# Quantum Griffiths singularity in three-dimensional MoTiN
superconducting films

Zi-Xiao Wang  Tianjin Key Laboratory of Low Dimensional Materials Physics and
Preparing Technology, Department of Physics, Tianjin University, Tianjin
300354, China  Tian-Yu Jing  Tianjin Key Laboratory of Low Dimensional
Materials Physics and Preparing Technology, Department of Physics, Tianjin
University, Tianjin 300354, China  Zi-Yan Han  Tianjin Key Laboratory of Low
Dimensional Materials Physics and Preparing Technology, Department of Physics,
Tianjin University, Tianjin 300354, China  Kuang-Hong Gao  Tianjin Key
Laboratory of Low Dimensional Materials Physics and Preparing Technology,
Department of Physics, Tianjin University, Tianjin 300354, China  Song-Ci Li
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing
Technology, Department of Physics, Tianjin University, Tianjin 300354, China
Zhi-Qing Li<EMAIL_ADDRESS>Tianjin Key Laboratory of Low Dimensional
Materials Physics and Preparing Technology, Department of Physics, Tianjin
University, Tianjin 300354, China

###### Abstract

Quantum Griffiths singularity (QGS) has been experimentally observed in a
range of two-dimensional (2D) superconducting systems. Although it is
theoretically suggested that the QGS also exists in three-dimensional (3D)
superconductors, there is almost no experimental support to the theoretical
prediction. In the present paper, we observe the occurrence of QGS in a series
of $\sim$80-nm-thick Mo0.8Ti0.2Nx ($0.84\lesssim x\lesssim 1.12$)
superconducting films near the field-driven superconductor-metal transition
(SMT). These films have a NaCl structure and are 3D with respect to the
superconductivity. For each film, the low-temperature magnetoresistance
isotherms, measured at magnetic fields being perpendicular or parallel to the
film plane, do not cross at a single point but in a wide region. The dynamical
critical exponents $z\nu_{\perp}$ (for perpendicular field) and
$z\nu_{\parallel}$ (for parallel field) obtained by analyzing the related
magnetoresistance isotherms increase with decreasing temperature and tend to
diverge as $T\rightarrow 0$ K. In addition, the effective resistivity data for
the perpendicular and parallel field in the vicinity of the SMTs both obey an
activated scaling based on the random transverse-field Ising model. The QGS in
the 3D Mo0.8Ti0.2Nx superconducting films originates from the slow dynamics of
the rare regions in these systems. We also fabricate a $\sim$80-nm-thick
(Mo0.8Ti0.2)2N1.06 superconducting film with face-centered cubic structure at
low nitrogen partial pressure. It is found that the low-temperature
magnetoresistance isotherms for the perpendicular (parallel) field cross at a
single point and the resistivity data for the perpendicular (parallel) field
in the vicinity of the field-induced SMT obey the power-law scaling deduced
from the dirty-boson model. Our results provide unambiguous experimental
evidence for the existence of QGS in 3D superconductors.

##  I Introduction

Quantum Griffiths singularity (QGS) has been discovered in a range of two-
dimensional (2D) superconducting systems and attracted great attention over
the past decade [1, 2, 4, 3]. Conceptually, the QGS refers to the phenomenon
that certain thermodynamic observables of a system are singular not just at
criticality but in a finite region in the vicinity of the quantum critical
point. As for the 2D superconducting systems, the main characteristic of QGS
is that the low-temperature magnetoresistance isotherms do not cross at a
point near the critical field but at multiple points, and the dynamical
critical exponent $z\nu$ obtained at each crossing point diverges as
$T\rightarrow 0$ K [1, 2, 4, 3]. Experimentally, the existence of QGS was
initially reported in three-dimensional (3D) magnetic materials, including 3D
heavy fermion systems and $f$-electron ferromagnetic alloys [5, 6, 7, 8].
Until 2015, the QGS was observed in 3-monolayer Ga films, providing
experimental evidence for the presence of QGS in 2D superconducting system
[1]. Thereafter, QGS was observed in many 2D superconducting systems, such as
LaAlO3/SrTiO3 interface [2], monolayer NbSe2 films [3], ion-gated ZrNCl and
MoS2 [4]. More recently, the existence of QGS has also been confirmed in 2D
InOx [9], WSi [10], $\beta$-W [11], LixMoS2 [12], and NbN films [13, 14]. In
addition, it has been reported that in four-monolayer crystalline PbTe2 films
the QGS can emerge not only in perpendicular fields but also in parallel
magnetic fields [15]. According to previous reports [1, 2, 4, 3], quenched
disorder is the main origin of the QGS. On the other hand, the theoretical
results indicate that the QGS can occur in both 2D and 3D systems [16, 19, 17,
20, 18, 21, 22], and the QGS has been observed in 3D magnetic materials [5, 6,
7, 8]. However, there is almost no experimental evidence for the existence of
QGS in 3D superconducting system thus far [23, 24]. Therefore, it is desirable
to explore whether there is QGS in 3D superconductors.

MoN-based films with NaCl structure could be a suitable system to realize the
QGS in 3D superconducting films. In 1980s, it was predicted that the
superconducting transition temperature $T_{\rm c}$ of MoN with NaCl structure
(so called $B$1-MoN) could be as high as 29 K [25, 26]. However, the predicted
high transition temperature has not yet been achieved in $B$1-MoN probably due
to its structural instability (the existence of many Mo or nitrogen vacancies,
dislocations and other defects) [27, 28, 29, 30, 31, 32]. In addition, it has
been found that adding Ti to $B$1-MoN can stabilize its structure [33].
Therefore, to explore whether the $T_{\rm c}$ of Ti stabilized _B_ 1-MoN is
enhanced, we fabricated a series of $B$1-Mo0.8Ti0.2Nx films with different
nitrogen contents $x$. Although the superconducting transition temperature of
the MoTiN films were not significantly improved as expected, it is found that
3D MoTiN films possess relatively high $T_{\rm c}$ and low upper critical
magnetic field (see in the following text). These features, together with the
nature of easily formed structural defects, could make MoTiN films be a suited
system for exploring the characteristics of 3D QGS. In the present paper, the
low-temperature electrical transport properties of a series of $\sim$80-nm-
thick Mo0.8Ti0.2Nx films with $x$ ranging from $\sim$0.53 to $\sim$1.36 were
studied. The QGS is found emerging in the $0.84\lesssim x\lesssim 1.12$ films.
We present and discuss the interesting observations in the following
subsections.

##  II Experimental method

Our MoTiN films with thickness $t\sim 80$ nm were grown on (100) MgO single
crystal substrates by the reactive magnetron sputtering method. A MoTi alloy
target with Mo/Ti ratio of $8:2$ and purity of 99.9% was selected as the
sputtering source. The base pressure of the chamber was less than $1\times
10^{-4}$ Pa. During the deposition, the sputtering power was set as 300 W, and
the substrate temperature and sputtering pressure are kept at 500 ∘C and 0.15
Pa, respectively. The deposition was carried out in a mixture of argon
(99.999%) and nitrogen (99.999%) atmosphere. To obtain MoTiN films with
different superconducting transition temperatures, the volume ratio of
nitrogen to argon was controlled in each deposition round. For the films used
in this paper, the volume ratios were set as $1:9$, $1:7$, $3:17$, $7:33$,
$1:4$, and $1:3$ (corresponding to nitrogen partial pressures $P_{\rm
N_{2}}\simeq 10.0\%$, 12.5%, 15.0%, 17.5%, 20.0%, and 25.0%), respectively.

The thicknesses of the film was controlled by growth rate and deposition time,
and further determined by the high-resolution transmission electron microscopy
(HRTEM) of the cross section of the film. The crystal structure was determined
by X-ray diffraction (XRD), including normal $\theta$-2$\theta$, $\phi$, and
$\omega$ scans. The composition of the film and valence of each element were
measured using a x-ray photoelectron spectrometer (Thermo Scientific Escalab
250Xi). Before the x-ray photoelectron spectroscopy (XPS) measurement was
carried out, the surface of each film was etched by Ar ions and the corrosion
depth was $\sim$20 nm. The microstructure of the films was characterized by
transmission electron microscopy (TEM, Tecnai G2 F20 S-Twin). The longitudinal
and Hall resistance vs temperature and magnetic field was measured using the
standard four-probe method in a physical properties measurement system
(PPMS-6000, Quantum Design) equipped with a 3He refrigerator. Hall-bar-shaped
films (1.0 mm wide, 10.0 mm long, and 3.0 mm distance between the two
electrodes) defined by mechanical masks were used in the measurements. For the
longitudinal resistance vs magnetic field at a fixed temperature and
longitudinal resistance vs temperature at a fixed field measurements, the
field was applied in directions being perpendicular and parallel to the film
plane, respectively. In the latter case, the field was also perpendicular to
the excitation current.

Figure 1: (a) XRD $\theta$-2$\theta$ scan patterns of MoTiN films deposited at
different nitrogen partial pressures. (b) $\phi$-scan spectrum of (220) plane
for the film deposited at $P_{\rm N_{2}}\simeq 17.5\%$. The inset in (b) is
the rocking curve of the (200) diffraction peak of the $P_{\rm N_{2}}\simeq
17.5\%$ film. (c) Cross-sectional HRTEM micrograph of the film deposited at
$P_{\rm N_{2}}\simeq 17.5\%$. (d) The enlarged HRTEM image near the MoTiN/MgO
interface for (c). Figure 2: XPS results of the films deposited at $P_{\rm
N_{2}}\simeq 10.0\%$, 17.5%, and 25.0%. (a), (b), and (c) are the spectra of
Mo-3$d$, Ti-2$p$, and N-1$s$ orbits for the $P_{\rm N_{2}}\simeq 10.0\%$ film.
(d), (e), and (f) are the spectra of Mo-3$d$, Ti-2$p$, and N-1$s$ orbits for
the $P_{\rm N_{2}}\simeq 17.5\%$ film. (g), (h), and (i) are the spectra of
Mo-3$d$, Ti-2$p$, and N-1$s$ orbits for the $P_{\rm N_{2}}\simeq 25.0\%$ film.

##  III Results and Discussions

In this section, we will first discuss the structures, atomic valences, and
compositions of the MoTiN films. Then we will explore the fundamental
transport properties of the films. Finally, we will study the characteristics
of the quantum phase transition driven by magnetic field. In the following
discussions, we use $\perp$ ($\parallel$) as the superscript or subscript of a
physical quantity to represent that the quantity is related to the field
perpendicular (parallel) to the film plane.

###  III.1 Crystal structure and atomic valence

Figure 1(a) shows the XRD $\theta$-$2\theta$ scan patterns of the films
deposited at different nitrogen partial pressures. In the spectra, the strong
peak centered at $\sim$43.20∘ and the weak peaks at $\sim$38.78∘ are the
diffractions of MgO (200) plane. The former corresponds to Cu $K_{\alpha}$
radiation, and the latter is related to Cu $K_{\beta}$ radiation. For each
film, besides the (200) peaks of the MgO substrate, only the (200) diffraction
of the face-centered cubic (fcc) MoTiN can be observed. The values of the
lattice constant $a$ can be evaluated using the position of (200)
diffractions. For the film deposited at $P_{\rm N_{2}}\simeq 10.0\%$, the
value of $a$ is 4.191 Å, which is close to that of $\gamma$-Mo2N [27]. For
other films, the values of $a$ increase from 4.228 to $4.272\,{\rm\AA}$ as
$P_{\rm N_{2}}$ increases from 12.5% to 25.0%. These values of the lattice
constants are comparable with those of $B$1-MoN in previous reports [34, 25,
26]. Since the ionic radius of Ti3+ is quite close to that of Mo3+ [35], it is
deduced that MoTiN films with $B$1-structure could be formed under
$12.5\%\lesssim P_{\rm N_{2}}\lesssim 25.0\%$. Figure 1(b) shows the
$\phi$-scan profile of (220) plane of the film deposited at $P_{\rm
N_{2}}\simeq 17.5\%$. Four uniformly distributed diffraction peaks can be
clearly observed, indicating that the MoTiN film is epitaxially grown on the
MgO substrate. The inset of Fig. 1(b) shows the rocking curve of the (200)
diffraction peak of the $P_{\rm N_{2}}\simeq 17.5\%$ film. The full width at
half maximum (FWHM) of the peak is 0.041∘, demonstrating that the film has
high crystalline quality. For other films, the $\phi$-scan profiles and
rocking curves are similar to those of the $P_{\rm N_{2}}\simeq 17.5\%$ film,
and the FWHMs of the peaks in the rocking curves are less than 0.12∘. The
lattice constant and FWHM of the rocking curve for each film are listed in
Table 1. Figure 1(c) shows the cross-sectional HRTEM micrograph of the $P_{\rm
N_{2}}\simeq 17.5\%$ film along the [001] axis. The MoTiN/MgO interface and
the surface of the MoTiN film can be clearly identified from the figure. The
thickness of the film estimated from the TEM image is $\sim$77 nm, which is
roughly identical to that obtained by the growth rate and deposition time.
Figure 1(d) shows the enlarged HRTEM image near the MoTiN/MgO interface for
the same film. The spacing $d$ between adjacent (100) [or (010)] planes
obtained from HRTEM is 0.213 nm, which is consistent with the result from XRD.
Combining with the XRD results, one further confirms that the $\langle
100\rangle$-oriented MoTiN films are epitaxially grown on the (100) MgO
substrates.

Table 1:  Relevant parameters of the Mo0.8Ti0.2Nx films. Here $a$ is the lattice constant, FWHM is the full width at half maximum in the rocking curve, $\rho(300\,{\rm K})$ is the resistivity at 300 K, $T_{\rm c}$ is the superconducting transition temperature, $n(10\,{\rm K})$ is the carrier concentration at 10 K, $k_{\rm F}\ell$ is the Ioffe-Regel parameter at 10 K, $B_{\rm c2}(0)$ is the upper critical magnetic field at 0 K, and $\xi_{\rm GL}$ is the Ginzburg-Landau coherence length. The superscripts $\perp$ and $\parallel$ represent the related quantities obtained for perpendicular and parallel field, respectively.  Film | $P_{\rm N_{2}}$ | $a$ | FWHM | $x$ | $\rho(300\,{\rm K})$ | $T_{\rm c}$ | $n(10\,{\rm K})$ | $k_{\rm F}\ell$ | $B_{\rm c2}^{\perp}(0)$ | $B_{\rm c2}^{\parallel}(0)$ | $\xi_{\rm GL}^{\perp}$ | $\xi_{\rm GL}^{\parallel}$
---|---|---|---|---|---|---|---|---|---|---|---|---
No. | (%) | (${\rm\AA}$) | (degree) |  | (m$\Omega$ cm) | (K) | ($10^{23}\,{\rm cm}^{-3}$) |  | (T) | (T) | (nm) | (nm)
1 | 10.0 | 4.191 | 0.048 | 0.53 | 1.245 | 3.30 | 2.53 | 2.78 | 5.21 | 7.38 | 7.96 | 6.68
2 | 12.5 | 4.228 | 0.046 | 0.84 | 1.267 | 3.75 | 2.47 | 2.69 | 6.05 | 8.21 | 7.38 | 6.34
3 | 15.0 | 4.232 | 0.115 | 1.05 | 1.284 | 3.13 | 2.41 | 2.55 | 5.23 | 7.27 | 7.94 | 6.73
4 | 17.5 | 4.255 | 0.045 | 1.12 | 1.904 | 2.15 | 2.03 | 1.63 | 3.49 | 5.73 | 9.72 | 7.59
5 | 20.0 | 4.257 | 0.043 | 1.20 | 1.942 |  | 1.97 | 1.56 |  |  |  |
6 | 25.0 | 4.272 | 0.049 | 1.36 | 1.942 |  | 1.85 | 0.76 |  |  |  |

Figure 2 shows the XPS results of the film deposited at $P_{\rm N_{2}}\simeq
10.0\%$, 17.5%, and 25.0% as examples. Among them, Fig. 2(a), 2(b), and 2(c)
are the spectra for the $P_{\rm N_{2}}\simeq 10.0\%$ film, Fig. 2(d), 2(e),
and 2(f) are the spectra for the $P_{\rm N_{2}}\simeq 17.5\%$ film, and Fig.
2(g), 2(h), and 2(i) are the spectra for the $P_{\rm N_{2}}\simeq 25.0\%$
film. For the Mo 3$d$ spectra, the characteristic peaks can be deconvoluted
into two peaks, originating from Mo 3$d_{\rm 3/2}$ and Mo 3$d_{\rm 5/2}$ of
fcc (MoTi)Nx (Moδ+, $2<{\rm\delta}<4$), respectively [36, 37]. The Ti 2$p$
spectra can also be deconvoluted into two peaks, whose positions are close to
those in fcc TiNx [38]. For the film deposited at $P_{\rm N_{2}}\simeq
17.5\%$, the binding energy of the Mo 3$d_{\rm 3/2}$ peak is 231.20 eV, which
is nearly identical to that of $B$1-MoN [28]. As the nitrogen partial pressure
is enhanced from 12.5% to 25.0%, the binding energies of Mo 3$d_{\rm 3/2}$
peaks gradually increase from 230.85 eV to 231.60 eV, which is similar to the
results in previous reports [28, 37]. The continuous shift of the Mo 3$d_{\rm
3/2}$ peaks to the high binding energy can be attributed to the increase of
the charge transfer from Mo to N with the increase of the amount of nitrogen
atoms in (MoTi)Nx lattice. For the film deposited at $P_{\rm N_{2}}\simeq
10.0\%$, the Mo 3$d_{\rm 3/2}$ peak is located at 230.60 eV, which is
decreased about 0.25 eV compared with that of the $P_{\rm N_{2}}\simeq 12.5\%$
film. The relative large reduction in binding energy means a great increase of
nitrogen vacancies in the $P_{\rm N_{2}}\simeq 10.0\%$ film. The variation
trend of the position of Ti 2$p_{3/2}$ peak is similar to that of the Mo
3$d_{\rm 3/2}$, and the position of Ti 2$p_{3/2}$ peak increases from 454.64
eV to 455.35 eV as $P_{\rm N_{2}}$ is enhanced from 10.0% to 25.0%. As for the
N $1s$ spectra, the binding energy of the peak decreases with increasing
$P_{\rm N_{2}}$, which is also the result of increasing charge transfer from
Mo to N. From the areas of the XPS peaks, one can estimate the atomic ratios
of Mo, Ti, and N in the films. For the films deposited at $P_{\rm N_{2}}\simeq
10.0\%$, 12.5%, 15.0%, 17.5%, 20.0%, and 25.0%, the atomic ratios of Ti to the
sum of Mo and Ti are 17.9%, 19.9%, 18.3%, 18.2%, 17.8%, and 18.8%,
respectively, i.e., the atomic ratio of Ti to Mo in each film is close to that
of the target. In the following discussion, the atomic ratio of Mo to Ti is
approximately written as $8:2$ for simplicity. The atomic ratio of the metals
(Mo and Ti) to nitrogen, denoted by the subscript $x$ in (MoTi)Nx, is also
obtained, and listed in Table 1. Inspection of Table 1 indicates that $x$
increases from 0.53 to 1.36 as the nitrogen partial pressure is enhanced from
10.0% to 25.0%. The ratio of metal to nitrogen atoms for the film deposited at
$P_{\rm N_{2}}\simeq 10.0\%$ is close to $2:1$, and that for the film
deposited at $P_{\rm N_{2}}\simeq 15.0\%$ is approximately $1:1$. For the
molybdenum-nitrogen system, the fcc $\gamma$-Mo2Ny with $0.78\lesssim
y\lesssim 1.08$ is known as a stable phase [39], while $B$1-MoN is believed to
be a metastable phase [40]. The XRD and XPS results discussed above indicate
that MoTi-N system can not only form fcc $\gamma$-(Mo0.8Ti0.2)2N phase but
also be stabilized in $B$1-Mo0.8Ti0.2Nx phase in a wide range of $x$. Our
results are consistent with those obtained by Sanjinés _et al_ [33].

Figure 3:  Normalized resistivity as a function of temperature for the MoTiN
films deposited at different nitrogen partial pressures. Inset: Resistivity vs
temperature at low temperature regime for films deposited at $P_{\rm
N_{2}}\lesssim 20.0\%$.

###  III.2 Fundamental transport properties

Figure 3 shows the normalized resistivity $\rho/\rho(300\,{\rm K})$ as a
function of temperature $T$ for the films deposited at different nitrogen
partial pressures. The resistivities of all the films slightly increase with
decreasing temperature above liquid helium temperature, i.e., the temperature
coefficient of the resistivity $(1/\rho)(\mathrm{d}\rho/\mathrm{d}T)$ for each
film is negative at high temperature regime. The films deposited at $P_{\rm
N_{2}}\lesssim 17.5\%$ enter into superconducting state below the
superconducting transition temperature $T_{\rm c}$, where $T_{\rm c}$ is
designated as the temperature at which the resistance drops to 90% of that in
normal state [$\rho(10\,{\rm K})$, see the inset of Fig. 3]. For the
$B$1-Mo0.8Ti0.2Nx films, the superconducting transition temperature increases
with decreasing nitrogen partial pressure, while the $T_{c}$ of the $P_{\rm
N_{2}}\simeq 10.0\%$ [$\gamma$-(Mo0.8Ti0.2)2N1.06] film is less than that of
the $P_{\rm N_{2}}\simeq 12.5\%$ [$B$1-Mo0.8Ti0.2N0.84] film. Using free-
electron model, one can obtain the product of the Fermi wave number and mean
free path of electrons $k_{\rm F}\ell$ via $k_{\rm
F}\ell=(\hbar/e^{2})(3\pi^{2})^{1/3}n^{-2/3}/\rho$, where $\hbar$ is the
Planck’s constant divided by $2\pi$, $e$ is the electronic charge, and $n$ is
the carrier concentration. The values of $k_{\rm F}\ell$ for the films at 10 K
are summarized in Table 1. Inspection of Table 1 indicates that the $k_{\rm
F}\ell$ is 0.76 for the $P_{\rm N_{2}}\simeq 25.0\%$ film, and varies between
1.56 and 2.78 for other films. According to the Ioffe-Regel criterion [41],
the $P_{\rm N_{2}}\simeq 25.0\%$ film just lies in the metal-insulator
transition region, while the other films are on the metal side of the metal-
insulator transition and in a weakly localized metallic state.

Figure 4: (a) The perpendicular upper critical magnetic field $B_{\rm
c2}^{\perp}$ vs temperature $T$ for the films deposited at $P_{\rm
N_{2}}\lesssim 17.5\%$. (b) The parallel upper critical magnetic field $B_{\rm
c2}^{\parallel}$ vs temperature $T$ for the films deposited at $P_{\rm
N_{2}}\lesssim 17.5\%$. The dashed curves are the least-squares fits to
$B_{\rm c2}^{i}(T)=B_{\rm c2}^{i}(0)[1-(T/T_{\rm c})^{2}]$ with $i=\perp$ in
(a) and $i=\parallel$ in (b).

Figure 4(a) and 4(b) show the temperature dependence of the perpendicular and
parallel upper critical fields for the $P_{\rm N_{2}}\lesssim 17.5\%$ films.
The experimental $B_{\rm c2}^{\perp}$-$T$ and $B_{\rm c2}^{\parallel}$-$T$
data are fitted to $B_{\rm c2}^{i}(T)=B_{\rm c2}^{i}(0)[1-(T/T_{\rm c})^{2}]$
[42], where $i=\perp$ and $\parallel$ stand for the perpendicular-field and
parallel-field situations, respectively, and $B_{\rm c2}^{i}(0)$ is the upper
critical magnetic field at 0 K. The fitted results are shown in Fig. 4 by the
dashed curves. The values of the adjusting parameters $B_{\rm c2}^{\perp}(0)$
and $B_{\rm c2}^{\parallel}(0)$ are obtained and listed in Table 1. Inspection
of Table 1 indicates that $B_{\rm c2}^{\parallel}(0)$ is only slightly greater
than $B_{\rm c2}^{\perp}(0)$ for each film, suggesting the films are 3D with
respect to superconductivity. Once the values of $B_{\rm c2}^{\perp}(0)$ and
$B_{\rm c2}^{\parallel}(0)$ are obtained, one can deduce the Ginzburg-Landau
(GL) coherence lengths $\xi_{\rm GL}^{\perp}$ and $\xi_{\rm GL}^{\parallel}$
for each film via $\xi^{i}_{\rm GL}=[\Phi_{\rm 0}/(2\pi B_{\rm
c2}^{i}(0))]^{1/2}$ [42], where $\Phi_{0}=h/2e$ is the flux quantum. The GL
coherence lengths $\xi^{\perp}_{\rm GL}$ and $\xi^{\parallel}_{\rm GL}$ for
each superconducting film are also listed in Table 1. Clearly, the coherence
length perpendicular to the film plane $\xi_{\rm GL}^{\perp}$ lies between
$\sim$7 and $\sim$10 nm, while the coherence length parallel to the film plane
$\xi_{\rm GL}^{\parallel}$ varies between $\sim$6 and $\sim$8 nm. For each
film, $\xi_{\rm GL}^{\perp}$ is comparable with $\xi_{\rm GL}^{\parallel}$ and
much less than the thickness of the film, which further confirms the 3D
superconductive characteristics of the MoTiN films.

Figure 5: (a) The resistivity as a function of temperature in different fields
(perpendicular to the film plane) for the film deposited at $P_{\rm
N_{2}}\simeq 15.0\%$. (b) The resistivity as a function of temperature in
different fields (parallel to the film plane) for the film deposited at
$P_{\rm N_{2}}\simeq 15.0\%$. The insets in (a) and (b) are the enlarged views
of the corresponding films near the superconducting transition region.

###  III.3 Quantum phase transition

In this subsection, we will focus on the quantum phase transitions in these 3D
superconducting films deposited at $P_{\rm N_{2}}\lesssim 17.5\%$. We first
explore the temperature dependent behavior of the resistivity evolution with
external field. Considering the results for the $P_{\rm N_{2}}\simeq 17.5\%$,
15.0%, 12.5%, and 10.0% films are similar, we only present and discuss the
results for a representative film deposited at $P_{\rm N_{2}}\simeq 15.0\%$.
Figure 5(a) shows the resistivity variation with temperature from 7.00 down to
0.50 K under different perpendicular fields for the $P_{\rm N_{2}}\simeq
15.0\%$ film. The inset of this figure is the close view of the
superconducting transition region. When the field is less than $\sim$6 T, the
film transforms from normal state to superconducting state upon cooling and
the superconducting transition temperature decreases with increasing field. As
the magnetic field is increased to 7 T, the resistivity slightly increases
with decreasing temperature over the whole measured temperature range.
Considering the $k_{\rm F}\ell$ of the film is 2.55 at 10 K, one can deduce
that the film transforms into a weakly localized metallic state under 7 T.
Inspection of the inset of Fig. 5(a) also indicates that the critical filed
for superconductor-metal transition (SMT) is in the vicinity of $\sim$6.0 T.
Figure 5(b) shows the resistivity vs temperature at different parallel fields
for the same film. Similar to the situation at perpendicular field, the
parallel field also drives the film from a superconducting state to a weakly
localized metallic state. The parallel critical filed for SMT lies in the
vicinity of $\sim$8.0 T, being slightly larger than the perpendicular critical
filed.

For field induced SMT or superconductor-insulator transition (SIT), the dirty-
boson model predicts that the field dependence of resistance curves at
different temperatures all cross at a single point with $B=B_{\rm c}$ ($B_{\rm
c}$ is the critical field) [43, 44, 45]. In addition, near the SMT (SIT) the
resistance at different fields and temperatures obeys a power-law scaling form
[47, 46]

$R(B,T)=R_{\rm c}f(\delta T^{-1/z\nu}),$ (1)

where $R_{\rm c}$ is the critical resistance,$f(x)$ is the scaling function
with $f(0)=1$, $\delta=|B-B_{\rm c}|$ is the distance from the critical field
$B_{\rm c}$, $\nu$ is the correlation length exponent, $z$ is the dynamical
critical exponent. Experimentally, the power-law scaling in Eq. (1) has been
observed in a range of 2D superconductors near the SIT or SMT. The values of
$z\nu$ were generally found to be $\approx$0.65 [48], $\approx$1.33 [43, 49],
and $\approx$2.33 [44, 45], which correspond to the universality class of
$(2+1)$D $XY$ model [50], classic percolation model [44], and quantum
percolation model [44], respectively. As mentioned in Sec. I, in some 2D
superconducting films, the magnetoresistance isotherms do not intersect at a
single point or narrow region, but rather in a wide region, indicating the
presence of QGS accompanying the SMT.

Figure 6: (a) Resistivity versus perpendicular field at different temperatures ranging from 0.50 to 1.80 K for the $P_{\rm N_{2}}\simeq 10.0\%$ film. (b) Normalized resistivity $\rho/\rho_{\rm c\perp}$ as a function of the scaling variable $|B_{\perp}-B_{\rm c\perp}|T^{-1/z\nu_{\perp}}$ measured at perpendicular field. (c) Resistivity versus parallel field at different temperatures ranging from 0.50 to 1.80 K for the $P_{\rm N_{2}}\simeq 10.0\%$ film. (d) Normalized resistivity $\rho/\rho_{\rm c\parallel}$ as a function of the scaling variable $|B_{\parallel}-B_{\rm c\parallel}|T^{-1/z\nu_{\parallel}}$ measured at parallel field. Table 2:  Relevant parameters of the Mo0.8Ti0.2Nx films deposited at $P_{\rm N_{2}}\simeq 12.5\%$, 15.0%, 17.5%. Here $C$ and $B_{\rm c}^{\ast}$ are the parameters in Eq. (3); $\tilde{B}_{\rm c}^{\ast}$, $\tilde{T}_{0}$, $u$, and $y$ are the characteristic field, characteristic temperature and parameters in Eq. (4). The subscripts $\perp$ and $\parallel$ represent the related quantities obtained for perpendicular and parallel field, respectively.  Film | $P_{\rm N_{2}}$ | $C_{\perp}$ | $B_{\rm c\perp}^{\ast}$ | $\tilde{B}_{\rm c\perp}^{\ast}$ | $\tilde{T}_{0\perp}$ | $u_{\perp}$ | $y_{\perp}$ | $C_{\parallel}$ | $B_{\rm c\parallel}^{\ast}$ | $\tilde{B}_{\rm c\parallel}^{\ast}$ | $\tilde{T}_{0\parallel}$ | $u_{\parallel}$ | $y_{\parallel}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
No. | (%) |  | (T) | (T) | (K) |  |  |  | (T) | (T) | (K) |  |
2 | 12.5 | 0.318 | 7.341 | 7.469 | 5.24 | 0.084 | 0.02 | 0.379 | 9.046 | 9.279 | 4.725 | 0.096 | 0.03
3 | 15.0 | 0.284 | 6.170 | 6.155 | 3.446 | 0.0256 | 0.008 | 0.434 | 7.962 | 8.075 | 3.314 | 0.063 | 0.012
4 | 17.5 | 0.291 | 4.595 | 4.682 | 2.703 | 0.064 | 0.16 | 0.482 | 6.287 | 6.499 | 3.606 | 0.205 | 0.24

To explore whether the QGS could occur in 3D superconductors, we measured the
magnetoresistance isotherms of the MoTiN films deposited at $P_{\rm
N_{2}}\lesssim 17.5\%$. Figure 6(a) shows the low-temperature
magnetoresistance isotherms of the (Mo0.8Ti0.2)2N1.06 film (i.e., the film
deposited at $P_{\rm N_{2}}\simeq 10.0\%$) measured at fields perpendicular to
the film plane. Clearly, the $\rho$-$B_{\perp}$ curves measured at different
temperatures almost cross at a single point. The field and resistivity
corresponding to the crossing point are $B_{\rm c\perp}\approx 6.04$ T and
$\rho_{\rm c\perp}\approx 1.423$ m$\Omega$ cm. Using the values of the
$\rho_{\rm c\perp}$ and $B_{\rm c\perp}$, we compare the
$\rho(B_{\perp},T)/\rho_{\rm c\perp}$ data with Eq. (1). As shown in Fig.
6(b), the $\rho/\rho_{\rm c\perp}$ vs $|B_{\perp}-B_{\rm
c\perp}|T^{-1/z\nu_{\perp}}$ collapse onto a single curve (two branches) as
$z\nu_{\perp}$ is taken the appropriate value. The optimized value of
$z\nu_{\perp}$ for the best collapse is $z\nu_{\perp}=0.84\pm 0.05$ using a
numerical minimization procedure. For the parallel field case, the low-
temperature magnetoresistance isotherms and related scaling curve are shown in
Fig. 6(c) and 6(d), respectively. Clearly, the results are similar to those
measured at the perpendicular field: the low-temperature magnetoresistance
isotherms cross at a single point and the $\rho(B_{\parallel},T)/\rho_{\rm
c\parallel}$ data obey the power-law scaling form of Eq. (1). Here the
optimized scaling parameter $z\nu_{\parallel}$ is $z\nu_{\parallel}=0.74\pm
0.02$, and the critical resistivity and field are $\rho_{\rm
c\parallel}\approx 1.456$ m$\Omega$ cm and $B_{\rm c\parallel}\approx 7.72$ T,
respectively. It should be noted that value of $B_{\rm c\parallel}$ is only
1.28 times as large as that of $B_{\rm c\perp}$, which also supports that the
(Mo0.8Ti0.2)2N1.06 film is 3D with respect to superconductivity. The features
of the magnetoresistance isotherms strongly suggest that the Cooper pairs in
the 3D (Mo0.8Ti0.2)2N1.06 film can also be described by a two-dimensional
boson system and the SMT in the film is caused by quantum fluctuations. In
fact, it is found that the $R(B,T)$ data of the 50-nm-thick Nb0.15Si0.85
superconducting films [51] and 3D BaPb1-xBixO3 ($0.24\leqslant x\leqslant
0.29$) superconductors [52] in the vicinity of the SIT can also be described
by Eq. (1).

For the films deposited at $P_{\rm N_{2}}\simeq 12.5\%$, 15.0%, 17.5%, the
features of the low-temperature magnetoresistance isotherms, which will be
presented below, are quite different from those of the (Mo0.8Ti0.2)2N1.06
film. The results for the three films deposited at $P_{\rm N_{2}}\simeq
12.5\%$, 15.0%, 17.5% are similar, hence, we only give and discuss the results
taken from the $P_{\rm N_{2}}\simeq 15.0\%$ film. We first explore what
happens when the field is perpendicular to the film plane. Figure 7(a) shows
$\rho$ as a function of $B_{\perp}$ at different temperatures from 0.50 to
3.10 K. Clearly, the magnetoresistance isotherms at low temperatures do not
cross at a single point or a narrow region, but cross at many points located
in a relatively large transition area. These crossing points form a continuous
boundary line of SMT. This phenomenon is very similar to that in 2D
superconductors with QGS [1, 2, 4, 3], and treated as a signature of QGS in 2D
superconductors. We analyze the magnetoresistance isotherms in a fashion
similar to those in 2D superconducting films with QGS. Assuming that
$\rho$-$B_{\perp}$ curves at three adjacent temperatures cross at one point,
we compare the $\rho$-$B_{\perp}$ data measured at three adjacent temperatures
with the transformed Eq. (1), $\rho(B,t)=\rho_{\rm c}f(\delta t)$ with $t$
being $t=(T/T_{0})^{-1/z\nu}$ and $T_{0}$ being the lowest temperature in each
group of $\rho$-$B_{\perp}$ data. Selecting the proper value of
$z\nu_{\perp}$, the $\rho$-$B_{\perp}$ data at the three adjacent temperatures
will be scaled onto two branches. Thus, the critical exponent $z\nu_{\perp}$
as a function of $B_{\rm c\perp}$, shown in Fig. 7(b), was obtained. In the
high temperature regime ($T\gtrsim 1.20$ K), $z\nu_{\perp}$ increases slowly
with increasing $B_{\rm c\perp}$, while in the low temperature regime
($T\lesssim 1.20$ K), $z\nu_{\perp}$ increases rapidly and tends to diverge as
$B_{\rm c\perp}\rightarrow B_{\rm c\perp}^{\ast}$, where $B_{\rm
c\perp}^{\ast}$ is the characteristic field. The $z\nu_{\perp}$ vs $B_{\rm
c\perp}$ data are compared with the equation [20]

$z\nu=C|B_{\rm c}-B_{\rm c}^{\ast}|^{-\upsilon\psi},$ (2)

where $C$ is a constant, the correlation length exponent $\upsilon=1.2$, and
the tunneling exponent $\psi=0.5$ [18, 19]. As shown by the solid curve in
Fig. 7(b), the $z\nu_{\perp}$ vs $B_{\rm c\perp}$ data of the film satisfy the
activated scaling described by Eq. (2). For the $P_{\rm N_{2}}\simeq 15.0\%$
film, the fitted value of $B_{\rm c\perp}^{\ast}$ is 6.170 T. The values of
$B_{\rm c\perp}^{\ast}$ for the other two films are listed in Table 2. We note
in passing that the theoretical curve of Eq. (2) would deviate from the
$z\nu_{\perp}$ vs $B_{\rm c\perp}$ data for each film if we set
$\upsilon\psi\simeq 0.33$, which is different from that in Ref. [24].

Figure 7: (a) Resistivity vs perpendicular field at different temperatures
ranging from 0.50 to 3.10 K for the film deposited at $P_{\rm N_{2}}\simeq
15.0\%$. (b) $B_{\rm c\perp}$ dependence of critical exponent $z\nu_{\perp}$.
The dashed vertical and horizontal lines represent the trajectories of $B_{\rm
c\perp}=B_{\rm c\perp}^{\ast}$ and $z\nu_{\perp}=1$, respectively. (c)
Temperature dependence of perpendicular critical field (solid triangles) and
$T^{\rm on}_{\rm c}$ at different fields (solid squares). The solid curve is
the least-square fit to Eq. (5) and the dashed curve is only the guide to
eyes. (d) The close view of the low temperature and high field region of (c).

Figure 7(c) shows the temperature dependence of perpendicular critical
magnetic field for the $P_{\rm N_{2}}\simeq 15.0\%$ film. Here the solid
triangles represent the critical field $B_{\rm c\perp}(T)$ at each crossing
point of two adjacent magnetoresistance isotherms, and the solid squares are
the superconducting onset temperature $T_{\rm c}^{\rm on}(B_{\perp})$ obtained
from the $\rho$-$T$ curves (the temperature for $\mathrm{d}\rho/\mathrm{d}T=0$
near the superconducting transition). It should be noted that the $T_{\rm
c}^{\rm on}(B_{\perp})$ data and the $B_{\rm c\perp}(T)$ data almost follow
the same trajectory above $\sim$1.10 K [see the dashed line in Fig. 7(c)],
which is also similar to that for 2D superconductors with QGS [1, 2, 3, 4].
The value of $B_{\rm c\perp}(T)$ gradually increases with decreasing
temperature from 3.10 to $\sim$1.10 K, and then rapidly increases with further
decreasing temperature [see Fig. 7(d)], i.e., the $B_{\rm c\perp}$ vs $T$ data
deviates from its original trajectory below $\sim$1.10 K. In 2D
superconductors with QGS, the temperature dependence of $B_{\rm c}$ in lower
temperature regime also deviates from its higher temperature trajectory, which
is considered as another signature for the occurrence of QGS [1]. The
similarities of the low-temperature magnetoresistance isotherms and the
temperature dependent behavior of $B_{\rm c}$ between the 3D Mo0.8Ti0.2Nx
superconducting films and 2D superconductors with QGS, together with the fact
that the $z\nu_{\perp}$ vs $B_{\rm c\perp}$ of the Mo0.8Ti0.2Nx films obey the
activated scaling law, strongly suggest the occurrence of QGS in the 3D
Mo0.8Ti0.2Nx films.

Figure 8: (a) The resistivity versus the parallel field at different
temperatures from 0.50 to 3.10 K for the $P_{\rm N_{2}}\simeq 15.0\%$ film.
(b) $B_{\rm c\parallel}$ dependence of critical exponent $z\nu_{\parallel}$
for the $P_{\rm N_{2}}\simeq 15.0\%$ film. The dashed vertical and horizontal
lines stand for the trajectories of $B_{\rm c\parallel}=B_{\rm
c\parallel}^{\ast}$ and $z\nu_{\parallel}=1$, respectively.

Figure 8(a) shows the magnetoresistance isotherms measured at fields being
parallel to the film plane for the $P_{\rm N_{2}}\simeq 15.0\%$ film. Similar
to those in the perpendicular fields, the low-temperature
$\rho$-$B_{\parallel}$ curves at different temperatures do not cross at a
single point or a narrow region, but at a wide range in which the field spans
from $\sim 6.7$ to $\sim 7.9$ T as the temperature decreases from 3.10 to 0.50
K. The features of the magnetoresistance isotherms strongly suggest that the
QGS may occur in the film as subjected to a parallel field. To verify the
speculation, we analyze the magnetoresistance isotherms measured at parallel
field, just like analyzing that for the perpendicular field mentioned above.
Then, $z\nu_{\parallel}$ as a function of the critical field $B_{\rm
c\parallel}$ is obtained, and shown in Fig. 8(b). From Fig. 8(b), one can see
that the variation of $z\nu_{\parallel}$ with $B_{\rm c\parallel}$ is quite
similar to that for $z\nu_{\perp}$ with $B_{\rm c\perp}$, i.e., the
$z\nu_{\parallel}$ increases with increasing $B_{\rm c\parallel}$ (or
decreasing temperature), and tends to diverge as $T\rightarrow 0$ K. The solid
curve in Fig. 8(b) is the theoretical prediction of Eq. (2), where the
optimized parameters $C_{\parallel}$ and $B_{\rm c\parallel}^{\ast}$ are 0.434
and 7.962 T, respectively. Inspection of Fig. 8(b) indicates that the
$z\nu_{\parallel}$ vs $B_{\rm c\parallel}$ data can also be well described by
Eq. (2), just like the situation in perpendicular fields mentioned above.
Thus, our results suggest that both the perpendicular and parallel fields can
drive the $12.5\%\lesssim P_{\rm N_{2}}\lesssim 17.5\%$ films into a QGS state
before entering into a weakly localized metallic state. Considering the values
of the characteristic fields $B_{\rm c\perp}^{\ast}$ and $B_{\rm
c\parallel}^{\ast}$ are comparable for each film, one can argue that the QGS
is observed in 3D MoTiN superconductors.

Figure 9: The activated scaling analysis with the irrelevant correction for
the $P_{\rm N_{2}}\simeq 15.0\%$ film. (a) Temperature dependence of the
irrelevant correction at different perpendicular fields. (b) The effective
resistivity $\tilde{\rho}$ vs the scaling variable
$\tilde{\delta}_{\perp}[\ln(\tilde{T}_{0\perp}/T)]^{1/\upsilon\psi}$. (c)
Temperature dependence of the irrelevant correction at different parallel
fields. (d) The effective resistivity $\tilde{\rho}$ vs the scaling variable
$\tilde{\delta}_{\parallel}[\ln(\tilde{T}_{0\parallel}/T)]^{1/\upsilon\psi}$.
Here $\upsilon\psi$ is taken as 0.6 for both the perpendicular and parallel
fields.

On the other hand, based on the random transverse field Ising model, Maestro
_et al_. [21] have proposed an activated scaling to describe the conductivity
of nanowires near SMT. Recently, it has been demonstrated that the activated
scaling can describe the variation of the resistance with the field and
temperature near the SMT for the 2D superconductors with QGS [9, 53].
According to Maestro _et al_. [21] and Lewellyn _et al_. [9], the activated
dynamics scaling can be written as

$R\left(\tilde{\delta},\ln\frac{\tilde{T}_{0}}{T}\right)=\Phi\left[\tilde{\delta}\left(\ln\frac{\tilde{T}_{0}}{T}\right)^{1/\upsilon\psi}\right],$
(3)

where $\tilde{\delta}=|B-\tilde{B}_{\rm c}^{\ast}|/\tilde{B}_{\rm c}^{\ast}$
is the relative distance from the critical field, $\tilde{B}_{\rm c}^{\ast}$
is the critical field for $T\rightarrow 0$ [54], $\upsilon$ is the correlation
length exponent, $\psi$ is the tunneling exponent, and $\tilde{T}_{0}$ is a
microscopic temperature scale associated with the quantum phase transition. In
Eq. (3), the influence of temperature was not considered. At nonzero
temperature, an irrelevant correction to the scaling should be included and
the scaling relation can be approximately expressed as [9, 53]

$\displaystyle R\left(\tilde{\delta},\ln\frac{\tilde{T}_{0}}{T},u\right)$
$\displaystyle\approx$
$\displaystyle\Phi\left[\tilde{\delta}\left(\ln\frac{\tilde{T}_{0}}{T}\right)^{\frac{1}{\upsilon\psi}}\right]$
(4) $\displaystyle+$ $\displaystyle
u\left(\ln\frac{\tilde{T}_{0}}{T}\right)^{-y}\Phi_{u}\left[\tilde{\delta}\left(\ln\frac{\tilde{T}_{0}}{T}\right)^{\frac{1}{\upsilon\psi}}\right],$

where $u$ is the leading irrelevant scaling variable, and $y>0$ is the
associated irrelevant exponent. Expanding Eq. (4) around $\tilde{\delta}=0$,
one can obtain

$B_{\rm c}(T)\approx\tilde{B}_{\rm
c}^{\ast}\left[1-u\left(\ln\frac{\tilde{T}_{0}}{T}\right)^{-\frac{1}{\upsilon\psi}-y}\right].$
(5)

Thus, defining the effective resistance $\tilde{R}$ as
$\tilde{R}=\Phi(\tilde{\delta}[\ln\tilde{T}_{0}/T]^{1/\upsilon\psi})$, one can
determine whether the activated scaling is satisfied by checking the
$\tilde{R}$ vs $\tilde{\delta}[\ln\tilde{T}_{0}/T]^{1/\upsilon\psi}$ curve.
Experimentally, it has been demonstrated that the low-temperature $\tilde{R}$
data at different temperatures and fields in the vicinity of $B_{\rm c}$ of
the 2D superconductors with QGS follow the prediction of Eq. (4) and collapse
onto a single curve (two different branches), which in turn is another
hallmark of the emergence of QGS.

To check whether the low-temperature magnetoresistance isotherms of the 3D
superconducting MoTiN films deposited at $12.5\%\lesssim P_{\rm N_{2}}\lesssim
17.5\%$ obey the activated scaling mentioned above, we compare the $B_{{\rm
c}i}$ vs $T$ and low-temperature $\rho(T,B_{i})$ data with Eq. (5) and Eq.
(4), respectively, where $i=\perp$ and $\parallel$. Firstly, the theoretical
predictions of Eq. (5) are fitted to the $B_{{\rm c}i}$ vs $T$ data of the
films by taking $\upsilon\psi=0.6$ [for the $P_{\rm N_{2}}\simeq 15.0\%$ film
in the perpendicular field, see the solid curve in Fig. 7(d) as an example].
Then the fitting parameters $B_{{\rm c}i}^{\ast}$, $\tilde{T}_{0i}$, $u_{i}$,
and $y_{i}$ can be determined (see Table 2). Secondly, fixing
$\tilde{\delta}_{i}[\ln\tilde{T}_{0i}/T]^{1/\upsilon\psi}$ at a certain value,
we obtain the value of $u_{i}\Phi_{u}$ in Eq. (4) via extracting the slope of
$\rho$ vs $(\ln\tilde{T}_{0i}/T)^{-y_{i}}$ of the films. Thus, the irrelevant
correction [second term on the right hand side of Eq. (4)] as a function of
temperature and the effective resistivity [the first term on the right hand
side of Eq. (4)] as a function of
$\tilde{\delta_{i}}[\ln\tilde{T}_{0i}/T]^{1/\upsilon\psi}$ for each film are
obtained, respectively. As an example, we present the temperature dependence
of the irrelevant correction $\Delta\rho_{\perp}$ [$\Delta\rho_{\parallel}$]
and the effective resistivity $\tilde{\rho}_{\perp}$
[$\tilde{\rho}_{\parallel}$] variation with the scaling parameter
$\tilde{\delta}_{\perp}(\ln\tilde{T}_{0\perp}/T)^{1/\upsilon\psi}$
[$\tilde{\delta}_{\parallel}(\ln\tilde{T}_{0\parallel}/T)^{1/\upsilon\psi}$]for
the $P_{\rm N_{2}}\simeq 15.0\%$ film in the temperature range of 0.50 to 1.10
K [1.20 K] in Fig. 9(a) [Fig. 9(c)] and 9(b) [9(d)], respectively. For this
film, $\Delta\rho_{\perp}$ ($\Delta\rho_{\parallel}$) is below 0.235 m$\Omega$
cm (0.106 m$\Omega$ cm), which is less than 15% (7%) of the resistivity at the
field and temperature regimes. In addition, Both $\Delta\rho_{\perp}$ and
$\Delta\rho_{\parallel}$ decrease with decreasing temperature, being
consistent with the theoretical predication. From Fig. 9(b) and 9(d) , one can
see that the $\tilde{\rho}_{i}$ vs
$\tilde{\delta}_{i}[\ln\tilde{T}_{0i}/T]^{1/\upsilon\psi}$ ($i=\perp$ and
$\parallel$) data collapse onto a single curve (two branches), satisfying the
activated scaling described by Eq. (4). Inspection of Table 2 indicates the
difference between $B_{{\rm c}i}^{\ast}$ [obtained from Eq. (2)] and
$\tilde{B}_{{\rm c}i}^{\ast}$ [obtained from Eq. (5)] is less than 3.4% for
each film. The results mentioned here confirm the emergence of QGS in these
$12.5\%\lesssim P_{\rm N_{2}}\lesssim 17.5\%$ films.

Theoretically, the QGS in 3D system is stable and originates from the quenched
disorder [19, 20, 21, 22], which is just like that in 2D system. For the _B_
1-type MoN film, it is reported that the fcc structure can be also stable when
a lot of vacancies, including both molybdenum and nitrogen vacancies, exist in
the fcc conventional cell [29, 55]. In the Mo0.8Ti0.2Nx films, $x$ only
represents the average atomic ratio of nitrogen to the sum of Mo and Ti and
the number of transition metal or nitrogen atoms in each fcc conventional cell
is generally less than 4 even if $x\simeq 1$, i.e., there are a lot of Mo (or
Ti) and nitrogen vacancies in the _B_ 1-Mo0.8Ti0.2Nx films. These vacancies
are the main source of the quenched disorder. At low temperature and high
field (in the vicinity of the critical point of the quantum phase transition),
the quenched disorder drives the films into an inhomogeneous superconducting
state, in which the superconducting islands or droplets (rare regions) coexist
with the disordered bulk of the system. The dynamics of the rare regions is
very slow and follows the activated scaling rather than the power-law
dynamical scaling. Thus, the sharp transition of the SMT is smeared and QGS
then emerges in these $12.5\%\lesssim P_{\rm N_{2}}\lesssim 17.5\%$
Mo0.8Ti0.2Nx films. For the fcc (Mo0.8Ti0.2)2N1.06 film, the low-temperature
magnetoresistance isotherms cross at a single point and obey the power-law
dynamical scaling. The $k_{\rm F}\ell$ of the (Mo0.8Ti0.2)2N1.06 film is the
largest among all the films, which means the disorder strength of this film is
the weakest. According to Vojta [56], the effects of quenched disorder on the
critical point can be classified into three classes. (1) The average disorder
strength decreases with increasing length scale, and the system tends to
homogeneous at large length scales. As a result, the influence of the quenched
disorder on the critical point becomes unimportant and can be neglected. (2)
The system is inhomogeneous at all length scales, but the relative strength of
inhomogeneities reaches to a finite value for large length scales. In this
system, the critical point still exhibits conventional power-law scaling.
However, the critical exponents become different from those of the clean
system. (3) The relative strength of inhomogeneities increases with increasing
length scale at any length scale. At these infinite-randomness critical
points, the activated scaling instead of the power-law scaling is satisfied.
As mentioned above, the fcc Mo2N is a stable phase [39] while the _B_ 1-MoN is
a metastable phase [40]. Thus, the strength of the inhomogeneities in the fcc
(Mo0.8Ti0.2)2N1.06 film could be asymptotically unimportant or approach a
finite value for large length scale. As a result, the critical point maintains
the power-law scaling on the whole.

##  IV Conclusion

A series of $\sim$80-nm-thick epitaxial MoTiN films with NaCl-type structure
were grown on (100) MgO single crystal substrates by reactive sputtering
method in an Ar and N2 mixture atmosphere, and their low-temperature
electrical transport properties were systematically studied. The composition
of the $P_{\rm{N_{2}}}\simeq 10.0\%$ film is (Mo0.8Ti0.2)2N1.06, while the
compositions of the $P_{\rm{N_{2}}}\simeq 12.5\%$, $P_{\rm{N_{2}}}\simeq
15.0\%$, and $P_{\rm{N_{2}}}\simeq 17.5\%$ films are Mo0.8Ti0.2N0.84,
Mo0.8Ti0.2N1.05, and Mo0.8Ti0.2N1.12, respectively. All the films reveal
superconducting properties at low temperatures and are 3D with respect to
superconductivity. A magnetic field, which is either perpendicular or parallel
to the film plane, can drive the films transforming from superconducting to
weakly disordered metal states. For the (Mo0.8Ti0.2)2N1.06 film, the low-
temperature magnetoresistance isotherms cross at one single point and the
resistivities near the SMT can be described by a power-law scaling deduced
from the dirty boson model, which is applicable for both the perpendicular and
parallel field. For the Mo0.8Ti0.2Nx ($0.84\lesssim x\lesssim 1.12$) films,
both the critical exponents $z\nu_{\perp}$ and $z\nu_{\parallel}$ diverge as
the systems approach the quantum critical points. In addition, both the
perpendicular and parallel effective resistivities obey an activated scaling
law deduced in framework of the random transverse-field Ising model. Our
results provide strong experimental evidences for the existence of QGS in 3D
superconductors.

###### Acknowledgements.

The authors are grateful to Prof. Juhn-Jong Lin for valuable discussion. This
work is supported by the National Natural Science Foundation of China through
Grant No. 12174282.

## References

  * [1] Y. Xing, H. M. Zhang, H. L. Fu, H. Liu, Y. Sun, J. P. Peng, F. Wang, X. Lin, X. C. Ma, Q. K. Xue, J. Wang, and X. C. Xie, Quantum Griffiths singularity of superconductor-metal transition in Ga thin films, Science 350, 542-545 (2015).
  * [2] S. C. Shen, Y. Xing, P. J. Wang, H. W. Liu, H. L. Fu, Y. W. Zhang, L. He, X. C. Xie, X. Lin, J. C. Nie, and J. Wang, Observation of quantum Griffiths singularity and ferromagnetism at the superconducting LaAlO3/SrTiO3(110) interface, Phys. Rev. B 94, 144517 (2016).
  * [3] Y. Xing, K. Zhao, P. Shan, F. Zheng, Y. Zhang, H. Fu, Y. Liu, M. Tian, C. Xi, H. Liu, J. Feng, X. Lin, S. Ji, X. Chen, Q. K. Xue, and J. Wang, Ising Superconductivity and quantum phase transition in macro-size monolayer NbSe2, Nano. Lett. 17, 6802 (2017).
  * [4] Y. Saito, T. Nojima, and Y. Iwasa, Quantum phase transitions in highly crystalline two-dimensional superconductors, Nat. Commun. 9, 778 (2018).
  * [5] M. C. de Andrade, R. Chau, R. P. Dickey, N. R. Dilley, E. J. Freeman, D. A. Gajewski, M. B. Maple, R. Movshovich, A. H. Castro Neto, G. Castilla, and B. A. Jones, Evidence for a common physical description of non-Fermi-liquid behavior in chemically substituted f-electron systems, Phys. Rev. Lett. 81, 5620-5623 (1998).
  * [6] A. H. Castro Neto, G. Castilla, and B. A. Jones, Non-Fermi liquid behavior and Griffiths phase in f-electron compounds, Phys. Rev. Lett. 81, 3531–3534 (1998).
  * [7] A. Steppke, R. Küchler, S. Lausberg, E. Lengyel, L. Steinke, R. Borth, T. Lühmann, C. Krellner, M. Nicklas, C. Geibel, F. Steglich, and M. Brando, Ferromagnetic quantum critical point in the heavy-fermion metal YbNi4(P1-xAsx)2, Science 339, 933 (2013).
  * [8] S. Ubaid-Kassis, T. Vojta, and A. Schroeder, Quantum Griffiths phase in the weak itinerant ferromagnetic alloy Ni1-xVx, Phys. Rev. Lett. 104, 066402 (2010).
  * [9] N. A. Lewellyn, I. M. Percher, J. Nelson, J. Garcia-Barriocanal, I. Volotsenko, A. Frydman, T. Vojta, and A. M. Goldman, Infinite-randomness fixed point of the quantum superconductor-metal transitions in amorphous thin films, Phys. Rev. B 99, 054515 (2019).
  * [10] X. Zhang, A. E. Lita, H. Liu, V. B. Verma, Q. Zhou, S. W. Nam, and A. Schilling, Size dependent nature of the magnetic-field driven superconductor-to-insulator quantum-phase transitions, Commun. Phys. 4, 100 (2021).
  * [11] C. Huang, E. Zhang, Y. Zhang, J. Zhang, F. Xiu, H. Liu, X. Xie, L. Ai, Y. Yang, M. Zhao, J. Qi, L. Li, S. Liu, Z. Li, R. Zhan, Y. Q. Bie, X. Kou, S. Deng, and X. C. Xie, Observation of thickness-tuned universality class in superconducting $\beta$-W thin films, Sci. Bull. 66, 1830 (2021).
  * [12] I. A. Verzhbitskiy, D. Voiry, M. Chhowalla, and G. Eda, Disorder-driven two-dimensional quantum phase transitions in LixMoS2, 2D Materials 7, 035013 (2020).
  * [13] T. Jing, Z. Han, Z. He, M. Shao, P. Li, and Z. Li, Quantum phase transition in NbN superconducting thin films, Phys. Rev. B 107, 184515 (2023).
  * [14] X. Wang, L. Wang, Y. Liu, F. Chen, W. Gao, Y. Wu, Z. Xu, W. Peng, Z. Wang, Z. Di, W. Li, G. Mu, and Z. Lin, Robust quantum Griffiths singularity above 1.5 K in nitride thin films, Phys. Rev. B 107, 094509 (2023).
  * [15] Y. Liu, S. Qi, J. Fang, J. Sun, C. Liu, Y. Liu, J. Qi, Y. Xing, H. Liu, X. Lin, L. Wang, Q. Xue, X. Xie, and J. Wang, Observation of In-Plane Quantum Griffiths Singularity in Two-Dimensional Crystalline Superconductors, Phys. Rev. Lett. 127, 137001 (2021).
  * [16] R. B. Griffiths, Nonanalytic Behavior Above the Critical Point in a Random Ising Ferromagnet, Phys. Rev. Lett. 23, 17 (1969).
  * [17] C. Pich, A. P. Young, H. Rieger, and N. Kawashima, Critical Behavior and Griffiths-McCoy Singularities in the Two-Dimensional Random Quantum Ising Ferromagnet, Phys. Rev. Lett. 81, 5916 (1998).
  * [18] I. A. Kovács, and F. Iglói, Renormalization group study of the two-dimensional random transverse-field Ising model, Phys. Rev. B 82, 054437 (2010).
  * [19] O. Motrunich, S. Mau, D. A. Huse, and D. S. Fisher, Infinite-randomness quantum Ising critical fixed points, Phys. Rev. B 61, 1160 (2000).
  * [20] T. Vojta, A. Farquhar, and J. Mast, Infinite-randomness critical point in the two-dimensional disordered contact process, Phys. Rev. E 79, 011111 (2009).
  * [21] A. Del Maestro, B. Rosenow, J. A. Hoyos, and T. Vojta, Dynamical Conductivity at the Dirty Superconductor-Metal Quantum Phase Transition, Phys. Rev. Lett. 105, 145702 (2010).
  * [22] B. Fan, A. M. García-García, Superconductivity at the three-dimensional Anderson metal-insulator transition, Phys. Rev. B 102, 184507 (2020).
  * [23] When the first draft of this paper was finished, we found a preprint, in which the QGS was also observed in a spinel oxide MgTi2O4, was announced in arXiv (arXiv identifier 2311.06710). The sample MgTi2O4 used in the preprint is a strongly localized system and the its normal state is near the mobility edge of Aderson localization, while our samples (MoTiN films) belong to weakly localized system and reveal weakly localized metallic characteristics in normal state. In addition, the origin of QGS in our MoTiN films is different from that in MgTi2O4 film.
  * [24] S. Qi, Y. Liu, Z. Wang, F. Chen, Q. Li, H. Ji, R. Li, Y. Li, J. Fang, H. Liu, F. Wang, K. Jin, X. C. Xie, and J. Wang, Quantum Griffiths singularity in three-dimensional superconductor to Anderson critical insulator transition, arXiv: 2311.06710.
  * [25] W. E. Pickett, B. M. Klein, and D. A. Papaconstantopoulos, Theoretical prediction of MoN as a high $T_{\rm c}$ superconductor, Physica B+C 107, 667 (1981).
  * [26] Y. Zhao, and S. He, $B$1-type MoN, a possible high $T_{\rm c}$ superconductor, Solid State Commun. 45, 281 (1983).
  * [27] H. Ihara, Y. Kimura, K. Senzaki, H. Kezuka, and M. Hirabayashi, Electronic structures of $B$1 MoN, fcc Mo2N, and hexagonal MoN, Phys. Rev. B 31, 3177 (1985).
  * [28] K. Inumaru, K. Baba, and S. Yamanaka, Structural distortion and suppression of superconductivity in stoichiometric $B$1-MoN epitaxial thin films, Phys. Rev. B 73, 052504 (2006).
  * [29] B. D. Ozsdolay, K. Balasubramanian, and D. Gall, Cation and anion vacancies in cubic molybdenum nitride, J. Alloys Compd. 705, 631 (2017).
  * [30] D. A. Papaconstantopoulos, and W. E. Pickett, Effects of disorder on high-temperature superconductivity in cubic MoN, Phys. Rev. B 31, 7093 (1985).
  * [31] Y. H. Shi, B. R. Zhao, Y. Y. Zhao, L. Li, and J. R. Liu, Superconducting and normal-state properties of MoNx thin films, Phys. Rev. B 38, 4488 (1988).
  * [32] G. L. W. Hart, and B. M. Klein, Phonon and elastic instabilities in MoC and MoN, Phys. Rev. B 61, 3151 (2000).
  * [33] R. Sanjinés, C. Wiemer, J. Almeida, and F. Lévy, Valence band photoemission study of the Ti-Mo-N system, Thin Solid Films 290, 334 (1996).
  * [34] G. Linker, R. Smithey, and O. Meyer, Superconductivity in MoN films with NaCl structure, J. Phys. F: Met. Phys. 14, L115 (1984).
  * [35] R. D. Shannon, Revised Effective Ionic Radii and Systematic Studies of Interatomie Distances in Halides and Chaleogenides, Acta Cryst. A 32, 751 (1976).
  * [36] G. Kim, T. Park, H. Chung, Y. Kim, M. Kwon, and J. Choi, Growth and characterization of chloronitroaniline crystals for optical parametric oscillators: I. XPS study of Mo-based compounds, Appl. Surf. Sci. 152, 35 (1999).
  * [37] Y. Wang, and R. Y. Lin, Amorphous molybdenum nitride thin films prepared by reactive sputter deposition, Mater. Sci. Eng. B 112, 42 (2004).
  * [38] M. Delfino, J. A. Fair, and D. Hodul, X-ray photoemission spectra of reactively sputtered TiN, J. Appl. Phys. 71, 6079 (1992).
  * [39] M. B. Kanoun, S. Goumri-Said, and M. Jaouen, Structure and mechanical stability of molybdenum nitrides: A first-principles study, Phys. Rev. B 76, 134109 (2007).
  * [40] H. Jehn, P. Ettmayer, The molybdenum-nitrogen phase diagram, J. Less Common. Met. 58, 85-98 (1978).
  * [41] A. F. Ioffe, A. R. Regel, Non-crystalline, amorphous, and liquid electronic semiconductors, Prog. Semicond. 4, 237 (1960).
  * [42] M. Tinkham, _Introduction to Superconductivity_ , 2nd ed (Dover Publications, Inc., Mineola, New York, 1996).
  * [43] A. Yazdani and A. Kapitulnik, Superconducting-Insulating Transition in Two-Dimensional $\alpha$-MoGe Thin Films, Phys. Rev. Lett. 74, 3037 (1995).
  * [44] M. A. Steiner, N. P. Breznay, and A. Kapitulnik, Approach to a superconductor-to-Bose-insulator transition in disordered films, Phys. Rev. B 77, 212501 (2008).
  * [45] N. P. Breznay, M. A. Steiner, S. A. Kivelson, and A. Kapitulnik, Self-duality and a Hall-insulator phase near the superconductorto-insulator transition inindium-oxide films, Proc. Natl. Acad. Sci. USA 113, 280 (2016).
  * [46] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Boson localization and the superfluid-insulator transition, Phys. Rev. B 40, 546 (1989).
  * [47] M. P. A. Fisher, Quantum phase transitions in disordered two-dimensional superconductors, Phys. Rev. Lett. 65, 923 (1990).
  * [48] H. Aubin, C. A. Marrache-Kikuchi, A. Pourret, K. Behnia, L. Berge, L. Dumoulin, and J. Lesueur, Phys. Rev. B 73, 094521 (2006).
  * [49] N. Mason and A. Kapitulnik, Dissipation Effects on the Superconductor-Insulator Transition in 2D Superconductors, Phys. Rev. Lett. 82, 5341 (1999).
  * [50] S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Continuous quantum phase transitions, Rev. Mod. Phys. 69, 315 (1997).
  * [51] H. Aubin, C. A. Marrache-Kikuchi, A. Pourret, K. Behnia, L. Bergé, L. Dumoulin, and J. Lesueur, Magnetic-field-induced quantum superconductor-insulator transition in Nb0.15Si0.85, Phys. Rev. B 73, 094521 (2006).
  * [52] P. Giraldo-Gallo, H. Lee, Y. Zhang, M. J. Kramer, M. R. Beasley, T. H. Geballe, and I. R. Fisher, Field-tuned superconductor-insulator transition in BaPb1-xBixO3, Phys. Rev. B 85, 174503 (2012).
  * [53] Z. Cui, L. Pan, J. Fang, S. Qi, Y. Xing, H. Liu, Y. Liu, and J. Wang, The activated scaling behavior of quantum Griffiths singularity in two-dimensional superconductors, J. Phys. D: Appl. Phys. 56, 374002 (2023).
  * [54] We use $\tilde{B}_{c}^{\ast}$ here to distinguish the characteristic field $B_{c}^{\ast}$ in Eq. (2).
  * [55] Y. Xiao, Z. Han, G. Zhang, and Y. Liang, Understanding the Role of Atomic Vacancies in the Stability and Hardening of Cubic Tungsten and Molybdenum Nitrides, J. Phys. C 127, 22984-22991 (2023).
  * [56] T. Vojta, Rare region effects at classical, quantum and nonequilibrium phase transitions, J. Phys. A: Math. Gen. 39, R143 (2006).