#  Effect of impurity scattering on percolation of bosonic islands and
reentrant superconductivity in Fe implanted NbN thin films

Rajdeep Adhikari<EMAIL_ADDRESS>Institut für Halbleiter-und-
Festkörperphysik, Johannes Kepler University, Altenbergerstr. 69, A-4040 Linz,
Austria  Bogdan Faina  Institut für Halbleiter-und-Festkörperphysik, Johannes
Kepler University, Altenbergerstr. 69, A-4040 Linz, Austria  Verena Ney
Institut für Halbleiter-und-Festkörperphysik, Johannes Kepler University,
Altenbergerstr. 69, A-4040 Linz, Austria  Julia Vorhauer  Institut für
Halbleiter-und-Festkörperphysik, Johannes Kepler University, Altenbergerstr.
69, A-4040 Linz, Austria  Antonia Sterrer  Institut für Halbleiter-und-
Festkörperphysik, Johannes Kepler University, Altenbergerstr. 69, A-4040 Linz,
Austria  Andreas Ney  Institut für Halbleiter-und-Festkörperphysik, Johannes
Kepler University, Altenbergerstr. 69, A-4040 Linz, Austria  A. Bonanni
<EMAIL_ADDRESS>Institut für Halbleiter-und-Festkörperphysik, Johannes
Kepler University, Altenbergerstr. 69, A-4040 Linz, Austria

(September 3, 2024)

###### Abstract

A reentrant temperature dependence of the thermoresistivity
$\rho_{\mathrm{xx}}(T)$ between an onset local superconducting ordering
temperature $T_{\mathrm{loc}}^{\mathrm{onset}}$ and a global superconducting
transition at $T=T_{\mathrm{glo}}^{\mathrm{offset}}$ has been reported in
disordered conventional 3-dimensional (3D) superconductors. The disorder of
these superconductors is a result of either an extrinsic granularity due to
grain boundaries, or of an intrinsic granularity ascribable to the electronic
disorder originating from impurity dopants. Here, the effects of Fe doping on
the electronic properties of sputtered NbN layers with a nominal thickness of
100 nm are studied by means of low-$T$/high-$\mu_{0}H$ magnetotransport
measurements. The doping of NbN is achieved $via$ implantation of 35 keV Fe
ions. In the as-grown NbN films, a local onset of superconductivity at
$T_{\mathrm{loc}}^{\mathrm{onset}}=15.72\,\mathrm{K}$ is found, while the
global superconducting ordering is achieved at
$T_{\mathrm{glo}}^{\mathrm{offset}}=15.05\,\mathrm{K}$, with a normal state
resistivity $\rho_{\mathrm{xx}}=22\,{\mu\Omega}\cdot{\mathrm{cm}}$. Moreover,
upon Fe doping of NbN, $\rho_{\mathrm{xx}}=40\,{\mu\Omega}\cdot{\mathrm{cm}}$
is estimated, while $T_{\mathrm{loc}}^{\mathrm{onset}}$ and
$T_{\mathrm{glo}}^{\mathrm{offset}}$ are measured to be 15.1 K and 13.5K,
respectively. In Fe:NbN, the intrinsic granularity leads to the emergence of a
bosonic insulator state and the normal-metal-to-superconductor transition is
accompanied by six different electronic phases characterized by a $N$-shaped
$T$ dependence of $\rho_{\mathrm{xx}}(T)$. The bosonic insulator state in a
$s$-wave conventional superconductor doped with dilute paramagnetic impurities
is predicted to represent a workbench for emergent phenomena, such as gapless
superconductivity, triplet Cooper pairings and topological odd frequency
superconductivity.

###### pacs:

72.25.Dc, 72.25.Mk, 76.50.+g, 85.75.-d

##  I Introduction

The doping of superconductors with non-magnetic and magnetic impurities paved
the way for understanding the physics of isotropic and anisotropic
superconductivity [1, 2, 3]. The effects of impurities on the electronic
properties of both conventional and unconventional superconductors were
explained in terms of electron lifetime and pairing symmetries due to
scattering by an ensemble of impurities [4, 5, 6, 1, 7, 8, 9, 10, 11, 12, 13,
14, 4]. According to the Cooper’s pairing model [15] and to the Bardeen-
Cooper-Schreiffer’s (BCS) theory, [16, 17], superconductivity is a result of
the instability of the Fermi surface against the pairing of time-reversed
quasiparticle states. Any perturbation that is unable to lift the Kramers
degeneracy of these states does not affect the superconducting transition
temperature. While non-magnetic or scalar impurities are not known to affect
the isotropic singlet $s$-wave order parameter in conventional BCS
superconductors, magnetic impurities efficiently break the Cooper pairs,
leading to the suppression of any long range superconducting order [1] or to
the emergence of gapless superconductivity [10, 6, 1]. Most theoretical
approaches [3, 12, 18] based on the BCS theory [16, 17] treat the interaction
of the impurity spin $\vec{S}$ with the spin $\vec{\sigma}$ of the conduction
electrons $via$ an exchage interaction $\vec{\sigma}\cdot{\vec{S}}$. For a
magnetic impurity embedded in the superconducting matrix, there is a coupling
between the local spin on the impurity site and the conduction electrons of
the matrix represented by the Hamiltonian:

$H_{\mathrm{imp}}=\sum_{\alpha\beta}\int
d\vec{r}J(\vec{r})\Psi^{\dagger}_{\alpha}(\vec{r})\hat{U}^{\dagger}_{\mathrm{i}mp}\Psi_{\beta}(\vec{r})$
(1)

where $\hat{U}^{\dagger}_{\mathrm{imp}}=J(\vec{r})\vec{\sigma}\cdot\vec{S}$
[14] and $J(\vec{r})$ is the exchange interaction between the impurity ion and
the conduction electrons while $\alpha$ and $\beta$ represent the particle
indices. Furthermore, it was also predicted that single magnetic impurities
lead to pair breaking and result in the formation of quasiparticle Yu-Shiba-
Rusinov (YSR) states within the energy gap and localized in the vicinity of
the impurity atom [19]. However, for dilute magnetic impurities with atomic
concentrations $\lesssim{10^{-4}}\%$ the spins can be considered randomly
oriented and uncorrelated [18]. Another approach to study the effect of
magnetic impurities on superconductivity is to map the nature of
superconductivity with a ferromagnetic background resulting from the proximity
of a superconducting thin film to magnetic layers, promoting the onset of
emergent phenomena, including spin-triplet Cooper pairing, Majorana fermions
and spin superfluids [20, 21]. The nature of superconductivity in strong spin-
exchange fields was extensively discussed by Fulde-Ferrel [22] and Larkin-
Ovchinnikov [23]. The existence of the Fulde-Ferrel-Larkin-Ovchinnnikov (FFLO)
pairing was reported for heavy fermion systems, organic superconductors and
recently for combinations of superconductors (S) and ferromagnets (F) in F/S/F
and S/F/S heterostructures [24, 25].

Within the family of the conventional superconductors, NbN with a bulk
superconducting transition temperature of $\sim$16 K, has been widely studied
both in the bulk crystal phase and as thin film [26, 27, 28, 29, 30, 31, 32,
33]. Magnetron sputtered NbN thin films are intensively investigated [31, 29,
30, 32, 28] in view of diverse relevant applications of superconducting NbN in
Josephson junctions, hot electron bolometers, single photon detectors [34, 35]
and in devices for quantum information and circuit quantum electrodynamics
[36]. Disordered NbN thin films grown by sputtering are used as the workbench
to investigate the Berezinskii-Kosterlitz-Thouless (BKT) phase transition [31,
37], superconductor-insulator transitions [38, 30], conductance fluctuation
[27], Andreev reflection and the Higgs-Anderson mechanism of superconductivity
[39, 40]. Doping of conventional superconductors like NbN with magnetic
impurities is expected to broaden the perspectives for hybrid structures-based
applications in superconducting spintronics [20], spin-orbitronics, in dark
matter detectors [41], integrated resonators and superconducting qubit
processors for quantum computation.

While superconducting layers of NbN are traditionally grown on conventional
substrates such as Si and MgO, the use of GaN and Al1-xGaxN templates and
substrates has emerged as an alternative [42, 43, 44, 45] and can serve as
basis for all nitride integrated superconductor/semiconductor devices [46].
The Si substrates are commonly employed to grow superconducting NbN films
intended for single photon detection and hot bolometer applications, due to
the advantages in device processing and relatively low losses at THz
frequencies. The significant lattice mismatch between Si and NbN generally
results in polycrystalline films with $T_{\mathrm{c}}\leq{10}$ K for layers
$\sim(5-10)$ nm thick. The lattice matched substrate MgO presents challenges
in device processing, due to its hydrophobic nature and sensitivity to
alkaline solutions that are used during the fabrication processes [42]. Since
the crystallographic orientation of the NbN films does not affect their
superconducting properties, the $c$-plane of hexagonal wurtzite
template/buffer layers is also suitable for NbN growth, provided that the
lattice parameter $a^{\mathrm{w}}_{\mathrm{hkl}}$ of the wurtzite template
matches the one of NbN along the (111) plane $i.e.$
$a^{\mathrm{NbN}}_{111}=\frac{a^{\mathrm{w}}_{\mathrm{100}}}{\sqrt{2}}$. Thus,
the epitaxial relation
$\left[111\right]\left(\mathrm{NbN}\right)\parallel\left[100\right]\left(\mathrm{GaN}\right)$
is established between NbN and $c$-plane wurtzite template[42, 45, 46].

The effect of dilute magnetic impurities on the superconducting properties of
NbN thin films is scarcely reported in literature. The ferromagnetic proximity
in the NbN/FeN bilayer system has been shown to lower the superconducting
transition temperature [47], and Gd ion implantation was reported to have
similar effects on the superconducting properties of sputtered NbN films [48].
In the following, the emergence of a bosonic insulator (BI) state in Fe
implanted sputtered NbN thin film using low-$T$/high-$\mu_{0}H$
magnetotransport studies is analysed.

##  II Samples and Experimental Details

Figure 1:  High resolution ($2{\theta}-{\omega}$) scan of #A and #B.

###  II.1 Growth and structure

The NbN films considered in this work are grown by reactive magnetron
sputtering with high purity Nb as target. Epitaxial single crystalline $\sim$
1 µm GaN grown on one-side polished $c$-plane Al2O3 (0001) substrates by means
of metal organic vapor phase epitaxy is taken as template for the deposition
of the NbN thin films. The GaN templates are cut into $\left(1\times 1\right)$
cm2 specimens for the sputtering of the NbN layers. The polycrystalline thin
films are grown in an ultrahigh vacuum (UHV) chamber with a base pressure of
$(2\times{10^{-9}})$ mbar. A high purity (99.99%) Nb target is employed for
the reactive magnetron sputtering process under a plasma consisting of Ar:N2
in the ratio 10:5 standard cubic centimeters per minute (sccm) with a power
$P=40$ W and at a constant substrate temperature
$T_{\mathrm{sub}}=500^{\circ}$C.

Table 1: Relevant parameters of the representative samples considered in this
work.

Sample | Material | Template | Nominal $d_{\mathrm{NbN}}$ (nm) | Ar:N2 ratio | P (W) | Tsub (°C) | Implanted ion |  $E_{\mathrm{ion}}$ (keV) | Dose (at/cm3)
---|---|---|---|---|---|---|---|---|---
A | NbN | wz-GaN | 100 | 10:5 | 40 | 500 | - | - | -
B | Fe:NbN | wz-GaN | 100 | 10:5 | 40 | 500 | Fe | 35 | $1\times 10^{14}$

The NbN films are implanted with ${}^{56}{\mathrm{Fe}}$ using powdered Fe3O4
as the source for Fe ions. The implantation is carried out at room temperature
at a base pressure of $(1\times{10^{-6}})$ mbar. Half of the surface of the
$\left(1\times 1\right)$ cm2 samples is covered with a brass mask and the
uncovered part of the NbN layers is exposed to the Fe ion beam for
implantation. The energy and ion dose are selected after simulating the
stoppage and range of ions in matter (SRIM). The simulation of the ion
distribution is considered for ion energies of 35 keV and 50 keV,
respectively. A low ion current $\sim$ 100 nA is required to ignite the plasma
and for the subsequent ion extraction. The ion current is kept constant by
tuning the filament current and ion beam focus during the entire implantation
process. The Fe implantation dose used in this work is kept constant at
$\left(1\times 10^{14}\right)$ atoms/cm2. After implantation, the samples are
not subjected to any thermal annealing, in order to avoid precipitation of
secondary Fe or FeN phases and also to preserve the defects generated by the
implanted ions. The $\left(1\times 1\right)$ cm2 samples are then cut into
$\left(5\times 5\right)$ mm2 specimens for further measurements and
characterization. In the following, a representative as-grown NbN layer sample
A (#A), is considered, while a 35 keV Fe implanted NbN layer is taken as
sample B (#B). Both the as-grown NbN and Fe implanted NbN are studied using
x-ray diffraction (XRD) and the high crystallinity of the sputtered films is
confirmed [49]. An overview of the relevant growth parameters for #A and #B is
provided in Table I.

Radial scans are acquired using a PANalytical X’Pert PRO Materials Research
Diffractometer (MRD) equipped with a hybrid monochromator with a divergence
slit of 1/4∘. The diffracted beam is collected by a solid-state PixCel
detector provided with a 9.1 mm anti-scatter slit. The
$\left(2\theta-\omega\right)$ scans for #A and #B are shown in Fig. 1. The
Bragg peaks of NbN(111) and NbN(222) are observed alongside the GaN(002) and
GaN(004) peaks, as indicated in Fig. 1. The obtained result also points at an
epitaxial relation
$a^{\mathrm{NbN}}_{111}=\frac{a^{\mathrm{w}}_{\mathrm{100}}}{\sqrt{2}}$ of the
NbN (111) on $c$-Al2O3 (0006) as reported in literature [42]. No secondary
Bragg peaks of precipitated or clustered Fe and Fe4N are detected, pointing
out the homogeneous incorporation of the implanted Fe in the cubic NbN crystal
lattice.

###  II.2 Magnetotransport

The low-$T$/high-$\mu_{0}H$ magnetotransport studies are performed on #A and
#B in a four probe van der Pauw geometry. Ohmic contacts to the $(5\times{5})$
mm2 specimens are achieved using electrically conducting Ag epoxy and are
bonded with Au wires of diameter 25 $\mu$m. The magnetotransport experiments
are carried out in a Janis Super Variable Temperature 7TM-SVM cryostat (Janis
Cryogenics, Westerville, OH, USA) equipped with a 7 T superconducting magnet.
A lock-in amplifier (LIA) ac technique is employed for measuring the
magnetotransport properties of the NbN and of the Fe:NbN thin films. The
constant current $I_{\mathrm{ac}}$ is sourced from a Stanford Research SR830
LIA $via$ a resistance decade box, while the longitudinal voltage
$V_{\mathrm{xx}}$ is measured in a phase-locked mode. The lock-in expand
function is employed to enhance the sensitivity of the LIA. All measurements
have been performed at a frequency of 127 Hz, while $I_{\mathrm{ac}}=1$ $\mu$A
for all measurements. The low input current minimizes the thermal drift due to
Joule heating of the samples. The magnetic fields are varied between $-7$ T
and $+7$ T for the magnetoresistance measurements.

Figure 2:  ZFC $\rho_{xx}$ as a function of $T$ for samples #A and #B.

All experimental set-ups including the LIA, the magnet power supply, and the
temperature controller are regulated by means of an indigenously developed
software. Prior to the measurements, the Ohmic characteristics of the Ag epoxy
contacts to the samples are tested with a high resolution Keithley 4200 SCS
source-measure unit (SMU). The longitudinal resistivity $\rho_{\mathrm{xx}}$
is estimated as a function of temperature $T$ and magnetic field $\mu_{0}H$
from the measured longitudinal voltage $V_{\mathrm{xx}}$. The evolution of
thermoresistivity $\rho_{\mathrm{xx}}(T)$ as a function of $T$ is estimated by
cooling the sample both in the absence and in the presence of a transverse
magnetic field $\mu_{0}H_{\perp}$ and of a longitudinal magnetic field
$\mu_{0}H_{\parallel}$. While $\mu_{0}H_{\perp}$ is applied parallel to the
surface normal of the sample, $\mu_{0}H_{\parallel}$ is applied perpendicular
to it. The zero field cooled (ZFC) $\rho_{\mathrm{xx}}-T$ behavior is studied
while the samples are cooled down for $\mu_{0}H_{\perp}=0$
($\mu_{0}H_{\parallel}=0$) and the field cooled (FC) for $\mu_{0}H_{\perp}\neq
0$ ($\mu_{0}H_{\parallel}\neq 0$).

##  III Results

The evolution of the ZFC $\rho_{\mathrm{xx}}$, measured as a function of $T$
in the interval $10\leavevmode\nobreak\ \mathrm{K}\leq{T}\leq
16.5\leavevmode\nobreak\ \mathrm{K}$ for samples #A and #B is reported in Fig.
2. The normal state resistivity $\rho_{\mathrm{xx}}$ of #B at
$T\geq{T_{\mathrm{loc}}^{\mathrm{onset}}}$ is recorded to be $\sim$40
$\Omega\cdot{\mathrm{cm}}$, while for #A, $\rho_{\mathrm{xx}}$ $\sim$22
$\Omega\cdot{\mathrm{cm}}$. The presence of Fe dopants in the NbN matrix is
identified to be the reason for the increase of $\rho_{\mathrm{xx}}$. As
evidenced in Fig. 2, the ZFC $\rho_{\mathrm{xx}}(T)$ of #B presents five
characteristic temperatures labeled by 1, 2, 3, 4 and 5. The characteristic
temperatures 1, 4 and 5 are found also for the ZFC $\rho_{\mathrm{xx}}(T)$ of
#A. These characteristic temperatures are marked as follows: 1:
$T_{\mathrm{loc}}^{\mathrm{onset}}$; 2: $T_{\mathrm{d}}$; 3:
$T_{\mathrm{peak}}$; 4: $T=T^{*}$ and 5: $T_{\mathrm{glo}}^{\mathrm{offset}}$.

It is also evidenced in Fig. 2, that the thermodynamic transition from the
normal state N to the superconducting state S occurs over a temperature range
$\Delta{T_{\mathrm{c}}}$ defined as:

$\Delta{T_{\mathrm{c}}}=\left(T_{\mathrm{loc}}^{\mathrm{onset}}-T_{\mathrm{glo}}^{\mathrm{offset}}\right)$
(2)

Figure 3:  ZFC and FC $\rho_{xx}$ as a function of $T$ under the application
of $\mu_{0}H_{\perp}$ = 0 T, 1 T, 3 T, 5 T and 7 T measured for samples (a) #A
and (b) #B. Transverse $\rho_{\mathrm{xx}}$ as a function of
$\mu_{0}H_{\perp}$ recorded at different $T$ for samples (c) #A and (d) #B.

In the as-grown NbN film #A, the dominant microstructural disorder originates
from the grain boundaries of the polycrystalline film, as reported for similar
polycrystalline and amorphous systems [50, 51, 52, 38, 28]. Such a
3-dimensional (3D) system with only a structural disorder is expected to
undergo a N/S transition with delayed onset of the global superconductivity.
These disordered superconductors are characterized by a finite non-zero value
of $\Delta{T_{\mathrm{c}}}$ [51]. The physical mechanism of this transition is
described in terms of the percolation of bosonic clusters - composed of Cooper
pairs - to form bosonic islands at the $T=T_{\mathrm{loc}}^{\mathrm{onset}}$
indicated in Fig. 2. These bosonic islands percolate to produce a network of
bosonic conduction channels, which in turn replaces the single particle
fermionic conduction channels of the N state of NbN. The grain boundaries
serve as scattering centers for the bosonic islands, leading to the observed
kink at $T=T^{*}$, as indicated in the ZFC $\rho_{\mathrm{xx}}(T)$ curve in
Fig. 2. At $T=T_{\mathrm{glo}}^{\mathrm{offset}}$, the global superconducting
phase sets in, as evidenced in Fig. 2.

However, in #B two more characteristic temperatures indicated by
$T_{\mathrm{d}}$ and $T_{\mathrm{peak}}$ are observed in the evolution of
$\rho_{\mathrm{xx}}(T)$, as presented in Fig. 2. The anomalous peak at
$T=T_{\mathrm{peak}}$ in the $\rho_{\mathrm{xx}}(T)$ points at the existence
of disorder induced electronic phase transitions for the N/S thermodynamic
phase transition in #B, due to the presence of the implanted Fe ions. In order
to understand the mechanism of the electronic processes involved in the
observed behavior of #B, FC $\rho_{\mathrm{xx}}(T)$ measurements are performed
on both #A and #B for $\mu_{0}H_{\perp}$ and $\mu_{0}H_{\parallel}$. The
behavior of $\rho_{\mathrm{xx}}(T)$ for #A and #B under $\mu_{0}H_{\perp}$ = 0
T, 1 T, 3 T, 5 T and 7 T is provided in Figs. 3(a) and 3(b), respectively. It
is observed, that with the increase of $\mu_{0}H_{\perp}$, for both samples #A
and #B $T_{\mathrm{loc}}^{\mathrm{onset}}$ and
$T_{\mathrm{glo}}^{\mathrm{offset}}$ are shifted to lower values compared to
the ones in the ZFC case. The kink in the $\rho_{\mathrm{xx}}(T)$ detected in
#A vanishes upon the application of $\mu_{0}H_{\perp}$. For #B, the anomalous
peak in $\rho_{\mathrm{xx}}(T)$ diminishes in amplitude, $T_{\mathrm{peak}}$
shifts to lower $T$ and vanishes for $\mu_{0}H_{\perp}=3$ T. In addition, the
resistivity minima at $T=T_{\mathrm{d}}$ and the kink at $T=T^{*}$ are also
found to quench for $\mu_{0}H_{\perp}\geq 3$ T. The resistivity
$\rho_{\mathrm{xx}}(\mu_{0}H)$ of #A and #B is recorded at different $T$ for
both $\mu_{0}H_{\perp}$ and $\mu_{0}H_{\parallel}$. In Figs. 3(c) and 3(d),
the $\rho_{\mathrm{xx}}(\mu_{0}H)$ of #A and #B measured for an applied
$\mu_{0}H_{\perp}$ are presented. In agreement with the results of
$\rho_{\mathrm{xx}}(T)$ measurements, the anomalous peak is also observed in
the measured $\rho_{\mathrm{xx}}(\mu_{0}H)$ recorded for $\mu_{0}H_{\perp}$.
The five characteristic temperatures observed in the ZFC
$\rho_{\mathrm{xx}}(T)$ behavior of #A and #B along with
$\Delta{T_{\mathrm{c}}}$ are summarized in Table 2.

Table 2: Characteristic temperatures of samples #A and #B recorded for ZFC $\rho_{\mathrm{xx}}(T)$. Sample |  $T_{\mathrm{loc,ZFC}}^{\mathrm{onset}}$ (K) |  $T_{\mathrm{d,ZFC}}$ (K) |  $T_{\mathrm{peak,ZFC}}$ (K) |  $T_{\mathrm{ZFC}}^{*}$ (K) |  $T_{\mathrm{glo,ZFC}}^{\mathrm{offset}}$ (K) |  $\Delta{\mathrm{T}}$ (K)
---|---|---|---|---|---|---
A | 15.72 | - | - | 15.27 | 15.05 | 0.67
B | 15.1 | 14.71 | 14.60 | 14.305 | 13.5 | 1.6

The $\rho_{\mathrm{xx}}(T)$ and $\rho_{\mathrm{xx}}(\mu_{0}H)$ have been also
measured for an applied $\mu_{0}H_{\perp}$. From the evolution of
$\rho_{\mathrm{xx}}(T)$ and $\rho_{\mathrm{xx}}(\mu_{0}H)$ measured at
different $T$ and as a function of $\mu_{0}H_{\perp}$ and
$\mu_{0}H_{\parallel}$, the characteristic temperatures namely -
$T_{\mathrm{loc,\perp/\parallel}}^{\mathrm{onset}}$,
$T_{\mathrm{d,\perp/\parallel}}$, $T_{\mathrm{peak,\perp/\parallel}}$,
$T_{\perp/\parallel}^{*}$ and
$T_{\mathrm{glo,\perp/\parallel}}^{\mathrm{offset}}$ \- are estimated. The
subscripts $\perp$ and $\parallel$ refer to $\mu_{0}H_{\perp}$ and
$\mu_{0}H_{\parallel}$, respectively.

Figure 4:  The behavior of the characteristic temperatures
$T_{\mathrm{loc}}^{\mathrm{onset}}$, $T_{\mathrm{d}}$, $T_{\mathrm{peak}}$,
$T=T^{*}$ and $T_{\mathrm{glo}}^{\mathrm{offset}}$ as a function of (a)
$\mu_{0}H_{\perp}$. (b) $\mu_{0}H_{\perp}$. (c) Evolution of
$T_{\mathrm{loc,\perp/\parallel}}^{\mathrm{onset}}$ recorded for #B.

The behavior of the characteristic temperatures of #B as a function of
$\mu_{0}H_{\perp}$ and $\mu_{0}H_{\parallel}$ is shown in Figs. 4(a) and 4(b),
respectively. While $T_{\mathrm{d,\parallel}}$ and
$T_{\mathrm{peak,\parallel}}$ are found to persist over the whole range of the
applied $\mu_{0}H_{\parallel}$ ($i.e.$
$0\,\mathrm{T}\leq{\mu_{0}H_{\parallel}}\leq 7\,\mathrm{T}$),
$T_{\mathrm{d,\perp}}$ and $T_{\mathrm{peak,\perp}}$ are suppressed for
$\mu_{0}H_{\perp}>{3\,\mathrm{T}}$. This anisotropy in the behavior of
$T_{\mathrm{d}}$ and $T_{\mathrm{peak}}$ for magnetic fields applied parallel
and perpendicular to the surface normal is not observed in similar disordered
systems such as B-doped diamond [50, 51] and $a$-InO [53] thin films. A
comparison between $T_{\mathrm{loc,\perp/\parallel}}^{\mathrm{onset}}$ and
$T_{\mathrm{glo,\perp/\parallel}}^{\mathrm{offset}}$ as a function of
$\mu_{0}H$ for #B is shown in Fig. 4(c). The similarity of these two
characteristic temperatures for #B indicates that the system is in a 3D regime
and a 3D to 2D crossover of dimensionality, as the one reported for ultrathin
TiN layers, can be ruled out [52]. The calculated full-width-and-half-maxima
of $T_{\mathrm{Peak}}$ and $\Delta{T_{\mathrm{c}}}$ estimated for samples #A
and #B recorded for $\mu_{0}H_{\parallel}$ and $\mu_{0}H_{\perp}$ are
summarized in Table 3.

Highly disordered superconducting thin films can show a reentrant resistive
behavior, either due to a dimensionality crossover [52] or due to the
application of a strong $\mu_{0}H$ [53]. The implantation of Fe into the NbN
matrix enhances the disorder of the system compared to that of the as-grown
NbN. The reentrant $T$ dependence of the normal state is referred to as a
$N$-shaped $T$ dependence [52], as the one recorded for the ZFC
$\rho_{\mathrm{xx}}(T)$ of #B. Such $N$-shaped $T$ dependence was reported in
high $T_{\mathrm{c}}$ superconductors (HTS) [54, 55, 56, 57, 58, 59, 60] and
ultrathin films of conventional superconductors [52, 53, 50, 51]. Even though
this phenomenon is ubiquitious, it still remains a subject of debate. In HTSs,
this effect is described in terms of scaling functions [59, 60] or attributed
to the emergence of the pseudogap phase [61].

Table 3: The full width and half maxima of the $T_{\mathrm{Peak}}$ and $\Delta{T_{\mathrm{c}}}$ estimated for samples #A and #B recorded for $\mu_{0}H_{\parallel}$ and $\mu_{0}H_{\perp}$. $\mu_{0}H_{\parallel/\perp}$ (T) | $\Delta{\mathrm{T}_{\mathrm{Peak}}^{\mathrm{A}}}(\mathrm{K})$ | $\Delta{\mathrm{T}_{\mathrm{Peak}}^{\mathrm{B}}}(\mathrm{K})$ | $\Delta{\mathrm{T}_{\mathrm{c}}^{\mathrm{A}}}(\mathrm{K})$ | $\Delta{\mathrm{T}_{\mathrm{c}}^{\mathrm{B}}}(\mathrm{K})$
---|---|---|---|---
| $\mu_{0}H_{\perp}$ | $\mu_{0}H_{\parallel}$ | $\mu_{0}H_{\perp}$ | $\mu_{0}H_{\parallel}$ | $\mu_{0}H_{\perp}$ | $\mu_{0}H_{\parallel}$ | $\mu_{0}H_{\parallel}$ | $\mu_{0}H_{\parallel}$
0 | - | - | 0.0653 | 0.0653 | 0.67 | 0.67 | 1.6 | 1.6
1 | - | - | 0.1376 | 0.1337 | 0.72 | 0.94 | 1.64 | 1.71
2 | - | - | 0.0858 | 0.1514 | - | - | 1.5 | 1.34
3 | - | - | - | 0.1392 | 0.74 | 0.94 | 1.25 | 1.54
4 | - | - | - | 0.0841 | - | - | 1.23 | 1.25
5 | - | - | - | - | 0.8 | 0.96 | 1.12 | 1.37
6 | - | - | - | - | - | - | 1.31 | 1.05
7 | - | - | - | - | 0.78 | 0.91 | 0.96 | 1.18
Figure 5:  ZFC $\rho_{\mathrm{xx}}$ as a function of $T$. The six electonic
phases and the phase boundaries of #B. are evidenced

In the case of ultrathin $<15$ nm layers of conventional superconductors,
quantum contributions due to a 3D to 2D dimensionality crossover have been
identified as the mechanism behind the observed $N$-shaped $T$ dependence of
$\rho_{\mathrm{xx}}$. In 3D films of disordered granular superconductors such
as B-doped diamond [50, 51], $a$-InO [53] or AlGe [62] this behavior is
interpreted within the framework of a N-BI-S transition originating from
granularity induced disorder.

In the ZFC $\rho_{\mathrm{xx}}(T)$ of #B a series of electronic phases can be
identified. These electronic phases are marked by EP-1, EP-2, EP-3. EP-4, EP-5
and EP-6 in Fig. 5. The phase boundaries of the adjacent electronic phases of
#B correspond to the characteristic temperatures
$T_{\mathrm{loc}}^{\mathrm{onset}}$, $T_{\mathrm{d}}$, $T_{\mathrm{peak}}$,
$T^{*}$ and $T_{\mathrm{glo}}^{\mathrm{offset}}$, as described earlier.

The evolution of the six electronic phases, $i.e.$ from EP-1 to EP-6, across
the phase boundaries is explained using an empirical model of competing
impurity scattering and a conduction network of fermionic conduction channel
and bosonic conduction channels across a N-BI-S phase transition. A schematic
of this empirical model is presented in Fig. 6. The Fe implantation dose of
$\left(1\times 10^{14}\right)$ at/cm3 results in a highly dilute Fe doped
system, restricting the Fe-Fe interaction to the paramagnetic limit. Since
after implantation #B is not thermally treated, the Fe ions are incorporated
into the NbN lattice randomly. Moreover, the incorporation efficiency of
implanted Fe ions is different for NbN grains with different crystallographic
orientations. The disorder of the Fe implanted NbN system stems from an
extrinsic granularity and an intrinsic one. The disorder due to the
polycrystalline texture and grain boundaries of the Fe:NbN lattice is the
extrinsic granularity. On the other hand, the electronic disorder due to the
randomly implanted Fe ions results in modulations of the chemical potential
and thus in intrinsic electronic granularity. The intrinsic granularity plays
a significant role in defining the observed electronic properties of #B.

The mechanisms leading to the specific electronic phases are:

(i) Phase EP-1: The normal electronic phase in the ZFC $\rho_{\mathrm{xx}}(T)$
of Fe:NbN sets in for $T>T_{\mathrm{loc}}^{\mathrm{onset}}$, as shown in Fig.
5. This electronic phase is represented in Fig. 6(a) and is referred to as
EP-1. For $T>T_{\mathrm{loc}}^{\mathrm{onset}}$, the electrical transport
takes place $via$ the percolation of thermally activated fermionic conduction
channels with normal electrons being the medium of transport. These fermionic
conduction channels are marked in the schematic diagram as solid lines and the
fermions are visualized by solid ellipses. The Fe impurities in the matrix are
represented by solid circles. The resistivity behavior of this phase is the
same as the one of the N phase of Fe:NbN.

(ii) Phase EP-2: As $T$ decreases, a superconducting gap opens at
$T=T_{\mathrm{loc}}^{\mathrm{onset}}$. A spatial inhomogeneity associated with
the superconducting gap due to the intrinsic granularity leads to the
formation of bosonic clusters. With a gradual decrease of $T$ below
$T=T_{\mathrm{loc}}^{\mathrm{onset}}$, the density of the bosonic clusters
increases and becomes locally phase locked, resulting in the initial decrease
of $\rho_{\mathrm{xx}}$ with decreasing $T$ through the percolation of the
phase locked bosonic clusters merging into bosonic islands. These islands are
responsible for the bosonic conduction channels for electronic transport in
the temperature range $T_{\mathrm{d}}<T<T_{\mathrm{loc}}^{\mathrm{onset}}$.
The bosonic conduction channels and the bosonic islands are represented in
Fig. 6(b) by connectors and dumb bells, respectively. In this regime, a
competition of the bosonic conduction channels with the fermionic conduction
channels leads to a decrease in $\rho_{\mathrm{xx}}$.

Figure 6:  Schematic of the empirical model to describe the N-BI-S transition
in #B. The connector lines connecting the ellipses and dumbbells represent
fermionic conduction channels and bosonic conduction channels, respectively.
The normal state fermions and the Cooper pairs are shown by ellipses and
dumbbells, respectively, while the Fe impurities are depicted by solid
circles. The electronic phase transitions taking place as the sample #B is
cooled from $T>T_{\mathrm{loc}}^{\mathrm{onset}}$ down to
$T\leq{T_{\mathrm{loc}}^{\mathrm{onset}}}$ are: (a) Phase EP-1 for
$T>T_{\mathrm{loc}}^{\mathrm{onset}}$. (b) Phase EP-2 for
$T_{\mathrm{d}}<T<T_{\mathrm{loc}}^{\mathrm{onset}}$. (c) Phase EP-3 for
$T^{\mathrm{peak}}<T<T_{\mathrm{d}}$. (d) Phase EP-4 for
$T_{\mathrm{Peak}}\leq{T}\leq{T^{*}}$. (e) Phase EP-5 for
$T^{*}\leq{T}\leq{T_{\mathrm{glo}}^{\mathrm{offset}}}$ and (f) Phase EP-6 for
$T\leq{T_{\mathrm{glo}}^{\mathrm{offset}}}$.

(iii) Phase EP-3: As the $T$ is further reduced, $i.e.$ for
$T<T_{\mathrm{d}}$, in the presence of the intrinsic granularity due to the
paramagnetic Fe impurities, the bosonic conduction channels dominate over the
fermionic conduction channels $via$ the removal of single fermions or normal
electrons from the fermionic conduction channels to fuel the bosonic
conduction channels. This process takes place by forming groups of phase-
locked bosonic clusters and producing isolated BIs. The consequent anomalous
increase of $\rho_{\mathrm{xx}}$ for $T^{\mathrm{peak}}<T<T_{\mathrm{d}}$ is
likely due to two mechanisms, $i.e.$

\- removal of the single fermions from the fermionic conduction channels to
estblish the percolation path for the bosonic conduction channels;

\- scattering of the bosonic clusters from the isolated and randomly
distributed Fe ions in the matrix.

The anomalous increase of $\rho_{\mathrm{xx}}$ continues up to
$T=T_{\mathrm{peak}}$ The electronic phase EP-3 is referred to as a bosonic
insulator phase and is represented in Fig. 6(c).

(iv) Phase EP-4: As $T$ is further lowered, the density of the bosonic
conduction channels increases $via$ the percolation of more bosonic islands,
leading to a decrease in $\rho_{\mathrm{xx}}$, as more and more phase locked
coherent bosonic clusters and islands percolate in the bulk of the system. The
dissipationless transport due to the coherent bosonic islands dominates over
the scattering from the paramagnetic Fe impurities, leading to the gradual
decrease of $\rho_{\mathrm{xx}}$ until $T=T^{*}$. The EP-4 phase is shown in
Fig. 6(d).

(v) Phase EP-5: At $T=T^{*}$, a kink in $\rho_{\mathrm{xx}}(T)$ is observed,
which is attributed to the scattering of the percolating bosonic islands due
the extrinsic granularity. This is also detected in the as-grown NbN #A, as
shown in Fig. 6(e).

(vi) Phase EP-6: At $T\leq{T_{\mathrm{loc}}^{\mathrm{onset}}}$, the global
phase coherence sets in between all the phase-locked bosonic islands and the
electrical transport is dominated solely by the bosonic conduction channels,
leading to a superconducting state of the Fe implanted NbN layer, as sketched
in Fig. 6(f).

The empirical model also describes the behavior of $\rho_{\mathrm{xx}}(T)$ for
applied $\mu_{0}H_{\perp}$ and $\mu_{0}H_{\parallel}$. The presence of
$\mu_{0}H_{\perp}$ breaks the Cooper pairing and shifts
$T_{\mathrm{loc}}^{\mathrm{onset}}$ to a lower $T$. This also suppresses the
onset of the bosonic insulator phase. In addition, the formation of the
Abrikosov vortices for $\mu_{0}H_{\perp}$ also plays a role in the suppression
of the BI phase and of the intrinsic granularity on the electronic transport
of #B.

For $\mu_{0}H_{\parallel}$ however, the fields required to break the Cooper
pairs are generally orders of magnitude stronger than the ones for
$\mu_{0}H_{\perp}$ and do not influence the formation of the BI state. As a
result both $T^{\mathrm{peak,\parallel}}$and $T_{\mathrm{d,\parallel}}$
persist up to the highest applied field $\mu_{0}H_{\parallel}=7\,\mathrm{T}$.

##  IV Conclusions

In conclusion, 100 nm thick crystalline films of NbN are deposited by reactive
magnetron sputtering on single crystalline GaN templates grown on epi ready
$c$-plane sapphire substrates. Paramagnetic Fe doping of the NbN layers is
achieved by ion implantation using Fe3O4 as the source for Fe ions. An
implantation energy of 35 keV and a dose of $\left(1\times 10^{14}\right)$
at/cm3 result in a highly dilute Fe doped system, restricting the Fe-Fe
interaction to the paramagnetic limit. Low-$T$/high-$\mu_{0}H$
magnetotransport measurements confirm, that the Fe doping does not suppress
the superconductivity of the sputtered NbN thin films, but decreases the
superconducting transition temperature. A reentrant resistive BI phase is
observed in the Fe doped NbN samples, which is explained by an empirical model
of a competition between the percolation of bosonic conduction channels at the
expense of fermionic conduction channels and by the scattering of the BI as a
result of intrinsic granularity due to the random Fe dopants. The observation
of a robust superconductivity in the dilute magnetic conventional
superconductor Fe:NbN mediated $via$ percolation of bosonic insulator states
is foreseen to violate the symmetry of electron-like and hole-like excitations
due to the formation of subgap bound Andreev states in the vicinity of
magnetic impurities, leading to giant thermoelectric effects [63]. A system
like the one reported in this work is expected to find applications in zero
biased thermoelectric bolometers with reduced power dissipation in large scale
multi-pixel arrays and in hybrid quantum interference devices (HyQUID) [64].
Moreover, these systems are the workbench for understanding quantum emergent
phenomena, including gapless superconductivity, triplet Cooper pairings, YSR
states and odd frequency superconductivity.

## Acknowledgements

The work was funded by the Austrian Science Fund (FWF) through Projects No.
P26830 and No. P31423.

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# Some results on the norm of finite groups

Mark L. Lewis, Zhencai Shen and Quanfu Yan

###### Abstract

Let $G$ be a finite group and $N_{\Omega}(G)$ be the intersection of the
normalizers of all subgroups belonging to the set $\Omega(G),$ where
$\Omega(G)$ is a set of all subgroups of $G$ which have some theoretical group
property. In this paper, we show that $N_{\Omega}(G)=Z_{\infty}(G)$ if
$\Omega(G)$ is one of the following: (i) the set of all self-normalizing
subgroups of $G$; (ii) the set of all subgroups of $G$ satisfying the
subnormalizer condition in $G$; (iii) the set of all pronormal subgroups of
$G$; (iv) the set of all $\mathscr{H}$-subgroups of $G$; (v) the set of all
weakly normal subgroups of $G$; (vi) the set of all $NE$-subgroups of $G$.

##  1 Introduction

All the groups considered in this paper are finite and $G$ denotes a finite
group.The notation and terminology used in this paper are standard, as in [5]
In 1935, R. Baer [1] considered the norm $N(G)$ of $G$, which is the
intersection of normalizers of all subgroups of $G$. In 1953, R. Baer [2]
considered the intersection of normalizers of all Sylow subgroups of $G$ and
proved that it coincides with the hypercenter $Z_{\infty}(G)$, where
$Z_{\infty}(G)$ is the terminal term of the upper central series of the group
$G$.

Let $\Omega(G)$ be the set of all subgroups of the group $G$ which have some
theoretical group property. Define $N_{\Omega}(G)$, called the $\Omega$-norm
$G$, be the intersection of the normalizers of all subgroups belonging to the
set $\Omega(G).$ In the case $\Omega(G)=\emptyset,$ we assume that
$G=N_{\Omega}(G).$ It is clear that the norm $N(G)$ is the $\Omega$-norm of
$G$ for the set $\Omega(G)$, which consists of all subgroups of $G$, and
$Z_{\infty}(G)$ is the $\Omega$-norm of $G$ for the set $\Omega(G)$, which
consists of all Sylow subgroups of $G$.

In this paper, we prove the following:

Theorem 1.1. Let $G$ be a group. Then $N_{\mathcal{SN}}(G)=Z_{\infty}(G),$
where $\mathcal{SN}(G)$ is the set of all self-normalizing subgroups of $G$.

Notice that the norm $N_{\mathcal{SN}}(G)$ can be viewed as the intersection
of all self-normalizing subgroups of $G.$ Similarly, we consider the
intersection of all self-centralizing subgroups of $G$ and get the following:

Theorem 1.2. Let $G$ be a group. Then $Z(G)$ coincides with the intersection,
denoted by $C_{\mathcal{SC}}(G)$, of all subgroups in $\mathcal{SC}(G)$, where
$\mathcal{SC}(G)$ is the set of all self-centralizing subgroups of $G$.

Remark 1.3. Notice that both $N_{\mathcal{SC}}(G)=Z_{\infty}(G)$ and
$N_{\mathcal{SC}}(G)=Z(G)$ are not always true. For example, let $G=Q_{16}$ be
the generalised quaternion group of order 16. Then
$Z(G)<N_{\mathcal{SC}}(G)=Z_{2}(G)<Z_{3}(G)=G.$ This is the reason why we
consider the intersection $C_{\mathcal{SC}}(G)$ of all self-centralizing
subgroups of $G$.

Theorem 1.4. Let $G$ be a group and $p$ be a prime. Let $\Omega_{p}(G)$ be the
set of all $p$-groups in $\Omega(G),$ where $\Omega(G)$ is one of the
following:
(i) the set of all pronormal subgroups of $G$.
(ii) the set of all $\mathscr{H}$-subgroups of $G$.
(iii) the set of all weakly normal subgroups of $G$.
(iv) the set of all subgroups of $G$ satisfying the subnormalizer condition in
$G$.
(v) the set of all $NE$-subgroups of $G$.
Then
(a) $N_{\Omega}(G)$ coincides with the intersection of normalizers of all
sylow subgroups of $G.$
(b) if $\Omega_{p}(G)$ is not the case (v), $N_{\Omega_{p}}(G)$ coincides with
the intersection of normalizers of all sylow $p$-subgroups of $G$.

##  2 Definitions

In this section, we recall some definitions and introduce some notation.

A subgroup $H$ of a group $G$ is called pronormal in $G$ if, for any $g\in G$,
there exists $x\in\langle H,H^{g}\rangle$ such that $H^{x}=H^{g}.$ Bianchi in
[4] introduced the concept of $\mathscr{H}$-subgroups as follows:

Definition 2.1. A subgroup $H$ of a group $G$ is an $\mathscr{H}$-subgroup of
$G$ if, for any $g\in G$, $N_{G}(H)\cap H^{g}\leq H.$

In [6], Müller introduces the weak normality:

Definition 2.2. A subgroup $H$ of $G$ is called weakly normal in $G$ if
$H^{g}\leq N_{G}(H)$ implies that $g\in N_{G}(H).$

In [9], Shirong Li introduced the definition of $NE$-subgroups of $G$.

Definition 2.3. A subgroup $H$ is called an $NE$-subgroup of $G$ if
$H=N_{G}(H)\cap H^{G},$ where $H^{G}$ is the normal closure of $H$ in $G.$

In [7], Mysovskikh introduced the following subgroup embedding property.

Definition 2.4. A subgroup $H$ of $G$ is said to satisfy the subnormalizer
condition in $G$ if for every subgroup $K$ of $G$ such that $H\unlhd K$, it
follows that $N_{G}(K)\leq N_{G}(H).$

For a prime $p$, we define the following sets of subgroups of $G$:

(i) $\mathcal{S}(G)$ is the set of all Sylow subgroups of $G$.
(ii) $\mathcal{SN}(G)$ is the set of all self-normalizing subgroups of $G$,
that is, $\mathcal{SF}(G)=\\{H\leq G|H=N_{G}(H)\\}.$
(iii) $\mathcal{SC}(G)$ is the set of all self-centralizing subgroups of $G$,
that is, $\mathcal{SC}(G)=\\{H\leq G|C_{G}(H)\leq H\\}.$
(iv) $\Sigma(G)$ is the set of all subgroups of $G$ satisfying the
subnormalizer condition in $G$.
(v) $\mathcal{S}_{p}(G)$ is the set of all Sylow $p$-subgroups of $G$.
(vi) $\Sigma_{p}(G)$ is the set of all $p$-subgroups of $G$ which belong to
$\Sigma(G)$.

##  3 Proofs of Main Results

To prove Theorem 1.1, we begin by the following obvious lemma.

Lemma 3.1. Let $G$ be a group. Assume that $N$ is normal in $G$ and $N\leq
H\leq G.$ Then $H\in\mathcal{SN}(G)$ if and only if $H/N\in\mathcal{SN}(G/N).$

Proof of Theorem 1.3. Notice that
$Z_{\infty}(G)=\cap_{P\in\mathcal{S}(G)}N_{G}(P)$ by Baer’s result and
$N_{G}(P)=N_{G}(N_{G}(P))\in\mathcal{SN}(G).$ Then $N_{\mathcal{SN}}(G)\leq
Z_{\infty}(G).$ Thus, we only need to show that $Z_{\infty}(G)\leq
N_{\mathcal{SN}}(G).$

We work by induction on the order of $G.$ If $Z(G)=1,$ then $Z_{\infty}(G)=1$
and so $N_{\mathcal{SN}}(G)=1$, we are done. Assume that $Z(G)>1.$ Then
$Z_{\infty}(G/Z(G))\leq N_{\mathcal{SN}}(G/Z(G))$ by induction. Notice that if
$H\in\mathcal{SN}(G),$ then $Z(G)\leq N_{G}(H)=H$. Hence
$H/Z(G)\in\mathcal{SN}(G/Z(G))$ if and only if $H\in\mathcal{SN}(G)$ by Lemma
3.1. It follows that

$\displaystyle N_{\mathcal{SN}}(G/Z(G))$ $\displaystyle=$
$\displaystyle\bigcap_{H/Z(G)\in\mathcal{SN}(G/Z(G))}H/Z(G)$ $\displaystyle=$
$\displaystyle\bigcap_{Z(G)\in H\in\mathcal{SN}(G)}H/Z(G)$ $\displaystyle=$
$\displaystyle\bigcap_{H\in\mathcal{SN}(G)}H/Z(G)$ $\displaystyle=$
$\displaystyle(\bigcap_{H\in\mathcal{SN}}H)/Z(G)$ $\displaystyle=$
$\displaystyle N_{\mathcal{SN}}(G)/Z(G).$

On the other hand, we have $Z_{\infty}(G)/Z(G)=Z_{\infty}(G/Z(G)).$ It follows
that $Z_{\infty}(G)/Z(G)\leq N_{\mathcal{SN}}(G)/Z(G)$ and so
$Z_{\infty}(G)\leq N_{\mathcal{SN}}(G),$ as wanted. $\Box$

The proof of Theorem 1.2 requires the following lemma.

Lemma 3.2. Let $A$ be a maximal abelian subgroup of a group $G$. Then
$A=C_{G}(A)$ and so $A$ is self-centralizing in $G$.

Proof. It is clear that $A\leq C_{G}(A).$ Assume that $A<C_{G}(A).$ Then there
is an element $x\in C_{G}(A)$ but $x\not\in A.$ Consider the group $\langle
A,x\rangle$ generated by $A$ and $x$. As $x\in C_{G}(A),$ we have $\langle
A,x\rangle$ is abelian, contrary to the maximal choice of $A$. Hence
$A=C_{G}(A)$, which is self-centralizing by definition. $\Box$

Proof of Theorem 1.2. Let $H$ be a self-centralizing subgroup of $G$. Notice
that $Z(G)\leq C_{G}(H)\leq H$ and so $Z(G)\leq C_{\mathcal{SC}}(G).$
Let $g\in C_{\mathcal{SC}}(G).$ Let $A$ be any maximal abelian subgroups of
$G$. Then by Lemma 3.2, $g\in A=C_{G}(A)$ and so $A\subseteq C_{G}(g).$ Notice
that for any element $x\in G$, we have $x\in\langle x\rangle$, which must be
contained in a maximal abelian subgroup of $G$. It follows that $G\subseteq
C_{G}(g)$ and so $g\in Z(G),$ as wanted. $\Box$

For Theorem 1.4, we begin with several lemmas.

Lemma 3.3. Let $G$ be a group. Suppose that $H$ satisfies the subnormalizer
condition in $G$ and $M$ is a subgroup of $G.$ Then
(1) If $H\leq M$, then $H$ satisfies the subnormalizer condition in $M$.
(2) If $H$ is subnormal in $G$, then $H$ is normal in $G$.

Proof. (1) Assume that $K$ is a subgroup of $M$ such that $H$ is normal in
$K.$ Since $H$ satisfies the subnormalizer condition in $G$, we have
$N_{G}(K)\leq N_{G}(H)$. Thus, $M\cap N_{G}(K)\leq M\cap N_{G}(H)$ and so
$N_{M}(K)\leq N_{M}(H)$, as wanted.

(2) Assume that $H\unlhd\unlhd G$. By induction on the length of a subnormal
series from $H$ to $G$, we may suppose that $H\unlhd K\unlhd G.$ Since $H$
satisfies the subnormalizer condition in $G$, we have $N_{G}(K)\leq N_{G}(H).$
Notice that $G=N_{G}(K).$ It follows that $G=N_{G}(H)$ and so $H\unlhd G,$ as
wanted.

Lemma 3.4. (1) If $H$ is a pronormal subgroup of a group $G$, then $H$ is a
weakly normal subgroup of $G$.
(2) If $H$ is a $\mathscr{H}$-subgroup of a group $G$, then $H$ is a weakly
normal subgroup of $G$.
(3) If $H$ a weakly normal subgroup of a group $G$, then $H$ satisfies the
subnormalizer condition in $G$.
(4) If $H$ an $NE$-subgroup of a group $G$, then $H$ satisfies the
subnormalizer condition in $G$.

Proof. For (1),(2) and (3), see [3]. Now we prove (4). Assume that $K$ is a
subgroup of $G$ such that $H$ is normal in $K.$ As $H$ an $NE$-subgroup of
$G$, we have $H=N_{G}(H)\cap H^{G}=N_{H^{G}}(H).$ Let $X=N_{G}(K).$ Consider
$H^{X}$, the normal closure of $H$ in $X$. It is easy to see that $H\leq
H^{X}\leq K$ and so $H=N_{H^{X}}(H)$ and $H\unlhd H^{X}.$ It follows that
$H^{X}\leq N_{H^{X}}(H)=H$ and so $H=H^{X}\unlhd X.$ Hence $X=N_{G}(K)\leq
N_{G}(H)$, which indicates that $H$ satisfies the subnormalizer condition in
$G$. $\Box$

Proposition 3.5. Let $G$ be a group. Then
$\mathcal{SN}(G)=\\{N_{G}(H)|H\in\Sigma(G)\\}.$

Proof. Let $S$ be a self-normalizing subgroup of $G$. Then $S=N_{G}(S).$
Assume that $K$ is a subgroup of $G$ such that $S$ is normal in $K.$ Clearly,
$N_{G}(K)=N_{G}(S)$ and so $H\in\Sigma(G).$ It follows that
$S=N_{G}(S)\in\\{N_{G}(H)|H\in\Sigma(G)\\}.$
Let $H\in\Sigma(G)$. We show that $N_{G}(N_{G}(H))=N_{G}(H).$ Notice that
$H\unlhd N_{G}(H)\unlhd N_{G}(N_{G}(H))$. Then by Lemma 3.3(1) and (2), we
have $H\unlhd N_{G}(N_{G}(H))$ and so $N_{G}(N_{G}(H))\leq N_{G}(H).$ Now it
is obvious that $N_{G}(N_{G}(H))=N_{G}(H)$ and so
$N_{G}(H)\in\mathcal{SN}(G).$ Hence,
$\mathcal{SN}(G)=\\{N_{G}(H)|H\in\Sigma(G)\\}.$ $\Box$

Proof of Theorem 1.4. (a) Assume first $\Omega(G)$ is one of the first four
cases. Then it is clear that $\Omega(G)$ contains all Sylow subgroups of $G$.
Hence, by Lemma 3.4(1), (2) and (3), we have
$\mathcal{S}(G)\subseteq\Omega(G)\subseteq\Sigma(G).$ Thus, $N_{\Sigma}(G)\leq
N_{\Omega}(G)\leq Z_{\infty}(G)$ by definition. It follows from Theorem 1.1
and Proposition 3.5, $N_{\Sigma}(G)=Z_{\infty}(G)$ and so
$N_{\Omega}(G)=Z_{\infty}(G).$
Now assume that $\Omega(G)$ is the set of all $NE$-subgroups of $G$. By
Theorem 1.1 and Lemma 3.4(4), we have $Z_{\infty}(G)\leq N_{\Omega}(G).$ Let
$X=N_{G}(P),$ where $P$ is any Sylow subgroup of $G.$ Then we have
$X=N_{G}(X)$, which indicates that $X$ is an $NE$-subgroup of $G.$ Hence,
$N_{\Omega}(G)\leq N_{G}(X)=X=N_{G}(P).$ By the arbitrariness of $P$, we have
$N_{\Omega}(G)\leq Z_{\infty}(G)$ and so $Z_{\infty}(G)=N_{\Omega}(G).$

(b) Notice that
$\mathcal{S}_{p}(G)\subseteq\Omega_{p}(G)\subseteq\Sigma_{p}(G).$ Hence
$N_{\Sigma_{p}}(G)\leq N_{\mathcal{S}_{p}}(G)$ by definition. Now we show that
$N_{\mathcal{S}_{p}}(G)\leq N_{G}(H)$ for all $H\in\Sigma_{p}(G).$
Let $H$ be a $p$-subgroup of $G$ satisfying the subnormalizer condition in
$G$. Then there exists a Sylow $p$-subgroup $P$ of $G$ containing $H$. Notice
that $H\unlhd\unlhd P\unlhd N_{G}(P).$ Then by Lemma 3.3(1) and (2), we have
$H\unlhd N_{G}(P)$ and so $N_{\mathcal{S}_{p}}(G)\leq N_{G}(P)\leq N_{G}(H).$
By the arbitrariness of $H$, we have $N_{\mathcal{S}_{p}}(G)\leq
N_{\Sigma_{p}}(G)$ and so (b) follows. $\Box$

Remark 3.6. Let $\Omega(G)$ is the set of all $NE$-subgroups of $G$. The
statement $N_{\Omega_{p}}(G)=N_{\mathcal{S}_{p}}(G)$ is not always true. For
example, let $G=A_{5}$ and $H$ be a Sylow 5-subgroup of $G$. Then we have
$H^{G}=G$ and so $N_{G}(H)\cap H^{G}=N_{G}(H)\cong D_{10},$ which has order
10. So $H$ is not an $NE$-subgroup of $G.$ It follows that
$\Omega_{5}(G)=\emptyset$ and so
$N_{\mathcal{S}_{5}}(G)<N_{G}(H)<G=N_{\Omega_{5}}(G).$

##  4 Applications

A group $G$ is said to be a $\mathscr{T}$-group if every subnormal subgroup of
$G$ is normal in $G$. In [3], A. Ballester-Bolinches proved that $G$ is a
solvable $\mathscr{T}$-group if and only if every subgroup of $G$ satisfies
the subnormalizer condition in $G.$ E. Shenkman in [8] proved that the norm
$N(G)$ is contained in the second center of $G$. Now it is easy to see the
following:

Corollary 4.1. Let $G$ be solvable $\mathscr{T}$-group. Then
$Z_{\infty}(G)=N(G)\leq Z_{2}(G),$ where $Z_{2}(G)$ is the second term of the
upper central series of the group $G$.

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