Transport properties in Manganite
Phan Van-Nham and Tran Minh-Tien
Transport properties in doped Manganite are studied by considering the simplified Double-exchange mechanism in the presence of diagonal disorder. It is modeled by a combination the Ising Double-exchange with Falicov-Kimball model. In this model, charge order ferromagnetic and transport quantities are presented by using Dynamical-mean field theory in the case of finite coupling Hund and disorder strenght. Those results suggest that, within this model the insulator phase can be stated in the range of below charge order temperature which is suitable with observation of experiment in manganites.
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\title{Transport properties in Manganite}

\author{Phan Van-Nham and Tran Minh-Tien}

\affiliation{Institute of Physics,
National Center for Natural Science and Technology, \\
P.O. Box 429, Boho, 10000 Hanoi, Vietnam.
}

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\begin{abstract}
Transport properties in doped Manganite are studied by considering the simplified
Double-exchange mechanism in the presence of diagonal disorder. It is modeled by a
combination the Ising Double-exchange with Falicov-Kimball model. In this model, charge order
ferromagnetic and transport quantities are presented by using Dynamical-mean field theory in the 
case of finite coupling Hund and disorder strenght. Those
results suggest that, within this model the insulator phase can be stated in the range of below 
charge order temperature which is suitable with observation of experiment in manganites.
\end{abstract}

%%Outline:

%\vskip2pc]

\maketitle

\section{Introduction}
An enormous interest in the family of the doped manganice oxides T$_{1-x}$D$_{x}$MnO$_3$ (
in which T is trivalent rare earth and D is divalent alkaline) in perovskite structure have been recent
renewed.\cite{ci1,ci2,ci3,ci4}As doping $x$ and temperature T varied, the manganites show a very rich phase diagram
involving phases with spin, charge and orbital order.\cite{ci1,ci2,ci3,ci4}In the cubic lattice, the five-fold degenerate
$3d$-orbitals of an isolated atom or ion are spilitted into a manifold of three lower energy ($d_{xy}$, $d_{yz}$, and
$d_{xz}$), usually referred to as $t_{2g}$ one mixing with the surrounding oxygens is included, and two higher
energy states (d$_{x^2-y^2}$ and d$_{3z^2-r^2}$) called $e_g$ state. Electron $t_{2g}$ are usually located but
electron $e_g$ are itinerant so they create band energy. The correlation between $t_{2g}$ and $e_g$ electron is
expressed by Double Exchange (DE) mechanism\cite{ci5} inwhich two motions involving electrons moving from the
oxygen atom to the Mn$^{4+}$ ions and other one from the Mn$^{3+}$ to the oxygen atom. The main feature of DE is
cooperative effect where the motion of an itinerant electron favors the ferromagnetic (FM) ordering of local spin
and, vice versa, the presence of the FM order facilitates the motions of the itinerant electron. The DE model
has been studied in "one orbital" model because it is not only simplify but also contains enough degree of
freedom to manifest the phenomena of DE mechanism which was envisioned in the early days of manganites studies.\\
Recent experiment observed in the Manganites have shown that the DE model alone can not explain the
quantitative features of the temperature dependence of the resistivity through the entire temperature range\cite{ci6}.
To improve the theoretical understanding in Manganites, there are two proposed resolutions of this problem. The
first one is a large Jahn-Teller lattice distortion which causes a metal-insulator transition (MIT) via strong
polaronic narrowing of the conduction electron band. However, the Jahn-Teller lattice distortion makes the
charge-ordering phase occurs before the FM transition temperature\cite{ci7}which is not suitable with experimental
observation at half-filling.\cite{ci2,ci8}In the second proposed resolution, the insulating behavior is caused
by a combination of both magnetic disorder and non-magnetic ionic disorder. The magnetic disorder arises from the
random DE factor cos($\theta /2$) in the electronic hoping ($\theta$ is the angle between local spins) which call
off-diagonal. The non-magnetic comes from the ionic doping of the D$^{2+}$ ion (i.e. from randomness at the
chemical substituting T$^{3+}$ by D$^{2+}$) which leads to a random local potential for the charge  carriers. This
disorder always presents in doped Manganite which called diagonal-disorder that can lead to Anderson localization
of the charge carriers.\cite{ci9}It is well known that, the diagonal-disorder can be modeled by the
Falicov-Kimball (FK) model.\cite{ci10} Although the FK model is simple, it contains a rich variety of phases which
are controlled by electron interaction.\cite{ci11,ci12}Incorporating the diagonal-disorder of FK type into
DE model, one may expect that a disorder-order phase transition could present. When phase transition occurs,
a CO-FM phase may be stabilized at low temperature\cite{ci13} and the behaviors of transport coefficients can be
more close with the experiment results. In this paper, the transport coefficients such as d.c. conductivity,
thermal conductivity and thermal power of system are being studied by the Dynamical mean-field theory (DMFT)\cite{ci14}.
The DMFT have been extensively used for investigating strongly correlated electron system.\cite{ci14}It is based
on the fact that, the self-energy depends only on frequency in the infinite dimension limit. Using DMFT we found
that the system stabilizes inhomogeneous ferromagnetic state at low temperature and gives us a fulfill picture of
behaviors of the transport coefficients.\\
This paper is organized as follows:\\



\section{Model and transport coefficients}
The system which we consider is described by the following
Hamiltonian

\begin{eqnarray}
H &=& - t\sum_{<ij>,\sigma} c^{\dagger}_{i\sigma} c^{\null}_{j\sigma}
- \mu \sum_{i\sigma}c^{\dagger}_{i\sigma} c^{\null}_{i\sigma}  - 2 J_{H} \sum_{i} S^{z}_{i} s^{z}_{i}
\nonumber \\
&&
 + U \sum_{i\sigma} n_{i\sigma} n_{fi}+E_{f} \sum_{i} n_{fi},
\label{hamil}
\end{eqnarray}
where $c^{\dagger}_{i\sigma}$($c^{\null}_{i\sigma}$) is the creation
(annihilation) operator for an itinerant electron with spin $\sigma$
at lattice site $i$. The first term in the Hamiltonian (\ref{hamil}) represents the hoping between the nearest
neighbors sites. $t$ is the hoping integral and is scaled with the spatial dimension $d$ in the limit $d\rightarrow \infty$ as \cite{ci14}.
$t=\frac{t^{\star}}{2\sqrt d}$. In the following we will take $t^{\star}=1$ as the unit of energy.\cite{ci14}
$S^{z}_{i}$ is the $z$ component of the local
spin at lattice site $i$, it takes values $-\frac{3}{2},-\frac{3}{2}+1,...,\frac{3}{2}$,
$s^{z}_{i}=\frac{1}{2}\sum_{\sigma}c^{\dagger}_{i\sigma}\sigma^z c^{\null}_{i\sigma}$ with $\sigma^z$ is the $z$-
component of Pauli matrix is the spin of itinerant electron.
$n_{fi}$ is a
classical variable that assumes the value $1$($0$) if site $i$ is occupied
by (is not occupied by) D$^{2+}$ ion. $U$ is the disorder strength and is mapped
onto the difference in the local potential which spilits energetical favor of Mn$^{3+}$ and Mn$^{4+}$ ion, the
expectation value $x=\sum_{i}\langle n_{fi}\rangle /N$ (N is the number od lattice sites) corresponds to the concentration of
unfavorable Mn$^{4+}$ sites.
The chemical potential $\mu$ controls
the carrier dopings, while $E_{n_f}$ controls the fraction of the sites
having the additional local potential. The condition $n+x=1$ where $n=\sum_{i\sigma}\langle n_{i\sigma}\rangle /N$ is electron doping is
used which determines $E_{f}$ for each doping $n$.
The first three terms in the Hamiltonian (\ref{hamil}) describe a
simplified DE model which contains only
the Ising-type interaction between the itinerant and local
electrons. Within the DMFT the Ising type of the Hund interaction
gives basically same results as the ones of the Hund interaction
of clasical spins.\cite{ci14}The simplication does not allow any spin-flip processes,
which can be important at low temperature where spin-wave excitations
may govern the thermodynamics of the system. However, within the DE model
the DMFT calculations
for quantum spins does not show significant differences from the one
for clasical spins.\cite{ci15,ci16}The last two terms in the Hamiltonian (\ref{hamil})
describe a binary randomness of the T-site substitution. They together
with the hoping term of itinerant electrons form the FK model.{\cite{ci17,ci18}}\\
 Transport coefficients are calculated within a Kubo-Greenwood formalism\cite{ci19}, in
which the dc-conductivity $\sigma $, thermal power $S$ and the thermal
conductivity $\kappa $ can be expressed with the relevant correlation
functions of the current operators. We have\cite{ci19} \\
\begin{eqnarray}
\sigma =\frac{e^2}{T}L^{11},
\label{sig}
\end{eqnarray}
\begin{eqnarray}
S=-\frac{1}{eT}\frac{L^{12}}{L^{11}},
\label{pow}
\end{eqnarray}
\begin{eqnarray} \kappa
=L^{22}-\frac{(L^{12})^2}{L^{11}}.
\label{kap}
\end{eqnarray}
where coefficients $L^{11}, L^{12}$ and $L^{22}$ are determined from the analytic continuation of
the relevant current-current correlation function at zero frequency, i.e.,\\
\begin{eqnarray}
L^{ij}=\lim_{\nu \rightarrow 0}T \textrm{Im} \frac{\overline{L}^{ij}(\nu)}{\nu}.
\end{eqnarray}
Here $\overline{L}^{ij}(\nu)$ is the relevant current-current correlation
functions\\
\begin{eqnarray} {\overline L}^{11}(i\nu_n)=\int_{0}^{\beta}d\tau
e^{i\nu_n\tau}\langle T_\tau {\bf j}(\tau){\bf j}(0)\rangle ,
\label{bl1}
\end{eqnarray}
\begin{eqnarray}
{\overline L}^{12}(i\nu_n)=\int_{0}^{\beta}d\tau e^{i\nu_n\tau}\langle
T_\tau {\bf j}(\tau){\bf j}_Q(0)\rangle ,
\label{bl2}
\end{eqnarray}
\begin{eqnarray}
{\overline L}^{22}(i\nu_n)=\int_{0}^{\beta}d\tau e^{i\nu_n\tau}\langle
T_\tau {\bf j}_Q(\tau){\bf j}_Q(0)\rangle .
\label{bl3}
\end{eqnarray}
Where
$\bf j$ and $\bf j_{Q}$ are the particle-current operator and the heat-current
operator, respectively. The particle-current operator is defined by taking
commutator of the Hamiltonian with the polarization operator $\sum_{i}{\bf
R}_{i}n_i$.\cite{ci19}In our model, there are two inequivalent sublattice A and B so
it is useful to divide Hamiltonian (\ref{hamil}) into four terms.
\begin{eqnarray}
H=H_A+H_B+H_{AB}+H_{BA},
\end{eqnarray}
with $H_{\alpha}=U\sum_{\sigma}n_{f\alpha}n_{\alpha\sigma}-2J\sum_{\sigma}S^{z}_{\alpha}s^{z}_{\alpha}+E_f\sum
n_{f\alpha}$ where $\alpha =A(B)$ for sublattice A(B) expresses the interactions of sublattice A(B) and $H_{AB}=
-t\sum_{\sigma}a_{\sigma}^{\dagger}b_{\sigma}$, $H_{BA}=-t\sum_{\sigma}b_{\sigma}^{\dagger}a_{\sigma}$ with
$a_{\sigma}^{\dagger},a_{\sigma}$ ($b_{\sigma}^{\dagger},b_{\sigma}$) are creation and annihilation operator
for sublattice A(B) respond to the hopping between A and B site. After some algebraic calculations we obtain.\\
\begin{eqnarray}
{\bf j}=\sum_{\bf q}v_{\bf q}(a_{\bf q\sigma }^{\dagger}b_{\bf q\sigma}+b_{\bf q\sigma }^{\dagger}a_{\bf q\sigma})
\label{jpar}
\end{eqnarray}
in momentum representation, where velocity $v_{\bf q}=\bigtriangledown_{\bf q} \epsilon (\bf q)$ and $\epsilon (\bf q)$ is
the dispersion of the non-interacting electrons. The heat-current ${\bf j}_{Q}$ can
be determined by the energy current ${\bf j}_E$ and the particle current ${\bf j}$, ${\bf j}_{Q}={\bf j}_{E}-\mu{\bf j}$.
The energy-current is defined by taking commutator of the Hamiltonian with the
energy polarization operator $\sum_i{\bf R}_ih_i$ (where $H=\sum_{i}h_i$). We
obtain \\

\begin{eqnarray}
{\bf j}_Q = \sum_{{\bf q}\sigma}{\bf
v_q}[\epsilon({\bf q}-\mu ](a_{\bf q\sigma }^{\dagger}b_{\bf q\sigma}+b_{\bf q\sigma }^{\dagger}a_{\bf q\sigma})
\nonumber \\
-\frac{1}{2}\sum_{{\bf q,q^{'}}\sigma}
W({\bf q-q')(v_{q'}+v_q)}(a_{\bf q\sigma }^{\dagger}b_{\bf q^{'}\sigma}+b_{\bf q\sigma }^{\dagger}a_{\bf q^{'}\sigma}),
\label{jq}
\end{eqnarray}
where $W({\bf q -q'}) = \frac{1}{N}\sum_{i}(J\sigma S_{i}^{z}-n_fU)e^{-i({\bf q -q'}){\bf R}_i}$.\\
Using the equation of motion (EOM) technique\cite{ci17} we can write the heat-current operator in the form

\begin{eqnarray}
{\bf j}_{Q}=\lim_{\tau^{'}\rightarrow \tau^{-}}\frac{1}{2}\sum_{\bf
q,\sigma}(\frac{\partial}{\partial \tau}-\frac{\partial}{\partial \tau^{'}}) v_{\bf q}
\nonumber \\
\times (a_{{\bf q}\sigma }^{\dagger}(\tau)b_{{\bf q}\sigma}(\tau^{'})+b_{{\bf q}\sigma }^{\dagger}
(\tau)a_{{\bf q}\sigma}(\tau^{'})).
\end{eqnarray}
Substubting ${\bf j}$ and ${\bf j}_{Q}$ from (\ref{jpar}) and
(\ref{jq}) into (\ref{bl1})(\ref{bl2}) and (\ref{bl3}), the transport
coefficients are able to be calculated in the canceling of the vertex
correction at infinite dimension limit\cite{ci16,ci20}
\begin{widetext}
\begin{eqnarray}
L^{11}=T\sum_{\sigma}\int d\epsilon \rho (\epsilon
)\int d\omega \left( -\frac{\partial f(\omega)}{\partial \omega}\right)
A_{A\sigma}(\epsilon ,\omega)A_{B\sigma}(\epsilon ,\omega),
\label{l11}
\end{eqnarray}
\begin{eqnarray}
L^{12}=T\sum_{\sigma}\int d\epsilon \rho (\epsilon )\int d\omega \left(
-\frac{\partial f(\omega)}{\partial \omega}\right) A_{A\sigma}(\epsilon ,\omega)A_{B\sigma}
(\epsilon ,\omega)\omega ,
\label{l12}
\end{eqnarray}
\begin{eqnarray}
L^{22}=T\sum_{\sigma}\int d\epsilon \rho (\epsilon )\int d\omega \left(
-\frac{\partial f(\omega)}{\partial \omega}\right) A_{A\sigma}(\epsilon ,\omega)A_{B\sigma}
(\epsilon ,\omega)\omega^{2}.
\label{l22}
\end{eqnarray}
\end{widetext}
where $f(\omega)=1/(\exp(\omega /T)+1)$ is Fermi-Dirac distribution function
and $A_{\alpha\sigma}(\epsilon ,\omega)=-\frac{1}{\pi}\textrm{Im}G_{\alpha\sigma}(\epsilon ,\omega)$ is
the spectral function of the Green function for the intinerant electron for sublattice A or B. In
that way, the transport coefficients are fully determined by the single
particle spectral function.\\
\section{DMFT in SDE within FK model}

We solve model (\ref{hamil}) by the DMFT.
The DMFT is based on the infinite dimension limit.
In the infinite dimension limit  the self energy is pure local and does not depend on momentum.
By using the standard technique\cite{ci14} the matrix Green function of the itinerant electrons satisfies
in the following form
\begin{equation}
\widehat{G}^{-1}_{\sigma}({\bf k}, i \omega)=
\left(\begin{array}{cc}
				\omega_{\bf k}+\mu -\Sigma^{A}_{\sigma}(\omega) &   -\epsilon({\bf k})\\
				-\epsilon({\bf k})	& \omega_{\bf k}+\mu -\Sigma^{B}_{\sigma}(\omega)\\
						\end{array}
               				\right),
\label{gmatrix}
\end{equation}
where $\varepsilon({\bf k})$ is the dispersion of free itinerant electrons on a hypercubic
lattice, $\Sigma^{A(B)}_\sigma(\omega)$ is the self energy for sublattice A(B).
The self energy is determined by solving an effective
single-site problem. The effective action for this
problem is
\begin{eqnarray}
S_{\alpha \small{\textrm{eff}}}=-\int d\tau \int
d\tau^{'}\sum_{\sigma}c_{\alpha \sigma}^{\dagger}(\tau){\cal
G}_{\alpha \sigma}^{-1}(\tau -\tau^{'})c_{\alpha \sigma}(\tau^{'})
\nonumber \\
-\int d\tau \sum_{\sigma n_{f \alpha}}[JS^z\sigma +\mu -Un_{f \alpha}]c_{\alpha\sigma}^{\dagger}(\tau)c_{\alpha\sigma}(\tau)+E_fn_{f\alpha},
\label{seff}
\end{eqnarray}
where $c_{\alpha\sigma}=a_{\sigma}(b_{\sigma})$ if $\alpha =A(B)$ and
similar to $c_{\alpha \sigma}^{\dagger}$, ${\cal G}_{\alpha \sigma}(\tau -\tau^{'})$ is the Green function of a
effective medium for each sublattice A or B. It plays as the base Green function of the effective problem.
The local Green function also satisfies the Dyson equation.\\
\begin{eqnarray}
G_{\alpha\sigma}^{-1}(\omega )={\cal G}_{\alpha\sigma}^{-1}(\omega )-\Sigma_{\alpha\sigma}(\omega ).
\label{g01}
\end{eqnarray}
The local Green function $G_{\alpha\sigma}(\omega )$ of the effective single-site problem is solely
determined by the partition function. It can be calculated by.\\
\begin{eqnarray}
G_{\alpha \sigma}(i\omega_n )=\frac{\delta {\cal Z}_{\alpha \small{\textrm{eff}}}}{\delta
{\cal G}_{\alpha \sigma}^{-1}(i\omega_n)},
\label{g1}
\end{eqnarray}
where $Z_{\small{\textrm{eff}}}$ is the partition function and $\omega_n=(2n+1)\pi T$ is the Matsubara frequency. Within
the effective single-site problem, the partition function becomes\\
\begin{eqnarray}
{\cal Z}_{\alpha \small{\textrm{eff}}}=\textrm{Tr}\int
Dc_{\alpha \sigma}^{\dagger}Dc_{\alpha \sigma}e^{-S_{\alpha \small{\textrm{eff}}}},
\label{zeff}
\end{eqnarray}
where the trace is
taken over $S_{z}$ and $n_{\alpha f}$. This partition function can be calculated exactly.
Indeed, the Dynamics of the local spin $S^z$ and impurity $n_{\alpha f}$ involved in the effective action
(\ref{seff}) independently, we could take the trace over $S^z$ and $n_{\alpha f}$ in partition
function(\ref{zeff}) independently. This is similar to DMFT solving the FK model\cite{ci21}, we obtain
\begin{widetext}
\begin{eqnarray}
{\cal Z}_{\alpha \small{\textrm{eff}}}=4\sum_{m,n_{\alpha f}}\exp \left\{\sum_{n,\sigma }(\ln [{\cal
G}_{\alpha \sigma}^{-1}(i\omega_n)+J_H\sigma m +\mu -Un_{\alpha f}]-\ln i\omega_n)e^{i\omega_n\theta^+}\right\}
e^{E_fn_{\alpha f}\beta},
\end{eqnarray}
where $m =-\frac{3}{2},-\frac{3}{2}+1,\dots \frac{3}{2}$ are projections of the local spin
$S_{z}$ on $z$ axit.\\
Using Eq.(\ref{g1}) we obtain the local Green function on real axis after taking analytical continuation\\
\begin{eqnarray}
G_{\alpha \sigma}(\omega )=\sum_{m,n_{\alpha f}}\frac{w_{m\alpha }}{{\cal G}_{\alpha \sigma}^{-1}(\omega )+J_H\sigma
m +\mu -Un_{\alpha f}},
\label{g02}
\end{eqnarray}
where
\begin{eqnarray}
w_{m\alpha }=\frac{1}{\cal Z}_{\small{\textrm{eff}}}\exp \left\{-\beta \int d\omega
f(\omega)\frac{1}{\pi}\textrm{Im}\sum_{\sigma }\ln [{\cal
G}_{\sigma}^{-1}(\omega)+J_H\sigma m +\mu -Un_{\alpha f}]\right\}.
\end{eqnarray}
\end{widetext}
Equation (\ref{g02}) is similar to the one with the classical local spin.\cite{ci15}It can be
considered as a generalized formula of the local Green function for any value of the local spin.
The self-consitent condition of the DMFT requires the local Green function of the original lattice. That mean,
\begin{equation}
G_{\alpha \sigma}(\omega) = \frac{1}{N} \sum_{\bf k} G_{\alpha \sigma}(\bf{k},\omega).
\label{greenlocal}
\end{equation}
From equation (\ref{gmatrix}) we obtain
\begin{equation}
G_{\alpha \sigma}(\omega) =\zeta_{\overline{\alpha}\sigma}\int d\epsilon \rho(\epsilon)\frac{2}
{\zeta_{A\sigma}\zeta_{B\sigma}-\epsilon^2},
\label{g03}
\end{equation}
where $\alpha =A,B$ and $\overline{\alpha}=B,A$, $\zeta_{\alpha\sigma}=\omega +\mu -\Sigma^{\alpha}_{\sigma}(\omega)$.\\
Eqs. (\ref{g01})(\ref{g02}), and (\ref{g03}) form the complete set of equations, which self-consistently determines
the self-energy and the Green function for each sublattice A and B.
\section{Numerical results}
In this section we present the solution of the self-consistent DMFT Eqs.(\ref{g01})(\ref{g02}), and (\ref{g03}).
The self-energy and the Green function are obtained by iterations.\cite{ci21}With the obtained Green function,
we calculate the d.c. resistivity, thermal conductivity, and thermal power by Eqs. (\ref{l11})(\ref{l12}), and (\ref{l22}).\\
In Fig.1a we present the magnetization $m_{A(B)}=\sum_{i\in A(B)}(n_{i\uparrow}-n_{i\downarrow})/N$ of sublattice A(B) as the
function of temperature and we also plot the dependence of charge-order parameter $\Delta=(n_A-n_B)$ (where
$n_{A(B)}=\sum_{i\in A(B)\sigma}n_{i\sigma}/N$) on temperature in a case where the filling of the localized and intinerant
electron is equal 0.5 with $J=8$ and various of disorder strengths $U$. In this paper, we only concentrate in the
case of normal manganites where the Hund coupling dominates with the disorder strenght that mean J$_H$ $\ll$ U. When $U$ is large, it shows that, below
critical temperature the magnetization of both-sublattices exists. They equal to each other until another critical
temperature where the charge-order parameter exist. That mean, at low temperature, system is in the charge board
CO-FM state. In this phase the charge-order coexists in the FM state which had been observed.\cite{ci8}But with the
small value of $U$, the CO-phase disappears, and only FM phase exists. With those coexistances of phases, the dependence
of transport coefficients on temperature behaves as expected. In Fig.1b the behaviors of resistivity in
temperature range is plotted.\\
It shows that, in high temperature range, the dependence of resistivity on temperature
shows the metallic behavior ($d\rho/dT>0$). Immediately below the transition temperature ferromagnetic $T_C$ the
resistivity declines with decreasing temperature due to the reduction of the magnetic contribution to electron
scattering. Hence, at low temperature system is in FM phase. But when temperature decreases below CO transition
temperature, the system changes the phase to insulator ($d\rho/dT<0$) in the large $U$ case. In the case of small $U$
our system is in metallic through the entire temperature range which is the same results as in the SDE model\cite{ci21}.\\
The kinks at $T_C$ and $T_{CO}$ temperature in the resistivity curve indicate the FM and CO transitions. The last two Fig.1(c and d) show the dependence
of the thermal conductivity and thermal power vs temperature. They behaviors express as expected. At low temperature
the thermal conductivity decreases and disappears when $T=0$. When temperature increases, the thermal conductivity
has a peak at the temperature which decreases with the decreasing of $U$, but with the large of $T$ the thermal
conductivity slowly increases and is reputed without depending on disorder strength $U$. The thermal power also
behaves as expected. At low temperature, it is negative that because of the priority of hole-carrier particles and shows a sharp peak.
When $T$ is increased, the thermal power changes the sign and increases. Those results had been obtained in the
FK model.\cite{ci18}\\

\begin{figure*}[htb]
       \centerline{
\psfig{figure=Tran_g08_n05.ps,width=0.685\textwidth,angle=-90}
}
\caption{\small  Temperature dependence of the magnetization m$_A$(m$_B$) for sublattice A(B)
and charge-order parameter $\Delta$(a), d.c. resistivity(b), thermal-conductivity(c), and thermal-power(d) in a case
of half-filling ($n_e=n_f=0.5$) and $J=8.0$ for different disorder strength $U$: $U=0.1$ (solid line), $U=0.2$
(dotted), $U=0.3$ (dashed line), $U=0.4$ (chain-dotted), and $U=0.5$ (double chain-dotted).}
\label{h2b}
\end{figure*}

\begin{figure*}[htb]
       \centerline{
\psfig{figure=DOS_g80_U04_n05.ps,width=0.685\textwidth,angle=-90}
}
\caption{\small  Density of State of initerant electron in the case of of half-filling ($n_e=n_f=0.5$), $J=8.0$,
and $U=0.5$ for various temperatures}
\label{h2b}
\end{figure*}

\begin{figure*}[htb]
       \centerline{
\psfig{figure=Tran_g08_n.ps,width=0.69\textwidth,angle=-90}
}
\caption{\small  Temperature dependence of the magnetization m$_A$(m$_B$) for sublattice A(B)
and charge-order parameter $\Delta$(a), d.c. resistivity(b), thermal-conductivity(c), and thermal-power(d) in a case
of half-filling ($n_e+n_f=1.0$) $U=0.5$, and $J=8.0$ but for different concentrations: $n=0.5$ (solid line), $n=0.6$
(dotted), and $n=0.7$ (dashed line).}
\label{h2b}
\end{figure*}

\begin{figure}[htb]
       \centerline{
\psfig{figure=DOS_g80_U_n05.ps,width=0.415\textwidth,angle=-90}
}
\caption{\small  Density of State of itinerant spin-up electron only in the case of of half-filling ($n_e=n_f=0.5$), $J=8.0$,
and some values of disorder strength: $U=0.1$ (diamond), $U=0.4$ (square), and $U=0.5$ (circle) (opened(filled) for
sublattice A(B)) at low temperature $T=0.001$.}
\label{h2b}
\end{figure}
\begin{figure}[htb]
       \centerline{
\psfig{figure=phase.ps,width=0.415\textwidth,angle=-90}
}
\caption{\small  The critical temperature $T_C$ (filled) and $T_{CO}$ (opened) as a function of disorder
strength $U$ for $J=8.0$ (square) and $J=6.0$ (circle).}
\label{h2b}
\end{figure}
Above results can be understood clearly if we plot the DOS of itinerant electron. In Fig.2 the DOS for half filling $n_f=n_e=0.5$ case
with $J=8, U=0.5$ for various temperature is expressed. At low temperature $(T<T_{CO})$ the DOS of each sublattice
and each spin direction are distincted, that mean our system is in CO-FM phase. At temperature $T_{CO}<T<T_C$ the
distinction between sublattice A and B disappears but ones for spin direction are still on distinction which
expresses the FM phase only. When the temperature is larger ($T>T_C$), all DOS of each sublattice
and each spin direction are concided which
decribed the Paramagnetic phase. In all temperatures above $T_{CO}$ the Fermi-level states in band energy so our
system is in metallic phase but with temperature is smaller than $T_{CO}$ the DOS are depleted at Fermi-level that
mean there occurs the gap which expresses the insulator phase.\\
The DOS for various $U$ of spin up only, at low temperature are also expressed in Fig.4. One can see that, for
weak coupling the DOS is essentially unchanged from the SDE model, but as $U$ is increased further, the gap of DOS
appears and its width increases. This result is precisely the kind of metal-insulator transition that Mott envisioned.\\
The similar results for various itinerant concentration $n_e$ also are plotted in the Fig.3. It shows that, when
$n_e$ increases the $T_{CO}$ and $T_C$ decrease which also is obtained in the infinite of Hund coupling J\cite{ci13}.
In Fig.5 the dependence of the critical temperature of the FM and CO phase transition on disorder strength are
presented. That means the disorder substantially decrease $T_C$ of the FM transition. At the same time, with
increasing U, critical temperature of the CO phase transition first increases, reaches its maximal value and decreases
later. This behavior of critical temperature of the CO phase transition is similar to the one in FK model\cite{ci12}.\\
In the large strength of $U$ the $T_{CO}$ and $T_C$ are close in on the same value but $T_{CO}$ still is smaller
than $T_C$. That means, the CO state is stabilized in the FM phase at low temperature. This results are observed in
experiment\cite{ci8}.\\
\section{Conclusion}
In this paper, we have considered the transport properties in the SDE model with the combination of diagonal-disorder
by employing the DMFT. It is found that the FM and CO states are stabilized at low temperature and expected behavior
of the transport coefficients are expressed respectively. Those results improve the one for SDE model so they describe
more suitable of the transport properties in the Manganites. Although, the manganites are too complicate a system
to be completely described by this simple model, the obtained transport properties qualitatively reproduce the ones
observed in the manganites.\\
{\bf Acknowledgement}\\
The authors grateful to Prof. Dr. Holger Fehske for the valuable discussions. This work is supported by the National Program of Basic Research  on National
Science, Project 4.1.2.\\

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\end{thebibliography}


\end{document}



cond-mat0310441.extracted_bib
\begin{thebibliography}{99}
\bibitem{ci1}M. B. Salamon and M. Jaime, Rev. Mod. Phys. {\bf 73}, 583
(2001).
\bibitem{ci2}E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep.
{\bf 344}, 1 (2001).
\bibitem{ci3}Y. A Izyumov and Y. N. Skryabin,  Physics- Uspekhi
 {\bf 44}, 109 (2001).
\bibitem{ci4}J. M. D. Coery, M. Viret, and S.Von
Molnar, Adv. Phys. {\bf 48}, 167
(1995).
\bibitem{ci5}F. Zhong, J. Dong, and Z. D. Wang, Phys. Rev. B {\bf 58}, 15310 (1998).
\bibitem{ci6}A. J. Millis, R. B. Littlewood, and B. I. Shraimon, Phys. Rev. Lett. {\bf 74}, 5144 (1995).
\bibitem{ci7}S. Yunoki, T. Hotta, and E. Dagotto, Phys. Rev. Lett. {\bf 84} 3714 (2000).
\bibitem{ci8}J. C. Loudon, N. D. Mathur, and P. A. Midgley, Nature {\bf 420}, 797 (2002).
\bibitem{ci9}R. Allub and B. Alascio, Solid State Commun. {\bf 99}, 66 (1996); Phys. Rev. B {\bf 55}, 14 113 (1997).
\bibitem{ci10}L. M. Falicov and J. C. Kimball, Phys. Rev. Lett. {\bf 22} 997 (1969).
\bibitem{ci11}T. Kennedy and E. H. Lieb, Physica A {\bf 138}, 320 (1986); {\em ibid.} {\bf 140}, 240 (1986).
\bibitem{ci12}U. Brandt and C. Mielsch, Z. Phys. B {\bf 63}, 45 (1986); {\em ibid}. {\bf 67}, 43 (1987),
{\em ibid}. {\bf 82}, 37 (1991).
\bibitem{ci13}Tran Minh-Tien, cond-mat/0302328.
\bibitem{ci14}A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. mod.
Phys. {\bf 68}, 13 (1996).
\bibitem{ci15}N. Furukawa, J. Phys. Soc. Ipn. {\bf 63}, 3214 (1994); {\em ibid}. {\bf 64}, 2754 (1995);
 {\em ibid}. {\bf 65}, 1174 (1996).
\bibitem{ci16}K. Nagai, T. Momoi, and K. Kubo, J. Phys. Soc. Jpn. {\bf 69}, 1837 (2000).
\bibitem{ci17}J. K. Freericks and V. Zlatic, cond-mat/0108500.
\bibitem{ci18}J. K. Freericks and V. Zlatic, cond-mat/0301188.
\bibitem{ci19}G. D. Mahan, "Many Particle Physics", 2nd edition, Plemum Press
(1990).
\bibitem{ci20}T. Pruschke, D. L. Cox, and M. Jarrell, Phys. Rev. B {\bf 47}
8553 (1993).
\bibitem{ci21}Phan Van-Nham and Tran Minh-Tien, Mod. Phys. Lett. B Vol. {\bf 17}, No. 1 (2003) 39-47.

\end{thebibliography}