I-V characteristics and diﬀerential conductance ﬂuctuations of Au nanowires
H. Mehrez, Alex Wlasenko, Brian Larade, Jeremy Taylor, Peter Gr¨ tter, and Hong Guo
u

arXiv:physics/0106027v1 [physics.comp-ph] 8 Jun 2001

Center for the Physics of Materials and Department of Physics, McGill University, Montreal, PQ, Canada H3A 2T8.

tance quanta. Pioneering experiments9,10 clearly showed
the correlation between conductance jumps and mechanical properties in the nano-contacts. These results conﬁrmed earlier predictions27 that the conductance variations are due to abrupt changes of nano-structure cross
section as a function of wire elongation. Extensive theoretical investigations on nano-structures have been published recently to analyze these systems. One major
theory focus is to calculate the zero bias conductance
through a ballistic quantum point contact. These calculations start by assuming various contact28–39 geometry, or by using more realistic atomic positions derived
from molecular dynamics simulations. The potential of
the constriction and/or interaction Hamiltonian is then
constructed40–51 from which the zero bias transport coeﬃcients are evaluated. While diﬀerent levels of approximations were employed in these theoretical analysis, density functional theory based ab initio analysis have also
been reported33,36,52 which provide self-consistent calculations of atomic nano-contacts.
While zero bias transport coeﬃcients have received a
great deal of attention, one must go beyond this limit
to understand the full nonlinear current-voltage (I-V)
characteristics of the nano-contacts, as this information
is essential for the understanding of real device operation. For example, due to the small cross section of these
systems, they are exposed to substantial current density
∼ 108 A/cm2 which may result in atomic rearrangement.
It is also expected that electron-electron (e-e) interaction
is enhanced due to the strong lateral conﬁnement possibly leading to Luttinger liquid behavior for the quasi
one dimensional atomic wires. Separating these diﬀerent eﬀects is experimentally challenging due to the many
variables which can aﬀect the results. This is probably
the origin of the existing controversy in explaining the experimentally observed non-linear I-V curves of the atomic
scale wires53–56 . From a theoretical point of view, this is
also a challenging problem: so far only two computationally accurate techniques exist which can treat systems
with open boundaries out of equilibrium due to external
bias33,57,58 . In the approach of Taylor et. al.57,58 , realistic atomic leads can be treated and the problem is solved
self consistently within LDA. Therefore, the leads, the
device (scattering region) and their couplings are incorporated without any preconditioned parameters.
To further shed light on the physics of quantum transport at molecular scale, we report in this paper our investigation on transport properties of Au nanostructures
both experimentally and theoretically. Experimentally,
we created stable Au nanowires using the mechanically
controllable break junction technique in air, and we si-

Electronic transport properties of Au nano-structure are
investigated using both experimental and theoretical analysis.
Experimentally, stable Au nanowires were created using mechanically controllable break junction in air, and simultaneous current-voltage (I-V) and diﬀerential conductance δI/δV
data were measured. The atomic device scale structures are
mechanically very stable up to bias voltage Vb ∼ 0.6V and
have a life time of a few minutes. Facilitated by a shape
function data analysis technique which ﬁnger-prints electronic
properties of the atomic device, our data show clearly diﬀerential conductance ﬂuctuations with an amplitude > 1% at
room temperature, and a nonlinear I-V characteristics. To
understand the transport features of these atomic scale conductors, we carried out ab initio calculations on various Au
atomic wires. The theoretical results demonstrate that transport properties of these systems crucially depend on the electronic properties of the scattering region, the leads, and most
importantly the interaction of the scattering region with the
leads. For ideal, clean Au contacts, the theoretical results indicate a linear I-V behavior for bias voltage Vb < 0.5V . When
sulfur impurities exist at the contact junction, nonlinear I-V
curves emerge due to a tunnelling barrier established in the
presence of the S atom. The most striking observation is that
even a single S atom can cause a qualitative change of the
I-V curve from linear to nonlinear. A quantitatively favorable comparison between experimental data and theoretical
results is obtained. We also report other results concerning
quantum transport through Au atomic contacts.

I. INTRODUCTION

Electron transport through atomic nano-contacts has
been an active research area for a decade both experimentally and theoretically. The scientiﬁc interest of these
systems is largely driven by their peculiar electronic and
transport behavior. The atomic nano-contacts are structures with low atomic coordination number and, as a
result, can behave very diﬀerently from the bulk counterpart. From a practical point of view, understanding the novel electronic and structural properties of the
atomic nano-contacts is an important step towards nanodevice fabrication and characterization. The ﬁrst set of
experiments on nano-contacts focused on their zero bias
conductance (G) using Scanning Tunnelling Microscopy
(STM)1–17 , Mechanically Controllable Break Junction
(MCBJ)18–24 , and Relay Contacts (RC)25,26 . In these
experiments a narrow constriction with a few atoms at
the cross section is formed. As the electrodes are pulled
apart, G is measured and found to change discontinuously forming plateaus with values close to n× G0 ; where
1
n is an integer and G0 = 2e2 /h ≃ 12.9KΩ is the conduc1

multaneously measure the I-V curve and the diﬀerential
conductance δI/δV . We found that our atomic scale Au
nano-contacts are mechanically very stable up to bias
voltage Vb ∼ 0.6V and have a life time of a few minutes
which is adequate for our measurements. As we are interested in features due to electronic degrees of freedom
of the nano-contacts, careful data analysis is needed because transport data can be aﬀected by many factors.
δI
By deﬁning the shape function, S = δV V , which ﬁngerI
prints electronic properties of the atomic device, our data
clearly shows diﬀerential conductance ﬂuctuations with
an amplitude > 1% at room temperature, and a nonlinear I-V characteristics. To understand these transport features, we carried out ab initio calculations on
various Au atomic wires bonded with atomic Au electrodes using the ﬁrst principles technique of Ref. 57,58.
Our calculations show that pure and perfect Au nanocontacts do not give the nonlinear I-V curves as measured in the experiments. However when Sulfur impurities are present near the wire-electrode contact region,
the nonlinear I-V curves emerge due to the tunnelling
barrier provided by the impurity atoms. The most striking observation is that even a single S atom can cause a
qualitative change of the I-V curve from linear to nonlinear. A quantitatively favorable comparison between
experimental data and theory results is then obtained.
Our combined theory-experiment investigation allows us
to conclude that transport through Au atomic wires is
strongly aﬀected by the properties of the wire-electrode
contacts.
The rest of the paper is organized as follows. In the
next section, experimental measurement and results are
presented. Section III presents the theoretical results
while Section IV discusses the transmission coeﬃcients
in more detail. We also discuss and compare previous
works with ours in section V, followed by a conclusion.

seem to vary signiﬁcantly in the details of their conductance behavior. However, for each polarity, one notices
that there seem to be similar details present in the DC
and diﬀerential conductance.
An important issue in order to understand these results
is the separation of eﬀects due to atomic rearrangement
in the nano structure from electronic properties. We address this problem by writing, very generally:
I(X, V ) ≡ g(X)f (X, V )

(1)

where the variable X symbolizes the eﬀects of atomic
structure, but no explicit knowledge of X is required. We
deﬁne g(X) as the unbiased conductance and f (X, V ) is
the normalized functional form of the voltage dependence
of the current. Therefore, in the zero bias limit V → 0,
f (X, V ) = V . We further deﬁne a new quantity called
the “shape function” (S),
S≡

δI V
δV I

.

(2)

Fig.2-a plots S corresponding to the data of Fig.1-b, for
both positive and negative bias voltages. The similarity
between the curves of the shape functions for both polarities, including ﬁne details, is in striking contrast to the
easily distinguishable conductance plots of Fig.1-b. By
its deﬁnition (Eq. 2), S depends only on f , the functional
form of I(X, V ), and is independent from the zero-bias
conductance g(X). This fact and the usefulness of the
shape function S can be seen by considering the following “Gedanken experiment”. Let’s assume that we measured I and δI/δV through a variable, ohmic potentiometer as a function of the applied voltage. Suppose there
were drastic changes in the temperature during the measurement, and some troublemaker stochastically turned
the knob of the variable potentiometer without telling
anyone. Glancing at the measurements of I(X, V ) and
δI/δV alone, one might wrongfully conclude that the potentiometer was exhibiting a nonlinear behavior. However, a plot of S would show that S(V ) = 1 (from Eq.2),
which would allow us to deduce that the I-V curve was
actually linear and thus in fact ohmic. One would also
conclude that the origin of the apparently nonlinear behavior was due to a change in the unbiased conductance
g(X), rather than due to a true nonlinearity in voltage.
Returning to the experimental results shown in Fig.2a, we note the overlap of the two curves of S for positive
and negative bias voltages; this points to a similar functional form for both bias polarities. Because of this, we
deduce that for this case f (X, V ), at most, depends very
weakly on X (especially at bias voltage Vb < 0.47V ). We

II. EXPERIMENTAL RESULTS

Atomic scale gold contacts were formed with a mechanically controllable break junction in air at room temperature. Once suitably stable atomic junctions were formed,
a slowly varying bias voltage was applied (typically a
0.1Hz triangle wave, 2Vp−p ) along with a small modulation voltage (typically 40kHz, 4mVrms ) across the
contact and a load resistor of 3KΩ. Current (I) and
diﬀerential conductance (δI/δV ) were measured with an
I-V preampliﬁer and a lock-in ampliﬁer. The experimental set up is shown in Fig.1-a. A typical measurement
through a stable Au nano-contact is presented in Fig.1b. We show our data for the diﬀerential conductance and
the DC conductance (G = I/V ). In the inset of Fig.1-b,
we also display a typical I-V measurement. This data was
taken over the course of a 5 seconds voltage sweep from
positive (dark lines) to negative (grey lines) bias voltage
(V ). Both polarities share a common overall shape, but

S

can therefore write f (X, V ) ≃ f0 (V ) ∝ e V dV , and the
unbiased conductance g(X) = I(X,V)) . Hence, as an exf0 (V
perimental voltage sweep typically takes 5 seconds, time
dependent atomic rearrangements (changes in X) manifest themselves as stochastic variations of g(X). Other
details of this data analysis technique and further discussion on the shape function are not within the scope of
2

details59,61 . This unavoidable averaging artifact keeps us
from making a more precise measurement of the shape,
amplitude and voltage characteristics of this ﬁne structure, but the true features should be sharper and more
pronounced than they are measured to be. From our
data, we observe “wiggles” on the voltage scale < 10mV
and with amplitude ∼ 1% of the signal. The width of
these ﬂuctuations is comparable to the modulation voltage, 4mVrms ∼ 10mVp−p ; hence, it is possible to have
features on a voltage scale < 10mV with amplitude which
is orders of magnitude larger than these 1% ﬂuctuations.
If one were to merely examine our I(V ) measurements,
similar results have been observed in air and at low
temperatures with RC25,55 and STM conﬁgurations56,62 .
The ﬁne details exposed by the analysis presented above
have not been discussed in the literature as they tend
to be hidden by the ﬂuctuations in the unbiased conductance g(X) (see Fig.2-c). The data presented here
represents the behavior of one junction. However, we emphasize that the general features presented in Fig.2 are
experimentally found to be independent from the zero
bias conductance g(V = 0) for values between 1 - 10 Go .
The ﬁne details of δf /δV are reproducible for a given
stable junction, but the speciﬁc details change for diﬀerent junctions. Although there seems to be a voltage scale
associated with these details (< 10mV ), Fourier analysis does not indicate any strong periodicity. It should
be noted that we have imposed a selection rule on the
junction type by studying device conﬁgurations that are
stable on the time scale of minutes. We have also observed junctions that exhibit linear behavior (S = 1)
as have others25,55,56 . These junctions however are not
stable over this long time scale. In comparison to their
nonlinear counterparts, linear junctions have smaller ﬂuctuations in g(X).
In the following we will discuss the physical origin of
the observed I(V ) characteristics. In this aspect we will
investigate the electronic eﬀects rather than the structural parameters which will be assumed static. This will
allow us to gain valuable insight into the voltage dependent conduction properties of nanoscale electrical contacts. We will thus compare our modeling to the normalized functional form of the current, f0 (V ), shown in
Fig.2-b, rather than the original I-V curve presented in
the inset of Fig.1-b. We will also explain the origin of the
ﬂuctuations in the normalized diﬀerential conductance
shown in Fig.2-d as well as the eﬀects of temperature
and modulation signal on their amplitude.

this paper and can be found elsewhere59 .
The normalized functional form of the voltage dependence of f0 (V ) for the measured data is shown in Fig.2b. We note that the curves for both polarities appear
on the top of each other and they are indistinguishable. However, as expected, the “trouble maker” in our
“Gedanken” experiment shows up as ﬂuctuations in g(X)
shown in Fig.2-c, indicated by the ﬂuctuations and by the
fact that positive and negative bias give diﬀerent traces
of g(X). Note that during the course of our bias sweep
from initial bias voltage to some bias value V = V1 , the
atomic structure has been ﬂuctuating and changed from
what we started with to something unknown. But if we
could freeze the structure at that moment and re-measure
the I − V curve for the ﬁxed structure, g(X) would be
its zero bias conductance. In other words, Fig. 2-c shows
a parametric plot of what the conductance g(X) would
be at zero bias at the point in time when this voltage
was measured experimentally; hence g(X) could also be
viewed as a function of time in this ﬁgure. Our analysis
show that the ﬂuctuations of g(X) which are less than
5% peak-peak, are attributed to changes in the atomic
conﬁguration of the junction. We have also found that
g(X) exhibits no time correlation and its Fourier Transform has a 1/f frequency behavior.
In Fig.2-d we plot the normalized diﬀerential conductance, δf /δV . Examining the curves in Fig.2-d, we can
see the same subtle ﬂuctuations in δf /δV as those found
in S (Fig.2-a). Both plots show broad and ﬁne details
that are uncorrelated to the ﬂuctuations of the unbiased
conductance g(X) (Fig.2-c). These plots reveal that it is
the ﬂuctuations of the g(X) which dominate the ﬂuctuation features in the measured I(X, V ) and δI/δV .
Over diﬀerent voltage ranges (0.2 − 0.35V, 0.1 −
0.35V, 0.1 − 0.5V ), δf /δV has a high correlation60 of
0.99 between both polarities. Over these same ranges,
g(X) had a weak correlation (−0.5, 0.25, −0.2, respectively) between both polarities which changed drastically
depending on the selected voltage range. We have also
calculated the correlations by shifting the voltage of one
polarity with respect to the other by ±10mV (the scale
of the ﬁne details) in increments of 1mV . Over all three
ranges, the correlation for δf /δV had a local maximum
for 0V shift. No correlation extrema were found in the
case of g(X). This quantiﬁes how similar the details in
both polarities of δf /δV are to each other, in contrast to
the more easily distinguishable curves for g(X). We thus
δf
conclude that the “wiggles” observed in δV are electronic
in nature and they are not variations in g(X).
The “wiggles” (magniﬁed in the inset of ﬁgure 2-d)
of δf /δV may actually be much more pronounced than
indicated in these plots. They are smeared out by unavoidable experimental constraints. Since we must add
a modulation signal to make our lock-in measurement of
δI/δV , we end up averaging over a range of V . This leads
to broadening and decrease in amplitude of these ﬁne

III. AB INITIO ANALYSIS OF THE I-V
CHARACTERISTICS

To provide a theoretical understanding of the experimental data presented above, we have calculated the I-V
characteristics of Au nano-contacts self consistently by
combining the density functional theory and the Keldysh

3

nonequilibrium Green’s functions. The method is based
on the newly developed ab initio approach to treat open
electronic systems under ﬁnite bias. For technical details of this method we refer interested readers to the
original papers57,58 . Very brieﬂy, our analysis uses an
s, p, d real space LCAO basis set57,58,63 and the atomic
cores are deﬁned by the standard nonlocal norm conserving pseudopotential.64 The density matrix of the device is constructed via Keldysh nonequilibrium Green’s
functions, and the external bias Vb provides the electrostatic boundary conditions for the Hartree potential which is solved in a three dimensional real space
grid. Once the density matrix is obtained, the KohnSham eﬀective potential Vef f (r; Vb ), which includes contributions from Hartree, exchange, correlation and the
atomic core, is calculated. This process is iterated until numerical convergence of the self-consistent density
matrix is achieved. In this way, we obtain the bias
dependent self-consistent eﬀective potential Vef f (r; Vb ),
from which we calculate57,58 the transmission coeﬃcient
T (E, Vb ) ≡ T (E, [Vef f (r, Vb )]), where E is the scattering
electron energy and T is a function of bias Vb through its
functional dependence on Vef f (r; Vb ).
In our analysis, the scattering states are deﬁned for
energy ranges between the left and right chemical potentials µL and µR , respectively. To solve for these states,
at a given energy E, we solve an inverse energy band
structure problem58 . We then group all states as left
and right propagating states depending on their group
velocity. For a scattering state coming from the left lead,
L
KL
ΨKn should start as a right propagating state ΦL n and it

stated. In addition to predicting the overall transport
properties of a device, our formalism enables us to study
transmission through each incoming Bloch state of the
leads separately, therefore allowing us to separate eﬀects
due to the leads and due to the scattering region. We
have used this formalism to calculate I-V characteristics
of structurally diﬀerent Au nano-contacts and compared
them with the experimental results described above.
A. Perfect Au nano-contacts

In a ﬁrst attempt to model our experiments, we
calculated the I-V characteristics of four Au atoms
(=“molecule”) in contact with Au(100) leads. The structure of the atomic device is illustrated in Fig.3-a. The
scattering region is bonded by two semi-inﬁnite Au leads
which extend to electron reservoirs at ±∞ where bias
voltage is applied and current is collected. The device
scattering region, indicated by D, is described by three
Au layers from the left lead, the four Au atoms in a chain,
and two layers of Au from the right lead. We have also
increased the two Au layers on the right side of the chain
to four to ensure that convergence is reached with respect
to the screening length. We note that long and thin gold
necks have directly been observed experimentally50 .In
this structure, the registry of the atomic chain with respect to lead surface layer can be diﬀerent. The most
common structures, which we analyze in this work are:
the hollow site, where the atomic chain faces the vacant
position in the lead layer as shown in the upper left inset of Fig.3-B; and the top site, where the atomic chain
and an atom from lead surface layer face each other as
illustrated in the upper right inset of this ﬁgure.
Our calculations show that in all cases charge transfer
between the Au chain and the leads is not important, being only ∼ 0.07−0.1 electrons per atom to the Au chain at
diﬀerent bias voltages. This corresponds to less than 1%
diﬀerence in electron population per atom and therefore
does not play any signiﬁcant role in the I-V characteristics of Au contacts. This is in contrast to a binary atomic
system such as carbon chains between Al(100) leads66 . In
addition, solving for the energy eigenvalues of the four
atom chain gives a HOMO-LUMO gap of 0.68eV , indicating that the molecule eigenstates should only have a
secondary eﬀect on the transport properties. Therefore,
the major eﬀect on the I-V characteristics is due to the
character of Bloch states in the leads and their couplings
to the molecule at the chain-lead interface.
In Fig.3-A, we show the results for a system with atoms
in the hollow site. At small bias Vb , we note that current
is a linear function of Vb with a slope G ≃ 0.94Go . The
linear function suggests that T (E, V ) = T0 with a weak
voltage dependence. In this regard, our self-consistent
calculation gives a result apparently similar to previous
theoretical work28–51 in the low bias regime. However, we
will show later, by addressing the origin of this “perfect

KL

gets reﬂected back as a left propagating state φL m with
L
L
reﬂection coeﬃcient rKm ,Kn in the left lead, and transmitted into the right lead as a right propagating state
KR
K R ,K L
φR m with transmission coeﬃcient tR m n . In the numerics, the scattering states are represented as a linear
combination of atomic orbitals inside the device region.
This allows us to write, for example, a left scattering
state as:
 kL
L
L
L
 ΦLn + φKm rKm ,Kn inside left lead

L
L
L
K
ΨKn = ψd n
inside device

R
 Km K R ,K L
φR t m n
inside right lead

A scattering state in the right lead can be written in a
similar fashion. For a symmetric two probe device, the
total transmission from the left lead is identical to the
one from the right lead65 . To calculate the total current
inside the device at a given bias voltage Vb applied to the
right lead, we use:
I(Vb ) =

2e +∞
dE T (E, Vb )[fL (E, µL = µ0 )
h −∞
−fR (E, µR = µ0 + eVb )]

(3)

where fL(R) is the Fermi function on the left (right)
lead evaluated at temperature T = 0K unless otherwise
4

linearity”, that the physical picture of a bias-independent
transmission coeﬃcient is not valid even for such a simple chain. We also note that the linear I-V characteristics
observed in these systems do not agree with our experimental data.
A major feature of Fig.3-A is the huge plateau at
Vb = 0.5V − 0.9V , as well as the ﬁne structures (or
sometimes negative diﬀerential resistance) observed for
larger voltages. In the lower inset of Fig.3-B, we show the
band structure of the Au(100) lead along the z−direction
(transport direction). Even-though at a given energy E
there are many Bloch states which are potential candidates for transporting current, our investigation found
that for E < 0.5eV there is only one state that is actually conducting (presented by a continuous line in the
inset). Once this state is terminated at E ≈ 0.5eV , the
current is saturated resulting in a large plateau until new
conducting states emerge at higher bias voltages. For
E > 0.9V , transport properties are more complex since
more states contribute to transmission. Under these circumstances, band crossing occurs more frequently and
T (E, Vb ) changes over small ranges of bias leading to the
small structures seen in the I-V characteristics of perfect
Au contacts. In fact these variations in T (E, Vb ) are the
origin of the ﬂuctuations in the normalized diﬀerential
conductance shown in Fig.2-d. They thus need to be attributed to the eﬀects of the leads’ band structure. In the
lower inset of Fig.3-A, we plot the theoretical δI/δV . The
magnitude of these conductance ﬂuctuations is of the order of 60% at zero temperature which is much larger than
the experimental ﬁnding. However, these ﬂuctuations are
reduced to 15% if the current is calculated using Eq.3 at
a temperature T = 300K (the temperature of our experiments). In addition, the ﬂuctuations further decrease
to ∼ 1% when current is averaged over the experimental
modulation voltage range of 4mV , completely consistent
with our experimental results of Fig.2-d.
The eﬀect of site registry is studied by placing the endatom of the Au chain at the top site of the leads. In
this situation the chain atoms are facing one atom of the
leads’ surface layers. This analysis is quite important, because it was shown46,47 that atoms at the junction change
registry from hollow to top sites resulting in bundle formation just before the nano structure breaks. The I-V
characteristics of these systems (shown in Fig.3-B) are
similar to the previous results, therefore no change is observed in the transport properties for the top site registry.
To further investigate the huge and peculiar plateau
occurring at Vb = 0.5V − 0.9V , we have plotted in Fig.4
the charge density at a given energy E, ρ(E, x, z) =
dyρ(E, x, y, z). We see clearly that at zero bias Vb =
0V , the device property turns from a perfect conductor at
E = 0eV (Fig.4-a) to an insulator at E = 0.68eV (Fig.4c) due to the termination of the conducting state. Here
we interpret the charge concentration as the eﬀective
bonding strength, or conductance probability. Applying
a bias voltage to the system drives it out of equilibrium,
and at Vb = 0.68V and E = 0.68eV , the charge in the

molecule redistributes, but the bonding is still very weak
as shown in Fig.4-d, with some molecular regions having
zero charge and resulting in the large plateau observed
in our I − V curve of Fig.3-A,B. This eﬀectively demonstrates the importance of both the energy and the voltage
dependence of the transmission coeﬃcient T (E, V ). This
point will be discussed in more detail in section V. A further point to notice is the diﬀerence of zero bias conductance, G ≃ 0.8G0 for the top-site device and G ≃ 0.94G0
for the hollow-site device. This diﬀerence is due to a
change in the coupling between the chain end-atoms and
the surface of the leads. To ensure the same nearest
neighbor separation distance for both cases, we end up
with four nearest neighbors for the hollow site registry
and only one nearest neighbor for the top site registry.
Under these circumstances, the hollow site has a better
coupling to the chain and hence a larger conductance.
A pure and perfect Au nano-contact, as studied in this
section, shows rich and interesting transport properties.
It also gives a good understanding of the origin of the
observed diﬀerential conductance ﬂuctuations as due to
coupling of the Au-chain to the leads’ band structure.
However, it has a linear I-V curve at small Vb < 0.5V,
rather than the experimentally observed nonlinear I-V
characteristics. How can a nano-contact produce a nonlinear I-V curve such as that of Fig.(1-b) ? The simplest
possibility to observe such a phenomenon is to have a
tunnelling barrier at the molecule-lead junction whose
eﬀect gradually collapses as a function of an increasing
bias voltage. Indeed, recent experimental ﬁndings indicate that perfectly linear I-V characteristics were reproducibly found in gold-gold nano-contacts in ultra high
vacuum56 , while nonlinear eﬀects emerge when the experiment was performed in the air. This suggests that
impurities play an important factor. Therefore, one has
to go one step further and investigate the eﬀect of impurity and disorder at the Au junction.
B. Au nano-contacts with S impurity

To simulate the eﬀect of an impurity at the contact, we
have replaced one of the Au atoms at the interface layer
with a sulphur atom. This is presented in the inset of
Fig.5. The choice of sulphur is motivated by the fact that
in our experimental labs located in down-town Montr´al,
e
sulphur is a non-negligible airborne pollutant (diesel exhausts); sulphur atoms bond actively with Au. We also
note that the band-width of sulphur, ∼ 10eV , is much
higher than that of Au (∼ 1eV ), thus a tunnelling barrier is expected to be provided by the presence of S atoms.
In this system, charge transfer to the atomic chain is still
small, and thus inadequate in explaining the experimentally observed I-V characteristics. We note that the S
atom suﬀers from an electron deﬁciency ∼ 4%. Also due
to the presence of the S atom, the coupling of the Bloch
states in the leads to the device scattering region is quite

5

due to the high concentration of S at the interface. Therefore, an experiment which dopes an Au contact with more
S impurity should show an enhancement of the I-V nonlinearity. This also gives a possible explanation of why
experimentally the nonlinear I-V ﬁtting parameters are
not universal53,54 : they dramatically depend on the contact structure as well as the impurity concentration.
Studying the eﬀect of disorder in nano-contacts is another important problem. To investigate this eﬀect, we
have randomized the contact layer in the left lead of a
pure Au nano-contact, as shown in the inset of Fig.6-B.
The distribution of disorder leads to a smaller distance
between the contact layer and the Au chain, resulting in a
better coupling. The current through this device is larger
than that of the ideal contact and is shown in Fig.6-B.
For this device, the slope of the current at zero bias is
G ∼ 0.915Go , and it slightly increases to G ∼ 0.92Go
at Vb ∼ 0.2V . The I-V curve shows very weak nonlinear
characteristics. This suggests that disorder alone may
create a tunnelling barrier which is overcome through the
application of a bias voltage. However, to observe the effect, there need to be conducting states in the scattering
region. For our pure Au device, only one single state
is conducting and the maximum zero-bias conductance
is G = Go . Therefore, at G ∼ 0.915Go , the conducting
channel is already open to near its maximum at zero bias,
hence it cannot be further enhanced in any signiﬁcant
way by applying a bias. The results in Fig.6-B show that
disorder is an important factor that allows more Bloch
states in the leads to couple with the scattering region
and contribute to transport properties. This is clearly
seen when we notice that the huge current plateau observed in Fig.3 essentially vanishes in disordered device.
Combining the eﬀect of disorder and S impurity is also
crucial. The latter enhances the tunnelling barrier and
the former may enhance the coupling of the device to the
leads. We have used the disordered structure studied in
the last paragraph and replaced one of the Au atoms at
the contact layer with an S atom, as shown in the insets
of Fig.6-C. The zero bias conductance for this device is
G ∼ 0.833Go. This number is larger than the one with
only a tunnelling barrier (ideal contact with S impurity),
but is smaller than the one with only a disordered contact (which has better coupling). The ﬁnal conductance
is due to a competition between both eﬀects. The I-V
curve for this device is presented in Fig.6-C, it shows very
weak nonlinear I-V characteristics: analysis of our data
show that the slope reaches G ∼ 0.85G0 at Vb ∼ 0.18V .
Therefore, there is still a weak nonlinear behavior, but it
is small due to the fast saturation of conducting channel
in the device.
From these results we can conclude that the I-V characteristics of an atomic junction is a complex phenomenon
in which the leads, eigenstates of the scattering region,
as well as impurity and disorder play major roles. In
particular, a tunnelling barrier created by impurities can
result in nonlinear I-V behavior of the nano-device. However, to be able to observe this eﬀect, conducting channels

diﬀerent as compared to the mono-atomic gold structure.
When the S atom is present, our calculations found that
all Bloch states are coupled to the scattering region, with
the highest transmitting mode still being the one corresponding to the conducting mode of the mono-atomic
gold system.
The I-V characteristics of the S doped Au nanocontacts is shown in Fig.5. We note that the I-V curve
for voltages up to 0.5V is very similar to the experimental
values with nonlinearity onset at non zero bias voltage.
We also note that the huge current plateau of the pure
Au device has now diminished because more states contribute to electronic transport. To compare these results
with the experimental measurement, we have used the
normalized functional form of the voltage, f0 (V ) shown
in Fig2-b, and multiplied it by the simulated zero bias
conductance (0.67G0 ). The result is shown as open circles in Fig.5. The qualitative and quantitative agreement between the theoretical and experimental results is
rather encouraging. In fact, this is the ﬁrst time that
experimental nonlinear I-V characteristics could be compared so well and so directly with ab initio self consistent
calculations. It is also a very surprising result because
a single S impurity can qualitatively alter transport in
these nano-contacts from linear to nonlinear.
The I-V curve in Fig.5 still shows the small features
similar to those found in pure and perfect Au contacts
(presented in Fig.3). These ﬁne details of the calculated
I-V curve would result in diﬀerential conductance ﬂuctuations similar to the ones shown in the experimental
data of Fig.2-d and the pure Au device of Fig.3-b. However, these ﬁne features as calculated are wider and occur
at higher bias voltages than the experimentally observed
ones. The former can be attributed to the zero temperature we used in our calculation. The absence of these
ﬁne features at smaller voltages is attributed to the small
size of the leads used in our theoretical modeling: close
to EF there are just a few states so that abrupt variation
of T (E, Vb ) at smaller Vb is less probable, resulting in a
smoother I-V curve at low Vb .
To understand other possible factors which can aﬀect
I-V characteristics of nano-contacts we have studied the
eﬀect of a larger number of S impurities, disorder, and
their combined eﬀects. The results of these calculations
are presented in the following subsection.
C. Contacts with several impurities and disorder

Including more S impurities at the contact enhances
the tunnelling barrier and may give rise to a smaller current with a larger nonlinear behavior. The result with
replacing two Au atoms at the interface by S atoms is
plotted in Fig.6-A. Indeed, as expected the I-V curve for
this system shows a larger nonlinear character. The nonlinearity starts at Vb ∼ 0.17eV . It is actually more nonlinear than that of the experimental data; this is mainly

6

tively. Due to the conductance channel saturation effect in the scattering region, the transmission coeﬃcient
has a very weak energy dependence (roughly constant).
This behavior also emerges in the I-V curve which shows
a very weak nonlinear behavior. We conclude that the
roughly linear I-V curves of Fig.3-B and Fig.6-B,C are
due to very diﬀerent origins. In the latter case it is due
to the channel saturation eﬀect in the scattering region,
whereas in the former it is due to a compensation between the eﬀects of increasing energy and bias voltage
on T (E, Vb ).

in the device need to be present, otherwise transmission
saturation is reached at small voltages and a linear I-V
characteristics is seen. Formally, it is the transmission
coeﬃcient T (E, Vb ) that is of crucial importance when
analyzing the eﬀect of the eigenstates of the leads and
the device, as well as the lead-device coupling.
In the next section we determine the behavior of
T (E, Vb ) and we will address the following questions: Is
the voltage independence of T (E, Vb ) an adequate picture? Which T (E, Vb ) behavior would result in nonlinear I-V characteristics? Can the major characteristics of
I(Vb ) be qualitatively estimated from simple arguments
or does one always need to perform an extensive ab-initio
simulations?

V. DISCUSSIONS

We have already shown in the previous sections that
transport at the molecular level is a complex phenomenon. To understand these systems, careful experimental work which separates electronic eﬀects from structural relaxations, as well as detailed calculations which
include the eﬀects of the molecule, leads and their coupling are required. In this section we discuss and compare our ﬁndings with previously published theoretical
concepts and experimental results.
Experimental work reported by Costa-Kr¨mer et.
a
al.53,54 have shown clear nonlinear I-V characteristics in
Au nano-contacts starting at bias Vb = 0.1V , and its
origin was attributed to strong e-e interactions. To rule
out the impurity eﬀect, the authors53 have used scanning electron microscopy to analyze contacts of diameter
∼ 300nm. They also used energy dispersive X-ray analysis and determined a contamination concentration below detection sensitivity. Experimental cleanness checks
performed for the large nano-contacts (∼ 300nm), however, can not be extrapolated to junctions of few atoms
in size due to the exquisite chemical sensitivity as demonstrated by our model. In other experiments55 with Au
relays, it was observed that conductance quantization
histogram survived for even larger bias, (1 × Go peak
persists for Vb ∼ 1.8V ). These data show clearly that in
such an experiment nonlinear I-V behavior cannot occur
at low bias (Vb ≃ 0.1V ). We believe that these junctions were formed between atomically clean gold contacts. These experiments were performed by forming
and breaking the contacts very quickly (∼ µ sc), thus
removing the impurity atoms from the Au junction even
if they existed67 . Recently, elegant experimental work by
Hansen et. al.56 found that contaminated Au nanostructure show nonlinear I-V characteristics, whereas experiments done with clean tip-sample in UHV show perfect
linearity for Vb < 0.7V . The nature of the contamination
was not determined.
A fundamental question that is to be addressed in this
section, is transport through an impurity the most likely
physical picture that can explain the observed nonlinear I-V characteristics. In the following we compare and
discuss some of the concepts described in the literature

IV. BEHAVIOR OF TRANSMISSION
COEFFICIENT T (E, VB )

For all the Au nano-contacts we have investigated theoretically, T (E, Vb = 0) increases as a function of E (for
E < 0.2eV ). A typical behavior is shown in Fig.7-a by
a dashed line for an ideal top site, pure and perfect Au
device. From this curve, it is clear that there is transmission enhancement as a function of E which should
result in a nonlinear I-V curve if T (E, Vb = 0) behaves
in the same way. In the same graph we have also plotted
T (E, Vb = 0.136V ). One can observe that the general energy dependence of E is still the same, namely increasing,
but there is a global shift of the curve downward. Therefore, an increase in transmission coeﬃcient as a function
of E is compensated by its decrease due to increased bias
voltage. The total eﬀect on the current is to produce a
linear I-V curve as shown in Fig.3-B. We also note that
this complete compensation between E and Vb is not universal and can be diﬀerent from one system to another.
In fact, even for the same device it can behave diﬀerently
at diﬀerent energy ranges. This results in diﬀerent features in the I-V curves such as plateaus, the “wiggles”,
etc. as we have discussed previously.
A similar analysis is done on a device with a S impurity. The results are shown in Fig.7-b. For this device
it is clear that the eﬀect of E on transmission at Vb = 0
is more pronounced. This is evidence that tunnelling is
an important factor. In addition to this, we note that
applying a bias voltage to the system causes a decrease
in the transmission coeﬃcient, but it is not a global decrease. In particular, a bias voltage of Vb = 0.136V actually increases transmission at small energies as shown in
Fig.7-b. Therefore, the combined eﬀect of E and Vb does
not cancel and it gives rise to the nonlinear I-V characteristics as seen in Fig.5. According to this analysis,
one can easily predict some aspects of the I-V curves just
by studying the transmission coeﬃcient T (E, Vb ) at two
diﬀerent bias voltages. Obviously, this helps to predict
I(Vb ) characteristics with much less computational eﬀort.
In Fig.7-c,d, we do a similar analysis on a device with
disorder and one with disorder plus impurity, respec7

that for these systems the resistance is due to contact
and thus the results are independent from the Luttinger
liquid behavior in the constriction. Therefore, in these
calculations69 transmission is found to be identical to
non-interacting particles. An important, single particle transmission, approximation is incorporated in this
model69 . However, at non-zero temperature or/and bias
voltage, a ﬁnite number of particles are injected into the
nano structure resulting in backscattering eﬀects. These
lead to charge accumulation at the interface, creating an
extra potential in addition to the original constriction potential. This additional potential is both Hartree (VH )
and exchange-correlation (Vxc ) in nature, and it is the
reason for the so called resistance dipole70 . Due to this
additional potential contribution, it was shown71 that for
1D systems the transmission coeﬃcient is renormalized.
It was further proposed that this charging eﬀect can even
close a conducting channel that is 90% transmissive54 .
This channel is gradually opened as Vb is increased for a
complete transmission at Vb ∼ 0.35V , thereby inducing
a nonlinear I-V curve. From a theory point of view, the
picture of charging induced nonlinear I-V characteristics
should overcome two further diﬃculties: that the renormalized transmission depends on an interaction parameter α which can not yet be determined for atomic wires;
and that previous calculations solved a 1D case with no
eﬀect of Vb on the transmission T (E, Vb ). In the rest of
this section, we follow the interesting idea of the charging eﬀect and analyze it in more detail to understand if
this eﬀect, which leads to channel closing53,54 can give
rise to nonlinear I-V curves of atomic devices. To start,
we follow the work of Yue et. al.71 by assuming a strong
interaction and write the renormalized transmission coeﬃcient as54,71 ,

relevant to this issue.
Free electron models have been used to describe the
behavior of nanostructures under external bias31,39 . In
these systems, depending on the potential proﬁle across
the device and the external bias voltage drop, various IV characteristics can be extracted. In passing, we note
that these calculations neglected the voltage dependent
coupling T (E, Vb ) (discussed in Fig.7). This is obviously
a major draw back for calculations at the molecular level,
as shown by our analysis. However, a voltage drop at the
contacts is still a reasonable approximation. In Fig.8,
we plot the Hartree potential across the four Au-atom
chain. It is clearly seen that the potential drops mostly
at the interfaces. However, assuming a uniform potential
across the constriction is not adequate due to the atomic
structure and the small variation of charge transfer as a
function of bias voltage.
A self consistent tight binding (TB) model has also
been implemented to ﬁnd the conduction dependence of
each eigenchannel in Au nano-contact as a function of
bias voltage68 . In these calculations, the TB parameters
are calculated from a bulk system and charge neutrality
of each atom of the nano-contact is enforced in order to
carry out the self consistent calculations. Although it
is not clear if the TB parameters determined from bulk
structures are directly transferable to nano-contacts with
atoms of low coordination number, especially when put
under a bias potential, our ab initio results show that
charge neutrality is a valid approximation for Au devices
without impurities since charge transfer is quite small. In
self consistent tight binding models, charge neutrality is
accomplished by locally adjusting the chemical potential.
For a zero bias calculation and nano-contacts with three
atoms, Ref. 68 shows that a local potential ∼ 3eV needs
to be added to the central atom to achieve charge neutrality. This seems to be a large value for Hamiltonian
correction. Therefore, we suggest that for pure Au nanostructures, zero bias ab initio calculations should be done
to extract the TB parameters (rather than from bulk).
These can then be used more safely with charge neutrality constraints to deduce I-V characteristics of nanocontacts. Since we have seen only small eﬀects of the bias
voltage on the atomic charge transfer for a S doped Au
structure in our ab initio calculations, we suggest that
for this particular structure it is possible to deduce the
charge distribution and the TB parameters from zero bias
calculations. Using these TB interaction parameters and
the zero bias charge at each atom as a constraint, the
I-V characteristics of these structures can then be solved
self-consistently. The results are less accurate compared
to a full ab initio calculation, but they should give better results than using the conventional TB parameters
derived from bulk systems.
There are other important theoretical calculations going beyond the single particle picture. Since the device
constriction is narrow, e-e interaction can be strong and
non-Fermi liquid behavior might have to be taken into
account. However, it was shown by Maslov and Stone69

T R (E) =

T0 (E/D0 )2α
R0 + T0 (E/D0 )2α

(4)

where, T0 , R0 are the transmission and reﬂection coeﬃcients of the non-interacting model such that T0 +R0 = 1,
and α is a parameter to describe e-e interaction in the
constriction. The parameter D0 is the energy range
near EF which contributes to renormalizing T0 and is
determined71 by D0 = ¯ vF /W , where vF is the Fermi
h
velocity of the system (108 cm/sc for Au) and W the
width of the nano-constriction54, W ∼ 10 − 20˚. ThereA
fore, we ﬁnd that for these devices D0 ∼ 1.0eV . From
Eq. 4, we compute the current I(Vb ) at zero temperature by integrating T R (E) from zero to eVb , assuming
no bias dependence of T R (E). The conductance is then
deduced by G = I(Vb )/Vb . We note that due to the ﬁnite length eﬀect of the constriction54 , L ∼ 100˚, the
A
renormalization eﬀect is cut oﬀ for bias voltages Vb < Vs ,
where Vs = 2π¯ vF /L ∼ 0.1D0 . Within this approach
h
and using the free parameters as speciﬁed, we have plotted in Fig.9-a the conductance of a single channel as a
function of bias voltage Vb . Qualitatively, the results
show that G increases with Vb due to the channel opening. However, quantitatively they do not give a complete
8

The self-consistently determined transmission coeﬃcient
T (E, Vb ) is shown to vary as a function of E and Vb .
This gives rise to diﬀerential conductance ﬂuctuations of
the order of 1% at temperature 300K taking into account the experimental averaging process. These ﬂuctuations are attributed mainly to the eﬀects of the lead
band structure. Most striking, however, is the fact that
a single impurity atom at the contact region can alter
I-V curves qualitatively in these devices: pure and perfect Au nano-contacts do not give the observed nonlinearity, while a sulfur doped device does. Importantly,
we have shown that the measured nonlinear I-V characteristics of Au nanowires can be quantitatively modeled
by impurity eﬀects which create a tunnelling barrier at
the nanostructure junction. This eﬀect, among others,
points to the vital importance for understanding contacts
in nano-electronic devices, and, perhaps, to exploit it for
the beneﬁt of device operation.
Acknowledgments: We gratefully acknowledge ﬁnancial support from NSERC of Canada and FCAR of
Quebec. H.M. thanks Dr. N. Agra¨ for a useful discusıt
sion on their experimental work.

channel opening at the experimental value of Vb ∼ 0.35V
(as suggested in Ref.54), if the channel is less than 10%
transmissive at Vb = 0. In fact, we found that a bias of
2V is needed to overcome the charging potential barrier,
thereby the nonlinearity in I-V curve can only set in at
much larger voltages. We also note that since the extra
charge due to backscattering is accumulated at the interface which has a larger cross section, we expect that
DFT and the local density approximation to Vxc should
work well. Our self-consistent calculations, with all the
charge transfer and re-arrangements accounted for, have
already partially included the backscattering eﬀects. Our
results show that it is possible to partially close a channel ∼ 60%–but not completely. Our experimental data
which shows zero bias conductance of ∼ 2.2Go , indicate
that at least one channel is 20% transmissive and that
channels are only partially closed, consistent with our theoretical work. Therefore, we can conclude at this point
that the physical picture of electron interactions to completely close and open a channel as a function of Vb , while
interesting, cannot explain our experimental data.
To further address the importance of backscattering
eﬀects71 , especially for our devices with realistic atomic
leads, we have investigated the dependence on the interaction parameter α in Eq.(4). Tuning this parameter can
dramatically change the renormalized transmission coefﬁcient as a function of Vb as shown in Fig.9-b. Therefore we have a wide range of this parameter for ﬁtting
our experimental data. However, if we change the Au
nano-contact by only a single sulphur atom, we should
not expect a very diﬀerent e-e interaction parameter α,
thereby this would predict an I-V curve not too diﬀerent
from the one without the sulphur. This suggests that
strong interaction alone would not predict strong sensitivity of the I-V curve dependence on small amount of
impurities, in contradiction to experimental results55,56 .
Transmission renormalization is important in 1D scattering problems where there is no charge in the leads. For
realistic atomic leads, the backscattered charge is a small
fraction of the original one, and its eﬀect should be well
screened. With all these considerations, we believe it is
unlikely that the dynamically established charging eﬀect
alone is large enough to cause the observed I-V nonlinearity for Au nano-contacts.

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16

10

FIG. 6. I-V characteristics of Au contacts with 2 S impurity
(A), disordered interface (B) and disordered interface with 1
S impurity (C). The insets illustrate the corresponding structures, with S represented as a dark circle; Au atoms in the
lead and the chain are shown with grey and light grey circles,
respectively. (A), (B) and (C) have zero-bias conductance of
0.55G0 , 0.915G0 and 0.833G0 , respectively.

Rev. B 49, 1966 (1994).

FIG. 1.
(a)-Schematic of the experimental set-up with
a mechanical controllable break junction (MCBJ) shown at
the centre with Au wires attached. (b)-DC conductance I/V
(lower) and diﬀerential conductance δI/δV (upper) plotted
vs V; Inset is the I-V curve. The dashed line, with slope
≃ 2.2G0 , corresponds to linear behavior and helps to recognize nonlinearity. These experimental results are measured
over a 5 second voltage sweep from positive (dark) to negative
(grey) bias.

FIG. 7. Scaled transmission coeﬃcient T (E) at Vb = 0
(dashed line) and Vb = 0.136V (solid line) for ideal contact
structure at the top site (a), 1 S impurity structure (b), disordered interface (c) and disordered interface with 1 S impurity
(d). The scaling factor is T (E = 0, Vb = 0).

FIG. 2. (a)-The Shape function (S,Eq.2) vs V calculated
from the data shown in Fig.1-b. (b)-Normalized functional
form of the voltage f0 (V ) with linear dashed line shown to
help view the onset of non linearity. (c)- Unbiased Conductance g(X) vs V . (d)-Normalised conductance, δf /δV vs V ;
with the inset corresponding to a magniﬁed section of the
same data to show more clearly ﬁne details. In all graphs,
dark (grey) lines correspond to positive (negative) bias.

FIG. 8. (a) Average Hartree potential (VH ) across the central cross section (∼ 9˚2 ) of the ideal contact at the top site;
A
dotted line at zero bias, and dashed line at Vb = 0.5V ; and
(b) their diﬀerence showing that most of the potential drops
symmetrically at the interfaces. Black circles correspond to
atom positions along the constriction for guidance

FIG. 9. Conductance calculated from renormalized transmission due to backscattering as a function of bias voltage; (a)
for T0 = 0.99, 0.9, 0.8 corresponding to solid, dashed and dotted line, respectively with interaction parameter α = 1.0 and
(b) for T0 = 0.9 and interaction parameter α = 0.1, 0.5, 1.0
corresponding to solid, dashed and dotted line, respectively.

FIG. 3. I-V characteristics of Au contacts with Hollow site
registry (A) and Top site registry (B); with zero-bias conductance 0.94G0 , 0.8G0 for (A) and (B), respectively. The
structure of the device is illustrated in (a) where LL, D and
RL correspond to Left Lead, eﬀective Device and Right Lead,
respectively. The Hollow site is shown in (c) and Top site (d)
where chain atoms are illustrated by dark circles. Inset (e)
shows the band structure of the lead along the transport direction, the conducting band is shown by solid line. Inset (b)
shows diﬀerential conductance ﬂuctuations of the Hollow site
as a grey dotted line, dark dotted line and dark continuous
line for T = 0K, T = 300K and for T = 300K with 4 mVrms
modulation voltage taken into account; respectively.

FIG. 4. Contour plots of surface charge density at particular bias voltage Vb (V ) and Energy E(eV ) indicated on
each ﬁgure. We use the same scale for all graphs to compare
the conductance probability for each conﬁguration. Dark circles correspond to atom positions along the constriction for
guidance.

FIG. 5. I-V characteristics of Au contact doped with S impurity (Solid line). The dashed line has a slope ≃ 0.67G0
and is shown to help view the onset of non linearity; Circles correspond to the experimental results obtained from
f0 (V ) (Fig.2-b) and multiplied by the zero bias conductance,
0.67G0 . The inset (a), shows the atoms registry, where dark
circles show the atomic chain, grey ones are Au atoms in the
leads’ surface and light grey is the S atom.

11

3.6

(a)

(b)

120

3.4

Conductance [Go]

80

3.2

60

I[µA]

100

40

3

20
0

2.8

2.6

0.1 0.2 0.3 0.4 0.5

δI/δV

2.4

I/V
2.2

0

0.1

0.2

0.3

Vb [V]

Fig.1

0.4

0.5

0.6

60
(A)

1.3

0.6

(a)

LL

40

1.2

D

RL

1.4

δI/δV [G0]

1.1

20
1

(a)
0

f0(V) [V]

0.4

I[µA]

S [Unit less]

0.5

0.3

1.2

0.2

1
0.8

0.1

(b)

0.6

0 1.1
1.12
0 Vb[V] 0.2 0.3 0.4 0.5 0.6
0.1

0.1 0.2 0.3 0.4 0.5 0.6

0

2.5

0

0.25

(b)

1.6
0.75

0.5

1

1.25

1.14

60

1.5

(B)

1.12

δf/δV [Unit less]

2.3

40

1.5

(c)

20
0

1.1

1.3
0.24

0.28

1.2
1.1
1

1

(d)

0.1 0.2 0.3 0.4 0.5 0.6 0.5

Vb [V]

Fig.2

1.4

(e)

E [eV]

2.2

2.1

(d)

(c)

I[µA]

g(X) [Go]

2.4

0

Vb [V]

0
0

0

0

0.25

0.5

1

0.75

Vb [V]
Fig.3

0.1 0.2 0.3 0.4 0.5 0.6

kza

2

3

1

1.25

V =0 , E=0
b

(b)

X-axis

(a)

V =0 , E=0.46
b
Z-axis (Transport direction)

(c) Y-axis

V =0 , E=0.68
b

Fig.4

(d)

V =0.68 , E=0.68
b

80
(a)

I[µA]

60

40

20

0

Fig.5

0

0.25

0.5

0.75

Vb [V]

1

1.25

80

80

80

(A)
(a)

70

(B)
(b)

70

(C)
70

60

60

50

50

50

40

40

40

30

I[µA]

60

30

(c)

30

20

20

(a’)

10
0

20

(b’)

10

0

Fig.6

0.25 0.5 0.75
Vb [V]

1

1.25

0

(c’)

10

0

0.25 0.5 0.75

Vb [V]

1

1.25

0

0

0.25

0.5

0.75

Vb [V]

1

1.25

1.2

1.2

Vb=0.0V
Vb=0.136V

1.15

1.15
1.1

1.05

1.05

1

1

0.95

T[E]

1.1

0.95

(a)
0.9

0

0.05

0.1

(b)
0.15

0.9

1.05

1

1

0.95

0.15

1.1

1.05

0.1

1.15

1.1

0.05

1.2

1.15

T[E]

1.2

0

0.95

(c)
0.9

0

0.05

E[eV]

Fig.7

0.1

(d)
0.15

0.9

0

0.05

E[eV]

0.1

0.15

8
7

VH[eV]

6
5
4
3
2
1

(a)

0
0.5

VδH[eV]

0.4
0.3
0.2
0.1

(b)
0

Z−axis (Transport direction)

Fig.8