text:
\section{Introduction}

The promised end of Moore's law \cite{Moore} brings attention to the so-called ``beyond complementary-metal-oxide semiconductor (CMOS)'' technologies. One of them is the superconductor technology distinguished by high energy efficiency and high clock frequencies \cite{Holmes,Tolp,Beil,IRDS2020}. It is considered to be especially suitable for ``cold electronics'' operating in the gradient between room temperature and temperature of cryogenic payloads like quantum computers, quantum internet, or scalable sensors \cite{DeBen}. Since the 1980s, superconducting circuits have been consistently developed reaching a fairly mature level at the end of the 1990s, showing an implementation of digital and mixed-signal devices \cite{Mukh1,Mukh2,Mukh3}.

However, further progress has slowed down. The main reason was the low integration density. It caused, e.g., implementations of superconducting memory to be of impractical storage capacity \cite{SQMem} that in turn impeded the realization of superconducting processors. Nevertheless, a decade later the researches were whipped up by US government investments aimed at the development of a notional prototype of a superconducting computer \cite{IARPA}. Its implementation is to show a prospective application of superconductor technology in the field of supercomputing. Unfortunately, along with significant overall progress in design and fabrication, we have to admit that the functional density of superconducting circuits is still quite low. The recently demonstrated benchmark circuits for the modern state-of the-art 150~nm of Massachusetts Institute of Technology Lincoln Laboratory (MIT LL) processes are the shift registers with $1.3 \times 10^7$ Josephson junctions (JJs) per square centimeter circuit density \cite{TolpygoASC}. This can be compared with the achieved $\sim 1.3 \times 10^{10}$~Tr/cm$^2$ density of transistors in the most advanced 5~nm process node of the metal–oxide–semiconductor field-effect transistor (MOSFET) technology.

One of the most difficult puzzles of the scaling is the reduction of the size of the Josephson junction \cite{Tolp}, which is a nonlinear element of superconducting circuits. In comparison with the modern transistor, the area of the junction is more than two orders of magnitude greater nowadays.

In this paper, we examine the most common types of Josephson junctions on their scalability. We argue that a planar SN-N-NS junction (S stays for a superconductor, and N - for a normal metal) with variable-thickness bridge geometry is promising for miniaturization. We calculate temperature dependencies of the supercurrent and the characteristic voltage of the SN-N-NS junction for various transparencies of the SN interfaces. Further, we consider limitations on the junction critical current providing stable  superconducting state in the S-electrodes. We also discuss the heat balance in the SN-N-NS structure and estimate the heating during junction operation. Finally, we discuss the achievable range of areas of the studied structures and possible ways of their fabrication.

\section{Scalability of Josephson junctions}

A workhorse of digital superconductor technology is a sandwich Superconductor/Insulator/Superconductor (SIS) Nb/Al-AlO$_x$/Nb Josephson junction. Due to the good wetting of Nb with Al, one can obtain a high uniformity of barrier transparency which provides just a few percents of the technological spread \cite{MITech} of its parameters: critical current, $I_c$, and normal state resistance, $R_n$. The routinely achieved critical current density of the junctions, $j_c = 0.1$~mA/$\mu$m$^2$, corresponds to a small insulator thickness still providing the homogeneous SI boundary. The critical current of Josephson junction should exceed the noise current, $I_n = (2\pi/\Phi_0) k_B T$ (where $k_B$ is the Boltzmann constant, $T$ is the temperature, $\Phi_0 = \pi\hbar/e$ is the flux quantum, $\hbar$ is the reduced Planck constant and $e$ is the electron charge), by about three orders of magnitude for low enough bit error rate. At the same time, the energy dissipation during junction switching is proportional to the critical current, $E_J \approx I_c\Phi_0$, that makes its high values undesirable. For standardly used helium temperature, $T = 4.2$~K, it turns to $I_c \approx 0.1$~mA, and area of the junction, $a = I_c / j_c \approx 1~\mu$m$^2$.

Since the sandwich type of tunnel Josephson junction possesses relatively high self-capacitance, $c \geq 60$~fF/$\mu$m$^2$, the resistive shunt, $R_s$, is always used to damp Josephson oscillations occurring after the junction switching. A workable operation regime is achieved with Steward-McCumber parameter value, $\beta_c = (2\pi/\Phi_0) j_c c R^2_s a^2 \approx 1$. Corresponding $R_s \approx 5$~$\Omega$ is commonly implemented with MoN$_x$ having resistance $\sim 5$~$\Omega$ per square \cite{MoNx}. Taking into account interconnect area with minimum wiring feature size (ca. 0.5-1~$\mu$m) the shunt about triples the total area of the junction. The parallel combination of the shunt and the internal resistance provides sum-mV characteristic voltage and sub-THz characteristic frequency of the junction .

A natural approach to scale the junction area below micron size is to increase the critical current density. Starting from $j_c = 0.5$~mA/$\mu$m$^2 $ the junction becomes self-shunted \cite{TolpSSJ,TolpSSJ1}. For example, the most functionally dense Random-Access-Memory (RAM) circuit fabricated recently \cite{STMem} were based on the Josephson junctions having $j_c = 0.6$~mA/$\mu$m$^2$ with no shunt resistors \cite{TolpSSJ1}, whilst minimum junction area was about $a \approx 0.2~\mu$m$^2$ ($\beta_c \approx 2$). However, the critical current density increase corresponds to the decrease of tunneling barrier thickness and according increase of its inhomogeneity in coordinate and momentum space. The influence of fluctuations in AlO$_x$ transparency (caused, e.g., by pin holes and localized states) on spread of Josephson junction critical currents becomes even more pronounced with the decrease of the junctions area . Since complex circuits require the uniformity of junctions, this scaling strategy is practically limited to the junction size of a few tenths of a micron.

Double-barrier, SINIS, Josephson structures were proposed to mitigate the difficulties of the scaling \cite{DBJJ,BCGKSR2001}. If the N-layer thickness, $d_n$, is much smaller then the decay length,
\begin{equation}
d_n\ll \xi_{n}=\sqrt{\hbar D_{n}/2 \pi k_B T_{c}}
\end{equation}
(dirty limit), where $T_{c}$ is the superconductor critical temperature, $D_{n}=v_{F}l/3$ is the diffusion coefficient, $v_{F}$ is the Fermi velocity and $l$ is the electron mean free path in the normal metal, then the cotunneling across INI weak link may prevail over sequential SIN$+$NIS tunneling process resulting in $I_{c}R_{n}$ product even larger than that of SIS junction with the same $I$ barrier \cite{kuprlukichevA,galaktionov_zaikin2002,SINISnote}. Simple estimations show \cite{DBJJ} that even the formation of small pin holes of a diameter $d_{ph}\ll \xi_{n}, ~d_n$, will not lead to large spread of the junction parameters if $\sqrt{d_{ph}\xi_{n}}\ll d_n$.

Unfortunately, due to difference in Al growth over Nb and AlO$_{x}$ surfaces, the implementation of equal barriers in Nb/AlO$_{x}$/Al/AlO$_{x}$/Nb structure appeared to be intractable task with increase of the barriers transparency \cite{BCIKSBGKR2003,TBGK2003}. The structure asymmetry led to localization of the weak link at one of the barriers that cancelled all the benefits, making this approach impractical for scaling \cite{TBGK2003}. The implementation of the much-needed symmetry of the transparency of the tunnel layers required the development of two different technological processes of aluminum oxidation. In the absence of such technology, the SINIS structures turned out to be useful only for implementation of Josephson voltage standards \cite{PhysC2002,Behr2005,Mueller2007}.

Another option is to artificially synthesize the interlayer material by doping semiconductor up to a degenerate state \cite{SDS,GKL1988,Kulikov1991,Baek2006,Baek2007,Olaya2008,Olaya2009,Chong2009,Muller2013,Olaya2014,Olaya2015,Cao2018,Olaya2019,Haygood2019,Olaya2019a}, e.g., using $\alpha$-Si, where Nb or W can be chosen as dopants.
The current transport in Nb/$\alpha$-Si/Nb (SDS, where D stays for doped semiconductor) junction is determined by elastic and inelastic resonance tunneling processes. The latter form quasi-one-dimensional channels with metallic conductivity and provide internal junction shunting. While $I_{c}R_{n}$ product of sub-millivolt level is readily attainable \cite{SDS,Olaya2015,Olaya2019,Haygood2019,Olaya2019a}, the inherent inhomogeneity of the barrier transparency prohibits the scaling of such SDS structure with preservation of the junction parameters reproducibility due to the probabilistic nature of the formation of resonant channels for the flow of normal and superconducting currents.

At nanometer scale, where area of Josephson junction is $a \sim 0.01~\mu$m$^2$, the Josephson junction critical current should be $j_c \sim 10$~mA/$\mu$m$^2$, taking into account the required critical current $I_c = 0.1$~mA at the working temperature $T = 4.2$~K. Such high values of the critical current density together with the considered inhomogeneity of the barriers of SIS and SDS junctions makes SNS junction to be the preferred type for scaling.

SNS structure obviously does not require shunt resistor. It is important to choose N-material so as to minimize the suppression of superconductivity in S-electrodes while maximizing the induction of superconductivity in N-interlayer for high values of $I_{c}R_{n}$ product and critical current, correspondingly. For transparent SN interface the proximity effect is described \cite{Ivanov1981,Kupriyanov1982} by the suppression parameter,
\begin{equation}
\gamma =\frac{\rho_{s}\xi_{s}}{\rho_{n}\xi_{n}},~   \xi_{s}=\sqrt{\frac{\hbar D_{s}}{2 \pi k_B T_{c}}},
\end{equation}
where $\xi_s$ is decay length in S-material and $\rho_{s,n}$ are normal resistivities of S- and N-materials. The desired value, $\gamma <1$, means that the amount of normal electrons diffusing per second from N to S is smaller than the same value of correlated electrons moving in the opposite direction. This case is commonly realized with pairs of materials like Nb (S) and Ti, Hf or Pd$_{x}$Au$_{1-x}$ (N), where $\xi_{n}\lesssim \xi _{s}\approx 10$~nm while resistivity of normal metals is larger or of an order of the one of niobium, $\rho _{n}\gtrsim \rho_{s}\approx 8~\mu \Omega ~$cm. For sandwich type  SNS junction and the N layer thickness larger than $\xi_{n}$, it is necessary \cite{Kupriyanov1982} to have $\gamma \lesssim 0.1$ for achieving a high value of $I_{c}R_{n}$ product at $T/T_{c}\approx 0.5$. However, this value of suppression parameter is difficult to implement. It is, for this reason, the experimentally obtained magnitudes of characteristic voltage is rather small \cite{DBJJ,SNS,PhysC2000SNS,Hagedorn2001,Schubert2001,Hagedorn2002,Dresselhaus2003,Hagedorn2003,schubertt2005,Hagedorn2006,Nagel2011}. Note that a simple decrease in the N layer thickness is not a solution of this problem. In addition to technological difficulties, the decrease in N layer thickness leads to spatial delocalization of the weak link region due to the depairing process in the vicinity of the interfaces \cite{Ivanov1981,Zubkov1983,Kupriyanov1992}.

\begin{figure}[t]
\includegraphics[width=0.8\columnwidth,keepaspectratio]{Fig1.eps}
\caption{Sketch of SN-N-NS Josephson junction with variable thickness bridge geometry.}
\label{Fig1}
\end{figure}

This problem is inherent for commonly used sandwich-type Josephson junctions. However, it can be circumvented by modification of the weak link region geometry with corresponding redistribution of current flowing therein \cite{Likharev1971,Likharev1975,Likharev1976,Zaikin2001,Vodolazov2020}. An example of such solution is the SN-N-NS junction in the form of variable thickness bridge \cite{Warlaumont1979,Beasley1981,Liengme1983,Nakano1987,Lukens1985,Baryshev1989a,Baryshev1989} presented in Fig.~\ref{Fig1}. Here a high current density in the bridge is lowered due to the current spread all over the area under the electrodes. The latter is owing to the finite transparency of SN interfaces \cite{Lukens1985,Zehnder1999,Pratt1999,Pratt2000}.

Metals with relatively large decay length are preferred to obtain high critical current value of SN-N-NS junction. Unfortunately, such metals generally possess relatively small resistivity, since $\xi_{n}\sim \sqrt{D_{n}}$ while $\rho_{n}\sim 1/D_{n}$. However, if the thickness of N layer is smaller than the decay length, $d_{n}\ll \xi_{n}$, then the suppression parameter is decreased \cite{Golubov1983} proportionally to their ratio,
\begin{equation}
\gamma_{M}=\gamma \frac{d_{n}}{\xi_{n}},
\end{equation}
corresponding to the decreased number of normal electrons diffusing from N to S. Thus, the SN-N-NS structure size ratio, $d_{n}\ll \xi_{n} \approx L_{b}$, seems to be optimal choice to provide both: high critical current and normal state resistance of the junction.

Below we consider SN-N-NS junction with specified ratio of geometrical parameters as the promising candidate for scaling, since the decay length, $\xi_n$, of the broadly utilized metals, like Al or Cu, lies in the range of a few tens of nanometers.

\section{Calculation of SN-N-NS junction supercurrent}

Our model of SN-N-NS junction contains a normal metal film connecting two massive superconducting electrodes of the length, $(L-L_b)/2$ each, located at the distance $\pm L_b/2$ from the center of this film, see Fig.~\ref{Fig1}. The total length of the junction is $L$. In the calculation of the critical current, we suppose that condition of dirty limit is fulfilled for all metals, the critical temperature of N material is equal to zero and its width, $W$, and thickness, $d_n$, are much smaller than Josephson penetration depth, $\lambda_J$ , and decay length, $\xi_n$, respectively.

The proximity effect in this system can be considered in the frame of Usadel equations \cite{Usadel}, which in the N film have the form,
\begin{equation}
\xi^{2}_n \frac{\partial }{\partial x}\left( G_{n}^{2}\frac{\partial \Phi_{n}}{\partial x}\right) +\xi^{2}_n\frac{\partial }{\partial y}\left( G_{n}^{2}\frac{\partial \Phi_{n}}{\partial y}\right) =\omega G_{n}\Phi_{n}.
\label{equTetaNF}
\end{equation}
Here $\Phi_{n},$ $G_{n}=\omega /\sqrt{\omega ^{2}+\Phi_n\Phi_{n}^{\ast }}$ are modified Usadel Green's functions, and $\omega =(2m+1)T/T_{c}$ are Matsubara frequencies, $m$ is integer. $\Phi_{n},  $ are normalized on $\pi T_c$.

We apply standard boundary conditions at the edges of the structure \cite{Suppl} and Kupriyanov-Lukichev boundary condition \cite{kuprlukichevA} at SN interfaces. The conditions, $\gamma_{M}\lesssim 0.3$, $d_{n}\ll \xi_{n}$ \cite{Golubov1983}, permit us to neglect the suppression of superconductivity in the S film due to proximity effect even in the case of fully transparent SN interfaces, and also allows considering functions $\Phi_{n}$ independent on the coordinate $y$, in the first approximation on the parameter $d_{n}/\xi_n$.

By integration of equations (\ref{equTetaNF}) over the coordinate $y$, we obtain the equations,
\begin{equation}
\frac{\gamma_{BM}\xi^{2}_n }{G_{n}}\frac{\partial }{\partial x}\left( G_{n}^{2}\frac{\partial \Phi_{n}}{\partial x}\right) -\left( G_{s}+\gamma_{BM}\omega\right) \Phi_{n}=-G_{s}\Phi_{s},  \label{EqUnderSr}
\end{equation}
for the region of electrodes, $|L_b/2|\leq |x|\leq |L/2|$, where $\gamma_{BM}=\gamma_{B}d_{n}/\xi_{n} $, $\gamma_{B}=R_{B}/\rho_{n}\xi_{n}$ is suppression parameter, $R_{B}$ is specific resistance of SN interface, $\Phi_{s}, $ are modified Usadel Green's functions normalized on $\pi T_c$, $G_{s}=\omega /\sqrt{\omega ^{2}+\Phi_s\Phi_{s}^{\ast }}$. In the area of the bridge we have the following equations,
\begin{equation}
\xi^{2}_n \frac{\partial }{\partial x}\left( G_{n}^{2}\frac{\partial \Phi_{n}}{\partial x}\right) =\omega G_{n}\Phi_{n}.  \label{EqOutS}
\end{equation}

The solution of the problem is simplified in the limit of a small gap between superconducting electrodes, $L_b\ll \xi_{n} $. We also suppose that in the practically interesting case, the length of superconducting electrodes is much larger than the characteristic scale, $L-L_b \gg \zeta$, which arise in the frame of analytical solution of the problem \cite{Suppl}. This characteristic scale is $\zeta \approx \sqrt{\gamma_{BM}} \xi_{n} $  in the limit of small $\gamma_{BM}$ for arbitrary normalized temperature, $t = T/T_c$. In the opposite limit of large $\gamma_{BM}$, it is  $\zeta \approx \xi_{n}$ at small temperatures and $\zeta \lesssim \xi_{n}$ at $t\gtrsim 0.5$.

Under these assumptions, we obtain \cite{Suppl} the following expression for the product of superconducting current, $J_s$, across the junction and its normal state resistance, $R_n$,

\begin{equation}
\frac{eR_{n}J_{s}}{2\pi k_B T_{c}}=t\sum_{\omega \geq 0}\frac{2\re\Phi_{n}\left(1+2\eta \right) }{\sqrt{\omega^{2}+(\re\Phi_{n})^{2}}}C,
\label{Js1}
\end{equation}
where $\eta = \sqrt{\gamma_{BM}} \xi_{n}/L_b$, C are the constants determined by matching solutions in the ranges of electrodes and the bridge \cite{Suppl}, and $\re\Phi_n$ is the real part of the functions $\Phi_n$, while
\begin{equation}
R_{n}=R_{nb}+2R_{sn} \label{Rjun}
\end{equation}
is the sum \cite{Karminskaya2010} of the resistance of the bridge, $R_{nb}$, and the resistances of two SN interfaces, $R_{sn}$,
\begin{equation*}
R_{nb}=\frac{\rho_{n}L_b}{d_{n}W},~R_{sn}=\frac{\rho_{n}\xi_{n}\sqrt{\gamma_{BM}}}{d_{n}W}.
\end{equation*}

In the limit of small suppression parameter,
\begin{equation*}
\sqrt{\gamma_{BM}} \ll L_b/\xi_{n} \ll 1
\end{equation*}
(rigid boundary conditions), the expression (\ref{Js1}) for supercurrent takes the following form,
\begin{equation}
\frac{eR_{n}J_{s}}{2\pi k_B T_{c}}=t\sum_{\omega \geq 0}\frac{2\Delta \cos \frac{\varphi }{2}}{\Omega_{1}}\arctan \frac{\Delta \sin \frac{\varphi }{2}}{\Omega_{1}},  \label{Js1KO}
\end{equation}
where $R_{n} = R_{nb}$, $\Delta$ is the magnitude of superconducting order parameter normalized on $\pi T_{c}$, $\varphi$ is superconducting phase difference across the junction, and
\begin{equation*}
\Omega_{1}=\sqrt{\Omega^{2}+\Delta^{2}\cos^{2}\frac{\varphi}{2}},~\Omega = \omega\left(1+\gamma_{BM}\sqrt{\omega^{2}+\Delta^{2}}\right).
\end{equation*}
For a vanishingly small suppression, $\gamma_{BM}\rightarrow 0$, the expression (\ref{Js1KO}) transforms into the formula obtained by Kulik and Omelyanchuk (KO-1) \cite{KO1}.

In the opposite limit of large $\gamma_{BM}$,
\begin{equation*}
\frac{L_b}{\xi_{n}} \ll \frac{\gamma_{BM}}{(1+\gamma_{BM})},
\end{equation*}
expression (\ref{Js1}) transforms into
\begin{equation}
\frac{eR_{n}J_{s}}{2\pi k_B T_{c}}=t\sum_{\omega \geq 0}\frac{\sqrt{2}\Delta^{2}\sin \varphi}{\Omega_{1}\sqrt{\left(\sqrt{\Omega^{2}+\Delta^{2}}+\Omega_{1}\right)\sqrt{\omega^{2}+\Delta^{2}}}},  \label{current}
\end{equation}
where $R_{n}$ is mainly determined by the resistance of SN interfaces.

\section{$I_cR_n$ product of SN-N-NS junction}

\begin{figure}[b]
	\includegraphics[width=1\columnwidth,keepaspectratio]{Fig2.eps}
	\caption{Normalized $I_cR_n$ product, $v_c = eI_cR_n/2\pi k_BT_c$, of SN-N-NS Josephson junction in dependence on (a) normalized temperature, $t$, and (b) 		suppression parameter, $\gamma_{BM}$, calculated using (\protect\ref{Js1}) (solid lines), (\ref{Js1KO}) (dotted lines, $\gamma_{BM} = 0.001,~0.01$, panel (a)) and (\protect\ref{current}) (dotted lines, $\gamma_{BM} = 0.3 ... 3$, panel (a) and panel (b)). Inset shows $v_c(\gamma_{BM})$ for $\gamma_{BM} \leq 1$. The upper solid line in	panel (a) corresponds to KO-1 expression \cite{KO1}.}
	\label{Fig2}
\end{figure}

Characteristic voltage of SN-N-NS junction can be obtained directly from expression (\ref{Js1}) and its limiting cases (\ref{Js1KO}), (\ref{current}). Fig.~\ref{Fig2}(a) shows normalized $I_cR_n$ product, $v_c = eI_cR_n/2\pi k_B T_c$  (where $I_c = \max[J_s(\varphi)]$), as a function of normalized temperature, $t$. The normalizing coefficient value is $2\pi k_B T_c/e \approx 5$~mV for niobium critical temperature, $T_c = 9.2$~K. The bridge length is $L_b = 0.1 \xi_n$. The curves obtained in both limits are well consistent. The presented dependencies are limited from above by the curve obtained using KO-1 expression \cite{KO1} for $\gamma_{BM} = 0$ as expected. The temperature corresponding to characteristic voltage drop by 20\% from its maximum value decreases from $t \approx 0.5$ at small suppression, $\gamma_{BM} \le 0.01$, to $t \le 0.2$ at $\gamma_{BM} \ge 1$.

Fig.~\ref{Fig2}(b) presents $v_c(\gamma_{BM})$ dependence for different temperatures in double logarithmic scale. Inset shows these dependencies for $\gamma_{BM} \leq 1$. The characteristic voltage increases with the temperature decrease, in accordance with Fig.~\ref{Fig2}(a). For the commonly used temperature, $t \approx 0.5$, the characteristic voltage drops from its maximum value, $v_{c\max} \approx 0.46$, by 20\% at $\gamma_{BM} \approx 0.06$, and becomes halved at $\gamma_{BM} \approx 0.25$. Starting from $\gamma_{BM} \approx 1$, the characteristic voltage $v_c(\gamma_{BM})$ falls as $\gamma_{BM}^{-3/2}$ with $v_c < 0.1$. The decrease of the characteristic voltage with the suppression parameter increase is slower with the temperature decrease.

In the considered approximation, the characteristic voltage is nearly independent on the length of the bridge up to $L_{b}$ $\lesssim \xi _{n}$ for arbitrary $\gamma _{BM}$. The critical current, $I_{c}$, is the larger the smaller are $L_{b}$ and $\gamma _{BM}$. The current flowing through the junction ($I \approx I_c$) must not destroy the superconductivity in its electrodes. Below, we formulate the restriction on geometrical parameters of SN-N-NS structure coming from this requirement.

\section{Limitation on the current in superconducting electrodes  }

In accordance with experimental data obtained in MIT LL \cite{TolpygoASC}, a Nb strip with thickness, $d_{s}=200$~nm, and width, $W=250$~nm, possesses the critical current density, $J_{cst}\approx 4.5 \times 10^7$~A/cm$^2$ so that the critical current is $I_{cst} \approx 22.5$~mA at $T=4.2$ K. With an increase of the width, the critical current grows proportional to $\sqrt{W},$ while with $W$ decrease below 250~nm, $I_{cst}$ falls linearly with $W$,
\begin{equation}
I_{cst} = J_{cst}(W-W_{0})d_s\label{denJdep}
\end{equation}
where $W_{0}\approx 50$ nm is a doubled thickness of the contaminated surface layer.

The linear dependence of the critical current on the width means that in the interesting for us range of the thickness, $W\lesssim 250~$~nm, the supercurrent is distributed uniformly over the film width. This fact allows us to seek the solution of the Usadel equations in the depth of the electrodes $
(\left\vert x\right\vert \lesssim L)$ in the form $\Phi _{s,n}(x,y)=\Phi
_{s,n}(y)\exp \left\{ ikx\right\} $ independent on coordinate $z$ (see Fig.~\ref{Fig1}). Here $k$ is an independent on $\omega $ and $x$ constant, which is proportional to superfluid velocity. By assuming further that the supercurrent flowing through the SN electrodes is significantly less than the depairing current, we find functions $\Phi _{s,n}(x)$ from the solution of the proximity effect problem between superconducting and thin normal films,
\begin{equation*}
\Phi _{s}(x)=\Delta ,~\Phi _{n}(x)=\frac{G_{s}\Delta }{G_{s}+\omega \gamma
_{BM}},~G_{s}=\frac{\omega }{\sqrt{\omega ^{2}+\Delta ^{2}}}.
\end{equation*}
For supercurrent densities, $J_{S}$ and $J_{N}$, in the S and N films we get
\begin{equation}
\frac{e\rho _{s}J_{S}}{2\pi k_B T_{c}}=kS_{1},~S_{1}=t\sum_{\omega =0}^{\infty }%
\frac{\Delta ^{2}}{\omega ^{2}+\Delta ^{2}},~  \label{denJinS}
\end{equation}%
\begin{equation}
\frac{e\rho _{n}J_{N}}{2\pi k_B T_{c}}=kS_{2},~S_{2}=t\sum_{\omega =0}^{\infty }%
\frac{\Delta ^{2}}{\omega ^{2}(1+\omega \gamma _{BM}G_{s}^{-1})^{2}+\Delta
^{2}},  \label{denJinN}
\end{equation}%
where parameter $k$ in (\ref{denJinS}), (\ref{denJinN}),
\begin{equation}
k=\frac{e\rho _{s} I_{SN}}{2\pi T_{c}d_{s}WS_1 \left( 1+q\right) }, \label{k}
\end{equation}%
is determined by the magnitude of the full current, $I_{SN}=W\left(
J_{S}d_{s}+J_{N}d_{n}\right) $, flowing through the SN electrodes, while $q$ in (\ref{k}) is the ratio of the currents flowing through the N and S films,
\begin{equation}
q= \frac{J_N d_n}{J_S d_s} =\frac{d_{n}\rho
_{s}}{d_{s}\rho _{n}}~\frac{S_{2}}{S_{1}},  \label{q}
\end{equation}
so that $1 + q = I_{SN}/J_S d_s W$.

At $T\approx 0.5T_{c}$, the Matsubara frequencies $\omega \ $are larger than $%
\Delta .$ We can neglect $\Delta $ in comparison with $\omega $ when
estimating the magnitudes of sums $S_{1,2}$ in (\ref{q}) and get,
\begin{equation*}
~q=\frac{d_{n}\rho _{s}}{d_{s}\rho _{n}(1+\gamma _{BM})^{2}}.~
\end{equation*}

The current flowing through the S film must be small in comparison with S-film critical current,
\begin{equation}
\frac{I_c}{I_{cst}}\frac{J_S d_s W}{I_{SN}} = \frac{I_c}{(1+q)I_{cst}} \lesssim \beta,  \label{demand}
\end{equation}
where $\beta \ll 1$ is the desired smallness coefficient. The magnitude of $I_{c}$ in (\ref{demand}) can be obtained from the data presented in Fig.~\ref{Fig2}, $I_c = v_c 2 \pi k_B T_c/e R_n$. Since $I_c \sim R_n^{-1} \sim d_n$ and $I_{cst} \sim d_s$, the restriction (\ref{demand}) transforms into
\begin{equation}
\frac{\mathcal{C}}{1+q}\frac{d_n}{d_s} \lesssim \beta,
\label{demandLarge}
\end{equation}
where the coefficient $\mathcal{C}$ for niobium is
\begin{equation}
\mathcal{C} = \frac{v_c \times 5~\text{mV} }{J_{cst} \rho_n (L_b + 2\xi_n \sqrt{\gamma_{BM}})}.
\label{Ccoef}
\end{equation}
We should note that for $\rho_n \approx \rho_s$ the ratio of the currents flowing through the N and S films is small, $q \ll 1$, and so $1 + q \approx 1$.

\section{Heat balance  }

\begin{figure}[b]
	\includegraphics[width=1\columnwidth,keepaspectratio]{Fig3a.eps}
	\caption{Sketch of an SN-N-NS Josephson junction with a variable thickness bridge geometry. The horizontal red, purple, and brown arrows indicate the areas where heat is transferred to the S electrodes, phonon subsystems in the normal film and substrate, respectively. The vertical yellow and red arrows schematically indicate the process of heat transfer from hot electrons to the electrodes and the phonon subsystem of a normal film. Vertical green arrows indicate the process of heat transfer into the substrate.}
	\label{Fig3}
\end{figure}

An important requirement to the design of circuits based on the SN-N-NS junction is that the normal component, $I_n$, of the total current must not lead to overheating of the SN-N-NS structure. Here we calculate the power dissipated in the junction during a clock period of a digital circuit and the power that can be absorbed by the junction environment. Then we evaluate the heating of niobium-copper SN-N-NS structure.

\subsection{Dissipated and absorbed power}

The heating effect arises due to Joule dissipation. The energy dissipated during the junction switching, $t_{sw} \approx \Phi_0/I_cR_n$, can be estimated as
\begin{equation}
E=\int_{0}^{t_{sw}}I_nVd \tau,  \label{power2}
\end{equation}
where $V=(\hbar /2e)\partial \varphi /\partial \tau$ is a voltage drop across the junction and $I_n = V/R_n$. By assuming that the phase increase rate is $\partial \varphi /\partial \tau \approx 2\pi/t_{sw}$ and that the junction is switching once in a clock period, $T_{clk}$, we obtain the dissipated energy and average dissipated power,
\begin{equation}
E \approx I_{c}\Phi _{0},~P=E/T_{clk}.  \label{disEn2}
\end{equation}

The energy $E$ is transferred to normal electrons located in the region of concentration of the electric field which includes the bridge between SN electrodes. The hot electrons may diffuse from the heated region on the characteristic time scale $\tau _{esc}\approx L_{b}^{2}/D_{n}.$ For a typical value of Fermi velocity, $v_{F}\approx 10^{6}$~m/s, electronic mean free path, $\ell \approx d_{n}\approx 10$~nm, and the bridge length, $L_b \approx \xi_n \approx 40$~nm, a rough estimation of $\tau _{esc}$ gives the value  $\sim 3\times 10^{-13}$~s. This is an order larger than the scattering time, $\tau_{sc} =\ell /v_{F}\approx 10^{-14}$~s.

Such a short value of $\tau _{esc}$ means that for a time equal to a clock period lying in picosecond time scale, hot electrons diffuse along the N film into the SN electrodes over a distance of the order of the current leakage length, $\xi_{n} \sqrt{\gamma _{BM}}$. This process is shown schematically by the red arrows in Fig.~\ref{Fig3}.

Within the standard model of electron energy relaxation, the electron-electron collision rate is assumed to be large so that the electrons maintain an equilibrium distribution characterized by the temperature, $T_{loc}$, which is different from the lattice temperature $T$. The hot electrons may relax the excess energy making use of the two channels schematically shown in Fig.~\ref{Fig3} by yellow and blue arrows.

The first channel for relaxation is the diffusion of hot electrons into superconducting electrodes. The heat conductance of the SN interface with small specific boundary resistance, $R_{B}$, is given by the expression \cite{Averin1996}
\begin{equation}
P_{sn}=\frac{2\Delta }{e^{2}R_{B}}\mathit{A}\sqrt{\frac{2\pi \Delta }{%
k_{B}T_{loc}}}\exp \left( -\frac{\Delta }{k_{B}T_{loc}}\right)
k_{B}\delta T,  \label{kappaSN}
\end{equation}where  $\delta T=T_{loc}-T,$  $\mathit{A}$ is the effective area through which the heat flow takes place. For a finite transparency of SN interface $%
(\gamma _{BM}\approx 1)$, the area  $\mathit{A}\approx 2\xi _{n}W.$

The second channel is the transfer of excess energy to phonons. In dirty metals, the rate of this process , $P_{e-ph},$ is given by \cite{Belzig}

\begin{equation}
P_{e-ph}=K_{\ast }\left( \frac{3\zeta (5)}{3\pi }\frac{c_{t}^{4}}{c_{l}^{4}}%
\frac{T_{loc}^{5}}{T_{\ast }^{5}}+\frac{4\pi ^{2}}{45}\frac{T_{loc}^{4}}{%
T_{\ast }^{4}}\right) ,~T_{loc}>T_{\ast },  \label{e-phpower}
\end{equation}%
\begin{equation}
K_{\ast }=\frac{N_{0}\tau_{sc} k_{B}T_{\ast }}{d_{fn}\ell ^{5}}\left( \frac{%
p_{F}^{2}}{m_e}\right) ^{2},~T_{\ast }=\frac{\hbar c_{t}}{k_{B}\ell }
\label{K}
\end{equation}%
where $N_{0}$ is the density of states at Fermy level, $c_{l},$ $c_{t},$ are longitudinal and transverse speeds of sound, $d_{fn}$ is the mass density, $p_{F}$ is Fermi momentum, $m_e$ is electron mass, $\zeta (z)$ is the Riemann zeta function. Expression (\ref{e-phpower}) can be rewritten via the thermal conductance per unit
volume $K(T)=dP_{e-ph}/dT_{loc}$
\begin{equation}
P_{e-ph}=\frac{K_{\ast }}{T_{\ast }}\left( \frac{15\zeta (5)}{3\pi }\frac{%
c_{t}^{4}}{c_{l}^{4}}\frac{T_{loc}^{4}}{T_{\ast }^{4}}+\frac{16\pi ^{2}}{45}%
\frac{T_{loc}^{3}}{T_{\ast }^{3}}\right) \delta T_{e-ph},  \label{e-phpower1}
\end{equation}where $\delta T_{e-ph} = $ $T_{loc}-T_{ph}$, $T_{ph},$ is the local temperature of phonon subsystem in the heated segment.

The process of energy transformation from hot electrons to phonons occurs approximately in a volume $\left( L_{b}+2\xi _{n}\right) d_{n}W$ marked by a violet arrow in Fig.~\ref{Fig3}.

Further phonon heat propagation along the N film can be described similar to the heat spread  in an infinite rod from the inner heated segment, $-\xi _{n}-L_{b}/2\leq x\leq \xi _{n}+L_{b}/2
$, which has a temperature $T_{loc}$ exceeding the temperature $T$ of its ends in the initial moment,
\begin{equation}
\begin{array}{c}
\delta T_{e-ph}(x,\tau)=\frac{\delta T_{e-ph}(x,0)}{2}\left[ \mathbf{erf}\left(
\frac{x+\xi _{n}+L_{b}/2}{2\sqrt{\mu _{n}^{2}\tau}}\right) -\right.  \\
-\left. \mathbf{erf}\left( \frac{x-\xi _{n}-L_{b}/2}{2\sqrt{\mu _{n}^{2}\tau}}%
\right) \right] ,%
\end{array}
\label{deltaT1}
\end{equation}
where $\mathbf{erf}(z)$ is the error function, $\mu _{n}^{2}=\lambda
_{n}/C_{vn}d_{fn},$ $\lambda _{n}$ and $C_{vn}$ are thermal conductivity and thermal capacity of the N metal.

During a clock period, $\delta T_{e-ph}$ decreases from its initial value down to
\begin{equation}
\delta T_{e-ph}=\delta T_{e-ph}(x,0)\mathbf{erf}\left( \frac{2\xi
_{n}+L_{b}}{4\sqrt{\mu _{n}^{2}T_{clk}}}\right).  \label{deltaT2}
\end{equation}
The product $\eta _{n}=\mu _{n}\sqrt{T_{clk}}$ determines the characteristic scale of the heat propagation.

The excess phonon temperature may further relax to the substrate (see green arrows in Fig.~\ref{Fig3}). The power transfer from the phonons in the normal metal to the substrate is given by the Kapitza coupling \cite{Swartz}
\begin{equation}
P_{K}(T_{n};T)=K_{k}A_{ph}\left( T_{ph}^{4}(x,t)-T^{4}\right) \approx
4K_{k}A_{ph}T^{3}\delta T_{ph-s},  \label{kapitsa}
\end{equation}
where $\delta T_{ph-s}=T_{ph}-T,$  $K_{k}$ depends on the materials, and $A_{ph}\approx \left( L_{b}+2\xi _{n}+2\eta _{n}\right) W$ is the effective interface area of the heat transfer marked by brown arrow in Fig.~\ref{Fig3}.

\subsection{Heating of SN-N-NS structures based on Nb/Cu material combination}

The SN-N-NS  variable thickness bridge proposed in this work can be fabricated using commonly used materials like Nb, MoRe, V as a superconductor and Cu, Au, Al as a normal metal. Below, we estimate the parameters of NbCu-Cu-CuNb structure in accordance with the restriction (\ref{demandLarge}) and then calculate its heating assuming the clock period, $T_{clk} = 40$~ps (clock frequency, $f_{clk} = 25$~GHz).

Substitution of the typical values \cite{VNbCu} of normal resistivity $\rho _{n}=3.7$~$\mu \Omega $ cm, decay length $\xi _{n}=37$ nm for Cu and $\rho _{s}=8$~$\mu \Omega $ cm for Nb into the obtained restriction on the critical current (\ref{demandLarge}) shows that the inequality can be fulfilled already for $\gamma_{BM} \geq 0.2$ at $T \approx 0.5 T_c$ with $\beta = 0.05$ if $L_b = \xi_n$, $d_n = 10$~nm, and $d_s \gtrsim 200$~nm. Here $\gamma_B = \gamma_{BM} \xi_n/d_n\approx 1$ is naturally implemented at the interface of the considered materials.

For $\gamma_{BM} = 0.5$ we obtain $v_c \approx 0.14$ (see Fig.~\ref{Fig2}) so that $I_cR_n = 0.7$~mV, $I_c \approx 0.53$~mA and $R_n \approx 1.3$~$\Omega$. The dissipated energy during a clock period is $E = I_c \Phi_0 \approx 1.1$~aJ and therefore the dissipated power is
\begin{equation}
P=E/T_{clk}\approx 2.7\times 10^{-8}~~\text{W}.
\label{powerEl}
\end{equation}
The specific boundary resistance is $R_{B} =
\gamma_{BM}\rho _{n}\xi^2 _{n}/d_n\approx 2.5\times 10^{-15}~\Omega~$m$^{2}$. By putting further $T_{loc}\approx T = 4.2$ K and $\Delta \approx 1.69k_{B}T_{c}$ in (\ref{kappaSN}), we get the power flowing to the S films across SN interfaces,
\begin{equation}
P_{sn}=K_{sn} \delta T,
\label{Psn}
\end{equation}where $K_{sn}\approx 2\times 10^{-7}$ W/K.

By substituting the typical Cu parameters, $v_{F}=1.57\times 10^{6}$~m/s, $
d_{fn}=8900$ kg/m$^{3},$ $c_{t}=4.8$ km/s, $c_{l}=2.3$ km/s, $T_{\ast }=1.8$~K, $\ell =10$ nm, $m_e = 9.1\times 10^{-31}$ kg, $E_{F}=1.13\times 10^{-18}$ J  into (\ref{e-phpower1}) and taking into account that the energy exchange between hot electrons and phonons occurs inside the volume of the order of $(L_b+2 \xi_n)d_nW$, we get the power transfer to phonons,
\begin{equation}
P_{e-ph}=K_{e-ph}\delta T_{e-ph},
\label{Peph1}
\end{equation}
where $K_{e-ph}\approx 1.9\cdot 10^{-9}$ W/K.

Further power transfer between the normal metal and the substrate phonons is given by the Kapitza coupling (\ref{kapitsa}). The coefficient $K_{k}$ at Cu/Si interface approximately equals \cite{Ketchen1985,Clarke1989,Martinis1993} to $100$ W m$^{-2}$K$^{-4}$. By taking \cite{Novitskiy,Phillips,Pobell} the thermal conductivity, $\lambda _{n}\approx 500$ W/m~K, and thermal capacity, $C_{vn}\approx 0.3$ J/kg, for Cu, we get $\eta_n\approx 70$~nm and (\ref{kapitsa}) reads
\begin{equation}
P_{k}=K_{kk}\delta T_{ph-s},
\label{Peph11}
\end{equation}
where $K_{kk}\approx1.5~10^{-9}$ W/K.

In accordance with the heat balance described in the previous section, the power dissipated in the junction can be absorbed by the S-electrodes and the phonons of the N-film with subsequent partial power transfer into the substrate. Since $K_{kk} \approx K_{e-ph} \ll K_{sn}$,
one can estimate the heating of the structure as follows,

\begin{equation}
\delta T=\frac{P}{K_{sn}}\approx 0.13~\text{K.}  \label{deltaTel-bath}
\end{equation}%
Such temperature increase can lead to a slight decrease in the value of the critical current by the amount
\begin{equation}
\left|\frac{\delta I_{c}}{I_{c}}\right|=\left|\frac{\partial v_c}{v_c \partial t}\right|
\frac{\delta T}{T_c}\approx 3.5\%,  \label{estimate}
\end{equation}see Fig.~\ref{Fig2}(a). The obtained deviation of the critical current seems acceptable in the view of standard optimization of the critical currents of Josephson junctions in digital circuits within the margins $\pm 20 \%$ and standard technological critical current spread, $\delta I_{ctec} \approx 3\%$.

The estimations (\ref{deltaTel-bath}), (\ref{estimate}) show that in the temperature range of interest, $T\approx 0.5T_{c}$, hot electrons effectively diffuse from the weak-link region into massive superconducting films, thereby eliminating the effect of the nonequilibrium state of the electronic subsystem on the mode of operation of SN-N-NS junction.  It is necessary to mention that the lowering of the operation temperature leads to exponential suppression of the heat transfer channel from hot electrons to S electrodes. This can lead to the noticeable difference between electron and phonon temperatures, especially in mK temperature range.

\section{Discussion}

In summary, we argue that SN-N-NS junctions with variable thickness bridge geometry are promising for miniaturization. The presented theoretical analysis shows that the junction $I_c R_n$ value reaches sub-millivolt level even when the rigid boundary conditions are not fulfilled at the SN interfaces. According to the obtained analytical expressions, the junction current-phase relation is close to a sinusoidal one at the operation temperature, $T/T_c \approx 0.5$. This allows the use of standard computer-aided design tools for superconducting digital circuit simulations.

The suppression parameter value, $\gamma_{BM} \approx 1$, can be taken as the upper threshold for the fabrication of Josephson junctions intended for operation at liquid helium temperature, $T \approx 4.2$~K. The corresponding normalized characteristic voltage, $v_c \approx 0.08$, turns into $I_cR_n \approx 0.4$~mV and characteristic frequency, $I_cR_n/\Phi_0 \approx 200$~GHz. Since the clock frequency of complex circuits is usually a fraction of the characteristic one, the chosen value of $\gamma_{BM} \lesssim 1$ provides the possibility of operation with frequencies up to several tens of GHz.

Based on available experimental data \cite{TolpygoASC} we argue that the optimal width of SN-N-NS structure lays nearby $250$~nm. For a pair of materials like Nb and Cu, for $W=250$~nm, $d_n=10$~nm and $L_b=40$~nm, the characteristic voltage, $I_cR_n \approx 0.7$~mV, is achieved for $I_c\approx 0.53$~mA and $R_n\approx1.3~\Omega$.

An increase of $W$ leads to undesirable inhomogeneity of the supercurrent distribution in the S electrodes. Its decrease results in the fast suppression of the S-film critical current. The use of additional technological steps aimed at the decrease of SN interface transparency may be required to decrease the junction critical current density in this case.

One of possible ways of controllable reduction of the SN interface transparency is the technique that has proved itself in the fabrication of Nb/Al-AlO$_{x}$/Nb tunnel junctions \cite{Tolp,TolpSSJ,TolpSSJ1,MITech}. Here a Nb/Al sandwich can be used as a normal metal. A thin nonsuperconducting Nb film plays the role of a buffer layer. The subsequent growth of Al on such a buffer can contribute to the formation of an atomically smooth free surface of Al which provides predominantly mirror reflection of conducting electrons. The required Nb/Al boundary transparency, $\gamma_{B}$, can be achieved either naturally \cite{Zehnder1999,Brammertz2001} or by using the aluminum oxidation regime previously used in the fabrication of high-$J_{c}$ tunnel junctions \cite{TBGK2003}.  Contrary to double barrier devices, in SN-N-NS junctions the two SN interfaces are oxidised under the same conditions thus keeping the symmetry of the structure.

The decrease of the transparency of the SN interfaces corresponds to a decrease in the characteristic voltage in proportion to $\gamma_{BM}^{-3/2}$ and in the critical current in proportion to $\gamma_{BM}^{-2}$. Ultimately, the width limitations is determined by the quality of the edges of the superconducting electrodes. In the presence of ``dead'' edges $\sim 25$~nm wide, the width can hardly be less than $150$~nm.

An effective area of SN-N-NS junction, $a_{JJ}$, can be determined according to the length within which the dominating heat transfer from hot electrons to superconducting films takes place, $L_{ht} = L_b + 2\xi_n$. Thus $a_{JJ} = 3\xi_n W$  for $L_b=\xi_n$ so that with $W=150$~nm we obtain $a_{JJ} \approx 18000$~nm$^2$. This is close to the area of semiconductor transistor, $a_t$, fabricated in the frame of a 40~nm technological process, under the assumption that $a_t \approx 50\lambda^2$, where $\lambda$ is the minimum feature size \cite{SemTrans}.

\begin{acknowledgments}
The authors are grateful to D.V. Averin for fruitful discussions. Theoretical analysis of the considered SN-N-NS junction (Sections I - V) was supported by grant No. 20-12-00130 of the Russian Science Foundation. The study of the heat balance (Section VI) was performed according to the Development program of the Interdisciplinary Scientific and Educational School of Lomonosov Moscow State University "Photonic and Quantum technologies. Digital medicine",  and  the plan of scientific research  of the SINP MSU .
\end{acknowledgments}

\section{Model of a SN-N-NS structure}

The model of SN-N-NS junction contains a normal metal film connecting two massive superconducting electrodes of the length, $L-L_b,$ located at the distance $\pm L_b/2$ from the centre of this film, see Fig.~\ref{Fig1}. The existence of the proximity effect between N and S materials leads to the induction of superconducting correlations in the N film and, therefore, the Josephson effect between the S-banks.

In the calculation of the critical current, we suppose that the condition of the dirty limit is fulfilled for all metals, the critical temperature of the N-material is equal to zero and its width, $W$, and thickness, $d_{n}$, is much smaller than Josephson penetration depth, $\lambda_{J}$, and decay length, $\xi_{n}=\sqrt{D_{n}/2\pi T_{c}}$, respectively.

\begin{figure}[b]
\includegraphics[width=0.6\columnwidth,keepaspectratio]{Fig1.eps}
\caption{Sketch of a SN-N-NS Josephson junction with a variable thickness bridge geometry.}
\label{Fig1}
\end{figure}

The proximity effect in this system can be considered in the frame of Usadel equations \cite{Usadel}, which in the N-film have the form
\begin{equation}
\frac{\partial }{\partial x}\left( G_{n}^{2}\frac{\partial \Phi _{n}}{\partial x}\right) +\frac{\partial }{\partial y}\left(G_{n}^{2}\frac{\partial \Phi_{n}}{\partial y}\right) =\omega G_{n}\Phi_{n}.
\label{equTetaNF}
\end{equation}
Here  $G_{n}=\omega /\sqrt{\omega^{2}+\Phi_n\Phi_{n}^{\ast }}$, $\Phi_{n}$ are modified Usadel Green's functions normalized on $\pi T_{c}$ , $\omega =(2m+1)T/T_{c}$ are Matsubara frequencies  normalized on $\pi T_{c}$ ($m$ is integer), $x$ and $y$ are coordinates along and across the N-film normalized on $\xi_{n}$. We place the origin of coordinates on the SN interface in the middle of the N-film, as it is shown in Fig.~\ref{Fig1}.

The equations (\ref{equTetaNF}) should be supplemented by the boundary conditions. Due to the symmetry of the problem, we can solve them for the positive $x$ only, making use of the boundary conditions
\begin{equation}
\frac{\partial }{\partial x}\re\Phi_{n}=0,\quad \im\Phi_{n}=0
\label{BcCenter}
\end{equation}
in the center of the bridge, $x=0$. Here $\re\Phi_{n}$ and $\im\Phi_{n}$ are the real and imaginary parts of the functions $\Phi_{n}.$

At the free N-interfaces $(0<x<L/2,$ $y=-d_{n})$, $(0<x\leq L_b/2,$ $y=0)$ the boundary conditions are
\begin{equation}
\frac{\partial }{\partial y}\Phi_{n}=0,  \label{BCfree}
\end{equation}%
and at $x=L/2$,
\begin{equation}
\frac{\partial }{\partial x}\Phi_{n}=0.  \label{BcInf}
\end{equation}%
At the SN interface $(L_b/2\leq x\leq L/2,~y=0)$ the functions $\Phi _{n}$ and the Usadel Green's functions in the S-electrode, $\Phi_{s},$ should be coupled
by the Kupriyanov-Lukichev boundary condition \cite{kuprlukichevA}
\begin{equation}
\gamma_{B}G_{n}\frac{\partial }{\partial y}\Phi_{n}=G_{s}\left( \Phi_{s}-\Phi_{n}\right),  \label{BCgammaB}
\end{equation}
where $\gamma_{B}=R_{B}/\rho_{n}\xi_{n}$ is suppression parameter, $R_{B}$ is specific resistance of the SN interface, $G_{s}=\omega\sqrt{\omega^{2}+\Phi_{s}\Phi_{s}^{\ast}}$.

The conditions \cite{Golubov1983}
\begin{equation}
\gamma_{M}\lesssim 0.3,\quad d_{n}\ll 1  \label{restr}
\end{equation}
permit us to neglect the suppression of superconductivity in the S-film due to the proximity effect even in the case of a fully transparent SN interface. We consider the Usadel Green's functions, $\Phi_{s}$, equal to their equilibrium values in a superconductor at a given temperature,
\begin{equation}
\Phi_{s}=\Delta \exp \{i\varphi /2\},\quad G_{s}=\frac{\omega }{\sqrt{\omega^{2}+\Delta^{2}}},  \label{tetaSinS}
\end{equation}
correspondingly. Here $\Delta $ is the magnitude of superconducting order parameter normalized on $\pi T_{c}$, $\varphi $ is the superconducting phase difference across the junction.

The conditions (\ref{restr}) also allow considering functions $\Phi_{n}$ independent of the coordinate $y$, in the first approximation on the
parameter $d_{n}$. By integration of the equation (\ref{equTetaNF}) over the coordinate $y$, and using the boundary conditions (\ref{BCfree}), (\ref{BCgammaB}), we obtain the equations
\begin{equation}
\frac{\gamma_{BM}}{G_{n}}\frac{\partial }{\partial x}\left( G_{n}^{2}\frac{\partial \Phi_{n}}{\partial x}\right) -\left( G_{s}+\gamma_{BM}\omega\right) \Phi_{n}=-G_{s}\Phi_{s},  \label{EqUnderSr}
\end{equation}
in the region $L_b/2\leq x\leq L/2$, where $\gamma_{BM}=\gamma_{B}d_{n}$.

In the area $0\leq x\leq L_b/2$ we have the equations
\begin{equation}
\frac{\partial }{\partial x}\left( G_{n}^{2}\frac{\partial \Phi_{n}}{\partial x}\right) =\omega G_{n}\Phi_{n}.  \label{EqOutS}
\end{equation}
The solution of equations (\ref{EqUnderSr}), (\ref{EqOutS}) is simplified in the limit of a small gap between the superconducting electrodes, $L_b\ll 1$.

The obtained functions $\Phi _{n}$ allows to calculate the supercurrent $J_{s}$ across the junction
\begin{equation}
\frac{eRJ_{s}}{2\pi T_{c}}=t\sum_{\omega \geq 0}\frac{G_{n}^{2}}{\omega ^{2}}%
Re\Phi _{n}\frac{\partial }{\partial x}Im\Phi _{n},\quad R=\frac{\rho
_{n}\xi _{n}}{d_{n}W},  \label{currentDef}
\end{equation}%
where $t = T/T_c$ is the normalized temperature.

\section{Solution of the problem}

\subsection{The bridge between the S-electrodes}

In the first approximation on $L_b \ll 1$, we may neglect the right-hand side of the equations (\ref{EqOutS}),
\begin{equation}
\frac{\partial }{\partial x}\left( G_{n}^{2}\frac{\partial \Phi_{n}}{\partial x}\right) =0,  \label{outS1}
\end{equation}
in the area $0\leq x\leq L_b/2$.

The functions $\re\Phi_{n}$ are independent of the coordinate $x$,
\begin{equation}
\re\Phi_{n}=\re\Phi_{n}(L_b/2),  \label{outS4}
\end{equation}
as it follows from the boundary conditions (\ref{BcCenter}), the condition of continuity of the functions $\Phi _{n}$ at $x=L_b/2$, and the equations (\ref{outS1}).

For the imaginary part of $\Phi_{n}$ we obtain
\begin{equation}
G_{n}^{2}\frac{\partial }{\partial x}\im\Phi_{n}=\frac{2}{L_b}\frac{\omega^{2}}{\sqrt{\omega^{2}+(\re\Phi_{n})^{2}}}C,  \label{outS6}
\end{equation}
\begin{equation*}
G_{n}^{2}=\frac{\omega^{2}}{\omega^{2}+(\re\Phi_{n})^{2}+\im\Phi_{n}^{2}},
\end{equation*}
where $C$ are integration constants. By taking into account that $\re\Phi_{n}$ are constants (\ref{outS4}), we may find the solution of (\ref{outS6}) in the form
\begin{equation}
\im\Phi_{n}=\sqrt{\omega^{2}+(\re\Phi_{n})^{2}}\tan \theta. \label{outS7}
\end{equation}
Substitution of (\ref{outS7}) into (\ref{outS6}) leads to
\begin{equation}
\frac{\partial \theta }{\partial x}=\frac{2}{L_b}C.  \label{outS8}
\end{equation}
This results in a linear dependence of the functions $\theta$ on the coordinate $x$,
\begin{equation}
\theta =\frac{2}{L_b}Cx,  \label{outS9}
\end{equation}
so the equations for the imaginary part of $\Phi_{n}$ are
\begin{equation}
\im\Phi_{n}=\sqrt{\omega^{2}+(\re\Phi_{n})^{2}}\tan \frac{2C}{L_b}x.
\label{out10}
\end{equation}

\subsection{The regions under the S-electrodes}

According to the equations (\ref{EqUnderSr}), for the real part of $\Phi_{n}$ under the S-electrodes we have
\begin{equation}
\xi_{ef}^{2}\frac{\partial }{\partial x}\left( G_{n}^{2}\frac{\partial }{\partial x}\re\Phi_{n}\right) - \re\Phi_{n}=-\delta \cos \frac{\varphi }{2},  \label{underS1}
\end{equation}
where
\begin{equation*}
\delta =\frac{G_{s}\Delta }{\left( G_{s}+\gamma_{BM}\omega \right) },~~~\xi_{ef}=\sqrt{\frac{\gamma_{BM}}{G_{n}\left( G_{s}+\gamma_{BM}\omega \right)}}.
\end{equation*}
The equations (\ref{underS1}) should be solved with the boundary conditions,
\begin{equation}
\frac{\partial }{\partial x}\re\Phi_{n}=0,
\end{equation}
at $x=L/2$ and $x=L_b/2.$

The solutions of this boundary problems are obviously constants independent of $x$,
\begin{equation}
\re\Phi_{n}=\delta \cos \frac{\varphi }{2}.  \label{ReFn}
\end{equation}

The equations for the imaginary parts of $\Phi_{n}$ can be written as follows,
\begin{equation}
\xi_{ef}^{2}\frac{\partial }{\partial x}\left( G_{n}^{2}\frac{\partial }{\partial x}\im\Phi_{n}\right) -\im\Phi_{n}=-\delta \sin \frac{\varphi }{2},
\label{underS2}
\end{equation}
where
\begin{equation}
G_{n}=\frac{\omega }{\sqrt{\omega^{2}+\left( \re\Phi_{n}\right)^{2}+\left( \im\Phi_{n}\right) ^{2}}}.  \label{underS3}
\end{equation}
By taking into account that $\re\Phi_{n}$ are constants (\ref{ReFn}), we again may find the solution of equation (\ref{underS3}) in the form (\ref{outS7}). Its substitution into (\ref{underS3}) leads to
\begin{eqnarray}
G_{n} &=&\frac{\omega \cos \theta }{\sqrt{\omega^{2}+\left(\re\Phi_{n}\right)^{2}}}.
\end{eqnarray}
This results in the following equation for the $\theta$ functions,
\begin{equation}
\frac{\gamma_{BM}\sqrt{\omega^{2}+\Delta^{2}}}{\Omega_{1}}\frac{\partial^{2}\theta }{\partial x^{2}}-\sin \theta =-\frac{\Delta \sin \frac{\varphi }{2}}{\Omega_{1}}\cos \theta,  \label{underS6}
\end{equation}
where
\begin{equation}
\Omega_{1}=\sqrt{\Omega^{2}+\Delta^{2}\cos^{2}\frac{\varphi}{2}},~~~\Omega = \omega \left( 1+\gamma_{BM}\sqrt{\omega^{2}+\Delta^{2}}\right). \label{underS7}
\end{equation}

\subsection{General solution}

The first integral of the equation (\ref{underS6}) gives
\begin{equation}
\frac{\gamma_{BM}\sqrt{\omega^{2}+\Delta^{2}}}{2\Omega_{1}}\left( \frac{\partial \theta }{\partial x}\right)^{2}+\cos \theta -\cos \Theta = \frac{\Delta \left( \sin \Theta -\sin \theta \right) \sin \frac{\varphi }{2}}{\Omega_{1}},  \label{un1} \end{equation}
where $\Theta$ are the magnitudes of the functions $\theta (x)$ at $x=L/2.$

By matching this solution at $x=L_b/2$ with the one (\ref{outS9}) in the area $0 \leq x\leq L_b/2$, we obtain an equation with respect to constants $C$,
\begin{equation}
\frac{2\eta^{2}\sqrt{\omega^{2}+\Delta^{2}}}{\Omega_{1}}C^{2}+\cos C-\cos \Theta = \frac{\Delta \left( \sin \Theta -\sin C\right) \sin \frac{\varphi }{2}}{\Omega _{1}}, \label{underS8}
\end{equation}
which, in turn, determines the product of the superconducting current, $J_{s}$, across the junction and its normal state resistance, $R_n$,
\begin{equation}
\frac{eR_{n}J_{s}}{2\pi T_{c}}=t\sum_{\omega \geq 0}\frac{2\re\Phi_{n}C}{\sqrt{\omega^{2}+(\re\Phi_{n})^{2}}}\left( 1+2\eta \coth \frac{L-L_b}{2\sqrt{\gamma_{BM}}}\right).  \label{Js}
\end{equation}
Here $\eta =\sqrt{\gamma_{BM}}/L_b$, while
\begin{equation}
R_{n}=R_{nb}+2R_{sn}
\end{equation}
is the sum \cite{Karminskaya2010} of the resistance of the bridge, $R_{nb}$, and the resistances of two SN interfaces, $R_{sn}$,
\begin{equation*}
R_{nb}=\frac{\rho _{n}L_b}{d_{n}W},~~~R_{sn}=\frac{\rho _{n}\xi _{n}\sqrt{\gamma_{BM}}}{d_{n}W}\coth \frac{L-L_b}{2\xi_{n}\sqrt{\gamma_{BM}}}.
\end{equation*}

From the equations (\ref{underS6}) and (\ref{un1}) it follows that the characteristic scale, $\zeta$, of spatial variation of the functions $\theta$ along the coordinate $x$ is
\begin{equation}
\zeta =\sqrt{\frac{\gamma_{BM}\sqrt{\omega^{2}+\Delta^{2}}}{2\sqrt{\omega^{2}\left( 1+\gamma_{BM}\sqrt{\omega^{2}+\Delta^{2}}\right)^{2}+\Delta^{2}\cos^{2}\frac{\varphi}{2}}}}.  \label{ScaleX}
\end{equation}
At $t\gtrsim 0.5$ the sum in (\ref{Js}) converges at $\omega >\Delta$. By neglecting $\Delta$ in comparison with $\omega$ in (\ref{ScaleX}) we obtain the following estimation for $\zeta$,
\begin{equation}
\zeta =\sqrt{\frac{\gamma_{BM}}{2\left( 1+\gamma_{BM}\omega \right) }}.
\label{scaleTc}
\end{equation}
At small temperatures, the sum in (\ref{Js}) converges at $\omega \approx \Delta $. According to (\ref{ScaleX}), we obtain
\begin{equation}
\zeta \approx \sqrt{\frac{\gamma_{BM}}{\sqrt{2}\sqrt{\left( 1+\gamma_{BM}\Delta \sqrt{2}\right)^{2}+\cos^{2}\frac{\varphi }{2}}}}.
\label{scale0}
\end{equation}
From (\ref{ScaleX}) - (\ref{scale0}) it follows that in the limit of small $\gamma_{BM}$ the characteristic scale is $\zeta $ $\approx $ $\sqrt{\gamma_{BM}},$ while for a large $\gamma_{BM}$ it is $\zeta \approx \Delta^{-1/2}$ $\approx 1$ at small temperatures and $\zeta \approx \omega^{-1/2}$ at $t\gtrsim 0.5$.

In a practically interesting case, the length of the superconducting electrodes is much larger than the characteristic scale,
\begin{equation}
L-L_b\gg \zeta.  \label{largeLS}
\end{equation}
Here the magnitude of the functions $\theta (x)$ at $x=L/2$ can be easily found from (\ref{un1}) as
\begin{equation}
\tan \Theta =\frac{\Delta \sin \frac{\varphi }{2}}{\sqrt{\Omega^{2}+\Delta^{2}\cos^{2}\frac{\varphi}{2}}},  \label{tanTeta}
\end{equation}
resulting in
\begin{equation}
\sin \Theta =\frac{\Delta \sin \frac{\varphi }{2}}{\sqrt{\Omega^{2}+\Delta^{2}}},  \label{sinTeta}
\end{equation}
\begin{equation}
\cos \Theta =\frac{\sqrt{\Omega^{2}+\Delta^{2}\cos^{2}\frac{\varphi }{2}}}{\sqrt{\Omega^{2}+\Delta^{2}}}.  \label{cosTeta}
\end{equation}
The expression (\ref{Js}) here has the form
\begin{equation}
\frac{eR_{n}J_{s}}{2\pi T_{c}}=t\sum_{\omega \geq 0}\frac{2\re\Phi_{n}\left( 1+2\eta \right) }{\sqrt{\omega^{2}+(\re\Phi_{n})^{2}}}C.
\label{Js1}
\end{equation}
Substitution of (\ref{sinTeta}) and (\ref{cosTeta}) into (\ref{underS8}) leads to the equations which are closed relatively the constants $C$. Their analytical solutions in the limits of small and large values of the suppression parameter $\gamma_{BM}$ are presented in the main text of the paper.

redpajama_set_name: RedPajamaArXiv


text:
Pick for Port
Interested in weighing in on who will lead the Port of Seattle?
The public has until today, June 23, to participate in an electronic survey on selecting the port's next executive director to replace Ted Fick who resigned in February.
"The Port of Seattle is one of our region's greatest public assets, and we want our next executive leader to have a strong public service ethic," said Port of Seattle Commissioner Courtney Gregoire, who co-chairs with Commissioner Fred Felleman the hiring selection process. "We encourage the public to be a part of this process from the very beginning."
The port commission hired Seattle executive search firm, Herd Freed Hartz, to conduct a regional search. The next executive director, who is expected to be selected in the fall, will lead a port with roughly 1,800 employees and a $1 billion budget. "The executive director will lead the port staff and execute the commission's vision that port assets generate strong financial returns while creating economic opportunities that are socially responsible and broadly shared throughout the region and protecting our environment," Gregoire said. "Selection of a publicly minded new executive committed to these priorities is essential to the port's long-term success in serving the diverse communities of King County."
The survey is available at https://www.surveymonkey.com/r/PortSearch.
New San Diego Commissioner
Michael Zucchet became the newest Port of San Diego commissioner when he took the oath of office June 15.
He joins the seven-member board, three of which represent San Diego, and one each for the cities of Chula Vista, Coronado, Imperial Beach and National City. Zucchet was appointed to represent San Diego.
Robert "Dukie" Valderrama, chairman of the Port of San Diego Board of Port Commissioners, welcomed him on behalf of the commission and said he looks forward to Zucchet's perspective and expertise.
"Mr. Zucchet brings a wide array of skills and interests that will benefit our communities, the environment, our businesses, and most importantly, our residents and visitors to the San Diego waterfront," Valderrama said.
Zucchet is general manager of the San Diego Municipal Employees Association, the union representing city workers. He previously worked for the Utility Consumers' Action Network, and for the San Diego City Fire Fighters as legislative and community affairs director. Additionally, he represented District 2 on the San Diego City Council.
Zucchet, who earned a master's degree in environmental economics and policy from Duke University, was a renewable energy economist with the Energy Information Administration of the United States Department of Energy in Washington, D.C. He also worked with Santa Barbara nonprofit Environmental Defense Center and was president of the San Diego League of Conservation Voters.
"Mr. Zucchet's public sector experience and background in environmental economics will serve us well as we approach the final stages of our Port Master Plan update," said Rafael Castellanos, vice chairman of the Port of San Diego Board of Port Commissioners. "His expertise will complement and increase the Commission's capability as we strive to balance the diversity of interests on our vibrant waterfront."
Labels: Michael Zucchet, Port of San Diego, San Diego Municipal Employees Association
Port of Portland Development
On June 14, the Port of Portland Commission approved the sale of 19 acres of Troutdale Reynolds Industrial Park (TRIP) to the Clayco real estate development, one of the nation's largest real estate developers whose clients include Pfizer, Coleman, Amazon and Walgreens.
Clayco's preliminary site concept features a 344,000-square-foot facility that can accommodate up to four industrial users for manufacturing, distribution and offices, according to the port.
The sale is the latest in the revitalization of TRIP, a 700-acre brownfield redevelopment. Earlier this year, the port approved the sale of 74 acres for an Amazon fulfillment center, bringing 1,500 jobs.
"This investment is another example of TRIP's evolution from a brownfield site into a thriving job center," said Keith Leavitt, port chief commercial officer. "We expect this development to follow suit in attracting nationally known companies that will bring quality jobs to our region."
Once home to the Reynolds Metals Co. aluminum plant, the industrial park had been declared a Superfund site in 1994 and remained inactive. When the Port purchased the property in 2007, it was annexed into the city of Troutdale and cleaned up by Alcoa. So far, the port and stakeholders have spent more than $130 million to buy and redevelop the site, including a 90-acre wetland mitigation area, according to the port.
"Within the dynamic Pacific Northwest market, the TRIP development is a premier site that enhances our industrial portfolio," Clayco Western Region Vice President John Banchero said.
"We see significant opportunity in the site with close proximity to the FedEx Ground hub, the Amazon fulfillment center, I-84 and the Troutdale Airport. When completed, the state-of-the-art facility and its location are guaranteed to attract today's top modern industrial, logistics and fulfillment users."
Labels: industrial park, logistics and fulfillment, Port of Portland
High Speed Bow Thrusters
The bow thrusters within San Francisco Water Emergency Transportation Authority's newest high-speed catamaran ferry came from marine electronics design and manufacturing company WESMAR.
The longtime marine electronics and thruster manufacturing company installed its WESMAR system into Hydrus, a 135-foot, 400-passenger ferry that was delivered to the transportation authority last month.
The ferry features two WESMAR hydraulic V2-12 dual prop, counter rotating bow thrusters, one in each hull. Each system includes three control stations in the pilothouse.
The system is designed to give extra power needed for high-speed ferries such as the Hydrus, which operates at a service speed of 27 knots.
The WESMAR's were installed for dockside maneuvering and to assist with on and off-loading, critical to achieving fast turnaround times, ensuring ferry runs stay on schedule.
"The Hydrus and the additional vessels coming on-line over the next few years will help us meet the increased need for ferry service and realize our strategic plan: a robust network of 44 vessels, serving five times today's ridership by 2035, with wait times of 15 minutes or less during peak commute hours," said Nina Rannells, executive director of the transportation authority. Hydrus will travel the Central Bay routes of San Francisco to Alameda, Oakland and Harbor Bay. It's the first of four ferries to join the fleet in three years. All will have the same design and be equipped with WESMAR systems, according to the company.
Kalama Methanol Permits Approved
The Port of Kalama has secured two key permit approvals from the Washington Department of Ecology that would allow for the construction of a proposed facility for manufacturing and exporting methanol.
The state department recently approved a conditional use permit and granted water quality certification for the project, which would be built on about 100 acres of industrial property at the north end of the port's marine industrial park.
The methanol plant and marine terminal project is allowable under the state's Shoreline Management Act and Cowlitz County's shoreline master program as long as it meets environmental protection standards, according to the state.
The facility is expected to produce up to one million metric tons of greenhouse gases a year to make methanol from natural gas. Under the conditional use permit, the port and Northwest Innovation Works will need to lessen greenhouse gas emissions that come from the methanol plant by 1.7 percent annually from the first year it is fully operational until emissions level off in 2035.The port says most of emissions associated with the condition are outside shoreline jurisdiction and added that the project would significantly lower GHG emissions by using Ultra-Low-Emission gas methanol instead of coal methanol.
The permit also requires putting in place on-site dredge disposal standards to protect water quality since the plant would be built near the Columbia River, though the methanol plant would have a system that recycles manufacturing process water to avoid discharges to the nearby river.
These permits are part of several more the port will need in order to construct the facility.
The marine terminal – which would be built, owned and operated by the Port of Kalama – would be used mainly by Northwest Innovation Works and available to other ships for lay berth use, according to the port. It would feature a dock, berth, loading equipment, utilities and a stormwater system. Methanol vessels coming to the terminal would be able to hook up to pollution-reducing shore power.
According to the state department, the facility would be able to produce 10,000 metric tons of methanol daily from natural gas once operational.
Labels: Marine Terminal, Methanol, Port of Kalama
New Long Beach Fireboat Stations
Long Beach harbor commissioners last week agreed to invest in the construction of two new fireboat stations by approving engineering design services for the projects.
Commissioners approved $50.1 million for Fireboat Station No. 15 to be built on the Main Channel and $51.6 million for Fireboat Station No. 20, which will be constructed in the port's Inner Harbor.
"These fireboat stations will safeguard the Port and our customers in this new era of big ships," said Port of Long Beach Executive Director Mario Cordero. "They will provide the best waterside response possible, maximize landside firefighting coverage and serve as a home for our new, state-of-the-art fireboats. With ships more than doubling in size in the last decade and cargo traffic breaking monthly records, this is a needed upgrade to Port safety."
Each fireboat station will feature living quarters, a garage for firefighting equipment and a full waterside wharf with enclosure for the fireboat. They will also include water- and energy-efficient technology and other environmental-friendly features.
Station No. 15 is expect to be completed in 2020, with Station No. 20 opening a year later.
Labels: fireboats, Port of Long Beach
Port of Los Angeles Suggests Air Quality Measures
An updated draft environmental impact report released Thursday by the Port of Los Angeles is suggesting a series of sustainable measures to curb "unavoidable significant impacts in air quality, greenhouse gas emissions and ground transportation" at China Shipping's terminal at Berths 97-109.
This supplemental draft updates a 2008 environmental impact report (EIR) that was created when China Shipping North America affiliate West Basin Container Terminal wanted to expand its terminal, which the port approved. The project was completed in 2013.
To offset the impacts, the EIR called for China Shipping to purchase cleaner cargo handling equipment and zero emissions trucks, and conduct a yearlong zero emissions demonstration with at least 10 pieces of cargo-handling equipment.
But according to the Daily Breeze, the L.A. port said in 2015 that many of the clean-air requirements have not yet happened.
This draft report evaluates the terminal's continued operation under adjusted mitigation measures, considering that cargo movement is higher now than it was in 2008.
Modified measures would include that ships must be able to use alternative maritime power (such as shore power) while docked at the port by January 1, participate in the vessel speed reduction program and replacing diesel-powered equipment with electric models.
The public has a chance to weigh in on the draft report until July 31. An open meeting will take place on July 18 at the Port of Los Angeles Administration Building, 425 S. Palos Verdes Street in San Pedro. For a copy of the draft EIR, visit https://www.portoflosangeles.org/EIR/ChinaShipping/DSEIR/dseir_china_shipping.asp
Labels: Air Quality, alternative marine power, Port of Los Angeles
Customs Offers Cross-Border Agreements
More than a dozen California and Washington-based businesses and agencies have been tentatively selected by the US Customs and Border Protection (CBP) for new reimbursable service agreements designed to promote economic growth in cross-border trade throughout the nation, the CBP announced recently.
These agreements – made possible through Section 481 of the Homeland Security Act, 2002 – allow selected entities to reimburse CBP for expanded services such as customs, agricultural processing, border security services and support services at ports of entry.
Those tentatively selected for these partnerships include:
APM Terminal Los Angeles (Los Angeles, Calif.);
California Cartage Company (Los Angeles, Calif.);
California United Terminals, Inc. (Los Angeles, Calif.);
Eagle Marine Services, Ltd (Los Angeles, Calif.);
FCL Logistics, LTD (Los Angeles, Calif.);
International Transportation Services, Inc. (Long Beach, Calif.);
Long Beach Container Terminal LLC (Long Beach, Calif.);
Port of Hueneme/Oxnard Harbor District (Hueneme, Calif.);
Price Transfer, Inc. (Long Beach, Calif.);
Price Transfer, Inc. (Los Angeles, Calif.);
Total Terminals International, LLC (Long Beach, Calif.);
Total Terminals International, LLC (Seattle, Wash.);
TraPac, LLC (Los Angeles, Calif.);
West Basin Container Terminal (Los Angeles, Calif.); and
Yusen Terminal LLC (Los Angeles, Calif.)
"With increasing demands placed on CBP operations across the nation, innovative solutions like the Reimbursable Services Program allow us to keep pace while ensuring the safety and security of the travelers and cargo arriving to the United States," said Acting Commissioner Kevin McAleenan. "The selection of these new partners reinforces CBP's commitment to supporting opportunities for economic advancement and increased service."
Labels: cross-border trade, Reimbursable Services Program, US Customs and Border Protection

redpajama_set_name: RedPajamaCommonCrawl


Our anatomically correct breast models are ideal for teaching how to identify signs of breast cancer, and the fundamentals of female breast anatomy, as well as promoting awareness. We offer a range of breast models which can be used for display or demonstration purposes, training tools for breast self-examination (BSE) or as references for women's health education and anatomy classes.
One of the common uses for breast models is to help educate and train individuals how to properly perform breast self-examinations (BSE) in order to detect signs of early breast cancer.
Models specifically designed for demonstrating the physical differences between a healthy breast and one that has a tumour, whether benign or malignant, are often modelled to be as realistic as possible. The tactile feedback it provides makes it a more effective tool for practising breast self-exams because it closely recreates how lumps will actually feel like beneath the skin and other underlying tissues.
A wearable breast model can be used by both men and women to realistically simulate a breast self-exam. Our catalogue features wearable breast models made by 3B Scientific, which can replicate the motions required to perform an actual exam on one's own body.
It can be setup to represent a normal healthy breast then quickly reconfigured to contain benign or malignant tumours, as well as other anomalies that one may encounter during a breast exam. This allows the patient or student to quickly compare and discern the differences between a healthy breast and one that has developed lumps. These wearable models can also be worn and used either standing up or lying down.
It is made out of 3B Scientific's SKINlike high-quality silicone, which is the same material used in a number of its realistic anatomic models, and has been dermatologically tested to provide an authentic tactile feel. It is also modelled after a full-sized female upper body with medium-sized breasts for added realism.
All wearable models come with a harness and stand. The more expensive model also comes with a carrying case for storage and convenient transportation.
In some situations, a wearable breast model may not be completely necessary or even feasible due to its size and relative bulk. In such cases, we offer a Standard Breast Exam model, also made by 3B Scientific.
This Standard Breast Exam Model features 3 life-like models of single breasts each one with different characteristics for easy comparison. One is a model of a healthy breast, while the other two simulate breasts with benign and malignant tumours in various stages of development.
These models feature the same 3B SKINlike high-quality silicon used in the wearable models. Due to each model's size, it can be more practical in a classroom or clinic setting where it can be passed around and examined with relative ease, compared to a model of a full female torso. It also comes with a moulded base for convenient storage.
This mounted set of left and right breast models is an invaluable teaching tool that helps students and patients alike get a glimpse of the internal anatomy and pathological differences that can lie beneath the skin.
The right breast represents a lactating breast and is medially divided into 2 halves with one half illustrating what a healthy breast looks like underneath, and the other half showing what a mastitis or gland inflammation looks like from both the outside, as well as inside.
The left breast represents a non-lactating breast and features several cuts and windows that show different medical conditions such as regional lymph nodes, cysts, fibroadenomas, fibrocystic breast disease and malignant tumours. It is separated into 3 different parts, which allows the model to illustrate multiple planes of the breast's internal anatomy.
The internal structures of these models have been textured to mimic the look of the different layers of subcutaneous tissue for added realism. The model itself is made out of tough and durable materials that can withstand years of use and in-depth study.
This activity set provides a number of useful tools to help facilitate discussions on women's health issues, pregnancy and basic health education. It features a fully detailed model of the lower part of a healthy and mature female torso with partially sectioned organs with a magnified cross section of the ovum to illustrate the concept of fertilisation.
Additional visual aids and study guides are also included to further discuss the inner workings of the female reproductive system. The guides were developed by 3B Scientific with different age groups in mind, ranging from pre-teens, teenagers to young adults.


text:
Wizards overcome Harden's 54, beat Rockets 135-131 in OT
Posted: Mon 10:54 PM, Nov 26, 2018
WASHINGTON (AP) - Even the Wizards seemed surprised by the way they won.
Bradley Beal said he couldn't have imagined allowing the opposing starting guards to combine for 90 points and still get a victory.
"No sir. Absolutely not. No sir," Beal said. "Not at all. It's amazing we pulled it off still."
John Wall scored a season-high 36 points, including six in overtime, and Washington overcame James Harden's season-best 54 points to outlast the Houston Rockets 135-131 on Monday night.
Harden, the NBA's leading scorer, also had 13 assists, while Eric Gordon added a season-high 36 points.
Wall had two baskets in the extra period and finished with 11 assists. Beal equaled his season high with 32 points for the Wizards.
Otto Porter hit a runner with 4:37 to play for the first basket in overtime and the Wizards remained ahead from there. Wall followed with a jumper to make it 129-125 and later made another jumper that pushed the lead to five. Beal's dunk extended it to 133-126 with 1:50 to play.
Harden had 44 points through three quarters but missed five of his seven shots in the fourth. In overtime, Harden whose career high is 60, was 1 of 2 from the field and had two free throws. He had 11 turnovers, three in overtime, and 13 assists.
"It's not even about myself," Harden said. "It's about our entire team. We had a really good start to the game and kind of let our guard down."
Harden scored 12 points in the first quarter, 11 in the second and 21 points in the third quarter. In the fourth, he began the quarter on the bench, and scored six as the Wizards and Rockets headed to overtime with the score knotted at 125.
Houston raced to a 27-10 lead, but Washington clawed its way back. Markieff Morris had 22 points and 10 rebounds for the Wizards.
Clint Capela had 17 points and 14 points for the short-handed Rockets, who played without Chris Paul for the second straight game and fell to 9-10.
Rockets: Paul was listed as being out with a sore left leg, though coach Mike D'Antoni classified the injury differently. "We've just got to make sure Chris gets 100 percent well from his strained hammy," D'Antoni said. ... G/F Gerald Green missed his third game with a sore right ankle. ... Houston added F Danuel House from the Rio Grande Valley Vipers of the G League. ... Made their first seven 3-point shots, then missed 10 straight.
Wizards: C Dwight Howard was out for the fourth straight game with a glute injury. "It's a slow process," coach Scott Brooks said. "He's doing everything he can." . Morris, F Kelly Oubre Jr. and Beal picked up technicals. ... Allowed 42 points in the first quarter, the fifth time this year they'd given up at least 40. ... Wall passed Wes Unseld for third place on the franchise career scoring list.
GUARDING HARDEN
Wall said that despite allowing 54 points to Harden, Beal did a fine job against him.
"It's what you dream of," Wall said. "It's what you live for, to be on the big stage, a big moment, going against a guy like that."
COACHING HARDEN
D'Antoni knows he's asking too much of Harden, who played a season-high 47 minutes.
"I mean, we're asking, especially James, you know what? Play (47) minutes, make every play," D'Antoni said. "There are going to be periods where he's just gassed."
HURT ROCKETS
Besides Paul and Green, the Rockets are without G Brandon Knight (left knee), C Nene (right calf) and F/C Zhou Qui. Knight and Nene haven't played at all this season, and Qui has played one minute.
It hasn't gotten Gordon down.
"We definitely want to turn it into a wins," Gordon said. "We're not discouraged. We're going to get this thing rolling here soon."
Rockets: Host the Mavericks on Wednesday
Wizards: Visit Pelicans on Wednesday as Washington plays six of its next seven on the road.
Former NBA commissioner David Stern has died at age 77, according to the league
Harden returns to lead Rockets past Nuggets 130-104
Doncic scores 24 on return, Mavericks hold off Spurs 102-98
Doncic scores 33, powers Mavericks past Pelicans, 118-97
Luka gets 27, Mavs snap Lakers' 10-game win streak 114-100
South Carolina uses 3-ball to down Texas A&M 81-67


text:
Шепелево — железнодорожная станция в Козельском районе Калужской области. Входит в состав Городского поселения «Город Сосенский».

Расположено примерно в 4 км к юго-востоку от города Сосенский.

Население

См. также
 Станция Шепелево.
 Карта N-37-13. Масштаб 1:200000.

Примечания

Населённые пункты Козельского района

redpajama_set_name: RedPajamaWikipedia


Last Lila option: in love with cactuses!
Today I'm showing you the very last jumpsuit Sylvia has sewn a couple of weeks ago. Wendy was the very first webshop that introduced the cactuses fabric and we immediately fel in love! It's a very nice quality to work with and the print is super cute for both boys and girls.
This jumpsuit has the basic bodice and an extra patch pocket for small treasures! The rest is kind of straight forward. :) Use the Lila pattern, pick an awesome fabric and ask the cutest girl ever to be your model for a day. This is the result. Hope you love it? I do! ;) Of course no beautiful pictures without Bjorn's help! I couldn't make a small selection so enjoy this cuteness overload.
The Lila jumpsuit is the perfect pattern for little girls and boys to play around. Super comfy and stylish!
Sylvia and I will launch the Lila dress and jumpsuit pattern on Monday. Tomorrow we'll be folding and filing the pattern packagings. This means the pattern will be available both as a paper and PDF version. Exciting!
Thanks Barbara! I'm glad you like our Lila pattern. Nore has the Wild boots too. She loves them!
oh super!!! Terug eentje om naar uit te kijken!!!
Bedankt voor de lieve feedback Hilde!
Mooi 😊 it is such a wonderful pattern. You ladies have done a terrific job!


text:
#import "NSObject-Protocol.h"

@protocol TSPObjectModifyDelegate <NSObject>
- (void)willModifyObject:(id)arg1 duringReadOperation:(_Bool)arg2;
@end

redpajama_set_name: RedPajamaGithub


Many people could resolve their disputes without a lawyer if courts were easier to use. By simplifying court procedures and making small claims courts more available, states can help people minimize or eliminate the cost of hiring a lawyer. Small claims courts, which use simplified procedures, are designed to be used without a lawyer. While some states allow small claims cases up to $25,000 to be brought in these user-friendly courts, others have much lower dollar limits. This can force people to walk away from meritorious cases because their case is too large for small claims court but too small to make hiring a lawyer cost-effective.


text:
Germany to pull the plug on three of its last six nuclear plants
Jan 1, 2022Jan 1, 2022 6:48 pm GMT
Germany will pull the plug on three of its last six nuclear power stations on Friday, another step towards completing its withdrawal from nuclear power as it turns its focus to renewables.
Source: www.reuters.com
asset-management
climate-change
nuclear-generation
Nathan Wilson on Jan 2, 2022
Here is more evidence that anti-nuclear activism is a major force in blocking decarbonization of our energy sector. For decades, environmental scientists have pleaded with us to reduce our fossil fuel consumption. Anti-nuclear activists have told us to instead reduce our use of safe, clean nuclear power; the result continues to be more fossil fuel use, more air pollution, and more CO2 emissions.
Our current nuclear fleet was funded and built by our parents and grandparents, and could easily last 80 years or more. Germans today are discarding that investment, and dooming their children to spend their money to replace them.
Sign in or Register to post comments
GEH Inks Contract to Build BWRX300 at OPG's Darlington Site
An Interview with Yaron Ben Nun – Nostromo Energy


text:
Q: How do I hookup a dual video card Crossfire rig to a single monitor? I just received a new Alienware box preinstalled with two nice shiny ATI 6950 graphics cards. My goal is to use both of them to give me really good resolution and frame rate on my single monitor.
There's zero documentation with the Alienware box and nothing I can find online.

*

*Do I just plug the monitor into one of the cards, and expect the two cards to do magic internally?

*The ATI Crossfire website indicates a need for an "external crossfirex connection cable". Do I need to buy one of these? If so, what kind of cable is it?

*The AMD Catalyst Control Center shows the second card as "Linked" but when I go to the AMD Overdrive section, the second card is at zero temperature, and clock, suggesting it's not actually doing anything. What do I need to do to make use of both cards? Is it something that doesn't kick in until I do something graphically intensive?

A: Just hook the monitor up to master card. The frames rendered by the secondary card is moved to the master card internally. You used to have to hook up a cable from the secondary card to the master card for crossfire, but that was a long time ago, and ATI has long since overcome that technological limitation.
Unless you are running a program, usually a game, for which crossfire is explicitly supported, the second card is not used. It is then put to sleep.

A: The master card is the topmost card, the one nearest the CPU.
Plug the video cards into that one (assuming you don't need more than 3 monitors, as each ATI card supports 3 video outputs). You can effectively ignore the lower video card when connecting monitors.

A: If the motherboard and the graphics cards are recent enough you probably won't need the external connection cable. I think I remember that SLI/Xfire still require a ribbon connector to operate. Regarding temperature etc does the card run in the box? if yes, the software may be just unable to acquire data. try an update maybe.

A: just to mention something obvious not mentioned yet, crossfire is for connecting two video cards together not monitors. You still connect the monitor to the one video card. The connection cable is usually a small short cable that attached to the up side west of the back panel ports between the two cards.
here is a good example image

redpajama_set_name: RedPajamaStackExchange


It's that time again! It's time for us to learn more about you, our readers, who are us, by asking you to fill out our Sprudge Reader Poll. These polls are a great alternative to us monitoring your browser cookie metadata for personal information, which is creepy; it also gives us a chance to ask you all kinds of questions that cookies can't answer, which is helpful.
This survey will help us grow, learn, and better serve you as a Coffee Media Publication. The questions are mostly painless and won't take longer than a minute or two to answer.
How Much Do You Spend On Coffee Each Year?


text:
NOWCAST KCCI News at Noon Weekday
$7.3 million expansion will add 13 new jobs
Updated: 12:27 PM CDT Oct 8, 2014
Tad Andersen
A $7.3 million expansion of a plant in Ankeny will create 13 new jobs, Ankeny city officials said.Mrs. Clark's Foods is planning the 69,000-square-foot expansion at 740 S.E. Dalbey Dr.The Ankeny City Council approved a development agreement Monday to provide $303,000 in tax rebates over five years to help the project.Mrs. Clark's Foods specializes in juices, condiments and dressings, as well as other custom manufactured foods. The new expansion will make room for manufacturing, refrigeration and warehouse space.City officials said the project is the latest in a series of large development announcements for Ankeny, which included U.S. AutoForce, Dick's Sporting Goods, Gordmans and Sam's Club.
ANKENY, Iowa —
A $7.3 million expansion of a plant in Ankeny will create 13 new jobs, Ankeny city officials said.
Mrs. Clark's Foods is planning the 69,000-square-foot expansion at 740 S.E. Dalbey Dr.
The Ankeny City Council approved a development agreement Monday to provide $303,000 in tax rebates over five years to help the project.
Mrs. Clark's Foods specializes in juices, condiments and dressings, as well as other custom manufactured foods. The new expansion will make room for manufacturing, refrigeration and warehouse space.
City officials said the project is the latest in a series of large development announcements for Ankeny, which included U.S. AutoForce, Dick's Sporting Goods, Gordmans and Sam's Club.


text:
Unscripted Submission Portal
Unscripted (Reality) content is handled differently than other creative projects submitted to traditional festivals, due to the nature of the material. Why? Well, it's [kinda] simple, most unscripted content shopped today is typically in early stages of development (most vulnerable stage for any developer), and not a finished or polished project like a feature or short film, for example. In some occasions a production company and/or individual EP may have a fully produced show and they are only looking for distribution, but that's rare. However, if that applies to you, then DIFF is not the right fit, and please DO NOT submit.
Accidental submissions will not be refunded.
Our aim with our unscripted category is to award concepts with further development. Our team at DIFF has created, optioned, and sold unscripted to virtually every network in every space. The top three (3) winners (all categories compete) will be offered an Option to develop the IP with one of our production partners at DIFF, who will then work with the Creator(s) to further develop the concept and pitch to prime time and streaming networks.
The Creator(s) will be invited to attend an/all pitch meetings in Los Angeles* in attempt to sell their show. DIFF and its production partners agree to pitch to no less than two (2) distribution entities, after which they reserve the right to dissolve the Option Agreement, and return the property rights back to the Creator(s).
You own my rights? WHAT!? This is what is called an "Option" on your work, should DIFF select your unscripted concept as one of our top three winners. It's best to think of it as a lease to the rights of your Reality TV show, while it is being shopped at the networks. During the Option period ("lease") — typically 30-90 days, depending on the project's CURRENT state of development, the production company will work on behalf of our Creator(s) to try and sell the show. Our Producers will also lead the pitch meeting(s), and be attached to the project as an Executive Producer; will be in your corner the entire way.
That said, there are a few extra steps that are required in order to safeguard both your interests as well as the DaVinci Film Festival moving forward. Please read our industry standard Rights Release TERMS & CONDITIONS carefully before submitting your UNSCRIPTED PROJECT.
ACCEPT OUR SUBMISSION POLICY


text:
Centrální depozitář cenných papírů, a. s. (do roku 2009 působící pod názvem UNIVYC, a. s.), je společnost působící v oboru vypořádání obchodů s cennými papíry (též označováno jako clearing) na českém kapitálovém trhu. Jejím hlavním úkolem je správa a vedení centrální evidence cenných papírů na území České republiky. Jediným jejím akcionářem je Burza cenných papírů Praha.

Služby CDCP spojené s vedením evidence cenných papírů a vypořádáním obchodů jsou poskytovány pouze prostřednictvím jeho účastníků.

Historie
Po uzavření Střediska cenných papírů (SCP) 26. června 2013 veškeré transakce s cennými papíry jdou od 7. července přes Centrální depozitář a jeho členy. To znamená, že drobní investoři už nemají mít účet na své jméno ve Středisku cenných papírů. Místo toho jim účty vedou obchodníci s cennými papíry.

V prosinci 2018 Centrální depozitář cenných papírů obdržel od České národní banky povolení k výkonu činnosti podle harmonizované evropské úpravy provedené nařízením CSDR.

V únoru 2020 dostal CDCP od ČNB povolení k vypořádání obchodů ve vypořádacím systému na účtech u zúčtovatele Národná banka Slovenska. Povolení dala Česká národní banka podle CSDR.

Činnost centrálního depozitáře
Centrální depozitář přiděluje identifikační označení v rámci mezinárodního systému číslování pro určení cenných papírů (zkratka ISIN) pro investiční nástroje, kterými jsou např. investiční cenné papíry, cenné papíry kolektivního investování, nástroje peněžního trhu, finanční rozdílové smlouvy, apod. Dále je centrální depozitář provozovatelem vypořádacího systému. Za splnění určitých podmínek může centrální depozitář vést samostatnou evidenci investičních nástrojů nebo vykonávat další podnikatelskou činnost, která však musí být zaregistrována u České národní banky.

Další činností firmy jsou například správa a řízení vkladů účastníků v Garančním fondu burzy, půjčování cenných papírů, úschova a správa cenných papírů, registrace cenných papírů pro účely vypořádání, vypořádání primárních emisí cenných papírů, zprostředkování služeb Střediska cenných papírů, vypořádání obchodů se zahraničními cennými papíry, vedení registru podílů na sběrném dluhopisu a vypořádání obchodů s podíly na sběrném dluhopisu.

Centrální depozitář má ze zákona informační povinnost k ČNB, kterou splňuje zasíláním informací a podkladů, které jsou potřebné pro výkon dohledu.

Účastníci centrálního depozitáře cenných papírů
Účastníkem centrálního depozitáře je osoba, která podává centrálnímu depozitáři příkaz k založení nebo zrušení majetkového účtu, k provedení změny na majetkovém účtu nebo k provedení služby. Konkrétní osobou (účastníkem), který podává příkaz, může být:
 banka,
 obchodník s cennými papíry,
 administrátor investičního fondu,
 organizátor regulovaného trhu,
 zahraniční centrální depozitář,
 Česká národní banka,
 Evropská centrální banka, a další.

Základní kapitál a akcie centrálního depozitáře cenných papírů
Centrální depozitář má ze zákona stanoven základní kapitál alespoň 100 000 000 Kč, přičemž emisní kurz jeho akcií musí být alespoň z jedné třetiny splacen peněžitými vklady.

Centrální depozitář nesmí vydávat prioritní akcie. Výše uvedené se váže i na jednu z mnoho podmínek pro povolení k činnosti centrálního depozitáře od ČNB. Česká národní banka udělí povolení k činnosti centrálního depozitáře, jestliže je žadatel akciovou společností, která má sídlo v České republice a vydává pouze listinné akcie na jméno na zaknihované akcie.

Přeměny centrálního depozitáře
Stejně jako u jiných kapitálových obchodních korporací je možná přeměna (fúze) i centrálního depozitáře, ale pouze s povolením České národní banky. Povolení je nutné také k uzavření smlouvy o převodu podniku centrálního depozitáře, nebo jeho části, smlouvy o zastavení podniku nebo jeho části nebo smlouvy o nájmu podniku centrálního depozitáře nebo jeho části.

Reference

Externí odkazy
 Centrální depozitář cenných papírů
 Seznam účastníků CDCP

Cenné papíry
Firmy v Praze
Burza


text:
Q: OpenGL ES 2.0 on android phones without GPU? Will a game run faster using OpenGL ES 2.0 on a Android device without GPU than using the Canvas? Or since it will run on CPU anyway there wont be any big differences in terms of performance?

A: I'm not sure you can run OpenGL ES 2.0 on an Android device without GPU. And if you could, performance would be abysmal —you'd be lucky to move a Pong-like game @ 10 FPS—. Emulating vertex and pixel programs on software is several orders of magnitude slower than running them on a GPU.


text:
SongsTube Nirvana
Nirvana "Rape Me"
About "Rape Me"
"Rape Me" is a song by the American rock band Nirvana, written by vocalist and guitarist Kurt Cobain. It is the fourth song on the band's third and final studio album, In Utero, released in September 1993.
"Rape Me" was released as the album's second single in December 1993, packaged as a double A-side with "All Apologies". It reached number 32 on the UK Singles Chart, and was the band's final single before Cobain's suicide in April 1994.
Top songs by Nirvana
Smells Like Teen Spirit
Here She Comes Now
I Hate Myself And I Want To Die
All Apologies
Do You Love Me
Even In His Youth
Big Long Now
Rape Me
Bad Moon Rising
I Want To Know Now
Where Did You Sleep Last Night
Ain't It A Shame
Heart—shaped Box
Aero Zeppelin
Endless, Nameless
Come On Death
Baba O' Riley
A-Z SONGS by Nirvana
More about Nirvana music
ⓘ INFO BIO DISCOGRAPHY
"Rape Me" video by Nirvana is property and copyright of its owners and it's embedded from Youtube.
Information about the song "Rape Me" is automatically taken from Wikipedia. It may happen that this information does not match with "Rape Me".
SONGSTUBE is against piracy and promotes safe and legal music downloading. Music on this site is for the sole use of educational reference and is the property of respective authors, artists and labels. If you like Nirvana songs on this site, please buy them on Itunes, Amazon and other online stores. All other uses are in violation of international copyright laws. This use for educational reference, falls under the "fair use" sections of U.S. copyright law.


Putative transcriptional activator. Control tissue-specific synthesis of anthocyanin pigments in various parts of the maize plant.
Below are the list of possible Anthocyanin regulatory Lc protein products. If you cannot find the target and/or product is not available in our catalog, please click here to contact us and request the product or submit your request for custom elisa kit production, custom recombinant protein production or custom antibody production. Custom ELISA Kits, Recombinant Proteins and Antibodies can be designed, manufactured and produced according to the reer's specifications.


What Is The Paleo Diet And How Does It Help?
It is often confusing as to what all to eat in the Paleo diet. There are a lot of fish, meats, poultry, vegetables, fruits, eggs, healthy fats from avocado, coconut etc that can be had in this diet. Actually the options are innumerable and your plate will always be filled with something interesting to eat.
Here's a great article from PaleoLeap that discusses paleo diet guidelines.
more Paleo Diet pages coming soon!


Award-winning journalist and author Tanya Talaga joined us in our office to dive deep into Indigenous stories from Canada and across the globe. She spoke about her national bestselling book Seven Fallen Feathers, which introduces us to seven Indigenous high school students who mysteriously died in Thunder Bay, as well as issues Indigenous communities in Canada and around the world are currently facing as a result of a shared history of cultural genocide.
"All Indigenous nations are connected that way, if you look at it, all children are taken from their language, their culture, their families, put into schools to be assimilated and from there you can see the social ills kind of falling forward," Tanya says.
In her powerful keynotes, Tanya shares Indigenous stories from across Canada and the world, humanizing the legacy of cultural genocide and sharing her hope for a more inclusive and equitable future.
In this next clip, Tanya takes a deep dive into Canada's history with their Indigenous populations, exploring how we're a nation built on broken treaties and a residential school system, and what Indigenous communities and Canada as a whole are experiencing now as a result of this.
Interested in learning more about Tanya and what she can bring to your next event? Email us at info@speakers.ca.