# Superconducting nitridized-aluminum thin films

Alba Torras-Coloma  Institut de Física d’Altes Energies (IFAE), The Barcelona
Institute of Science and Technology (BIST), Bellaterra (Barcelona) 08193,
Spain  Departament de Física, Universitat Autònoma de Barcelona, 08193
Bellaterra, Spain  Leyre Martínez de Olcoz  Institute of Microelectronics of
Barcelona (IMB-CNM), Spanish National Research Council (CSIC), Cerdanyola
08193, Spain  Eva Céspedes  Institute of Microelectronics of Barcelona (IMB-
CNM), Spanish National Research Council (CSIC), Cerdanyola 08193, Spain  Elia
Bertoldo  Institut de Física d’Altes Energies (IFAE), The Barcelona Institute
of Science and Technology (BIST), Bellaterra (Barcelona) 08193, Spain  David
López-Núñez  Institut de Física d’Altes Energies (IFAE), The Barcelona
Institute of Science and Technology (BIST), Bellaterra (Barcelona) 08193,
Spain  Barcelona Supercomputing Center, Barcelona 08034, Spain  Departament de
Física Quàntica i Astrofísica and Institut de Ciències del Cosmos, Universitat
de Barcelona, Barcelona 08028, Spain  Sagar Paul  PHI, Karlsruhe Institute of
Technology, 76131 Karlsruhe, Germany  Wolfgang Wernsdorfer  PHI, Karlsruhe
Institute of Technology, 76131 Karlsruhe, Germany  IQMT, Karlsruhe Institute
of Technology, 76344 Eggenstein-Leopoldshafen, Germany  Gemma Rius
<EMAIL_ADDRESS>Institute of Microelectronics of Barcelona (IMB-CNM),
Spanish National Research Council (CSIC), Cerdanyola 08193, Spain  P. Forn-
Díaz<EMAIL_ADDRESS>Institut de Física d’Altes Energies (IFAE), The
Barcelona Institute of Science and Technology (BIST), Bellaterra (Barcelona)
08193, Spain  Qilimanjaro Quantum Tech SL, Barcelona, Spain

(August 31, 2024)

###### Abstract

We report the direct observation of superconductivity in nitridized-aluminum
thin films. The films are produced by sputtering deposition of aluminum in a
controlled mixture of nitrogen diluted in argon. The concentration of applied
nitrogen directly determines the properties of the superconducting thin films.
We observe samples displaying critical temperatures up to 3.38$\pm$0.01 K and
resilience to in-plane magnetic fields well above 1 T, with good
reproducibility of the results. This work represents an unambiguous
demonstration of tunable superconductivity in aluminum-based nitridized thin
films. Our results put forward nitridized aluminum as a promising material to
be employed in superconducting quantum circuits for quantum technology
applications.

##  I Introduction

Superconducting quantum circuits are a leading platform for building
applications in quantum technologies, particularly in the field of quantum
computation [1, 2]. The performance of these circuits is closely tied to the
quality of the materials employed, establishing, e.g., limitations on qubit
coherence [3, 4] and on resonator quality [5, 6]. One of the leading loss
mechanisms is surface dielectric loss in the form of microscopic two-level
defects, ubiquitous in both the substrate-metal and metal-air interfaces of
the superconducting quantum devices [7]. Among the strategies to reduce
surface dielectric losses, nitrogen-based superconductors, such as NbN, TiN,
or NbTiN, have become of great interest in the quantum computing community of
superconducting circuits in the last decade, particularly as materials to
build resonators and qubit capacitors [8, 9, 10, 11, 12, 13]. It is considered
that the presence of nitrogen on the surface and within the superconducting
thin films helps reducing the impact of two-level system defects that is
attributed to native oxides or impurities [14, 15], such as those typically
found in aluminum and niobium thin films [16]. In addition, a high degree of
disorder is typically attributed to very thin ($\lesssim 20\,\mathrm{nm}$)
nitrogen-based superconductors which explains their characteristic large
kinetic inductance [17]. Superconducting thin films with high kinetic
inductance are key to produce high-impedance circuits with low losses, a very
relevant feature required in high-coherence superconducting circuits [18] as
well as in single photon detectors [19, 20], state-of-the-art parametric
amplifiers [21, 22], compact inductance elements and superinductors [23], and
kinetic inductance detectors [24, 25].

Motivated by the properties exhibited by nitrogen-based superconductors for
quantum circuits, we present here NitrAl (short term for nitridized aluminum),
a new superconducting material obtained by depositing aluminum via DC-
magnetron sputtering in a controlled atmosphere of nitrogen and argon. To our
knowledge, this material has not been analyzed in depth as a superconducting
material and only minimal evidence exists on its superconductivity [26, 27].

Crystalline aluminum nitride (AlN) is a well-known wide band-gap material with
hexagonal crystal lattice commonly used in microelectronics, for example, as a
nanomechanical resonator [28]. In view of its piezoelectric properties it is a
suitable candidate for acoustic-wave resonating structures [29]. Recently, AlN
has been employed as a bulk acoustic resonator coupled to a superconducting
qubit [30]. However, the strong piezoelectricity displayed by AlN results in a
loss mechanism through phonon dissipation [28]. Despite this detrimental
effect, thin-film dielectric AlNx has been used as the junction tunnel barrier
in both flux [31] and transmon qubits [32], exhibiting long coherence times.
The approach followed by both these works consisted on depositing thin films
($\sim$2 nm) in the cubic phase of AlN which does not display piezoelectricity
due to its inversion symmetry. On the other hand, tunnel junctions made of AlN
have been sputtered for bolometer applications using an inductively coupled
plasma source yielding high quality, highly transparent junctions [33]. These
results indicate that the sources of loss evidenced in wide-bandgap hexagonal
AlN can be unimportant in other phases of AlN, existing as e.g. disordered or
non-stoichiometric thin films, which may then be used as parts of
superconducting circuits.

In contrast to aluminum nitrogen-based superconductors, an oxygen-based
superconductor has already been extensively investigated in the form of
granular aluminum (grAl) [34]. GrAl consists of aluminum grains embedded in an
aluminum oxide matrix and can display superconductivity. Actually, this
material presents several properties that make it attractive for a variety of
applications, such as in quantum computing circuits [35] or employed in
astronomy detectors [36]. GrAl displays a higher critical temperature
($T_{\mathrm{c}}$) than pure Al, up to 3 K for nitrogen-cooled substrates
during deposition [37], and can attain very high room temperature resistivity
values, which correlate into superinductor behavior in the superconducting
state [38].

Our work presents the unambiguous signatures of superconductivity exhibited by
a range of NitrAl thin films produced at different concentrations of applied
nitrogen (N2). We observe the impact of nitrogen concentration on the $T_{c}$
of the material, as well as the response under large magnetic fields. Our
results indicate that NitrAl may be used as a superinductor in quantum circuit
applications, as well as in experiments where magnetic field compatibility is
desired, such as in semiconductor spin qubits [39] and topological-based
condensed matter systems [40].

##  II Thin Film Materials and Methods

In this work, we have focused on the dependence of the resulting NitrAl
properties as function of the N2 concentration, while keeping the rest of
parameters at a constant value. However, other deposition parameters can
significantly influence the properties of the films, such as the working
pressure or the power value at the target, which will be considered in future
studies.

The Al-based thin films are grown by sputtering technique at the Clean Room of
the IMB-CNM-CSIC. The equipment is a DC magnetron sputtering from Karl Suss
GmbH, model KS-800 HR. The thin film deposition conditions have been developed
for substrates at the 4-inches wafer scale. The substrates consist in
commercially-available silicon wafers 111The properties of the silicon wafers
used for this study are the following. Type/dopant: p/Bor; resistivity:
4-40$\text{\,}\mathrm{\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$; orientation:
$\langle 100\rangle\pm 1$$\text{\,}\mathrm{\SIUnitSymbolDegree}$; diameter:
$100.0\pm 0.3$$\text{\,}\mathrm{mm}$; thickness: $525\pm
15$$\text{\,}\mathrm{\SIUnitSymbolMicro m}$; single sided polished; primary
flat: $(101)\pm 1$$\text{\,}\mathrm{\SIUnitSymbolDegree}$, $32.5\pm
2.5$$\text{\,}\mathrm{mm}$. covered with a high quality insulating silicon
dioxide layer, typically 500 nm in thickness deposited also in-house. A wet
oxidation technique with a processing temperature of 1100 C was applied for
obtaining such a silicon dioxide layer. The metal target is supplied as 99.99%
Al, while the processing gases N2 and Ar are of high purity ($>5.5$). The
sputtering chamber base pressure is $2\times 10^{-7}$ mbar.

The processing parameters for the sputtering deposition, common for all the
tests reported in this work, have been established as 60 sccm Ar flow, working
pressure $P_{w}=5\times 10^{-3}$ mbar, and power to the target $P_{t}=1000$ W,
i.e. those being considered as sputtering conditions within a conventional
range. The addition of N2 is kept in a regime where non-stoichiometric AlN is
expected to be obtained, similar to other works using TiN [42]. Specifically,
the effect of diluted N2 in the sputtered Al-based thin film properties and
functionality has been tested and fully evaluated for N2/Ar fractional flow
mainly in the range of 0% to 15%, which correspond to N2 flow range 0 to 9
sccm (Table 1). In this N2 flow range, the actual deposition rate is
determined to be inversely proportional to the applied N2 flow (Table 1),
therefore, the corresponding adjustment of the processing time for each N2
flow value has been applied. A common nominal or target thickness of $t=100$
nm for the Al and NitrAl thin films is set. Thin film deposition parameters
and methodology reported in this study have been developed and consolidated
according to microelectronics standards, to ensure reproducibility in the
processing and reliability in the implementation, as well as thin film
characteristics, including thickness homogeneity and deposition rate control,
and low values of sheet resistance and roughness.

We point out that the actual amount of atomic nitrogen incorporated in the
NitrAl thin films produced at variable N2 flow values could not be reported in
this work. Instead, we characterize each nitrogen concentration with the N2 %
fractional flow value used in the fabrication process.

Table 1: Range of N2 flow values and the parameters used in the fabrication of NitrAl samples reported. The following parameters are set constant for all samples: Ar flow = 60 sccm, power set to the target $P_{t}=1000$ W, nominal thickness deposition $t=100$ nm, Ar working pressure $P_{w}=5\times 10^{-3}$ mbar, sputtering chamber base pressure $P_{b}=2\times 10^{-7}$ mbar. N2(sccm) |  N2/Ar(%) | Rate(nm/s)
---|---|---
0.0 | 0.00 | 1.91
1.5 | 2.50 | 1.78
2.0 | 3.33 | 1.73
3.0 | 5.00 | 1.64
4.0 | 6.67 | 1.55
5.0 | 8.33 | 1.46
6.0 | 10.00 | 1.37
7.0 | 11.67 | 1.28
8.0 | 13.33 | 1.19
9.0 | 15.00 | 1.10

Once the thin films are produced at wafer scale, each sample is obtained by
wafer cleavage in a square geometry with approximate dimensions
$$7\text{\,}\mathrm{mm}$\times$7\text{\,}\mathrm{mm}$$, and fixed to a
commercial chip carrier using vacuum grease where it is wire bonded with Al
wires. A pair of wire bonds are placed directly on the NitrAl thin films in
each of the corners of the sample to improve the electrical contact. Each
sample is tested at room temperature by the Van der Pauw technique in this
4-probe configuration. The resistance of the film is obtained by fitting the
resulting current-voltage (IV) curve measured with a source-meter instrument.
To calculate the sheet resistance, we multiply the resistance obtained from
the fit by the Van der Pauw constant: $\pi/\ln{2}$ [43].

After the room temperature validation, each sample is mounted either on the
mixing chamber stage or on the still plate of a dry dilution refrigerator with
a base temperature of 20 mK. Standard magnetic shielding is employed both at
room temperature and at low temperature consisting in high-$\mu$ materials.
Accurate temperature control is implemented in the determination of
$T_{\mathrm{c}}$ values of the thin films. As the $T_{\mathrm{c}}$ of most
samples lies in the 1-3.5 K range, specific procedures to stabilize the
temperature in this range are required (see Sec. A of the Supplementary
Material for details).

In order to capture the entire evolution of resistance with temperature,
$R(T)$, the resistance of each thin film is monitored as the system cools
down/warms up. The range of swept currents is gradually adjusted to avoid
excessive heat dissipation when the samples are in the normal state. Once in
the superconducting state, the current is adjusted to determine the switching
current value at base temperature. Then, the current range is reduced
significantly below the switching current value in order to perform the
$T_{\mathrm{c}}$ measurements.

Magnetic field-dependent IV-characteristics measurements of NitrAl samples are
performed in a different dilution refrigerator in the KIT laboratory equipped
with a vector magnet, as described in [44]. Using this experimental setup,
perpendicular fields $B_{\mathrm{z}}$ up to 0.1 T and in-plane fields
$B_{\mathrm{x}}$ up to 1.4 T can be applied without manually rotating the
sample, while the bath temperature can be varied down to 30 mK. In this case,
the approximate dimensions of the samples are reduced to
$$1\text{\,}\mathrm{mm}$\times$3\text{\,}\mathrm{mm}$$ to fit the special chip
carrier used. The resistance values are obtained from the IV-characteristics
measured in lock-in configuration about zero constant bias current $I_{b}$ and
$10\text{\,}\mathrm{\SIUnitSymbolMicro A}$ modulation amplitude, as a function
of the temperature and the magnetic field applied in the different directions.
More details can be found in Sec. D1 of the Supplementary Material.

##  III Functional characteristics of the NitrAl thin films

###  III.1 Electrical properties at room temperature

In Fig. 1(a) we show the resistivity values at room temperature
$\rho_{\mathrm{RT}}$ for the range of thin films produced, plotted as a
function of the ratio of N2/Ar flow in the process of sputtering deposition,
and for two different fabrication runs (red and black markers). The blue
triangle markers represent direct determination of $\rho_{\mathrm{RT}}$
averaged over several points randomly chosen across the 4” wafer scale, for
which dispersion is found to be about or below 3-5%. The resistivity
estimations at the sample level based on the Van der Pauw technique at chip
level agree with the wafer-scale averaged values.

The obtained data show that the addition of N2 flow in the sputtering process
clearly impacts the thin film resistivities for all flow values, with a
significant increase with respect to the 0% reference thin film (pure Al)
whose value is close or nearly comparable to bulk Al. The results for the two
different fabrication runs of sputtered NitrAl thin films show significant
reproducibility. $\rho_{\mathrm{RT}}$ steadily rises up to an order of
magnitude with respect to pure Al in the range 2.5%–5%. Then, between 5% and
8.33% $\rho_{\mathrm{RT}}$ flattens. Beyond 10%,$\rho_{\mathrm{RT}}$ continues
to rise, following a divergent dependence, finally displaying an electrically
insulating characteristic at 15% N2 flow, with $\rho_{\mathrm{RT}}\sim
1.5\leavevmode\nobreak\ \Omega\,$cm, even visible in the surface color of the
sample (see Supplementary Material E for more details).

Figure 1: (a) Resistivity $\rho_{\mathrm{RT}}$ measured for each NitrAl film
at room temperature as function of N2 flow, relative to the fixed Ar flow used
in the sputtering process. The samples labeled in black (red) were produced in
the first (second) run of fabrication. The blue triangles correspond to the
averaged values over a wafer scale. The Al reference values are marked with a
gray dashed line for reference. (b)
$RRR\equiv\rho_{\mathrm{RT}}/\rho_{\mathrm{4K}}$ as function of N2 flow,
$\rho_{\mathrm{4K}}$ being the resistivity at $4\text{\,}\mathrm{K}$.

###  III.2 Electrical properties at low temperatures

In Fig. 1(b), we display the residual resistivity ratio,
$RRR\equiv\rho_{\mathrm{RT}}/\rho_{\mathrm{4K}}$ with $\rho_{\mathrm{4K}}$ the
resistiviy measured at $4\text{\,}\mathrm{K}$, as function of N2 flow down to
4 K. The decrease in $RRR$ as N2 is incorporated is a clear indicator of
increased disorder and impurities in the thin film [45]. Between 2.5% up to
10%, $RRR\gtrsim 1$. Above 10%, $RRR$ tends to decrease below 1. This
behaviour is reminiscent of grAl devices [46], and it is in contrast with,
e.g., NbN disordered films, where the conductivity of the material gradually
decreases as disorder increases [47]. Exemplary IV curves for several samples
at different temperatures are shown in Fig. 6 of the Supplementary Material.

An abrupt increase of resistivity at higher N2 flows [Fig. 1(a)] is compatible
with a Superconductor-to-Insulator Transition (SIT), either driven by
localization of conduction electrons (Mott) or by disorder (Anderson) [48].
The values of $\rho_{\mathrm{RT}}$ displayed by the NitrAl samples
investigated are in the same order of magnitude as grAl devices previously
reporting a Mott insulator-like phase transition [46, 38], as well as
disordered NbN and NbTiN thin films displaying an Anderson-like phase
transition [47, 49].

Figure 2 shows the normalized sheet resistance curves $R_{s}/R_{RT,s}$
obtained for different NitrAl samples in the range between
$19\text{\,}\mathrm{mK}$ and 300 K. As the N2 flow is increased, we observe
two different regimes which we classify according to the change in slope with
temperature on the different curves, as follows:

  * •

Metallic-like $(\mathrm{d}R_{\mathrm{s}}/\mathrm{d}T>0)$: In this regime the
resistance monotonically decreases as temperature decreases ($RRR>1$). For the
studied set of NitrAl films, the metallic regime comprises samples produced
with N2 flows between $0\%$ and $8.33\%$ or, equivalently, for samples with
$\rho_{\mathrm{RT}}\lesssim$76\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{\SIUnitSymbolOhm}\,\mathrm{c}\mathrm{m}$$.

  * •

Transition point $(\mathrm{d}R_{\mathrm{s}}/\mathrm{d}T\sim 0)$: it is
observed around the sample processed with $10\%$ N2 flow, which corresponds to
$\rho_{\mathrm{RT}}\sim
115$–$151\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{\SIUnitSymbolOhm}\,\mathrm{c}\mathrm{m}$
$(RRR\sim 1)$.

  * •

Insulator-like $(\mathrm{d}R_{\mathrm{s}}/\mathrm{d}T<0)$: this regime
comprises samples with increasing resistance as temperature decreases
($RRR<1$). We include in this category samples produced with $11.67\%$ and
$13.3\%$ of N2 flow, or equivalently, with
$\rho_{\mathrm{RT}}\gtrsim$396\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{\SIUnitSymbolOhm}\,\mathrm{c}\mathrm{m}$$.
For this group of samples, we can also spot two distinctive features. For
films produced with $11.67\%$ of N2 fractional flow, the increase in
resistance is quasi-linear for decreasing temperatures. On the other hand, the
most resistive samples ($13.3\%$) show an accelerated, exponential-like
increase of resistance as temperature is lowered.

Figure 2: Normalized sheet resistance curves for several NitrAl samples. The
samples with N2 flow in the range 0% to 6.67% exhibit metallic-like behavior
with $RRR>$1\. By contrast, the samples with N2 flow in the range 11.67%-13%
exhibit insulator-like behavior with $RRR<1$. In fact, the sample 13.3% (r2)
remains resistive down to the base temperature of our refrigerator (19 mK).
The transition point between the two regimes appears around the sample with
10% N2 flow with $RRR\sim 1$.

Below 4 K, all NitrAl samples in the 2.5%-13.3% N2 flow range show a normal-
to-superconducting state transition at varying $T_{\mathrm{c}}$ values. Figure
3(a) shows the suppression of resistance at the transition temperature
$T_{\mathrm{c}}$ for samples with different N2 flow fractions. We take the
definition of $T_{\mathrm{c}}$ as the point where the resistance drops $50\%$
with respect to the onset point of $R(T)$ [50]. We did not observe the
superconducting transition above $19\text{\,}\mathrm{mK}$ for sample $13.33\%$
produced in the second run of fabrication (r2), indicating that the SIT may
lie close to that process N2 flow value.

In several samples we have observed an increase of 20%-30% of the resistance
prior to the transition to the superconducting state. Such a peak appears only
for particular combination of current and voltage probes. Annex B.2 gives more
details about this feature, which we attribute to the disordered nature of the
thin films.

Figures 3(b)-(c) summarize the obtained critical temperatures as a function of
N2 flow and $\rho_{\mathrm{RT}}$, respectively. Overall, the trend of
$T_{\mathrm{c}}$ values changes non-monotonically, with most samples exceeding
the $T_{\mathrm{c}}$ of pure Al (dashed gray-line). The highest
$T_{\mathrm{c}}=3.38\pm$0.01\text{\,}\mathrm{K}$$ is observed at 5% N2 flow
222It is likely that and even higher $T_{\mathrm{c}}$ than the one observed at
5% lies in a N2 flow not sampled by the discrete set of values used in this
work., while the 13.33% N2 flow sample lies below Al with
$T_{\mathrm{c}}=0.86\pm$0.02\text{\,}\mathrm{K}$$. The general shape of the
$T_{\mathrm{c}}$ curve is dome-like, both in percentage and resistivity
$\rho_{\mathrm{RT}}$. However, the samples with $6.67\%$ and $8.33\%$ N2 flow
do not seem to evenly follow the trend of a dome. On one hand, they display a
drop in $T_{\mathrm{c}}$ in the center of the dome, and, on the other hand,
their values of $\rho_{\mathrm{RT}}$ versus $T_{\mathrm{c}}$ in Fig. 3(c) do
not follow the rest of samples. A more in-depth study of NitrAl materials at
the microscopic level would be required to interpret these particular
features.

Figure 3: (a) Sheet resistance measurement near the normal metal-to-
superconductor transition for different samples. Note the decrease in slope at
the transition temperature for increasing N2 flow. (b)-(c) Critical
temperatures of all NitrAl samples determined by resistivity measurements as
function of (b) the fraction of N2 flow during fabrication and (c) the
resistivity measured at room temperature [Fig. 1(a)]. Samples produced in the
first (second) run of fabrication are shown in black squares (red circles).
The $T_{\mathrm{c}}$ of aluminum (0% reference sample) is shown with a gray
dashed line for reference.

We note that the dome-shape of $T_{\mathrm{c}}$ versus $\%\ch{N2}$ in Fig.
3(b) roughly defines three regions: increasing $T_{\mathrm{c}}$ (0%-5%),
saturation of $T_{\mathrm{c}}$ (5%-10%) and decreasing $T_{\mathrm{c}}$
(10%-13.33%), which approximately correspond with the three regimes described
in Fig. 1(a) of $\rho_{\mathrm{RT}}$ versus %N2/Ar. This correspondence points
to a possible connection between $\rho_{\mathrm{RT}}$ and $T_{\mathrm{c}}$
driven by the same mechanism, likely the disorder and/or localization of the
superconducting wave function [48]. Indeed, from the data displayed in Fig. 2,
the insulator-like behavior begins at the 10% samples.

Three different regimes of $T_{\mathrm{c}}$ versus $\rho$ have also been
observed in other disordered superconductors such as grAl [52]. In the case of
grAl it is argued that a competitive effect occurs with increasing
concentration of oxygen between the phase stiffness $J$ of the superconducting
condensate, the charging energy $E_{\mathrm{C}}$ for a single electron to
tunnel between aluminum grains, and the superconducting gap $\Delta$ [46]. In
the metallic regime, a single wave function is formed by all electrons which
is characterized by a phase stiffness $J$, orders of magnitude above
$E_{\mathrm{C}}$ and $\Delta$ [that would correspond to $\rho_{\mathrm{RT}}$
values below
$\sim$40\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{\SIUnitSymbolOhm}\,\mathrm{c}\mathrm{m}$$
for our NitrAl thin films, Fig. 3(c)]. At the dome maximum, the phase
stiffness $J$ falls below the Coulomb repulsion
[$\rho\sim$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{\SIUnitSymbolOhm}\,\mathrm{c}\mathrm{m}$$
in Fig. 3(c)]. At this point, the transition towards an insulator begins [48].
When $J\sim\Delta$, the Coulomb repulsion localizes the wave function in the
grains and the material arrives at the SIT
[$\rho>$1000\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{\SIUnitSymbolOhm}\,\mathrm{c}\mathrm{m}$$
in Fig. 3(c)].

This qualitative description of grAl appears to fit well to our observations
with NitrAl despite that we have no evidence of the latter having a granular
structure. Yet, there exist some differences. Generally, in disordered and
granular superconductors the SIT is expected for resistances close to the
quantum of resistance $R_{\mathrm{Q}}=h/(2e)^{2}\approx
6.4\leavevmode\nobreak\ \text{k}\Omega$. For the present study, the
$R_{\mathrm{4K,s}}$ threshold value for the suppression of superconductivity
is $700.8\pm 0.9$ $\Omega/\square$, which is almost an order of magnitude
smaller than $R_{\mathrm{Q}}$. We note that the observations of Ref. [46] are
made on thinner films (20 nm thick), which likely enhance the effects by
increasing the film resistance in the normal state. For instance, we do not
have evidence of features appearing at $\sim 1.2$ K reminiscent of pure Al for
the insulating films, which might derive from the core-shell structure of
grAl. The thickness-dependent effects and the microscopic material description
of NitrAl will be the subject of future works, where these features will be
explored explicitly in more detail.

We are in the process of building a microscopic model of NitrAl, necessary to
draw quantitative conclusions from the data in Fig. 3 to understand the
dominant energy scales in each range of $\rho_{\mathrm{RT}}$, and to determine
the type of SIT.

The values of $T_{\mathrm{c}}$ and $\rho_{\mathrm{4K}}$ can be used to
estimate the sheet kinetic inductance ($L_{\mathrm{k,s}}$) of the films by
applying the Mattis-Bardeen formula for the complex conductivity in the local,
dirty limit at low frequency $(hf\ll k_{\mathrm{B}}T)$ and in the low
temperature limit $(T\ll T_{\mathrm{c}})$ [53],

$L_{\mathrm{k,s}}=0.18\frac{\hbar R_{\mathrm{4K,s}}}{k_{B}T_{\mathrm{c}}},$
(1)

where $R_{\mathrm{4K,s}}$ is the normal state sheet resistance at
$4\text{\,}\mathrm{K}$, $\hbar$ is the reduced Planck constant and
$k_{\mathrm{B}}$ stands for the Boltzmann constant. The values of sheet
kinetic inductance obtained for the NitrAl samples range between $1.31\pm
0.01$ pH/$\square$ (0% N2) and $422.48\pm 8.22$ pH/$\square$ (13.3% N2). The
complete set of values for $L_{\mathrm{k,s}}$, together with $T_{\mathrm{c}}$,
$\rho_{\mathrm{RT}}$ and other sample parameters, are shown in Table 2. This
preliminary estimation of $L_{\mathrm{k,s}}$ points towards NitrAl as a
material with high$L_{\mathrm{k,s}}$, especially for the most resistive
samples (11.67%-13.33% N2 flow) that reach values close to other reported
superinductive materials such as grAl [54].

Table 2: Observed parameters for the different NitrAl samples measured: sample label, N2/Ar fractional flow (%), fabrication run number, nominal thickness $t$, critical temperature $T_{\mathrm{c}}$, resistivity at room temperature $\rho_{\mathrm{RT}}$, and at $4\text{\,}\mathrm{K}$ $\rho_{\mathrm{4K}}$, estimated sheet kinetic inductance $L_{\mathrm{k,s}}$ using Eq. (1).  Sample |  N2/Ar (%) | Run |  $t$ ($\text{\,}\mathrm{nm}$) |  $T_{\mathrm{c}}$($\text{\,}\mathrm{K}$) |  $\rho_{\mathrm{RT}}$ ($\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$) |  $\rho_{\mathrm{4K}}$ ($\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$) |  $L_{\mathrm{k,s}}$ (pH/$\square$)
---|---|---|---|---|---|---|---
A |  $0$ \- Al ref. | 2 | $100.0$ | $1.25\pm 0.01$ | $2.92\pm 0.04$ | $1.05\pm 0.05$ | $0.12\pm 0.01$
B | $2.50$ | 2 | $100.0$ | $2.30\pm 0.01$ | $25.66\pm 0.03$ | $21.82\pm 0.03$ | $1.31\pm 0.01$
C | $3.33$ | 2 | $100.0$ | $2.57\pm 0.08$ | $36.67\pm 0.03$ | $32.28\pm 0.11$ | $1.72\pm 0.06$
D | $5.00$ | 1 | $100.0$ | $3.38\pm 0.01$ | $50.01\pm 0.03$ | $46.27\pm 0.03$ | $1.88\pm 0.01$
E | $5.00$ | 2 | $100.0$ | $3.07\pm 0.02$ | $42.07\pm 0.13$ | $37.50\pm 0.04$ | $1.68\pm 0.01$
F | $6.67$ | 2 | $100.0$ | $2.85\pm 0.04$ | $29.25\pm 0.03$ | $22.17\pm 0.03$ | $1.07\pm 0.02$
G | $8.33$ | 2 | $100.0$ | $2.43\pm 0.01$ | $55.88\pm 0.03$ | $48.50\pm 0.03$ | $2.74\pm 0.01$
H | $10.0$ | 1 | $83.0$ | $2.89\pm 0.12$ | $115.97\pm 0.16$ | $115.63\pm 0.03$ | $6.61\pm 0.27$
I | $10.0$ | 2 | $100.0$ | $2.74\pm 0.01$ | $150.26\pm 0.03$ | $156.91\pm 0.03$ | $7.88\pm 0.01$
J | $11.67$ | 1 | $97.1$ | $2.19\pm 0.01$ | $317.52\pm 0.03$ | $370.11\pm 0.03$ | $23.91\pm 0.08$
K | $11.67$ | 2 | $100.0$ | $2.02\pm 0.01$ | $444.13\pm 0.03$ | $458.59\pm 0.03$ | $35.30\pm 0.10$
L | $13.33$ | 1 | $94.6$ | $0.91\pm 0.02$ | $1312.25\pm 0.03$ | $2641.66\pm 0.05$ | $422.48\pm 8.22$
M | $13.33$ | 2 | $100.0$ | - | $2177.01\pm 0.03$ | $7008.47\pm 0.03$ | -

Additionally, the critical currents $I_{\mathrm{c}}$ have been measured at
base temperature for each N2 flow value. Since these measurements were
performed on thin films with no device pattern, the results only provide a
qualitative understanding of the behavior of $I_{\mathrm{c}}$ in NitrAl films.
The resulting values are shown in Fig. 4. Note that when the samples from both
fabrication runs are sorted by $\rho_{\mathrm{RT}}$ (Fig. 4b) they follow an
overall consistent trend, which is in contrast to the deviations between the
$I_{\mathrm{c}}$ from different fabrication runs for a given N2 flow value
(Fig. 4a). For samples in the range of 8.33%-13.33% the critical current lies
below that of Al, possibly indicating that the superconducting state is
weakening due to the increase in disorder/localization.

Figure 4: Critical currents $I_{\mathrm{c}}$ of NitrAl as a function of (a)
the fraction of N2 flow during fabrication and (b) the resistivity
$\rho_{\mathrm{RT}}$ at room temperature of the samples.

###  III.3 Magnetic field response

Figure 5 shows the extracted fields at which superconductivity is destroyed
$B_{\mathrm{c}}$ for perpendicular $B_{\mathrm{z,c}}$ (a) and in-plane
$B_{\mathrm{x,c}}$ (b) externally applied magnetic fields as function of
temperature.

Figure 5: Magnetic fields at which superconductivity is destroyed as function
of the normalized temperature in the (a) $z$-direction, $B_{\mathrm{z,c}}$
(perpendicular to the chip) and (b) the $x$ direction, $B_{\mathrm{x,c}}$
(parallel to the chip) for different NitrAl samples produced in the second run
of fabrication. The temperatures have been normalized to the critical
temperature of each sample. In (a), the reference Al sample shows no
temperature dependence as the smallest applicable field of 0.01 T already
drives it in the normal state at base temperature.

At base temperature, the Al reference ($0\%$ N2 flow) has a critical field
below $0.01\text{\,}\mathrm{T}$ for $B_{\mathrm{z}}$ and
$0.06\text{\,}\mathrm{T}$ for $B_{\mathrm{x}}$, as expected for thin-film Al
[55]. For the subset of NitrAl samples displayed in Fig. 5(a), the available
range of the magnet in $B_{\mathrm{z}}$ is only capable of driving the system
out of the superconducting state close to the critical temperature of the
NitrAl samples. On the other hand, for $B_{\mathrm{x}}$ [Fig. 5(b)] the
increase of resilience of NitrAl to magnetic fields with respect to the Al
reference sample is high but less pronounced. We observe an increase of almost
a factor 7 in the $B_{\mathrm{x,c}}$ for sample $3.3\%$. For the $10\%$
sample, the available range of the magnet in$B_{\mathrm{x}}$ is not enough to
destroy the superconducting state at base temperature.

Measurements of IV-characteristics as function of the applied magnetic fields
and temperature can be found in Fig. 12 of the Supplementary Material. In
particular, for the 10% sample a gradual increase of resistance is observed
for the largest applied fields close to the switching current of the sample.
This effect is reminiscent of type-II superconductivity where such a
resistance at large applied fields close to the switching current of the
sample is due to vortex flow induced by the applied current [55]. In light of
this connection with type-II superconductors, the observed field at which
superconductivity is destroyed may be identified as
$B_{\mathrm{c}}=\mu_{0}H_{\mathrm{c2}}$. Additional measurements under
magnetic fields are needed to study the type of superconductivity displayed by
NitrAl, but those are beyond the scope of this initial work.

The results obtained for NitrAl as function of the magnetic field follow the
observations in other disordered superconductors [56, 57], where an increase
in resistivity and superconducting gap with respect to the pure superconductor
is accompanied by an increase in critical field.

Overall, there is a clear increase in the field at which superconductivity is
destroyed in both axes $B_{\mathrm{z}},B_{\mathrm{x}}$ for increasing
$\rho_{\mathrm{RT}}$ in the NitrAl samples. This magnetic field resilience
makes NitrAl an interesting candidate to be used in circuits where high
magnetic fields are required, such as, e.g., those manipulating electron spins
[39] or hybrid superconductor-semiconductor structures [58]. The behaviour of
NitrAl thin films under strong magnetic fields is comparable to that displayed
in general by disordered superconductors, including grAl [59], NbN [47], and
NbTiN [60].

##  IV Conclusions

We have reported in this work an in-depth characterization of the
superconducting properties of nitridized aluminum -NitrAl-, a new
superconducting material that we put forward to be introduced in quantum
circuits for quantum technology-related applications. By adjusting the
relative concentration of nitrogen with respect to argon in the sputtering
process to form the NitrAl films, we are able to control its superconducting
and room temperature characteristics. For a wide parameter range, NitrAl
samples display properties superior to pure Al, such as higher critical
temperature (up to 3.38$\pm$0.01 K) and larger fields at which
superconductivity is destroyed (above 1 T for in-plane fields). The
controllability of the fabrication process, the range of reported critical
temperatures and the extent of measured resistivities open the possibility to
use NitrAl as a superinductor material, as well as a superconductor with
adjustable $T_{\mathrm{c}}$.

Further studies are required to infer the microscopic properties of NitrAl,
particularly to discern whether the likely presence of native oxide is
responsible for some of the properties observed. In addition, its behavior
needs to be investigated as a material for high-frequency circuits in the form
of resonators or even superconducting qubits, particularly the losses
exhibited by NitrAl under these conditions.

## Data availability statement

Data available upon request or in the following link:
https://doi.org/10.5281/zenodo.10194875.

## Acknowledgements

The authors would like to thank Karl Jacobs from I. Physikalisches Institut,
Universitaet zu Koeln, for providing the first set of NitrAl samples and for
insightful discussions. We would also like to acknowledge Fabio Henriques for
helping with the temperature measurements, Juan Jiménez for developing
software for data analysis, and fruitful discussions with E. Driessen, T.
Klapwijk, and A. Endo. We acknowledge funding from the Ministry of Economy and
Competitiveness and Agencia Estatal de Investigación (RYC2019-028482-I,
RYC-2016-21412, FJC2021-046443-I, PCI2019-111838-2, PID2021-122140NB-C31,
PID2021-122140NB-C32, TED2021-130292B-C44), the European Commission (FET-Open
AVaQus GA 899561, QuantERA), and program ‘Doctorat Industrial’ of the Agency
for Management of University and Research Grants (2020 DI 41; 2020 DI 42).
IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.
This study was supported by MICIIN with funding from European Union
NextGenerationEU(PRTR-C17.I1) and by Generalitat de Catalunya.

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## Supplementary Material

###  A IV current caracteristics

The dilution refrigerator is in a stable configuration with the pulse tube ON
while the 4K, Still, and mixing chamber (MXC) plates are thermally connected.
No 3He/4He mixture is circulating. A small quantity (20-50 mbar) of mixture
(total $\sim 500$ mbar) is introduced in the system. When all plates are at
the same temperature, the MXC and Still plates are thermally isolated, while
the Still remains in thermal contact with the 4K plate. In this configuration,
the MXC can be stabilized using its heater to temperatures below the ones
reachable by only employing the pulse tube (3-4K). If the Still is then
isolated from the 4K plate and more mixture is condensed in a controlled way,
the system can slowly reach temperatures $\sim 1$ K or lower. The temperature
of the mixing chamber and the Still can be modified by using their individual
heaters and/or by restoring the thermal connection between the two plates. The
whole procedure can be sped up or slowed down by controlling the amount of
mixture in circulation and the heat introduced by the Still and MXC heaters.

Figure 6 shows different IV current characteristics for different NitrAl
samples. The figure is a zoom into the range used for the low temperature
$T_{\mathrm{c}}$ measurements. There is a recurring offset in the voltage axis
related to the source-meter setup.

Figure 6: Current-Voltage (IV) characteristics for sample with 2.5% N2 (a),
and 13.3% N2 (b). Notice the difference in voltage ranges used for both
samples.

###  B Resistance curves

####  B.1 Normalized resistance curves

In Figure 7 we provide the normalized sheet resistance curves not shown in the
main text.

Figure 7: Normalized sheet resistance curves for several NitrAl samples.

####  B.2 Resistance peak near the superconducting transition

In several samples we have observed an increase of 20%-30% of the resistance
prior to the transition to the superconducting state. Such a peak appears only
for the combination of current and voltage probes giving slightly higher
resistance. We tested different probe configurations (see Figure 8) for two
samples near the superconducting transition. In Figure 9 we plot the
resistance versus temperature for different probe configurations labeled as
CA, CB, CC following the nomenclature of Figure 8. We did not include the in-
line configuration C for sample G-8.33% due to the low resistance and noise in
the measurement.

Figure 8: Different contact locations for the four probe configuration used to
test the origin of the anomalous peak near the superconducting transition.

With the first configuration (A) we measured slightly larger resistances and
we observed the spike before the superconducting transition. Rotating the
contacts (B) provided slightly lower resistance and we were not able to
reproduce the rapid increase in the R(T) curve near Tc. Finally, with the in-
line configuration (C), no spike was visible. A similar phenomenon has been
observed in other disordered and homogeneous thin films measured in 4-probe
configuration [61, 62]. The anomalous peak can be attributed to local
variations of Tc and current redistribution effects on the sample. Actually,
the critical temperatures measured in different 4-probe configurations differ
slightly by 0.01K.

Figure 9: Resistance versus temperature for NitrAl (a) 8.33% in two different
4-probe configurations and (b) 11.67% in three different 4-probe
configurations.

###  C Sheet kinetic inductance and London penetration depth

Using Eq. (1), in Fig. 10 we plot the resulting sheet kinetic inductance as
function of (a) the N2 flow and (b) as function of the room temperature
resistivity.

Figure 10: Estimated sheet kinetic inductance using Eq. (1) for different N2
flow values (a), and as function of room temperature resistivity $\rho$ (b).

Figure 10(a) shows a similar trend as $\rho$ at room temperature, with an
initial jump with respect to pure Al, an intermediate regime with constant
values, and a final exponential increase above 10%.

Similarly, in Figure 11 we plot the London penetration depth estimated using
the following expression [63],

$\lambda_{\mathrm{L}}\simeq$105\text{\,}\mathrm{nm}$\sqrt{\frac{\rho_{\mathrm{4K}}($\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$)}{T_{c}($\text{\,}\mathrm{K}$)}}.$
(2)

Figure 11: London penetration depth estimated for different NitrAl samples as
a function of (a) N2 flow values, and (b) room temperature resistivity $\rho$.

###  D Magnetic field measurements

####  D.1 Experimental setup

The samples are connected to the I-V measurement setup by wire bonding, in a
simple four probe arrangement. An ADWIN gold data acquisition system and
NanoQt interface was used to acquire amplified voltage data as well as apply
bias voltage through a high resistor to obtain a bias current through the
NitrAl thin-film samples. Such a simple configuration works as a constant
current source, since the NitrAl film resistances used in these measurements
at room temperature are very small ($<30\leavevmode\nobreak\ \Omega$) compared
to the high-resistance bias resistor ($R_{b}=10$ k$\Omega$). The maximum bias
current $I_{b}$ was limited to 1 mA by the maximum voltage $V_{b}$ range of
$\pm 10\leavevmode\nobreak\ V$ from the ADWIN output. A Femto commercial
voltage amplifier was used to with a gain 100/ 1000/ 10000 depending on the
magnitude of the small voltage at the sample.

In addition to the direct I-V measurements, dV/dI at each bias current were
measured by an integrated lock-in code in the ADWIN interfaced with NanoQt. A
small oscillating bias current modulation of amplitude
$10\text{\,}\mathrm{\SIUnitSymbolMicro A}$ and frequency 91 Hz was added about
each $I_{b}$ to measure sample resistance (dV/dI) in the lock-in
configuration. The in-phase and 90 degree phase shifted components of this
oscillating signal were used as a reference to measure the in-phase and out of
phase components of the response voltage, which were filtered by a fourth
order filter with time constant set to 50 ms (about 5 oscillations at signal
frequency). The squared sum of these in-phase and out of phase components
provides $R^{2}$. The bias current $I_{b}$ was varied very slowly such that
during a lock-in data acquisition, the $I_{b}$ change can be ignored. The
dV/dI obtained from lock-in measurement and from the slope of direct I-V data
were compared to find that both methods provide similar results. However, the
lock-in measurements show less noise, hence the resistance measured only in
lock-in configuration are presented.

####  D.2 IV-characteristics under magnetic field

As a complement to the data shown in Sec. IV of the main text, we show here
raw data of the IV-characteristics. Note that the maximum current of the
source employed is 1 mA.

Figure 12: VI-characteristics as a function of the temperature and the
magnetic field in the $x$-direction, $B_{\mathrm{x}}$ (in-plane with the chip)
for different NitrAl samples produced in the second run of fabrication.

###  E Optical properties

We can also connect the resistivity at room temperature with the surface
appearance of the samples (Fig. 13). A qualitative difference in the surface
color of the films is observed between conductive and insulating samples.
These differences might indicate a change in the optical properties and
optical responses of the films. In particular, for conducting samples the
surface is shiny and silver-looking, close to the one of aluminum, while
samples above $13.3\%$ appear to be silver-green. A quantitative analysis of
the optical response of the thin films at room temperature will be included in
a future work.

Figure 13: NitrAl films produced with different $\ch{N2}/\ch{Ar}$ flows. A
clear difference in color between the insulating sample ($15\%$), of dark
green, and the conductive one ($11.67\%$), of dark gray, can be observed.


$X_{1}:(N_{1},L_{1})=X_{2}:(N_{1},L_{1})$ for
$(N_{1},L_{1})\in\mathcal{T}_{1}$. Let also $c_{2}$ a mapping on
$\mathcal{F}(\mathcal{G}_{2})$ so that $c_{2}(Y_{1})=c_{2}(Y_{2})$ if and only
if $Y_{1}:(N_{2},L_{2})=Y_{2}:(N_{2},L_{2})$ for
$(N_{2},L_{2})\in\mathcal{T}_{2}$. If by an application of Lemma $8$ we get a
$T\in\mathcal{S}_{\infty}(U)$ such that the first alternative holds, then we
have that $\mathcal{T}_{1}\upharpoonright T=\mathcal{T}_{2}\upharpoonright T$.

###### Proof.

The proof is by induction on the rank $\alpha$ of the uniform families
$\mathcal{F}(\mathcal{G}_{1})$ and $\mathcal{F}(\mathcal{G}_{2})$, which is
identical by Lemma $11$. If $\alpha\in\omega$, then by the discussion before
Lemma $6$ we have that for any $(N_{0},L_{0}),(N_{1},L_{1})\in\mathcal{T}_{1}$
and any two members of their strong subtree envelopes
$X_{0}\in\mathcal{C}^{T}_{(N_{0},L_{0})}$ and
$X_{1}\in\mathcal{C}^{T}_{(N_{1},L_{1})}$ one has:
$\iota_{b^{\alpha},X_{0}}\circ\iota^{-1}_{b^{\alpha},X_{1}}(N_{1})=(N_{0})$
and $|L_{0}|=|L_{1}|$. Similarly for $\mathcal{T}_{2}$. Suppose that
$\mathcal{T}_{1}\upharpoonright T\neq\mathcal{T}_{2}\upharpoonright T$. Let
$(N_{1},L_{1})\in\mathcal{T}_{1}\upharpoonright T$, so that if $|N_{1}|>1$
then for every $t,t^{\prime}\in N_{1}$, the absolute value of the difference
$|t|-|t^{\prime}|$ is greater than $1$. For any
$X\in\mathcal{C}^{T}_{(N_{1},L_{1})}$ consider $c_{1}(X)$. Since we have the
first alternative of Lemma $8$ on hold, we must have that $c_{2}(X)=c_{1}(X)$
for $X\in\mathcal{C}^{T}_{(N_{2},L_{2})}$,
$(N_{2},L_{2})\in\mathcal{T}_{2}\upharpoonright T$ as well. That must be true
for all the members of $\mathcal{C}^{T}_{(N_{1},L_{1})}$, which implies that
$\mathcal{C}^{T}_{(N_{1},L_{1})}=\mathcal{C}^{T}_{(N_{2},L_{2})}$. As a result
for $X\in\mathcal{C}^{T}_{(N_{1},L_{1})}$ and
$Y\in\mathcal{C}^{T}_{(N_{2},L_{2})}$ we have that $L_{X}=L_{Y}$. If
$N_{1}=\\{t\\}$, then $N_{2}=\\{t\\}$ as well, otherwise if $N_{2}=\\{s\\}$,
then $X(0)=t\neq s=X(0)$, a contradiction. Suppose that $|N_{1}|>1$ and let
$t\in\\{(N_{1}\setminus N_{2})\cup(N_{2}\setminus N_{1})\\}$ is of minimal
height. Suppose that $t\in N_{1}$. Since $X\in\mathcal{C}^{T}_{(N_{1},L_{1})}$
there exists $n\in|X|$ such that $t\in X(n)$. Choose a
$Y\in\mathcal{C}^{T}_{(N_{2},L_{2})}$ such that $t\notin Y(n)$. Notice that
$Y\notin\mathcal{C}^{T}_{(N_{1},L_{1})}$, a contradiction. If now
$N_{1}=N_{2}=\emptyset$ then we must have $L_{1}=L_{2}$, other wise for every
$X^{\prime}\in\mathcal{C}^{T}_{(N_{1},L_{1})}$ and
$Y^{\prime}\in\mathcal{C}^{T}_{(N_{2},L_{2})}$ we would have that
$L_{X}^{\prime}\neq L_{Y}^{\prime}$ contradicting that
$\mathcal{C}^{T}_{(N_{1},L_{1})}=\mathcal{C}^{T}_{(N_{2},L_{2})}$. Finally if
$N_{2}=\emptyset$ and $N_{1}\neq\emptyset$ pick $t\in N_{1}$ so that for any
other $t^{\prime}\in N_{1}$, we have that $l=|t|\geq|t^{\prime}|$. Pick a
$Y\in\mathcal{C}^{T}_{L_{2}}$ so that $t\notin Y$. This is always possible
since our node-level set $(N_{2},L_{2})$ is only a level set. Then
$Y\notin\mathcal{C}^{T}_{(N_{1},L_{1})}$, a contradiction of
$\mathcal{C}^{T}_{(N_{1},L_{1})}=\mathcal{C}^{T}_{(N_{2},L_{2})}$. As a
consequence $\mathcal{C}^{T}_{(N_{1},L_{1})}=\mathcal{C}^{T}_{(N_{2},L_{2})}$
implies that for any $X^{\prime}\in\mathcal{C}^{T}_{(N_{1},L_{1})}$,
$Y^{\prime}\in\mathcal{C}^{T}_{(N_{2},L_{2})}$ both finite strong subtrees of
height $\alpha<\omega$, we have that
$\iota_{b^{\alpha},X^{\prime}}\circ\iota^{-1}_{b^{\alpha},Y^{\prime}}(N_{2})=(N_{1})$
and $|L_{1}|=|L_{2}|$. Therefore $\mathcal{T}_{1}\upharpoonright
T=\mathcal{T}_{2}\upharpoonright T$.

Assume that the assertion of our proposition holds for $\beta<\alpha$ uniform
families and consider the case of $\alpha\geq\omega$ uniform families
$\mathcal{F}(\mathcal{G}_{1})$ and $\mathcal{F}(\mathcal{G}_{2})$. For any
node $t\in T$, $\mathcal{F}(\mathcal{G}_{1})(t)\upharpoonright T$ and
$\mathcal{F}(\mathcal{G}_{2})(t)\upharpoonright T$ are both uniform families
of rank less than $\alpha$. The inductive hypothesis applies to give us
$\mathcal{T}^{t}_{1}\upharpoonright T=\mathcal{T}^{t}_{2}\upharpoonright T$.
That being true for every $t\in T$ implies that
$\mathcal{T}_{1}\upharpoonright T=\mathcal{T}_{2}\upharpoonright T$. ∎

Now the family of node-level sets $\mathcal{T}$ that will satisfy the
conditions of Definition $16$ is defined as follows: For a node $s_{0}\in T$
if there exists a node $s_{1}\in T$ so that the first alternative of the above
statement holds, then $\mathcal{T}^{s_{0}}\subset\mathcal{T}$. If for all
$s_{1}\in T$ we have the second alternative holding then
$s_{0}\cup\mathcal{T}^{s_{0}}:=\\{(s_{0}\cup
N,L):(N,L)\in\mathcal{T}_{s_{0}}\\}\subset\mathcal{T}$. Similarly for $\phi$
i.e. if $\mathcal{T}^{s_{0}}\subset\mathcal{T}$ then
$\phi\upharpoonright\mathcal{T}^{s_{0}}=\phi_{s_{0}}$. If now
$s_{0}\cup\mathcal{T}^{s_{0}}\subset\mathcal{T}$, then
$\phi\upharpoonright(s_{0}\cup\mathcal{T}^{s_{0}})=\phi_{s_{0}}\upharpoonright\mathcal{T}^{s_{0}}$.

This completes the inductive step and the proof of Theorem $7$.

We give a proof now of our second remark.

######  Proposition 3.

In the contact of Definition $16$, by taking the union of all the strong
subtree envelopes of all node-level sets of the family $\mathcal{T}$ and by
passing to an infinite strong subtree if necessary, we obtain a uniform family
$\mathcal{F}(G)$ of rank less than or equal to the rank of $\mathcal{G}$.

###### Proof.

We give a proof by induction on the rank of $\mathcal{G}$. Suppose that the
rank of $\mathcal{G}$ is finite. As we have seen above from the discussion
before Lemma $6$, if $(N_{1},L_{1})$, $(N_{2},L_{2})\in\mathcal{T}$, then
$X_{1}\in\mathcal{C}^{U}_{(N_{1},L_{1})}$ and
$X_{2}\in\mathcal{C}^{U}_{(N_{2},L_{2})}$ are isomorphic and have height equal
to $n$. By taking the union of all the strong subtree envelopes of all members
of $\mathcal{T}$, we get a family $\mathcal{F}(\mathcal{G})$ of finite strong
subtrees of $U$ with height equal to $n$. By applying Corollary $1$ we get a
strong subtree $T$ of $U$ so that the second statement of this corollary
holds. To see that suppose we get $T\in\mathcal{S}_{\infty}(U)$ such that
$\mathcal{S}_{n}(T)\cap\mathcal{F}(\mathcal{G})=\emptyset$. But
$\mathcal{G}\upharpoonright T$ is also a uniform family and the mapping $c$
restricted on that family is canonical. Pick an
$X\in\mathcal{G}\upharpoonright T$ and consider $Y\in\mathcal{C}^{T}_{f(X)}$.
Note that $Y\in\mathcal{F}(\mathcal{G})\cap\mathcal{S}_{n}(T)$, a
contradiction. Notice that the elements of any strong subtree envelop have
height $n$. Therefore we get a uniform family $\mathcal{F}(G)$ of rank $n$. In
fact we get the unique uniform family of rank $n$ on $T$.

Assume now that the rank of $\mathcal{G}$ is $\omega$. By definition
$\mathcal{G}(t)$ is of rank $n$, for some $n\in\omega$. By above
$\mathcal{F}(\mathcal{G})(t)$ is of rank less than or equal to $n$. Consider
the coloring $c^{\prime}:\mathcal{S}_{1}(U)\to\omega$ defined by $c(t)=n$ if
and only if the rank of $\mathcal{F}(\mathcal{G})(t)$ is $n$. By Theorem $5$
we get a strong subtree $T$ of $U$ such that either the coloring is constant
and equal to $n_{0}\in\omega$, one-to-one, or is constant on each level, i.e.
$c(t)=c(s)$ if and only if $|t|=|s|$. In the first case the rank of
$\mathcal{G}_{0}$ is $n_{0}$. In the last two cases the rank of
$\mathcal{F}(G)$ is $\omega$.

Suppose now that the rank of $\mathcal{G}$ is $\alpha$, for $\alpha>\omega$
and for all $\beta<\alpha$ our proposition holds. By definition
$\mathcal{G}(t)$ is of rank $\beta<\alpha$, so the inductive hypothesis
applies and we proceed as in the above paragraph. ∎

Finally we show that our definition of a canonical coloring is the appropriate
one.

######  Proposition 4.

Let $c$ be a canonical coloring of a uniform family $\mathcal{G}$ on $U$ and
let $(\mathcal{T}_{0},f_{0})$, $(\mathcal{T}_{1},f_{1})$ be two pairs that
satisfy the conditions $1$ and $2$ of the Definition $16$. Then there exists
$T\in\mathcal{S}_{\infty}(U)$ so that:

$\mathcal{T}_{0}\upharpoonright T=\mathcal{T}_{1}\upharpoonright T$ and
$f_{0}=f_{1}$ on $\mathcal{G}\upharpoonright T$

###### Proof.

By definition $f_{i}:\mathcal{G}\to\mathcal{T}_{i}$ is such that
$c(X_{0})=c(X_{1})$ if and only if $f_{i}(X_{0})=f_{i}(X_{1})$, for $i\in 2$.
Let $\mathcal{G}_{0}$ be the uniform family resulting by taking the union of
all the strong subtree envelopes of the node-level sets in $\mathcal{T}_{0}$
and $\mathcal{G}_{1}$ the one resulting from $\mathcal{T}_{1}$. We remind the
reader here that both uniform families are assumed to be defined on $U$
instead of one of its infinite strong subtrees. By an application of Lemma $8$
on $(\mathcal{G}_{0},c_{0})$ and $(\mathcal{G}_{1},c_{1})$, we get
$T\in\mathcal{S}_{\infty}(U)$ such that the first statement of the lemma
holds. We also notice that both ranks of $\mathcal{G}_{0}$ and
$\mathcal{G}_{1}$ must be equal by Lemma $11$. By Proposition $2$ we have that
$\mathcal{T}_{0}\upharpoonright T=\mathcal{T}_{1}\upharpoonright T$.

We claim that $\mathcal{T}_{0}\upharpoonright T=\mathcal{T}_{1}\upharpoonright
T$, implies that $f_{0}$ agree with $f_{1}$ on $\mathcal{G}\upharpoonright T$.
To see this suppose that for $X\in\mathcal{G}$ we have that
$f_{0}(X)=(N_{0},L_{0})\neq f_{1}(X)=(N_{1},L_{1})$. Let
$X_{0}\in\mathcal{C}^{T}_{(N_{0},L_{0})}$ and
$X_{1}\in\mathcal{C}^{T}_{(N_{1},L_{1})}$. Then $c_{0}(X_{0})\neq
c_{0}(X_{1})$ and $c_{1}(X_{0})\neq c_{1}(X_{1})$. But then $c(X)\neq c(X)$, a
contradiction.

∎

The inductive step of Theorem $8$ is identical with the inductive step of
Theorem $7$. Therefore we extended the result of Milliken completing the
research along the line of P.Erdös and R. Rado.

Next we mention a possible application of our canonical result. Suppose that
$\mathcal{U}$ and $\mathcal{V}$ are ultrafilters on index-sets $X$ and $Y,$
respectively. Let $\mathcal{V}\leq_{RK}\mathcal{U}$ denote the fact that there
is a map $F:X\rightarrow Y$ such that $\mathcal{V}=\\{M\subseteq
Y:F^{-1}(M)\in\mathcal{U}\\}.$ Put $\mathcal{U}\equiv_{RK}\mathcal{V}$
whenever $\mathcal{V}\leq_{RK}\mathcal{U}$ and
$\mathcal{U}\leq_{RK}\mathcal{V}.$ This is equivalent to saying that there is
a bijection between a set in $\mathcal{U}$ and a set in $\mathcal{V}$ that
transfers one ultrafilter into the other. There is a coarser pre-ordering
between ultrafilters that is of a considerable recent interest. This is the
_Tukey ordering_ which says that $\mathcal{V}\leq_{T}\mathcal{U}$ if there is
a monotone map $F^{\prime}:\mathcal{U}\rightarrow\mathcal{V}$ whose range
generates $\mathcal{V},$ i.e., every element of $\mathcal{V}$ is refined by
$F^{\prime}(M)$ for $M\in\mathcal{U}.$ Recall that a (non-principal)
ultrafilter $\mathcal{U}$ on $\mathbb{N}$ is _selective_ if for every map
$f:\mathbb{N}\rightarrow\mathbb{N}$ there is $M\in\mathcal{U}$ such that the
restriction $f\upharpoonright M$ is either one-to-one or constant.

In [Ra-To], S. Todorcevic has used the Theorem $2$ to prove the following
result.

######  Theorem 9 ([Ra-To]).

Tukey predecessors of a selective ultrafilter on $\mathbb{N}$ are exactly its
countable transfinite Fubini powers modulo, of course, the Rudin-Keisler
equivalence.

In section $4$ we established that $(\mathcal{S}_{\infty}((U_{i})_{i\in
d}),\subseteq,r)$ forms a topological Ramsey space. It turns out that every
topological Ramsey space has the corresponding notion of a selective
ultrafilter (see [Mij]). Since we proved the analogue of the Pudlák-Rödl
result for the space of $\mathcal{S}_{\infty}(U)$ of strong subtrees, we
really believe that our Theorem $7$ can be used to characterize the Tukey
predecessors of ultrafilters on $\mathcal{S}_{1}(U)$ that are selective
relative to the space $\mathcal{S}_{\infty}(U)$.

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