# Large Microwave Inductance of Granular Boron-Doped Diamond Superconducting
Films

Bakhrom Oripov<EMAIL_ADDRESS>Quantum Materials Center, Physics
Department,
University of Maryland, College Park, MD 20742, USA  Dinesh Kumar  Nano
Functional Materials Technology Centre and Materials Science Research Centre,
Department of Physics, Indian Institute of Technology (IIT) Madras, Chennai
600036, India  Cougar Garcia  Quantum Materials Center, Physics Department,
University of Maryland, College Park, MD 20742, USA  Materials Science and
Engineering Department,
University of Maryland, College Park, MD 20742, USA  Patrick Hemmer  Quantum
Materials Center, Physics Department,
University of Maryland, College Park, MD 20742, USA  T. Venkatesan  NUSNNI
NanoCore, National University of Singapore, 117411, Singapore  Neocera LLC,
Beltsville, MD 20705, USA  M. S. Ramachandra Rao  Nano Functional Materials
Technology Centre and Materials Science Research Centre,
Department of Physics, Indian Institute of Technology (IIT) Madras, Chennai
600036, India  Steven M. Anlage  Quantum Materials Center, Physics Department,
University of Maryland, College Park, MD 20742, USA  Materials Science and
Engineering Department,
University of Maryland, College Park, MD 20742, USA

###### Abstract

Boron-doped diamond granular thin films are known to exhibit superconductivity
with an optimal critical temperature of $T_{c}=7.2K$. Here we report the
measured complex surface impedance of Boron-doped diamond films in the
microwave frequency range using a resonant technique. Experimentally measured
inductance values are in good agreement with estimates obtained from the
normal state sheet resistance of the material. The magnetic penetration depth
temperature dependence is consistent with that of a fully-gapped s-wave
superconductor. Boron-doped diamond films should find application where high
kinetic inductance is needed, such as microwave kinetic inductance detectors
and quantum impedance devices.

Diamond is the hardest known natural material and an excellent thermal
conductor despite also being electrically insulating with a large band gap.
Doping diamond can create either semiconducting or metallic states.Yokoya  _et
al._ (2005); Watanabe  _et al._ (2012) It was discovered from high pressure
and high temperature (HPHT) synthesis techniques that bulk single crystal
diamond can be doped with Boron (hole doping) to the point that
superconductivity is observed with $T_{c}\cong 2.3\;K$.Sidorov and Ekimov
(2010) The first thin films of Boron-doped diamond (BDD) were single-crystal
like and had a transition temperature that increased with Boron doping, but
only slightly exceeded that obtained in bulk.Bustarret  _et al._ (2004) Later
it was discovered that higher B concentrations could be obtained by preparing
the films with energetic and non-equilibrium methods, including microwave
plasma-assisted chemical vapor deposition (MPCVD),Takano  _et al._ (2004)
which resulted in $T_{c}\cong 4.2\;K$. Upon further refinement, materials that
have become known as nano-crystalline diamond (NCD) prepared by MPCVD showed
even higher transition temperatures. A remarkable dome-shaped $T_{c}$ vs.
doping curve has emerged from extensive studies of boron doped granular
diamond films, with a peak transition temperature of $T_{c}\cong 7.2\;K$.Kumar
_et al._ (2018a) Meanwhile Moussa and Cohen have predicted that the transition
temperature of B-doped diamond will grow to 55 K with B concentration in the
range of 20% to 30%.Moussa and Cohen (2008) There are also predictions of high
transition temperature superconductivity in metastable crystals of Boron-
Carbon compounds based on electronic and vibrational numerical
calculations.Saha  _et al._ (2020) Recently superconductivity with a zero-
resistance $T_{c}=$ 24 K has been discovered in 27 at% B-doped rapidly
quenched Carbon.Bhaumik  _et al._ (2017)

The single-crystal-like B-doped diamond films show clear signs of BCS s-wave
superconducting behavior in STM tunneling experiments, with no sub-gap states
and a peak in the density of states consistent with
$\Delta(0)/k_{B}T_{c}=1.74\;$.Sacépé  _et al._ (2006) However it was noted
from scanning tunneling spectroscopy methods that Boron-doped diamond NCD
films displayed modulation of the superconducting order parameter between
grains, strong superconducting fluctuations, and substantial tunneling between
grains in the superconducting state, all consistent with a strong granular
nature to superconductivity.Willems  _et al._ (2010) It was noted that diamond
grains with different exposed surfaces may absorb Boron to a widely varying
extent, resulting in locally inhomogeneous superconducting properties in NCD
films.Dahlem  _et al._ (2010); Zhang  _et al._ (2011) It was argued that doped
NCD films are near a superconductor-to-insulator transition driven by
competition between Coulomb blockade in the grains and the superconducting
proximity effect that encourages long-range transport. The NCD films have
residual resistance ratio values less than unity Zhang  _et al._ (2011) and
show large values of resistivity ($\rho_{n}=2-40\;m\Omega\;cm$) just above the
transition temperature.Dahlem  _et al._ (2010); Kumar  _et al._ (2018a) As one
consequence of this complex microstructure, the doped NCD films show local
values of $\Delta(0)/k_{B}T_{c}$ substantially less than the weak-coupling
limit value of 1.76.Dahlem  _et al._ (2010)

For some time it has been known that the kinetic inductance of superconductors
in narrow thin film wires can exceed the geometrical inductance of the
structure.Little ; Meservey and Tedrow (1969) The use of low carrier density
materials, disordered superconductors, and granular materials has further
enhanced the effective inductance of patterned film structures. These high
effective inductance materials have enabled compact microwave resonators,Zhang
_et al._ (2019) and a new generation of highly sensitive photon detectors
based on lumped-element microwave resonators Zmuidzinas (2012). They also find
use as high-impedance (exceeding $R_{Q}=h/(2e)^{2}\simeq 6.5\;k\Omega$)
structures for quantum circuits, current-tunable delay lines Anlage,
Snortland, and Beasley (1989); Annunziata  _et al._ (2010), phase
shifters,Bockstiegel  _et al._ (2016) etc. Superconducting materials of
interest include NbNx Peltonen  _et al._ (2013), InOx,Fiory, Hebard, and
Glaberson (1983) NbTiN,Kawamura  _et al._ (1999) TiN,Leduc  _et al._ (2010)
and granular Aluminum (grAl).Cohen and Abeles (1968); Maleeva  _et al._ (2018)
For a BCS superconductor in the dirty limit, one can relate the
superconducting inductance to the normal state sheet resistance $R_{n}$ and
superconducting gap $\Delta$ as $L_{s}=\hbar R_{n}/(\pi\Delta)$, with
$R_{n}=\rho_{n}/t$, where $t<\lambda$ is the film thickness and $\lambda$ is
the magnetic penetration depth.Zmuidzinas (2012) Hence there has been a search
for new materials with large normal state sheet resistances that do not
compromise the superconducting properties (in terms of $T_{c}$ and $\Delta$)
so that the material will retain the low microwave loss properties
($\sigma_{1}\ll\sigma_{2}$) associated with robust superconductors.

In this paper we utilize the parallel plate resonator (PPR) technique to
directly measure the superconducting inductance and microwave losses of a new
class of granular materials, namely granular Boron-doped diamond films.

Boron doped diamond films (films "A","B", and "C") were deposited using the
hot filament chemical vapor deposition (HFCVD) technique Kumar, Chandran, and
Ramachandra Rao (2017); Abdel-Hafiez  _et al._ (2017); Kumar  _et al._
(2018b). The silicon substrate temperature was maintained at 850 °C and
pressure of the chamber was $\sim$ 7 Torr. The gases used during the
depositions were CH4 (80 sccm), H2 (3000 sccm) and B(CH3)3 with the flow rate
maintained in such a way that the B/C ratio was approximately 10,000 ppm. The
Hall concentration ($n_{h}$) and onset critical temperature
($T_{c,DC(Onset)}$) of film "A" were found to be $3.0\times{10}^{21}{cm}^{-3}$
and 7.2 K, respectively. The film was deposited for 3 h and the thickness was
found to be 1.5 $\mu m$ by cross-sectional electron microscopy. The other
three films, covering a range of $T_{c}$ values, were grown under similar
conditions with the full list of properties listed in Table.1 of the the
Supplemental Material. Sup

Figure 1:  Temperature dependence of the resonant frequency $f_{0}(T)$
extracted from PPR measurement on Boron-doped diamond film "A" (black squares)
and fits to the data using Eq.(1) (solid red line). Inset shows the quantity
$f_{0}(0)\Delta f_{0}(T)/f_{0}^{2}(T)$ vs temperature, which is proportional
to $\Delta\lambda(T)/\lambda(0)$. Solid blue line is the fit to the data using
Eq.(2) with $\Delta(0)=924.38\pm 76.60\mu eV$.

The surface impedance measurements were done by means of the parallel plate
resonator (PPR) technique.Taber (1990); Pambianchi  _et al._ (1994); Talanov
_et al._ (2000) Two nominally identical films approximately $8\times
6\;mm^{2}$, prepared by dicing a single sample, are placed face-to-face and
sandwiching a Sapphire dielectric spacer of thickness $d=75\;\mu m$ or
$430\;\mu m$ (see inset of Fig.2). The PPR is placed in a metallic enclosure
and cooled to cryogenic temperatures. Two microwave coaxial cables make
capacitive coupling to the PPR and induce a series of microwave resonances,
and the complex transmitted signal $S_{21}$ vs. frequency is measured from
$T=25\;mK$ to near $T_{c}$. The measurements are performed under vacuum in a
BlueFors dilution refrigerator and the PPR assembly is firmly anchored to the
mixing chamber plate. The measurements are repeated at several microwave
powers and only the power-independent data is considered to exclude the
influence of nonlinearity and microwave heating. These data are analyzed to
extract the resonant frequencies $f_{0}$ and quality factors $Q$ of the PPR
modes.Petersan and Anlage (1998) The temperature of the PPR is systematically
varied and $Q(T)$ and $f_{0}(T)$ of the resonator are measured. As discussed
below, these quantities are converted to the complex surface impedance, from
which the surface resistance and magnetic penetration depth of the film can be
determined.Taber (1990); Pambianchi  _et al._ (1994); Talanov  _et al._ (2000)

The resonant frequency data for a PPR mode at frequency $f\simeq 7.39\;GHz$
measured on film "A" is shown in Fig.1. A similar PPR resonant frequency data
measured at $f\simeq 8.08\;GHz$ measured on film "B" is shown in Fig.2. The
first thing to note is the unusually large range of frequency shift of the
modes, on the order of several hundred MHz. The fractional frequency shift is
5 times larger than previously published data.Taber (1990) Given the large
dielectric spacer thickness this implies that the change in the penetration
depth $\Delta\lambda(T)$ is unusually large for this material.

Figure 2:  Temperature dependence of the resonant frequency $f_{0}(T)$
extracted from PPR measurement on Boron-doped diamond film "B" (black squares)
and fits to the data using Eq.(1) (solid red line). Inset shows a schematic of
the Parallel Plate Resonator and coaxial microwave cables with capacitive
coupling. The directions of rf magnetic field and currents for a
representative mode are shown. Inset figure not to scale.

The measured temperature dependence of the resonant frequency $f_{0}$ stems
from the temperature dependence of the penetration depth $\lambda(T)$, and can
be written as,Taber (1990); Talanov  _et al._ (2000)

$\frac{f_{0}(T)}{f_{0}(0)}=\frac{\left[1+2\frac{\lambda(0)}{d}coth\left(\frac{t}{\lambda(0)}\right)\right]^{1/2}}{\left[1+2\frac{\lambda(T)}{d}coth\left(\frac{t}{\lambda(T)}\right)\right]^{1/2}}\quad.$
(1)

where $t$ is the thickness of the two (assumed identical) superconducting
samples and $f_{0}(0)$ is the resonant frequency at $T=0$. Here, we use the
BCS s-wave temperature-dependence of the penetration depth in all fits.

Equation (1) is fit to the $f_{0}(T)$ data (Fig.1) using the LMFIT function in
Pythons’s SciPy library LMf , where the ’basinhopping’ optimization method bas
was utilized. An estimate of the zero temperature penetration depth
$\lambda(0)=2.189\pm 0.006\;\mu m$ and the critical temperature
$T_{c}=6.717\pm 0.001\;K$ for film "A" was extracted. Note that it is often
observed that the $T_{c}$ obtained from fitting to microwave frequency shift
data is lower than the zero resistance $T_{c,DC}$ value,Anlage  _et al._
(1989) especially in granular materials.Muller  _et al._ (1980); Ramachandran
_et al._ (1992) The same procedure is repeated for the other films and the
extracted parameters are summarized in Table 1.

An order of magnitude estimate for the zero temperature penetration depth for
a homogeneous Boron-doped diamond material can be obtained using the London
screening length $\lambda_{L}=\sqrt{\frac{m}{\mu_{0}ne^{2}}}=97.1nm$ where
$n=n_{h}=3.0\times 10^{21}cm^{-3}$ is the doping density of sample "A" and
$e$, $m$ are the un-renormalized electron charge and mass. The measured
penetration depth is much larger than $\lambda_{L}$ hence it’s value can not
be explained by low carrier density alone and the effects of granularity
should be considered.

Figure 3: Measured nonlinear inductance of sample "B". The shift in resonance
frequency $\Delta f$ at $8.11GHz$ as a function of the average circulating
photon number $<n>$ was measured at $T=100\,mK$ and is shown on a linear
scale. The red line represents a linear fit to the data with the slope
corresponding to the self-Kerr coefficient of $K_{11}=15.25\,mHz/Photon$.

The low temperature penetration depth $\lambda(T)$ of a homogeneous s-wave
superconductor has an exponential dependence on temperature Prozorov and
Giannetta (2006):

$\frac{\Delta\lambda(T)}{\lambda(0)}=\sqrt{\frac{\pi\Delta(0)}{2k_{B}T}}exp\left({-\frac{\Delta(0)}{k_{B}T}}\right),$
(2)

where $\Delta(0)$ is the zero temperature BCS superconducting gap. The inset
of Fig.1 shows the low temperature ($T<T_{c}/3$) behaviour of the frequency
shift for sample "A". Here the change in penetration depth
$\Delta\lambda(T)/\lambda(0)\propto\Delta f_{0}(T)/f_{0}^{2}(T)$ with $\Delta
f_{0}(T)\equiv f_{0}(T)-f_{0}(0)$ is plotted. Our data indicates that
$\Delta\lambda(T)$ is exponentially activated in temperature, consistent with
the full energy gap expected for an s-wave superconductor. This indicates that
granularity has not introduced sub-gap states into the superconductor.

Eqs.(1) and (2) were fit to the low temperature portion of the measured
frequency shift (see solid blue line in the inset of Fig.1) with the value of
$\lambda(0)=2.189\;\mu m$ fixed and $\Delta(0)$ as a fitting parameter, which
yielded $\Delta(0)=924.38\pm 76.60\mu eV$. Using $T_{c}=6.717K$ extracted from
the full temperature fit to the same data, the superconducting gap to critical
temperature ratio is calculated to be $\Delta(0)/k_{B}T_{c}=1.597$ somewhat
less than 1.764 which is expected from weak-coupled BCS theory. However we
believe that our film is a heavily doped granular material, and not a doped
single crystal. Previous work on granular Boron-doped diamond films by
scanning tunneling spectroscopy showed a range of values for the gap ratio
($0.63<\frac{\Delta(0)}{k_{B}T_{c}}<1.54$) Dahlem  _et al._ (2010). The
authors of that work attributed this variability to an inverse proximity
effect whereby the granular aspect of the sample created a range of gap values
at the nanoscale.

Table 1:  Summary of the properties extracted from the PPR measurement for all samples. Sample | Film thickness $t$ | dielectric spacer | $f_{0}(0)$ | $\lambda(0)$ | $T_{c}(RF)$ |  $R_{eff}(0)$111Upper limit estimate of resistive losses. See Supp.Mat.Sup  | $\lambda_{BCS}^{dirty}(0)$
---|---|---|---|---|---|---|---
A | $1.5\mu m$ |  $430\mu m$ sapphire | 7.39 GHz | $2.19\pm 0.01\mu m$ | 6.72 K | 64.22 $m\Omega$ | $3.1\mu m$
B | $3.5\mu m$ |  $75\mu m$ sapphire | 8.08 GHz | $3.81\pm 0.03\mu m$ | 2.21 K | 127.16 $m\Omega$222These results were obtained in cooldowns where Cryoperm magnetic shielding was utilized. | $7.3\mu m$
C | $4.0\mu m$ |  $75\mu m$ sapphire | 6.02 GHz | $3.82\pm 0.07\mu m$ | 1.54 K | 12.24 $m\Omega$ | $4.8\mu m$
A | $1.5\mu m$ |  $75\mu m$ sapphire | 5.30 GHz 333This mode is different from the $f_{0}(0)=7.39GHz$ mode listed above. The extracted values for $\lambda(0)$ and $T_{c}$ depend on the mode. This suggests that the sample may be inhomogeneous since the two modes send rf current through different parts of the films. |  $4.02\pm 0.02\mu m$ | 5.68 K  | 1.37 $m\Omega$ | $3.4\mu m$

The nonlinearity of the material can be quantified through the self-Kerr
coefficient $K_{11}$, Maleeva  _et al._ (2018) which is defined as the amount
of resonant frequency shift per added photon. For this measurement, the sample
"B" was kept at base temperature of $T=100mK$ and the resonance was measured
while sweeping the excitation power. The circulating photon number in the
resonator was roughly estimated as $<n>=4QP_{in}/(hf_{0}^{2})$, where $P_{in}$
is the power entering the resonator, $Q$ is the loaded quality factor and $h$
is Planck’s constant. Fig. 3 shows the shift in resonance frequency as a
function of the average circulating photon number. A linear fit to the data
yields the slope of $K_{11}=15.25\,mHz/photon$. This value can be
substantially enhanced by decreasing the thickness of the film.

The extracted zero temperature penetration depth values for the Boron-doped
diamond films are significantly larger than the values reported for other
conventional superconductors, which range between $50nm-500nm$. Greytak and
Wernick (1964); Anlage and Wu (1992); Prozorov  _et al._ (2000); Hashimoto
_et al._ (2009) A possible explanation for the large value of the penetration
depth is that these films are highly granular, giving rise to a large
effective screening length due to a combination of Meissner and Josephson
screening. This is consistent with numerous previous studies showing granular
behaviour in Boron-doped diamond films. Dubrovinskaia  _et al._ (2006);
Willems  _et al._ (2010); Dahlem  _et al._ (2010); Zhang  _et al._ (2013); Li
_et al._ (2014)

The effective penetration depth of polycrystalline and nanocrystalline
superconductors can be estimated using the Laminar Model, Hylton and Beasley
(1989) which predicts that the effective penetration depth of such
superconductors strongly depends on the coupling strength between the grains,
approaching the intrinsic value of the penetration depth of the grains in the
strong coupling limit. For an MCD or NCD film in the small grain and weak
coupling limit one finds an effective penetration depth of
$\lambda_{eff}=\sqrt{\frac{\hbar\rho_{barrier}}{\pi\mu_{0}\Delta}}\sqrt{\frac{t_{barrier}}{a}}$,
where $t_{barrier}$ and $\rho_{barrier}$ are the thickness and resistivity of
the barrier material between the grains, and $a$ is the grain size.Sup  This
can be compared to the dirty limit BCS expression of
$\lambda_{BCS}^{dirty}=\sqrt{\frac{\hbar\rho_{n}}{\pi\mu_{0}\Delta}}$, where
$\rho_{n}$ is the resistivity of the assumed-homogeneous film.Zmuidzinas
(2012) Depending on the $t_{barrier}$ and $\rho_{barrier}$ parameter values
(which are unknown), in the MCD and NCD cases one can find values of the
effective screening length greater than, or less than, the standard BCS dirty
limit value. Table I contains estimates of $\lambda_{BCS}^{dirty}$ for our
films, and they are on the same order of magnitude of the measured screening
length values, consistent with our estimate of $\lambda_{eff}$

Similarly large values of penetration depth are observed in other amorphous
and granular superconductors: $\lambda(0)=0.39\mu m$ in NbN Pambianchi  _et
al._ (1994), $\lambda(0)>0.51\mu m$ in Mo3Si Wördenweber, Kes, and Tsuei
(1986), $\lambda(0)=0.575\mu m$ in TiN Vissers  _et al._ (2010),
$\lambda(0)=0.645\mu m$ in Mo-Ge Missert (1989), $\lambda(0)>0.65\mu m$ in
Nb3Ge Wördenweber, Kes, and Tsuei (1986) and $\lambda(0)=1.2\mu m$ in granular
aluminium film Cohen and Abeles (1968). Such large values of penetration depth
make Boron-doped diamond films an appealing material to be used in
applications where large inductance in needed, such as microwave kinetic
inductance detectors (MKID) or superconducting microresonator bolometers
Zmuidzinas (2012).

The effective surface resistance of these films is obtained from the quality
factor $Q(T)$ through $R_{eff}(T)=\pi\mu_{0}f_{0}(T)d/Q(T)$. Taber (1990);
Talanov  _et al._ (2000) The effective loss data (see Fig. 3 in the Supp.
Mat., Ref. Sup, ) shows substantial residual loss in the zero temperature
limit, summarized in Table 1. A fit to the effective surface resistance vs.
temperature is shown in the Supp. Mat. (Ref.Sup, ). There are several
contributions to the measured $Q$, including coupling loss, radiation loss
from the sides of the PPR and through the films, enhanced Ohmic loss due to
the finite thickness of the films, and losses in the grain boundaries, Hylton
_et al._ (1988) which have been observed to be rich in Boron Kumar  _et al._
(2018a) and trans-polyacetylene.Kumaran  _et al._ (2015) Disentangling all of
these effects in beyond the scope of this paper.

Boron-doped diamond has also been proposed as a platform to build future
hybrid quantum devices Coleman  _et al._ (2020), thanks to its unique
properties. By controlling the dopant type and concentration in Diamond films
one can create an insulator, a semiconductor (both p and n type), an
optically-transparent electrode,Wächter  _et al._ (2016) or a superconductor,
all using the same starting material. Along these lines, Watanabe et al.
demonstrated very uniform vertical SNS Josephson junctions solely with BDD by
carefully controlling the doping concentration during the deposition of the
thin-film Watanabe  _et al._ (2012). This and recent successful demonstrations
of nanosize and microsize patterning processes for Boron-doped diamond films
Kageura  _et al._ (2018) indicates that the prospect of building complex
superconducting devices using Boron-doped diamond films is possible. Finally
we note that quantum devices based on nitrogen vacancy centers in diamond
offer new opportunities for superconducting quantum devices.Kubo  _et al._
(2011)

In this work we present the first measurements of complex surface impedance of
Boron-doped diamond films using a microwave resonant technique. We demonstrate
that these samples have very large penetration depth which is consistent with
the local-limit estimates that are appropriate for granular superconductors.
Furthermore, we provide additional evidence that Boron-doped diamond is an
s-wave superconductor with few sub-gap states and shows promise for high
kinetic inductance applications.

We thank Chung-Yang Wang for assistance with PPR measurements. B.O. and S.M.A.
acknowledge support from the U.S. Department of Energy/ High Energy Physics
through Grant No.DESC0017931 (support for B.O.) and DOE/Basic Energy Sciences
through Grant No.DESC0018788 (measurements). M.S.R. and D.K. would like to
acknowledge the funding and extended supported from the Department of Science
and Technology (DST), New Delhi, that facilitated the establishment of “Nano
Functional Materials Technology Centre” (Grant $\\#$s: SRNM/NAT/02-2005,
DST/NM/JIIT-01/2016 (G); SR/NM/NT-01/2016).

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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there is a fixed-point free involution $g$ of the board $B$. Suppose that $g$
is such that for each $s\in\mathcal{S}$ there is some $f\in\mathcal{F}$ for
which $f\subset g[s]$.

Then, the second player in $\mathcal{G}$ does not have a winning strategy.

Hence, we can informally interpret the involution $g$ as a symmetry (or
better, a reassemblage) of the board that allows the first player to act as
the second player on the new board.

###### Proof of Proposition A.3.

Let the first player make an arbitrary initial move, which is advantageous,
and then play in the following way; when the second player chooses a vertex
$v\in B$, then the first player, in the next turn, marks $g(v)$.

Note that $s\in\mathcal{S}$ is infinite by strict non-openness of
$\mathcal{G}$, and that the second player will mark all the vertices of $s$,
each after finitely many turns. Thus, the first player will eventually mark
all the vertices of $g[s]$.

Therefore, the first player will also win by marking all the vertices of some
$f\in\mathcal{F}$ such that $f\subset g[s]$, which is a contradiction. ∎

######  Remark A.4.

Observe that we cannot relax the assumption of strict non-openness in
Proposition A.3.

Otherwise, the second player could pick a finite $s\in\mathcal{S}$, which
preferably does not contain the initial move of the first player, and then
could proceed to mark all the vertices in $s$, so that the first player would
be able to fully satisfy his winning condition $f\subset g[s]$ only with his
move that follows the last move of the second player; that is a move too late,
so that the second player actually wins.

Note that strengthening the condition by imposing that the involution $g$ is
such that for each $s\in\mathcal{S}$ there is some $f\in\mathcal{F}$ for which
$f\subsetneq g[s]$ is still not enough; there could be an appropriate order in
which the second player could mark the vertices of a finite $s\in\mathcal{S}$
that defeats the opponent.

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