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\begin{document}

\title{Ballistic transport in disordered graphene}

\classification{73.63.-b, 73.22.-f}

\author{A.~Schuessler}{
  address={Institut f\"ur Nanotechnologie, Forschungszentrum Karlsruhe,
 76021 Karlsruhe, Germany}
}

\author{P.~M.~Ostrovsky}{
  address={Institut f\"ur Nanotechnologie, Forschungszentrum Karlsruhe,
 76021 Karlsruhe, Germany},
  altaddress={L.~D.~Landau Institute for Theoretical Physics RAS,
 119334 Moscow, Russia}
}

\author{I.~V.~Gornyi}{
  address={Institut f\"ur Nanotechnologie, Forschungszentrum Karlsruhe,
 76021 Karlsruhe, Germany},
  altaddress={A.F.~Ioffe Physico-Technical Institute,
 194021 St.~Petersburg, Russia} 
}

\author{A.~D.~Mirlin}{
  address={Institut f\"ur Nanotechnologie, Forschungszentrum Karlsruhe,
 76021 Karlsruhe, Germany},
  altaddress={Inst. f\"ur Theorie der kondensierten Materie,
 Universit\"at Karlsruhe, 76128 Karlsruhe, Germany},
  altaddress={Petersburg Nuclear Physics Institute,
 188300 St.~Petersburg, Russia}
}


\begin{abstract}
An analytic theory of electron transport in disordered graphene in a ballistic
geometry is developed. We consider a sample of a large width $W$ and analyze
the evolution of the conductance, the shot noise, and the full statistics of the
charge transfer with increasing length $L$, both at the Dirac point and at a
finite gate voltage. The transfer matrix approach combined with the disorder
perturbation theory and the renormalization group is used. We also discuss the
crossover to the diffusive regime and construct a ``phase diagram'' of various
transport regimes in graphene.
\end{abstract}

\keywords{}

\maketitle

\section{Introduction}

Recent successes in manufacturing of atomically thin graphite samples
\cite{Novoselov04, NovoselovPNAS, Novoselov05, Zhang05} (graphene) have
stimulated intense experimental and theoretical activity \cite{Geim07, RMP07}.
The key feature of graphene is the massless Dirac type of low-energy electron
excitations. This gives rise to a number of remarkable physical properties of
this system distinguishing it from conventional two-dimensional metals. One of
the most prominent features of graphene is the ``minimal conductivity'' at the
neutrality (Dirac) point. Specifically, the conductivity \cite{Novoselov05,
Zhang05,Kim} of an undoped sample is close to $e^2/h$ per spin per valley,
remaining almost constant in a very broad temperature range --- from room
temperature down to 30mK.

A number of numerical simulations of electron transport in disordered graphene
\cite{Nomura07, Rycerz07, Bardarson07, Koshino07, San-Jose07, Lewenkopf08,
Tworzydlo08} confirmed the absence of localization in the presence of long-range
random potential. The main quantity studied numerically in most of these works
is the conductance $G$ of a finite-size graphene sample with a width $W$ much
larger than the length $L$. This setup allows one to define the ``conductivity''
$\sigma\equiv G L/W$ even for  ballistic samples with $L$ much shorter than the
mean free path $l$. Remarkably, in graphene at the Dirac point, such ballistic
``conductivity'' has a universal value $e^2/\pi h$ in the clean case
\cite{Katsnelson06, Tworzydlo06}. This setup was studied experimentally in
Refs.\ \cite{Morpurgo06,Miao07,Danneau07, Danneau08} and the ballistic value
$e^2/\pi h$ was indeed observed for large aspect ratios. This geometry of
samples is particularly advantageous for the analysis of evolution from the
ballistic to diffusive transport.

A complete description of the electron transport through a finite system
involves not only the conductance but also higher cumulants of the distribution
of transferred charge. The second moment is related to the current noise in the
system. The intensity of the shot noise is characterized by the Fano factor $F$.
For clean graphene, this quantity was studied in Ref.\ \cite{Tworzydlo06}.
Surprisingly, in a short and wide sample ($W\gg L$) the Fano factor takes
the universal value $F = 1/3$, that coincides with the well-known result for a
diffusive metallic wire \cite{BeenakkerRMP}. This is at odds with usual clean
metallic systems, where the shot noise is absent ($F=0$). The Fano factor
$F=1/3$ in clean graphene is attributed \cite{Tworzydlo06} to the fact that the
current is mediated by evanescent rather than propagating modes. Furthermore,
the whole distribution of transmission eigenvalues for the massless Dirac
equation in a clean sample at the Dirac point agrees with that of mesoscopic
metallic wires in the diffusive regime \cite{Ludwig07}.

The effect of disorder on the shot noise was studied numerically in Refs.\
\cite{San-Jose07, Lewenkopf08}, where the value of the Fano factor $F
\approx 0.3$ was found across the whole crossover form ballistics to diffusion.
The Fano factor close to $1/3$ was also observed at the Dirac point
experimentally \cite{Danneau07, Danneau08}. When the chemical potential was
shifted away from the Dirac point, the Fano factor decreased, then showed an
intermediate shoulder at $F\approx 0.15$, and finally approached zero for
largest gate voltages (carrier concentrations).

While both diffusive and clean limits have been addressed analytically, only
numerical and experimental results for the intermediate regime of ballistic
transport through disordered samples have been available so far \cite{foot-1D}.
The aim of this paper is to fill this gap. We develop the analytic theory of
electron transport in disordered graphene in the ballistic geometry ($L\ll W,l$)
and calculate the full statistics of the charge transfer for both zero (the
Dirac point) and large concentration of carriers. We also discuss the crossover
to diffusive regime and construct the overall ``phase diagram'' of transport
regimes.

A number of experimental results show that the dominant disorder in graphene
scatters electrons within the same valley. First, this disorder model is
supported by the odd-integer quantization \cite{Novoselov05, Zhang05, Geim07} of
the Hall conductivity, $\sigma_{xy} = (2n+1) 2e^2/h$, representing a direct
evidence \cite{OurQHE} in favor of smooth disorder which does not mix the
valleys. The analysis of weak localization also corroborates the dominance of
intra-valley scattering \cite{Savchenko}. Furthermore, the observation of the
linear density dependence \cite{Geim07} of graphene conductivity away from the
Dirac point implies that the relevant disorder is due to charged impurities
and/or ripples \cite{Nomura07, Ando06, Nomura06, Khveshchenko, OurPRB}, that is
a particular type of long-range disorder. Finally, apparent absence of
localization at the Dirac point down to very low temperatures \cite{Novoselov05,
Zhang05,Kim} can be explained only by some special symmetry of disorder. In
particular, a lack of valley mixing leads to emergence of a topological term in
the corresponding field theory (unitary or symplectic $\sigma$-model)
\cite{OurPRL}. The peculiar topological properties protect the system from
localization \cite{OurPRL, OurEPJ, OurQHE, Ryu07, Koshino07}. It is worth
mentioning that a topologically protected metallic state emerging in graphene
with long-range random potential also arises at a surface of a three-dimensional
$Z_2$ topological insulator \cite{Hasan08, Schnyder08}.

Motivated by the experimental observations, we will adopt the single-valley
model of graphene. More specifically, we will consider potential disorder that
scatters within a single valley and neglect intervalley scattering events.
A more general study allowing for other intravalley scattering mechanisms
(random vector potential and random Dirac mass) can be found in Ref.\
\cite{Schuessler}.


\section{Transfer-matrix technique}
\label{Sec:Tmatrix}

We start with introducing our model and the general formalism of transfer matrix
technique. For graphene, this approach was employed in Refs.\
\cite{Katsnelson06, Tworzydlo06, Titov06, CheianovFalko, Titov07,
San-Jose07}.

The single-valley massless Dirac Hamiltonian of electrons in graphene has the
form (see, e.g., Ref.\ \cite{RMP07})
\begin{equation}
 H
  = v_0 \bm{\sigma} \mathbf{p} + V(x,y).
 \label{ham}
\end{equation}
Here $\bm{\sigma} \equiv \{\sigma_x, \sigma_y\}$ is the two-dimensional vector
of Pauli matrices acting on the electron pseudospin degree of freedom
corresponding to the sublattice structure of the honeycomb lattice and the
Fermi velocity is $v_0 \approx 10^8$ cm/s. The random part $V(x,y)$ is in
general a $2 \times 2$ matrix in the sublattice space. We will consider only the
case of potential disorder, when the operator $V(x,y)$ is trivial in the
pseudospin space. For a more general case see Ref.\ \cite{Schuessler}. Below we
set $\hbar = 1$ and $v_0 = 1$ for convenience.

We will calculate transport properties of a rectangular graphene sample with
the dimensions $L \times W$. The contacts are attached to the two sides of the
width $W$ separated by the distance $L$. We fix the $x$ axis in the direction of
current, Fig.\ \ref{Fig:sample}, with the contacts placed at $x=0$ and $x=L$.
We assume $W \gg L$, which allows us to neglect the boundary effects related to
the edges of the sample that are parallel to the $x$ axis (at $y=\pm W/2$).

\begin{figure}
 \centerline{\includegraphics[width=0.95\columnwidth]{sample.eps}}
 \caption{Schematic setup for two-terminal transport measurements. Graphene
sample of dimensions $L \times W$ is placed between two parallel contacts. We
assume $W\gg L$ throughout the paper.}
 \label{Fig:sample}
\end{figure}

Following Ref.\ \cite{Tworzydlo06}, metallic contacts are modelled as
highly doped graphene regions described by the same Hamiltonian (\ref{ham}). In
other words, we assume that the chemical potential $E_F$ in the contacts is
shifted far from the Dirac point. In particular, $E_F \gg \epsilon$, where
$\epsilon$ is the chemical potential inside the graphene sample counted from the
Dirac point. (All our results are independent of the sign of energy, thus we
assume $\epsilon > 0$ throughout the paper.) A large number of propagating modes
exists in the leads, all belonging to the circular Fermi surface of radius $p_F
= E_F/v_0$. These modes are labelled by the momentum $p_n = 2\pi n/W$ in $y$
direction with $|n| < N = W p_F/2\pi$. Particular boundary conditions at $y =
\pm W/2$ shift the quantized values of $p_n$ by a constant of order $1/W$.
However, this constant has no significance in the limit $W \gg L$ when many
channels participate in electron transport.

We will use the  mixed momentum-coordinate representation, with the wave
function $\Psi_n(x)$ bearing a vector index $n$ in the space of transverse
momenta supplemented by a 2-spinor structure in pseudospin (sublattice) space.
The eigenstates of the clean Hamiltonian $H_0=v_0\bm{\sigma} \mathbf{p}$ have
the direction of pseudospin  parallel to the electron momentum. It is convenient
to perform the unitary rotation \cite{Titov07} in the pseudospin space $\psi =
\mathcal{L} \Psi$ with $\mathcal{L} = (\sigma_x + \sigma_z)/\sqrt{2}$ which
transforms $\sigma_x$ to the diagonal form: $\mathcal{L} \sigma_x
\mathcal{L}^\dagger = \sigma_z$. Hence the two components of the rotated spinor
correspond to right- and left-propagating waves, $\psi = \{ \psi^R, \psi^L \}$.
In terms of the new function $\psi_n(x)$, the Schr\"odinger equation $H \Psi =
\epsilon \Psi$ acquires the form \cite{San-Jose07, Titov07}
\begin{equation}
 \frac{\partial \psi_n}{\partial x}
  = (\sigma_x p_n +i \sigma_z \epsilon) \psi_n
    -i\sigma_z \sum_m U_{nm}(x) \psi_m.
 \label{psi_evolution}
\end{equation}
The matrix $U_{nm}(x)$ represents the operator $V(x,y)$ in the mixed
momentum-coordinate representation
\begin{equation}
 U_{nm}(x)
  = \int \frac{dy}{W}\, e^{-i(p_n - p_m) y} V(x,y).
\end{equation}

A standard description of electron propagation involves the transfer matrix
$\mathcal{T}$ which relates left- and right-moving waves at the point $x = L$
to those at $x = 0$:
\begin{equation}
 \begin{pmatrix}
  \psi^R(L) \\
  \psi^L(L)
 \end{pmatrix}
  = \mathcal{T} \begin{pmatrix}
      \psi^R(0) \\
      \psi^L(0)
    \end{pmatrix},
 \quad
 \mathcal{T}
  = \begin{pmatrix}
     {t^\dagger}^{-1} & r t^{-1} \\
     -t^{-1} r & t^{-1}
    \end{pmatrix}.
 \label{Tdefinition}
\end{equation}
The elements $t$ and $r$ are matrices in channel space formed by transmission
and reflection amplitudes respectively. This description is convenient due to
the simple multiplicativity property: $\mathcal{T}(x_3, x_2) \mathcal{T}(x_2,
x_1) = \mathcal{T}(x_3, x_1)$.

By definition, the transfer matrix $\mathcal{T}(x_2, x_1)$ yields a solution
to the Schr\"odinger equation (\ref{psi_evolution}) in the form $\psi(x_2) =
\mathcal{T}(x_2, x_1) \psi(x_1)$. Transfer matrix itself, as a function of
its first argument, obeys the same Schr\"odinger equation with the initial
condition $\mathcal{T}(x,x) = \mathbb{1}$. In a clean sample the solution
depends only on the difference $x_2 - x_1$ and is diagonal in channel space:
\begin{equation}
 \mathcal{T}^{(0)}_{nm}(x_2,x_1)
  = \delta_{nm}
    \exp\bigl[ (\sigma_x p_n + i \sigma_z \epsilon) (x_2-x_1) \bigr].
 \label{T0}
\end{equation}

In order to include disorder as a perturbation, it is convenient to cast the
Schr\"odinger equation (\ref{psi_evolution}) into an integral form. In terms of
transfer matrix the integral equation reads
\begin{multline}
 \mathcal{T}(x_2, x_1)
  = \mathcal{T}^{(0)}(x_2, x_1) \\
    -i \int_{x_1}^{x_2} dx\,
      \mathcal{T}^{(0)}(x_2, x) \sigma_z U(x) \mathcal{T}(x, x_1).
 \label{evol}
\end{multline}

The transport statistics of the sample is expressed in terms of transmission
eigenvalues $T_n$ --- the eigenvalues of the matrix $t^\dagger t$. One can
extract these transmission eigenvalues from the upper left element of the
transfer matrix (\ref{Tdefinition}). The first two moments of the transferred
charge distribution determine the conductance (by Landauer formula) and the Fano
factor \cite{BeenakkerRMP}
\begin{equation}
 G
  = \frac{4e^2}{h} \mathop{\mathrm{Tr}}(t^\dagger t),
 \qquad
 F
  = 1 - \frac{\mathop{\mathrm{Tr}}(t^\dagger
t)^2}{\mathop{\mathrm{Tr}}(t^\dagger t)}.
 \label{GFn}
\end{equation}
The factor $4$ in the expression for the conductance accounts for the spin and
valley degeneracy.


\section{Clean graphene}
\label{Sec:clean}

We will first analyze transport properties of a clean graphene strip. In the
``short and wide'' geometry ($W\gg L$) we are considering, the total number of
channels participating in charge transfer is large. This allows us to replace
summation over channels by integration. From now on, we will identify
channels by the dimensionless momentum $p=p_n L$ in $y$ direction.

The transfer matrix $\mathcal{T}^{(0)}$, and hence its upper-left block
${t^\dagger}^{-1}$, are diagonal in channels. Using the explicit form of the
clean graphene transfer matrix, Eq.\ (\ref{T0}), one calculates the transmission
eigenvalues \cite{Titov07}
\begin{equation}
 T_p
  = (t^\dagger t)_{pp}
  = \left[
      1 + \frac{p^2 \sinh^2 \sqrt{p^2 - (\epsilon L)^2}}{p^2 - (\epsilon L)^2}
    \right]^{-1}.
 \label{Tclean}
\end{equation}

For the conductance and Fano factor we obtain from Eq.\ (\ref{GFn})
\begin{equation}
 G
  = \frac{2e^2 W}{\pi h L} \int dp\; T_p,
 \qquad
 F
  = 1 - \frac{\int dp\; T_p^2}{\int dp\; T_p}.
 \label{GF}
\end{equation}
The result of numerical integration of Eq.\ (\ref{GF}) is shown in Fig.\
\ref{Fig:GF}. A detailed analytical analysis of the two limiting cases of small
and large energies is presented below.

\begin{figure}
 \centerline{\includegraphics[width=0.95\columnwidth]{gf.eps}}
 \caption{Energy dependence of the (a) conductance and the (b) Fano factor of
the clean sample with $W \gg L$. Solid lines show numerical results. Low energy
asymptotics Eq.\ (\protect\ref{GFsmallE}) is plotted by dashed lines while
dotted lines correspond to high energy limit Eqs.\ (\protect\ref{Gosc}) and
(\protect\ref{Fosc}). Asymptotical curves provide a very good approximation to
the exact result in the whole range of energies.}
 \label{Fig:GF}
\end{figure}


\subsection{Low energies: $\epsilon L \ll 1$}
\label{Sec:CleanLowEnergies}

In the limit of low energies, $\epsilon L \ll 1$, we describe transport
properties with the help of a distribution function $P(T)$ of transmission
eigenvalues (\ref{Tclean}). This distribution function provides a measure in the
space of channels,
\begin{equation}
 P(T)\, dT
  = \frac{W\, d|p|}{\pi L}.
 \label{PTdef}
\end{equation}
According to (\ref{Tclean}), there is one-to-one correspondence between the
transmission eigenvalue $0\le T \le 1$ and the absolute value $|p|$ of the
momentum; an extra factor of $2$ in the right-hand side of
Eq.\ (\ref{PTdef}) accounts for the double degeneracy between channels with
momenta $p$ and $-p$. 

In the low energy limit, we calculate the distribution function $P(T)$ in the
form of a power series in the small parameter $\epsilon L$. In order to perform
this calculation, we invert the function $T_p$ given by Eq.\ (\ref{Tclean})
keeping  terms of the second order in $\epsilon L$ and then substitute the
result into Eq.\ (\ref{PTdef}).
\begin{multline}
 P(T)
  = \frac{W}{2\pi L} \frac{1}{T \sqrt{1-T}} \\
    \times \left[
      1 + (\epsilon L)^2 \left(
        \frac{\sqrt{1 - T}}{\mathop{\mathrm{arccosh}}^3\frac{1}{\sqrt{T}}}
        - \frac{1 + T}{2 \mathop{\mathrm{arccosh}}^2\frac{1}{\sqrt{T}}}
      \right)
    \right].
 \label{PTe}
\end{multline}

At zero energy, the function $P(T)$ reproduces the well-known Dorokhov result
\cite{Dorokhov83} for a diffusive wire. This is, in particular, the reason for
the $1/3$ Fano factor in graphene \cite{Tworzydlo06}. The fact that the clean
graphene sample is characterized by exactly the same form of the transmission
distribution as a generic diffusive wire is highly nontrivial. We will show
below (Secs.\ \ref{Sec:first-correction} and \ref{Sec:RG}) that this remarkable
correspondence remains valid in the ballistic regime when leading disorder
effects are incorporated.

Using the distribution (\ref{PTe}), we obtain the following results for the
conductance and the Fano factor of clean graphene at low energies, $\epsilon L
\ll 1$,
\begin{align}
 G
  &=\frac{4 e^2}{\pi h}\; \frac{W}{L} \left[
     1 + c_1 (\epsilon L)^2
   \right],
 \quad
 F
  =\frac{1}{3} \left[
     1 + c_2 (\epsilon L)^2
   \right],
\label{GFsmallE}\\
 c_1
  &=\frac{35 \zeta(3)}{3 \pi^2} - \frac{124 \zeta(5)}{\pi^4}
  \approx 0.101,
 \label{c1}\\
 c_2
  &=-\frac{28 \zeta(3)}{15 \pi^2} - \frac{434 \zeta(5)}{\pi^4}
    +\frac{4572 \zeta(7)}{\pi^6}
  \approx -0.052.
 \label{c2}
\end{align}
At the Dirac point ($\epsilon=0$), Eq.\ (\ref{GFsmallE}) reproduces the earlier
analytical results of Refs.\ \cite{Katsnelson06, Tworzydlo06}. Low energy
asymptotics is shown with dashed lines in Fig.\ \ref{Fig:GF}.


\subsection{High energies: $\epsilon L \gg 1$}
\label{Sec:clean-high-energy}

When the Fermi energy $\epsilon$ in the sample is far from the Dirac point,
many conducting ($T\sim 1$) channels are opened. In this regime, the
conductivity and higher moments of the transmission distribution are essentially
linear in $\epsilon$ with small oscillating corrections (see Fig.\
\ref{Fig:GF}). These oscillations are due to interference effects: conductance
is relatively enhanced and the noise is suppressed when a channel exhibits
resonant transmission with $T$ close to $1$. This phenomenon is similar to the
Fabry-Perot resonances.

At high energies, it is convenient to deal with the generating function of
transmission moments defined as
\begin{equation}
 \mathcal{F}(z)
  = \sum_{n=1}^\infty z^{n-1} \mathop{\mathrm{Tr}} (t^\dagger t)^n
  = \mathop{\mathrm{Tr}} \left[
      t^{-1} {t^\dagger}^{-1} - z
    \right]^{-1}.
 \label{Fdef}
\end{equation}
In order to calculate $\mathcal{F}(z)$, we substitute Eq.\ (\ref{Tclean}) into
Eq.\ (\ref{Fdef}). The integrand oscillates rapidly in the interval $-\epsilon L
< p < \epsilon L$. This interval of momenta contains all open (well-conducting)
channels and thus provides the main contribution to the generating function.
We introduce a new variable $u$, such that $p = \epsilon L\sqrt{1 - u^2}$. In
terms of this new variable the integral acquires the form
\begin{equation}
 \mathcal{F}(z)
  = \frac{W \epsilon}{\pi} \int_0^1 \frac{u\, du}{\sqrt{1-u^2}} \left[
      \cos^2(u \epsilon L) + \frac{\sin^2(u \epsilon L)}{u^2} - z
    \right]^{-1}.
\end{equation}

Trigonometric functions in the integrand rapidly oscillate. To take advantage
of this property, we represent the integrand as a sum over Fourier harmonics
$\cos(n u\epsilon L)$. The first and the second terms of such Fourier
expansion are
\begin{multline}
 \mathcal{F}(z)
  = \frac{W \epsilon}{\pi \sqrt{1-z}}
    \int_0^1 \frac{u^2\, du}{\sqrt{(1 - u^2)(1- z u^2)}} \Biggl[
      1 \\
      +\frac{2 \bigl(u \sqrt{1 - z} - \sqrt{1-z u^2} \bigr)^2}{1 - u^2}
        \cos(2 u \epsilon L)
    \Biggr].
 \label{Fzosc-integral}
\end{multline}

The first term of the above expression provides the main (linear in $\epsilon$)
contribution to the generating function. It is expressed in terms of complete
elliptic integrals. The second term is suppressed due to oscillations of the
integrand. We estimate its contribution using the saddle-point method. The
result of this calculation reads
\begin{equation}
 \mathcal{F}(z)
   = W \epsilon \left[
       \frac{K(z) - E(z)}{\pi z \sqrt{1-z}}
       +\frac{\sin(2 \epsilon L - \pi/4)}
       {8 \sqrt{\pi} (1 - z)^2 (\epsilon L)^{3/2}}
     \right].
 \label{Fzosc}
\end{equation}
The conductance and the Fano factor are then calculated by expanding
$\mathcal{F}(z)$ in small $z$, see Eq.\ (\ref{Fdef}).
\begin{align}
 G
  &= \frac{e^2}{h}\; W \epsilon \left[
      1 + \frac{\sin (2\epsilon L - \pi/4)}{2\sqrt{\pi} (\epsilon L)^{3/2}}
    \right], \label{Gosc} \\
 F
  &= \frac{1}{8} \left[
      1 - \frac{9 \sin (2\epsilon L - \pi/4)}{2\sqrt{\pi} (\epsilon L)^{3/2}}
    \right]. \label{Fosc}
\end{align}
These results are in a good agreement with the high-energy behavior of $G$ and
$F$ calculated numerically, see Fig.\ \ref{Fig:GF}.

Let us emphasize that transport properties of the system at high energies
depend on the particular model of the contacts \cite{Schomerus07, Blanter07}.
In our calculation we assume that the boundaries between graphene and the leads
are sharp. This model is well justified if the actual extension $d$ of the
transitional region at the interface is small compared to the electron
wavelength in graphene. This condition is violated at very high energy,
$\epsilon \gg 1/d$, when the boundary becomes adiabatically smooth. This, in
particular, leads to the vanishing Fano factor because the semiclassically
propagating electrons are either transmitted or reflected without any
uncertainty.

Our results for the energy dependence of the conductance and the Fano factor in
clean graphene are in agreement with the findings of Refs.\
\cite{Tworzydlo06, Titov07}, where the sum over transmission channels was
evaluated numerically for a finite (but sufficiently large) ratio $W/L$.
Experimentally, such a ballistic setup was studied in Refs.\
\cite{Danneau07, Danneau08}. Most of the experimental observations
reasonably agree with our results. The ``conductivity'' $GL/W$ (which is equal
to $4e^2/\pi h$ at the neutrality point, as expected for a ballistic sample)
increases roughly linearly with energy $\epsilon$. The Fano factor has a value
close to $1/3$ at the Dirac point and decreases when one moves away from the
Dirac point, showing a tendency to saturate at $F\approx 0.15$, which is not
far from the value 1/8 we have obtained in the high-energy regime. Measurements
on other samples reveal that very far from the Dirac point the Fano factor
decreases again, reaching a value as low as 0.02. Apparently, the intermediate
plateau corresponds to the high-energy regime $L^{-1}\ll \epsilon \ll d^{-1}$
investigated in our work, while the vanishing of the Fano factor at still
higher electron concentrations corresponds to the ultra-high-energy range,
$\epsilon \gg d^{-1}$.


\section{Including disorder: Perturbative treatment}
\label{Sec:first-correction}

So far, we have considered the transport properties of a clean graphene sample.
In the present section we include disorder on the level of the leading
perturbative correction. We assume the Gaussian statistics for the random
potential $V(x,y)$,
\begin{equation}
 \begin{gathered}
 \langle V(x, y) \rangle
  = 0, \\
 \langle V(x, y) V(x', y') \rangle
   = 2\pi \alpha \delta(x - x') \delta(y - y').
 \end{gathered}
 \label{Gauss}
\end{equation}
The dimensionless disorder strength $\alpha$ is similar to the parameter
$\alpha_0$ used in Ref.\ \cite{OurPRL}.

The Gaussian white-noise disorder (\ref{Gauss}) is realized when the scattering
is due to impurities in the substrate separated by a thick (compared to the
lattice constant) clean spacer layer from the graphene plane. The intervalley
matrix elements of the disorder potential are then exponentially suppressed and
can be safely neglected. A more realistic case of long-range charged impurities
with $1/r$ potentials can also be treated perturbatively within the Gaussian
model, but with an energy-dependent scattering amplitude \cite{OurPRB}.

In the present section we will calculate the first disorder correction to the
transport properties of a graphene sample. Specifically, we will find a
linear-in-$\alpha$ contribution to the distribution function $P(T)$ at low
energies and to the generating function $\mathcal{F}(z)$ at high energies.


\subsection{Low energies: $\epsilon L \ll 1$}

We have already calculated the lowest order correction to the distribution
function $P(T)$ due to small energy [see Eq.\ (\ref{PTe})]. Now we are going to
find the lowest disorder correction at exactly zero energy. To the main order,
these two contributions merely add up.

At zero energy, there are no propagating modes in graphene, all the channels are
evanescent. The generating function $\mathcal{F}(z)$ defined by Eq.\
(\ref{Fdef}) at $\epsilon = 0$ simplifies to
\begin{equation}
 \mathcal{F}_0(z)
  = \frac{W}{2\pi L} \int \frac{dp}{\cosh^2 p - z}
  = \frac{W}{\pi L}\, \frac{\arcsin\sqrt{z}}{\sqrt{z - z^2}}.
\label{F0z}
\end{equation}

In order to find the lowest disorder correction to the generating function, we
solve the evolution equation (\ref{evol}) perturbatively up to second order in
random potential. Then we single out the transmission matrix $t$ from
$\mathcal{T}$ and substitute it into Eq.\ (\ref{Fdef}). After averaging over
Gaussian distribution (\ref{Gauss}) the correction to the generating function
acquires the form
\begin{equation}
 \mathcal{F}_\text{dis}(z)
  = \frac{2 \alpha W}{\pi L}\, \frac{\arcsin\sqrt{z}}{\sqrt{z - z^2}}.
\end{equation}

Remarkably, the disorder correction retains the functional form of
$\mathcal{F}(z)$ at zero energy changing only the overall prefactor. We will
discuss the consequences of this fact in Sec.\
\ref{Sec:single-parameter-scaling}. This implies that the transmission
distribution $P(T)$ also preserves its Dorokhov form. Together with the energy
correction from Eq.\ (\ref{PTe}), we find the following result
\begin{multline}
 P(T)
  = \frac{W}{2\pi L} \frac{1}{T \sqrt{1-T}} \Biggl[
      1 + 2 \alpha \\
      +(\epsilon L)^2 \left(
        \frac{\sqrt{1 - T}}{\mathop{\mathrm{arccosh}}^3\frac{1}{\sqrt{T}}}
        -\frac{1 + T}{2 \mathop{\mathrm{arccosh}}^2\frac{1}{\sqrt{T}}}
      \right)
    \Biggr].
 \label{PTUB}
\end{multline}


\subsection{High energies: $\epsilon L \gg 1$}
\label{high-ballistic}

The transport properties of a clean graphene sample at high energies were
considered in Sec.\ \ref{Sec:clean-high-energy}. The main contribution to the
conductance and to higher moments is proportional to $\epsilon L$ and comes from
the band of fully opened channels with $|p_n| < \epsilon$. Carrying out the same
perturbative calculation as for the zero-energy case, we get a complicated
double integral of a rapidly oscillating function. After averaging over
oscillations, the integral simplifies and yields the disorder correction to the
generating function
\begin{equation}
 \mathcal{F}_\text{dis}(z)
  = -\alpha W L \epsilon^2
    \frac{[(1 - z)K(z) - E(z)]^2}{\pi z^2 (1 - z)}.
 \label{F0Fdis}
\end{equation}

Expanding at $z = 0$, we readily calculate disorder corrections to the
conductance and Fano factor. Combining these corrections with the results for
clean sample, Eqs.\ (\ref{Gosc}) and (\ref{Fosc}), we obtain
\begin{align}
 G
  &= \frac{e^2}{h}\; W \epsilon \biggl[
      1 + \frac{\sin (2\epsilon L - \pi/4)}{2\sqrt{\pi} (\epsilon L)^{3/2}}
      -\frac{\pi}{4}\, \alpha \epsilon L
    \biggr], \label{Gelarge} \\
 F
  &= \frac{1}{8} \biggl[
      1 - \frac{9\sin (2\epsilon L - \pi/4)}{2\sqrt{\pi} (\epsilon L)^{3/2}}
      +\frac{3\pi}{4}\, \alpha \epsilon L
    \biggr]. \label{Felarge}
\end{align}
We see that at high energies any disorder suppresses conductance and enhances
noise at the level of the lowest perturbative correction.

At sufficiently high energies, $\alpha \epsilon L \gtrsim 1$, disorder
correction becomes comparable to the clean result. This implies a crossover to
the diffusive regime, where the perturbative approach developed in the present
section fails, see Sec.\ \ref{Sec:RG}.


\section{Renormalization group and overall phase diagram}
\label{Sec:RG}

In the previous section we have calculated the lowest disorder correction to
transport properties of a ballistic graphene sample. In the present section we
will discuss the resummation of higher-order contributions.

The second-order and all higher terms contain logarithmic divergences and thus
become important when system is still in the ballistic regime, $L\ll l$.
These logarithms are intrinsic for two-dimensional Dirac fermions subjected to
disorder and were extensively studied in various contexts using renormalization
group technique. \cite{Dotsenko, Ludwig, NersesyanTsvelik, Bocquet,
AltlandSimonsZirnbauer, Guruswamy} Application of such a renormalization group
(RG) to disordered graphene was developed in Refs.\ \cite{Aleiner06,
OurPRB}.

We will use two-loop perturbative RG for the disorder strength $\alpha$ and
one-loop for the energy $\epsilon$. Higher terms in the RG equations are
non-universal and depend on the particular regularization scheme. A
comprehensive derivation of the two-loop RG equations can be found in, e.g.,
Ref.\ \cite{Bondi}.
\begin{equation}
 \frac{\partial\alpha}{\partial \ln \Lambda}
  = 2 \alpha^2 + 2 \alpha^3, \qquad
 \frac{\partial\epsilon}{\partial \ln \Lambda}
  = \alpha \epsilon.
 \label{RG}
\end{equation}
Here $\Lambda$ is the running scale parameter in the real space, it has the
dimension of length. Bare values of energy and disorder coupling, which are the
initial conditions for RG equations, correspond to the scale of the order of
lattice spacing $a$. After renormalization procedure, we obtain renormalized
values of the parameters at the scale $\Lambda$ and also a new effective
bandwidth $1/\Lambda$.

The renormalization proceeds until one of the following events happens: (i) the
running scale $\Lambda$ reaches the system size $L$, (ii) the disorder coupling
becomes of the order unity, or (iii) the renormalized energy reaches the
bandwidth. We will discuss these three possibilities below. Once the
renormalization has been performed, we can calculate observables by simply
applying the perturbation theory. The results of previous section for transport
characteristics thus remain applicable with bare parameters replaced by their
renormalized values.

Let us first consider the zero energy limit when the only parameter of the model
is the disorder strength $\alpha$. Disorder is enhanced in course of
renormalization. The RG should be stopped when the renormalized value of
$\alpha$ becomes of order of unity, so that the perturbative expansion of the
beta function fails. The corresponding scale is the zero-energy mean free path
\begin{equation}
 l_0
  = a \sqrt{\alpha} e^{1/2 \alpha}.
 \label{l0}
\end{equation}
At scales shorter than the mean free path $l_0$, the renormalized value of
$\alpha$ is given by
\begin{equation}
 \alpha(\Lambda)
  = \frac{1}{2 \ln(l_0/\Lambda) + \ln \ln (l_0/\Lambda)}.
 \label{alpha0renorm}
\end{equation}
As long as $L\ll l_0$, and thus $\alpha(L)\ll 1$, we can describe the transport
properties by the distribution function (\ref{PTUB}) with the renormalized value
$\alpha(L)$ and $\epsilon = 0$.

A small but non-zero energy does not change the qualitative behavior of the
system, as long as the RG flow is terminated by the system size. We refer to
this situation as ``ultraballistic regime''. The energy gets renormalized
according to Eq.\ (\ref{RG}),
\begin{equation}
 \epsilon(\Lambda)
  = \frac{\epsilon}
    {\sqrt{\alpha[2 \ln(l_0/\Lambda) + \ln \ln (l_0/\Lambda)]}}.
 \label{erenorm}
\end{equation}
The value $\epsilon(L)$ is to be substituted into Eq.\ (\ref{PTUB}) along with
the renormalized value of $\alpha(L)$. This yields the full description of
transport properties for the system in the ultraballistic regime. In particular,
the conductance and the Fano factor are
\begin{align}
 G
  &=\frac{4 e^2}{\pi h}\; \frac{W}{L} \left[
      1 + \frac{2\alpha
      +c_1 (\epsilon L)^2}{\alpha[2 \ln(l_0/L) + \ln \ln (l_0/L)]}
    \right],
 \label{GUB} \\
 F
  &=\frac{1}{3} \left[
      1 + \frac{c_2 (\epsilon L)^2}{\alpha[2 \ln(l_0/L) + \ln \ln (l_0/L)]}
    \right],
 \label{FUB}
\end{align}
with the constants $c_{1,2}$ given by Eqs.\ (\ref{c1}) and (\ref{c2}).

When the initial (bare) value of energy is increased, the renormalized energy
eventually becomes comparable to the effective bandwidth $1/\Lambda$ before the
running scale $\Lambda$ reaches $L$ (and still before the disorder coupling
$\alpha(\Lambda)$ reaches unity). The length scale at which $\epsilon(\Lambda) =
1/\Lambda$ plays the role of the effective Fermi wavelength $\lambda$. (Indeed,
in the absence of disorder, energy is not renormalized and $\lambda =
1/\epsilon$.) Using Eq.\ (\ref{erenorm}), we find
\begin{equation}
 \lambda
  = \frac{1}{\epsilon} \sqrt{2 \alpha \ln(\epsilon/\gamma)},
 \label{lambda}
\end{equation}
where $\gamma$ is the characteristic disorder-induced energy scale
\begin{equation}
 \gamma
  = \sqrt{\alpha}/l_0
  = \Delta e^{-1/2\alpha},
 \label{gamma}
\end{equation}
$\Delta = 1/a$ is the initial bandwidth of the model, and we assumed that
$\epsilon \gg \gamma$. According to Eqs.\ (\ref{alpha0renorm}) and
(\ref{lambda}), the renormalized value of the coupling constant at the scale of
the Fermi wave length is
\begin{equation}
 \alpha(\lambda)
  = \frac{1}{2\ln(\epsilon/\gamma)}.
\end{equation}

\begin{figure}
 \centerline{\includegraphics[width=0.95\columnwidth]{phase.eps}}
 \caption{Schematic ``phase diagram'' of various transport regimes in the
graphene sample with random potential. The lines indicate crossovers between
corresponding regimes. The shortest sample exhibits ultraballistic
transport with the conductance and Fano factor given by Eqs.\
(\protect\ref{GUB}) and (\protect\ref{FUB}) respectively. When the length of the
sample exceeds Fermi wave length (\protect\ref{lambda}), ballistic results
(\protect\ref{GB}) and (\protect\ref{FB}) apply. In a sample longer than the
mean free path (\protect\ref{MFP}), diffusive regime establishes with the Drude
conductivity (\protect\ref{Drude}) and the Dorokhov distribution of transmission
eigenvalues (\protect\ref{Dorokhov}). The conductivity experiences symplectic
antilocalization in this case.}
 \label{Fig:phase}
\end{figure}

In Fig.\ \ref{Fig:phase} we show the phase diagram of various transport regimes.
If $\epsilon \lesssim \gamma$ and $L\ll l_0 $ or, alternatively, $\epsilon
\gtrsim \gamma$ and $L\ll \lambda$, the renormalization terminates by the system
size, $\Lambda=L$, and the system is in the ultraballistic regime discussed
above [see Eqs. (\ref{GUB}) and (\ref{FUB})]. If $\epsilon \gg \gamma$ and
$\lambda \ll L \ll l$, the renormalization stops at $\Lambda = \lambda$ and the
running scale does not reach $L$. We refer to this case as ``ballistic regime'',
since the system size is still smaller than the mean free path $l$,
\begin{equation}
 l
  = \frac{\lambda}{\pi \alpha(\lambda)}
  = \frac{\sqrt{\alpha}}{\pi \epsilon} [2 \ln (\epsilon/\gamma)]^{3/2}.
 \label{MFP}
\end{equation}
This value \cite{OurPRB} of the mean free path is a result of the standard Born
approximation with renormalized coupling constant $\alpha(\lambda)$. Note that
for the model with random scalar potential, the transport mean free path, which
determines diffusion coefficient, is twice longer, $l_{\mathrm{tr}} = 2 l$.

In the ballistic regime, the renormalized energy is such that $\epsilon(\lambda)
L = L/\lambda \gg 1$. This means that we have to use the high-energy results of
Sec.\ \ref{high-ballistic}. In particular, with the renormalized parameters, the
conductance and the Fano factor, Eqs.\ (\ref{Gelarge}) and (\ref{Felarge}),
become
\begin{align}
 G
  &= \frac{e^2}{h}\; \frac{W}{\lambda} \left[
      1 + \frac{\sin (2 L/\lambda - \pi/4)}{2\sqrt{\pi} (L/\lambda)^{3/2}}
      -\frac{L}{4 l}
    \right],
 \label{GB}\\
 F
  &= \frac{1}{8} \left[
      1 - \frac{9\sin (2 L/\lambda - \pi/4)}{2\sqrt{\pi} (L/\lambda)^{3/2}}
      +\frac{3 L}{4 l}
    \right].
 \label{FB}
\end{align}
In the expressions (\ref{GB}) and (\ref{FB}) there are two corrections to the
leading term. The first (oscillating) correction exists in the clean limit and
is small provided $L \gg \lambda$. The second correction due to disorder is
small only if $L \ll l$. This imposes the natural upper bound on the ballistic
regime: if the system size exceeds the mean free path, electron transport
becomes diffusive. In this case, the system is naturally characterized by
the conductivity $\sigma$, which determines the conductance via the Ohm's law,
$G=\sigma W/L$. The Drude expression for the conductivity reads \cite{Aleiner06}
\begin{equation}
 \sigma
  = \frac{4 e^2}{\pi h \alpha(\lambda)}
  = \frac{8 e^2}{\pi h }\, \ln (\epsilon/\gamma).
 \label{Drude}
\end{equation}
The distribution function of transmission eigenvalues in the diffusive regime is
the same as in a usual quasi-one-dimensional metallic sample \cite{Dorokhov83}
\begin{equation}
 P(T)
  = \frac{W}{2\pi L}\, \frac{g}{T \sqrt{1 - T}},
 \label{Dorokhov}
\end{equation}
with the dimensionless conductivity $g = (\pi h / 4 e^2) \sigma$. Taking into
account interference effects leads to $L$ dependence of $g$ in this formula, as
we are going to discuss.


\section{Single parameter scaling at zero energy}
\label{Sec:single-parameter-scaling}

Remarkably, the transmission distribution function at zero energy appears to be
the same in
ultraballistic and diffusive limits. In both cases it has the form of Dorokhov
distribution (\ref{Dorokhov}) with the parameter
\begin{equation}
 g
  = \begin{cases}
      1 + 2 \alpha(L), & \text{ultraballistic}, \\
      \dfrac{\pi h}{4 e^2}\; \sigma(L), & \text{diffusive},
    \end{cases}
\end{equation}
which has the meaning of the dimensionless conductivity. In the ultraballistic
regime, the scaling of $g$ is induced via renormalization of $\alpha$ according
to equation (\ref{RG}) while in the diffusive limit, $g\gg 1$ acquires
antilocalization corrections characteristic for a disordered system of
symplectic symmetry. This allows us to infer a unified scaling law covering both
limiting cases
\begin{equation}
 \frac{\partial \ln g}{\partial \ln L}
  = \begin{cases}
      (g - 1)^2 - (g - 1)^3/2, & g-1 \ll 1, \\
      1/g, & g \gg 1.
    \end{cases}
 \label{scaling-g}
\end{equation}
The scaling function (\ref{scaling-g}) is depicted in Fig.\ \ref{Fig:scaling}.
It is qualitatively similar to the numerical results of Ref.\
\cite{Bardarson07}.

\begin{figure}
 \centerline{\includegraphics[width=0.95\columnwidth]{beta.eps}}
 \caption{Unified scaling function (\protect\ref{scaling-g})
for both ultraballistic and diffusive regimes at zero energy in the case of the
random potential disorder.}
 \label{Fig:scaling}
\end{figure}

We stress that the ultraballistic asymptotics of the beta function in Eq.\
(\ref{scaling-g}) is only valid for Gaussian white-noise statistics of random
potential. The interpolation between the two asymptotics of the beta function in
Fig.\ \ref{Fig:scaling} implicitly assumes a smooth crossover between the two
regimes (in particular, without any intermediate fixed points), as suggested by
numerical simulations. \cite{Bardarson07, Koshino07, San-Jose07, Tworzydlo08}

The scaling function (\ref{scaling-g}) characterizes the evolution of the
dimensionless conductivity with increasing $L$.  Whether the full distribution
function $P(T)$ retains exactly its form (\ref{Dorokhov}) (parameterized by $g$
only) in the crossover remains an open question. Strictly speaking, what we know
at the moment is that this form of $P(T)$ emerges (i) in the clean limit, (ii)
in the ultraballistic regime within the first order in $\alpha(L)$, and (iii)
in the diffusive regime. On this basis, one could speculate that this might be
an exact statement for the whole crossover.


\section{Summary}
\label{Sec:summary}

In this paper, we have analyzed transport properties of a graphene sample
in the ``wide and short'' geometry, $W \gg L$, with disorder effects restricted
to a random potential scattering within a single valley. Starting from the clean
limit and using the transfer-matrix technique, we have analyzed the evolution of
the transmission distribution $P(T)$ and, in particular, of the conductance $G$
and the Fano factor $F$, with increasing system size $L$. To take the randomness
into account, we have developed a perturbative treatment of the transfer-matrix
equations supplemented by an RG formalism describing the renormalization of
disorder couplings. This has allowed us to get complete analytical description
of the transport properties of graphene in the ultraballistic ($L\ll \lambda$)
and ballistic ($\lambda \ll L \ll l$) regimes. We have also constructed ``phase
diagrams'' of different transport regimes (ultraballistic, ballistic,
diffusive).


\begin{theacknowledgments}
We are grateful to M.~Titov, A.W.W.~Ludwig, P.~San-Jose, E.~Prada, and
H.~Schomerus for valuable discussions. The work was supported by the Center for
Functional Nanostructures of the Deutsche Forschungsgemeinschaft. The work of
I.V.G. was supported by the EUROHORCS/ESF EURYI Awards scheme. I.V.G. and
A.D.M. acknowledge kind hospitality of the Isaac Newton Institute for
Mathematical Sciences of the Cambridge University during the program
``Mathematics and physics of Anderson localization: 50 years after'', where a
part of this work was performed.
\end{theacknowledgments}


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