#  $\textrm{H}_{3}$S is a high-$\kappa$ superconductor with columnar pinning
defects

Miodrag L. Kulić  Institute for Theoretical Physics, Goethe-University
Frankfurt am Main, Germany

###### Abstract

Recently, the existence of superconductivity in $\mathrm{H_{3}S}$ and other
high-pressure hydrides is called into question, because of some flawed
magnetic measurements. In Ref. Kulic-RTSC-v1 ) the SCPC model is proposed,
where strong pinning of vortices is due to long columnar defects in
$\mathrm{H_{3}S}$ \- with lengths $\mathrm{L}$ of the order of vortex lengths
$\mathrm{L_{v}}$. Two relevant type of experiments in magnetic fields are
explained by this model: (1) Reduction (with respect to standard
superconductors) of the thermal broadening of resistance ($\mathit{TBR}$) in
magnetic field $\mathrm{h=H/H_{c2}}$, $\mathrm{\mathrm{\delta t_{c}(h)}}$, is
governed by the small parameter $\mathrm{C=}$$\mathrm{\xi_{0}}/\mathrm{L}$, i.
e. $\mathrm{\mathrm{\delta t_{c}^{scpc}(h)}\sim C^{1/2}h^{1/2}}$ with
$\mathrm{C\sim 10^{-3}}$. This gives $\mathrm{\mathrm{\delta
t_{c}^{scpc}(h)}\apprle 0.01}$ for $\mathrm{h\lesssim 0.01}$ and
$\mathrm{L}\sim 1$ $\mu m$, which is in satisfactory agreement with
measurements in $\mathrm{H_{3}S}$ Eremets-Meissner-2022 . The $\mathit{TBR}$
measurements give, that in $\mathrm{H_{3}S}$ there is a large shift of the
irreversible line $\mathrm{B_{irr}(T)}$ towards the $\mathrm{B_{c2}(T)}$ line,
with $\mathit{\mathrm{B_{ir}^{(H_{3}S)}\sim C^{-1}(1-t)^{2}}}$ instead of
$\mathrm{B_{ir}\sim(1-t)^{3/2}}$ ($\mathrm{t=T/T_{c}}$) - in standard
superconductors with point defects. (2) In ZFC (zero field cooled) experiments
on penetration of the magnetic field ($\mathit{PMF}$) in $\mathrm{H_{3}S}$
Troyan , the latter reaches the center of a superconducting disk at much
larger external fields, i. e. for $\mathrm{B_{0}\gg B_{c1}}$. The later is due
to the pronounced pinning of vortices, but not to the Meissner effect. The
calculated $\mathrm{T}$\- dependence of the penetrated, perpendicular
$\mathrm{B_{\perp}(T)}$ and parallel $\mathrm{B_{\parallel}}(\mathrm{T})$,
magnetic field into the sample is in satisfactory agreement with the
experimental results for $\mathrm{H_{3}S}$ in Troyan , where
$\mathrm{B_{\perp}>B_{\parallel}}$. Problems related to measurements of the
Meissner effect in $\mathrm{H_{3}S}$ are also discussed. The SCPC model,
applied on the measurements in the bulk $\mathrm{H_{3}S}$ sample, predicts,
that the latter is a high-$\kappa$ superconductor with $\xi_{0}\approx(15-20)$
Å , $\lambda_{0}\approx(1-2)\times 10^{3}$Å ; $\kappa\approx(50-100)$,
$\mathrm{\mathrm{\mu_{0}}H_{c1}(0)\approx}(18-60)$ $\mathrm{m\mathrm{T}}$,
$\mathrm{\mathrm{\mu_{0}}H_{c0}}$$\approx(0,6-1,1)$ $\mathrm{\mathrm{T},}$
$\mathrm{\mathrm{\mu_{0}}H}_{c2}(0)\approx(80-140$) $\mathrm{T}$.

##  I Introduction

The first almost room-temperature superconductor was reported in 2015 in
sulphur-hydrides ($H_{3}S$) with $\mathrm{T_{c}}$$\approx 203$ $\mathrm{K}$
under high pressure $P\approx 150$ $GPa(\approx 1.5$ $\mathrm{Mbar})$, which
is based on resistivity measurements Drozdov-2015 . This finding opens a new
frontier in physics and a number of other $\mathrm{HP}$-hydrides were
predicted before these were synthesized thereafter. Let us mention some of
them with $\mathrm{T_{c}}$$\geq 200$ $\mathrm{K}$ \- such as
$\mathrm{LaH_{10}}$with $T_{c}\approx 250$ $\mathrm{K}$,
$\mathit{P\mathrm{\approx 190}}$ $\mathrm{GPa}$ LaH10 ; $\mathrm{LaYH}_{x}$
with $\mathrm{T_{c}}$$\mathrm{=253}$ $\mathrm{K}$, $\mathrm{P}$$\approx 190$
$\mathit{GPa}$ Semenok . In all of them $\mathrm{T_{c}}$ goes down by
increasing magnetic field, compatible with the standard theory of
superconductivity. There are also reports on the room temperature
superconductivity in the CSH hydride - a superconductor based on C, S and H,
with $\mathrm{T_{c}}$$\mathrm{=287}$ $\mathrm{K}$ at $\mathrm{P}$$\approx 267$
$\mathrm{GPa}$ Snider . However, this result was not yet confirmed, neither
experimentally nor theoretically by other groups.

There is also theoretical support for the (almost) room-temperature
superconductivity in HP-hydrides, which are based on the calculated critical
temperature $\mathrm{T_{c}}$ in the microscopic Migdal-Eliashberg theory for
superconductivity - which is due to the electron-phonon interaction. Moreover,
the theoretical prediction of superconductivity in HP-hydrides Duan  is a rare
example in the physics of superconductivity, that the theory goes ahead of
experiments, by predicting $\mathrm{T_{c}(\approx 200}$ $\mathrm{K)}$ in
$\mathrm{H_{3}S}$. Note, that in Drozdov-2015  the $\mathrm{H_{2}S}$ structure
is proposed, while in Mazin-H3S  it is argued, that at high pressures the
phase diagram favors decomposition of $\mathrm{H_{2}S}$ into $\mathrm{H_{3}S}$
and pure $\mathrm{S}$.

A main proof for superconductivity is the existence of the Meissner effect,
where the magnetic field $\mathrm{H<H_{c1}}$ \- applied above
$\mathrm{T_{c}}$, is $\mathit{expelled}$ from the sample at temperatures below
$\mathit{\mathrm{T_{c}}}$ \- the field cooled ($\mathrm{FC}$) experiment.
However, until now the Meissner effect is not proved experimentally in HP-
hydrides, which caused justified criticism on this subject in Hirsch-1-1
-Hirsch-Marsiglio-arXiv V5-2022-1 . The failure to measure the Meissner effect
in $\mathrm{H_{3}S}$ is due to the following reasons: ($\mathit{i}$) Magnetic
measurements in small samples are delicate; ($\mathit{ii}$) The existence of
some extrinsic paramagnetic effects in samples. In that respect, some
contradictory experimental results and their inconsistent theoretical
interpretations \- given in Eremets-Meissner-2022 ,Drozdov-2015 ,Minkov-
Meissner-2021 , were among the reasons that Hirsch and Marsiglio even called
into question the existence of superconductivity in $\mathrm{HP}$-hydrides
Hirsch-1-1 -Hirsch-Marsiglio-arXiv V5-2022-1 . In the following, we are going
to show that some magnetic measurements in HS can refute this skepticism.

The content of the article is following. In Section $\mathrm{II}$ the SCPC
model - firstly introduced in Kulic-RTSC-v1  for hard type-II superconductors
with strong columnar pinning centers, is further elaborated and applied to the
magnetic measurements in the $\mathrm{H_{3}S}$ superconductor. This model
proposes, that the vortex pinning is due to long columnar defects
($\mathrm{L\sim L_{v}\gg\xi_{0}}$) \- with the radius of the cross-section
$\mathrm{r\sim\xi_{0}}$. In that case, the core and electromagnetic pinning
contribute almost equally to the elementary pinning energy $\mathrm{U_{p}}.$
This is an optimal situation for pinning in a superconductor, since there is a
maximal gain in the condensation and electromagnetic vortex energy . This
gives a maximal pinning force if the superconductivity is fully suppressed in
the columnar defects, i.e. for $\Delta=0$ inside a defect. It seems that
$\Delta$ is finite in $\mathrm{H_{3}S}$, but $(\Delta<\Delta_{0})$ and the
critical current density is smaller than the maximal one, i.e.
$\mathrm{j_{co}=\mathrm{\eta_{col}j_{c0}^{max}}}$ with
$\mathrm{\eta_{col}<1}$. In Section $\mathrm{III}$ the reduction of the
temperature broadening of resistance ($\mathit{TBR}$) in a magnetic field,
$\delta t_{c}^{exp}(h)$ (with $\mathrm{h=H/H_{c2}}$), in HP-hydrides is
studied in the SCPC model and applied to $\mathrm{H_{3}S}$. The temperature
dependence of the irreversibility line with
$\mathit{\mathrm{B}_{\mathrm{ir}}\sim\mathrm{C^{-1}(1-t)^{2}}}$ is predicted
within the SCPC model also in Section $\mathrm{III}$. In Section $\mathrm{IV}$
the SCPC model is applied in studying the penetration of the magnetic field
($\mathrm{\mathit{PMF}}$) into the center of the $\mathit{\mathrm{H_{3}S}}$
sample. The $\mathrm{\mathrm{PMF}}$ measurements are analyzed in the Bean
critical state model, first for a long superconducting cylinder (the parallel
configuration $\mathrm{B_{\parallel}}$approximately realized in Troyan  ) and
then for thin disks (perpendicular configuration $\mathrm{B_{\perp}}$). It is
shown, that the critical magnetic field $\mathbf{B_{\mathrm{p}}}$ at which it
penetrates into the center of the sample is larger for the perpendicular
geometry than for the parallel one, i.e. $\mathrm{B_{p\perp}>B_{p\parallel}}$.
The comparison of the theory and the experimental results for
$\mathrm{\mathit{PMF}}$ in $\mathrm{H_{3}S}$ Troyan  gives a large critical
current $\mathrm{j}$c0$\apprge 10^{7}$ $\mathrm{A/cm^{2}}$.

In Section $\mathrm{\mathrm{I}V}$ we discuss some experimental controversies
related to the Meissner effect and FC experiments in $\mathrm{H_{3}S}$, which
display a large residual paramagnetic magnetization. The latter fact does not
fit into the classical theory of the Meissner effect. Note, that the
paramagnetic signal can be present in some magnetic superconductors,
superconductors with intrinsic magnetic moments in Tallon-Ru -EuFeIrAs  or
extrinsic ones Pan-FS . Finally, in Section $\mathrm{V}$ the obtained results
for $\mathrm{TBR}$ and $\mathrm{PMF}$ in the SCPC model for $\mathrm{H_{3}S}$
are summarized and discussed. Here, the crucial difference between pinning
forces in HTSC-cuprates and HP-hydridesis also discussed. Note, in the
following we use the notation $\xi_{0}\approx\xi(T\ll T_{c})$ and
$\lambda_{0}\approx\lambda(T\ll T_{c})$

##  II Strong vortex pinning by columnar defects in H3S

###  II.1 Single vortex pinning

To increase the critical current density it is desirable to have extended
(long columnar) pinning defects - the $\mathit{correlated}$
$\mathit{disorder}$. In this case the pinning potential is correlated over the
extended size of the defect $\mathrm{L}$$\sim\mathrm{L_{v}}$.
($\mathrm{L_{v}}$ is the vortex length.) This means that the superconductivity
is suppressed in a large volume
$\mathrm{V_{col}}$$=\pi\xi_{0}^{2}$$\mathrm{L_{v}}$ and therefore a single
vortex prefers sitting on this defect. Note, that the maximal pinning energy
$\mathrm{U_{p}^{max}=\epsilon_{p}^{max}V_{col}}$ is reached when the
superconductivity is fully suppressed in the core of defects, i.e. when
$\Delta=0$ in dielectric columnar defects. This might be not the case in
$\mathrm{H_{3}S}$, where the real pinning energy density is
$\mathrm{\epsilon_{p}=\eta_{col}\cdot\epsilon_{p}^{max}}$ with the reducing
factor $\mathrm{\eta_{col}}<1$, because the gap inside the columnar defect is
finite, $\Delta<\Delta_{0}$. In that case
$\mathrm{\eta_{col}=\epsilon_{p}/\epsilon_{p}^{max}\sim(1}-\Delta^{2}/\Delta_{0}^{2})$.
As a result, the critical current density for a single vortex (sitting on such
a defect) is given by
$\mathrm{j_{c}}=\eta_{\mathrm{col}}\cdot\mathrm{j}_{c}^{\mathrm{max}}$.

Note, that in HTSC-cuprates long columnar pinning defects are made
artificially by irradiating $\mathrm{YBa_{2}Cu_{3}O_{7}}$ single crystalline
samples of small platelets of $\mathrm{1}$$\times 1\times 0.02$
$\mathrm{mm^{3}}$ with different doses of $\mathrm{580}$ $\mathrm{MeV}$
$\mathrm{{}^{116}Sn^{30+}}$ ions Civale -Blatter-RMP . The density of the ion
doses are $\mathrm{D\approx 5\times 10^{10},}1.5\times 10^{11},2.4\times
10^{11}$$\mathrm{ions/cm^{2}}$, which are equivalent to the corresponding
vortex densities with the magnetic induction $\mathrm{B_{\phi}\approx 1,3,5}$
$\mathrm{T}$ \- $\mathrm{B_{\phi}}$ is the matching field. Since the
ionization energy-loss rate was $\mathrm{2.7}$ $\mathrm{keV/\textrm{\r{A}}}$
they produce long tracks with the length $\mathrm{L\sim 20-30}$ $\mu m$ and
diameter $\mathrm{\sim 50\textrm{ \r{A}}}$. In that case $\mathrm{j_{c}(\sim
j_{c0})}$ is of the order $\mathrm{1.5}$$\times 10^{7}$$\mathrm{A/cm^{2}}$ at
$\mathrm{T=5}$ $\mathrm{K}$ and $\mathrm{10^{6}A/cm^{2}}$at $\mathrm{T=77}$
$\mathrm{K.}$ These current densities are much larger than those for point
defects, where the pinning is due to oxygen vacancies. In the columnar case
the irreversible line $\mathrm{B_{ir}(T)}$ (the line below which the pinning
is pronounced) lies higher than for weak pinning with point defects Civale .

In the following it is argued, that the columnar pinning defects dominate in
$H_{3}S$ \- see blue colored cylinders in $\mathit{Fig.}$ $\mathit{2}$, with
the averaged distance $\mathrm{d_{\phi}=(\varPhi_{0}/B_{\phi})^{1/2}}$
Mrktchyan-Scmidt -Kulic-CD . For an optimal pinning the superconductivity
should be fully destroyed inside the dielectric columnar defect with radius
$\mathrm{r\gtrsim\xi}$ Mrktchyan-Scmidt -Kulic-CD . In this case, both
mechanisms of pinning - the $\mathit{core}$ and the $\mathit{electromagnetic}$
one, are operative with almost same pinning energy Kulic-CD . The maximal
pinning energy is given by $\mathit{\mathrm{U_{p}}^{\mathrm{scpc}}}$$\approx
2\pi\xi^{2}L_{\mathrm{v}}(H_{c}^{2}/8\pi)\equiv
L_{\mathrm{v}}\mathrm{\epsilon_{p}^{max}}$ with
$\mathrm{\epsilon_{p}^{max}}\approx\Phi_{0}^{2}/32\pi^{2}\mathrm{\lambda^{2}(T)}$.
Since the elementary pinning force (per unit vortex length)
$f_{p}=$$\mathit{\epsilon_{p}^{\mathrm{max}}/\xi}$ is balanced in the critical
state by the Lorenz force (per unit length) $\mathit{\mathrm{f_{L}(\equiv
F_{L}/L_{v})=j_{c}\Phi_{0}/c}}$, the critical current density
$\mathrm{j_{c}^{max}}$ in the $\mathrm{SCPC}$ model is given by Mrktchyan-
Scmidt -Kulic-CD

$j_{c}^{\mathrm{max}}\approx\frac{c\Phi_{0}}{32\pi^{2}\lambda^{2}\xi}.$ (1)

In anisotropic superconductors $\mathrm{j_{c}^{max}}$ is given in Kulic-CD .
(In SI units
$\mathrm{j_{c,SI}^{max}\approx\Phi_{0}/8\pi\mu_{0}\lambda^{2}\xi}$, where
$\lambda$$\mathrm{(T)}\approx\lambda_{0}(1-t)^{-1/2}$ and
$\xi(T)\approx\xi_{0}(1-t)^{-1/2}$ are the Ginzburg-Landau penetration depth
and coherence length, respectively). In $\mathrm{H_{3}S}$ one has
$\lambda\approx(1-2)\times 10^{3}\textrm{\r{A}}$ and
$\xi_{0}\sim(15-20)\textrm{ \r{A}}$ Drozdov-2015 , which gives for
$\mathrm{j_{c0}^{max}}$ in the range
$\mathrm{j_{c0}^{max}}$$\approx(0.8-3)\times
10^{8}\mathrm{A}/\mathrm{cm^{2}}$. In the following, the critical current
density $\mathrm{j_{c0}}$ will be estimated from experiments on the magnetic
penetration in $\mathrm{H_{3}S}$ Troyan , where it is found
$\mathrm{j_{c0}\approx(1,3-1,5)\times}10^{7}$$\mathrm{\mathrm{\mathrm{\textrm{\r{A}}}/cm^{2}}}$,
which gives for the reducing factor $\mathrm{\eta\approx(0.1-0,3)}$.

###  II.2 Crossover field $\mathit{\mathrm{\mathit{B}_{rb}}}$ for the single
vortex to vortex bundles pinning

The generic $\mathrm{H-T}$ phase diagram for the high temperature
superconductors with columnar pinning defects is shown in $\mathit{Fig.}$
$\mathit{1}$ \- see Blatter-RMP , and some magnetic properties of
$\mathrm{H_{3}S}$ will be discussed in the framework of this phase diagram. In
$\mathbf{II.B}$ the single-vortex pinning is considered, which occurs at small
magnetic field $\mathrm{B<B_{\phi}}$. When the inter-vortex distance
$\mathrm{a_{v}\approx(\Phi_{0}/B)^{1/2}}$ is larger than the average distance
between columnar defects
$\mathrm{d_{\phi}}=(\Phi_{0}/\mathrm{B_{\phi}})^{1/2}$ and the single vortex
pinning energy is larger than the inter-vortex energy, than the vortices
accommodate freely to the pinning sites. However, when $\mathrm{B>B_{\Phi}}$
the inter-vortex interaction starts to be important for pinning and dynamical
properties (such as the vortex creep). In that case, vortex bundles are
pinned. The maximal crossover field $\mathrm{B_{rb}^{max}}$ \- see
$\mathit{Fig.}$ $\mathit{1}$, separates the single vortex regime from the
vortex bundle regime and it is obtained by comparing the energy of the elastic
shear deformation (of the order $\mathrm{u\sim d_{\Phi}}$) with the maximal
pinning potential $\mathrm{\epsilon_{p}^{max}}$ per unit length, i.e.
$\epsilon_{\mathrm{shear}}=\mathrm{c_{66}(d_{\phi}/a_{v})^{2}a_{v}^{2}}$$\backsimeq\mathrm{\epsilon}_{p}^{\mathrm{max}}$.
The shear modulus is given by
$\mathrm{c_{66}\approx\mathrm{\mathrm{\varPhi_{0}B/(8\pi\lambda)^{2}}}}$what
gives for $\mathrm{B_{rb}}$

$\mathrm{B_{rb}<B_{rb}^{max}\lesssim\frac{4\mathrm{\epsilon_{p}^{max}}}{\epsilon_{0}}B_{\phi}},$
(2)

where $\epsilon_{0}=\Phi_{0}^{2}/16\pi^{2}\lambda^{2}$ Blatter-RMP . Since
$\mathrm{\epsilon_{p}^{max}=\epsilon_{0}/2}$ it gives, that for
$\mathrm{B<B_{br}<B_{rb}^{max}\approx 2B_{\Phi}}$ the columnar defects
outnumber the vortices and the single-vortex pinning prevails. Note, that the
real crossover field is $\mathrm{B_{rb}<B_{rb}^{max}}$. Since,
$\mathrm{\eta_{col}<1}$ this inequality holds for $\mathrm{T\ll T_{c}}$, where
$\xi(T)\approx\xi_{0}$ and $\lambda(T)\approx\lambda_{0}$. It is argued
bellow, that in order to explain the $\mathrm{TBR}$ experiment in
$\mathrm{H_{3}S}$ the regime $\mathrm{B>B_{rb}}$ is also realized.

Figure 1: The generic $H-T$ phase diagram for superconductors with
$\mathit{columnar}$ $\mathit{defects}$. The doted melting line
$\mathrm{B_{m}(T)}$ of the pure sample is transformed into a Bose-glass
transition line $\mathrm{B_{BG}(T)}$. Also shown are the various pinning
regimes with a single-vortex/single-rod pinning region at low fields
$\mathrm{\mu_{\mathrm{0}}H<B_{rb}(T)<2B_{\Phi}}$. For $\mathrm{T<T_{dp}^{r}}$
the fluctuations of a vortex from one defect to another are suppressed, while
for $\mathrm{T>T_{dp}^{r}}$ the pinning potential is exponentially reduced. At
$\mathrm{T>T_{dl}}$ the individual flux lines are pinned collectively by an
assembly of columnar defects (rods) at high temperatures. Above the crossover
line $\mathrm{B_{rb}(T)}$, the largest energy in the problem is the inter-
vortex interaction and pinning involves vortex bundles - taken from Blatter-
RMP .

Note, that in cuprates with columnar defects, pinning properties are highly
anisotropic with the maximal critical current for the magnetic field aligned
along the columnar defect (and for $\mathrm{j_{c}\perp\mathbf{H}}$). The
latter is confirmed for irradiated $\mathrm{YBa_{2}Cu_{3}O_{7}}$ Civale . It
seems, that this is not the case for $\mathrm{H_{3}S}$, where
$\mathrm{j_{c0}^{\perp}\approx j_{c0}^{\parallel}}$ and similar columnar
densities of defects are realized along and perpendicular to the sample
surface, i. e. $\mathrm{B_{\phi}^{\perp}\simeq B_{\phi}^{\parallel}}$ Troyan .
This implies that the columnar defects in $\mathrm{H_{3}S}$ are oriented
along, both, perpendicular and parallel, axes almost equally - see
$\mathit{Fig.}$ $\mathit{2}$.

###  II.3 Thermal vortex depinning from columnar defects

For any high temperature superconductor thermal fluctuations of vortex lines
are important at higher temperatures, because of smoothing of the pinning
potential. This significantly lowers $\mathrm{j_{c}(T)}$ at and above some
depinning temperature $\mathrm{T_{dp}^{r}}$, when the effective pinning
potential (per unit vortex length) $\epsilon_{p}(T)=\epsilon_{p}(0)\varphi(T)$
becomes small, since $\mathrm{\varphi(T>T_{dp}^{r})\ll 1}$. For the sake of
clarity - there are two types of thermal motion of vortex lines in the
presence of pinning centers : ($\mathit{i}$) Phonon-like, with small amplitude
fluctuations affecting an individual pinning potential -
$\mathit{intravalley}$ fluctuations, thus smoothing the pinning potential and
reducing $\mathrm{j_{c}(T)}$ significantly near $\mathrm{T_{dp}^{r}}$.
($\mathit{ii}$) The second kind of thermal motion is related to large
$\mathit{intervalley}$ thermal fluctuations, which cause jumping of vortices
from one to another pinning center (valley). These are mainly responsible for
the vortex-creep phenomena \- which is not studied here. Here, we deal with
type ($\mathit{i}$) thermal effects in the regime of the single-vortex
pinning. Above and at $\mathrm{T_{dp}^{r}}$ the amplitude of the thermal
fluctuations $\left\langle\mathrm{u}^{2}\right\rangle_{\mathrm{th}}^{1/2}$
increases beyond the extent of the vortex core,
$\left\langle\mathrm{u^{2}}\right\rangle_{\mathrm{th}}>\xi^{2}$. In that case,
the vortex experiences smaller averaged pinning potential. The calculation of
$\mathrm{T_{dp}^{r}}$ is sophisticated and based either on: ($\mathit{i}$) the
analogy of the vortex statistical physics with columnar defects and the
quantum 2D-Bose gas placed in a random pinning potential, or
($\mathit{i\mathit{i}}$) on the statistical physics of vortices Blatter-RMP .
It turns out that for $\mathrm{r\gtrsim\sqrt{2}\xi_{0}}$ the depinning
temperature $\mathrm{T_{dp}^{r}}$ is approximately given by the self-
consistent equation $\mathrm{T_{dp}^{r}\approx
r\cdot\sqrt{\epsilon_{p}(T_{dp})\epsilon_{0}(T_{dp})}}$, where
$\mathrm{\epsilon}_{p}(T)=\eta_{\mathrm{col}}\cdot\Phi_{0}^{2}/32\pi^{2}\mathrm{\lambda^{2}(T)}$
and $\epsilon_{0}=\Phi_{0}/16\pi^{2}\mathrm{\lambda^{2}}(T)$ Blatter-RMP .

##  III Temperature broadening of resistance in the magnetic field in
$\mathrm{H_{3}S}$

In Refs. Hirsch-1-1 -Hirsch-Marsiglio-arXiv V5-2022-1  it was claimed that in
order to explain the temperature broadening of the resistance in magnetic
field ($\mathit{TBR}$), $\delta t_{c}(h)\equiv(T_{c}-T_{c}(h))/T_{c}$, in HP-
hydrides and in the framework of physics of soft superconductors, it is
necessary to invoke an unphysically large critical current density
$\mathrm{j_{c0}>10^{9}}$$\mathrm{A/cm^{2}.}$ In the case of the questionable
CSH superconductor even much larger critical current is needed, i.e.
$\mathrm{j_{c0}>10^{11}}$$\mathrm{A/cm^{2}}$\- where $\delta t_{c}(h)$ is
field independent Snider . In the following it is argued, that the reduction
of $\delta t_{c}(h)$ in $\mathrm{H_{3}S}$ can be explained by invoking the
SCPC model - which assumes that $\mathrm{H_{3}S}$ is a $\mathit{hard}$
$\mathit{type-II}$ $\mathit{superconductor}$ with elongated intrinsic columnar
pinning defects. The geometry of the experiment is shown in $\mathit{Fig.}$
$\mathit{2}$.

Let us briefly introduce the reader into the subject of $\mathit{TBR}$ in
$\mathrm{H_{3}S}$, which is based on the Tinkham theory for $\mathit{TBR}$
Tinkham-1  and the SCPC model Kulic-RTSC-v1 .

Figure 2: Schematic view of the experiment of the magnetic flux trapping and
penetration in the $H_{3}S$ disk-like sample Kulic-RTSC-v1 : $\mathit{D=30\mu
m;}P=5\mu m$ for the perpendicular geometry $\mathrm{\mathbf{H}_{0}\perp D}$.
According to the SCPC-model, long columnar defects ($\mathit{blue}$
$\mathit{cylinders}$) strongly pin and trap vortices making huge magnetization
hysteresis and the critical current density $\mathit{j_{c}\sim\delta M}$. The
non-superconducting 119Sn film ($\mathit{yelow}$ with $\mathit{d=20\mu
m;\textrm{ }}p=2.6$$\mu m$) is implemented in the experiment for the detection
of the penetrated magnetic field Troyan . Similar analyzes holds for the
parallel geometry $\mathrm{\mathbf{H}_{0}\parallel D}$.

Namely, in all superconductors dissipationless current can flow in the vortex
state with pinning defects. However, when the pinning energy is small,
especially for $\mathrm{T}$ near $\mathrm{T_{c}}$, vortices jump easily from
one center to another under temperature fluctuations, thus giving rise to a
vortex motion and dissipation of energy - called flux creep Anderson . These
jumps are activation-like and proportional to the escape probability (from the
pinning center) $\mathit{exp\mathrm{(-U_{p}/T)}}$ . Since for point defects
$\mathit{\mathrm{U}_{\mathrm{p}}^{\mathrm{pd}}}$$\sim\xi^{3}$, then this
energy barrier is small in superconductors with small $\xi$, what is, for
instance, the origin of a pronounced dissipation in HTSC-cuprates (with oxygen
vacancies). In that sense, the long pinning defects with the energy $U$p $\sim
L_{v}\xi^{2}$ make this barrier much higher, thus suppressing the dissipation
effects significantly. In magnetic fields much higher than the lower critical
field, i.e. for $\mathit{\mathrm{H_{c1}\ll H<H_{c2}}}$, one has
$\mathit{\mathrm{B\approx\mu_{0}H}}$ and the vortex distance is given by
$\mathrm{\mathit{\mathrm{a}\approx(\mathbf{\mathrm{\Phi_{0}/B}})^{1/2}}<\lambda}$.
In that case bundles of vortices, each with the surface $\sim a^{2}$, are
pinned Yeshurun  with the pinning energy of the bundle
$\mathit{\mathrm{U_{p}}}$$\sim\mathrm{L_{v}}a^{2}$. The Tinkham $\mathit{TBR}$
theory Tinkham-1  applied to such problems is based on the Ambegaokar and
Halperin theory for thermally activated phase motion in Josephson junctions
Ambegaokar . As the result, it gives for the resistance of the superconductor
(with small transport current) Tinkham-1

$R/R_{N}\approx\left[I_{0}(\gamma/2)\right]^{-2},\gamma=U_{p}/T,$ (3)

where $\mathrm{R_{N}}$ is the resistance of the normal state at
$\mathit{T}_{c}$ and $\mathit{I_{0}}$ is the modified Bessel function.

In the $\mathrm{SCPC}$ model with $\mathit{columnar}$ $\mathit{pinning}$
centers with $\mathit{\mathrm{L\approx L_{v}}}$ and near $\mathit{T_{c}}$ one
obtains for $\gamma$

$\gamma^{\mathrm{scpc}}=\beta_{K}\left(\frac{L_{v}}{\xi_{0}}\right)\frac{(1-t)^{2}}{h}(2\pi\xi_{0}^{2}\frac{j_{c0}\Phi_{0}}{cT_{c}}),$
(4)

where $\mathit{\mathrm{h}\approx\mathrm{H/H_{c2}(0)}},$$\mathrm{\delta
t_{c}\equiv(1-t)\equiv 1-T/T_{c}}$ and $\beta_{K}\approx 1$. In the case of
$\mathit{point}$ $\mathit{defects}$ one has for $\gamma$

$\gamma^{\mathrm{pd}}\approx\frac{(1-t)^{3/2}}{h}(2\pi\xi_{0}^{2}\frac{j_{c0}^{\mathrm{pd}}\Phi_{0}}{cT_{c}}).$
(5)

Here, the critical current for weak pinning
$\mathrm{j_{c0}^{pd}=\eta_{pd}\cdot j_{c0}}$ is defined via the weak pining
energy $\mathrm{U_{p}^{pd}\approx\eta_{pd}H_{c}^{2}\xi^{3}}$ with
$\eta_{\mathrm{pd}}<1$ , while in the SCPC model one has
$\mathrm{j_{c0}=\eta_{col}\cdot j_{co}^{max}}$ \- see $\mathrm{Eq.}$
$\mathrm{(1)}$. The critical current density for weak pinning centers is
usually of the order $\mathrm{j_{c0}^{pd}\sim 10^{6}}$ $\mathrm{A/cm^{2}}$.
Note, that $\gamma^{\mathrm{scpc}}\sim(\mathrm{1-t})^{2}/h$ for long columnar
defects, while $\gamma^{pd}\sim(\mathrm{1-t})^{3/2}/h$ for point defects. The
latter fact gives rise to the different temperature dependence of the
irreversible line $\mathrm{H_{irr}(T)}$ in HTSC (with oxygen vacancies as
point defects) and $\mathrm{H_{3}S}$ \- see more below. From these expressions
it is seen, that the relative barrier in the case of long columnar pinning
centers is larger by the factor $\mathit{\mathrm{(L_{v}/\xi_{0})\gg 1}}$,
compared with the one for point-like randomly distributed defects - used in
Hirsch-1-1 -Hirsch-Marsiglio-arXiv V5-2022-1 . This means that
$\mathrm{j}_{c0}^{\mathrm{pd}}$ for point defects is replaced by much larger
quantity $\mathit{\mathrm{(L_{v}/\xi_{0})\times j_{c0}}}$ in the SCPC-model -
where one has $\mathit{\mathrm{(L_{v}/\xi_{0})\times
j_{c0}}}$$\gg$$\mathrm{j_{c0}}$.

Let us compare the prediction of the standard theory for weak pinning (with
point-like defects) applied to $\mathrm{H_{3}S}$ \- done in Hirsch-1-1
-Hirsch-Marsiglio-arXiv V5-2022-1 . In the case when the resistance is
measured at the 10 % level, i.e. $\mathit{R/R_{N}=\mathrm{0.1}}$, it gives for
$\mathit{I_{0}\mathrm{(\gamma/2)}\approx\mathrm{3.2}}$ and $\gamma\approx
5(=\gamma^{scpc}=\gamma^{pd}$ in both cases. In the field
$\mathit{\mathrm{B\approx\mu_{0}H\approx 1}}\mathrm{T}$ and for
$\mathit{\mu_{0}\mathrm{H_{c2}\mathrm{(0)}}\approx\mathrm{100}}$ $\mathrm{T}$
one has $\mathrm{\mathrm{\mathit{\mathrm{h}\approx\mathrm{0.01}}}}$. We assume
also the following realistic parameters for the $\mathrm{HP}$-hydrides:
$\xi_{0}\approx 20\mathrm{\textrm{ \r{A}}}$ and $\mathit{\mathrm{T}_{c}}$$\sim
200$ $\mathrm{K}$, $\Phi_{0}=2\times 10^{-7}G\times cm^{2}$. If one takes
$\mathrm{j_{c0}^{pd}=}$$\alpha\mathrm{j}_{c0}\approx\alpha\times 10^{7}$
$\mathrm{A/cm^{2}}$ with $\alpha\sim 0.1$, then by using Eq.(5) one obtains

$\delta t_{c}^{pd}\equiv\frac{\delta
T_{c}^{pd}(h=0.01)}{T_{c}}\approx\frac{0.1}{\alpha}.$ (6)

This is a too large value ($\delta t_{c}^{pd}>0.1$) - see the red line in
$\mathit{Fig.}$ $\mathit{3}$, compared to the experimental one $\delta
t_{c}^{\mathrm{exp}}\apprle 10^{-2}$ Eremets-Meissner-2022 . Based on Eq.(5)
one concludes that in order to explain the experimental value $\delta
t_{c}^{exp}(h=0.01)\apprle 10^{-2}$ in $\mathrm{H_{3}S}$ in the framework of
the weak pinning theory one needs a mach larger (effective) critical current
$\mathrm{j_{c0}^{pd}}>3\cdot$$10^{8}$ $\mathrm{A/cm^{2}}$. However, the latter
value is far beyond the range of the weak pinning theory.

However, the experimental value $\delta t_{c}^{\mathrm{exp}}\apprle 10^{-2}$
can be explained by the SCPC model with the long columnar pinning defects,
where the $\delta t_{c}^{\mathrm{scpc}}(h)$ depends on the large factor
$\mathrm{(L/\xi_{0})\times j_{c0}}$, thus making
$\delta\mathrm{t}_{c}^{\mathrm{scpc}}(h)$ small. In the case of
$\mathrm{H_{3}S}$ where $\gamma_{00}=2\pi$$\xi_{0}^{2}$$\mathrm{\times
j_{co}}$$\Phi_{0}/cT_{c}$ and $\mathrm{j_{c0}}$ is in the range
$\mathit{\mathrm{j_{c0}}}>10^{7}$$A/cm^{2}$ (in CGS units
$\mathrm{\mathit{\mathrm{j}_{\mathrm{\mathrm{c}o}}}}$$>3\times 10^{16}\textrm{
}\mathrm{esu/cm^{2}}\textrm{)}$ and for
$\mathit{\mathrm{L}_{v}\sim\mathrm{(0.5-1)\textrm{ }}\mu m}$ one obtains

$\delta
t_{c}^{\mathrm{scpc}}(h)\approx\left(\frac{5\xi_{0}h}{L_{v}\gamma_{00}}\right)^{1/2}\lesssim
0.01.$ (7)

The obtained result for $\mathit{TBR}$ in $\mathrm{Eq.}$ $\mathrm{(7)}$ in the
SCPC-model for $\mathrm{h\approx 0.01}$ and $\mathrm{j_{c0}\sim
10^{7}}$$\mathrm{A/cm^{2}}$ is in satisfactory agreement with the experimental
values shown in $\mathit{Fig.}$ $\mathit{3}$ \- the blue line. Since $\delta
t_{c}^{\mathrm{scpc}}\ll$$\delta t_{c}^{pd}$ this means that the SCPC model is
able to describe $\mathit{TBR}$ in the $\mathrm{H_{3}S}$ superconductor.

Figure 3: Broadening of the superconducting transition ($\mathit{TBR}$)
$\delta t_{c}(h)\equiv\Delta T/T_{c}$ under external magnetic fields in
different superconducting HP-hydrides derived in Ref. Hirsch-1-1  \- points
and solid lines and the experimental values for TBR in $\mathrm{H_{3}S}$
extracted in Eremets-Meissner-2022  \- $\mathit{blue}$ $\mathit{points}$.
$\mathit{Blue}$ $\mathit{line}$ \- the theoretical line predicted by the SCPC-
model $\delta\mathrm{t^{scpc}}_{c}(h)\sim h^{1/2}$ ($\mathrm{h=H/H_{c2}}$,
$\mathrm{H_{c2}\approx 100}$$\mathrm{T}$) for $\mathit{j_{c0}}\approx
10^{7}$$A/cm^{2}$ and $\mathrm{L\approx 0.5}$$\mu m$. The model is more
suitable for low $\mathrm{h\leq 0.01}$\- see text. $\mathit{Red}$
$\mathit{line}$ \- the prediction of the standard model for $\mathit{TBR}$
with weak-pinning $\delta t_{c}^{pd}(h)\sim h^{2/3}$ and for
$\mathit{j_{c0}}\approx 10^{7}$$A/cm^{2}$ is inadequate for $\mathrm{H_{3}S}$
.

Note, that there is $\mathit{TBR}$ also in the zero magnetic filed
($\mathrm{H=0}$), i.e. there is an intrinsic $\mathit{TBR}$, $\delta T_{0}\neq
0$ , which should be taken into account in the analyzes of experiments. This
was done in Eremets-Meissner-2022  and Hirsch-1-1 , where one has $\delta
T_{c}(h)$$\approx$$\delta T_{\mathrm{tot}}(h)-\delta T_{c0}$ and $\delta
T_{\mathrm{tot}}$ is the total $\mathit{TBR}$. The extracted experimental
results for $\delta t_{c}(h)(\equiv\delta T_{c}(h)/T_{c})$ in
$\mathrm{H_{3}S}$ are shown in $\mathit{Fig.}$ $\mathit{3}$ Eremets-
Meissner-2022  \- blue circles, for $\mathrm{h}$ within the range of
$\mathrm{0<h<0.15}$.

We stress again, that in order to explain the much smaller $\mathit{TBR}$ ,
$\delta t_{c}(h)$, in $\mathrm{H_{3}S}$ (and in other HP-hydrides) than the
standard Tinkham theory predicts for the weak pinning - see Eq.(5), the
authors of Hirsch-1-1 -Hirsch-Marsiglio-arXiv V5-2022-1  assume an
unrealistically large critical current density
$\mathrm{\mathit{\mathrm{j}^{\mathrm{pd}}}}$$\sim(10^{9}-10^{11})\mathrm{A/\mathrm{cm^{2}}}$.
The latter value is much larger than the experimental one
$\mathit{\mathrm{j_{c0}^{exp}}}$$\apprge 10^{7}A/\mathrm{cm^{2}}$. The second
possibility is to call into question the existence of superconductivity in HP-
hydrides. This (second) possibility is accepted in Hirsch-1-1 -Hirsch-
Marsiglio-arXiv V5-2022-1 . However, in the proposed SCPC model
$\mathrm{j_{c0}}$ in the formula for $\delta t_{c}(h)$ is in fact replaced by
the much larger quantity $\mathit{\mathrm{(L_{v}/\xi_{0})\times j_{c0}}}$ \-
see Eq.(4), which for $\mathit{\mathrm{(L_{v}/\xi_{0})\sim 10^{3}}}$ gives
realistic values for
$\mathrm{\mathrm{\mathrm{\mathit{\mathrm{j}_{c\mathrm{0}}}}\apprge 10^{7}}}$
$\mathrm{A/cm^{2}}$. This analysis confirms our claim, that there is no reason
to call into question the existence of the conventional superconductivity in
$\mathrm{H_{3}S}$.

##  IV penetration of the magnetic field in $\mathrm{H_{3}S}$

In order to confirm the existence of superconductivity in a sample, two kinds
of experiments in an external magnetic field are necessary: (i) At
$\mathrm{T>T_{c}}$ the sample is placed in a magnetic field
$\mathbf{\mathrm{H<H_{c1}}}$(or $\mathrm{H_{c}}$ in type-I superconductors)
and than cooled below $\mathrm{T_{c}}$ \- the $\mathit{FC}$ (field cooled)
experiment. If the magnetic field is $\mathit{expelled}$ from the sample the
$\mathit{Meissner}$ $\mathit{effect}$ is realized. Its existence means a
definitive proof for superconductivity in the sample; (ii) In the
$\mathit{zero}$ $\mathit{field}$ $\mathit{cooled}$ ($\mathit{ZFC}$) experiment
the sample is first cooled to $\mathrm{T<}\mathrm{T_{c}}$ at zero external
field ($\mathrm{H=0}$) and then is the field $\mathrm{H<H_{c1}}$ applied. In
this case, the magnetic field is $\mathit{excluded}$ from the sample - the so
called $\mathit{diamagnetic}$ $\mathit{shielding}$ $\mathit{mode}$. The latter
effect is in fact due to the well known Lenz law in electrodynamics. In the
Meissner state of type-I superconductors currents in a bulk sample decay
exponentially from the sample surface and the average current over the bulk
sample is zero. In an ideal type-II superconductor with regular vortex lattice
the average circulating current (around the vortices) is also zero. A net
average (transport) current can exist only due to distortions of the vortex
lattice in the presence of pinning defects. As a result, the transport current
can flow in bulk type-II superconductors, $j(r)\neq 0$ , and the field can
penetrate into the sample at much larger distance than the penetration depth
$\lambda$. If the pinning is strong (and the critical current large) such
superconductors are called hard superconductors.

In Section III the theoretical (in the SCPC-model) and experimental results
for $\mathit{TBR}$ suggest, that $\mathrm{H_{3}S}$ (and other HP-hydrides) is
a hard type-II superconductor, with large Ginzburg-Landau parameter
$\kappa(=\lambda/\xi)=(50-100)$ and high (low-temperature) critical current
density $\mathit{\mathrm{j}_{c\mathrm{0}}}$$\apprge 10^{7}A/\mathrm{cm^{2}}$.
Next questions related to $\mathit{\mathrm{H_{3}S}}$ are: ($\mathit{i}$) What
is the prediction of the SCPC model for the penetration of magnetic field
($\mathit{PMF}$) into a superconducting sample; ($\mathit{ii}$) How big is the
magnetic hysteresis observed in $\mathrm{H_{3}S}$ Troyan ? For simplicity,
these questions are analyzed quantitatively in the framework of the Bean
critical state model - first for a long cylinder and than for a disk with
small thickness $\mathrm{P(\ll D)}$. Afterwards, the theory is compared with
the experiment in $\mathrm{H_{3}S}$ Troyan . In the following, the results are
presented either in the CGS or in SI units - according to the existing
magnetic measurements. $\mathrm{B^{0}}$, $\mathrm{B_{0}}$ and
$\mathrm{B_{ext}}$ are equivalent labeling for the applied magnetic field
(induction). Some problems related to the $\mathit{FC}$ Meissner effect are
discussed at the end of this Section.

###  IV.1 Experimental results for the ZFC field penetration in
$\mathrm{H_{3}S}$

The $\mathit{ZFC}$ penetration depth is studied experimentally in
$\mathrm{\mathrm{H_{3}}S}$ by the novel NRS technique Troyan  \- the geometry
of the ZFC experiment is shown in $\mathit{Fig.}$ $\mathit{2}$. The experiment
measures the time evolution of the scattered radiation re-emitted by the 119Sn
film - $\mathit{Fig.}$ $\mathit{3}$ in Ref. Troyan . In this ZFC experiment a
non-superconducting thin 119Sn film is placed inside the $\mathrm{H_{3}S}$
disk-sample (see $\mathit{Fig.}$ $\mathit{2}$ \- yellow color) being used as a
sensor of the magnetic field. The internal magnetic field on the 119Sn sensor
is monitored by the nuclear resonance scattering of synchrotron radiation.
When the magnetic field penetrates in the 119Sn non-superconducting film, the
experimental curve for the radiation intensity as a function of time shows
$\mathit{quantum}$ $\mathit{beats}$, due to the interference of the split
119Sn nuclear levels by the magnetic field. These beats are indeed observed in
the field $\mathrm{B^{0}=0.68}$ $\mathrm{T}$ at $\mathrm{T>T_{c}}$ \- when the
sample is non-superconducting and the perpendicular magnetic field penetrates
completely into the 119Sn film - $\mathit{Fig.}$ $\mathit{3}$ in Ref. Troyan .
Note, that since $\mathrm{H_{3}S}$ is a type II superconductor one expects
that if the sample is free of pinning centers, then for
$\mathrm{B^{0}\gg\mu_{0}H_{c1}=}(18-60)$ $\mathrm{mT}$ the magnetic field
should enter into the whole sample in the form of (almost) homogeneously
distributed vortices - thus showing quantum beats in the 119Sn film. However,
in the $\mathit{perpendicular}$ $\mathit{geometry}$ of the experiment
($\mathrm{\mathbf{H}\perp D}$ in $\mathit{Fig.}$ $\mathit{2}$) at low
temperatures and in the field $\mathrm{B^{0}=0.68}$ T Troyan  the vortices do
not show up in the center of the 119Sn film up to some temperature, i.e. there
are no quantum beats - $\mathit{Fig.}$ $\mathit{3}$ in Troyan . This means
that the vortices are strongly pinned in the region outside of the 119Sn film.
Strong pinning of vortices in a superconductor implies, that the magnetization
of the sample is strongly hysteretic, as revealed experimentally in
$\mathrm{H_{3}S}$ \- see Eremets-Meissner-2022 ,Drozdov-2015 ,Minkov-
Meissner-2021 .

Figure 4: The experimental temperature dependence of the magnetic field on the
sensor 119Sn film (placed) inside $\mathrm{H_{3}S}$ (see $\mathit{Fig.}$
$\mathit{2}$) at 153 GPa shown by $\mathit{blue}$ $\mathit{triangles}$. The
external field at the reference non-superconducting sample 119Sn in
$\mathrm{H_{2}}$ at 150 GPa is shown by $\mathit{red}$ $\mathit{dots}$ \- see
Troyan . ($\mathbf{A\mathrm{)}}$ and $\mathbf{\mathrm{(}B}$) are measurements
in the perpendicular ($\mathrm{H_{ext}^{\perp}\perp D}$) and parallel
($\mathrm{H_{ext}^{\parallel}\parallel D}$) geometry of the external magnetic
field, respectively. Dashed lines are guides to the eye. Taken from Troyan .

Two geometries were studied in Troyan : ($\mathbf{\mathrm{\mathit{A}}}$) The
$\mathit{perpendicular}$ $\mathit{geometry}$ \- where the field is
perpendicular to the disk surface ($\mathrm{\mathbf{H}\perp D}$ in
$\mathit{Fig.}$ $\mathit{2}$); ($\mathit{\mathbf{\mathrm{B}})}$ The
$\mathit{parallel}$ $\mathit{geometry}$ \- the field is parallel to the disk
surface ($\mathrm{\mathbf{H}\parallel D}$). As it is seen in $\mathit{Fig.}$
$\mathit{4A}$ the average field inside the 119Sn sensor in the perpendicular
geometry is approximately zero for $\mathrm{T<80}$ $\mathrm{K}$ and reaches
the value $\mathrm{B_{ext}=}$$0.68$ $\mathrm{T}$ at around
$\mathrm{T}$$\approx 120$ $\mathrm{K}$. In the parallel geometry the field
inside the 119Sn film is finite for $\mathrm{T>0}$ (but smaller than
$\mathrm{B_{ext}=}$$0.68$ $\mathrm{T}$) starting to increase at
$\mathrm{T>80}$ $\mathrm{K}$ saturating to $\mathrm{B_{ext}=}$$0.68$
$\mathrm{T}$ at $\mathrm{T}$$\gtrsim 140$ $\mathrm{K}$ \- $\mathit{Fig.}$
$\mathit{4B}$. In the next Subsection these experimental results are explained
by the theory which is based on the Bean critical state model and the SCPC
model with high critical current density $\mathrm{j_{c0}\apprge
10^{7}}$$\mathrm{A/cm^{2}}$.

###  IV.2 Penetration of the magnetic field in a long cylinder with pinning
centers

In Section III it is argued, that the experimental results on the thermal
broadening of resistance ($\mathit{TBR}$) in magnetic filed in
$\mathrm{H_{3}S}$, $\delta t_{c}^{\mathrm{exp}}(h)$, give much smaller value
than those predicted by the Tinkham theory for standard superconductors with
weak pinning, i. e. $\delta t_{c}^{\mathrm{exp}}(h)$$\ll$ $\delta
t_{c}^{pd}(h)$. However, the SCPC model with the columnar pinning centers and
large critical currents $\mathrm{j_{c}(0)}>10^{7}$$\mathrm{A/cm^{2}}$ explains
this property, i. e. $\mathrm{\delta t_{c}^{scpc}(h)\approx\delta
t_{c}^{exp}(h)}.$ Note, that the large value of $\mathrm{j_{c}(0)}$ inevitably
causes large magnetization hysteresis, which is seen experimentally in
$\mathrm{H_{3}S}$ Eremets-Meissner-2022 ,Drozdov-2015 ,Minkov-Meissner-2021 .
The critical current density $\mathrm{j_{c}(0)}$ is a measure of the strength
of pinning forces. Below, it is shown that the large value of
$\mathrm{j_{c}(0)}$ is compatible with experiments for the penetration of the
magnetic field into thin $\mathrm{H_{3}S}$ film Troyan . To maximally simplify
the problem a long cylinder placed in the external field
$\mathrm{B_{0}=\mu_{0}H_{0}}$ is first considered (in absence of external
transport currents), thus escaping demagnetization effects. This case is also
used in studying the field penetration in the parallel geometry of
$\mathrm{H_{3}S}$, i.e. when $\mathrm{\mathbf{B}_{0}\parallel D}$.

In an homogeneous system in thermodynamic equilibrium, in an ideal type II
superconductor (without pinning centers), vortices are homogeneously
distributed over the bulk sample. In that case the macroscopically averaged
(over the sample) local magnetic induction is constant, i.e.
$\mathbf{\mathbf{\mathrm{\mathbf{\overline{B}}_{\mathrm{eq}}}}}$$(\mathbf{r}$)$\mathrm{=const}$
and the macroscopic local magnetization current density (averaged over the
vortex unit cell) is zero, i.e.
$\mathrm{\mu_{0}\mathbf{\bar{j}_{\mathrm{eq}}}\mathrm{(\mathbf{r})=rot}\overline{\mathbf{B}}_{eq}}$=$\mathrm{0,}$
as well as the Lorentz force per unit volume of the vortex lattice
$\mathbf{f^{\mathrm{eq}}\mathrm{{}_{L}}=\mathrm{\mathbf{\bar{j}_{\mathrm{eq}}}}}\times\mathbf{\overline{B}_{\mathrm{eq}}}$$\mathrm{=0}$.
However, in the presence of pinning centers
$\overline{\mathbf{B}}$$\mathrm{(\mathbf{r})}$ is $\mathit{inhomogeneous}$ and
$\bar{\mathrm{\mathbf{j}}}$$\mathrm{(\mathbf{r})}$$\neq 0$, thus producing
finite Lorentz force (per unit volume) on vortices
$\mathbf{f\mathrm{{}_{L}}=\mathrm{\mathbf{\bar{j}}}}\times\mathbf{\overline{B}}$$\neq
0$. When the vortices are pinned there is a pinning force (per unit volume)
$\mathbf{f}$p which counteracts the Lorentz force. In the static case the
vortices are not moving and the condition $\mathbf{f}$${}_{L}=-\mathbf{f}_{p}$
is fulfilled everywhere in the sample. So, by increasing the applied field the
vortices are so rearranged that locally the maximal critical current
$\mu_{0}\mathrm{\mathbf{j_{\mathrm{c}}}}$$(B)=\mathrm{rot}\mathbf{\overline{B}}$
is achieved.

The simplest, but very useful, model for the critical state is $\mathit{the}$
$\mathit{Bean}$ $\mathit{critical}$ $\mathit{state}$ $\mathit{model}$ Bean-1 ,
which assumes that $\mathrm{j_{c}}$ $\mathit{is}$ $\mathit{independent}$
$\mathit{on}$ $\mathrm{B}$. In the case of a $\mathit{long}$
$\mathit{cylinder}$ $L$$\gg R$ (no demagnetization effects) one has
$\mathrm{d\overline{B}/dr}=\pm\mu_{0}\overline{j}_{c}$ and
$\mathrm{\overline{B}(r)}$ is given by

$B(r,B_{0})=B_{0}-B^{*}(T)(1-\frac{r}{R}).$ (8)

It is seen from Eq.(8) that for an applied filed $\mathrm{0\leq B_{0}<B^{*}}$
the field $\mathrm{B(r)}$ penetrates up to the point
$\mathrm{r_{B_{0}}}=(1-\mathrm{B}_{0}/\mathrm{B^{*}}$$)R$, where
$\mathrm{B(r_{B_{0}},B_{0})=0}$. At $\mathrm{B_{0}=B^{*}}$ $\mathit{the}$
$\mathit{field}$ $\mathit{reaches}$ $\mathit{the}$ $\mathit{center}$
$\mathit{of}$ $\mathit{the}$ $\mathit{cylinder}$, i.e. at
$\mathrm{r_{B^{*}}=0}$ one has $\mathrm{B(r_{B^{*}}=0,B^{*})=0}$ \- see
$\mathit{Fig.}$ $\mathit{5}$. In the experiment Troyan  \- schematically given
in $\mathit{Fig.}$ $\mathit{2}$, one has $\mathrm{D=30}$ $\mathrm{\mu m}$ and
by assuming that $\mathrm{L\gg D}$ one has $1.8$
$\mathrm{\mathrm{T<}B^{*}(j_{c0})}$$\mathrm{<18}$ $\mathrm{T}$. This range for
$\mathrm{B^{*}(j_{co})}$ is due to $\mathrm{j_{c0}}$ in the range
$\mathrm{j_{c0}}\approx(10^{7}-10^{8})$ $\mathrm{A/cm^{2}}$.

However, the applied field in the experiment by Troyan et al. Troyan  was
fixed to $\mathrm{B_{0}=0.68}$ $\mathrm{T}$ , i.e.
$\mathrm{B_{0}}$$\mathrm{<B^{*}(T\ll T_{c})}$. This means that at very low
temperature ($\mathrm{T}$$\ll T_{c}$) the field does not penetrate into the
center of the non-superconducting 119$\mathrm{Sn}$ film, if the sample would
be a long cylinder with $\mathrm{L(=P)\gg D}$. Note, that the magnetic field
reaches the front of the 119S film at $\mathrm{r_{B_{0}^{S}}=10}$$\mu m$, i.e.
for $\mathrm{B_{0}^{S}=(1/3)B^{*}}$ where $\mathrm{B_{0}^{S}}$$\approx(0.6-6$)
$\mathrm{\mathrm{T}}$, for
$\mathrm{j_{c0}\approx(10^{7}-}10^{8}$$\mathrm{A/cm^{2}}$. Therefore, in the
case of a long cylinder with $\mathrm{L(=P)\gg D}$ at low $\mathrm{T}$ with
$\mathrm{j_{c0}\approx 1.5\times 10^{7}}A/\mathrm{cm^{2}}$ placed in the
external field $\mathrm{B_{0}=0.68}$ $\mathrm{T}$ , the field would not be
able to penetrate neither the front of the 119S film (where
$\mathrm{r_{B_{0}^{S}}=10}$$\mathrm{\mu m}$) nor the center
($\mathrm{r_{B^{*}}=0}$) of the sample, since
$\mathrm{B_{0}<B_{0}^{S}}<\mathrm{B^{*}}$.

We stress again, that the experiment in Ref. Troyan  was performed on a
finite-size sample in two geometries: ($\mathrm{A}$) the
$\mathit{perpendicular}$ geometry when the magnetic field is perpendicular to
the thin cylinder, i.e. $\mathrm{\mathbf{B}_{0}\perp D}$ \- see
$\mathit{Fig.}$ $\mathit{2}$, and ($\mathrm{B}$) in the $\mathit{parallel}$
$\mathit{geometry}$ when the field is parallel to the diameter $\mathrm{D}$ of
the sample, i.e. to the surface of the thin disk with $\mathrm{P\ll D.}$ In
the next Subsection it is shown, that in the case ($\mathrm{A}$) finite size
effects are very important, while in the case ($\mathrm{B}$) they are less
pronounced and the problem can be, in the first approximation, treated as a
long cylinder, since $\mathrm{P\ll D}$.

Figure 5: The Bean critical state model for a long cylinder with radius
$\mathrm{R\ll L}$: $(a)$ distribution of local averaged field
$\mathit{\mathrm{\overline{B}(r)}}$ in the external field $\mathrm{B^{0}}$
parallel to the surface of the cylinder - $\mathit{up}$ $\mathrm{arrow}$ for
increased $\mathrm{B^{0}}$ from $\mathrm{0}$ to $\mathrm{B_{max}^{0}}$ and
$\mathit{down}$ $\mathit{arrow}$ for decreased $\mathrm{B^{0}}$ from
$\mathrm{B_{max}^{0}}$ to $\mathrm{B^{0}}.$ For $\mathrm{B^{0}=B^{*}}$ the
field penetrates to the center of the sample, where
$\mathit{\mathrm{\overline{B}(0)=0}}$; $(b)$ the hysteresis loop for the
magnetization $\mathrm{M_{ir}}=<\mathrm{\overline{M}}>$ (averaged over the
cylinder) - the virgin (initial) line in $\mathit{bold}$; (c) distribution of
the current density in the superconducting cylinder in an axial field:
$\mathit{left}$ \- in the increasing film from $\mathrm{B^{0}=0}$ to
$\mathrm{B^{0}<B^{*}}$ the screening current flows between $\mathrm{r_{f}}$
and $\mathrm{R}$ where $\mathit{\mathrm{\overline{B}(r_{f})=0}}$;
$\mathit{middle}$ \- for $\mathrm{B^{0}>B^{*}}$ it flows in the whole sample;
$\mathit{right}$ \- in decreasing field from $\mathrm{B_{max}^{0}}$ to
$\mathrm{B}^{0}$ the current is inverted in the external sheath. From Mangin .

###  IV.3 Theory for the penetration of the magnetic field in
$\mathrm{H_{3}S}$ \- perpendicular geometry

In the perpendicular geometry
$\mathrm{\mathbf{B}_{0}(=\mu_{0}\mathbf{H}_{0})\perp D}$ of the thin disk with
$\mathrm{L\ll R}$ (i.e. $\mathrm{P\ll D}$ in the experiment of Ref. Troyan  \-
see $\mathit{Fig.}$ $\mathit{2}$) there is a geometric barrier when the sample
shape (due to geometric spikes) is different from an ellipsoid. It turns out,
that in that case the magnetic field penetrates into the sample in an
inhomogeneous way Brandt . The theory Brandt  predicts that in the
perpendicular geometry the flux is penetrated up the center in the external
field $\mathrm{B_{0\perp}=B_{p\perp}}$ with

$B_{p\perp}(T)=B^{*}(T)\frac{P}{D}\mathrm{ln}\left[\frac{D}{P}+\sqrt{(1+\frac{D^{2}}{P^{2}}}\right],$
(9)

where $\mathrm{B^{*}(T)=\mu_{0}j_{c}(T)D/2}$. In order to explain the
temperature dependence of the field in the 119Sn sensor Troyan  \- see
$\mathit{Fig.}$ $\mathit{4A}$, we define an $\mathit{effective}$
$\mathit{field}$ on the 119Sn sensor

$B_{Sn}^{\perp}(T)\equiv B_{0}-B_{p\perp}(T).$ (10)

(Here, $\mathrm{T}$ is the absolute temperature.) By definition, for
$\mathrm{\mathrm{\mathit{B_{\mathrm{0}}\leq B_{p\perp}(T)}}}$ one has
$\mathrm{\mathrm{\mathit{\mathrm{B}_{Sn}^{\perp}=\mathrm{0}}}}$, i.e. the
field does not reach the center of the sample. When, $\mathrm{T\ll T_{c}}$ ,
$\mathrm{j_{c0}\approx(1.4-1.5)\times 10^{7}}\textrm{\r{A}}/\mathrm{cm^{2}}$
\- see $\mathit{Fig.}$ $\mathit{6A}$, and $\mathrm{(P/D)\approx 1/6}$ \- see
$\mathit{Fig.}$ $\mathit{2}$, one has $\mathrm{B_{p\perp}\approx
1}$$\mathrm{T}$ which gives that

$\mathrm{B_{0}(=0.68}T\mathrm{)<}B_{\mathrm{p\perp}}.$ (11)

This means, that the external field is not strong enough to push the pinned
vortices to the center of the sample. However, in order to study the
temperature dependence of $\mathrm{\mathit{B_{Sn}^{\perp}(T)}}$ it is
necessary to know the temperature dependence of the current density
$\mathrm{\mathit{\mathrm{j}_{c}\mathrm{\mathrm{(}T})}}$. In that respect, one
should take into account two effects: ($\mathit{a}$) The (mean field)
temperature dependence of $\lambda(T)$ and $\xi(T)$ since
$\mathit{\mathrm{j_{c}(T)}\sim\mathrm{\lambda^{-2}(T)\xi^{-1}(T)}}$. In the
temperature range $\mathrm{0<T\apprle T_{c}/2}$ we mimic
$\mathrm{\mathrm{\mathit{\mathrm{\lambda^{-2}(T)\xi^{-1}(T)}\sim\mathrm{(1-T^{2}/T_{c}^{2})^{3}}}}}$;
($\mathit{b}$) The temperature fluctuations cause depinning effects (see
$\mathbf{II.C}$ ) which decrease $\mathit{\mathrm{j}_{c}\mathrm{(T)}}$
additionally - even bringing it almost to zero at some $\mathrm{T_{dp}^{r}}$
\- see $\mathit{Fig.}$ $\mathit{1}$ and $\mathit{Fig.}$ $\mathit{6}$. For the
sake of simplicity, $\varphi\mathrm{(T)}$ is described approximately by the
linear function
$\mathrm{\mathrm{\varphi(T)}\approx(T_{dp}^{r}-T)/(T_{dp}^{r}-T_{kink})}$ for
$\mathit{T\geq T_{kink}}$. As the result, one obtains
$\mathrm{\mathit{\mathrm{j_{c}(T)=j_{c0}\cdot\varphi(T)}\mathrm{(1-T^{2}/T_{c}^{2}}})^{3}}$.
The fit of the experimental curve in $\mathit{Fig.}$ $\mathit{6A}$ is obtained
for $\mathit{\mathrm{T_{kink,\perp}\approx 101}}$$\mathit{\mathrm{K}}$
and$\mathrm{\mathrm{\mathit{T_{dp,\perp}^{r}\approx\mathrm{122}}}}$$\mathit{K}$
which gives the theoretical curve for $\mathrm{\mathit{B_{Sn}^{\perp}(T)}}$
shown in $\mathit{Fig.}$ $\mathit{6A}$.

It is important to point out, that the theoretical curve
$\mathrm{\mathit{B_{Sn}^{\perp}(T)}}$ (in $\mathit{Fig.}$ $\mathit{6A}$) fits
the experimental results (in $\mathit{Fig.}$ $\mathit{4A}$) if the current
density is high, i. e.
$\mathrm{\mathrm{\mathrm{\mathrm{j_{c0}}\approx\mathrm{1.4\times
10^{7}}}\mathit{\mathrm{A/cm^{2}}}}}$. This value for $\mathrm{j_{c0}}$ is
compatible with the one obtained from the $\mathit{TBR}$ measurements in
$\mathrm{H_{3}S}$, where
$\mathrm{\mathit{\mathrm{j_{c0}}\gtrsim\mathrm{10^{7}}}\mathit{\mathrm{A}/\mathrm{cm^{2}}}}$.
Both findings confirm the assumption of the SCPC model, that the intrinsic
defects in $\mathrm{H_{3}S}$ are approximately elongated columns, which pin
vortices strongly, thus giving high critical current densities in this
material.

Figure 6: The temperature dependence of the effective magnetic field
$\mathit{B_{Sn}(T)}$ in the 119Sn sensor in the combined Bean model and the
SCPC-model. For $\mathrm{T_{kink}}$ and $\mathrm{T_{dep}(\equiv T_{dp}^{r})}$
see text. $\mathbf{A}$ \- $\mathit{B_{Sn}^{\perp}(T)}$ in the perpendicular
geometry; $\mathbf{B}$ \- $\mathit{B_{Sn}^{\parallel}(T)}$ in the parallel
geometry. The various curves are for different current densities
$\mathrm{\mathrm{\mathit{\mathrm{j_{c0}}=\alpha\times\mathrm{10^{7}}}}\mathit{A/cm^{2}};\mathrm{\alpha}=\mathrm{\mathit{\mathrm{1},\mathrm{1.2}},\mathit{\mathrm{1.3},\mathrm{1.5},\mathrm{2.0}}}}$

###  IV.4 Theory for the penetration of the magnetic field in
$\mathrm{H_{3}S}$ \- parallel geometry

Similarly, when the field is parallel to the thin disk, i.e.
$\mathrm{\mathit{B^{\parallel}(T)\parallel D}}$ \- see $\mathit{Fig.}$
$\mathit{1}$, the effective field on the 119Sn sensor is given by

$B_{Sn}^{\parallel}(T)\equiv B_{0}-B_{p\parallel}(T).$ (12)

In this geometry and with $\mathrm{P\ll D}$ \- see $\mathit{Fig.}$
$\mathit{2}$, the system can be approximated by a long cylinder with
$\mathrm{R\approx P/2}$ and $\mathrm{L\approx D}$. In that case the formula
Eq.(8) holds approximately, which gives
$\mathrm{\mathit{\mathrm{B_{p\parallel}(T)\approx}\mathrm{\mu_{0}j_{c}(T)P/2}}}$.
For assumed value
$\mathrm{\mathrm{\mathrm{\mathrm{j_{c0}}\approx\mathrm{1.4\times 10^{7}}}}}$
$\mathrm{A/cm^{2}}$one obtains $\mathrm{B_{p\parallel}(T\ll T_{c})\approx
0.45}$ $\mathrm{T}$, which is smaller than the applied field
$\mathrm{B_{0}=0,68}$ $\mathrm{T}$, i.e.

$\mathrm{B_{p\parallel}(T\ll T_{c})<B_{0}(=0,68}\mathrm{T})$ (13)

This means, that $\mathrm{B_{Sn}^{\parallel}(T\ll T_{c})>0}$ and the external
field $\mathrm{B_{0}}$ is large enough to push vortices into the center of the
119Sn film - see $\mathit{Fig.}$ $\mathit{6B}$, which is in agreement with the
experimental result in $\mathit{Fig.}$ $\mathit{4B}$. The fit of $\varphi(T)$
(note, that $\mathrm{j_{c}(T)\sim\varphi(T)}$) is analogous to the
perpendicular case but with slightly different parameters
$\mathrm{T_{kink,\parallel}\approx 122}$$\mathrm{K}$ and
$\mathrm{T_{dp,\parallel}^{r}\approx 142}$$\mathrm{K}$ . Note, that there is a
qualitative difference between $\mathrm{B_{Sn}^{\perp}(T)}$ and
$\mathrm{B_{Sn}^{\parallel}(T),}$ since $\mathrm{B_{Sn}^{\perp}(T)}$ is zero
up to some finite temperature - see $\mathit{Fig.}$ $\mathit{4}$ and
$\mathit{Fig.}$ $\mathit{6}$, and $\mathrm{B_{Sn}^{\parallel}}(T)$ is finite
even at $\mathrm{T=0}$$\mathrm{K}$. This is due to different sample
dimensions, i.e. that $\mathrm{P\ll D}$ \- see $\mathit{Fig.}$ $\mathit{2}$.
These results are in a qualitative and semi-quantitative agreement with the
experimental results in Troyan .

It is necessary to point out following items: ($\mathit{a}$) It is seen in
$\mathit{Fig.}$ $\mathit{5}$, that in the Bean critical state model the
irreversible magnetization for a long cylinder in fields
$\mathrm{B_{0}>B^{*}}$ is constant (field independent). Since the measurements
of the magnetic moment of the $\mathrm{H_{3}S}$ sample in Troyan  are given in
the CGS unit $\mathrm{emu}$ we use it also here. After averaging of
$\mathrm{\overline{M}=(\overline{B}-H)/4\pi}$ over the sample one has
$\mathrm{\overline{M}_{ir}(emu/cm^{3})\approx}(R/30)\mathrm{j}_{c}(\mathrm{A/cm^{2}})$
and the total magnetic moment
$\mathrm{\mu}_{\mathrm{ir}}=$$\bar{\mathrm{M_{ir}}}\cdot\mathrm{\mathrm{V_{s}}}$.
Note, that $\mathrm{\bar{M}_{ir}=(\bar{M}_{+}-\bar{M}_{-})/2=\bar{M}_{-}}$
since in the Bean model $\mathrm{\bar{M}_{+}=-\bar{M}}$ holds for
$\mathrm{B_{0}>B^{*}}$. For the volume of the disk $\mathrm{V_{s}\approx
0.8\times P\cdot D^{2}}$ one obtains the trapped magnetic moment in the sample
to be of the order $\mu\approx(0.3-2)\times 10^{-5}$ $\mathrm{emu}$ for
$\mathrm{B_{0}>B_{p\perp}}$ and for $\mathrm{j_{c0}\approx(1.4-10)\times
10^{7}}$$\mathrm{A/cm^{2}}$. The calculated $\mathit{trapped}$ magnetic moment
in the $\mathrm{H_{3}S}$ sample of Ref.Troyan  is far beyond the SQUID
sensitivity threshold \- which is $\sim 10^{-8}$ $\mathrm{emu}$ Troyan . This
means that the measurements of the trapped magnetic flux of vortices (but
without the extrinsic moments) are realizable. ($\mathit{b}$) In order to
explain the field penetration in the $\mathrm{H_{3}S}$ sample of Ref. Troyan
it comes out that the perpendicular ($\mathbf{H\perp}$$\mathrm{D}$) critical
current density $\mathrm{j_{c0}^{\perp}}$ must be approximately equal to the
parallel ($\mathbf{H\parallel\mathrm{D}}$) one $\mathrm{j_{c0}^{\parallel}}$,
i. e. $\mathrm{j_{c0}^{\perp}}\approx\mathrm{j_{c0}^{\parallel}}$. This means,
that in the $\mathrm{H_{3}S}$ sample of Ref. Troyan  the columnar defects are
oriented along both directions, the parallel and perpendicular one, in a
similar way - schematically shown in $\mathit{Fig.}$ $\mathit{2}$. Having in
mind the cubic-like structure of $\mathrm{H_{3}S}$ this kind of pinning
isotropy is an acceptable assumption; ($\mathbf{c}$) In order to explain the
$\mathrm{\mathit{TBR}}$ and $\mathit{PMF}$ measurements in thin
$\mathrm{H_{3}S}$ samples in the SCPC model, it comes out that the bulk
$\mathrm{H_{3}S}$ sample is a high-$\kappa$ type II superconductor with the
parameters: $\xi_{0}\approx(15-20)$ Å, $\lambda_{0}\approx(1-2)\times 10^{3}$Å
; $\kappa\approx(50-100)$, $\mathrm{\mu_{0}H_{c1}}\approx(18-60)$
$\mathrm{m\mathrm{T}}$,
$\mathrm{\mathrm{\mu_{0}}H_{c0}}$$\mathrm{\approx(0.6-1.1)}$
$\mathrm{\mathrm{T},}$ $\mathrm{\mathrm{\mu_{0}}H_{c2}}\approx(80-140)$
$\mathrm{T}$. These values are very different from those obtained in Eremets-
Meissner-2022 , Minkov-Meissner-2021 , where $\xi_{0}\sim 20$ Å,
$\lambda_{0}\sim(1.3-2)\times 10^{2}$Å ; $\kappa\sim 7-10$,
$\mathrm{\mathrm{\mu_{0}}H_{c0}}$$\mathrm{\sim 6}$ $\mathrm{\mathrm{T},}$
$\mathrm{\mathrm{\mu_{0}}H_{c1}\sim 1}$ $\mathrm{T}$,
$\mathrm{\mu_{0}H_{c2}}\approx(80-140$) $\mathrm{T}$. The values of the
parameters predicted in the SCPC model are compatible with those obtained in
the magnetic measurements and also with the microscopic theory - see the
discussion below.

To this point, Hirsch and Marsiglio have recently realized Hirsch-optics-arXiv
, that their previous interpretation Hirsch-1-1 -Hirsch-2-1  of the Troyan’s
measurements in $\mathrm{H_{3}S}$ Troyan  in terms of pinning-free
superconductors with unphysically high $\mathrm{j_{c0}}$$\sim
10^{11}\mathrm{A}/\mathrm{cm^{2}}$ may be inadequate. Namely, due to the
pronounced magnetization hysteresis in Drozdov-2015  they speculated the
presence of pinning forces in $\mathrm{H_{3}S}$ \- with the critical current
density $\mathrm{j_{c0}}\sim 10^{7}\mathrm{A}/\mathrm{cm^{2}}$. However, they
did not realize that the $\mathit{TBR}$ and $\mathit{PMF}$ effects in
$\mathrm{H_{3}S}$ are due to the strong pinning by the elongated (columnar)
defects - as the SCPC model predicts Kulic-RTSC-v1 .

To conclude this Section - the magnetic measurements of the penetration of the
magnetic field $B_{0}=0.68$ $\mathrm{T}$ in the $\mathrm{H_{3}S}$ sample
Troyan can be naturally explained in the framework of the SCPC model and the
Bean critical state model. This approach also explains naturally the high
critical current density in the $\mathrm{H_{3}S}$ samples, which is of the
order $\mathrm{j_{c0}\approx(1.4-1.5)\times 10^{7}}$ $\mathrm{A/cm^{2}}$.
Thereby, the finite-size effects in the thin $\mathrm{H_{3}S}$ disk of Ref.
Troyan  are taken into account. This analyzes tells us, that there is no need
to call into question the existence of superconductivity in $\mathrm{H_{3}S}$
(and in other HP-hydrides), as it is claimed in Hirsch-1-1 -Hirsch-Marsiglio-
arXiv V5-2022-1 .

###  IV.5 Meissner effect in $\mathrm{H_{3}S}$

The Meissner effect is an important hallmark of the superconducting state. It
is realized in the so called $\mathrm{FC}$ ($\mathit{field}$
$\mathit{cooled}$) $\mathit{experiment}$, when the sample (ideally without
pinning defects) is in the normal state ($\mathrm{T>T_{c}}$) and placed in a
magnetic field $\mathrm{H_{0}<H_{c1}}$(for type-II superconductors). The
latter penetrates into the normal metallic sample fully, i. e. one has
$\mathit{B\approx\mu_{0}\mathrm{H_{0}}}$. However, if the sample is then
cooled down into the superconducting state ($\mathrm{T<T_{c}}$) the magnetic
field will be $\mathit{expelled}$ from the bulk sample, i. e. one has
$\mathrm{\overline{B}=0}$ in an ideal non-magnetic superconductor. The
Meissner effect should not be confused with the $\mathrm{ZFC}$
($\mathrm{\mathit{\mathrm{zero}-\mathrm{f}\mathrm{ield}\mathrm{\textrm{
}cooled}}}$) $\mathit{experiment}$, where a nonmagnetic bulk metallic sample
is first cooled into the superconducting state at $\mathrm{T<T_{c}}$ in the
zero field ($\mathrm{H=0}$) and thereafter magnetic field $\mathrm{H<H_{c1}}$
is turned on. As a result, the magnetic field (induction) is
$\mathit{excluded}$ from the bulk sample, i.e. $\mathrm{B=\mu H=0}$ with
$\mu=\mu_{0}(1+\chi)=0$. The $\mathrm{ZFC}$ phenomenon is also called
$\mathit{diamagnetic}$ $\mathit{shielding}$. However, the $\mathrm{ZFC}$
effect would also be realized in a $\mathit{perfect}$ $\mathit{metal}$ (with
$\varrho=0$) - if it existed in nature, where it is due to the classical Lenz
law of the electrodynamics. From this analyzes comes out, that the magnetic
susceptibility of an ideal bulk superconductor is diamagnetic $\chi=-1$ in the
SI system ($\mathrm{4\pi\chi=-1}$ in the $\mathrm{CGS}$) in both types of
experiments. Note, that when the FC experiment is done in a perfect metal, the
magnetic field is not expelled from the sample at $\mathrm{T<T_{c}}$, i.e. the
magnetic flux stays frozen in the sample with the same value as in the normal
state, i.e. $\mathrm{\mathit{\mathrm{B}\approx\mu_{0}\mathrm{H_{0}}}}$. So,
for a definite proof of the Meissner effect in superconductors one should
perform the FC experiment.

Figure 7: $\mathit{Up}$ \- The $\mathrm{FC}$ and $\mathrm{ZFC}$ magnetization
($\mathrm{M}$) in $\mathrm{H_{3}S}$ \- from Eremets-Meissner-2022
,Drozdov-2015 . $\mathit{Down}$ \- the volume susceptibility
$\chi_{\mathrm{V}}$ of the weak ferromagnetic superconductor
$\mathrm{RuSr_{2}GdCu_{2}O_{8}}$; $\mathbf{a)}$ \- ZFC (zfc) measurements of
$\chi_{\mathrm{V}}$at $\mathrm{H^{ex}=6.5}$ $\mathrm{Oe}$; doted line is
$\chi_{V}$ in the non-superconducting state. $\mathrm{T_{M}\approx 137}$
$\mathrm{K}$ is the magnetic critical temperature; $\mathrm{T^{ms}\approx 30}$
$\mathrm{K}$ is the transition temperature to the spontaneous vortex state
($\mathrm{SVS}$) which exists at $\mathrm{T^{ms}<T<T_{c}}$. Inset: enlarged
scale of $\chi_{\mathrm{V}}$ around the superconducting transition
$\mathbf{\mathrm{T_{c}}\approx\mathrm{45}}$$\mathrm{K}$ ; $\mathbf{b)}$ FC
measurements of $\chi_{\mathrm{V}}$ for external fields
$H^{ex}\approx(0.5-2.5)$ $\mathrm{Oe}$. Inset: The volume fraction of the
Meissner state $\mathrm{f}\approx 40\%$ at $\mathrm{H^{ex}=0,5}$$\mathrm{Oe}$
\- from Tallon-Ru .

In that respect, several inconsistent experimental and theoretical results
related to the magnetization measurements in $\mathrm{H_{3}S}$ (and
$\mathrm{LaH_{10}}$) were published (in the period 2015-2022) in Eremets-
Meissner-2022 ,Drozdov-2015 ,Minkov-Meissner-2021 . These results are strongly
criticized in Hirsch-1-1 -Hirsch-Marsiglio-arXiv V5-2022-1 . We shall not
elaborate this criticism in details, but enumerate few, in our opinion, main
points by Hirsch and Marsiglio Hirsch-1-1 -Hirsch-Marsiglio-arXiv V5-2022-1 .
These are: $\mathit{\mathrm{(\mathbf{\mathrm{\mathit{1}}})}}$ The Meissner
effect is not observed experimentally in $\mathrm{H_{3}S}$ (and other HP-
hydrides) Eremets-Meissner-2022 ,Drozdov-2015 ,Minkov-Meissner-2021 ). Namely,
in the FC measurements only some parasitic paramagnetic susceptibility is
observed, i. e. $\chi_{FC}(\mathrm{T<T_{c}})>0$, and there are no signs of the
flux expulsion. However, in the ZFC experiment there is a clear diamagnetic
shielding (at $\mathrm{T<T_{c}}$) but existing on the background of the
parasitic paramagnetic signal. Most probably, the origin of this paramagnetic
signal is not intrinsic, since there is no reliable reason for strong
paramagnetic effects in $\mathrm{H_{3}S}$. To this point, there is also a
pronounced paramagnetic contribution in the FC measurements in the HTSC
superconductor $\mathrm{RuSr_{2}GdCu_{2}O_{8}}$ \- with
$\mathrm{T_{c}}$$\approx 45$ $\mathrm{K}$, where a weak ferromagnetism appears
below $\mathrm{T_{M}\approx 137}$ $\mathrm{K}$ Tallon-Ru . At $\mathrm{T\geq
T^{ms}\approx 30}$ $\mathrm{K}$ a spontaneous vortex state ($\mathrm{SVS}$)
appears, where $\mathrm{H_{c1}(T)<M}$ and $\mathrm{M}$ is the spontaneous
magnetization. In $\mathit{Fig.}$ $\mathit{7}$ it is seen, that in the FC
measurements $\chi_{v}$ is paramagnetic, even for $\mathrm{T<T_{c}}$, i. e.
$\chi_{\mathrm{V}}(T)>0$ . At $\mathrm{T^{ms}\approx 30}$ $\mathrm{K}$ the
susceptibility $\chi_{V}$ decreases suddenly and the sample is in the bulk
Meissner state at $\mathrm{T<T^{ms}}$ with the volume fraction
$\mathrm{f=\mid(\chi_{V}(0)-\chi_{V}(T^{ms})/(1+\chi_{V}(T^{ms}))\mid\approx
40}\%$ at $\mathrm{H^{ex}}=0,5$ Oe - see Inset in $\mathit{Fig.}$
$\mathit{7b}$. This tells us that pinning centers are present in the
$\mathrm{RuSr_{2}GdCu_{2}O_{8}}$ sample.

Having in mind this analyzes, one expects that in the FC measurements in
$\mathrm{H_{3}S}$ a sudden decrease of $\chi_{V}$ should be realized below
$\mathrm{T_{c}}$. However, so far, no such results have been published with
clearly realized Meissner effect in HP-hybrides. The paramagnetic signal in
the $\mathrm{H_{3}S}$ sample is most probable due to some extrinsic properties
of the diamond anvil cell (or of the rest of the sulfur atoms, since
$\mathrm{H_{2}S}$ is decomposed into $\mathrm{H_{3}S}$ and $\mathrm{S}$) and
this fact deserves further studies. $\mathrm{(\mathit{2})}$ Additional
critical remarks, given in Hirsch-1-1 -Hirsch-Marsiglio-arXiv V5-2022-1 , are
related to some controversial findings in Minkov-Meissner-2021 , that in the
$\mathrm{H_{3}S}$ (and $\mathrm{LaH_{10}}$) sample the critical fields
$\mathrm{H_{c1}(0)}$ and $\mathrm{H_{c}(0)}$ are too large, while the
penetration depth $\lambda_{0}$ is too small. If this is true, the large value
of the critical field $\mathrm{\mathrm{\mu_{0}}H_{c1}(0)}$ in Minkov-
Meissner-2021  would give very high critical current density in
$\mathrm{H_{3}S}$ (and $\mathrm{LaH_{10}}$), i. e. $\mathrm{j}_{c0}\sim
10^{10}$ $\mathrm{A/cm^{2}}$. This is much higher value than the depairing
current density $\mathrm{j_{dep}\simeq 5\times 10^{8}}$$\mathrm{A/cm^{2}}$,
what is in fact impossible. The large value for $\mathrm{H_{c}(0)}$ (in Ref.
Minkov-Meissner-2021 ) is incompatible with the microscopic theory of
superconductivity, which relates the condensation energy to
$\mathrm{H_{c}(0)}$ by $\mathrm{(N(E_{F})\Delta^{2})=H_{c}^{2}(0)/8\pi}$.
Here, $\mathrm{N(E_{F})}$ is the density of states (per unit volume) at the
Fermi surface. For $\mathrm{2\Delta\approx 3,5}$$\mathrm{T_{c}}$ and
$\mathrm{\mathrm{\mu_{0}}H_{c}(0)\sim 10}$ $\mathrm{T}$ (in Minkov-
Meissner-2021 ) one obtains that $\mathrm{N(E_{F})}\approx$$\mathrm{30\times
N^{DFT}(E)}$, while the density functional theory gives
$\mathrm{N^{DFT}(E_{F})\approx 0,2}$ $\mathrm{states/spin\times
eV\times}\textrm{(\r{A}})^{3}$ Hirsch-1-1 -Hirsch-Marsiglio-arXiv V5-2022-1 .
Therefore, this highly overestimated value for the density of states, which
are extracted from the magnetic measurements in $\mathrm{H_{3}S}$ Eremets-
Meissner-2022 ,Drozdov-2015 ,Minkov-Meissner-2021 , is an highly unacceptable
value.

Let us discuss the origin of these too large values for the bulk critical
fields $\mathrm{\mathrm{\mu_{0}}H_{c1}(\sim 1}$ $\mathrm{T})$ and
$\mathrm{\mathrm{\mu_{0}}H_{c0}(\sim 10}$ $\mathrm{T)}$ obtained in Minkov-
Meissner-2021 . The latter result is based on an inadequate experimental
definition of $\mathrm{H_{c1}}$. Namely, it is determined from the onset field
$\mathrm{\mathrm{\mu_{0}}H_{p}(0)(\sim 0.1}$$\mathrm{T)}$ of the deviation of
$\mathrm{M(H)}$ from the linear dependence by assuming that
$\mathrm{H_{c1}=H_{p}(0)/(1-N)}$. In the measurements in Ref. Minkov-
Meissner-2021  the demagnetization factor is $\mathrm{N\approx 0.96}$, what
gives too large value for $\mathrm{\mathrm{\mu_{0}}H_{c1}(T=0)\approx 2}$
$\mathrm{T}$. That this procedure is not well defined, i. e.
$\mathrm{H_{p}(0)}$ is not related to $\mathrm{H_{c1}}$, can be seen in
$\mathit{Fig.}$ $\mathit{5b}$ where $\mathrm{M(H)}$ curve in the Bean critical
state model, where the saturation field $\mathrm{H^{*}(P,D)\gg H_{c1}(0)}$. In
the perpendicular geometry of the experiment Troyan  one has $\mathrm{P=5}$
$\mathrm{\mu m}$ and $\mathrm{D=30}$ $\mathrm{\mu m}$ , $\mathrm{N\sim 0.7}$
and according to $\mathit{Eq.}$ $\mathit{(9)}$ one has
$\mathrm{\mathrm{\mu_{0}}H_{p,\perp}(0)\approx 1}$ $\mathrm{T}$ what is much
larger than the real $\mathrm{H_{c1}(T=0)}$. If we would apply the same
procedure for obtaining $\mathrm{H_{c1}}$ in the $\mathrm{H_{3}S}$ sample, as
it was done in Ref. Minkov-Meissner-2021 , we would obtain also an unrealistic
value $\mathrm{\mathrm{\mu_{0}}H_{c1}(T=0)\approx 3}$ $\mathrm{T}$ , instead
of the realistic one $\mathrm{\mathrm{\mu_{0}}H_{c1}(0)}\approx(18-60)$
$\mathrm{m\mathrm{T}}$.

##  V Summary and discussion

Recently, the authors o Refs. Hirsch-1-1 -Hirsch-Marsiglio-arXiv V5-2022-1
raised important questions on the reliability of magnetic measurements in
high-pressure hydrides (HP-hydrides). Their skepticism goes so far, that they
tend to conclude that superconductivity does not actually exist in HP-hydrides
Hirsch-1-1 -Hirsch-Marsiglio-arXiv V5-2022-1 . This attitude is mostly related
to the FC magnetic measurements, which are until now unable to prove
unambiguously the existence of the Meissner effect in small samples of
$\mathrm{H_{3}S}$. On the other side, some ZFC measurements, in small samples
of $\mathrm{H_{3}S}$, are experimentally more reliable, because these are not
related to the flux trapping effects. There are also difficulties to explain
some experimental results of magnetic measurements in $\mathrm{H_{3}S}$ \-
done in Eremets-Meissner-2022 ,Drozdov-2015 ,Minkov-Meissner-2021 , if they
are treated by the standard theory of superconductivity with weak pinning of
vortices. The latter approach is mainly accepted in Hirsch-1-1 -Hirsch-
Marsiglio-arXiv V5-2022-1 , thus reducing the possibility for explaining
experiments in $\mathrm{H_{3}S}$, such as: ($\mathit{i}$) $\mathrm{TBR}$ \-
the temperature broadening of resistance in magnetic field, and
($\mathit{ii}$) $\mathit{PMF}$ \- the penetration of the magnetic field into
the center of the sample.

In order to explain these two kinds of experiments - which can not be
explained by the weak pinning theory, Ref. Kulic-RTSC-v1  introduces the SCPC
model - which holds for superconductors with strong and elongated pinning
defects. Moreover, even the quantitative explanation of these two phenomena
(in $\mathrm{H_{3}S}$ ) is possible by assuming that the pinning centers are
in the form of long columnar defects, which are “isotropically” distributed
over the sample, i. e. with the same values of the critical current densities
flowing perpendicular and parallel to the sample surface. In the framework of
the SCPC model it is possible to explain these two kind of experiments. In the
following we summarize the obtained results: $\mathbf{1}$. The large reduction
of $TBR$, $\mathrm{\mathrm{\delta t_{c}^{scpc}(h)}}$, is due to the long
columnar defects $\mathrm{L}$$\approx\mathrm{L_{v(ortex)}}$$\gg\xi_{0}$ with
the radius $\mathrm{r}\sim\xi_{0}$. In such a case, both the core and the
electromagnetic pinning are operative. These cause high densities of the
critical current (at $\mathrm{T\ll T}_{c}$) - of the order
$\mathrm{j_{c0}=(10^{7}-10^{8})}$ $\mathrm{A/cm^{2}}$. The SCPC model
predicts, that in $\mathrm{H_{3}S}$ the temperature width of $\mathit{TBR}$ is
governed by the small parameter $\mathrm{C=}$$\xi_{0}/\mathrm{L_{col}}$, i.e.
$\mathrm{\mathrm{\delta t_{c}^{scpc}(h)}\sim C^{1/2}h^{1/2}}$ with
$\mathrm{C\sim 10^{-3}}$ and $\mathrm{h=H/H_{c2}}$. This gives
$\mathrm{\mathrm{\delta t_{c}^{scpc}(h)}\apprle 0.01}$ for $\mathrm{h\lesssim
0.01}$ and $\mathrm{L}\sim 1$ $\mathrm{\mu m}$, which is in satisfactory
agreement with experimental results in $\mathrm{H_{3}S}$ Eremets-Meissner-2022
, as shown in $\mathit{Fig.}$ $\mathit{3}$. It is also seen that the SCPC
model fits the experimental results much better than the model with weak
pinning by small defects. The SCPC model also predicts, that the magnetic
irreversible field is governed by $\mathrm{C^{-1}}$, i.e.
$\mathit{\mathrm{H_{irr}\sim(1-t)^{2}L/\xi_{0}}}$. The irreversibility line is
not only significantly increased compared to HTSC-cuprates, but the
temperature dependence, as a measure of the strength of pinning forces, is
given by $\mathit{\mathrm{(1-t)^{2}}}$ instead of
$\mathit{\mathrm{(1-t)^{3/2}}}$ \- characteristic for materials with point-
like defects. Measurements of the irreversible line $\mathrm{H_{irr}}$ in
$\mathrm{H_{3}S}$ are desirable.

These columnar defects cause high density of the critical current and also a
large magnetization hysteresis $\Delta M$ in $\mathrm{H_{3}S}$. This property
opens a possibility for making powerful high-field superconducting magnets.
For instance, by making (if possible) a long superconducting cylinder of
$\mathrm{H_{3}S}$ (with $\mathrm{L\gg R}$) the magnetic hysteresis in that
case is given by $\Delta M\sim\mathrm{j_{c0}\times R}$, where $\mathrm{R}$ is
the radius of the superconducting cylinder. For instance, in $\mathrm{H_{3}S}$
for $\mathrm{j_{c0}}>10^{7}$$\mathrm{A/cm^{2}}$ and for $\mathrm{R\sim 0.6}$
$\mathrm{cm}$ one has $\mathrm{\mathrm{\mu_{0}}M}\sim 100$ $\mathrm{T}$ at the
temperature $\mathrm{T<50}$ $\mathrm{K}$.

$\mathbf{2.}$ $\mathrm{\mathit{PMF}}$ in $\mathrm{H_{3}S}$ can be naturally
explained by the SCPC model, where the strong columnar pinning of vortices
dominates. The experiment measures $\mathit{\mathrm{\mathit{PMF}}}$ in the
applied field $\mathrm{\mathrm{B_{0}}=\mathrm{\mu}_{0}H_{0}=0.68}$
$\mathrm{T}$ Troyan . If the field is penetrated in the center of the sample,
then the quantum beats should appear in the 119Sn sensor. It turns out that in
the perpendicular geometry, when the field is perpendicular to the sample
surface ($\mathrm{B_{0}\perp D}$ \- see $\mathit{Fig.}$ $\mathit{2}$), the
magnetic field does not penetrate to the center, while in the parallel case
($\mathrm{B_{0}\parallel D}$ \- see $\mathit{Fig.}$ $\mathit{2}$) it
penetrates partially even for $\mathrm{T\ll T_{c}}$ Troyan . These
experimental results are naturally explained in the SCPC model, where the
“isotropically” distributed strong columnar pinning defects make a large
current density $\mathrm{j_{c0}^{\perp}\approx
j_{c0}^{\parallel}\approx(1.3-1.5)}$$\times 10^{7}$ $\mathrm{A/cm^{2}}$. It is
also predicted, that the depinning temperature $\mathrm{T_{dp}^{r}}$ \- where
the critical current density is strongly weakened ($\mathrm{j_{c}(T)\ll
j_{c0}}$) and the field $\mathrm{B_{0}}$ is fully penetrated into the
$\mathrm{H_{3}S}$ sample, is of the order
$\mathrm{T_{dp}^{r}\sim(100-120)}$$\mathrm{K}$.

Moreover, in order to explain the $\mathit{TBR}$ and $\mathit{PMF}$
experiments by the SCPC model it comes out that $\mathrm{H_{3}S}$ is a
high-$\kappa$ superconductor with bulk physical parameters:
$\xi_{0}\approx(15-20)$ Å, $\lambda_{0}\approx(1-2)\times 10^{3}$Å ;
$\kappa\approx(50-100)$, $\mathrm{\mathrm{\mu_{0}}H_{c1}(0)\approx(18-60)}$
$\mathrm{m\mathrm{T}}$,
$\mathrm{\mathrm{\mu_{0}}H_{c0}\approx}$$\mathrm{(0.6-1.1)}$
$\mathrm{\mathrm{T},}$
$\mathrm{\mathrm{\mathrm{\mu_{0}}H}}_{c2}\approx(80-140)$ $\mathrm{T}$. These
values of parameters are also compatible with the microscopic theory of
superconductivity. Finally, it is natural to raise the question - what is the
origin of these columnar pinning defects in $\mathrm{H_{3}S}$? Serious
candidates are single edge dislocations or their bundles, what is a matter of
further research. To conclude - the SCPC model, which assumes the existence of
strongly acting columnar pinning centers, is able to explain important $TBR$
and $\mathit{PMF}$ measurements in a high-$\kappa$ $\mathrm{H_{3}S}$
superconductor. Therefore, there is no need for calling into question the
existence of superconductivity in $\mathrm{H_{3}S}$ Hirsch-1-1 -Hirsch-
Marsiglio-arXiv V5-2022-1 .

###### Acknowledgements.

The author would like to thank Dirk Rischke and Radoš Gajić for permanent
support and to Igor Kulić for discussions and support.

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