Escape from a zero current state in a one dimensional array of Josephson junctions
K.~Andersson
A long one dimensional array of small Josephson junctions exhibits Coulomb blockade of Cooper pair %%@ tunneling. This zero current state exists up to a switching voltage, $V_{sw}$, where there is a %%@ sudden onset of current. In this paper we present histograms showing how $V_{sw}$ changes with %%@ temperature for a long array and calculations of the corresponding escape rates. Our analysis of %%@ the problem is based on the existence of a voltage dependent energy barrier and we do not make any %%@ assumptions about its shape. The data divides up into two temperature regimes, the higher of which %%@ can be explained with Kramers thermal escape model. At low temperatures the escape becomes %%@ independent of temperature.
EscapeFromCBCPT.TEX
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\begin{document}

\preprint{APS/123-QED}

\title{Escape from a zero current state in a one dimensional array of Josephson junctions}

\author{K.~Andersson}
\email{karina@nanophys.kth.se}
\author{D.~B.~Haviland}
\affiliation{Nanostructure physics, KTH\\
AlbaNova universitetscentrum\\
S-106 91 Stockholm, SWEDEN }

\date{\today}

\begin{abstract}
A long one dimensional array of small Josephson junctions exhibits Coulomb blockade of Cooper pair %%@
tunneling. This zero current state exists up to a switching voltage, $V_{sw}$, where there is a %%@
sudden onset of current. In this paper we present histograms showing how $V_{sw}$ changes with %%@
temperature for a long array and calculations of the corresponding escape rates. Our analysis of %%@
the problem is based on the existence of a voltage dependent energy barrier and we do not make any %%@
assumptions about its shape. The data divides up into two temperature regimes, the higher of which %%@
can be explained with Kramers thermal escape model. At low temperatures the escape becomes %%@
independent of temperature.
\end{abstract}

\pacs{73.23.Hk, 74.50.+r, 73.23.-b, 05.40.Jc}
\maketitle

Escape from a metastable state can be used to describe a wide variety of physical phenomenon, from %%@
solid state diffusion \cite{diFoggio:diffusionHydro:82} to magnetic switching %%@
\cite{Sun:magnSwitch:01}, and has for a long time attracted attention from both a theoretical and %%@
an experimental point of view (for a review see e.g. \cite{hanggi:50yrs:90}). Thermal escape over %%@
an energy barrier was studied theoretically by Kramers in the 1940s \cite{kramers:escape:40}. More %%@
recently, the influence of quantum tunneling on the escape process has been an important means of %%@
studying the quantum behavior of macroscopic variables. In this context, escape from the zero %%@
voltage state of a Josephson junction has been thoroughly studied %%@
\cite{martinis:phasedifference:87, devoret:QuantumEffectRev:92}. In this paper we adopt the Kramers %%@
model to analyze the escape from the zero \emph{current} state of a one dimensional array of %%@
Josephson junctions exhibiting a Coulomb blockade of Cooper pair tunneling. We find that the escape %%@
process is well described by thermal activation at higher temperatures, but at lower temperatures %%@
the escape process becomes independent of temperature. 


In the Kramers thermal activation model, the rate of escape $\Gamma$, over an energy barrier %%@
$\Delta U$, is given by, 
\begin{equation}
  \Gamma=\kappa{\frac{\omega _p}{2 \pi}}\exp({-{\frac{\Delta U}{k_BT}}})
  \label{eq:ThermEsc}
\end{equation}
where $k_B$ is Boltzmann's constant and $\omega _p$ is the attempt frequency, or the frequency of %%@
small oscillations around the metastable minimum. The dimensionless transmission factor $\kappa$ %%@
depends on the shape of the barrier and the damping \cite{hanggi:50yrs:90}. This model has been %%@
applied to single Josephson junctions, where the energy barrier is a one dimensional tilted %%@
washboard potential, and the tilt is controlled by bias current applied to the junction, $\Delta %%@
U(I)$ \cite{fulton:swCurr:74,martinis:phasedifference:87,devoret:QuantumEffectRev:92}. Long %%@
Josephson junctions (modeled by parallel arrays of single junctions) have also been studied, and in %%@
this case the escape is from a metastable minimum in a multi-dimensional space %%@
\cite{castellano:escLongJJ:96}. All Josephson junction systems studied with this technique thus far %%@
exhibit a zero voltage state with a finite supercurrent, above which the system switches to a %%@
finite voltage state. A voltage across the junction develops when the superconducting phase, %%@
$\phi$, across the junction escapes over the Josephson potential energy barrier to a free-running %%@
state ($V\propto \dot{\phi}$). The escape can be due to thermal activation or quantum tunneling of %%@
the junction phase through the energy barrier. The important quantities for the escape problem are %%@
the height of the Josephson potential and the attempt frequency, which are given in terms of other %%@
junction parameters such as normal state resistance, superconducting energy gap and junction %%@
capacitance. This escape description applied to the Josephson junction is based on the classical %%@
dynamics of the Josephson phase, as embodied in the Josephson effect. 

In this letter we analyze the Coulomb blockade of Cooper pair tunneling, a phenomena which is in %%@
many ways dual to the Josephson effect \cite{agren:kinInd:01, agren:licThesis:00}. In %%@
one-dimensional arrays of small capacitance Josephson junctions, the coupling of the Josephson %%@
phase to dissipation can be manipulated so that there are very large quantum fluctuations of the %%@
phase \cite{haviland:SItrans:00}. In this case, the number of Cooper pairs becomes a classical %%@
variable, and transport through the array is described by a Coulomb blockade, where a zero current %%@
state persists until a switching voltage is exceeded. One- and two-dimensional arrays of small %%@
capacitance Josephson junctions have been well studied in the context of the %%@
superconductor-insulator transition \cite{fazio:phaseTrans:01}. Here we focus on analysis of %%@
transport properties of a 1D array deep in the insulating regime. We do this by studying %%@
fluctuations of the switching voltage and analyizing them in the context of the Kramers model. This %%@
is the first time to our knowledge that such an experimental study has been carried out on Cooper %%@
pair tunneling in the insulating regime.

The array is made of Al, with Al$_2$O$_3$ tunnel barriers, on a SiO$_2$ surface using the standard %%@
two angle evaporation technique \cite{andersson:licThesis:99}. Electron beam lithography was used %%@
to define the mask pattern, which made 400 junctions in series. Each junction had dimensions %%@
$100$nm$ \times 150$nm, and each island was connected by two junctions in parallel, forming a %%@
Superconducting QUantum Interference Device (SQUID). The charging energy $E_C$ of the SQUID was %%@
$E_C\equiv e^2/2C=59\mu$eV, where we have assumed a specific capacitance of %%@
$45fF/\mu$m$^2$\cite{delsing:arraysprb:94}. The junctions normal state resistance $R_n$ controls %%@
the Josephson energy at zero magnetic field, $E_{J0}=(R_Q/R_n) \Delta/2$ where $\Delta=200 \mu$eV %%@
is the superconducting energy gap of Al. For the array presented here we had $R_n=4.6$k$\Omega$ per %%@
SQUID, or $E_{J0}=142\mu$eV. 

The advantage of the SQUID geometry is that we can tune the Josephson coupling by applying a %%@
magnetic flux to the SQUID loops $E_J=E_{J0}\mid \cos(\pi BA/\Phi_0)\mid$, where $B$ is the %%@
magnetic field perpendicular to the SQUID loop, $A$ is the effective area of the loop (in this case %%@
$0.85\times 0.2 \mu $m$^2$) and $\Phi _0=h/2e=2.06\times 10^{-15}$Wb is the superconducting flux %%@
quantum. 

The arrays were measured in a dilution refrigerator with well filtered measurement lines. The chip %%@
is mounted on a socket which is in turn mounted in a RF tight copper can. The leads into the can %%@
are 1 meter of Thermocoax which attenuates at high frequency \cite{zorin:thermocoax:95}. The can %%@
and filtered leads are mounted at the lowest temperature. The bonding pads and contact leads on the %%@
chip were fabricated of Au to within $50\mu$m of the array edge. Low noise and high input impedance %%@
amplifiers were used to measure the voltage and current. 

In Fig.\ \ref{fig:CBCPT} we see a typical current-voltage (I-V) characteristic of a 1D array in the %%@
Coulomb blockade state. When increasing the bias voltage across the array the Coulomb blockade %%@
state is interrupted by a sudden onset of current at the switching voltage, $V_{sw}$. Subsequently %%@
lowering the bias voltage across the array, the Cooper pairs are retrapped at a voltage $V_r < %%@
V_{sw}$, giving rise to a hysteresis in the current-voltage characteristics. The retrapping voltage %%@
remains the same at fixed magnetic field, whereas the switching voltage fluctuates. Both $V_{sw}$ %%@
and $V_r$ depend on $E_J$ and are modulated with magnetic field having a period associated with one %%@
flux quantum in each SQUID loop, as can be seen in the inset of Fig.\ \ref{fig:CBCPT}. This %%@
modulation is clear evidence that the switching voltage is associated with the Cooper pair %%@
tunneling in the array.
\begin{figure}[b]
\center 
\includegraphics[width=0.9\columnwidth]{Fig1.eps} 
\caption{\label{fig:CBCPT}Current voltage characteristics of a typical long one dimensional array %%@
of Josephson junctions. Voltage bias gives a hysteretic $I-V$ curve, with a switching voltage, %%@
$V_{sw}$, and the retrapping voltage, $V_r$. The arrows on the curve indicate the direction of the %%@
sweep. The inset shows how $V_r$ is modulated with B.} 
\end{figure}

By making $10^4$ sweeps of the bias voltage and each time registering the switching voltage %%@
$V_{sw}$, we obtain a switching histogram representing the probability distribution $P(V)$ from %%@
which the escape rate can be calculated through the formula \cite{fulton:swCurr:74},
\begin{equation}
  \Gamma(V)={\frac {1}{\Delta V}}{\frac{dV}{dt}}\ln{\left({\sum_{v\geq %%@
V}P(v)\text{\Huge/}{\sum_{v\geq V+\Delta V}P(v)}}\right)}
\label{eq:CalcG}
\end{equation}
where $dV/dt$ is the ramp rate of the voltage and $\Delta V$ is the width of the bins in the %%@
histogram.

To establish that the switching process is indeed sensitive to the applied voltage, we have varied %%@
the sweep rate by changing the frequency of our triangular sweep between $5$ and $200$Hz, while %%@
keeping the amplitude of the sweep constant at $1.2$mV. For lower sweep rates the histogram shifted %%@
to lower voltages as we are less likely to reach higher voltages before escape occurs. However, %%@
when the escape rate $\Gamma(V)$ was the calculated according to (\ref{eq:CalcG}) and comparison %%@
was made for different sweep rates, we found see that for sweep rates lower than $0.12$V/s, the %%@
escape rate at a given voltage was essentially unaffected by the sweep rate. We also found that for %%@
sweep rates higher than $0.12$V/s, the escape rate $\Gamma(V)$ becomes sensitive to the sweep rate. %%@
This limitation on sweep rate was set by the biasing and current measurement circuit together with %%@
the stray capacitance of the cryostat leads. From this analysis we conclude that the switching %%@
process, when measured with sweep speed below $0.12$V/s, is truly a voltage dependent escape %%@
problem. We also investigated if the switching process is affected by the time spent in the %%@
dissipative (finite current) state by putting in a delay and waiting at zero voltage between %%@
successive ramps. The result of this investigation was that the switching histogram for fixed sweep %%@
rate and temperature remained the same, independent of delay time. 

\begin{figure}[t]
\center 
\includegraphics[width=0.9\columnwidth]{Fig2.eps} 
\caption{\label{fig:Hist}A selection of histograms containing $10^4$ switching events, each at %%@
different temperatures. As the temperature increases the histogram shifts towards lower voltages. %%@
At high temperatures it is evident how the retrapping current affects the histogram as $V_{sw}$ %%@
gets closer to $V_r$. The inset shows how the peak of the histogram moves with temperature.} 
\end{figure}

Figure \ref{fig:Hist} shows a selected number of histograms at different temperatures. The inset of %%@
Fig.\ \ref{fig:Hist} shows how the voltage value at the peak of the histogram changes versus %%@
temperature. At temperatures above about $130$mK, the displacement of the peak position is %%@
proportional to the temperature. We also see that the width of the distribution increases as the %%@
temperature is decreased. The retrapping voltage put a lower limit on the switching voltage and %%@
thereby sets a maximum temperature of about $180$mK. Therefore we limit ourselves to temperatures %%@
between $170$mK and the base temperature of the cryostat, about $50$mK. 

\begin{figure}[t]
\center 
\includegraphics[width=0.9\columnwidth]{Fig3.eps} 
\caption{\label{fig:lnG}$\ln \Gamma$ versus the inverse temperature at $1.1$mV. Two temperature %%@
regimes are seen, the higher of which follows the thermal escape model.} 
\end{figure}

From the histograms we can determine the escape rate $\Gamma(V)$ via equation (\ref{eq:CalcG}). In %%@
Fig.\ \ref{fig:lnG} we have plotted $\Gamma(V=1.1$mV) versus $1/T$. Here we see that in the high %%@
temperature region between $170$mK and $130$mK, the data is well described by Kramers' model of %%@
thermally activited escape, equation (\ref{eq:ThermEsc}). However at low temperatures, below ca %%@
$130$mK, the escape rate crosses over to a different behavior, becoming independent of temperature %%@
at the lowest temperatue attainable. By extrapolating the linear fit in the low temperature region %%@
shown in Fig.\ \ref{fig:lnG}, we find the intersection with the $\ln \Gamma$-axis and make an %%@
estimate of the factor, $\kappa \omega _p$ of the order of $10^9-10^{10}$Hz. An alternative way to %%@
find $\kappa \omega _p$, which uses the data at all voltages, is to extract the energy barrier %%@
$\Delta U(V)$ from the escape rate $\Gamma(V)$ via equation (\ref{eq:ThermEsc}). We then adjust %%@
$\kappa \omega _p$ so that the data at all temperatures fall on to a single $\Delta U(V)$ curve. We %%@
find that in the high temperature region between $170$mK and $130$mK, the data collapse on to one %%@
curve for $\kappa \omega _p=80$GHz as is shown in Fig.\ \ref{fig:dUvsT}.

We note that this method of determining $\kappa \omega _p$ differs from the traditional analysis of %%@
escape from the superconducting state of Josephson junctions. In the superconducting case we have a %%@
well established model for the energy barrier, and to an excellent approximation, escape from the %%@
cubic potential can be assumed. The cubic potential results in a straight line plot of %%@
$(\ln(\omega_p/2\pi\Gamma))^{2/3}$ versus the bias current \cite{martinis:phasedifference:87}. If %%@
we follow this analysis and plot $(\ln(\omega_p/2\pi\Gamma))^{2/3}$ versus bias voltage in the %%@
temperature range $130-170$mK using the value $\kappa \omega _p=80$GHz, we also find a straight %%@
line. However, the limited width of the distribution and the small temperature range make it %%@
difficult to say if the data follows the theoretical predictions (slope $\propto T^{-2/3}$ and %%@
intersect with the x-axis at $V_0$). We can, however, conclude that the qualitative behavior is %%@
correct.


\begin{figure}[t]
\center 
\includegraphics[width=0.9 \columnwidth]{Fig4.eps} 
\caption{\label{fig:dUvsT}Selected curves showing how all data can be reduced to a single $\Delta %%@
U(V)$ curve in eq. (\ref{eq:ThermEsc}), with $\kappa \omega _p=80$GHz. The low temperature curves %%@
with the open symbols are fitted with an effective temperatures indicated in the parenthesis.} 
\end{figure}


Thus the data are well described by the thermal activation model in the high temperature regime %%@
with $\kappa \omega _p=80$GHz. Fixing the value of $\kappa \omega _p=80$GHz, we can then analyze %%@
the low temperature regime ($T<130$mK) by adjusting the temperature in equation (\ref{eq:ThermEsc}) %%@
to find an effective temperature. Following this procedure, all data for all voltage and tempeature %%@
can be reduced to one $\Delta U(V)$ as shown for a few selected temperatures in Fig.\ %%@
\ref{fig:dUvsT}. The effective temperature, $T_{eff}$, is plotted versus the thermometer %%@
temperature in Fig.\ \ref{fig:TeffVsT}. Here we can see how the data can be described by the %%@
thermal escape model if an effective temperature is assumed that reaches a minimum value of $122$mK %%@
as the cryostat is cooled towards base temperature. 

 
\begin{figure}[b]
\center 
\includegraphics[width=0.9\columnwidth]{Fig5.eps} 
\caption{\label{fig:TeffVsT}The effective temperature, $T_{eff}$, is plotted versus the thermometer %%@
temperature, T. At high temperatures $T_{eff} \equiv T$. At low temperatures $T_{eff}$ was inferred %%@
by mapping all data to the same $\Delta U(V)$ curve.} 
\end{figure}

There can be several reasons for deviation from the thermal activation model at low temperatures. %%@
The escape temperature of $122$mK may be a result some source of heating of the sample, so that the %%@
sample is not in equilibrium with our thermometer. We can rule out self-heating by the current in %%@
the sample \cite{kautz:selfheating:93} because we are measuring escape from a zero current state, %%@
where there is no dissipation in the sample. The low temperature escape may also be influenced by %%@
RF and microwave noise reaching the sample through the filtered measurement leads. Another %%@
possibility is that the escape process at low temperatures is determined by macroscopic quantum %%@
tunneling (MQT). 

MQT rates have been calculated for the cubic potential at $T=0$ \cite{devoret:MQT:85},
\begin{equation}
  T_{esc} \approx {\frac{\hbar\omega _p/k_B}{7.2(1+0.87/Q)}}
\end{equation}
where Q is the damping factor. Assuming moderate to weak damping ($Q \gg 1$) and $\kappa \sim 1$ we %%@
can calculate an MQT escape temperature $T_{esc}\sim 85$mK, which is in reasonable agreement with %%@
the effective escape temperature found with our method of analysis. However, similar measurements %%@
and analysis on a slightly shorter array found a much lower $\kappa \omega _p \approx 10$MHz. This %%@
would result in $T_{esc}\sim 10 \mu$K in the MQT model, in poor agreement with the effective escape %%@
temperature $T_{eff}(0)\approx 100$mK found in the data analysis.

In conclusion, we have studied the onset of finite current in long, one-dimensional Josephson %%@
junction arrays in the Coulomb blockade state. By analyzing the temperature dependence of %%@
fluctuations of the switching voltage for the onset of finite current, we have shown that this %%@
onset can be described by the Kramers model for thermally activated escape over an energy barrier, %%@
where the barrier height depends on the bias voltage. The data fall into two temperature regimes.  %%@
At high temperature thermal activation applies, but at low temperature an effective temperature %%@
must be assumed to fit the data to the Kramers model. The low temperature behavior could be the %%@
result of external noise causing the escape over the barrier, or possibly, macroscopic quantum %%@
tunneling through the barrier.

We would like to acknowledge helpful discussions with R.~Kautz and T.~Duty, and financial support %%@
from the NFR, G\"oran Gustafsson Foundation and the EU project SQUBIT.  

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\end{thebibliography}

\end{document}


cond-mat0204369.extracted_bib
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\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

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