Density-functional studies on spin,
charge, and orbital ordering
R. Vidya, P. Ravindran, A. Kjekshus, and
H. Fjellvåg
Center for Materials Science and
Nanotechnology, Department of Chemistry,
University of Oslo, P.O.Box 1033 Blindern,
N-0315 Oslo, Norway.
E-mail: vidya.ravindran@kjemi.uio.no
Abstract:
Spin, charge, and orbital orderings are influenced
by electron/hole doping, cation radii, oxygen
stoichiometry, temperature, magnetic field etc. In
order to understand the effect of oxygen content
and cation radii, we have studied spin, charge,
and orbital ordering in RBaMn2O5+δ (R = Y, La;
0, 0.5) by density functional theory as
implemented in the full potential linearizedaugmented plane-wave method. For δ = 0 the
ground state is found to be ferrimagnetic whereas
the variants with oxygen content δ = 1/2 give rise
to an antiferromagnetic ground state, all in
perfect agreement with experimental findings.
The charge and orbital ordering are analyzed with
the help of the energy-projected-density matrices
of the d electrons. Very different ordering
patterns have emerged for the different phases
indicating that both cation radii and oxygen
stoichiometry play an important role in deciding
spin-, charge-, and orbital-ordering.
1. Introduction
Transition-metal (TM) oxides have been
intensively studied in order to elucidate
relationships between structure, bonding,
electronic, and magnetic properties motivated
both by intrinsic interest in the subject and
also due to the extraordinary technological
potential of these materials. Perovskite-type
oxides (with general formula ABO3; A = rare/alkaline-earth, B = TM) are the most exotic,
and exhibit a wide spectrum of physical
properties such as superconductivity,
magnetoresistance (MR), various spin
ordering (SO) features, ferroelectricity,
thermoelectricity etc.
MR is the relative change in the
electrical resistance of a material produced
by the application of a magnetic field.
MR = [Δρ/ρ(0)] = |ρ (H) ─ ρ (0) |/ ρ (0)
where ρ (H) and ρ (0) are the resistance at a
given temperature in the presence and
absence of a magnetic field (H), respectively.
MR based sensors, read heads, and memories
are already commercially available in the
market. Large MR [known as colossal
magnetoresistance (CMR)] has attracted
wide attention, in particular for perovskite
oxides. The CMR materials reveal some
remarkable features such as charge ordering
(CO), orbital ordering (OO), in addition to
opening up new avenues for research [1].
The structure of perovskite oxides is
highly compact and sensitive to the size of
the cationic constituents. Consequently,
efforts to go beyond the tolerance limits for
the geometrical requirements challenge the
structural stability and cause structural
distortions. Details of the structural
arrangement and interplay of various
contributions (like the oxygen environment,
cation substitution) may have a considerable
impact. This results in the preference of
certain constituents for certain sites and
formation of an ordered state which in
general leads to the lowering of the structural
symmetry. One of the most-prominent
structural distortions occurring in perovskitelike oxides is the co-operative Jahn-Teller
distortion (JTD) [2]. In a TM oxide, where
each orbital of the TM constituent has
different anisotropy, the displacements of the
surrounding O atoms are intimately coupled
to the configuration of the d orbitals. For
example, when the two apical O atoms in an
octahedron move toward the central atom,
the degeneracy of the eg orbitals is lifted.
1.1. Charge ordering
Charge ordering is a phenomenon observed
in solids wherein certain electrons de-facto
become localized as a result of ordering of
cations with different charges on specific
lattice sites [3], thus often rendering the
material insulating. The basic requirement
for CO to take place is the presence of an
element (usually a TM constituent) in
different valence states. The incomplete d
shells of TMs in oxides do not represent
universally stable configurations, and the
cations tend to exist in various valence states
depending on the actual constituents and
external factors like temperature, pressure
etc. The combination of all such effects
determines the degree of the localization of
the valence electrons.
The valence of any element is
determined by the involvement of the
valence electrons in bonding and magnetism.
In addition, crystal-field effects play an
important role in determining the splitting of
the d levels, and consequently in the choice
between different valence states for the TM
constituents. (In an effort to determine
valence states of TM constituents, we make
use of theoretically-calculated magnetic
moments as well as site- and orbitalprojected density of states (DOS) in our
studies.)
One of the parameters used to describe
CO is the transfer integral t between TM-d
and O-p states. When the t value is high,
electrons are imagined to be able to hop from
one TM atom to another via an appropriately
arranged intervening O atom, giving rise to
metal-like couplings between electrons
known as double-exchange interaction.
However, when t is small, the electrons get
localized on the TM sites and this ultimately
results in CO. Thus the oxygen content and
population of the oxygen orbitals relative to
the Mn-d orbitals govern the metal-insulator
transitions and play an important role in the
SO, CO, and OO mechanisms.
1.2. Orbital Ordering
Orbital ordering among the TM oxides is
concerned with the preferential occupation of
electrons in specific d orbitals. The OO gives
rise to the anisotropy of the electron-transfer
interaction. This favors or disfavors the
double-exchange interaction and the superexchange [ferromagnetic (F) or antiferromagnetic (AF)] interaction. Together
with the charge and spin degrees of freedom,
OO determines all the rich variety of the
properties of transition-metal oxides.
1.3. Charge and orbital ordering in
RBaMn2O5+δ:
Many interesting features like CMR, metalinsulator transitions, and SO, CO, and OO
are associated with each other in double
perovskite variants with the general formula
RAMn2O6 (R = rare-earth, A = alkaline-earth
element). In proper cation-ordered double
perovskites, alternate stacking of layers of R
and A is formed along the c axis. An
interesting aspect of these phases with R = Y
and La is, their ability to form stable oxygendeficient variants; the oxygen vacancies are
normally formed in the R layers of such
structures. In systems with R = Y, La and A =
Ba, the mismatch between the RO and BaO
layers is smallest for the combination La/Ba
and largest for Y/Ba. Hence systems
comprising Y and La constituents should be
good candidates to analyze the effect of
cation radius on CO and OO phenomena.
In the present phases with δ = 0, the
equal amounts of Y/La and Ba lead to equal
amounts of Mn atoms in the formal ionic
valence states 2+ and 3+, thus maximizing
the Coulomb stabilization energy of the CO
state. YBaMn2O5 and LaBaMn2O5 crystallize
in tetragonal structures (space group P4/nmm)
[4,5]. Oxygen-vacant Y/La and oxygen-full
Ba layers are formed alternatively along the c
axis. Two crystallographically different Mn
atoms (Mn1 and Mn2) are situated in squarepyramidal co-ordination with their oxygen
neighbors. The extra oxygen atoms in the δ =
0.5 cases partially occupy the earlier oxygenempty Y/La layers. Therefore half of the
square-pyramids gain an extra oxygen atom
and change their co-ordination polyhedra to
octahedra and the crystal structures become
orthorhombic.
2. Computational Details
In the present calculations we have made use
of the density-functional-theory (DFT)
approach implemented in the full-potential
linearized-augmented plane-wave (WIEN2k)
method [6] in a fully-relativistic version
(including spin-orbit coupling). The Brillouin
zone (BZ) integration was done with a
modified tetrahedron method [7] using
approximately 200 k points in the irreducible
wedge of BZ, depending on the crystal
structure. Exchange and correlation effects
are treated under the generalized-gradientapproximation (GGA) [8]. In order to
account for the strong correlation effects, we
have also performed GGA+U calculations for
all phases. More details are provided
elsewhere [9,10].
identify occupation of a particular orbital in a
certain energy range, as well as obtain the
correct orientation of the orbitals. The OO
and CO patterns obtained using this more
appropriate procedure are shown in Fig. 1.
3. Results and Discussion
3.1. RBaMn2O5
We have performed complete structural
optimization for these phases in various
magnetic configurations and found that the
AF configuration has the lowest total-energy.
The calculated magnetic moments on Mn1
and Mn2 atoms in YBaMn2O5 are 2.94 and
3.79 μB, respectively (2.99 and 3.88 μB for
those in LaBaMn2O5). These two phases
have approximately 1 μB total moments in
the AF configuration, and accordingly take a
ferrimagnetic ground state. The calculated
moments and ferrimagnetic ground state are
in perfect agreement with experimental
findings [4,5].
The RBaMn2O5 phases are semiconducting with an energy gap (Eg) of 0.88
and 0.86 eV for R = Y and La, respectively.
The total DOS obtained from GGA+U
calculations with U = 4.0eV and J = 0.95eV
shows that Eg for YBaMn2O5 changes to
1.49eV and that for LaBaMn2O5 changes to
0.32 eV. The topology of DOS curves for
Mn1 and Mn2 are different in these phases.
Together with different magnetic moments,
they clearly imply a mixed-valence situation
for Mn atoms.
In the present study we make use of
the energy-projected occupation-density
matrix of d electrons to evaluate the CO and
OO. Using this approach we are able to
Fig. 1. Orbital ordering pattern obtained from the
occupation-density matrices of d states close to
the Fermi level in (a) YBaMn2O5 and (b)
LaBaMn2O5. Charge ordering pattern obtained
from the occupation-density matrices for
majority-spin d orbitals in the entire valence band
in (c) YBaMn2O5 and (d) LaBaMn2O5.
From Figs. 1a. and 1b we find no
significant difference between the OO
patterns of YBaMn2O5 and LaBaMn2O5. The
dz2 orbital on Mn1 and dz2 and dx2─y2 orbitals
on Mn2 occur close to the Fermi level (EF).
Therefore, these orbitals order in both
RBaMn2O5 phases according to a pattern
known as F-orbital order. As the degree of
filling of the dx2─y2 orbital on Mn2 in
LaBaMn2O5 is slightly larger than that in
YBaMn2O5, the radial distribution of this
orbital is depicted as considerably larger in
Fig.1b).
The atomic arrangements of the
RBaMn2O5 phases have Mn-Ob-Mn (Ob =
basal oxygen) bond angles deviating from
180o in the square-pyramid base plane which
hinder the transfer of electrons and tend to
localize more electrons on the Mn2 site than
on the Mn1 site, thereby leading to a CO
state. As each Mn atom is surrounded by five
alio-valent Mn atoms as nearest neighbors, a
checker board-type CO is established (see
Fig.1c,d).
3.2. RBaMn2O5.5
When the oxygen atoms half-fill the R layer
in the δ = 1/2 phases, half of the square
pyramids have become octahedra. Owing to
the oxygen vacancies still present in the R
layers, strain in the RMnO2.5 layer induces
short Mn2-Oa (apical) distances along the c
direction. At the same time, stretching of the
BaMnO3 layer produces long Mn1-Oa
distances. Consequently JTD of the Mn2
octahedra extends along [010] and this
perturbation plays [11] an important role in
the ordering phenomena of YBaMn2O5.5 and
LaBaMn2O5.5.
An explicit magnetic structure for
YBaMn2O5.5 is hitherto not reported, but
LaBaMn2O5.5 is reported [11] to take a
magnetic structure with ferromagnetic spin
ladders (SL) along the b axis that are AF
coupled along the a and c axes. In order to
establish the correct magnetic structure for
YBaMn2O5.5 and confirm the experimentallydeduced magnetic structure of LaBaMn2O5.5,
we have performed complete structural
optimization for these phases in various
magnetic configurations. Both YBaMn2O5.5
and LaBaMn2O5.5 take up the b-SL-type
arrangement as the ground-state magnetic
configuration. As our calculation correctly
reproduces the experimental ground state for
LaBaMn2O5.5, we believe that YBaMn2O5.5
should also take up the b-SL-type magnetic
configuration; however this needs further
experimental verifications. The magnetic
moments on Mn atoms with squarepyramidal (Mn1) and octahedral (Mn2)
configurations in YBaMn2O5.5 are 3.24 and
3.36 μB and those in LaBaMn2O5.5 have 3.15
and 3.25 μB.
In the ground-state b-SL-type AF
configuration, both the R = Y and La phases
have semiconducting behavior with an Eg of
0.58 eV for YBaMn2O5.5 and 0.54 eV for
LaBaMn2O5.5. The Eg of YBaMn2O5.5 and
LaBaMn2O5.5 increase to 1.25 and 0.95 eV,
respectively, on introduction of Coulomb
correlation effects (GGA+U). The DOS
features of Mn atoms in these phases show
some topological differences, however they
are not significantly distinct as found in the
RBaMn2O5 phases.
Fig.2. Orbital ordering pattern obtained from the
occupation-density matrices of d states close to
the Fermi level in (a) YBaMn2O5.5 and (b)
LaBaMn2O5.5. Charge ordering pattern obtained
from the occupation-density matrices for
majority-spin d orbitals in the entire valence band
in (c) YBaMn2O5.5 and (d) LaBaMn2O5.5.
The dxy, dyz, and some dxz states have
significant presence close to the EF. Thus
these orbitals determine the OO on Mn1 in
YBaMn2O5.5. On the other hand, mainly dz2
together with a few dx2─y2 states are
predominantly present close to the EF on
Mn2 and accordingly these orbitals
determine OO at this site. However, due to
the JTD, the dz2 (d3y2─r2) orbital is rotated and
lies along the b axis. The corresponding OO
patterns in the YBaMn2O5.5 phase are
displayed in Fig. 2a.
The dz2 orbital orders on Mn1 in
LaBaMn2O5.5 along the c axis. The states
close to EF on Mn2 have predominantly dz2
character together with a certain weft of
dx2─y2 character. Similar to the case in
YBaMn2O5.5, the dz2 orbital is rotated and
lies along the b-axis (Fig. 2b) due to the JTD
effect.
It may be noted that even though the
overall electronic-structure features are
almost similar for YBaMn2O5.5 and
LaBaMn2O5.5, the difference in OO patterns
leads to a different degree of distortion of the
co-ordination polyhedra in the R = Y and La
phases. In spite of the fact that the OO
patterns exhibit noticeable differences in the
YBaMn2O5.5 and LaBaMn2O5.5 phases, the
degree of filling of the valence band is same
for both phases. This can be inferred from the
almost similar CO patterns as displayed by
the filling of majority-spin channel in Fig. 2c,
and 2d.
4. Conclusions
In order to evaluate the combined effect of
oxygen content and size of the R constituents
on spin, charge, and orbital orderings, we
have carried out accurate electronic band
structure calculations on RBaMn2O5+δ (R =
Y, La; δ = 0, 1/2). While ferrimagnetic
ground states have been correctly established
for the RBaMn2O5 phases, an experimentally
inferred spin-ladder arrangement along the b
direction for LaBaMn2O5.5 has been correctly
reproduced by our calculations. Similarly, the
magnetic structure for YBaMn2O5.5 is
predicted to be the same as that of
LaBaMn2O5.5. The electronic structure of
these phases is found to be semiconducting,
however with a decreased band gap for the
δ = 1/2 phases.
The overall magnetic and electronic
properties change as a function of the oxygen
content. The influence of the oxygen content
on the Mn-O framework together with
valence changes appear to be the origin for
this relationship, notably through facilitation
or obstruction of charge transfer and/or
exchange interactions. We have also
established that the size of R constituent
plays a role in determining the shape and the
related distortions of the coordination
polyhedra such as elongation or shortening of
particular Mn-O bond(s) which in turn
influences the occupancy or vacancy of a
particular d orbital(s). Therefore, intricate
details of charge and orbital ordering (such as
occupation/localization of a particular orbital
in a particular energy range and the
associated features) are found to vary with
the size of the R constituent.
Acknowledgement
The authors are grateful to the Research
Council of Norway for financial support and
computer time in Norwegian supercomputer
facilities.
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