Electronic Supplementary Material (ESI) for Nanoscale.
This journal is © The Royal Society of Chemistry 2022

SUPPLEMENTARY INFORMATION
S1- Preparation and characterization of superconducting tip.
STM tips made of superconducting materials are widely used to increase theenergy resolution in STS measurements beyond the thermal energy limit [25], [39]. Lead(Pb) and Niobium (Nb) are the most used materials, mainly for practical reasons as they
are BCS superconductors with high critical temperature. SC STM tips are normally
produced by either etching wire of the mentioned materials or by indenting W (or PtIr)
tips into samples of the superconducting material of choice [25,38,40-42].
The material employed in this article for the superconducting (SC) tips is Niobium,a conventional s-wave superconductor with a bulk transition temperature Tc of 9.3 K andcritical magnetic field of Bc=0.198 T. The STM tips were produced by electrochemical
etching commercial (99.9% pure) polycrystalline Nb wire of 0.25mm width [49]. A lengthof 2mm of the wire was submerged into a 22% HCl solution and then etched by applying
a 25Hz ac voltage of 16V between the wire and a counter electrode of graphite. Etchedtips were mounted into a tip holder and transferred to the preparation chamber where
they were sputtered for 5 hours at 3 keV in order to remove oxides and yield the widest
possible gap.
The resulting SC tips have an average gap of Δ=1.05 mV at 1.2 K, Fig. 1S(a),and are fully capable of attaining atomic resolution as shown in Fig. 1S(b) where boththe atomic periodicity and the standing waves of Cu(111) are resolved.

Figure S1. (a) Scanning tunnelling spectroscopy showing the Nb tip density of states at 1.2 K, ∆= 1.05 mV calculated by fitting the spectra. STS parameters: V=10 mV, I=2 nA, Vmod=100 𝜇V. (b)STM image of Cu(111) single crystal showing routinary atomic resolution and standing waves
produced by a nearby defect. Image parameters: V= 10 mV, I=3 nA, T=1.2 K. (c) Temperature-1-series: set of tunnelling spectra each recorded at a different temperature, the critical temperature
is close to 8.4 K. STS parameters: V=10 mV, I=2 nA, Vmod=100 𝜇V. (d) Magnetic field series:
tunnelling spectra of the Nb STM tip showing the evolution of the SC gap with increasing magnetic
field at T=1.2 K, critical magnetic field close to 1.2 T. STS parameters: V=10 mV, I=2 nA, Vmod=
100 𝜇V.

S2- Reversible functionalization
As already mentioned in the main text, the molecule can be released from thetip via voltage pulses above a certain threshold on molecule free areas. If the voltage
pulse applied ∆𝑉 ≥ +2 V the functionalization of the tip may be reversed, this is depictedin Fig. S2(a) and (b). Here starting with a functionalized tip, blue spectrum, a voltage
pulse of +2 V is applied over a given gr/Ir area of the sample (where the bias voltage isapplied to the sample). The superconducting gap of the Nb tip is recovered, redspectrum, and some clusters that resemble the shape of a TCNQ are scanned over at
the top right part of the STM image in Fig. 2S(b).

Figure S2. (a) STM topography image of a clean gr/Ir(111) area, where the spectrum of the TCNQ
functionalized tip (blue) has been performed. Defect highlighted with green circle for reference.
(b) STM topography image of the same clean gr/Ir(111) area after a +2 V pulse has been
performed at the center of the image. The pulse is performed against the sample. STS performed
after the voltage pulse shows that the pristine superconducting gap of the Nb tip is recovered. At
the top of the STM image a TCNQ molecule maybe distinguished. Pulse parameters: V=15 mV,
I=0.5 nA, V=+2 V and t=100 ms. Image parameters: V=1 V, I=0.08 nA, 20x20 nm. STSparameters: V=10 mV, I=0.5 nA, Vmod=150 µV.

-2-S3- Control experiment with un-functionalized Nb tip
This experiment is analogous to the ones presented in Fig. 2 and Fig. 3 of themain text, it is carried out over a gr/Ir area that is completely molecule free and the only
difference is that in this case the tip is not functionalized with a TCNQ molecule. The
experiment serves as a control experiment to confirm that if no molecule is present nosub-gap states will emerge no matter how close the tip is brought towards the sample,
that in this case is 800 pm. As expected, no sub-gap state emerges throughout the whole
length of the experiment, also no contact is stabilized between tip and sample in thesense that the current is always below the quantum of conductance G0.
b)Norm. dI/dV [Arb.Unts]

a)

0−200

1.00.50−0.5−10010Sample Voltage [mV]−400−600−800−10−50510Sample Voltage [mV]c)
G 0 15mV

Norm. dI/dV [Arb.Unts]

10 −6Tunneling Current [A]

1.5 ΔZ=+100 pm

10 −810 −1010−1210−14−800

Norm. dI/dV [Arb.Unts]

Relative Z offset [pm]200−600−400−2000200

Relative Z offset [pm ]2 ΔZ=-600 pm

10−10010Sample Voltage [mV]
3ΔZ=-800 pm

210−10010Sample Voltage [mV]Figure S3. (a) Relative tip-sample distance dependence of STS spectra. The green-yellow colour
scale represents the normalized dI/dV intensity as a function of bias voltage (horizontal axis) andrelative tip-sample distances (vertical axis) of a non-functionalized Nb tip. The SC gap maybe
observed throughout the whole experiment. Stabilization parameters throughout all the process:
V=15 mV, I=0.5 nA, Vmod=100 V. (b) Three selected dI/dV spectra extracted from the stackedplot, each representative of the different regions in (a). (c) I vs Z curve acquired with the same-3-

SC tip and covering the same 1100 pm Z offset range shown in (a). The conductance quantum,G0, is marked with a green dashed line. Stabilization parameters: V=15 mV, I=0.5 nA.

S4- Kondo fits from magnetic field series
Figure S4 shows the spectra of two Kondo resonances that correspond to thetwo functionalized tips depicted in Fig. 2 of the main manuscript when a magnetic fieldof 1.5 T is applied to quench the SC state of the Nb tip. Both spectra are fitted with theFano function [35]:𝜌[𝐸] =(1+𝐸0 ‒ 𝐸)
( )

𝑞+2

Γ

𝐸0 ‒ 𝐸

2Γ

Where q is the so-called asymmetry factor and Γ the intrinsic half width at half
maximum (HWHM). The thermal broadening and the broadening produced by the lockin modulation are accounted for by doing the convolution of the Fano function with thefermi function and with a semi-sphere [60]. Finally, to account for the broadening
produced by the external magnetic field, the expected Zeeman splitting is subtracted
from the value of HWHM. In the case of a magnetic impurity of spin ½ the Zeeman
splitting is given by [54].
2∆ = 2(𝑔𝜇𝐵𝐵𝑆𝑒𝑓𝑓)
Where g is the Landé factor, 𝜇𝐵 is the Bohr magneton in electronvolts, B themagnetic field and 𝑆𝑒𝑓𝑓 the effective spin of the magnetic impurity. This yields a value of
0.14 meV in our case.

-4-Figure S4. (a) Same tunnelling spectrum shown in Fig. 4(a) corresponding to a magnetic impurity
in the strong coupling regime, the spectrum is fitted with a Fano function (red). Stabilization
parameters: V=15 mV, I=0.5 nA, Vmod=150 V and T=1.2 K. (b) STS showing the Kondo
resonance corresponding to the functionalized tip in the quantum phase (QP) transition shown in
Fig. 4(b), the spectrum is also fitted with a Fano function shown in red. Stabilization parameters:
V=15 mV, I=0.5 nA, Vmod=150 V and T=1.2 K

S5- Temperature evolution of Kondo resonances.
The temperature dependence of the intrinsic half width half maximum (HWHM) of
two Kondo resonances are studied in the range from 1.2 K to ~17 K. For the lowest
temperatures, a 1.5 T magnetic field was applied in order to quench the SC state of thetip, and the corresponding Zeeman splitting was accounted for, as explained in theprevious section S4. However, note that the splitting always falls within the given error
bar. The temperature evolution represented in green corresponds to the Kondo
resonance of a TCNQ molecule in the strong coupling regime, in this case the data
comes from a different functionalized tip that the one shown in the Fig.2 of the main text.
The temperature dependence of the HWHM for the Kondo resonance represented inblue corresponds to a TCNQ molecule in the quantum phase (QP) transition, in this casethis is the same functionalized tip as the one shown in the magnetic field series of Fig. 2of the main text.
The intrinsic HWHM Γ is expected to be temperature dependent and broaden
progressively with temperature, this is a signature of a Kondo system. In the case offinite temperatures its behaviour may be described by [14,16-18]:Γ=1(𝛼𝑘𝐵𝑇)2 + (2𝑘𝐵𝑇𝐾)22-5-Looking at this expression a linear growth is expected when T ≫ TK with a slope
given by 𝛼, a constant that depends on the system studied, for T ≪ TK the width isexpected to saturate (Γ = 𝑘𝐵𝑇𝐾). Then the data presented in the graph below may befitted with the above expression to calculate the Kondo temperature (TK) for the magnetic
impurities in the two coupling regimes. The calculated Kondo temperature for themagnetic impurity in the strong coupling regime is TK=23.8 K (𝛼=2.96), in the case of themagnetic impurity in the QP transition TK=10.8 K (𝛼=1.65).

Figure S5. Intrinsic HWHM of the kondo resonances plotted against temperature. There are two
sets of data for molecules in different coupling regimes, one in the strong coupling regime (green)
and the other in the quantum phase transition (blue). Both sets of data show the expected
evolution for the HWHM of a Kondo resonance with temperature, yielding a Kondo temperature
TK of 23.8 K and 10.8 K respectively.

S6- Relationship between the YSR and Kondo screening energy scale
Matsuura’s universal relation describes the competition between the Kondo
screening and the induced Yu-Shiba-Rushinov bound states. When a magnetic impurity
is placed on a superconductor the spin of the magnetic impurity becomes underscreened as there is a decrease in the number of itinerant electrons that can participate
in the Kondo cloud. Matsuura predicted that the weaker the Kondo screening the morestrongly bound the sub-gap states would be. Therefore, if the energy value of the subgap states (EBS) is plotted against the energy of the Kondo screening channel (kBTK) thedata should follow the relation predicted by Matsuura (green dashed line in Fig.S6)[30,31].
In order to complete the Matsuura plot, we need to know for the same functionalized
tip: EBS and kBTK. To calculate EBS the spectra at B=0 T are fitted with a Cauchy function.
The energy associated to the screening channel, kBTK, has been calculated by fitting theHWHM of the corresponding Kondo resonances (B=1.5 T) and then using the equation
shown in section S5 when 𝑇→0 𝐾 [14,62]:-6-Γ𝑇 = 0 =1
4 𝑘 𝐵𝑇𝐾 𝑘 𝑇
2= 𝐵 𝐾In the case of the system studied in this work the QP transition occurs for avalue of 0.99, that could seem to contradict the expected value of 0.3 reported ontheoretical works [30,31] [62-65]. However, this deviation originates from the alternative
definitions of TK used in experimental and theoretical works, as pointed out in[25,29,61,62]. The most common expression for TK used in theoretical works is [62-65]:𝜋𝜀𝑥(𝜀𝑥 + 𝑈)

𝑇𝐾 = 0.29 𝑈Γ 𝑒2Γ𝑈Where in this case TK depends not only the hybridization value, but also on theenergy position of the sub-gap states 𝜀𝑥 and on the Coulomb energy U. It is argued inrefs [62] and [66] that the values of TK calculated from this equation and the one used inthis work may vary by a factor close to 4. Taking this into account, our result for the QP
transition in Fig. S3 would be translated to a 0.25 ratio, which is in better agreement withthe theoretical prediction of 0.3 corresponding to the alternative definition of TK.
We propose to use both the external magnetic field and the procedure presented
in this manuscript to modify the coupling regime of the magnetic impurity in combination
as a new way to study Matsuura’s universal relation. In principle the whole spectrum of
the interaction strength of a magnetic molecule or adatom could be accessed bymanipulating it onto a SC tip apex and performing spectroscopy experiments whilechanging the tip-sample distance (∆Z). Then, an experiment could be envisioned like theones presented in Fig. 2 and Fig. 3 of the main text but with the magnetic field turned on,in this way the width of the Kondo resonances (to calculate kBTK) that corresponds to theposition all the EBS for the different couplings of the magnetic impurity could bemeasured. Thus, covering the whole green curve presented in the figure below.
1.0

0.99

0.8
0.6
0.4

EBS/Δ

0.2
WEAK COUPLING

0STRONG COUPLING

−0.2
−0.4
−0.6
−0.8
−1.0

0.1

1

10kBTK/ΔFigure S6. Relation between energy scales of the screening channel and magnetic induced
bound states energy obtained at 1.2 K. EBS is the position of the larger bound state, EBS < 0 for amolecule in the strong coupling regime. The energy scale of the Kondo channel, kBTK, is-7-calculated by quenching the SC state of the tip with the external magnetic field and fitting thecorresponding Kondo resonances with the Fano function.

S7- Reversibility of Z offset experiments
All the Z offset experiments presented in the manuscript are completely reversible.
We ensured this by repeating each experiment back and forth several times, both theI(Z) curves and the STS spectra. A selection of data demonstrating this is presented infigure S7 below.

-8-Figure S7. Reversibility in Z-offset experiments. Set of 9 consecutive I(Z) curves corresponding
to the experiments presented in figure 3 (panel A) and figure 4 (panel C) of the main text. Panels
B and D show three consecutive STS spectra alternating between the two Z-offset values
highlighted in panels A and C, respectively.

-9-