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Journal of the Physical Society of Japan 75.5 (2006): 051009
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Typeset with jpsj2.cls <ver.1.2> Full Paper
Theoretical Aspects of Charge Ordering in Molecular Conductors
Hitoshi Seo
1 ∗
, Jaime Merino
2
, Hideo Yoshioka
3
, and Masao Ogata
4
1
Non-Equilibrium Dynamics Project, ERATO-JST, c/o KEK, Tsukuba 305-0801, Japan
2
Departamento de F
´
isica Te´orica de la Materia Condensada, Universidad Aut´onoma de Madrid, Madrid 28049, Spain
3
Department of Physics, Nara Women’s University, Nara 630-8506, Japan
4
Department of Physics, Faculty of Science, University of Tokyo, Tokyo 113-0033, Japan
(Received February 4, 2008)
Theoretical studies on charge ordering phenomena in quarter-filled molecular (organic) con-
ductors are reviewed. Extended Hu bbard models includ ing not only the on-site but also the
inter-site Coulomb repulsion are constructed in a straightforward way from the crystal struc-
tures, which serve for individual study on each material as well as for their systematic under-
standings. In general the inter-site Coulomb interaction stabilizes Wigner crystal-typ e charge
ordered states, where the charge localizes in an arranged manner avoiding each other, and can
drive the system insulating. The variety in the lattice structures, represented by anisotropic
networks in not only the electron hopping but also in the inter-site Coulomb repu lsion, brings
about diverse problems in low-dimensional strongly correlated systems. Competitions and/or
co-existences between th e charge ordered state and other states are discussed, such as metal,
superconductor, and the dimer-type Mott insulating state which is another typical insulating
state in molecular conductors. Interplay with magnetism, e.g., antiferromagnetic state and spin
gapped state for example due to the spin-Peierls transition, is considered as well. Distinct sit-
uations are pointed out: influences of the coupling to the lattice degree of freedom and effects
of geometrical frustration which exists in many molecular crystals. Some related t opics, such
as charge order in transition metal oxides and its role in new molecular conductors, are briefly
remarked.
KEYWORDS: molecular conductors, charge ordering, strongly correlated electron system, low-dimensional
systems, metal-insulator transition, superconductivity, magnetism, electron-lattice coupling,
geometrical frustration
1. Introduction
The charge ordering (CO) phenomenon is actively
studied in the resea rch field of charge trans fer ty pe
molecular conductors,
1
since it plays a key role in their
physical properties. Following its clear obser va tion in
DI-DCNQI
2
Ag,
2
metal-insulator transitions in many of
these materials a re now understood as due to CO. It has
been found not only in newly synthesized compounds but
also in systems which have been well known for years,
where, however, its existence has been veiled until re-
cently. Typical examples are two representative families
of this field, the Bechgaard salts TM
2
X
3
(TM = TMTSF
or TMTTF) a nd ET
2
X
4
(ET = BEDT-TTF) where X
takes different atoms or molecules, both being studied for
more than 20 years. The CO transition was revealed re-
cently and now beca me one of the most important issues
in these sys tems.
These materials are members of the so-called 2:1 salts
expressed as A
2
B, to which interest of this field has been
conducted. Numerous compounds with such a 2:1 com-
position have been synthesized and fo und to exhibit a
rich variety of properties.
5–7
Many of them show elec-
tron conduction at room temperature, where the car-
riers are due to a charge transfer from cations B
+
or
anions B
−
, resulting in an average valence of −1/2 or
+1/2 for A molecules, respectively. The B ion has closed
shell in most cases then the valence band near the Fermi
energy is composed of the frontier orbital, LUMO or
∗
E
-mail address: seo@post.kek.jp
HOMO, of the A molecule, which is quarter-filled as
a whole in terms of electr ons or holes. The variety in
their properties has b e e n revealed to be originated from
the diversity of aniso tropic lattices resulting in differ-
ent non-interacting band structures, to gether with strong
correla tio n effects experienced by electrons among this
HOMO/LUMO band determining the low energy prop-
erties.
8
CO is a typical consequence of such strong cor-
relation, namely, large electron-electron Coulomb repul-
sion compared to the kinegic energy, especially due to
the long-range nature of this Coulomb forc e . In fact, it is
now ubiqitously found in A
2
B compounds as well as in
other strongly correlated electron systems such as tr an-
sition metal oxides.
In this article, we review theoretical aspects of CO
mainly aiming at following points: In what kind of situ-
ation a re they formed? In what situation do they melt?
How does the spin degree of freedo m act? Are there any
supe rconducting (SC) state near the CO phase ?
Such theoretical works on CO have been done from
early days, motivated by experiments. For example, the
metal-insulator transition in a classical trans itio n metal
oxide Fe
3
O
4
, the magnetite, was proposed to be due to
CO by Verwey,
9
although its existence is still contro-
versial to date.
10
Another trigger was an early molec-
ular conductor TTF-TCNQ, where an incommensurate
4k
F
charge-density-wave (CDW) is observed, but only
in a diffusive manner therefore long ranged order is not
achieved.
11, 12
Here we refer to the term CO as the phe-
2 J. Phys. Soc. Jpn. Full Paper H. Seo, J. Merino, H. Yoshioka, and M. Ogata
nomenon due to strong C oulomb interaction, sometimes
called as “Wigner crystal on lattice”.
13
This should be
distinguished with other transitions resulting in peri-
odic modulations of the charge density, such as the 2k
F
CDW (the Peierls-Fr¨olich state) driven by the nesting
of the Fermi surface together with the electron-lattice
coupling,
12
which is ess entially a phenomenon at weak
correla tio n.
The observations of CO transition in A
2
B molecular
systems have stimulated many theor e tical studies ad-
hered to these compounds. Experiments have fortunately
appeared around when resea rchers in this field started
to realize that effects of electron corr e lation could be
modeled in a straightforward way and that r ather simple
models would successfully describe their physical prop-
erties.
14
That is, each constituent A molecule is repre-
sented by a “site” and only the frontier orbital is con-
sidered. The non-interacting band str uctur e s near the
Fermi level are well reproduced by the extended H¨uckel
tight-binding s cheme.
15, 16
Then, the Coulomb interac-
tion between electrons in this orbital is taken into ac-
count.
17, 18
We call such model as the extended Hubbard
model (EHM), written as fo llows:
H
EHM
= −
X
hijiσ
t
ij
c
†
iσ
c
jσ
+ h.c.
+
X
i
Un
i↑
n
i↓
+
X
hiji
V
ij
n
i
n
j
. (1)
Here, hiji denotes pair s of the lattice sites (i.e.,
molecules) i and j, σ is the spin index which takes ↑
and ↓, n
iσ
and c
†
iσ
(c
iσ
) denote the number operator and
the creation (annihilation) operator for the electron of
spin σ at the ith site, respectively, and n
i
= n
i↑
+ n
i↓
.
The transfer integrals, t
ij
, reflect the anisotropy resulting
from the particular spatial extent of the frontier orbital,
calculated, e.g., by the extended H¨uckel metho d or from
tight-binding fitting of first principle calculations. The
Coulomb inte ractions of not only on-s ite U but als o inter-
site V
ij
are considered, the latter being crucial for the CO
as we will see later explicitly. In the [A
2
]
−
B
+
([A
2
]
+
B
−
)
systems the non-inte racting band as a whole is quarter-
filled in terms of e lec trons (holes), namely, there ex ists
one electron (ho le) per two sites on ave rage.
Because of this clea r way of constructing microscopic
models from crystal structures, results of theoretical
works could be checked back in the experiments, and such
interplay has greatly accelerated the research. Through-
out these we have learned that, although the bas ic pic-
ture of CO is rather classical and essentially known from
the early days, the physics ther e in is rich and diverse.
The main reason for such diversity is the variety in the
geometry of lattice structures of the materials where it
is realized. More specifically, relative positions betwee n
molecules not only reflect directly on the anisotropy in
t
ij
controlling the band structure, but also affect drasti-
cally the nature of the CO state itself through V
ij
. This
is in contrast with the c ase of the Mott insulating state in
half-filled systems where the driving force is the on-site
term U , which is a character of the atom/molecule itself;
in the case of the dimer-type Mott insulator in A
2
B sys-
tems (see later), it is the “on-dimer” Coulomb repulsion,
U
dimer
.
17, 19, 20
The research of CO is still continuing and rather
rapidly growing. New phenomena are uncove red, even in
the above mentioned compounds, a nd now under exten-
sive investigatio ns . For example a pressure-tempera tur e
phase diagram of DI-DCNQI
2
Ag has been explored
where an anomalous temperature dependence of metal-
lic resis tivity ρ ∝ T
3
is found just beyond the border of
the CO phase.
21
A peculiar interplay between CO and
magnetic properties found in TMTTF
2
X compounds un-
der pressure
22
requires a reformulation of the generic
phase diagram of TM
2
X.
23
Problems of CO system on
anisotropic triangular lattice structures characteristic of
ET
2
X compounds have ma ny new aspects.
4, 24
We can-
not offer comprehensive explanation for each of these
cases here as many a re still not yet resolved theoreti-
cally, but we hope that this review, by explaining the
present status, would provide a base for tackling such
new physics and lead the readers toward challenges.
The organization of this paper is as follows. Since most
of the A
2
B compounds have low-dimensional structures,
we will devide this review into one-dimensional (1D) and
two-dimensional (2D) problems, while in reality there
exist finite interchain/interlayer interactions; they are
quasi-1D and quasi-2D systems. Theoretical results de-
voted for the quasi-1D compounds are described in § 2.
This starts with studies on purely 1D electronic models
and then additional effects, such as interchain interac-
tion and coupling to the lattice degree of freedom, are
considered. The 1D case can be studied in more con-
trolled ways, analytically and numerically, than in the
study of the quasi-2D compounds, which we discuss in
§ 3. There, theoretical works on 2D electronic models
are still in pro gress and influence of additional effects,
e.g., co upling to the lattice, is no t fully unders tood yet.
Studies aimed at SC states nea r the CO phase in 2D
models, motivated by its observations, are reviewed as
well. A problem of CO systems under geometrical frus-
tration which is expected to be relevant to many molec-
ular conductors will be pointed out in § 2 and 3. In § 4
related topics will be mentioned, such as analogous CO
states observed in transition metal oxides. Possible ro le s
of CO in other molecular systems will be added as well,
as perspectives. A summary is given in § 5.
Some of the experimental studies on CO are mentioned
in this review but many refere nce s including important
ones are left out; we refer to other rev ie ws from exper-
imental standpoints
1, 25
which would be complementary
to this article. One can find many review articles on the
properties of molecular compounds in general,
1, 5–7
espe-
cially we refer to refs. 8 and 18 fo r pap e rs from theoreti-
cal but more systematic po int of views including the CO
systems as well.
2. Quasi-One-dimensional Systems
Early theo retical works on CO in 1D models have been
performed motivated by the observation of 4k
F
CDW
in TTF-TCNQ as mentioned in § 1, where the impor-
tance of the long-ra nge Coulomb interaction was empha-
J. Phys. Soc. Jpn. Full Paper H. Seo, J. Merino, H. Yoshioka, and M. Ogata 3
sized.
13, 26–30
However for the quasi-1D A
2
B systems, be-
fore CO was found, analyses were mainly concentrated
on the Hubbard-type models only considering the on-site
Coulomb energy U , but some did discuss the relevance
of the inter-site V
ij
.
31–33
For example, Mila
33
estimated
that the nearest-neighbor Coulomb r e pulsion is appre-
ciable in TM
2
X, such as more than one third of U , by
comparing calculations on the dimerized version of 1D
EHM (see e q. (2)) with optical measurements.
34
After
CO was found, many further works devoted to the 1D
EHM and its variants have been c arried out, which we
will review in this section.
2.1 Charge ordering in quasi-one-dimensional systems
When the first observation of CO in DI-DCNQI
2
Ag
was made by
13
C-NMR,
2
in an independent theoreti-
cal work based on mean-field (MF) approximation, Seo
and Fukuyama
35
proposed that CO due to the nearest-
neighbor Coulomb repulsion might exist in TMTTF
2
X.
This was bas e d on a comparison between self-consistent
solutions at zero temperatur e obtained by the standard
MF treatment applied to the appropriate dimerized 1D
EHM and the spin structure in the antiferromagnetic
phase suggested by a
1
H-NMR measure ment.
36
Soon af-
ter, it was found that members of TMTTF
2
X indeed
show CO, directly s e e n in a
13
C-NMR measurement.
3
Moreove r a divergence of the dielectric constant at the
CO transition temperature is obser ved, suggesting a fer-
roelectric state.
37
This is a co nsequence of CO, which we
will mention later in this section.
These systems are members of DCNQI
2
X (X: mono -
valent metal cation X
+
, e.g., Ag and Li)
38, 39
and TM
2
X
(X: monovalent anion X
−
, e.g., PF
6
, AsF
6
, SCN, and
Br),
23
respectively, both having quasi-1D structures. DC-
NQI stands for the R
1
R
2
-DCNQI molecule where R
1
, R
2
are s ustituents such as CH
3
, Br, I, etc. Here, quasi-
1D “structure” implies twofold meanings: in their crys-
tal structures the DCNQI/TM mole c ules assemble in a
stacking manner, and, in their electronic structures the
transfer integrals in the inte rchain direction, t
inter
, are
one order of magnitude smaller than those in the intra-
chain direction, t
intra
.
The netwo rks of DCNQI/TM molecules are schemati-
cally shown in Fig. 1. In DCNQI
2
X the interchain cou-
plings are three-dimensional (3D) and uniform, while
TM
2
X they are quasi-2D and the TM molecules are con-
nected in a rather complicated way (the β-type structure;
see § 3.1). For DCNQI
2
X systems, |t
intra
| ≃ 0.15 ∼ 0.25
eV, |t
inter
| ≃ 0 .01 ∼ 0.03 eV,
40, 41
while for TM
2
X sys-
tems: in TMTSF compounds |t
intra
| ≃ 0.2 ∼ 0.4 eV,
|t
inter
| ≃ 0 .01 ∼ 0.05 eV and in TMTTF compounds
|t
intra
| ≃ 0.1 ∼ 0.25 eV, |t
inter
| ≃ 0.01 ∼ 0.03 eV.
16, 42–44
These values depend on the actual salts and different
methods of calculation also provide varied estimations.
We note that the distances between mo lec ules does
not simply correspond to the degree of anisotropy in
t
ij
, since the anisotropic shape of the frontier orbitals
makes the dependence of t
ij
to the relative configuration
of molecules rather complicated in general.
16, 45
In con-
trast, V
ij
obeys more or less a monotonic function of the
a2
a1
b
p1
p2
c
a
c
a
~
b
~
(b) TM
2
X
a
b
(a) DCNQI
2
X
Fig. 1. (Color online) Schematic representation of the molecular
networks in (a) DCNQI
2
X and (b) TM
2
X. The crystallographic
axes are shown; in (a) we take ˜a = a + b,
˜
b = a − b where a and
b are the axes in the tetragonal cell. The interchain interactions
are three-dimensional in (a), whereas it is two-dimensional in (b)
since interlayer couplings are one order of magnitude smaller due
to the anion layers. The indices (for TM
2
X, taken from ref . 16)
are not only for t
ij
but also for different values of V
ij
, although
the degree of anisotropy can be different (see text).
distance b e c ause it is the Coulomb interaction. There-
fore the interchain part of V
ij
is not necessarily small
compared to the intrachain one even in these quasi-1D
systems.
46, 47
However, let us neglect the interchain in-
teractions first and discuss this issue later in § 2.4 and
2.6.
Estimations of the Coulomb energies in these molecu-
lar systems are difficult at present. Qua ntum chemistry
calculations are performed for a n isolated molecule or
clusters of them, which provide unrealisticly large values
since the scre e ning effect in solids is left out, while such
estimations from the first principle in these molecular
solids are still yet to be done. However it is believed to
be of the order of U ≃ 1 eV from different measurements
and estimates, which gives U/|t
intra
| ≃ 5 for DCNQI
2
X
and TMTTF
2
X, while for TMTSF
2
X smaller values o f
U/|t
intra
| ≃ 3. The estimated values for the intrachain
Coulomb energy V
intra
are again ambiguous but many
provide rather large values: the ratio V
intra
/U in a range
about 0.2 ∼ 0.6.
33, 46–48
A crucial difference betwe e n these two families is that
the stacking of DCNQI molecules is uniform while that
of TM moluecules is slightly dimerized, as seen in Fig. 1.
Thus a minimal effective model to investigate the oc c u-
rance of CO in these compounds is the 1 D dimer iz e d
EHM, represented as,
H
1D
= −t
X
iσ
(1 + (−1)
i
δ
d
)(c
†
i+1σ
c
iσ
+ h.c.)
+U
X
i
n
i↑
n
i↓
+ V
X
i
n
i
n
i+1
, (2)
where i is the site index along the chain. Here, the trans-
fer integrals allow dimerization as alternating t(1 + δ
d
)
and t(1−δ
d
); in the DCNQI compounds δ
d
= 0, while for
the TM co mpo unds values of t
a1
and t
a2
16, 42–44
(for the
indices see Fig. 1) read δ
d
. 0.1. The inter-site Coulomb
repulsion between neighboring sites is set to be uniform
as V , which is an approximation for TM
2
X. Again, the
value of δ
d
does not directly result in a similar value of
dimerization in V
intra
. In fact, q uantum chemistry calcu-
4 J. Phys. Soc. Jpn. Full Paper H. Seo, J. Merino, H. Yoshioka, and M. Ogata
(a)
(b)
V
U
t (1+
δ
d
) t (1-
δ
d
)
J
J
Fig. 2. (Color online) Two limiting cases of strongly correlated
insulators in quarter-filled system
8, 18
: (a) the charge ordered
insulating state and (b) the dimer Mott insulating state. Black
dots and colored area represent the lattice sites and the localized
carriers, respectively, while the thickness of the bonds show the
difference in the transfer integrals. The arrows show the param-
agnetic localized spins.
lations for TM
2
X
46, 47
provide the dimerization parame-
ters in V
a1
and V
a2
to be less than 1 %. For the DCNQI
compounds the electron filling is one quarter while it is
three quarter for the TM comp ounds, but in this 1D
model the two situations are equiva lent since electron-
hole s ymmetry holds. Note that other effects may break
this symmetry then we s hould treat the two cas es sepa-
rately.
It is useful to describe two limiting case s for the insu-
lating states in this model at quarter- filling due to strong
Coulomb interaction,
8, 18
which are shown in Fig. 2.
These states, in a broad sense, a re realized in a wide
range of A
2
B compounds, not only in quasi-1D but also
in quasi-2D systems discussed in § 3. One is the Wigner
crystal-type CO state stabilized by V in the presence
of strong U, and the o ther is the Mott insulating state
stabilized by U in the pre sence of strong dimerization
δ
d
: a dimer Mott insulator. The spin degree of freedom
in such insulating states would behave as an S = 1/2
localized spin chain, where the spins are located on e v-
ery other site for the CO state whereas on every dimer
for the dimer Mott insulating state. Roughly speak ing,
charge localization is determined by large energy scales
such as U and V , while s pin proper ties are determined
by a s maller energy scale of the or der o f the Heisenberg
coupling, J, acting between these loca lize d spins. We will
discuss their detailed properties in the following subsec -
tions.
The MF so lutions mentioned ab ove are consistent with
such two limiting cases. This is seen in the obtained
spin and charge patterns for the two cases , schematically
shown in Fig. 3, and actually many works have been
performed on H
1D
based on such MF results.
49
How-
ever we should keep in mind that this MF treatment
cannot correctly describe such insulators at strongly cor-
related regime in general, for example the paramag netic
insulating phase at temperatures above the magnetically
ordered phases observed in experiments cannot be repro-
duced. Moreover in purely 1D models the role o f quantum
fluctuations which is left out in MF is crucial, and we will
see that it considerably modifies the MF results.
We note that the c ations/anions are at crystallo-
graphically equivalent positions from the DCNQI/TM
molecules at high temper atures so that, essentially, they
(a)
(b)
Fig. 3. (Color online) Typical mean-field solutions in one-
dimensional dimerized extended Hubbard model at quarter-
filling,
35
H
1D
in eq. (2): (a) the charge ordered antiferromag-
netic insulating state for δ
d
= 0 and large U and V , and, (b) the
dimer-type antiferromagnetic insulating s tate for δ
d
6= 0, large
U, and V = 0. The size of the colored circles and the arrows
represents the charge density and the amount of spin moment
on each site. The char ge densities are (a) dis proportionated al-
ternatively as, 1/2 + δ, 1/2 − δ, 1/2 + δ, 1/2 − δ, where δ is the
amount of charge disproportionation, and (b) uniform at 1/2.
do not contribute to the electronic pro perties. However,
in the case of non-centrosymmetrical anions in TM
2
X,
the anions s how a disorder-to-order transition by lower-
ing tempera tur e and generate a potential with longe r pe-
riodicity than the original unit cell, resulting in modifica-
tions in the one-particle properties.
50
We do not discuss
such cas e s in this paper since its relation with the CO
due to electron correlation is still obscure, but the fact
that the a nion ordering produces charge density mod-
ulations suggests the importance of the a nions in some
cases. Actually possible roles of the X unit in stabilizing
CO states through its coupling to the electron system
will be addressed in § 2.5.
2.2 One-dimensional extended Hubbard model
The quarter-filled 1D EHM, H
1D
in eq. (2) with δ
d
=0,
was first considered by Ovchinnikov
26
who discussed that
in the U/t = ∞ limit a CO insulating ground sta te is re-
alized for V > 2t (= V
cr
), as in the fo llowing. In this limit,
double occupancy is strictly prohibited so that every site
is either occupied or unoccupied. Furthermore the spin
degree of fr e e dom is frozen since the “order” of electrons
cannot be changed due to the 1D nature. Then the charge
degree of freedom is equivalent to a spinless fermion (SF)
model at half-filling with the nearest neighbor Coulomb
interaction V , i.e., the inter acting SF mo del, and the spin
degree of freedom a cts freely as Curie spins.
This SF model, by identifying occupied site as up spin
and unoccupied site as down spin, ca n be mapped onto a
1D S = 1/2 (pseudo-)spin model through Jordan-Wigner
transformation. Quarter-filling in the original electronic
system corresponds to half-filling in the SF model where
half of the sites would be occupied, and to total magnetic
moment of zero in the e ffective spin model. The inter-site
Coulomb interaction between SF transforms to an anti-
ferromagnetic interaction, J
z
S
z
i
S
z
j
with J
z
= V , while
the kinetic enery ter m becomes as J
xy
(S
x
i
S
x
j
+S
y
i
S
y
j
) with
J
xy
= 2t. This is an XXZ model, where a T = 0 phase
transition from a gapless “XY” state (J
z
< J
xy
) to an
antife rromag netic “Ising ” state (J
z
> J
xy
) is known.
51
Transforming back to the SF model, these correspond
to a metallic Tomonaga-Luttinger liquid (TLL) and the
CO insulating state, repectively; the critical point is the
![Fig. 4. Ground state phase diagram of the one-dimensional extended Hubbard model at quarter-filling on the plane of U/t and V/t.63 CO stands for charge order and TLL for TomonagaLuttinger liquid. SC represents the superconducting phase. Contour map for the Tomonaga-Luttinger parameter Kρ is shown. The bold line represents the boundary for the metal-insulator transition. The shaded area is the region with an exponentially small gap. [By courtesy of S. Ejima.]](/figures/fig-4-ground-state-phase-diagram-of-the-one-dimensional-1mlttnnd.webp)
![Fig. 5. Spin susceptibility χ calculated by exact diagonalization as a function of V for several values of U at quarter filling (t = 1).53 Solid squares represent exact results in the one-dimensional Hubbard model and arrows indicate the points above which the charge gap opens. [After ref. 53.]](/figures/fig-5-spin-susceptibility-kh-calculated-by-exact-31j8w516.webp)
![Fig. 11. Ground state phase diagram of the extended Hubbard model at quarter-filling on the zigzag ladder, determined by the density matrix renormalization method.103 U/t1 is fixed at 10 while t2 is set to 0. Sandwitched by the two insulating CO phases with different charge patterns, the Tomonaga-Luttinger liquid (TLL) metallic phase is extended toward the frustrated line V1 = 2V2. The contour lines indicate the TLL parameter Kρ. [Reprinted figure with permission from S. Ejima et al., Phys. Rev. B 72 (2005) 033101. Copyright (2005) by the American Physical Society.]](/figures/fig-11-ground-state-phase-diagram-of-the-extended-hubbard-3m76hjf4.webp)
