# Electronic phase separation: recent progress in the old problem

M.Yu. Kagan<EMAIL_ADDRESS>K.I. Kugel  A.L. Rakhmanov  National
Research University Higher School of Economics, Moscow 101000, Russia  Kapitza
Institute for Physical Problems, Russian Academy of Sciences, Moscow 119334,
Russia  Institute for Theoretical and Applied Electrodynamics, Russian Academy
of Sciences, Moscow 125412, Russia  Moscow Institute of Physics and Technology
(National Research University), Dolgoprudny, Moscow region 141700, Russia

(today)

###### Abstract

We consider the nanoscale electronic phase separation in a wide class of
different materials, mostly in strongly correlated electron systems. The phase
separation turns out to be quite ubiquitous manifesting itself in different
situations, where the itineracy of charge carriers competes with their
tendency toward localization. The latter is often related to some specific
type of magnetic ordering, e.g. antiferromagnetic in manganites and low-spin
states in cobaltites. The interplay between the localization-induced lowering
of potential energy and metallicity (which provides the gain in the kinetic
energy) favors an inhomogeneous ground state such as nanoscale ferromagnetic
droplets in an antiferromagnetic insulating background. The present review
article deals with the advances in the subject of electronic phase separation
and formation of different types of nanoscale ferromagnetic (FM) metallic
droplets (FM polarons or ferrons) in antiferromagnetically ordered (AFM),
charge-ordered (CO), or orbitally-ordered (OO) insulating matrices, as well as
the colossal magnetoresistance (CMR) effect and tunneling electron transport
in the nonmetallic phase-separated state of complex magnetic oxides. It also
touches upon the compounds with spin-state transitions, inhomogeneous phase-
separated state in strongly correlated multiband systems, and electron polaron
effect. A special, attention is paid to the systems with the imperfect Fermi
surface nesting such as chromium alloys, iron-based pnictides, and AA stacked
graphene bilayers.

††journal: Physics Reports

Keywords: electronic phase separation, strongly correlated electron systems,
magnetic polarons, orbital ordering, spin-state transitions, imperfect
nesting, graphene-based materials

###### Contents

  1. 1 Introduction. Spontaneously formed nanoscale inhomogenieties in different materials
  2. 2 Magnetic polarons and related transport phenomena
    1. 2.1 Magnetic polarons in magnetic semiconductors and manganites
    2. 2.2 Electron transport in inhomogeneous phase-separated manganites
      1. 2.2.1 Electrical conductivity in the phase-separated systems
      2. 2.2.2 Comparison with experimental data
      3. 2.2.3 Magnetoresistance of phase-separated manganites
      4. 2.2.4 Specific features of the magnetic susceptibility in the phase-separated state
      5. 2.2.5 $1/f$ voltage noise in the systems with phase separation
  3. 3 Some model systems. Internal structure of magnetic polarons
    1. 3.1 FM polarons in layered manganites
    2. 3.2 FM polarons in doped anisotropic AFM systems
      1. 3.2.1 Shape of FM droplets in anisotropic 2D case
      2. 3.2.2 FM droplets in the case of frustrated triangular lattice
    3. 3.3 Bound magnetic polarons with extended spin distortions
    4. 3.4 Magnetic structure of a coated ferron
  4. 4 Charge ordering and phase separation
    1. 4.1 Introduction
    2. 4.2 The simplest model for charge ordering
    3. 4.3 Phase separation in the Verwey model
    4. 4.4 Phase separation into two large clusters (complete phase separation)
    5. 4.5 Nanoscale phase-separated state within the charge-ordered insulating host
    6. 4.6 Phase separation in the extended double exchange model
    7. 4.7 Concluding remarks
  5. 5 Phase separation in strongly correlated systems with two electron bands
    1. 5.1 Phase separation in the two-band Hubbard model
      1. 5.1.1 General equations
      2. 5.1.2 Phase separation
    2. 5.2 Role of the interband hybridization
    3. 5.3 Several remarks: beyond the Hubbard I approximation
    4. 5.4 Electron–lattice interaction and the phase separation
    5. 5.5 Characteristic size and geometry of inhomogeneities
    6. 5.6 Electronic phase separation in high-$T_{c}$ superconducting cuprates and pnictides
    7. 5.7 Phase separation in two-dimensional electron gas
    8. 5.8 Electronic phase separation in magnetic oxides with Jahn–Teller ions
    9. 5.9 Electron polaron effect and anomalous resistivity in the two-band Hubbard model
      1. 5.9.1 Introductory remarks
      2. 5.9.2 Electron polaron effect and the tendency toward phase separation
      3. 5.9.3 Temperature dependence of the electrical resistivity
  6. 6 Spin-state transitions and magnetic instabilities
    1. 6.1 Introduction
    2. 6.2 Spin states of cobalt ions
    3. 6.3 Effects of electron delocalization and spin-state transitions: LS–LS ground state for isolated ions
    4. 6.4 The case of HS–HS ground state for isolated ions
    5. 6.5 Phase diagrams
    6. 6.6 Conclusions
  7. 7 Effects of orbital degrees of freedom, different shapes of polarons
    1. 7.1 Introduction
    2. 7.2 Symmetrical model
    3. 7.3 Anisotropic model
    4. 7.4 Inhomogeneities in the orbitally ordered structures
    5. 7.5 Simplified two-band model with the orbital ordering
      1. 7.5.1 Formulation of the problem
      2. 7.5.2 Homogeneous states and phase separation
    6. 7.6 A model with charge carriers occupying $e_{g}$ orbitals: 2D case
      1. 7.6.1 A model
      2. 7.6.2 Antiferro-orbital ordering near half filling
      3. 7.6.3 Orbitally disordered state
    7. 7.7 Conclusions
  8. 8 Systems with imperfect nesting and phase separation
    1. 8.1 Introduction
    2. 8.2 Phase separation in the Rice model for chromium and its alloys
    3. 8.3 The effect of magnetic field on the phase separation in systems with imperfect nesting
    4. 8.4 Phase separation in the organic metals under pressure
    5. 8.5 Phase separation in iron pnictide superconductors
  9. 9 Phase separation in graphene-based systems
    1. 9.1 AA graphene bilayer
      1. 9.1.1 Single-electron spectrum
      2. 9.1.2 Symmetry-breaking phases
      3. 9.1.3 Phase separation
    2. 9.2 Twisted bilayer graphene
    3. 9.3 Phase separation of hydrogen atoms adsorbed on graphene
  10. 10 Conclusions. Some additional problems

##  1 Introduction. Spontaneously formed nanoscale inhomogenieties in
different materials

The phase separation is a universal and ubiquitous phenomenon, which is
present not only in electronic liquid in metals, semiconductors [1], heavy-
fermion [2, 3] and high-$T_{c}$ superconductors, but also in different types
of mixtures (Fermi–Bose, Fermi–Fermi, and Bose–Bose mixtures), in bismuthates
(BaKBiO), liquid and solid solutions of 3He in 4He, mixtures of 6Li and 7Li,
40K and 87Rb isotopes, in spin-polarized (imbalanced) Fermi gases, and in two
kinds of Bose gases and other atomic condensates in magnetic traps and on
optical lattices, as well as in low-dimensional organic compounds and in mixed
states (different pasta-like structure) in nuclear physics [4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17]. In condensed matter physics, it usually
manifests itself either in the form of electronic phase separation (with the
formation of paramagnetic spin bags or strings, ferrons, or stripes [18, 19,
20], etc.) or the phase separation in ionic subsystems (e.g. charge density
wave formation in superconducting cuprates such as LaBaCuO and LaSrCuO for
some specific values of hole doping [4, 5, 6, 21, 22]). While the latter
phenomena usually require a reconstruction of the crystal lattice or formation
of the superstructures (e.g. doubling of the lattice period), the former one
does not necessarily correspond to the instability of the crystal lattice. In
our review, we are dealing with the electronic phase separation. Moreover, we
will mostly consider the nanoscale phase separation in the electronic
subsystem and only briefly discuss other types of electronic phase separation
(such as a total phase separation giving rise to two large clusters, formation
of stripes, etc.). We will also focus on strongly correlated electron systems
with a strong interplay between kinetic and potential energy of electrons. The
gain in kinetic energy favors delocalization and metallicity in the electron
subsystem. At the same time, the potential energy promotes localization and
insulating behavior (as e.g. in the Hubbard model at strong onsite Coulomb
repulsion $U$ or in the $t-J$ model in the limit of large $J/t$ ratios [23,
24, 25]).

We stress again that the instability of the ground state with respect to the
nanoscale phase separation is an important feature characteristic of numerous
strongly correlated electron systems. It corresponds, in particular, to the
creation of small ferromagnetic (FM) droplets inside different insulating host
materials. The host materials can be antiferromagnetic [26, 27, 28, 29, 30,
31, 32, 33], charge-ordered (CO) [34, 35, 36, 37], or orbitally ordered (OO)
ones [38, 39] The FM droplets are randomly distributed within these hosts and
are usually metallic ones. The shape of the droplets is usually a spherical or
an ellipsoidal one [26, 27, 30]. We can say that the droplets are indeed real
nanoscale objects looking like raisins in a cake. Their radius is of the order
of $R$ = 10–15 Å, which corresponds to 2–4 interatomic distances). Inside
droplets, there are typically 30–100 local spins, which are FM oriented and
owing in fact to one additional conduction electron or hole. Hence, a FM
droplet exhibits a large magnetic moment of the order of 15–50 Bohr magnetons.
Usually, the phase-separated state in complex magnetic oxides (such as
manganites, cobaltites, and nickelates) is a superparamagnetic one. This fact
is due to the residual magneto-dipole interaction between the FM droplets.
Meanwhile, the electron transport here is governed by the tunneling of
conduction electrons between the neighboring metallic droplets through the
tunneling barriers, which are formed in the insulating host. The tunneling
usually corresponds to the regime of Coulomb blockade [40, 41, 42, 43, 44, 45,
46, 47, 48], and it is spin-dependent, which could be important for the
applications in spintronics.

In this review article, we will consider different types of free and bound
magnetic polarons arising in AFM, CO, and OO matrices both in the isotropic
and anisotropic cases on regular (cubic, quadratic) and frustrated
(triangular) lattices. We ananalyze the thermodynamic and transport (kinetic)
characteristics of the phase-separated state including magnetization and
magnetic susceptibility, resistivity, magnetoresistance, and the spectrum of
$1/f$-noise in it [49, 50, 51, 52].

As we have already mentioned, in cobaltites, for example, the phase separation
is usually accompanied by the transitions between high- and low-spin states
[53]. For cobaltites, the electronic phase separation was considered not only
in 3D and 2D systems but in quasi-one-dimensional magnets such as BaCoO3 as
well (see [54, 55, 56, 57]). These papers deal with the creation of magnetic
polarons in spin chains with Heisenberg AFM interaction upon doping. For such
1D systems, it was possible to demonstrate the formation of rather long-range
spin distortions around the polaronic core. The long-range tail of slowly
decaying spin distortions (usually they decay in a power-law fashion) is also
formed around the ferrimagnetic core of the magnetic polaron bound by donor
impurity. This type of spin distortions arise in manganites at very low doping
levels in 2D and 3D cases in the model, which combines the Hund’s rule and
Heisenberg type magnetic interactions (as well as the one-site magnetic
anisotropy) with electron–ion Coulomb attraction potential between Sr (or Ba)
impurity ion and conduction electron [58, 59].

Another type of interesting mesoscopic phenomena in strongly correlated
electron systems corresponds to the electron polaron effect (EPE). EPE usually
takes place in two-band systems with strongly differing band masses. This
situation is typical of many mixed valence electron systems [60, 61, 62, 63,
64, 65], including uranium-based heavy fermion compounds. The most general
model, which should be used here, is a two-band Hubbard model with one narrow
band [66, 67, 68, 69, 70]. The EPE usually describes the strong enhancement of
a heavy mass due to many body effects. They are predominantly related to the
dressing of a heavy particle in a coat of low energy (infrared) electron–hole
pairs of light particles and lead to a strong additional narrowing of the
heavy particle bandwidth. The EPE has a lot of similarities with the physics
of infrared divergences [71, 72, 73] for the Brownian motion of a heavy
particle in the Fermi liquid of light particles [74, 75]). Another close
analogy here is that with the Andeson physics of the orthogonality catastrophe
[76, 77], which describes a 1D chain of $N$ electrons in the presence of one
impurity in the system.

The electron polaron effect not necessarily leads to the nanoscale phase
separation (though this scenario is also possible, at least, at the level of
the strong density redistribution for the large initial mismatch between the
densities of heavy and light components in agreement with the predictions of
[78]).

At the same time, in all the cases (with and without phase separation), the
electron polaron effect leads to the anomalous temperature dependence of the
resistivity in 3D and especially in quasi 2D (layered) case providing possible
alternative explanation to the mechanism of spin dependent tunneling
conductivity when we try to explain resistivity characteristics in manganese
silicides [79] or layered manganites. In manganese silicides MnSi and in the
stoichiometric alloys Mn1-xFexSi, we have a very peculiar power-law (almost of
the square root type) dependence of the resistivity $R(T)$ (for temperatures
in the range from 30 to 250 K) and magnetization $M(H)$ (in moderate and high
magnetic fields from 2–4 T up to 50 Tesla) in the paramagnetic phase.

In our review, we also consider the role of a more standard electron–phonon
polaron effect as well as a long-range part of Coulomb interaction in the
formation of different inhomogeneities and nanoscale phase-separated
structures. Usually the electron polaron effect and polaron effect of the
phonon origin play in concert leading to the multiplicative effect for the
bandwidth narrowing. In this connection, it is important to mention the giant
oxygen isotope effects in manganites [80, 81, 82] (see also [83] and
references therein) and in cobaltites [84], which are closely related both to
the polaron effect and phase separation. Moreover, the oxygen isotope exchange
can even induce the metal–insulator transition in manganites [85, 86, 87].
Electron–phonon (electron–ion) interactions are responsible for the creation
of different types of superstructures in manganites and other Jahn–Teller
systems [88, 89, 90]. In manganites, two conductive bands (delocalized
orbitals) of $e_{g}$ electrons are additionally split by sufficiently large
Jahn–Teller gap, while three localized $t_{2g}$ orbitals form the local spin
$S=3/2$ at them in accordance with the generalized Hund’s rule.

For the physics of phase separation and the formation of different anisotropic
structures (stripes, zig-zags, spin ladders, etc.), a very important role is
played by the orbital degrees of freedom (often related to the Jahn–Teller
type effects), as well as by the relativistic (magneto-dipole or spin-orbit)
interactions. The anisotropic magneto-dipole interaction could be also a
source for the formation of anisotropic elongated structures (such as stripes)
in many strongly correlated systems [88]. Another possibility for a strongly
anisotropic situation and formation of stripes arises when we take into
account the interplay between nanoscale phase separation and charge ordering.
At the same time, the formation of zig-zags and spin ladders [91, 92] takes
place even when we consider stable crystallographic distortions.

An important role of orbital degrees of freedom also manifests itself in
multiorbital systems such as transition metals compounds with several $e_{g}$
and $t_{2g}$ orbitals for $d$ electrons [38]. In such a case, it is possible
to formulate a spin–orbital model, which adequately describes entangled
spin–orbital excitations in the system [93]. The phase separation in
Jahn–Teller systems with localized and itinerant electrons was analyzed in
detail in [94].

Finally, in the last sections of this review, we consider the systems with the
imperfect nesting such as chromium-based alloys, iron-based superconductors,
stacked AA graphene bilayers, hydrogenized graphene, and graphene-based
heterostructures [95]. These systems are inherently unstable toward the
electronic phase separation [96, 97, 98, 99] already in the framework of the
simplest Rice model [100], which was originally proposed for the description
of spin density waves in chromium-based alloys. This model captures the band
structure effects in itinerant antiferromagnetism and contains repulsive
Coulomb interaction between almost spherical electron and hole pockets of
different radii (imperfect nesting).

An idealized graphene monolayer as well as an idealized AB graphene bilayer
are also the subjects of anomalous Kohn–Luttinger type of superconductivity
[101, 102, 103]. The same type of anomalous superconductivity and nanoscale
phase separation with the formation of small metallic droplets inside the CO
matrix (similar to phase separation in the Verwey model [104, 105] for
manganites) occurs in the strong-coupling limit of the Shubin–Vonsovsky model
[106] for strong Coulomb interaction at nearest-neighbor sites, $V\gg t$, and
$n=1/2$ (near the quarter filling) [107]. The Shubin–Vonsovsky model can be
important for cuprates, iron-based superconductors, and idealized monolayer
and bilayer graphene. The electron polaron effects are clearly pronounced in
monolayer and bilayer graphene as well as in the other quasi-2D and 3D (Weyl)
semimetals with the Dirac spectrum such as bismuth, for example [108, 109,
110]. It is interesting here to study more thoroughly a combined effect of
strong Coulomb correlations and disorder for the conductivity in this class of
materials.

One of the main goals of our review is to describe the recent progress in the
subject of nanoscale phase separation. It is quite surprising that a lot of
intriguing results on this subject obtained during about fifteen last years
have not been reflected yet in comprehensive review articles. Indeed, on the
wave of the so called “manganite boom” at the end of nineties, there appeared
several reviews and books [27, 26, 28, 111, 112, 113] related mostly to the
phase separation in manganites, and nearly nothing after that. In this paper,
we are trying to fill somehow such evident “blank spot”.

The structure of the review is as follows: it consists of 10 Sections
including Introduction and Conclusions. In the next Section 2, we consider
ferromagnetic polarons (ferrons) in magnetic semiconductors and manganites as
well as the colossal magnetoresistance, the tunneling conductivity, and
$1/f$-noise spectrum in the phase-separated nonmetallic state of these
systems. In Section 3, we analyze the shape and internal structure of free and
bound magnetic polarons, which arise in different model systems in 3D, 2D, and
1D cases on regular and frustrated lattices. In Section 4, we discuss the
interplay between charge ordering and nanoscale (as well as large-scale) phase
separation in the Verwey and Shubin–Vonsovsky models. In Section 5, we
consider charge density redistribution in strongly correlated multiband
systems, especially having in mind the possible application to cuprates and
heavy-fermion systems as well as to the systems near the topological Lifshitz
phase transitions. In this section, we also consider electron–lattice
interactions, nonuniform distortions, and the formation of the superstructures
in manganites and other Jahn–Teller systems. At the end of this section, we
consider EPE and anomalous resistivity in the two-band model with one narrow
band. In Section 6, we analyze magnetic instabilities and spin-state
transitions in cobaltites and other complex magnetic oxides. In Section 7, we
discuss the important effects of orbital degrees of freedom in transition
metals and oxides as well the formation of the orbital polarons, stripes, and
zig-zag structures in the framework of degenerate two-band Hubbard model and
orbital $t-J$ model. In Section 8, we describe the formation of the electronic
phase separation in chromium alloys, iron-based superconductors, and other
systems with the imperfect nesting. In Section 9, we are dealing with the
phase separation in graphene-based systems with a special emphasis on AA
graphene bilayers and hydrogenated graphene. In Section 10, we present our
conclusions and discuss some interesting unsolved problems.

##  2 Magnetic polarons and related transport phenomena

###  2.1 Magnetic polarons in magnetic semiconductors and manganites

Historically, nanoscale phase separation with the formation of small FM
droplets inside AFM insulating matrices was first predicted at the end of
nineteen sixties in seminal papers of Mott, Nagaev, and Kasuya on magnetic
semiconductors [114, 115, 116, 117, 118]. In this context, it is important to
mention also the papers of Krivoglaz [119] on heterophase inhomogeneities (so
called fluctuons) arising, e.g., in paramagnetic insulating matrices at finite
temperatures, as well as a very important paper by de Gennes [120] on the
double exchange model, canted spin states, and other type of (bound) magnetic
polarons (or magnetic impurities), which can arise in such model. Later on,
analogous predictions for bound magnetic polarons on regular and frustrated
lattices were made in [58, 59, 121], in the framework of the specific model,
which effectively combines exchange interaction, magnetic anisotropy, and
attractive electron–ion Coulomb interaction for manganites in the limit of
very small doping (see Section 3 for more details). The nanoscale electronic
phase separation in manganites and other systems exhibiting the phenomena of
the colossal magnetoresistance (CMR) [122, 123, 124, 125, 112], was
experimentally confirmed by several groups [126, 127, 128, 129, 130, 131,
132], which utilized different experimental techniques including elastic and
inelastic neutron scattering, NMR, STM, electron diffraction, dark image
spectroscopy, and so on. Let us recall that the CMR phenomenon represents a
colossal decrease of resistivity (often by a factor of 100–10000) in
moderately strong magnetic fields $H=2-4$ T. Physically it is often related to
the increase in the size of magnetic inhomogeneities (magnetic polarons or
ferrons) in the applied magnetic field, which, in its turn, increases the
metallic fraction in the sample. As a result, we have a strong decrease in the
resistivity. The magnetoresistance is negative in this case

$MR=\frac{\rho(H)-\rho(0)}{\rho(H)}\,.$ (1)

Let us remind that according to the Mott–Nagaev–Kasuya approach, the radius of
a 3D spherical droplet in the absence of magnetic field is (see the details
below)

$R_{pol}=a\left(\frac{\pi t}{2zJ_{ff}S^{2}}\right)^{1/5}\,,$ (2)

where $a$ is the period of the lattice, $z$ is the number of nearest spins in
the given lattice, $S$ is the value of a local spin, $J_{ff}$ is the
Heisenberg (AFM) exchange interaction between local spins at the neighboring
sites, and $t$ is the hopping matrix element between the neighboring sites for
conduction electrons. In the 2D case for FM droplets of the circular shape,
the power 1/5 in Eq. (2) is replaced by the power 1/4. The droplet radius in
Eq. (2) can be found from the minimization of the energy of the nanoscale
phase-separated state, which reads [28]

$E_{pol}=-tna\left(z-\frac{\pi^{2}a^{2}}{R^{2}}\right)+\frac{4}{3}\pi\frac{zJ_{ff}S^{2}}{2}\left(\frac{R}{a}\right)^{3}n-\frac{1}{2}zJ_{ff}S^{2}\left[1-\frac{4}{3}\pi\left(\frac{R}{a}\right)^{3}\right]n\,.$
(3)

From the minimization condition $dE_{pol}/dR=0$, we find the optimum droplet
radius given by Eq. (2). The structure of a spherical FM droplet within the
AFM matrix is illustrated in Fig. 1

Figure 1:  FM polarons with one conduction electron inside the insulating AFM
matrix [28].

Let us now show that the homogeneous canted state for the typical values of
parameter $\alpha=t/J_{ff}S^{2}$ of the order of 80–100 (which is typical of
manganites) is less favorable in energy than the (inhomogeneous) nanoscale
phase-separated state. Moreover, in a wide range of parameters, the
homogeneous canted state has a negative compressibility
$\kappa=d^{2}E/dn^{2}$, which signals the instability of the ground state
toward the electronic phase separation. This instability can be easily
demonstrated for the classical canted state proposed by de Gennes. In the
classical picture, an electron cannot move being surrounded by of AFM-ordered
local spins, and that is why, it is favorable for the electron to cant one
sublattice relative to another for having a possibility of hopping to a
neighboring site. As a result, the homogeneous canted state has the energy

$E=-tn\cos\frac{\theta}{2}+\frac{1}{2}zJ_{ff}S^{2}\cos\theta\,,$ (4)

where $\theta$ is the canting angle between two AFM sublattices and the
arising effective hopping integral $t_{eff}=t\cos(\theta/2)$ is related to the
spinor nature of the electron wave function [133]. Minimizing the energy in
Eq. (4) with respect to $\cos(\theta/2)$, we get

$\cos\frac{\theta}{2}=\frac{tn}{2J_{ff}S^{2}}\,,$ (5)

and, consequently,

$E=-\frac{zt^{2}n^{2}}{4J_{ff}S^{2}}-\frac{zJ_{ff}S^{2}}{2}\,.$ (6)

As a result, the compressibility of the classical canted state reads

$\kappa=-\frac{zt^{2}}{2J_{ff}S^{2}}<0\,.$ (7)

It is obviously negative, thus suggesting a tendency toward the phase
separation.

It is important to stress that the energies of both the nanoscale phase-
separated state (3) and the homogeneous canted state (5) can be considered as
different approximations to the exact energy of the ground state in the
double-exchange model, which is specified by the Hamiltonian

$\hat{H}=-J_{H}\sum_{i}{\bf S}_{i}{\bf\sigma}_{i}-t\sum_{\langle
i,j\rangle}Pc_{i\sigma}^{{\dagger}}c_{i\sigma}P+J_{ff}\sum_{\langle
i,j\rangle}{\bf S}_{i}{\bf S}_{j}\,,$ (8)

where $J_{H}$ is a strong onsite Hund’s rule coupling,
${\bf\sigma}=c^{{\dagger}}{\bf\tau}c/2$ is the spin operator of conduction
electrons, ${\bf\tau}$ are Pauli matrices, $J_{ff}$ is the AFM Heisenberg
exchange between local spins at the neighboring sites, ${{\dagger}c}$ and $c$
are creation and annihilation operators, $W=2zt$ is the electron bandwidth. In
manganites, projection operator $P$ is responsible for the single occupation
of $e_{g}$ orbitals, for which a strong onsite Coulomb repulsion $U$ prevents
two conduction electrons from occupying the same site. In actual manganites,
we have $J_{H}\simeq 1$ eV, $t\simeq 0.3$ eV, $J_{ff}\simeq 0.001$ eV, $z=6$
for the simple cubic lattice, $S=3/2$ and thus we are in the strong coupling
limit of the FM Kondo lattice model (FM KLM) $J_{H}\gg t\gg J_{ff}S^{2}$,
which in the literature is just called the double-exchange model of de Gennes.
In this limit, a conduction electron and local spin form together a coupled
state with a total spin $S_{tot}=S+1/2$ at one site. Both in the energy of the
phase-separated state (3) and the energy of the classical canted state (5), we
omit the large constant term $J_{H}S/2$ (related to the parallel alignment of
the local and conduction electron spins) since this term does not change the
difference between the energies of these states. The result for $t_{eff}$ in
the classical canted state was derived in [133] just in the strong coupling
limit of the FM Kondo lattice model.

In the quantum canted state introduced by Nagaev [115], there are two bands
$t_{+}$ and $t_{-}$ for the motion of conduction electrons corresponding to
two different projections of the total spin $S_{tot}^{z}=S+1/2$ and
$S_{tot}^{z}=S-1/2$. The de Gennes classical description of the canted state
corresponds to the limit $t_{-}\rightarrow 0$, whereas, in a quantum treatment
for the canting angle $\theta=\pi$ (which corresponds to the collinear AFM
structure), both hopping integrals are equal to each other
$t_{-}=t_{+}=t/\sqrt{2S+1}$. In this point lies the main difference between
quantum and classical canted state. In the quantum picture, the hopping
integral for large values of local spin $S\ll 1$ is small only as $/\sqrt{S}$,
while in the de Gennes classical limit, it is small as $1/S$. In quantum
description, the physical mechanism allowing the electron hopping in the AFM
environment (even for the angle $\theta=\pi$, where it is classically
forbidden) is related to the possibility of coherent string-type motion (see
[134]). This type of motion occurs when an electron hops from the site with a
total spin projection $S_{tot}^{z}=S-1/2$ to the neighboring site (where it
forms the state with the total spin projection $S_{tot}^{z}=S-1/2$), then hops
back to the site where it forms the state with $S_{tot}^{z}=S+1/2$, and so on.
As a result of this motion, the coherent band of the width $W=2zt/\sqrt{2S+1}$
is formed. However, even in the quantum case, it is possible to show that, at
least for substantially large values of the parameter $\alpha=1/J_{ff}S^{2}$,
the nanoscale phase-separated state is more favorable in energy [135]. The
homogeneous string-like motion in collinear AFM environment in exact quantum
treatment (which includes the spatial variations of the canting angle
$\theta(x)$ and its gradients) can be stabilized only when
$\alpha<\alpha_{c}$, where $\alpha_{c}$ is the threshold value of the order of
75 (see [30] and Fig. 2). For cuprates, we have $\alpha=1/J_{ff}S^{2}\approx
2-3$ (for $S=1/2$) and instead of FM polarons, the AFM strings discussed in
[19] and [20] can be stabilized in the system at low hole doping.

Figure 2:  (Color online) Comparison of the Nagaev–Mott solution (taking into
account the quantum effects) and the exact numerical solution for the energy
of a FM polaron [30].

In the applied magnetic field, an effective Heisenberg exchange decreases due
to the Zeeman splitting and reads

$J_{ff}^{eff}S^{2}=J_{ff}S^{2}-g\mu_{B}HS\,$ (9)

where $g$ is the gyromagnetic ratio and $\mu_{B}$ is the Bohr magneton.
Correspondingly, the ferron radius in Eq. (2) and its volume

$\Omega=\frac{4\pi}{3}\left(\frac{R_{pol}}{a}\right)^{3}\,,$ (10)

also increase due to the decrease in the effective exchange.

In the case of electron-doped system, the central spin $\sigma=1/2$ of the
conduction electron in a FM droplet is surrounded by local spins $S=3/2$,
whereas in case of hole doping (typical for manganites), the central unpaired
spin $S=3/2$ is surrounded by paired spins $S_{tot}=2$.

The similar effect of the FM polaron growth in magnetic field takes place for
the so called temperature ferrons, which arise in the paramagnetic (PM) phase.
In 3D systems, their radius in the absence of magnetic field is given by

$R_{T}\approx a\left(\frac{\pi t}{2T\ln(2S+1)}\right)^{1/5}\,,$ (11)

and can be obtained from the minimization of the corresponding change in the
free energy $\Delta F$. This change takes place due to the formation of the FM
droplets above the Curie temperature $T_{C}$ in the PM matrices and can be
written as [28]

$\Delta F=-tn\left(z-\frac{\pi^{2}a^{2}}{R^{2}}\right)+\frac{4}{3}\pi
T\left(\frac{R}{a}\right)^{3}n\ln{(2S+1)}\,.$ (12)

In the denominator of Eq. (11), instead of the Heisenberg exchange term
$J_{eff}S^{2}$, we have the term $T\ln{(2S+1)}$, which determines the
contribution of the spin entropy to $\Delta F$. This term also decreases under
effect of the applied field due to the Zeeman effect, which reduces the spin
degeneracy, thus increasing again the radius and the volume of the 3D
spherical droplet.

It should be emphasized again that in the case of the nanoscale phase
separation, electron transport is described by the spin-dependent tunneling of
conduction electron between the neighboring metallic FM droplets separated by
AFM insulating barriers. When we increase the concentration of droplets, we
get closer to the percolation threshold. In this case, the average distance
between the droplets becomes strongly reduced and we can effectively consider
the system as a network of the tunneling contacts (or bridges). This limiting
case is described by the physics of the tunneling magnetoresistance (TMR)
[136]. If we apply an external magnetic field, we can control the contact
resistance. In this way, we can control the electric current flowing across
contact and hence, can read and write the information. This is a main goal and
the main challenge for the applications in spintronics [137].

In the system of 3D spherical droplets, the percolation threshold is achieved
at the critical density of charge carriers

$n_{c}=1/\Omega=\frac{3}{4\pi}\left(\frac{a}{R_{pol}}\right)^{3}\,.$ (13)

In real 3D manganites, the percolation threshold corresponds to the critical
density of about 0.16. For larger densities, the ferrons start to overlap
forming the infinite FM metallic cluster. The CMR phenomenon is especially
pronounced for the charge carrier density within the $n=0.2-0.4$ range (see
Fig. 3 illustrating the global phase diagram of manganites). Here, we have two
phase transitions, which occur simultaneously at the Curie temperature
$T_{C}$, namely the metal–insulator and FM–PM transitions. That is why, even
within this density range, for temperatures above $T_{C}$, the electrical
resistivity is mainly determined by the presence of temperature ferrons inside
the PM matrix. At the same time, the temperature ferrons overlap at the
critical density

$\delta_{c}=\frac{3}{4\pi}\left(\frac{a}{R_{T}}\right)^{3}\,.$ (14)

Substitution of Eqs. (2) and (11) to Eqs. (13) and (14) yields for the ratio

$\frac{n_{c}}{\delta_{c}}\varpropto\left[\frac{T\ln{(2S+1)}}{zJ_{ff}S^{2}}\right]^{3/5}\varpropto\left[\frac{T_{C}\ln{(2S+1)}}{T_{N}}\right]^{3/5}\,$
(15)

near the triple point between FM, AFM, and PM phases. Here, $T_{C}$ and
$T_{N}$ are the Curie and Néel temperatures, respectively. For the hole-doped
manganites, $T_{C}\backsim T_{N}\approx 120-150$ K and $\ln{(2S+1)}\approx
1.6$ for $S=2$. Thus, $\delta_{c}<n_{c}$ in agreement with the experimental
data.

Figure 3:  Schematic illustration of the global phase diagram of manganites
[18]. PSI and PSII are different kinds of the phase-separated state: FM
metallic droplets in the AFM and charge ordered (CO) insulating matrices,
respectively.

###  2.2 Electron transport in inhomogeneous phase-separated manganites

####  2.2.1 Electrical conductivity in the phase-separated systems

Let us consider in more detail the transport properties of the phase-separated
manganites. As we will show below, the electrical resistivity in the nanoscale
phase-separated state reads

$\rho(T)=BT\exp\frac{A}{2T}\,,$ (16)

where $A$ is the energy corresponding to the Coulomb repulsion of two
conduction electrons occupying the same FM metallic droplet. In the model of
spherical droplets, $A$ is given by

$A\approx\frac{e^{2}}{\varepsilon R_{pol}}\,,$ (17)

where $\varepsilon$ is the static dielectric constant. In manganites, it can
be of the order of 10–20, so the typical values of $A$ are of the order of
(3–4)$\times 10^{3}$ K. In Eq. (17), the droplet radius $R_{pol}$ is given by
Eq. (2).

The temperature-independent factor $B$ in Eq. (16) in the spherical droplet
model can be written as [40]

$B=\frac{1}{128\pi e^{2}\omega_{0}l^{5}n^{2}k}\,,$ (18)

where $e$ is the electron charge, $n$ is the concentration of the FM droplets,
$l$ is the characteristic tunneling length, $\omega_{0}$ is a frequency
corresponding to the characteristic energy of an electron in a droplet, and
$k$ is the number of itinerant charge carriers in the droplet. Let us
emphasize that in small droplets with the radius given by Eq. (2), we have
$k=1$, whereas in large droplets usually observed in transport measurements
[138, 139, 140, 141, 142], we are dealing with $k\gg 1$. A typical size of a
large droplet can be (7–8)$a$ and hence, the number of charge carriers can
reach such values as $k=50-100$. Thus, effectively we can visualize a large
droplet as an object which can be divided into many Wigner–Seitz spheres of
small radius, each sphere containing one charge carrier to minimize the
Coulomb energy.

Let us try to understand qualitatively the results of Eqs. (16)–(18). For this
purpose, we should following [40, 41] consider a sample of volume $V_{S}$ in
external electric field $\mathbf{E}$. We assume that the sample is in the
insulating state and there are $N$ metallic droplets (polarons) embedded in
the sample. Then, for the density of the droplets, we find $n=N/V_{S}$. In the
simplest case, upon doping, the number of droplets is equal to the number of
electrons (or holes). We neglect the conductivity of the insulating phase and
thus consider the case, when we have charge carriers only inside the droplets.
In this case, a charge transfer is governed by the motion of the (heavy)
droplets as a whole or (if the droplets are considered to be immobile) by the
electron tunneling between the droplets. The first mechanism is usually
ignored because the motion of a droplet as a whole is related to a strong
reconstruction of the local spins surrounding the droplet. Hence a droplet
acquires a large effective mass. Moreover, quite often the droplets are
localized at dislocations and at other lattice defects. As a result, we can
consider only the tunneling of electrons between almost immobile droplets.

If the size of the droplet is small enough, the polaron typically posses only
one electron (or hole) in the ground state of the droplet. However, if we take
the tunneling processes into account, then we should consider also the
droplets with more than one electron (e.g., with two electrons) as well as
empty droplets (assuming that their lifetimes are quite large). Let the energy
of the empty droplet be zero, $E(0)=0$. In this case, the energy of a one-
electron droplet is $E(1)\approx t(a/R_{pol})^{2}$. This estimate corresponds
to the kinetic energy of electron localization inside a spherical droplet of
radius $R_{pol}$. Accordingly, for the energy of a two-electron droplets, we
can give an estimate $E(2)\approx 2E(1)+U$, where $U$ is the energy of
interaction between two electrons. Here, we neglect the surface energy. By
doing this, we assume that the surface energy for the FM droplets (or ferrons)
is small according to the estimates given in [135, 29, 30]. Therefore, we
obtain $E(2)+E(0)>2E(1)$. This means that to form the two-electron droplets,
we need to overcome a potential barrier with the energy $A=E(2)-2E(1)\approx
U$. It is easy to understand that $U$ is governed by the Coulomb repulsion
between two electrons localized at one droplet. In this way, we come to the
estimate for $A$ given by Eq. (17). We assume that we are working in the low-
doping limit, where the mean distance between the droplets is $n^{-1/3}\gg
R_{pol}$ (the polarons do not overlap). Thus, in accordance with our estimates
and transport measurements (see below), the typical values of $A\simeq$
3500–3700 K are larger than the average Coulomb energy
$e^{2}n^{1/3}/\varepsilon_{0}$. Moreover, $A\gg T$ for all realistic
temperatures. That is why, we can neglect the creation of the droplets with
the number of electrons larger than two. It is possible to demonstrate that
even if the many-electron droplets are stable, their contribution to
conductivity is strongly reduced by the Coulomb interaction and hence, it is
very small for low droplet densities (when the system is far from the
percolation threshold).

Let us evaluate factor $B$ in Eq. (18), which is essentially temperature
independent [44]). It is convenient to introduce parameters $N_{1}$, $N_{2}$,
and $N_{3}$ for the numbers of droplets with one electron, two electrons, and
of empty droplets, respectively. According to our model, we assume that the
total number of the droplets in the system is constant. Thus, $N_{2}=N_{3}$
and $N_{1}+2N_{2}=N$, and the partition function of the system is given by

$Z=\sum_{m=0}^{N}{C_{N}^{m}C_{N-m}^{m}\exp{(-m\beta)}},\,\,\,\,\,\beta=A/T\,,$
(19)

where $C_{N}^{m}$ are the binomial coefficients. Using the Stirling formula,
replacing the summation by the integration, and calculating the corresponding
integral by the steepest descent method, we obtain the following expressions
for the average values of $N_{1}$, $N_{2}$, and $N_{3}$

$\displaystyle\overline{N_{2}}=\overline{N_{3}}=N\exp{\left(-\frac{A}{2T}\right)}\,,$
$\displaystyle\overline{N_{2}}=N-2\overline{N_{2}}=N\left[1-2\exp{\left(-\frac{A}{2T}\right)}\right]\,.$
(20)

The electron tunneling implies one of the four possible events (see Fig. 4)
(i) two droplets with one electron are converted into the droplet with two
electrons and an empty droplet, (ii) process, which is inverse to the previous
one, (iii) droplets with one and two electrons exchange their positions, (iv)
the same for the droplet with one electron and an empty droplet. The
corresponding total current density is given by the sum of the contributions
of all these processes $j=j_{1}+j_{2}+j_{3}+j_{4}$. It was shown in [40] that
all these processes equally contribute to the total electrical current.

Figure 4:  Elementary tunneling processes [40]. The arrows and $\sigma_{i}$
denote spin projections.

Let us consider, for example, the contribution of the first two processes

$j_{1,2}=en_{1,2}\langle\sum_{i}{v_{1,2}^{i}}\rangle=en_{1,2}\langle\sum_{i}{\frac{r^{i}\cos{\theta^{i}}}{\tau_{1,2}(r^{i},\theta^{i})}}\rangle\,,$
(21)

In Eq. (21), $n_{1,2}=N_{1,2}/V_{S}$ denote the densities of one-electron and
two-electron droplets, $\langle...\rangle$ corresponds to the statistical and
time averages, $\langle v^{i}\rangle$ is the average velocity of electrons
along the electric field ${\mathbf{E}}$, $r^{i}$ is the distance between the
droplets involved in these processes, $\theta^{i}$ is the angle between vector
${\mathbf{E}}$ and the direction of the electron motion, and
$\tau_{1,2}(r^{i},\theta^{i})$ are characteristic times associated with these
processes. The conventional expressions for $\tau_{1,2}$ (which are derived in
[40] using the detailed balance equation) have the following form

$\tau_{1,2}=\omega_{0}^{-1}{1,2}\exp{\left(\frac{r}{l}\pm\frac{A}{2T}-\frac{eEr\cos{\theta}}{T}\right)}\,,$
(22)

where $l$ is the tunneling length and $\omega_{0}$ is a characteristic magnon
(or depolarization) frequency. Let us stress that when we are far from the
percolation threshold, the average in (21) is reduced to the space average for
velocity $v_{i}$ multiplied by the number of droplets, to which the hopping is
allowed. These numbers are equal to $N_{1}$ for process (i) and respectively
to $N_{2}$ for process (ii). If we assume that the external electric field is
small, $eEl/T\ll 1$ and moreover, the repulsion energy is large, $A\gg T$,
then in the first order in $E$, we have

$\langle\sum_{i}{v_{1,2}^{i}}\rangle=\frac{eE\omega_{0}}{T}N_{1,2}\exp{\left(\mp\frac{A}{2T}\right)}\langle
r^{2}\cos^{2}{\theta}\exp{\left(-\frac{r}{l}\right)}\rangle_{V}\,,$ (23)

where $\langle...\rangle_{V}$ denotes the average over the sample volume

$\langle...\rangle_{V}=V_{S}^{-1}\int{...d^{3}{\mathbf{r}}}\,.$ (24)

Performing the integration, we get

$j_{1,2}=\frac{32\pi
e^{2}E\omega_{0}l^{5}n_{1,2}^{2}}{T}\exp{\left(\mp\frac{A}{2T}\right)}\,.$
(25)

Analogously, it is possible to demonstrate rather straightforwardly that for
other tunneling processes, the corresponding current densities read

$j_{3,2}=\frac{32\pi e^{2}E\omega_{0}l^{5}n_{1}n_{2}}{T}\,.$ (26)

Comparing Eqs. (26) and (26), we can see that in the last one, the exponential
factor is missing and the factors $n_{1,2}^{2}$ are replaced by $n_{1}n_{2}$.

From Eqs. (2.2.1), (25), and (26), we can obtain the expression for dc
conductivity

$\sigma=1/\rho=\frac{128\pi
e^{2}n^{2}\omega_{0}l^{5}}{T}\exp{\left(-\frac{A}{2T}\right)}k\,$ (27)

For small droplets described by Eq.(2), we can show that only one-electron
droplets are stable, while the empty droplets and two-electron droplets decay.
The decay time of the empty droplets is governed by the inverse depolarization
frequency. At the same, time an empty droplet can get an electron from the
one-electron or two-electron droplets, located nearby. This process involves a
time scale, which can be estimated as (see [37]) in agreement with Eqs. (16)
and (18). For small droplets described by Eq. (2), we can show that, in fact,
only one-electron droplets are stable, whereas empty droplets and two-electron
droplets decay. The decay time of empty droplets is governed by the inverse
depolarization frequency $1/\omega_{0}$. At the same time, an empty droplet
can get an electron from one-electron or two-electron droplets, located
nearby. This process involves a time scale $\tau_{0}$, which can be estimated
as (see [40])

$\tau_{0}=\frac{\exp{\left(-\frac{A}{2T}\right)}}{8\pi\omega_{0}l^{5}n}\,.$
(28)

The similar estimate is valid for an electron, which leaves a two-electron
droplet. The validity of this picture is determined by the inequality
$\tau_{0}\ll\omega_{0}^{-1}$. Thus, our approach requires inequality $A\gg T$
and not too small droplet density $n$.

####  2.2.2 Comparison with experimental data

In Fig. 5, we present the plots for resistivity $\rho(T)$ in six families of
manganites using the experimental data reported in [138, 139, 140, 141, 142].
As can be seen from these data, the samples differ in chemical composition,
the type of the crystal structure, the magnitude and the low temperature
behavior of the resistivity (some of the samples behave as metals, some as the
insulators, whereas the resistivity at a given temperature varies more than in
two orders of magnitude). However, in the high-temperature range above the
Curie temperature, the resistivity exhibits an identical behavior for all the
samples governed by the universal relation (27). This relation is typical of
many tunneling systems [114] and is represented by solid curve in Fig. 5.

Figure 5: Temperature dependence of electrical resistivity for (a)
(La1-yPry)0.7 Ca0.3MnO3 with 16O $\rightarrow$ 18O oxygen isotope substitution
for $y=1$ (squares) and $y=0.75$ (triangles); (b) Pr0.71Ca0.29MnO3; (c) a
layered manganite with the chemical composition (La0.4Pr0.6)1.2 Sr1.8Mn2O7;
and (d) La0.8Mg0.2MnO3 [44].

Using the experimental data and Eq. (27), we can determine the value of the
Coulomb barrier energy $A$ and make a reasonable estimate for the
characteristic depolarization frequency $\omega_{0}\simeq 10^{13}-10^{14}$ Hz.
We can also determine the important factor $l^{5}n^{2}k$ which enters Eq. (18)
for the temperature-independent coefficient $B$ (see Table 1).

Table 1:  Coulomb energy $A$, resistivity $\rho$ at 200 K, and product $l^{5}n^{2}k$ for several families of manganites [44] Samples |  $A$ (K) |  $\rho$(200 K) | $l^{5}n^{2}k$ | Data source
---|---|---|---|---
|  |  $\Omega\cdot$cm | cm-1 |
La1-yPry)0.7 Ca0.3MnO3 | 3650 | 1.25 | $2\times 10^{5}$ | Fig. 5a (Babushkina et al., 2003 [138])
Pr0.71Ca0.29MnO3 | 3500 | 0.57 | $3\times 10^{5}$ | Fig. 5b (Fisher et al., 2003 [139])
(La0.4Pr0.6)1.2 Sr1.8Mn2O7 | 3600 | 1.5 | $1.5\times 10^{5}$ | Fig. 5c (Wagner et al., 2002 [140])
La0.8Mg0.2MnO3 | 3700 | 283 | $1\times 10^{3}$ | Fig. 5d (Zhao et al., 2001 [141])

####  2.2.3 Magnetoresistance of phase-separated manganites

Now we can proceed to the analysis of the magnetoresistance in phase-separated
manganites. Using Eq. (9) for the effective Heisenberg exchange, we can obtain
the following expression for the droplet radius in experimentally accessible
magnetic fields

$R_{pol}(H)=R_{pol}(0)(1+bH)\,,$ (29)

where in the 3D case, the coefficient $b$ is

$b=\frac{g\mu_{B}}{5J_{ff}S}\,.$ (30)

For the corresponding decrease in the Coulomb barrier height $A$, we find

$A(H)=A(0)(1-bH)\,.$ (31)

Thus, using eq/ (16), we find that the magnetoresistance in the nanoscale
phase-separated state is negative and can be written as

$MR=1-\exp{\left(\frac{AbH}{2T}\right)}\,.$ (32)

It is shown in [44] that such exponential behavior really takes place in
strong magnetic fields $H>10$ Tesla. Since the gyromagnetic ratio $g\sim 10$
is typical of manganites, this behavior corresponds just to the magnetic field
range where the argument of exponential in Eq. (32) becomes larger than unity.
At the same time, at low and moderate magnetic fields, the magnetoresistance
is quadratic in the field (in agreement with the textbooks on solid state
physics [143]) and is given by the relation

$MR\propto-H^{2}/T^{n}\,,$ (33)

where the exponent $n$ in Eq. (33) is either 1, 2, or 5 depending on the range
of magnetic fields under consideration, the value of magnetic anisotropy, and
so on [42, 43, 44]. The magnetoresistance in this case is determined by the
effects of the spin-dependent tunneling. The value $n=5$ is typical of the
magnetoresistance in manganites in the range of low magnetic fields and high
temperatures. In this case, we have [44]

$|MR|=5\times
10^{-3}\frac{\mu_{B}^{3}S^{5}J_{FM}^{2}N_{eff}^{3}Z^{2}g^{3}H_{a}}{T^{5}}H^{2}=5\times
10^{-3}\frac{M_{0}^{3}J_{FM}^{2}Z^{2}S^{2}H_{a}}{T^{5}}H^{2}\,,$ (34)

where we introduced the magnetic moment of a droplet

$M_{0}=g\mu_{B}SN_{eff}\,.$ (35)

In Eq. (34), $N_{eff}$ is the number of Mn atoms inside the droplet, $S=2$ is
the total spin of Mn3+ ions in manganites under the hole doping, and $Z$ is
the number of nearest neighbors of a Mn ion. Let us stress that for 3D cubic
manganites with $Z=6$, $H_{a}$ is the effective magnetic anisotropy field of a
droplet. At the same time, an effective FM exchange integral $J_{FM}$ is
specified by the well-known relationship of the molecular field theory [144]

$T_{C}=S(S+1)ZJ_{FM}/3\,.$ (36)

The behavior of the magnetoresistance given by Eq. (34) was also
experimentally observed in different families of manganites [138, 139, 140,
141, 142] (see Fig. 6).

Figure 6: Temperature dependence of the absolute value of magnetoresistance.
Solid curve show the results of the theoretical calculations using relation
(34), $|MR|\propto 1/T^{5}$ . (a) $|MR|/H^{2}$ ratio for (La1-yPry)0.7
Ca0.3MnO3 at $y=1$ and $y=0.75$; (b) Pr0.71Ca0.29MnO3 at $H=2$ T; (c) layered
manganite (La0.4Pr0.6)1.2 Sr1.8Mn2O7 at $H=1$ T; and (d) La0.8Mg0.2MnO3 for
$H=1.5$ T [44].

The solid curves in Fig. 6 were calculated for the value of $T_{C}=120$ K. At
the same time, the fitting parameter $N_{eff}$ was determined from the
relation $k=N_{eff}x$, where $x$ is the atomic fraction of the dopant. The
values of $N_{eff}$, $x$, and $k$ are presented in Table 2.

Table 2:  Effective number $N_{eff}$ of manganese atom and the number of electrons $k$ in a droplet, and the dopant fraction $x$ for several different families of manganites, estimated using Eq. (34) and experimental data presented in Fig. 6 Samples | $N_{eff}$ | $x$ | $k$ | Data source
---|---|---|---|---
La1-yPry)0.7 Ca0.3MnO3 | 250 | 0.35 | 75 | Fig. 6a (Babushkina et al., 2003 [138])
Pr0.71Ca0.29MnO3 | 200 | 0.29 | 58 | Fig. 6b (Fisher et al., 2003 [139])
(La0.4Pr0.6)1.2 Sr1.8Mn2O7 | 250 | 0.4 | 100 | Fig. 6c (Wagner et al., 2002 [140])
La0.8Mg0.2MnO3 | 265 | 0.2 | 53 | Fig. 6d (Zhao et al., 2001 [141])

In brief, the derivation of Eq. (34) is as follows. The conductivity of the
system is controlled by the “spin” contribution $\Sigma(H)$ to $\sigma$, that
is $\sigma(H)=\sigma_{0}\langle\Sigma(H))\rangle$. Therefore, we have
$|MR|=\langle\Sigma(H)\rangle-1$. Assuming the interaction between the
droplets to be negligibly small and using an expression (35) for the effective
magnetic moment, we can represent the free energy of the droplet in a magnetic
field in the following form

$U(H)=U(0)-M(H\cos{\theta}+H_{a}\cos^{2}{\Psi})\,.$ (37)

In Eq. (37), $\theta$ is the angle between the external magnetic field
${\mathbf{H}}$ and the magnetic moment ${\mathbf{M}}$, and $H_{a}$ stands for
the field of magnetic anisotropy. Finally, $\Psi$ corresponds to the angle
between the axis of the uniaxial anisotropy and ${\mathbf{M}}$. Let
${\mathbf{H}}$ be parallel to the $z$ axis and let the anisotropy axis lie in
the ($x,z$) plane and have the angle $\beta$ with vector ${\mathbf{H}}$. In
this configuration, we find

$\cos{\Psi}=\sin{\theta}\sin{\beta}\cos{\phi}+\cos{\theta}\cos{\beta}\,.$ (38)

Tn Eq. (38), $\phi$ denotes the angle between the $x$ axis and the projection
of magnetic moment ${\mathbf{M}}$ onto the ($x,y$) plane.

In the classical limit, a given orientation of vector ${\mathbf{M}}$
corresponds to the probability

$P(H,\theta,\varphi)=C(H)\exp{\frac{M(H\cos{\theta}+H_{a}\cos^{2}{\Psi(\theta,\varphi)}}{T}}\,,$
(39)

where $C(H)$ is the normalization factor. The eigenstates for a conduction
electron correspond to the conservation of the spin projection $s\pm 1/2$ onto
the effective field direction in FM correlated region. Let the electron
interact with $Z$ magnetic moments of the droplet. The energy of this
interaction is $E_{s}=-J_{FM}SZs$. Since the value of $J_{FM}SZ$ product is
according to Eq. (34) is of the order of the Curie temperature, $E_{s}$ is
much larger than the energy of interaction between the electron spin and
magnetic field for realistic magnetic fields $H<100$ T. In this situation, the
direction of the effective field is parallel to the direction of the magnetic
moment ${\mathbf{M}}$. Therefore, the probability for the projection of the
electron spin to be equal to $s$ can be written as

$P_{s}=\frac{\exp{(-E_{s}/T)}}{2\cosh{(E_{s}/T)}}\,.$ (40)

In the process of the transfer from droplet 1 to droplet 2, an electron moves
in the effective field, which forms an angle $\nu$ with the effective field in
the initial state. This angle is naturally defined via the obvious relation
$\cos{\nu}=\cos{\theta_{1}}\cos{\theta_{2}}+\sin{\theta_{1}}\sin{\theta_{2}}\cos{(\varphi_{1}-\varphi_{2})}$,
where subscripts 1 and 2 refer to the droplet numbers. Then, the work
performed in the process of electron transfer from droplet 1 to droplet 2 is
$\Delta E_{s}(1-\cos{\nu})$ . Accordingly, the probability of this transfer is
proportional to $\exp{(-\Delta E_{s}/T)}$. Taking into account all the
probability factors introduced above, we obtain the final expression for the
“spin contribution”

$\langle\Sigma(H)\rangle=\int_{0}^{2\pi}{d\varphi_{1}}\int_{0}^{2\pi}{d\varphi_{2}}\int_{0}^{\pi}{\sin{\theta_{1}}d\theta_{1}}\int_{0}^{\pi}{\sin{\theta_{2}}d\theta_{2}}P(\theta_{1},\phi_{1})P(\theta_{2},\phi_{2})\sum_{s=\pm
1/2}{P_{s}\exp{\left(\frac{-\Delta E_{s}}{T}\right)}}\,.$ (41)

The high-temperature expansion of the expression (41) for
$T\gg\\{\mu_{B}gSN_{eff}H,\mu_{B}gSN_{eff}H_{a}\\}$, yields formula (34).

####  2.2.4 Specific features of the magnetic susceptibility in the phase-
separated state

At the high-temperature limit, the susceptibility of nanoscale phase-separated
state is given by the Curie–Weiss law [44]

$\chi(T)=\frac{n(\mu_{B}gSN_{eff})^{2}}{3(T-\Theta)}=\frac{nM_{0}^{2}}{3(T-\Theta)}\,,$
(42)

where $\Theta$ is the Curie–Weiss constant.

The linear temperature dependence for the inverse susceptibility is also
confirmed in experiments [138, 139, 140, 141, 142] for several different
families of manganites (see Fig. 7, where solid curves illustrate the fitting
procedure based on Eq. (42)). Using these experimental results [138, 139, 140,
141, 142] and theoretical results of Eq. (42), we can evaluate the Curie–Weiss
constant $\Theta$, the density of ferrons $n$, the FM phase fraction
$p=nN_{eff}a^{3}$ (where for all the samples, we used the value $a=3.9$Åas a
lattice constant), as well as the effective tunneling length $l$ (see Table
3).

Figure 7: Temperature dependence of the inverse magnetic susceptibility of (a) (La1-yPry)0.7 Ca0.3MnO3 for $y=1$; (b) Pr0.71Ca0.29MnO3; and (c) (La0.4Pr0.6)1.2 Sr1.8Mn2O7 [44]. Table 3:  Curie–Weiss constant $\Theta$, density of ferrons $n$, FM phase fraction $p$, and effective tunneling length $l$ in different families of manganites Samples |  $\Theta$, K |  $n$, cm-3 | $p$ |  $l$, Å | Data source
---|---|---|---|---|---
La1-yPry)0.7 Ca0.3MnO3 | 55 | $1.8\times 10^{18}$ | 0.03 | 24 | Fig. 7a (Babushkina et al., 2003 [138])
Pr0.71Ca0.29MnO3 | 105 | $6.0\times 10^{18}$ | 0.07 | 17 | Fig. 7b (Fisher et al., 2003 [139])
(La0.4Pr0.6)1.2 Sr1.8Mn2O7 | 255 | $2.5\times 10^{18}$ | 0.04 | 19 | Fig. 7c (Wagner et al., 2002 [140])
La0.8Mg0.2MnO3 | 150 | $0.6\times 10^{18}$ | 0.01 | 14 | Zhao et al., 2001 [141]

Let us emphasize that effectively we can regard a small FM droplet as a qubit
since the direction of the magnetic moment of the droplet can be parallel or
antiparallel to the easy axis. If we apply a high magnetic field, we can
strongly polarize FM droplets and thus can write down the information. In our
case, an entangled two-qubit state (which is very important for quantum
calculations) is organized by a pair of two droplets located nearby. The
droplets in the pair are connected with the tunneling bridges. Their
interaction has a magneto-dipole origin. Thinking about the applications in e
spintronics, we can say that the network of FM droplets or nanocomposites
(multilayers) Fe/Cr,Co/Cu, which is connected by the tunneling bridges, can
serve as the basic element for the memory device [137].

Returning back to qubits, note that a popular nowadays magnetic principle of
their realization is based on the quantum spin tunneling in the system of
large magnetic molecules [145] and thus is very similar to the spin-dependent
tunneling of conduction electron between the neighboring FM droplets in our
case.

In addition, there are interesting possibilities for the applications of CMR
manganites in the nanoscale phase-separated state for optoelectronics and
nanoplasmonics. Possible application for the fabrication of the new optical
devices can be related to a strong change in the infrared behavior of the
transmission coefficient in the phase-separated state, when we apply an
external magnetic field to our system. This change in the behavior gives us a
possibility to construct an infrared radiation modulator based on such
manganites [146].

Another possible application of the phase-separated manganites in the
nanoplasmonics [147] was first proposed in [148]. The main idea here is to
construct on the surface of a bulk metal an artificial layer with small
metallic droplets embedded in the insulating charge ordered (CO) matrix. Let
us consider an incident electromagnetic wave coming at the surface from the
vacuum. It is very interesting then to study the angular and frequency
dependence of the transmission ($T$) and reflection ($R$) coefficients in this
geometry assuming that metallic droplets form a periodic superstructure within
the surface layer. The reflection and transmission coefficients will have the
resonance peaks of different shapes for some critical angles including the
angles of total internal reflection. It is important to stress that the
critical angles themselves will correspond to the resonance excitation of the
surface plasmon modes here. Thus, we can calculate the spectrum of surface
plasmons and find the resonance frequencies. The experimental realization of
this idea can lead to the development of a new generation of the devices for
Raman spectroscopy, which will have a strongly enhanced sensitivity.

####  2.2.5 $1/f$ voltage noise in the systems with phase separation

The difficulties, which we have on the way of the creation of magnetic devices
are mostly associated with a very large amplitude of the $1/f$ noise in the
phase-separated state. In [52], we find the following expression for the
intensity of the $1/f$ noise spectrum (or the Hooge constant ) in the
framework of the droplet model [40, 41, 42, 43])

$\alpha_{H}=\frac{\langle\delta
U^{2}\rangle_{\omega}V_{s}\omega}{U_{dc}^{2}}=2\pi^{2}l^{3}\ln^{2}{\left(\frac{\bar{\omega}}{\omega}\right)}\,.$
(43)

In Eq. (43), $\langle\delta U^{2}\rangle_{\omega}$ is the spectral density of
voltage fluctuations, $U_{dc}$ is the applied voltage, $\langle\delta
U^{2}\rangle_{\omega}/U_{dc}^{2}=\delta\sigma^{2}\rangle_{\omega}/\sigma^{2}$
and $\bar{\omega}=\omega_{0}\exp{(A/2T)}$. To derive Eq. (43), we should use
the important relation [40]

$\delta\sigma=\sigma\frac{\delta n_{2}}{{\bar{n}_{2}}}[1-2\exp{(-A/2T)}]$ (44)

between conductivity fluctuations and the fluctuations of the density of two-
electron droplets. We use then the general prescription from the textbooks on
statistical physics (see, e.g. [50, 149]), which relates fluctuation spectrum
to the thermal average

$\langle\delta n_{2}^{2}\rangle_{\omega}=\langle\delta
n_{2}^{2}\rangle_{T}\langle\sum_{i}{\frac{2\tau(r^{i})}{1+\omega^{2}\tau^{2}}}\rangle\,,$
(45)

where neglecting the effect of electric field on the relaxation time, we have
$\tau(r)=\omega_{0}^{-1}\exp{(r/l-A/2T)}$. In Eq. 45), we calculate the sum
over the pairs which include empty and two-electron droplets. Correspondingly,
in Eq. 45), $r^{i}$ is the intersite distance within the pair. In our case,
all the pairs contribute to the noise spectrum. That is why, in Eq. (45), we
can safely proceed from the average to the spatial integration, in which the
dominant contribution is governed by small distances

$\langle\delta n_{2}^{2}\rangle_{\omega}=8\pi{\bar{n}_{2}}\langle\delta
n_{2}^{2}\rangle_{T}\int_{0}^{\infty}{\frac{\tau(r)}{1+\omega^{2}\tau^{2}(r)}r^{2}dr}\,.$
(46)

As a result, we obtain the final expression (43) for the noise spectrum, which
in the broad range of frequencies has almost the $1/f$ form.

Substituting in Eq. (43) the parameters typical of manganites, we obtain
$\alpha_{H}\simeq 10^{-16}$ cm3 at temperatures 100–200 K and frequencies in
the 1–1000 s-1 range. The value of Hooge constant $\alpha_{H}$ in phase-
separated state is is by 3–5 orders of magnitude higher than that in
homogeneous semiconductors. The application of external magnetic field or the
proximity to the percolation threshold can lead to an additional increase in
the Hooge constant [45].

##  3 Some model systems. Internal structure of magnetic polarons

###  3.1 FM polarons in layered manganites

In the previous section, we mostly considered isotropic 3D manganites (such as
La1-xCaxMnO3 or La1-xSrxMnO3), which typically have a cubic perovskite
structure. In addition to this family of manganites, a serious attention was
also focused on the family of layered manganites with a structure specified by
La2-xCaxMnO4, La3-xCaxMn2O7, and so on (see e.g., [26, 28]). In the
literature, these compounds are usually referred to as the Ruddelsden–Popper
phases. The first member of this family La2-xCaxMnO4 structurally is close to
the well-known high-$T_{c}$ cuprate La2-xCuxMnO4, whereas the second one
La3-xCaxMn2O7 contains two neighboring MnO planes.

In the absence of doping, at low concentration of Ca or Sr, the layered
manganites usually exhibit the so-called A-type AFM structure. In this
structure (see Fig. 8), spins in the conducting MnO layer are aligned
ferromagnetically but the FM moments of neighboring layers are antiparallel to
one another. At finite doping, the physics of layered manganites is similar to
the physics of high-$T_{c}$ compounds. Indeed, the basic magnetic and
transport properties of layered manganites are determined by 2D metallic MnO
layers, while the LaO and SrO layers play a role of charge reservoir similar
to that in cuprates.

Figure 8:  Spin and charge arrangement in layered manganites [28].

At low doping levels, the main properties of layered manganites are captured
by the anisotropic double-exchange model. The Hamiltonian of this model can be
written as

$\hat{H}=-J_{H}\sum_{ia}{\mathbf{S}_{ia}\mathbf{\sigma}_{ia}}-t_{\parallel}\sum_{\langle
i,j\rangle
a\sigma}{c_{ia\sigma}^{{\dagger}}c_{ja\sigma}}-t_{\perp}\sum_{i,a\sigma}{c_{ia\sigma}^{{\dagger}}c_{ia+1\sigma}}-J_{FM}\sum_{\langle
i,j\rangle
a}{\mathbf{S}_{ia}\mathbf{S}_{ja}}+J_{ff}\sum_{ia}{\mathbf{S}_{ia}\mathbf{S}_{ia+1}}\,,$
(47)

where subscript $a$ denotes the layer number, $i$ and $j$ refer to a lattice
site within the layer, and $\langle i,j\rangle$ means the summation over
neighboring sites. Accordingly, $t_{\parallel}$ and $t_{\perp}$ are the
intralayer and interlayer hopping integrals, respectively. Finally, $J_{FM}$
describes the FM exchange between local spins inside the layer, whereas
$J_{ff}$ takes into account the AFM exchange between local spins of the
neighboring layers. We again consider the case of a large local spin $S\gg 1$
and work in the strong coupling limit, where the following set of inequalities
is satisfied

$J_{H}S\gg\\{t_{\parallel},t_{\perp}\\}\gg\\{J_{FM}S^{2},J_{ff}S^{2}\\}\,.$
(48)

We start the analysis of the different possible states in the layered
manganites with Hamiltonian (47) in the strong coupling limit (48) with the
homogeneous classical canted state. In this state, a conduction electron
travelling parallel to the layer should occupy the bottom of the conduction
band. At the same time, its motion perpendicular to the layer should be an 1D
analogue of the de Gennes classical canted state. Then, it is reasonable to
assume that the spectrum of conduction electrons in the layered case has the
form

$\varepsilon=-4t_{\parallel}-2t_{\perp}\cos{\frac{\theta}{2}}\,,$ (49)

where $\theta$ is the canting angle between the FM moments of the neighboring
metallic MnO layers and $z=4$ is the number of the nearest neighbors in a MnO
layer on a square lattice. The spectrum in Eq. (49) corresponds to the total
spin at one site equal to $S_{tot}=S+1/2$ and the total spin projection
$S_{tot}^{z}=S+1/2$. At the same time, it is possible to show (see [26, 28,
115, 150, 116]) that unlike the isotropic situation, the second conduction
band, which corresponds to the total spin projection $S_{tot}^{z}=S-1/2$, has
the spectrum

$\varepsilon_{-}=-\frac{-4t_{\parallel}-2t_{\perp}\cos{\frac{\theta}{2}}}{2S+1}\,.$
(50)

Thus, $\varepsilon_{-}/\varepsilon\propto 1/(2S+1)\ll 1$, so we can ignore the
second band and consider the classical de Gennes canted state. Minimization of
the classical canting energy with respect to the parameter $\cos{(\theta/2)}$
leads for the layered case to the following result

$\cos{\frac{\theta}{2}}=\frac{t_{\perp}n}{2J_{ff}S^{2}}\,,$ (51)

where $n$ is the number of electrons per site. It is possible to demonstrate
that the compressibility of the canted state ($\kappa^{-1}=d^{2}E/dn^{2}$) is
negative. Moreover, the nanoscale phase-separated state with small FM droplets
inside AFM matrix is more favorable in energy.

We will assume that ferrons have an ellipsoidal shape (Fig. 9). Similarly to
the isotropic case, the energy of a FM polaron can be written as

$E_{pol}=E_{0}-4t_{\parallel}n-t_{\perp}n\left(2-\frac{\pi^{2}a^{2}}{R_{\perp}^{2}}\right)+2J_{ff}S^{2}n\Omega\,,$
(52)

where $\Omega$ is the volume of FM droplet, $R_{\perp}$ is the droplet radius
in the direction perpendicular to the layers, and $E_{0}$ is given by

$E_{0}=-J_{ff}S^{2}-2J_{FM}S^{2}\,.$ (53)

The volume of ellipsoidal droplet is

$\Omega_{ell}=\frac{4}{3}\pi\frac{R_{\perp}^{2}R_{\parallel}}{a^{3}}\,.$ (54)

It is possible to show that in the anisotropic double-exchange model, the
optimum energy of the system corresponds to

$R_{\parallel}=R_{\perp}\sqrt{\frac{t_{\parallel}}{t_{\perp}}}\gg
R_{\perp}\,.$ (55)

Figure 9:  FM droplet of the ellipsoidal shape in layered manganites [28].

Relation (55) is in agreement with the neutron scattering experiments of
Hennion et al. [126]. Using this relation, we can write the expression for the
droplet volume in the following form

$\Omega_{ell}=\frac{4}{3}\pi\left(\frac{R_{\perp}}{a}\right)^{3}\frac{t_{\parallel}}{t_{\perp}}\,.$
(56)

The minimization of the droplet energy (52) with respect to parameter
$R_{\perp}$ yields

$R_{\perp}=a\left(\frac{\pi
t_{\perp}^{2}}{4J_{ff}S^{2}t_{\parallel}}\right)^{1/5}\,.$ (57)

Correspondingly, the optimum volume of an ellipsoidal droplet is

$\Omega_{ell}=\frac{4}{3}\pi\left(\frac{\pi
t_{\perp}^{2}}{4J_{ff}S^{2}t_{\parallel}}\right)^{3/5}\frac{t_{\parallel}}{t_{\perp}}\,.$
(58)

It is interesting to compare the energies of the droplets of different shapes
and thus to get the feeling what shape of the droplet is the optimum one for
layered manganites. Let us illustrate this point considering also the droplets
of the cylindrical shape [28, 150].

The volume of a cylindrical droplet is (see Fig. 10)

$\Omega_{cyl}=\frac{\pi\rho^{2}L}{a^{3}}\,.$ (59)

The energy of cylindrical droplet is given by

$E_{pol}=E_{0}-t_{\parallel}n\left(4-\frac{9\pi^{2}a^{2}}{16\rho^{2}}\right)-t_{\perp}n\left(2-\frac{\pi^{2}a^{2}}{L^{2}}\right)+2J_{ff}S^{2}n\Omega\,,$
(60)

where $E_{0}$ is defined in (53).

Figure 10:  FM droplet of the cylindrical shape in layered manganites [28,
150].

The coefficient $9\pi^{2}/16$ in the expression for the energy shift from the
bottom of the band (which is just $-4t_{\perp}n$) is determined by a zero of
the Bessel function $J_{0}=0$ for $x\approx 3\pi/4$, where we have introduced
the dimensionless variable $x=\rho/a$. It corresponds to the energy of the
lowest bound state of an itinerant electron in the 2D cylindrically symmetric
potential well.

The minimization of the droplet energy in Eq. (60) with respect to both $\rho$
and $L$ yields

$\displaystyle\rho$ $\displaystyle\approx$ $\displaystyle
0.53L\sqrt{\frac{t_{\parallel}}{t_{\perp}}}\,,$ $\displaystyle L$
$\displaystyle\approx$ $\displaystyle 0.98a\left(\frac{4\pi
t_{\perp}^{2}}{t_{\parallel}J_{ff}S^{2}}\right)^{1/5}\,.$ (61)

As a result, the optimum volume of the cylindrical droplet reads

$\Omega_{cyl}\approx\pi\frac{L^{2}}{a^{3}}\frac{t_{\parallel}}{4t_{\perp}}\approx
0.26\pi\left(\frac{4\pi
t_{\perp}^{2}}{t_{\parallel}J_{ff}S^{2}}\right)^{3/5}\frac{t_{\parallel}}{t_{\perp}}\,.$
(62)

The direct comparison of the optimum energies of ellipsoidal and cylindrical
droplets shows that

$\frac{E_{ell.pol}-\tilde{E}_{0}}{E_{cyl.pol}-\tilde{E}_{0}}\approx\frac{\Omega_{ell}}{\Omega_{cyl}}\approx
0.96\,,$ (63)

where $\tilde{E}_{0}=E_{0}-4t_{\perp}n-2t_{\parallel}n$.

Hence, we come to the conclusion that the optimum shape of the droplet for
layered manganites is an ellipsoidal one. Physically, this is quite reasonable
since the optimum shape of FM polaron (its “ground state” to some extent)
repeats the form of the electronic spectrum in the layered case, which is
elliptically symmetrical and is given by $\varepsilon(p)\propto
t_{\parallel}p_{\parallel}^{2}t_{\perp}p_{\perp}^{2}$. Of course, after the
dilatation transformation of the momenta
$p_{\parallel}=\tilde{p}_{perp}\frac{}{}\sqrt{\frac{t_{\parallel}}{t_{\perp}}}$
to the new reference frame, an electronic spectrum becomes spherical. In the
layered case, the percolation threshold, which corresponds to the overlap of
the droplets, takes place at the critical density

$\tilde{n}_{c}=\frac{1}{\Omega_{ell}}=\frac{3}{4\pi}\left(\frac{4J_{ff}S^{2}t_{\parallel}}{\pi
t_{\perp}^{2}}\right)^{3/5}\frac{t_{\perp}}{t_{\parallel}}\,.$ (64)

For all densities $n<\tilde{n}_{c}$, the energy of the droplet state is lower
than all the optimum energies of the homogeneous state.

###  3.2 FM polarons in doped anisotropic AFM systems

We are now ready to face a more general situation and to consider the shape
and the size of FM droplets in anisotropic two-dimensional (2D) or 3D cases
when the electron hopping integrals along $x$, $y$, and $z$ directions,
$t_{x}$, $t_{y}$, and $t_{z}$, as well as the AFM exchange integrals $J_{x}$,
$J_{y}$, and $J_{z}$ are different. We find that in analogy with the situation
in layered manganites, the most favorable shape of the FM droplet is again an
ellipsoidal one. The optimum energy and the optimum volume of the droplet are
expressed in terms of the universal averaged parameters
$\overline{J}=(J_{x}+J_{y}+J_{z})S^{2}$ and $t_{eff}=(t_{x}t_{y}t_{z})^{1/3}$
in 3D or respectively $\overline{J}=(J_{x}+J_{y})S^{2}$ and
$t_{eff}=(t_{x}t_{y})^{1/2}$ in 2D anisotropic AFM systems [32]. These results
are, in particular, interesting in relation to neutron scattering experiments
[126, 151, 152, 153], which prove the existence of FM droplets of different
shapes in perovskite and layered manganites. Additional experimental evidence
of the existence of nanoscale FM droplets in layered manganites comes from
scanning tunneling microscopy with atomic resolution [154]. The ferrons in
low-doped layered manganites often proved to be “platelets” characteristic of
2D rather than spherical droplets characteristic of the 3D case [152, 153].

####  3.2.1 Shape of FM droplets in anisotropic 2D case

In the anisotropic 2D case, the system is described by the anisotropic FM
Kondo-lattice model in the double-exchange limit

$\hat{H}=-J_{H}\sum_{i}{\mathbf{S}_{i}\mathbf{\sigma}_{i}}+\sum_{\langle
i,j\rangle_{a}}{J_{a}\mathbf{S}_{ia}\mathbf{S}_{ja}}-\sum_{\langle
i,j\rangle_{a}\sigma}{t_{a}c_{ia\sigma}^{{\dagger}}c_{ja\sigma}}\,,$ (65)

where $\langle i,j\rangle_{a}$ denotes the neighboring sites in the square
lattice along the $a$ direction, $a=\\{x,y\\}$, and
$J_{H}\gg\\{t_{x},t_{y}\\}\gg\\{J_{x},J_{y}\\}$.

The analysis of the volume and energy of the FM droplets of different shapes
in anisotropic 2D case is very similar to the analysis of the energy and
volume of the ellipsoidal and cylindrical droplets in the layered manganites.
It uses again the dilatation transformation of the coordinates (which restores
the cylindrical symmetry in the new reference frame in 2D) alongside with the
properties of the zeroth order Bessel function $J_{0}$ (the value of its first
zero) for the lowest binding state of an electron in the cylindrical potential
well. As a result, the most energetically favorable shape of the droplet in
the anisotropic 2D case is an elliptical one with the volume

$\Omega=\frac{j_{0,1}}{2}\left(\frac{\pi
t_{eff}}{\overline{J}}\right)^{1/2}\,,$ (66)

where $j_{0,1}=0.2404\approx 3\pi/4$ is the first zero of the zeroth order
Bessel function $J_{0}(x)$ corresponding to the solution of the Schrödinger
equation in polar coordinates and $t_{eff},\overline{J}$ are the universal
averaged parameters mentioned above.

Similarly, in the 3D anisotropic case the most favorable shape of the FM
droplet is an ellipsoidal one. Its optimum volume is

$\Omega=\frac{\pi^{8/5}2^{1/5}}{3}\left(\frac{t_{eff}}{\overline{J}}\right)^{3/5}\,,$
(67)

and is again determined by the ratio $t_{eff}/\overline{J}$ of the universal
averaged parameters.

####  3.2.2 FM droplets in the case of frustrated triangular lattice

Just in the same manner, we can find the volume and the most favorable shape
of the FM droplet on the 2D frustrated triangular lattice (with the same
magnitude of AFM exchanges and hopping integrals along the sides of triangle)
for the planar spin configuration. Similarly to the situation described by Eq.
(66), the most favorable shape of the ferron corresponds to the circle with an
area [121]

$\Omega=j_{0,1}\left(\frac{\pi t}{3J}\right)^{1/2}\,.$ (68)

###  3.3 Bound magnetic polarons with extended spin distortions

In Section 2, we have mentioned an old idea, which belongs to de Gennes about
the magnetic polarons of a different type, which resemble the magnetic
impurity and can produce a long-range tail of spin distortions due e.g., to
the magnetodipole interaction. This type of bound magnetic polarons was
introduced in [58, 59, 121], based on the model for an electron, which is
bound at a donor impurity by the Coulomb attractive potential. The Hamiltonian
of such a model has the following form

$\hat{H}=\hat{H}_{el}+J_{ff}\sum_{\langle
i,j\rangle}{[\mathbf{S}_{i}\mathbf{S}_{j}+S^{2}]}-K\sum_{i}{[(S_{i}^{x})^{2}-S^{2}]}\,,$
(69) $\hat{H}_{el}=-t\sum_{\langle
i,j\rangle\sigma}{c_{i\sigma}^{{\dagger}}c_{j\sigma}}--J_{H}\sum_{i}{\mathbf{S}_{i}\mathbf{\sigma}_{i}}-V\sum_{i,\sigma}{\frac{c_{i\sigma}^{{\dagger}}c_{i\sigma}}{|\mathbf{n}-\mathbf{n}_{0}|}}\,,$
(70)

where $K$ is the constant of magnetic anisotropy. The last term in Eq. (70)
describes the Coulomb interaction between conduction electrons and the
impurity ion located in the middle of the main diagonal of the unit cell
(e.g., in the 3D case between the lattice sites $\mathbf{n}=(0,0,0)$ and
$\mathbf{n}=(1,1,1)$, i.e. $\mathbf{n}_{0}=(1/2,1/2,1/2)$ and similarly,
$\mathbf{n}_{0}=(1/2,1/2)$ in the 2D case, see Fig. 11.

Figure 11:  Nonmagnetic Ca(Sr) donor impurity is located in the middle of the
quadratic unit cell in 2D [33].

We work in the limit of strong Coulomb interaction and strong Hund’s rule
coupling

$V\gg J_{H}S\gg t\gg J_{ff}S^{2},KS^{2}\,.$ (71)

In this range of parameters, the radius of electron localization is of the
order of interatomic distance $a$. The magnetic distortions outside the
electron localization region (the ferron core), as it was shown in [58, 59,
121], decrease with the distance $r$ as $1/r^{4}$ and $1/r^{2}$ in the 3D
cubic and 2D square lattices, respectively. At the same time, they decay even
slower, as $1/r$ in the 2D frustrated triangular lattice (for the planar spin
configuration).

In both 3D and 2D cases, these “coated” magnetic polarons correspond to the
ground state of the system at very small doping levels. At the same time, in
the 1D case for the AFM chain of interacting neighboring spins with donor
impurities located just in the middle (between the nearest neighbor spins )
the situation is quite different. Here, as it was shown in [54, 55, 155], the
“coated” magnetic polarons with slowly decaying tails of spin distortions
(shown e.g., in Fig. 12) correspond usually to the excited state of the
system, while the ground state corresponds to the more standard (“bare” or
rigid) magnetic polarons with very rapidly decreasing tails of spin
distortions. In other words, we can say that in rigid ferrons, the
intermediate region where the canting angle $\nu$ changes from zero (FM
domain) to $\pi$ (AFM domain) is narrow (of the order of the interatomic
distance $a$), so the radius of magnetic polaron $R_{pol}$ is a well-defined
quantity. Moreover, the tails of magnetic distortions exponentially decrease
for the distances $r>R_{pol}$ (outside the ferron).

Figure 12:  Magnetic structure with slowly decaying spin distortions between
two donor impurities corresponding to the coated magnetic polaron in the 1D
AFM chain [155].

Let us now focus on in the 3D and 2D cases. At low electron or hole doping
($n\ll 1$ or $x\ll 1$), we can consider a one-particle problem. In the limit
of strong Hund’s rule coupling $J_{H}\rightarrow\infty$, the spin
$\mathbf{\sigma}_{n}$ of a conduction electron at site $n$ should be parallel
to the local spin $\mathbf{S}_{n}$. If $S\gg 1$, it is sufficient to consider
the local spin as a classical vector, i.e., to represent it in the 3D case as
[58, 59, 121]

$\mathbf{S}_{n}=S(\sin{\theta_{n}}\cos{\varphi_{n}},\sin{\theta_{n}}\sin{\varphi_{n}},\cos{\theta_{n}})=S\mathbf{e}_{n}\,,$
(72)

where $\mathbf{e}_{n}$ is the unit vector, $\theta_{n}$ and $\varphi_{n}$ are
polar and azimuthal angles of $\mathbf{e}_{n}$.

For the planar structure, when all local spins lie in ($x,y$) plane, we have
$\theta_{n}=\pi/2$ and

$\mathbf{S}_{n}=S(\cos{\varphi_{n}},\sin{\varphi_{n}},0),\,\,(-\pi<\varphi_{n}\leq\pi)\,.$
(73)

In the limit of strong electron–impurity coupling $V\rightarrow\infty$, the
electron wave function $\Psi_{n}$ will be nonzero only at sites nearest to the
impurity. Let us focus on the 2D case (the 3D case is considered in a similar
way). In this case, we can assume that $\Psi_{n}\neq 0$ only at four sites
nearest to the impurity: $\mathbf{n}_{1}=(1,1)$, $\mathbf{n}_{2}=(1,0)$,
$\mathbf{n}_{3}=(0,1)$, and $\mathbf{n}_{4}=(0,0)$ in the square lattice (see
Fig. 11) and similarly, $\Psi_{n}$ is nonzero at eight nearest sites for the
3D cubic lattice, and three nearest sites at the frustrated triangular
lattice. The wave function of the conduction electron in the explicit form can
be written as [58, 59, 121]

$|\Phi\rangle=\frac{1}{\sqrt{2}}\sum_{n}{\Psi_{n}\left(\hat{c}_{n\uparrow}^{{\dagger}}+\exp{(-i\varphi_{n})}\hat{c}_{n\downarrow}^{{\dagger}}\right)}|0\rangle\,,$
(74)

where $|0\rangle$ is a vacuum state. The energy of a conduction electron in
the state $|\Phi\rangle$ is

$E_{el}[\Phi]=\langle\Phi|\hat{H}_{el}|\Phi\rangle=-t\sum_{\langle
n,m\rangle}{[T_{mn}\Psi_{n}^{*}\Psi_{m}+T_{mn}^{*}\Psi_{n}\Psi_{m}^{*}]}-\frac{1}{2}J_{H}S-V\sum_{n}{\frac{|\Psi_{n}|^{2}}{|\mathbf{n}-\mathbf{n}_{0}|}}\,,$
(75)

where $\sum_{n}{|\Psi_{n}|^{2}}=1$,

$\displaystyle T_{mn}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1+\exp{(i(\varphi_{n}-\varphi_{m}))}\right]=\cos{\frac{\nu_{nm}}{2}}\exp{(i\omega_{nm})}\,,$
$\displaystyle\nu_{nm}$ $\displaystyle=$
$\displaystyle\varphi_{n}-\varphi_{m}\,,\,\,\,\omega_{nm}=\frac{\nu_{nm}}{2}\,.$
(76)

In Eq. (3.3), $\nu_{nm}$ is the angle between the local spins $\mathbf{S}_{n}$
and $\mathbf{S}_{m}$ (canting angle) and $\omega_{nm}$ is the Berry phase
[156, 157].

The electron energy in the ground state for fixed planar spin configuration
(at given angles $\varphi_{n}$) can be found from the minimization of Eq. (75)
with respect to $\Psi_{n}$. As a result, we have

$E_{el}(\varphi_{n})=\frac{1}{2}J_{H}S-\frac{V}{b}-t\varepsilon(a_{nm})\,,$
(77)

where $b=1/\sqrt{2}$ for the square lattice,
$a_{nm}=\cos{\frac{\nu_{nm}}{2}}$, and

$\varepsilon(a_{nm})=\frac{1}{\sqrt{2}}\\{a_{12}^{2}+a_{23}^{2}+a_{34}^{2}+a_{41}^{2}+[((a_{12}-a_{34})^{2}+(a_{23}+a_{41})^{2})((a_{12}+a_{34})^{2}+(a_{23}-a_{41})^{2})]^{1/2}\\}^{1/2}\,.$
(78)

Correspondingly, for the triangular lattice, we have $b=1/\sqrt{32}$ and

$\varepsilon(a_{nm})=g(a_{nm})+\frac{a_{12}^{2}+a_{23}^{2}+a_{31}^{2}}{3g(a_{nm}}\,,$
(79)

where

$g(a_{nm})=\\{a_{12}a_{23}a_{31}+\frac{1}{3\sqrt{3}}[(a_{12}^{2}+a_{23}^{2}+a_{31}^{2})^{3}-27a_{12}^{2}a_{23}^{2}a_{31}^{2}]^{1/2}\\}^{1/3}\,.$
(80)

The electron energy $E_{el}$ has a minimum when all neighboring local spins
surrounding an impurity are parallel (or as we will see later, almost
parallel) to each other. Hence, at this level of approximation, we have a
bound magnetic polaron state, which can be described by a ferromagnetic core
of radius $b$ (with four local spins at the square lattice and three local
spins at the triangular one) embedded into the antiferromagnetic host. When in
the next subsection, we include magnetic degrees of freedom into account, then
after the minimization of the total energy, we will observe that for the most
energetically favorablel “coated” magnetic polaron, its magnetic core will not
be fully polarized and the AFM background around it will be substantially
distorted on rather large distances.

###  3.4 Magnetic structure of a coated ferron

Magnetic contribution to the total energy of a coated ferron can be found in
the general case of 3D spins by the minimization of the total energy with
respect to the angles $\theta_{n}$ and $\varphi_{n}$. To do this, it is
convenient, first, to perform the following transformation of angles:
$\varphi_{n}\rightarrow\varphi_{n}+\pi,\theta_{n}\rightarrow\pi-\theta_{n}$
for one of the sublattices of the cubic (or square) lattice. For planar spin
configuration in the ($x,y$) plane, after the transformation, $\theta_{n}$ is
still equal to $\pi/2$. For the 2D triangular lattice and planar spin
configuration, the corresponding transformation reads
$\varphi_{n}\rightarrow\varphi_{n}\pm 2\pi/3$ for two of the three magnetic
sublattices. As a result of this transformation, the AFM order becomes FM and
_vice versa_. Such a transformation allows us to work with continuously
changing orientation of spins outside the ferron core. Moreover, all the
angles belong to the range from 0 to $\pi/2$ for cubic and square lattices and
to the range from 0 to $2\pi/3$ for the triangular lattice.The total energy is

$E=E_{el}+E_{mag}\,,$ (81)

where $E_{el}$ is given by Eq. (77) and

$E_{mag}=J_{ff}S^{2}\sum_{\langle
n,m\rangle\sigma}{(1-\cos{\nu_{nm}})}-KS^{2}\sum_{n}{(\sin^{2}{\theta_{n}}\cos^{2}{\varphi_{n}}-1)}\,.$
(82)

After this transformation, we should replace $a_{nm}$ in Eqs. (78)–(80) by
$\alpha_{nm}=\sin{\frac{\nu_{nm}}{2}}$.

Minimization of the total energy with respect to the angles $\theta_{n}$ and
$\varphi_{n}$ yields two possible types of the solutions for the bound
magnetic polaron. The first one corresponds to the magnetic polaron with
completely polarized spins inside its core, which is embedded in the purely
AFM background. The total magnetic moment of the polaron is parallel to the
easy axis. We refer this trivial solution as a “bare” magnetic polaron. Its
energy for the 2D square lattice is

$E_{pol}^{0}=-2t+16J_{ff}S^{2}\,,$ (83)

and

$E_{pol}^{0}=-3t+48J_{ff}S^{2}\,,$ (84)

for the 3D cubic lattice [58, 59, 121]. The corresponding expression for the
energy of a “bare” magnetic polaron on the 2D triangular lattice is more
cumbersome. Its explicit form is given in [121].

There exists another solution corresponding to the magnetic polaron state with
magnetic moment perpendicular to the easy axis. We call this solution a
“coated” magnetic polaron. Let us consider the planar spin configuration in
the ($x,y$) plane on the 2D square lattice. Then, from the symmetry
requirements, it is clear that the angles $\varphi_{n},\,\,n=1,...,4$ for the
local spins inside the core of magnetic polaron should satisfy the conditions

$\varphi_{1}=\varphi_{3}=\varphi_{0},\,\,\varphi_{2}=\varphi_{4}=-\varphi_{0}\,,$
(85)

where $0<\varphi_{0}<\leq\pi/2$ (see Fig. 13)

Figure 13:  Spin symmetry inside the ferron after the lattice transformation
[33].

For the 3D cubic lattice and the 2D triangular lattice, the angles
$\varphi_{n}$ satisfy, respectively the conditions

$\displaystyle
1=\varphi_{3}=\varphi_{5}=\varphi_{7}=\varphi_{0},\,\,\varphi_{2}=\varphi_{4}=\varphi_{6}=\varphi_{8}=-\varphi_{0}\,,\textrm{for
the 3D cubic lattice},$
$\displaystyle\varphi_{1}=0,\,\varphi_{2}=-\varphi_{3}=\varphi_{0},\,0<\varphi_{0}\leq
2\pi/3\,,\textrm{for the 2D triangular lattice}\,.$ (86)

Minimization of the total energy of a coated magnetic polaron with respect to
$\varphi_{n}$ yields the following set of nonlinear equations

$\sum_{\mathbf{\Delta}}{\sin{(\varphi_{\mathbf{n}+\mathbf{\Delta}}-\varphi_{\mathbf{n}})}}-\frac{\kappa}{2}\sin{2\varphi_{0}}=\frac{t}{2J_{ff}S^{2}}\sum_{i}{\delta_{\mathbf{nn}_{i}}(-1)^{i}\cos{\varphi_{i}}}\,,$
(87)

where $\mathbf{n}=(n_{x},n_{y}),\,\kappa=2K/J_{ff}$, $\delta_{\mathbf{nm}}$ is
the Kronecker symbol, and $\mathbf{\Delta}$ takes the values ($\pm 1,0$) and
(0, $\pm 1$).

Equations (87) with conditions (85) were solved numerically for the cluster
containing $40\times 40$ sites. The further growth in the number of sites in
the cluster does not change the obtained results. The initial angle
$\varphi_{0}$ was also found. The calculated magnetic structure is shown in
Fig. 14.

Figure 14:  “Coated” magnetic polaron on the square lattice before (left
panel) and after (right panel) transformation of angles in one of sublattices.
The magnetic structure is calculated by solving Eq. (87) at $t/J_{ff}S^{2}=50$
and $\kappa=5\times 10^{-3}$. At these values of parameters, we have
$\varphi_{0}=85^{\circ}$ [58].

We see from this figure that in contrast to the “bare” ferron, the “coated”
magnetic polaron produces spin distortions of the AFM background outside the
region of electron localization (the core region). The “coated” magnetic
polaron has a magnetization lower than its saturation value
($\varphi_{0}<\pi/2$ ). Moreover the “coat” has a magnetic moment opposite to
that of the core (sites 1–4 in Fig. 14 for the square lattice). To obtain
analytical estimations for the spatial distribution of the spin distortions,
it is important to find an approximate solution of Eq. (87) in the continuum
limit. To do this, we should treat the angles $\varphi_{n}$ as the values of
the continuous function $\varphi(\mathbf{r})$ at points
$\mathbf{r}=\mathbf{n}-\mathbf{n}_{0}$. Assuming that outside the magnetic
polaron, the following condition for spatial variations of the angles
$|\varphi_{\mathbf{n}+\mathbf{\Delta}}-\varphi_{\mathbf{n}}|\ll 1$ is met, we
can expand $\varphi(\mathbf{r}+\mathbf{\Delta})$ in the Taylor series up to
the second order in $\mathbf{\Delta}$

$\varphi(\mathbf{r}+\mathbf{\Delta})\approx\varphi(\mathbf{r})+\Delta^{\alpha}d_{\alpha}\varphi(\mathbf{r})+\frac{1}{2}\Delta^{\alpha}\Delta^{\beta}d_{\alpha}d_{\beta}\varphi(\mathbf{r})\,.$
(88)

Substituting this expansion in Eq. (87), we find that the function
$\varphi(\mathbf{r})$ outside the magnetic polaron should satisfy the 2D sine-
Gordon equation

$\Delta\varphi-\frac{\kappa}{2}\sin{2\varphi}=0\,.$ (89)

In the range of the parameters under study, $K<J_{ff}$, that is, $\kappa<1$,
we can linearize this equation. As a result we obtain

$\Delta\varphi-\kappa\varphi=0\,.$ (90)

This equation should be solved with the boundary conditions at infinity
$\varphi(\mathbf{r})_{\mathbf{r}\rightarrow\infty}\rightarrow 0$ and with some
boundary conditions at the surface of the magnetic polaron. We model the
magnetic polaron by a circle of radius $R_{pol}=a/\sqrt{2}$, where $a$ is the
intersite distance, and choose the Dirichlet boundary conditions (see Fig. 15)

$\varphi(\mathbf{r})|_{r=b}=\tilde{\varphi}(\zeta)\,,$ (91)

where we introduced the polar coordinates ($r,\zeta$) in the ($x,y$) plane.

Figure 15:  Dirichlet boundary conditions at the surface of magnetic polaron
for the 2D square lattice. We model the core of magnetic polaron by the circle
of radius $R_{pol}=a/\sqrt{2}$ [33].

The function $\tilde{\varphi}(\zeta)$ can be found in the following way. For
the square lattice, $\tilde{\varphi}(\zeta)$ should meet symmetry conditions
(85) at the points $\zeta_{i}=\pi(2i-1)/4$

$\tilde{\varphi}(\zeta)=\varphi_{i}\,,\,\,\,i=1,...,4\,.$ (92)

Since the function $\tilde{\varphi}(\zeta)$ is periodic, it can be expanded
into Fourier series

$\tilde{\varphi}(\zeta)=\sum_{m=0}^{\infty}{[a_{m}\cos{m\zeta}+b_{m}\sin{m\zeta}]}\,.$
(93)

In Eq. (93), we neglect the terms with $m>2$ , which allows us to keep the
minimum number of terms to satisfy conditions (85). It follows from Eq. (92)
that $a_{0}=a_{1}=a_{2}=b_{1}=0$. Finally, we obtain

$\tilde{\varphi}(\zeta)=\varphi_{0}\sin{2\zeta}\,.$ (94)

The solution of Eq.(90) with boundary condition (91) yields

$\varphi(\mathbf{r})=\frac{\varphi_{0}}{K_{2}(R_{pol}/r_{0})}K_{2}\left(\frac{r}{r_{0}}\right)\sin{2\zeta}\,,\,\,\,r_{0}=\frac{a}{\sqrt{\kappa}}\,,$
(95)

where $K_{2}(x)$ is the Macdonald function of the second order. Within the
$|\mathbf{r}|<r_{0}$ range, the function $\varphi(\mathbf{r})$ behaves as
$R_{pol}^{2}/r^{2}$, whereas at large distances it decreases exponentially,
$\varphi(\mathbf{r})\propto\exp{(-r/r_{0})}$. Thus, $R_{pol}$ and $r_{0}$ play
role of the size of the core and the radius of the “coat”, respectively.
Moreover, in the case of small anisotropy ($\kappa\ll 1$), the distance, where
function $\varphi(\mathbf{r})$ has a power-law behavior, can be large enough
in agrement with the de Gennes conjecture [120]. The function
$\varphi(\mathbf{r})$ at $\zeta=\pi/4$ and the numerical results for
$\varphi_{n,n}$ are plotted in Fig. 16.

Figure 16:  Decrease with the distance of an angle $\varphi(\mathbf{r})$
between misaligned spins and easy axis for a coated magnetic polaron on 2D
square lattice [58]. The circles correspond to the numerical solution of the
discrete problem on a square cluster at $t/J_{eff}S^{2}=50$ and
$\kappa=5\times 10^{-2}$ ($r_{0}=4.5a$). The solid curve corresponds to
analytical solution (95) in the continuum limit. At small distances,
$\varphi(\mathbf{r})$ decreases proportionally to $1/r^{2}$, while at large
distances $r\gg r_{0}$, it decreases exponentially.

For the 3D cubic lattice, the similar analytical solution of Eq. (90) with
symmetry conditions (3.4), the Dirichlet boundary conditions, and a spherical
core of magnetic polaron with the radius $R_{pol}=a\sqrt{3}/2$ yields

$\varphi(x,y,z)=\varphi_{0}\frac{R_{3}(r/r_{0})}{R_{3}(R_{pol}/r_{0})}\frac{3\sqrt{3}xyz}{r^{3}}\,,$
(96)

where $r_{0}=a/\sqrt{\kappa}$ and

$R_{3}(\rho)=\left(1+\rho+\frac{2\rho^{2}}{5}+\frac{\rho^{3}}{15}\right)\frac{e^{-\rho}}{\rho^{4}}\,,\quad\rho=\frac{r}{r_{0}}\,.$
(97)

It follows from Eqs. (96) and (97) that $\varphi(\mathbf{r})$ decreases as
$(R_{pol}/r)^{4}$ at $r<r_{0}$. At distances $r$ exceeding the characteristic
radius $r_{0}$, the spin distortions decay exponentially
$\varphi(\mathbf{r})\propto\exp{(-r/r_{0})}$ as in the 2D case.

Finally, for the 2D triangular lattice an analytical solution of Eq. (90) with
symmetry conditions (3.4) and the Dirichlet boundary conditions for polar
coordinates and a circular core of magnetic polaron with the radius
$R_{pol}=a/\sqrt{3}$ is given by [121]

$\varphi(\mathbf{r})=\frac{2\varphi_{0}}{\sqrt{3}}\frac{K_{1}(r/r_{0}^{*})}{K_{1}(R_{pol}/r_{0}^{*})}\sin{\zeta}\,,$
(98)

where $r^{*}=a\sqrt{4\kappa/3}=r_{0}\sqrt{3}/2$ and $K_{1}(x)$ is the
Macdonald function of the first order. Within the $r<r_{0}^{*}$ range, the
function $\varphi(\mathbf{r})$ slowly decreases as $R_{pol}/r$.. At the same
time, at large distances, it decreases exponentially again,
$\varphi(\mathbf{r})\propto\exp{(-r/r_{0}^{*})}$ [121].

Let us compare now the energies of the “coated” magnetic polarons with the
“bare” ones. For the 2D square lattice, the energy of the “bare” magnetic
polaron is given by Eq. (83) and the energy of the “coated” magnetic polaron
is given by Eqs. (81) and (82), where $\theta_{n}=\pi/2$ and $\varphi_{n}$ are
the solutions of Eqs.(87). In the continuum approximation, we expand
$\cos{\nu_{nm}}$ and $\cos^{2}{\varphi_{n}}$ in the Taylor series up to the
second order outside the ferron core. Changing the summation by the
integration over $r$, we can represent the total energy of the “coated”
magnetic polaron as

$E=E_{el}+E_{M}=\left[-\frac{1}{2}J_{H}S-V\sqrt{2}-2t\sin{\varphi_{0}}\right]+8J_{ff}S^{2}\left(1+\frac{\kappa}{4}\right)\sin^{2}{\varphi_{0}}+\frac{J_{ff}S^{2}}{2}\int_{r\geq
r_{0}}{d^{2}r[(\nabla\varphi)^{2}+\kappa\varphi^{2}]}\,,$ (99)

In Eq. (99), the last two terms correspond to the magnetic part of the total
energy. The first one of them comes from the summation over spins in the core
of magnetic polaron. Substituting the solution for the function
$\varphi(\mathbf{r})$ from Eq. (95) into Eq. (99) and performing the
integration, we get

$E_{pol}=8J_{ff}S^{2}\left(1+\frac{\kappa}{4}\right)\sin^{2}{\varphi_{0}}+\frac{J_{ff}S^{2}\varphi_{0}^{2}}{2}I(R_{pol}/r_{0})-2t\sin{\varphi_{0}}\,,$
(100)

where

$I(\rho_{0})=2\pi\left[1+\frac{\rho_{0}K_{1}(\rho_{0})}{2K_{2}(\rho_{0})}\right]$
(101)

and $\rho_{0}=R_{pol}/r_{0}$.

Since $R_{pol}<r_{0}$, the value of the function $I(\rho_{0})$ is close to
$2\pi$. The optimum angle $\varphi_{0}$ is found by the minimization of the
energy in Eq. (100) and is given by

$\cos{\varphi_{0}}-\frac{4J_{ff}S^{2}}{t}\left(1+\frac{\kappa}{4}\right)\sin{2\varphi_{0}}-\frac{J_{ff}S^{2}\varphi_{0}^{2}}{2t}I(R_{pol}/r_{0})=0\,.$
(102)

Hence,

$\varphi_{0}\approx\frac{\pi}{2}\left[1-\frac{\pi J_{ff}S^{2}}{t}\right]\,.$
(103)

At $J_{ff}S^{2}/t=50$ correspnfing to Fig. 16, estimate (103) gives us just
the required value $\varphi_{0}\approx 85^{\circ}$. The dependence of energy
difference between “coated” and “bare” magnetic polarons $\Delta
E=E_{pol}-E_{pol}^{0}$ at the 2D square lattice on the parameter of anisotropy
$\kappa$ is shown in Fig. 17. As it follows from this figure, “coated”
polarons are more favorable if the anisotropy parameter $\kappa$is not too
large. Otherwise, “bare” magnetic polarons are more favorable.

Figure 17:  Energy difference between “coated” and “bare” magnetic polarons
_v_ s $\kappa=2K/J_{jj}$ calculated at $J_{ff}S^{2}/t=50$. Solid curve
corresponds to the numerical calculations , whereas dashed curve is calculated
using Eq. (100) [58]

It can be shown in the same manner that for 3D cubic lattice and 2D triangular
lattice, the energies of the “coated” magnetic polarons for not very high
values of the magnetic anisotropy can be also lower then the energies of the
corresponding “bare” magnetic polarons [58, 59, 121].

##  4 Charge ordering and phase separation

###  4.1 Introduction

In this section, we present the results of our group [28, 33, 34, 35, 107], as
well as of other groups [4, 5, 6, 37] on the charge ordering (CO) and
formation of nanoscale metallic droplets in CO insulating matrices in
framework of the Verwey [104, 105] and Shubin–Vonsovsky models [106]. We will
also consider ferromagnetic metallic droplets, which arise in CO and
antiferromagnetic insulating matrices in more extended models [28, 33, 34, 35]
combining the double exchange magnetic interactions with the strong intersite
Coulomb interactions. Finally, we will comment on the large-scale phase
separation and formation of stripes and zig-zag structures typical of these
models and confirmed by different experimental techniques including the
electron diffraction and dark-image electron microscopy [132, 158, 159].

Historically, the problem of CO in magnetic oxides has attracted attention of
theorists since the discovery of the Verwey localization transition in
magnetite Fe3O4 at the end of the 1930s [104]. The early theoretical
description of this phenomenon is given, for example, in [160]. At the end of
1990s, this problem was reexamined in a number of papers in relation to the
colossal magnetoresistance (CMR) phenomena in manganites and related compounds
[161, 162, 163]. Mechanisms stabilizing the CO state can be different: the
Coulomb repulsion of charge carriers (the energy minimization requires keeping
the charge carriers as far away as possible from each other, similarly to the
Wigner crystallization) or the electron–lattice interaction leading to the
effective repulsion of electrons at the nearest-neighbor sites [60, 61, 62,
63] if the electron bandwidth is sufficiently small. In the opposite case,
large electron kinetic energy stabilizes the homogeneous metallic state. In
actual materials, in contrast to the Wigner crystallization, the underlying
lattice periodicity determines the preferential type of CO. Thus, in the
simplest bipartite lattice, to which belong, for example, both cubic
(La1-xSrxMnO3) and layered (La2-xSrxMnO4, La2-2xSr1+2xMn2O7) hole-doped
manganites, the optimum conditions for the formation of CO state exist for
$x=1/2$. At this value of $x$, the concentrations of Mn3+ and Mn4+ ions are
equal and the simple checkerboard arrangement is possible (see Fig. 18). A
remarkable experimental fact is that even at $x\neq 1/2$ (in underdoped
manganites with $x<1/2$), only the simplest version of CO is experimentally
observed with the alternating checkerboard structure of the occupied and empty
sites in the basal plane. In other words, this structure corresponds to the
doubling of the unit cell, whereas the more complicated structures with a
longer period (or even incommensurate structures) do not actually appear in
this case.

Figure 18:  Verwey type of charge ordering at the doping level $x=1/2$ with
the checkerboard distribution of Mn3+ (crosses) and Mn4+ (open circles) ions,
which here play the role of electrons and holes, respectively [33].

Then, a natural question arises, namely, how the extra or missing electrons
can be redistributed for an arbitrary doping level such that the
superstructure remains the same as for $x=1/2$? To answer this question, the
experimentalists introduced the concept of the incipient CO state
corresponding to the distortion of a long-range charge ordering by microscopic
metallic clusters [158, 159, 164]. In fact, the existence of this state
implies a kind of phase separation. In this section, we show that the
interplay between the CO and the tendency toward the phase separation plays a
very important role in the physics of manganites and other related compounds
(magnetite [104], cobaltites [165], and nickelates [166]).

###  4.2 The simplest model for charge ordering

We start our theoretical considerations of the CO and phase separation with
the simplest Verwey model, which includes the Coulomb repulsion of electrons
at the neighboring sites and the intersite electron hopping. The Hamiltonian
of the Verwey model is

$\hat{H}^{\prime}=-t\sum_{<i,j>}{c_{i}^{{\dagger}}c_{j}}+V\sum_{<i,j>}{n_{i}n_{j}}-\mu\sum_{i}{n_{i}}\,,$
(104)

where $t$ is the hopping integral, $V$ is the nearest-neighbor Coulomb
repulsion (a similar nearest-neighbor repulsion can be obtained via the
interaction with the breathing type optical phonons), $\mu$ is the chemical
potential, $c_{i}^{{\dagger}}$ and $c_{j}$ are respectively the electron
creation and annihilation operators at site $i$,
$n_{i}=c_{i}^{{\dagger}}c_{i}$. The symbol $\langle i,j\rangle$ denotes as
usual the summation over the nearest-neighbor sites. In Eq. (104), we omit
spin and orbital indices for simplicity, thus considering one orbital for
spinless electrons and emphasizing the most robust effects related to the
nearest-neighbor Coulomb repulsion. We also assume that the double occupancy
does not occur in this model because of the strong onsite repulsion
$U\to\infty$ between electrons.

The explicit inclusion of the arbitrary (either weak or strong) onsite Hubbard
repulsion $U$ in the model together with an account taken of spin degrees of
freedom transforms the Verwey model [104] to the Shubin–Vonsovsky model [106]
(or, using a more common term, the extended Hubbard model) with the
Hamiltonian [107]

$\hat{H}^{\prime}=-t\sum_{<i,j>\sigma}{c_{i\sigma}^{{\dagger}}c_{j\sigma}}+U\sum_{i}{n_{i\uparrow}n_{i\downarrow}}+\frac{V}{2}\sum_{<i,j>}{n_{i}n_{j}}-\mu\sum_{i\sigma}{n_{i\sigma}}\,,$
(105)

where $n_{i,\sigma}=c_{i\sigma}^{{\dagger}}c_{i\sigma}$ is the electron
density at site $i$ with spin projection $\sigma=|\uparrow,\downarrow\rangle$,
and $n_{i}=\sum_{\sigma}{n_{i\sigma}}$. We will show later in this section
(see also [33, 107]) that in the strong coupling limit of the Shubin–Vonsovsky
model ($U\gg V\gg W=2zt$ and densities $n\neq 1/2$), we obtain the same
results for the radius of the nanoscale metallic droplets in the CO matrix as
in the strong-coupling limit of the Verwey model ($V\gg W$) after the evident
substitution $V\to V/2$.

In the Shubin–Vonsovsky model, the Verwey localization transition at densities
$n\to 1/2$ is an additional one to the more standard Mott–Hubbard localization
transition [23, 167] at $U\gg W$ and $n\to 1$.

In the main part of this section (see Eqs. (104) and (105)), we usually speak
about electrons. However, in application to manganites, for example, we mostly
have in mind underdoped systems like La1-xSrxMnO3 with the hole doping level
lower than 1/2 ($x<1/2$). For them, we must therefore think in terms of holes.
All the theoretical treatment definitely remains the same (from the very
beginning, we could define the operators in the Hamiltonian of the Verwey
model (104) as the operators of holes). We hope that this does not lead to any
misunderstanding.

In what follows, we consider the simplest case of square (2D) and cubic (3D)
lattices, where the simple two-sublattice ordering occurs (in manganites for
$x=1/2$). This is the case for layered manganites, whereas in 3D perovskite
manganites, this ordering occurs as we already mentioned only in the basal
plane: the ordering is not checkerboard “in-phase”) along the $c$-axis
direction. A more complicated model is apparently needed to account for this
behavior.

If the bandwidth $W$ is even smaller than the Coulomb repulsion between
electrons located at more distant sites ($V\gg V_{2}\gg V_{3}\gg\dots\gg
V_{m}\gg W$ , where $V_{m}\sum_{i}{n_{i}n_{m+i}}$ accounts for the Coulomb
interaction between electrons at a distance $ma$ apart), then the variation of
the charge carrier density will give rise to the whole cascade of localization
phase transitions, which differ in their ground state structure. For instance,
if $V_{2}\gg W$ is included to the model, the system will exhibit an
additional phase transition to a localized state at electron densities $n\to
1/3$. A more general localization criterion for narrow-band systems
($\mathbf{a}\triangledown V(\mathbf{r})\gg W$ was originally proposed in [168,
169].

Let us return to the simplest model with only the nearest-neighbor Coulomb
repulsion (Eq. (104)). For densities $n=1/2$, the Verwey model was analyzed in
many papers [160, 161, 162, 170, 171]). In this section, we follow the
treatment of [160] (see also [34, 35, 33]). As mentioned above, the Coulomb
repulsion stabilizes the CO in the form of a checkerboard arrangement of the
occupied and the empty sites, whereas the first term (band energy) opposes
this tendency. At arbitrary values of the electron density $n$, we first
consider a homogeneous CO solution and use the same ansatz as in [160],
namely,

$n_{i}=n[1+(-1)^{i}\tau]\,.$ (106)

This expression implies doubling the lattice periodicity with the local
densities

$n_{1}=n(1+\tau),\quad n_{2}=n(1-\tau)\,.$ (107)

at the neighboring sites. At $n=1/2$ for a general form of the electron
dispersion without nesting, the CO state exists only for a sufficiently strong
repulsion $V>2t$ [160]. The order parameter is $\tau<1$ for finite $V/2t$ , so
the average electron density differs from zero or unity even at $T=0$.

We use the coupled Green’s function approach [160]), which yields [34, 35]

$\displaystyle(E+\mu)G_{1}-t_{k}G_{2}-zVn(1-\tau)G_{1}$ $\displaystyle=$
$\displaystyle 1/2\pi\,,$
$\displaystyle(E+\mu)G_{2}-t_{k}G_{1}-zVn(1+\tau)G_{2}$ $\displaystyle=$
$\displaystyle 0\,.$ (108)

In Eq. (4.2), $G_{1}$ and $G_{2}$ are the Fourier transforms of the normal
lattice Green functions $G_{ij}=\langle\langle
c_{i}c_{j}^{{\dagger}}\rangle\rangle$ for the sites $i$ and $j$ belonging
respectively to the same or different sublattices, $z$ is the number of
nearest neighbors, and $t_{k}$ is the Fourier transform of the hopping matrix
element. In deriving Eq. (4.2), we performed a mean-field decoupling and
replaced the averages $\langle c_{i}^{{\dagger}}c_{i}\rangle$ by the onsite
densities from Eq. (106). The solution of Eq. (4.2) gives the following
spectrum

$E+\mu=Vnz\pm\sqrt{(Vn\tau z)^{2}+t_{k}^{2}}=Vnz\pm\omega_{k}\,.$ (109)

The spectrum defined by Eq. (109) resembles the spectrum of superconductor
and, hence, the first term under the square root is analogous to the
superconducting gap squared. In other words, we can introduce the CO gap by
the formula

$\Delta=Vn\tau z\,.$ (110)

It depends on the density not only explicitly, but also via the density
dependence of $\tau$. We thus obtain

$\omega_{k}=\sqrt{\Delta^{2}+t_{k}^{2}}\,.$ (111)

Using Eqs. (4.2), (110), and (111), we can find the Green’s functions

$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle\frac{A_{k}}{E+\mu-
Vnz-\omega_{k}+i0}+\frac{B_{k}}{E+\mu-Vnz+\omega_{k}+i0}\,,$ $\displaystyle
G_{2}$ $\displaystyle=$
$\displaystyle\frac{t_{k}}{4\pi\omega_{k}}\left[\frac{1}{E+\mu-
Vnz-\omega_{k}+i0}+\frac{1}{E+\mu-Vnz+\omega_{k}+i0}\right]\,.$ (112)

where

$A_{k}=\frac{1}{4\pi}\left(1-\frac{\Delta}{\omega_{k}}\right),\quad
B_{k}=\frac{1}{4\pi}\left(1+\frac{\Delta}{\omega_{k}}\right)\,.$ (113)

After a standard Wick transformation $E+i0\to iE$ in the expression for
$G_{i}$, we find the densities in the following form

$\displaystyle n_{1}$ $\displaystyle=$ $\displaystyle
n(1+\tau)=\int{\left[\left(1-\frac{\Delta}{\omega_{k}}\right)f_{F}(\omega_{k}-\mu+Vnz)+\left(1+\frac{\Delta}{\omega_{k}}\right)f_{F}(-\omega_{k}-\mu+Vnz)\right]\frac{d^{D}\mathbf{k}}{2\Omega_{BZ}}}\,,$
$\displaystyle n_{2}$ $\displaystyle=$ $\displaystyle
n(1-\tau)=\int{\left[\left(1+\frac{\Delta}{\omega_{k}}\right)f_{F}(\omega_{k}-\mu+Vnz)+\left(1-\frac{\Delta}{\omega_{k}}\right)f_{F}(-\omega_{k}-\mu+Vnz)\right]\frac{d^{D}\mathbf{k}}{2\Omega_{BZ}}}\,,$
(114)

where

$f_{F}(y)=\frac{1}{e^{y/T}+1}$

is the Fermi distribution function and $\Omega_{BZ}$ is the volume of the
first Brillouin zone. Adding and subtracting the equations for $n_{1}$ and
$n_{2}$, we obtain the final system of equations for $n$ and $\mu$

$\displaystyle n$ $\displaystyle=$
$\displaystyle\int{\left[f_{F}(\omega_{k}-\mu+Vnz)+f_{F}(-\omega_{k}-\mu+Vnz)\right]\frac{d^{D}\mathbf{k}}{2\Omega_{BZ}}}\,,$
$\displaystyle 1$ $\displaystyle=$ $\displaystyle
Vz\int{\frac{1}{\omega_{k}}\left[f_{F}(-\omega_{k}-\mu+Vnz)-f_{F}(\omega_{k}-\mu+Vnz)\right]\frac{d^{D}\mathbf{k}}{2\Omega_{BZ}}}\,.$
(115)

For low temperatures ($T\to 0$ ) and $n\leq 1/2$, it is reasonable to assume
the filling of only the lowest levels (negative $\mu-Vnz$). Therefore,

$f_{F}(\omega_{k}-\mu+Vnz)=0$

and

$f_{F}(-\omega_{k}-\mu+Vnz)=\theta(-\omega_{k}-\mu+Vnz)$

is the step function.

It can be shown that for $n=1/2$, the system of equations (4.2) yields
identical results for all $-\Delta\leq\mu-Vnz\leq\Delta$. This means that
$n=1/2$ is a point of indifferent equilibrium. For infinitely small deviations
from $n=1/2$, i.e. for densities $n=1/2-0$, the chemical potential should be
defined as

$\mu=-\Delta+\frac{Vz}{2}=\frac{Vz}{2}(1-\tau)\,.$ (116)

If we consider the strong coupling $V\gg 2t$ and assume a constant density of
states inside the band, then for $n=1/2$, we can find [34, 35]

$\tau=1-\frac{2W^{2}}{3V^{2}z^{2}}$ (117)

with $z=6$ for the simple cubic lattice and, therefore,

$\mu=\frac{W^{2}}{3Vz}\,.$ (118)

For the density $n=1/2$, the CO gap $\Delta$ appears for an arbitrary
interaction strength $V$. This is due to the existence of nesting in our
simple model. In the weak coupling case $V\ll 2t$ and with perfect nesting, we
have

$\Delta\propto W\exp{\left(-\frac{W}{Vz}\right)}\,,$ (119)

and $\tau$ is exponentially small. For $V\gg W=2tz$, or accordingly for $V\gg
2t$, it follows that $\Delta\approx Vz/2$ and $\tau\to 1$ (Eq. (110)). In the
general case, when there is no nesting, the CO exists only if the interaction
strength $V$ exceeds a certain critical value of the order of the bandwidth
$W$ [160]. In what follows, we restrict ourselves to the physically more
instructive strong-coupling case $V\gg 2t$.

For the constant density of states, the integrals in Eq. (4.2) can be taken
explicitly and the system of equations (4.2) can be easily solved for
arbitrary $n$. However, in the strong-coupling case $V\gg 2t$ and for small
density deviations from 1/2 ($\delta\ll 1$), the results are not very
sensitive to the form of the electron dispersion.

When $n=1/2-\delta$, the chemical potential is $\mu=\mu(\delta,\tau)$, and we
have two coupled equations for $\mu$ and $\tau$. As a result, we obtain [34]

$\mu(\delta)\approx
Vnz(1-\tau)-\frac{4W^{2}}{Vz}\delta^{2}\approx\frac{W^{2}}{3Vz}+\frac{4W^{2}}{3Vz}\delta+O(\delta^{2})\,.$
(120)

The energy of CO state is therefore given by

$E_{CO}(\delta)=E_{CO}(0)-\frac{W^{2}}{3Vz}\delta-\frac{2W^{2}}{3Vz}\delta^{2}+O(\delta^{2})\,,$
(121)

where

$E_{CO}(0)=-\frac{W^{2}}{6Vz}$ (122)

is the energy precisely corresponding to the density $n=1/2$. Moreover,
$|E_{CO}(0)|\ll W$. At the same time, the CO gap $\Delta$ is given by

$\Delta\approx\frac{Vz}{2}\left[1-2\delta-\frac{2W^{2}}{3V^{2}z^{2}}(1+4\delta)\right]\,.$
(123)

The dependence of the chemical potential $\mu$ and the total energy $E$ on
$\delta$ in Eqs. (120) and (121) is in fact related to the linear decrease of
the CO gap $\delta$ with the deviation from half-filling.

For $n>1/2$, the energy of the CO state starts to increase rapidly due to a
large contribution of the Coulomb repulsion (the upper Verwey band is
partially filled at $n>1/2$). For $n>1/2$, in contrast to the case of $n<1/2$,
each extra electron placed into the checkerboard CO state necessarily has the
occupied nearest-neighbor sites, increasing the total energy by $Vz|\delta|$.
For $|\delta|=n-1/2>0$, we then have

$E_{CO}(\delta)=E_{CO}(0)+\left(Vz-\frac{W^{2}}{3Vz}\right)|\delta|-\frac{2W^{2}}{3Vz}\delta^{2}+O(\delta^{3})\,.$
(124)

Accordingly, the chemical potential is given by

$\mu(\delta)=Vz-\frac{W^{2}}{3Vz}-\frac{4W^{2}}{3Vz}|\delta|+O(\delta^{2})\,.$
(125)

It undergoes a stepwise change equal to $Vz$ at $\tau\to 1$. The gap $\Delta$
is symmetric for $n>1/2$ and is given by

$\Delta\approx\frac{Vz}{2}\left[1-2|\delta|-\frac{2W^{2}}{3V^{2}z^{2}}(1+4|\delta|)\right]\,.$
(126)

We could make the entire picture symmetric with respect to $n=1/2$ by shifting
all the one-electron energy levels and the chemical potential by $Vx/2$, i.e.
defining $\mu^{\prime}=\mu-Vz/2$. In terms of $\mu^{\prime}$, Eqs. (120) and
(125) can be written as

$\displaystyle\mu^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{Vz}{2}+\frac{W^{2}}{3Vz}+\frac{4W^{2}}{3Vz}\delta,\quad
n<1/2\,,$ $\displaystyle\mu^{\prime}$ $\displaystyle=$
$\displaystyle\frac{Vz}{2}-\frac{W^{2}}{3Vz}-\frac{4W^{2}}{3Vz}|\delta|,\quad
n>1/2\,.$ (127)

Similarly to the situation in semiconductors, we have $\mu^{\prime}=0$
precisely at the point $n=1/2$, which means that the chemical potential lies
in the middle of the band gap (see Fig. 19). At densities $n=1/2-0$ , the
chemical potential $\mu^{\prime}=-Vz/2$ coincides with the upper edge of the
filled Verwey band. We get just the same band structure in the strong-coupling
limit of the Shubin–Vonsovsky model [33].

Figure 19:  Band structure of the Verwey and Shubin–Vonsovsky models at
$n=1/2$ [34, 107]. The lower Verwey band is completely filled. The upper
Verwey band is empty. The chemical potential $\mu^{\prime}=0$ lies in the
middle of the gap with the width of $2\Delta$.

###  4.3 Phase separation in the Verwey model

We now check the stability of the CO state in the Verwey model. At the
densities close to $n=1/2$, the dependence of the energy on the charge density
has the form illustrated in Fig. 20.

Figure 20:  Energy of the CO state versus charge density for $n\to 1/2$. The
energy has a kink at $n=1/2$ [34].

Figure 20 clearly demonstrates an instability of the CO state. Indeed, the
most remarkable implication of Eqs. (120)–(4.2) is that the compressibility
$\kappa$ of the homogeneous CO state is negative for the densities different
from 1/2 ,

$\kappa^{-1}\propto\frac{d\mu}{dn}=\frac{d^{2}E}{d\delta^{2}}=-\frac{4W^{2}}{3Vz}<0\,,$
(128)

where $\delta=1/2-n$. As we already mentioned, this is a manifestation of the
tendency toward the phase separation characteristic of the CO system with
$\delta\neq 0$. The presence of a kink in the $E_{CO}(\delta)$ curve
determined by Eqs. (121) and (124) implies that one of the states. into which
the system might separate, would correspond to the checkerboard CO state with
$n=1/2$, whereas the other would have a certain density $n^{\prime}$ smaller
or larger than $n$. This conclusion (based on very general considerations
[149]) resembles that discussed in [135, 162, 172], although the detailed
physical mechanism is different. The possibility of the phase separation in
model (104) away from half-filling was also reported earlier in for the
infinite-dimensional case. In what follows, we focus on the situation with
$n<1/2$. The case when $n>1/2$ apparently has certain special properties – the
existence of stripe phases etc. [158, 159], the detailed origin of which
should will be analyzed separately.

It is easy to understand the physics of phase separation in our case. As
follows from Eq. (123), the CO gap decreases linearly with the deviation from
the half-filling. Correspondingly, the energy of the homogeneous CO state
rapidly increases, and it is more favorable to “extract” extra holes from the
CO state, putting them into one part of the sample, while creating the “pure”
checkerboard CO state in the other part. The energy loss due to this
redistribution of holes is overcompensated by the gain provided by the better
CO.

However, the hole-rich regions would not be completely “empty” similarly to
pores or voids (clusters of vacancies) in crystals: we can gain extra energy
by “dissolving” a certain amount of electrons there. In doing this, we
decrease the band energy of the electrons due to their delocalization. Thus,
the second phase would be a metallic one. The simplest state of this kind is a
homogeneous metal with the electron density $n_{met}$. This density, as well
as the relative volumes of the metallic and CO phases, can be easily
calculated by minimizing the total energy of the system. The energy of the
metallic part of the sample, $E_{met}$, in the 3D case is given by

$E_{met}=-tzn_{met}+ct(n_{met})^{5/3}+\frac{Vz}{2}(n_{met})^{2}\,,$ (129)

where $c$ is a constant.

Minimizing Eq. (129) with respect to $n_{met}$, we find the equilibrium
electron density in the metallic phase. For the strong-coupling case $V\gg
2t$, we obtain the following simple expression (neglecting the relatively
small correction provided by the term which contains $n_{met}^{5/3}$)

$n_{met}^{(0)}=t/V\,.$ (130)

The optimum energy of the metallic phase is given by

$\left(E_{met}\approx-\frac{t^{2}z}{2V}=-\frac{W^{2}}{8Vz}\right)>\left(E_{CO}=-\frac{W^{2}}{6Vz}\right)\,,$
(131)

and it is larger than the energy of CO state.

###  4.4 Phase separation into two large clusters (complete phase separation)

In accordance with the general treatment in statistical physics, the system
with the density $n_{met}^{(0)}<n<1/2$ separates fully on two large clusters
(Fig. 21): a metallic one with the density $n_{met}^{(0)}$ and a CO one with
the density 1/2. For arbitrary $n$, the relative volumes $\Omega_{met}$ and
$\Omega_{CO}$ of these clusters can be found from the Maxwell construction

$\frac{\Omega_{met}}{\Omega_{CO}}=\frac{1/2-n}{n-n_{met}^{(0)}}\,,$ (132)

which implies that the metallic cluster occupies the part $\Omega_{met}$ of
the total volume $\Omega=\Omega_{met}+\Omega_{CO}$ given by

$\frac{\Omega_{met}}{\Omega}=\frac{1/2-n}{1/2-n_{met}^{(0)}}=\frac{\delta}{\delta_{0}}\,,$
(133)

where we introduced deviations from the half-filling, $\delta=1/2-n$ and
$\delta_{0}=1/2-n_{met}^{(0)}$.

Accordingly, the energy of the fully separated state
$E_{comp.sep.}=E_{met}(\Omega_{met}/\Omega)+E_{CO}(\Omega_{CO}/\Omega)$ takes
the form

$E_{comp.sep.}\approx-\frac{W^{2}}{8Vz}\frac{\delta}{\delta_{0}}-\frac{W^{2}}{6Vz}(1-\frac{\delta}{\delta_{0}})\,.$
(134)

The metallic cluster will occupy the entire sample volume when the total
electron density $n$ is less than $n_{met}^{(0)}$, while CO cluster occupies
all the volume for $n=1/2$.

Figure 21:  Complete phase separation into two large clusters (metallic and CO
ones) [28].

###  4.5 Nanoscale phase-separated state with small metallic droplets within
the charge-ordered insulating host material

We now consider a different possibility for the phase-separated state, namely
the possibility, which is closer to the nanoscale phase separation in
manganites and magnetic semiconductors, described in Section 2. To be more
specific, we will analyze metallic droplets of a the small radius in the CO
insulating host. The expression for the energy of this state in the 3D cubic
case can be written in the form [28, 34]

$E_{drop}=-tn_{drop}\left(z-\frac{\pi^{2}a^{2}}{R^{2}}\right)-\frac{W^{2}}{6Vz}\left[1-\frac{4}{3}\pi\left(\frac{R}{a}\right)^{3}n_{drop}\right]\,.$
(135)

It corresponds to the situation illustrated in Fig. 22 and describes the
process, in which small metallic droplets with one conduction electron are
formed in the CO matrix.

Figure 22:  Simplest nanoscale phase separation scenario involving the
formation of small metallic droplets with one conduction electron embedded
into a CO host [28, 107] .

In expression (135), similarly to the situation considered in Section 2, $R$
is the radius of a droplet, and $a$ is the interatomic spacing. The first term
in Eq. (135) describes the kinetic energy gain due to the localization of an
electron inside the droplet. The second term is the energy of CO in the
insulating region of the sample. The parameter $n_{drop}$ in Eq. (135) is the
concentration of metallic droplets. Similarly to the situation of FM metallic
droplets in AFM matrix, a surface energy term can be incorporated into the
total energy of the system. This term, however, is of the order of
$W^{2}R^{2}/V$ for a metallic droplet in a CO matrix, so that for $R\gg a$, it
is always small compared to the bulk energy term (which is proportional to
$R^{3}$ in the 3D case) and it will be neglected in the following discussion.
The minimization of energy in (135) with respect to the droplet radius $R$
results in the following condition

$\frac{R}{a}\propto\left(\frac{V}{t}\right)^{1/5}\,.$ (136)

Therefore, at the critical concentration of droplets

$n_{drop}^{(c)}=\frac{3}{4\pi}\left(\frac{a}{R}\right)^{3}\propto\left(\frac{t}{V}\right)^{3/5}$
(137)

the metallic droplets begin to overlap and the sample as a whole undergoes a
transition to the metallic state. Expression (135) is in fact the analogue of
the Nagaoka theorem [173] for the Verwey model. Similarly, in the 2D case, the
exponent 1/5 in Eq. (136) for the droplet radius is replaced by 1/4 and the
exponent 3/5 in the expression for the critical density in Eq. (137) is
replaced by 1/2 [33, 107].

Just the same expression for the critical density corresponding to the phase
separation was obtained in [107] for the strong-coupling limit of the
Shubin–Vonsovsky model, whereas for low electron densities $n\ll 1$ the ground
state of the model corresponds to the triplet $p$-wave superconductivity,
which is governed by the Kohn–Luttinger mechanism [101, 102, 107]. The
qualitative phase diagram for the phase-separation regions in the strong-
coupling limit of the 2D Shubin–Vonsovsky model is presented in Fig. 23.

Figure 23:  Qualitative phase diagram of the nanoscale phase-separated state
in the strong-coupling limit of the 2D Shubin–Vonsovsky model [33].

If we use Eq. (137) for the critical droplet concentration, the total energy
(135) of the system can be rewritten as

$E_{drop}=-tn_{drop}\left[z-\tilde{c}(n_{drop}^{(c)})^{2/3}\right]-\frac{W^{2}}{6Vz}\left[1-\frac{n_{drop}}{n_{drop}^{(c)}}\right]\,,$
(138)

where $\tilde{c}$ is a numerical coefficient of the order of unity.

The ratio $n_{drop}/n_{drop}^{(c)}$ in Eq. (138) describes the relative volume
$\Omega_{drop}/\Omega$ occupied by droplets. This volume can be found from the
condition for the conservation of the total number of particles in the system,
which reads

$N=\frac{1}{2}(\Omega-\Omega_{drop})+n_{drop}\Omega_{drop}=n\Omega\,,$ (139)

where $n$ is the average density of the conduction electrons. From condition
(139), the required relation (which resembles relation (133) for complete
phase separation) follows immediately and has the form

$\frac{\Omega_{drop}}{\Omega}=\frac{1/2-n}{1/2-n_{drop}^{(c)}}=\frac{\delta}{\delta_{c}}\,,$
(140)

where again $\delta=1/2-n$ and $\delta_{c}=1/2-n_{drop}^{(c)}$ . From Eq.
(139), it follows that $n_{drop}=0$ at $n=1/2$. Accordingly, at the critical
density $n_{drop}=n_{drop}^{(c)}$, we have
$\delta=\delta_{c}=1/2-n_{drop}^{(c)}$. Finally, the energy of the phase-
separated state can be written as

$E_{drop}=-t\frac{\delta}{\delta_{c}}n_{drop}^{(c)}[z-\tilde{c}(n_{drop}^{(c)})^{2/3}]-\frac{W^{2}}{6Vz}\left(1-\frac{\delta}{\delta_{c}}\right)\approx-\frac{W}{2}\frac{\delta}{\delta_{c}}n_{drop}^{(c)}-\frac{W^{2}}{6Vz}\left(1-\frac{\delta}{\delta_{c}}\right)\,.$
(141)

Comparison of energy (141) with that of the CO state given by (121) shows that
for the densities $n_{drop}^{(c)}<n<1/2$, we have

$E_{drop}-E_{CO}\approx-\frac{W}{2}\frac{\delta}{\delta_{c}}n_{drop}^{(c)}<0\,.$
(142)

Thus, a nanoscale phase-separated state with small metallic droplets inside a
CO matrix is more favorable than a homogeneous CO state. The direct comparison
of the energy for nanoscale phase-separated state (141) and the energy for the
complete separation into two large clusters (134) shows that for
$n_{met}^{(0)}<n_{drop}^{(c)}$, we have $E_{drop}<E_{comp.sep}$ and the
nanoscale phase separation is more favorable than complete one, as well.
Physically, it is related to a kind of Nagaoka theorem for the Verwey model.
The point is that even when metallic droplets overlap, each conduction
electron occupies a sphere of the radius $R/a\sim(V/t)^{1/5}$ and thus
effectively it is located at the larger distance from the other electrons than
it would be in the case of complete phase separation. Therefore, at the
complete phase separation, the energy of Coulomb interactions between
electrons turns out to be higher than for metallic droplets of a small radius.

In addition to the one-electron metallic droplets, a nanoscale phase
separation scenario of another type can be organized [28, 33] (see Fig. 24).
In this scenario, a metallic droplet can be formed by replacing one electron
with a hole in the center of the droplet. Note, however, that the energy of
such an undermelted CO state (a resonance valence bond or RVB state for the
Verwey model) is much more difficult to evaluate than that for a one-electron
droplet, and this problem is not considered in the present review.

Figure 24:  Nanoscale phase-separation scenario with an undermelted CO state
inside a metallic droplet embedded into the CO matrix [28].

Concluding this subsection, we can summarize the main results as follows. In
the strong-coupling limit, for $V\gg 2t$ and $(t/V)^{3/5}<n<1/2$, the 3D
Verwey model is unstable toward the nanoscale phase separation. Moreover, in
the simplest case, the most energetically beneficial scenario of the phase
separation corresponds to small one-electron metallic droplets inside the CO
insulating host. Similar results for the nanoscale phase separation, we get in
the strong-coupling limits of the 2D Verwey model [33] and the 2D
Shubin–Vonsovsky model [106] for the densities $(t/V)^{1/2}<n<1/2$.

###  4.6 Phase separation in the extended double exchange model with the
nearest-neighbor Coulomb interaction

Let us now return to the basic double exchange model (or the ferromagnetic
Kondo-lattice model) considered in Section 2 (see Eq. (8) there) and add the
term with nearest-neighbor Coulomb interaction to this model. The Hamiltonian
of this extended model reads

$\hat{H}^{\prime}=-J_{H}\sum_{i}{\mathbf{S}_{i}\bm{\sigma_{i}}-t\sum_{<i,j>\sigma}{Pc_{i\sigma}^{{\dagger}}c_{j\sigma}}P}+J_{ff}\sum_{<i,j>}{\mathbf{S}_{i}\mathbf{S}_{j}}+V\sum_{<i,j>}{n_{i}n_{j}}-\mu\sum_{i\sigma}{n_{i\sigma}}\,.$
(143)

It is physically reasonable to consider this model in the strong-coupling
limit

$J_{H}S>V>W>J_{ff}S^{2}\,.$ (144)

In the absence of the Coulomb term it is exactly the conventional double
exchange model.

Let us consider the possibility of the nanoscale phase separation in the
extended model (143). The simplest scenario of the phase separation in this
model corresponds in analogy with Section 2 and previous subsection to small
FM metallic droplets (with one conduction electron inside) embedded into the
AFM CO insulating host [28, 135, 29] (see Fig. 25).

Figure 25:  Formation of metallic FM droplets of small radius embedded into
the AFM CO insulating host material ] [28].

The energy of this state in the 3D case is given by the expression

$\displaystyle E$ $\displaystyle=$ $\displaystyle-
tn_{drop}\left(z-\frac{\pi^{2}a^{2}}{R^{2}}\right)+\frac{1}{2}zJ_{ff}S^{2}\frac{4}{3}\pi\left(\frac{R}{a}\right)^{3}n_{drop}$
(145)
$\displaystyle-\frac{1}{2}zJ_{ff}S^{2}\left[1-\frac{4}{3}\pi\left(\frac{R}{a}\right)^{3}n_{drop}\right]-\frac{W^{2}}{6Vz}\left[1-\frac{4}{3}\pi\left(\frac{R}{a}\right)^{3}n_{drop}\right]\,.$