# How cold is the junction of a millikelvin scanning tunnelling microscope?

Taner Esat  Peter Grünberg Institut (PGI-3), Forschungszentrum Jülich, 52425
Jülich, Germany  Jülich Aachen Research Alliance (JARA), Fundamentals of
Future Information Technology, 52425 Jülich, Germany  Xiaosheng Yang  Peter
Grünberg Institut (PGI-3), Forschungszentrum Jülich, 52425 Jülich, Germany
Jülich Aachen Research Alliance (JARA), Fundamentals of Future Information
Technology, 52425 Jülich, Germany  Farhad Mustafayev  Peter Grünberg Institut
(PGI-3), Forschungszentrum Jülich, 52425 Jülich, Germany  Jülich Aachen
Research Alliance (JARA), Fundamentals of Future Information Technology, 52425
Jülich, Germany  Helmut Soltner  Zentralinstitut für Engineering, Elektronik
und Analytik (ZEA-1), Forschungszentrum Jülich, 52425 Jülich, Germany  F.
Stefan Tautz  Peter Grünberg Institut (PGI-3), Forschungszentrum Jülich, 52425
Jülich, Germany  Jülich Aachen Research Alliance (JARA), Fundamentals of
Future Information Technology, 52425 Jülich, Germany  Experimentalphysik IV A,
RWTH Aachen University, 52074 Aachen, Germany  Ruslan Temirov  r.temirov@fz-
juelich.de Peter Grünberg Institut (PGI-3), Forschungszentrum Jülich, 52425
Jülich, Germany  University of Cologne, Faculty of Mathematics and Natural
Sciences, Institute of Physics II, 50937 Cologne, Germany

###### Abstract

We employ a scanning tunnelling microscope (STM) cooled to millikelvin
temperatures by an adiabatic demagnetization refrigerator (ADR) to perform
scanning tunnelling spectroscopy (STS) on an atomically clean surface of
Al(100) in a superconducting state using normal-metal and superconducting STM
tips. Varying the ADR temperatures between 30 mK and 1.2 K, we show that the
temperature of the STM junction $T$ is decoupled from the temperature of the
surrounding environment $T_{\mathrm{env}}$. Simulating the STS data with the
$P(E)$ theory, we determine that $T_{\mathrm{env}}\approx 1.5$ K, while the
fitting of the superconducting gap spectrum yields the lowest $T=77$ mK.

scanning tunnelling microscopy, scanning tunnelling spectroscopy, low
temperatures, superconductivity, Josephson tunnelling

##  I Introduction

The quantum effect of electron tunnelling lies in the core of scanning
tunnelling microscopy (STM). STM junctions operated in a highly controlled
manner under ultra-high vacuum (UHV) and low-temperature conditions enable
high-resolution imaging, and precise manipulation of surface nanostructures
Wagner and Temirov (2015). However, further explorations of quantum-coherent
phenomena in such nanostructures Chen, Bae, and Heinrich (2022) may demand
even better control over the STM junction and its environment because of the
utmost sensitivity of quantum tunnelling to the microscopic properties of the
tunnelling junction and its environment. Since the temperature is one of the
most critical environmental parameters, STM setups operating at ultra-low
temperatures, i.e. well below 1 K, are being developed Song  _et al._ (2010a);
Misra  _et al._ (2013); Assig  _et al._ (2013); Roychowdhury  _et al._ (2014);
Machida, Kohsaka, and Hanaguri (2018); Balashov, Meyer, and Wulfhekel (2018);
von Allwörden  _et al._ (2018); Wong  _et al._ (2020); Schwenk  _et al._
(2020); Esat  _et al._ (2021); Fernández-Lomana  _et al._ (2021).

The motivation for conquering the millikelvin (mK) temperature range with STM
is two-fold: First, the ultra-low temperatures stabilize emergent quantum
ground states with exceptional properties Machida  _et al._ (2019); Nuckolls
_et al._ (2020); Kamber  _et al._ (2020). Second, upon cooling the junction,
the width of the Fermi-Dirac distribution in its electrodes shrinks,
increasing the energy resolution of scanning tunnelling spectroscopy (STS)
and, in that way, also providing better access to low-energy excitations with
longer lifetimes Song  _et al._ (2010b); Yazdani, da Silva Neto, and Aynajian
(2016); Feldman  _et al._ (2017).

The problem of determining and later controlling the STM junction temperature
is, however, anything but trivial. An obvious reason is that a direct
temperature measurement typically performed with macroscopic sensors is
technically impossible in such a microscopic junction. Another complication is
that the junction permanently exchanges energy with its environment. These
energy exchange processes affect the tunnelling rates appreciably Grabert and
Devoret (1992), necessitating the characterization of the environmental
temperature, too.

One indirect way of deducing the junction temperature is to perform an STS
experiment, i.e., to measure the dependence of the tunnelling current $I$ on
the applied bias voltage $V$. Most generally,

$I=e[\Gamma^{\ +}(V)-\Gamma^{\ -}(V)],$ (1)

where $e$ is the elementary charge and $\Gamma^{\ \pm}$ is the tunnelling rate
in the direction along (against) the applied bias voltage. For an STM junction
interacting with its electromagnetic environment Grabert and Devoret (1992),
$\Gamma^{\ +}$ can be calculated from

$\begin{split}\Gamma^{\
+}(V)=\frac{4\pi}{\hbar}\iint_{-\infty}^{\infty}dEdE^{\prime}n_{\mathrm{T}}(E)n_{\mathrm{S}}(E^{\prime}+eV)\times\\\
f_{\mathrm{T}}(E)[1-f_{\mathrm{S}}(E^{\prime}+eV)]|M(E,E^{\prime}+eV)|^{2}P(E-E^{\prime}),\end{split}$
(2)

where $n_{\mathrm{T,S}}$ are the densities of states (DOS) and
$f_{\mathrm{T,S}}(E)=1/[1+\mathrm{exp}(E/k_{\mathrm{B}}T_{\mathrm{T,S}})]$ are
the Fermi-Dirac distribution functions of the tip (T) and sample (S)
electrodes. $k_{\mathrm{B}}$ is the Boltzmann constant and $M$ is Bardeen’s
matrix element accounting for the overlap between the single-electron
wavefunctions in the tip and the sample Bardeen (1961); Voigtländer (2015);
Gottlieb and Wesoloski (2006). Apart from the last factor under the integral,
Eq. 2 is identical to the classical result by Bardeen for the tunnelling
current in a junction isolated from the environment and connected to an ideal
voltage source Bardeen (1961); Grabert and Devoret (1992).

The environment enters Eq. 2 in the form of the so-called $P(E)$ function,
which describes the probability for the tunnelling electron to exchange the
energy $E$ with the environment Grabert and Devoret (1992). $P(E-E^{\prime})$
substitutes the delta function $\delta(E-E^{\prime})$ in Bardeen’s original
expression for the non-interacting junction, thus accounting for the fact that
also inelastic tunnelling processes occur. Note that it is the thermal
dependence of $f_{\mathrm{T,S}}(E)$ that allows extracting the temperature of
the tip $T_{\mathrm{T}}$ and the sample $T_{\mathrm{S}}$ from the STS data if
$n_{\mathrm{T,S}}(E)$ and $P(E)$ are known.

The problem with $n_{\mathrm{T,S}}(E)$ is that it generally results from a
complicated and often irregular atomic structure of the tip or sample,
respectively. For evaluating $T_{\mathrm{T,S}}$, it is therefore desirable to
use materials with a well-defined DOS, on which atomic defects have little
influence. It turns out that the best class of materials that satisfy this
condition are the Bardeen-Cooper-Schrieffer (BCS) superconductors with their
DOS given by the well-known expression Tinkham (1996)

$n^{\mathrm{BCS}}(E)=n_{0}\mathrm{Re}\bigg{[}\frac{1}{\sqrt{E^{2}-\Delta^{2}}}\bigg{]},$
(3)

in which $n_{0}$ defines the DOS in the normal state and $\Delta$ is the width
of the superconducting gap. $n^{\mathrm{BCS}}$ features two spectroscopic
singularities – the so-called quasiparticle peaks – situated symmetrically
around $E_{\mathrm{F}}$ at the lowest quasiparticle excitation energy
$\Delta$. Since according to Eq. 3 the quasiparticle peaks are
spectroscopically very sharp, the effect of their temperature-dependent
broadening can help the accurate evaluation of $T_{\mathrm{T,S}}$ in a mK STM
junction.

In a tunnelling junction comprising a normal metal tip and a superconducting
surface, when the temperature is substantially lower than the critical
temperature of the superconducting transition $T_{\mathrm{c}}$, Eq. 2 for
$\Gamma^{\ +}(V)$ simplifies to

$\begin{split}\Gamma^{\
+}(V)=\frac{1}{e^{2}R}\iint_{-\infty}^{\infty}dEdE^{\prime}n_{\mathrm{S}}^{\mathrm{BCS}}(E^{\prime}+eV)\times\\\
f_{\mathrm{T}}(E)P(E-E^{\prime}),\end{split}$ (4)

losing its dependence on $f_{\mathrm{S}}(E)$ and hence $T_{\mathrm{S}}$. Also
note that under the assumption $n_{\mathrm{T}}(E)=const$, and
$M(E,E^{\prime})=const$, in the relevant range of energies, they factor out
and together with $n_{0}$ are absorbed into $R$, the high-bias, or normal-
state resistance of the junction. According to Eq. 4, tunnelling between a
normal-metal tip and a superconducting surface yields $T_{\mathrm{T}}$ if the
$P(E)$ function is known. Conversely, tunnelling between a superconducting tip
and a normal-metal surface yields $T_{\mathrm{S}}$. Interestingly, in a
junction with both electrodes being superconducting the tunnelling current,
i.e, the Josephson current of Cooper pairs, is completely independent of
$T_{\mathrm{T,S}}$ if $T_{\mathrm{T,S}}\ll T_{\mathrm{c}}$, and is given by

$I(V)=\frac{\pi eE_{\mathrm{J}}}{\hbar}\big{[}P(2eV)-P(-2eV)\big{]},$ (5)

which thus provides the most direct way of determining the $P(E)$ function
experimentally Grabert and Devoret (1992). In Eq. 5 $E_{\mathrm{J}}=\hbar
I_{\mathrm{c}}/2e$ is the Josephson energy and $I_{\mathrm{c}}$ the critical
Josephson current.

The first attempt at determining $P(E)$ of an STM junction is due to Ast and
coworkers Jäck  _et al._ (2015); Ast  _et al._ (2016). Applying the $P(E)$
theory as sketched in the Methods section, they successfully simulated the
experimental STS data collected with a dilution refrigerator operated mK STM
Assig  _et al._ (2013) on various tunnelling junctions Jäck  _et al._ (2015);
Ast  _et al._ (2016); Jäck  _et al._ (2016, 2017); Senkpiel  _et al._ (2020a,
b, 2022). They also demonstrated that the $P(E)$ function causes broadening of
the STS data and thus imposes a fundamental limitation on the resolution of
STS experiments. Furthermore they proposed that the effective capacitance of
the STM junction $C$ is an important factor determining the degree of that
broadening Ast  _et al._ (2016).

Up to now, the $P(E)$ theory has been applied to STM, under the assumption
that the junction is well thermalized with its environment, or in other words,
the temperature of the junction is equal to that of its environment:
$T_{\mathrm{T,S}}=T_{\mathrm{env}}$. It is, however, possible to envision a
situation where this condition does not hold Martinis and Nahum (1993);
Siewert, Nazarov, and Falci (1997).

Here we demonstrate that the junction of our mK STM is much colder than the
surrounding environment $T_{\mathrm{T}}\ll T_{\mathrm{env}}$ by performing
variable temperature STS of the Al(100) surface in its superconducting state,
using both normal-metal and superconducting STM tips. First, we measured the
temperature-dependent Josephson conductance and recovered the $P(E)$ function
of the environment together with its temperature $T_{\mathrm{env}}$ from these
data. Then we employed the obtained $P(E)$ function and $T_{\mathrm{env}}$ for
evaluating the temperature $T_{\mathrm{T}}$ of the STM tip, which we consider
to be an upper estimate of the junction temperature. Specifically, we obtained
$T_{\mathrm{T}}$ by fitting the STS of the superconducting gap measured with a
normal metal tip. In doing so, we additionally expose a potential problem of
the commonly used fitting approach and suggest how to avoid it. Finally, we
argue that the comparatively warm environment of our mK STM junction results
in photon-assisted tunnelling that is responsible for the in-gap conductance
seen in the STS of the superconducting gap.

Figure 1: The PtIr tip used in the experiments. (a) Model of the tip based on
the SEM images (b-c). (b-c) SEM images of the tip. (d) Simulated surface
charge density in the junction biased by the voltage of 1 V. (inset)
Distribution of the potential in the junction. The dashed white line
represents the outline of the tip. (e) Dependence of the junction capacitance
$C$ on the length $L$ of the tip wire. (inset) Model of the STM head used for
simulating $C$. Figure 2: Josephson conductance spectra measured with a
superconducting tip (cf. text) on a superconducting Al(100) surface. Higher
temperature data are shifted downwards for clarity. The dotted lines are eye
guides exhibiting the temperature dependence of the Josephson conductance
peak.

##  II Results and discussion

###  II.1 Capacitance of the tip-sample junction

Considering the broadening effect of the tip-surface junction capacitance $C$
on STS Ast  _et al._ (2016), we attempted to increase $C$ by using blunter
PtIr tips. The model of the tip geometry exhibited in Fig. 1a was extracted
from the scanning electron microscopy images shown in Fig. 1b-c: We
approximated the tip apex shape by a sphere 90 $\mu$m in radius and crowned by
a cone with 120∘ opening angle.

Placing the model tip over a flat metal surface, we simulated the charge
distribution in the tip-surface junction with a commercial software employing
the boundary element method (BEM) BEM . Fig. 1d shows the simulated surface
charge resulting from a potential difference of 1 V applied to the surface
while the grounded tip is located at a distance of 1 nm. Although the induced
charge density drops quickly with the distance from the tip apex, a more
careful analysis shows that the density of charge that accumulates on the
shaft of the tip, i.e., far from the apex, is not negligible.

Because the energy resolution of a mK STM may be as high as 10 $\mu$eV which
corresponds to $\approx 2.5$ GHz Schwenk  _et al._ (2020), the dimensions
contributing to the junction capacitance may reach the scale of several cm.
We, therefore, considered the complete STM head Esat  _et al._ (2021) for
evaluating $C$. A simplified model that captures the essential features of our
STM head (Fig. 1e) pictures the junction capacitor as a cylindrical metallic
cavity with a radius of $7\ \mathrm{mm}$ and a height of 7 mm inside which the
tip, i.e., the second electrode, is located. The bottom of the cavity
coincides with the surface plane of the sample; hence, the distance from the
tip to the cavity bottom is 1 nm. Systematically changing the length of the
tip wire $L$ between 2 and 6 mm, we obtained the plot shown in Fig. 1e, which
demonstrates that $C$ depends on $L$ and hence the correct evaluation of $C$
needs to account for such additional elements as the tip holder and possibly
also the tunnelling current wire connected to the tip. Instead of performing
more elaborate simulations, we pick $C=100$ fF as a ballpark value for our
further analysis.

###  II.2 Temperature of the environment, $T_{\mathrm{env}}$

According to $P(E)$ theory, the environment of the STM junction is essentially
the source of the fluctuations of the junction’s phase $\tilde{\phi}$ (see Eq.
9 in the Methods section), which couple to the charge degree of freedom $Q$ of
the junction and thus affect the tunnelling process. The use of the
fluctuation-dissipation theorem enables one to represent the fluctuating
environment by a dissipative impedance $R_{\mathrm{env}}$ connected in series
to the tunnelling junction with resistance $R$ and capacitance $C$ as shown in
the lumped element model Grabert and Devoret (1992) in Fig. 3. Because the
spectrum of fluctuations depends on the temperature $T_{\mathrm{env}}$ of the
environment (see Eq. 10), an analysis of experimental tunnelling spectra
yields the value of $T_{\mathrm{env}}$ Martinis and Nahum (1993). As was
mentioned in the introduction, Josephson tunnelling spectroscopy provides the
most direct access to the $P(E)$ function of the environment and hence
$T_{\mathrm{env}}$.

Figure 3: Lumped element representation of the tunnelling junction and its
environment. The tunnelling junction with tunnelling resistance $R$ and
temperature $T$ is shunted by the junction capacitance $C$ and connected to a
purely dissipative impedance $R_{\mathrm{env}}$ whose temperature is
$T_{\mathrm{env}}$.

We exploited the unique capability of our instrument to perform temperature-
dependent STS and recorded 91 differential conductance $dI/dV(V)$ spectra of
the STM junction in the Josephson regime while increasing the temperature
$T_{\mathrm{ADR}}$ of the adiabatic demagnetization refrigerator from 30 to
$600\ \mathrm{mK}$. The spectra were acquired automatically with a rate of one
spectrum per every 30 min. The measurement was done on an atomically clean
superconducting Al(100) surface using a superconducting tip. The latter was
prepared by gentle indentations of the PtIr tip (see Fig. 1) into the Al(100)
surface. Visual inspection of the spectra in Fig. 2 reveals that the Josephson
conductance peak, located at zero bias Grabert and Devoret (1992) and visible
in all spectra up to $T_{\mathrm{ADR}}\approx$ 500 mK, shows no noticeable
thermal dependence, except for its eventual disappearance above 500 mK, which
most likely is a natural consequence of the loss of superconductivity in the
tip. This finding is surprising, considering that $T_{\mathrm{ADR}}$
experiences an almost 20-fold increase.

In an attempt to rationalize the absence of the thermal dependence in the
data, we recall that $P(E)$, which according to Eq. 5 defines the $I(V)$ and
hence the $dI/dV(V)$ spectra of the Josephson tunnelling junction, depends on
$T_{\mathrm{env}}$ rather than $T_{\mathrm{T,S}}$ Martinis and Nahum (1993);
Siewert, Nazarov, and Falci (1997). Thus, the absent temperature broadening
indicates that $T_{\mathrm{env}}$ is decoupled from $T_{\mathrm{T,S}}$. Indeed
a situation were $T_{\mathrm{env}}\gg T_{\mathrm{T,S}}$ could naturally be
realized due to an insufficient radiation shielding of the junction from the
higher temperature stages of the cryostat Hergenrother  _et al._ (1994, 1995).

Figure 4: The black data series depicts the Josephson $I(V)$ spectrum acquired
simultaneously with the lowest temperature $dI/dV$ curve from Fig. 2 measured
at 34 mK and fitted with $T_{\mathrm{env}}=1.5$ K (red curve cf. text).
Measured by the positions of the $I(V)$ extrema in this curve, the broadening
due to environmental effects is $35$ $\mu$V. The blue dashed line shows the
$I(V)$ spectrum of a junction in a well-thermalized environment obtained with
the same parameters as for the black curve (cf. text) but
$T_{\mathrm{env}}=0.1$ K. The corresponding value of environmental broadening
is $9\ \mu$V. Gray data correspond to the $I(V)$ spectrum of the Josephson
junction measured at 34 mK with the high-frequency filter of the bias line
Esat  _et al._ (2021) unplugged. The data were fitted with
$T_{\mathrm{env}}=2.3$ K, keeping all other parameters unchanged (cf. text).
Measured by the positions of the $I(V)$ extrema in this curve, the broadening
due to the environmental effects is $44$ $\mu$V.

To test the plausibility of the $T_{\mathrm{env}}\gg T_{\mathrm{T,S}}$
scenario, we used the $P(E)$ theory to fit the experimental $I(V)$ curve
acquired simultaneously with the lowest temperature $dI/dV(V)$ spectrum from
Fig. 2 recorded at $T_{\mathrm{ADR}}=34$ mK. As Fig. 4 shows, presumably due
to the finite size effects in our superconducting tip, we registered a
considerable quasiparticle tunnelling background in the $I(V)$ data. To keep
the number of fitting parameters small, we simply neglected the presence of
the quasiparticle background, by constraining the fitting to the [-25,25]
$\mu$V interval of the bias voltage.

According to the Eqs. 5, 8, 10-13, in the simplest case where the effective
impedance of the circuit can be set to a real value $R_{\mathrm{env}}$, the
simulation of the experimental $I(V)$ data from Fig. 4 needs four fit
parameters: $C$, $R_{\mathrm{env}}$, $E_{\mathrm{J}}$ and $T_{\mathrm{env}}$.
Making an unconstrained fit with these parameters is, however, impossible as
$C$ and $T_{\mathrm{env}}$ are coupled to each other, as can be spotted in Eq.
12d defined in terms of $T/C$ ratio. Therefore, we fix $C=100$ fF in
accordance with the BEM simulations of the STM junction discussed above, while
freely varying the remaining three parameters $R_{\mathrm{env}}$,
$E_{\mathrm{J}}$ and $T_{\mathrm{env}}$. Note that, in principle,
$E_{\mathrm{J}}$ could be obtained from the experimental data using the
Ambegaokar-Baratoff formula Ambegaokar and Baratoff (1963); Jäck  _et al._
(2016). However, here we used $E_{\mathrm{J}}$ as a fit parameter, because our
data did not yield a reliable estimate of the superconducting gap
$\Delta_{\mathrm{T}}$ in the tip, necessary for the evaluation of
$E_{\mathrm{J}}$.

Figure 5: The $P(E)$ function (red line) calculated for $T_{\mathrm{env}}=1.5$
K, $R_{\mathrm{env}}=31$ $\Omega$, $E_{\mathrm{J}}=8\ \mu$eV and $C=100$ fF.
The black curve displays $n^{\mathrm{BCS}}(E)$ for $\Delta=173\ \mu$eV. The
height of the quasiparticle peaks is limited by the finite size of the energy
scale used in the calculation.

Fig. 4 exhibits the experimental $I(V)$ spectrum of the Josephson junction
measured at 34 mK (black curve). The curve clearly demonstrates the signature
of the Josephson super current in the presence of thermal fluctuations in the
environment Joyez  _et al._ (1999). We fit the experimental curve by
calculating the $P(E)$ function according to the numerical procedure proposed
by Ingold and Grabert Ingold and Grabert (1991) and also used by Ast and
coworkers Ast  _et al._ (2016). The fit (red curve) yields
$T_{\mathrm{env}}=1.5$ K, $R_{\mathrm{env}}=31$ $\Omega$ and
$E_{\mathrm{J}}=8\ \mu$eV. The $P(E)$ function corresponding to the fit is
shown in Fig. 5. The value of $T_{\mathrm{env}}$ obtained from the fit
indicates that the scenario in which a cold mK STM junction finds itself
inside a much warmer environment is indeed feasible. One could even speculate
that the obtained value of $T_{\mathrm{env}}$ indicates the presence of
thermal radiation from the 1 K stage of the cryostat. To visualize the effect
that the environment has on the tunnelling, we plotted in Fig. 4 another
$I(V)$ (blue dashed) curve simulated with $T_{\mathrm{env}}=0.1$ K, i.e. for
the case of a well-shielded junction with $T_{\mathrm{env}}\approx
T_{\mathrm{T,S}}$. Fig. 4 shows that a better shielding of the junction from
the hot radiation could improve the spectroscopic resolution considerably.

The thermal radiation arriving from the hotter stages of the cryostat affects
the tunnelling by coupling to the junction via its electrodes acting like an
antenna Ast  _et al._ (2016). However, the high-frequency noise from the room
temperature electronics should also couple to the junction, producing an
effective increase of $T_{\mathrm{env}}$. We exploited a convenient
possibility to demonstrate this effect experimentally and re-measured the
$I(V)$ spectrum of the Josephson junction at $T_{\mathrm{ADR}}=34$ mK with the
Pi-filter of the bias line (see Methods section) unplugged. The spectrum
measured without the filter shown as the grey curve in Fig. 4 was fitted with
$T_{\mathrm{env}}=2.3$ K, keeping all other parameters fixed. Thus, unplugging
the filter changes $T_{\mathrm{env}}$ and worsens the experimental resolution
from 35 $\mu$V to 44 $\mu$V, as measured by the extrema of the Josephson
current feature in Fig. 4.

###  II.3 Temperature of the tip

Having determined the influence of the junction environment, we now turn to
the problem of estimating $T_{\mathrm{T,S}}$, which, as was mentioned above,
are the temperatures appearing in the Fermi-Dirac distributions of the tip and
the sample electrodes. Generally speaking, the tip and the sample can have
different temperatures, i.e. $T_{\mathrm{T}}\neq T_{\mathrm{S}}$. Therefore,
to fully characterize the junction temperature, it could be necessary to
perform two different experiments. As mentioned above, $T_{\mathrm{S}}$ can be
obtained by STS with a superconducting tip on a normal metal sample, while
$T_{\mathrm{T}}$ follows from STS with a normal metal tip on a superconducting
sample. Assuming that the tip should have a higher temperature in our STM
design, we only evaluate $T_{\mathrm{T}}$ as the upper bound on the junction
temperature.

We remark that despite a consensus about the utility of $n^{\mathrm{BCS}}(E)$
for estimating the junction temperature, there is no general agreement on the
details of the analysis, with most variations occurring at the stage of
fitting the experimental data. The problem is that using Eqs.3 and 4 does not
usually produce satisfactory fits, and as a consequence one has to employ
expressions for the superconducting DOS that include additional parameters.
The most commonly used expression of that type Song  _et al._ (2010a); Assig
_et al._ (2013); Roychowdhury  _et al._ (2014); Machida, Kohsaka, and Hanaguri
(2018); Balashov, Meyer, and Wulfhekel (2018); von Allwörden  _et al._ (2018);
Wong  _et al._ (2020); Esat  _et al._ (2021) was introduced by Dynes et al.
Dynes, Narayanamurti, and Garno (1978) to account for the finite quasiparticle
lifetime in a superconductor. It follows from Eq. 3 by introducing a
phenomenological parameter $\gamma$

$n_{\mathrm{S}}^{\mathrm{D}}=\mathrm{Re}\bigg{[}\frac{E-i\gamma}{\sqrt{(E-i\gamma)^{2}-\Delta^{2}}}\bigg{]}.$
(6)

Although $n_{\mathrm{S}}^{D}$ indeed results in better fits of the STS data,
the physical significance of $\gamma$ remains unclear, which makes the fitting
procedure ambiguous.

Figure 6: (a) Raw $dI/dV(V)$ spectrum of the superconducting Al(100) surface
measured with a normal-metal tip at 44 mK and fitted with Eq. 4, where
$n_{\mathrm{S}}^{\mathrm{BCS}}$ was substituted by the Dynes expression for
$n_{\mathrm{S}}^{\mathrm{D}}$ defined in Eq. 6. For values of the fit
parameters cf. text. (b) The $I(V)$ curve measured simultaneously with the
$dI/dV(V)$ spectrum of panel (a) and fitted using the same parameter set used
for calculating the fit in (a) complemented by the high-voltage or normal-
state resistance of the junction $R=1.2$ M$\Omega$.

Besides the Dynes expression in Eq. 6, there have been attempts to fit the
experimental STS of superconducting gaps with the so-called Maki formula,
which considers the effects of magnetic scattering Assig  _et al._ (2013);
Machida, Kohsaka, and Hanaguri (2018). Inspection of the original work of Maki
Maki (1964) reveals, however, that their approach applies to dirty
superconductors, for which the mean free path of an electron is smaller than
the coherence length of a Cooper pair. Because our clean single-crystal
Al(100) sample does not fulfill this condition, we only analyze our data using
the Dynes expression in Eq. 6.

In Fig. 6a we present a $dI/dV$ spectrum of the superconducting gap measured
on Al(100) with a normal-metal PtIr tip at $T_{\mathrm{ADR}}=$ 44 mK. As
expected for this temperature, the curve features a fully developed
superconducting gap and two sharp conductance spikes situated symmetrically
around zero bias. First, following the standard approach we fit the spectrum
with Eq. 4 in which we substitute $n_{\mathrm{BCS}}$ with
$n_{\mathrm{S}}^{\mathrm{D}}$ and use $\Delta$ and $\gamma$ (see Eq. 6) as two
independent fit parameters, while the Fermi-Dirac distribution in Eq. 4
contributes $T_{\mathrm{T}}$ as the third fit parameter. The $P(E)$ function
in Eq. 4 introduces no additional fit parameters, as it is calculated with the
fixed set of parameters determined earlier from the Josephson junction data
analysis. Note that in using Eq. 4 we neglect the small effect of the 4 $\mu$V
lock-in bias modulation (see Methods).

Figure 7: Illustration of how the $I(V)$ function simulated using the Dynes
expression in Eq. 6 reacts to changes in $T$ and $\gamma$. The respective
parameters are listed in the insets.

Fig. 6a shows the fit of the superconducting gap obtained with $\Delta=175\
\mu$eV, $\gamma=2.4\ \mu$eV, and $T_{\mathrm{T}}=99$ mK. Notice that the value
of $T_{\mathrm{T}}$ comes out substantially lower than the one we reported
earlier Esat  _et al._ (2021), which occurs mostly due to the inclusion of the
$P(E)$ broadening effect into the consideration. The conventional fitting
approach stops here, without giving detailed attention to the value of
$\gamma=2.4\ \mu$eV, which seems to be an order of magnitude too high, if
compared, e.g., to the data reported for mesoscopic junctions Pekola  _et al._
(2010).

However, the unusually high $\gamma$ causes a problem manifesting itself as a
high in-gap conductance visible by closer inspection of the fit in Fig. 6a.
Technically, the problem arises because the least-square fitting routine
minimizes the total sum of absolute quadratic deviations
$\sum_{i}\sqrt{x^{2}_{i}-y_{i}^{2}}$ between the data $x_{i}$ and the fit
$y_{i}$ calculated at every experimental point $i$. Because the minimized
deviation is absolute, the points where the $x_{i}$ are large contribute more
to its sum. Conversely, the points inside the gap, where the $x_{i}$ are
small, produce a smaller contribution; hence, the deviations between the fit
and the data inside the gap are less significant for minimizing that sum.

To exhibit the extent of the problem clearly, we show in Fig. 6b a log-log
plot of the $I(V)$ curve measured simultaneously with the $dI/dV$ data in Fig.
6a. The latter was fitted with a curve that has been generated with the same
parameters as in Fig. 6a, but complemented with the high-voltage (i.e. normal-
state) resistance $R$ = 1.2 $\mathrm{M\Omega}$ of the junction. Fig. 6b
confirms that the fitting of the experimental data inside the gap needs
improvement.

Figure 8: (a) Fit of the $I(V)$ curve from Fig. 6b, achieved with the modified
fitting routine (cf. text). (b) $dI/dV(V)$ spectrum corresponding to the
$I(V)$ curve of panel (a) plotted on a logarithmic scale to show the quality
of the fit. The red curve is generated with the same parameter set as in panel
(a). The blue curve is the fit from Fig. 6a. (inset) Quasiparticle peak
feature plotted on a linear scale.

Before we fix the problem with the fitting, it is instructive to inspect how
the Dynes-generated $I(V)$ curve responds to changes in the parameters
$\gamma$ and $T$. Fig. 7a shows that for small $T$ and $\gamma$, the value of
$\gamma$ defines primarily the zero-bias conductance $G_{V=0}$ of the
junction. $T$, on the other hand, as Fig. 7b shows, controls the sharpness of
the transition between the low and high conductance regimes. Interestingly, as
$T$ increases, it smears the gap edge towards zero bias and thus also raises
the zero-bias conductance of the junction. Comparing Fig. 6b and Fig. 7a-b, we
see that the value of $\gamma$ that we obtained from the first fitting attempt
is indeed too high.

To fit the in-gap data correctly, we first change to fitting $I(V)$ instead of
$dI/dV(V)$ curves. Besides that, we minimize the sum of relative quadratic
deviations $\sum_{i}\sqrt{\big{(}\frac{x_{i}-y_{i}}{x_{i}+y_{i}}\big{)}^{2}}$.
The modified fit obtained with $\Delta=173\ \mu$eV, $\gamma=0.5\ \mu$eV,
$T_{\mathrm{T}}=78$ mK and $R_{\mathrm{T}}=1.2\ \mathrm{M\Omega}$ in Fig. 8a
exhibits a much better agreement in the gap. Nevertheless, the improvement in
the gap comes at the expense of the fit quality at the quasiparticle peaks,
where, as the inset of Fig. 8b shows, the new fit overshoots the experiment
appreciably.

Concluding that the Dynes-generated curves cannot simultaneously fit all the
features of the experimental superconducting gap spectrum, we come to the
question of which part is more relevant for our analysis. We propose that the
in-gap part of the spectrum is of greater importance because the quasiparticle
peaks, as was also recently suggested by Schwenk et al.Schwenk  _et al._
(2020), may experience an additional non-intrinsic energy-dependent
broadening. One could further speculate that the observed broadening may stem
from non-equilibrium effects in the junction, e.g., local heating of the tip
by the current of tunnelling electrons Nahum, Eiles, and Martinis (1994). If
this indeed was the case, the temperature increase due to such heating could
be about 20 mK, as Fig. 8b suggests.

Figure 9: The experimentally measured temperature dependence of $dI/dV(V)$ (a)
and $I(V)$ (b) of the normal metal - superconductor tunnelling junction. The
data were measured in a fully automatic mode, taking a spectrum every 30
minutes during ca. 48 hours. The noise visible in the $dI/dV(V)$ data comes
from the few curves that were affected by small instabilities of the junction,
caused by rapid pressure variations in the He exhaust line.

Developing the argument further and recalling Fig. 7b, which shows how the
zero-bias conductance $G_{V=0}(T)$ reacts to the increasing temperature, we
wonder whether $G_{V=0}(T)$ alone could provide a good estimate of the
temperature in the tip. To explore this question, we measured the junction
characteristics, varying $T_{\mathrm{ADR}}$ between 44 mK and 1.2 K. As
expected, the data plotted in Fig. 9a-b clearly show a gradual disappearance
of the superconducting gap upon increasing $T_{\mathrm{ADR}}$. The
experimental $G_{V=0}(T_{\mathrm{ADR}})$ plot in Fig. 10a extracted from the
data in Fig. 9 reveals two regimes: Below $T_{\mathrm{ADR}}\sim 200$ mK
$G_{V=0}$ stays almost constant, with its value determined by $\gamma$. At
higher $T_{\mathrm{ADR}}$, $G_{V=0}$ increases quickly, indicating that the
smearing of the gap reaches zero bias, similar to Fig. 7b.

To simulate the $G_{V=0}(T_{\mathrm{ADR}})$ curve with Eq. 6, we assumed that
$\gamma$ is independent of the temperature and used the value $\gamma=0.56\
\mu$eV. Second, we calculated $\Delta(T)$ according to a well-known analytic
formula Gasparovic, Taylor, and Eck (1966) by taking the critical temperature
$T_{\mathrm{c}}$ of Al to be 1.2 K and using $\Delta(T=0)$ = 173 $\mu$eV as
obtained from the fit in Fig. 8a. Plotting the calculated $G_{V=0}(T)$ curve
in Fig. 10a next to the experiment, we find a striking agreement between both
curves if the experimental temperature scale $T_{\mathrm{ADR}}$ is shifted by
$T_{\mathrm{shift}}$ = 45 mK towards higher temperatures.

The data from Fig. 10a can be also plotted to show a $T_{\mathrm{shift}}$ for
each individual experimental data point, which results in the
$T_{\mathrm{shift}}(T_{\mathrm{ADR}})$ curve plotted in Fig. 10b. Apparently,
our $G_{V=0}(T)$ analysis yields a reasonably precise estimate of
$T_{\mathrm{shift}}$ in the range where $T_{\mathrm{ADR}}\gtrsim 200$ mK. Due
to the flattening of the $G_{V=0}(T)$ curve, however, the precision for
$T_{\mathrm{ADR}}<200$ mK deteriorates, but it seems that one could
nevertheless safely assume that $T_{\mathrm{shift}}$ remains roughly constant
towards lower-temperatures. Interestingly, as $T_{\mathrm{ADR}}$ increases,
$T_{\mathrm{shift}}$ tends to diminish, which could happen due to increasing
heat conductivity of the materials, leading to a better thermalization of the
tip.

Looking back at Fig. 8 that displays the $I(V)$ curve measured at
$T_{\mathrm{ADR}}=44$ mK and recalling that $T_{\mathrm{T}}=78$ mK extracted
from its fit, we can now compare this latter temperature to the
$T_{\mathrm{T}}=T_{\mathrm{ADR}}+T_{\mathrm{shift}}=44\ \mathrm{mK}+45\
\mathrm{mK}=89$ mK predicted by the analysis of $G_{V=0}(T)$. Pondering the
discrepancy between the predicted values, we note that due to the mentioned
inaccuracy of the $G_{V=0}(T)$ approach below 200 mK, the result from the fit
of the full $I(V)$ curve should probably be given more credence. Note,
however, that unlike the $G_{V=0}(T)$ analysis, the full $I(V)$ curve fitting
will generally work much less reliably at higher temperatures due to the
washing out of the gap.

Figure 10: (a) Thermal dependence of the zero-bias conductance $G_{V=0}$
obtained from the experiment (grey dots) and simulated (red line). The black
dots belong to the same experiment as the gray ones, but are shifted by 45 mK
to higher temperatures. (b) A shift between $T_{\mathrm{ADR}}$ and the
temperature of the tip $T_{\mathrm{T}}$ was obtained from the data in (a). Red
horizontal line marks the value $T_{\mathrm{shift}}=45$ mK.

Finally, we conclude our analysis by discussing $\gamma$, the microscopic
origin of which, especially for temperatures far below $T_{\mathrm{c}}$, is
not yet well-understood. Recently, Pekkola et al. Pekola  _et al._ (2010)
proposed that $\gamma$ may reflect the strength of photon-assisted tunnelling
that occurs at energies far below the gap edge. Assuming that our environment
has a blackbody spectrum with $T_{\mathrm{env}}=1.5$ K, we obtain $2.7\times
T_{\mathrm{env}}=4.25$ K as the average energy of a photon. This is
substantially higher than $\Delta/k_{\mathrm{B}}\approx 2\ \mathrm{K}$, which
points towards the possibility of photon assisted tunnelling.

According to Eqs.6 and 4 the dimensionless parameter $\gamma/\Delta$ is
directly related to the ratio of the junction conductance at zero bias to its
conductance at high bias, $\gamma/\Delta\simeq R/R_{0}$. Due to the presence
of the superconducting gap, tunnelling at zero bias can only occur if an
electron absorbs energy larger than $\Delta$ from the environment. Because the
probability of such an event is defined by the $P(E)$ function for
$E\leq-\Delta$, Di Marco et al. Di Marco  _et al._ (2013) proposed to evaluate
$\gamma/\Delta$ as

$\gamma/\Delta\simeq 2\int^{-\Delta}_{-\infty}dE\ n^{\mathrm{BCS}}(E)P(E),$
(7)

Using the parameters determined previously (by fitting to experimental data,
see above), the two functions under the integral in Eq. 7 are plotted in Fig.
5. Evaluating the integral numerically yields $\gamma/\Delta=4\times 10^{-4}$,
while the experimental fit in Fig. 8 provides $\gamma/\Delta=3\times 10^{-3}$.
We regard this discrepancy as rather small, taking the phenomenological
character of the procedure for evaluating the $P(E)$ function into account.
Thus, the data favour the idea of Pekola et al. Pekola  _et al._ (2010) that
$\gamma$ of the Dynes fit reflects the strength of the photon-assisted
tunnelling. Note that this effect could naturally occur in our STM junction,
because it is not shielded from the radiation of the 1 K stage of the cryostat
Esat  _et al._ (2021).

In conclusion, our temperature-dependent STS data acquired with the
Al/vacuum/Al(100) Josephson junction between 30 and 600 mK demonstrated no
appreciable thermal broadening effects. We propose that this occurs due to the
difference between the temperature $T_{\mathrm{env}}$ of the environment and
the junction temperature $T_{\mathrm{T,S}}$ with $T_{\mathrm{env}}\gg
T_{\mathrm{T,S}}$. Fitting the Josephson STS data, we obtained the $P(E)$
function that characterizes the probability of the energy exchange between the
environment and the junction. According to the fit, the temperature of the
junction’s environment is $T_{\mathrm{env}}=1.5$ K, substantially higher than
the temperature of the ADR cryostat in which we performed the measurement.
Although our data provide a qualitatively clear picture of a cold STM junction
embedded in a hot environment, the obtained value of $T_{\mathrm{env}}$ needs
to be treated with caution, due to the phenomenological character of the
approach to calculating the $P(E)$ function and the crudeness with which we
evaluated the junction capacitance $C$.

Using the tunnelling spectrum of the superconducting gap measured with a
normal-metal tip on the Al(100) surface, we have demonstrated the potential
fallibility of the usual $dI/dV(V)$ fitting, which does not pay sufficient
attention to the low-conductance data inside the gap. We showed that
minimizing the relative deviation helps achieving a much better fit of the in-
gap data, albeit at the expense of the fit quality of the quasiparticle peaks.
However, as we argued, the quasiparticle peaks may be subject to additional
non-intrinsic and energy-dependent broadening mechanisms. Therefore, their
accurate fitting may not be necessary for determining the intrinsic parameters
such as the junction’s temperature and the Dynes $\gamma$. Using a modified
fitting routine, we found that at $T_{\mathrm{ADR}}=44$ mK the Fermi-Dirac
distribution temperature in the tip was $T_{\mathrm{T}}=77$ mK and
$\gamma=0.5\ \mu$eV or $\gamma/\Delta=0.003$.

We also proposed a new way of determining the temperature of the STM junction
by analyzing the temperature scaling of the zero-bias conductance $G_{V=0}(T)$
only. Using this approach, we reconstructed $T_{\mathrm{T}}$ by mapping out
its deviation $T_{\mathrm{shift}}$ from the cryostat temperature
$T_{\mathrm{ADR}}$. Our data showed that $T_{\mathrm{shift}}\approx 45$ mK for
$T<600$ mK. According to this method, the lowest $T_{\mathrm{T}}$ that we can
currently achieve in our setup is $26\ \mathrm{mK}+45\ \mathrm{mK}=71\
\mathrm{mK}$.

Finally, our data suggest that the value of the Dynes parameter $\gamma$
obtained from fitting the superconducting gap, in agreement with the proposal
of Pekkola et al. Pekola  _et al._ (2010); Di Marco  _et al._ (2013), reflects
the strength of the photon-assisted tunnelling caused by the radiation
arriving at the STM junction from the hotter stages of the cryostat. This
finding emphasizes the importance of good radiation shielding and possibly
low-temperature high-frequency filtering in the design of a mK STM.

##  III Methods

###  III.1 Experimental setup

We performed the experiments in a mK STM, the details and performance of which
were reported recently Esat  _et al._ (2021). Thus, we only briefly list its
essential characteristics here. The STM head is thermally well-anchored to a
base plate of an adiabatic demagnetization refrigerator (ADR) that resides
inside a UHV chamber. The ADR reaches a minimum temperature of 26 mK, measured
by a calibrated RuOx sensor positioned on the base plate of the ADR right next
to the STM head. We refer to the temperature reading of that thermometer as
$T_{\mathrm{ADR}}$. ADR enables STM/STS experiments at varying
$T_{\mathrm{ADR}}$ conditions spanning the range between 30 mK and 1-2 K.
STM/STS can be performed in the presence of a magnetic field of up to $8\
\mathrm{T}$ applied in the direction perpendicular to the sample surface. To
reduce the high-frequency noise in the junction we use Pi-filters on the bias
and all five high-voltage piezo lines.

###  III.2 STS

The STS data discussed here were acquired on the clean surface of an Al(100)
single crystal clamped to a custom-made flag-type sample holder. The sample
holder body was fabricated from tungsten, while the clamping comprised a
molybdenum foil and a set of molybdenum screws fixing the foil to the sample
holder body. Due to the superconductivity of molybdenum, we made sure that the
foil contained no closed holes to prevent flux trapping at low temperatures.
The sample surface was prepared in UHV by repeated cycles of Ar+ ion
sputtering and subsequent annealing at 400∘C. We used low-energy electron
diffraction (LEED) and Auger electron spectroscopy (AES) to verify the surface
quality.

The Al(100) sample was superconducting during all of the STS measurements
reported here. Interestingly, we could only observe the superconductivity by
applying a compensating B field of 8-10 mT to the sample magnet. We speculate
that the compensation field was necessary to neutralize the effect of the
magnetic flux caught by one of the superconducting coils during the ADR run
Esat  _et al._ (2021).

The superconductivity in the sample always appeared abruptly upon ramping up
the compensating B field. Moreover, the tunnelling spectrum of the
superconducting gap did not change when the compensating sample B field was
further varied. Instead, it disappeared and reappeared erratically. We
rationalize this behavior by assuming that our sample was mainly in an
intermediate state characterized by an ordered structure of macroscopically
large superconducting domains separated by normal-metal areas Tinkham (1996).
In such a state, slight variations of the sample B field can cause
rearrangements of the domain structure, which, in turn, produce abrupt changes
in the type of the domain (superconducting vs. normal) faced by the tip.

The Josephson junction STS data were measured at 30 mK $\leq
T_{\mathrm{ADR}}\leq 600$ mK with a compensating field of 8 mT kept throughout
the measurement series. For the measurement, the PtIr tip was made
superconducting by indenting it gently into the Al(100) surface until the
Josephson conductance peak at zero bias (see Fig. 2) appeared in the
conductance spectra. The series of spectra featuring the superconducting gap
of the Al(100) surface (Fig. 9) was acquired with a clean PtIr tip at 40 mK
$\leq T_{\mathrm{ADR}}\leq 1.2$ K. The lower temperature data were collected
with the compensating field of 8 mT. At $T_{\mathrm{ADR}}\approx 460\
\mathrm{mK}$, the sample superconductivity disappeared abruptly, and we
regained it by tuning the compensating field to 10 mT, the value at which all
higher temperature data were then collected.

STS was performed using the internal lock-in of the Nanonis software. The
parameters of the lock-in were: modulation frequency 187.7 Hz, modulation
amplitude 2 $\mu$V for the Josephson (Fig. 2) and 4 $\mu$V for the
superconducting gap spectroscopy (Fig. 9). The tunnelling current was measured
using a fixed gain $(10^{10})$ amplifier from NF Corporation NF . All
experimental data are displayed as raw data without any additional
postprocessing. Each presented STS curve is a single sweep spectrum acquired
in less than 220 s.

###  III.3 Fitting STS data with the $P(E)$ theory

The $P(E)$ theory which was originally developed for mesoscopic tunnelling
junctions Grabert and Devoret (1992), derives the $P(E)$ function from the
junction’s equilibrium phase-phase correlation function
$J(t)\equiv\langle[\tilde{\phi}(t)-\tilde{\phi}(0)]\tilde{\phi}(t)\rangle$,

$P(E)=\frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}dt\
\mathrm{exp}\bigg{[}J(t)+\frac{i}{\hbar}Et\bigg{]}.$ (8)

According to the Wiener-Khinchin theorem (see Appendix of the ref. Clerk  _et
al._ (2010)), $P(E)$ is a noise power spectral density of the junction’s
charge shift operator $e^{-i\tilde{\phi}}$, where the phase $\tilde{\phi}(t)$
is defined as

$\tilde{\phi}(t)=\frac{e}{\hbar}\left(\int_{-\infty}^{t}dt^{\prime}U(t^{\prime})-Vt\right),$
(9)

with $U=Q/C$ being the momentary voltage across the junction with capacitance
$C$ charged by $Q$, and $V$ is the constant voltage applied by an external
source considered to be ideal.

In their pioneering attempt to apply the $P(E)$ theory to the STM, Ast and
coworkers extended the theory by pointing out that the fluctuations of the
phase seen by the STM junction split into two contributions Ast  _et al._
(2016): $J(t)=J_{0}(t)+J_{\mathrm{N}}(t)$. According to the fluctuation-
dissipation theorem, $J_{0}(t)$ is related to the imaginary part of the
response function $\chi(\omega)=(e/\hbar)^{2}Z_{t}(\omega)/i\omega$, where
$Z_{t}(\omega)$ is the total impedance of the circuit consisting of the the
junction capacitance $C$ and the environmental impedance $Z(\omega)$ via
Grabert and Devoret (1992)

$\begin{split}J_{0}(t)=2\int^{\infty}_{0}\frac{d\omega}{\omega}\frac{\mathrm{Re}[Z_{\mathrm{t}}(\omega)]}{R_{\mathrm{K}}}\times\\\
\times\bigg{[}\coth\bigg{(}\frac{1}{2}\frac{\hbar\omega}{k_{\mathrm{B}}T}\bigg{)}[\cos(\omega
t)-1]-i\sin(\omega t)\bigg{]}.\end{split}$ (10)

The newly introduced term $J_{\mathrm{N}}(t)$ determines thermal voltage noise
on the junction capacitor $C$. Splitting $J(t)$ into the two contributions
helps express the $P(E)$ function as a convolution of two parts

$P(E)=\int dE^{\prime}P_{0}(E-E^{\prime})P_{\mathrm{N}}(E^{\prime}),$ (11)

where $P_{\mathrm{N}}$ related to $J_{\mathrm{N}}(t)$ takes a simple Gaussian
form

$P_{\mathrm{N}}(E)=\frac{1}{\sqrt{4\pi
E_{\mathrm{C}}k_{\mathrm{B}}T}}\exp\bigg{[}-\frac{E^{2}}{4E_{\mathrm{C}}k_{\mathrm{B}}T}\bigg{]},$
(12)

with $E_{\mathrm{C}}=e^{2}/2C$ being the charging energy of the STM junction
capacitor $C$. Note that for the Josephson current carried by Cooper pairs
$E_{\mathrm{C}}=(2e)^{2}/2C$.

According to Eqs. 10 and 12 both $P_{0}(E)$ and $P_{\mathrm{N}}(E)$ depend on
the temperature of the junction environment $T_{\mathrm{env}}$ Martinis and
Nahum (1993). Therefore, calculating $P(E)$ and using it in Eq. 5 for fitting
the experimental $I(V)$ characteristics of a Josephson junction should yield
$T_{\mathrm{env}}$.

While obtaining $P_{\mathrm{N}}(E)$ is straightforward, the calculation of
$P_{0}(E)$, according to Eq. 10, becomes a task of finding
$Z_{\mathrm{t}}(\omega)$. Currently, there are no well-established schemes
which allow doing this reliably for the case of an STM junction. Therefore, we
follow the procedure developed for mesoscopic tunnelling junctions Grabert and
Devoret (1992); Ast  _et al._ (2016). That approach starts by representing the
junction and its environment as a lumped-element circuit with the tunnelling
junction shunted by a capacitance $C$ and connected to an ideal voltage source
via an environmental impedance $Z(\omega)$. Then the expression for
$Z_{\mathrm{t}}(\omega)$ can be written as Grabert and Devoret (1992)

$Z_{\mathrm{t}}(\omega)=\frac{1}{i\omega C+1/Z(\omega)}.$ (13)

In the simplest case $Z(\omega)$ only has a dissipative part that is
independent of $\omega$, i.e. $Z(\omega)=R_{\mathrm{env}}$ Grabert and Devoret
(1992); Hergenrother  _et al._ (1995); Pekola  _et al._ (2010); Di Marco  _et
al._ (2013). The value $R_{\mathrm{env}}$ determines the strength of coupling
between the junction and its environment, with $R_{\mathrm{env}}\ll
R_{\mathrm{Q}}\equiv h/e^{2}$ determining the weak coupling regime in which
most tunnelling electrons leave the environmental modes, except those close to
$\omega=0$, undisturbed, the $P(E)$ function is thus peaked at $E=0$, and the
charge transferred through the junction is removed almost instantaneously by
the voltage source Grabert and Devoret (1992). According to Eq. 5, fitting of
the $I(V)$ curves measured with a Josephson junction then needs four fit
parameters: $C$, $R_{\mathrm{env}}$, $E_{\mathrm{J}}$ and $T_{\mathrm{env}}$.

##  IV Acknowledgments

We thank Christian Ast and Gianluigi Catelani for fruitful discussions.
Funding: T.E., R.T. and F.S.T. acknowledge support from the German Federal
Ministry of Education and Research through the funding program 13N16032.

##  V Author contributions

T.E., F.S.T., and R.T. conceived the experiment. T.E. and R.T. performed the
measurements with the technical help from X.Y. and F.M.. T.E. and R.T.
analyzed the experimental data. H.S. performed the BEM calculations of the STM
junction capacitance. R.T. wrote the manuscript with significant input from
F.S.T.. All authors discussed the results and reviewed the manuscript.
Competing interests: The authors declare no competing interests. Data and
materials availability: All data presented in this paper are publicly
available through the Jülich Dataset Respository with the identifier xxx.

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# A new family of $q$-analogue of Genocchi numbers and polynomials of higher
order

Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY<EMAIL_ADDRESS>,  Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY<EMAIL_ADDRESS>and  Jong Jin Seo
Department of Applied Mathematics, Pukyong National University, Busan 608-737,
Republic of KOREA<EMAIL_ADDRESS>
###### Abstract.

The new $q$-Euler polynomials was introduced by T. Kim in “$q$-Generalized
Euler numbers and polynomials, Russian Journal of Mathematical Physics, Vol.
13, No. 3, 2006, pp. 293-308” by means of the following generating function:

$\sum_{j=0}^{\infty}\frac{z^{j}}{\left[j\right]_{q}!}E_{j,q}\left(x\right)=\frac{\left[2\right]_{q}}{e_{q}\left(z\right)+1}e_{q}\left(xz\right)\text{.}$

In this work, we consider the generating function of Kim’s $q$-Euler
polynomials and introduce new generalization of $q$-Genocchi polynomials and
numbers of higher order. Also, we give surprising identities for studying in
Analytic Numbers Theory and especially in Mathematical Physics. Moreover, by
applying $q$-Mellin transformation to generating function of $q$-Genocchi
polynomials of higher order and so we define $q$-Hurwitz-Zeta type function
which interpolates of this polynomials at negative integers.

2010 Mathematics Subject Classification. 11S80, 11B68.

Keywords and phrases. Genocchi numbers and polynomials, $q$-Genocchi numbers
and polynomials of higher order, $q$-Mellin transformation, $q$-Hurwitz-Zeta
function, $q$-Gamma function, $q$-Exponential function.

.$\sim$25th International Conference of the Jangjeon Mathematical
Society$\sim$

##  1\. Introduction

Throughout this work, we assume that $q\in\mathbb{C}$ with $\left|q\right|<1$.
The $q$-integer of $x$ is defined by $\left[x\right]_{q}=\frac{1-q^{x}}{1-q}$
and note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$. The $q$-derivative
is defined by F. H. Jackson as follows:

(1)
$D_{q}f\left(x\right)=\frac{d}{d_{q}x}f\left(x\right)=\frac{f\left(x\right)-f\left(qx\right)}{\left(1-q\right)x}\text{.}$

Taking $f\left(x\right)=x^{n}$ in (1), it becomes as follows:

$D_{q}x^{n}=\frac{x^{n}-\left(qx\right)^{n}}{\left(1-q\right)x}=\left[n\right]_{q}x^{n-1}\text{
and }\left(\frac{d}{d_{q}x}\right)^{n}f\left(x\right)=\left[n\right]_{q}!$

where
$\left[n\right]_{q}!=\left[n\right]_{q}\left[n-1\right]_{q}\cdots\left[1\right]_{q}$.
Now, we give definitions of two kinds of $q$-exponential functions as follows:

For any $z\in\mathbb{C}$ with $\left|z\right|<1$,

(2)
$e_{q}\left(z\right)=\sum_{l=0}^{\infty}\frac{z^{l}}{\left[l\right]_{q}!}\text{
and
}E_{q}\left(z\right)=\sum_{l=0}^{\infty}q^{\binom{l}{2}}\frac{z^{l}}{\left[l\right]_{q}!}.$

By (2), it is not difficult to show that
$\left[l\right]_{\frac{1}{q}}!=q^{-\binom{l}{2}}\left[l\right]_{q}!$. Then, we
have the following

(3) $e_{\frac{1}{q}}\left(z\right)=E_{q}\left(z\right)\text{.}$

For the $q$-commuting variables $x$ and $y$ such that $yx=qxy$, we know that

(4) $e_{q}\left(x+y\right)=e_{q}\left(x\right)e_{q}\left(y\right)\text{.}$

The $q$-integral was defined by Jackson as follows:

(5)
$\int_{0}^{x}f\left(\xi\right)d_{q}\xi=\left(1-q\right)x\sum_{l=0}^{\infty}f\left(q^{l}x\right)q^{l}$

provided that the series on the right hand side converges absolutely.

In particular, if $f(\xi)=\xi^{n}$, then we have

(6)
$\int_{0}^{x}\xi^{n}d_{q}\xi=\frac{1}{\left[n+1\right]_{q}}x^{n+1}\text{.}$

The definitions of $q$-integral and $q$-derivative imply the following
formula:

(7) $D_{q}\left(\int_{0}^{x}f\left(t\right)d_{q}t\right)=f\left(x\right)$

and

(8)
$D_{q}\left(f\left(x\right)g\left(x\right)\right)=f\left(x\right)D_{q}\left(g\left(x\right)\right)+g\left(qx\right)D_{q}\left(f\left(x\right)\right)\text{.}$

For more informations of Eqs. (1-8), you can refer to [19-23].

The ordinary Euler numbers and polynomials are defined via the following
generating function:

$e^{E\left(x\right)t}=\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!}=\frac{2}{e^{t}+1}e^{xt}\text{,
}\left|t\right|<\pi\text{ }$

where the usual convention about replacing $E^{n}\left(x\right)$ by
$E_{n}\left(x\right)$ (see [1], [2], [3], [5], [6], [13]).

In [6], the new $q$-generalization of Euler polynomials are introduced by T.
Kim as follows:

$\sum_{j=0}^{\infty}\frac{z^{j}}{\left[j\right]_{q}!}E_{j,q}\left(x\right)=\frac{\left[2\right]_{q}}{e_{q}\left(z\right)+1}e_{q}\left(xz\right)\text{.}$

By using the above generating function, Kim gave some interesting and
fascinating properties for new $q$-generalization of Euler numbers and
polynomials. We note that these polynomials are used to study in Analytic
Numbers Theory. So, in the next section, we shall introduce generating
function of $q$-Genocchi numbers and polynomials of higher order.
Additionally, we shall give their applications.

##  2\. New $q$-Genocchi numbers and polynomials of higher order

In this section, we introduce generating function for $q$-Genocchi polynomials
of higher order by using Kim’s method in [6]. Thus, we now start as follows:

(9)
$S_{q}\left(t:\alpha\right)=\sum_{n=0}^{\infty}\frac{t^{n}}{\left[n\right]_{q}!}G_{n,q}^{\left(\alpha\right)}\text{.}$

Here $G_{n,q}^{\left(\alpha\right)}$ is called as the $q$-Genocchi numbers of
higher order. By using $q$-derivative operator, we compute as follows:

(10)
$S_{q}\left(\frac{d}{d_{q}x}:\alpha\right)x^{k}=\sum_{n=0}^{\infty}\frac{1}{\left[n\right]_{q}!}G_{n,q}^{\left(\alpha\right)}\left(\frac{d}{d_{q}x}\right)^{n}x^{k}=\sum_{n=0}^{k}\binom{k}{n}_{q}G_{n,q}^{\left(\alpha\right)}x^{k-n}\text{,}$

where

$\binom{k}{n}_{q}=\frac{\left[k\right]_{q}\left[k-1\right]_{q}\cdots\left[k-n+1\right]_{q}}{\left[n\right]_{q}!}\text{.}$

Similarly, by (10), we develop as follows:

$\displaystyle S_{q}\left(\frac{d}{d_{q}x}:\alpha\right)e_{q}\left(tx\right)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{G_{j,q}^{\left(\alpha\right)}}{\left[j\right]_{q}!}\left(\frac{d}{d_{q}x}\right)^{j}\sum_{k=0}^{\infty}\frac{x^{k}}{\left[k\right]_{q}!}t^{k}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{t^{j}}{\left[j\right]_{q}!}G_{j,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$ $\displaystyle S_{q}\left(x,t:\alpha\right)\text{.}$

From this point of view, we can also consider the $q$-Genocchi polynomials of
higher order in the form:

(11)
$\sum_{n=0}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}G_{n,q}^{\left(\alpha\right)}\left(x\right)=\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\alpha}e_{q}\left(zx\right)\text{.}$

As $q\rightarrow 1$ and $\alpha=1$ in Eq. (11), we easily reach the following

$\lim_{q\rightarrow
1}G_{n,q}^{\left(1\right)}\left(x\right)=G_{n}\left(x\right)$

which $G_{n}\left(x\right)$ is known as ordinary Genocchi polynomials (for
details, see [5], [8], [14], [15], [17], [18]).

By (9) and (11), we readily see that

$\sum_{j=0}^{\infty}\frac{z^{j}}{\left[j\right]_{q}!}G_{j,q}^{\left(\alpha\right)}\left(x\right)=\sum_{j=0}^{\infty}\left(\sum_{n=0}^{j}\binom{j}{n}_{q}x^{j-n}G_{n,q}^{\left(\alpha\right)}\right)\frac{z^{j}}{\left[j\right]_{q}!}\text{.}$

By comparing the coefficients of $\frac{z^{j}}{\left[j\right]_{q}!}$ on both
sides of the above equation, then we obtain the following theorem.

######  Theorem 2.1.

For any $j\in\mathbb{N}$, we have

$G_{j,q}^{\left(\alpha\right)}\left(x\right)=\sum_{n=0}^{j}\binom{j}{n}_{q}x^{j-n}G_{n,q}^{\left(\alpha\right)}\text{.}$

By applying $q$-derivative operator to (11), then we see that

$\sum_{n=1}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}\left\\{\frac{d}{d_{q}x}G_{n,q}^{\left(\alpha\right)}\left(x\right)\right\\}=z\sum_{n=0}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}G_{n,q}^{\left(\alpha\right)}\left(x\right)\text{.}$

By comparing the coefficients of $z^{n}$ on both sides of the above equation,
we arrive the following theorem.

######  Theorem 2.2.

For any $n\in\mathbb{N}^{\ast}=\left\\{0,1,2,3,\ldots\right\\}$, we have

$\frac{d}{d_{q}x}G_{n,q}^{\left(\alpha\right)}\left(x\right)=\left[n\right]_{q}G_{n-1,q}^{\left(\alpha\right)}\left(x\right)\text{.}$

For $q$-commuting variables $x$ and $y$ ($yx=qxy$), we note that

$\displaystyle\sum_{l=0}^{\infty}\frac{z^{l}}{\left[l\right]_{q}!}G_{l,q}^{\left(\alpha\right)}\left(x+y\right)$
$\displaystyle=$
$\displaystyle\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\alpha}e_{q}\left(z\left(x+y\right)\right)$
$\displaystyle=$ $\displaystyle
e_{q}\left(zy\right)\left(e_{q}\left(zx\right)\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\alpha}\right)$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\left(\sum_{j=0}^{l}\binom{l}{j}_{q}y^{l-j}G_{j,q}^{\left(\alpha\right)}\left(x\right)\right)\frac{z^{l}}{\left[l\right]_{q}!}\text{.}$

As a result, we procure the following theorem.

######  Theorem 2.3.

For any $n\in\mathbb{N}^{\ast}$, we have

$G_{n,q}^{\left(\alpha\right)}\left(x+y\right)=\sum_{j=0}^{n}\binom{n}{j}_{q}y^{n-j}G_{j,q}^{\left(\alpha\right)}\left(x\right)\text{.}$

By expression (11), we compute as follows:

$\displaystyle\sum_{l=0}^{\infty}\frac{z^{l}}{\left[l\right]_{q}!}G_{l,q}^{\left(\alpha+\beta\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\alpha}\right]\left[\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\beta}e_{q}\left(zx\right)\right]$
$\displaystyle=$
$\displaystyle\left[\sum_{j=0}^{\infty}\frac{z^{j}}{\left[j\right]_{q}!}G_{j,q}^{\left(\alpha\right)}\right]\left[\sum_{k=0}^{\infty}\frac{z^{k}}{\left[k\right]_{q}!}G_{k,q}^{\left(\beta\right)}\left(x\right)\right]$

by using Cauchy product on the above equation, we derive that

(13)
$\sum_{l=0}^{\infty}\frac{z^{l}}{\left[l\right]_{q}!}\left(\sum_{n=0}^{l}\binom{l}{n}_{q}G_{n,q}^{\left(\alpha\right)}G_{l-n,q}^{\left(\beta\right)}\left(x\right)\right)\text{.}$

Comparing the coefficients of Eqs. (2) and (13), then we present the following
theorem.

######  Theorem 2.4.

For $l\in\mathbb{N}^{\ast}$, then we have

$G_{l,q}^{\left(\alpha+\beta\right)}\left(x\right)=\sum_{n=0}^{l}\binom{l}{n}_{q}G_{n,q}^{\left(\alpha\right)}G_{l-n,q}^{\left(\beta\right)}\left(x\right)\text{.}$

Jackson are defined the $q$-analogue of the Gamma function by

(14)
$\Gamma_{q}\left(x\right)=\frac{\left(q;q\right)_{\infty}}{\left(q^{x};q\right)_{\infty}}\left(1-q\right)^{1-x}\text{,
}x\neq 0,-1,-2,\cdots.$

which have the following properties:

$\Gamma_{q}\left(x+1\right)=\left[x\right]_{q}\Gamma_{q}\left(x\right)\text{,
}\Gamma_{q}\left(1\right)=1\text{ and }\lim_{q\rightarrow
1^{-}}\Gamma_{q}\left(x\right)=\Gamma\left(x\right)\text{,
}\Re\left(x\right)>0\text{.}$

It has the $q$-integral representation as follows:

(15)
$\Gamma_{q}\left(s\right)=\int_{0}^{\frac{1}{1-q}}t^{s-1}E_{q}\left(-qt\right)d_{q}t\text{.}$

When $\frac{\log\left(1-q\right)}{\log q}\in\mathbb{Z}$, becomes

(16)
$\Gamma_{q}\left(s\right)=\int_{0}^{\infty}t^{s-1}E_{q}\left(-qt\right)d_{q}t\text{.}$

The $q$-Mellin transformation of a suitable function $f$ on $\mathbb{R}_{q,+}$
is defined by

(17)
$M_{q}\left(f\right)\left(s\right)=\int_{0}^{\infty}t^{s-1}f\left(t\right)d_{q}t$

(for details of Eqs. (14-17), see [10], [20], [21]).

In [23], the novel $q$-differential operator was defined by Rubin as follows:

(18)
$\partial_{q}\left(f\right)\left(x\right)=\frac{f\left(q^{-1}x\right)+f\left(-q^{-1}x\right)-f\left(qx\right)+f\left(-qx\right)-2f\left(-x\right)}{2\left(1-q\right)x}\text{.}$

By (18), we note that

(19) $\lim_{q\rightarrow
1}\partial_{q}\left(f\right)\left(x\right)=f{\acute{}}\left(x\right)\text{.}$

By applying Rubin’s $q$-differential operator to the generating function of
$q$-Genocchi numbers and polynomials of higher order, we compute as follows:

$\displaystyle\sum_{n=0}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}\partial_{q}G_{n,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\partial_{q}\left\\{\left(\frac{\left[2\right]_{q}z}{e_{q}\left(z\right)+1}\right)^{\alpha}e_{q}\left(xz\right)\right\\}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\frac{1}{2\left(1-q\right)}\sum_{l=0}^{n}\binom{n}{l}_{q}\left\\{\begin{array}[]{c}q^{-l}+\left(-1\right)^{l}q^{-l}-q^{l}\\\
+\left(-1\right)^{l}q^{l}+2\left(-1\right)^{l}\end{array}\right\\}x^{l-1}G_{n-l,q}^{\left(\alpha\right)}\right)\frac{z^{n}}{\left[n\right]_{q}!}\text{.}$

By comparing the coefficients of $\frac{z^{n}}{\left[n\right]_{q}!}$ on both
sides of the above equation. Then, we state the following theorem.

######  Theorem 2.5.

Let $T_{q}\left(l\right)=$
$q^{-l}+\left(-1\right)^{l}q^{-l}-q^{l}+\left(-1\right)^{l}q^{l}+2\left(-1\right)^{l}$,
then we get

(21)
$\partial_{q}G_{n,q}^{\left(\alpha\right)}\left(x\right)=\frac{1}{2\left(1-q\right)}\sum_{l=0}^{n}\binom{n}{l}_{q}T_{q}\left(l\right)x^{l-1}G_{n-l,q}^{\left(\alpha\right)}\text{.}$

By (21), we readily derive the following

$\displaystyle\partial_{q}G_{n,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(1-q\right)}\sum_{l=0}^{\left[\frac{n}{2}\right]}\binom{n}{2l}_{q}\left\\{q^{-2l}+1\right\\}G_{n-2l,q}^{\left(\alpha\right)}$
$\displaystyle+\frac{1}{\left(q-1\right)}\sum_{l=0}^{\left[\frac{n-1}{2}\right]}\binom{n}{2l+1}_{q}\left\\{q^{2l+1}+1\right\\}G_{n-1-2l,q}^{\left(\alpha\right)}\text{.}$

Here $\left[.\right]$ is Gauss’ symbol. Consequently, we derive the following
theorem.

######  Theorem 2.6.

For any $n\in\mathbb{N}^{\ast}$, we have

$\displaystyle\partial_{q}G_{n,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(1-q\right)}\sum_{l=0}^{\left[\frac{n}{2}\right]}\binom{n}{2l}_{q}\left\\{q^{-2l}+1\right\\}G_{n-2l,q}^{\left(\alpha\right)}$
$\displaystyle+\frac{1}{\left(1-q\right)}\sum_{l=0}^{\left[\frac{n-1}{2}\right]}\binom{n}{2l+1}_{q}\left\\{q^{2l+1}+1\right\\}G_{n-1-2l,q}^{\left(\alpha\right)}\text{.}$

From (21) and (2.6), we conclude as follows:

######  Corollary 2.7.

The following identity

$\displaystyle\sum_{l=0}^{n}\binom{n}{l}_{q}T_{q}\left(l\right)x^{l-1}G_{n-l,q}^{\left(\alpha\right)}$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{\left[\frac{n}{2}\right]}\binom{n}{2l}_{q}\left\\{2q^{-2l}+2\right\\}G_{n-2l,q}^{\left(\alpha\right)}$
$\displaystyle+\sum_{l=0}^{\left[\frac{n-1}{2}\right]}\binom{n}{2l+1}_{q}\left\\{2q^{2l+1}+2\right\\}G_{n-1-2l,q}^{\left(\alpha\right)}$

is true.

By (21), we have the following corollary.

######  Corollary 2.8.

For any $n\in\mathbb{N}^{\ast}$, we get

$\lim_{q\rightarrow
1}\partial_{q}G_{n,q}^{\left(\alpha\right)}\left(x\right)=nG_{n-1}^{\left(\alpha\right)}\left(x\right)$

where $G_{n}^{\left(\alpha\right)}\left(x\right)$ are called Genocchi numbers
and polynomials of higher order which are defined by the following generating
function:

$\sum_{n=0}^{\infty}G_{n}^{\left(\alpha\right)}\left(x\right)\frac{t^{n}}{n!}=\left(\frac{2t}{e^{t}+1}\right)^{\alpha}e^{xt}\text{
(see \cite[cite]{[\@@bibref{}{Jolany}{}{}]},
\cite[cite]{[\@@bibref{}{Rim}{}{}]}).}$

We now give a $q$-analogue of D. Miliĉic’s Lemma (see [22, page 1, Lemma
1.2.1]).

######  Lemma 2.9.

Let $a_{n,q}$, $n\in\mathbb{N}^{\ast}:=\left\\{0,1,2,3,...\right\\}$, be
complex numbers such that $\sum_{n=0}^{\infty}\left|a_{n,q}\right|$ converges.
Let

$\lambda=\left\\{-n\mid n\in\mathbb{N}^{\ast}\text{ and }a_{n,q}\neq
0\right\\}\text{.}$

Then,

$g_{q}\left(z\right)=\sum_{n=0}^{\infty}\frac{a_{n,q}}{\left[z+n\right]_{q}}$

converges absolutely for $z\in\mathbb{C}-\lambda$ and uniformly on bounded
subsets of $\mathbb{C}-\lambda$. The $q$-function is a meromorphic function on
complex plane with simple poles at the points in $\lambda$ and
Res$\left(g_{q},-n\right)=a_{n,q}$ for any $-n\in\lambda$.

###### Proof.

By using similar method in lecture notes of D. Miliĉic in [22], it is clear
that if $\left|\left[z\right]_{q}\right|<R$, we see

$\left|\left[z+n\right]_{q}\right|=\left|\left[z\right]_{q}+q^{zn}\left[n\right]_{q}\right|\geq\left|\left[n\right]_{q}-R\right|$

for all $\frac{1-q^{n}}{1-q}>R$. Then, we get

$\left|\frac{1}{\left[z+n\right]_{q}}\right|\leq\frac{1}{\left[n\right]_{q}-R}$

for $\left|\left[z\right]_{q}\right|<R$ and $\left[n\right]_{q}>R$. It follows
that $\left[n_{0}\right]_{q}>R$, we have

$\left|\sum_{n=n_{0}}\frac{a_{n,q}}{\left[z+n\right]_{q}}\right|\leq\sum_{n=0}^{\infty}\frac{\left|a_{n,q}\right|}{\left|\left[z+n\right]_{q}\right|}\leq\sum_{n=0}^{\infty}\frac{\left|a_{n,q}\right|}{\left[n\right]_{q}-R}\leq\frac{1}{\left[n_{0}\right]_{q}-R}\sum_{n=n_{0}}^{\infty}\left|a_{n,q}\right|$

Hence, the series $\sum_{n>R}\frac{a_{n,q}}{\left[z+n\right]_{q}}$ converges
absolutely and uniformly on the disk $\left\\{z\mid\left|z\right|<R\right\\}$
and defines there a meromorphic function. It follows that

$\sum_{n=0}^{\infty}\frac{a_{n,q}}{\left[z+n\right]_{q}}$

is a meromorphic function on that disk with the simple poles at the points of
$\lambda$ in $\left\\{z\mid\left|z\right|<R\right\\}$. Then, for any
$-n\in\lambda$, we have

$g_{q}\left(z\right)=\frac{a_{n,q}}{\left[z+n\right]_{q}}+\sum_{-j\in\lambda-\left\\{n\right\\}}^{\infty}\frac{a_{j,q}}{\left[z+j\right]_{q}}=\frac{a_{n,q}}{\left[z+n\right]_{q}}+\vartheta\left(z\right)$

where $\vartheta\left(z\right)$ is holomorphic at $-n$. This also shows that

$Res\left(g_{q},-n\right)=a_{n,q}.$

Thus, we successfully complete the proof of lemma.

We now want to indicate that $\Gamma$ extends to a meromorphic function by
using Lemma 2. 9. That is, we discover the following

$\Gamma_{q}\left(z\right)=\int_{0}^{\infty}t^{z-1}E_{q}\left(-qt\right)d_{q}t=\int_{0}^{1}t^{z-1}E_{q}\left(-qt\right)d_{q}t+\int_{1}^{\infty}t^{z-1}E_{q}\left(-qt\right)d_{q}t\text{.}$

Then, the second integral converges for any complex $z$ and represents an
entire function. On the other hand, the $q$-exponential function is entire,
and we have

$\displaystyle\int_{0}^{1}t^{z-1}E_{q}\left(-qt\right)d_{q}t$ $\displaystyle=$
$\displaystyle\int_{0}^{1}t^{z-1}\left\\{\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}q^{\binom{j+1}{2}}}{\left[j\right]_{q}!}t^{j}\right\\}d_{q}t$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}q^{\binom{j+1}{2}}}{\left[j\right]_{q}!}\left\\{\int_{0}^{1}t^{z+j-1}d_{q}t\right\\}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}q^{\binom{j+1}{2}}}{\left[j\right]_{q}!}\frac{1}{\left[z+j\right]_{q}}$

for any $z\in\mathbb{C}$. Now also, we can write as follows:

$\Gamma_{q}\left(z\right)=\int_{1}^{\infty}t^{z-1}E_{q}\left(-qt\right)d_{q}t+\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}q^{\binom{j+1}{2}}}{\left[j\right]_{q}!}\frac{1}{\left[z+j\right]_{q}}$

for any $z$ in the right half plane. From Lemma 2. 9, the right hand-side of
the above identity defines a meromorphic function on the complex plane with
simple poles at $z=-j,$ $j\in\mathbb{N}^{\ast}$. Then, we have the following
theorem.

######  Theorem 2.10.

For any $j\in\mathbb{N}^{\ast}$, we derive the following

(23)
$\text{{Res}}\left(\Gamma_{q},-j\right)=\frac{\left(-1\right)^{j}q^{\binom{j+1}{2}}}{\left[j\right]_{q}!}\text{.}$

As $q\rightarrow 1$ into (23), we easily derive that

$\lim_{q\rightarrow
1}\text{{Res}}\left(\Gamma_{q},-j\right)=\frac{\left(-1\right)^{j}}{j!}$

which it is residue of Euler’s Gamma function (see [22]).

Now also, by applying $q$-Mellin Transformation to generating function of
$q$-Genocchi polynomials of higher order, then we compute as follows:

$\displaystyle\Im_{q}\left(z,x:\alpha\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma_{\frac{1}{q}}\left(z\right)}\int_{0}^{\infty}t^{z-\alpha-1}\left\\{\left(-1\right)^{\alpha}S_{q}\left(x,-t:\alpha\right)\right\\}d_{\frac{1}{q}}t$
$\displaystyle=$
$\displaystyle\sum_{l_{1},l_{2},...,l_{\alpha}=0}^{\infty}\left(-1\right)^{l_{1}+l_{2}+...+l_{\alpha}}\left\\{\frac{1}{\Gamma_{\frac{1}{q}}\left(z\right)}\int_{0}^{\infty}t^{z-1}E_{\frac{1}{q}}\left(-\frac{t}{q}\left(qx+q\sum_{k=1}^{\alpha}l_{k}\right)\right)d_{\frac{1}{q}}t\right\\}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}^{\alpha}\sum_{l_{1},l_{2},...,l_{\alpha}=0}^{\infty}\frac{q^{-z}\left(-1\right)^{l_{1}+l_{2}+...+l_{\alpha}}}{\left(l_{1}+l_{2}+...+l_{\alpha}+x\right)^{z}}\text{.}$

So, we now introduce definition of $q$-Hurwitz-Zeta type function as follows:

######  Definition 1.

For any $z\in\mathbb{C}$, then we define

$\Im_{q}\left(z,x:\alpha\right)=\left[2\right]_{q}^{\alpha}\sum_{l_{1},l_{2},...,l_{\alpha}=0}^{\infty}\frac{\left(-1\right)^{l_{1}+l_{2}+...+l_{\alpha}}}{\left(qx+q\sum_{k=1}^{\alpha}l_{k}\right)^{z}}\text{.}$

Via the above definition, we derive interpolation functions for $q$-Genocchi
numbers and polynomials of higher order at negative integers with the
following theorem.

######  Theorem 2.11.

The following equality holds:

$\Im_{q}\left(-n,x:\alpha\right)=\frac{q^{-n}G_{n+\alpha,q}^{\left(\alpha\right)}\left(x\right)}{\left[\alpha\right]_{q}!\binom{n+\alpha}{\alpha}_{q}}\text{.}$

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