# Recyclable Organic Bilayer Piezoresistive Cantilever for Torque Magnetometry
at Cryogenic Temperatures

###### Abstract

Flexible sensors made from organic bilayer films of molecular conductor on
polymeric matrix have attracted many interest due to their simple fabrication
with high potential for being scaled up, and for their high-performing multi-
functionality at room temperatures. In particular, the piezoresistive property
of the organic bilayer film is among one of the highest ever reported,
allowing its utilization in various sensing applications. In this work, we
present the study of the flexural piezoresistivity of an organic bilayer film
based on $\beta-\rm(BEDT-TTF)_{2}I_{3}$ on polycarbonate matrix from room
temperatures down to cryogenics temperatures. Non-trivial temperature
dependent profile of the gauge factor is revealed, including enhancement of
the gauge factor from $\sim 18$ at room temperatures to $\sim 48$ at 4.3 K. An
organic bilayer cantilever magnetometer is developed and demonstrated to
measure magnetic properties of a single crystalline organic superconductor
$\kappa-\rm(BEDT-TTF)_{2}Cu(N(CN)_{2})Br$ at temperatures down to $\sim 2.75$
K and magnetic fields up to 5 T. The high-performing bilayer devices can be
fabricated in a very simple manner, and they are robust and recyclable.

###### keywords:

Organic bilayer film, Recycable, Gauge factor, Cantilever magnetometer,
Cryogenic temperatures, $\kappa-\rm(BEDT-TTF)_{2}Cu(N(CN)_{2})Br$

Eden Steven*, Danica Krstovska*, Daniel Suarez, Tasya Berliana, Eric
Jobiliong, Eun Sang Choi

E. Steven
Emmerich Research Center, Pluit Karang Permai 12 B9 Barat no 1 and 2, DKI
Jakarta, 14450, Indonesia
E-mail<EMAIL_ADDRESS>
D. Krstovska
Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius
University, Arhimedova 3, 1000 Skopje, North Macedonia
<EMAIL_ADDRESS>
D. Suarez, E. S. Choi
National High Magnetic Field Laboratory and Department of Physics, 1800 East
Paul Dirac Drive, Tallahassee, FL 32310, USA
T. Berliana
Department of Electrical Engineering, Universitas Kristen Krida Wacana,
Tanjung Duren Raya No.4, DKI Jakarta, 11470, Indonesia
E. Jobiliong
Faculty of Science and Technology, Universitas Pelita Harapan, M.H. Thamrin
Boulevard 1100 Lippo Village, Tangerang 15811, Indonesia

##  1 Introduction

The role of electronic sensors encompasses almost every aspect of life,
including but not limited to the development of smart energy saving systems,
robotics, health and environmental monitors, and instrumentations for
scientific discoveries. In some cases, it is advantageous for the sensors to
exhibit mechanical flexibility, in addition to their intrinsic sensing
properties. For example, flexible sensors based on silk are used for
monitoring brain activity due to their softness and flexibility, therefore,
allowing the sensors to conform to the target sensing area effectively.[1] In
other cases, carbon nanotube-based flexible sensors are used to develop smart
electronic gloves and health monitoring technologies[2, 3]. Conducting
polymers have also been used in consumer electronics, for example as touch
sensors[4, 5].

One of the classes of conducting polymers is known as the organic bilayer
films (BL). This class of conducting polymer is based on charge transfers
salts, a rich system of materials known for its rich physics[6] and multi-
functionalities[7]. The bilayer system is composed of an electrically
conducting active layer and an insulating layer, typically of polycarbonate[8,
9], poly-lactic acid (PLA) [10] polymers, and more[11]. Several charge
transfer salts have been demonstrated to serve as the active materials
including but not limited to those that are based on BEDT-TTF[8, 9], BEDO-
TTF[12, 13], MDF-TSF[10] donor molecules with various possible acceptors such
as $\rm I_{3}$[8, 9], $\rm I_{x}Br_{1-x}$[14], $\rm I_{x}Br_{3-x}$[15].
Depending on the synthesis protocols, multiple phases of these materials can
be made by tuning its electronic properties. For example, $\alpha-\rm(BEDT-
TTF)_{2}I_{3}$/polycarbonate film is semiconducting [11, 16] whereas
$\beta-\rm(BEDT-TTF)_{2}I_{3}$/polycarbonate film is metallic[9, 11]; in some
cases, under ideal growth conditions, $\beta-\rm(BEDT-
TTF)_{2}I_{3}$/polycarbonate film also shows superconductivity[17].

The tunability of bilayer films have been exploited for various devices. For
example, the electrical resistance of $\alpha’-\rm(BEDT-
TTF)_{2}I_{x}Br_{3-x}$/polycarbonate films is extremely sensitive to
temperature[15], allowing resistive temperature monitoring with 0.005
${}^{\circ}\rm C$ accuracy between room temperatures and 60 ${}^{\circ}\rm C$,
thus capable of non-contact radiant heat sensing[18]. In the $\rm(BEDO-
TTF)_{2.4}I_{3}$/polycarbonate bilayer films, rectification and
electrochromism have been observed [13], suggesting potential applications for
flexible diodes and displays. High piezoresistivity has been reported for
$\alpha-$ and $\beta-\rm(BEDT-TTF)_{2}I_{3}$/polycarbonate films that have a
gauge factor of $\sim 10$ and $\sim 18$ at room temperatures, respectively[16,
19], leading to applications for breathing and blood pulse monitoring[19] and
pressure sensors for extreme pressure up to 500 bar[20].

Despite the above mentioned advances, there are vast opportunities for even
further device developments of new technologies that would give rise to a more
efficient and eco-friendly devices. For example, studies of the
piezoresistivity of organic bilayer films have been mainly focused on devices
under longitudinal straining/stressing[11, 16, 19]. Furthermore, most of the
devices are developed for applications at room temperatures[11, 13, 15].

Here we report the physical characterizations and device applications of the
$\beta-\rm(BEDT-TTF)_{2}I_{3}$/polycarbonate, hereafter
$\beta-\rm(ET)_{2}I_{3}$/PC, bilayer film under flexural deformations. The
flexural characteristics are investigated from room temperatures down to
cryogenic temperatures, revealing $\sim 270\%$ increase in the gauge factor at
4.3 K compared to that at room temperatures. The enhanced electromechanical
property is exploited for developing torque magnetometers based on the organic
bilayer film cantilevers. The organic bilayer film cantilevers are simple to
produce, robust, can be easily cut to various shapes and sizes, and
recyclable. As a proof-of-concept, torque magnetometry is carried out using
the organic bilayer film cantilever to study superconductivity and flux
dynamics in an organic superconductor $\kappa-\rm(BEDT-
TTF)_{2}Cu[N(CN)_{2}]Br$[7, 21] at cryogenics temperatures down to $\sim 2.75$
K and at magnetic fields up to 5 T.

##  2 Results and Discussion

###  2.1 Interface robustness and recyclability of organic bilayer films

Prior works have demonstrated that the organic bilayer films exhibit robust
electrical properties and long term stability at ambient conditions[18, 19].
However, it was not clear whether the organic bilayer film would retain its
robustness under flexural deformations. One of the possible problems for
flexible sensors is the sliding/buckling of the active layer when flexed due
to insufficient adhesion between the active layer and the base substrate.
Irreversible damage could occur due to the sliding, buckling, or in the worst
scenario, being peeled off.

To evaluate the interfacial adhesion between the active and base layer, we
have carried out a 180-degree peel test (Figure 1) by subjecting the active
layer to a tape adhesive. No observable damage was seen after the peel test at
a peel rate of 50 mm/min and strength of $\sim 100$ N/m. A more rigorous
peeling and shearing using the tape adhesive was also carried out which shows
no observable damage to the active layer (Supporting Information Video S1).
This simple fact is appreciable when we consider other types of flexible
sensors where the active layer is often separately applied, glued or laminated
on the flexible base substrate[2]. In these cases, it is critical crucial to
avoid physically disturbing the active layer or to implement additional
protective measures such as additional laminates. However, such measures could
prove problematic due to possible reduction in performance, interfacial
mismatch and other problems. Unlike the typical flexible sensors where the
active sensing layer is applied, sprayed, glued or laminated onto a flexible
base [2], the organic bilayer film’s active sensing layer is grown from within
the base substrate[8, 9], and therefore is inherently anchored to the base
substrate like as a plant is rooted in soil.

Figure 1: 180-degree peel test of $\beta-\rm(ET)_{2}I_{3}$/PC bilayer films.
(a) Experimental setup illustration. The bilayer film is first laminated onto
G10 substrate using a double sided tape. The G10 was vertically clamped at a
fixed position. A layer of tape adhesive (6 mm wide) was adhered to the top
surface of the bilayer film using 100 kPa pressure. (b) Photograph of the
sample setup. (c) Peel strength vs time during the peel test.

Organic bilayer films are fabricated through a vapor oxidation reaction of
homogenously dissolved molecular crystals in a polymeric matrix. Essential to
the method is the composition of the vapor which is used to dope the film. The
vapor contains a solvent which acts to swell the film as well as a dopant. In
the case of the $\rm(ET)_{2}I_{3}$/PC bilayer films, the base substrate is
homogenously mixed ET in PC, the dopant is iodine, and dichloromethane is the
swelling agent.

As the iodine/dichloromethane reaches one of the surface of the ET/PC films,
due to concentration gradient in the swelling side, ET crystals migrate and
accumulate on the surface where they are then doped with the iodine. As a
result, a self-assembly of $\rm(ET)_{2}I_{3}$ micro-crystallites occurs
forming physically and electrically interconnected network of polycrystalline
surface. After the doping process, as the swelling subsides, the micro-
crystallite network is further densified forming a robust conducting active
layer on one side of the film, and insulating on the other side of the film.
Thus, in all of these processes, the active layer and the base substrate are
always in union.

It is also worth noting that, we have discovered that the organic bilayer film
can be re-dissolved and reconstituted from the same ingredients it is formed,
i.e., from the same ET and PC materials. Then, it can be re-doped into the
$\beta-\rm(ET)_{2}I_{3}$/PC bilayer film repeatedly without observable
degradation. We show in Figure 2, the photograph of the recycled bilayer film
under an optical microscope showing similar profiles over six recycling
processes. A similar order of sheet resistances of the bilayer in each process
is observed. These observations suggest a simple recycling process; collecting
unused, broken, torn, and mistakenly cut organic bilayer sensors and gathering
them into a single vessel to be recycled into a brand new sheet of sensors for
new uses, thus minimizing the hazard of such sensors to be an environmental
plastic pollutant (Supporting Information Figure S1).

Figure 2: Recycling and reconstitution of the $\beta-\rm(ET)_{2}I_{3}$/PC
organic bilayer sensor. (a) Recycling process of the organic bilayer film. The
dissolved solution is stable at room temperatures. Enclosed glass vial can be
used for long term storage. Various shapes of film can be re-casted and re-
doped depending on the applications. (b) Optical photographs of re-casted and
re-doped film over 6 cycles. The films exhibit sheet resistances similar in
order of magnitude.

###  2.2 Temperature dependent gauge factor of the $\beta-\rm(ET)_{2}I_{3}$/PC
bilayer film

The temperature dependent gauge factor of $\beta-\rm(ET)_{2}I_{3}$/PC film is
obtained through temperature dependent 4-point resistance measurement at
various constant flexed conditions. The full description of the method is
given in the Materials and Methods section. Briefly, the film is mounted
without being clamped to a linear stage by resting the film against
rectilinear edges (Figure 3a). By decreasing the linear stage distance, one
can induce higher flex to the film. After obtaining multiple temperature
dependent resistance curves at various flex conditions, the resistance versus
strain at various temperatures was obtained.

The inset in Figure 3a shows an overlay of optical photographs of the films at
various flex by controlling the linear stage distance. The radius of curvature
is extracted digitally (Supporting Information Figure S2) which is then used
to calculate the corresponding linear strain on the film using the standard
equation $\varepsilon=t/2r$, where $\varepsilon$ is the linear strain, $t$ is
the thickness of the film ($\sim 25$ $\mu$m), and $r$ is the radius of
curvature. In Figure 3b, the room temperature resistance corresponding to each
radius of curvature is plotted, which shows reversible linear dependence
during flexing and relaxing processes as expected for elastic behavior. The
gauge factor is determined to be $\sim 17.5$ which is consistent with the
result obtained using linear straining method reported in previous works[11,
19].

Figure 3: Room temperature flexing gauge factor determination. (a) Photograph
of the sample setup. Inset: side-view optical photograph of the
$\beta-\rm(ET)_{2}I_{3}$/PC film under various flex. (b) Normalized resistance
(against the resistance of flat film at room temperature $\sim 297$ K) vs
linear strain as the film is being flexed (red circles) and relaxed (blue
rectangles). Black line represents a linear fit whose slope corresponds to a
gauge factor of $\sim 17.5$.

There are several interesting notes concerning the benefits of determining the
gauge factor of thin films via flexural deformation compared to that in
typical linear straining methods. First, since the strain relates only to the
thickness and radius of curvature, it is not necessary to measure the initial
length of the film thus minimizing the errors that may come from inaccurate
electrical contact distance measurement. Second, since the film is not clamped
down (not possible for linear straining method), there is much less risk of
sample damage due to over clamping or slippage due to under-clamping.
Furthermore, as we cool the film down to cryogenics temperatures, the linear
straining method could be affected by thermal contraction which induces
internal stress to the film if not compensated.

This is especially important for the PC film since at 4.2 K it contracts up to
$1.5\%$ of its initial length at room temperatures[22]. If such film was
clamped down on a linear stage, the internal strain induced by the cooling
down process would result in an irreversible damage as the contraction would
go past its elastic limit of $\sim 1\%$ strain[19]. In our setup, the film is
not clamped but held under natural tension as it is flexed. This allows
thermal contraction of the PC film to occur freely. As the film is cooled
down, the length of the film decreases which actually reduces the degree of
flex of the film (i.e., increases the radius of curvature). In our gauge
factor measurement, we have taken into account the thermal contraction effect
of the film, and compensated the increasing radius of curvature during the
cool down process. The full detail of the thermal expansion correction is
given in the Supporting Information Section S3.

The temperature dependent resistances of the $\beta-\rm(ET)_{2}I_{3}$/PC film
at various flex are shown in Figure 4a and 4b. Seven curves are obtained and
sliced to plot the resistance vs strain at various temperatures from which the
gauge factor was extracted from the linear slope. Thus, the gauge factor is
obtained at various temperatures and shown in Figure 4c and 4d.

Figure 4:  Temperature dependent electromechanical properties of
$\beta-\rm(ET)_{2}I_{3}$. Resistance vs temperature data at various strain:
(a) from room temperature and (b) from 80 K down to 4.3 K. (c, d) Temperature
dependent gauge factor extracted from data in (a) and (b), respectively.
Thermal contraction correction has been applied for both data shown in (c) and
(d). A kink is observed at $\sim 55$ K from the temperature dependent gauge
factor. We note from the resistance vs temperature data, that no observable
features are seen at $\sim 55$ K. The black line represents the exponential
fit line.

The obtained results allow to make several interesting conclusions. First,
note that at low temperatures, there is an upturn in the resistance as the
sample is cooled down below $\sim 10$ K. In a perfect single crystal of
$\beta-\rm(ET)_{2}I_{3}$ full metallic behavior was reported[7]. The lack of
the full metallic behavior in the bilayer film may suggest either slightly
incomplete transformation of the $\beta-\rm(ET)_{2}I_{3}$ from the
$\alpha-\rm(ET)_{2}I_{3}$ crystallites or imperfections due to grain
boundaries between the micro-crystallites in the active layer. This proved to
be advantageous for application of the device under consideration as described
in the text below. Second, a significant enhancement of gauge factor at lower
temperatures for the $\beta-\rm(ET)_{2}I_{3}$/PC bilayer film has been
revealed. Indeed, the gauge factor increases linearly between room temperature
down to about 120 K, with gauge factor to temperature ratio of 0.102/K. Then
it transitions into a monotonic plateau state with only a slight gauge factor
increase between $\sim 120$ K to $\sim 60$ K. At $\sim 60$ K, a drop in the
gauge factor is observed which is correlated with the formation of the
thermally activated state; the gauge factor is exponentially increasing as the
temperature is further lowered down to $\sim 4.3$ K (the lowest temperature in
our measurements) according to the following equation, $\rm
GF$$=y_{0}+Ae^{-T/B}$, where $\rm GF$ is the gauge factor, $T$ is the
temperature and $y_{0},A$ and $B$ are fitting constants with values of 37.125,
15.244 and 13.510 K, respectively. Overall, from temperatures 300 K down to
4.3 K, an incredible increase of a $\sim 270\%$ in the gauge factor of the
bilayer film is observed (from $\sim 18$ up to $\sim 48$, respectively).

###  2.3 Fabrication and device setup of the organic bilayer cantilever
magnetometer

The high piezoresistivity of the $\beta-\rm(ET)_{2}I_{3}$/PC bilayer film at
low temperatures opens up new venues for applications at cryogenics
temperatures. Here the utilization of the bilayer film as cantilever
magnetometers is elaborated.

Conventionally, piezoresistive highly doped Si-based cantilevers are used in
torque magnetometry. The gauge factor of the device is $\sim 140$. Typical
arrangement consists of two levers, a reference and sensing levers, both
incorporated into a Wheatstone bridge circuit. A sample under test is then
glued at the end of the sensing lever where the bridge is balanced and then
exposed to an external magnetic field. As the sample experiences torque and/or
force by the external magnetic fields, the cantilever deflects according to
the magnetic properties of the sample. The deflection is detected and recorded
as differential bridge voltages. The Si-based device is highly sensitive and
reliable for a wide range of measurements. However, fabrication complexities
of the device are inherent in the Si-based devices. The fabrication process of
such cantilever requires specialized facilities and complex lithographic
processing. Also, from practical perspective, sample mounting can be a
challenge, especially for a larger-sized sample. Due to limited sample
mounting space, ($0.3\sim 0.5$ mm) x (0.1 mm), care must be taken in order to
avoid the sample mounting adhesive from contaminating the piezoresistive
element as that will impair the sensitivity. Furthermore, as in any Si-based
technologies, the levers are extremely rigid and brittle, rendering the lever
prone to damages due to unexpectedly large magnetic forces or torques of the
sample or unintended stress during sample mounting.

Unlike the Si-based cantilevers, the fabrication process of the organic
bilayer cantilever is very simple. As shown above (Figure 1), the robustness
of the organic bilayer film allows to simplify the fabrication process to only
a few low-technology steps, avoiding usually needed processes in the
lithographic methods such as evaporative deposition, spin casting, washing,
bonding, laminating, encapsulation and more.

First, a piece of the organic bilayer film is generated. From this a smaller
piece is cut to a desired shape using common tools such as scissors, blades or
lasers, on which electrodes are finally attached by using carbon paste. In
Figure 5, a setup of a prototype of the organic bilayer cantilever device is
shown that is cut to a desired size by using scissors. Two pieces of the
cantilever film ($\sim 1.5$ mm by $\sim 4$ mm) are prepared and mounted on a
G10 substrate using PVA (polyvinyl alcohol) glue such that about 3 mm of the
lever is protruding outward from the G10 base. The electrodes are established
on each of the levers using carbon paste and 24 micron gold wires, after
which, they are incorporated into a Wheatstone bridge configuration (Figure 5b
inset). The sample under test is mounted at the tip of the sensing lever using
a thin layer of silicone vacuum grease.

Figure 5: Prototype of the $\beta-\rm(ET)_{2}I_{3}$/PC organic bilayer
cantilever devices. (a) Schematic illustration of the setup. (b) Photographs
of the setup. The bilayer cantilever is $\sim 1.5$ mm wide and 4 mm long,
mounted on a G10 substrate using PVA (polyvinyl alcohol) glue. Inset:
Wheatstone bridge circuit; RS is resistance of the sensing lever where the
sample is mounted; RREF is resistance of the reference lever; R1 and R2 are
variable resistances for balancing the bridge. Sample under test is
$\kappa-\rm(ET)_{2}Cu(N(CN)_{2})Br$ and it is mounted on the tip of the
sensing lever using thin layer of grease.

###  2.4 Testing the organic bilayer cantilever magnetometer to monitor
superconductivity and flux relaxation in $\kappa-\rm(ET)_{2}Cu(N(CN)_{2})Br$

To evaluate the performance of the organic bilayer cantilever at low
temperatures and high magnetic fields, we have used a single crystal of
$\kappa-\rm(ET)_{2}Cu(N(CN)_{2})Br$ as the test sample (Figure 5b). This
material is among the organic superconductors with the most robust
superconductivity[7, 21]. In Figure 6a, the field dependent torque
measurements at various temperatures are presented with field direction
perpendicular to the conducting plane of the crystal. Both field up and down
sweep data are obtained which show strong hysteretic torque signals that are
characteristic of type II superconductors. A systematic increase in the peak
signal and the critical field of the hysteresis is also observed with lowering
the temperature. This behavior is expected for a superconductor as the
superconductivity becomes more robust at low temperatures.

To further check whether the signal observed is coming from the sample or from
the lever itself, we have also obtained the background differential signal of
the Wheatstone bridge without sample attached. The result is shown in Figure
6b. It is clearly seen that the background differential signal of the levers
is generally monotonic without hysteretic features.

Additional measurement is as well performed for the purpose of confirmation
that the magnetoresistance of a single lever does not show special features
that may complicate sample data interpretation (Supporting Information Figure
S4). However, in single crystal $\beta-\rm(ET)_{2}I_{3}$, both
superconductivity ($T_{c}\sim 1.4$ K) and quantum oscillations are known to
occur[7]. In the considered bilayer film, these features are not observed most
likely due to the same reasons that lead to the appearance of an upturn in the
film’s resistance at low temperatures (Figure 4). Indeed, in single crystal
$\beta-\rm(ET)_{2}I_{3}$, superconductivity is observed below $\sim 7$ T while
above this field a beating pattern is present in the SdH quantum oscillations.
The beating pattern has also been revealed in the dHvA quantum oscillations
with a higher amplitude than that registered for the SdH oscillations. The
fact that these properties are not present in the $\beta-\rm(ET)_{2}I_{3}$/PC
film suggests that the formation of the grain boundaries in the film’s active
layer induces disorder in the $\rm I^{-3}$ anions causing a change in the
phonon modes responsible for the formation of the Cooper pairs. This, on the
other hand, changes the prerequisites for the appearance of superconductivity
and therefore it is absent in the film. Apart from that, the additional
disorder significantly increases the electron scattering that may lead to a
suppression of the fast magnetic quantum oscillations in the film to an extent
that they cannot be experimentally resolved. Thus, the presence of a slight
imperfection in the organic bilayer film turns out beneficial for the intended
purpose.

The true signal from the sample is then obtained by subtracting the data shown
in Figure 6b from those given in Figure 6a. The true signal is shown in Figure
6c. By comparing the raw and true signal one finds that the overall features
in both signals appear to be very similar.

Figure 6: Torque vs field sweeps of $\kappa-\rm(ET)_{2}Cu(N(CN)_{2})Br$ single
crystal at various temperatures. (a) As recorded sample plus background
signals. (b) As recorded background signal. (c) Extracted sample signal after
background subtraction. The loop indicates superconductivity of the sample.

Most strikingly, the torque signal of the bilayer cantilever changes very
smoothly at low temperatures as indicated by the absence of sharp step-like
jumps in the data shown in Figure 6. Such a smooth deflection is also
reflected when flux relaxation measurements were performed at various
temperatures (Figure 7). The flux relaxation measurement is carried out at
various constant temperatures by first ramping up the magnetic field to 1 T at
a rate of 0.02 T/s after which the field is held constant and the voltage
differential signal is monitored over time, corresponding to the flux
relaxation of the sample. The flux relaxation follows a typical exponential
relaxation behavior as evident from Figure 7. The extracted relaxation time
constant $\tau$, is also confirmed to be decreasing with increasing
temperatures (inset in Figure 7). This completes the validation that the
organic bilayer cantilever is a viable torque magnetometer that works at
cryogenics temperatures and high fields.

Figure 7: Flux relaxation measurements of $\kappa-\rm(ET)_{2}Cu(N(CN)_{2})Br$.
The magnetic field is first swept from 0 to 1 T with a 0.02 T/s rate and then
the field is held constant. The curves show the voltage differential signal
while holding the field at various temperatures. The black solid line is the
exponential fit to the curves. Inset: Extracted time constant $\tau$ vs
temperature.

##  3 Conclusion

This work highlights the simplicity, versatility, robustness, recyclability
and effectiveness of the bilayer films as flexural devices. The deep anchoring
of the $\beta-\rm(ET)_{2}I_{3}$ crystallites into the polycarbonate base
renders the material highly compliant to flexural deformations. Combined with
migration and then the densification of the crystallite network during the
vapor annealing, a robust grain boundary is achieved resulting in a stable
electromechanical response down to cryogenic temperatures. Electromechanical
characterization of the active sensing layer down to cryogenic temperatures is
carried out via iso-flexural method. The thermal contraction corrected profile
reveals a non-trivial enhancement of the temperature dependent profile of the
gauge factor with an observed state transition at around 60 K where the gauge
factor becomes exponentially activated as the temperature is lowered. A
remarkable increase in the gauge factor from $\sim 18$ to $\sim 48$ is found
at temperatures of $\sim 297$ K to $\sim 4.3$ K, respectively. Using the
organic bilayer film as cantilevers, magnetic torque measurements are
performed on the organic superconductor $\kappa-\rm(ET)_{2}Cu(N(CN)_{2})Br$
sample, where the signatures of superconductivity are clearly observed.
Finally, this class of flexible sensors is also unique in that it can easily
be re-solubilize and re-doped into pristine film without observable
degradation, an aspect that is increasingly becoming important to consider.

##  4 Experimental Section

$\beta-\rm(ET)_{2}I_{3}$/polycarbonate bilayer film fabrication BEDT-TTF (ET)
crystals were homogenously dissolved at 130 ${}^{\circ}\rm C$ in polycarbonate
(PC) matrix at ET:PC ratio of 2:98 using Anisole as the solvent with
solid:solvent ratio of 1.5:50 (w/v), for example, 1.5 gram of solids in 50 mL
solvent. The ET/PC solution was then drop casted on a glass petri dish and
then dried in an oven or on a hotplate at 130 ${}^{\circ}\rm C$, for $\sim 20$
minutes until dry. Separately, a saturated $\rm I_{2}$ solution in
dichloromethane were prepared and incubated in a glass trough with a petri
dish as a cover at 25 ${}^{\circ}\rm C$ for 45 minutes to generate a stable of
iodine/dichloromethane vapor. Subsequently, the petri dish cover was replaced
by the petri dish containing the ET/PC dried film and the incubation at 23
${}^{\circ}\rm C$ continued for another 3-4 minutes where the
iodine/dichloromethane vapor swells the surface of the ET/PC film, causing ET
crystals to migrate to the surface where they react with the iodine forming a
densified polycrystalline layer of $\alpha-\rm(ET)_{2}I_{3}$[11]. Then, the
$\alpha-\rm(ET)_{2}I_{3}$/PC film was peeled off from the dish and annealed at
150 ${}^{\circ}\rm C$ for 15 minutes in order to allow for
$\alpha-\rm(ET)_{2}I_{3}$ to convert into $\beta-\rm(ET)_{2}I_{3}$. To prevent
iodine from escaping, the film was sandwiched between two glass slides during
the annealing process.

Temperature dependent gauge factor determination A cryostat equipped with a
metallic linear sliding stage was used to mount the film sample. A 10 mm by 2
mm film was cut and placed resting on a rectilinear edge on the two sides of
the stage without being clamped. The degree of flexing of the sample can be
increased by decreasing the distance between the two ends of the stage. A four
probe electrical configuration was employed via the use of carbon paste and
thin 12 micron gold wires in order to ensure minimal impact of wire stiffness
on the flexing of the film sample. The distance between the inner two contacts
is 7 mm. Electrical resistance measurements were done using a combination of
constant current source (Keithley 6221) and nanovoltmeter (Keithley 2182A). To
obtain the gauge factor at various temperatures, multiple resistance vs
temperature curves are obtained at various constant flex. First, the sample is
flex to a desired level. Then, a scale-calibrated photograph is obtained from
the side of the stage, where the radius of curvature of the flexed film is
extracted using IgorPro image digitization and mathematical fittings. The
sample was then cooled down from room temperature down to 4.3 K, after which
the process repeats for the next constant flex conditions. In total, 12
different sets of resistance vs temperature curves are obtained at various
degree of flex that corresponded to linear strain of up to $\sim 0.42$. By
slicing the resistance vs temperature curves, we were able to plot and linear
fit the resistance vs strain curves, obtained at various temperatures, from
which the temperature dependent gauge factor was determined.

Thermal contraction consideration and correction Thermal contraction of the
polycarbonate film affects the radius of curvature as the film is cooled down.
To account for this, we have included a temperature dependent correction
factor for the length of the film as it has been cooled down based on the data
provided by G. Schwarz[22]. The thermal contraction of the film lowers the
degree of flex that corresponds to an increase in the radius of curvature of
the film. The details of the calculations are given in Supporting Table S1.
Thermal contraction of the linear sliding stage may affect the measurement.
However, since the stage was made of phosphor bronze, it has a much less
pronounced thermal contraction compared to that of polycarbonate. For example,
at most, cooling down to $\sim 10$ K would contract the stage by $\sim 0.3\%$
of its initial lengths at room temperature[23], whereas for polycarbonate it
would contract by $\sim 1.5\%$ under similar conditions[22]. Thus, the thermal
contraction effects of the stage is not considered in this work.

Torque magnetometry of $\kappa-\rm(ET)_{2}Cu(N(CN)_{2})Br$ Two pieces of films
were cut and mounted side by side on a substrate where the films were allowed
to extend freely beyond the substrate to serve as levers. The two levers were
setup in a Wheatstone bridge configuration, where one served as the reference
lever and the other served as the sample lever. A single crystal of the
organic superconductor $\kappa-\rm(ET)_{2}Cu(N(CN)_{2})Br$ was mounted under
the lever on the insulating side of the bilayer film using a minimal amount of
grease. A lock-in-amplifier (Stanford Research SR830) was used to provide
driving voltage and to measure the differential voltage from the bridge
circuit at various magnetic fields and temperatures. The measurements were
carried out in the National High Magnetic Field Laboratory (NHMFL) millikelvin
facility.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the
author.

Acknowledgements

We thank Emmerich Research Center and National High Magnetic Field Laboratory
for the support and access to the lab facilities. A portion of this work was
performed at the National High Magnetic Field Laboratory, which is supported
by National Science Foundation Cooperative Agreement No. DMR-1644779 and the
State of Florida. We also acknowledge NSF-DMR 1309146 for financial support.

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Consider nodes $A$ and $M$, and node sets $Z$ and $W=\{W_1,\dots,W_k\}$ in an ADMG $\g = (V,E)$, such that $V=\{V_1,\dots,V_q\}$ is generated from a linear structural equation model compatible with $\g$. Let $T=(Z,W)$ and consider the residual $\delta_{a.w}=A-\beta_{aw}W$. If the coefficient for $\epsilon_m$ in the equation for $\delta_{a.w}$ is non-zero, then there are two cases: First, if $M \in W$ then there exists a path $p$ from $A$ to $M$ that ends with an edge pointing into $M$ and which is open given $W$. Second, if $M \notin W$, then there exists a causal path from $M$ to some node $M' \in \{A\} \cup W$ such that the coefficient of $\epsilon_{m'}$ is non-zero and there exists a path from A to M that is open given W.

Consider also the difference of residuals $\delta_{a.w} - \delta_{a.zw}=\beta_{at}T - \beta_{aw}W$. If the coefficient for $\epsilon_m$ in the equation for $\delta_{a.w} - \delta_{a.zw}$ is non-zero, then there exist two cases: First, if $M \in W$ then there exists a path $p$ from $Z$ to $M$ that ends with an edge pointing into $M$ and which is open given $W$. Second, if $M \notin W$, then there exists a causal path from $M$ to some node $M' \in (Z \cup W)$ such that the coefficient of $\epsilon_{m'}$ is non-zero and there exists a path from M to Z that is open given W.

The first statement is a generalization of Lemma B.4 of 16, which is for directed acyclic graphs, to acyclic directed mixed graph. Using Lemma <ref>, it can be shown with the exact same arguments as in the proof of Lemma B.4 which we now give for completeness.

We first rewrite the random variable $\delta_{a.w}$ as a linear combination of the errors from the underlying linear structural equation model as follows:

\begin{align*}
\delta_{a.w}
&=\sum_{V_j \in V}\tau_{av_j} \epsilon_{v_j}
-\sum_{V_j \in V}\beta_{aw} \tau_{wv_j} \epsilon_{v_j}\\
&=\sum_{V_j \in V} \gamma_{v_j} \epsilon_{v_j},
\end{align*}

with $\gamma_{v_j}=\tau_{av_j}-\beta_{aw} \tau_{wv_j}$ the coefficient corresponding to $\epsilon_{v_j}$.

We now prove the first half of the first statement of the Lemma by contraposition. So suppose $M \in W$ and assume that there is no path from $A$ to $M$ that is open given $W$ and ends with an edge pointing into $M$. Under these assumption Lemma <ref> holds and therefore

\begin{align*}
\gamma_M=\tau_{aw_j}-\beta_{aw} \tau_{ww_j}=0,
\end{align*}

where we assume that $M=W_j$.

We now prove the second half of the first statement of the Lemma. So suppose $M \notin W$ and let $W'=\{W_1,\dots,W_k,A\}$. Then

\begin{align*}
\gamma_M
&=\tau_{am}-\beta_{aw} \tau_{wm}\\
&=\sum_{W_i \in W'}\tau_{w_im.w'_{-i}}(\tau_{aw_j}-\beta_{aw} \tau_{ww_j})\\
&=\sum_{W_i \in W'}\tau_{w_im.w'_{-i}}\gamma_{w_j},
\end{align*}

where we use that by Lemma B.7 of 16,
$\tau_{w_im} = \sum_{W_j \in W'} \tau_{w_jm.w'_{-j}} \tau_{w_iw_j}$.
Lemma B.7, although stated for directed acyclic graphs, also holds for acyclic directed mixed graphs as all of its arguments only depend on the linear structural equation model via the matrix of edge coefficients $\mathcal{A}$. Therefore, $\gamma_M$ is only non-zero if there exists a $W_j \in W$ such that $\tau_{w_jm.w'_{-j}} \neq 0$ and $\gamma_{w_j} \neq 0$, that is, if there exists a causal path $p$ from $M$ to some $W_j \in W'$ (that does not contain other nodes in $W'$) and the coefficient for $\epsilon_{w_j}$ is non-zero. The latter implies that there exists a path $p'$ from $A$ to $W_j$ that is open given $W$ and whose last edge points into $W_j$ by what we have already shown in this proof.

It remains to show that there exists a path from $M$ to $A$ that is open given $W$. If $W_j=A$, $p$ is such a path so assume this is not the case. Consider the first node $I$ on $p$ where $p$ and $p'$ intersect and let $q=p(M,I) \oplus p'(I,A)$. If $I=M$ or $I=A$, $q$ is a subpath of $p'$ respectively $p$ and therefore open given $W$. Suppose first that $I \in W$. Then $I=W_j$, as $W_j$ is the only node in $W$ on $p$. But then $W_j$ is a collider by the fact that the edge on $p'$ points into $W_j$ and $p$ is causal. Therefore, $q$ is open given $W$. If $I \notin W$ then by the fact that any node on $p$ has $W_j$ as a descendant by nature of $p$, $W_j \in \de(I,\g)$ and therefore $q$ is open irrespective of whether $I$ is a collider on $q$ or not.

For the second statement of the lemma we will show that if the coefficient for $M$ in the equation for $\delta_{z.w}$, denoted $\gamma'_m$ is zero, so is the one in the equation for $\delta_{a.w} - \delta_{a.zw}$, denoted $\gamma_m$. We can write

\begin{align*}
\gamma_m
&= \beta_{az.w} \tau_{zm} + \beta_{aw.z} \tau_{wm}  - \beta_{aw} \tau_{wm} \\
&= \beta_{az.w} \tau_{zm} + \beta_{aw.z} \tau_{wm}  - (\beta_{aw.z} + \beta_{az.w} \beta_{zw}) \tau_{wm} \\
&= \beta_{az.w} \tau_{zm} -  \beta_{az.w} \beta_{zw} \tau_{wm}
\end{align*}

where we use Lemma <ref> to plug in $\beta_{aw}=\beta_{aw.z} + \beta_{az.w} \beta_{zw}$. This coefficient is zero whenever

\[
\gamma'_{m} = \tau_{zm} - \beta_{zw} \tau_{wm} = 0.
\]

But since we can apply the first half of the Lemma to $\gamma'_{m}$, our statement follows.

Consider node sets $Z$ and $W=\{W_1,\dots,W_k\}$ in an acyclic directed mixed graph $\g=(V,E)$, such that $V$ is generated from a linear structural equation model  compatible with $\g$. Let $W_j$ be some node in $W$, such that all paths from $Z$ to $W_j$ that end with an edge of the form $\rightarrow$ or $\leftrightarrow$ are blocked by $W$, then

\begin{align}
\tau_{zw_j}=\beta_{zw} \tau_{ww_j}.
\label{eq: cancellation}
\end{align}

We show our claim in two steps. First we construct an enlarged linear structural linear equation model, such that there exists valid adjustment sets with respect to $(W_j,Z)$ and $(W_j,W_{-j})$. We use these to replace the terms $\tau_{zw_j}$ and $\tau_{ww_j}$ in Equation (<ref>) with ordinary least squares coefficients. Second, we apply Lemma <ref> to the ordinary least squares coefficients corresponding to $\tau_{zw_j}$ and then simplify the resulting equation by using Lemma <ref> to arrive at Equation (<ref>).

We first enlarge the underlying linear structural equation model as follows. For every node in $V \in \mathrm{sib}(W_j,\g)$ add a node $\epsilon_v$, a directed edge from $\epsilon_v$ to $V$ and a bidirected edge from $\epsilon_v$ to $W_j$. Let $\g'=(V',E')$ denote this enlarged model and its causal graph. Consider the set $P'=pa(W_j,\g)\cup sib(W_j,\g')$ and two sets $A \subseteq \de(W_j,\g)\setminus W_j$ and $B \subseteq V \setminus \de(W_j,\g)$. By construction the set $P'\cup B$ does not contain any descendants of $W_j$. In addition, any non-causal path from $W_j$ to $A$ that starts with an edge into $W_j$, must contain a node in $P'\cup B$ that is a non-collider. The latter is due to the fact that by the construction of $\g'$ no node in $\mathrm{sib}(W_j,\g')$ may be a collider on any path. Therefore, $P'\cup B$ is a valid adjustment set with respect to $(W_j,A)$. In particular, $\tau_{zw_j}=\beta_{zw_j.p'}$ and $\tau_{w_{-j}w_j}=\beta_{w_{-j}w_j.p'}$, where $W_{-j}=W \setminus W_j$. Applying this to equation  (<ref>) we obtain

\begin{align*}
\tau_{zw_j}
&=\beta_{zw_j.p'} \\
&=\beta_{zw_j.w_{-j}p''} + \beta_{zw_{-j}.w_jp''} \beta_{w_{-j}w_j.p'} \\
&=\beta_{zw.p''} \tau_{ww_j},
\end{align*}

where we use Lemma <ref> in the second step and where $P''$ is some maximal subset of $P'$, such that the covariance matrix of $(W,P'')$ has full rank. The covariance matrix $(W_j,P')$ is of full rank by our construction of $\g'$. By assumption $P' \perp_{\g'} Z \mid W$, as any path that would contradict this statement could be extended to a path from $Z$ to $W_j$ whose last edge points into $W_j$ and that is open given $W$. As a result $\beta_{zw.p''}=\beta_{zw}$ and our claim follows.

Consider a random variable $X$, and two random vectors $Z$ and $W$. Then

\[E\{(\delta_{x.w}-\delta_{x.zw})^2\}=\sigma_{xx.w}-\sigma_{xx.zw},\]

where $\delta_{x.w}=X - \beta_{xw} W$ and $\delta_{x.zw}=X - \beta_{xw.z} W - \beta_{xz.w} Z$.

\begin{align*}
&= E(\delta_{x.w})^2-2 E(\delta_{x.w} \delta_{x.zw}) + E(\delta_{x.zw}^2)\\
&= \sigma_{xx.w}-2 E(\delta_{x.w} \delta_{x.zw}) + \sigma_{xx.zw}.
\end{align*}

\begin{align*}
E(\delta_{x.w} \delta_{x.zw})
&= E\{(X-\beta_{xw}W)(X-\beta_{xw.z}W-\beta_{xz.w}Z)^\top\}\\
&= \sigma_{xx} - \Sigma_{xw}\beta^\top_{xw.z} - \Sigma_{xz}\beta^\top_{xz.w} - \beta_{xw} \Sigma_{wx} + \beta_{xw}\Sigma_{ww}\beta^\top_{xw.z} + \beta_{xw}\Sigma_{wz}\beta^\top_{xz.w} \\
&= (\sigma_{xx} - \beta_{xw} \Sigma_{ww} \beta^\top_{wx}) - \beta_{xw} \Sigma_{ww} \beta^\top_{xw.z} - \beta_{xz}\Sigma_{zz}\beta^\top_{xz.w}  + \\
& \quad \quad \beta_{xw}\Sigma_{ww}\beta^\top_{xw.z} + \beta_{xw}\Sigma_{ww}\beta_{zw}^\top\beta^\top_{xz.w}\\
&= \sigma_{xx.w} - \beta_{xz}\Sigma_{zz}\beta^\top_{xz.w}
+ \beta_{xw}\Sigma_{ww}\beta_{zw}^\top\beta^\top_{xz.w} \\
&= \sigma_{xx.w} - (\beta_{xz}\Sigma_{zz} - \beta_{xw}\Sigma_{ww}\beta_{zw}^\top)\beta^\top_{xz.w} \\
&= \sigma_{xx.w} - \{(\beta_{xz.w} + \beta_{xw.z}\beta_{wz})\Sigma_{zz} - (\beta_{xw.z}+\beta_{xz.w}\beta_{zw})\Sigma_{ww}\beta_{zw}^\top\}\beta^\top_{xz.w} \\
&= \sigma_{xx.w} - \beta_{xz.w}(\Sigma_{zz} - \beta_{zw}\Sigma_{ww}\beta_{zw}^\top)\beta^\top_{xz.w} \\
&= \sigma_{xx.w} - \beta_{xz.w}\Sigma_{zz.w}\beta^\top_{xz.w} \\
&= \sigma_{xx.zw}.
\end{align*}

As a result,

\begin{align*}
&= \sigma_{xx.w} - \sigma_{xx.zw}
\end{align*}

Let $A,B,C$ and $D$ be mean 0 random vectors with finite variance, such that $\Sigma_{e_1e_1}$, with $E_1=(B,C,D)$, is not of full rank but $\Sigma_{e_2e_2}$ and $\Sigma_{e_3e_3}$, with $E_2=(B,C)$ and $E_3=(B,D)$, are of full rank.

\[
\beta_{ab.c} = \beta_{ab.c'd} + \beta_{ad.bc'} \beta_{db.c},
\]

where $C'$ is any maximally sized subset of $C$, such that $\Sigma_{e_4e_4}$, with $E_4=(B,C',D)$ is of full rank.

Regressing $A$ on $B,C'$ and $D$, yields the equation

\[
A = \beta_{ab.c'd} B + \beta_{ac'.bd} C' + \beta_{ad.bc'} D + \epsilon_{a.bc'd}.
\]

Regressing $D$ on $B,C$, yields the equation

\[
D = \beta_{db.c} B + \beta_{dc.b} C + \epsilon_{d.bc}.
\]

Plugging the latter equation into the former yields

\begin{equation*}
A = (\beta_{ab.c'd} + \beta_{ad.bc'}\beta_{db.c})B + \beta_{ac'.bd} C' + \beta_{ad.bc'} \beta_{dc.b} C + \beta_{ad.bc'}\epsilon_{d.bc} + \epsilon_{a.bc'd}.
\end{equation*}

We can rewrite this equation as

\[
A = (\beta_{ab.c'd} + \beta_{ad.bc'}\beta_{db.c})B + \tilde{\beta} C + \tilde{\epsilon},
\]

where $\tilde{\beta}$ is some real-valued vector of the same length as the vector $C$ and $\tilde{\epsilon}=\beta_{ad.bc'}\epsilon_{d.bc} + \epsilon_{a.bc'd}$. As $E\{\epsilon_{a.bc'd}(B,C',D)\}=0$ it follows that it also holds that $E\{\epsilon_{a.bc'd}(B,C,D)\}=0$ by our assumptions on $C$. We can therefore conclude that $E\{\tilde{\epsilon}(B,C)\}=0$. Our claim then follows by the uniqueness of the ordinary least squares regression coefficient vector $(\beta_{ab.c},\beta_{ac.b})$.

§.§ Proof of Theorem <ref>

Consider nodes $X$ and $Y$ in an acyclic directed mixed graph $\g$ such that $\de(X,\g)=\{X,Y\}$. Let $(Z_1,W_1)$ and $(Z_2,W_2)$ be valid conditional instrumental sets relative to $(X,Y)$ in $\g$. Let $W_{1\setminus 2}=W_1 \setminus W_2$ and $W_{2\setminus 1}=W_2 \setminus W_1$. If the following four conditions hold

\begin{align}
& W_{1\setminus 2} \perp_{\tilde{\g}} Y \mid W_2, \tag{a} \label{app:condition1}\\
& \text{i) } W_{1\setminus 2} \perp_{\g} Z_2 \mid W_2 \text{ or ii) }W_{1\setminus 2} \setminus Z_2 \perp_{\g} X \mid Z_2 \cup W_2, \tag{b}  \label{app:condition2}\\
& \text{i) } W_{2\setminus 1} \setminus Z_1 \perp_{\g} Z_1\mid W_1 \text{ and } W_{2\setminus 1} \cap Z_1 \perp_{\g} X \mid W_1 \cup (W_{2\setminus 1}\setminus Z_1) \text{ or} \tag{c} \label{app:condition3}\\
& \quad \text{ii) } W_{2\setminus 1} \perp_{\g} X \mid  W_1, \nonumber\\
& Z_1 \setminus (Z_2 \cup W_{2\setminus 1}) \perp_{\g} X \mid Z_2 \cup W_1 \cup W_{2\setminus 1}, \tag{d} \label{app:condition4}
\end{align}
then for all linear structural equation models compatible with $\g$ such that $\Sigma_{xz_1.w_1}\neq 0$,
a.var(\hat{\tau}_{yx}^{z_2.w_2}) \leq a.var(\hat{\tau}_{yx}^{z_1.w_1}).

Fix some linear structural equation model such that $\Sigma_{xz_1.w_1}\neq 0$. By the m-separation assumptions we can apply both Lemma <ref> and Lemma <ref> to conclude that $\sigma_{\tilde{y}\tilde{y}.w_1} \geq \sigma_{\tilde{y}\tilde{y}.w_2}$ and $\sigma_{xx.w_1} - \sigma_{xx.z_1w_1} \leq \sigma_{xx.w_2} - \sigma_{xx.z_2w_2}$.

Given that $\Sigma_{xz_1.w_1} \neq 0$, it holds that $\sigma_{xx.w_1}-\sigma_{xx.z_1w_1}=\Sigma_{xz_1.w_1}\Sigma_{z_1z_1.w_1}^{-1}\Sigma^\top_{xz_1.w_1}>0$. Therefore, $\sigma_{xx.w_2}-\sigma_{xx.z_2w_2} = \Sigma_{xz_2.w_2}\Sigma_{z_2z_2.w_2}^{-1}\Sigma^\top_{xz_2.w_2}>0$ which implies that $\Sigma_{xz_2.w_2} \neq 0$ also holds. As a result, Theorem <ref> applies to both $(Z_1,W_1)$ and $(Z_2,W_2)$, so we can conclude that $	a.var(\hat{\tau}_{yx}^{z_1.w_1})=\sigma_{\tilde{y}\tilde{y}.w_1}/(\sigma_{xx.w_1} - \sigma_{xx.z_1w_1})$ and $a.var(\hat{\tau}_{yx}^{z_1.w_1})=\sigma_{\tilde{y}\tilde{y}.w_2}/(\sigma_{xx.w_2} - \sigma_{xx.z_2w_2})$. The claim than follows by the fact that $\sigma_{\tilde{y}\tilde{y}.w_1} \geq \sigma_{\tilde{y}\tilde{y}.w_2}$ and $\sigma_{xx.z_1w_1} \leq \sigma_{xx.z_2w_2}$ as shown already.

Consider mutually disjoint nodes $X$ and $Y$, and node sets $W_1$ and $W_2$ in an acyclic directed mixed graph $\g=(V,E)$ such that $V$ is generated from a linear structural equation model compatible with $\g$ and $\de(X,\g)=\{X,Y\}$. Let $\tilde{Y} = Y - \tau_{yx}X$. If
W_1 \setminus W_2\perp_{\tilde{\g}} Y \mid W_2,
then $\sigma_{\tilde{y}\tilde{y}.w_2} \leq \sigma_{\tilde{y}\tilde{y}.w_1}$.

As we assume that $W_{1 \setminus 2}  \perp_{\tilde{\g}} Y \mid W_2$ it follows that $W_{1 \setminus 2}  \ci \tilde{Y} \mid W_2$ by Lemma <ref>. It follows that

\begin{align*}
\sigma_{\tilde{y}\tilde{y}.w_1} \geq \sigma_{\tilde{y}\tilde{y}.w_{1\setminus 2} w_2} = \sigma_{\tilde{y}\tilde{y}.w_2}
\end{align*}

Consider nodes $X$ and $Y$ in an acyclic directed mixed graph $\g=(V,E)$, such that $V$ is generated from a linear structural equation model compatible with $\g$ and $\de(X,\g)=\{X,Y\}$. Let $(Z_1,W_1)$ and $(Z_2,W_2)$ be two pairs of disjoint node sets in $\g$. Let $W_{1\setminus 2}=W_1 \setminus W_2$ and $W_{2\setminus 1}=W_2 \setminus W_1$. If the following three conditions hold

\begin{align}
& W_{1\setminus 2} \perp_{\g} Z_2 \mid W_2 \text{ or } W_{1\setminus 2} \setminus Z_2 \perp_{\g} X \mid Z_2 \cup W_2 \text{ and }  \label{efficiency:ind2} \tag{a} \\
& \text{i) } W_{2\setminus 1} \setminus Z_1 \perp_{\g} Z_1\mid W_1 \text{ and } W_{2\setminus 1} \cap Z_1 \perp_{\g} X \mid W_1 \cup (W_{2\setminus 1}\setminus Z_1) \label{efficiency:ind3} \tag{b} \\
& \quad \quad \text{or ii) } W_{2\setminus 1} \perp_{\g} X \mid  W_1 \text{ and}  \nonumber  \\
& Z_1 \setminus (Z_2 \cup W_{2\setminus 1}) \perp_{\g} X \mid Z_2 \cup W_1 \cup W_{2\setminus 1},  \label{efficiency:ind4} \tag{c}
\end{align}
then $\sigma_{xx.w_2} - \sigma_{xx.w_2z_2} \geq \sigma_{xx.w_1} - \sigma_{xx.w_1z_1}$.

Let $A=W_{1\setminus 2}$ and $C=W_{2\setminus 1}$ to unclutter the notation.
By Condition (<ref>) it follows that

\begin{align*}
\sigma_{xx.w_1} - \sigma_{xx.z_1w_1} \leq \sigma_{xx.w_1c} - \sigma_{xx.z_1w_1c'},
\end{align*}

with $C'=C \setminus Z_1$. This follows either by first applying Lemma <ref> with $B=C'$ and then using that if $ C \cap Z_1 \perp_{\g} X \mid (W_1 \cup C')$, $\sigma_{xx.w_1c'} =  \sigma_{xx.w_1c}$ or by applying the latter argument to all of $C$ using the alternative assumption $C \perp_{\g} X \mid W_1$. By Condition (<ref>), it holds that $\sigma_{xx.z_1w_1c'} \geq \sigma_{xx.z_1z''_2w_1c'} = \sigma_{xx.z'_2w_1c}$, where $Z'_2 = Z_2 \setminus W_1$ and $Z''_2 = Z_2 \setminus (W_1 \cup Z_1)$.

\begin{align*}
\sigma_{xx.w_1c} - \sigma_{xx.z_1w_1c'} \leq \sigma_{xx.w_1c} - \sigma_{xx.z'_2w_1c}.
\end{align*}

Finally, by Condition (<ref>) it follows that

\begin{align*}
\sigma_{xx.w_1c} - \sigma_{xx.z'_2w_1c} = \sigma_{xx.aw_2} - \sigma_{xx.z'_2aw_2} \leq \sigma_{xx.a'w_2} - \sigma_{xx.z_2w_2} \leq \sigma_{xx.w_2} - \sigma_{xx.z_2w_2},
\end{align*}

with $A'=A \cap Z_2$.
This follows either by applying Lemma <ref> with $B=A\setminus Z_2$ or by the fact that if $(A \setminus Z_2) \perp_{\g} X \mid Z_2 \cup W_2$, then $\sigma_{xx.z'_2w_2a} =  \sigma_{xx.z_2w_2}$ holds.

Consider a node $X$ and two disjoint node sets $Z$ and $W$ in an acyclic directed mixed graph $\g=(V,E)$ such that the distribution of $V$ is compatible with $\g$. Let $ W = A \cup B$ be a partition of $W$, i.e. $A \cap B = \emptyset$, and suppose that $A \perp_{\g} Z \mid B$. Then

\[
\sigma_{xx.ab} - \sigma_{xx.zab} = \sigma_{xx.b} - \sigma_{xx.zb}.
\]

Using Lemma <ref> we can conclude that

\[\sigma_{xx.ab} - \sigma_{xx.zab} = \beta_{xz.ab} \sigma_{zz.ab} \beta^\top_{xz.ab} =
\beta_{xz.b} \sigma_{zz.b} \beta^\top_{xz.b} = \sigma_{xx.b} - \sigma_{xx.zb}\]

§.§ Proof of Proposition <ref>

Consider nodes $X,Y$ and $N$ in an acyclic directed mixed graph $\g$ such that $\de(X,\g) = \{X,Y\}$. Let $(Z,W)$ and $(Z,W\cup N)$ be distinct valid conditional instrumental sets relative to $(X,Y)$ in $\g$.
If $(Z\cup N,W)$ is not a valid conditional instrumental set relative to $(X,Y)$ in $\g$, then for all linear structural equation models compatible with $\g$ such that $\Sigma_{xz.w}\neq 0$,
$a.var(\hat{\tau}_{yx}^{z.wn}) \leq a.var(\hat{\tau}_{yx}^{z.w}).$

We will show our claim by showing that $(Z,W)$ and $(Z,W\cup N)$ being valid conditional instrumental sets while $(Z\cup N,W)$ is not, implies that $N \perp_{\g} Z \mid W$. Using this we can apply Theorem <ref> with $Z_1=Z_2=Z,W_1=W$ and $W_2=W \cup  N$ to conclude the proof.

We first show that $N \not\perp_{\tilde{\g}} Y \mid W$, as we use this later. The fact that $(Z,W\cup N)$ is a valid conditional instrumental set implies that $(Z\cup N,W)$ fulfills Condition (i) of Theorem <ref>. Similarly, the fact that $(Z,W)$ is a valid conditional instrumental set, implies that $(Z\cup N,W)$ also fulfills Condition (ii). Therefore $(Z\cup N,W)$ not being a valid conditional instrumental set, requires a violation of Condition (iii), i.e., a path $p$ from $N$ to $Y$ in $\tilde{\g}$ that is open given $W$.

We now show that $N \perp_{\g} Z \mid W$ by contradiction, so suppose that $N \not\perp_{\g} Z \mid W$, i.e., there exists a path $p'$ from some node $A \in Z$ to $N$ open given $W$ in $\g$. By Lemma <ref>, $p'$ may not contain the edge $X \rightarrow Y$ and be open given $W$. Therefore $p'$ is also a path in $\tilde{\g}$. Further, $\de(M,\g) \setminus \de(M,\tilde{\g})=Y$ for any node $M$ and as a result $p'$ must also be open given $W$ in $\tilde{\g}$. We have already shown that there exists a path $p$ from $N$ to $Y$ open given $W$ in $\tilde{\g}$. Let $I$ be the first node on $p'$ where these two paths intersect and consider the concatenated path $q = p'(A,I) \oplus p(I,Y)$. We will now sequentially discuss the cases that $I=A,I=Y,I=N,I\neq N$ is a non-collider on $q$ and $I\neq N$ is a collider on $q$. We will show that in all these cases $q$ is open given $W$ or $W \cup  N$ and that the existence of the paths $p$ and $p'$ therefore contradicts our assumption that both $(Z,W)$ and $(Z,W\cup N)$ are valid conditional instrumental sets.

If $I=A$ or $I=Y$, then $q$ is a subpath of $p'$ or $p$, respectively, and as a result is trivially open given $W$. Suppose now that $I=N$. Then if $N$ is a collider, $q$ is open given $W \cup N$. If not, $q$ is open given $W$. Suppose now that $I\neq N$ is a non-collider on $q$. Then it also has to be a non-collider on either $p$ or $p'$. Therefore, $I \notin W$ and $q$ is open given $W$. Suppose finally that $I\neq N$ is a collider on $q$. Then $q$ is open given $W$ if $\de(I, \g) \cap W \neq \emptyset$. If $\de(I, \g) \cap W = \emptyset$, on the other hand, then $p(I,N)$ must be of the form $I \rightarrow \dots \rightarrow N$. Otherwise some node in $\de(I,\g)$ would have to be a collider $C$ on $p(I,N)$ with $\de(C,\g) \subset \de(I,\g)$, which would contradict our assumption that $p$ is open given $W$. Thus, $N \in \de(I,\g)$ and $q$ is an open path given $W \cup  N$ in this case.

Consider a node $X$ in an acyclic directed mixed graph $\g$ such that $\de(X,\g)=X$. Let $W$ be a node set in $\g$ such that $X \notin W$. Then any path in $\g$ that contains $X$ as a non-endpoint node is blocked by $W$.

Consider a path $p$ in $\g$ that contains $X$ as a non-endpoint node. Since $\de(X,\g)=X$ there is no edge of the form $X \rightarrow N$ in $\g$. Therefore, $X$ must be a collider on $p$. Since, $W \cap \de(X,\g) = \emptyset$ it follows that $p$ is blocked by $W$.

§.§ Proof of soundness of Algorithm <ref>

Consider nodes $X$ and $Y$ in an acyclic directed mixed graph $\g$ such that $\de(X,\g) = \{X,Y\}$. Let $(Z,W)$ be a valid conditional instrumental set relative to $(X,Y)$ in $\g$. Then applying Algorithm <ref> to $(\g,X,Y,Z,W)$ minimizes $a.var(\hat{\tau}_{yx}^{z'.w'})$ at each step.

Consider a step of Algorithm <ref> given a valid tuple $(Z',W')$ and node $N$. We will now show that in all possible cases Algorithm <ref> uses $N$ in a manner that greedily minimizes the asymptotic variance.

Suppose that $(Z'\cup N,W')$ is valid and $N \not\perp_{\g} X \mid W' \cup Z'$. Then by Corollary <ref>, $(Z'\cup N,W')$ is more efficient than $(Z',W')$. Further, if $(Z',W'\cup N)$ is valid, it is less efficient than $(Z'\cup N,W')$ by Corollary <ref>. Therefore, adding $N$ to $Z'$ greedily minimizes the asymptotic variance.

Suppose now that $(Z'\cup N,W')$ is not valid, $(Z',W'\cup N)$ is valid and $N \not\perp_{\tilde{\g}} Y \mid W'$. In this case, $(Z',W'\cup N)$ is more efficient than $(Z',W')$ by Proposition <ref>. Therefore, adding $N$ to $W'$ greedily minimizes the asymptotic variance.

Suppose now that $(Z\cup N,W')$ is valid, $N \perp_{\g} X \mid W \cup Z$, $(Z',W'\cup N)$ is valid and $N \not\perp_{\tilde{\g}} Y \mid W'$. Since $(Z\cup N,W')$ being valid requires that $N \not\perp_{\tilde{\g}} Y \mid Z' \cup W'$ this case cannot occur.

It remains to show that in all other cases, discarding $N$ greedily minimizes the asymptotic variance.

Suppose that neither $(Z'\cup N,W')$ nor $(Z',W'\cup N)$ are valid. Then it is trivial that discarding $N$ greedily minimizes the asymptotic variance.

Suppose that $(Z'\cup N,W')$ is not valid, $(Z',W'\cup N)$ is valid and $N \perp_{\tilde{\g}} Y \mid W'$. The fact that $(Z'\cup N,W')$ is not valid and $(Z',W')$ is, implies that $N \not\perp_{\tilde{\g}} Y \mid W'$ and therefore this case cannot occur.

Suppose that $(Z'\cup N,W')$ is valid, $N \perp X \mid W' \cup Z'$ and $(Z',W'\cup N)$ is not valid. We can apply Theorem <ref> with $(Z_1,W_1)=(Z'\cup N,W')$ and $(Z_2,W_2)=(Z',W')$, to conclude that $(Z',W')$ is more efficient than $(Z'\cup N,W')$. Therefore, discarding $N$ greedily minimizes the asymptotic variance.

Suppose that $(Z'\cup N,W')$ is valid, $N \perp X \mid W' \cup Z', (Z',W'\cup N)$ is valid and $N \perp_{\tilde{\g}} Y \mid W'$. By Corollary <ref>, $(Z' \cup N,W')$ is more efficient than $(Z',W'\cup N)$. We can apply Theorem <ref> with $(Z_1,W_1)=(Z'\cup N,W')$ and $(Z_2,W_2)=(Z',W')$, to conclude that $(Z',W')$ is more efficient than $(Z'\cup N,W')$. Therefore, discarding $N$ greedily minimizes the asymptotic variance and this concludes all possible cases.

§.§ Proof of Theorem <ref>

Consider nodes $X$ and $Y$ in an acyclic directed mixed graph $\g$ such that $\de(X,\g) = \{X,Y\}$. Let $W^o= \mathrm{dis}^+_{X}(Y,\g) \setminus \{X,Y\}$ and $Z^o=\mathrm{dis}^+_Y(X,\g) \setminus (\{X,Y\} \cup W^o)$. Then the following two statements hold: (i) if $Z^o \neq \emptyset$  then $(Z^o, W^o)$ is a valid conditional instrumental set relative to $(X,Y)$ in $\g$; (ii) if $Z^o \cap \{\pa(X,\g) \cup \mathrm{sib}(X,\g)\} \neq \emptyset$ then $(Z^o, W^o)$ is also graphically optimal relative to $(X,Y)$ in $\g$.

We first show that if $Z^o\neq \emptyset$ then $(Z^o,W^o)$ is a valid conditional instrumental set. By Lemma
<ref>, $Z^o \perp_{\tilde{\g}} Y \mid W^o$. Further, by construction of $Z^o$ and our assumption that $Z^o=\emptyset$ there exists a path $p$ from any node in $Z^o$ to $X$ consisting entirely of colliders in $Z^o\cup W^o$. Hence, $Z^o \not\perp_{\g} X \mid W^o$.

We now consider the second statement of the theorem. First we will show that for any valid conditional instrumental set $(Z,W)$ such that $W^o \setminus W =\emptyset$, we can use Theorem <ref> with $(Z_1,W_1)=(Z,W)$ and $(Z_2,W_2)=(Z^o,W^o)$ to conclude that

\[
a.var(\hat{\tau}_{yx}^{z^o.w^o}) \leq a.var(\hat{\tau}_{yx}^{z.w}).
\]

By Lemma <ref>, $W \setminus W^o \perp_{\tilde{\g}} Y \mid W^o$ and as a result Condition (<ref>) of Theorem <ref> holds. By Lemma <ref>, $W \setminus (W^o \cup Z^o) \perp_{\g} X \mid Z^o \cup W^o$ and therefore Condition (<ref>) of Theorem <ref> holds. By Lemma <ref>, $W \setminus (W^o \cup Z^o) \perp_{\g} X \mid Z^o \cup W^o$ and $Z \setminus Z^o \perp_{\g} X \mid Z^o \cup W^o$, where we use that $Z \cap W^o=\emptyset$ since $Z \cap W=\emptyset$. By the weak union property it follows that $Z \setminus Z^o \perp_{\g} X \mid Z^o \cup W$ and therefore Condition (<ref>) of Theorem <ref> holds. Finally, Condition (<ref>) is void by the fact that $W^o\setminus W = \emptyset$. We can therefore suppose that $W^o\setminus W \neq \emptyset$ for the remainder of the proof.

Let $(Z,W)$ be a valid conditional instrumental set in $\g$, such that $W^o \setminus W \neq \emptyset$ and consider the ratio of asymptotic variances

\[
\frac{a.var(\hat{\tau}_{yx}^{z^o.w^o})}{a.var(\hat{\tau}_{yx}^{z.w})}
=\frac{\sigma_{\tilde{y}\tilde{y}.w^o}(\sigma_{xx.w} -\sigma_{xx.wz})}{\sigma_{\tilde{y}\tilde{y}.w}(\sigma_{xx.w^o} -\sigma_{xx.w^oz^o})}.
\]

We will now construct a Gaussian linear structural equation model such that this ratio is smaller than 1. To keep our notation readable we make the dependence of our conditional variances on the underlying Gaussian linear structural equation model explicit only when necessary.

Let $F'$ be the set of all edges that are the first on any proper path from $W^o$ to $X$ that is open given $Z^o$.
Similarly, let $F''$ be the set of all edges that are first on any proper path from $W^o$ to $X$ that is open given the empty set.
We then define the following class of Gaussian linear structural equation models. Let $(\mathcal{A},\Omega)$ be a Gaussian linear structural equation model compatible with $\g$ such that$\Omega$ is strictly diagonally dominant. Let $(\mathcal{A}_{F}(\epsilon),\Omega_{F}(\epsilon))$ be the Gaussian linear structural equation model, such that the edge coefficients and error covariances corresponding to edges in $F=F'\cup F''$ are replaced with the value $\epsilon > 0$.
Recall that by Lemma <ref> the limit for any conditional covariance as a function of $\epsilon$, with $\epsilon \rightarrow 0$ exists and is the corresponding conditional covariance in the model $(\mathcal{A}_{F}(0),\Omega_{F}(0))$. Clearly, $(\mathcal{A}_{F}(0),\Omega_{F}(0))$ is also compatible with the truncated acyclic directed mixed graph $\mathcal{G}'$ with the edges in $F$ removed and we can therefore invoke additional m-separation statements as implied by $\mathcal{G}'$.
Choose $(\mathcal{A},\Omega)$ such that the model is faithful to $\g$, the truncated model with $\tilde{Y}$ instead of $Y$ to $\tilde{\g}$ and the same is true for the model $(\mathcal{A}_{F}(0),\Omega_{F}(0))$ with respect to $\g'$ and $\tilde{\g}'$. Such a model exists by the fact that a faithfulness violation corresponds to a non-trivial polynomial in the non-zero parameters of $(\mathcal{A},\Omega)$ evaluating to zero [34]. As a long as we impose a finite number of faithfulness constraints the resulting set of parameters for which the resulting model violates one of our constraints therefore has Lebesgue measure $0$.

We first make two observations that hold for any Gaussian linear structural equation model compatible with $\mathcal{G}$. Let $A = W \setminus W^o$ and $C = W^o \setminus W$. By Lemma <ref>, $\sigma_{\tilde{y}\tilde{y}.w^o}=\sigma_{\tilde{y}\tilde{y}.aw^o}=\sigma_{\tilde{y}\tilde{y}.wc}$. Therefore,

\begin{equation}
0 < \frac{\sigma_{\tilde{y}\tilde{y}.w^o}}{\sigma_{\tilde{y}\tilde{y}.w}}=\frac{\sigma_{\tilde{y}\tilde{y}.wc}}{\sigma_{\tilde{y}\tilde{y}.w}}
=\frac{\sigma_{\tilde{y}\tilde{y}.w}-\sigma_{\tilde{y}c.w} \Sigma^{-1}_{cc.w} \sigma^\top_{\tilde{y}c.w}}{\sigma_{\tilde{y}\tilde{y}.w}} = 1 - q, \quad q \geq 0.
\label{eq:residual variance}
\end{equation}

\begin{equation}
\frac{\sigma_{xx.w} -\sigma_{xx.wz}}{\sigma_{xx.w^o} -\sigma_{xx.w^oz^o}}
\leq \frac{\sigma_{xx.w} -\sigma_{xx.w'z'z^o}}{\sigma_{xx.w^o} -\sigma_{xx.w^oz^o}}
\leq \frac{\sigma_{xx} -\sigma_{xx.w'z'z^o}}{\sigma_{xx.w^o} -\sigma_{xx.w^oz^o}},
\label{eq:instrumental strength}
\end{equation}

with $W'=W \setminus Z^o$ and $Z'=Z \setminus Z^o$.

By assumption on $Z^o$, there exists a path from some node in $Z^o$ to $X$ consisting entirely of colliders in $Z^o$. Clearly no edge in this path is adjacent to any node in $W^o$ and therefore no edge is in $F$. Thus, $Z^o \not\perp_{\mathcal{G}'} X \mid W^o$. By construction $(\mathcal{A}_{F}(0),\Omega_{F}(0))$ is faithful to $\mathcal{G}'$. Therefore,

\begin{align*}
\lim_{\epsilon \rightarrow 0} \sigma_{xx.w^o}\{\mathcal{A}_{F}(\epsilon),\Omega_{F}(\epsilon)\} &- \sigma_{xx.w^oz^o}\{\mathcal{A}_{F}(\epsilon),\Omega_{F}(\epsilon)\} \\  &=\sigma_{xx.w^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}- \sigma_{xx.w^oz^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}> 0.
\end{align*}

This allows us to conclude that

\begin{align*}
&\lim_{\epsilon \rightarrow 0} \frac{a.var[\hat{\tau}_{yx}^{z^o.w^o}\{\mathcal{A}_{F}(\epsilon),\Omega_{F}(0)\}]}{a.var[\hat{\tau}_{yx}^{z.w}(\{\mathcal{A}_{F}(\epsilon),\Omega_{F}(\epsilon)\}]}
=  \\
& \quad \quad \quad \frac{\sigma_{\tilde{y}\tilde{y}.w^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}}{  \sigma_{\tilde{y}\tilde{y}.w}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}}
\frac{[\sigma_{xx.w}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\} -\sigma_{xx.wz}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}]}{[\sigma_{xx.w^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\} -\sigma_{xx.w^oz^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}]}.
\end{align*}

By a similar argument $\sigma_{\tilde{y}c.w}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}  \neq 0$ and therefore by Equation (<ref>)

\begin{align*}
\frac{\sigma_{\tilde{y}\tilde{y}.w^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}}{  \sigma_{\tilde{y}\tilde{y}.w}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}} &= 1-\frac{ \sigma_{\tilde{y}c.w}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\} \Sigma^{-1}_{cc.w}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\} \sigma^\top_{\tilde{y}c.w}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}}{\sigma_{\tilde{y}\tilde{y}.w}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}} \\
&= 1-q^0 < 1.
\end{align*}

Combined with Equation (<ref>) we have

\begin{align}
\lim_{\epsilon \rightarrow 0} \frac{a.var(\hat{\tau}_{yx}^{z^o.w^o})}{a.var(\hat{\tau}_{yx}^{z.w})}
&= (1-q^0)
\frac{\sigma_{xx.w}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\} -\sigma_{xx.wz}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}}{\sigma_{xx.w^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\} -\sigma_{xx.w^oz^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}}
\nonumber \\
&\leq (1-q^0)
\frac{\sigma_{xx}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\} -\sigma_{xx.w'z'z^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}}{\sigma_{xx.w^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\} -\sigma_{xx.w^oz^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}} \label{eq: fraction}.
\end{align}

We now show that by construction of $F$, $W^o \perp_{\mathcal{G}'} X$ and $S \perp_{\mathcal{G}'} X \mid Z^o$ for any $S \subseteq V \setminus (Z^o \cup \{X,Y\})$ in order to further simplify Equation (<ref>).

Consider first the claim that $W^o \perp_{\mathcal{G}'} X$. Let $p$ be a path in $\g'$ from $W^o$ to $X$. Clearly, $p$ is also a path in $\g$ that does not contain any edges in $F$. By the fact that $F''$ contains all first edges on paths from $W^o$ to $X$ that are open given the empty set it follows that $p$ is blocked in $\g$. By Lemma <ref>, it follows that $p$ is also blocked in $\g'$ and our claim follows.

Consider now the claim that $S \perp_{\mathcal{G}'} X \mid Z^o$ for any $S \subseteq V \setminus (Z^o \cup \{X,Y\})$.
To do so we show that any proper path from some node $N \in S$ to $X$ that is open given $Z^o$ in $\g$ contains an edge in $F$. This suffices by the arguments given in the previous paragraph.

Let $p$ be proper path from some node $N \in S$ to $X$ that is open given $Z^o$ in $\g$. By Lemma <ref>, $p$ may not contain $Y$ and be open given $Z^o$, since $\de(Y,\g)=Y$ and $Y \notin Z^o$. Further, any other node adjacent to $X$ in $\g$ is either in $Z^o$ or $W^o$. If the latter is the case we are done, so suppose that $p$ contains a subsegment $p(Z,X)$ chosen to be of maximal possible length, such that every node in $p(Z,X)$, except for $X$ is in $Z^o$. Let $M$ be the node before $Z$ on $p$ and consider $p(M,X)$. By assumption, $p(M,X)$ may not contain a non-collider and thus must be of the form

\[
M \rightarrow Z \leftrightarrow \dots \leftrightarrow X \text{ or } M \leftrightarrow Z \leftrightarrow \dots \leftrightarrow X.
\]

In either case, to not contradict our assumption that $p(Z,X)$ is of maximal length, $M \in W^o$ must hold. Thus the edge $M \rightarrow Z$, respectively $M \leftrightarrow Z$, is in $F''$ and our claim follows.

Since $W^o \perp_{\mathcal{G}'} X$,

\[\lim_{\epsilon \rightarrow 0}\sigma_{xx.w^o}\{\mathcal{A}_{F}(\epsilon),\Omega_{F}(\epsilon)\} = \sigma_{xx.w^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}=\sigma_{xx}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}.\]

Since $S \perp_{\mathcal{G}'} X \mid Z^o$ for any $S \subseteq V \setminus (Z^o \cup \{X,Y\})$,

\[\lim_{\epsilon \rightarrow 0}\sigma_{xx.w'z'z^o}\{\mathcal{A}_{F}(\epsilon),\Omega_{F}(\epsilon)\}=\sigma_{xx.w'z'z^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}=\sigma_{xx.z^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}\]

\[\lim_{\epsilon \rightarrow 0}\sigma_{xx.w^oz^o}\{\mathcal{A}_{F}(\epsilon),\Omega_{F}(\epsilon)\}= \sigma_{xx.w^oz^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}=\sigma_{xx.z^o}\{\mathcal{A}_{F}(0),\Omega_{F}(0)\}.\]

Using these limit statements, we can conclude that

\begin{align*}
\lim_{\epsilon \rightarrow 0} \frac{a.var[\hat{\tau}_{yx}^{z^o.w^o}\{\mathcal{A}_{F}(\epsilon),\Omega_{F}(\epsilon)\}]}{a.var[\hat{\tau}_{yx}^{z.w}\{\mathcal{A}_{F}(\epsilon),\Omega_{F}(\epsilon)\}]}
&\leq (1-q^0) < 1.
\end{align*}

Therefore, there exists an $\epsilon > 0$, such that the Gaussian linear structural equation model $(\mathcal{A}_{F}(\epsilon),\Omega_{F}(\epsilon))$ is compatible with $\g$ and the asymptotic variance provided by $(Z,W)$ is larger than the one provided by $(Z^o,W^o)$.

Consider nodes $X$ and $Y$ in an acyclic directed mixed graph $\g$ with node set $V$ such that $\de(X,\g) =\{X,Y\}$. Let $W^o = \mathrm{dis}^+_{X}(Y,\g) \setminus \{X,Y\}$, then for any $S \subseteq V \setminus (W^o \cup \{X,Y\})$,

\[
S \perp_{\tilde{\g}} Y \mid W^o.
\]

Let $p$ be a proper path from some node $A \in S$ to $Y$ in $\tilde{\g}$.  We now show that $p$ is blocked by $W^o$.

By Lemma <ref>, $p$ is blocked if it contains $X$ so assume this is not the case. Every node adjacent to $Y$ in $\tilde{\g}$ is either in $W^o$ or $X$. Therefore $p$ contains at least one node $W \in W^o$, such that $p(W,Y)$ consists, except for $Y$, of nodes in $W^o$. Choose $W \in W^o$, such that $p(W,Y)$ is of maximal possible length. Let $N \notin W^o$ be the node adjacent to $W$ on $p$. We will now show that $p(N,Y)$ contains at least one non-collider by contradiction. Suppose that all nodes on $p(N,Y)$ are colliders, i.e., $p(N,Y)$ is of the form

\[
N \leftrightarrow W \leftrightarrow \dots \leftrightarrow Y  \text{ or }
N \rightarrow W \leftrightarrow \dots \leftrightarrow Y.
\]

Since $p$ does not contain $X$, $N \neq X$. By the existence of $p(N,Y)$ it follows that $N \in W^o$ which contradicts our assumption that $p(W,Y)$ is maximal. Therefore, $p(W,Y)$ contains a non-collider. As all nodes on $p(W,Y)$ are in $W^o$, it follows that $p(W,Y)$ and therefore $p$ are blocked by $W^o$. Since this is true for any proper path it follows that $S \perp_{\tilde{\g}} Y \mid W^o$.

Consider nodes $X$ and $Y$ in an acyclic directed mixed graph $\g$ with node set $V$ such that $\de(X,\g)=\{X,Y\}$. Let $W^o = \mathrm{dis}^+_{X}(Y,\g)\setminus \{X,Y\}$ and $Z^o=\mathrm{dis}^+_Y(X,\g) \setminus (\{X,Y\} \cup W^o)$, then for any $S \subseteq V \setminus (W^o \cup Z^o \cup \{X,Y\})$

\[
S \perp_{\g} X \mid W^o \cup Z^o.
\]

Follows by the same arguments as given for Lemma <ref>.

Consider a node set $W$ and a path $p$ in an acyclic directed mixed graph $\g$, such that $p$ is blocked by $W$ in $\g$. Let $\g'$ be a graph obtained by removing edges from $\g'$ such that $p$ is also a path in $\g'$. Then $p$ is blocked by $W$ in $\g'$.

Since $p$ is blocked by $W$ in $\g$, it must i) contain a non-collider that is in $W$ or ii) a collider $C$ such that $W \cap \de(C,\g) =\emptyset$. If i) holds in $\g$ then it clearly also holds in $\g'$. The same is true for ii), by the fact that $\de(C,\g') \subseteq \de(C,\g)$.

§ ADDITIONAL EXAMPLES AND RESULTS

§.§ Additional example for Section <ref>

[Valid conditional instrumental sets]

To illustrate Theorem <ref> and Lemma <ref> we consider the graph $\g$ from Fig. <ref>. We are interested in classifying all valid conditional instrumental sets relative to $(V_4,V_6)$ in $\g$. By Condition (i) of Theorem <ref>, no valid conditional instrumental set $(Z,W)$ may contain any nodes in $\mathrm{forb}(V_4,V_6,\g)=\{V_4,V_5,V_6,V_7\}$. Therefore, we only need to consider sets $Z,W \subseteq \{V_1,V_2,V_3\}$.
For Condition (ii) to hold, there must exist a path from $Z$ to $X$ that is open given $W$ in $\g$. For any $V_i$ with $i\in\{1,2,3\}$ there exist three paths to $X$ in $\g$: the path ending with the edge $V_1 \rightarrow V_4$, the one ending with the edge $V_5 \leftarrow V_4$ and the one ending with the edge $V_5 \leftrightarrow V_6$. The latter two always include $V_6$ as a collider and as $\de(V_6,\g)=\{V_6,V_7\}$ this implies that they are both always blocked by any $W \subset \{V_1,V_2,V_3\}$. The first path on the other hand does not contain a collider and is therefore only closed if there exists a node $V_j \in W$ with $j< i$. Therefore, Condition (ii) holds if and only if there exists an $i \in \{1,2,3\}$, such that $V_i \in Z$ and for all $j < i, V_j \notin W$. Finally, for Condition (iii) to hold there must not be a path from $Z$ to $Y$ that is open given $W$ in the graph $\tilde{\g}$, which is the graph $\g$ but with the edge $V_4 \rightarrow V_5$ removed. For any $V_i$ with $i\in\{1,2,3\}$ there exist two paths to $Y$ in $\tilde{\g}$: the path ending with the edge $V_3 \rightarrow V_6$ and the one ending with the edge $V_4 \leftrightarrow V_6$. The latter always includes $V_4$ as a collider and as $\de(V_4,\tilde{\g})=\emptyset$ it is always blocked by any $W \in \{V_1,V_2,V_3\}$. The first path on the other hand does not contain a collider and is therefore only closed if there exists a node $V_j \in W$ with $j > i$. Therefore, Condition (iii) holds if and only if for all $i \in \{1,2,3\}$, such that $V_i \in Z$ there exists a $j > i$ such that $V_j \in W$.
It follows that there are five possible valid conditional instrumental sets relative to $(V_4,V_6)$ in $\g$: $(V_1,V_2)$, $(V_1,V_3)$, $(\{V_1,V_2\},V_3)$, $(V_1,\{V_2,V_3\})$ and $(V_2,V_3)$.

§.§ Additional results for Section <ref>

The asymptotic variance formula for the two-stage least squares estimator from Theorem <ref> elegantly mirrors the asymptotic variance formula for the ordinary least squares estimator, which is

\begin{equation*}
\end{equation*}

We can use this to derive the following simple proof for the known result that, in cases where it is possible to do either, it is more efficient to estimate a total effect with the ordinary least squares estimator than with the two-stage least squares estimator. Interestingly, the proof shows that this gain in efficiency depends primarily on the instrumental strength $\sigma_{xx.w}- \sigma_{xx.zw}$.

Let $X,Y$ be nodes in an ADMG $\g=(V,E)$, such that $V$ is generated from a linear structural equation model compatible with $\g$. Let $(Z,W)$ be a conditional instrumental set relative to $(X,Y)$ in $\g$ and let $W$ be an adjustment set relative to $(X,Y)$ in $\g$. Then

\[
a.var(\hat{\tau}_{yx}^{z.w}) \geq 	a.var(\hat{\beta}_{yx.w}).
\]

By the assumption that $W$ is a valid adjustment set it follows that $\beta_{yx.w}=\tau_{yx}$. Therefore, $\beta_{\tilde{y}w}=\beta_{yw}-\tau_{yx}\beta_{xw}=\beta_{yw.x}$ by Lemma <ref>. As a results $\sigma_{\tilde{y}\tilde{y}.w} = \sigma_{yy.xw}$. Further, by the non-negativity of conditional variances $\sigma_{xx.w}\geq \sigma_{xx.w}- \sigma_{xx.zw}$. Combined this implies that

\[
\frac{a.var(\hat{\beta}_{yx.w})}{a.var(\hat{\tau}_{yx}^{z.w})} =
\frac{\sigma_{xx.w}-\sigma_{xx.zw}}{\sigma_{xx.w}} = 1 - \frac{\sigma_{xx.zw}}{\sigma_{xx.w}} \leq 1.
\]

§.§ Additional examples for Section <ref>

$Z$     $W$

$A$     $\{B,D\},\{C,D\},\{B,C,D\}$

$B$     $C,\{A,C\},\{C,D\},\{A,C,D\}$

$D$     $B,C,\{A,B\},\{A,C\},\{B,C\},\{A,B,C\}$

$\{A,B\}$     $C,\{C,D\}$

$\{A,D\}$     $B,C,\{B,C\}$

$\{B,D\}$     $C,\{A,C\}$

$\{A,B,D\}$     $C$

List of all valid conditional instrumental sets in the graph from Fig. <ref> of the Main.

[Neutral conditioning]
Consider the graph $\g$ from Fig. <ref> of the Main. We are interested in tuples of the form $(D,W)$ and $(D,W')$, with $W \subseteq \{A,C\}$ and $W' = W \cup B$. By Example <ref> of the Main, any such tuple is a valid conditional instrumental set relative to to $(X,Y)$ in $\g$. Further, as $B \perp_{\g} D \mid W$ we can apply Theorem <ref> with $W_1=W,W_2=W'$ and $Z_1=Z_2=D$ (Condition (<ref>) i) holds, Conditions (<ref>), (<ref>) and (<ref>) are void). As $B \perp_{\tilde{\g}} Y \mid W$ and $B \perp_{\g} D \mid W$ we can also apply Theorem <ref> with the roles of $W$ and $W'$ reversed, i.e., $W_1=W',W_2=W$ and $Z_1=Z_2=D$ (Conditions (<ref>) and (<ref>) i) hold, Conditions (<ref>) and (<ref>) are void). We can therefore conclude that adding $B$ to any $W \subseteq \{A,C\}$ has no effect on the asymptotic variance.

The reason conditioning on $B$ has no effect, is that $B \ci \tilde{Y} \mid W$ and $B \ci D \mid W$. Therefore, $\sigma_{\tilde{y}\tilde{y}.wb}=\sigma_{\tilde{y}\tilde{y}.w}$ and $\sigma_{xx.w}-\sigma_{xx.wd}=\sigma_{xx.wb}-\sigma_{xx.wbd}$ (see Lemma <ref>). Conditioning on $B$ has no effect on either the residual variance or the conditional instrumental strength.

[Beneficial conditioning 2]

Consider the graph $\g$ from Fig. <ref> in the Main and the two tuples $(\{B,D\},C)$ and $(\{B,D\},\{A,C\})$. By Example <ref>, both tuples are valid conditional instrumental sets relative to $(X,Y)$ in $\g$. Further, as $A \perp_{\g} X \mid  C$ we can apply Theorem <ref> with $W_1=C,W_2=\{A,C\}$ and $Z_1=Z_2=\{B,D\}$ (Condition (<ref>) ii) holds, Conditions (<ref>),(<ref>) and (<ref>) are void). We can therefore conclude that conditioning on $A$ can only improve the asymptotic variance.

The reason conditioning on $A$ is beneficial, is that $A \ci \tilde{Y} \mid C$ and $A \ci X \mid C$ but $A \notci X \mid \{B,C,D\}$. Therefore, $\sigma_{\tilde{y}\tilde{y}.ac} = \sigma_{\tilde{y}\tilde{y}.c}, \sigma_{xx.ac}=\sigma_{xx.c}$ but $\sigma_{xx.abcd} \leq \sigma_{xx.bcd}$. Interestingly, this also implies that $(\{B,D\},\{A,C\})$ has the exact same residual variance and conditional instrumental strength as the valid tuple $(\{A,B,D\},C)$: $\sigma_{\tilde{y}\tilde{y}.c}$ and $\sigma_{xx.c} - \sigma_{xx.abcd}$, respectively. Conditioning on the covariate $A$ is therefore beneficial because adding it to the conditioning set has the same effect as adding it to the instrumental set and the latter is beneficial by Corollary <ref>.

(Optimal tuple in Fig. <ref> of the Main)
Consider the graph $\g$ in Fig. <ref> of the Main. We first characterize all valid conditional instrumental sets relative to $(X,Y)$ in $\g$. As $\f{\g}=\{X,Y\}$, we only need to consider disjoint node sets $Z$ and $W$ that are subsets of $\{A,B,C,D\}$. Further, $Z \perp_{\tilde{\g}} Y \mid W$ if and only if $C \notin Z$ and therefore $Z \subseteq \{A,B,D\}$. Finally, $Z \perp_{\g} X \mid  W$ if and only if $\{B,D\} \cap Z = \emptyset$ and $D \in W$.  Therefore, any valid tuple with respect to $(X,Y)$ in $\g$ has to be of the form $(Z,W)$, with $Z \subseteq \{A,B,D\}$ non-empty, $W \subseteq \{A,B,C,D\} \setminus Z$ if $Z \neq \{A\}$, and $W \subseteq \{B,C\}$ if $Z = \{A\}$. There are $34$ tuples of this form.

Let $(Z,W)$ be any of these $34$ valid tuples. Suppose that $Z\cup W \neq \{A,B,C,D\}$. By the same argument as in Example <ref> we can use Proposition <ref> and Corollary <ref> to obtain a more efficient tuple $(Z',W')$, such that $Z' \cup W' = \{A,B,C,D\}$. We can also apply Corollary <ref> with $S = \{A,B,D\} \setminus Z'$ and conclude that the valid conditional instrumental set $(\{A,B,D\},C)$ is at least as efficient as $(Z',W')$. Since we began from an arbitrary valid tuple, this shows that $(\{A,B,D\},C)$ provides the smallest attainable asymptotic variance among all valid tuples. However, as
$A \perp_{\g} X \mid \{B,C,D\}$
we can apply Theorem <ref> with $W_1 = W_2 = \{C\}$, $Z_1=\{A,B,D\}$ and $Z_2=\{B,D\}$ to conclude that $(\{B,D\},C)$ is at least as efficient as $(\{A,B,D\},C)$. Therefore, the smaller tuple $(\{B,D\},C)$ is also asymptotically optimal.

§.§ Additional example for Section <ref>

[>=stealth',shorten >=1pt,auto,node distance=0.7cm,scale=.9, transform shape,align=center,minimum size=3em]
[state] (x) at (0,0) $X$;
[state] (y) [right =of x] $Y$;
[state] (w1) [left =of x] $A$;
[state] (w2) [above =of y] $B$;

[->]   (x) edge    (y);
[<->]   (x) edge  [bend left]  (y);
[->]   (w1) edge     (x);
[<->]   (w2) edge    (y);
[->]   (w1) edge    (w2);;

[>=stealth',shorten >=1pt,auto,node distance=0.7cm,scale=.9, transform shape,align=center,minimum size=3em]
[state] (x) at (0,0) $X$;
[state] (y) [right =of x] $Y$;
[state] (w1) [left =of x] $C$;
[state] (w2) [above =of y] $B$;
[state] (z) [above =of w1] $A$;

[->]   (x) edge    (y);
[<->]   (x) edge  [bend left]  (y);
[<->]   (w1) edge     (x);
[<->]   (w2) edge    (y);
[->]   (w1) edge    (w2);
[->]   (z) edge    (w1);

subfig:Zo empty and subfig:Zo not optimal Causal acyclic directed mixed graphs from Section <ref>.

To illustrate the role of the two conditions on $Z^o$ made in Theorem <ref>, consider the acyclic directed mixed graphs in Fig. <ref> and <ref>, denoted here as $\g$ and $\g'$, respectively. Again, we are interested in estimating $\tau_{yx}$. In $\g$, $Z^o=\emptyset$, even though $\{A,\emptyset\}$ is a valid conditional instrumental set relative to $(X,Y)$ in $\g$. It is therefore an example, where $(Z^o,W^o)$ is not a valid conditional instrumental set, even though one exists. In $\g'$, $Z^o=A$ and $W^o=\{B,C\}$, which is a valid conditional instrumental set relative to $(X,Y)$ in $\g'$. However, $Z^o \cap \{\pa(X,\g') \cup \mathrm{sib}(X,\g')\}=\emptyset$. As a result in many linear structural equation model compatible with $\g'$ the instrumental strength of $(A,\{B,C\}), \sigma_{xx.bc} - \sigma_{xx.abc}$, will be small.

§ SUPPLEMENTARY MATERIAL FOR SIMULATIONS

We now provide some additional information regarding the simulations in our main paper. Reproducible code is made available at .

For completeness we provide violin plots for all alternative valid conditional instrumental sets in our simulation study from Section <ref>. In addition we also also add violin plots for the ratio of the theoretical asymptotic standard deviation of $(Z^o,W^o)$ to the alternative valid tuple per Theorem <ref>. These plots arguably represent the case for $n=\infty$, except for the biased ordinary least squares estimator.

The plots for $\g_1$ are given in Fig. <ref>. We dropped the alternative valid tuple $(\{B,D\},\{A,C\})$ from the figure because it has the exact same asymptotic variance as the optimal tuple $(\{A,B,D\},C)$ (see Example <ref>). As a result adding it would heavily distort the violin plots.

The plots for $\g_2$ are given in Fig. <ref> and <ref>. We dropped the alternative valid tuple $(\{A,B,D\},C\})$ from the figure because it has the exact same asymptotic variance as the optimal tuple $(\{B,D\},C)$ (see Example <ref>). As a result adding it would heavily distort the violin plots.

Violin plot of the ratios of the root mean squared error, respectively the asymptotic standard deviation, for $(Z^o,W^o)$ to the one for the tuple $(Z,W)$ given on the X-axis. For the labels on the X-axis we use OLS to denote the ordinary least squares regression of $Y$ on $X$. The black dots mark the geometric mean of the ratios.

Violin plot of the ratios of the root mean squared error, respectively the asymptotic standard deviation, for $(Z^o,W^o)$ to the one for the tuple $(Z,W)$ given on the X-axis. For the labels on the X-axis we use OLS to denote the ordinary least squares regression of $Y$ on $X$. The black dots mark the geometric mean of the ratios.

Violin plot of the ratios of the root mean squared error, respectively the asymptotic standard deviation, for $(Z^o,W^o)$ to the one for the tuple $(Z,W)$ given on the X-axis. For the labels on the X-axis we use OLS to denote the ordinary least squares regression of $Y$ on $X$. The black dots mark the geometric mean of the ratios.

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