# Crossover From Cooper-Pair Hopping to Single-Electron Hopping in
Pbx(TiO2)1-x Granular Films

Zhi-Hao He  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  Zi-Wu Wang  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

The electrical transport properties of Pbx(TiO2)1-x ($x$ being the Pb volume
fraction) granular films are investigated experimentally. The charging energy
of the Pb granules is reduced to less than the superconducting gap of Pb
granules for the low temperature insulating films by using high-$k$ dielectric
TiO2 as the insulating matrix. For the insulating films in the vicinity of the
superconductor-insulator transition, Cooper-pair hopping governs the low-
temperature hopping transport. For these films, the low-temperature
magnetoresistance is positive at low field and the resistivity vs temperature
for Cooper-pair hopping obeys an Efros-Shklovsii-type variable-range-hopping
law. A crossover from Cooper-pair-dominated hopping to single-electron-
dominated hopping is observed with decreasing $x$. The emergence of single-
electron-dominated hopping in the more insulating films is due to the
causation that the intergrain Josephson coupling becomes too weak for Cooper
pairs to hop between adjacent superconducting Pb granules.

Disordered superconductors have attracted great attention over the past few
decades since a range of novel phenomena, such as superconductor-insulator
transition in two-dimensional superconducting films [1, 2, 3, 4, 5, 6],
anomalous metallic state [7, 8, 9, 10, 11, 12], macroscopic quantum tunneling
[13, 14, 15, 16], and disorder-induced inhomogeneity (emergence of
superconducting islands or puddles) [17, 18, 19, 20], are related to the
interplay between the superconductivity and localization. Among the disordered
superconductors, granular superconductor is an important experimentally
accessible model system, which offer a unique testing ground for studying
tunable combined effects of disorder, Coulomb interactions, and
superconductivity [21]. The knowledge on these effects is important in
understanding the properties of high-temperature superconductors (the
disorder-induced granularity was evidenced in high-temperature
superconductors) [22, 23, 24, 25] as well as in understanding the behaviors of
devices related to Josephon junctions [26, 27, 28]. In granular
superconductors, the ground state of the system is ultimately determined by
the competition of Josephson coupling energy $E_{J}$ and charging energy
$E_{c}$. The granular superconducting arrays with $E_{J}<E_{c}$ are
insulators, while those with $E_{J}>E_{c}$ are superconductors at low
temperature [21]. In the insulating regime and below the local superconducting
transition temperature $T_{c}^{L}$, Cooper pairs are localized on the granules
with sufficiently large size by Coulomb repulsion. Thus an interesting
question is whether the electric transport below $T_{c}^{L}$ is dominated by
the hopping of Cooper pairs or electrons at the insulating regime.
Theoretically, it has been proposed that the current through a superconducting
grain is sensitive to the relation between the superconducting gap $\Delta$
and the charging energy $E_{c}$. For $E_{c}>\Delta$, the transport is governed
by single-electron hopping, while the hopping transport is mediated by Cooper
pairs in the opposite case $E_{c}<\Delta$ [29, 30, 31]. When a magnetic field
is applied, the Cooper-pair hopping should lead to a positive
magnetoresistance due to the suppression of the order parameter [4]. However,
the Cooper-pair hopping has not been observed experimentally in granular
superconductors as yet.

To satisfy the condition for Cooper-pair hopping, $E_{c}<\Delta$, one need to
choose a suitable superconductor-insulator granular system with relative small
$E_{c}$ and large $\Delta$. For a superconductor-insulator granular composite,
the charging energy of a grain can be estimated via $E_{c}\simeq
e^{2}/(4\pi\epsilon_{0}\tilde{\kappa}a)$, where $e$ is the electronic charge,
$\epsilon_{0}$ is the permittivity of free space, $\tilde{\kappa}$ is the
effective dielectric constant and proportional to the permittivity of the
insulator constituent $\epsilon_{r}$, and $a$ is the mean grain size [32].
Therefore, an effective way of reducing the charging energy is to select an
high-$k$ dielectric as the insulating matrix of the granular composites. In
this Letter, we successfully realized the Cooper-pair hopping transport and
observed the Efros-Shklovsii (ES)-type variable range hopping (VRH)
conductivity of Cooper pairs in the insulating regime (near the
superconductor-insulator transition) of Pbx(TiO2)1-x ($x$ being the Pb volume
fraction) granular films. A crossover from Cooper-pair hopping to single-
electron hopping is also observed as the films are extended to a more
insulating region. It is noted that the room-temperature permittivity of TiO2
is $\sim$10 ($\sim$20) times as large as that of Al2O3 (SiO2) [33], and the
zero-temperature superconducting gap of Pb is 1.37 meV [34], being larger than
the gaps of most superconducting metals.

The Pbx(TiO2)1-x films with $x$ ranging from $\sim$0.45 to $\sim$0.69 were
deposited on glass substrates at room temperature by co-sputtering Pb and TiO2
targets in Ar atmosphere. The details of the deposition procedures are similar
to those described in Ref. [35]. The thicknesses of the films ($\sim$400 to
$\sim$600 nm) were measured using a surface profiler (Dektak, 6M). The Pb
volume fraction $x$ in each film was obtained from the energy-dispersive x-ray
spectroscopy analysis (EDS; EDAX, model Apollo X). The microstructure of the
films was characterized by transmission electron microscopy (TEM; Tecnai G2
F20). The resistance as well as the current vs voltage were measured using the
standard four probe method and the temperature and magnetic field environments
were provided by a physical property measurement system (PPMS-6000, Quantum
Design). For films with large $x$, both the current source and the voltmeter
were provided by the model 6000 PPMS controller, while for the small-$x$
films, a Keithley 236 and a Keithley 2182A were used as current source and
voltmeter, respectively. The narrow rectangle shape films (1 mm$\times$10 mm),
defined by mechanical masks, were used for the transport measurement.

The TEM results indicate that the films are granular composites in which the
Pb granules are embedded in the amorphous TiO2 matrix. The mean size of Pb
granules decreases with decreasing $x$, e.g., the mean grain size is $\sim$6.1
nm for the $x\simeq 0.69$ film, decreases to $\sim$5.1 nm when $x\simeq 5.2$,
and further decreases to $\sim$4.3 nm as $x$ is reduced to 0.45. Since the
thickness of the film is at least 50 times larger than the mean size of Pb
granules, the Pb granules form three-dimensional granular arrays.

Figure 1: Resistivity as a function of temperature for the Pbx(TiO2)1-x films
($0.45\lesssim x\leq 0.69$) at zero magnetic field.

Figure 1 shows the temperature dependence of resistivity for the Pbx(TiO2)1-x
films from $\sim$15 down to $\sim$2 K. Clearly, global superconductivity
appears in the films with $x\gtrsim 0.59$ at low temperatures. For the
$x\simeq 0.56$ film, the resistivity slightly increases with decreasing
temperature initially, then sharply decreases about 6 orders of magnitude from
$\sim$7.0 to $\sim$2.4 K, and finally tends to be saturated with further
decreasing temperature. Similar dissipation effect of $\rho(T)$ was also
observed in Pbx(SiO2)1-x films with $0.60\lesssim x\lesssim 0.72$ and have
been explained by a model that includes both thermally activated phase slips
and quantum phase slips [35]. For the $x\simeq 0.55$ and 0.52 films, the
temperature dependence of resistivity exhibits a reentrant behavior, i.e., the
resistivity increases with decreasing temperature, starts to decreases with
decreasing temperature at $\sim$7.0 K, reaches to its minimum, and then
increases with further decreasing temperature. According to Parendo _et al._
[36], in granular films of amorphous bismuth the drop in resistivity is due to
the occurrence of superconductivity on the isolated grains below the
superconducting transition temperature of the grain, while the increase in
resistivity at low temperature is caused by the opening of the energy gap on
the superconducting grains coupled by quasiparticle tunneling (hopping). Then
if the charging energy $E_{c}$ is less than the superconducting gap $\Delta$,
the Cooper pairs do not have to decompose into electrons, and could conduct
directly among the grains via ES-VRH [31]. (For the Pbx(TiO2)1-x films, which
mechanism makes the resistivity rise at low temperature will be amply studied
below.) For the $x\lesssim 0.50$ films, the resistivity increases with
decreasing temperature and increases more rapidly below $\sim$7.0 K. We will
focus our attention on the four films with $x\lesssim 0.52$ in the following
discussion.

Table 1:  Relevant parameters for the Pbx(TiO2)1-x films with different $x$. Here $x$ is volume fraction of Pb, $\rho$N is the zero-field resistivity at 15 K, $a$ is mean diameter of Pb granules, $E_{c}$ is the average charging energy, $t$ is the film thickness, $\xi^{\rm{CPH}}$ is the Cooper-pair localization length for the ES hopping, $T_{0}$ is the characteristic temperature in Eq.(1). $x$ |  $\rho$N | $a$ | $E_{c}$ | $t$ | $T_{0}$ | $\xi^{\rm{CPH}}$
---|---|---|---|---|---|---
| ($\Omega$ cm) | (nm) | (meV) | (nm) | (K) | (nm)
0.52 | $5.18$ | 5.1 | 0.64 | 562 | 169.7 | 0.9
0.50 | $6.51$ | 4.8 | 0.72 | 501 | 97.4 | 1.7
0.47 | $5.35\times 10^{2}$ | 4.5 | 0.83 | 376 |  |
0.45 | $1.08\times 10^{3}$ | 4.3 | 0.91 | 405 |  |

Figure 2 (a) and (b) show the normalized resistivity $\rho(H)/\rho(0)$ as a
function of magnetic field $\mu_{0}H$ (with $\mu_{0}$ being the permeability
of free space) at different temperatures for the $x\simeq 0.52$ and 0.50
films, where $\rho(H)$ and $\rho(0)$ are the resistivities at $H$ and zero
field, respectively. Below $\sim$4 K, the resistivity increases with
increasing field at low field, peaks at $H_{\rm P}(T)$, and then decreases
with further increasing field above $H_{\rm P}(T)$. The field $H_{\rm P}(T)$
is sample dependent and decreases with increasing temperature for each film.
For a single Pb grain, the critical size for existing superconductivity is
$\sim$2.0 nm [37], thus the granules remain superconducting in our insulating
Pbx(TiO2)1-x films. The behaviors of low-temperature magnetoresistances
strongly suggest that the transport process is governed by Cooper-pair hopping
below $H_{\rm P}(T)$: the magnetic field can destroy the Josephson coupling
between the superconducting grains and suppress the amplitude of
superconducting order parameter in each individual grain; when the magnetic
field is relative small, the order parameter amplitude is almost unchanged,
while the intergrain Josephson coupling is gradually suppressed with
increasing field [4], which reduces the hopping probability of Cooper pairs
between the superconducing grains and thus leads to a positive
magnetoresistance. As the field is gradually close to $H_{\rm P}(T)$, the
order parameter amplitude is suppressed and the Cooper pairs start to be
broken into single electrons. This opens the channels for electron hopping,
and will slow down the enhancement in the resistivity. Denoting the Cooper-
pair hopping and electron hopping conductances as $G_{\rm CH}(H)$ and $G_{\rm
EH}(H)$, respectively [the total conductance of the film $G(H)$ is the sum of
$G_{\rm CH}(H)$ and $G_{\rm EH}(H)$], one can readily obtain that $G_{\rm
CH}(H)$ decreases with increasing $H$ whereas $G_{\rm EH}(H)$ increases with
increasing $H$ at a certain temperature. At $H_{\rm P}(T)$, $\partial G_{\rm
CH}/\partial H=-\partial G_{\rm EH}/\partial H$, then $\rho(H)$ reaches its
maximum; when the field is greater than $H_{\rm P}(T)$, $\partial G_{\rm
EH}/\partial H$ would be greater than $-\partial G_{\rm CH}/\partial H$, which
results in the reduction of resistivity with increasing field above $H_{\rm
P}(T)$. Since the Cooper pairs would become more stable with decreasing
temperature, the magnetoresistance peak will shift to a higher field with
decreasing temperature, which is what we see in Fig. 2 (a) and (b). These
observations strongly suggest that Cooper-pair hopping dominates the transport
process at the low field regime.

Figure 2:  The normalized resistivity $\rho(H)/\rho(0)$ vs magnetic field at
different temperatures for (a) $x\simeq 0.50$ and (b) $x\simeq 0.52$
Pbx(TiO2)1-x films, and (c) Pb0.50(SiO2)0.50 film. The field $H_{\rm P}(T)$ at
which the resistivity reaches the maximum is indicated by the arrow. (d)
Resistivity vs $1/T^{1/2}$ for the $x\simeq 0.50$ and $x\simeq 0.52$
Pbx(TiO2)1-x films at zero field. The inset shows the zero-field $I$-$V$
curves at different temperatures for the $x\simeq 0.50$ Pbx(TiO2)1-x film.

We evaluate the charging energies of the insulating Pbx(TiO2)1-x films using
$E_{c}=e^{2}/(4\pi\epsilon_{0}\tilde{\kappa}a)$. The effective dielectric
constant can be estimated via $\tilde{\kappa}=\epsilon_{r}(a+s)/s$ with $s$
being the intergrain spacing between neighboring grains [32]. Assuming the Pb
particles form a simple cubic lattice, one can roughly obtained $s$ via
$x=a^{3}/(a+s)^{3}$. Taking $\epsilon_{r}$ of TiO2 as $\sim$86 [33], we obtain
the values of $E_{c}$ for the films in the dielectric regime and list them in
Table I. Clearly, the charging energy increases with decreasing $x$, and
satisfies the condition for Cooper-pair hopping ($E_{c}<\Delta_{0}$ with the
zero-temperature gap $\Delta_{0}\simeq 1.37$ meV for the Pb granules) for the
$x\lesssim 0.52$ films. For comparison, we give the low temperature
$\rho(H)/\rho(0)$ vs $\mu_{0}H$ data in Fig. 2(c) for a Pb0.50(SiO2)0.50 film
with thickness of $\sim$1 $\mu$m and fabricated previously [35]. This film is
in insulating regime, and its resistivities at 2.0, 3.0, and 4.0 K are 1086,
987 and 722 $\Omega$ cm, respectively. For this film, taking $a$ and
$\epsilon_{r}$ as 7.9 nm and 4.5 [33], respectively, one can obtain
$E_{c}\simeq$ 8.4 meV, which is much greater than the energy gap $\Delta_{0}$
of Pb granules. Inspection of Fig. 2(c) indicates that the magnetoresistance
at each temperature below $\sim$4 K is negative over the whole measured field
range (from zero to 8.0 T). In fact, for the insulating Pbx(SiO2)1-x films
near the superconductor-insulator transiton, such as $x\simeq 0.53$ to 0.51,
the low-temperature magnetoresistances are all negative [35]. Thus, the
positive magnetoresistance in Pbx(TiO2)1-x ($x\simeq 0.50$, 0.52, and 0.55)
films and negative magnetoresistance in the Pbx(SiO2)1-x film strongly suggest
that the physical quantities governing the charge types of hopping in the
insulating regime of granular superconductors are $E_{c}$ and $\Delta_{0}$.

Figure 2(d) shows the logarithm of resistivity as a function of $T^{-1/2}$ for
the $x\simeq 0.50$ and 0.52 films, as indicated. Clearly, the logarithm of
resistivity varies linearly with $T^{-1/2}$ from $\sim$4.7 to $\sim$2.7 K for
the two films, which suggests that ES-type Cooper-pair-hopping mechanism
governs the conduction process in this temperature regime. In the ES-type
Cooper-pair-hopping process, the resistivity obeys [31]

$\rho(T)=\rho_{01}\exp\left(\frac{T_{0}}{T}\right)^{1/2},$ (1)

with

$T_{0}=\frac{b(2e)^{2}}{4k_{B}\pi\epsilon_{0}\widetilde{\kappa}\xi^{\rm
CPH}},$ (2)

where $\rho_{01}$ is a prefactor, $b$ is a numerical factor of the order of
unity, $k_{B}$ is the Boltzmann constant, and $\xi^{\rm
CPH}=a/\ln(8\bar{E}/\pi\bar{g}\bar{\Delta})$ is the Cooper-pair localization
length for the ES hopping. In the expression of the localization length,
$\bar{g}$ and $\bar{E}\sim E_{c}$ are the typical values of the conductance
and Coulomb correlation energy between neighboring grains, respectively. The
experimental data are least-squares fitted to Eq. (1), and the results are
plotted as solid lines in Fig. 2 (d). Setting $b=1$, one can readily obtain
the values of $\xi^{\rm CPH}$, which are 1.7 and 0.9 nm for the $x\simeq 0.50$
and 0.52 films, respectively. According to Lopatin and Vinokur, $\xi^{\rm
CPH}$ should be smaller than the mean grain size $a$ in the insulating state,
and be approximately equal to $a$ when the film turns to a superconducting
state [31]. Thus, the value of $\xi^{\rm CPH}$ for each film is reasonable.

Inspection of Fig. 2 indicates that the ln $\rho$ vs $T^{-1/2}$ curves deviate
from the ES law and tend to saturate with further decreasing temperature below
$\sim$2.7 K. During the $\rho$-$T$ measurements, the applied current keeps at
10 nA, around which no electroresistance effect is observed [see the inset of
Fig. 2(d)]. The saturation trend at low temperatures is, therefore, irrelevant
to a negative electroresistance effect [38]. According to Delsing _et al_.
[26] and Henl _et al_. [39], the quantum fluctuations of charge would become
important at low temperatures and result in a finite conductance. Thus, in the
Pbx(TiO2)1-x ($x\simeq 0.50$ and 0.52) films the saturation trend of
resistivity could be the comprehensive effect of the ES-hopping and quantum
fluctuations of Cooper pairs. We note in passing that the behaviors of $\rho$
vs $\mu_{0}H$ and $\rho$ vs $T^{-1/2}$ for the $x\simeq 0.55$ film are similar
to those for the $x\simeq 0.52$ film.

Figure 3: The normalized resistivity $\rho(H)/\rho(0)$ vs magnetic field at
3.0 and 5.0 K for (a) $x\simeq 0.47$ and (b) $x\simeq 0.45$ Pbx(TiO2)1-x
films.

Figure 3(a) and 3(b) shows the low-temperature magnetoresistances of the
$x\simeq 0.47$ and 0.45 Pbx(TiO2)1-x films, respectively. Interestingly, the
low-temperature magnetoresistances of the two films are all negative,
indicating that single-electron hopping dominates the hopping transport. In
the insulating Pbx(TiO2)1-x granular films, the average separation between two
adjacent Pb granules increases with decreasing $x$, which will greatly enhance
the normal-state intergrain resistance $R_{N}$. On the other hand, the
Josephson energy $E_{J}$ between neighbor grains is
$E_{J}=R_{Q}\Delta/2R_{N}$, where $R_{Q}=h/4e^{2}$ is the quantum resistance
of Cooper pair [21]. The normal-state resistivitie (at 15 K) of the $x\simeq
0.47$ (0.45) film is 535 (1080) $\Omega$ cm, which is $\sim$2 orders of
magnitude greater than that of the $x=0.50$ film. Then the Josephson energies
of the 0.47 and 0.45 films are reduced $\sim$2 orders of magnitude compared to
that of the $x\simeq 0.50$ film. In this situation, the intergrain Josephson
coupling becomes too weak for Cooper pairs to hop between the adjacent
superconducting Pb granules. As a result, the electrical transport process
undergoes a transition from a Cooper-pair-dominated hopping to a single-
electron-dominated hopping.

Figure 4:  Resistivity vs $T^{-1}$ from 15 down to 2.8 K at zero magnetic
field for the $x\simeq 0.47$ and 0.47 Pbx(TiO2)1-x films.

Figure 4 shows the zero-field resistivity (in a logarithmic scale) as a
function of $T^{-1}$ for the $x\simeq 0.47$ and 0.45 Pbx(TiO2)1-x films, as
indicated. Clearly, $\log_{10}\rho$ ($\ln\rho$) varies linearly with $T^{-1}$
from $\sim$5.0 down to $\sim$2.8 K. For single-electron hopping, the
concentration of single-electron excitations follows
$n\propto\exp(-\Delta/k_{B}T)$. Thus, the temperature dependence of
resistivity obeys [5]

$\rho(T)=\rho_{02}\exp\left(\frac{\Delta}{k_{B}T}\right),$ (3)

where $\rho_{02}$ is a prefactor. The experimental $\rho(T)$ data are least-
squares fitted to Eq. (3), and the fitted results are presented in Fig. 4 by
solid lines. For the $x\simeq 0.47$ (0.45) Pbx(TiO2)1-x film, the fitted value
of $\Delta$ is 1.55 (1.62) meV, which is close to the zero-temperature
superconducting gap $\Delta_{0}$ (1.37 meV) of Pb obtained via single-electron
tunneling spectra measurements.

In summary, we successfully realized Cooper-pair hopping in insulating
Pbx(TiO2)1-x ($x\simeq 0.50$, 0.52, and 0.55) granular films by using the
high-$k$ dielectric TiO2 as the insulating matrix. For these films, the low-
temperature magnetoresistances are positive under low field, and the
temperature dependent behavior of resistivity obeys the ES-type VRH law in
low-temperature region. When $x$ is reduced to $\lesssim 0.47$, a crossover
from Cooper-pair-dominated hopping to single-electron-dominated hopping is
observed. These $x\lesssim 0.47$ films possess negative magnetoresistance at
low temperatures and the resistivity vs temperature obeys a law of thermal-
activation form. For the $x\lesssim 0.47$ Pbx(TiO2)1-x films, the intergrain
Josephson coupling is too weak for Cooper pairs to hop between the adjacent
superconducting Pb granules, and the low-temperature transport can only be
mediated by single-electron hopping.

This work is supported by the National Natural Science Foundation of China
through Grant No. 12174282.

## References

  * [1] B. G. Orr, H. M. Jaeger, A. M. Goldman, and C. G. Kuper, Global phase coherence in two-dimensional granular superconductors, Phys. Rev. Lett. 56, 378 (1986).
  * [2] H. M. Jaeger, D. B. Haviland, B. G. Orr, and A. M. Goldman, Onset of superconductivity in ultrathin granular metal films, Phys. Rev. B 40, 182 (1989).
  * [3] A. Yazdani and A. Kapitulnik, Superconducting-Insulating Transition in Two-Dimensional $\alpha$-MoGe Thin Films, Phys. Rev. Lett. 74, 3037 (1995).
  * [4] H. Q. Nguyen, S. M. Hollen, M. D. Stewart, Jr., J. Shainline, A. Yin, J. M. Xu, and J. M. Valles, Jr, Observation of Giant Positive Magnetoresistance in a Cooper Pair Insulator, Phys. Rev. Lett. 103, 157001 (2009).
  * [5] V. F. Gantmakher and V. T. Dolgopolov, Superconductor-insulator quantum phase transition, Phys. Usp. 53, 1 (2010).
  * [6] G. Sambandamurthy, L. W. Engel, A. Johansson, and D. Shahar, Superconductivity-Related Insulating Behavior, Phys. Rev. Lett. 92, 107005 (2004).
  * [7] D. Ephron, A. Yazdani, A. Kapitulnik, and M. R. Beasley, Observation of Quantum Dissipation in the Vortex State of a Highly Disordered Superconducting Thin Film, Phys. Rev. Lett. 76, 1529 (1996).
  * [8] N. Mason and A. Kapitulnik, Dissipation Effects on the Superconductor-Insulator Transition in 2D Superconductors, Phys. Rev. Lett, 82, 5341 (1999).
  * [9] Y. Qin, C. L. Vicente, and J. Yoon, Magnetically induced metallic phase in superconducting tantalum films, Phys. Rev. B 73, 100505(R) (2006).
  * [10] C. Yang, Y. Liu, Y. Wang, L. Feng, Q. He, J. Sun, Y. Tang, C. Wu, J. Xiong, W. Zhang, X. Lin, H. Yao, H. Liu, G. Fernandes, J. Xu, J. M. Valles Jr, J. Wang, Y. Li, Intermediate bosonic metallic state in the superconductor-insulator transition, Science 366, 1505 (2019).
  * [11] N. P. Breznay, and A. Kapitulnik, Particle-hole symmetry reveals failed superconductivity in the metallic phase of two-dimensional superconducting films, Sci. Adv. 3, e1700612 (2017).
  * [12] P. Phillips and D. Dalidovich, The Elusive Bose Metal, Science 302, 243 (2003).
  * [13] N. Giordano, Evidence for macroscopic quantum tunneling in one-dimensional superconductors, Phys. Rev. Lett. 61, 2137 (1988).
  * [14] N. Giordano, Dissipation in a one-dimensional superconductor: Evidence for macroscopic quantum tunneling, Phys. Rev. B 41, 6350 (1990).
  * [15] M. Tian, J. Wang, J. S. Kurtz, Y. Liu, M. H. W. Chan, T. S. Mayer, and T. E. Mallouk, Dissipation in quasi-one-dimensional superconducting single-crystal Sn nanowires, Phys. Rev. B 71, 104521 (2005).
  * [16] A. Belkin, M. Belkin, V. Vakaryuk, S. Khlebnikov, and A. Bezryadin, Formation of Quantum Phase Slip Pairs in Superconducting Nanowires, Phys. Rev. X 5, 021023 (2015).
  * [17] A. Ghosal, M. Randeria, and N. Trivedi, Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave Superconductors, Phys. Rev. Lett. 81, 3940 (1998)
  * [18] A. Ghosal, M. Randeria, and N. Trivedi, Inhomogeneous pairing in highly disordered s-wave superconductors, Phys. Rev. B 65, 014501 (2001).
  * [19] Y. Dubi, Y. Meir and Y. Avishai, Nature of the superconductor-insulator transition in disordered superconductors, Nature 449, 876 (2007).
  * [20] B. Sacépé, C. Chapelier, T. I. Baturina, V. M. Vinokur, M. R. Baklanov, and M. Sanquer, Disorder-Induced Inhomogeneities of the Superconducting State Close to the Superconductor-Insulator Transition, Phys. Rev. Lett. 101, 157006 (2008).
  * [21] I. S. Beloborodov, A. V. Lopatin, V. M. Vinokur, and K. B. Efetov, Granular electronic systems, Rev. Mod. Phys. 79, 469 (2007).
  * [22] S. H. Pan, J. P. O’Neal, R. L. Badzey, C. Chamon, H. Ding, J. R. Engelbrecht, Z. Wang, H. Eisaki, S. Uchida, A. K. Gupta, K.-W. Ng, E. W. Hudson, K. M. Lang and J. C. Davis, Microscopic electronic inhomogeneity in the high-Tc superconductor Bi2Sr2CaCu2O8+x, Nature 413, 282 (2001).
  * [23] T. Cren, D. Roditchev, W. Sacks and J. Klein, Nanometer scale mapping of the density of states in an inhomogeneous superconductor, Europhys. Lett. 54, 84 (2001).
  * [24] R. F. Jardim, L. Ben-Dor, D. Stroud, and M. B. Maple, Granular behavior in polycrystalline Sm2-xCexCuO4-y compounds, Phys. Rev. B 50, 10080 (1994).
  * [25] E. A. Early, C. C. Almasan, R. F. Jardim, and M. B. Maple, Double resistive superconducting transition in Sm2-xCexCuO4-y, Phys. Rev. B 47, 433 (1993).
  * [26] P. Delsing, C. D. Chen, D. B. Haviland, Y. Harada, and T. Claeson, Charge solitons and quantum fluctuations in two-dimensional arrays of small Josephson junctions, Phys. Rev. B 50, 3959 (1994).
  * [27] H. S. J. van der Zant, W. J. Elion, L. J. Geerligs, and J. E. Mooij, Quantum phase transitions in two dimensions: Experiments in Josephson-junction arrays, Phys. Rev. B 54, 10081 (1996)
  * [28] Y. Takahide, R. Yagi, A. Kanda, Y. Ootuka, and S. Kobayashi, Superconductor-Insulator Transition in a Two-Dimensional Array of Resistively Shunted Small Josephson Junctions, Phys. Rev. Lett. 85, 1974 (2000).
  * [29] D. V. Averin and Y. V. Nazarov, Single-electron charging of a superconducting island, Phys. Rev. Lett. 69, 1993 (1992).
  * [30] K. A. Matveev, M. Gisselfält, L. I. Glazman, M. Jonson, and R. I. Shekhter, Parity-induced suppression of the Coulomb blockade of Josephson tunneling, Phys. Rev. Lett. 70, 2940 (1993).
  * [31] A. V. Lopatin and V. M. Vinokur, Hopping transport in granular superconductors, Phys. Rev. B 75, 092201 (2007).
  * [32] J. Zhang and B. I. Shklovskii, Density of states and conductivity of a granular metal or an array of quantum dots, Phys. Rev. B 70, 115317 (2004).
  * [33] W. M. Haynes, CRC handbook of chemistry and physics: a ready-reference book of chemical and physical data, 98th ed. (CRC, 2017).
  * [34] W. L. McMillan and J. M. Rowell, Lead Phonon Spectrum Calculated from Superconducting Density of States, Phys. Rev. Lett. 14, 108 (1965).
  * [35] X. Z. Duan, Z. H. He, Y. Yang, and Z. Q. Li, Hopping conductance and macroscopic quantum tunneling effect in three dimensional Pbx(SiO2)1-x nanogranular films, Phys. Rev. B. 99, 094204 (2019).
  * [36] K. A. Parendo, K. H. S. B. Tan, and A. M. Goldman, Evolution towards superconductivity in granular films of bismuth, Phys. Rev. B. 76, 100508(R) (2007).
  * [37] W.-H. Li, C. C. Yang, F. C. Tsao, and K. C. Lee, Quantum size effects on the superconducting parameters of zero-dimensional Pb nanoparticles, Phys. Rev. B. 68, 184507 (2003).
  * [38] R. P. Barber, Jr., S. Y. Hsu, J. M. Valles, Jr., R. C. Dynes, and R. E. Glover III, Negative magnetoresistance, negative electroresistance, and metallic behavior on the insulating side of the two-dimensional superconductor-insulator transition in granular Pb films, Phys. Rev. B 73, 134516 (2006).
  * [39] B. Hen, V. Shelukhin, E. Greenberg, G. K. Rozenberg, A. Kapitulnik and A. Palevski, Superconductor-insulator transitions in three-dimensional indium-oxide at high pressures, J. Phys.: Condens. Matter 34, 135402 (2022).


# Matching Powers: Macaulay2 Package

Antonino Ficarra  Antonino Ficarra, Department of mathematics and computer
sciences, physics and earth sciences, University of Messina, Viale Ferdinando
Stagno d’Alcontres 31, 98166 Messina, Italy<EMAIL_ADDRESS>
###### Abstract.

We introduce the Macaulay2 package MatchingPowers. It allows to compute and
manipulate the matching powers of a monomial ideal. The basic theory of
matching powers is explained and the main features of the package are
presented.

###### Key words and phrases:

monomial ideals, matching powers, monomial ideals, edge ideals

###### 2020 Mathematics Subject Classification:

Primary 13F20; Secondary 13F55, 05C70, 05E40.

##  1\. Basic Properties of Matching Powers

Let $S=K[x_{1},\dots,x_{n}]$ be the standard graded polynomial ring with
coefficients over a field $K$. Let $I$ be a monomial ideal of $S$. The $k$th
matching power of $I$, denoted by $I^{[k]}$ is the monomial ideal generated by
the products $u_{1}\cdots u_{k}$ such that $u_{1},\dots,u_{k}$ is a monomial
regular sequence of length $k$ contained in $I$. The biggest $k$ such that
$I^{[k]}\neq 0$ is called the monomial grade of $I$ and it is denoted by
$\nu(I)$. The concept of matching powers was first introduced in the
squarefree case in [6] and later extended to all monomial ideals in [3].

One of the main motivations for considering matching powers comes from graph
theory. Let $G$ be a finite simple graph on vertex set $[n]=\\{1,\dots,n\\}$.
The edge ideal of $G$ is the squarefree monomial ideal $I(G)$ generated by the
monomials $x_{e}=x_{i}x_{j}$ such that $e=\\{i,j\\}$ is an edge of $G$. Let
$e_{1},\dots,e_{k}$ be $k$ edges of $G$, then $x_{e_{1}}\cdots x_{e_{k}}$ is a
generator of $I(G)^{[k]}$ if and only if $\\{e_{1},\dots,e_{k}\\}$ is a
$k$-matching of $G$. This observation justifies the choice to name the powers
$I^{[k]}$ the matching powers of $I$.

Similarly, for a non-squarefree example, we can consider edge ideals of
weighted oriented graphs. Recall that a (vertex)-weighted oriented graph
$\mathcal{D}=(V(\mathcal{D}),E(\mathcal{D}),w)$ consists of an underlying
graph $G$, with $V(\mathcal{D})=V(G)=[n]$, on which each edge is given an
orientation and it is equipped with a weight function
$w:V(G)\rightarrow\mathbb{Z}_{\geq 1}$. The weight of a vertex $i\in V(G)$,
denoted by $w_{i}$, is the value $w(i)$ of the weight function at $i$. The
directed edges of $\mathcal{D}$ are denoted by pairs $(i,j)\in E(\mathcal{D})$
to reflect the orientation, hence $(i,j)$ represents an edge directed from $i$
to $j$. The edge ideal of $\mathcal{D}$ is defined as the ideal

$I(\mathcal{D})\ =\ (x_{i}x_{j}^{w_{j}}\ :\ (i,j)\in E(\mathcal{D})).$

As before, given $k$ generators
$u_{\ell}=x_{i_{\ell}}x_{j_{\ell}}^{w_{j_{\ell}}}$ of $I(\mathcal{D})$, with
$e_{\ell}=(i_{\ell},j_{\ell})\in E(\mathcal{D})$, for $\ell=1,\dots,k$, the
product $u_{1}\cdots u_{k}$ is a generator of $I(\mathcal{D})^{[k]}$ if and
only if $\\{e_{1},\dots,e_{k}\\}$ is a $k$-matching of the underlying simple
graph $G$ of $\mathcal{D}$.

In the remaining part of this section, we summarize some of the main results
proved in [3, Section 1].

Let $G(I)$ be the unique minimal set of monomial generators of $I$. The
initial degree of $I$ is defined as $\textup{indeg}(I)=\min\\{\deg(u):u\in
G(I)\\}$. For a monomial $u\in S$, the $x_{i}$-degree of $u$ is defined as
$\deg_{x_{i}}(u)=\max\\{j:x_{i}^{j}\ \textup{divides}\ u\\}$. Whereas, the
support of $u$ is the set $\operatorname{supp}(u)=\\{i:\deg_{x_{i}}(u)>0\\}$.
Following [7], we define the bounding multidegree of $I$ as

${\bf deg}(I)\ =\ (\deg_{x_{1}}(I),\dots,\deg_{x_{n}}(I)),$

with

$\deg_{x_{i}}(I)\ =\ \max_{u\in G(I)}\deg_{x_{i}}(u),\ \ \textup{for all}\ \ \
1\leq i\leq n.$

Furthermore, if ${\bf a}=(a_{1},\dots,a_{n})\in{\mathbb{Z}}_{\geq 0}^{n}$, we
set $|{\bf a}|=a_{1}+\dots+a_{n}$.

For a monomial $u=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\in S$, we set
$u^{\wp}=\prod_{i=1}^{n}(\prod_{j=1}^{a_{i}}x_{i,j})$ in the polynomial ring
$K[x_{i,j}:1\leq i\leq n,1\leq j\leq a_{i}]$. The polarization of $I$ is
defined to be the squarefree ideal $I^{\wp}$ of $S^{\wp}=K[x_{i,j}:1\leq i\leq
n,1\leq j\leq\deg_{x_{i}}(I)]$ whose minimal generating set is
$G(I^{\wp})=\\{u^{\wp}:u\in G(I)\\}$.

######  Theorem 1.1.

Let $I\subset S$ be a monomial ideal. The following facts hold.

  1. (i)

$I^{[k]}=(u_{1}\cdots u_{k}\ :\ u_{i}\in G(I),\
\operatorname{supp}(u_{i})\cap\operatorname{supp}(u_{j})=\emptyset,\ 1\leq
i<j\leq k)$.

  2. (ii)

$I^{[k]}\neq 0$ if and only if $1\leq k\leq\nu(I)$.

  3. (iii)

$(I^{[k]})^{\wp}=(I^{\wp})^{[k]}$ for all $1\leq k\leq\nu(I)$.

  4. (iv)

$\operatorname{depth}(S/I^{[k]})=\operatorname{depth}(S^{\wp}/(I^{\wp})^{[k]})-|{\bf
deg}(I)|+n$, for all $1\leq k\leq\nu(I)$.

  5. (v)

$\operatorname{depth}(S/I^{[k]})\geq\textup{indeg}(I^{[k]})-1+(n-|{\bf
deg}(I)|)$.

Based on inequality (v), we define the normalized depth function of $I$ as

$g_{I}(k)\ =\ \operatorname{depth}(S/I^{[k]})+|{\bf
deg}(I)|-n-(\textup{indeg}(I^{[k]})-1),$

for all $1\leq k\leq\nu(I)$. In the squarefree case, this is the function
introduced in [6].

As a consequence of Theorem 1.1 it was proved that [3]

######  Corollary 1.2.

We have $\nu(I)=\nu(I^{\wp})$ and $g_{I}(k)=g_{I^{\wp}}(k)$ for all $k$.

It was conjectured in [6] that $g_{I}$ would be non-increasing for all
squarefree monomial ideals $I$. Because of Corollary 1.2, it the above
conjecture was true, then $g_{I}$ would be non-increasing for all monomial
ideals $I$ [3, Proposition 1.8]. However, very recently, Fakhari [18]
constructed a family of monomial ideals $I$ such that the difference
$g_{I}(2)-g_{I}(1)$ may be arbitrary large, thus disproving the above
conjecture.

It is an open question whether for any sequence of non-negative integers
$a_{1},\dots,a_{g}$, there exists a monomial ideal $I$ such that $\nu(I)=g$
and $g_{I}(k)=a_{k}$ for all $k=1,\dots,g$.

Currently, the study of matching powers of monomial ideals and of various kind
of edge ideals [13, 14, 15] is a very active area of research [1, 2, 3, 4, 5,
6, 9, 16, 17, 18].

In the next two sections, we illustrate and explain how to use the Macaulay2
[10] package MatchingPowers. In Section 2, we describe the main functions of
the package. In Section 3, an example is presented, that demonstrates how to
use the package.

##  2\. The Package

The purpose of the package MatchingPowers is to provide the tools to
manipulate and calculate the matching powers of a monomial ideal $I$.

The multidegree of a monomial $u\in S$ is defined to be the vector

${\bf deg}(u)\ =\ (\deg_{x_{1}}(u),\dots,\deg_{x_{n}}(u)).$

Whereas, for ${\bf a}=(a_{1},\dots,a_{n})\in{\mathbb{Z}}_{\geq 0}^{n}$, we set
${\bf x^{a}}=\prod_{i}x_{i}^{a_{i}}$. In particular, $u={\bf
x}^{{\bf\deg}(u)}$.

Given two monomial ideals $I,J\subset S$, we define the matching product of
$I$ and $J$ to be the monomial ideal

$I*J\ =\ (uv\ :\ u\in I,v\in J,\ u,v\ \textup{is a regular sequence}).$

It is not difficult to see that

$I*J\ =\ (uv\ :\ u\in G(I),\ v\in G(J),\
\operatorname{supp}(u)\cap\operatorname{supp}(v)=\emptyset).$

Moreover, $I^{[2]}=I*I$ and by induction $I^{[k]}=I^{[k-1]}*I$ for all $k\geq
2$.

Finally, we recall that a graded ideal $I\subset S$ is linearly related if $I$
is generated in single degree, say $d$, and the first syzygy module of $I$ is
generated $d+1$.

The next table collects the functions available in the package. By $I$ and $J$
we denote two monomial ideals of $S$, ${\bf a}\in{\mathbb{Z}}_{\geq 0}^{n}$ an
integral vector, ${\bf x^{a}}$ a monomial.

Functions | Description
---|---
toMonomial$(S,{\bf a})$ |  Computes the monomial ${\bf x^{a}}$ if ${\bf a}\in{\mathbb{Z}}^{n}_{\geq 0}$
toMultidegree$({\bf x^{a}})$ |  Computes the multidegree ${\bf a}$ of ${\bf x^{a}}$
boundingMultidegree$(I)$ |  Computes $\textbf{deg}(I)$
matchingProduct$(I,J)$ |  Computes the matching product of $I$ and $J$
matchingPower$(I,k)$ |  Computes the $k$th matching power of $I$
monomialGrade$(I)$ |  Computes the monomial grade $\nu(I)$
gFunction$(I)$ |  Computes the normalized depth function $g_{I}$
isLinearlyRelated$(I)$ |  Checks if $I$ is linearly related
Table 1. List of the functions of MatchingPowers.

##  3\. An Example

Consider the weighted oriented graph $\mathcal{D}$ depicted below.

$a$$x_{4}$$x_{3}$$x_{2}$$x_{1}$$x_{5}$$b$$x_{7}$$x_{6}$$x_{8}$$x_{9}$$c$$x_{11}$$x_{10}$$x_{12}$

We assign the weights as follows. For $i=1,\dots,12$, we let $w(x_{i})=1$,
whereas we set $w(a)=w(b)=w(c)=3$. Thus the edge ideal of $\mathcal{D}$ is

$I(\mathcal{D})=(x_{1},x_{2},x_{3},x_{4},x_{5})(a^{3})+(x_{5},x_{6},x_{7},x_{8},x_{9})(b^{3})+(b,x_{10},x_{11},x_{12})(c^{3}).$

Then, $\nu(I(\mathcal{D}))=3$, and by using the package we checked that
$I(\mathcal{D})^{[3]}$ is linearly related. Indeed, this also follows from [3,
Theorem 4.2].

`i1: S = QQ[x_1..x_12,a..c];`

`i2: I = ideal(x_1,x_2,x_3,x_4,x_5)*ideal(a^3)`

`+ ideal(x_5,x_6,x_7,x_8,x_9)*ideal(b^3)`

`+ ideal(b,x_10,x_11,x_12)*ideal(c^3);`

ii3: loadPackage "MatchingPowers"

ii4: matchingPower(I,1) == I

io4: true

ii5: isLinearlyRelated I

io5: false

ii6: J = matchingPower(I,3);

ii7: isLinearlyRelated J

io7: true

ii8: monomialGrade I

io8: 3

A good companion to the MatchingPowers package is the HomologicalShiftIdeals
package [8]. Using this latter package, one can check that
$I(\mathcal{D})^{[3]}$ is a polymatroidal ideal, which again follows from [3,
Theorem 4.2].

## References

  * [1] M. Bigdeli, J. Herzog, R. Zaare-Nahandi, On the index of powers of edge ideals, Comm. Algebra, 46 (2018), 1080–1095.
  * [2] M. Crupi, A. Ficarra, E. Lax, Matchings, squarefree powers and Betti splittings, 2023, preprint https://arxiv.org/abs/2304.00255
  * [3] N. Erey, A. Ficarra, Matching Powers of monomial ideals and edge ideals of weighted oriented graphs (2023), preprint arxiv.org/abs/2309.13771
  * [4] N. Erey, T. Hibi, Squarefree powers of edge ideals of forests, Electron. J. Combin., 28 (2) (2021), P2.32.
  * [5] N. Erey, J. Herzog, T. Hibi, S. Saeedi Madani, Matchings and squarefree powers of edge ideals, J. Comb. Theory Series. A, 188 (2022).
  * [6] N. Erey, J. Herzog, T. Hibi, S. Saeedi Madani, The normalized depth function of squarefree powers, Collect. Math. (2023). https://doi.org/10.1007/s13348-023-00392-x
  * [7] A. Ficarra, Homological shifts of polymatroidal ideals, preprint arxiv:2205.04163 (2022).
  * [8] A. Ficarra. HomologicalShiftIdeals, Macaulay2 Package, 2023, preprint arXiv:2309.09271.
  * [9] A. Ficarra, J. Herzog, T. Hibi, Behaviour of the normalized depth function, Electron. J. Comb., 30 (2), (2023) P2.31.
  * [10] D. R. Grayson, M. E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2.
  * [11] H.T. Hà, K.N. Lin, S. Morey, E. Reyes, R.H. Villarreal, Edge ideals of oriented graphs, International Journal of Algebra and Computation, 29 (03), (2019), 535–559.
  * [12] J. Herzog, T. Hibi. _Monomial ideals_ , Graduate texts in Mathematics 260, Springer, 2011.
  * [13] J. Martínez-Bernal, Y. Pitones and R. H. Villarreal, Minimum distance functions of graded ideals and Reed-Muller-type codes, J. Pure Appl. Algebra 221 (2017), 251–275.
  * [14] C. Paulsen and S. Sather-Wagstaff, Edge ideals of weighted graphs, J. Algebra Appl., 12 (2013), 1250223-1-24.
  * [15] Y. Pitones, E. Reyes, J. Toledo, Monomial ideals of weighted oriented graphs, Electron. J. Combin., 26 (2019), no. 3, Research Paper P3.44.
  * [16] S. A. Seyed Fakhari, On the Castelnuovo-Mumford regularity of squarefree powers of edge ideals, 2022, available at arxiv.org/abs/2303.02791
  * [17] S. A. Seyed Fakhari, On the Regularity of squarefree part of symbolic powers of edge ideals, 2023, available at arxiv.org/abs/2207.08559
  * [18] S. A. Seyed Fakhari, An Increasing normalized depth function, 2023, available at arxiv.org/abs/2309.13892.