# A simple strategy to measure the contact resistance between metals and doped
organic films

Anton Kirch  Axel Fischer  Robert Werberger  Shayan Miri Aabi Soflaa  Dresden
Integrated Center for Applied Physics and Photonic Materials (IAPP) and
Institute of Applied Physics, Technische Universität Dresden, Germany
Nöthnitzer Straße 61, 01187 Dresden, Germany  Karolina Maleckaite  Dresden
Integrated Center for Applied Physics and Photonic Materials (IAPP) and
Institute of Applied Physics, Technische Universität Dresden, Germany
Nöthnitzer Straße 61, 01187 Dresden, Germany  Center of Physical Sciences and
Technology, Sauletekio av. 3, LT-10257 Vilnius, Lithuania  Paulius Imbrasas
Johannes Benduhn  Sebastian Reineke<EMAIL_ADDRESS>Dresden
Integrated Center for Applied Physics and Photonic Materials (IAPP) and
Institute of Applied Physics, Technische Universität Dresden, Germany
Nöthnitzer Straße 61, 01187 Dresden, Germany

(August 27, 2024)

###### Abstract

Charge injection from electrodes into doped organic films is a widespread
technology used in the majority of state-of-the-art organic semiconductor
devices. Although such interfaces are commonly considered to form Ohmic
contacts via strong band bending, an experiment that directly measures the
contact resistance has not yet been demonstrated. In this study, we use a
simple metal/doped organic semiconductor/metal stack and study its voltage-
dependent resistance. A transport layer thickness variation proves that the
presented experiment gains direct access to the contact resistance of the
device. We can quantify that for an operating current density of
$10\text{\,}\mathrm{m}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$ the
investigated material system exhibits a voltage drop over the metal/organic
interface of about $200\text{\,}\mathrm{m}\mathrm{V}$, which can be reduced by
more than one order of magnitude when employing an additional injection layer.
The presented experiment proposes a simple strategy to measure the contact
resistance between any metal and doped organic film without applying numerical
tools or elaborate techniques. Furthermore, the simplistic device architecture
allows for very high, homogeneous, and tunable electric fields within the
organic layer, which enables a clear investigation of the Poole-Frenkel
effect.

Contact resistance, doped organic semiconductor, charge carrier injection,
Poole-Frenkel effect

††preprint: APS/123-QED

##  I Introduction

Doped organic layers enable efficient charge carrier injection and extraction
in various emerging semiconductor devices and have a huge impact on their
overall performance. They enable reduced voltage losses at the contact
interfaces of photovoltaic (PV) devices [1, 2, 3, 4], account for the low
operating voltage and reasonable charge balance of light-emitting diodes
(LEDs) using organic injection layers [5, 6, 7], and enhance the switching
speed of organic transistors [8, 9]. Thus, physical understanding and easy
measurement routines are crucial to evolving this technology.

Any charge injection or extraction between electrodes and organic
semiconductors experiences a resistance at the contact interface. Its
mechanism is commonly treated in the picture of a metal-semiconductor bilayer
rooting in classical semiconductor physics and is standard content in
pertinent textbooks [10, 11, 12]. Even though amorphous organic semiconductors
do not form energy band structures in the sense of their crystalline inorganic
counterparts, the model of valence and conduction band is often applied to the
distribution of molecular orbitals in amorphous organic films for the sake of
simplicity and delivers a readily elaborate physical understanding. In this
picture, charge carriers face a potential barrier caused by the difference
between the metal work function and the semiconductor’s energy levels. Tuning
this injection barrier is commonly achieved by energy level alignment, i.e. by
introducing injection layers [13, 14, 15], or energy band bending [11, 16,
17].

The latter option is mediated via blending the organic semiconductors with
dopant molecules [18, 19, 20] and provides an enhanced tunneling probability
for charge carriers, which is even increased by an image charge potential [21,
22]. Depending on the intended majority charge carrier, an electron or hole
transport layer requires n-type or p-type doping, respectively. Organic
molecules, such as F4-TCNQ and F6-TCNNQ, or halide materials like FeCl3 are
commonly used as p-type dopants. Alkali metals like Li or Cs, on the other
hand, are very popular n-type dopants [23].

Under standard operating conditions, the interface between metal and doped
organic semiconductor is commonly treated as Ohmic and only poses a
significant contact resistance below certain bias voltages [24, 25, 26].
Naturally, the question arises what the term “Ohmic” refers to. And indeed,
its definition is not intuitive and sometimes even confused with a connection
to Ohm’s law. In classical semiconductor physics, the term “Ohmic” describes a
contact that poses negligible resistance relative to other resistances in the
device regardless of the applied bias polarity and may exhibit non-linear
behavior [10, 11]. Is it actually correct that an Ohmic contact can be assumed
at common organic LED or PV operating conditions? How much voltage drop does
the contact resistance cause? The answer is not straightforward and depends on
the application intention of the respective device. For high-current
applications, such as future electrically driven organic lasers, these voltage
losses will be severe [27]. Also, considering the resistance contribution of
contacts may develop the understanding of OLED operating voltages deviating
from their theoretical limits [28, 29], can pave tracks enhancing transistor
performance [30], and provide further insight into voltage losses in organic
or perovskite PV devices [31].

While there have been extensive research efforts to understand metal/doped
organic semiconductor contacts in detail [31, 32, 33], this study presents a
direct experimental method to measure the contact resistance of such material
combinations. It is a particular challenge to isolate the pure contact
behavior, as it is commonly overlaid with the charge transport characteristics
of the investigated device. To overcome this problem, we reduce the device
complexity as much as possible: A doped organic semiconductor with variable
thickness is sandwiched between two equally thick silver electrodes and its
voltage-dependent device characteristics are studied. As a result, the
resistance contributions are reduced to the two contacts and a transport
layer, whose influence is controlled via a thickness variation. At the same
time, we drastically decrease the active area and use a 4-wire crossbar setup
to measure high current densities unaffected by parasitic series resistance or
self-heating, which would otherwise be detrimental to the experimental
outcome. By thoroughly studying the presented model system, we can prove that
the manufactured devices follow classical semiconductor physics to a fair
extend and that the introduced experimental strategy gains direct access to
the contact characteristics of a metal-organic semiconductor interface.

##  II Results

###  II.1 Device architecture

Figures 1(a) and 1(b) present the device architecture with a p-doped organic
semiconductor (m-MTDATA:F6-TCNNQ, $4\text{\,}\mathrm{w}\mathrm{t}\mathrm{\char
37\relax}$) evaporated between two silver electrodes. The doping ratio was
chosen high enough to yield a low charge carrier injection/extraction barrier,
but also low enough to sustain a medium doping efficiency [34, 19]. The
semiconductor thickness is varied from $50\text{\,}\mathrm{n}\mathrm{m}$ to
$400\text{\,}\mathrm{n}\mathrm{m}$ to study the impact of the charge transport
resistance. This strategy is similar to the conception of a transmission line
method (TLM) experiment often performed with transistors [35, 36, 37]. The
crucial differences are, first, a drastically reduced device complexity
sporting no dependence on geometry or charge carrier accumulation. Second, the
semiconductor thickness in our experiments is orders of magnitudes below a
horizontal organic transistor channel length. The transport resistance
becomes, as shall be presented, almost insignificant for the thin devices used
in our experiments. This enables a direct and voltage-dependent contact
resistance measurement without extrapolation to zero semiconductor thickness
as performed in TLM.

Figure 1: (a) Device design: A simple metal/doped organic semiconductor/metal
stack with a variation of semiconductor thickness. (b) Energy level diagram at
short circuit: The p-doping induces band bending and depletion regions at the
interfaces. Energy levels taken from Refs. [38, 39, 40]. (c) Energy level
diagram under applied bias. (d) Equivalent circuit model: The physical
characteristics of the presented stack can be described by a simple series of
contact and transport equivalent circuits, each represented by a parallel RC
element.

The minimal transport layer is chosen to be $50\text{\,}\mathrm{n}\mathrm{m}$,
as films below this thickness become increasingly prone to short-circuit
failures. Furthermore, any spatial overlap of the two depletion regions must
be avoided to keep the device physics clear. With typical values for our
material system like the relative permittivity of a typical organic
semiconductor $\epsilon_{\mathrm{r}}\approx 3$ [23], the intrinsic potential
barrier at the interface $V_{\mathrm{BI}}=$0.5\text{\,}\mathrm{e}\mathrm{V}$$,
and the ionized acceptor concentration
$N_{\mathrm{A}}=$1\text{\times}{10}^{18}\text{\,}\mathrm{c}\mathrm{m}^{-3}$$,
the depletion width $W$ of one contact can be estimated at $V=0$ to range
around $4\text{\,}\mathrm{n}\mathrm{m}$.

$\displaystyle
W=\sqrt{\frac{2\epsilon_{0}\epsilon_{\mathrm{r}}(V_{\mathrm{BI}}-V)}{eN_{\mathrm{A}}}}\approx$4\text{\,}\mathrm{n}\mathrm{m}$$
(1)

The depletion width $W$ additionally depends on the applied bias, as
schematically indicated in Fig. 1(c). Still, a minimum transport layer
thickness of $50\text{\,}\mathrm{n}\mathrm{m}$ safely rules out depletion
region interference.

The utilization of a single doped organic film guarantees a very homogeneous
current density and electric field distributions within the device, which is
not compromised by additional intrinsic layers or organic-organic interfaces
in the film [41, 42]. This is proven by current density over voltage ($JV$)
measurements for different sample areas, which were all found to coincide
closely, cf. SI Sect. 4. For the experiments presented throughout this study,
we used the smallest realized cross-section area of
$0.09\text{\,}\mathrm{m}\mathrm{m}^{2}$ to resolve the lowest possible current
densities and to prevent self heating-induced inhomogeneities scaling with
active area dimension [43].

###  II.2 Resistance measurements

Figure 2: Panels (a) and (b) show direct current $JV$ characteristics scanned
back and forth at $T=$300\text{\,}\mathrm{K}$$ for a thickness variation of
the organic semiconductor film. Panel (a) also presents its $RV$
representation. Panels (c) and (d) show direct current $JV$ characteristics
for the $l=$50\text{\,}\mathrm{n}\mathrm{m}$$ device at various temperatures.
Panels (a) and (c) present experimental results, panels (b) and (d) data from
the equivalent circuit modeling.

A four-wire $JV$ scan is run back and forth in a Peltier element-based
cryostat under vacuum (below
$1\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$) to ensure stable thermal
conditions. Details about the four-wire setup are introduced in Ref. [44]. The
technique enables more accurate measurements excluding external series
resistances that jeopardize a precise capacitance analysis. This is essential
for observing the intended characteristics.

Figure 2(a) presents the measurement results for all four devices at a
constant temperature $T=$300\text{\,}\mathrm{K}$$. The blue-shaded lines show
the $JV$ characteristics and the grayscale curves the same data set in
resistance representation $R=V/J$. Only a very slight hysteresis, which is
scaling with increasing current density, is detected. It can be identified
more clearly in Fig. S2. Electrothermal feedback, which causes a deviation
between backward and forward $JV$ measurement directions, can therefore be
rendered unimportant for the overall data analysis [44].

For $V<$300\text{\,}\mathrm{m}\mathrm{V}$$, the $JV$ curves of the four
different transport layer thicknesses differ only slightly. This indicates
that they are governed by the contact resistance between metal and doped
organic semiconductor at low voltages. If the transport resistance posed a
major contribution, the transport layer variation by a factor of 8 would cause
a respective line separation. This is not the case. While the
$400\text{\,}\mathrm{n}\mathrm{m}$ device shows a slight current density and
resistance deviation, the three thinnest devices can merely be kept apart.

For $V>$300\text{\,}\mathrm{m}\mathrm{V}$$, the resistance of all devices
plummets. Also, the lines start separating drastically according to their
transport layer thickness, i.e. the contact resistance drops below the
transport resistance which starts to govern the device characteristics.

Figure 1(d) presents an equivalent circuit that can be used to better
understand this behavior. Any of the three introduced device layers, i.e.
injection, transport, and ejection layer, needs to comprise a resistive
component. The circuit model also emphasizes that our devices sport two
contact resistances, one at each metal/organic film interface. The experiment,
therefore, measures the sum of a forward and reverse contact resistance
contribution.

Ejection and injection interfaces each need to be represented by a variable
resistor $R_{\mathrm{inj}}$ and $R_{\mathrm{ejec}}$, as all devices are
limited by their contact resistances at low bias voltages. Instead of an
accurate but complex interface analysis, only the governing physical concepts
are included. This is sufficient to understand the device physics, as we see
that our model successfully represents all experimental trends. Accurate
quantitative tracing of the experimental results is not our intention here.
The charge carrier injection and ejection currents can be understood as
Schottky contacts and are described by a tunneling term [45] and an ideal
diode equation [10], which is reduced by the effective, i.e. image force-
reduced, potential barrier at the interface $V_{\mathrm{BI}}$.

$\displaystyle
j_{\mathrm{Diode}}=j_{\mathrm{0,S}}\exp\left(\frac{-eV_{\mathrm{BI}}}{n_{\mathrm{i}}k_{\mathrm{B}}T}\right)\left[\exp\left(\frac{eV}{n_{\mathrm{i}}k_{\mathrm{B}}T}\right)-1\right]$
(2) $\displaystyle
j_{\mathrm{Tun}}=\sigma_{\mathrm{0,R}}V\exp\left(\frac{-eV_{\mathrm{BI}}}{n_{\mathrm{i}}k_{\mathrm{B}}T}\right)\left[\exp\left(c\frac{\sqrt{|V|}}{k_{\mathrm{B}}T}\right)-1\right]$
(3)

The parameter $k_{\mathrm{B}}$ depicts the Boltzmann constant, $T$ the device
temperature, $n_{\mathrm{i}}$ the diode ideality factor, $e$ the elementary
charge, $c$ an arbitrary tunneling scaling factor, $V$ the voltage drop across
the respective resistor element, $j_{\mathrm{0,S}}$ the scale current density,
and $\sigma_{\mathrm{0,R}}$ the scale conductivity. Please note that
$\sigma_{\mathrm{0,R}}\cdot V$ again yields a current density, which is the
current density of the tunneling term. The global model parameters are given
in Tab. 1.

The symmetric architecture yields a symmetric device behavior, as shown in the
SI Fig. S2. So, both contacts need to be described by the same formula. We
choose a simple superposition, which only differs in the respective signs for
injection and ejection current densities.

$\displaystyle
j_{\mathrm{inj}}=j_{\mathrm{Tun}}(V_{\mathrm{inj}})-j_{\mathrm{Diode}}(-V_{\mathrm{inj}})$
(4) $\displaystyle
j_{\mathrm{ejec}}=-j_{\mathrm{Tun}}(-V_{\mathrm{ejec}})+j_{\mathrm{Diode}}(V_{\mathrm{ejec}})$
(5)

Both contact elements are susceptible to the voltage drop across the
respective interface and decrease exponentially, as found by the experimental
data. An increasing bias causes a more pronounced band bending at the
injection interface, which increases the tunneling probability. Also, the
Schottky barrier at the ejection interface is reduced [46]. Both processes
make the total contact resistance drop with increasing bias voltage.

The transport resistance must also be represented by a variable resistor
element $R_{\mathrm{trans}}$, since the devices depend on the thickness of the
organic film at elevated bias voltages. Due to the high doping concentration
throughout the entire organic semiconductor film, space charge-limited effects
cannot play a pivotal part and the transport resistance is assumed to be
purely Ohmic. Please be referred to the SI Sect. 2 for the detailed reasoning.
The current density passing $R_{\mathrm{trans}}$ reads

$\displaystyle
j_{\mathrm{trans}}=e\,N_{\mathrm{A}}\mu\frac{V_{\mathrm{trans}}}{d},$ (6)

with $d=l-W_{\mathrm{inj}}-W_{\mathrm{ej}}$ being the thickness of the non-
depleted transport layer and $l$ the total thickness of the doped organic
film, cf. Fig. 1 (c), $V_{\mathrm{trans}}$ the voltage drop across the
transport layer, and $N_{\mathrm{A}}$ temperature-activated ionized dopant
concentration, equal to the effective charge carrier density, cf. SI Sect. 1
and the parameters in Tab. 1.

$\displaystyle
N_{\mathrm{A}}=N_{\mathrm{A,0}}\exp\left(\frac{-E_{\mathrm{A,a}}}{k_{\mathrm{B}}T}\right)$
(7)

As temperature- and field-dependent experiments are carried out and alter the
device characteristics, the mobility term $\mu=\mu(T,V)$ must contain a
temperature and an electric field dependency to successfully model the
experimental results. $R_{\mathrm{transport}}$ has to drop with increasing
bias, since the $JV$ curves bend upwards when separated according to their
transport layer thickness. This cannot be explained by linear resistance
characteristics. The behavior at high electric fields, here exceeding
$100\text{\,}\mathrm{k}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$, requires a
field-induced mobility enhancement, commonly known as Poole-Frenkel (P-F)
effect [45, 11].

$\displaystyle\mu=\mu_{\mathrm{0}}\exp\left(\frac{-E_{\mathrm{\mu,a}}}{k_{\mathrm{B}}T}\right)\exp\left(\beta\sqrt{\frac{|V_{\mathrm{trans}}|}{d}}\right)$
(8)
$\displaystyle\beta=\frac{e}{k_{\mathrm{B}}T}\sqrt{\frac{e}{\pi\epsilon_{\mathrm{r}}\epsilon_{0}}}$
(9)

The parameter $\beta$ is defined in Ref. [47], $k_{\mathrm{B}}$ is the
Boltzmann constant, $T$ the device temperature, $\epsilon_{\mathrm{r}}$ the
relative permittivity of the organic film, and $E_{\mathrm{\mu,a}}$ the
mobility activation energy. For both dependencies, temperature and electric
field, we used simplistic approaches. We are well aware that both dependencies
have to be modeled with more complexity if a good agreement between
experimental and model results is expected. Our model fails at high electric
fields, as can be seen by the increasing deviations between Figs. 2(a) and
(b). This accounts for an oversimplification of the P-F term in the equivalent
circuit. From the simulated results in Fig. 2(b), comprising both a scenario
with and without P-F term, one can estimate the true P-F contribution to be in
between the two realized modeling scenarios, i.e. a reduction of $\beta$ would
yield better agreements. Pursuing this, however, is not the intention of our
investigation. It is yet worth to point out that our presented experimental
approach allows studying the P-F effect in organic semiconductors with utmost
precision. While stacked films in complete semiconductor devices induce a
varying electric field in the vertical direction, the geometry realized here
ensures a very homogeneous field strength throughout the transport layer.

The thinner the transport layer, the smaller its resistance contribution. A
thinner device follows the pure contact resistance characteristics up to
higher current densities than a thick device. Hence, the isolated contact
resistance can be estimated best from the
$l=$50\text{\,}\mathrm{n}\mathrm{m}$$ sample. Figure 2(b) presents the model
scenario with vanishing transport influence, $R_{\mathrm{trans}}=0$, indicated
in red. It is, therefore, safe to deduce that the contact resistance
characteristics must be nonlinear and stays reasonably close to the
$l=$50\text{\,}\mathrm{n}\mathrm{m}$$ device up to about
$100\text{\,}\mathrm{m}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$. This
range is sufficient to quantify the contact resistance for most organic
electronics applications.

Both contact resistance and transport characteristics are susceptible to
temperature, which is presented in Fig. 2(c) and (d). That is, the mobility in
eq. (8) is enhanced by an Arrhenius term, as are the tunneling and diode
characteristics in eqs. (2) and (3). The experimental trend of temperature-
induced conductivity is replicated successfully by the model.

Figure 3: Impedance characteristics at $V=0$. Panel (a) and (b) show data at
$T=$300\text{\,}\mathrm{K}$$ for a thickness variation of the organic
semiconductor. Panels (c) and (d) show data for the
$l=$100\text{\,}\mathrm{n}\mathrm{m}$$ device at various temperatures. Panels
(a) and (c) are experimental results, (b) and (d) from the equivalent circuit
modeling.

The simulation of the three resistance contributions displayed in Fig. 1(d) is
evaluated with LTspice and allows to quantify each of them separately. Figure
S9 presents the modeled contributions and makes it plain that the injection
and ejection resistances govern the device behavior at low voltages. For
$V>$100\text{\,}\mathrm{m}\mathrm{V}$$ all three resistances start to drop
dramatically. The two contact resistance contributions $R_{\mathrm{inj}}$ and
$R_{\mathrm{ejec}}$, however, drop faster than the transport resistance, which
takes over the device characteristics at increasing bias. Figure S9 also
implies that injection and ejection follow roughly the same trend at low
voltages, owing to their mutual exponential nature. Thus, an estimation to
identify either the injection (important for OLEDs) or ejection resistance
(important for PV) from the measurements would be $R_{\mathrm{inj}}\approx
R_{\mathrm{ejec}}$ at low voltages.

###  II.3 Impedance measurements

To gather further evidence that the pure contact behavior is disclosed in the
presented devices, impedance measurements are performed. Figures 3(a) and (c)
present the experimental capacitance density over frequency characteristics
($Cf$) at zero bias voltage $V=0$ with varying thickness and temperature,
respectively. Please find the details on capacitance density evaluation from a
four-wire measurement in the SI Sect. 5. Again, it is very important to employ
a four-wire measuring technique to keep parasitic series resistance out of the
evaluation.

Figure 4: Impedance characteristics at $T=$300\text{\,}\mathrm{K}$$ and
$f=$1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ for a thickness variation of
the transport channel. Panel (a) presents experimental data and panel (b)
results from the equivalent circuit presented in Fig. 1 (d).

The experimental results present two distinct capacitance density plateaus.
This behavior can be understood if treating any of the three distinct layers
within the device not only as having a resistive representation but as an RC
element, cf. Fig. 1(d). At $V=0$, the depletion regions of injection and
ejection have the same width $W$ and, treated like plate capacitors in the
equivalent circuit, also the same capacitance density
$C_{\mathrm{inj}}=C_{\mathrm{ejec}}$.

$\displaystyle
C_{\mathrm{inj}}=\frac{\epsilon_{0}\epsilon_{\mathrm{r}}}{W_{\mathrm{inj}}}$
(10) $\displaystyle
C_{\mathrm{ejec}}=\frac{\epsilon_{0}\epsilon_{\mathrm{r}}}{W_{\mathrm{ejec}}}$
(11)

At low frequencies, the reactance of the transport capacitance
$C_{\mathrm{trans}}$ is high and the charge carrier flow is mediated rather
via $R_{\mathrm{trans}}$. As a result, the device capacitance is governed by
the injection and ejection capacitances. At room temperature, with no bias
voltage, negligible transport resistance, and low frequencies, the device
capacitance density can be estimated as

$\displaystyle C(f\ll$1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$)$
$\displaystyle\approx\left(\frac{1}{C_{\mathrm{inj}}}+\frac{1}{C_{\mathrm{ejec}}}\right)^{-1}$
$\displaystyle\stackrel{{\scriptstyle
V=0}}{{=}}\,\frac{C_{\mathrm{contact}}}{2},$ (12)

with $C_{\mathrm{inj}}=C_{\mathrm{ejec}}=C_{\mathrm{contact}}$ due to device
symmetry at $V=0$. From Fig. 3(a), however, it is apparent that eq. (12) holds
merely for low transport resistances, i.e. very thin transport layers. The
thicker the transport layer, the higher its resistance. As a result, even at
low frequencies the transport capacitance mediates a significant share of the
device current and reduces the total capacitance density. Thus, the
capacitance at low frequencies is decreasing with increasing transport layer
thickness and the pure contact capacitance can only be measured for transport
layers striving toward zero thickness, i.e. the
$l=$50\text{\,}\mathrm{n}\mathrm{m}$$ device.

At high frequencies, the transport capacitance becomes dominant as its
reactance drops below the resistance of the transport layer. Since the
depletion region is smaller than the transport layer thickness $d$ of all
devices, cf. eq. (1), the contact capacitance is expected to be higher than
the transport capacitance. As a result, the total capacitance density falls to
a second plateau at high frequencies. This plateau is governed by the device
geometry (organic film thickness) and is called the geometrical capacitance
density $C_{\mathrm{geo}}$.

$\displaystyle C(f\gg$1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$)$
$\displaystyle=C_{\mathrm{geo}}$
$\displaystyle=\left(\frac{1}{C_{\mathrm{inj}}}+\frac{1}{C_{\mathrm{trans}}}+\frac{1}{C_{\mathrm{ejec}}}\right)^{-1}$
(13)

The transport capacitance density $C_{\mathrm{trans}}$ is a function of the
transport layer thickness $d$ and is, as a simplistic approach, also
represented by a plate capacitor in the equivalent circuit given in Fig. 1(d).

$\displaystyle C_{\mathrm{trans}}=\frac{\epsilon_{0}\epsilon_{\mathrm{r}}}{d}$
(14)

To support the just-presented understanding, the impedance spectroscopy was
also carried out with a fixed organic film thickness
($l=$100\text{\,}\mathrm{n}\mathrm{m}$$) for different temperatures. Figure
3(c) presents the transition interval between the contact and the geometrical
capacitance density plateau as being temperature-dependent. This can be mainly
understood by considering $R_{\mathrm{trans}}=R_{\mathrm{trans}}(V,T)$ as a
function of temperature according to eqs. (6) and (8). At low temperatures,
$R_{\mathrm{trans}}$ is high and the current is mediated earlier via the
transport capacitance, whose reactance does not scale directly with
temperature. For increasing temperatures, $R_{\mathrm{trans}}$ decreases and
the cut-off frequency $f_{\mathrm{G}}$ for the transport RC element increases.

$\displaystyle f_{\mathrm{G,trans}}(V,T)=\frac{1}{2\pi\cdot
R_{\mathrm{trans}}(V,T)\cdot C_{\mathrm{trans}}(V)}$ (15)

Figures 3(b) and (d) present the results from the equivalent circuit model
performed with LTspice. The qualitative trends of the experiments are
reproduced and even quantitatively the results are similar. The transition
characteristics in the experiment, however, are sloping more gently than in
the model. The equivalent circuit follows the Schottky assumption of three
distinct layers having an abrupt change from depleted to neutral semiconductor
characteristics. This is a simplification that allows modeling the two
different, frequency-dependent operating regimes. In the real device, however,
the transition between depleted and neutral regions is smooth and hence the
impedance characteristics are smeared out with respect to the model.

###  II.4 Depletion zone variation

The impedance spectroscopy experiments discussed above were performed at zero
bias voltage $V=0$. Here, the contact capacitances at either side of the
device must be equal, according to the device’s symmetry. The depletion width
$W$, however, depends on the voltage drop over the respective layer, cf. eq.
(1) and Fig. 1(c). The idea of the subsequent measurement is to investigate
the influence of voltage on the contact capacitances.

To access the isolated contact capacitance characteristics as well as
possible, the experiment is performed at
$f=$1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ and room temperature, as
deduced from the last section. Figure 4(a) presents the capacitance density of
all four devices over the applied bias voltage. Using the global set of model
parameters displayed in Tab. 1, a short-circuit depletion width of about
$W\approx$4\text{\,}\mathrm{n}\mathrm{m}$$ at each interface can be expected.
According to eq. (1) and as indicated in Fig. 1(c), $W_{\mathrm{inj}}$
increases and $W_{\mathrm{ejec}}$ decreases under forward bias. When
considering the contact-limited capacitance at
$1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, where the transport capacitance
density $C_{\mathrm{trans}}$ is assumed to play no pivotal role, the
capacitance density trend can be understood to follow

$\displaystyle
C(f=$1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$,V)=\left(\frac{\epsilon_{\mathrm{r}}\epsilon_{0}}{W_{\mathrm{inj}}(V_{\mathrm{inj}})+W_{\mathrm{ejec}}(V_{\mathrm{ejec}})}\right),$
(16)

with both $W_{\mathrm{inj}}$ and $W_{\mathrm{ejec}}$ following eq. (1). For
low applied bias, both voltage dependencies apparently cancel each other, as
around $V=0$ their behavior can be approximated as being linear. Beyond an
applied bias of about $100\text{\,}\mathrm{m}\mathrm{V}$, the linear
approximation fails, and the denominator of eq. (16) starts to grow. The total
capacitance density of the devices drops. The same overall trend can be
reproduced by the equivalent circuit simulation, presented in Fig. 4(b).

###  II.5 Demonstration scenario for injection enhancement quantification

Throughout the sections, an experimental setting was introduced that evidently
provides direct access to the contact resistance of a given interface between
a metal and a doped organic film. As an outline towards potential application
scenarios, this section presents a short study on the effect of additional
interface layers that reduce the contact resistance.

Figure 5: Panel (a) shows the investigated stack designs comprising
$100\text{\,}\mathrm{n}\mathrm{m}$ of p-doped organic semiconductor and either
none, one, or two contact enhancement layers of MoO3
($0.5\text{\,}\mathrm{n}\mathrm{m}$). Panel (b) presents their respective $JV$
characteristics at $T=$300\text{\,}\mathrm{K}$$ and illustrates the effect on
the measured contact resistance.

The subsequent experiment uses an equivalently simple stack of
$100\text{\,}\mathrm{n}\mathrm{m}$ of Spiro-TTB doped with C60F36
($4\text{\,}\mathrm{w}\mathrm{t}\mathrm{\char 37\relax}$) sandwiched between
two silver electrodes. Again, we performed a thickness-dependent $JV$ analysis
and found like for the m-MTDATA:F6-TCNNQ devices that the resistance behavior
for $J<$10\text{\,}\mathrm{m}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$$ is
not influenced by the organic layer thickness up to
$l=$200\text{\,}\mathrm{n}\mathrm{m}$$, cf. SI Sect. 6. That is, direct access
to the device’s contact resistance is guaranteed. Now, either at the top, the
bottom, or both contact interfaces a thin layer
($0.5\text{\,}\mathrm{n}\mathrm{m}$) of molybdenum oxide ($\ce{MoO3}$) is
added, cf. Fig. 5(a), which is commonly used to reduce the contact resistance
of p-doped organic semiconductor devices [48, 49, 4, 50, 15]. The data in Fig.
5(b) proves that adding one injection interlayer clearly reduces the contact
resistance of the respective interface. There appears to be no significant
difference between employing the enhancement layer in the top or bottom
configuration. Most likely, the non-enhanced contact governs the contact
resistance characteristics here. Using MoO3 on both interfaces reduces the
contact resistance at
$10\text{\,}\mathrm{m}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$ by more
than one order of magnitude from about
$30\text{\,}\Omega\,\mathrm{c}\mathrm{m}^{2}$ to below
$3\text{\,}\Omega\,\mathrm{c}\mathrm{m}^{2}$. A detailed investigation of this
material combination can be found in the SI Sect. 6. In such a fashion,
contact resistance enhancing strategies can easily be evaluated.

##  III Conclusion

This article presents a strategy to measure the isolated contact resistance
between metals and doped organic semiconductors up to reasonable device
operating conditions. For the investigated model structure of a p-doped film
(m-MTDATA:F6-TCNNQ, $4\text{\,}\mathrm{w}\mathrm{t}\mathrm{\char 37\relax}$)
between to silver electrodes, the two metal-organic interfaces cause a
potential drop of about $400\text{\,}\mathrm{m}\mathrm{V}$ at
$10\text{\,}\mathrm{m}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$. At this
current density, one contact poses therefore a resistance of
$R_{\mathrm{contact}}\approx$20\text{\,}\Omega\,\mathrm{c}\mathrm{m}^{2}$$.
The presented conception requires neither complex experiments nor
sophisticated theoretical treatment but is based entirely on an easy-to-
perform $JV$ scan. As a proof of concept, we perform a thorough evaluation for
the mentioned hole-only device configuration. We employ resistance and
impedance measurements while scanning a variety of device temperatures and
organic film thicknesses and successfully compare the results to a simple
equivalent circuit realized in LTspice. The model uses one global set of
parameters replicating all experimental findings. This proves that the
measured device characteristics directly disclose the contact behavior of the
device up to about
$100\text{\,}\mathrm{m}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$ for the
given system, which is sufficient for most operating scenarios in LEDs or
solar cells. We further present how our measurement concept can help other
groups evaluate their contact enhancement strategies by investigating the
influence of molybdenum oxide as an injection layer in a second material
system comprising Spiro-TTB doped with C60F36 and silver electrodes. The
presented device architecture turns out to achieve extremely strong and
homogeneous electric fields, which allows to specifically study the Poole-
Frenkel effect in organic semiconductors.

##  IV Methods

###  IV.1 Device fabrication

The hole-only devices were fabricated by thermal evaporation under high vacuum
(Kurt J. Lesker Company, evaporation pressure
$<$1\text{\times}{10}^{-6}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$$)
on
$$2.5\text{\,}\mathrm{c}\mathrm{m}$\times$2.5\text{\,}\mathrm{c}\mathrm{m}$$
glass substrates (Schott Borofloat33 glass, Prince Optics) of
$1.1\text{\,}\mathrm{m}\mathrm{m}$ thickness. The substrates were cleaned in
an ultrasonic bath with acetone, ethanol, and deionized water. Aluminum
(Chempur) and silver (M&K GmbH Jena), $50\text{\,}\mathrm{n}\mathrm{m}$ each,
are used as bottom and top electrodes. Only the silver contacts are drawn in
Fig. 1 for simplicity. The p-layer consists of m-MTDATA (Synthon, sublimation
cleaned) doped with F6-TCNNQ (Novaled AG, sublimation cleaned) at a ratio of
$4\text{\,}\mathrm{w}\mathrm{t}\mathrm{\char 37\relax}$, or Spiro:TTB (Lumtec,
sublimation cleaned) doped with C60F36 (Ionic Liquids Technologie GmbH,
sublimation cleaned) also at $4\text{\,}\mathrm{w}\mathrm{t}\mathrm{\char
37\relax}$. The thickness and deposition rates were monitored using a quartz
crystal microbalance. Please find the full names of the materials and reverse
$JV$ characteristics in the SI Sect. 1 and 3.

To prohibit air and moisture contamination, the device stacks were
encapsulated under nitrogen atmosphere after fabrication. The encapsulation
glass (Sodalime glass, AMGTECH Korea) comprises a small cavity above the
pixels that prevents direct contact between sensitive materials and the
encapsulation glass. It was attached to the substrate using an epoxy resin
(XNR5516Z-L and XNR5590, Nagase Europa GmbH).

###  IV.2 Device evaluation

The devices were placed in a Peltier element-equipped cryostat that is
controlled by a temperature controller (Belektronig, HAT control). The air
pressure inside the cryostat was reduced to below
$1\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ using a pre-vacuum
pumping system (Trivac D16B, Germany) to keep a steady temperature and
preventing unwanted air convection. Every measurement was performed using the
four-wire method run by a dual-channel SMU (Keithley 2602). Impedance
measurements were performed using an LCR meter (Hewlett Packard 4284A
precision LCR meter) and a home-built switching matrix to switch between SMU
und LCR and target individual device pixels. Temperature controller, LCR
meter, and SMU were run and controlled by the software tool SweepMe! [51],
which enables automated measurement protocols.

###  IV.3 Equivalent circuit modeling

The software LTspice (Linear Technology) was used for modeling the equivalent
circuit presented in Fig. 1(d) with one fixed set of parameters to achieve all
model results, cf. Tab. 1. In contrast to an analytical Python code, LTspice
can self-consistently iterate solutions to model voltage dependent impedance
measurements, as presented in Fig. 4.

Table 1: Global parameter set used for equivalent circuit modeling with LTspice. Diode ideality factor | $n_{\mathrm{i}}$ | 1
---|---|---
Tunneling scaling factor | $c$ | $2\text{\times}{10}^{-1}$
Tunneling scale conductivity | $\sigma_{\mathrm{0,R}}$ | $4\text{\times}{10}^{3}\text{\,}\mathrm{m}\mathrm{S}\,\mathrm{c}\mathrm{m}^{-2}$
Diode scale current density | $j_{\mathrm{0,S}}$ | $5\text{\times}{10}^{2}\text{\,}\mathrm{m}\mathrm{A}\,\mathrm{c}\mathrm{m}^{-2}$
Interface potential barrier | $V_{\mathrm{BI}}$ | $0.35\text{\,}\mathrm{V}$
Scale mobility | $\mu_{\mathrm{0}}$ | $2\text{\times}{10}^{-1}\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{V}^{-1}\mathrm{s}^{-1}$
Scale doping concentration | $N_{\mathrm{A,0}}$ | $10\text{\times}{10}^{18}\text{\,}\mathrm{c}\mathrm{m}^{-3}$
Relative permittivity | $\epsilon_{\mathrm{r}}$ | $3.5$
Doping activation energy | $E_{\mathrm{A,a}}$ | $25\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$
Mobility activation energy | $E_{\mathrm{\mu,a}}$ | $0.3\text{\,}\mathrm{e}\mathrm{V}$

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## Acknowledgements

This work was supported by the German Research Foundation (DFG) within the
project HEFOS (Grant No. FI 2449/1-1) and EFOD (Grant No. RE 3198/6-1). The
authors thank Andreas Wendel and Tobias Günther for manufacturing the devices.

##  V Author contributions

A.F. conceptualized the study. A.K. and A.F. analyzed the data and performed
the LTspice implementation. A.K. prepared the graphs and mainly wrote the
manuscript. A.F., R.W., P.I. and K.M. set up and conducted the experiments.
J.B. and S.M. contributed major ideas to the physical understanding. S.R.
supervised the project. All authors discussed the results, reviewed the
manuscript, and contributed to the text.

##  VI Competing interests

Dr. Axel Fischer is co-founder of “SweepMe! GmbH” which provided the
measurement software “SweepMe!” (sweep-me.net). The name of the program is
given in the manuscript. The other authors declare no conflicts of interest.

##  VII Materials and Correspondence

All material, code and data is available on reasonable request.


# Constraining the optical potential in the search for $\eta$-mesic
${}^{4}\mbox{He}$

M. Skurzok<EMAIL_ADDRESS>P. Moskal  N. G. Kelkar  S. Hirenzaki
H. Nagahiro  N. Ikeno  Institute of Physics, Jagiellonian University, prof.
Stanisława Łojasiewicza 11, 30-348 Kraków, Poland  Departamento de Fisica,
Universidad de los Andes, Cra. 1E, 18A–10, Bogotá, Colombia  Department of
Physics, Nara Women’s University, Nara 630-8506, Japan  Research Center for
Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan  Department
of Life and Environmental Agricultural Sciences, Tottori University, Tottori
680-8551, Japan

###### Abstract

A consistent description of the $dd\rightarrow$ ${}^{4}\mbox{He}\eta$ and
$dd\rightarrow$ (4He$\eta$)${}_{bound}\rightarrow X$ cross sections was
recently proposed with a broad range of real ($V_{0}$) and imaginary
($W_{0}$), $\eta$-4He optical potential parameters leading to a good agreement
with the $dd\rightarrow$ ${}^{4}\mbox{He}\eta$ data. Here we compare the
predictions of the model below the $\eta$ production threshold, with the WASA-
at-COSY excitation functions for the $dd\rightarrow$ ${}^{3}\mbox{He}N\pi$
reactions to put stronger constraints on $(V_{0},W_{0})$. The allowed
parameter space (with $|V_{0}|<\,\sim$ 60 MeV and $|W_{0}|<\,\sim$ 7 MeV
estimated at 90% CL ) excludes most optical model predictions of $\eta-^{4}$He
nuclei except for some loosely bound narrow states.

###### keywords:

mesic nuclei, optical model, nuclear potential

††journal: Journal of LaTeX Templates

##  1 Introduction

Mesic nuclei are currently one of the hottest topics in nuclear and hadronic
physics, both from experimental [1, 2, 3, 4, 5] and theoretical points of view
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].
This exotic nuclear matter is supposed to consist of a nucleus bound via the
strong interaction with a neutral meson such as the $\eta$, $\eta^{\prime}$,
$K$ or $\omega$. Although, its existence has been predicted over thirty years
ago, it still remains to be one of the undiscovered nuclear objects. Some of
the most promising candidates for such bound states are $\eta$-mesic nuclei,
postulated by Haider and Liu in 1986 [26] following the coupled channel
calculations by Bhalerao and Liu [27] which reported an attractive
$\eta$-nucleon interaction. Current studies of hadron- and photon-induced
production of the $\eta$ meson resulting in a wide range of values of the
$\eta N$ scattering length, $a_{\eta N}$, indicate the interaction between the
$\eta$ meson and a nucleon to be attractive and strong enough to create an
$\eta$-nucleus bound system even in light nuclei [7, 8, 9, 10, 28, 29, 30].
However, experiments performed so far have not brought a clear evidence of
their existence [31, 32, 33, 34, 35, 36, 37, 38]. They provide only signals
which might be interpreted as indications of the $\eta$-mesic nuclei. The
interested reader can find recent reviews on the $\eta$ mesic bound states
searches in Refs [4, 5, 14, 16, 39, 40, 41, 42, 43, 44].

Some of the promising experiments related to $\eta$-mesic nuclei have been
performed with the COSY facility [45]. The most recent of these involves the
measurement of the $dd\rightarrow$ ${}^{3}\mbox{He}n\pi^{0}$ and
$dd\rightarrow$ ${}^{3}\mbox{He}p\pi^{-}$ reactions which has been performed
by the WASA-at-COSY Collaboration. Due to the lack of theoretical predictions
for cross sections below the $\eta$ production threshold, the data have been
analyzed assuming that the signal from the bound state has a Breit-Wigner
shape [1, 2]. However, a better guidance for the shape of the cross sections
for the $dd\rightarrow$ (${}^{4}\mbox{He}$-$\eta)_{bound}\rightarrow$
${}^{3}\mbox{He}N\pi$ processes is provided by a theoretical model described
in Ref. [6] in the excess energy range relevant to the $\eta$-mesic nuclear
search. Given that the model is the very first attempt to provide a consistent
description of the data below and above the $\eta$ meson production threshold,
the authors used a phenomenological approach with an optical potential for the
$\eta$-4He interaction. The available data on the $dd\rightarrow\,^{4}$He
$\eta$ reaction is reproduced quite well for a broad range of optical
potential parameters for which the authors predict the cross section spectra
corresponding to $\eta$-4He bound state formation in the subthreshold region.
In this article we present a comparison between this new theoretical model and
experimental data collected by WASA-at-COSY in order to further constrain the
range of the allowed $\eta$-4He optical potential parameters. The latter, as
we shall see, narrows down the search for $\eta$-mesic helium to a region of
small binding energies and widths.

##  2 Theoretical model

The formalism presented in Ref. [6] predicted for the first time, the
formation rate of the $\eta$-mesic ${}^{4}\mbox{He}$ in the deuteron-deuteron
fusion reaction within a model which reproduced the data on the
$dd\rightarrow\,^{4}$He $\eta$ reaction quite well. The authors determined the
total cross sections for the $dd\rightarrow$
(${}^{4}\mbox{He}$-$\eta)_{bound}\rightarrow$ ${}^{3}\mbox{He}N\pi$ reaction
based on phenomenological calculations. The calculated total cross section
$\sigma$ consists of two parts: conversion $\sigma_{conv}$ and escape
$\sigma_{esc}$ part. The conversion part, determined for different parameters
$V_{0}$ and $W_{0}$ of a spherical $\eta$-${}^{4}\mbox{He}$ optical potential
$V(r)=(V_{0}+iW_{0})\frac{\rho_{\alpha}(r)}{\rho_{\alpha}(0)}$, is equal to
the total cross section in the subthreshold excess energy region where the
$\eta$ meson is absorbed by the nucleus (its energy is not enough to escape
from the nucleus), while the $\eta$ meson escape part contributes to the
excess energy region above the threshold for $\eta$ production and can be
calculated as $\sigma_{esc}=\sigma-\sigma_{conv}$. Fig. 1 shows the example of
a calculated total cross section for $\eta$-${}^{4}\mbox{He}$ optical
potential parameters ($V_{0},W_{0}$)=$-$(70,20) MeV.

We should mention here that the above theoretical calculations (which are
being used in the present work) were done assuming the one-nucleon absorption
of the $\eta$ meson since the strength of the multi-nucleon absorption
processes is not well known. Based on the experimental data on the $pn\to
d\eta$ and $pN\to pN\eta$ reactions, the strength of the $\eta$ meson
absorption by a two-nucleon pair at the nuclear center was estimated in [46]
to be 4.2 MeV and 0.2 MeV for the spin triplet and singlet nucleon pairs,
respectively. This strength can be larger for 4He because of the higher
central density as mentioned in [47]. The values of the $W_{0}$ parameters in
the present work could be compared with these numbers to get a rough estimate
of the ratio of the one- and two-body absorption probability at the nuclear
center. The two body absorption potential is expected to provide an additional
contribution to the conversion cross section. However, it is only the one-
nucleon absorption cross section which should be compared with the present
data since multi-nucleon absorption processes would contribute to different
final states not considered in the present work. Thus, the present analysis of
experimental data from Ref. [1] based on the theoretical calculation assuming
the one-body absorption seems reasonable.

Figure 1: Calculated total cross section of the $dd\rightarrow$
(${}^{4}\mbox{He}$-$\eta)_{bound}\rightarrow$ ${}^{3}\mbox{He}N\pi$ reaction
for the formation of the ${}^{4}\mbox{He}$-$\eta$ bound system plotted as
function of the excess energy Q for $\eta$-${}^{4}\mbox{He}$ optical potential
parameters ($V_{0},W_{0}$)=$-$(70,20) MeV. The black solid line denotes the
total cross section $\sigma$, while the red dashed line denotes the conversion
part $\sigma_{conv}$.

The spectrum has been normalized in the sense that the escape part reproduces
the measured cross sections for the $dd\rightarrow$ ${}^{4}\mbox{He}\eta$
process [48, 49, 50]. Moreover, the flat contribution in the conversion
spectrum, considered to be a part of the background, has been subtracted
(taking minimum value of the $\sigma_{conv}$ in the excess energy range from
-20 to 15 MeV).
Since the signal from the $\eta$-mesic bound system is expected below the
threshold for the $\eta$ meson production, authors focused here only on the
conversion part of the cross sections. An example of the calculated
$\sigma_{conv}$ is shown in Fig. 2 for potential parameters
($V_{0},W_{0}$)=$-$(70,20) MeV. Authors of Reference [6] concluded that as a
next step it would be important to compare these theoretical results with the
experimental data, convoluting the theoretical cross sections with the
experimental resolution functions. In this article we present results of such
a comparison. The details are presented in Section 4 which will be preceded by
a brief description of the experimental conditions.

Figure 2: Calculated conversion part of the cross section of the
$dd\rightarrow$ (${}^{4}\mbox{He}$-$\eta)_{bound}\rightarrow$
${}^{3}\mbox{He}N\pi$ reaction for the formation of the
${}^{4}\mbox{He}$-$\eta$ bound system plotted as a function of the excess
energy Q for $\eta$-${}^{4}\mbox{He}$ optical potential parameters
($V_{0},W_{0}$)=$-$(70,20) MeV. The cross section is scaled by fitting the
escape part to the existing $dd\rightarrow$ ${}^{4}\mbox{He}\eta$ data and the
flat contribution is subtracted as well. The red dashed line shows the
theoretical spectrum while the black solid line shows the spectrum after
binning (details in Sec. 3).

##  3 Experimental data

Recent measurements at WASA-at-COSY, dedicated to search for $\eta$-mesic
${}^{4}\mbox{He}$ nuclei were carried out using the unique ramped beam
technique allowing for the beam momentum to be changed slowly and continuously
around the $\eta$ production threshold in each of the acceleration cycles [1,
2, 42, 44]. This technique allows to reduce systematic uncertainties with
respect to separate runs at fixed beam energies [2, 34, 51]. The
${}^{4}\mbox{He}$-$\eta$ bound states were searched by studying the excitation
functions for $dd\rightarrow$ ${}^{3}\mbox{He}n\pi^{0}$ and $dd\rightarrow$
${}^{3}\mbox{He}p\pi^{-}$ processes in the excess energy range $Q$ from -70
MeV to 30 MeV. The obtained excitation functions do not reveal any direct
narrow structure below the $\eta$ production threshold, which could be
considered as a signature of the bound state. Therefore, only the upper limit
of the total cross section for the $\eta$-mesic ${}^{4}\mbox{He}$ formation
was determined.

In the first approach, the upper limits of the total cross sections for both
processes were estimated at a 90% confidence level (CL) fitting simultaneously
the excitation functions with a sum of a Breit-Wigner and a second order
polynomial function corresponding to the bound state signal and background,
respectively. Moreover, the isospin relation between $n\pi{}^{0}$ and
$p\pi{}^{-}$ pairs was taken into account. The corresponding data analysis is
presented in detail in Ref. [1]. The analysis resulted in the value of the
upper limit in the range from 2.5 to 3.5 nb for the $dd\rightarrow$
(${}^{4}\mbox{He}$-$\eta)_{bound}\rightarrow$ ${}^{3}\mbox{He}n\pi^{0}$
process and from 5 to 7 nb for the $dd\rightarrow$
(${}^{4}\mbox{He}$-$\eta)_{bound}\rightarrow$ ${}^{3}\mbox{He}p\pi^{-}$
reaction. Systematic uncertainty, contributed mainly from the assumption of
the Fermi momentum of the $N^{*}$ resonance inside ${}^{4}\mbox{He}$ [13], to
be equal to that of a nucleon in 4He [52], varies from 42% to 46% for both
reactions.

These experimental results are revisited in the next section in the light of a
new theoretical model [6] which reproduces the $dd\rightarrow\,^{4}$He$\eta$
cross section data and with the same $\eta$-4He optical potential predicts the
cross sections for $dd$ fusion with the formation of an $\eta$-mesic 4He below
the $\eta$ production threshold. The objective of the present analysis is
twofold: to provide (i) stronger constraints on the optical potential
parameters which are already capable of reproducing the $\eta$ production data
and (ii) to improve the upper limits on the cross sections found in [1] using
a theoretical model (constrained by the above threshold data) for the possible
bound state, rather than the simple Breit-Wigner form used in [1].

##  4 Comparison between theory and data: results and discussion

As mentioned in the previous section, we performed the analysis which allows
to compare excitation functions measured for $dd\rightarrow$
${}^{3}\mbox{He}n\pi^{0}$ and $dd\rightarrow$ ${}^{3}\mbox{He}p\pi^{-}$
processes [1] with the theoretical predictions presented in Ref. [6]. For this
purpose, theoretical conversion spectra were convoluted with the experimental
resolutions of the excess energy $Q$. The COSY beam is characterized by a high
momentum resolution of up to $\frac{\Delta p}{p}\approx 1\cdot 10^{-4}$
resulting in the resolution of $\Delta Q$ of about 70 keV in the energy range
of interest. This is about 70 times smaller than the binning of the spectra
used by the WASA-at-COSY collaboration [1]. Hence, we bin the theoretical
predictions in the same way as data, dividing the spectra into 20 intervals
each of 5 MeV width. We assume also that the reconstruction efficiency in the
WASA-at-COSY experiment is in a good approximation independent of the excess
energy $Q$ as was proven in Ref. [53]. An example of the theoretical spectrum
after the binning procedure is presented in Fig. 2 as a black histogram.

In the next step, the experimental excitation functions for $dd\rightarrow$
${}^{3}\mbox{He}n\pi^{0}$ and $dd\rightarrow$ ${}^{3}\mbox{He}p\pi^{-}$
reactions were fitted simultaneously with a sum of binned theoretical function
(signal) and a second order polynomial (background). The $n\pi^{0}$,
$p\pi^{-}$ isospin relation was taken into account. The fitting functions can
be presented as follows:

$\sigma_{n\pi^{0}}(Q)=\frac{1}{3}A\cdot Theory(Q)+B_{1}Q^{2}+C_{1}Q+D_{1}$ (1)
$\sigma_{p\pi^{-}}(Q)=\frac{2}{3}A\cdot Theory(Q)+B_{2}Q^{2}+C_{2}Q+D_{2}$ (2)

for $dd\rightarrow$ ${}^{3}\mbox{He}n\pi^{0}$ and $dd\rightarrow$
${}^{3}\mbox{He}p\pi^{-}$, respectively. $Theory(Q)$ denotes the theoretical
function after binning with the amplitude normalized to unity, while
$B_{1,2}Q^{2}+C_{1,2}Q+D_{1,2}$ is a polynomial of the second order. The fit
was performed for theoretical spectra obtained for different optical potential
parameters ($V_{0},W_{0}$) [6]. During the fit, the amplitude $A$ of the
theoretical spectrum and polynomial coefficients were treated as free
parameters. As an example, the excitation functions with the fit results for
optical potential parameters ($V_{0},W_{0}$)=-(70,20)MeV are presented in Fig.
3.

Figure 3: Excitation function for $dd\rightarrow$ ${}^{3}\mbox{He}n\pi^{0}$
(upper panel) and $dd\rightarrow$ ${}^{3}\mbox{He}p\pi^{-}$ reaction (lower
panel) determined as described in Ref. [1]. The red solid line represents a
fit with theoretical prediction for potential parameters
($V_{0},W_{0}$)=-(70,20) MeV combined with a second order polynomial. The blue
dotted line shows the second order polynomial describing the background while
blue solid line shows the signal contribution. The experimental data [1] are
indicated with black squares.

The performed fit delivers the amplitudes $A$ for $dd\rightarrow$
${}^{3}\mbox{He}N\pi$ consistent with zero within 2$\sigma$ for all sets of
$V_{0},W_{0}$ parameters, which is given in Table 1. Therefore, the upper
limit of the total cross section was determined, like in Ref. [1], at the
confidence level 90% based on standard deviation of the amplitude $\sigma_{A}$
($\sigma^{CL=90\%}_{upp}$=1,64$\cdot\sigma_{A}$). $\sigma^{CL=90\%}_{upp}$
values are presented for different parameters $V_{0},W_{0}$ in Table 1.

$V_{0}$ | $W_{0}$ | A (fit) [nb] |  $\sigma^{CL=90\%}_{upp}$ [nb]
---|---|---|---
-30 | -5 | -5.0$\pm$3.9 | 6.5
-30 | -20 | -2.2$\pm$3.5 | 5.8
-30 | -40 | 0.2$\pm$3.8 | 6.3
-50 | -5 | 0.1$\pm$3.8 | 6.3
-50 | -20 | 3.3$\pm$4.1 | 6.8
-50 | -40 | 6.0$\pm$4.2 | 6.9
-70 | -5 | 6.4$\pm$4.5 | 7.4
-70 | -20 | 7.9$\pm$4.5 | 7.4
-70 | -40 | 7.5$\pm$3.7 | 6.1
-100 | -5 | 6.3$\pm$4.5 | 7.4
-100 | -20 | 6.9$\pm$3.9 | 6.4
-100 | -40 | 5.3$\pm$3.1 | 5.2
Table 1: Results obtained from the fit of theoretical spectra to experimental
data. Table includes: optical potential parameters (first and second columns),
amplitude obtained from the fit with its statistical uncertainty (third
column) and upper limit of the total cross section for the $dd\rightarrow$
(${}^{4}\mbox{He}$-$\eta)_{bound}\rightarrow$ ${}^{3}\mbox{He}N\pi$ process at
CL=90% (fourth column).

Obtained $\sigma^{CL=90\%}_{upp}$ is weakly sensitive to the $V_{0},W_{0}$
parameters, varying from 5.2 to 7.4 nb. Taking into account the systematic
uncertainties of about 44% estimated in Ref. [1], the values increase, varying
from about 7.5 to 10.7 nb. Therefore, in the contour plot shown in Fig. 4, we
exclude the region where the cross section is above 10.7 nb (light shaded
area). Dark shaded area shows the systematic error. The latter estimate is
based on a calculation [13] for the $N^{*}$ momentum distribution for a given
set of $\pi NN^{*}$ and $\eta NN^{*}$ coupling constants. If we take into
account the calculations in [13] using all available values of the coupling
constants, the allowed region in the $V_{0}$-$W_{0}$ plane can extend as far
as the red line shown in Fig. 4. The coloured dots shown in the figure are the
results of some optical model calculations which will be discussed in the next
section.

Figure 4: Contour plot of the theoretically determined conversion cross
section in $V_{0}-W_{0}$ plane [6]. Light shaded area shows the region
excluded by our analysis, while the dark shaded area denotes systematic
uncertainty of the $\sigma^{CL=90\%}_{upp}$. The red line extends the allowed
region based on a new estimate of errors (see text for details). Dots
correspond to the optical potential parameters corresponding to the predicted
$\eta$-mesic 4He states.

##  5 Optical model predictions of $\eta$-mesic 4He

After constraining the region of the optical potential ($V_{0}$, $W_{0}$)
parameter space allowed by the cross section data below the $\eta$ production
threshold, let us now examine the possibility for the existence of
$\eta$-mesic helium nuclei predicted within the optical model. To start with,
we notice that all states predicted in Table 1 of [6] by solving the Klein
Gordon equation with the optical potential of the present work, are excluded.
On the other hand, since a wide range of $V_{0}$, $W_{0}$ values in [6] do
reproduce the $dd\rightarrow\eta\,^{4}$He data, it seems worthwhile to
investigate other optical model predictions in literature.

The authors in [10] for example, compare their results using a few body
formalism with existing optical model calculations by using the following form
of the $\eta$-4He potential with the complex $\eta$-nucleon scattering
amplitude $F_{\eta N}$ chosen from two different models in literature [54,
55]:

$V(r)=-{6\pi\over\mu_{\eta N}}\,F_{\eta
N}\,(r_{0}\sqrt{\pi})^{-3}\exp(-{r^{2}\over r_{0}^{2}}).$ (3)

Replacing the parameter, $r_{0}=1.267$ fm, as given in [10] and rewriting the
above equation for the potential as,
$V(r)=[V_{0}+iW_{0}]\,\exp(-{r^{2}}/{r_{0}^{2}})$, we identify the strengths
$V_{0}$ and $W_{0}$ and list them in Table 2 for the different cases listed in
Table 4 of Ref. [10].

$\eta N$ model | $\delta\sqrt{s}[MeV]$ |  $B_{\eta 4He}$ [MeV] |  $\Gamma$ [MeV] | -$V_{0}$ [MeV] | -$W_{0}$ [MeV]
---|---|---|---|---|---
GW [54] | 0 | 25.1 | 40.8 | 175.7 | 54.2
| -32.4 | 1.03 | 2.35 | 89.7 | 8.6
CS [55] | 0 | 6.39 | 21 | 125.87 | 29.35
| -19.2 | - | - | 69.15 | 5.046
Table 2: Strength of the optical potentials corresponding to the $\eta$-4He
states given in [10]. $\delta\sqrt{s}=\sqrt{s}-\sqrt{s_{th}}$ with $\sqrt{s}$
being the energy available in the center of mass of the $\eta N$ system.
$B_{\eta 4He}$ and $\Gamma$ are the binding energies and widths of the
$\eta$-4He states.

The $\eta N$ amplitude of [54] (GW), was obtained within a K-matrix
description of the $\pi N$, $\pi\pi N$, $\eta N$ and $\gamma N$ coupled
channels. The authors fitted the $\pi N\to\pi N$, $\pi N\to\eta N$, $\gamma
N\to\pi N$ and $\gamma N\to\eta N$ data in the energy range of about 100 MeV
on either side of the $\eta$ threshold. Ref. [55] presented the $\eta N$
amplitudes calculated within a chirally motivated separable potential model
with the parameters of the model fitted to $\pi N\to\pi N$ and $\pi N\to\eta
N$ data. A comparison of the $V_{0}$ and $W_{0}$ values in Table 2 with the
allowed region of the $V_{0}-W_{0}$ plane leads us to the conclusion that all
the bound states listed in Table 2 are excluded by our analysis.

Having excluded the optical potential predictions of unstable bound states in
literature, we turn to examine the special case of an unstable state centered
at zero energy. The case of a zero energy bound state (or zero energy
resonance), sometimes referred to as a transition state [56] has been widely
studied in literature [57] in the context of different physical situations and
has also been observed in ultracold atoms [58]. Let us recall some basic
facts: a bound state corresponds to a pole in the S-matrix for $E<0$. A
resonance corresponds to a pole at positive energies. A state at $E=0$ (which
is usually referred to as a zero energy bound state in case the angular
momentum $l>0$ and zero energy resonance otherwise) leads to a scattering
length, $a\to\infty$, i.e., the scattering length has a pole when $E=0$. Ref.
[56] has examined the occurrence of such states for a class of potentials of
the form $V(r,r_{0})=-{g\over r^{s}}\,f\biggl{(}{r\over
r_{0}}\biggr{)}\,\,(g>0,\,\,r_{0}>0)$, which include the Gaussian, exponential
and Hulthen among others. For the Gaussian optical potential of the present
work, we identify $g$ with $V_{0}$, $s=0$ and $f=\exp({-r^{2}/r_{0}^{2}})$.
Analytical as well as numerical solutions of the Schroedinger equation for
these potentials are provided in Ref. [56]. It is shown that the existence of
the transition state solution depends on a critical parameter given by
$\beta=2\mu\,V_{0}\,r_{0}^{2}/\hbar^{2}$, numerical values of which are listed
in a table for several values of $l$. Taking their value of $\beta=2.684$ in
case of the Gaussian potential with $l=0$, $\mu$ the reduced mass of
$\eta$-4He and with $r_{0}=1.267$ fm, we find $V_{0}=-68.04$ MeV. Putting back
this value in the expression, $V_{0}=-[6\pi/\mu_{\eta N}]\,\Re eF_{\eta
N}\,(r_{0}\sqrt{\pi})^{-3}$, arising from (3), we determine $\Re eF_{\eta
N}=0.364$ fm. This value of $\Re eF_{\eta N}$ corresponds to the subthreshold
energies of $\sqrt{s}$ = 1418.2 and 1467 MeV of the GW and CS $\eta$N
amplitudes respectively (see Fig. 1 in Ref. [10]). The imaginary parts of the
amplitudes can be seen from the same figure in [10] (at the corresponding
energies) to be $\Im mF_{\eta N}=0.0167$ fm and $\Im mF_{\eta N}=0.0245$ fm
for the GW and CS models respectively. The imaginary part of the optical
potential can now be determined using, $W_{0}=-[6\pi/\mu_{\eta N}]\,\Im
mF_{\eta N}\,(r_{0}\sqrt{\pi})^{-3}$.

Thus, in case of the zero energy resonance, we find the optical potential
parameters, ($V_{0}$, $W_{0}$) to be (-68.04, -3.12) MeV and (-68.04, -4.55)
MeV for the GW and CS $\eta$-nucleon interactions respectively. Repeating the
exercise for a different value of the Gaussian parameter, $r_{0}=1.373$ fm as
in [6], the potential parameters are found to be (-58.01, -3.2) MeV and
(-58.01, -4.9) MeV for the GW and CS $\eta$-nucleon interactions respectively.

The above method of first considering the E=0 state of a real Gaussian
potential to determine $V_{0}$ and then finding $W_{0}$ seems a posteriori
justified considering the small values of $W_{0}$ (as compared to $V_{0}$)
obtained. Indeed a similar procedure of first finding the binding energy by
considering only the real part of the potential and later finding
$\Gamma=-2<\Psi|\Im mV_{\eta A}|\Psi>$ using perturbation theory where $\Psi$
is the solution of the real Hamiltonian has been used in [10] too.

Finally, motivated by the above discussion, a renewed search for the
$\eta$-4He states within the model of [6] is performed. At the edge of the
allowed region in Fig. 4, very narrow and weakly bound states of $\eta$-4He,
with binding energies and widths in the range of $\sim$ 2 - 230 keV and $\sim$
8 - 64 keV respectively are found by solving the Klein Gordon equation as in
[6]. These states correspond to the optical potential parameters $|V_{0}|$ in
the range from 58 to 65 MeV and $W_{0}$ = 0.5 MeV (red dots in Fig. 4). For
values of $|V_{0}|<$ 58 MeV, no bound states are found. We should mention
here, however, that some of the potential parameters are not consistent with
the experimental data on the $\eta$ production cross section above threshold
as reported in Ref. [6], especially for the cases with weak absorption. Hence
we think that a systematic analysis including both the escape and the
conversion cross sections covering the above- and below-threshold region is
necessary in order to investigate the weak absorptive potential region.

##  6 Subthreshold considerations and uncertainties

The $\eta$-nucleus optical potentials are in principle energy dependent and
would depend strongly for example on the energy at which the elementary
$\eta$N amplitude, $F_{\eta N}$, of Eq. (3) is evaluated. In the case of
$\eta$-mesic nuclei, the $\eta$N interaction happens at subthreshold energies
and $F_{\eta N}$ should be evaluated at an energy shifted by an amount
$\delta$ below threshold. The importance of taking such a downward shift into
account has been discussed with different points of view in literature [59,
60, 61, 62]. The authors in [59, 60] (and references therein) provide a
detailed analysis of this topic and derive an expression for $\delta$ which
depends on the nuclear binding energy per nucleon as well as the real part of
the optical potential itself. Refs. [61, 62], however, provide a simpler
method with $\delta$ given by the average binding of the target nucleons.

Since the experimental analysis of the present work relies on the input from
the theoretical calculations in [6] where the above effects were not taken
into account explicitly, we shall now try to estimate the uncertainties on
$\sigma_{upp}$ (shown by the red mesh in Fig. 5) introduced by this omission.
To obtain this estimate, we evaluate the optical potential parameters $V_{0}$,
$W_{0}$ using Eq. (3) by comparing them with the form
$V(r)=[V_{0}+iW_{0}]\exp(-{r^{2}\over r_{0}^{2}})$. Thus, as observed in the
previous section, $V_{0}=-[6\pi/\mu_{\eta N}]\,\Re eF_{\eta
N}\,(r_{0}\sqrt{\pi})^{-3}$ and $W_{0}=-[6\pi/\mu_{\eta N}]\,\Im mF_{\eta
N}\,(r_{0}\sqrt{\pi})^{-3}$. Evaluating $F_{\eta N}$ at threshold and at 7 MeV
(binding energy per nucleon for 4He) and 30 MeV below threshold, we obtain the
optical potential parameters given in Table 2 for different models of $F_{\eta
N}$ [54, 55, 63, 64, 65] in literature. The values of $\sigma_{upp}$
corresponding to the parameters $V_{0}$, $W_{0}$ in Table 2 are read off from
Fig. 5 and listed in Table 3. Even if the optical potential parameters do
change a lot depending on the choice of the energy at which $F_{\eta N}$ is
evaluated, the upper limits on the cross sections do not seem to be very
sensitive to this change. Depending on the model, the change in the upper
limits can be between 0 - 6 %. Given that the upper limits $\sigma_{upp}$
determined in the present analysis are not very sensitive to the parameters
$V_{0}$, $W_{0}$ (see Table 1 and Fig. 5), such a small uncertainty was
expected.

Figure 5: Upper limits on the cross sections, $\sigma_{upp}$ in nb, as a function of the optical potential parameters $V_{0}$ and $W_{0}$. The red mesh represents the values determined in the present analysis (as in Table 1). The symbols are the values of $\sigma_{upp}$ corresponding to $V_{0}$, $W_{0}$ given in Table 2 for the different $\eta$N models. The black symbols correspond to $\sigma_{upp}$ for $V_{0}$, $W_{0}$ evaluated using $F_{\eta N}$ at a subthreshold $\eta$N centre mass energy of $\sqrt{s}$ \- 7 MeV and the blue symbols with $F_{\eta N}$ at threshold.  $F_{\eta N}$ |  $\delta$=0 |  |  |  $\delta$=-7 |  |  |  $\delta$=-30 |  |
---|---|---|---|---|---|---|---|---|---
| -$V_{0}$ | -$W_{0}$ | $\sigma_{upp}$ | -$V_{0}$ | -$W_{0}$ | $\sigma_{upp}$ | -$V_{0}$ | -$W_{0}$ | $\sigma_{upp}$
CS [55] | 97.7 | 21.9 | 6.5 | 72.5 | 10 | 6.88 | 44.3 | 2.7 | -
M2 [63] | 54.9 | 28.8 | 6.6 | 45.2 | 22 | 6.59 | 26.6 | 13 | -
KSW [64] | 68.4 | 32.6 | 6.57 | 56.8 | 22 | 6.67 | 38.7 | 13 | 6.5
IOV [65] | 42.8 | 37.8 | 6.55 | 36.35 | 27.8 | 6.46 | 20.16 | 16.5 | -
GW [54] | 139 | 43.6 | - | 104 | 23.7 | - | 71.7 | 8.1 | 6.95
Table 3: Optical potential parameters $V_{0}$ and $W_{0}$ (in MeV) evaluated
using (3) with the $\eta$-N amplitude $F_{\eta N}$ evaluated at $\delta=0$
(threshold), $\delta$ = -7 MeV and $\delta$ = -30 MeV, with
$\delta=\sqrt{s}-\sqrt{s_{th}}$. The upper limits on the cross sections listed
in this table are read from the mesh (representing the $\sigma_{upp}$ (in nb)
determined in the present analysis) in Fig. 5 at the values of $V_{0}$ and
$W_{0}$ in this table.

##  7 Summary and Conclusions

We performed an analysis in order to constrain the $\eta$-4He optical
potential parameters by comparing a recently developed theoretical model for
$\eta$-${}^{4}\mbox{He}$ bound state production in $dd\rightarrow$
${}^{3}\mbox{He}N\pi$ reactions [6] with the experimental data collected by
WASA-at-COSY [1]. Convoluting the theoretical cross section with experimental
resolutions, we estimated the upper limits of the total cross sections for the
formation of the $\eta$-mesic Helium nuclei in $dd\rightarrow$
${}^{3}\mbox{He}N\pi$ processes at a 90% confidence level. Comparison of the
determined upper limits for the creation of $\eta$-mesic nuclei via the
$dd\rightarrow$ ${}^{3}\mbox{He}N\pi$ process with the cross sections obtained
in Ref. [6] excludes a wide range of $\eta$-${}^{4}\mbox{He}$ optical
potential parameters. With the values of $|V_{0}|$ and $|W_{0}|$ being
restricted to be less than 60 MeV and 7 MeV respectively, most predictions of
$\eta$-mesic helium states seem to be excluded within the present analysis.
Extremely narrow and loosely bound states within the model of [6] seem however
to appear in the allowed region of the optical potential parameters.

In spite of some shortcomings such as the absence of the explicit inclusion of
the strong energy dependence of the $\eta$N interaction [60] and the fact
that, in principle, the $\eta$-helium nuclei should be treated within a few
body formalism [8, 10, 66, 67], it is worth noting that in the decades long
search for $\eta$-mesic nuclei, the present work is indeed a first attempt to
combine the experimental data below $\eta$ production threshold with
predictions from a theoretical model which can reproduce the existing data
above threshold too. There exist approaches such as the coupled channels
generalization of the optical potential [24] which can bring out interesting
aspects related to the existence of the $\eta$-mesic helium. Hence, it is
hoped that the optical model analysis of the present work should provide
guidance in narrowing down the search for $\eta$-mesic ${}^{4}\mbox{He}$.

##  8 Acknowledgements

We acknowledge the support from the Polish National Science Center through
grant No. 2016/23/B/ST2/00784 and the Faculty of Science, Universidad de los
Andes, Colombia, through project number P17.160322.007/01-FISI02. This work
was partly supported by JSPS KAKENHI Grant Numbers JP16K05355 (S.H.) and
JP17K05443 (H.N.) in Japan.

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