# Quantum transport properties of tantalum-oxide resistive switching filaments

Tímea Nóra Török  Department of Physics, Institute of Physics, Budapest
University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest,
Hungary.  Institute of Technical Physics and Materials Science, Centre for
Energy Research, Konkoly-Thege M. út 29-33, 1121 Budapest, Hungary.  Péter
Makk  Department of Physics, Institute of Physics, Budapest University of
Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary.  MTA-
BME Correlated van der Waals Structures Momentum Research Group, Műegyetem
rkp. 3., H-1111 Budapest, Hungary.  Zoltán Balogh  Department of Physics,
Institute of Physics, Budapest University of Technology and Economics,
Műegyetem rkp. 3., H-1111 Budapest, Hungary.  ELKH-BME Condensed Matter
Research Group, Műegyetem rkp. 3., H-1111 Budapest, Hungary.  Miklós Csontos
Institute of Electromagnetic Fields, ETH Zurich, Gloriastrasse 35, 8092
Zurich, Switzerland.  András Halbritter∗ Department of Physics, Institute of
Physics, Budapest University of Technology and Economics, Műegyetem rkp. 3.,
H-1111 Budapest, Hungary.  ELKH-BME Condensed Matter Research Group, Műegyetem
rkp. 3., H-1111 Budapest, Hungary.

###### Abstract

Filamentary resistive switching devices are not only considered as promising
building blocks for brain-inspired computing architectures, but they also
realize an unprecedented operation regime, where the active device volume
reaches truly atomic dimensions. Such atomic-sized resistive switching
filaments represent the quantum transport regime, where the transmission
eigenvalues of the conductance channels are considered as a specific device
fingerprint. Here, we gain insight into the quantum transmission properties of
close-to-atomic-sized resistive switching filaments formed across an
insulating Ta2O5 layer through superconducting subgap spectroscopy. This
method reveals the transmission density function of the open conduction
channels contributing to the device conductance. Our analysis confirms the
formation of truly atomic-sized filaments composed of 3 – 8 Ta atoms at their
narrowest cross-section. We find that this diameter remains unchanged upon
resistive switching. Instead, the switching is governed by the redistribution
of oxygen vacancies within the filamentary volume. The set/reset process
results in the reduction/formation of an extended barrier at the bottleneck of
the filament which enhances/reduces the transmission of the highly open
conduction channels.

Keywords: random matrix theory, superconducting subgap spectroscopy, resistive
switching, memristor

##  I Introduction

Resistive switching (RS) memory devices (a.k.a. “memristors”) are identified
as a major hardware platform enabling the power-efficient implementation of
artificial intelligence. Pioneering applications employ large memristor
crossbar arrays in artificial neural networks, which implement machine
learning approaches at the hardware level at unprecedented speed and energy
efficiency. [1, 2, 3, 4] The neuromorphic functionalities of single memristors
arise from the dynamical properties of a previously unavailable operation
regime, where the active device volume can reach truly atomic dimensions. [5,
6, 7, 8, 9] This ultra-small active region is manifested by close-to-atomic-
sized metallic filaments which can be formed or ruptured by appropriate
voltage signals. Such “artificial synapses” are utilized to encode synaptic
weights through their analog tunable filamentary conductance.

Beyond the synaptic functionalities of RS devices, the _metallic_ nature of
the filamentary volume yields a completely different _quantum transport
regime_ compared to semiconductor heterostructures: (i) The large electron
density ($\approx 1$ electron/lattice atom) results in orders of magnitude
larger Fermi energies ($E_{F}^{\mathrm{metal}}\approx 3-10\,$eV [10]) than in
semiconducting systems. Therefore, quantum transport is realized at room
temperature due to the $E_{F}\gg k_{B}T$ condition. (ii) Unlike in
semiconductor heterostructures, the metallic Fermi wavelength
($\lambda_{F}^{\mathrm{metal}}\approx 0.3-0.6\,$nm) falls in the regime of the
lattice constant. Accordingly, the electrons experience a potential landscape
which is rough on the scale of their wavelength. This results in far more
complex quantum transport properties than universal conductance quantization
in semiconductor quantum point contacts, [11] where a smooth, transparent
quantum channel is realized. This complex quantum transport behavior has been
widely investigated in pure metallic atomic-sized nanowires, [12, 13, 14] but
has remained poorly explored in filamentary RS devices.

In this paper we investigate the quantum transport properties of RS filaments
formed across a Ta2O5 thin film due to the voltage induced redistribution of
oxygen vacancies.[15, 16] Ta2O5 is a prominent representative of transition
metal oxides which are widely considered as a competitive material platform
for neuromorphic hardware architectures. [17, 18, 19, 20, 21, 22, 23] In order
to track the atomistic changes in the filament structure taking place upon
resistive switching, we apply superconducting subgap spectroscopy.[24, 14, 25,
12, 26, 5] This method enables the analysis of the distribution of the
$\tau_{i}$ quantum transmission eigenvalues of the conduction channels
contributing to the conductance of the RS filament. The total filamentary
conductance is obtained via the Landauer formula [27] as
$G=G_{0}\cdot\sum_{i=1}^{M}\tau_{i}$, where $G_{0}=2e^{2}/h$ is the quantum
conductance unit and $M$ is the number of open conductance channels. We focus
on RS filaments, where $M$ is sufficiently large so that the transmission
eigenvalues can be described by a probability density function
$\rho^{\textrm{Ta}_{2}\textrm{O}_{5}}(\tau)$. We determine
$\rho^{\textrm{Ta}_{2}\textrm{O}_{5}}(\tau)$ in the high (HCS) and low (LCS)
conductance states and compare to that of pure Ta nanowires,
$\rho^{\textrm{Ta}}(\tau)$. Whereas a close-to-atomic-scale metallic nanowire
is responsible for the conduction in both systems, the formation and the
environment of this metallic wire is fundamentally different. The reference Ta
nanowires represent a pure metallic system which is thinned by mechanical
elongation in vacuum conditions. In contrast, the oxide-based RS filaments are
embedded in an insulating, amorphous Ta2O5 matrix. The evolution of the RS
junction relies on the voltage-induced displacement of oxygen vacancies, and
thereby the formation/destruction of oxygen-deficient conducting filamentary
regions. Consequently, our analysis reveals a fundamentally different
conductance channel distribution for these two systems. In pure Ta nanowires
$\rho^{\textrm{Ta}}(\tau)$ resembles the universal probability density
function $\rho^{\mathrm{RMT}}(\tau)\sim\tau^{-1}(1-\tau)^{-1/2}$ in a broad
diameter range. The latter formula was derived for mesoscopic diffusive wires
by random matrix theory.[28] In this case point-like scattering centers are
considered which are, however, bypassed by certain conductance channels
yielding almost perfect transmission ($\tau_{i}\approx 1$) for a subset of the
channels. In contrast, in RS filaments the formation of an extended barrier
across the narrowest section of the filament is revealed upon the reset
transition while the filament diameter is hardly affected. This barrier is
attributed to the accumulation of oxygen ions. It reduces the transmission of
the highly opened conduction channels, yielding a completely different
transmission distribution than that of pure Ta nanowires.

##  II Results and Discussion

Experimental approach. For our comparative analysis we established atomic-
sized RS filaments in Ta2O5 and pure Ta atomic wires by an STM (scanning
tunneling microscopy) point contact setup and a mechanically controllable
break junction (MCBJ) setup, respectively (see Fig. 1a,c). In order to apply
the method of superconducting subgap spectroscopy, both systems were cooled to
a base temperature of $T\approx 1.3\,$K, i.e. well below the superconducting
phase transition of tantalum ($T_{C}=4.48\,$K). The current-voltage [$I(V)$]
characteristics of a statistical ensemble of junctions were recorded in the
superconducting state and fitted by simulated $I(V)$ curves relying on model
transmission distributions. In the following, we first describe our
experimental approach. Next, we discuss the applied model transmission
distributions and the fitting procedure. Finally, we describe and discuss the
results of our analysis.

Figure 1: Characterization of RS filaments in Ta2O5 and pure Ta atomic wires.
(a) Illustration of a Ta2O5 RS device realized in an STM point-contact
arrangement. (b) Representative room temperature RS current-voltage
characteristics using a conventional PtIr tip. The hysteretic switching trace
opens up as the amplitude of the triangular driving voltage signal is
increased (see the color-scale). The arrow indicates the direction of the
hysteresis. The inset shows the low-bias conductance of the HCS (blue) and LCS
(red) at each voltage amplitude demonstrating multilevel programming (average
and standard deviation of 10 consecutive switching cycles are shown for each
voltage). (c) Illustration of pure Ta atomic wires established by mechanical
rupture using the three point bending arrangement in a MCBJ setup. (d) A
representative conductance histogram of pure Ta nanowires obtained by
collecting thousands of conductance versus electrode separation traces during
the repeated opening and closing of the junction. The cryogenic vacuum at
liquid helium temperature grants the extreme cleanliness during the in-situ
rupture. (e) An exemplary conductance versus electrode separation trace.

RS was studied in an STM point-contact setup, where PtIr or Ta tips were
approached and touched to Ta2O5/Ta thin film samples in a controlled manner.
The PtIr tips were applied in routine room temperature characterization
measurements, whereas Ta tips were used in the low temperature measurements to
have superconducting electrodes at both sides of the RS filament. The
resistive switching layer was prepared by sputtering $\sim$30 nm thick
stoichiometric Ta2O5 on top of a $30\,$nm thick Ta layer (see Fig. 1a). The
approach of the STM tip was stopped at a setpoint resistance of
$1\,$G$\Omega$. Afterwards, the $I(V)$ characteristics with typical voltage
driving amplitudes of $V^{0}_{\mathrm{drive}}\approx 1-2\,$V readily exhibited
resistive switching without a dedicated high-voltage electroforming process.
The set process was terminated by a series resistor of
$R_{S}=300-3600\,\Omega$. Accordingly, the bias voltage on the sample is
calculated as $V_{\textrm{bias}}=V_{\textrm{drive}}-I\cdot R_{S}$. The details
of the switching were fine-tunable either by the gentle manipulation of the
STM tip or the variation of the driving voltage amplitude. Room temperature
resistive switching characteristics at fixed tip position and varying driving
amplitude are exemplified in Fig. 1b. Positive bias refers to a higher
potential acting on the thin film with respect to the STM tip. The inset
demonstrates the low-bias conductance values in the LCS and HCS of the
hysteretic $I(V)$ characteristics demonstrating fine analog tunability in the
$4~{}G_{0}-7~{}G_{0}$ range. Our measurements on several resistive switching
junctions focused on the similar, $\approx 1~{}G_{0}-15~{}G_{0}$ conductance
regime, where the quantum transport properties can be compared to those of
atomic-sized pure tantalum nanowires. Measurements utilizing Ta STM tips
exhibited similar characteristics both at room temperature (see Fig. S1 in the
SI) and at cryogenic temperatures (see Fig. 3 and its discussion). Note, that
the above conductance regime is also ideal for resistive switching synapses in
artificial neural networks, where high-accuracy hardware-based vector-matrix
multiplication operation requires linear $I(V)$ characteristics below the
resistive switching threshold voltage. [1, 2, 29]

As a reference system, we have established pure Ta atomic-sized wires in a
MCBJ arrangement. In this setup a notched Ta wire with $0.1\,$mm diameter is
broken in a three point bending configuration (see Fig. 1c) using the
combination of a coarse stepper motor actuation and a fine piezo control. To
characterize the such established Ta nanowires, thousands of conductance vs.
electrode separation traces (see an example trace in Fig. 1e) are recorded
along the repeated opening and closing of the junction. The conductance data
of these traces are used to construct a conductance histogram (Fig. 1d). Note,
that both the voltage controlled RS filaments (see Fig. 1b and a further
example covering the $\approx 1~{}G_{0}-3~{}G_{0}$ regime in Fig. S2 of the
SI) and the mechanically actuated pure Ta nanowires exhibit a fine analog
tunability of the conductance in the entire investigated conductance range
lacking any conductance quantization features.

The $I(V)$ characteristics of the superconductor (S) - nanoconstriction (c) -
superconductor (S) junctions were acquired according to the same protocol as
in our previous work on the subgap analysis of niobium-oxide resistive
switching filaments.[5] The latter work focused on the investigation of RS
filaments with truly single-atom cross-section dominated by a single
conductance channel. Here, we investigate somewhat larger RS filaments with
significantly more open conductance channels and analyze the probability
density of the transmission eigenvalues, $\rho(\tau)$. We exploit that in an
ScS junction the current contributions of multiple Andreev reflection
processes strongly depend on the corresponding transmission eigenvalues.
Consequently, the superconducting $I(V)$ characteristics provide a sensitive
diagnostic tool to investigate $\rho(\tau)$ (see the computed ScS $I(V)$
curves in Fig. S3 of the SI).

Figure 2: Model approach for the analysis of the quantum conductance channel
distributions. Panel (a) illustrates possible model geometries: (i) a
ballistic quantum wire (black frame), (ii) a diffusive wire without any
extended barrier (brown frame), (iii) a diffusive wire with weaker/stronger
extended barrier at the bottleneck (blue/red frame). The corresponding model
transmission density functions are illustrated in panel (b): the black Dirac
delta peaks illustrate the case of quantized conductance channels in a
ballistic quantum wire. The brown curve shows the $\rho^{\mathrm{RMT}}(\tau)$
density function for a diffusive wire. The gray curves including the blue/red
example curves represent the density functions in case of an extended barrier
with variable strength, as labeled by the $\alpha$ parameters. Panel (c) shows
the calculated ScS $I(V)$ curves at $G=10\,$G0 for $\rho^{\mathrm{RMT}}(\tau)$
(brown), for a filament with a weaker/stronger extended barrier (blue/red
curves corresponding to the blue/red density functions highlighted in panel
(b)), and for a highly transparent junction with perfectly open conductance
channels (black). In the latter case an ideally infinite zero bias slope is
expected. The apparent finite slope is a result of the finite energy
resolution included in our model. Panels (d) and (e) respectively illustrate a
pure Ta atomic wire and an oxide-based resistive switching filament. In (d)
the five colored lines connecting two neighbor atoms illustrate the typical
transport through 5 non-vanishing inter-atomic conductance channels in a
$d$-valent metal. The transmission eigenvalues of such 5 inter-atomic channels
strongly scatter between zero and unity transmission, giving rise to a
diffusive-like transport. In the oxide-based RS filaments (e) the relocation
of oxygen atoms play a key role in the switching process, also influencing the
transport properties. The eventual formation of an extended barrier (as
illustrated by the orange region) is attributed to the voltage-controlled
accumulation of oxygen.

Model analysis of the transmission distributions. Our modeling approach for
the fitting of the superconducting $I(V)$ characteristics is illustrated in
Fig. 2. As a reference, we rely on the
$\rho^{\mathrm{RMT}}(\tau)\sim\tau^{-1}(1-\tau)^{-1/2}$ universal probability
density function of the transmission eigenvalues (see brown curve in Fig. 2b).
This density function was derived for phase coherent mesoscopic diffusive
wires with a large number of open conductance channels ($M\gg 1$). The
corresponding model system is illustrated on the second panel of Fig. 2a
(brown frame), where the gray background symbolizes the diffusive nature of
the transport. The RMT density function is bimodal:
$\rho^{\mathrm{RMT}}(\tau)$ is peaked at both the low-transmission ($\tau\ll
1$) and high-transmission ($\tau\approx 1$) ends while the intermediate region
also exhibits a finite weight. The presence of completely open conductance
channels, which was indeed confirmed by shot noise measurements,[30] seems
counter-intuitive due to the diffusive nature of the transport, i.e., the
short mean free path compared to the system size. However, it can be shown,
that point-like defects can be bypassed by certain conductance channels,
yielding almost perfect channel transmission.[28] It was recognized, that this
universal density function also well approximates the transmission
distribution in Pb atomic-sized wires,[31] where the disordered nature is
attributed to the interplay of non-perfect crystalline ordering, non-perfect
transmission especially through the highly oriented $p$ orbitals, and surface
scattering. We anticipate similar behavior in transition metals, like the
investigated pure Ta nanowires, where the electron transmission through two
neighbor atoms is shared among $\approx 5$ conductance channels, as
illustrated by the 5 colored lines between two neighbor atoms in Fig. 2d. This
expectation relies on the tight-binding model of a single-atom nanowire[32,
13] identifying the number of channels with the number of valence orbitals at
the central atom (i.e. 1 $s$-type and 5 $d$-type channels in a transition
metal), and predicting the closing of one channel due to symmetry reasons.
This specific behavior was experimentally verified in Nb and Ta junctions,
which have very similar electron structure and conduction properties.[14, 26]
Note, that the above considerations imply $\approx 10-15$ non-vanishing
conductance channels for a filament with only $2-3$ atoms in its narrowest
cross-section. Such a large channel number justifies the applicability of a
transmission density function.

Our goal is to experimentally verify the universal transmission density
function in pure Ta atomic wires, and to identify and understand possible
deviations from this in oxide-based RS filaments. Since the (re-)distribution
of oxygen impurities (see Fig. 2e) plays important roles both in the transport
properties and in the resistive switching process, such a quantitative
comparison will enable to draw conclusions about the atomic-scale evolution of
the filament upon resistive switching. To this end, we consider possible model
transmission density functions both with _more transmissive_ and _less
transmissive_ characteristics (see Figs. 2a,b). In the former case the most
extreme situation is illustrated: a fully ballistic junction (left panel in
Fig. 2a, where the white background symbolizes the ballistic nature). This
scenario corresponds to the case of universal conductance quantization, where
all open conductance channels exhibit perfect transmission, as demonstrated by
the black density function with Dirac-delta peaks at $\tau=0$ and $\tau=1$ in
Fig. 2b. The situation where the overall transmission is lowered compared to
the RMT case can be considered according to the model of Nazarov and
coworkers,[33] where an extended barrier is formed at the bottleneck of the
filament. This results in an enhanced back-scattering in all the channels,
yielding a suppression of the peak at $\tau\approx 1$ in the
$\rho^{\mathrm{RMT}}(\tau)$ transmission distribution. This model relies on
the series connection of a diffusive wire with $R_{W}$ resistance and an
extended barrier with $R_{B}$ resistance. The correspondingly derived
transmission density function depends on the $\alpha=R_{W}/R_{B}$ parameter,
as demonstrated by the gray curves (including the example curves highlighted
in red and blue) in Fig. 2b. The corresponding model systems are illustrated
on the right two panels of Fig. 2a, where the orange region symbolizes the
extended barrier. Note, that the $\alpha\rightarrow\infty$ limit recovers the
universal $\rho^{\mathrm{RMT}}(\tau)$ transmission density function.

In order to experimentally determine the relevance of the above described
scenarios in our devices, we compare the measured $I(V)$ traces to simulated
model $I(V)$ traces corresponding to a given transmission density function as
follows. By choosing a certain model transmission density function
$\rho(\tau)$ normalized to unity ($\int\rho(\tau)\mathrm{d}\tau$=1), the
average transmission probability of the open channels is determined as
$\overline{\tau_{\rho}}=\int\tau\rho(\tau)\mathrm{d}\tau$. Relying on this
average transmission, the number of open channels at a certain $G$ conductance
is estimated as
$M_{\rho,G}=\mathrm{round}\left((G/G_{0})/\overline{\tau_{\rho}}\right)$. The
transmission eigenvalues of an atomic filament/wire with $G$ conductance are
simulated by generating $M_{\rho,G}$ random numbers according to the chosen
$\rho(\tau)$ probability density function. Having these random transmission
eigenvalues at hand, the $I(V)$ curve of the hypothetical ScS junction
characterized by the particular $\tau_{i}$ values is simulated according to
the multiple Andreev reflection theory.[34, 35, 31] For the sake of a
meaningful comparison to experimental $I(V)$ data, an ensemble averaging is
performed by repeating the above approach for a large number of independent
sets of random transmission eigenvalues. Thereby an average ScS $I(V)$
characteristic can be modeled.

Additionally, our analysis also takes into account the finite energy
resolution of the measurements arising from the proximity effect[5] by
smoothing the modeled $I(V)$ curves with a Gaussian kernel according to the
actual energy resolution (see further details in Fig. S3 of the SI).
Furthermore, our numerical analysis applies a bottom cutoff at $\tau=0.03$,
i.e., channels with smaller transmission are not considered as open. With this
physically reasonable choice $\rho^{\mathrm{RMT}}(\tau)$ yields
$\overline{\tau_{\rho}}=0.39$, which reproduces the expected $M=5$ open
channels for a single-atom junction corresponding to the conductance histogram
peak in Fig. 1d. The value of $\overline{\tau_{\rho}}$ and $M$ obviously
depend on the choice of the cutoff, but the results of the subgap analysis are
insensitive to the precise cutoff value (see Fig. S4 of the SI).

Figure 2c shows the such simulated $I(V)$ characteristics for hypothetical ScS
junctions with $G=10\,G_{0}$ conductance and various transmission density
functions. The brown curve illustrates the case of the universal
$\rho^{\mathrm{RMT}}(\tau)$ density function corresponding to the brown curve
in Fig. 2b. The blue and red curves correspond to junctions including extended
barriers with $\alpha=0.62$ and $\alpha=0.12$, respectively (see the blue and
red $\rho(\tau)$ curves in Fig. 2b). As a reference, the black curve
corresponds to a junction satisfying universal conductance quantization, where
all the open conductance channels have $\tau_{i}=1$ transmission. The ScS
$I(V)$ trace shows markedly different behavior for the above, fundamentally
different transmission densities. By normalizing the $I(V)$ curves with the
$\Delta$ superconducting gap and the $G_{N}$ normal state conductance (see the
axes in Fig. 2c) a unity slope line is obtained in the normal state for any
filament type. Superconductivity, however, enhances the contribution of open
channels ($\tau_{i}\approx 1$) and suppresses the contribution of weakly
transmitting channels ($\tau_{i}\ll 1$). Accordingly, a junction with highly /
poorly transmitting channels (black / red) exhibits a considerably larger /
smaller slope than unity around zero bias ($eV\ll 2\Delta$). As another
dominant difference, the highly transmitting junction also exhibits a
considerable excess current, i.e., the high-bias linear section of the $I(V)$
curve extrapolates to a significant positive current at zero bias.[36] The
universal $\rho^{\mathrm{RMT}}(\tau)$ density yields an $I(V)$ curve in-
between these two extremities, with moderately larger than unity slope at zero
bias, and a moderate excess current at high bias.

Results of the subgap spectroscopy measurements. Figure 3 displays the subgap
analysis of the transmission density functions based on the ScS $I(V)$ curves
measured on RS filaments (Fig. 3c) and pure Ta atomic wires (Fig. 3d). As a
reference, first the data of the latter system is analyzed. The dotted line
shows the measured ScS $I(V)$ curve, which very much resembles the simulated
$I(V)$ curve according to the universal $\rho^{\mathrm{RMT}}(\tau)$ density
function (brown dashed line). A slightly better agreement is obtained
according to Nazarov’s theory,[33] using the best fitting $\alpha=1.6$
parameter (green dashed line). The corresponding transmission density function
(green line in Fig. 3b) is indeed positioned very close to the
$\rho^{\mathrm{RMT}}(\tau)$ universal density function (brown curve in Fig.
3b). This analysis supports our preconception, that Ta atomic wires are well
described by the universal transmission density function. For this particular
junction the $G=8.33\,$G0 conductance and the universal transmission density
function yields $M\approx 21$ open channels, indicating a junction with
$\approx 4$ atoms at the narrowest cross-section according to the $5$
channels/atom estimate.

Figure 3: Subgap analysis of the conductance channels in a representative RS
filament and a pure Ta atomic wire. (a) Typical resistive switching curve of
Ta(tip)/Ta2O5/Ta STM junctions at cryogenic temperature. The arrow indicates
the direction of the hysteresis. (b) The red, blue and green lines demonstrate
the best fitting transmission densities for the LCS and HCS of the RS filament
in panel (c) and for the pure Ta atomic wire in panel (d), respectively. As a
reference, the light brown curve shows $\rho^{\mathrm{RMT}}(\tau)$. (c) ScS
$I(V)$ curves for there HCS (top dotted line, $G=4.03\,$G0) and the LCS
(bottom dotted line, $G=2.02\,$G0) of the RS $I(V)$ curve in panel (a). The
blue and red curves represent the best fitting model $I(V)$ curves, wheres the
light brown curve shows the model $I(V)$ curve for $\rho^{\mathrm{RMT}}(\tau)$
as a reference. For all the colored curves the light colored background region
represents the standard deviation of the $I(V)$ curves for the same density
function, but various random transmission sets. (d) The measured ScS $I(V)$
curve (dotted line, $G=8.33\,$G0) and the best fitting model $I(V)$ curve
(green dashed line) for the reference pure Ta atomic wire. The measured $I(V)$
curve very much resembles the model $I(V)$ curve for
$\rho^{\mathrm{RMT}}(\tau)$ (light brown dashed line).

Figure 3a exemplifies a cryogenic temperature ($T\approx 1.3\,$K) resistive
switching curve, where the HCS and LCS are shown in blue and red colors,
respectively. Resistive switching occurs on the voltage-scale of $1\,$V,
whereas the superconducting features are detected on a much smaller voltage
scale, $V\approx 2\Delta/e\approx 1.33\,$mV. [37] The top and bottom dotted
lines in Fig. 3c show the measured ScS $I(V)$ curves in the latter voltage
range for the HCS and the LCS, respectively. The blue and red lines represent
the best fitting model $I(V)$ curves according to Nazarov’s theory, using
$\alpha=0.51$ (HCS, blue curve) and $\alpha=0.12$ (LCS, red curve). The
corresponding transmission density functions are shown in Fig. 3b by blue and
red lines, respectively. These density functions, especially the red curve for
the LCS significantly differ from the light brown universal density function,
$\rho^{\mathrm{RMT}}(\tau)$. This significant difference is also demonstrated
on the level of the ScS $I(V)$ curves in Fig. 3c. The brown curve shows the
reference average $I(V)$ curve corresponding to $\rho^{\mathrm{RMT}}(\tau)$
such that the light brown region marks the standard deviation of the $I(V)$
curve for different random number transmission sets. Similarly, the light blue
and light red regions for the blue and red model $I(V)$ curves demonstrate the
standard deviation of these $I(V)$ curves for different random number sets,
though for the red curve this standard deviation is so small, that it is not
visible on the curve. It is clear, that the deviation of the HCS and LCS
$I(V)$ traces from the reference brown curve calculated with
$\rho^{\mathrm{RMT}}(\tau)$ is significantly larger than the standard
deviation of the model $I(V)$ curves.

Figure 4a displays the results of the statistical analysis performed for a
larger amount of oxide-based RS filaments (blue circles for the HCS and red
circles for the LCS) as well as for several pure Ta atomic wires (green
triangles) covering the conductance range of $\approx 1-13\,$G0. The $\alpha$
parameter of the model density functions is extracted by fitting the ScS
$I(V)$ curves likewise in Fig. 3. The black curve represents a quantitative
boundary as a function of the conductance. Below this boundary the standard
deviation of the best fitting model $I(V)$ curve does not overlap with the
standard deviation of the model $I(V)$ curve corresponding to
$\rho^{\mathrm{RMT}}(\tau)$. That is, below this boundary line the density
function best describing the measured $ScS$ $I(V)$ curve significantly differs
from $\rho^{\mathrm{RMT}}(\tau)$. Figure 4 evidences that all the measurements
on pure Ta atomic wires / oxide-based RS filaments show non-significant /
significant difference compared to the $\rho^{\mathrm{RMT}}(\tau)$ density
function. Furthermore, it is also clear that in the LCS of the RS filaments
(red circles) the deviation from $\rho^{\mathrm{RMT}}(\tau)$ is even more
pronounced than in the HCS (blue circles).

Figure 4: Analysis of the barrier strength for a statistical ensemble of pure
Ta nanowires and RS filaments. (a) The $\alpha$ values extracted from the ScS
$I(V)$ curve fitting are presented for multiple pure Ta atomic wires (green
triangles) as well as the HCS (blue dots) and LCS (red dots) states of various
RS filaments. The black line demonstrates the border, above/below which the
deviation from the $\rho^{\mathrm{RMT}}(\tau)$ universal transmission density
function is insignificant/significant. Panels (b) and (c) respectively
demonstrate the average transmission ratio and the channel number ratio as a
function of the conductance ratio for all the RS transitions in panel (a). The
orange/black dotted line illustrate the cases, where solely the
transmission/channel number changes during reset.

Discussion. Finally, we discuss the results of our superconducting subgap
analysis. First, we have confirmed our preconception, that the quantum
transport in pure Ta atomic-sized nanowires is well described by
$\rho^{\mathrm{RMT}}(\tau)$. Moreover, this universal transmission density is
valid in the entire investigated conductance range (see Fig. 4). As the Ta
wire is elongated, the number of the open channels decreases but the
characteristic distribution of the transmission eigenvalues and, thereby, the
$\overline{\tau_{\rho}}\approx 0.39$ average transmission is left similar. The
$\rho^{\mathrm{RMT}}(\tau)$ density function represents a broad variation of
the transmission eigenvalues, with some highly open channels. We attribute the
latter to the transmission through $s$-type orbitals, whereas the $d$-type
orbitals rather contribute to the broad variation of $\tau_{i}$. Despite the
few open channels, $\rho^{\mathrm{RMT}}(\tau)$ does not yield any sign of
conductance quantization features, as demonstrated by the simulated
conductance histogram in Fig. S4 of the SI. This is in sharp contrast to noble
metals and alkali metals, where a significantly more transmissive transmission
density function [38] is realized due to the dominance of the highly
delocalized $s$-electrons and, thus, conductance quantization is observed.[13,
14]

In filamentary RS devices the formation of atomic-sized metallic filaments is
anticipated, where quantum transport properties might resemble those of pure
metallic nanowires. Our analysis, however, highlights significant differences.
The superconducting excess current[36] and the leading subgap contribution to
the ScS $I(V)$ curve are quadratically suppressed for $\tau\ll 1$, i.e. these
superconducting features are extremely sensitive to the presence of
transparent channels, even if they represent a minority in the entire
distribution. This enables the precise identification of a weak barrier
described by $\alpha$, from which the average transmission values and the
corresponding number of open channels can be determined. For the
representative switching curve in Fig. 3a,c this analysis yields
$\overline{\tau_{\rho}}=0.25$ and $M=16$ in the HCS and
$\overline{\tau_{\rho}}=0.12$ and $M=17$ in the LCS. These results evidence a
markedly different scenario compared to pure Ta nanowires: in the RS filament
the reset transition leaves the number of open channels nearly unchanged.
Instead, the resistance change occurs due to a significant decrease in the
average transmission. Such a process is consistent with the atomistic picture
of the RS mechanism where the electric field driven redistribution of oxygen
ions at the filament bottleneck (see the schematic illustration in Fig. 2e)
results in a stronger / weaker transport barrier in the LCS / HCS whereas the
overall diameter of the filament stays similar. The unchanged filament
diameter and modulated oxygen vacancy concentration upon the reset transition
is also in qualitative agreement with continuum model simulations performed in
the limit of nanometer-scale RS filaments in TaOx-based memristors.[16] Our
analysis yields an estimate on the filament size assuming that the $5$
channels between two neighboring Ta atoms is a reasonable assumption for the
oxide-based RS filaments as well. According to this assumption, the HCS and
LCS in Fig. 3a,c correspond to an RS filament with $\approx 3$ Ta atoms in the
narrowest cross-section, with smaller/larger oxygen content in the HCS/LCS
bottleneck.

To perform similar analysis on various RS junctions, we plot the
$\overline{\tau}_{\rho}^{\mathrm{HCS}}/\overline{\tau}_{\rho}^{\mathrm{LCS}}$
(Fig. 4b) and the $M^{\mathrm{HCS}}/M^{\mathrm{LCS}}$ (Fig. 4c) ratios as a
function of the $G^{\mathrm{HCS}}/G^{\mathrm{LCS}}$ conductance ratio for the
RS data of Fig. 4a. The orange/black dotted lines illustrate the cases, where
solely the transmission/solely the channel number changes along the RS. It is
clear that the former process is dominant in the data, i.e. the reset
transition leaves the junction size similar, and rather the transmission
decreases significantly due to the formation of an oxygen-based barrier at the
bottleneck. Relying on the extracted $M$ values the conductance regime
displayed in Fig. 4a can be attributed to RS filaments composed of 3 – 8 Ta
atoms in their narrowest cross-section.

##  III Conclusions

In conclusion, we utilized superconducting subgap spectroscopy to reveal the
microscopic structure of atomic-sized oxide-based RS filaments in their high
and low conductance states. We experimentally determined the transmission
density function of the open conduction channels supported by the filamentary
region. Our results were analyzed in comparison to the reference system of
pure Ta atomic wires. In the latter case the conductance histogram lacks any
conductance quantization features due to the presence of partially
transmitting $d$ channels. Instead, the transmission density function exhibits
a broad distribution which is well described by the universal
$\rho^{\mathrm{RMT}}(\tau)$ density function. The presence of some highly
transparent channels is characteristic to this universal transmission
distribution, which is sensitively detected by superconducting subgap
spectroscopy. In sharp contrast, the highly transparent conduction channels in
RS filaments are blocked upon the reset transition. This implies that during
reset an extended potential barrier arises at the narrowest cross-section of
the filament due to the voltage-induced accumulation of oxygen ions. Meanwhile
the filament diameter is preserved at an estimated configuration involving a
few Ta atoms. This finding demonstrates that the reset process is
fundamentally different from the atom-by-atom thinning of metallic nanowires,
resulting in markedly different quantum transport properties. Furthermore, our
analysis demonstrates the merits of quantum transport measurements in the
microscopic understanding of atomistic processes in filamentary resistive
switching devices.

## Author Contributions

T. N. T. performed the RS measurements and all the data analysis. T. N. T. and
P. M. contributed to the measurements on pure Ta atomic wires. M. C. prepared
the Ta2O5 layer. Z. B. contributed to MCBJ sample preparation and supervised
the low-temperature measurements. A. H. conceived the idea of the project and
supervised the project. The manuscript was prepared by A. H., T. N. T. and M.
C. All authors contributed to the discussion of the results and the
manuscript.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

This research was supported by the Ministry of Culture and Innovation and the
National Research, Development and Innovation Office within the Quantum
Information National Laboratory of Hungary (Grant No.
2022-2.1.1-NL-2022-00004), and the NKFI K119797 and K143169 grants. T. N. T.
acknowledges the support of the KDP-2020 963575 Cooperative Doctoral Programme
of the Ministry for Innovation and Technology. Z.B. acknowledges the support
of the Bolyai János Research Scholarship of the Hungarian Academy of Sciences
and the ÚNKP-22-5 New National Excellence Program of the Ministry for
Innovation and Technology from the source of the National Research,
Development and Innovation Fund. The cryogenic measurements were supported by
the helium liquefaction plant of the BUTE (VEKOP 2.3.3-15-2017-00015). The
authors are grateful to G. Zaránd for useful discussions and to G. Rubio-
Bollinger for the fitting program based on the theory of multiple Andreev
reflections.

## References

  * Yang et al. [2013] Yang, J. J.; Strukov, D. B.; Stewart, D. R. Memristive devices for computing. _Nat. Nanotechnol._ 2013, _8_ , 13–24.
  * Zidan et al. [2018] Zidan, M. A.; Strachan, J. P.; Lu, W. D. The future of electronics based on memristive systems. _Nat. Electron._ 2018, _1_ , 22–29.
  * Ambrogio et al. [2018] Ambrogio, S.; Narayanan, P.; Tsai, H.; Shelby, R. M.; Boybat, I.; di Nolfo, C.; Sidler, S.; Giordano, M.; Bodini, M.; Farinha, N. C. P.; Killeen, B.; Cheng, C.; Jaoudi, Y.; Burr, G. W. Equivalent-accuracy accelerated neural-network training using analogue memory. _Nature_ 2018, _558_ , 60–67.
  * Xia and Yang [2019] Xia, Q.; Yang, J. J. Memristive crossbar arrays for brain-inspired computing. _Nat. Mater._ 2019, _18_ , 309–323.
  * Török et al. [2020] Török, T. N.; Csontos, M.; Makk, P.; Halbritter, A. Breaking the Quantum PIN Code of Atomic Synapses. _Nano Lett._ 2020, _20_ , 1192–1200.
  * Milano et al. [2022] Milano, G.; Aono, M.; Boarino, L.; Celano, U.; Hasegawa, T.; Kozicki, M.; Majumdar, S.; Menghini, M.; Miranda, E.; Ricciardi, C.; Tappertzhofen, S.; Terabe, K.; Valov, I. Quantum Conductance in Memristive Devices: Fundamentals, Developments, and Applications (Adv. Mater. 32/2022). _Adv. Mater._ 2022, _34_ , 2270235.
  * Aono and Hasegawa [2010] Aono, M.; Hasegawa, T. The Atomic Switch. _IEEE Proc._ 2010, _98_ , 2228–2236.
  * Cheng et al. [2019] Cheng, B.; Emboras, A.; Salamin, Y.; Ducry, F.; Ma, P.; Fedoryshyn, Y.; Andermatt, S.; Luisier, M.; Leuthold, J. Ultra compact electrochemical metallization cells offering reproducible atomic scale memristive switching. _Communications Physics_ 2019, _2_ , 28.
  * van Ruitenbeek et al. [2016] van Ruitenbeek, J. M.; Morales-Masis, M.; Miranda, E. In _Resistive Switching: From Fundamentals of Nanoionic Redox Processes to Memristive Device Applications Ch. 7. Quantum Point Contact Conduction_ ; Ielmini, D., Waser, R., Eds.; Wiley-VCH, Weinheim, 2016; pp 197–224.
  * Kittel [2005] Kittel, C. _Introduction to Solid State Physics, 8 th edition_; John Wiley & Sons, Inc., New York, 2005.
  * van Wees et al. [1988] van Wees, B. J.; van Houten, H.; Beenakker, C. W. J.; Williamson, J. G.; Kouwenhoven, L. P.; van der Marel, D.; Foxon, C. T. Quantized conductance of point contacts in a two-dimensional electron gas. _Phys. Rev. Lett._ 1988, _60_ , 848–850.
  * Agraït et al. [2003] Agraït, N.; Yeyati, A. L.; van Ruitenbeek, J. M. Quantum Properties of Atomic-Sized Conductors. _Phys. Rep._ 2003, _377_ , 81–279.
  * Cuevas and Scheer [2010] Cuevas, J. C.; Scheer, E. _Molecular electronics: an introduction to theory and experiment_ ; World Scientific Series in Nanoscience and Nanotechnology; World Scientific: Singapore, 2010.
  * Scheer et al. [1998] Scheer, E.; Agraït, N.; Cuevas, J. C.; Yeyati, A. L.; Ludoph, B.; Martín-Rodero, A.; Bollinger, G. R.; van Ruitenbeek, J. M.; Urbina, C. The signature of chemical valence in the electrical conduction through a single-atom contact. _Nature_ 1998, _394_ , 154–157.
  * Park et al. [2013] Park, G.-S.; Kim, Y. B.; Park, S. Y.; Li, X. S.; Heo, S.; Lee, M.-J.; Chang, M.; Kwon, J. H.; Kim, M.; Chung, U.-I.; Dittmann, R.; Waser, R.; Kim, K. In situ observation of filamentary conducting channels in an asymmetric Ta2O5-x/TaO2-x bilayer structure. _Nat. Commun._ 2013, _4_ , 2382.
  * Marchewka et al. [2016] Marchewka, A.; Roesgen, B.; Skaja, K.; Du, H.; Jia, C.-L.; Mayer, J.; Rana, V.; Waser, R.; Menzel, S. Nanoionic Resistive Switching Memories: On the Physical Nature of the Dynamic Reset Process. _Adv. El. Mater._ 2016, _2_ , 1500233.
  * Yang et al. [2010] Yang, J. J.; Zhang, M.-X.; Strachan, J. P.; Miao, F.; Pickett, M. D.; Kelley, R. D.; Medeiros-Ribeiro, G.; Williams, R. S. High switching endurance in TaOx memristive devices. _Appl. Phys. Lett._ 2010, _97_ , 232102.
  * Torrezan et al. [2011] Torrezan, A. C.; Strachan, J. W.; Medeiros-Ribeiro, G.; Williams, R. S. Sub-nanosecond switching of a tantalum oxide memristor. _Nanotechnology_ 2011, _22_ , 485203.
  * Strachan et al. [2011] Strachan, J. P.; Torrezan, A. C.; Medeiros-Ribeiro, G.; Williams, R. S. Measuring the switching dynamics and energy efficiency of tantalum oxide memristors. _Nanotechnology_ 2011, _22_ , 505402.
  * Lee et al. [2011] Lee, M.-J.; Lee, C. B.; Lee, D.; Lee, S.; Chang, M.; Hur, J.; Kim, Y.-B.; Kim, C.-J.; Seo, D.; Seo, S.; Chung, U.-I.; Yoo, I.-K.; Kim, K. A Fast, High-Endurance and Scalable Non-Volatile Memory Device Made from Asymmetric Ta2O5-X/TaO2-X Bilayer Structures. _Nat. Mater._ 2011, _10_ , 625–630.
  * Graves et al. [2017] Graves, C. E.; Dávila, N.; Merced-Grafals, E. J.; Lam, S.-T.; Strachan, J. P.; Williams, R. S. Temperature and field-dependent transport measurements in continuously tunable tantalum oxide memristors expose the dominant state variable. _Appl. Phys. Lett._ 2017, _110_ , 123501.
  * Rao et al. [2022] Rao, M.; Song, W.; Kiani, F.; Asapu, S.; Zhuo, Y.; Midya, R.; Upadhyay, N.; Wu, Q.; Barnell, M.; Lin, P.; Li, C.; .i Wang,; Xia, Q.; Yang, J. J. Timing Selector: Using Transient Switching Dynamics to Solve the Sneak Path Issue of Crossbar Arrays. _Small Science_ 2022, _2_ , 2100072.
  * Csontos et al. [2022] Csontos, M.; Horst, Y.; Olalla, N. J.; Koch, U.; Shorubalko, I.; Halbritter, A.; Leuthold, J. Picosecond Time-Scale Resistive Switching Monitored in Real-Time. _arXiv:2209.06732_ 2022,
  * Scheer et al. [1997] Scheer, E.; Joyez, P.; Esteve, D.; Urbina, C.; Devoret, M. H. Conduction Channel Transmissions of Atomic-Size Aluminum Contacts. _Phys. Rev. Lett._ 1997, _78_ , 3535–3538.
  * Ludoph et al. [2000] Ludoph, B.; van der Post, N.; Bratus’, E. N.; Bezuglyi, E. V.; Shumeiko, V. S.; Wendin, G.; van Ruitenbeek, J. M. Multiple Andreev reflection in single-atom niobium junctions. _Phys. Rev. B: Condens. Matter Mater. Phys._ 2000, _61_ , 8561.
  * Makk et al. [2008] Makk, P.; Csonka, S.; Halbritter, A. Effect of hydrogen molecules on the electronic transport through atomic-sized metallic junctions in the superconducting state. _Phys. Rev. B: Condens. Matter Mater. Phys._ 2008, _78_ , 045414.
  * Landauer [1970] Landauer, R. Electrical resistance of disordered one-dimensional lattices. _Philos. Mag._ 1970, _21_ , 863–867.
  * Beenakker [1997] Beenakker, C. W. J. Random-matrix theory of quantum transport. _Rev. Mod. Phys._ 1997, _69_ , 731–808.
  * Li et al. [2018] Li, C. et al. Analogue signal and image processing with large memristor crossbars. _Nat. Electron._ 2018, _1_ , 52–59.
  * Henny et al. [1999] Henny, M.; Oberholzer, S.; Strunk, C.; Schönenberger, C. 1/3-shot-noise suppression in diffusive nanowires. _Phys. Rev. B: Condens. Matter Mater. Phys._ 1999, _59_ , 2871–2880.
  * Riquelme et al. [2005] Riquelme, J. J.; de la Vega, L.; Yeyati, A. L.; Agraït, N.; Martin-Rodero, A.; Rubio-Bollinger, G. Distribution of conduction channels in nanoscale contacts: Evolution towards the diffusive limit. _Europhysics Letters (EPL)_ 2005, _70_ , 663–669.
  * Cuevas et al. [1998] Cuevas, J. C.; Yeyati, A. L.; Martín-Rodero, A. Microscopic Origin of Conducting Channels in Metallic Atomic-Size Contacts. _Phys. Rev. Lett._ 1998, _80_ , 1066–1069.
  * Nazarov [1994] Nazarov, Y. V. Limits of universality in disordered conductors. _Phys. Rev. Lett._ 1994, _73_ , 134–137.
  * Cuevas et al. [1996] Cuevas, J. C.; Martín-Rodero, A.; Levy Yeyati, A. Hamiltonian approach to the transport properties of superconducting quantum point contacts. _Phys. Rev. B: Condens. Matter Mater. Phys._ 1996, _54_ , 7366–7379.
  * Averin and Bardas [1995] Averin, D.; Bardas, A. ac Josephson Effect in a Single Quantum Channel. _Phys. Rev. Lett._ 1995, _75_ , 1831–1834.
  * Senkpiel et al. [2022] Senkpiel, J.; Drost, R.; Klöckner, J. C.; Etzkorn, M.; Ankerhold, J.; Cuevas, J. C.; Pauly, F.; Kern, K.; Ast, C. R. Extracting transport channel transmissions in scanning tunneling microscopy using superconducting excess current. _Phys. Rev. B: Condens. Matter Mater. Phys._ 2022, _105_ , 165401.
  * Levy and Rudnick [1963] Levy, M.; Rudnick, I. Ultrasonic Determination of the Superconducting Energy Gap in Tantalum. _Phys. Rev._ 1963, _132_ , 1073–1080.
  * Vardimon et al. [2016] Vardimon, R.; Matt, M.; Nielaba, P.; Cuevas, J. C.; Tal, O. Orbital origin of the electrical conduction in ferromagnetic atomic-size contacts: Insights from shot noise measurements and theoretical simulations. _Phys. Rev. B: Condens. Matter Mater. Phys._ 2016, _93_ , 085439.