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\begin{document}

\title{Effect of magnetic field on transport in granular materials.}

\classification{\texttt{73.63.-b, 73.23.Hk, 61.46.Df}}

\keywords {grains, intergrain tunnelling, Coulomb blockade, Hall
resistivity}


\author{K.B. Efetov}{
  address={Theoretische Physik III, Ruhr-Universit\"{a}t Bochum, Germany}
}

\author{M.Yu. Kharitonov}{
  address={Material Science Division, Argonne National Laboratory, Argonne, IL, 60439, USA}
}



\begin{abstract}
We review properties of electron transport in granular materials
in the presence of a weak magnetic field. In addition to
longitudinal responses we present results for the Hall
conductivity and resistivity. We demonstrate that, at sufficiently
high temperatures when Coulomb blockade effects can be neglected,
the Hall resistivity does not depend on the tunnelling amplitude
between the grains and gives information about the interior of the
grains. At lower temperatures this quantity acquires a logarithmic
in temperature contribution in all dimensions of the array of the
grains. In the limit of very low temperatures the dependence of
the Hall resistivity on temperature is similar to the one for
homogeneously disordered metals.
\end{abstract}

\maketitle



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% MAINMATTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}
Hall resistivity (HR) of metals and semiconductors gives a very
important information about their properties. According to the
classical Drude-Boltzmann theory HR
\begin{equation}
\rho _{xy}=H/(nec)  \label{eq:rxy}
\end{equation}%
does not depend on the mean free path and allows one to
experimentally determine the carrier concentration $n$. At
sufficiently low temperatures quantum effects (e.g. Coulomb
interaction and weak localization) set in (see, e.g.
\cite{AA,LR}), giving corrections to Eq.~(\ref{eq:rxy}).

Recently, much attention from both experimental and theoretical
sides has been paid to granular systems (see a review
Ref.\cite{BELVreview} and references therein). Although various
physical quantities have been calculated in different regimes
\cite{ET}, Hall transport in granular matter has not been
addressed theoretically in all these works. The absence of a
theoretical description is apparently one of the reasons, why
measurements of the Hall resistivity have not become a standard
tool for characterization of the granular metals, although they do
not seem to be very difficult.

Trying to apply the conventional theory of disordered metals to
the granular systems, the following questions can be asked:

To what extent is the formula (\ref{eq:rxy}) applicable for
granular metals? How is the carrier concentration extracted from
Eq.~(\ref{eq:rxy}) related to the actual carrier concentration
inside the grains? How can quantum effects change HR of the
granular system?

In this talk, theory of the Hall effect in granular system in the
metallic regime is presented and these questions are answered.

In the metallic regime, when the intergrain tunnelling conductance $%
G_{T}=(2e^{2}/\hbar )g_{T}$ is large, $g_{T}\gg 1$ (further we set $\hbar =1$%
), the granular system as a whole is roughly speaking a good
conductor and its properties are quite similar to those of
ordinary homogeneously disordered metals (HDMs). At the same time,
the granularity of the system brings a new physical aspect,
namely, confinement of electrons inside the grains. In a system
with \textquotedblleft well-pronounced\textquotedblright\
granularity electron traverses each grain many times before it
escapes to a neighboring grain due to the
tunnelling. This is ensured by the condition that the tunnelling escape rate $%
\Gamma $ is much smaller than the Thouless energy $E_{Th}$:
\begin{equation}
\Gamma \ll E_{Th},  \label{eq:granular1}
\end{equation}%
or, equivalently, the conductance $G_{0}=(2e^{2}/\hbar )g_{0}$ of
the grain is much larger than the tunnelling conductance $G_{T}$:
\begin{equation}
g_{0}\gg g_{T},  \label{eq:granular2}
\end{equation}%
since $\Gamma =g_{T}\delta $ and $E_{Th}\propto g_{0}\delta $ ($%
\delta $ is the mean level spacing of the grain).


\begin{figure}[tbp]
\includegraphics[width=.48\textwidth]{fig.eps}
\caption{ Granular system and classical picture of Hall
conductivity. The Ohmic current $I_y=G_T V_y$ running through the
grain in the $y$ direction causes the Hall voltage drop $V_H=R_H
I_y$ between its opposite banks in the $x$ direction. In the left
part, a diffuson giving the main contribution into the Hall
conductivity is represented} \label{fig:fig}
\end{figure}

At the same time, the conditions (\ref{eq:granular1},
\ref{eq:granular2}), leading to new physics absent in HDMs,
simplify calculations. For example, in the limit $g_{T}\ll g_{0}$
the main contribution to the classical resistivity $\rho
_{xx}^{(0)}=\left( \sigma _{xx}^{(0)}\right) ^{-1}$ comes from the
tunnel barriers between the grains rather than from scattering on
impurities inside the grains and the longitudinal conductivity
(LC) equals
\begin{equation}
\sigma _{xx}^{(0)}=G_{T}a^{2-d},  \label{eq:sigxx0}
\end{equation}%
where $a$ is the size of the grains and $d$ is the dimensionality
of the system.

Formally, the conditions (\ref{eq:granular1}, \ref{eq:granular2})
enable one to consider only the zero space harmonics for phases
and potentials inside the grains \cite{ET,BELVreview}. Therefore
when studying the longitudinal transport one may neglect electron
dynamics inside the grains, which is a significant simplification.

For the Hall transport, however, the situation appears to be more
complicated. The Hall current originates from the transversal
drift in crossed magnetic and electric fields \emph{inside} the
grains. From simple classical considerations (Fig.~\ref{fig:fig})
one obtains, that the Hall conductivity(HC) $\sigma _{xy}^{(0)}$ in
the leading in $%
g_{T}/g_{0}$ order is
\begin{equation}
\sigma _{xy}^{(0)}=G_{T}^{2}R_{H}a^{2-d}.  \label{eq:sigxy0}
\end{equation}%
where $R_{H}$ is the Hall resistance of the grain. The Hall resistance $%
R_{H} $ should be obtained from the solution of a classical
electrodynamics problem for the distribution of the electric
potential inside the grain. We come to the situation when one is
forced to take the intragrain electron
dynamics into account, no matter how well the condition Eq.~(\ref%
{eq:granular2}) is satisfied. In other words, the zero space
harmonics approximation is no longer suitable for description of
the Hall transport and one should take into account higher
harmonics.

However, purely classical approach to the problem based on the
classical electrodynamics, giving a quick answer
Eq.~(\ref{eq:sigxy0}), does not allow to include into
considerations quantum effects (such as the screening of the
Coulomb interaction and the weak localization) that come into play
at sufficiently low temperatures and can significantly affect
transport properties.

In this paper we review results obtained by a method based on
diagrammatic technique that allowed us to take the intragrain
electron dynamics into account by considering non-zero modes of
standard two-particle propagators
(\textquotedblleft diffusons\textquotedblright ) inside the grain. %
The suggested procedure accounts for the finiteness of the ratio
$g_{T}/g_{0}$ and reproduces the solution of the classical
electrodynamics problem for the conductivity of a granular medium.
The generality of our approach allows us, in principle, to study
both LC and HC of the granular system for arbitrary ratio
$g_{T}/g_{0}$ and for arbitrary type of the intragrain electron
dynamics, either ballistic or diffusive. Non-zero modes of
two-particle propagators are eventually related to the
longitudinal $G_{0}^{-1}$ and Hall $R_{H}$ resistances of the
grain. We apply our method to the problem of Hall transport for
which considering intragrain dynamics is inevitable. Neglecting
quantum effects, we do recover the classical formula
Eq.~(\ref{eq:sigxy0}). Diagrammatic approach allows us to include
quantum effects of the Coulomb interaction and weak localization
straightforwardly into the developed scheme. We study the
influence of the Coulomb interaction on HC and HR by calculating
first order corrections. A more detailed description of our work
can be found in the publications \cite{KE}.


\section{Model}

We consider a quadratic ($d=2$) or cubic ($d=3$) lattice of equal
in form and size metallic grains coupled to each other by tunnel
contacts (Fig.~\ref{fig:fig}). At the same time, they may be
different microscopically, which means that the grains may have
different impurities on the surface and inside them.

To provide more explicit results and simplify the calculations we
assume that the intragrain electron dynamics is diffusive, i.e,
the bulk mean free path $l$ in the grains is much smaller than the
size $a$ of the grain, $l\ll a$. In this case details of electron
scattering off the grain boundary are irrelevant. However, our
approach is also perfectly applicable to the case of ballistic ($%
l\sim a$) intragrain disorder, when the surface scattering becomes
important. The main results are valid for both diffusive and
ballistic grains.

In the metallic regime, $g_{T}\gg 1$, quantum effects of the
Coulomb interaction can be considered as a perturbation with the
small parameter $1/g_{T}$.

We write the Hamiltonian describing the system as
\begin{equation}
\hat{H}=\hat{H}_{0}+\hat{H}_{t}+\hat{H}_{c}.  \label{a1}
\end{equation}

In Eq. (\ref{a1}), the first term $\hat{H}_{0}$,
\begin{equation}
\hat{H}_{0}=\sum_{\mathbf{i}}\int d\mathbf{r}_{\mathbf{i}}\psi ^{\dagger }(%
\mathbf{r}_{\mathbf{i}})\left[ \xi \left( \mathbf{p}_{\mathbf{i}}-\frac{e}{c}%
\mathbf{A}(\mathbf{r}_{\mathbf{i}})\right)
+U(\mathbf{r}_{\mathbf{i}})\right] \psi (\mathbf{r}_{\mathbf{i}}),
\label{a2}
\end{equation}%
is the Hamiltonian of isolated grains, $\xi (\mathbf{p})=\mathbf{p}%
^{2}/(2m)-\epsilon _{F}$, $\mathbf{A}(\mathbf{r}_{\mathbf{i}})$ is
the vector potential describing uniform magnetic field $\mathbf{H}=H\mathbf{e}%
_{z}$ directed along the $z$ axis, $U(\mathbf{r}_{\mathbf{i}})$ is
the random disorder potential of the grains, $\bf{i}=(i_{1},\ldots
,i_{d})~\in ~\bf{Z}^{d}$ is an integer vector numerating the
grains. The integration over $\mathbf{r}_{\mathbf{i}}$ is
performed over the volume of the grain $\mathbf{i}$. Since we do
not deal with
spin-related phenomena in this paper, we omit spin indices of the operators $%
\psi (\mathbf{r}_{\mathbf{i}})$. Accounting for spin degeneracy in
calculations is simple: each electron loop comes with the factor
$2$. We consider white-noise disorder and perform averaging using
Gaussian distribution with the variance
\begin{equation}
\langle
U(\mathbf{r}_{\mathbf{i}})U(\mathbf{r}_{\mathbf{i}}^{\prime
})\rangle _{U}=\frac{1}{2\pi \nu \tau _{0}}\delta (\mathbf{r}_{\mathbf{i}}-%
\mathbf{r}_{\mathbf{i}}^{\prime }),  \label{eq:UU}
\end{equation}%
where $\nu $ is the density of states in the grain at the Fermi
level per one spin projection.

The tunnelling Hamiltonian $\hat{H}_{t}$ in Eq. (\ref{a1}) is
given by
\begin{equation}
\hat{H}_{t}=\sum_{\langle \mathbf{i},\mathbf{j}\rangle }(X_{\mathbf{i},%
\mathbf{j}}+X_{\mathbf{j},\mathbf{i}})  \label{eq:Ht}
\end{equation}%
where $X_{\mathbf{i},\mathbf{j}}$ is the operator describing
tunnelling from the grain $\mathbf{j}$ to the grain $\mathbf{i}$,
the summation is taken over the neighboring grains connected by a
tunnel contact, such that each contact is counted only once.

For studying Hall effect the geometry of the grains and contacts
is essential, therefore we write the tunnelling operators $X_{\mathbf{i}\mathbf{j%
}}$ in the coordinate representation:
\begin{equation}
X_{\mathbf{i},\mathbf{j}}=\int d\mathbf{s}_{\mathbf{i}}d\mathbf{s}_{\mathbf{j%
}}\,t(\mathbf{s}_{\mathbf{i}},\mathbf{s}_{\mathbf{j}})\psi ^{\dagger }(%
\mathbf{s}_{\mathbf{i}})\psi (\mathbf{s}_{\mathbf{j}}),
\label{eq:X}
\end{equation}%
where the integration is carried out over two surfaces of
the contact: one of them ($\mathbf{s}_{\mathbf{i}}$) is in the $%
\mathbf{i}$-th grain, whereas the other
($\mathbf{s}_{\mathbf{j}}$) in the $\mathbf{j}$-th grain. Such a
form implies that the tunnelling occurs from a close vicinity of
the contact but not from the bulk of the grain. This is a natural
assumption, because we consider the limit of a good metal in the
grains, such that the electron wave length is short. Fast
oscillations of the wave functions in the grains result in a fast
decay of the overlap of
the wave functions of different grains outside the contacts. Since $\hat{H}%
_{t}^{\dagger }=\hat{H}_{t}$, we have
$X_{\mathbf{i},\mathbf{j}}^{\dagger
}=X_{\mathbf{j},\mathbf{i}}$ and $t^{\ast }(\mathbf{s}_{\mathbf{i}},\mathbf{s%
}_{\mathbf{j}})=t(\mathbf{s}_{\mathbf{j}},\mathbf{s}_{\mathbf{i}})$.

The amplitudes $t(s_{\mathbf{i}},s_{\mathbf{j}})$ in Eq.~(%
\ref{eq:X}) describe probability of the electron tunnelling
from a point $s_{\mathbf{j}}$ to another point $\mathbf{s}_{%
\mathbf{i}}$ on the other side of the contact. It is natural to
assume that the electrons effectively tunnel from the point $\mathbf{s}_{%
\mathbf{j}}$ to the points $\mathbf{s}_{\mathbf{i}}$ in the vicinity of $%
\mathbf{s}_{\mathbf{j}}$ of atomic size only.

Therefore $t(\mathbf{s}_{\mathbf{i}},\mathbf{s}_{\mathbf{j}})$
should decay rapidly on atomic scale as a function of %distance
$\mathbf{s}_{\mathbf{i}}-\mathbf{s}_{\mathbf{j}}$. The tunnelling amplitude $%
t(s_{\mathbf{i}},s_{\mathbf{j}})$ can fluctuate
as a function of $\mathbf{s}_{\mathbf{i}}$ for fixed $\mathbf{s}_{\mathbf{i}%
}-\mathbf{s}_{\mathbf{j}}$ due to irregularities of the contact on
atomic scale.

To effectively model this behavior of the tunnelling amplitudes we
will consider $t(\mathbf{s}_{\mathbf{i}},\mathbf{s}_{\mathbf{j}})$
as Gaussian random variables and average over them with the
variance
\begin{equation}
\langle t(\mathbf{s}_{\mathbf{i}},\mathbf{s}_{\mathbf{j}})t(\mathbf{s}_{%
\mathbf{j}},\mathbf{s}_{\mathbf{i}})\rangle _{t}=t_{0}^{2}\delta (\mathbf{s}%
_{\mathbf{i}}-\mathbf{s}_{\mathbf{j}}),  \label{eq:tt}
\end{equation}%
where $\delta (\mathbf{s}_{\mathbf{i}}-\mathbf{s}_{\mathbf{j}})$
is an atomic scale $\delta $-function on the contact surface,
$t_{0}^{2}$ has the meaning of the tunnelling probability per unit
area of the contact.

 The assumption of the short
electron wave length enables us to
neglect contributions coming from the regular parts $\langle t(\mathbf{s}_{%
\mathbf{i}},\mathbf{s}_{\mathbf{j}})\rangle _{t}$ of the
tunnelling amplitudes.

The third term in Eq. (\ref{a1}) stands for the Coulomb
interaction between electrons. In principle, one has to start with
the bare Coulomb interaction between the electrons
\begin{equation}
\hat{H}_{c}=\frac{1}{2}\sum_{\mathbf{i},\mathbf{j}}\int d\mathbf{r}_{\mathbf{%
i}}d\mathbf{r}_{\mathbf{j}}\,\psi ^{\dagger
}(\mathbf{r}_{\mathbf{i}})\psi
^{\dagger }(\mathbf{r}_{\mathbf{j}})\frac{e^{2}}{|\mathbf{r}_{\mathbf{i}}-%
\mathbf{r}_{\mathbf{j}}|}\psi (\mathbf{r}_{\mathbf{j}})\psi (\mathbf{r}_{%
\mathbf{i}}).  \label{eq:Hc}
\end{equation}%
Proceeding with the calculations one has to take into account the
screening of the Coulomb interaction by electron
motion. One should distinguish between the \emph{intragrain} and \emph{%
intergrain} electron motion. In the static limit, one may model
the Coulomb interaction, Eq. (\ref{eq:Hc}), by an effective
charging energy $E_{\mathbf{i}\mathbf{j}}$. In this approximation,
one considers the interaction of the total charges of the grains.
Accounting for the tunnelling leads to a screened form of the
charging energy interaction \cite{BEAH}, which is sufficient for
studying \emph{intergrain} transport. However,
\emph{coordinate-dependent} interaction modes inside each grain
arising from the \emph{intragrain} motion are necessary to get the
correct classical expression for the ``bare'' (without
interactions) Hall resistance $R_{H}$ of a single grain.


\section{Results}

The model introduced in the previous section was studied using a
diagrammatic technique developed for the granular systems. The
details of this method can be found in Refs. \cite{KE}. In
principle, the diagrams are analogous to those used for
description of homogeneously disordered metals \cite{AA,LR},
although specifics of the granularity is definitely important.
 One should distinguish between diagrams containing expressions
 oscillating at the Fermi length in the space and those that vary
 in space smoothly. Only the latter give an essential contribution
 into the conductivity. Several examples of such diagrams are
 represented in Fig. \ref{fig:eloops}

\begin{figure}[tbp]
\includegraphics{eloops.eps}
\caption{Diagrams for the current-current correlation function.
``Prohibited'' diagrams (a) and (b) contain oscillating at Fermi length $%
\protect\lambda_F$ functions, which after the integration of the
contacts surfaces give 0. (c) The only type of ``allowed''
diagram, that does not contain oscillating functions and gives
nonvanishing contribution.} \label{fig:eloops}
\end{figure}

We perform calculations for magnetic fields $H$ such that $\omega
_{H}\tau _{0}\ll 1$, where $\omega _{H}=eH/mc$ is the cyclotron
frequency and $\tau _{0}$ is the electron scattering time inside
the grain. Since the effective mean free path $l=v_{F}\tau _{0}$
does not exceed the grain size $a$, and typically $a\approx
10-100nm$, the condition $\omega _{H}\tau _{0}\ll 1$ is well
fulfilled even for experimentally very high fields $H$. We also
assume that the granularity of the system is \textquotedblleft
well-pronounced\textquotedblright , i.e. the conditions, Eq.~(\ref%
{eq:granular1}, \ref{eq:granular2}), are satisfied.

First, we neglect quantum effects of the Coulomb interaction and obtain Eq.~(%
\ref{eq:sigxy0}) for Hall conductivity $\sigma _{xy}^{(0)}$ in the
lowest non-vanishing order in $g_{T}/g_{0}$. This result obtained
by diagrammatic methods is of completely classical origin provided
the tunnelling contact is viewed as a surface resistor with
conductance $G_{T}$. The HR of the system, following from
Eq.~(\ref{eq:sigxx0}, \ref{eq:sigxy0}),
\begin{equation}
\rho _{xy}^{(0)}=\frac{\sigma _{xy}^{(0)}}{(\sigma _{xx}^{(0)})^{2}}%
=R_{H}a^{d-2}=\frac{H}{n_{d}^{\ast }ec}  \label{eq:rhoxy0}
\end{equation}%
is given by the Hall resistance of a single grain $R_{H}$ that
depends on \emph{the geometry of the grain} but \emph{not on the
intragrain
disorder}.

Eq.~(\ref{eq:rhoxy0}) defines the effective carrier density $%
n_{d}^{\ast }$ of the granular medium. For a three-dimensional
(3D) ($d=3$, many granular layers) array $n_{3}^{\ast }=An$
differs from the electron density $n$ in the grain by a numerical
factor $A$, $0<A\leq 1$, determined by the grain geometry. For
grains of a simple geometry (e.g. having reflectional symmetry in
all three dimensions) this factor is given by the ratio of the
largest cross section area $S$ to the cross section area of the
lattice cell $a^{2}$: $A=S/a^{2}$. So, $A=1$ for cubic grains
($S=a^{2}$), and $A=\pi /4$ for spherical grains ($S=\pi a^{2}/4
$). For a two-dimensional(2D) ($d=2$, granular monolayer) array
the 3D result must be multiplied by the thickness of the layer
$a$: $n_{2}^{\ast }=aAn$.

The result, Eq.~(\ref{eq:rhoxy0}), for the Hall resistivity $\rho
_{xy}^{(0)}$ is quite \emph{universal}. It is valid even if 1) the
tunnelling conductances $G_{T}$ fluctuate from contact to contact:
HR is simply independent of the distribution of $G_{T}$. 2) the
mean free path $l$ fluctuates from contact to contact. Therefore
Eq.~(\ref{eq:rhoxy0}) is applicable to real granular arrays where
such fluctuations are always present.

Next, we calculate the first-order correction to HC $\sigma
_{xy}^{(0)},$ Eq.~(\ref{eq:sigxy0}), due to Coulomb interaction.
We find significant corrections for temperatures $T<g_{T}E_{c}$
not exceeding the inverse $RC$ time $g_{T}E_{c}$ of the system
($E_{c}=e^{2}/a$ is the charging energy of the grain), whereas for
$T>g_{T}E_{c}$ the relative corrections are of the order of
$1/g_{T}$ or smaller.

Three types of corrections to HC can be identified:

\begin{equation}
\sigma _{xy}=\sigma _{xy}^{(0)}+\delta \sigma _{xy}^{TA}+\delta
\sigma _{xy}^{EC}+\delta \sigma _{xy}^{AA}  \label{eq:sigxy}
\end{equation}%
The first one, $\delta \sigma _{xy}^{TA}$, can be attributed to
the
renormalization of the individual tunnelling conductances $G_{T}$ (%
\emph{tunnelling anomaly} (TA) \cite{AA,TA1,TA2}) in the granular
medium and has the form:
\begin{equation}
\frac{\delta \sigma _{xy}^{TA}}{\sigma _{xy}^{(0)}}=-\frac{1}{\pi g_{T}d}\ln %
\left[ \frac{g_{T}E_{c}}{\max (T,\Gamma )}\right] .
\label{eq:dsigxyTA}
\end{equation}%
This correction renormalizes the tunnelling conductances $G_{T}$ in Eq.~(\ref%
{eq:sigxy0}) but does not affect the Hall resistance of the grain
$R_{H}$.

The second correction $\delta\sigma_{xy}^{EC}$ corresponds to the
process of virtual electron diffusion through the grain:
\begin{equation}
\frac{\delta\sigma^{EC}_{xy}}{\sigma^{(0)}_{xy}} =\frac{c_d}{4 \pi g_T } \ln %
\left[\frac{\min(g_T E_c,E_{Th})}{\max(T,\Gamma)}\right].
\label{eq:dsigxyEC}
\end{equation}
where $c_d$ is a numerical factor. Contrary to $%
\delta\sigma_{xy}^{TA}$, the correction $\delta\sigma_{xy}^{EC}$
is
suppressed at temperatures greater than the Thouless energy of the grain $E_{%
Th}$. Physically, it is analogous to the \emph{elastic
cotunneling} (EC)\cite{EC} process for weakly coupled grains.

For $T\>\Gamma $ both the corrections are $\ln T$-dependent. This
dependence saturates at temperature $T\sim \Gamma $, such that
both $\delta \sigma _{xy}^{TA}$ and $\delta \sigma _{xy}^{EC}$
remain logarithmically large constants at $T<\Gamma $.

These two corrections are specific for granular systems. They
arise from spatial scales of the order of the grain size $a$ and
are absent in homogeneously disordered metals (HDMs). The
logarithmic form of the corrections is due to the screened Coulomb
interaction in granular systems obtained in \cite{BEAH}. They have
the same functional form in 2D and
3D but the coefficients are not universal, being lattice-dependent: $1/d$ and $%
c_{d}$ are the results for the cubic (3D) or quadratic (2D)
lattice, respectively, and we assume these lattice symmetries in
our calculations.

The third correction $\delta \sigma _{xy}^{AA}$ is analogous to
the one present in HDMs. It might be significant at $T\ll \Gamma $
only, when the thermal length $L_{T}=a\sqrt{\Gamma /T}\gg a$
exceeds the size of
the grain ($\Gamma a^{2}$ is the effective diffusion coefficient at scales $%
\gg a$). However, we find that this correction \emph{vanishes
identically} due to the symmetry in quasimomentum space both in 2D
and 3D:
\begin{equation}
\delta \sigma _{xy}^{AA}=0  \label{eq:dsigxyAA}
\end{equation}%

It can be instructive to compare the results for a granular system
with those for a HDM. For the quantities arising from spatial
scales exceeding the size of the grain $a$ one expects the
correspondence, because at such scales the microscopic structure
of the system becomes irrelevant.

Indeed, the result Eq.~(\ref{eq:dsigxyAA}) for $\delta \sigma
_{xy}^{AA}$ agrees with the one obtained for HDMs in Refs.
\cite{AKLL}.

The quantity directly measured in experiments is the Hall
resistivity
\begin{equation}
\rho _{xy}=\frac{\sigma _{xy}}{\sigma _{xx}^{2}}=\rho
_{xy}^{(0)}+\delta \rho _{xy},
\end{equation}
where $\rho _{xy}^{(0)}$ is the bare HR, Eq.~(\ref{eq:rhoxy0}),
and $\delta \rho _{xy}$ is the Coulomb interaction correction,
\begin{equation}
\frac{\delta \rho _{xy}}{\rho _{xy}^{(0)}}=\frac{\delta \sigma
_{xy}}{\sigma _{xy}^{(0)}}-2\frac{\delta \sigma _{xx}}{\sigma
_{xx}^{(0)}}
\end{equation}%
The interaction corrections to LC were studied in Refs.
\cite{ET,BELVreview} and the following result has been obtained:
\begin{equation}
\sigma _{xx}=\sigma _{xx}^{(0)}+\delta \sigma _{xx}^{TA}+\delta
\sigma _{xx}^{AA}.
\end{equation}%
($\delta \sigma _{xx}^{TA}$ and $\delta \sigma _{xx}^{AA}$ correspond to $%
\delta \sigma _{1}$ Eq.~(2b) and $\delta \sigma _{2}$ Eq.(2c) in Ref.~%
\cite{BELVreview}, respectively). The correction $\delta \sigma
_{xx}^{TA}$
is due to the tunnelling anomaly and renormalizes the tunnelling conductance $%
G_{T}$ in Eq.~(\ref{eq:sigxx0}). Its Hall counterpart is $\delta
\sigma _{xy}^{TA}$. The correction $\delta \sigma _{xx}^{AA}$ is
analogous to that for HDM first obtained by Altshuler and
Aronov(AA) \cite{AA}. Its Hall counterpart is $\delta \sigma
_{xy}^{AA}$. The AA correction does not diverge at large scales in
3D, and is significant in 2D at temperatures $T\ll \Gamma $
\cite{BELVreview}:
\begin{equation}
\frac{\delta \sigma _{xx}^{AA}}{\sigma _{xx}^{(0)}}=-\frac{1}{4\pi ^{2}g_{T}}%
\ln \frac{\Gamma }{T},\mbox{ }T\ll \Gamma ,d=2
\label{eq:dsigxxAA}
\end{equation}



Since the TA effects lead to the renormalization of the tunnelling
conductance $G_{T}$ only, it cannot affect the HR $\rho
_{xy}^{(0)}$, Eq.~(\ref{eq:rhoxy0}), that does not contain
$G_{T}$.

 Indeed, we have
\begin{equation}
\frac{\delta \sigma _{xy}^{TA}}{\sigma _{xy}^{(0)}}=2\frac{\delta
\sigma _{xx}^{TA}}{\sigma _{xx}^{(0)}}
\end{equation}%
and the correction to HR from TA effect equals zero. Therefore,
the total correction $\delta \rho _{xy}$ to HR is

\begin{equation}
\frac{\delta \rho _{xy}}{\rho _{xy}^{(0)}}=\frac{\delta \sigma _{xy}^{EC}}{%
\sigma _{xy}^{(0)}}-2\frac{\delta \sigma _{xx}^{AA}}{\sigma
_{xx}^{(0)}}, \label{eq:drhoxy}
\end{equation}

where $\delta \sigma _{xy}^{EC}$ is given by Eq.~(\ref{eq:dsigxyEC}), and $%
\delta \sigma _{xx}^{AA}$, Eq.~(\ref{eq:dsigxxAA}), is significant in 2D at $%
T\ll \Gamma $. In a wide temperature range $\Gamma <T<\min
(g_{T}E_{c},E_{Th})$, the essential $T$-dependent correction comes
both in 2D and 3D from EC effect, Eq.~(\ref{eq:dsigxyEC}), only:

\begin{equation}
\frac{\delta \rho _{xy}}{\rho _{xy}^{(0)}}=\frac{\delta \sigma _{xy}^{EC}}{%
\sigma _{xy}^{(0)}}=\frac{c_{d}}{4\pi g_{T}}\ln \left[ \frac{\min
(g_{T}E_{c},E_{Th})}{T}\right] .  \label{eq:drhoxy2}
\end{equation}%

The temperature behavior of $\delta \rho _{xy}$,
Eq.~(\ref{eq:drhoxy}), is shown in Fig.~\ref{fig:drhoxy}.

Another effect occurring at similar temperatures is weak
localization (WL). The WL corrections to LC were studied in
Refs.~\cite{BelUD,BVG,BCTV}. In 2D, the essential logarithmic
contribution arises from spatial scales greater than the grain
size $a$, when the inverse dephasing time is small, $1/\tau _{\phi
}<\Gamma $ (if $1/\tau _{\phi }\propto T/g_{T} $ \cite{BelUD,BVG},
this corresponds to $T<T_{WL}\equiv
g_{T}\Gamma $). However, we find \cite{KE} that the first-order in $%
1/g_{T}$ WL correction to HR \emph{vanishes identically} both in
2D and 3D in correspondence with the result for HDMs
\cite{Fukuyama, AKLL, Khodas}:
\begin{equation}
\delta \rho _{xy}^{WL}=0.
\end{equation}%
Therefore weak localization effects do not change our results,
Eqs.~(\ref{eq:rhoxy0}, \ref{eq:drhoxy}), for HR.

\begin{figure}[tbp]
\includegraphics[width=0.37\textwidth]{drhoxy.eps}
\caption{ Temperature dependence of the total correction to Hall
resistivity
$\protect\delta\protect\rho_{xy}(T)=\protect\delta\protect\rho_{xy}^{EC}(T)+%
\protect\delta\protect\rho_{xy}^{AA}(T)$,
Eq.~(\protect\ref{eq:drhoxy}).} \label{fig:drhoxy}
\end{figure}


Now we shortly summarize our findings. At temperatures $T>\min
(g_{T}E_{c},E_{Th})$ the Hall resistivity $\rho _{xy}=\rho
_{xy}^{(0)}$ is given by Eq.~(\ref{eq:rhoxy0}) and is independent
of both the intragrain and tunnel contact disorder. Measuring
$\rho _{xy}$ at such $T$ and using Eq.~(\ref{eq:rhoxy0}) one can
extract its effective carrier density $n_{d}^{\ast }$ that is an
important characteristics of the granular system.

At temperatures $\Gamma<T<\min (g_{T}E_{c},E_{Th})$, the Coulomb
interaction
leads to $\ln T$-dependent corrections to the Hall resistivity $%
\rho _{xy}$. Comparison of Eqs. (\ref{eq:rhoxy0}, \ref{eq:drhoxy})
with experimental data may serve as a good check of the theory
developed here. The temperature dependence of the total correction
to the the Hall conductivity is represented in Fig.
\ref{fig:drhoxy}
\section{Discussion}

In conclusion, we presented theory of the Hall conductivity of
granular metals. In spite of its importance this question has not
been addressed before. It turned out that considering only zero
intragrain space harmonics that was very successful in describing
the longitudinal conductivity \cite{ET,BELVreview} is not
sufficient for computation of the Hall conductivity and we
considered also higher harmonics.
 Proceeding in this way we have shown that at
high enough temperatures the Hall resistivity is given by the
classical expression, from which one can extract the effective
carrier density of the system. At lower temperatures, charging
effects give a logarithmic temperature dependent contribution to
the Hall resistivity that has the form of Eq. (\ref{eq:drhoxy2}).

 We emphasize, however, that Eq. (\ref{eq:drhoxy2}
 gives the first correction to the resistivity and the result is only valid
 when this correction is small. Therefore, the result of the
 calculation for the Hall conductivity (resistivity) is less
 accurate than the one obtained for the longitudinal conductivity
 in Ref. \cite{ET} using a renormalization group analysis. In the latter
 method, the logarithmic contribution could become very close in its value to
 the main part.
   In order to reach similar accuracy when calculating the Hall
 conductivity one should find a way to write proper
 renormalization group equations. This is not easy using the
 present diagrammatic approach and more sophisticated methods
 are needed.

The logarithmic dependence $\rho _{xx}=a+b\ln T$ of granular
metals has been observed experimentally \cite{expRxx}, and $\rho
_{xy}$ can also be measured (see e.g. \cite{expRxy,expRxy2}). The
authors of Ref.~\cite{expRxy}
reported that HR $\rho _{xy}$ of their granular samples was \emph{independent%
} of annealing temperature, although the latter did change the
grain size $a$ and LR $\rho _{xx}$ (i.e. $G_{T}$, see
Eq.~(\ref{eq:sigxx0})). This fact supports our result
Eq.~(\ref{eq:rhoxy0}) for $\rho _{xy}^{(0)}$. Our theory
may also be applied to indium tin oxide(ITO) materials (see e.g. \cite{ITO}%
).

We hope that more experiments on this subject will be done in the
nearest future and that the measurement of the Hall resistivity
will evolve into a very important method of characterization of
the granular materials.




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\end{document}
