# decoupling environmental modes from tunneling electrons in a partially wet
phase molecular mechanically controllable break junction

C.J. Muller

This research has, in its entirety, been privately conducted and funded by the
author Correspondence: PO Box 60, 6580 AB Malden, The Netherlands

###### Abstract

A quantum effect at ambient conditions is presented. A benzene dithiol (BDT)
molecule or a tetrahydrofuran (THF) molecule is used as a barrier molecule
bridging the gold electrodes from a Mechanically Controllable Break junction.
It has been known for a long time that the environment of a junction couples
to the conduction. With a microscopic layer of fluid it is actually possible
to influence this coupling. The electrode-molecule-electrode configuration is
lined with a microscopic layer of THF fluid, also known as the “partially wet
phase”. This is effectively decoupling the electrode-molecule-electrode system
from environmental modes outside of this system. The effects on coherence and
quantum interference are exceptional.

##  1 Introduction

More than 60 years ago von Hippel vonhippel1956source1a,*vonhippel1959source1b
laid out his vision where he saw a world using the intrinsic properties of
atoms and molecules:

"One could build materials from their atoms and molecules for the purpose at
hand … . He can play chess with elementary particles according to their
prescribed rules until engineering solutions become apparent."

It was a time where radio tubes were being replaced by integrated circuits and
transistors. In contrast to the logical and, already in those days, destined
successful path of ongoing miniaturization for IC’s, von Hippel was advocating
a bottom up approach which he called Molecular Engineering. With this view von
Hippel laid the foundation for a field which we now call Molecular Electronics
scheer2017molecularelectronics,*datta1997electronictransport,*datta2005quantumtransport.

During that same time frame, although seemingly unrelated at the time, Gorter
[6] established an important finding on the resistivity-temperature curve of
granular Sn particle films:

"The granularity of the transported charge manifests itself in the conductance
as a result of the Coulomb repulsion of individual electrons. The transfer by
tunneling of one electron between two initially neutral regions, of mutual
capacitance $C$, increases the electrostatic energy of the system by an amount
of $e^{2}/2C$. At low temperatures and small applied voltages, conduction is
suppressed because of the charging energy."

This phenomenon is now known as the Coulomb blockade of single-electron
tunneling [7]. Although the physics of granular films and systems need to
account for a distribution in particle-size and thus capacitance, theoretical
elements used in the early descriptions
gorter1951source2,zeller1969tunneling,*zeller1968superconductivity,lambe1969chargequantization
are still present in today’s theory for more well defined systems for example
a single junction.

A single small-capacitance junction, consisting of superconducting material,
was predicted to show a new type of voltage oscillation in 1975 [11]
transposing the current and voltage character as compared to the normal AC
Josephson effect. This is reminiscent of Bloch electrons in a perfect crystal
with lattice constant $a$, moving through the energy band representation and
being reflected by the Brillouin zone at $2\pi/a$ to the opposing Brillouin
zone at $-2\pi/a$, physically identical in nature. A conduction electron under
the influence of a constant electric field $E$, will start to oscillate in
real space, as a result of interaction with the periodic lattice potential
with a Bloch frequency: $f_{B}=aeE/(2\pi\hbar)$. In real life this effect is
not observed in crystals, as the conduction electrons are scattered before
they can complete the cycle in the Brillouin zone in $k$ space.

In superconducting small-capacitance current biased junctions the elementary
frequency equals $f_{SC}=I/(2e)$, with a corresponding band-zone edge at
$Q=+e$ or $-e$ in the energy band representation, $Q$ being the charge on the
capacitor. The origin of these predicted voltage oscillations, is the
continuous charging until the level reaches $+e$ or $-e$ after which a tunnel
event occurs and the recharging starts again. At the tunnel event the system
jumps from the band-zone edge to the opposing zone edge, both physically
identical in nature, and the cycle starts again.

The charging of the capacitor with a continuous charge $Q$, smaller than a
$2e$ Couper pair charge is physically possible. A moving charge carrier can
create some influence charge far smaller as compared to the unit charge of a
charge carrier on the small capacitor while moving to the location where it
will eventually tunnel, hence treating $Q$ as a classical variable.

The interest in this system originates from the uncertainty relation between
$\varphi$, the superconductor order parameter, and $Q[\varphi,Q]=i2e$. When
$\varphi$ is well defined, the system is described by the AC Josephson effect,
it implies that $Q$ is undefined, which is equivalent to a large capacitance
system. In the other extreme, for small capacitance systems, theoretically $Q$
is well defined and $\varphi$ is undefined, leading to the small-capacitance
effects as described above.

In the mid 1980’s several authors of the first hour [12, 13, 14, 15]
theoretically analyzed small-capacitance effects in both normal metal and
superconducting metal junctions, they predicted that both the superconducting
as well as the normal metal small capacitance junction would each have their
own type of voltage oscillation. The normal metal type single electron
tunneling (SET) oscillations will have a frequency $f_{N}=I/e$, the tunnel
event occurs at the zone edge $+e/2$ or $-e/2$ in the energy band
representation. A rigorous theoretical underpinning of these predicted effects
[16, 17] initiated a widespread experimental research, for extensive early
reviews see [18, 19]. Defining a phase
$\varphi_{S}(t)=(e/\hbar)\int^{t}_{-\infty}V(t^{\prime})\differential
t^{\prime}$ being the integral over the voltage across the junction, Schön
[20] demonstrated the commutation rule $[\varphi_{S},Q]=ie$ for normal metal
junctions. It was well recognized that with a small-capacitance junction we
might have a system with the possibility to complete the full cycle through
the energy band representation and arrive at a novel quantum effect.

Experiments on small capacitance single junctions became possible due to
advanced e-beam lithography capabilities in the 80’s. A capacitance of the
order of ${10}^{-15}\text{\,}\mathrm{F}$ leads to a charging energy equivalent
to $1\text{\,}\mathrm{K}$, a temperature well accessible by cryogenic
techniques. Results showed that the direct electrical environment of the
junction played a crucial role in the occurrence of a Coulomb blockade gap in
the current-voltage (IV)-curve. Parasitic capacitance of the electrical leads
connecting to the junction is in the $\mathrm{p}\mathrm{F}$ range, much larger
than the junction capacitance, leading to a high capacitance-value as seen by
the junction and thus easily destroying any $E_{C}=e^{2}/2C$ charging-energy
effects. This was prompting experiments where the junction was decoupled from
the leads by inclusion of large resistors, with values much larger than
$R_{Q}=h/e^{2}$ in the electrical leads at very close proximity,
$\mathrm{\SIUnitSymbolMicro m}$’s, to the junction [21].

An alternative elegant way to decouple a small capacitance from the leads, is
by implementing two high resistive tunnel junctions, with tunnel resistances
$R_{T}\gg R_{Q}$, on either side of a conductive island [22], very similar to
Gorter’s use of granular films where small Sn particles are connected via
oxidized tunnel barriers. These structures are known as "dots". The low
capacitance central island part of the dot is perfectly isolated from
environmental modes in the leads by the two high impedance tunnel junctions.
The coupling or tunnel rates of these model systems to the leads can be tuned
by the characteristic resistance of the tunnel junctions. The capacitance can
be low as it scales with the size of the dot. The condition $R_{T}\gg R_{Q}$
for the two tunnel junctions ensures that after tunneling to the dot, the
electron is localized there. The dot system thus exhibits both the wave
character, tunneling to and from the dot, as well as the particle character of
an electron as there can only be an integer number of electrons on the dot. If
$R_{T}$ becomes smaller than $R_{Q}$ the charge becomes delocalized along the
system and charging effects will be suppressed.

There are a number of dot regimes possible [23]. A metallic grain of
$\mathrm{\SIUnitSymbolMicro m}$ size holds billions of electrons occupying
billions of states up to the Fermi energy. The coulomb repulsion by putting 1
additional electron on this grain requires this electron to have an excess
energy of $e^{2}/2C$, before being able to enter the grain. The Fermi
wavelength is of the order of the inter atomic distance, much smaller as the
particle size, ensuring the metal electron states are as in bulk metal, finely
dispersed with energy separations much smaller than the $e^{2}/2C$ value.

A different regime holds in a semiconductor dot also of
$\mathrm{\SIUnitSymbolMicro m}$ dimensions. The Fermi wavelength in
semiconductors is of the order of $100\text{\,}\mathrm{nm}$. Now the charging
energy as well as finite size effects in this dot interact with one-another.
When the charging energy of these dots become comparable to the energy level
spacing it is predicted that the spin degeneracy is lifted and a regular
renormalized level spacing is the resultant [24]. Because next to the charging
effects the finite size effects of these types of dots start to become a
crucial factor in the description these dots are called quantum dots.

Where Gorter meets von Hippel we arrive at the ultimate quantum dot. One
molecule attached via two tunnel junctions to the leads. Note that this dot
diameter is smaller by more than a factor of 1000, as compared to the quantum
dot regime detailed above. HOMO and LUMO orbitals are interacting with the
charging energy and are expected to play a role in the conduction. Free
molecules have a discrete set of electronic levels, typical level spacing is
in the $\mathrm{eV}$ range. Molecules exhibit vibrational and rotational
degrees of freedom, where the typical vibrational energies are in the
$0.1\text{\,}\mathrm{eV}$ range and the rotational energies are in the
$\mathrm{meV}$ range. One of the early and most spectacular successes of
quantum theory is the answers it provided to questions related to molecular
structure and spectra. Now we are faced with questions related to a single
connected molecule. Will such a system still be described by an electrostatic
capacitance? As the molecule only measures a few
$\mathrm{\SIUnitSymbolAngstrom}$ it is not safe to assume the screening length
in the molecule to be smaller as compared to the size of the molecule. It
cannot be assumed to be an electrostatic capacitance. For simplicity, this
system is treated as if it can be described by a single electrostatic
capacitance value.

In all of the above described systems, the environment of the junction has
played an important role ever since the first experiments on small capacitance
junctions. It is important to define the quantum mechanical environment [25,
26, 27]. Next to the system defining modes, determined for example by junction
capacitance, inductance, resistance, the tunneling electron can couple to
other modes. These modes can couple to the junction via the leads (lead
environmental modes) or via the vacuum (vacuum environmental modes). The
vacuum environmental modes reflect the coupling between tunneling electrons
and modes in the medium enclosing the electrode-molecule-electrode
configuration. These modes are present in every enclosing medium, even in
vacuum, hence “vacuum environmental modes” even though the enclosing medium
may technically not be a vacuum.

In reality the different types of modes are hard to distinguish, a certain
environment encompasses a coupling of various modes and mode types [25, 26,
27]. Quantum dots with $R_{T}\gg R_{Q}$ are well decoupled from lead
environmental modes. The decoupling from vacuum environmental modes is however
from an experimental point of view difficult to influence. It is important to
realize that a tunneling electron can couple to vacuum environmental modes and
loss or gain energy to or from these modes. With increasing frequency, the
coupling to vacuum environmental modes becomes easier as high frequency
processes experience very low impedance in vacuum, as the impedance of vacuum
equals $377\text{\,}\mathrm{\SIUnitSymbolOhm}$ only, far less than $R_{Q}$.

A study of partially wet phase molecular junctions is presented, where a
molecule is bridging the gap between two electrodes of a MCB junction. In the
ideal situation, the system defining modes are the only relevant modes for
tunneling electrons to couple to. Ideally all lead environmental modes and
vacuum environmental modes are decoupled from the electrode-molecule-electrode
system. Below it will be shown that nature provides for such a system.

##  2 Experimental Setup and Way of Work

Experiments are executed at ambient conditions. The classical notched filament
MCB junction muller1992source1a,*muller1992source1b,*muller1992source1c has
been used. A phosphor bronze bending beam is laminated with a
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick insulating Kapton layer,
glued with Stycast epoxy to the bending beam. Filament material is either made
up of 99.99 $\mathrm{\char 37\relax}$ hard temper Au ($\text{Au}^{\text{H}}$)
or $99.99\text{\,}\mathrm{\char 37\relax}$ soft temper Au
($\text{Au}^{\text{S}}$) glued with Stycast epoxy to the Kapton insulated
bending beam. A glass liquid cell is mounted surrounding the junction area
using a liquid rubber. Fig. 1 shows the bending beam assembly. Two types of
cells have been used. One with a $4.5\text{\,}\mathrm{mm}$ inner diameter
($\phi_{\text{in}}^{4.5}$), which releases the solution upon pivoting of the
setup to the cell opening, where the solution can be absorbed. The other cell
inner diameter measures $2\text{\,}\mathrm{mm}$ ($\phi_{\text{in}}^{2}$). This
cell needs to be drained with a paper tip due to the capillary forces holding
the fluid in the cell upon tilting.

Figure 1: Bending beam assemblies with mounted cells, the beam measures
$1\times 5\times 24$ $\mathrm{mm}$. Photo a and c show the
$\phi_{\text{in}}^{2}$ cell with a typical length of $26\text{\,}\mathrm{mm}$
. The inside of the cell shows the junction in the center, the liquid rubber
is applied from the outside ensuring a tight and flexible seal. Photo b shows
the $\phi_{\text{in}}^{4.5}$ cell with a typical length of
$22\text{\,}\mathrm{mm}$. Copper leads complete the connections.

In the measurements no dependency could be detected on the temper of the gold
filament. Some dependency was detected related to the inner diameter of the
cell, $\phi_{\text{in}}^{2}$ or $\phi_{\text{in}}^{4.5}$. For completeness the
temper of the gold and the inner cell diameter are listed for reported data of
every bending beam assembly.

The experimental setup makes use of a pivoting MCB setup, enabling the
partially wet phase [31], as detailed in ref [32]. In this setup the MCB
bending beam assembly can be pivoted to a desired angle, to either submerge
the junction in fluid, or to drain the junction area from fluid while the
adjusted junction remains intact during pivoting. For bending beam assemblies
using the $\phi_{\text{in}}^{4.5}$ cell, the pivoting was used to drain the
cell, after draining the cell was placed in vertical position again. For
bending beam assemblies using the $\phi_{\text{in}}^{2}$ cell the setup was
maintained in a fixed vertical cell position.

The IV curves are recorded with a HP4155 Semiconductor Parameter Analyzer. The
junctions are voltage biased, one side of the junction is kept at zero the
other side was voltage scanned, while measuring the current. The HP4155
exhibits 4 independent signal monitoring units (SMU’s) of which 2 are required
for a measurement. The reproducibility of the obtained results has been
validated on 2 independent sets of SMU’s. The HP4155/measurement setup
attached to the MCB setup has been regularly tested for linearity with a
$50\text{\,}\mathrm{M\SIUnitSymbolOhm}$ resistor connected at the device
location.

Double shielded cables are used to connect the HP4155 to the device; the last
few centimeters to the junction are unshielded. No additional filtering is
used. The piezo HV supply, normally in use for fine tuning of a MCB junction,
is not used for the reported experiments. IV traces (single direction,
indicated by small arrows) consist of 1000 data points, recorded in 45
seconds. The time between consecutive data points remains
$45\text{\,}\mathrm{ms}$ where partial traces are shown. The scans which are
recorded faster are indicated. On some occasions two consecutive scans with
opposing scan direction are shown, in these cases also each scan consists of
1000 data points and there is no delay at the point of scan reversal.

In the 90‘s Tour was pioneering the synthesis of di-thiolate molecules [33]
which exhibit 2 opposing thiol anchor groups, enabling the initial MCB BDT
experiments at Yale [34]. In the below experiments commercially available BDT
molecules (99+ $\mathrm{\char 37\relax}$ pure) are dissolved in 99.85+
$\mathrm{\char 37\relax}$ anhydrous ($<50$ $\mathrm{p}\mathrm{p}\mathrm{m}$
$\text{H}_{2}\text{O}$) tetrahydrofuran stabilized with 200-250
$\mathrm{p}\mathrm{p}\mathrm{m}$ butylated hydroxyl-toluene.

Two different types of experiments have been performed. In one type a BDT THF
solution is used with a concentration of approximately
$1\text{\,}\mathrm{m\textsc{m}}$, where the intent is to create a THF
partially wet phase electrode-BDT molecule-electrode structure. The other
experiment type made use of the pure THF fluid with the intention to create a
THF partially wet phase electrode-THF molecule-electrode structure. In each
type of experiment the solution/fluid is injected into the cell prior to
breaking the filament. The molecules and THF fluid are handled under inert
Argon gas. Standard cannulation techniques for handling air sensitive reagents
have been used. Only at injecting the BDT/THF solution or the pure THF fluid
into the cell, it is exposed to air. The following experimental way of work
has been followed in the described experiments:

  * •

The MCB setup is pivoted such that the liquid cell is in upright, vertical
position.

  * •

The liquid cell is injected with either the $1\text{\,}\mathrm{m\textsc{m}}$
BDT solution or the THF fluid by a glass syringe with stainless needle, while
the bending beam assembly filament is still unbroken.

  * •

The motor is switched on while monitoring the conductivity of the filament,
once the filament is broken the motor is immediately switched off. Based on
the spindle velocity as well as the bending beam assembly attenuation factor
it is estimated that the electrode separation at this stage is in the order of
nanometers.

  * •

In this situation the junction is completely immersed in fluid and thus in the
completely wet phase. Approximately 2-5 minutes settling time is provided for
wetting the newly formed electrodes and for molecules to adhere, prior to
draining.

  * •

The bending beam assembly is drained, either by pivoting the setup
($\phi_{\text{in}}^{4.5}$ cell) or by a paper tip ($\phi_{\text{in}}^{2}$
cell). Draining is performed in such a way that it takes about half an hour to
get to the completely dry phase.

  * •

Directly after draining the setup is pivoted back to the original position in
case the $\phi_{\text{in}}^{4.5}$ cell has been used; IV measurements can be
recorded while the junction area slowly dries. Alternatively a current
measurement can be taken (for example at $-5\text{\,}\mathrm{V}$) every few
minutes where full IV measurements are only started once the partially wet
phase has been reached.

  * •

Once the setup is in the original position, it is not to be touched any more.
The drying process progresses the junction through the following phases:

    * –

The completely wet phase, once broken and still immersed in fluid and also
just after draining

    * –

The partially wet phase, where a layer thickness of only 1 or a few THF
molecules will line the electrode-molecule-electrode system.

    * –

The completely dry phase, defined as completely dry from THF.

  * •

After reaching the completely dry phase the MCB junction has to be discarded
of and needs to be replaced with a new (unbroken filament) bending beam
assembly.

A limited number of experiments are carried out in the H2O partially wet
phase. In order to arrive at this phase a regular household humidifier is put
together with the MCB setup and a hair-hygrometer in an acrylate plastic
enclosure.

The majority of the experiments are performed in the THF partially wet phase,
limiting measurement time to a few minutes only before entering the completely
dry phase. During this time the IV traces still change as a result of the
continuous evaporation of THF molecules. Reproducing the results within the
same bending beam assembly is therefore inherently not possible. For this
reason only results are shown which have been demonstrated in multiple bending
beam assemblies. The periodicity of the presented data, the validated data on
two SMU sets, the specific reproducing line shape of the oscillations and in
some cases the demonstrated progression of the effects over a short time
lapse, all should provide confidence in the presented data despite the lack of
reproducibility within the same bending beam assembly.

##  3 Experimental Results

###  3.1 Field emission oscillations

Once the bending beam assembly filament is broken in pure THF, IV curves are
as indicated in Fig. 2a. The IV curves show a cyclic voltammetry component,
exhibiting a non-zero current at $V=0$. Also structure in the IV curve is
observed to reproduce at a voltage value which switches polarity once the scan
direction is reversed. Towards higher voltages a more or less linear part
shows in the IV curve which exhibits none or only minor hysteresis.

Figure 2: Panel 2a shows a typical IV curve after a junction is broken and
immersed in THF fluid, AuH, $\phi_{\text{in}}^{4.5}$. Panel 2b shows data from
a bending beam assembly in the drying process after being immersed in a
BDT/THF solution, revealing enhanced conductivity and also similarity in shape
to the curve shown in panel a, AuH, $\phi_{\text{in}}^{4.5}$. In panel c and d
the conductivity of the two traces in panel b are shown. Clear voltammetry
behavior is visible.

This IV curve is very similar in shape to an IV curve of a junction immersed
in a $1\text{\,}\mathrm{m\textsc{m}}$ BDT/THF solution of which the conduction
at $10\text{\,}\mathrm{V}$ is approximately $100\text{\,}\mathrm{nA}$ see the
inset of Fig. 3. These IV curves of devices submerged in fluid are independent
on electrode gap distance unlike the exponential influence on the tunnel
current for a vacuum gap. In this case we cannot draw any positional
conclusion from these curves as to how large the gap between the electrodes
actually is. Crashing the electrodes into each other leads to a metallic
constriction. The resistance of the smallest possible one atom point-contact
amounts to approximately
$h/2e^{2}\approx$12\text{\,}\mathrm{k\SIUnitSymbolOhm}$$, which would show as
a vertical line through the origin on the scale in Fig. 2a.

Figure 3: Typical drying behavior of a junction submersed in a BDT/THF
solution after the solution has been drained. The arrows indicate the
continuous increase in conductivity over time. The inset shows the effect of
the draining on the junction; a steep decrease in conduction at the point in
time where the cell is drained, AuH, $\phi_{\text{in}}^{4.5}$.

The limits of cyclic voltammetry will be investigated under the continuous
reduction of the amount of fluid, until we reach a single layer of fluid
molecules, solidly adhered to the electrodes. In Fig. 3 this is demonstrated
during the drying process for a BDT/THF solution. The effect of the draining
process can be registered during a measurement, indicated in the inset of Fig.
3 as a steep decrease of the current. In general a current reduction at
draining of about 50-65 $\mathrm{\char 37\relax}$ is observed. After draining,
the wet electrodes and junction area will slowly start to dry which lasts
approximately half an hour. The conductance of the junction increases
continuously during the drying process. The IV curves can be recorded during
this process, as the duration to take a trace is short as compared to the
duration of the drying process. The IV curves in Fig. 3 are recorded at fixed
time intervals for a positive scan direction during the drying process. Just
after draining the current is approximately $50\text{\,}\mathrm{nA}$ at
$10\text{\,}\mathrm{V}$. Increases during the drying process to values of
$3\text{\,}\mathrm{\SIUnitSymbolMicro A}$ at $10\text{\,}\mathrm{V}$ have been
measured, prior to a sudden conductivity decrease upon entering the completely
dry phase.

A law of nature seems to hold for two electrodes spaced at nanometers distance
during the drying process. Be it soft temper Au or hard temper Au used as the
MCB electrode material, be it broken in air and after that immersed in fluid
or broken directly in a THF environment, the shown electrode behavior upon the
drying process is the same. Once the excess fluid has been drained, the
continuous increase of the conductivity is attributed to a constantly
decreasing distance between the two closest points on the opposing electrodes
due to the lasting evaporation of the fluid on and between the electrodes.
Apparently there is enough elasticity in the nanometers separated electrodes
such that the exact initial separation does not matter. The continuously
decreasing amount of fluid between the electrodes pulls the electrodes ever
closer together. This process continues until the fluid is getting depleted,
and a molecule is squeezed between the electrodes, blocking any further
movement. The adhesion forces of the molecules (either BDT or THF) to the gold
electrode surface, together with the partially wet phase layer which is lining
the entire electrode-molecule-electrode system are such that a metal-molecule-
metal junction is favored over a metallic junction.

The above also holds for electrodes broken in air where a metallic microscopic
contact is re-established by reversing the motor action and subsequently
exposing the junction to the solution. Once the solution is submitted the
microscopic metallic contact is “broken” because fluid gets in between the
electrodes.

Fig. 2b shows data from a bending beam assembly in the drying process after
being immersed in a BDT/THF solution. Although the conductivity has increased
by a factor of 10 compared to the value just after draining, there is still a
large similarity with the curve in 2a immersed in pure THF. The conductance of
the IV traces in 2b is shown in 2c and d. Typical voltammetry behavior is
observed at the structure between 1-2 $\mathrm{V}$ which reproduces in the
opposing scan direction at opposite polarity. In addition an increase in noise
towards higher voltages shows.

This is consistent with Fig. 4, which shows partially wet phase data of a
junction which also has been immersed in a BDT/THF solution. In the
conductance data in panel b and c also an increase in noise exists for
positive polarity in panel b. Conductance oscillations show in both panels b
and c, for negative voltages in excess of $-4\text{\,}\mathrm{V}$. In panel c
the conductance oscillations are bounded by a smooth envelope, sometimes
called “wave package” or “frequency beating”. Oscillations in the conductance
at voltages in excess of the work-function $\Phi^{\text{Au}}$, approximately
$5\text{\,}\mathrm{V}$ for gold, are well known and are usually explained in
the context of field emission resonance
gundlach1966berechnung,binnig1982helv,kolesnychenko2000field,*kolesnychenko2002experimental.

The IV curve in the positive scan direction shows a steep decrease in
conduction which is typical for entering the completely dry phase. In the
completely dry phase the THF is fully evaporated. The remaining molecules on
and between the electrodes are strongly adhering to the electrodes and are now
exposed to water and air molecules. The current is quickly reduced from a
maximum value just prior to entering the completely dry phase to $\mathrm{pA}$
values in a matter of minutes. In this situation the electrodes are still in
contact with undefined molecules between them acting as glue. After entering
the completely dry phase the bending beam assembly can no longer be used as
the electrodes are contaminated with undefined molecules and have been exposed
to air and water molecules.

Figure 4: Panel a shows an IV curve in the partially wet phase of a junction
submitted to the BDT/THF solution. During the increasing voltage scan the
transition to the completely dry phase occurs at positive voltages. Panel b
and c present the associated conductance showing clear oscillations at
negative voltages. Panel b also shows an increase in noise towards larger
voltages, AuH, $\phi_{\text{in}}^{4.5}$.

Fig. 5 presents data in the partially wet phase from a bending beam assembly
which has been exposed to the THF fluid. The IV curve over the full
$-10\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ range is shown in the upper
inset, an enlargement of the central section is shown in the lower inset. Also
this IV curve shows pronounced oscillations in this case however the current
oscillates, showing in the IV curve as negative differential resistance parts.
This is uncommon in devices where the electrodes are made from metal and where
the density of states in the electrodes is expected to be fairly constant at
the Fermi energy. It is posed that partially wet phase molecular junctions
created in the way described above should not be defined by a fairly constant
density of states in the electrodes. Current oscillations can also be observed
in Fig. 6a and Fig. 6b which details IV traces of two different bending beam
assemblies in the partially wet phase after being exposed to a THF/BDT
solution. The insets show the corresponding large scale IV curve for both
bending beam assemblies. Fig. 6a and Fig. 6b show again “wave package” or
“frequency beating” where alternating parts of the IV curve do and do not show
current oscillations.

Figure 5: IV curve from a junction in the partially wet phase which has been
exposed to the THF fluid. Oscillations in the current show over a substantial
voltage range. The upper inset shows the entire IV curve over a
$-10\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ range. The lower inset shows the
central part of the IV curve, AuS, $\phi_{\text{in}}^{2}$. Figure 6: Current
oscillations are common in the partially wet phase as shown in panel a and b,
where both junctions have been submitted to the BDT/THF solution. The insets
in panel a and b show the entire IV curve over the
$-10\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ range. An abrupt transition to
the completely dry phase close to $8\text{\,}\mathrm{V}$ is demonstrated by
the inset in panel a. Panel a: AuS, $\phi_{\text{in}}^{2}$, panel b: AuS,
$\phi_{\text{in}}^{4.5}$.

For Fig. 6a the oscillations occur at negative bias, for Fig. 6b they occur at
positive bias. In both cases the oscillations are occurring at voltages in
excess of $\absolutevalue{4}$ $\mathrm{V}$. The inset of Fig. 6a shows an
abrupt transition to the completely dry phase near $8\text{\,}\mathrm{V}$.
Although gradual transitions to the completely dry phase also occur, it shows
that defining the point in time where the partially wet phase ends is not
difficult.

Normally the partially wet phase exists just prior to entering the completely
dry phase. The transition between an increasing and a decreasing conductance
is an indication that the completely dry phase nears. In addition, the
partially wet phase in general shows oscillations in the current at larger
($eV>\Phi^{\text{Au}}$) voltages.

###  3.2 Oscillations in a THF barrier THF partially-wet-phase junction

In this section all devices have been immersed in pure THF fluid, after
draining and drying the results are recorded in the partially wet phase. The
bridging THF molecule(s) between the electrodes are considered constituting
the tunnel junction barrier. The THF layer lining the device provides for the
partially wet phase.

The experiments presented in the following sections zoom in at a high voltage
offset with a $1\text{\,}\mathrm{mV}$ resolution for a $1\text{\,}\mathrm{V}$
scan and a $2.5\text{\,}\mathrm{mV}$ resolution for a $2.5\text{\,}\mathrm{V}$
scan. Due to the nature of the experiment and the short duration of the
partially wet phase no data for the large scale
$-10\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ total IV curve have been
measured. In general the IV curve at large
$-10\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ scale for THF barrier junctions
can be expected to be similar in shape to the one shown in Fig. 5. Data from
Fig. 7 panel a, b and c are measured for the same junction with about
$1\text{\,}\mathrm{min}$ intervals. In panels b and c an increasing and
decreasing voltage scan are part of the same scan, in panel a only the
increasing scan is shown. The curve in panel a starts with a steep decrease in
current, this “switch on” effect is followed by a linear increase. Initially
the drying process increased the conductivity going from panel a to panel b as
can be observed by the current levels. Going from panel b to panel c the
overall conduction has started to decrease. A clear pattern of oscillations is
revealed. In a timescale of several minutes the pattern changes with the
overall trace changing as a result of the drying process. The amplitude of the
oscillations is more or less constant for the larger stretches in panel b and
c. Panel c in addition shows that the period of the oscillations is not
constant. The lower curve shows periods from $21\text{\,}\mathrm{mV}$ at
$9\text{\,}\mathrm{V}$ to $12\text{\,}\mathrm{mV}$ at $10\text{\,}\mathrm{V}$.
In the upper curve the period varies from $14\text{\,}\mathrm{mV}$ at
$9.35\text{\,}\mathrm{V}$ to $12\text{\,}\mathrm{mV}$ at
$10\text{\,}\mathrm{V}$. Parts of Fig. 7 have been enlarged in Fig. 9. The
line-shapes of the oscillations in the different panels have a sharp “V
shaped” minimum in common. The top part of the line-shape is rounded. The
line-shape in Fig. 9b2 is more or less vertically oriented, approximately 2
minutes later the increasing voltage scan in Fig. 9c show line-shapes leaning
to the left while the decreasing scan shows line-shapes leaning to the right.

Figure 7: A THF barrier junction in the partially wet phase progressing over
time. In panel b oscillations are seen to be developing near
$10\text{\,}\mathrm{V}$. In panel c, about 2.5 minutes later, these
oscillations are developed at $10\text{\,}\mathrm{V}$, AuS,
$\phi_{\text{in}}^{2}$. Figure 8: Panel a and b show the “V-shaped”
oscillations in the partially wet phase. Structure of only
$2\text{\,}\mathrm{mV}$ wide is seen to reproduce in the oscillations. The
lower curve in panel b is offset $-1\text{\,}\mathrm{nA}$ for clarity, AuS,
$\phi_{\text{in}}^{2}$. Figure 9: The corresponding panels in Fig. 7 are
enlarged. Apart from panel b1 which shows the developing oscillations near
$10\text{\,}\mathrm{V}$, the panels show sharp defined “V-shaped” oscillations
of similar amplitude. The period in panel a is seen to gradually reduce for
increasing voltage.

A scan from $9\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ with a
$1\text{\,}\mathrm{mV}$ resolution in an alternative bending beam assembly is
shown in Fig. 8. The line shape obtains a period of $12\text{\,}\mathrm{mV}$
at $10\text{\,}\mathrm{V}$. An enlargement of the
$9.75\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ range from panel a is shown in
panel b, where the increasing voltage scan has been offset with
$-1\text{\,}\mathrm{nA}$ for clarity. Similar to Fig. 9 also this line has a
sharp “V shaped” minimum. A spike of a single data point stands out on one of
the flanks and reproduces over the stretch of the oscillations. The line-shape
in the increasing voltage scan is continuing, current-axis mirrored, in the
decreasing voltage scan inclusive of the $12\text{\,}\mathrm{mV}$ features
which are mirrored as well.

A scan which has been taken in 5 seconds in one direction is presented Fig.
10. The increasing and decreasing voltage scan have been recorded in one turn
without interruption. A clear "switch on" effect is also present in this IV
curve. The amplitude is of the same order as the oscillations in Fig. 7 the
period however is about $30\text{\,}\mathrm{mV}$, three times as large as
compared to Fig. 7b.

Figure 10: A deviating scan speed of $1\text{\,}\mathrm{V}$ in 5 seconds has
been used in the partially wet phase of this assembly to capture these
"V-shaped" oscillations which extend over the entire voltage range, AuS,
$\phi_{\text{in}}^{2}$.

###  3.3 Oscillations in a BDT barrier THF partially-wet-phase junction

The junctions in this section have all been immersed in a
$1\text{\,}\mathrm{m\textsc{m}}$ BDT/THF solution. All results are obtained in
the partially wet phase, after the draining and drying process. The goal is to
capture an island in the form of a phenyl ring from the BDT molecule, coupled
via two barriers to the opposing electrodes.

In Fig. 11 an increasing and decreasing voltage trace is shown for two bending
beam assemblies, one in panel 11a the other in panel 11b. The lower IV curves
with a decreasing voltage are offset $-0.2\text{\,}\mathrm{\SIUnitSymbolMicro
A}$ for clarity. The increasing and decreasing voltage scan are both part of
the same scan. Clear discontinuous current steps are present in the observed
structure. For the increasing scans a negative differential resistance section
follows after each step, this section has some curvature before the next step
occurs. For the decreasing scans the behavior is simply a continuation of the
behavior in the increasing scan but mirrored in the current-axes. The
curvature of the segments gets somewhat steeper towards higher voltages, which
coincides with smaller voltage increments in subsequent steps. The insets show
the IV curve over the full $-10\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ range
the lower curve is also offset with $-0.2\text{\,}\mathrm{\SIUnitSymbolMicro
A}$. The structure at positive bias does not reproduce at negative bias in
both shown traces.

Also for the high voltage offset BDT barrier junctions the scans are not
accompanied by the $-10\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ scan. These
scans are similar to the ones shown in the insets of Fig. 6 and Fig. 11. Fig.
12 presents data from a trace between $8\text{\,}\mathrm{V}$ and
$9\text{\,}\mathrm{V}$. Repetitive structure is present in both the increasing
and the decreasing voltage scan. Panels b and c provide an enlargement, in
panel c the lower IV curve has been offset $-3\text{\,}\mathrm{nA}$ for
clarity. Over 40 oscillations in the upper curve and over 30 oscillations in
the lower curve are counted. Similarly as in Fig. 11 the shape of the
oscillation for the decreasing voltage scan is current-axes mirrored of the
shape in the increasing scan. The oscillation shape has also some resemblance
to the line shapes shown in Fig. 11, there is a distinct discontinuous step
present in every period. The periodicity of the oscillation changes from
$12\text{\,}\mathrm{mV}$ at $8.4\text{\,}\mathrm{V}$ to
$14\text{\,}\mathrm{mV}$ at $10\text{\,}\mathrm{V}$.

Figure 11: IV curves in the partially wet phase of two assemblies both exposed
to the BDT/THF solution. Similarities in panel a and b are discontinuous
current jumps trailed by a negative differential resistance in the increasing
voltage scan and trailed by a positive differential resistance in the
decreasing voltage scan. The insets show the increasing and decreasing scans
over the entire $-10\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ range. All
decreasing voltage scans are offset by $-200\text{\,}\mathrm{nA}$ for clarity,
panel a: AuH, $\phi_{\text{in}}^{4.5}$, panel b: AuS,
$\phi_{\text{in}}^{4.5}$.

In Fig. 13 data from 2 additional bending beam assemblies a/b and c/d show a
similar result as indicated in Fig 12. A regular pattern of peaks and
discontinuous current jumps is clearly visible. Also in this case asymmetry in
the peaks indicates a negative differential conductance trailing the current
jump for the increasing scan and a positive differential conductance trailing
the jump for the decreasing scan.

Figure 12: A high voltage trace of a BDT/THF junction in the partially wet
phase, the panels a-c show an increasing enlargement. Similarities to the
structure in Fig. 11 are current discontinuities, trailing negative and
positive differential resistance after a discontinuity depending on scan
direction and current-axis mirrored structure upon scan reversal. In panel c
the current of the lower curve is offset by $-3\text{\,}\mathrm{nA}$ for
clarity, AuS, $\phi_{\text{in}}^{2}$.

The complete $910\text{\,}\mathrm{V}$ scale of the data in Fig. 13a and b are
presented in Fig. 14. Over 100 peaks are counted in this scan, the periodicity
changes from $24\text{\,}\mathrm{mV}$ near the start of the scan to
$6\text{\,}\mathrm{mV}$ at $10\text{\,}\mathrm{V}$.

A markedly different line-shape is observed at the start of the oscillations
in Fig. 15a with a scan extending from
$7.5\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$. The initial line shape exhibits
a very sharp “V” shaped minimum and a somewhat rounded top as observed in the
previous section. For increasing voltages exceeding $8.75\text{\,}\mathrm{V}$
a “shoulder” is developing which is shifting with respect to the sharp bottom
of the line as shown in Fig. 15b. Fig. 15c shows the development of a smooth
line-shape into a more chaotic one, although the repetition of the sharp
minima reflects the presence of the periodicity all the way to
$10\text{\,}\mathrm{V}$.

Figure 13: Panels a, b and c, d tie pair-wise to two bending beam assemblies.
Increasing and decreasing scans replicate the main observations from Fig. 12.
The lines indicating the slope are a guide to the eye. Panel a, b: AuS,
$\phi_{\text{in}}^{2}$, panel c, d: AuH, $\phi_{\text{in}}^{4.5}$.  Figure 14:
A high voltage scan over a $1\text{\,}\mathrm{V}$ range, current
discontinuities show as narrow spikes. The periodicity exceeding
$9.65\text{\,}\mathrm{V}$ is regular as was shown in Fig. 13a and b. The lower
curve has been offset by $-5\text{\,}\mathrm{nA}$ for clarity.  Figure 15:
Although a BDT/THF solution has been used, the line shape resembles a THF
barrier junction. Panels b and c show the increasing IV scan direction. The
oscillations start of as “V-shaped” but quickly get more disorderly towards
higher voltages, AuH, $\phi_{\text{in}}^{4.5}$. Figure 16: The partially wet
phase abruptly ends in the decreasing voltage scan at
$9.3\text{\,}\mathrm{V}$. The irregular oscillations show typical BDT barrier
behavior. Once the partially wet phase seizes to exists a smooth structure-
less IV curve is the result, AuH, $\phi_{\text{in}}^{4.5}$.

The transition to the completely dry phase is indicated in Fig. 16. A steep
decrease in conduction, with a lot of structure, is abruptly transformed in a
line without any structure. This is the point where the partially wet phase is
expected to transit abruptly to the completely dry phase as for example also
noted in the inset of Fig. 6a.

###  3.4 Large voltage range molecular spectra

In the previous sections it has been shown that some of the high voltage fine
structure of the overall line-shape for THF barrier and BDT barrier junctions
in the THF partially wet phase reproduce. Is it possible to also reproduce
partially wet phase molecular spectra over a large voltage range?

Figure 17: Large voltage scale molecular IV spectra of bending beam assembly a
and b. Curves have been offset by $-0.4\text{\,}\mathrm{\SIUnitSymbolMicro A}$
for clarity. The inset shows how IV curve a has progressed after 15 minutes,
just prior to entering the completely dry phase, a,b: AuS,
$\phi_{\text{in}}^{4.5}$.

Devices in Fig. 17 have been immersed in the BDT/THF solution in the
$\phi_{\text{in}}^{4.5}$ cell. Two consecutive voltage scans for bending beam
assemblies a and b in the partially wet phase are presented. The IV curves are
offset by $0.4\text{\,}\mathrm{\SIUnitSymbolMicro A}$ with respect to each
other. For the increasing scans two discontinuities are visible at negative
bias for which the amplitude and exact positions differs. The observed
structure at positive bias shows some resemblance to what was shown in Fig.
11. In this case there are plateaus, there is structure on the plateaus, there
are discontinuous steps as well but the repetition is more irregular and less
in number as compared to the structure in Fig. 11. For the decreasing voltage
scans the structure at positive bias resembles that in Fig. 11 in that the
discontinuous steps are trailed by a positive differential resistance, however
the repetition is more irregular and there are far less steps. For the scans
in negative direction there is an inflexion point at approximately
$-2\text{\,}\mathrm{V}$ and $-5\text{\,}\mathrm{V}$. The inset shows how the
IV curve from assembly a has progressed over time after approximately 15
minutes from recording the IV curves in the main figure. The structure has
disappeared and the current extends to $3\text{\,}\mathrm{\SIUnitSymbolMicro
A}$ at $10\text{\,}\mathrm{V}$.

Figure 18: IV curves in the H2O partially wet phase. Compliance was set at
$\pm$3\text{\,}\mathrm{\SIUnitSymbolMicro A}$$, the curves are offset with
$3\text{\,}\mathrm{\SIUnitSymbolMicro A}$ for clarity. The IV curves are very
similar and are stable over hours. The sub-gap current is dependent on the
voltage scan speed as indicated by curve b2 which was obtained with a double
scan-speed as compared to b1. The insets provide an enlargement of the central
section for curves a and b1, a,b: AuS, $\phi_{\text{in}}^{2}$.

IV traces of two bending beam assemblies, a and b, with an H2O partial wet
phase are shown in Fig. 18. These junctions are created by starting off with a
bending beam assembly immersed in a BDT/THF solution. The procedure in order
to arrive at a molecular junction in the THF partially wet phase as described
above has been followed to the point where during the formation of the
junction the ambient H2O humidity was increased to $95\text{\,}\mathrm{\char
37\relax}$ by using a humidifier within a contained environment. The
$95\text{\,}\mathrm{\char 37\relax}$ humidity level was reached in a matter of
seconds, the humidifier was switched off and the system was allowed to
gradually transit to an ambient humidity level of $60\text{\,}\mathrm{\char
37\relax}$ in a few hours’ time. A gradual transition from the default BDT IV
curve (see for example insets Fig. 6, Fig. 11) to the IV curves shown in Fig.
18 took place. The IV curves have been recorded in a continuous scan mode
between $-5\text{\,}\mathrm{V}$ and $5\text{\,}\mathrm{V}$, the current level
compliance was set at $-3\text{\,}\mathrm{\SIUnitSymbolMicro A}$ and
$3\text{\,}\mathrm{\SIUnitSymbolMicro A}$. Once reaching their final shape the
IV curves a and b1 were stable for hours. Doubling the voltage scan speed, see
curve b2, left the IV curve as is however the sub-gap structure near
$-1\text{\,}\mathrm{V}$ and $1\text{\,}\mathrm{V}$, approximately doubled in
height. The insets show an enlargement of the central section of curves a and
b1. Fig. 18 shows the only data concerning H2O partially wet phase junctions.

##  4 Theoretical Background and Discussion

###  4.1 One-dimensional conduction

From the data in section 3 it is clear that the conductance of the devices
under study are of the order of, or well below ${10}^{-3}$ $2e^{2}/h$. This
indicates that the physics is described by transmission and reflection, deep
in the Landauer regime landauer1957one,*landauer1981two,*landauer1989three.
Fig. 19 provides a schematic representation of the junctions which are
studied.

Figure 19: A schematic representation of the junctions under study. The
molecule is connected via two barriers to a quantum wire on either side.

The central molecule is connected via tunnel barriers to the two quantum
wires, extending in the x-direction with a length $L_{x}$, a width $L_{z}$ and
a height $L_{y}$. For a free electron gas the energy of the conduction modes
in the quantum wires is described by:
$E_{n,p}=E_{c}+n^{2}\varepsilon_{y}+p^{2}\varepsilon_{z}+(\hbar
k_{x})^{2}/2m$, likewise the relation of the valence modes is given by:
$E_{n,p}=E_{v}-n^{2}\varepsilon_{y}-p^{2}\varepsilon_{z}-(\hbar
k_{x})^{2}/2m$, where $E_{c}-E_{v}$ is the gap between the lowest conduction
band and the highest valence band, $m$ is the electron mass and
$\varepsilon_{y}$, $\varepsilon_{z}$ are provided by:
$(\pi\hbar)^{2}/2mL^{2}_{y}$, $(\pi\hbar)^{2}/2mL^{2}_{z}$ respectively, $n$ ,
$p$ are integers $>0$. Neglecting spin-degeneracy and taking $L_{z}=L_{y}$
equals $8.6\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ or 3 Au atom diameters
leads to $0.5\text{\,}\mathrm{eV}$ for $\varepsilon_{y}$, $\varepsilon_{z}$.
The density of states of the quantum wires is provided in Fig. 20a where the
estimate for $E_{c}-E_{v}$ is $4\text{\,}\mathrm{eV}$, $n$, $p$ runs from 1 to
4 and all modes are counted. It is noted that the inputted numbers are rough
estimates to be able to provide a qualitative picture of the conduction
through a single molecule. In Fig. 20b only one conduction mode is populated,
the lowest $n=1$, $p=1$ mode is responsible for the conduction in this case
the higher energy conduction modes are empty. Finally in Fig. 20c the filled
valence modes are allowed to interact and couple, leading to a smoother
density of states as indicated qualitatively. Derived from first principles
Grigoriev et al. [42] arrived at a gap pinned to the Fermi level for a BDT
molecule bridged between two Au electrodes, which supports the described
experimental observations.

Figure 20: a. One dimensional density of states for a free electron gas,
estimated input-parameters: $L_{z}=L_{y}$ equals
$8.6\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ leads to
$\varepsilon_{y}=\varepsilon_{z}=$0.5\text{\,}\mathrm{eV}$$,
$E_{c}-E_{v}=$4\text{\,}\mathrm{eV}$$. Panel b shows the filling of the modes
up to and including the lowest conduction sub-band. In panel c the valence
sub-bands are allowed to interact and couple together.

For the voltage biased systems under study it is notable that almost the
entire voltage will be biased across the molecule and associated barriers. The
voltage over the quantum wire typically amounts to less than one or a few
$\mathrm{mV}$ even if the molecule is biased at $10\text{\,}\mathrm{V}$. This
is due to the large difference in the total conduction with respect to
$2e^{2}/h$ of conduction per mode in the quantum wire. The density of states
of the quantum wire in Fig. 20c is voltage independent and will simply shift
“up and down” with an increasing or decreasing voltage bias over the contacts,
even if this amounts to $10\text{\,}\mathrm{V}$. Is $10\text{\,}\mathrm{V}$
not an extremely high voltage to bias a single molecule with? Molecules
exhibit efficient tunnel barriers, capable of sustaining high $\mathrm{eV}$
molecular energy levels. In addition, a single molecule can consist of
multiple barriers [42], for example a bridging BDT molecule exhibits 4 tunnel
barriers reducing the bias per barrier:

  * •

Barrier 1 from the 1st Au electrode to the 1st S atom

  * •

Barrier 2 from the 1st S atom to the phenyl ring

  * •

Barrier 3 from the phenyl ring to the 2nd S atom

  * •

Barrier 4 from the 2nd S atom the 2nd Au electrode

Finally, the experiment allows and caters for this. Dissipation will in
accordance with the Landauer theory take place far away from the molecule and
quantum wires, in the electrode banks where energetic electrons will be able
to get into thermal equilibrium with the lattice.

What is the experimental evidence in support of one dimensional conduction? To
answer this question the process of how to arrive at partially wet phase
junctions is detailed. The process is split into two: a formation process
followed by a nucleation process. During the formation process excess fluid is
slowly evaporating, BDT molecules are attaching to surfaces and the two
electrodes are slowly getting ever closer together, continuously increasing
the conductivity. The conduction still increases over time for a THF partially
wet phase BDT junction, after the formation process has stopped. This increase
is attributed to physi-sorption of one or more molecules to the bridging
molecule. The increasing size of the central island leads to a denser energy
level spacing thus increasing conductivity while still maintaining a single
point of tunneling related to both electrodes. This process continues until
the nucleated island becomes too big and starts to touch one of the two
electrodes, creating multiple conduction points and thus losing the one
dimensionality.

The data in Fig. 17 have been carefully selected to be similar at a specific
point in the nucleation process from a data set of 20 different bending beam
assemblies. The data exhibit structure in the IV curve which disappears
altogether into a higher conductivity curve by waiting 15 minutes see the
inset in Fig. 17. Even though the a and b curves are not identical, they do
show similar features for the increasing as well as the decreasing curves. In
general the $\phi_{\text{in}}^{4.5}$ cell shows more nucleation as compared to
the $\phi_{\text{in}}^{2}$ cell as defined by the conductivity of the final IV
curve just prior to entering the completely dry phase. The
$\phi_{\text{in}}^{4.5}$ cell can conduct
$3\text{\,}\mathrm{\SIUnitSymbolMicro A}$ at $10\text{\,}\mathrm{V}$ for a BDT
barrier junction, for the $\phi_{\text{in}}^{2}$ cell a
$1\text{\,}\mathrm{\SIUnitSymbolMicro A}$ level is the norm. This is
attributed to an easier access of the junction to air molecules in the
$\phi_{\text{in}}^{4.5}$ cell.

The data in Fig. 18 stand out as halfway the BDT THF formation process the H2O
humidity was increased to $95\text{\,}\mathrm{\char 37\relax}$. It is
astonishing that this system reorganizes itself to a stable molecular junction
for hours with an H2O partial wet phase layer. The collapse into a completely
dry phase is not observed, nor is the nucleation phase observed. Apparently
the H2O partially wet phase layer protects the bridging BDT molecule from
other molecules nucleating on it. The fact that the H2O partially wet phase
remains intact indefinitely at ambient conditions is to be expected [31].
These data have not been selected. The same reproducible enduring end-state IV
curves indicated in Fig. 18 result from this experiment in 4 out of 8 bending
beam assemblies.

The sub-gap current peak in the curves in Fig. 18 shows voltammetry behavior:
the current peak at approximately 1V in the positive direction reproduces in
the opposing scan direction at reverse polarity. What is the source of this
current? As the electrodes are not in the wet phase there can be no chemical
electrode surface reaction responsible. The dipole moment of the H2O molecule
is expected to play a determining role. At a certain voltage bias the dipole-
moment of all the partially wet phase H2O molecules on the biased electrode
will direct themselves with their electropositive or electronegative side
towards the electrode, depending on the polarity of the bias. These aligned
dipoles will hold some opposite charge, residing on the electrodes, captive.
Once the dipole is flipped by a changing polarity on the electrode all the
charge that was held captive becomes available and creates a displacement
current proportional to $\derivative*{V}{t}$. From Fig. 18 it is obvious that
the flipping of the dipoles occurs in a range around $-1\text{\,}\mathrm{V}$
and $1\text{\,}\mathrm{V}$, releasing about ${10}^{-6}\text{\,}\mathrm{C}$ as
obtained from integrating the sub-gap current peak over the time it was
measured in. Doubling the voltage scan rate in curve 2b in Fig. 18 also
approximately doubles the measured sub-gap current which is consistent with an
independent constant total captive charge, released upon flipping of the
dipoles. Partially wet phase voltammetry can be used as a means of detection
of H2O in for example the THF partially wet phase layer, by studying the
presence of structure at $-1\text{\,}\mathrm{V}$ and at
$1\text{\,}\mathrm{V}$.

The two main experimental findings in support of 1D conduction are:

  1. 1.

Obtaining the same IV curves for multiple BDT barrier H2O partially wet phase
junctions is a strong indication that the same singular molecular junction is
created. This is indicative that also the THF barrier and BDT barrier THF
partially wet phase junctions are of singular nature.

  2. 2.

The regular occurrence of negative differential resistance in the IV curves in
section 3.2 and 3.3 also is a strong indication of 1D conduction.

###  4.2 Grundlach oscillations

Oscillations in the conductance as a function of energy in a planar metal
tunnel junction biased with a voltage larger than the work functions of the
electrodes were predicted by Grundlach in 1966 [35]. Grundlach oscillations,
or also field emission resonances, can be viewed as electron standing waves in
the resonator bounded by the barrier created by the field on one side and the
surface potential step on the other side see Fig. 21. Gundlach oscillations
have been observed in planar junctions [43], in high vacuum STM junctions [36]
as well as in MCB junctions at cryogenic temperatures [37]. Solving the 1
dimensional Schrödinger equation with a trapezoidal barrier and a surface
potential step at electrode 2 leads to standing waves at specific energy
eigenvalues $E_{n}$ in the well:

$E_{n}=\left(\frac{3\pi\hbar
e}{2\sqrt{2m}}\right)^{2/3}\varepsilon^{2/3}\left(n-\frac{1}{4}\right)^{2/3}\;n=1,2,...$
(1)

Figure 21: Explanation of the Grundlach resonances. Standing waves are able to
build in the trapezoidal resonator. The one dimensional density of states peak
at the Fermi level is ideally suited to probe these states, explaining the
observed current oscillations in IV curves.

An effective electric field inside the barrier is represented by
$\varepsilon$. At the surface potential step in electrode 2 there is no
barrier, the energy spectrum is continuous and electrons can enter electrode 2
at any energy above $\Phi^{\text{Au}}$. Fig. 21 shows the junction conductance
to peak whenever the bias voltage reaches one of the levels $E_{n}$.

$eV=\Phi^{\text{Au}}+\left(\frac{3\pi\hbar
e}{2\sqrt{2m}}\right)^{2/3}\varepsilon^{2/3}\left(n-\frac{1}{4}\right)^{2/3}$
(2)

In Fig. 21 the one dimensional density of states from Fig. 20c is included for
electrode 1 and 2. For the vast majority of the data obtained from partially
wet phase junctions, created in the way described above, current oscillations
are detected. This is consistent with a one dimensional density of states.
With an increasing bias the oscillations in the current in Fig. 5 for example,
are explained in a natural way. Whenever the peak at the Fermi level coincides
with an energy state the current is maximized, if the Fermi level is in
between states the current is reduced. At some bias greater than $E_{c}-E_{v}$
the valence band states starts to gradually contribute to the current as well.

In Fig. 22 the peak number n and peak energy values are displayed for data
from Fig. 4c and Fig. 5. The work-function from gold $\Phi^{\text{Au}}$ as
well as the effective electric field $\varepsilon$ can be directly determined
from the interception with the energy axis and the slope of the graph
respectively. Values are indicated in the graph, the values for $\varepsilon$
are low, work-function values are in line with what is reported in [37].

Figure 22: Energy of the observed peak in the IV curve versus peak index
revealing the work-function $\Phi^{\text{Au}}$ as well as the effective
electric field $\varepsilon$

The “beating” or “wave package” visible in the oscillations in Fig. 4c have
been observed in various experiments [44, 37]. Fig. 6 seems to indicate a
number of these wave packages however it is not only the amplitude being
modulated also the period of the oscillations shows modulation. We expect on
the basis of Fig. 21 the periods closest to zero to be the largest, with an
increasing bias away from zero the periods get smaller. In contrast we observe
in Fig. 6a a decreasing period getting closer to zero and for Fig. 6b an
increasing period for voltages increasing towards $8\text{\,}\mathrm{V}$. Both
IV curves Fig. 6a and Fig. 6b show regions where there seems to be destructive
interference for a substantial voltage range. The full explanation of this
behavior is unclear at present.

###  4.3 What is the effect of the partially wet phase?

This is explained in the framework of the Fermi Golden Rule for tunnel
junctions and the more generic form which takes into account energy exchange
between tunneling electrons and the environment. The Golden Rule describes a
transition rate between an initial state $i$ and a final state $f$ which
correlates with the strengths of the coupling between the states $i$ and $f$.
The Golden Rule is used successful in a large variety of physical transitions:

$\vec{\Gamma}_{i\to
f}=\frac{2\pi}{\hbar}\left|\matrixelement{i}{Ht}{f}\right|^{2}\delta\left(E_{f}-E_{i}-\Delta
E\right)$ (3)

The delta function represents the condition for the energy of the initial and
final states: $E_{f}-E_{i}=\Delta E$. Applied to the tunnel barrier the Golden
Rule states that the transition rate $\vec{\Gamma}_{i\to f}$ represents the
transition from an electron in state $i$ on one side of the barrier to state
$f$ on the other side of the barrier. For elastic tunneling, where the energy
of the electron prior to and after tunneling is identical, equation 3 is
described in the following well-known form, at a voltage $V$ for the forward
rate, $\vec{\Gamma}(V)$, only:

$\vec{\Gamma}(V)=\frac{1}{e^{2}R_{T}}\int^{\infty}_{-\infty}\differential
Ef(E)\left[1-f(E+eV)\right]$ (4)

Usually it is assumed that the electrodes are large on an atomic dimension, to
ensure the electrodes contain a large number of free electrons and possess
quasi continuous energy spectra. The tunnel resistance $R_{T}$ is defined by:
$R_{T}=\hbar/(2\pi e^{2}\left|\matrixelement{i}{Ht}{f}\right|^{2}D^{2})$, $D$
represents the density of states in the left and right electrode and is
expected to be approximately the same and constant,
$f(E)=(1+\exp(E\beta))^{-1}$ is the Fermi Dirac distribution at inverse
temperature $\beta=1/k_{B}T$. In a full quantum mechanical description the
tunneling electrons are allowed to interact and exchange energy with the
environment. For example the system defining modes related to the charging
energy may be excited by exchange of energy quanta $e^{2}/2C$, just as other
environmental modes may be excited by energy quanta absorption from the
tunneling electron. Likewise the energy of the tunneling electron can be
increased by transfer of energy quanta from the environment to it. Under the 2
conditions that $RT\gg h/2e^{2}$ (weakly coupled electrodes) and charge
equilibrium is established: i.e. the time between tunnel events is larger than
the charge relaxation time, equation 4 takes on the following form, for the
forward rate, $\vec{\Gamma}(V)$, [25, 26, 27]

$\begin{split}\vec{\Gamma}(V)=&\frac{1}{e^{2}R_{T}}\int^{\infty}_{-\infty}\differential
E\int^{\infty}_{-\infty}\differential E^{\prime}\big{\\{}f(E)\\\
&\quad\left[1-f(E^{\prime}+eV)\right]P(E-E^{\prime})\big{\\}}\end{split}$ (5)

The corresponding physics can be described by a tunneling electron in an
initial state with energy E having a multitude of probabilities of tunneling
to a multitude of final states, thereby transferring an energy
$(E-E^{\prime})$. If $(E-E^{\prime})$ is positive the electron transfers
energy to the environment, if it is negative the electron absorbs energy from
the environment. Quantum mechanics teaches us that we can only know the
probability $P(E-E^{\prime})$ of the tunneling electron ending up in a final
state $E^{\prime}$, thus $\int^{\infty}_{-\infty}P(E)\differential E=1$. For
every initial state tunneling to a final state integration over
$P(E-E^{\prime})$ needs to be performed. $P(E)$ is provided for a number of
different environments in reference [26].

At this stage a “gedankenexperiment” related to the influence of the partially
wet phase is postulated. It is assumed that charge is able to accumulate at
the THF partially-wet-phase/air interface. The charge is able to freely move
at this interface, without being able to flow to the underlying metal
conductor. Under these conditions the function $P(E)$ in equation 5 is
dramatically impacted since the conducting layer at the partially-wet-
phase/air interface effectively prohibits any interaction of tunneling
electrons with vacuum environmental modes. The device, enclosed by the
partially wet phase layer, provides for the only surviving modes that
tunneling electrons can couple to. In other words, the device is caged. A
miniature Faraday cage prevents tunneling electrons inside the cage from
interacting with low impedance vacuum environmental modes outside of the cage.

###  4.4 Continuous Q charging of a THF barrier THF partially-wet-phase
junction

A THF barrier/partially-wet-phase junction is schematically shown in Fig. 23c.
The shield lining the junction indicates the junction is in the partially wet
phase and thus shielded from vacuum environmental modes. The well-known
unshielded counterpart of this junction is shown in Fig. 23a.

Figure 23: Schematic representation of a: a single junction, b: a double
junction, c: a single junction in the partially wet phase, d: a double
junction in the partially wet phase.

A sudden switch to a large voltage bias often leads to a switch on effect as
demonstrated in Fig. 7a. The current adapts over seconds or 10’s of seconds to
the sudden high voltage bias. Other presented graphs may not show this as
these graphs may be recorded during continuous scanning after the first
“switch on” scan. The switch on effect is attributed to a redistribution of
charge at the THF partially-wet-phase/air interface. The partially wet phase
layer is finite, it will stop at the cell wall and charge on it will try to
counter the electrode charge, acting as a capacitor $C_{d}$. The reported
systems may show a displacement current $I_{d}=C_{d}\derivative*{V}{t}$, or a
continuous increasing current as in Fig. 8 or decreasing current as in Fig. 7b
and c as a consequence of the stage the drying-process is in. The resulting
drift in the measurements is a given, it does however not prevent the study of
the processes taking place inside the junction, which in general correlate to
fast timescales in relation to the scale at which the drift takes place.

The charging energy $E_{CH}$ to charge the junction capacitance $C$ with one
additional electron on top of the $n-1$ electrons already on the capacitor
relates to:

$E_{CH}(n,Q)=\frac{(Q-ne)^{2}}{2C},\quad n-\tfrac{1}{2}<Q/e<n+\tfrac{1}{2}$
(6)

$Q$ is representing the total charge on the capacitor. This equation has been
used for the single electron-box decoupled capacitor [45] it will in general
define a decoupled capacitor.

Figure 24: Panel a shows the free energy or the charging energy from equation
6 as a function of $Q$. Panel b and c provide data segments from Fig. 10 and
7b respectively. Arrows indicate the scan direction. The continuous line
reflects equation 9 with $C=$5.5\text{\times}{10}^{-18}\text{\,}\mathrm{F}$$,
$\alpha=$2.3\text{\times}{10}^{-3}$$ ($I_{ex}=$13\text{\,}\mathrm{nA}$$) for
panel b and $C=$1.5\text{\times}{10}^{-17}\text{\,}\mathrm{F}$$,
$\alpha=$3.2\text{\times}{10}^{-3}$$ ($I_{ex}=$6.5\text{\,}\mathrm{nA}$$) for
panel c.

The energy $E_{CH}$ is drawn in Fig. 24a as a function of $Q$ for various
occupation electron numbers $n$. With increasing $Q$, the electron number
related to the lowest energy state will increase. This is a continuous
process, at the degenerate points there is a change-over to adding an
additional electron, only increasing $Q$ at a fractional electron charge. The
curve in Fig. 24a equals the free energy $\Delta F$ of the system, the energy
able to do work. Adding electrons one by one in a continuous manner reveals
the nature of the $e/C$ periodicity.

Only taking into account the contribution from the density of states peak at
higher biases ($eV>\Phi^{\text{Au}}$), the current through the molecular
junction in a one dimensional conduction channel as indicated in Fig. 19
follows from equation 3 which modifies to:

$\vec{\Gamma}_{i\to
f}=\frac{2\pi}{\hbar}\left|\matrixelement{i}{Ht}{f}\right|^{2}\delta\left(E_{f}-E_{i}-\Delta
F\right)$ (7)

Where $\Delta F$ is the free energy catering for the change from state $i$ to
state $f$. The matrix element $\matrixelement{i}{Ht}{f}$ ties to both states
$i$ and $f$, it represents the strength of the coupling between the two states
hence $\left|\matrixelement{i}{Ht}{f}\right|^{2}$ is proportional to $\Delta
F$:

$\left|\matrixelement{i}{Ht}{f}\right|^{2}=\alpha\Delta F,\quad 0<\alpha<1$
(8)

It is possible that a number of states constitute an ensemble responsible for
the observed quantum effect. For simplicity, transitions between only two
states are considered below. Disregarding a linear term in $V$, the current
contribution $I_{\text{1D}}=-e\vec{\Gamma}_{i\to f}$ yields:

$\begin{split}&I_{\text{1D}}=-\frac{2\pi\alpha
e}{\hbar}\frac{(Q-ne)^{2}}{2C}\\\
&n-\tfrac{1}{2}<Q/e<n+\tfrac{1}{2}\end{split}$ (9)

The delta function in equation 7 equals unity for $E_{f}-E_{i}=\Delta F$.
Equation 9 explains the origin and periodicity of the oscillations shown in
Fig. 7-10 it also explains the “V shaped” minima in these oscillations. The
sharp minima coincide with the degenerate points at $Q=(n\pm 1/2)e$ of the
free energy in Fig. 24a. A section of Fig. 7b and Fig. 10 is shown in panel
24c and 24b respectively. The current of these sections have been aligned to
the voltage-axis with the top of the oscillations by subtracting a linear
function. The capacitance is deduced from the periodicity $\Delta=e/C$, from
the current extrema $I_{ex}$, at $(n\pm 1/2)e$, $\alpha$ can be derived. For
panel b $I_{ex}=$13\text{\,}\mathrm{nA}$$,
$C=$5.5\text{\times}{10}^{-18}\text{\,}\mathrm{F}$$,
$\alpha=$2.3\text{\times}{10}^{-3}$$. For panel c
$I_{ex}=$6.5\text{\,}\mathrm{nA}$$ and
$C=$1.5\text{\times}{10}^{-17}\text{\,}\mathrm{F}$$,
$\alpha=$3.2\text{\times}{10}^{-3}$$.

The following two processes in this junction are dependent on voltage. The
value of Q on the junction electrodes, and thus the free energy $\Delta F(Q)$,
can be continuously varied via $Q=CV$. Where the electrodes are in closest
proximity tunneling will take place, the peak density of states pinned to the
Fermi level will be located at an energy $E=eV$. In scanning the voltage the
density of states peak probes the free energy, $\Delta F(Q)$, of the system
via coupling to subsequent $Q$ states in a continuous manner.

The following view emerges for the origin of this quantum effect: the energy
$E$ of the tunneling electron is provided to de-populate a filled $Q$ state,
to occupy that state while the remaining energy $E^{\prime}$ is provided to
the depopulated electron. At a specific $Q$ state, the energy values for $E$,
$E^{\prime}$, and $\Delta F(Q)$ are each separately identical for every
subsequent tunnel event, enabling a coherent time correlated flow of tunneling
electrons all sequentially involved in the same processes with the same energy
transfers. Due to the existing THF partially wet phase layer, there is no
possibility for low impedance vacuum environmental modes to couple into this
process, which would break up the coherence and destroy this quantum effect.

In Fig. 8 a minute feature with a width of about 1-2 $\mathrm{mV}$ is observed
to reproduce in the oscillations. The features appear approximately half way
in $Q$-space between the top of the oscillation and the degenerate points. In
analogy to the physics of SET multi junction devices this feature is
attributed to an excited state in close vicinity of the junction, contributing
to the conduction by opening an additional conduction channel. In this case
the exited state is coupled via a $Q$ state, being $e/C$ periodic. The
coupling takes place via the free energy $\Delta F$ in equation 6. The
excitation energy can be obtained directly by taking $\Delta F$ at the $Q$
state where the excitation is positioned, $0.5\text{\,}\mathrm{meV}$ in this
case. The capacitance related to the oscillations is
$1.3\text{\times}{10}^{-17}\text{\,}\mathrm{F}$, see table 1.

Fig. 9c presents a somewhat tilted oscillation as compared to the upright
oscillations in Fig. 9b. For the increasing scan oscillations tilt to the
left, for the decreasing scan a tilt to the right is observed. This change has
occurred in the 2 minutes timespan between the two scans. The mechanism behind
it is unclear at present. Coupling of the point contact inductance to the
charging modes may play a role. From panel 7b a capacitance of
$1.5\text{\times}{10}^{-17}\text{\,}\mathrm{F}$ was derived. $E_{C}$ from this
contact equals $\hbar\omega$ with $\omega=(LC)^{-1/2}$ for
$L=$3\text{\times}{10}^{-10}\text{\,}\mathrm{H}$$. The
${10}^{-10}\text{\,}\mathrm{H}$ order of magnitude for an inductance related
to point-contacts is well known from superconducting rings interrupted by a
point-contact [46].

###  4.5 Continuous Q charging of a BDT barrier THF partially-wet-phase
junction

An ultimate quantum dot is created, where the BDT conducting phenyl ring is
weakly connected to the two electrodes. This double junction is schematically
presented in Fig. 23d. The shield lining the structure indicates that it is in
the partially wet phase and thus shielded from vacuum environmental modes. The
well-known unshielded counterpart of this SET device is shown in Fig. 23b.

The characteristic features in the IV curves in Fig. 11-14 are the
discontinuities or current jumps. The large periods trailing the
discontinuities in Fig. 11 are rare, observed in only $5\text{\,}\mathrm{\char
37\relax}$ of the bending beam assemblies. The smaller periods trailing
discontinuities (Fig. 12-14) are observed in more than half of the bending
beam assemblies. Even though the occurrence differs the similarity in shape
and slope is clear. Sometimes the periods are too small to clearly attribute a
slope to, they appear as oscillations. For those discontinuities where a slope
can be attributed it attains without exception a negative differential
resistance for increasing voltages and a positive differential resistance for
decreasing voltages.

The difference with the THF barrier junction is the conducting island which
can only contain an integer number of electrons. The interesting physics of
such a dot is that with an increasing Fermi level of the source electrode, the
highest filled level of the dot will in general not be aligned to the Fermi
level. The electrodes can obtain any $Q$ value, the dot can only host $n$
electrons, $n$ being an integer. The capacitance as seen from the island
consists of a capacitance to the source as well as a capacitance to the drain.
Below the total capacitance $C$ as seen from the island is used.

Also for these BDT barrier junctions $e/C$ periodicity is reflected. For
devices where the single electron levels are small compared to $e^{2}/C$ the
spin degeneracy of the levels is lifted and the charging energy regulates the
spacing [24]. With an increasing $E_{F}$ at some point an electron occupies an
additional single electron level on the dot with an energy $e^{2}/C$ above the
previous level. During the increase of $E_{F}$ and prior to the “electron to
island tunneling” the free energy $\Delta F$ increases as well. The Fermi
level increased above the last filled level in the dot. An imbalance is
created in the energy spectrum of the dot, less than $e^{2}/C$. As a
consequence $\Delta F$ increases with this amount. This lasts until the free
energy can be reduced by the amount $e^{2}/C$, at that stage an electron
tunnels to the dot. By reducing $E_{F}$ the situation is not invariant as also
in this case the free energy is increased. In this case $E_{F}$ is below the
highest filled dot level. When $E_{F}$ gets $e^{2}/C$ below it an electron
leaves the dot. At that point the free energy can be reduced by $e^{2}/C$
which it gained in this cycle.

This process is detailed in Fig. 25 and Fig. 26 where in panel a the free
energy is provided for the charging of the capacitor as well as for the charge
imbalance on the dot for both increasing and decreasing $E_{F}$. Both
contributions are separately indicated to the far left in Fig. 25a and the far
right in Fig. 26a. The remaining curves present the combined contribution to
the free energy. A section of Fig. 11 a and b is shown in Fig. 25b and Fig.
26b respectively, the data have been aligned to the voltage-axis with the top
of the periodic structure by subtracting a linear function. The positive and
negative differential resistance parts in the IV discontinuities naturally
find their origin from detailing the total free energy $\Delta F$, which
increases saw tooth wise both for increasing and decreasing voltage scans. The
minimum of $\Delta F$ was set at 0. The magnitude of $\Delta F$ is $2E_{C}$.
The function was scaled to match the current extrema in Fig. 25c and Fig. 26c.

Figure 25: In panel a the two contributions to the free energy, charging
energy and charge imbalance, are indicated to the far left as a function of
$Q$. The remaining curves express the combined contribution. Data from a
segment of Fig. 11a are shown in panel b as well as the inverted and scaled
free energy curves after aligning to zero. Arrows indicate the scan direction.
Figure 26: Panel a shows the two separate contributions to the free energy at
the far right as a function of $Q$. Data from a segment of Fig. 11b are shown
in panel b as well as the inverted and scaled free energy curves after
aligning to zero. Arrows indicate the scan direction.

The capacitance is derived from the $e/C$ periodicity, the current extrema are
obtained from the data providing: $I_{ex}=$75\text{\,}\mathrm{nA}$$,
$C=$7.5\text{\times}{10}^{-19}\text{\,}\mathrm{F}$$ and
$I_{ex}=$130\text{\,}\mathrm{nA}$$,
$C=$9.1\text{\times}{10}^{-19}\text{\,}\mathrm{F}$$ for Fig. 25 and Fig. 26
respectively. The equations in section 4.4 are derived for a single junction,
caution is required interpreting a derived $\alpha$ for a double junction
system.

Based on the free energy it is established that every $e^{2}/C$ cycle an
electron tunnels to or from the island depending on an increasing or
decreasing $E_{F}$, since the free energy can be lowered by this energy
amount. The resulting discontinuities are observed in the IV curves as current
increases. In Fig. 14 for example over 100 electrons are tunneling in
individual $e/C$ voltage cycles to the island for an increasing voltage in the
range from $9\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$. Why is the level
spacing of the single electron levels so small in a molecule where we expect
$\mathrm{eV}$ levels rather than $\mathrm{meV}$ levels? At this stage this is
not clear. It is possible that the interaction of the BDT molecule with the
lining THF molecules leads to many more levels as compared to the original BDT
molecule.

The data in Fig. 15 show many similarities with the smooth THF barrier THF
partially-wet-phase junction oscillations with sharp “V-shaped” minima. No
discontinuous current jumps are visible in panel b. Even though the BDT
molecule solution has been used, it shows the unsuccessful attempt to create a
quantum dot. It also shows that in being unsuccessful, most likely a THF
barrier THF partially-wet-phase junction has been created instead. It is
possible that not the entire electrode surface has been covered with BDT
molecules or that some molecules went down with both thiol groups on one
electrode, the resulting IV curve is very similar to the THF barrier THF
partially wet phase junctions. Panel b and c show a shoulder in the
oscillations appearing and increasingly causing the regular oscillation to
break up. Similarly to Fig. 8 this is attributed to excited states coupled to
a $Q$ state, entering the $e/C$ window.

Fig. 16 provides insight in the criticality of the partially wet phase. The IV
curve shows a somewhat irregular structure, for the increasing voltage scan
the structure shows at least for a number of discontinuities a negative
differential resistance. Electrons are tunneling onto the dot. In the
decreasing voltage scan electrons are tunneling from the dot. This effect
disappears abruptly once the partially wet phase has disappeared in the lower
curve at $9.25\text{\,}\mathrm{V}$. What happens to the hundreds of electrons
that may be stored on the dot at this point? The nature of the quantum effect
disappears once vacuum environmental modes start to mingle in the conduction.
The conduction becomes incoherent single electron effects are no longer
visible. The hundreds of electrons may remain on the dot, however the
intricate and subtle difference of adding or subtracting them one by one, gets
lost.

###  4.6 Capacitance reviewed

This section is deviating from the electrostatic capacitance assumption.
Defining capacitance at the microscopic level is complex. In one view [47] the
characteristic time $\tau_{t}$ the electron takes to traverse the barrier
while tunneling is determining the distance $\tau_{t}c$ it surveys, c is the
charge propagation speed. Nazarov posed a different view [48] where the time
$\tau_{N}$, defined by $\tau_{N}=\hbar/eV$, is the relevant parameter for the
junction-lead system. This leads to a distance $\tau_{N}c$, being surveyed by
the electron. The high bias voltages up to $10\text{\,}\mathrm{V}$ in the
shown junction results will lead to extremely small $\tau_{N}c$ distances as
compared to normal measurements in the sub $0.1\text{\,}\mathrm{V}$ range.

Figure 27: A THF barrier junction in the partially wet phase where during one
segment of the IV curve the voltage scan rate increased from
$22\text{\,}\mathrm{mV}\text{\,}{\mathrm{s}}^{-1}$ to
$200\text{\,}\mathrm{mV}\text{\,}{\mathrm{s}}^{-1}$. The periods of
$7.1\text{\,}\mathrm{mV}$ are consequently enlarged to
$66.6\text{\,}\mathrm{mV}$ while the amplitude remains similar. In addition a
small oscillation is present on the large period oscillation. This is also
shown in the inset of panel a. Panel b is an enlargement of panel a, AuS,
$\phi_{\text{in}}^{2}$.

Fig. 27 and Fig. 28 show data of two bending beam assemblies THF barrier
junctions in the partially wet phase. In these IV curves the scan speed was
modified from the standard fairly low
$22\text{\,}\mathrm{mV}\text{\,}{\mathrm{s}}^{-1}200\text{\,}\mathrm{mV}\text{\,}{\mathrm{s}}^{-1}$
for a few indicated voltage sections. Fig. 27b is an enlargement of 27a, the
IV curve in Fig. 28b was recorded just after the curve in Fig. 28a. In Fig. 27
oscillations in the $22\text{\,}\mathrm{mV}\text{\,}{\mathrm{s}}^{-1}$
sections are visible. Increasing the scan-speed 9 fold is increasing the
period of the oscillations by a factor of 9 as well. For Fig. 28 a similar
behavior is shown. In this figure the black parts appear noisy, however by
zooming in it is clear that this is reflecting small oscillations, see for
example the inset. The 9 fold increase in scan-speed also leads to an
approximate 9 fold increase in period in both panels a and b, where the period
of the smaller oscillations has been measured next to the increased scan-rate
section. It is notable that the amplitude of the large period oscillations
remains very similar to the amplitude of the original small period
oscillations. The above described behavior on scan-speed change and amplitude
has also been measured in BDT barrier junctions in the partially wet phase.

Figure 28: A THF barrier junction in the partially wet phase. Panel a and b
show two IV traces recorded after each other where for segments of the traces
the scan-speed was enlarged. The inset of panel a shows an enlargement of a
0.1 voltage section, AuS, $\phi_{\text{in}}^{2}$

Table 1 summarizes the results from Fig. 27 and Fig. 28 as well as from all
other figures where periodic oscillations have been measured. The capacitance
has been derived from the periodicity $e/C$. Fig. 29 shows the derived
capacitance as a function of the used scan rate in the experiment. This graph
shows that by increasing the scan-rate by a factor of 20 the capacitance is
reduced by a factor of 20, consistent with the findings in Fig. 27 and Fig. 28
where the period increased by a similar fraction as the scan-rate increase.

Practically the capacitance can be described by $C_{0}\exp(-8\text{SR})$ with
SR the scan-rate in $\mathrm{V}\text{\,}{\mathrm{s}}^{-1}$ and
$C_{0}=$2\text{\times}{10}^{-17}\text{\,}\mathrm{F}$$. The data in the
preceding sections are still valid, however now it is known that it would have
been different with a different scan-rate.

Figure 29: Capacitance values from table 1 as a function of the used voltage
scan-speed in the measurement. The line represents an exponential fit to the
data points; $C=C_{0}\exp(-8\text{SR})$ with SR the scan-rate in
$\mathrm{V}\text{\,}{\mathrm{s}}^{-1}$ and
$C_{0}=$2\text{\times}{10}^{-17}\text{\,}\mathrm{F}$$.

Essentially the scan-rate of
$22\text{\,}\mathrm{mV}\text{\,}{\mathrm{s}}^{-1}$ is very low and close to
the $C_{0}$ value in a static situation. The dependence of period and
capacitance on scan-rate may also explain the rare occurrence of curves as
indicated in Fig. 11 which were measured at the highest scan-rate.

Table 1: Summarizing capacitance values and scan-rates of periodicity reflected in the various Figures. The last point is added and not reported in any of the Figures. Figure | Period | Capacitance | Scan Rate | $\derivative*{Q}{t}$
---|---|---|---|---
| ($\mathrm{mV}$) | ($\mathrm{F}$) | ($\mathrm{mV}\text{\,}{\mathrm{s}}^{-1}$) | ($\mathrm{e}\mathrm{/}\mathrm{s}$)
10 | 29.2 | $5.5\text{\times}{10}^{-18}$ | 200 | 6.9
7b | 10.9 | $1.5\text{\times}{10}^{-17}$ | 22 | 2.0
11a | 212.0 | $7.5\text{\times}{10}^{-19}$ | 444 | 2.1
11b | 176.0 | $9.1\text{\times}{10}^{-19}$ | 444 | 2.5
8 | 12.5 | $1.3\text{\times}{10}^{-17}$ | 22 | 1.8
12 | 13.5 | $1.2\text{\times}{10}^{-17}$ | 22 | 1.6
13ab | 9.2 | $1.7\text{\times}{10}^{-17}$ | 22 | 2.4
13cd | 7.6 | $2.1\text{\times}{10}^{-17}$ | 22 | 2.9
15bc | 28.2 | $5.7\text{\times}{10}^{-18}$ | 56 | 2.0
27 | 66.6 | $2.4\text{\times}{10}^{-18}$ | 200 | 3.0
27 | 7.1 | $2.2\text{\times}{10}^{-17}$ | 22 | 3.1
28a | 3.2 | $5.0\text{\times}{10}^{-17}$ | 22 | 6.9
28a | 29.1 | $5.5\text{\times}{10}^{-18}$ | 200 | 6.9
28b | 45.4 | $3.5\text{\times}{10}^{-18}$ | 200 | 4.4
28b | 5.0 | $3.2\text{\times}{10}^{-17}$ | 22 | 4.4
not rep. | 9.7 | $1.6\text{\times}{10}^{-17}$ | 56 | 5.8

It appears that increasing the scan-rate reduces the $\tau_{N}c$ distance, the
horizon of the electron. This is reducing the capacitance. The scan-rate
increase, coinciding with a proportional capacitance reduction, leaves
$\derivative*{Q}{t}$ constant, within the accuracy of the experiment. Fig. 29
shows $\derivative*{Q}{t}$ for the junctions indicated in table 1, the data
points from Fig. 27 and Fig. 28 are pairwise highlighted, indicating that
$\derivative*{Q}{t}$ remains constant at a scan-rate increase. The amplitude
of the oscillations remains constant at a scan-speed change. The amplitude
correlates to the free energy in equation 8 which in turn correlates with the
charging energy in equation 6. The capacitance behavior at scan-speed changes
related to the periodicity appears to be different from the capacitance
behavior related to the free energy scale.

Summarizing this section, the high voltage is plausible to play a role as to
why single junctions are decoupled from the leads and display small
capacitance effects. The measured periodicity/capacitance is dependent on the
scan-rate. At scan-speed changes $\derivative*{Q}{t}$ remains constant.

Figure 30: $\derivative*{Q}{t}$ as a function of scan-speed for the various
junctions indicated in table 1. The data points from the three IV curves in
Fig. 27 and Fig. 28 are pairwise highlighted as they originate from the same
IV curve.

###  4.7 General discussion

One of the striking observations is related to temperature. Why is it possible
to observe effects in a single junction at room temperature where single
electron tunneling is regularly studied at dilution fridge temperatures? Why
is it possible to reproduce a 1-2 $\mathrm{mV}$ feature, measure sharp
$\mathrm{mV}$ width “V-shaped” curves at room temperature where $k_{B}T$
equals $25\text{\,}\mathrm{meV}$? The answers to these questions will probably
have the same basis as the existence of this quantum effect. One aspect will
be that the tunneling electrons are simply not capable of interacting with any
other modes than a few device defining modes.

Confining charge carriers to a plane is not new; the two dimensional electron
gas (2DEG) has advanced our understanding of science over the past 4 decades.
Bandgap engineering in a GaAs-AlGaAs heterojunction can create a potential
trap for electrons in the surface potential perpendicular to the interface
leading to a 2DEG at the interface of the electronically different materials
GaAs and AlGaAs. Here, nature provides for a perfect conductive layer, lining
the molecular device via the THF partially wet phase. In this case the charge
gas follows a contour-shaped liner layer. The surface potential-trap is
perpendicular to the THF partially wet phase layer/air interface, see Fig. 31.
The trapped charge carriers are free to move in the plane of the lining layer,
in the same way as in a 2DEG. Based on the presented experiments, the fact
that the partially wet phase is a requirement to be able to observe these
effects, the analogy to the physical principle of a 2DEG at material
interfaces, as well as the explanation and derivation of the oscillations and
line shapes in the IV curves, the only possible conclusion is that the earlier
posed “gedankenexperiment” is in actual fact the reality.

Figure 31: An impression of the BDT barrier junction in the partially wet
phase, not to scale. The blue/purple layer represents the THF partially wet
phase layer the red liner represents the finite width of the surface potential
trapping charge carriers at the THF/air interface.

The above does not imply that every partially wet phase layer will have
floating charge carriers, in fact for H2O it was shown that the partially wet
phase layer is able to hold charge at the electrode side captive due to the
dipole moment of the H2O molecule. It is simply not known at this stage how
charge at the H2O/air side will behave.

The high voltage and the THF partially wet phase seem to be a combined
requirement for the observation of the reported quantum effects in the THF
barrier and BDT barrier junctions. There are multiple experimental findings
revealing that the $Q$-states are real: $\derivative*{Q}{t}$ has been shown to
be constant for a junction. The IV line shape of the THF barrier junction
shows behavior as predicted by the free energy. Excited states couple to
specific $Q$-states and reproduce over the oscillations with $e/C$
periodicity. The BDT barrier junction shows expected behavior with respect to
tunneling to and from an island depending on scan direction. The voltage scan-
rate couples to the inner quantum mechanics of the junction, leading to scan-
rate dependent IV curves. Based on the above as well as the combined presented
results we feel confident stating that the $Q$-side of the commutator has been
reached. This implies that the Schön phase, $\varphi_{S}$, has become
undefined and $Q$ needs to be treated as a classical variable.

The effects of the partially wet phase on fundamental tunnel properties are
probably not restricted to molecular junctions only. Small capacitance solid
state tunnel junctions and compiled multi-junction systems, lined with a
single molecular layer of a well-chosen gas, at cryogenic/dilution-fridge
temperatures are potentially interesting devices. Almost half a century ago
Kulik and Shekhter predicted their voltage oscillations [11]. We are now on
the verge of discovering the full potential of these quantum effects.

In the end Molecular Electronics will be unified with the ongoing device
miniaturization by the partially wet phase.

##  5 Conclusions

A method to produce molecular junctions in the partially wet phase has been
presented. In the partially wet phase the conduction of these systems is
essentially one dimensional. The connection to the “known” has been
established via observed Grundlach oscillations. Reproducible and long-term
enduring single molecular junctions at ambient conditions can be realized. At
high voltages a quantum effect related to continuous $Q$ charging has been
demonstrated. The THF barrier junction reflects $e/C$-periodicity of the free
energy. In the BDT barrier junction IV curve discontinuities reflect electron
tunneling to or from an island. For a particular junction $\derivative*{Q}{t}$
is constant and independent of the voltage scan-rate. $Q$ needs to be treated
as a classical variable.

The final conclusion is that the THF partially wet phase is responsible for
shielding interactions between tunneling electrons and vacuum environmental
modes.

## Acknowledgements

We would like to thank F.L. Muller for many stimulating, encouraging and
constructive discussions.

Not because it was easy, but because he had a conviction it was worthwhile
pursuing, von Hippel had a visionary belief in molecular electronics. Up to
this date we are discovering new physics in this field, progressing von
Hippel’s chess game to an interesting next phase.

## References

  * [1] A Von Hippel.  Molecular engineering.  Science, 123(3191):315–317, 1956.
  * [2] A Von Hippel et al.  Molecular science and molecular engineering.  Published jointly by the Technology Press of MIT and J. Wiley, 1959.
  * [3] JC Cuevas and E Scheer.  Molecular Electronics: An Introduction to Theory and Experiment.  World Scientific, 2010.
  * [4] S Datta.  Electronic transport in mesoscopic systems.  Cambridge university press, 1997.
  * [5] S Datta.  Quantum transport: atom to transistor.  Cambridge university press, 2005.
  * [6] CJ Gorter.  A possible explanation of the increase of the electrical resistance of thin metal films at low temperatures and small field strengths.  Physica, 17(8):777–780, 1951.
  * [7] KK Likharev.  Correlated discrete transfer of single electrons in ultrasmall tunnel junctions.  IBM Journal of Research and Development, 32(1):144–158, 1988.
  * [8] HR Zeller and I Giaever.  Tunneling, zero-bias anomalies, and small superconductors.  Physical Review, 181(2):789, 1969.
  * [9] I Giaever and HR Zeller.  Superconductivity of small tin particles measured by tunneling.  Phys. Rev. Lett., 20(26):1504, 1968.
  * [10] J Lambe and RC Jaklevic.  Charge-quantization studies using a tunnel capacitor.  Phys. Rev. Lett., 22(25):1371, 1969.
  * [11] IO Kulik and RI Shekhter.  Kinetic phenomena and charge-discreteness effects in granulated media.  Zhur. Eksper. Teoret. Fiziki, 68(2):623–640, 1975.
  * [12] E Ben-Jacob and Y Gefen.  New quantum oscillations in current driven small junctions.  Physics Letters A, 108(5-6):289–292, 1985.
  * [13] E Ben-Jacob, Y Gefen, K Mullen, and Z Schuss.  Squid’85, H Lübbig and HD Hahlbohm, eds., 1985.
  * [14] DV Averin and KK Likharev.  Squid’85, H Lübbig and HD Hahlbohm, eds., 1985.
  * [15] F Guinea and G Schön.  Coherent charge oscillations in tunnel junctions.  EPL (Europhysics Letters), 1(11):585, 1986.
  * [16] DV Averin and KK Likharev.  Coulomb blockade of single-electron tunneling, and coherent oscillations in small tunnel junctions.  Journal of low temperature physics, 62(3):345–373, 1986.
  * [17] KK Likharev and AB Zorin.  Theory of the bloch-wave oscillations in small josephson junctions.  Journal of low temperature physics, 59(3):347–382, 1985.
  * [18] DV Averin and KK Likharev.  Single electronics: A correlated transfer of single electrons and cooper pairs in systems of small tunnel junctions.  In Modern Problems in Condensed Matter Sciences, volume 30, pages 173–271. Elsevier, 1991.
  * [19] G Schön and AD Zaikin.  Quantum coherent effects, phase transitions, and the dissipative dynamics of ultra small tunnel junctions.  Physics Reports, 198(5-6):237–412, 1990.
  * [20] G Schön.  Quantum shot noise in tunnel junctions.  Physical Review B, 32(7):4469, 1985.
  * [21] AN Cleland, JM Schmidt, and J Clarke.  Charge fluctuations in small-capacitance junctions.  Phys. Rev. Lett., 64(13):1565, 1990.
  * [22] TA Fulton and GJ Dolan.  Observation of single-electron charging effects in small tunnel junctions.  Phys. Rev. Lett., 59(1):109, 1987.
  * [23] LP Kouwenhoven, CM Marcus, PL McEuen, S Tarucha, RM Westervelt, and NS Wingreen.  Electron transport in quantum dots.  In Mesoscopic electron transport, pages 105–214. Springer, 1997\.
  * [24] CWJ Beenakker, H van Houten, and AAM Staring.  Granular nanoelectronics.  Plenum Press, New York, page 359, 1991.
  * [25] MH Devoret and H Grabert.  Single charge tunneling: Coulomb blockade phenomena in nanostructures.  Plenum Press, New York, 1992.
  * [26] GL Ingold and Yu V Nazarov.  Single charge tunneling: Coulomb blockade phenomena in nanostructures.  Plenum Press, New York, 1992.
  * [27] MH Devoret, D Estève, H Grabert, GL Ingold, H Pothier, and C Urbina.  Effect of the electromagnetic environment on the coulomb blockade in ultrasmall tunnel junctions.  Phys. Rev. Lett., 64(15):1824, 1990.
  * [28] CJ Muller.  An experimental study on mechanically controllable break junctions.  PhD thesis, University Leiden, 1992.
  * [29] CJ Muller, JM Van Ruitenbeek, and LJ De Jongh.  Experimental observation of the transition from weak link to tunnel junction.  Physica C: Superconductivity, 191(3-4):485–504, 1992.
  * [30] CJ Muller, JM Van Ruitenbeek, and LJ De Jongh.  Conductance and supercurrent discontinuities in atomic-scale metallic constrictions of variable width.  Phys. Rev. Lett., 69(1):140, 1992.
  * [31] D Bonn, J Eggers, J Indekeu, J Meunier, and E Rolley.  Wetting and spreading.  Reviews of modern physics, 81(2):739, 2009.
  * [32] CJ Muller.  A pivoting mechanically controllable break junction setup enabling partially wet phase mcb-junctions, 2021.
  * [33] JM Tour, L Jones, DL Pearson, JJS Lamba, TP Burgin, GM Whitesides, DL Allara, AN Parikh, and S Atre.  Self-assembled monolayers and multilayers of conjugated thiols,. alpha.,. omega.-dithiols, and thioacetyl-containing adsorbates. understanding attachments between potential molecular wires and gold surfaces.  Journal of the American Chemical Society, 117(37):9529–9534, 1995\.
  * [34] CJ Muller, BJ Vleeming, MA Reed, JJS Lamba, R Hara, L Jones, and JM Tour.  Atomic probes: a search for conduction through a single molecule.  Nanotechnology, 7(4):409–411, dec 1996.
  * [35] KH Gundlach.  Zur berechnung des tunnelstroms durch eine trapezförmige potentialstufe.  Solid-State Electronics, 9(10):949–957, 1966.
  * [36] G Binnig and H Rohrer.  Scanning tunneling microscopy.  Helv. Phys. Acta, 55:726–735, 1982.
  * [37] O Yu Kolesnychenko, Yu A Kolesnichenko, OI Shklyarevskii, and H van Kempen.  Field-emission resonance measurements with mechanically controlled break junctions.  Physica B: Condensed Matter, 291(3-4):246–265, 2000.
  * [38] O Kolesnychenko.  Experimental investigations of complex electronic states at metal surfaces.  PhD thesis, Radboud University Nijmegen, 2002.
  * [39] R Landauer.  Spatial variation of currents and fields due to localized scatterers in metallic conduction.  IBM Journal of research and development, 1(3):223–231, 1957.
  * [40] R Landauer.  Can a length of perfect conductor have a resistance?  Physics Letters A, 85(2):91–93, 1981.
  * [41] R Landauer.  Conductance determined by transmission: probes and quantised constriction resistance.  Journal of Physics: Condensed Matter, 1(43):8099, 1989.
  * [42] A Grigoriev, J Sköldberg, G Wendin, and Ž Crljen.  Critical roles of metal-molecule contacts in electron transport through molecular-wire junctions.  Physical Review B, 74(4):045401, 2006.
  * [43] J Maserjian.  Tunneling in thin mos structures.  Journal of Vacuum Science and Technology, 11(6):996–1003, 1974\.
  * [44] JH Coombs and JK Gimzewski.  Fine structure in field emission resonances at surfaces.  Journal of Microscopy, 152(3):841–851, 1988.
  * [45] LP Kouwenhoven, G Schön, and LL Sohn.  Introduction to mesoscopic electron transport.  In Mesoscopic Electron Transport, pages 1–44. Springer, 1997.
  * [46] MC Koops, GV Van Duyneveldt, and R de Bruyn Ouboter.  Direct observation of the current-phase relation of an adjustable superconducting point contact.  Phys. Rev. Lett., 77(12):2542, 1996.
  * [47] M Buttiker and R Landauer.  Traversal time for tunneling.  IBM journal of research and development, 30(5):451–454, 1986.
  * [48] Yu V Nazarov.  Anomalies of current voltage characteristics of tunnel-junctions.  Zhur. Eksper. Teoret. Fiziki, 95(3):975–984, 1989.