Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid
We present a model approach to describe charge fluctuations and different charge phases in
 strongly correlated 3d oxides. As a generic model system one considers that of centers each with
 three possible valence states M⁰, described in frames of S 1 pseudospin (isospin) formalism...
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| Cite this: | Charge states of strongly correlated 3d oxides: from typical
 insulator to unconventional electron-hole Bose liquid / A.S. Moskvin // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 314-327. — Бібліогр.: 34 назв. — англ. |
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| author | Moskvin, A.S. |
| author_facet | Moskvin, A.S. |
| citation_txt | Charge states of strongly correlated 3d oxides: from typical
 insulator to unconventional electron-hole Bose liquid / A.S. Moskvin // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 314-327. — Бібліогр.: 34 назв. — англ. |
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| container_title | Физика низких температур |
| description | We present a model approach to describe charge fluctuations and different charge phases in
strongly correlated 3d oxides. As a generic model system one considers that of centers each with
three possible valence states M⁰, described in frames of S 1 pseudospin (isospin) formalism by
an effective anisotropic non-Heisenberg Hamiltonian which includes both two types of single particle
correlated hopping and the two-particle hopping. Simple uniform mean-field phases include
an insulating monovalent M⁰ phase, mixed-valence binary (disproportionated) M phase, and
mixed-valence ternary («under-disproportionated») M⁰, phase. We consider two first phases in
more details focusing on the problem of electron-hole states and different types of excitons in
M⁰ phase and formation of electron-hole Bose liquid in M phase. Pseudospin formalism provides
a useful framework for revealing and describing different topological charge fluctuations, in particular,
like domain walls or bubble domains in antiferromagnets. Electron-lattice polarization effects
are shown to be crucial for the stabilization of either phase. All the insulating systems such as
M0 phase are subdivided to two classes: stable and unstable ones with regard to the formation of
self-trapped charge transfer (CT) excitons. The latter systems appear to be unstable with regard to
the formation of CT exciton clusters, or droplets of the electron-hole Bose liquid. The model approach
suggested is believed to be applied to describe a physics of strongly correlated oxides such
as cuprates, manganites, bismuthates, and other systems with charge transfer excitonic instability
and/or mixed valence. We shortly discuss an unconventional scenario of the essential physics of
cuprates which implies their instability with regard to the self-trapping of charge transfer excitons
and the formation of electron-hole Bose liquid.
|
| first_indexed | 2025-12-07T18:58:39Z |
| format | Article |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3, p. 314–327
Charge states of strongly correlated 3d oxides: from typical
insulator to unconventional electron-hole Bose liquid
A.S. Moskvin
Department of Theoretical Physics, Ural State University, Ekaterinburg 620083, Russia
E-mail: alexandr.moskvin@usu.ru
Received July 31, 2006
We present a model approach to describe charge fluctuations and different charge phases in
strongly correlated 3d oxides. As a generic model system one considers that of centers each with
three possible valence states M0,� described in frames of S � 1 pseudospin (isospin) formalism by
an effective anisotropic non-Heisenberg Hamiltonian which includes both two types of single par-
ticle correlated hopping and the two-particle hopping. Simple uniform mean-field phases include
an insulating monovalent M0 phase, mixed-valence binary (disproportionated) M� phase, and
mixed-valence ternary («under-disproportionated») M0,� phase. We consider two first phases in
more details focusing on the problem of electron-hole states and different types of excitons in
M0 phase and formation of electron-hole Bose liquid in M� phase. Pseudospin formalism provides
a useful framework for revealing and describing different topological charge fluctuations, in par-
ticular, like domain walls or bubble domains in antiferromagnets. Electron-lattice polarization ef-
fects are shown to be crucial for the stabilization of either phase. All the insulating systems such as
M0 phase are subdivided to two classes: stable and unstable ones with regard to the formation of
self-trapped charge transfer (CT) excitons. The latter systems appear to be unstable with regard to
the formation of CT exciton clusters, or droplets of the electron-hole Bose liquid. The model ap-
proach suggested is believed to be applied to describe a physics of strongly correlated oxides such
as cuprates, manganites, bismuthates, and other systems with charge transfer excitonic instability
and/or mixed valence. We shortly discuss an unconventional scenario of the essential physics of
cuprates which implies their instability with regard to the self-trapping of charge transfer excitons
and the formation of electron-hole Bose liquid.
PACS: 71.10.–w Theories and models of many-electron systems;
71.28.+d Narrow-band systems; intermediate-valence solids;
71.30.+h Metal–insulator transitions and other electronic transitions;
74.72.–h Cuprate superconductors — high-Tc and insulating parent compounds.
Keywords: mixed-valence, electron correlations, pseudospins, oxides, cuprate superconductors, manganites.
1. Introduction
The discovery of the high-Tc superconductivity in
doped cuprates [1], observation of many unconven-
tional properties in doped manganites with their co-
lossal magnetoresistance, bismuthates with high-Tc' s
nickellates and many other oxides [2] shows that we
deal with a manifestation of novel strongly correlated
states with a local charge instability, mixed valence,
«metal–dielectric» duality, strong coupling of differ-
ent (charge, spin, orbital, structural) degrees of free-
dom and non-Landau behavior of quasiparticles. All
this has generated a flurry of ideas, models and scenar-
ios of the puzzling transport phenomena and stimulated
the intensive studies of various correlation effects and
charge transfer (CT) phenomena in strongly correlated
systems derived in either way from insulators unstable
with regard to the CT fluctuations. Despite intense ef-
fort, the behavior of strongly correlated 3d oxides re-
mains poorly understood and we are still far from a
comprehensive understanding of the underlying phys-
ics. Moreover, it seems that there are missing qualita-
tive aspects of the problem beyond the simple Hubbard
scenario that so far escaped the identification and the
recognition. Firstly it concerns strong electron-lattice
© A.S. Moskvin, 2007
polarization effects which may be subdivided into elec-
tron-lattice interaction itself [3,4], and a contribution
of an electronic background that is electronic subsys-
tem which is not incorporated into effective Hubbard
model Hamiltonian [5,6]. These effects are of great im-
portance for the ground state electronic and crystalline
structure, and can seriously modify the doping response
of 3d oxide up to the crucial change of the seemingly
natural ground state. This question has not received the
attention it deserves. It should be emphasized that tra-
ditional Fr�hlich approach to the electron-lattice cou-
pling implies the description of linear effects whereas
the charge fluctuations in the insulator do imply
strongly nonlinear electron-lattice coupling with the
predominance of polarization and relaxation effects,
and another energy scale. Electron-lattice effects may
be directly incorporated into effective Hubbard model.
Assuming the coupling with the local displacement
(configurational) coordinate Q in the effective poten-
tial energy we arrive at a generalized Peierls—Hubbard
model [7]. From the other hand, the taking account of
similar effects in the kinetic energy results in a general-
ized Su—Schrieffer—Heeger (SSH) model [8]. The
correlation effect of an electronic background was
shown [5,6] to be of primary importance for atomic sys-
tems with filled or almost filled electron shells. Namely
such a situation is realized in oxides with O2�(2p6)
oxygen ions. In particular, the effect results in a corre-
lated character of a charge transfer that seems to be one
of the main features for 3d oxides.
Many strongly correlated 3d oxides reveal anoma-
lous sensitivity to a small nonisovalent substitution.
For example, only 2% Sr2� substituted for La3� in
La2CuO4 result in a dramatical suppression of long-
range copper antiferromagnetism, while it is sup-
pressed with isovalent Cu2� substitution by Zn2� at a
much higher concentration close to the site dilution
percolating threshold. Simultaneously, the transport
properties of La2�xSr xCuO4 system reveal uncon-
ventional insulator–metal duality starting from very
low dopant level [9]. Most likely, all this points to a
charge phase instability intrinsic for parent 214 sys-
tem which somehow evolves with nonisovalent substi-
tution due to a well developed charge potential
inhomogeneity and/or hole doping effect. The prob-
lem seems to be closely related with the hidden
multistability intrinsic to each solid [7,10]. If the
ground state of a solid is pseudo-degenerate, being
composed of true and false ground states with each
structural and electronic orders different from others,
one might call it multi-stable. In this connection it is
worth noting the text-book example of BaBiO3 system
where we unexpectedly deal with the dispropor-
tionated Ba3� + Ba5� ground state instead of the con-
ventional lattice of Ba 4� cations [11]. The bismuthate
situation can be viewed also as a result of a condensa-
tion of CT excitons, in other words, the spontaneous
generation of self-trapped CT excitons in the ground
state with a proper transformation of lattice parame-
ters. At present, a CT instability with regard to
disproportionation is believed to be a rather typical
property for a number of perovskite 3d transi-
tion-metal oxides such as SrFeO3, LaCuO3, RNiO3
[12], moreover, in solid state chemistry one consider
tens of disproportionated systems [13]. Novel princi-
ples must be developed to treat such CT unstable sys-
tems with their dramatical non-Fermi-liquid behavior.
In particular, we have to change the current paradigm
of the metal-to-insulator (MI) transition to that of an
insulator-to-metal (IM) phase transition. These two
approaches imply essentially different starting points:
the former starts from a rather simple metallic-like
scenario with inclusion of correlation effects, while
the latter does from strongly correlated atomic-like
scenario with the inclusion of a charge transfer. Elec-
tron-lattice polarization effects accompanying the
charge transfer appear to be of primary importance to
stabilize either phase state. One should emphasize that
the theoretical description of such systems is one of
the challenging problems in solid state physics. The
electronic states in strongly correlated 3d oxides man-
ifest both significant correlations and dispersional fea-
tures. The dilemma posed by such a combination is the
overwhelming number of configurations which must
be considered in treating strong correlations in a truly
bulk system. One strategy to deal with this dilemma is
to restrict oneself to small 3d-metal-oxygen clusters,
creating model Hamiltonians whose spectra may rea-
sonably well represent the energy and dispersion of
the important excitations of the full problem. Indeed,
such clusters as CuO4 in quasi-2D cuprates, MnO6 in
manganite perovskites are basic elements of crystal-
line and electronic structure. Despite a number of
principal shortcomings, including the boundary condi-
tions, the breaking of local symmetry of boundary at-
oms, sharing of common anions for nn clusters etc., the
embedded molecular cluster method provides both, a
clear physical picture of the complex electronic struc-
ture and the energy spectrum, as well as the possibil-
ity of quantitative modelling.
Hereafter, we develop a model approach to describe
different charge fluctuations and charge phases in
strongly correlated 3d oxides with main focus on the
correlated CT effects. As an illustrative model system
we address a simple mixed-valence system with three
possible stable nondegenerate valent states of a cat-
ion-anionic cluster, hereafter M: M M0, � , forming
the charge (isospin) triplet. The M0 valent state is as-
Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 315
sociated with the conceptually simple one like CuO4
6�
in insulating copper oxides (CuO, La2CuO4,
YBa2Cu3O6, Sr2CuO2Cl2, ...) or MnO6
9� in man-
ganite LaMnO3 or BiO6
9� in bismuthates. It is worth
noting that such a model is the most relevant one to
describe different cuprates where novel concepts
should compete with a traditional Hubbard model ap-
proach in a hole representation implying the vacuum
state formed by M � (CuO4
7�) centers, and some con-
centration of holes. That is why overall the paper we
refer the insulating cuprates to illustrate the main
concepts of the approach developed. Our algebra is
based on the S � 1 pseudo-spin formalism to be the ef-
fective tool for the description of the essential physics
both of insulators unstable with regard to the CT fluc-
tuations and related mixed-valence systems. Such an
approach provides the universal framework for a uni-
fied description of these systems as possible phase
states of a certain parent multi-stable system. In addi-
tion, we may make use of powerful methods developed
in the physics of spin systems. The model system of
M0,� centers is described in frames of S � 1
pseudo-spin formalism.
Our main goal is to describe different charge phases
of the model system and a scenario of evolution of visi-
bly typical insulator to unconventional electron-hole
Bose liquid which reveals many unexpected properties,
including superconductivity. The paper is organized as
follows: In Sec. 2 we address the effects of electron-lat-
tice polarization and relaxation. In Sec. 3 we introduce
the S � 1 pseudospin formalism to describe the model
mixed-valence systems, present the effective pseudospin
Hamiltonian and possible mean-field phase states. In
Sec. 4 we analyse an eh-representation of different exci-
tations in a monovalent M0 phase, discuss a CT instabil-
ity, and nucleation of electroh-hole (EH) droplets. Elec-
tron-hole Bose liquid is discussed in Sec. 5 with a main
focus on topological phase separation effect that accom-
panies the deviation from half-filling. Implications for
cuprates are discussed in Sec. 6.
2. Electron-lattice polarization effects
2.1. Correlation effects of electronic background
The correlation problem becomes of primary impor-
tance for atoms/ions near Coulomb instability when
the one-electron gluing cannot get over the destruc-
tive effect of the electron-electron repulsion. Such a
situation seems to realize in oxides where Hirsch et al.
[5] have proposed an instability of O2–(2p6) elec-
tronic background. The main suggestion in their the-
ory of «anionic metal» concerns the occurrence of the
nonrigid degenerate structure for a closed electron
shell such as O2�(2p6) with the internal purely corre-
lation degrees of freedom. In other words, one should
expect sizeable correlation effects not only from un-
filled 3d or oxygen 2p shells, but from completely
filled O 2p6 shell! In order to relevantly describe such
a nonrigid atomic background and its coupling to the
valent hole one might use a concept of the well-known
«shell-droplet» model for nuclei after Bohr and
Mottelson [14]. In accordance with the model a set of
completely filled electron shells which form an atomic
background or vacuum state for a hole representation
is described by certain internal collective degrees of
freedom and a number of physical quantities such as
electric quadrupole and magnetic moments. Valent
hole(s) moves around this nonrigid background with
strong interaction inbetween. Such an approach
strongly differs from the textbook one that implies a
rigid atomic orbital basis irrespective of varying fill-
ing number and external potential.
None of the effective many-body Hamiltonians that
are most widely used to study the effect of electron
correlation in solids such as the Hubbard model, the
Anderson impurity and lattice models, the Kondo
model, contain this very basic and fundamental aspect
of electron correlation that follows from the atomic
analysis [15]. The Hubbard on-site repulsion U be-
tween opposite spin electrons on the same atomic or-
bital is widely regarded to be the only important
source of electron correlation in solids. It is a clear
oversimplification, and we need in a more realistic
atomic models to describe these effects, especially for
atoms in a specific external potential giving rise to a
Coulomb instability. To this end we have proposed a
generalized nonrigid shell model (see Ref. 6). The
model represents a variational method for the many-
electron atomic configurations with the trial parame-
ters being the coordinates of the center of the one-par-
ticle atomic orbital. The resulting displacement of the
atomic orbitals allows a simple interpretation of the
electron density redistribution stemmed from taking
into account the electron-electron repulsion, and the
symmetry of a system can be readily used for the con-
struction of the trial many-electron wave function.
The model is a generalization of the well-known shell
model by Dick and Overhauser [16] widely used in
lattice dynamics. In frames of the model the ionic con-
figuration with filled electron shells is considered to
be composed of an outer spherical shell of 2(2l + 1)
electrons and a core consisting of the nucleus and the
remaining electrons. In an electric field the rigid shell
retains its spherical charge distribution but moves
bodily with respect to the core. The polarizability is
made finite by a harmonic restoring force of spring
constant k which acts between the core and shell. The
shells of two ions repel one another and tend to be-
316 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
A.S. Moskvin
come displaced with respect to the ion cores because of
this repulsion. The respective displacement vector ap-
pears to be a simplest collective coordinate which
specifies the change of the electron-nucleus attraction.
It should be noted that such a displacement does not
imply any variation in electron-electron repulsion and
respective correlation energy.
However, a simple shell model can be easily gener-
alized to take account of correlation effects. To this
end we must consider the displacements of separate
one-electron orbitals to form the set of the variational
parameters in a correlation function. Then we can in-
troduce both the displacement of the center of «grav-
ity» for filled shell and a set of the relative displace-
ments of separate one-electron orbitals with regard to
each other. The former form an «acoustical» mode and
are described in frames of conventional shell model,
while the latter form different novel «optical» modes.
Such a seemingly naive nonrigid shell picture can pro-
vide both the microscopic substantiation of the con-
ventional shell model and its generalization. More-
over, this nonrigid shell model points to a physically
clear procedure to account for the correlation effects.
Indeed, the «optical» displacement mode is believed
to minimize the electron-electron repulsion. It is
worth noting that the optical displacement mode
yields some sort of a hidden order parameter.
2.2. Electron-lattice relaxation effects
The minimal energy cost of the optically excited
disproportionation or electron-hole formation in insu-
lating cuprates is 2.0–2.5 eV [17]. However, the ques-
tion arises, what is the energy cost for the thermal exci-
tation of such a local disproportionation? The answer
implies first of all the knowledge of relaxation energy,
or the energy gain due to the lattice polarization by the
localized charges. The full polarization energy R in-
cludes the cumulative effect of electronic and ionic
terms, associated with the displacement of electron
shells and ionic cores, respectively [3]. The former term
Ropt is due to the nonretarded effect of the electronic
polarization by the momentarily localized electron-hole
pair given the ionic cores fixed at their perfect crystal
positions. Such a situation is typical for lattice response
accompanying the Franck–Condon transitions (optical
excitation, photoionization). On the other hand, all the
long-lived excitations, i.e., all the intrinsic thermally
activated states and the extrinsic particles produced as
a result of doping, injection or optical pumping should
be regarded as stationary states of a system with a de-
formed lattice structure. These relaxed states should be
determined from the condition that the system energy
has a local minimum when account is taken of the inter-
action of the electrons and holes with the lattice defor-
mations. At least, it means that we cannot, strictly
speaking, make use of the same energy parameters to
describe the optical (e.g., photoexcited) hole and ther-
mal (e.g., doped) hole.
For the illustration of polarization effects in
cuprates we apply the shell model calculations to look
specifically at energies associated with the localized
holes of Cu3� and O � in «parent» La2CuO4 com-
pound. It follows from these calculations that there is
a large difference in the lattice relaxation energies for
O � and Cu3� holes. The lattice relaxation energy,
��Rth
� , caused by the hole localization at the O-site
(4.44 eV) appears to be significantly larger than that
for the hole localized at the Cu-site (2.20 eV). This in-
dicates the strong electron-lattice interaction in the
case of the hole localized at the O-site and could sug-
gest that the hole trapping is more preferential in the
oxygen sublattice. In both cases we deal with the sev-
eral eV-effect both for electronic and ionic contribu-
tions to relaxation energy. Moreover, such an estima-
tion seems to be typical for different insulators [3,4].
Transition metal oxides with strong electron and
lattice polarization effects need in a revisit of many
conventional theoretical concepts. In particular, we
should modify the usual Hubbard model as it is done,
for instance, in a «dynamic» Hubbard model by
Hirsch [15] or a modified Peierls—Hubbard model [7]
with a classical description of the anharmonic
core/shell displacements. Depending on the parame-
ters of the hole-configurational coupling and corre-
lated hopping the modified Peierls—Hubbard model
[7] can stabilize the «disproportionated» or charge or-
dered (CO) electron phase with the on-site filling
numbers n � 0, and n � 2 thus leading to the «nega-
tive-U» effect.
3. Model mixed-valence system
3.1. Pseudospin operators
The problem of the multi-stability of solids looks
rather trivial when one say about the orbital and/or
spin degrees of freedom. Usually in such a case we start
from the lattice of coupled orbital and/or spin
momenta described by the relevant spin-Hamiltonian
that implies the variety of possible collective orbital
and/or spin orderings that compete with each other
under different external conditions. In other words, the
multi-stability accompanies the basic degeneracy inher-
ent to a certain atom, ion, or center with a nonzero or-
bital and/or spin momentum. Such an outlook is be-
lieved to be easily extended to systems with charge
degree of freedom which can be represented to be a
system of either centers which possible charge states
form a pseudo-multiplet. Below we address a simple
Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 317
model of a mixed-valence system with three possible
stable valent states of a cation-anionic cluster, hereaf-
ter M: M M0, � , forming the charge (isospin) triplet.
Similarly to the neutral-to-ionic electronic-structural
transformation in organic charge-transfer crystals (see
paper by T. Luty in Ref. 10) the system of charge trip-
lets can be described in frames of the S � 1 pseudospin
formalism. To this end we associate three charge states
of the M center with different valence: M M0, � with
three components of S � 1 pseudospin (isospin) triplet
with MS � � �0 1 1, , , respectively. The S � 1 spin algebra
includes three independent irreducible tensors �Vq
k of
rank k � 0 1 2, , with one, three, and five components, re-
spectively, obeying the Wigner—Eckart theorem [18]
� � ��SM V SMS q
k
S| � |
� �
�
�
�
�
�
� � ��
�
( ) | | � | |1 S M
S S
kS
S k S
M q M
S V S . (1)
Here we make use of standard symbols for the Wigner
coefficients and reduced matrix elements. In a more
conventional Cartesian scheme a complete set of the
nontrivial pseudospin operators would include both
S and a number of symmetrized bilinear forms
{ } ( )S S S S S Si j i j j i� � , or spin-quadrupole operators,
which are linearly coupled to Vq
1 and Vq
2, respectively:
V S S Sq q z
1
0� �; , S S iSx y� �
�
1
2
( );
V Sz0
2 2 23� �( ),S V S S S Sz z� � �� �1
2 ( ),
V S� ��2
2 2.
To describe different types of pseudospin ordering in
a mixed-valence system we have to introduce eight or-
der parameters: two diagonal order parameters � �Sz
and � �Sz
2 , and six off-diagonal order parameters � �Vq
k
(q � 0). Two former order parameters can be termed
as valence and ionicity, respectively. The off-diago-
nal order parameters describe different types of the
valence mixing. Indeed, operators Vq
k (q � 0) change
the z-projection of pseudospin and transform the
| SMS � state into | SM qS � � one. In other words,
these can change valence and ionicity. It should be
noted that for the S � 1 pseudospin algebra there are
two operators: V�1
1 and V�1
2 , that change the
pseudospin projection by
1, with slightly different
properties: � � � �
� �� �0 1 1 0 1| � | | � | ,S S� � but
� � � � � �
� � � �� � � �0 1 1 0 1| ( )| | ( )| .S S S S S S S Sz z z z�
Three spin-linear (dipole) operators �
, ,S123 and five inde-
pendent spin-quadrupole operators { � , � } �S Si j ij� 2
3
2S �
given S � 1 form eight Gell-Mann operators being the gen-
erators of the SU(3) group [19]. The generalized spin-1
model can be described by the Hamiltonian bilinear on the
SU(3)-generators �( )k
� � �
,,
( ) ( )H Jkm
k mi
i
k
i
m� �
�
���
1
8
�
�
� � . (2)
Here i,� denote lattice sites and nearest neighbors, re-
spectively. This is a S � 1 counterpart of the S /� 1 2
model Heisenberg Hamiltonian with three generators of
the SU(2) group or Pauli matrices included instead of
eight Gell-Mann matrices. In frames of a classical, or
mean-field description of the S � 1 quantum pseudospin
system we start with trial on-site functions [19]:
Ψ Ψj j j� �( ( ) ( ))c , where j labels a lattice site and the
spin functions � in a Cartesian basis are used:
� z � �|10 and � x y /, (| | )∼ 11 1 1 2�
� � ; c a b� � i is
a vector order parameter. The linear (dipole) pseudospin
operator within | , ,x y z� basis is represented by a simple
matrix: � � � �� � �i j k ijkS i| | , and for the order para-
meters one easily obtains: � � � � �� [ ];S a b2 � � �{ � � }S Si j
� � �2( )� ij i j i ja a b b given the normalization cons-
traint a b2 2 1� � . Thus, for the case of spin-1 system
the order parameters are determined by two classical vec-
tors (two real components of one complex vector
c a b� � i ). The two vectors are coupled, so the minimal
number of dynamic variables describing the S � 1 spin
system appears to be equal to four.
3.2. Effective pseudospin Hamiltonian
Effective pseudospin Hamiltonian for our model
mixed-valence system should incorporate a large num-
ber of contributions that describe different long- and
short-range coupling between M0,� centers, single-ion
and two-ion terms. Single-site terms can be subdi-
vided into single-ion anisotropy and pseudo-Zeeman
interaction. Bilinear and biquadratic two-site terms
can be subdivided into diagonal interactions like
«density-density», and off-diagonal terms that de-
scribe charge fluctuations conserving the total charge
of the system, such as one-electron (hole) and
two-electron (hole) transport. An effective
pseudospin Hamiltonian of the model mixed-valence
system which takes into account the main part of
aforementioned contributions can be represented as
follows
� ( )
,
H S h S v S S
i
i iz i iz ij
i j
iz jz� � � �� �
�
� 2 2 2
� �
�
�V S Sij
i j
iz jz
,
[ ( )( )
,
D S S S Sij
i j
i j i j
1
�
� � � �� � �
318 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
A.S. Moskvin
� � � �� � � �
�
� � � ��D T T T T t S S S Sij i j i j ij
i j
i j i j
( )
,
( )] (2 2 2 2 2 ),
(3)
where T S S S Sz z� � �� �( ). Two first single-ion
terms describe the effects of bare pseudospin split-
ting, or the local energy of M0,� centers. Interest-
ingly, the parameter � can be related with correlation
Hubbard parameter U: U � 2�. The second term may
be associated with a pseudomagnetic field hi , in par-
ticular, a real electric field. It is easy to see that it de-
scribes an electron/hole assymetry. The third and
fourth terms describe the effects of long- and short-
range inter-ionic interaction including screened Cou-
lomb and covalent coupling. If to apply the familiar
spin terminology, the first term in (3) represents a
single-ion anisotropy, the second does the Zeeman
term, the fourth and fifth do the anisotropic
Heisenberg exchange, and the third and sixth do the
biquadratic spin-quadrupolar coupling.
One should note that despite many simplifications,
the effective Hamiltonian (3) is rather complex, and
represents one of the most general forms of the
anisotropic S � 1 non-Heisenberg Hamiltonians. For the
system there are two classical (diagonal) order para-
meters: � � �S nz being a valence, or charge density with
electro-neutrality constraint � � � �
i
i
i
izn S 0, and
� � �S nz p
2 being the density of polar centers M � , or
«ionicity». In addition, there are two unconventional
off-diagonal order parameters: «fermionic» � ��S and
«bosonic» � ��S2 ; the former describes a phase ordering
for the disproportionation reaction, or the single-particle
transfer, while the latter does for exchange reaction, or
for the two-particle transfer. Indeed, the �S� operator
creates a hole and is fermionic in nature, whereas the �S�
2
does a hole pair, and is bosonic in nature.
3.3. Single and two-particle transport
The last three terms in (3) representing the one-
and two-particle hopping, respectively, are of primary
importance for the transport properties, and deserve
special interest. Two types of one-particle hopping are
governed by two transfer integrals D( , )12 , respec-
tively. The transfer integral � � �t D Dij ij ij( )( ) ( )1 2
specifies the probability amplitude for a local
disproportionation, or the eh-pair creation: M0 �
� � ��M M M0 �; and the inverse process of the
eh-pair recombination: M M M M� � � �� 0 0,
while the transfer integral t D Dij ij ij
� � � �( )( ) ( )1 2 speci-
fies the probability amplitude for a polar center trans-
fer: M M M M� �� � �0 0 , or the motion of the
electron (hole) center in the matrix of M0 centers or
motion of the M0 center in the matrix of M � centers.
It should be noted that, if �� �tij 0 but � �tij 0, the
eh-pair is locked in two-site configuration. The two-
electron (hole) hopping is governed by transfer inte-
gral tij , or a probability amplitude for the exchange
reaction: M M M M� �� � �� � , or the motion of
the electron (hole) center in the matrix of hole (elec-
tron) centers. It is worth noting that in Hubbard-like
models all the types of one-electron (hole) transport
are governed by the same transfer integral:
� � �� �t t tij ij ij , while our model implies independent
parameters for a disproportionation/recombination
process and simple quasiparticle motion in the matrix
of M0 centers. In other words, we deal with a «corre-
lated» single particle transport [5].
3.4. Generic MFA phases
First of all we would like to emphasize the differ-
ence between classical and quantum mixed-valence
systems. Classical (or chemical) description implies
the neglect of the off-diagonal purely quantum CT ef-
fects: D t( , )12 0� � , hence the valence of any site re-
mains to be definite: 0 1,
, and we deal with a system
of localized polar centers. In quantum systems with a
nonzero charge transfer we arrive at quantum
superpositions of different valence states resulting in
an indefinite on-site valence and ionicity which effec-
tive, or mean values � �Sz and � �Sz
2 can vary from –1 to
+1 and 0 to +1, respectively.
Making projection of the effective pseudospin
Hamiltonian for the system onto a space of on-site
states like �( )j , we obtain an energy functional which
equivalent to a classical energy of the two coupled
vector ( , )a b fields defined on the common lattice.
Thus, in the framework of the pseudospin S � 1centers
model when the collective wave function is repre-
sented to be a product of the site functions, the quan-
tum problem is reduced to a classical variation prob-
lem for a minimum of the energy for two coupled
vector fields. All the MFA phases one may subdivide
into those with a definite and indefinite ionicity,
respectively. There are two MFA phases with definite
ionicity
3.4.1. Insulating monovalent M0 phase with � � �Sz
2 0
The M0 phase is specified by a simple uniform ar-
rangement of a and b vectors parallel to z axis:
a | b || | Oz . In such a case the on-site wave function is
specified by unit vector (a, or b) parallel to z axis. It is
a rather conventional ground-state phase for various
charge transfer insulators such as oxides with a posi-
tive magnitude of � parameter (U � 0). All the centers
have the same bare M0 valence state. In other words,
the M0 phase is characterized both by definite site
ionicity and valence. So, all the order parameters turn
into zero: � � � � � � � � � � � �� �S S S Sz z
2 2 0. This is an
Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 319
«easy-plane» phase for the pseudospins, but an
«easy-axis» one for the a and/or b vectors. This phase
is a typical one for the ground state of insulating tran-
sition metal oxides, or Mott—Hubbard insulators. It
is worth noting that in frame of conventional band
model approach the M0 phase, e.g., in parent cup-
rates, is associated with a metallic half-filled hole
band system.
3.4.2. Mixed-valence binary (disproportionated)
M � phase with � � �Sz
2 1
This phase usually implies an overall
disproportionation M M M M0 0� � �� � that seems
to be realizable if � parameter becomes negative one
(negative U � 0 effect). It is a rather unconventional
phase for insulators. All the centers have the «ionized»
valence state, one half the M � state, and another half the
M � one, though one may in common conceive of devia-
tion from fifty-fifty distribution. A simplified «chemical»
approach to M � phase as to a classical disproportionated
phase is widely spread in solid state chemistry [13]. In
contrast with the M0 phase the M � phase is specified by
a planar orientation of a and b vectors (a b, �Oz ) with a
varied angle in between. There is no fermionic transport:
� � ��S 0, while the bosonic one may exist, and, in com-
mon, � � � � � ��
�S a b
i a b2 0cos( ) ( )� �
e . This is an
«easy-axis» phase for the pseudospins, but an «easy-
plane» one for the a and b vectors. The mixed valence
M � phase as a system of strongly correlated electron and
hole centers is equivalent to the lattice hard-core Bose
system with an inter-site repulsion, or electron-hole Bose
liquid (EHBL) in contrast with EH liquid in conven-
tional semiconductors like Ge, Si where we deal with a
two-component Fermi liquid. Indeed, one may address
the electron M � center to be a system of a local boson
(e2) localized on the hole M � center: M M e� �� � 2.
In accordance with this analogy we assign three well
known molecular-field uniform phase states of the M �
binary mixture:
i) charge ordered (CO) insulating state with
� � �
Sz 1, a b� , and zero modulus of bosonic off-diago-
nal order parameter: | |� � ��S2 0;
ii) Bose-superfluid (BS) superconducting state
with � � �Sz 0, a and b being collinear, � � ��S i2 2e
;
iii) mixed Bose-superfluid-charge ordering (BS +
+ CO) superconducting state (supersolid) with
0 1� � � �| |Sz , a and b being oriented in xy plane, but
not collinear, � � � � � ��
�S a b
i a b2 0cos( ) ( )� �
e . In
addition, we should mention the high-temperature
nonordered (NO) Bose-metallic phase with �� �� �Sz 0.
Rich phase diagram of M � binary mixture with uncon-
ventional superfluid and supersolid regions looks
tempting, however, actually, their stabilization requires
strong suppression of Coulomb repulsion between elec-
tron (hole) centers. Despite significant screening effect,
the stabilization of uniform BS or BS + CO supercon-
ducting state as a result of a disproportionation reaction
in a bare insulator [13] seems to be unrealistic.
3.4.3. Mixed-valence ternary («under-disproportio-
nated») M0,� phase
For two preceding cases the order parameter � �Sz
2 ,
or ionicity relates to its limiting values (0 or 1, respec-
tively). For the MFA phase with indefinite ionicity,
or mixed-valence ternary («under-disproportiona-
ted») M0,� phase, 0 12� � � �Sz , that is we have a mix-
ture of the M M0, � centers. From the viewpoint of
the classical a b, vectors formalism the phase corre-
sponds to the arbitrarily space-oriented l a b� �[ ] vec-
tor. Both off-diagonal order parameters, fermionic
� ��S and bosonic � ��S2 are, in common, non-zero, al-
beit with some correlation in between. So, for the
ternary system one expects a coexistence of fermionic
and bosonic transport. It should be noted that a com-
plete pseudospin description of the two last model
mixed-valence systems implies a two-sublattice ap-
proximation to be a minimal model compatible with a
sign of Coulomb interaction and a respective tendency
towards the checkerboard-like charge ordering.
4. Insulating monovalent M0 phase
4.1. Electron, hole, and electron-hole excitations
Starting from monovalent M0 phase as a vacuum
state |0� we introduce an electron-hole representation
where Mi
� center is derived as a result of an electron
creation � |†ai 0�, and Mi
� center is derived as a result of a
hole creation � |†bi 0�. Then we transform pseudospin
Hamiltonian (3) into that of a system of effective elec-
trons and holes
� ( ) ( � � � � )† †H n n t a a b bi
h
i
i
e
ij
ij
i j i j� � � � �� �
�
�
[ ( )( ( )
,
V ij n n V ij n nhh
i j
i
h
j
h ee
i
e
j
e� � �
� � � �� �
�
V ij n n n n t a b a beh
i
h
j
e
j
h
i
e
ij i j i j
ij
( )( )] ( � � � � )† †
� .
(4)
where Vii � � to prohibit two-particle occupation of a
single site. Here we suppose h � 0 that provides an
electron-hole symmetry and neglect the two-particle
transport. The first two sums in (4) represents the sin-
gle particle (electrons/holes) terms, the third one does
the interparticle coupling, while the fourth one descri-
bes the creation and annihilation (recombination) of
eh-pairs. It is worth noting that the latter terms
320 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
A.S. Moskvin
describe some sort of eh-coupling. In terms of a
pseudospin analogy the electrons and holes are asso-
ciated with pseudospin �Sz �
1 deviations for an
easy-plane magnet, localized or delocalized (pseudospin
wave).
The behavior of electron/hole system crucially de-
pends on the relation between two transfer integrals
� ��t tij ij, . Below we address two distinct limiting situa-
tions:
I. �� �tij 0: Forbidden recombination/creation re-
gime. In this case we deal with the bands of well de-
fined electrons and holes with a charge gap Eg
e h, �
� � �� z tnn| | for both types of carriers. Optical gap for
unbound eh-pairs is E Eg
eh
g
e h� 2 , . However, in such a
case we may expect for the formation of Wannier
excitons, or eh-pairs bound due to a screened Coulomb
eh-coupling. In terms of pseudospin formalism the
Wannier excitons may be regarded as two pseudospin
waves having formed a quasilocalized state due to a long-
range antiferromagnetic Ising exchange: V S Sij iz jz .
II. tij
� � 0: Regime of localized electrons and holes
with a dimerization effect and well defined nn
eh-pairs, or CT excitons. In such a case the charge gap
is Eg
e h, � � for both types of localized quasiparticles.
The �� �tij 0 hopping results in a dimerization effect
with a quantum renormalization of the vacuum state
and indefinite ionicity, the formation of two types of
localized eh-pairs, or CT excitons of even S and odd P
type. In frames of our nn approximation the CT
excitons are localized. Optical gap is determined by the
energy of P type CT exciton: Eg
eh � �2� �, with � be-
ing a covalent correction to the ground state energy.
Thus we arrive at two limiting types of monovalent
M0 insulators with a dramatic difference in behavior
of electrons and holes, as well as electron-hole pairs.
In type I insulators ( � �� ��t tij ij ) we deal with well de-
fined bands of electrons and holes forming Wannier
excitons, while in type II insulators ( �� �� �t tij ij ) we
deal with localized electrons and holes which can form
nn eh-pairs, or CT excitons. The most part of 3d oxides
are characterized by an antiferromagnetic spin back-
ground that implies strong spin reduction of one-parti-
cle transfer integrals �tij . In other words, these, seem-
ingly, belong to type II insulators, where spin-singlet
CT excitons can move through the lattice freely with-
out disturbing the antiferromagnetic spin background,
in contrast to the single hole motion. So, it seems that
the situation in antiferromagnetic 3d insulators differs
substantially from that in usual semiconductors or in
other bandlike insulators where, as a rule, the effec-
tive mass of the electron-hole pair is larger than that
of an unbound electron and hole. The Wannier
excitons are formed due to an eh-coupling, while the
CT excitons are formed due to a kinetic cutoff, or a
specific feature of correlated hopping, in other words,
the former have a potential, while the latter a kinetic
nature. It is worth noting that both M centers within
P type CT excitons have a certain ionicity in contrast
to S type CT excitons which can mix with bare
M M0 0 ground state. CT excitons form peculiar
quanta of a disproportionation reaction and may be
viewed to be a minimal droplet of electron-hole Bose
liquid. In general, eh-excitations in M0 phase consist
of superpositions of pairs of free electrons and holes,
and CT excitons. One expects two types of super-
positions: CT exciton-like and band-like. The former
have a nearly localized character, while the latter have
a nearly itinerant one.
The nature of the optical excitations accompanied
by creation of electron-hole pairs in 3d oxides is not
fully understood. One of the central issues in the
analysis of electron-hole excitations is whether low-ly-
ing states are comprised of free charge carriers or
excitons. A conventional approach implies that if the
Coulomb interaction is effectively screened and weak,
then the electrons and holes are only weakly bound
and move essentially independently as free charge-car-
riers. However, if the Coulomb interaction between
electron and hole is strong, excitons are believed to
form, i.e., bound particle-hole pairs with strong corre-
lation of their mutual motion. To distinguish bound
and unbound electron-hole states one might use the
density-density correlation function [20] C i j( , ) �
� � � � � � � � �( � � )( � � ) ,n n n ni i j j which measures a correla-
tion of charge fluctuations on site i to a charge fluctu-
ations on site j. A negative value correlates an excess
(deficit) of charge with deficit (excess), or elec-
tron-hole distribution. In frame of pseudospin ap-
proach this correlation function measures the longitu-
dinal (| | z) short-range antiferromagnetic fluctuations
C i j S S S Siz jz iz jz( , ) � � � � .� � � � � �� �
4.2. Charge transfer instability and CT exciton
self-trapping
Electron and hole in a CT exciton in type II M0 in-
sulator are strongly coupled both in between and with
the lattice. In contrast with conventional wide-band
semiconductors where the excitons dissociate easily
producing two-component electron-hole gas or plasma
[22], small CT excitons both free and self-trapped are
likely to be stable with regard the eh-dissociation. To
illustrate the principal features of CT exciton
self-trapping effect we addressed recently [21] a sim-
plified two-level model of a two-center MM cluster in
which a ground state and a CT state are associated
with a pseudospin 1/2 doublet, | � and |!�, respec-
tively. In addition, we introduced some configura-
tional coordinate Q associated with a deformation of
Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 321
the cluster, or respective anionic background [6]. For
lower branch of adiabatic potential (AP) in the sys-
tem we have either a single minimum point or the
two-well structure with two local minimum points,
leading to a «bistability» effect which is of primary
importance for our analysis. Indeed, these two points
may be associated with two (meta)stable charge states
with and without CT, respectively, which form two
candidates to struggle for a ground state. It is easy to
see, that for large values of the transfer integral the
system does not manifest bistability. Thus one con-
cludes that all the systems such as copper oxides may
be divided to two classes: CT stable systems with the
only lower AP branch minimum for a certain charge
configuration, and bistable, or CT unstable systems
with two lower AP branch minima for two local
charge configurations one of which is associated with
self-trapped CT excitons resulting from self-consistent
charge transfer and electron-lattice relaxation [4].
4.3. Nucleation of EH droplets and phase separa-
tion effects in CT unstable M0 phase
The AP bistability in CT unstable insulators points to
tempting perspectives of their evolution under either ex-
ternal impact. Metastable CT excitons in the CT unsta-
ble M0 phase being the disproportionation quanta pres-
ent candidate «relaxed excited states» to struggle for
stability with ground state and the natural nucleation
centers for electron-hole liquid phase. What way the CT
unstable M0 phase can be transformed into novel phase?
It seems likely that such a phase transition could be real-
ized due to a mechanism familiar to semiconductors with
filled bands such as Ge and Si where given certain condi-
tions one observes a formation of metallic EH liquid as a
result of the exciton decay [22]. However, the system of
strongly correlated electron M � and hole M � centers
appears to be equivalent to an electron-hole Bose liquid
in contrast with the electron-hole Fermi liquid in con-
ventional semiconductors. The Wannier excitons in the
latter wide-band systems dissociate easily producing
two-component electron-hole gas or plasma [22], while
small CT excitons both free and self-trapped are likely
to be stable with regard to the eh-dissociation. At the
same time, the two-center CT excitons have a very large
fluctuating electrical dipole moment | |d eRMM∼ 2 and
can be involved into attractive electrostatic dipole-di-
pole interaction. Namely this is believed to be important
incentive to the proliferation of excitons and its clus-
terization. The CT excitons are proved to attract and
form molecules called biexcitons, and more complex
clusters where the individuality of the separate exciton
is likely to be lost. Moreover, one may assume that like
the semiconductors with indirect band gap structure, it
is energetically favorable for the system to separate into
a low-density exciton phase coexisting with the micro-
regions of a high-density two-component phase
composed of electron M � and hole M � centers, or EH
droplets. Indeed, the excitons may be considered to be
well defined entities only at small content, whereas at
large densities their coupling is screened and their over-
lap becomes so considerable that they loose individuality
and we come to the system of electron M � and hole M �
centers, which form a electron-hole Bose liquid. An in-
crease of injected excitons in this case merely increases
the size of the EH droplets, without changing the free
exciton density.
Homogeneous nucleation implies the spontaneous
formation of EH droplets due to the thermodynamic
fluctuations in exciton gas. Generally speaking, such a
state with a nonzero volume fraction of EH droplets and
the spontaneous breaking of translational symmetry can
be stable in nominally pure insulating crystal. However,
the level of intrinsic nonstoihiometry in 3d oxides is sig-
nificant (one charged defect every 100–1000 molecular
units is common). The charged defect produces random
electric field, which can be very large (up to108 V/cm)
thus promoting the condensation of CT excitons and the
inhomogeneous nucleation of EH droplets. Deviation
from the neutrality implies the existence of additional
electron, or hole centers that can be the natural centers
for the inhomogeneous nucleation of the EH droplets.
Such droplets are believed to provide the more effective
screening of the electrostatic repulsion for additional
electron/hole centers, than the parent insulating phase.
As a result, the electron/hole injection to the insulating
M0 phase due to a nonisovalent substitution as in
La2�xSrxCuO4, Nd2�xCexCuO4, or change in oxy-
gen stoihiometry as in YBa2Cu3O6�x , La2CuO4��,
La2Cu1�xLixO4, or field-effect is believed to shift the
phase equilibrium from the insulating state to the un-
conventional electron-hole Bose liquid, or in other
words induce the insulator-to-EHBL phase transition.
This process results in an increase of the energy of the
parent phase and creates proper conditions for its com-
peting with others phases capable to provide an effective
screening of the charge inhomogeneity potential. The
strongly degenerate system of electron and hole centers
in EH droplet is one of the most preferable ones for this
purpose. At the beginning (nucleation regime) an EH
droplet nucleates as a nanoscopic cluster composed of
several number of neighboring electron and hole centers
pinned by disorder potential. Charged defects support-
ing the EH droplet nucleation promote the formation of
metastable («superheated») clusters of parent phase. It
is clear that such a situation does not exclude the
self-doping with the formation of a self-organized collec-
tive charge-inhomogeneous state in systems which are
near the charge instability. EH droplets can manifest it-
322 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
A.S. Moskvin
self remarkably in various properties of the 3d oxides
even at small volume fraction, or in a «pseudoimpurity
regime». Insulators in this regime should be considered
as phase inhomogeneous systems with, in general, ther-
mo-activated mobility of the inter-phase boundaries. On
the one hand, main features of this «pseudoimpurity re-
gime» would be determined by the partial intrinsic con-
tributions of the appropriate phase components with
possible limitations imposed by the finite size effects.
On the other hand, the real properties will be deter-
mined by the peculiar geometrical factors such as a vol-
ume fraction, average size of droplets and its dispersion,
the shape and possible texture of the droplets, the geo-
metrical relaxation rates. These factors are tightly cou-
pled, especially near phase transitions for either phase
(long range antiferromagnetic ordering for the parent
phase, the charge ordering and other phase transforma-
tions for the EH droplets) accompanied by the variation
in a relative volume fraction.
Numerous examples of the unconventional behavior
of the 3d oxides in the pseudo-impurity regime could be
easily explained with taking into account the inter-phase
boundary effects (coercitivity, the mobility threshold,
non-ohmic conductivity, oscillations, relaxation etc.)
and corresponding characteristic quantities. Under in-
creasing doping the «pseudo-impurity regime» with a
relatively small volume fraction of EH droplets
(nanoscopic phase separation) can gradually transform
into a macro- (chemical) «phase-separation regime»
with a sizeable volume fraction of EH droplets, and fi-
nally to a new EH liquid phase. Our scenario can readily
explain photo-induced effects in pseudo-impurity phase
[21]. Indeed, the illumination of a material with light
leads to the generation of eh-pairs that will proliferate
and grow up to be a novel nonequilibrium EH droplet or
simply to be trapped in either EH droplet with the rise
in its volume fraction. The excitation energy appears to
be lower when exciton is closer to the EH droplet.
Therefore once the excitation transfer is finite, the opti-
cal excitation is attracted to the nearest neighbour of the
EH clusters so that this cluster expands effectively un-
der the light irradiation. In other words, the
photoexcitation would result in an increase of the EH
droplet volume fraction, that is why its effect in optical
response resembles that of chemical doping. After
switching off the light the droplet phase would relax to
the thermodynamically stable state. Furthermore, such a
simple model can immediately explain the persistent
photoconductivity (PPC) phenomena, found in insulat-
ing YBCO system [23], where the oxygen reodering pro-
vides the mechanism of a long-time stability for the EH
droplets. In PPC phenomena, an illumination of a mate-
rial with light leads to a long-lived photoconductive
state. During the illumination of underdoped YBCO
near the insulator–metal transition, the material may
even become superconducting.
5. Electron-hole Bose liquid
The pseudospin Hamiltonian (3) for the mixed-va-
lence binary (disproportionated) M � phase, or elec-
tron-hole Bose liquid can be mapped onto the Hamil-
tonian of hard-core Bose gas on a lattice (Bose—Hubbard
model) [24]:
H t B B B BBG ij i j j i
i j
� � � �
�
� ( )† †
� �
�
� �V N N Nij i j
i j
i"
, (5)
where N B Bi i i� † , " is a chemical potential derived
from the condition of fixed full number of bosons
N Nl
i
N
i� � � �
�1
, or concentration n N /Nl� # [ , ]01 . The
tij denotes an effective transfer integral, Vij is an
intersite interaction between the bosons. Here B B†( )
are the Pauli creation (annihilation) operators; N is a
full number of sites. From the other hand this
Hamiltonian is equivalent to a system of spins
S /� 1 2 exposed to an external magnetic field in the z
direction. Indeed, the charge ( , )e h , or M � doublet
one might associate with two possible states of the
charge pseudospin (isospin) s /� 1 2: |� �1 2/ and
|� �1 2/ for electron M � and hole M � centers, respec-
tively. Then the effective Hamiltonian can be written
as follows [24–26]:
H J s s s sBG
xy
i j
i j j iij
� � �
�
� � � �� ( )
� �
�
� �J s s h sij
z
i j
i
z
j
z
i
z
i
, (6)
where
J tij
xy
ij� 2 , J Vij
z
ij� , h V s Bij
j j i
� � �
�
��"
( )
, ,
1
2
s B s B B s s isz
i i
x y� �� � � � � �
1
2
1
2
1
2
† †, , ( ).�
The model exhibits many fascinating quantum phases
and phase transitions. Early investigations predict at
T � 0 charge order (CO), Bose superfluid (BS) and
mixed (BS + CO) supersolid uniform phases with an
Ising-type melting transition (CO–NO) and Koster-
litz–Thouless-type (BS–NO) phase transitions to a
nonordered normal fluid (NO) in 2D systems [24].
Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 323
5.1. Topological phase separation in 2D EH Bose
liquid away from half-filling
Above we focused on the homogeneous phase states
of the mixed-valence system. Main short-length scale
charge fluctuations in M0 and M � systems are associ-
ated with a thermal exciton creation, or annihilation
due to a reaction: ( ) ( )M M M M0 0� $ �� � .
Amongst the long-length scale charge fluctuations in a
model system we would like to address the topological
defects in quasi-2D systems such as cuprates, in par-
ticular, different bubble-like entities like skyrmions,
or another out-of-plane vortices. Namely these one can
play the main role in a nucleation of unconventional
charge phases.
One of the fundamental hot debated problems in
bosonic physics concerns the evolution of the charge
ordered ground state of 2D hard-core BH model
(hc-BH) with a doping away from half-filling which
in our case relates closely with the doping response of
nominal insulating phase. The most recent quantum
Monte Carlo (QMC) simulations [27] found two sig-
nificant features of the 2D Bose–Hubbard model with
a screened Coulomb repulsion: the absence of su-
persolid phase at half-filling, and a growing tendency
to phase separation (CO + BS) upon doping away
from half-filling. The physics of the CO + BS phase
separation in Bose–Hubbard model is associated with
a rapid increase of the energy of a homogeneous CO
state with doping away from half-filling due to a large
«pseudospin-flip» energy cost. Hence, it appears to be
energetically more favorable to «extract» extra bosons
(holes) from the CO state and arrange them into finite
clusters with a relatively small number of particles.
Such a droplet scenario is believed to minimize the
long-range Coulomb repulsion.
Magnetic analogy allows us to make unambiguous
predictions as regards the doping of BH system away
from half-filling. Indeed, the boson/hole doping of
checkerboard CO phase corresponds to the magnetiza-
tion of an antiferromagnet in z direction. In the uni-
form easy-axis lz phase of anisotropic antiferromagnet
the local spin-flip energy cost is rather big. In other
words, the energy cost for boson/hole doping into
checkerboard CO phase appears to be big one due to a
large contribution of boson-boson repulsion. How-
ever, the magnetization of the anisotropic anti-
ferromagnet in an easy axis direction may proceed as a
first order phase transition with a «topological phase
separation» due to the existence of antiphase domains.
The antiphase domain walls provide the natural nucle-
ation centers for a spin-flop phase having enhanced
magnetic susceptibility as compared with small if any
longitudinal susceptibility thus providing the advan-
tage of the field energy. Namely domain walls would
specify the inhomogeneous magnetization pattern for
such an anisotropic easy-axis antiferromagnet in rela-
tively weak external magnetic field. As concerns the
domain type in quasi-two-dimensional antiferromag-
net one should emphasize the specific role played by
the cylindrical, or bubble domains which have finite
energy and size. These topological solitons have the
vortex-like in-plane spin structure and resemble classi-
cal, or Belavin–Polyakov skyrmions [28]. Although
some questions were not completely clarified and re-
main open until now, the classical and quantum
skyrmion-like topological defects are believed to be a
genuine element of essential physics both of ferro- and
antiferromagnetic 2D easy-axis systems. The magnetic
analogy seems to be a little bit naive, however, it
catches the essential physics of doping the hc-BH sys-
tem. Recently [26] we have shown that the doping, or
deviation from half-filling in 2D EH Bose liquid is ac-
companied by the formation of multi-center topologi-
cal defect such as charge order bubble domain(s) with
Bose superfluid and extra bosons both localized in
domain wall(s), or a topological CO + BS phase sepa-
ration, rather than an uniform mixed CO + BS super-
solid phase. The number of such entities in a multi-
granular texture nucleated with doping has to nearly
linearly depend on the doping. Generally speaking,
each individual bubble may be characterized by its po-
sition, nanoscale size, and the orientation of U(1) de-
gree of freedom. In contrast with the uniform states
the phase of the superfluid order parameter for a bub-
ble is assumed to be unordered. In the long-wave-
length limit the off-diagonal ordering can be described
by an effective Hamiltonian in terms of U(1) (phase)
degree of freedom associated with each bubble. Such a
Hamiltonian contains a repulsive, long-range Cou-
lomb part and a short-range contribution related to
the phase degree of freedom. The latter term can be
written out in the standard for the XY model form of
a so-called Josephson coupling [25,29]
H JJ ij
i j
i j� � �
�
�
,
cos( ),� � (7)
where � �i j, are global phases for micrograins centered
at points i j, , respectively, Jij is Josephson coupling pa-
rameter. Namely the Josephson coupling gives rise to
the long-range ordering of the phase of the superfluid
order parameter in such a multi-center texture. Such a
Hamiltonian represents a starting point for the analysis
of disordered superconductors, granular superconduc-
tivity, insulator–superconductor transition with � �i j,
array of superconducting islands with phases � �i j, .
To account for Coulomb interaction and allow for quan-
tum corrections we should introduce into effective
Hamiltonian the charging energy [29]
324 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
A.S. Moskvin
H q n C ni
i j
ij jch � � � �1
2
2 1
,
( ) ,
where ni is a number operator for particles bound in
i-th micrograin; it is quantum-mechanically conjugated
to �: n i /i i� � % %� , ( )C ij
�1 stands for the capacitance
matrix, q for a particle charge. Such a system appears to
reveal a tremendously rich quantum-critical structure
[30,31]. In the absence of disorder, the T � 0 phase dia-
gram of the multi-bubble system implies either triangu-
lar, or square crystalline arrangements with possible
melting transition to a liquid. The critical properties of
a two-dimensional lattice without any internal degree of
freedom are successfully described applying the BKT
(Berezinsky—Kosterlitz—Thowless) theory to disloca-
tions and disclinations of the lattice, and proceeds in
two steps. The first implies the transition to a liq-
uid-crystal phase with a short-range translational order,
the second does the transition to isotropic liquid. For
such a system provided the bubble positions fixed at all
temperatures, the long-wave-length physics would be
described by an (anti)ferromagnetic XY model with
expectable BKT transition and gapless XY spin-wave
mode. The low-temperature physics in a multi-bubble
system is governed by an interplay of two BKT transi-
tions, for the U(1) phase and positional degrees of free-
dom, respectively [31]. Dislocations lead to a mismatch
in the U(1) degree of freedom, which makes the dislo-
cations bind fractional vortices and lead to a coupling
of translational and phase excitations. Both BKT tem-
peratures either coincide (square lattice) or the melting
one is higher (triangular lattice) [31].
Quantum fluctuations can substantially affect
these results. Quantum melting can destroy U(1) or-
der at sufficiently low densities where the Josephson
coupling becomes exponentially small. In terms of our
model, the positional order corresponds to an incom-
mensurate charge density wave, while the U(1) order
does to a superconductivity. In other words, we arrive
at a subtle interplay between two orders. The super-
conducting state evolves from a charge order with
T TC m& , where Tm is the temperature of a melting
transition which could be termed as a temperature of
the opening of the insulating gap (pseudogap!?). The
normal modes of a dilute multi-bubble system include
the pseudospin waves propagating inbetween the bub-
bles, the positional fluctuations, or quasiphonon mo-
des, which are gapless in a pure system, but gapped
when the lattice is pinned, and, finally, fluctuations
in the U(1) order parameter.
The orientational fluctuations of the multi-bubble
system are governed by the gapless XY model [30].
The relevant model description is most familiar as an
effective theory of the Josephson junction array. An
important feature of the model is that it displays a
quantum-critical point. The low-energy collective ex-
citations of a multi-bubble liquid includes an usual
longitudinal acoustic phonon-like branch. The liquid
crystal phases differ from the isotropic liquid in that
they have massive topological excitations, i.e., the
disclinations. One should note that the liquids do not
support transverse modes, these could survive in a liq-
uid state only as overdamped modes. So that it is rea-
sonable to assume that solidification of the bubble lat-
tice would be accompanied by a stabilization of
transverse phonon-like modes with its sharpening be-
low melting transition. In other words, an instability
of transverse phonon-like modes signals the onset of
melting. The phonon-like modes in the bubble crystal
have much in common with usual phonon modes, how-
ever, due to electronic nature these can hardly be de-
tected if any by inelastic neutron scattering. A generic
property of the positionally ordered bubble configura-
tion is the sliding mode which is usually pinned by the
disorder. The depinning of sliding mode(s) can be de-
tected in a low-frequency and low-temperature optical
response.
6. Implications for cuprates
The unconventional behavior of cuprates strongly
differs from that of ordinary metals and merely resem-
bles that of doped semiconductor. The copper oxides
start out life as insulators in contrast with BCS super-
conductors being conventional metals. In our view, the
essential physics of the doped cuprates, as well as other
strongly correlated oxides, is driven by a self-trapping of
the CT excitons, both one-center, and two-center [17].
Such excitons are the result of self-consistent charge
transfer and lattice distortion with the appearance of the
«negative-U» effect [3,4]. Thus, three types of CuO4
centers CuO4
567, , � should be considered in cuprates on
equal footing. As regards the self-trapped CT excitons
(STE) in cuprates, we have some straightforward exper-
imental indications. A key characteristic of the STE is its
luminescence: STE are short-lived luminescent states of
excited crystals. The observation of photoluminescence
(PL) near 2.0–2.4 eV in La2CuO4, near 1.3 and 2.4 eV
in YBa2Cu3O6, near 178 195 2 06. , . , . eV in PrBa2Cu3O6
[32] is a direct evidence of strongly localized long-lived
states related to self-trapped excitons or their deriva-
tives. Thus, cuprates are believed to be unconventional
systems which are unstable with regard to a self-trap-
ping of the low-energy charge transfer excitons with a
nucleation of electron-hole droplets being actually the
system of coupled electron CuO4
7� and hole CuO4
5� cen-
ters having been glued in lattice due to a strong elec-
tron-lattice polarization effects.
Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 325
The system of strongly correlated electron CuO4
7�
and hole CuO4
5� centers appears to be equivalent to an
electron-hole Bose liquid (EHBL) in contrast with the
electron-hole Fermi liquid in conventional semiconduc-
tors. A simple model description of such a liquid im-
plies a system of local singlet bosons with a charge of
q e� 2 moving in a lattice formed by hole centers. Local
boson in our scenario represents the electron counter-
part of Zhang–Rice singlet, or two-electron configura-
tion b Ag g1
2 1
1 . Naturally, that conventional electron
CuO4
7� center represents a relaxed state of composite
system: «hole CuO4
5� center plus local singlet boson»,
while the «nonretarded» scenario of a novel phase is as-
sumed to incorporate the unconventional states of elec-
tron CuO4
7� center up to its orbital degeneracy. In
cuprates we deal with the electron/hole injection to
the insulating parent phase due to a nonisovalent sub-
stitution as in La2�xSr xCuO4, Nd2�xCe xCuO4, or
change in oxygen stoihiometry as in YBa2Cu3O6�x ,
La2CuO4��, La2Cu1�xLi xO4. Such a substitution
provokes the nucleation of EH droplets and shifts the
phase equilibrium from the insulating state to the un-
conventional electron-hole Bose liquid, or, in other
words, induces the insulator-to-EHBL phase transi-
tion. Hence, the formation of EHBL in cuprates can be
considered as the first order phase transition. The dop-
ing in cuprates gradually shifts the EHBL state away
from half-filing. It is clear that the EHBL scenario
makes the doped cuprates the objects of bosonic phys-
ics. There are numerous experimental evidence that
support the bosonic scenario for doped cuprates [33].
In this connection, we would like to draw attention to
the little known results of comparative high-tempera-
ture studies of thermoelectric power and conductivity
which unambiguously reveal the charge carriers with
q e� 2 , or two-electron(hole) transport [34]. The
well-known relation % % � � �' (/ k/qln const with
| | | |q e� 2 is fulfilled with high accuracy in the limit of
high temperatures ( ∼ 700–1000 K) for different
cuprates (YBa2Cu3O6+x, La3Ba3Cu6O14+x,
(Nd2/3Ce1/3)4(Ba2/3Nd1/3)4Cu6O16+x).
Conclusions
We have developed a model approach to describe
charge fluctuations and different charge phases in
strongly correlated 3d oxides. In frames of S � 1
pseudospin formalism different phase states of the sys-
tem of the metal-oxide M centers with three different
valent state M0,� are considered on the equal footing.
Simple uniform mean-field phases include an insulat-
ing monovalent M0 phase, mixed-valence binary
(disproportionated) M � phase, and mixed-valence
ternary («under-disproportionated») M0,� phase. We
consider two first phases in more details focusing on
the problem of electron/hole states and different
types of excitons in M0 phase and formation of elec-
tron-hole Bose liquid in M � phase.
Our consideration was focused mainly on a number
of issues seemingly being of primary importance for
the various strongly correlated oxides such as cup-
rates, manganites, bismuthates, and other systems
with CT instability and/or mixed valence. These in-
cludes two types of single particle correlated hopping
and the two-particle hopping, CT excitons, elec-
tron-lattice polarization effects which are shown to be
crucial for the stabilization of either phase, topologi-
cal charge fluctuations, nucleation of droplets of the
electron-hole Bose liquid and phase separation effect.
We emphasize an important role of self-trapped CT
excitons in typical Mott—Hubbard insulators as can-
didate «relaxed excited states» to struggle for stabil-
ity with ground state and natural nucleation centers
for unconventional electron-hole Bose liquid which
phase state include the superfluid. Pseudospin formal-
ism has appeared to be very efficient to reveal and de-
scribe different aspects of essential physics for
mixed-valence system. All the insulating systems such
as M0 phase may be subdivided to two classes: stable
and unstable ones with regard to the formation of
self-trapped CT excitons. The latter systems appear to
be unstable with regard the formation of CT exciton
clusters, or droplets of the electron-hole Bose liquid.
The model approach suggested is believed to provide a
conceptual framework for an in-depth understanding
of physics of strongly correlated oxides such as
cuprates, manganites, bismuthates, and other systems
with charge transfer excitonic instability and/or
mixed valence. We shortly discuss an unconventional
scenario of the essential physics of cuprates that im-
plies their instability with regard to the self-trapping
of charge transfer excitons and the formation of elec-
tron-hole Bose liquid.
Author acknowledges the stimulating discussions
with V. Vikhnin, A.V. Mitin, S.-L. Drechsler, T.
Mishonov, R. Hayn, I. Eremin, M. Eremin, Yu.
Panov, V.L. Kozhevnikov and partial support by
CRDF Grant No. REC-005, RFBR grants Nos.
04-02-96077, 06-02-17242, and 06-03-90893.
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Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 327
|
| id | nasplib_isofts_kiev_ua-123456789-127739 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T18:58:39Z |
| publishDate | 2007 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Moskvin, A.S. 2017-12-27T14:47:32Z 2017-12-27T14:47:32Z 2007 Charge states of strongly correlated 3d oxides: from typical
 insulator to unconventional electron-hole Bose liquid / A.S. Moskvin // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 314-327. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 71.10.–w, 71.28.+d, 71.30.+h, 74.72.–h https://nasplib.isofts.kiev.ua/handle/123456789/127739 We present a model approach to describe charge fluctuations and different charge phases in
 strongly correlated 3d oxides. As a generic model system one considers that of centers each with
 three possible valence states M⁰, described in frames of S 1 pseudospin (isospin) formalism by
 an effective anisotropic non-Heisenberg Hamiltonian which includes both two types of single particle
 correlated hopping and the two-particle hopping. Simple uniform mean-field phases include
 an insulating monovalent M⁰ phase, mixed-valence binary (disproportionated) M phase, and
 mixed-valence ternary («under-disproportionated») M⁰, phase. We consider two first phases in
 more details focusing on the problem of electron-hole states and different types of excitons in
 M⁰ phase and formation of electron-hole Bose liquid in M phase. Pseudospin formalism provides
 a useful framework for revealing and describing different topological charge fluctuations, in particular,
 like domain walls or bubble domains in antiferromagnets. Electron-lattice polarization effects
 are shown to be crucial for the stabilization of either phase. All the insulating systems such as
 M0 phase are subdivided to two classes: stable and unstable ones with regard to the formation of
 self-trapped charge transfer (CT) excitons. The latter systems appear to be unstable with regard to
 the formation of CT exciton clusters, or droplets of the electron-hole Bose liquid. The model approach
 suggested is believed to be applied to describe a physics of strongly correlated oxides such
 as cuprates, manganites, bismuthates, and other systems with charge transfer excitonic instability
 and/or mixed valence. We shortly discuss an unconventional scenario of the essential physics of
 cuprates which implies their instability with regard to the self-trapping of charge transfer excitons
 and the formation of electron-hole Bose liquid. Author acknowledges the stimulating discussions
 with V. Vikhnin, A.V. Mitin, S.-L. Drechsler, T.
 Mishonov, R. Hayn, I. Eremin, M. Eremin, Yu.
 Panov, V.L. Kozhevnikov and partial support by
 CRDF Grant No. REC-005, RFBR grants Nos.
 04-02-96077, 06-02-17242, and 06-03-90893. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Новые электронные материалы и системы Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid Article published earlier |
| spellingShingle | Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid Moskvin, A.S. Новые электронные материалы и системы |
| title | Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid |
| title_full | Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid |
| title_fullStr | Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid |
| title_full_unstemmed | Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid |
| title_short | Charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole Bose liquid |
| title_sort | charge states of strongly correlated 3d oxides: from typical insulator to unconventional electron-hole bose liquid |
| topic | Новые электронные материалы и системы |
| topic_facet | Новые электронные материалы и системы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/127739 |
| work_keys_str_mv | AT moskvinas chargestatesofstronglycorrelated3doxidesfromtypicalinsulatortounconventionalelectronholeboseliquid |