g
t
2g
e
g
t
2g
e
g
t
2g
e
g
t
2g
e
g
t
2g
e
g
t
2g
e
g
t
2g
e
g
LM
O
Int
e
rf
ace S
M
O
Mn−1
Mn0
Mn1
Mn2
D
OS (
states
/
eV cell
)
Energy (eV)
FIG. 5. Spin-up 共↑兲 and spin-down 共↓兲 Mn-d DOS for the n
=3 superlattice. Up and down spins are with respect to the local
magnetic moment of the Mn atom. The labeling of the Mn atoms is
as in Fig. 4. The projected Mn-3 densities 共not shown here兲 are
similar to the Mn-2 densities as the bulk limit has already been
reached.
ELECTRONIC AND MAGNETIC STRUCTURE OF THE… PHYSICAL REVIEW B 79, 054428 共2009兲
054428-5
