The Amplitude of Non-Equilibrium Quantum Interference in Metallic Mesoscopic Systems
C. Terrier
T. Nussbaumer and C. Sch\"{o}nenberger
We study the influence of a DC bias voltage $V$ on quantum interference corrections to the measured differential conductance in metallic mesoscopic wires and rings. The amplitude of both universal conductance fluctuations (UCF) and Aharonov-Bohm effect (ABE) is enhanced several times for voltages larger than the Thouless energy. The enhancement persists even in the presence of inelastic electron-electron scattering up to $V\sim 1$\,mV. For larger voltages electron-phonon collisions lead to the amplitude decaying as a power law for the UCF and exponentially for the ABE. We obtain good agreement of the experimental data with a model which takes into account the decrease of the electron phase-coherence length due to electron-electron and electron-phonon scattering.
Terrier3.tex
\documentclass[nocite,nobm]{epl}

\begin{document}

\title{The Amplitude of Non-Equilibrium Quantum Interference in Metallic Mesoscopic Systems}
%
\author{C. Terrier, 
D. Babi\'{c}\footnote{Present address: Department of Physics, Faculty of Science, 
University of Zagreb, Croatia.}, 
C. Strunk\footnote{Present address: 
Institute for Experimental and Applied Physics, 
University of Regensburg, Germany.}, 
T. Nussbaumer and C. Sch\"{o}nenberger}
\institute{
Institute of Physics, University of Basel, Klingelbergstrasse 82, 
CH-4056 Basel, Switzerland 
}
%
\pacs{73.23.-b}{Electronic transport in mesoscopic systems}
\pacs{72.10.-d}{Theory of electronic transport; scattering mechanisms}

\maketitle

\begin{abstract}

We study the influence of a DC bias voltage $V$ on
quantum interference corrections to the 
measured differential conductance
in metallic mesoscopic wires and rings.
The amplitude of both universal conductance fluctuations (UCF) 
and Aharonov-Bohm effect (ABE) is enhanced
several times for voltages larger than the Thouless energy. 
The enhancement persists even in the presence of
inelastic electron-electron scattering up to 
$V\sim 1$\,mV.
For larger voltages electron-phonon collisions lead to
the amplitude decaying as a power law for the UCF and 
exponentially for the ABE.
We obtain good agreement of the experimental data
with a model which takes into account the decrease
of the electron phase-coherence length due to 
electron-electron and electron-phonon scattering.


\end{abstract}


If the size of a conductor is of order of the
electron phase-coherence length $L_\varphi (T)$ the wave character
of electrons leads to experimentally observable quantum interference contributions to 
the conductance $G$. 
These are the aperiodic and periodic fluctuations $\delta G$ of
the conductance around its average value. In the former case,
called universal conductance fluctuations, the interference pattern is 
formed as a superposition of 
contributions from a continuous range of possible interference paths. 
Averaged over either impurity configuration,
magnetic field or energy, the root-mean-square (rms) conductance 
fluctuation $\delta G_{rms}$ is
of the order $\sim e^2/h$ \cite{leealt}. The periodic fluctuations, known as the
Aharonov-Bohm  effect \cite{ab}, are observed
if the interference is imposed by geometry of a sample, most commonly 
in the form of a loop. 
If the loop is threaded by a magnetic
flux $\phi$,  $\delta G (\phi)$ exhibits periodic oscillations with a 
period $h/e$, and
$\delta G_{rms}$ is again $\sim e^2/h$. 
Both the UCF \cite{skocpol} and the ABE \cite{umbach}
are suppressed by ensemble averaging if
independent phase-coherent units are connected 
in series.
The above behaviour is characteristic of the linear-response regime, 
{\it i.e.} of $eV \ll k_B T$ or $eV \ll E_c$, where $E_c$ is the coherence energy (Thouless energy)
determined by the size of the conductor.
In non-equilibrium ($eV \gg E_c, k_B T$) the fluctuations are expected to 
be remarkably different, as 
predicted theoretically by Larkin and Khmel'nitski\u{\i} (LK) \cite{lk}. 
If inelastic processes can be neglected
the rms fluctuation $\delta g_{rms}$ of the {\it differential conductance} $g$ 
increases with $V$ according to
$\delta g_{rms} \sim (e^2/h) \sqrt{V/V_c}$, where $V_c=E_c/e$. 
This, at first sight surprising result, can be understood as follows.  At $V \gg V_c$
the relevant energy range for the transport subdivides into $N=V/V_c$ 
uncorrelated energy  intervals, each  contributing to the fluctuations of the current by an amount
$\sim (e^2/h) V_c$. Incoherent superposition of these contributions leads to 
$\delta g_{rms}$ being $N^{1/2}$ times larger than $e^2/h$.
Inelastic scattering at large voltages destroys quantum interference.
The enhancement of
$\delta g_{rms}(V)$ is thereby suppressed and 
$\delta g_{rms}$
eventually decreases with increasing voltage \cite{lk}.
Experimental studies  of both the
UCF \cite{webb1,kaplan,ralph,schafer} and ABE \cite{webb1,ct,haus} under
non-equilibrium conditions have been done, but did not result
in a satisfactory understanding of $\delta g (V)$.
Moreover, \cite{ralph} reported on a voltage-independent $\delta g_{rms}$,
\cite{webb1} on a decrease of $\delta g_{rms}(V)$ for $V > V_c$,
whereas in \cite{schafer,ct,haus}  it was found that $\delta g_{rms}(V)$   
shows a non-monotonic behaviour. 
We also note that explanations \cite{ct,haus} of the non-monotonic behaviour of 
$\delta g(V)$, observed in the present work as well, were at a qualitative level.

In this paper we report on the non-equilibrium UCF and ABE in diffusive 
gold samples measured
over a wide voltage range of $V \gg k_B T / e, V_c$ 
(strongly $V$-dominated energy averaging)
that covers 
both the low-voltage $(V/V_c)^{1/2}$ 
enhancement of $\delta g_{rms}$ and its suppression at large voltages.
Emphasis is put on the 
decay of $\delta g_{rms}^{UCF}(V)$ and  $\delta g_{rms}^{ABE}(V)$.
It is shown that $\delta g_{rms}(V)$ decays as a power law for the UCF and 
exponentially for the ABE. A quantitative comparison between simple models and
the experiments allows to extract the voltage-dependent phase-coherence length $L_{\varphi}(V)$ 
and to discuss the nature of inelastic scattering processes out
of equilibrium.

The samples were produced by electron-beam lithography 
and evaporation of 99.99 \% pure gold.
The substrate was silicon covered with 400 nm of SiO$_2$. 
Three 20 nm thick samples of different planar geometries were 
prepared \cite{samples}: (1) 
for the ABE measurements
a ring of average diameter 1 $\mu$m and line width 0.09 $\mu$m (sample S$_{ABE}$),
with a resistance of 25.8 $\Omega$ at 0.3 K; 
(2)  for the UCF measurements a $L_w = 1.5$ $\mu$m long and 0.13 $\mu$m wide wire
(sample S$_{UCF}$), with a resistance of 18.4 $\Omega$ at 0.3 K;
(3)  for the weak-localisation (WL) measurements a  98 $\mu$m long and 0.17 $\mu$m wide wire
(sample S$_{WL}$).
Sample S$_{WL} $ was made relatively long in order to suppress the UCF.
All samples were produced under identical conditions
(the same source of gold and the same evaporation parameters). The diffusion
constants $D$ are consequently very similar:
116 cm$^2$/s for  samples S$_{UCF}$ and S$_{WL}$, and 
110 cm$^2$/s for sample S$_{ABE}$.
The measurements were carried out in a $^3$He cryostat 
with cryogenic rf-filtering, using a low-frequency ($37$\,Hz) lock-in technique 
to measure the differential conductance.
Typical voltage resolution was $\sim 0.3$ nV.
Non-equilibrium UCF and ABE were measured at $T=300$ mK by superimposing
a comparatively large DC bias voltage $V_{DC}$ to the small excitation voltage
$V_{AC}$. The following hierarchy of energies was 
always maintained: $eV_{DC} \gg k_B T \geq eV_{AC} > E_c$. 
Ensemble averaging was achieved by measuring the UCF and ABE over a
magnetic-field range of $\sim 2$\,T, largely exceeding the correlation field.
Typical sweep rates were 0.1 mT/s.
Fig.1a displays raw data of a differential-conductance measurement  on sample S$_{ABE}$,
taken at $V_{DC}=0.5$ mV. Both types of the fluctuations are present. In Fig.1b we show the Fourier
transform of the same data, exhibiting a well-defined peak at the position corresponding to
$\phi=h/e$. 
From the magnetoconductance traces we have extracted 
$\delta g_{rms}^{ABE}$ and $\delta g_{rms}^{UCF}$ for sample S$_{ABE}$, as shown
in Fig.2 (discussed in more detail later), and $\delta g_{rms}^{UCF}$ for sample S$_{UCF}$.
The values of $\delta g_{rms}^{UCF}$ have been determined as standard deviations of the 
whole differential-conductance traces. The amplitudes  $\delta g_{rms}^{ABE}$ have been calculated by 
averaging the periodic part of the fluctuations over  $\sim 500$ periods, after the coarse background 
has been removed numerically.

The WL was measured on sample S$_{WL}$ in the linear-response regime ($V_{DC}=0$) and in the 
temperature range 0.3 - 10 K.
By fitting a one-dimensional WL expression \cite{altshulerwl} to 
the low-field magnetoconductance data 
we obtained the linear-response $L_\varphi$ as a function of temperature, as shown in the inset to Fig.3.   
Similarly to published work \cite{mohanty,gougam},
$L_\varphi (T)$ saturates at low temperatures (below 1 K), 
which has been attributed to impurity mediated 
inelastic electron-electron scattering \cite{glazman}.
At high temperatures $L_\varphi$
follows a $L_\varphi \propto T^{-q}$ dependence with $q \approx 1.2$ that 
suggests dephasing by electron-phonon interaction.
At $T= 300$ mK, $L_\varphi = 3$ $\mu$m and 
$\tau_\varphi = \sqrt{L_\varphi^2 / D} =  0.77$ ns,
implying that in the linear-response regime the diffusion of electrons is coherent throughout the samples
$S_{UCF}$ and $S_{ABE}$.
$L_\varphi$ has to be compared further with the
effective length of the sample $L_{c0}$ which depends on the particular geometry 
and the coherence phenomenon investigated. For the UCF in sample S$_{UCF}$,
$L_{c0} = L_w = 1.5$ $\mu$m. In the case of ABE in sample S$_{ABE}$,
$L_{c0} = C_r \approx 3.14$ $\mu$m, where $C_r$ is the circumference of the ring.
Finally, for the UCF in sample S$_{ABE}$, $L_{c0}$
is calculated as  the distances ring - voltage contacts plus $C_r /2$, which
gives $\approx 2.2$ $\mu$m. 
Thus, $L_\varphi$(300 mK) is always bigger or very similar to 
$L_{c0}$, so that the linear-response coherence
energy is set by the time of electron diffusion through the sample
according to $E_{c0} = eV_{c0} = \hbar D / L_{c0}^2$.

With increasing temperature or applied voltage the coherence length decreases,
and once it becomes shorter than $L_{c0}$ the interference
contributions are suppressed. 
The effect
of temperature was demonstrated by Milliken {\it et al.} \cite{milliken}.
Their method was to determine  $L_\varphi (T)$ from the WL and to use this to
describe the decay of {\it equilibrium} $\delta G$, which turned out to be a power 
law for the UCF and exponential for the ABE. Our approach is complementary: 
we keep the bath temperature constant and investigate the voltage dependence
of the {\it non-equilibrium}
$\delta g$.  While Milliken {\it et al.} observed
a monotonically-decreasing $\delta G (T)$, in our case $\delta g (V)$ is
a non-monotonic function, as shown  in Fig.2.
For small $V_{DC}$ (but still much larger than $V_{c0}$), 
$\delta g_{rms}$ is enhanced by $V_{DC}$. At higher voltages
$\delta g_{rms}$ decreases with $V_{DC}$, {\it faster for the} ABE
{\it than for the} UCF,
which is in qualitative agreement with the results of Milliken {\it et al.}

Let us first turn to the details of non-equilibrium UCF. For convenience we
denote the phase-coherence length at a finite bias voltage by $L_{\varphi}(V)$, while
$L_{\varphi}(T)$ referes to the linear-response value. 
If $V_{DC} \gg V_{c0}, k_B T/e$ and
$L_{\varphi} (V_{DC}) > L_{c0}$, the LK enhancement of $\delta g_{rms}$ holds and one
expects that $\delta g_{rms} \propto \sqrt{V_{DC}/V_{c0}}$. 
Once $L_{\varphi} (V_{DC})$  becomes substantially
smaller than $L_{c0}$ inelastic processes start to be important for most of 
the electrons, and the sample effectively 
subdivides into $L_{c0}/L_ {\varphi}$ uncorrelated phase-coherent  sections in series. 
The LK enhancement in a single section is still valid if the voltage
drop $V^{sec}_{DC} = V_{DC} (L_{\varphi}/L_{c0})$  over a section exceeds the 
corresponding coherence voltage $V_c^{sec} = V_{c0} (L_{c0}/L_{\varphi})^2$.
Following the LK arguments, $\delta g_{rms}$ corresponding to a single
section is proportional to 
$(V^{sec}_{DC} / V_c^{sec})^{1/2} = [ (V_{DC}/V_{c0}) (L_{\varphi}/L_{c0})^3]^{1/2}$.
Incoherent addition of these contributions of sections in series 
leads to the {\it differential conductance} being suppressed by a factor
$(L_{\varphi}/L_{c0})^{1/2}$, and hence
%
\begin{equation}
%
\delta g_{rms}^{UCF} = A_{UCF} \frac{e^2}{h}     \sqrt{ \frac{V_{DC}}{V_{c0}} } 
\left( \frac{L_{\varphi} (V_{DC})}{ L_{c0}} \right) ^2 \; ,
%
\end{equation}
%
where $A_{UCF}$ is a constant prefactor. This expression is 
different from that resulting from the linear-response ensemble averaging, 
and valid if $V_{DC} > V_{c0}$ and $L_{\varphi} < L_{c0}$ \cite{remark}. It enables an
unambiguous determination of $L_{\varphi} (V_{DC})$. 
Since the form of $L_{\varphi} (V_{DC})$ is not known {\it a priori}, 
we cannot determine $A_{UCF}$ and $L_{\varphi} (V_{DC})$
simultaneously from a direct fit to the data. 
To avoid this problem we fix $A_{UCF}$  
by setting $L_\varphi= L_{c0}$
for the first measured point which satisfies $V_{DC} \gg V_{c0}$,
and then simply extract
$L_{\varphi} (V_{DC})$ for all the other points from the measured data. 
For the two sets of UCF data, {\it i.e.} for samples S$_{UCF}$ and
S$_{ABE}$, we have obtained excellent agreement in $L_{\varphi}(V_{DC})$, 
as we show in Fig.3 by open 
(S$_{ABE}$) and full (S$_{UCF}$) squares. Normalised to the 
corresponding  values of $L_{c0}$
both results collapse onto a single curve. 
The phase-coherence length deduced from our non-equilibrium
experiment $L_{\varphi}(V)$ has a qualitatively similar functional form
as $L_{\varphi}(T)$ obtained from the WL measurements
(apart from the saturation of $L_\varphi (T)$, since
saturation of $L_\varphi (V)$ is not expected 
for $eV \gg k_B T$ \cite{glazman}). Two regimes
with different power-law dependences are discernible.
In the region of increasing $\delta g_{rms}(V_{DC})$ there is a power law 
$L_{\varphi}(V_{DC}) \propto V_{DC}^{-s}$, where $s = 0.19 \pm 0.02$. 
For decreasing $\delta g_{rms} (V_{DC})$, {\it i.e.} where $L_{\varphi}(V_{DC})$ 
is considerably smaller than $L_{c0}$,
the dependence changes to $L_{\varphi} \propto V_{DC}^{-p}$ with 
$p = 0.57 \pm 0.03$. Inserting thus found $L_{\varphi}$ back into Eq.1 results in 
the dotted line in Fig.2.

We now discuss the non-equilibrium ABE. Here the LK theory cannot be 
directly applied to extract $\delta g_{rms}$. 
Subdivision of a ring into phase-coherent sections makes no sense in this case, as only those
electrons which stay coherent over the whole length $C_r$ contribute to the interference. However, since the
same physics governs the UCF and ABE, 
$\delta g_{rms}^{ABE}$ is expected to be $\propto \sqrt{V_{DC}/V_{c0}}$
in the regime where inelastic scattering is absent.
This is indeed the case, as shown in Fig.2.
The form of the suppression of  $\delta g_{rms}^{ABE}$ can be inferred from a theoretical analysis of
DiVincenzo and Kane \cite{dk}. They have shown that inelastic processes influence the ABE in two ways.
First, the fluctuations are suppressed as $\exp( - \beta C_r / L_{\varphi})$, where $\beta$ is 
of order unity in the range $1 < C_r / L_{\varphi} < 3$ 
and smaller in the case of a true exponential decay occurring in
the asymptotic limit $C_r / L_{\varphi} \rightarrow \infty$. Second, the proper form of $E_c$ in the presence
of inelastic processes is given by 
$E_c^{in} = \hbar D / C_r^{2-\alpha} L_{\varphi}^\alpha = E_{c0} (C_r/L_{\varphi})^\alpha$ 
with $\alpha \approx 1.3$. This renormalisation of the coherence energy is a consequence of the statistics
for those electrons that diffuse around a ring without being scattered inelastically \cite{dk}.
Roughly, the probability distribution for an electron to contribute to the ABE has a maximum
at a value of the electron traversal time $\sim D /C_r L_{\varphi}$ \cite{dk}.
Combining the above arguments we can write for $L_{\varphi} < C_r$:
%
\begin{equation}
%
\delta g_{rms}^{ABE} = A_{ABE} \frac{e^2}{h}     \sqrt{ \frac{V_{DC}}{V_{c0}} } 
\left(  \frac{L_{\varphi}(V_{DC})}{C_r} \right) ^{0.65} 
e^ { - \beta C_r / L_{\varphi}(V_{DC}) } \; .
%
\end{equation} 
%
Good agreement of the ABE data with both Eq.2 and the UCF data 
is obtained for $\beta \approx 1.1$, 
as we show in Fig.2 by the solid curve and in Fig.3 by crosses.

Below we discuss how the observed power law dependences
in $L_\varphi (V_{DC})$ can be linked to microscopic scattering processes.
From Figs.2,3 we can distinguish two 
regimes (denoted by {\rm I} and {\rm II} in Fig.3): 
{\rm (I)} $\delta g_{rms} (V_{DC})$ increases with  $V_{DC}$ and
$L_{\varphi}(V_{DC}) \propto V_{DC}^{-0.19}$, and {\rm (II)},
$\delta g_{rms}(V_{DC})$ decreases with $V_{DC}$ while
$L_{\varphi}(V_{DC}) \propto V_{DC}^{-0.57}$. 

The phase-coherence length is determined by the dephasing rate
$\tau_{\varphi}^{-1}$ via $L_{\varphi}=\sqrt{D\tau_{\varphi}}$.
$\tau_{\varphi}^{-1}$ can be expressed as an average of the
inelastic scattering rate $\tau_{in}^{-1}(\epsilon)$, ($\epsilon$
is the energy exchanged in the interaction) of electrons over the
accessible energy range given by the width of the electron
distribution function $f$ \cite{gougam}. If the electron-electron
scattering is dominant, the scattering rate is energy dependent as
described by the kernel function
$K(\epsilon)=\kappa_{\eta}\epsilon^{-\eta}$. The energy interval
$E$ accessible for the scattering is set by $k_BT$ in equilibrium
and $eV_{DC}$ in non-equilibrium. At high energies, where
single scattering events determine $\tau_\varphi$, a simple
argument \cite{gougam} leads to $\tau_{\varphi,ee}^{-1}(E) \propto
E^{1/\eta}$ \cite{saturation}. Two choices for $\eta$ are currently under debate:
$\eta=3/2$ for the disorder enhanced Coulomb interaction \cite{gougam} and
$\eta=2$  for magnetic impurity mediated interaction
\cite{glazman,pierre}. This results in
$L_{\varphi,ee}^{-1}(V_{DC}) \propto
V_{DC}^{1/3}$ for $\eta = 3/2$, and $L_{\varphi,ee}^{-1}(V_{DC}) \propto V_{DC}^{1/4}$ 
for $\eta = 2$.
Hence the kernel exponent $\eta=2$ appears more consistent with
our experimental observation of $L_\varphi^{-1} \propto
V_{DC}^{0.19}$. This is supported by the observed saturation of
the equilibrium $\tau_\varphi$ for temperatures below \mbox{1 K}.

We emphasise that - independently of the precise interaction
mechanism - the rather weak decrease of $L_{\varphi,ee}$ with
increasing $V$ is insufficient to suppress the LK enhancement of
$\delta g_{rms}$. This is seen as follows. We recall that the
condition for the appearance of the LK enhancement in a
phase-coherent section of the wire is $V_{DC}^{sec} > V_c^{sec}$, which can be written as
$V_{DC}/V_{c0} > (L_{c0}/L_{\varphi})^3$. Since $L_{\varphi,ee}
\propto V_{DC}^{-1/4}$, the right-hand side of the above
inequality increases more slowly with $V_{DC}$ than the left-hand
side, and the condition for the enhancement is maintained over
the whole voltage range of increasing $\delta g_{rms}  (V_{DC})$.

While the regime {\rm I} is determined by electron-electron scattering
and a non-equilibrium electron distribution function $f$, 
the stronger decay of $L_{\varphi}(V)$ in the regime {\rm II} 
is caused by electron-phonon scattering. It can be very well
described by equilibrium properties if we assume local
thermal equilibrium with an elevated electron temperature
$T_{el} (V_{DC})$.
Electron-phonon scattering leads to 
$\tau_{\varphi,ep} \propto T_e^{-m}$, where $m$ equals 3 for very clean samples,
4 for strongly disordered samples, and 2 - 3 for samples of intermediate degree of 
disorder \cite{schmid}.
Since $T_e$ is related to the applied voltage by
$T_e \propto V^{2/(2+m)}$ \cite{anderson}, one obtains 
$\tau_{\varphi,ep} \propto V^{-2m/(2+m)}$ 
and $L_{\varphi,ep} \propto V^{-m/(2+m)}$. Our result 
$L_{\varphi}(V_{DC}) \propto V_{DC}^{-0.57 \pm 0.03}$ gives
$m = 2.6 \pm 0.3$, which agrees well with the value $2.5$ obtained
in a noise measurement on similar samples \cite{henny3}.
Moreover, our WL result
($L_{\varphi,ep}(T)\propto T^{-m/2}$) is $m = 2.4$,
in excellent agreement
with the above values.
We also note that even though $T_e$
rises up to $\sim 10 - 15$ K at high $V_{DC}$ the energy
averaging given by the LK approach remains essentially 
unaffected because $T_e$ increases 
more slowly than the voltage itself. 
Although our phenomenological analysis leads to a
consistent interpretation of our data,
further theoretical work is needed to fully understand
the dephasing in the case of strong non-equilibrium.

In conclusion, we have measured the universal conductance fluctuations 
and Aharonov-Bohm effect in mesoscopic gold
samples under highly non-equilibrium conditions of large applied bias voltages.
The rms fluctuation $\delta g_{rms}$ of 
the differential conductance initially increases with voltage
$\propto \sqrt{V/V_c}$,
which demonstrates the validity of the theoretical 
prediction by Larkin and Khmel'nitski\u{\i}.
This increase is followed by a decay of $\delta g_{rms}$
at higher voltages, where inelastic scattering becomes 
substantial. The amplitude decays as a power law for the universal conductance
fluctuations and exponentially in the case of the Aharonov-Bohm
effect. The decrease of the phase-coherence length with
increasing voltage is in 
good agreement with the inferred $L_{\varphi}(V)$
dependences for electron-electron and 
electron-phonon scattering.
In particular, the electron-electron collisions
are not sufficient to suppress the enhancement mechanism.
This work was supported by the Swiss National Science Foundation.

%
% FIG1
%
\begin{figure}
\onefigure[width=8cm]{Terrierfig1.eps}
\caption{(a) A differential-magnetoconductance trace of sample S$_{ABE}$, taken at 
$V_{DC} = 0.5$ mV. 
Both the periodic (expanded view) and aperiodic fluctuations are present. 
(b) Fourier
transform of (a), showing a well-defined peak at the position corresponding to
$\phi = h/e$.}
\label{f.1}
\end{figure}

%
% FIG 2
%
\begin{figure}
\onefigure[width=8cm]{Terrierfig2.eps}
\caption{Plot of $\delta g_{rms}^{ABE}$ (crosses) and $\delta g_{rms}^{UCF}$ (open squares)
{\it vs.} $V_{DC}$ for sample S$_{ABE}$,
with $\delta g_{rms}$ normalised to the 
values $\delta g_{rms}^0$ at the first $V_{DC} \neq 0$ points (which also satisfy the criterion
$V_{DC} \gg V_{c0}$). The lines are plots of Eq.1 (dashed line)
and Eq.2 (solid line) discussed in the text.}
\label{f.2}
\end{figure}
%
%FIG3
%
\begin{figure}
\onefigure[width=8cm]{Terrierfig3.eps}
\caption{Phase-coherence length $L_{\varphi}(V_{DC})$, 
normalised to the characteristic lengths $L_{c0}$
defined in the text,
deduced from the non-equilibrium UCF and ABE measurements.
Crosses: ABE in sample S$_{ABE}$.
Open squares: UCF in sample S$_{ABE}$. Full squares: UCF in sample S$_{ABE}$.
Below $V_{DC} \sim 1$ mV, $L_{\varphi} \propto V_{DC}^{-0.19 \pm 0.02}$, and
above $L_{\varphi} \propto V_{DC}^{-0.57 \pm 0.03}$.
Inset:  Linear-response coherence length $L_\varphi (T)$ found from the WL measurements on
sample S$_{WL}$.}
\label{f.3}
\end{figure}

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\bibitem{remark}
Strictly speaking, the conditions are $V_{DC}\gg V_{c0}$ and $L_{\varphi}\ll L_{c0}$.

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\bibitem{saturation}
The saturation of $\tau_\varphi (T)$ at very low energies is not covered by the simple
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\Name{ Pierre F., Pothier H., Esteve D., Devoret M. H., Gougam A. B. \and Birge N. O.}
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Magnetic Scattering, Pecs, Hungary}
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\Year{2001}

\bibitem{schmid}
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\bibitem{henny3}
  \Name{Henny M.} PhD Thesis, University of Basel, 1998. 




\end{thebibliography}

\end{document}


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\bibitem{samples}
The results for these samples are representative of the data obtained by measurements 
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\bibitem{glazman}
  \Name{Kaminski A. \and Glazman L. I.}
  \REVIEW{Phys. Rev. Lett.}{86}{2001}{2400}.

\bibitem{milliken}
  \Name{Milliken F. P., Washburn S., Umbach C. P., Laibowitz R. B. \and Webb R. A.}
  \REVIEW{Phys. Rev. B}{36}{1987}{4465}.

\bibitem{remark}
Strictly speaking, the conditions are $V_{DC}\gg V_{c0}$ and $L_{\varphi}\ll L_{c0}$.

\bibitem{dk}
  \Name{DiVincenzo D. P. \and  Kane C. L.}
  \REVIEW{Phys. Rev. B}{38}{1988}{3006}.  

\bibitem{saturation}
The saturation of $\tau_\varphi (T)$ at very low energies is not covered by the simple
argument of Ref. 15.

\bibitem{pierre} 
\Name{ Pierre F., Pothier H., Esteve D., Devoret M. H., Gougam A. B. \and Birge N. O.}
in \Book {Proceedings of the NATO Advanced Research Workshop on Size-Dependent
Magnetic Scattering, Pecs, Hungary}
\Publ{Kluwer, Dodrecht, The Netherlands}
\Year{2001}

\bibitem{schmid}
  \Name{Schmid A.}
  in \Book{Localization, Interaction and Transport Phenomena}
  \Editor{Kramer B., Bergmann G. \and Bruynsereade Y.}
  %\Vol{9}
  \Publ{Springer, Berlin}
  \Year{1985}
%  \Page{666}.

\bibitem{anderson}
  \Name{Anderson P. W., Abrahams E. \and Ramakrishnan T. V.}
  \REVIEW{Phys. Rev. Lett.}{43}{1979}{718}.

\bibitem{henny3}
  \Name{Henny M.} PhD Thesis, University of Basel, 1998. 




\end{thebibliography}