Charge Ordering in Half-Doped Manganites: Weak Charge Disproportion and
  Leading Mechanisms
Charge Ordering in Half-Doped Manganites: Weak Charge Dis-
proportion and Leading Mechanisms
Dmitri Volja1,2, Wei-Guo Yin1 (a) and Wei Ku1,2 (b)
1 Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York
11973, USA
2 Physics Department, State University of New York, Stony Brook, New York 11790, USA
PACS 75.47.Lx – Manganites (magnetotransport materials)
PACS 71.45.Lr – Charge-density waves - collective excitations
PACS 71.10.Fd – Lattice fermion models (Hubbard model, etc.)
PACS 71.30.+h – Metal-insulator transitions and other electronic transitions
Abstract. - The apparent contradiction between the recently observed weak charge disproportion
and the traditional Mn3+/Mn4+ picture of the charge-orbital orders in half-doped manganites is
resolved by a novel Wannier states analysis of the LDA+U electronic structure. Strong electron
itinerancy in this charge-transfer system significantly delocalizes the occupied low-energy “Mn3+”
Wannier states such that charge leaks into the “Mn4+”-sites. Furthermore, the leading mechanisms
of the charge order are quantified via our first-principles derivation of the low-energy effective
Hamiltonian. The electron-electron interaction is found to play a role as important as the electron-
lattice interaction.
Introduction. – The exploration of interplay among
distinct orders lies in the heart of condensed matter
physics and materials science, as this interplay often give
rises to tunable properties of practical applications, such
as exotic states and colossal responses to external stimuli.
Manganese oxides such as La1−xCaxMnO3, which host
rich charge, orbital, spin, and lattice degrees of freedom,
have thus attracted great attention [1]. In particular, the
vastly interesting colossal magnetoresistance (CMR) ef-
fect for x ∼ 0.2 − 0.4 exemplifies rich physics originating
from proximity of competing orders. In a slightly more
doped system (x = 0.5), all these orders coexist in an in-
sulating state [2], providing a unique opportunity for a
clean investigation of the strength and origin of each or-
der. Therefore, the study of half-doped manganites is key
to a realistic understanding of the physics of manganites
in general and the CMR effect in particular [3, 4].
The peculiar multiple orders in half-doped manganites
have long been understood in the Goodenough model [2]
of a Mn 3+/4+ checkerboard charge order (CO) with the
occupied Mn3+ eg orbitals zigzag ordered in the CE-type
antiferromagnetic background [2]. Pertaining to CMR, it
is broadly believed that a key is the emerging of nanoscale
E-mail: wyin@bnl.gov
E-mail: weiku@bnl.gov
charge-ordered insulating regions of Goodenough type at
intermediate temperatures, which could melt rapidly in
the magnetic field [4, 5].
Nonetheless, the simple yet profound Goodenough pic-
ture has been vigorously challenged thanks to recent ex-
perimental observations of nearly indiscernible charge dis-
proportion (CD) in a number of half-doped manganites
[6–13]. Such weak CD has also been observed in first-
principles computations [14, 15] and charge transfer be-
tween Mn and O sites was reported as well [15]. In
essence, these findings have revived a broader discussion
on the substantial mismatch of valence and charge in most
charge-transfer insulators. Indeed, extensive experimen-
tal and theoretical effort has been made in light of the
novel Zener-polaron model [9,33] in which all the Mn sites
become equivalent with valence being +3.5. Amazingly,
most of these investigations concluded in favor of two
distinct Mn sites as predicted in the Goodenough model
[11–13,16, 17], calling for understanding the emergence of
weak CD within the 3+/4+ valence picture.
Another closely related crucial issue is the roles of dif-
ferent microscopic interactions in the observed charge-
orbital orders, in particular the relevance of electron-
electron (e-e) interactions in comparison with the well-
accepted electron-lattice (e-l) interactions. For example,
http://arxiv.org/abs/0704.1834v2
D. Volja et al.
∆n, the difference in the electron occupation number be-
tween Mn3+ and Mn4+ states, was shown to be small in an
e-e only picture [18]; this is however insufficient to explain
the observed weak CD, since the established e-l interac-
tions will cause a large ∆n [19]. Moreover, despite the
common belief that e-l interactions dominate the general
physics of eg electrons in the manganites [2, 20], a recent
theoretical study [21] showed that e-e interaction plays an
essential and leading role in ordering the eg orbitals in the
parent compound. It is thus important to quantify the
leading mechanisms in the doped case and uncover the ef-
fects of the additional charge degree of freedom in general.
In this Letter, we present a general, simple, yet quanti-
tative picture of doped holes in strongly correlated charge-
transfer systems, and apply it to resolve the above con-
temporary fundamental issues concerning the charge order
in half-doped manganites. Based on recently developed
first-principles Wannier states (WSs) analysis [21,22,34] of
the LDA+U electronic structure in prototypical Ca-doped
manganites, the doped holes are found to reside primarily
in the oxygen atoms. They are entirely coherent in short
range [23], forming a Wannier orbital of Mn eg symme-
try at low-energy scale. This hybrid orbital, together with
the unoccupied Mn 3d orbital, forms the effective “Mn eg”
basis in the low-energy theory with conventional 3+/4+
valence picture, but simultaneously results in a weak CD
owing to the similar degree of mixing with the intrinsic
Mn orbitals, thus reconciling the current conceptual con-
tradictions. Moreover, our first-principles derivation of the
low-energy interacting Hamiltonian reveals a surprisingly
essential role of e-e interactions in the observed charge or-
der, contrary to the current lore. Our theoretical method
and the resulting simple picture provide a general frame-
work to utilize the powerful valence picture even with weak
CD, and can be directly applied to a wide range of doped
charge-transfer insulators.
Small CD vs. 3+/4+ valence Picture. – To pro-
ceed with our WSs analysis, the first-principles electronic
structure needs to reproduce all the relevant experimental
observations, including a band gap of ∼ 1.3 eV, CE-type
magnetic and orbital orders, and weak CD, as well as two
distinct Mn sites. We find that the criteria are met by the
LDA+U (8 eV) [14,24] band structure of the prototypical
half-hoped manganite La1/2Ca1/2MnO3 based on the re-
alistic crystal structure [25] supplemented with assumed
alternating La/Ca order. Hence, a proper analysis of this
LDA+U electronic structure is expected to illustrate the
unified picture of weak CD with the Mn3+/Mn4+ assign-
ment, which can be easily extended to other cases. In
practice, we shall focus on the most relevant low-energy
(near the Fermi level EF) bands—they are 16 “eg” spin-
majority bands (corresponding to 8 “spin-up” Mn atoms
in our unit cell) spanning an energy window of 3.2 eV,
as clearly shown in Fig. 1(a). For short notation, the Mn
bridge- and corner-sites in the zigzag ferromagnetic chain
are abbreviated to B- and C-sites, respectively.
Mn spin minority
Mn spin majority
2 2x y−2 23z r−
Oxygen
Fig. 1: (Color online) (a) LDA+U Band structures (dots). The
(red) lines result from the Wannier states analysis of the four
occupied spin-majority Mn 3d-derived bands. (b) An occupied
B-site (“Mn3+”) Wannier orbital in a spin-up (up arrow) zig-
zag chain, showing remarkable delocalization to the neighbor-
ing Mn C-sites. (c) Low-energy Mn atomic-like Wannier states
containing in their tails the integrated out O 2p orbitals.
The simplest yet realistic picture of the CO can be ob-
tained by constructing occupation-resolvedWSs (ORWSs)
from the four fully occupied bands, each centered at one
B-site as illustrated in Fig. 1(a). This occupied B-site eg
Wannier orbital of 3x2 − r2 or 3y2 − r2 symmetry (so for-
mal valency is 3+) contains in its tail the integrated out
O 2p orbitals with considerable weight, indicative of the
Charge Ordering in Half-Doped Manganites
charge-transfer nature [15]. Moreover, this “molecular or-
bital in the crystal” extends significantly to neighboring
C-sites on the same zig-zag chain. Therefore, although by
construction the C-site eg ORWSs (not shown) are com-
pletely unoccupied (so formal valency is 4+), appreciable
charge is still accumulated within the C-site Mn atomic
sphere owing to the large tails of the two occupied OR-
WSs centered at the two neighboring B-sites. Integrating
the charges within the atomic spheres around the B- and
C-site Mn atoms leads to a CD of mere 0.14 e, in agree-
ment with experimental 0.1− 0.2 e [7–12]. In this simple
picture, one finds a large difference in the occupation num-
bers of the ORWSs at the B- and C- sites (∆n = 1), but
a small difference in real charge. That is, the convenient
3+/4+ picture is perfectly applicable and it allows weak
CD, as long as the itinerant nature of manganites is incor-
porated via low-energy WSs rather than standard “atomic
states.”
In comparison, to make connection with the conven-
tional atomic picture and to formulate the spontaneous
symmetry breaking with a symmetric starting point (c.f.
the next section), we construct from the 16 low-energy
bands “atomic-like” WSs (AWSs) of Mn d3z2−r2 and
dx2−y2 symmetry, as shown in Fig. 1(b). In this pic-
ture, both B- and C-site AWSs are partially occupied with
∆n = 0.6. Now weak CD results from large hybridization
with O 2p orbitals, which significantly decreases the charge
within the Mn atomic sphere. Obviously, this picture is
less convenient for an intuitive and quantitative under-
standing of weak CD compatible with the 3+/4+ picture
than the previous one, as the latter builds the information
of the Hamiltonian and the resulting reduced density ma-
trix into the basis. On the other hand, it indeed implies
that strong charge-transfer in the system renders it highly
inappropriate to associate CD with the difference in the
occupation numbers of atomic-like states.
Furthermore, we find that the above conclusions are
generic in manganites by also looking into the two end lim-
its of La1−xCaxMnO3 (x = 0, 1). As shown in Fig. 2, Mn d
charge (within the Mn atomic sphere) is found to change
only insignificantly upon doping, in agreement with ex-
periments [7, 26]. This indicates that doped holes reside
primarily in the oxygen atoms, but are entirely coherent
in short-range and form additional effective “Mn eg” or-
bitals in order to gain the most kinetic energy from the
d-p hybridization, as shown in Fig. 1(a)-(b), in spirit sim-
ilar to hole-doped cuprates [23]. This justifies the present
simplest description of the charge-orbital orders with only
the above Mn-centering eg WSs [27].
Leading Mechanisms. – To identify the leading
mechanisms of the charge-orbital orders in a rigorous for-
malism, we proceed to derive a realistic effective low-
energy Hamiltonian, Heff , following our recently devel-
oped first-principles WS approach [21, 34]. As clearly
shown in Fig. 1, the low-energy physics concerning charge
and orbital orders is mainly the physics of one zig-zag FM
0.0 0.5 1.0
spin minority
spin majority
total
C-site
C-site
C-site
B-site
B-site
B-site
 Level of Ca doping
Fig. 2: The calculated d charge within the Mn atomic sphere
for La1−xCaxMnO3 (x = 0, 1/2, and 1). The results are shown
in terms of (i) B-site (filled symbols), C-site (empty symbols),
and site-average (half-filled symbols); (ii) spin-majority (up tri-
angles), spin-minority (down triangles), and total charge (dia-
monds). The lines are guides to eye.
chain, since electron hopping between the antiferromag-
netically arranged chains is strongly suppressed by the
double-exchange effect [18, 28, 29]. Our unbiased first-
principles analysis of the 16-band one-particle LDA+U
Hamiltonian in the above AWS representation reveals
[18, 28, 29, 36]
Heff = −
〈ij〉γγ′
jγ′diγ + h.c.)− Ez
+ Ueff
ni↑ni↓ + V
niQ1i + T
i Q2i + T
i Q3i) (1)
in addition to the elastic energy K({Qi}). Here d†iγ and
diγ are electron creation and annihilation operators at site
i with “pseudo-spin” γ defined as | ↑〉 = |3z2 − r2〉 and
| ↓〉 = |y2−x2〉 AWSs, corresponding to pseudo-spin oper-
ator T x
i↑di↓+d
i↓di↑)/2 and T
i↑di↑−d
i↓di↓)/2.
ni = d
i↑di↑ + d
i↓di↓ is the electron occupation number.
The in-plane hoppings are basically symmetry related:
= t/4, t
= 3t/4, t
3t/4 where the signs
depend on hopping along the x or y direction. Ez stands
for the oxygen octahedral-tilting induced crystal field. Ueff
and V are effective on-site and nearest-neighbor e-e in-
teractions, respectively. g is the e-l coupling constant.
Qi = (Q1i, Q2i, Q3i) is the standard octahedral-distortion
vector, where Q1i is the breathing mode (BM), and Q2i
and Q3i are the Jahn-Teller (JT) modes [18,19,28,29,36].
In Eq. (1) the electron-lattice couplings have been con-
strained to be invariant under the transformation of the
cubic group [36].
The effective Hamiltonian are determined by match-
ing its self-consistent Hartree-Fock (HF) expression with
D. Volja et al.
Table 1: Contributions of different terms to the energy gain
(in units of meV per Mn) due to the CO formation in self-
consistent mean-field theory. BM (JT) denotes the contribu-
tion from electronic coupling to the breathing (Jahn-Teller)
mode. K denotes that from the elastic energy.
Qi Total Ueff V t BM JT K
0 -13 -11 -15 12 0 0 0
realistic -127 -22 -42 71 -42 -113 24
HLDA+U [21, 34] owing to the analytical structure of the
LDA+U approximation [30]. An excellent mapping re-
sults from t = 0.6 eV, Ez = −0.08 eV, Ueff = 1.65 eV,
V = 0.44 eV, and g = 2.35 eV/Å. These numbers are
close to those obtained for undoped LaMnO3 [21] (exclud-
ing V , which is inert in the undoped case), and indicates
that the spin-majority eg electrons in the manganites are
still in the intermediate e-e interaction regime with compa-
rable e-l interaction. Note that Ueff should be understood
as an effective repulsion of corresponding Mn-centered WS
playing the role of “d” states, rather than the “bare” d-d
interaction U = 8 eV [21]. Furthermore, the rigidity of the
low-energy parameters upon significant (x = 0.5) doping
verifies the validity of using a single set of parameters for
a wide range of doping levels, a common practice that is
not a priori justified for low-energy effective Hamiltoni-
ans. Clearly, the observed optical gap energy scale of ∼ 2
eV originates mainly from Ueff instead of the JT splitting
widely assumed in existing theories [5, 20].
Now based on the AWSs, CO is measured by ∆n =
〈ni∈B〉 − 〈nj∈C〉 = 0.6. It deviates from unity because the
kinetic energy ensures that the ground state is a hybrid of
both B- and C-site AWSs, like in the usual tight-binding
modeling based on atomic orbitals. However, since the
AWSs considerably extend to neighboring oxygen atoms,
the actual CD is much smaller than ∆n.
With the successful derivation of Heff , the microscopic
mechanisms of the charge-orbital orders emerge. First of
all, note that the kinetic term alone is able to produce the
orbital ordered insulating phase [18,29]: Since the intersite
interorbital hoppings of the Mn eg electrons along the x
and y directions have opposite signs, the occupied bonding
state is gapped from the unoccupied nonbonding and an-
tibonding states (by t and 2t, respectively) in the enlarged
unit cell. As for orbital ordering, the Mn dy2−z2 (dx2−z2)
orbital on any B-site bridging two C-sites along the x (y)
direction is irrelevant, as the hopping integrals involving
it is vanishing. That is, only the d3x2−r2 (d3y2−r2) orbital
on that B-site is active and the B-sites on the zigzag FM
chains have to form an “ordered” pattern of alternating
(3x2 − r2)/(3y2 − r2) orbitals. However, the kinetic term
alone give only ∆n = 0. Clearly, CO is induced by the
interactions, Ueff , V , or g.
To quantify their relative importance for CO, we calcu-
late their individual contributions to the total energy gain
with respect to the aforementioned ∆n = 0 but orbital-
ordered insulating state in the self-consistent mean-field
theory. The results are listed in Table 1. The first row
obtained without lattice distortions provides a measure of
the purely electronic mechanisms for CO. Interestingly, al-
though the tendency is weak (−13 meV), e-e interactions
all together are sufficient to induce a CO, consistent with
the results of our first-principles calculations. The second
row is obtained for the realistic lattice distortions, which
shows a dramatic enhancement of CO by the JT distor-
tions (−113 meV), given that only half of the Mn atoms
are JT active. Together with a −42 meV gain from the
BM distortion, the e-l interactions overwhelm the cost of
the kinetic (71 meV) and elastic (24 meV) energy by −63
meV, further stabilizing the observed CO. Nevertheless,
the −64 meV energy gain from the overall e-e interac-
tions accounts for half of the total energy gain, illustrating
clearly their importance to the realization of the resulting
∆n = 0.6. Indeed, a further analysis reveals that e-l cou-
plings alone (Ueff = V = 0) produce ∆n ∼ 0.3, only half
of the realistic ∆n, manifesting the necessity of including
e-e interactions.
Further insights can be obtained by considering the in-
dividual microscopic roles of these interactions acting to
the kinetic-only starting point. First, infinitesimal Ueff or
g can induce CO, as a result of exploiting the fact that
the B- (C-) site has one (two) active eg orbital: (i) Ueff
has no effect on the B-sites; therefore, Ueff pushes the eg
electrons to the B-sites, in order to lower the Coulomb
energy on the C-sites [18]. This is opposite to its normal
behavior of favoring charge homogeneity in systems such
as straight FM chains (realized in the C-type antiferro-
magnet). (ii) It is favorable to cooperatively induce the
(3x2 − r2)/(3y2 − r2)-type JT distortions on the B-sites
and the BM distortions on the C-sites in order to mini-
mize elastic energy [2]. These lattice distortions lower the
relative potential energy of the active orbitals in the B-
sites, also driving the eg electrons there. Hence, Ueff and
e-l interactions work cooperatively in the CO formation.
Unexpectedly, we find that V alone must be larger than
Vc ≃ 0.8 eV to induce CO. This is surprising in compar-
ison to the well-known Vc = 0 for straight FM chains.
The existence of Vc is in fact a general phenomenon in
a “pre-gapped” system. Generally speaking, in a sys-
tem with a charge gap, ∆0, before CO takes place (e.g.
the zigzag chain discussed here), forming CO always costs
non-negligible kinetic energy due to the mixing of states
across the gap. As a consequence, unlike the first order
energy gain from Ueff and g, the second order energy gain
from V is insufficient to overcome this cost until V is large
enough (of order ∆0). In this specific case, V = 0.44 eV is
insufficient to induce CO by itself, but it does contribute
significantly to the total energy gain once CO is triggered
by Ueff or g, as discussed above.
It is worth mentioning that the contribution from the
EzTz term is neglected from Table 1 because of the small
coefficient of Ez ≃ 8 meV, consistent with the previous
Charge Ordering in Half-Doped Manganites
study for undoped La3MnO3 [21]. In perovskites, the tilt-
ing of the oxygen octahedra could yield the Jahn-Teller-
like distortion of GdFeO3 type, which can be mathemat-
ically described by the EzTz or ExTx term. However, in
the perovskite manganites these terms are shown here and
in Ref. [21] to be negligibly small. In addition, in half-
doped manganites, the pattern of the orbital order is pre-
dominantly pined by the zig-zag pathway of the itinerant
electrons and thus the effect of tilting is less relevant. On
the other hand, the EzTz is very effective to explain the
zig-zag orbital ordering of (x2 − z2)/(y2 − z2) pattern in
half-doped layered manganites, such as La0.5Sr1.5MnO4
Ref. [35], where pseudo-spin-up (the 3z2 − r2 orbital) is
favored by a much stronger Ez due to the elongation of
the oxygen octahedral along the c-axis. Described via
the pseudo-spin angel, θ = arctan(Tx/Tz) [21], Ez signif-
icantly reduces θ from ±120◦ [i.e., (3x2 − r2)/(3y2 − r2)]
to ±30◦ [i.e., (x2 − z2)/(y2 − z2).
The present results would impose stringent constraints
on the general understanding of the manganites. For ex-
ample, Zener polarons were shown to coexist with the CE
phase within a purely electronic modeling of near half-
doped manganites [31]. However, to predict the realistic
phase diagram of the manganites, one must take into ac-
count the lattice degree of freedom. In fact, when propos-
ing the CE phase, Goodenough considered its advantage
of minimizing the strain energy cost. Note that the pre-
vious HF theory indicated a decrease of the total energy
by 0.5 eV per unit cell with Zener-polaron-like displace-
ment [15]. The present LDA+U calculations reveal an
increase of the total energy by 1.07 eV per unit cell with
the same Zener-polaron-like displacement. This discrep-
ancy is quite understandable from the characteristics of
the LDA functional, which favors covalent bond, while the
HF approximation tends to over localize the orbital and
disfavor bonding. To resolve the competition between the
CE phase and the Zener polaron phase, real structural
optimization is necessary and will be presented elsewhere.
As another example, in the absence of e-l interactions the
holes were predicted to localize in the B-site-type region
(i.e., the straight segment portion of the zigzag FM chain)
in the CxE1−x phase of doped E-type manganites [32].
However, since the B-sites are susceptible to the JT dis-
tortion, they are more likely to favor electron localization
instead; future experimental verification is desirable.
Summary. – A general first-principles Wannier func-
tion based method and the resulting valence picture of
doped holes in strongly correlated charge-transfer systems
are presented. Application to the charge order in half-
doped manganites reconciles the current fundamental con-
tradictions between the traditional 3+/4+ valence picture
and the recently observed small charge disproportion. In
essence, while the doped holes primarily resides in the oxy-
gen atoms, the local orbital are entirely coherent following
the symmetry of Mn eg orbital, giving rise to an effective
valence picture with weak CD. Furthermore, our first-
principles derivation of realistic low-energy Hamiltonian
reveals a surprisingly important role of electron-electron
interactions in ordering charges, contrary to current lore.
Our theoretical method and the resulting flexible valence
picture can be applied to a wide range of doped charge-
transfer insulators for realistic investigations and interpre-
tations of the rich properties of the doped holes.
∗ ∗ ∗
We thank E. Dagotto for stimulating discussions and V.
Ferrari and P. B. Littlewood for clarifying their Hartree-
Fock results [15]. The work was supported by U.S. Depart-
ment of Energy under Contract No. DE-AC02-98CH10886
and DOE-CMSN.
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	Introduction. –
	Small CD vs. 3+/4+ valence Picture. –
	Leading Mechanisms. –
	Summary. –
