text:
\section{Introduction}

Ever since its inception \cite{avi}, molecular rectification continues
to be of great practical interest. While rectification could arise
from asymmetries in the intrinsic molecular structure or vacuum
barriers at the ends, there are multiple experiments \cite{rei1,jpark,rscott}
that exhibit pronounced asymmetries in current voltage (I-V) or
conductance voltage (G-V) characteristics due to unequal coupling
with contacts. Fig.~\ref{f0} shows that the nature of contact-induced
asymmetry is qualitatively different depending on the nature of the
molecule-contact bonding. For molecules strongly coupled with the
contacts with adiabatic charge addition, equal current plateaus are
reached over unequal SCF voltage widths (Fig.~\ref{f0}a), leading
to prominent conductance asymmetries \cite{rei1}. The origin of this
asymmetry is the different average charging energies that generate {\it{
unequal
mean-field potentials}} for opposite bias voltages \cite{rasymm}.
Reducing the contact-molecular coupling drives the system into CB, where the
intermediate open-shell
current values are also asymmetric \cite{rralph2} (Fig.~\ref{f0}b). This asymmetry
has a different physical origin rooted in its many-body excitations, driven
by the {\it{unequal number of discrete charge
addition and removal channels}} at opposite bias. It is thus clear that
the physics of rectification can differ widely depending on
the strength of the electron-electron interaction.

\begin{figure}[ht]
\centerline{\epsfig{figure=expt.ps,width=3.0in}}
\caption{(Color online) Experiments showing (a) comparable currents reached over unequal
voltage widths \cite{rei1} in the SCF limit; (b) unequal currents with comparable
widths in the CB limit \cite{rralph2}.} \label{f0}
\end{figure}

The non-equilibrium Green's function (NEGF) formalism is widely
established for treating quantum transport in the SCF regime, for a
whole variety of materials from nanoscale silicon transistors to
molecules, nanowires, nanotubes and spintronic elements. The ability
to incorporate sophisticated quantum chemical models
\cite{max_dft,damle2} through averaged potentials makes the NEGF-SCF
scheme particularly attractive in the community. What is not widely
appreciated is that this approach does not readily translate to the
CB regime, even qualitatively \cite{rbhasko,rbhaskob,rbhaskoc}. The
CB regime, observed in molecules with weak contact coupling
\cite{jpark,rscott}, manifests clear signatures of single-electron
charging such as suppressed zero-bias conductances and abrupt jumps
in current. Although several approximate treatments
\cite{max_dft_2,rpal,ssan,pals} have been suggested to handle CB
within effective one-electron potentials, the inherent difficulty
arises from the fact that the open-shell current levels depend on
full exclusion statistics in its many-body Fock space. Even for a
minimal single-orbital model, it is easy to establish that while the
open-shell current plateau widths depend on the correlation
strengths, their heights are independent of correlation and, in that
sense, universal \cite{rbhasko}. The transport problem in CB maps
onto a rather difficult combinatorial problem in many-body space
that is hard to capture {\it{a-priori}} through a one-particle SCF
potential, or improve upon phenomenologically.

It seems likely that in the limit of weak coupling to contacts a
proper treatment of the above excitations will require solving a set
of master equations directly in the Fock space of the molecular
many-body Hamiltonian \cite{rbhasko, rhettler}. A significant
penalty is the increased computational cost that requires
sacrificing the quantum chemical sophistication of ab-initio models
in lieu of an exact treatment of the Coulomb interaction in simpler,
phenomenological models. Within such an exactly diagonalizable
model, one can capture transport features quite novel and unique to
the CB regime, such as inelastic cotunneling, gate-modulated current
rectification and Pauli spin blockade
\cite{rbhasko2,rbhasko,rsiddiqui}. The presence of contact asymmetry
makes these features even more intriguing, while somewhat
simplifying the analysis by effectively driving the system into
equilibrium with the stronger contact.

In this paper, we first identify the origin of current asymmetry
with a minimal system of a single spin degenerate energy doublet,
employing NEGF in the SCF limit and master equations for sequential
tunneling \cite{rralph} in the CB limit (we refer to this as a
``Fock-space" CB model). We then extend this ``Fock-space model" to
a general molecular Hamiltonian to explain how multiple orbitals in
CB allow simultaneous sequential tunneling into excited states,
making the conductance peak heights vary with gate voltage. We also
explain the origin of exchange in conductance peak asymmetry by the
neutral and singly charged molecule. While this Fock space model
involves a computationally intensive, exactly diagonalized many-body
Hamiltonian about the charge degeneracy point arising from the
different excitation spectra accessed, in many cases a simpler
approximation works. Our analysis of the experimental trends in the
CB regime reveals that many of the novel signatures such as gate
dependence of conductance peaks and asymmetry flipping can be
explained using a {\it{simpler}} ``orthodox model" that involves an
incoherent sum of Fock space excitations. We conclude by pointing
out some limitations of such a simplified approach, as well as
possible applications that may need careful attention to the
detailed excitation spectrum.

\section{Origin of Current Asymmetries -- The essential physics}
As mentioned earlier, there are two distinct physical limits of transport. In the
SCF limit, contact broadenings $\Gamma$ are greater than or comparable with the
single electron charging $U$. In the opposite CB limit $U\gg\Gamma$ and single-electron
charging dominates. Conductance asymmetries in both regimes
of transport have been experimentally observed in molecular conduction. While there are
ways to handle each regime separately, treatments are inherently perturbative, with an
approximate treatment of correlation (in terms of $U/\Gamma$) for the SCF regime, and
an approximate treatment of broadening (in terms of $\Gamma/U$) for the CB regime. The
lack of a small parameter in the intermediate coupling regime ($U \sim \Gamma$) makes
the exact treatment of transport, even for a simple model system, potentially
intractable \cite{rbhaskob}.

\begin{figure}
\centerline{\epsfig{figure=fig1.ps,width=3.8in,height=3.8in}}
\caption{(Color Online) Origin of Current Asymmetries: a) In the SCF
regime (stars), when $\gamma_L \gg\gamma_R$ for $V>0$, the left
contact keeps the HOMO $\epsilon_{HOMO}$ level filled to its neutral
charge making conduction possible, while for $V<0$, the left contact
empties the electron leading to the level floating out of the bias
window, creating {\it{unequal plateau widths}} for opposite bias. b)
In the CB regime (bold solid line) however, for $V<0$, charge
removal process by the right contact is rate determining and occurs
in two different ways, while for $V>0$, charge addition process is
rate determining and occurs in only one way. This leads to
{\it{asymmetry in plateau heights}} along the two bias directions.
c) While the transition between the two limits is hard to accomplish
correctly, a phenomenological broadening through an artificial
enhanced temperature (circles) illustrates how the open-shell
intermediate CB plateau morphs into a higher effective broadening,
restoring the SCF result in the limit of $\Gamma \gg U$} \label{f1}
\end{figure}

The origin of asymmetric I-Vs can be easily elucidated with a
minimal model for current conduction through a spin degenerate,
filled (closed-shell) molecular level doublet. We assume equal
{\it{capacitive}} couplings but unequal {\it{resistive}} couplings
to the contacts, so that the molecular level shifts by half the
applied Laplace potential and the current onsets arise symmetrically
around zero bias. In the SCF limit, contact asymmetry results in
equal currents adiabatically smeared out over a larger voltage width
along one bias direction than the other. This charging based
asymmetry has been experimentally seen \cite{rei1}, and can be
intuitively rationalized as follows. Consider a spin degenerate
energy level, a highest occupied molecular orbital (HOMO) for
example, that is fully occupied at equilibrium. For asymmetric
contact couplings $\gamma_L \gg \gamma_R$, charge addition dominates
for positive bias on the right contact and removal for negative
bias, as shown in Fig.~\ref{f1}(a). For positive bias the energy
level is maintained at neutrality by the dominant left contact and
the current flow through the level is determined by the removal
rate. Along reverse bias, in contrast, charge removal by the left
contact drives the system away from neutrality towards a net
positive charge, whose Coulomb cost floats the level out of the bias
window. This means that a larger bias is needed to fully conduct
through the level, dragging out the I-V in that direction. The
direction of the asymmetry flips if conduction is through the lowest
unoccupied molecular orbital (LUMO) instead. Notably the peak
currents and initial plateau onsets remain the same along both
directions, but their complete saturations are delayed by different
amounts.

The origin and manifestation of current asymmetry is qualitatively different
in the CB limit, where charge addition or removal is abrupt and in
integer amounts. Given asymmetric contact couplings ($\gamma_L\gg\gamma_R$),
the left contact adds (removes) an electron as soon as the right
contact removes (adds) it, so that the rate determining step becomes
the dynamics of the weaker right contact. For positive bias, charge
removal can happen in two ways, from $\uparrow\downarrow$ to
$\uparrow$ and $\downarrow$, while for opposite bias the right
contact can add a spin in only one way, either $\uparrow$ or
$\downarrow$ to $\uparrow \downarrow$. This scheme of charge
transfer (Fig.~\ref{f1}b) leads to twice the current step for
positive bias than negative \cite{rralph,rralph2}.

An important question is whether one can smoothly transition from
the CB to the SCF asymmetry by progressively increasing the
broadening. While this is hard to do exactly owing to the inherent difficulty involved in
the broadening many-particle states \cite{rWacker}, for the purpose of
illustration one can add various degrees of approximate broadening
\cite{rappenzeller}. We choose to do this by increasing the
temperature and incorporating this through Boltzmann factors in the
many-body occupancies \cite{rgerhard}. As seen in Fig.~\ref{f1}(c),
this approximate treatment morphs the CB asymmetry into the very
different version seen for the SCF limit. {For negative bias on the
weaker contact, `shell-filling' \cite{rzung} of the HOMO level with a net
positive charge creates a CB plateau that is missing in its positive
bias `shell-tunneling' counterpart. This extra CB plateau gets
encased by the broadened manifold, leading to the postponed
conduction seen in the SCF limit. It is worth mentioning though that
the correspondence is only qualitative and is observed to worsen for
higher onset voltages, underscoring the inadequacy of thermal
effects and possibly other phenomenological ways to incorporate
broadening in correlated systems, particularly in the
near equal coupling, non-equilibrium limit which combines shell
tunneling with shell filling.

\section{Coulomb Blockade Formalism: Fock-space vs Orthodox}
In this paper, we focus mainly on the CB regime. The object of this
section is to present the formalism in detail, beginning with the
one that employs the Fock space. Here, one needs to keep track not
only of various ground state charge configurations, but also various
excitations within each charge state. Following that we derive the
orthodox model that integrates out the effect of excitations in an
incoherent way. A discussion of merits and de-merits of such a
simplification comprises the subsequent sections.
\subsection{Fock-Space Master Equation}
The starting point is a molecular many-body Hamiltonian
\begin{eqnarray}
\hat{H} &=& \sum_{m} \epsilon_{m} n_{m}
+ \sum_{m \neq p} t_{m p} c_{m}^{\dagger} {c_p} \nonumber\\
&+& \sum_{m} U_{m m} n_{m\uparrow} n_{m\downarrow}
+ \frac{1}{2} \sum_{m \neq p} U_{m p} n_{m} n_{p},
\label{eq:mbh}
\end{eqnarray}
where $m, p$ denote the spin-charge basis functions within
a tight binding formulation, with $\epsilon, t$ and $U$ denoting
on-site, hopping and charging terms, respectively. Exactly
diagonalizing this Hamiltonian yields a large spectrum of closely
spaced excitations in every charged molecular configuration. The
lead molecule exchange processes are accounted for within the
sequential tunneling approximation \cite{timm,rbraig}. This results
in a set of master equation that involve transition rates
$R_{(N,i)\rightarrow(N\pm 1,j)}$ between states differing by a
single electron. If one neglects off-diagonal coherences, the master
equation \cite{rbraig} is cast in terms of the occupation
probabilities $P^N_i$ of each N electron many-body state
$|N,i\rangle$ with total energy $E^N_i$. The end result is a set of
independent equations defined by the size of the Fock space
\cite{rralph}
\begin{equation}
\frac{dP^N_i}{dt} =-\sum_j\left[R_{(N,i)\rightarrow(N\pm 1,j)}P^N_i - R_{(N\pm
1,j)\rightarrow(N,i)}P^{N\pm 1}_j\right]
\label{ebeenakker}
\end{equation}
along with the normalization equation $\sum_{i,N}P^N_i =
1$. For weakly coupled dispersionless contacts, parameterized using
bare-electron tunneling rates $\gamma_{\alpha}$ ($\alpha$: left/right
contact) within a Golden Rule treatment, we define the transition-resolved
rate constants \begin{eqnarray}
\Gamma_{ij\alpha}^{Nr} &=& \gamma_\alpha|\langle
N-1,j|c^{}_\alpha|N,i\rangle|^2\nonumber\\ \Gamma_{ij\alpha}^{Na}
&=& \gamma_\alpha|\langle N+1,j|c^{\dagger}_\alpha|N,i\rangle|^2,
\end{eqnarray} where $c^\dagger_\alpha,c^{}_\alpha$ are the
creation/annihilation operators for an electron on the left or right
molecular end atom coupled with the corresponding electrode. The
transition rates are given by
\begin{eqnarray}
R_{(N,i)\rightarrow(N-1,j)} &=& \sum_{\alpha=L,R}\Gamma_{ij\alpha}^{Nr}\left[1-f(\epsilon^{Nr}_{ij}-\mu_\alpha)\right]
\nonumber\\
R_{(N-1,j)\rightarrow(N,i)} &=& \sum_{\alpha=L,R}\Gamma_{ij\alpha}^{Nr}f(\epsilon^{Nr}_{ij}-\mu_\alpha).
\label{rates}
\end{eqnarray}
for the removal levels $(N,i \rightarrow N-1,j)$, and replacing $(r
\rightarrow a,   f \rightarrow 1-f)$ for the addition levels $(N,i
\rightarrow N+1,j)$. $\mu_\alpha$ are the contact electrochemical
potentials, $f$ is the corresponding Fermi function, with single
particle removal and addition transport channels $\epsilon^{Nr}_{ij}
= E^N_i - E^{N -1}_j$, and $\epsilon^{Na}_{ij} = E^{N+1}_j - E^N_i$.
Finally, the steady-state solution to Eq.~(\ref{ebeenakker}) is used
to get the left terminal current
\begin{equation}
I =\pm\frac{e}{\hbar}\sum_{N,ij}\left[R^L_{(N,i)\rightarrow(N\pm
1,j)} P^N_i - R^L_{(N\pm 1, j)\rightarrow(N,i)}P^{N\pm 1}_j \right]
\label{curr}
\end{equation}
where $R^L$ includes the contributions to $R$ from the left contact
alone. In the equation above, states corresponding to a removal of
electrons by the left electrode involve a negative sign. We usually
calculate current in a
break-junction configuration with equal electrostatic coupling with
the leads, $\mu_{L,R} = E_F \mp eV_d/2$.
\subsection{The Orthodox Model}
In the previous sub-section, the computational complexity of the
master equation defined in Eq.~(\ref{ebeenakker}) arises from the
need to keep track of not only charge $N$, but also all
configurational degrees of freedom $i$. Thus the evaluation of
transition rates defined in Eq.~(\ref{rates}) and eventually current
(Eq.~\ref{curr}) depends on our knowledge of various many-electron
wavefunctions $|N, i \rangle$ and total energies $E^{N}_{i}$. The
``orthodox'' theory of single-electron tunneling, however, does not
distinguish between excitation levels \cite{rsold,rkorot}. Note also
that in the orthodox theory, the junctions are generally denoted by
$(1,2)$ instead of $(L,R)$. Eq.~(\ref{curr}) then becomes
\begin{eqnarray}
I & = &
\pm\frac{e}{\hbar}\sum_{N}\left[R_{N\rightarrow{N\pm1}}^{(1)}-
R_{N\rightarrow{N\mp1}}^{(1)}\right] P^{N} \\
& = & \pm\frac{e}{\hbar}\sum_{N}\left[R_{N\rightarrow{N\mp1}}^{(2)}-
R_{N\rightarrow{N\pm1}}^{(2)}\right] P^{N} \label{eq:curr_master}
\end{eqnarray}
We have simplified the second term in the summation using a simple
change of variables.  Let us assume incoherent processes that
introduce an additional exclusion term in the transition process
(appendix). Then a ``golden-rule'' calculation gives
\begin{eqnarray}
&R_{N\rightarrow{N+1}}^{(2)}&=\int_{-\infty}^{\infty}{\displaystyle{2\pi}}\Gamma_{2}^{Na}
D_{2}(E-\mu_{2})f(E-\mu_{2})\nonumber\\
&\times& D_{m}(E-\mu)\left[1-f(E-\mu)\right]dE
\mbox{, } \label{goldenrule}
\end{eqnarray}
where $D_{2}(E)$ and $D_{m}(E)$ are the densities of states of the
right and middle electrodes, and $\mu$ is the Fermi energy of the
molecular system.  If the level separation between one-electron
levels is very small, like in a metallic quantum dot, the densities
of states are approximately constant for the calculation of the
removal and addition rates. By additionally assuming
$\Gamma_{\alpha}^{Na}$ is energy-independent, Eq.~\eqref{goldenrule}
simplifies:
\begin{equation}
R_{N\rightarrow{N\pm1}}^{(j)} =\frac{\hbar}{R_{j}e^{2}} \left(
\frac{-\Delta
    E_{j}^{\pm}}{1-exp(\Delta E_{j}^{\pm} / kT)} \right) \mbox{, }
\label{gamma}
\end{equation}
where $R_j$ is the junction resistance. $\Delta E_{j}^{\pm}$ is the
transition energy corresponding to adding or removing an electron at
the $j^{th}$ contact; from simple electrostatics,
\begin{eqnarray}
\Delta E_{1}^{\pm} = \Delta U^{\pm} \mp \frac{eC_2}{C_{\Sigma}}
            V_D \mp \frac{eC_{G}}{C_{\Sigma}}V_{G} , \nonumber \\
            \Delta E_{2}^{\pm} =\Delta U^{\pm} \pm
            \frac{eC_1}{C_{\Sigma}}V_D \mp \frac{eC_{G}}{C_{\Sigma}}V_{G},
\label{etr}
\end{eqnarray}
where $C_{1,2,G}$ and $C_{\Sigma}$ are the terminal and total
capacitances respectively, $V_D$ and $V_G$ are drain and gate voltages respectively, and $\Delta U^{\pm}$ is the Coulomb offset
for the addition or removal of an electron. Notice that we now have a
simpler analytical and closed form solution to our set of master
equations, using our trick of summing over excitations. In doing so,
we also brought in simpler circuit parameters such as junction
resistance and capacitance that aid in our analytical understanding
of threshold voltages and current magnitudes.

When considering the ``orthodox'' Coulomb Blockade theory in the
regime of strong contact asymmetry ($R_2 \gg R_1$), it is
illuminating - and not overly limiting - to consider what happens at
very low temperatures. Following the analysis of Hanna and Tinkham
\cite{Han91}, at low temperatures the ensemble distribution of
electrons on the middle electrode can be described by a delta
function $\delta_{n,n_0},$ where $n_0$ is the most probable number
of electrons. The signs of $n_0$ and $Q_0$ in the following
equations will differ from \cite{Han91} because we are using a
negative charge carrier convention. The delta function probability
density reduces Eq.~(\ref{eq:curr_master}) to $I(V_D,V_G) =
e/\hbar[R^{(2)}_{n_{0}\rightarrow{n_0\mp1}}-R^{(2)}_{n_{0}\rightarrow{n_0\pm1}}]$.
For low bias the Coulomb cost of electrons tunneling across the
contacts is high, resulting in a zero-conductance region limited by
the positive and negative threshold voltages $V_{CB}^{(+)}$ and
$V_{CB}^{(-)}$, respectively. Outside of this region the transition
rates simplify:
\begin{eqnarray}
 I(V_D,V_G) = & \displaystyle\frac{1}{R_{2}C_{\Sigma}}[&-(n_{0}e-Q_0)+C_{1}V_{D}-C_{G}V_{G}\nonumber\\
  & & -\frac{e}{2}sgn(V_{D}-V_{CB}^{(-)})]\mbox{ ,}
\label{eq:approxIV}
\end{eqnarray}
where $sgn$ denotes the Heaviside sign function. The linearity of
Eq.~\eqref{eq:approxIV} with drain voltage is only interrupted when
new levels enter into the bias window, causing $n_0$ to change by
$\pm1$, which in turn causes the current to``jump" in value.

\begin{figure}
\centerline{\epsfig{figure=fig2.ps,width=3.0in,height=3.0in}}
\caption{(Color Online) Illustration of Orthodox Model:  The
orthodox theory parameters define a set of $I(V,n_0)$ curves,
represented by the dashed lines.  Generally $n_0$ is initially zero.
As a voltage is applied, the current follows the $I(V,n_0=0)$ curve
until $n_0$ changes.  If $|V_{CB}|<|V_{n_0=0}|$, as is the case
here, the current will rise linearly out of the zero-conductance
region; otherwise, there will be a ``jump onset.''  For this
simulation, $R_{1}=10$ M$\Omega$, $R_{2}=6$ G$\Omega$, $C_{1}=8$ aF,
$C_{2}=5.3$ aF, $Q_{0}=0$, and $T=2$ K.} \label{fig4}
\end{figure}

  We thus have an intuitive picture of how I-V ($I-V_D$) curves are constructed
in the orthodox theory.  For given system parameters (R, C, etc.)
and gate voltage, there is a set of $I(V,n_0)$ curves for different
values of $n_0$, as dictated by Eq.~\eqref{eq:approxIV}. The dashed
lines in Fig.~\ref{fig4} correspond to three members of such a set.
In general, both $n_0$ and $I$ are initially zero.  As a drain
voltage is applied, I(V) remains on the $I(V,n_{0}=0)$ curve until
$n_0$ changes, at which point I(V) jumps to the $I(V,n_{0}=\pm1)$
curve. Generalizations of Eqs.~(5,6a) from Hanna and Tinkham
\cite{Han91} allow one to specify the Coulomb Blockade threshold
voltages,
\begin{subequations}
    \begin{align}
       V_{CB}^{(+)} = C_{1}^{-1}(e/2-n_{0}e+Q_0+C_{G}V_{G})\\
       V_{CB}^{(-)} = C_{1}^{-1}(-e/2-n_{0}e+Q_0+C_{G}V_{G})\mbox{,}
    \end{align}
    \label{vcbonsets}
\end{subequations}
and the voltages at which the system transitions from $n_0$ to
$n_{0} \pm 1$ electrons,
\begin{subequations}
    \begin{align}
        V_{n0}^{(+)}=C_{2}^{-1}(e/2+n_{0}e-Q_0-C_{G}V_{G})\\
        V_{n0}^{(-)}=C_{2}^{-1}(-e/2+n_{0}e-Q_0-C_{G}V_{G})\mbox{.}
    \end{align}
    \label{vn0onsets}
\end{subequations}
In Eqs.~(\ref{vcbonsets},\ref{vn0onsets}) the positive superscripts
refer to positive onset voltages, and the negative superscripts are
similarly defined. In Fig.~\ref{fig4}, for example, we can see that
the CB threshold voltage is reached (at $10$ mV) before $n_0$
increases (at $15$ mV), resulting in a linear onset of the current.
If, however, $|V_{n0}^{(\pm)}|$ were smaller than
$|V_{CB}^{(\pm)}|$, then there would be a ``jump'' onset at the
zero-conductance region threshold.

Let us now compare the orthodox and Fock space model approaches to
transport in the CB regime and apply them within the context of
experimental trends.

\section{CB Asymmetries: Gate Dependent Rectification}
One of the simplest consequences of asymmetric contact coupling is
rectification; in other words, a bias direction dependence in the
I-V characteristics. To calibrate with experiments, we not only
concern ourselves with rectification {\it{per-se}}, but also how it
is influenced by a gate. In fact, experiments \cite{jpark,rscott}
showcase gate dependences of the rectification properties that are
arguably more interesting than the rectifications themselves. These
experiments (see for example \cite{jpark,rscott}) show the following
gate-able features: (i) a gate-dependent shift of conductance peak
onsets and (ii) a gate-dependent modulation of the corresponding
conductance peak heights. In addition, there is (iii) a prominent
exchange in conductance peak asymmetry for gate voltage variations
about the charge degeneracy point in the stability diagram
\cite{rscott}. We will argue that much of the relevant physics has
to do with the way the molecule accesses various electronic
excitations under bias, which would require going beyond our
one-orbital model to a multi-orbital system. Charge addition or
removal causes jumps in the I-V, while charge redistribution
(excitation) leads to closely spaced plateaus that merge onto a
linear ramp when summed incoherently. In the rest of the section, we
will explain how each CB model (Fock-space and orthodox)
successfully captures the gate modulation of the asymmetric I-Vs, as
summarized schematically in Figs.~\ref{fig:CB1} and \ref{fig:CB2}.

\subsection{Gate Modulation of current onsets and heights}

\begin{figure}
\centerline{\epsfig{figure=fig3.ps,width=3.8in,height=3.8in}}
\caption{(Color Online) Origin of peak asymmetry and variation with gate voltage.
(a) For gate voltages that place the contact Fermi energy $\mu_F$ in
the $N$ electron blockade region, the levels align such that $\mu_F
> \epsilon^{Nr}_{00}$. A state transition diagram (b) shows the
addition and removal of up (down) spins resulting in transitions
$\epsilon^{Nr}_{00}$ (bold double arrow) between ground states of
neutral (light orange) and positively charged species (light blue).
Also shown in state transition diagram are transitions (dashed
double arrow) between various configurations of neutral excited
state (deep orange) and positively charged ground state, labeled
$\epsilon^{Nr}_{10}$. (c) The resulting G-V shows clear asymmetry in
conductance peak height due to there being more ways to add an
electron. Increasing gate bias increases the number of excitations
available, giving a pronounced peak modulation with gate voltage.
The inset shows the corresponding I-V characteristics
\cite{rbhasko,jpark,rscott}} \label{fig:CB1}
\end{figure}

The onset of conduction is determined by the offset between the
equilibrium Fermi energy and the first accessible transition energy,
marked $\epsilon^{Nr}_{00}$ in Fig.~\ref{fig:CB1}
(following the nomenclature in section II A).
This can be varied by varying the gate voltage, thereby accounting
for the variation in conductance gap with gate bias (Fig.~\ref{fig:CB1}c). While the
current step and corresponding conductance peak are generated by this
threshold transition, there follows a quasi-ohmic rise in current
leading to a subsequent constant non-zero conductance in the G-V.
This feature arises from the sequential access of several closely spaced
transport channels under bias, arising
from excitations within the $N$ and $N-1$ electron subspaces \cite{rbhasko}.
While net charge addition and removal come at large Coulomb prices, excitations
involve charge reorganization within the Fock space that cost much smaller
correlation energies.

The presence of multiple orbitals generates several configurations
of excited states, creating more accessible transport channels
within the bias window. For example, in Fig.~\ref{fig:CB1}(a)
conduction occurs simultaneously via the $\epsilon^{Nr}_{10}$ and
$\epsilon^{Nr}_{00}$ removal channels. $\epsilon^{Nr}_{10}$
corresponds to a transition between the first excited state `1' of
the $N$-electron neutral species, and the ground state `0' of the
$N-1$ electron cationic species. We show four possible
configurations corresponding to the transport channel
$\epsilon^{Nr}_{10}$ and the corresponding I-V
(Fig.~\ref{fig:CB1}b). Increasing the gate bias increases both the
threshold for current conduction and the number of such excited
state channels accessed by the contacts, thereby altering the height
of the corresponding conductance peak with gate bias
(Fig.~\ref{fig:CB1}c).

The previous paragraph illustrates the origin of gate-modulated
current as rationalized by the {\it{Fock-space CB model}}. One can
also explain this within the {\it{simpler orthodox model}}, which
ignores the identities of the resolved excitations by incoherently
summing over them. Under the approximations of contact asymmetry and
low temperature, the rate $R^{(j)}_{N\rightarrow{N\pm1}}$ is linear
in the transition energies $\Delta E_{j}^{\pm}$ that increase with
drain voltage. With increasing gate bias one needs a larger
corresponding drain bias to overcome the zero-conductance regime. At
this higher drain voltage the coupling has a greater value and,
consequently, the current magnitude is larger. Physically, the drain
voltage-dependence of the coupling represents a linear approximation
of the excitation spectra.  Even though the orthodox model
indiscriminately sums the excitations within the $N$ and $N-1$
subspaces, the fact that it captures them at all allows it to
qualitatively capture the modulation of current height.

\begin{figure}
\centerline{\epsfig{figure=fig4.ps,width=3.8in,height=3.8in}}
\caption{(Color Online) Experiment from J. Park \textit{el al.}
\cite{jpark} showing gate-rectification properties in the Coulomb
Blockade regime: a) Experimental traces, b) Orthodox fit with
parameters $C_1 = 0.624$ aF, $C_2 = 0.486$ aF, $C_G = 0.0708$ aF,
$R_1 = 1$ M$\Omega$, $R_2 = 75$ M$\Omega$, $Q_0=-0.05e$ and $T=2.2$
K. c) Fit from Fock-space model.} \label{fig:fig7}
\end{figure}

Figs.~\ref{fig:fig7}(a) and \ref{fig:Zhitenev}(a) show experimental
evidence of gate-modulation of current onsets and heights.  Fig.
\ref{fig:fig7}(a), from J. Park \textit{et al.}, shows a simple
experiment for which the negative bias onset is set by moving from
the $N$ to $N-1$ electron subspace, while the positive bias onset
starts where the $N$ electron excitation spectrum moves into the
bias window. Figs.~\ref{fig:fig7}(b,c) show the abilities of both
the orthodox model and the Fock-space model to capture the
gate-modulated features of the experiment.
Fig.~\ref{fig:Zhitenev}(a), by Zhitenev \textit{et al.}, is a
similar experiment with slightly more complex features. Curves (d-h)
show the same negative bias onset as seen in Fig.~\ref{fig:fig7},
but as the gate voltage is further decreased, the negative onset
changes to a linear onset, representing access to the $N$ electron
excitation spectrum.  The positive bias shows that decreasing gate
voltage brings the $N+1$ level closer to the bias window, and for
curves (a-c) the positive bias type consequently becomes a ``jump''
onset. In spite of these new degrees of freedom that must be
captured, the orthodox theory still models very accurately, as seen
in Fig.~\ref{fig:Zhitenev}(b). It is worth noting that the x-axis in
Fig.~\ref{fig:Zhitenev} is $V_{SD}$, rather than $V_{DS}$, which
must be accounted for when using Eqs.~(\ref{gamma}-\ref{vn0onsets})
of the orthodox theory. A discussion of the extraction of orthodox
parameters from the experimental curves in Figs.~\ref{fig:fig7} and
\ref{fig:Zhitenev} is included in the appendix.

Mathematically, it is straightfoward to understand the dependence of
current height on gate voltage within the orthodox theory.  Using
the experiment of J. Park. \textit{et al.} (Fig.~\ref{fig:fig7}a) as
an example, we see that there is a jump onset for negative bias
voltages. Therefore, $\left|V_{n0}^{(-)}\right| <
\left|V_{CB}^{(-)}\right|$. At $V_{n0}^{(-)}$, the I-V transitions
to the $I(V,n0=-1)$ curve, so to find the current height at the
onset voltage we can use Eq.~\eqref{eq:approxIV} for $n_0=-1$.
Inserting $V_{n0}^{(-)}$ for $V_D$, one finds:
\begin{equation}
I(V_{n0}^{-})= -\frac{C_{1}+C_{2}}{R_{2}C_{2}C_{\Sigma}} \left(Q_0
+e/2+C_{G}V_{G} \right)
\end{equation}
Clearly the magnitude of the current at the onset voltage increases
with gate voltage (Fig.~\ref{fig:fig7}b), matching the experimental
result seen in Fig.~\ref{fig:fig7}(a).

\begin{figure}
\begin{center}
    \mbox{
      \subfigure{{\epsfig{figure=zhitenevExpt.ps,width=1.6in,height=1.5in}}} \quad
      \subfigure{{\epsfig{figure=zhitenevSim.ps,width=1.6in,height=1.5in}}} \quad
      }
\caption{(Color Online) Asymmetric CB results showing (a) experiment
\cite{zhitenev} and (b) orthodox theory with parameters $C_1=3.70$
aF, $C_2 = 3.24$ aF, $C_G=0.061$ aF, $R_1=2$ M$\Omega$, $R_2 = 210$
M$\Omega$, $Q_0=0.175e$, and $T=4.2$ K.} \label{fig:Zhitenev}
\end{center}
\end{figure}

The accuracy of the orthodox simulation would imply at the very
least that individual excitations do not play an important role in
the transport characteristics seen.  Because the experimental I-V
has a strongly linear dependence on drain voltage seen in
Eq.~(\ref{eq:approxIV}), it seems that the experiments may have
measured transport through a metallic particle, which has a
relatively featureless density of states, as is assumed in the
orthodox model.

\subsection{Peak Exchange}

Experiments exhibit a characteristic flipping of conductance peak
asymmetry around the charge degeneracy point A in the stability
diagram (Figs.~\ref{fig:CB2}a,b). Figs.~\ref{fig:CB2}(e,f) show
typical calculated G-Vs in this regime, featuring conductance peak
asymmetries with respect to voltage bias, arising due to asymmetric
contact couplings ($\gamma_L \gg \gamma_R$). Within the Fock-space
model, this can be explained by enumerating the channels for adding
and removing electrons under bias (Figs.~\ref{fig:CB2}b,d), with the
weaker right contact once again setting the rate limiting step. The
dominant transport channel $\epsilon^{Nr}_{00}$ corresponds to
electronic transitions between the neutral and cationic ground
states \cite{rbhasko}, states which SCF theories do take into
account. In the CB limit, however, there are additional electronic
excitations that are accessible with very little Coulomb cost. These
states are responsible for the peak asymmetry exchange observed in
these experiments, as we will now explain.

The origin of this asymmetry can be understood with a simple model
system: in our case, a quantum dot with 8 spin-degenerate levels and
$N=4$ electrons in its ground state. When the Fermi energy lies to
the immediate right of the charge degeneracy point as shown in
Fig.~\ref{fig:CB2}(a), only transitions between the $N$ and $N-1$
electron states (4 and 3) are allowed, with the weaker right contact
setting the rate-limiting step. For positive bias on the right
contact an electron can be removed from the 4-electron to the
3-electron ground state in two ways (Fig.~\ref{fig:CB2}b)). For
negative bias, however, the electron removed by the left contact can
be replenished by the right contact back into the 4-electron ground
state, but also into one of many possible excited states
$\epsilon^{4r}_{i0}=E^{4}_{i}- E^{3}_{0}$, (i $>$ 0). Since there
are more ways to bring the electron back (6 shown here), the
conductance is larger for negative bias (Fig.~\ref{fig:CB2}e). The
situation changes dramatically for a different position of the Fermi
energy (Fig.~\ref{fig:CB2}c) in the stability diagram lying to the
left of the charge degeneracy point A with three electrons at
equilibrium. For negative bias the right contact adds an electron
from the 3 to the 4-electron ground state, while for negative bias
it returns it to the $j^{th}$ 3-electron excited state through
transitions $\epsilon^{4r}_{0j}=E^{4}_{0}-E^{3}_{j}$. There now are
more ways to remove than add charge (Fig.~\ref{fig:CB2}d), so the
asymmetry flips (Fig.~\ref{fig:CB2}f).

\begin{figure}
\centerline{\epsfig{figure=fig5.ps,width=3.2in}}
\caption{(Color Online) Origin of exchange in asymmetry of conductance peaks: (a),
(b) Schematic of vicinity of charge degeneracy point A in Coulomb
diamond. Also shown are the corresponding energy diagrams at
threshold. Notice a different set of threshold transport channels
between (a) and (b). (c) and (d) State transition diagrams illustrating
the different excitation spectra accessed on either side of the
charge degeneracy point A. (e) and (f) G-V plots for scenarios (a) and
(b) Notice the clear peak-exchange as a result of accessing a
different excitation spectra in either case.} \label{fig:CB2}
\end{figure}

Analogous to the {\it{Fock-space model}}, the {\it{orthodox model}}
also captures gate-dependent peak exchange, in spite of its
approximate treatment of excitations.  The origin of the asymmetry
is once again transitions between the $N$ to $N-1$ electron regimes.
In Fig.~\ref{fig4}, we can see that such a change results in a jump
onset that has a much higher conductance value than a linear onset.
Moving from the $I(V,n0)$ curve to the $I(V,n0-1)$ curve, therefore,
essentially captures the excitations of the $N-1$ electron spectrum
that are pivotal to the argument in the preceding paragraph.  The
fact that the conductance peak switches across zero bias only means
that the zero-bias state of the system changes from $N$ to $N-1$
electrons, which the orthodox method clearly captures.

Peak exchange has been reported experimentally \cite{rscott}, as
seen in Fig.~\ref{fig:fig6}(a).  Fig.~\ref{fig:fig6}(b) shows an
orthodox simulation of the experiment.  One can see close
qualitative as well as quantitative agreement between the
experimental data and the theoretical simulation.  The conductance
peaks have similar magnitudes, and the exchange of the peak
asymmetry occurs at $-3.75$ $V$ in both graphs. The evident validity
of the orthodox theory in this case demonstrates its ability to
capture excitation features, as long as the features can be linearly
approximated.

\begin{figure}
\centerline{\epsfig{figure=fig6.eps,width=3.5in,height=2.0in}}
\caption{(Color Online) (a) Experimental trace demonstrating peak
asymmetry exchange \cite{rscott}. Compare with Fock-space model
results in Fig.~\ref{fig:CB2}(e,f). (b) Orthodox simulation
reproducing peak exchange observed in the experiments. The
parameters are: $R_{1}=35$ M$\Omega$, $R_{2}=350$ M$\Omega$,
$C_{1}=0.673$ aF, $C_{2}=0.612$ aF, $C_{G}=0.0135$ aF,
$Q_{0}=-0.18e$, and $T=4.2$ K.} \label{fig:fig6}
\end{figure}

\section{Limitations of the Orthodox Model} In the previous sections,
we saw that the orthodox model has been fairly successful in
reproducing the three key trends in asymmetric CB transport; namely,
(i) rectification, (ii) gate-modulation of rectification and (iii)
exchange of rectification. The agreement between experiment and
theory are quantitatively quite close, indicating that the
Fock-space model's handling of well-resolved excitations is perhaps
an overkill, especially considering its substantially greater
computational complexity. It is tempting to conclude that molecules
with redox-active centers only exhibit incoherent sums of
excitations that do not manifest well-resolved features. In this
section, we point out examples where {\it{the discrete excitation
spectrum can indeed play a noticeable role in molecular transport
experiments, making an orthodox theoretical treatment quite
inadequate}}.

Molecules are unique in that they can potentially exhibit both
charge and size quantization \cite{rbhaskob}. Charge quantization is
enforced by $U
> \Gamma$, which amounts to a large contact resistance compared to
the resistance quantum (the correspondence arises by relating the broadening
$\Gamma$ to an RC time-constant through the uncertainty principle,
using the capacitance $C$ to set the single-electron charging energy $U$). At the same time, the
small sizes of molecules make their spectra discrete. While transport
examples showcasing discrete molecular
signatures are relatively rare, they are pretty commonplace in the
literature of inorganic quantum dots or `artificial molecules' \cite{rbanin}.
Part of the reason is the lower broadening in these dots, both from
the contact as well as incoherent scattering (which is larger in
organic molecules owing to their conformational flexibility).
Recently, a novel negative differential resistance (NDR) has been
reported in inorganic double quantum dots \cite{tar}, that has been
explained via the formation of an excited triplet state
\cite{tar,rbhasko2}. Other quantum dot experiments routinely show
Coulomb diamonds with very well resolvable excitation lines
\cite{sampaz} such as due to cotunneling \cite{cotunel} and Kondo correlations \cite{kondo}.

Well resolved excitation lines in the conductance
spectra manifest as varying plateaus in the I-V characteristics.
Consider the low bias ($V_D <500$ mV) I-V characteristics of the
plot from J-O. Lee  {\it{et al.}} \cite{rdekker}, seen in
Fig.~\ref{fig:fig8}. A striking feature is the existence of a plateau
at onset followed by several other plateaus, sometimes
even merging into a quasi ohmic rise as a result of several
unresolvable plateaus. This feature, observed in multiple experiments from
other experimental groups, can be easily explained within the
Fock-space model by keeping track of individual molecular excitations.
For example, it is well known that the gap between ground and first
excited states, involving charge addition or removal, is greater than
the gap between subsequent excitations involving charge reorganization.
In such a case it is clearly seen that a brief plateau occurs at
threshold that persists until the first excitation is been accessed,
as discussed in detail in \cite{rbhasko}.

This leads to an important
point. Normally small plateaus (few tens of mV) are naturally associated
with vibronic modes whose energy scales lie in that range. Coulomb
interactions are usually assumed to be larger in energy, because the
energy to add or remove an electron is much larger. However, the correlation
energy to reorganize charges could actually be
very small, are therefore quite capable of explaining
a range of plateau widths seen experimentally, as our calculations show.

The orthodox theory, on the other hand, cannot even qualitatively
match the experimental data in Fig.~\ref{fig:fig8}, owing to its
{\it{inability to incorporate size quantization effects and the associated
discrete spectra}}.  From Eq.~\ref{eq:approxIV}, it is clear
that outside of the zero-conductance region and excluding jumps due
to changes in $n_0$, conductance values in orthodox theory must
{{remain constant}}, with a value of $R_{1}/R_{2}C_{\Sigma}$. {\it{The
orthodox theory can capture a plateau, and it can also capture a
linear rise, but it does not seem to capture both in the same I-V curve.}}
Fig.~\ref{fig:fig8}(c) shows the best attempt at modeling the experimental
data in Fig.~\ref{fig:fig8}(a) within orthodox theory; {{the plateau
followed by a linear rise seems hard to duplicate}}.

\begin{figure}
\centerline{\epsfig{figure=fig7.ps,width=3.8in,height=3.8in}}
\caption{(Color Online) Limitation of Orthodox Model: a) Experimental trace \cite{rdekker} showing fine structure. b)
The fine structure in I-V's, result from keeping track of excitations explicitly within
the Fock-space CB model. c) An orthodox calculation merely maintains the same slope between charge
addition jumps, thus may not reproduce any fine structure.}
\label{fig:fig8}
\end{figure}

A second limitation of the orthodox model comes in its
{\it{treatment of gate voltage}}. Even though it effectively modeled
the data in Figs.~\ref{fig:fig7}, \ref{fig:Zhitenev} and
\ref{fig:fig6}, there again exist experimental features that the
orthodox theory cannot even qualitatively model.  Looking at Eqs.
\eqref{vcbonsets} and \eqref{vn0onsets} one can see that a change in
gate voltage causes a {\it{translation}} in the Coulomb Blockade
threshold voltages; a similar effect, albeit in the opposite
direction, is seen for the voltage limits at which $n_0$ changes.
Fig.~\ref{fig:fig5} shows how an orthodox I-V curve changes with
gate voltage for the four possible onset combinations - which come
from having linear or jump onsets at positive and negative bias. The
scaling of the conductance gap in Figs.~\ref{fig:fig5}(a,b)
successfully explained the experiments in Figs.~\ref{fig:fig7} and
\ref{fig:fig6}. However, for experiments with symmetric onsets
(Figs.~\ref{fig:fig5}c,d), one can see that the orthodox theory
predicts an overall translation in the I-V curve with gate voltage.
One would think that changing the gate voltage would only shift the
conducting level closer to or further from the bias window, thereby
narrowing or widening the I-V curve, respectively. Indeed,
experimental evidence demonstrates such narrowing \cite{rralph2},
and the Fock-space model captures that quite easily. The orthodox
theory was unable to match this experimental trend even
qualitatively.

\begin{figure}
\centerline{\epsfig{figure=fig8.ps,width=3.8in,height=3.8in}}
\caption{(Color Online) Gate-voltage dependence of I-V curves in the orthodox
theory.  Blue arrows indicate the direction of movement of the onset voltages
with increasing gate voltage.  The four plots correspond to the four
possible onset combinations: (a) a jump onset at negative bias and a
linear onset at positive bias, for which the I-V conductance
gap {\it{widens}} with increasing gate voltage; (b) a linear onset then
a jump onset, with the gap {\it{narrowing}} around zero-bias; (c)
two linear onsets, which corresponds to a {\it{translation}} of the I-V
curve, and (d) two jump onsets, which correspond to a {\it{translation}}
in the opposite direction of (c).} \label{fig:fig5}
\end{figure}

\section{Discussions}
It may seem that the proper treatment of well-resolved excitations
is rather academic with respect to molecular experiments. However,
we believe that there can be more important experimental features
that need a proper quantitative theory as transport spectroscopy of
molecules under becomes feasible \cite{ucla}. In fact, a possible
explanation for molecular NDRs \cite{tour,kiehl} could necessitate
keeping track of excitations in a donor-acceptor molecular system
\cite{rbhasko2}; in particular, the notably different lifetimes of
ground and excited states, analogous to issues relevant to the
double quantum dot literature. Small molecules could function as
tunable quantum dots with high single-electron charging energies.
Molecular quantum dots coupled to transistor channels can be
important for the detection, characterization and manipulation of
individual spin qubits \cite{ucla}, the transistor conductance
providing a way to achieve electronic read-out \cite{rabi}. The
large charging energies could allow redox-active molecules to
operate as storage centers for memory \cite{misra}. Finally,
rectification is important to avoid parasitic pathways in cross-bar
logic based architectures \cite{stan}.

The accurate treatment of well-resolved excitations is crucial to
molecular operation in this scattering regime. But the price paid
for this accuracy is the loss of simplicity associated with orthodox
theory. Instead, we will need a major improvement in computational
algorithms to handle the exponential scaling of the many-body Fock
space, such as partial configuration interaction (CI) with frontier
orbitals coupled to the standard master equations. Major challenges
involve the proper inclusion of the contact boundary conditions, and
doing justice to broadening \cite{rWacker}. Also, one could
incoherently sum selected excitations and resolve the relevant ones,
if we have some prior idea which ones could be important. {\it{Spin
unrestricted approaches may capture some of the salient effects,
although they are unlikely to capture the current heights (which
depend on difficult parameter-independent combinatorial arguments),
and even more significantly, the number of current plateaus (since
an energy-independent one-electron potential does not generate
enough poles).}} Needless to say, there is enormous room for
theoretical activity and device potential in this domain.

It is worth emphasizing that the wide success and popularity of the
SCF-NEGF approach often leads researchers to automatically assume
that the method will work in the CB regime \cite{revers,rpal}. The
NEGF method is indeed prolific in its flexibility to incorporate
contact microstructure, molecular chemistry, broadening,
intereference and electrostatic details. However, the method, as
widely implemented, is fundamentally limited by the need to work
with a one-electron potential, at best incorporating many-body
effects as corrections to this potential. While this may be
sufficient to capture equilibrium properties like total energy,
transport measurements can potentially probe the rich excitation
spectra that are hard to incorporate readily into the average
electronic potential, even parametrically. For a typical
self-energy, there simply are not enough poles in the one-electron
Green's function to do justice to the many more many-electron
transitions characterizing transport, not to mention the
complications of the full counting statistics in Fock space under
non-equilibrium that show up as effective
{\it{interaction-independent}} degeneracies of the open-shell
levels.

In this paper, we have discussed contact asymmetry as a paradigm to
delve into various experimental features, outlining the qualitative
difference and cross-over between the weakly correlated SCF and the
strongly correlated CB regimes. An understanding of the asymmetry
was provided using a basic model, and then extended to conjugated
molecular systems. Novel experimental-trends were identified in the
CB-regime of transport. Two different approaches, the Fock-space CB
and the orthodox CB, were discussed in detail. While a simpler
orthodox approach captures the salient asymmetric effects, the
Fock-space approach may be essential in examples where the interplay
between specific excitations governs the principle operating physics
and possibly also interesting device applications envisaged.

\section{Acknowledgments}
We would like to thank S. Datta, L. Harriott, G. Scott and H-W.
Jiang for useful discussions. This project was supported by the
DARPA-AFOSR grant.

\section{Appendix I: Extracting parameters for orthodox model}
The low temperature, strong contact asymmetry approximation to the
orthodox theory, as introduced in section III, is most useful
because it simplifies the process of simulating experimental data.
Whereas the general orthodox formalism requires best-fit
computational techniques, Eqs.~(\ref{eq:approxIV}-12) allow for a
straightforward analytic calculation of the system circuit
parameters.  Consider Fig.~(\ref{fig:fig7}) as an example of data to
be simulated with the orthodox theory.  To start, consider the $V_G
= -0.85 V$ I-V curve (light blue).  We have a linear onset at
approximately $19$ $mV$, and a ``jump'' onset at, say, $-24$ $mV$.
Plugging these values into Eqs. (11a) and (12b), respectively, and
solving finds:
\begin{eqnarray}
C_{1}(19mV)=C_{2}(24mV) \mbox{, and} \\
C_1=\frac{e+2Q_{0}+2C_{G}V_{G}}{2(19mV)} \mbox{.}
\end{eqnarray}
Consideration of a second I-V curve in the data gives us further
information; here we will use the $V_G=-0.55 V$ (green) curve, which
has a positive onset voltage of approximately $53$ mV. After
plugging the correct onset and gate voltage values into Eq. (11a)
for both I-V curves considered, one can solve for a new, independent
equation:
\begin{equation}
C_{1}=\frac{\Delta V_{G}}{(53-19)mV}C_{G}=8.82C_{G}
\end{equation}
We now have three equations and four unknowns ($C_1$, $C_2$, $C_G$,
and $Q_0$).  There are now two approaches one can take.  The first,
which was applied in the simulation shown in Fig. 4(b), is to simply
take a small, reasonable value of $Q_0$, such as $-0.05e$, and solve
the remaining equations for the other variables.  However, this will
not always work, for a reason that is somewhat subtle.  It is
important to note in Fig. 4(a) that not only does the $V_G=-0.85 V$
I-V curve have a linear onset at $19$ $mV$, but it also does not
have a jump onset at positive bias until at least $100$ $mV$.
Setting Eq. (12a) greater than $100$ $mV$ provides a fourth
restriction.  Indeed, for the experiment in 4(a) this is trivially
satisfied; Fig.~\ref{fig:Zhitenev}, on the other hand, is an example
of data for which such a fourth equation is necessary for a
successful simulation.

A final parameter one can obtain is $R_2$.  From
Eq.~\eqref{eq:curr_master} we know that the conductances of the
linear portions of orthodox curves have the value
$C_{1}/R_{2}C_{\Sigma}$.  Having already calculated all of the
capacitance values it is thus straightforward to find the slope of
the experimental I-V curve and find $R_2$.  $R_1$ cannot be found so
easily, but its relative unimportance in the regime of contact
asymmetry means that a simple estimate of $R_{1} \sim R_{2}/100$
will usually suffice.

redpajama_set_name: RedPajamaArXiv


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Falcons Primary School

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text:
Pontevedra és un municipi i una ciutat del sud-oest de Galícia, capital de la província de Pontevedra i de la comarca homònima. És una ciutat principalment de serveis, ja que la gran majoria del teixit industrial de la zona es troba a la ciutat veïna de Vigo, que és la més poblada de la província. Té una població de 82.934 habitants (INE 2013) repartits en una extensió de 117 km², sent el sisè municipi més poblat de Galícia.

La ciutat, reconeguda com a capital de les Rías Baixas, es troba a la vora del riu Lérez, en un estuari que forma la ria de Pontevedra.

Història

Llegenda grega
Sobre el naixement de la ciutat de Pontevedra existeix una llegenda d'origen renaixentista que diu que el responsable de la fundació de la vila va ser Teucro, un dels herois de la Guerra de Troia. La llegenda diu que va arribar a aquestes terres després de ser rebutjat pel seu pare Telamó i va fundar un assentament amb el nom d'Helenes i després d'això es va casar amb Helena la filla del Rei Putrech qui en aquell moment dirigia l'exèrcit grec cap a la ciutat d'Atenes.

Origen romà
La veritat és que diversos estudis arqueològics no demostren existència de vida humana abans de l'època romana. El veritable germen de la ciutat és, llavors, l'assentament de Turocqua, fundat en la riba sud del riu Lérez. Aquest assentament romà prosperaria i seria conegut com a Dvos Pontes (Dos ponts, el construït per a fundar Turocqua i un posterior) o per Pontus Veteri (Pont Vell, en referència al primer pont).

Anys daurats
Des de l'Edat Mitjana i fin a la fi del  la ciutat es va convertir en un dels principals ports de Galícia, sent el seu burg un dels més populosos. A partir d'aquest segle la vila entraria en decadència a causa de l'abandó per part de la Corona de Castella primer i de les successives guerres després.

Del XIX a l'actualitat
No seria fins al  quan amb la creació de les províncies i designació de Pontevedra com a capital, la ciutat ressorgís, en aparèixer noves institucions i serveis que van portar població i diners. A principis del , abans de la Guerra Civil espanyola Pontevedra era un formiguer social i cultural, però després de la guerra i els anys d'autarquia, la indústria es va decantar per Vigo, ciutat oberta al mar i amb un dels millors ports del món. Actualment Pontevedra viu una segona joventut, veient com la seva població augmenta en detriment de les altres grans nuclis de Galícia, que veuen com els seus habitants es traslladen als municipis limítrofs.

Geografia
El municipi de Pontevedra té una superfície de 117 km² i una llargada de nord a sud de gairebé 20 km. S'estén per la part sud de la ria de Pontevedra, on ocupa les valls fluvials del Lérez i del riu dos Gafos. Cap al sud, arriba fins a la desembocadura del riu Verdugo, a Ponte Sampaio, ja a la ria de Vigo.

La ciutat està situada al fons de la ria de Pontevedra, a la desembocadura del Lérez. Està assentada en un turó rocós relativament aplanat per l'erosió que, tot i que no és gaire alt, fa que el Lérez el rodegi pel nord abans d'obrir-se a la ria. Per aquest motiu els únics carrers amb un cert pendent són els que baixen a la ribera. La situació resulta estratègica, ja que és el primer punt, venint des del mar, a través de la ria, pel qual es pot creuar de nord a sud.

Limita al nord amb els municipis de Meis, Barro, Moraña i Campo Lameiro; a l'est amb Cotobade i Ponte Caldelas; al sud amb Soutomaior, Vilaboa i Marín, i a l'oest amb Poio i amb la ria de Pontevedra.

Clima
El clima de Pontevedra és suau, amb una temperatura mitjana anual de 15 °C i una amplitud tèrmica de 10 °C, i plujós (1.600-1.800 mm), especialment a final de tardor i l'hivern. A més, està exposat a una sèrie de tempestes atlàntiques durant la tardor i l'hivern, que amb freqüència tenen una certa violència.

Parròquies
El municipi està format pel nucli urbà i per 18 parròquies, la majoria rurals:

Demografia
L'any 2014 tenia una població de 82.946 habitants, sent el segon municipi més poblat de la seva província per darrere de Vigo i el sisè de Galícia. Entre el cens de 1857 i l'anterior augmenta el terme municipal perquè incorpora el de Verducido. Entre el cens de 1877 i l'anterior annexiona Alba, Mourente i Salcedo, entre el de 1950 i l'anterior incorpora a Xeve, i en el cens de 1960 annexiona Ponte Sampaio.

Política

En l'actualitat, l'alcalde de l'ajuntament de Pontevedra és el metge Miguel Anxo Fernández Lores, del Bloque Nacionalista Galego, que governa en coalició amb el PSdeG. Els partits més rellevants a nivell local, a més del BNG són el PPdeG, el portaveu del qual és Jacobo Moreira, i el PSdeG amb Antón Louro.

L'ajuntament de Pontevedra s'estructura en diferents regidories: de cultura i normalització lingüística, de serveis municipals (esports, joventut, transports i il·luminació pública), de medi ambient, de benestar social i participació ciutadana, de promoció econòmica, turisme i comerç i policia local, i d'obres. L'ajuntament celebra plens ordinaris cada mes, tot i que habitualment se celebren plens extraordinaris, amb l'objectiu de tractar temes i problemes que afecten al municipi.

La corporació municipal té 25 membres. D'ells, 14 formen la junta de govern en funcions (composta pels 11 regidors del BNG i els 3 del PSdeG) i els altres 11 són del PPdeG.

Economia
La major part dels ciutadans de Pontevedra treballen en el sector serveis: la ciutat, com a capital de província que es compta amb nombroses oficines administratives i instal·lacions que requereixen molts funcionaris. A més és destacable la presència d'una important base militar al municipi veí de Vilaboa, però que a efectes econòmics repercuteix gairebé en exclusiva en la ciutat, on fan vida la majoria dels militars.

Actualment Pontevedra és un referent en la indústria forestal, i compta amb la fàbrica de pasta de paper més important d'Espanya. A més s'està esperant l'aprovació d'una paperera la qual cosa suposaria el tancament del cicle productiu. En el cas de la fàbrica de la pasta de paper, ha estat condemnada per delicte ecològic, el que ha fet més impopular la seva presència a la zona; arribant a celebrar-se manifestacions exigint el seu trasllat (l'última, el 2005, va reunir a 50.000 persones a Pontevedra)

La importància del sector primari disminueix cada vegada més a causa de l'envelliment de la població rural del municipi i de què la seva població està emigrant cap al nucli urbà.

Comunicacions

El nucli antic de la ciutat i bona part de l'Eixample estan formats per carrers per vianants i només poden accedir-hi vehicles de residents i serveis. L'any 2010, Pontevedra va ser la primera capital de província d'Espanya en reduir a 30 quilòmetres per hora la velocitat màxima en el seu nucli urbà. La ciutat s'ha convertit en els últims anys en un referent de la reforma urbana. A més, el fet que el Lérez travessi la ciutat ha propiciat la construcció de nombrosos ponts, com els d'A Barca, As Correntes, O Burgo, Santiago i Os Tirantes.

La ciutat està molt ben comunicada amb l'exterior. Per carretera, la via més important és l'autopista de l'Atlàntic (AP-9), que travessa tota la comunitat des de Ferrol fins a Vigo passant per la ciutat. Altres carreteres importants són la N-550, que té un recorregut paral·lel a l'AP-9, la N-541, que uneix la ciutat amb Ourense, la PO-546 cap a Marín, la PO-308 i el corredor CG-4.8 cap a O Salnés, la PO-531 cap a Vilagarcía de Arousa i la PO-532 cap a Ponte Caldelas.

Per ferrocarril, l'estació de Pontevedra es troba dins l'eix atlàntic d'alta velocitat, que uneix la ciutat amb les principals ciutats gallegues de la façana atlàntica, i també té serveis de llarga distància a Madrid i Alacant. A més serveix d'enllaç ferroviari amb el port de Marín. Hi ha també un petit baixador, anomenat Pontevedra-Universidade, que dona servei al campus.

L'estació d'autobusos de Pontevedra connecta la ciutat amb multitud de destinacions autonòmiques, estatals i internacionals. El servei interurbà d'autobús que comunica amb Marín, gestionat per l'empresa Monbus, és conegut com a "trole", per l'antic troleibús que feia el mateix recorregut.

Llocs d'interès

La major part dels monuments es troben a la zona monumental, en el recinte que antigament estava emmurallat. Actualment la zona és només accessible a vianants. Cal destacar:

 L'Alameda, gran zona verda on hi ha la Creu dels Caiguts i el Monuments als Herois de Ponte Sampaio. També hi ha el Parc Municipal das Palmeiras, anomenat així per les seves altes palmeres.
 La Casa Consistorial, construïda sobre restes de les antigues cases consistorials (segles XV i XVI).
 El Pazo de la Diputació Provincial de Pontevedra.
 Les runes de San Domingos de Pontevedra.
 La Basílica de Santa Maria la Major, del .
 La Capella de la Verge Pelegrina.
 La Praza do Teucro, plaça rodejada de cases senyorials dels segles  i XVIII. A prop hi ha la Capella d'O Nazareno, el Teatre Principal, el Liceu Casino i la Casa das Campás, construcció civil més antiga de la ciutat.
 El Pazo de Mugartegui, a la Praza da Pedreira.

Educació
La ciutat compta amb diversos col·legis i instituts tant públics com concertats i privats. Existeix també un petit campus depenent de la Universitat de Vigo, situat en la riba nord del Lérez a la sortida cap a La Corunya, on es troben les Escoles Universitàries d'Enginyeria Forestal i Fisioteràpia; i la Facultat de Ciències Socials i Comunicació. Fora del campus estan la Facultat de Ciències de l'Educació amb un Pavelló d'Esports propi, l'Escola Universitària d'Infermeria en l'Hospital Provincial i la Facultat de Belles Arts en una antiga caserna al costat de l'Albereda. Dintre de la Facultat de Belles Arts s'ha implantat un nou Graduat en Disseny i Moda. A més existeixen diversos centres de Formació Professional en els quals es poden cursar la gran majoria de cicles formatius. També compta amb un Conservatori Professional de Música de nova construcció en una de les zones en expansió de la ciutat cap a l'est.

Festes

El dia 20 de gener se celebra la festivitat de San Sebastián, patró de la ciutat, i el 18 de desembre, la de la Virxe da O, patrona de la ciutat.

No obstant, les festes més importants són les de la Virxe Peregrina, patrona de la província, que se celebren entre el segon dissabte i el tercer diumenge del mes d'agost acompanyades d'una festa taurina. El segon diumenge d'agost és el dia d'A Peregrina i se celebra una processó que acaba amb l'actuació del grup de dansa Duos Pontes amb el seu tradicional ball de cintes, deixant a l'aire un colom al final d'aquest.

També tenen repercussió les festes de Santiaguiño (25 de juliol, a la parròquia d'O Burgo), la romeria de San Benitiño de Lérez (11 de juliol, a la parròquia de Lérez) i la Feira Franca (primer dissabte de setembre).

Ciutats agermanades
Pontevedra està agermanada amb les següents localitats:

Fills il·lustres
 Manuel Cuña Novás (1924-1992), poeta i polític socialista
 Manuel Quiroga Losada (1892-1961), violonista
 Carlos Ulloa Varela (1849-1887) cantant d'òpera.
 Huberto González (1868-1923) violinista.

Referències

Enllaços externs
 Ajuntament de Pontevedra

redpajama_set_name: RedPajamaWikipedia


text:
Q: Why is this throwing a java.util.InputMismatchException? Here is my code:
    public static void readFile() throws FileNotFoundException {
        Scanner scan = new Scanner(file);
        scan.useDelimiter("/|\\.");
        scan.nextLine();

        GrocerieList.foodList.add(scan.next());
        GrocerieList.foodAmount.add(scan.nextInt());

        System.out.println(GrocerieList.foodList);  // this is for testing
        System.out.println(GrocerieList.foodAmount); // also for testing
    }

This is the code for foodAmount:
static ArrayList<Integer> foodAmount = new ArrayList<>();

The file that holds holds my input is a simple .txt and the input looks like this:
eggs
bread.
3
6.

Now there shouldn't be a mismatch in my mind because GrocerieList.foodAmount.add(scan.nextInt()); clearly picks up two integers?!

A: scan.useDelimiter("[/|\\.\n]");

fixes it.
if you also add a scan.nextLine() after adding to the foodList to consume the period.
See Scanner delimiter not working as expectedwith input file (Java)
That fixes the error. It remains true that you are currently reading and storing only bread and only reading the 3 and not the 6.

redpajama_set_name: RedPajamaStackExchange


text:
About fpdsouthflorid
So far fpdsouthflorid has created 11 blog entries.
Harris Deutch Letter Requesting Supplemental Funding for Public Defense (4.8.20)
Text of the letter is below; you may read the original letter in PDF format. The Honorable Nancy Pelosi Speaker House of Representatives Washington, DC 20515 The Honorable Mitch McConnell Senate Majority Leader United States Senate Washington, DC 20510 The Honorable Kevin McCarthy Minority Leader House of Representatives Washington, DC 20515 The Honorable [...]
fpdsouthflorid2020-04-09T16:40:35+00:00April 8th, 2020|
Assistant Federal Public Defender, Miami Office
The Federal Public Defender for the Southern District of Florida is accepting applications for the position of Assistant Federal Public Defender in our Miami office. The Federal Public Defender, a branch of the United States Courts, an Equal Opportunity Employer, operates under the authority of the Criminal Justice Act, 18 U.S.C. §3006A, to [...]
fpdsouthflorid2018-04-10T14:44:34+00:00April 10th, 2018|
United States v. Maxi, No. 15-13182
In United States v. Maxi, No. 15-13182 (Apr. 5, 2018) (Martin, Jordan, Walker), the Court affirmed the defendants' drug-trafficking convictions. As to one defendant, the Court affirmed the denial of his motion to suppress the search of a duplex. The Court first concluded that the defendant had standing to challenge the search because, [...]
United States v. Carroll, No. 16-16652
In United States v. Carroll, No. 16-16652 (Apr. 5, 2018) (Wilson, Dubina, Goldberg), the Court affirmed in part and reversed in part the defendant's child pornography convictions. First, the Court concluded that a warrant affidavit supported the conclusion that there was probable cause to believe that child pornography would be found in the [...]
United States v. Harris, No. 16-17646
In United States v. Harris, No. 16-17646 (Jill Pryor, Hull, Proctor) (Apr. 2, 2018), the Court affirmed the defendant's witness tampering and obstruction convictions over four evidentiary challenges. First, the Court rejected the defendant's Rule 403 argument that it was unduly prejudicial for the government to introduce evidence of a murder for which [...]
fpdsouthflorid2018-04-10T14:58:21+00:00April 2nd, 2018|
United States v. Lee, No. 16-16590
In United States v. Lee, No. 16-16590 (Apr. 2, 2018) (Martin, Jordan, Ginsburg) (per curiam), the Court re-affirmed its many circuit precedents holding that Florida robbery -- of all varieties and all years of conviction -- satisfies the elements clause of the ACCA and Sentencing Guidelines. The panel recognized that the defendant's [...]
United States v. Green, No. 14-12830
In United States v. Green, No. 14-12830 (Sept. 29, 2017) (Julie Carnes, Jordan, Robreno), the Court revised its earlier panel opinion from December 2016 in light of the recent en banc decision in Vail-Bailon. Relying on Vail-Bailon, it held that the defendant's Florida felony battery conviction satisfied the elements clause. The panel removed [...]
fpdsouthflorid2018-03-06T22:04:37+00:00September 29th, 2017|
Griffith v. United States, No. 15-11877
In Griffith v. United States, No. 15-11877 (Sept. 26, 2017) (Ed Carnes, Rosenbaum, Dubina), the Court concluded that the district court erred by failing to hold an evidentiary hearing a 2255 motion alleging ineffective assistance of counsel. The motion alleged that trial counsel was ineffective by failing to argue that some waste materials [...]
Beeman v. United States, No. 16-16710
In Beeman v. United States, No. 16-16710 (Sept. 22, 2017) (Julie Carnes, Edmondson, Kathleen Williams), the Court held that an initial 2255 movant bringing an ACCA Johnson claim failed to meet his burden of proof, because he could not prove that it was more likely than not that the sentencing court relied on [...]
fpdsouthflorid2018-03-06T22:06:22+00:00September 22nd, 2017|
United States v. Williams, No. 16-16444
In United States v. Williams, No. 16-16444 (Sept. 20, 2017) (Tjoflat, Hull, William Pryor) (per curiam), the Court affirmed the denial of the defendant's motion to suppress. First, the Court concluded that the search of an outbuilding adjacent to the defendant's residence was lawful, because the search was a reasonable entry pursuant to [...]


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Send me a message below and I will be in touch as soon as possible.


text:
The Surge 2 Review: Better Than The Original?
in Best Games, Game News, Gaming
by Andre Kurniawan 53 Views 0 Comments
The nature is a unique view you can see among all the repetitive city streets and rooftops.
The Surge 2 is still a satisfying grind.
The Surge 2 features the ability to create loadouts and swap them at will.
The Surge 2 Review Verdict
Creating a good sequel can be hard. It has to draw fans interest like the original did, and also brings new ideas to the game. Unfortunately, there are sequels which have great quality but doesn't give good impression like the prequel. One of them is the recently released sci-fi action game, The Surge 2.
The Surge 2 is able to retain what worked in the prequel and improve it with new updates. However, some environmental design and parts of the plot eliminate some of the original's appeal.
The Surge 2 takes place in the ruined Jericho City, a city close to the CREO facility, after a mysterious plane crash which unleashed the sentient machines called Nanites and made the city cordoned off. All of the five main areas in the game have their own different look. But if we look at the details, we can see that they are sharing the same urban wasteland concept.
Instead of their actual design, the five different areas are divided by color and lighting. We can see how the port is rusty and wet and the hospital is dark. Some situations in the game bring several unique feels to each area, but overall they still look similar. The only area which can give different impression to the environment is Gideon's Rock, with its hills and rocky canyons.
However, The Surge 2 wins on the mechanics and gameplay aspects. The combat is still fun and challenging, full of suspense, and the grinding is still satisfying like the prequel. You'll see that you know the drill when you play this game. Getting through enemies, find some stuff, kill a boss, get back to a safe zone or find your way to a new medbay, leveling up and upgrading your gears, and so on. Yeah, it's kind of repetitive, but an action RPG fan must know that those are part of the enjoyment.
The Surge 2 is a nice blend of arcade strategy hack-and-slash concept like Dark Souls with some sci-fi vibes. It also introduces the ability to create loadouts and swap them at will. You can use certain equipment for certain situations without bothering with the inventory to much. Another cool addition is the drone attachment that allows you to spray graffiti tags around the world. Similar to notes in Bloodborne or Dark Souls, it can alert players to the presence of hidden enemies or treasure.
The Surge 2 leans more heavily into the RPG stuff than its prequel. But it's something of a bummer that I didn't find the main story of The Surge 2 to be particularly engaging. It felt like the story was trying to raise the stakes too high when compared to the relatively small scale of the original.
The Surge 2 manages to recapture those what made the original Souls-style sci-fi adventure fun. However, the drawbacks made the sequel an overall lesser game than the original. Uninspired environmental design and overly ambitious story choices rob it of the same level of satisfaction, even with the mechanical improvements. That said, if you're just here to have fun hacking and slashing, you're probably gonna have a great time. But it will probably spend most of your playthrough remembering the prequel.
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