# Collapse of Metallicity and High-$T_{c}$ Superconductivity
in the High-Pressure phase of FeSe0.89S0.11

Pascal Reiss<EMAIL_ADDRESS>Clarendon Laboratory, Department of Physics,
University of Oxford, Oxford, UK  Max Planck Institute for Solid State
Research, Stuttgart, Germany  Alix McCollam  High Field Magnet Laboratory
(HFML-EMFL), Radboud University, 6525 ED Nijmegen, The Netherlands  Zachary
Zajicek  Clarendon Laboratory, Department of Physics, University of Oxford,
Oxford, UK  Amir A. Haghighirad  Clarendon Laboratory, Department of Physics,
University of Oxford, Oxford, UK  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, Karlsruhe, Germany  Amalia I.
Coldea<EMAIL_ADDRESS>Clarendon Laboratory, Department of
Physics, University of Oxford, Oxford, UK

###### Abstract

We investigate the high-pressure phase of the iron-based superconductor
FeSe0.89S0.11 using transport and tunnel diode oscillator studies. We
construct detailed pressure-temperature phase diagrams that indicate that
outside of the nematic phase, the superconducting critical temperature reaches
a minimum before it is quickly enhanced towards 40 K above 4 GPa. The
resistivity data reveal signatures of a fan-like structure of non-Fermi liquid
behaviour which could indicate the existence of a putative quantum critical
point buried underneath the superconducting dome around 4.3 GPa. Further
increasing the pressure, the zero-field electrical resistivity develops a non-
metallic temperature dependence and the superconducting transition broadens
significantly. Eventually, the system fails to reach a fully zero-resistance
state despite a continuous finite superconducting transition temperature, and
any remaining resistance at low temperatures becomes strongly current-
dependent. Our results suggest that the high-pressure, high-$T_{c}$ phase of
iron chalcogenides is very fragile and sensitive to uniaxial effects of the
pressure medium, cell design and sample thickness which can trigger a first-
order transition. These high-pressure regions could be understood assuming a
real-space phase separation caused by concomitant electronic and structural
instabilities.

11footnotetext: See Supplemental Material at [URL will be inserted by
publisher] for further experimental data and analyses.

In the quest of seeking higher and higher superconducting transition
temperatures, the application of large hydrostatic pressures is an important
tool. While the highest transition temperatures close to room temperatures
were observed in conventional hydride superconductors under enormous pressures
in the mega-pascal range [1, 2, 3], much lower pressures in the giga-pascal
range are sufficient to boost superconductivity in unconventional
superconductors, most notably in the copper-based [4] and in the iron-based
[5] systems.

The family of the iron-chalcogenide FeSe has emerged as an enormously
versatile system in which superconductivity can be tuned not only by
hydrostatic pressure, but also by uniaxial strain, isovalent and charge
doping, surface dosing, and chemical intercalation [6, 7, 8, 9, 10, 11, 12,
13, 14]. The richness of this system partially stems from its instability
towards an electronic nematic phase at ambient conditions, and its proximity
to a magnetic phase which is stabilized under high pressures [8]. Remarkably,
it is possible to disentangle the electronic nematic and magnetic phases
through a careful combination of iso-electronic doping of FeSe1-xSx with
hydrostatic pressures [15], which allows one to study the contributions of
their corresponding order parameter fluctuations to the superconducting
pairing independently.

In particular, applied pressure studies of FeSe0.89S0.11 have provided unique
access to an isolated nematic quantum critical point [16, 17]. Towards higher
pressures, multiple studies identified a strongly enhanced superconducting
phase with the transition temperature $T_{c}$ approaching values of $\approx
35$ K. However, the nature of this high-pressure phase and of the underlying
electronic structure remains unclear, and previous studies produced partially
contradictory results. In particular, a resistivity study reported enhanced
superconductivity up to 8 GPa, and the emergence of a seemingly competing
spin-density magnetic phase in the pressure range $2-4$ GPa [15]. In contrast,
a subsequent $ac$ susceptibility study did not find signatures of any magnetic
order but observed a complete loss of the superconducting shielding at $4$ GPa
[18]. Another recent transport and NMR study detected weak signatures for
magnetism above $3$ GPa, and a gradual suppression of superconductivity beyond
$4$ GPa [19]. In CuxFe1-xSe only magnetism was suppressed under pressure but
superconductivity remained strong [20], whereas in thin flakes of FeSe, an
unsual suppression of the magnetic and superconducting phases under pressure
with decreasing flake thickness was observed [21]. Similarly, the sample
thickness and the choice of pressure medium were found to strongly affect the
boundaries of the superconducting phase in bulk FeSe above $5$ GPa, likely due
to additional uniaxial stress along the $c$ direction [22].

In order to assess the nature of this enigmatic high-pressure phase in detail,
we investigate the high-pressure regime of FeSe0.89S0.11 using transport and
tunnel-diode oscillator (TDO) probes inside opposing Diamond Anvil Cells
(DACs) up to $\approx 7$ GPa. We find that the superconducting transition
temperature $T_{c}$ shows a minimum around $\approx 1$ GPa, before it
increases towards $40$ K (onset) above $\approx 4$ GPa. Above $p_{s}\approx 4$
GPa, we observe significant changes in the samples properties as the room-
temperature normal state resistivity displays a sudden increase with pressure,
whereas the low-temperature resistivity develops a marked fan of non-Fermi
liquid behavior centered at $p_{s}$. The superconducting transition broadens
significantly above $p_{s}$ until it becomes incomplete as it fails to induce
a zero-resistance state even at the lowest temperatures, where we detect an
unusually small critical current density. Our study suggests that the high-
pressure phase of single-crystalline FeSe0.89S0.11 suffers a structural
instability above $p_{s}$, likely to an orthorhombic symmetry, as reported on
FeSe [8, 23]. As a result of the first-order nature of the structural
transition, the system shows a pressure-induced cross-over from a metallic-
like to a insulating-like behaviour which harbours a phase coexistence between
superconducting/metallic and non-superconducting/semi-metallic domains. Within
this picture, the observation of a quantum critical fan as a typical signature
of a second-order electronic instability is unusual and it is interrupted by
the first-order transition.

Figure 1: Evolution of the zero-field properties of FeSe0.89S0.11. (a) The
temperature and pressure dependencies of the zero-field resistivities of
samples A (thin lines) and sample N (thick line). The pressure transmitting
medium for both samples is Glycerol. (b) First derivative of the low-
temperature regime of panel (a) focusing on the superconducting transitions.
Here, the resistivities are normalized against their values at $T=100$ K and
their derivatives shifted vertically for clarity. (c) Low-temperature
evolution of the resonant frequency of a TDO circuit comparing the effects of
either Glycerol as pressure transmitting medium (TDO 1, ‘Gly’, thin lines), or
a 4:1 mixture of Methanol:Ethanol (TDO 2, ‘M:E’, thick lines). (d) First
derivative of the TDO resonant frequency. $T_{s}$ indicates the position of
the nematic transition visible at lowest pressures. Vertical dashed lines in
panels (c) and (d) indicate the pressure-independent superconducting
transitions of In, Sn and Pb, which originate from the solder joints outside
the pressure cell. (e,f) Definition of the superconducting critical
temperatures indicated by arrows. The onset and offset temperatures,
$T_{c}^{\text{on}}$ and $T_{c}^{\text{off}}$, are defined where the derivative
of the resistivity or the TDO resonant frequency recovers their high- and low-
temperature values, respectively. The peak in the derivative defines
$T_{c}^{\text{p}}$, and $T_{c}^{0}$ is the temperature with truly zero
resistance. At high pressures $p\gtrsim 4$ GPa, an additional low-temperature
tail in resistivity leads to $T_{c}^{0}\ll T_{c}^{\text{off}}$.

##  I Methods

Single crystals of FeSe1-xSx with $x=0.11$ sulfur substitution were grown
using the KCl/AlCl3 chemical vapor transport method as described elsewhere
[24]. High-pressure measurements were carried out using an opposing Diamond
Anvil Cell (DAC) with a design similar to Refs. 25 and 26, using $800$ $\mu$m
bevelled culets. BeCu gaskets with an initial thickness between $400$-$450$
$\mu$m were pre-indented to $\approx 80$ $\mu$m, and were subsequently drilled
to produce a pressure chamber with a diameter of $400$ $\mu$m. A thin layer of
Stycast 1266:Alumina mixture was applied as gasket insulation, and the gasket
was drilled again with a diameter of $350$ $\mu$m. A total of five single
crystals, all roughly $150\times 70\times 30~{}\mu\text{m}^{3}$, in dimensions
were subjected to high-pressure transport measurements inside the DAC using
Glycerol as pressure medium. Samples A and B were measured using a standard 4
contact configuration with the voltage contacts placed onto one of the longest
sample edges to determine the longitudinal resistivity. For samples C and D,
the voltage contacts were placed on opposing edges of the sample suitable for
a Hall effect measurement. Sample N was measured using a 5 contact setup,
allowing simultaneous longitudinal and Hall effect measurements. For
comparison in the low-pressure limit, sample E was measured under ambient
pressure only, equally with a 5 contact layout. Sample P was measured using a
5 contact layout inside a piston cylinder cell (PCC) using Daphne Oil 7373 as
pressure transmitting medium, as previously reported in Refs. 16 and 17
(called Sample A there). All transport measurements were carried out using the
AC LockIn technique with a low frequency and a low excitation current
$I_{p}=1$ mA within the $(ab)$ plane, unless stated otherwise. A further two
single crystals were studied using the Tunnel Diode Oscillator (TDO)
technique, using either Glycerol (TDO1) or a 4:1 mixture of Methanol:Ethanol
(TDO2) as pressure media. The latter ensures hydrostatic conditions up to $10$
GPa, well beyond the highest pressure reported here, whereas the former shows
non-hydrostatic behavior from 4-6 GPa onwards [27, 28, 25, 26]. In the TDO
studies, we identify pressure-independent signatures associated with the
superconducting transition temperatures of Pb, Sn and In, which occur in the
solder joints outside the pressure cell. When they become superconducting, the
quality factor of the $LC$ circuit changes which affects the resonant
frequency $f$. For the DACs, we used the Ruby fluorescence technique at room
temperature to determine the pressure inside the cell using multiple small
Ruby chips. The reported pressures here correspond to the average before and
after cooldown, and error bars indicate the difference which was usually below
$0.1$ GPa. For the piston cylinder cell used for sample P, the pressure was
determined from the superconducting transition temperature of Sn at low-
temperatures. In order to compare the low-$T$ pressure scale of sample P
reported before [16, 17] with the room-temperature pressure scales of the
other samples reported here, the pressures determined for sample P were
shifted by $+0.2$ GPa for $p<1$ GPa and $+0.1$ GPa for $p<2$ GPa to account
for the pressure losses of Daphne 7373 during cool-downs. All high-pressure
measurements were carried out on compression. Pressures were changed at room-
temperature and the cells were allowed to relax until no change in pressure
and the applied load could be resolved. Cooling rates were $0.5$ K/min except
for sample N where higher rates were used.

For a quantitative assessment of the evolution of superconductivity, we define
four different superconducting transition temperatures, as exemplified in
Figs. 1(e) and (f). Firstly, by fitting the derivatives of the resistivity and
the TDO resonant frequency, $\partial\rho/\partial T$ or $\partial f/\partial
T$, respectively, the onset temperature $T_{c}^{\text{on}}$ is extracted from
the crossing of tangents fitted to the high-temperature normal state data and
the leading edge of the superconducting transition. Secondly, the peak
temperature $T_{c}^{\text{p}}$ where the first derivatives show a maximum or
minimum, respectively, is extracted from the crossing of tangents fitted to
the leading and trailing edges of the transition. Thirdly, the offset
temperature $T_{c}^{\text{off}}$ is defined analogously to
$T_{c}^{\text{on}}$, but using the trailing edge and the low-temperature data.
Fourthly, we extract the true zero-resistance temperature $T_{c}^{0}$ when the
signal falls below the noise level.

##  II Normal state and superconducting properties in zero magnetic field

###  II.1 Phase Diagram

Previous studies have established that at ambient pressure FeSe0.89S0.11
becomes superconducting below a sharp transition temperature $T_{c}\approx
10.4$ K inside a nematic electronic phase below $T_{s}\approx 60$ K [11, 29,
16, 30, 17, 19]. Upon application of hydrostatic pressure, the nematic
transition temperature is quickly suppressed and a nematic quantum critical
point is revealed around $p_{c}\approx 0.6$ GPa as determined at low
temperatures (corresponding to approximately $0.8$ GPa at the room-temperature
pressure). Quantum oscillations have revealed the presence of a Lifshitz-like
transition across the border of the nematic phase while the effective masses
of the quasiparticles do not display divergent behaviour [30, 16].

Figures 1(a-d) show the temperature dependence of the zero-field resistivity
and TDO resonant frequency for different single crystals of FeSe0.89S0.11 up
to $p\approx 7$ GPa. At the lowest pressure accessible with the Diamond Anvil
Cell, the TDO data for $p=0.6(1)$ GPa shows a weak anomaly around $25$ K, best
visible as a dip in the temperature derivative shown in Fig. 1(d). We
associate this anomaly with the nematic phase transition, in good agreement
with previous reports using piston cylinder cells [16]. Moreover, Figs. 1(a-d)
reveal clear signatures of superconducting phase transitions as either a sharp
drop in the resistance, or a surge in the TDO resonant frequency.
Interestingly, above $4.2$ GPa the superconducting transitions broaden
significantly and become incomplete as the sharp drop in the resistance does
not lead to a zero-resistance state anymore, as shown in Fig. 1(a). Therefore,
in order to quantify the broadening and the loss of a fully superconducting
phase, we will follow the evolution of four different superconducting
transition temperatures, corresponding to the onset ($T_{c}^{\text{on}}$),
sharpest drop ($T_{c}^{\text{p}}$), offset ($T_{c}^{\text{off}}$) and true
zero-resistance ($T_{c}^{0}$), as indicated in Figs. 1(e-f) and the Methods
section.

Figure 2 summarizes the evolution of the different superconducting transition
temperatures in a detailed pressure-temperature phase diagram for all single
crystals of FeSe0.89S0.11 studied here and in previous reports [18, 15, 29,
16, 19]. At the lowest pressures, the superconducting transition temperatures
$T_{c}$ drop from $\approx 10$ K at ambient pressure towards a minimum
$\approx 6$ K at a pressure of $\approx 1$ GPa, independent of the pressure
technique or medium employed. In the intermediate pressure regime between
$1\,\text{GPa}<p\lesssim 4\,\text{GPa}$, superconductivity is strongly
enhanced and all critical temperatures continue to follow each other closely,
reaching values close to $35$ K at 4.0 GPa. However, the highest values for
$T_{c}$ are obtained for pressure techniques which ensure a more hydrostatic
environment, such a piston cylinder cells or cubic anvil cells [15, 29, 16].
In contrast, studies performed using opposing anvil cells, as in this report
and Ref. 18, display transition temperatures up to $10$ K lower, or shift the
high-pressure, high-$T_{c}$ phase to higher pressures by $\Delta p\approx 1$
GPa, as indicated in Fig. 2(b).

In the high pressure regime above $\approx 4$ GPa, the critical temperatures
display divergent trends. Fig. 2 shows that the onset temperature
$T_{c}^{\text{on}}$ keeps increasing for all samples studied and reaches a
maximum close to $40$ K around $p\approx 6$ GPa, remarkably similar to the
maximum superconducting temperature of $38.3$ K detected in FeSe at $6.3$ GPa
[10]. In contrast, the zero-resistance critical temperature $T_{c}^{0}$ and
the temperature $T_{c}^{\text{p}}$, marking the sharpest drop in the
resistivity or the surge in the TDO resonant frequency, respectively, become
strongly sample dependent. On average, $T_{c}^{0}$ drops quickly and vanishes
around $p\approx 5$ GPa, consistent with a previous report [19], whereas
$T_{c}^{\text{p}}$ disappears around $p\approx 6$ GPa. These distinct pressure
dependencies of the critical temperatures $T_{c}^{0}$, $T_{c}^{\text{p}}$ and
$T_{c}^{\text{on}}$ suggest that the loss of superconductivity does not imply
a closure of the gap, but rather an inhomogeneous and/or filamentary nature of
superconductivity. This finding will be corroborated by the critical current
studies that indicate a strongly reduced superconducting volume fraction, as
discussed in Section III. Moreover, the alteration of the superconducting
phase is consistent with a previous $ac$ susceptibility study which identified
a weakened diamagnetic shielding in the same pressure regime [18].

Figure 2: Pressure dependence of the different superconducting transition
temperatures $T_{c}$. (a) The pressure dependencies of the onset temperature
$T_{c}^{\text{on}}$, the temperature of the steepest change of the resistance
or the TDO signal $T_{c}^{\text{p}}$, and the zero-resistance temperature
$T_{c}^{0}$. These transition temperatures start deviating from each other
beyond $\approx 4$ GPa. The open (white) symbols correspond to $T_{c}^{0}$
from a previous report for FeSe0.91S0.09 in Ref. [19]. A strong current
dependence is observed for the data point marked by an arrow ($T_{c}^{0}$ of
sample N at $p=5.2$ GPa), as discussed in the main text. Thick solid lines are
guide-to-the-eyes and the dashed line is a linear extrapolation. (b)
Comparison of the offset superconducting transition temperatures
$T_{c}^{\text{off}}$ using different high-pressure methods, as reported here
and previously. CAC = Cubic Anvil Cell, DAC = opposing Diamond Anvil Cell, PCC
= Piston Cylinder Cell, Meth:Eth = 4:1 mixture of Methanol:Ethanol, $\rho$ =
transport study, $\chi_{ac}$ = AC susceptibility study [18, 15, 29, 16, 19].

Mirroring the evolution of the superconducting transition temperatures with
pressure, the overall behaviour of the resistivity is also very sensitive to
the applied pressure in different regimes. To better quantify the relevant
pressure ranges, Fig. 3(a) shows the isothermal evolution of the resistivity
for all samples studied here. Initially, and at room temperature, the
resistivity shows a continuous reduction for pressures $p\lesssim 5$ GPa which
appears consistent with an increasing electronic bandwidth as the orbital
overlaps increase. In contrast, the low-temperature resistivity shows a more
nuanced pressure dependence. At the lowest pressures, below $1$ GPa, a marked
drop of the low-temperature resistivity is observed as the nematic phase is
suppressed. This trend can be explained by a growing Fermi surface and a
reduced quasiparticle mass, based on our previous high-pressure transport and
quantum oscillation studies up to $2.2$ GPa [16]. Additionally, scattering off
nematic and/or spin fluctuations as well as nematic domain boundaries within
the nematic phase may further add to the low-pressure, low-temperature
resistivity, but all effects get suppressed with increasing pressure [31, 32].
Upon further increasing the pressure, the low-temperature resistivity shows an
upturn around $\approx 1.2$ GPa (sample P) or $\approx 2.2$ GPa (samples A-N).
The corresponding pressure difference $\Delta p\approx 1$ GPa between the
resistivities of sample P and samples A-N is consistent with the pressure
difference of their superconducting transition temperatures, as discussed
above. Interestingly, for an intermediate temperature $T\approx 100$ K, the
resistivity appears essentially pressure independent between $\approx 1$ GPa
and $\approx 4$ GPa, which corresponds to the visible crossing point of the
resistivity curves in Fig. 1(a), similar to previous reports [16, 19]. Such a
crossing could correspond to an electronic crossover from incoherent to
coherent transport predicted for a Hund’s metal, or to a change in scattering
or electronic correlations tuned by pressure [33].

Towards highest pressures, the evolution of the resistivity changes
significantly at all temperatures. Figure 3(a) shows an upturn in the
resistivity as a function of pressure at $p_{s}(T)\approx 4.5-5$ GPa. Since
$p_{s}$ can be identified up to room temperature, we interpret it as a
signature of a pressure induced structural transition. Figure 3(b) shows the
residual resistivity just below the offset temperature $T_{c}^{\text{off}}$
which illustrates that the zero-resistivity superconducting state is lost
above $p_{s}(T_{c}^{\text{off}})\approx 4.2$ GPa. At higher pressures, the
residual resistivity increases sharply as $\propto(p-p_{s})^{2}$, signaling
the emergence of a low-temperature resistive tail below the main
superconducting transition. Eventually, the resistivity displays incomplete
superconducting transition beyond $p\approx 5$ GPa, as discussed above. This
unusual loss of superconductivity is similar to previous studies of FeSe,
where it was associated with a structural transition into a orthorhombic or
possibly hexagonal phase [34, 8]. Therefore, the evolution of the normal and
superconducting behavior at $p=p_{s}$ appear related and could be driven by a
structural transition.

Figure 3: Correlation between resistivity increase and loss of
superconductivity. (a) Absolute resistivity as a function of pressure at
selected fixed temperatures. The horizontal bar indicates a pressure offset of
$\Delta p\approx 1$ GPa between the data measured on sample P using a piston
cylinder cell (open triangles) and the other samples using Diamond Anvil Cells
(filled symbols). (b) Residual resistivity just below $T_{c}^{\text{off}}$.
Solid lines are a guides to the eye. The dashed lines indicate the relevant
pressure scale where kinks in the data appear.

Importantly, the pressure $p_{s}\approx 4.5$ GPa, at which there are
substantial changes in the resistive and superconducting properties of
FeSe0.89S0.11, is similar to the hydrostatic limit of the employed pressure
medium Glycerol [28]. In order to test the influence of the pressure medium,
we have performed a comparative study using the TDO technique with Glyercol
and a 4:1 mixture of Methanol:Ethanol, which ensure hydrostatic conditions up
to $10$ GPa [27, 28]. Figures 1(c) and (d) show the evolution of the resonant
frequencies and their first derivatives, respectively, for both pressure media
as a function of pressure and temperature. Since an estimation of the sample
resistivity from the TDO frequency depends on precise knowledge of the
experimental set up (sample and coil dimensions and any parasitic capacitance
unavoidable in a pressure cell), we focus on the evolution of the
superconducting transition only. Figs. 1(c) and (d) reveal that
superconductivity is lost for both pressure media, albeit at a pressure about
$1$ GPa larger when using Methanol:Ethanol compared to Glycerol. Therefore,
the loss in superconductivity and the increase in resistivity are intrinsic to
high-pressure FeSe0.89S0.11, but the exact pressure is affected by the
transmitting medium.

Figure 4: Non-Fermi liquid behavior in pressurized FeSe0.89S0.11 above the
superconducting dome. (a) Longitudinal resistivity $\rho_{xx}(T)$ of samples
A, B and N plotted against $T^{0.75}$ for pressures close to the maximum of
$T_{c}^{0}$. Dashed lines are linear fits and curves are vertically offset for
clarity. (b) Resistivity exponent $k$ as obtained by a the logarithmic
derivative of $\partial\rho/\partial T$ of sample B as a function of
temperature and pressure. A clear fan with $k\approx 0.75$ is observed above
the superconducting dome reaching up to $\approx 130$ K, whereas $k\approx
1.5$ outside the fan at low pressures. Dashed lines indicate the boundaries of
the fan and their intersection at a pressure $p_{Q}$ at zero temperature. (c)
Resistivity exponent $k$ obtained by fitting the experimental data using
$\rho_{xx}(T)=\rho_{0}+aT^{k}$ over the fixed temperature interval $40-100$ K,
outside the nematic phase. The horizontal dashed line indicates the best
estimate for $k$ at the minimum and its experimental uncertainty. The vertical
dashed line indicates the extrapolated critical pressure $p_{Q}$ as obtained
from panel (b). The finite size horizontal bar indicates a pressure offset
$\Delta p\approx 1$ GPa between the data obtained from sample P measured
inside a PCC, and the other samples measured inside DACs. (d) The pressure
dependence of the Fermi liquid temperature, $T_{\text{FL}}$, and the
superconducting onset temperature $T_{c}^{\text{on}}$. The inset shows the
residual resistivity $\rho_{0}$ obtained from fits using $k=0.75$ which
changes sign at $p_{Q}$. Vertical error bars indicate the standard deviation
of $T_{\text{FL}}$ and $\rho_{0}$ from varying $k\pm\delta k$ with $\delta
k=0.1$. Solid lines are guides-to-the-eye and the vertical dashed lines
represents $p_{Q}$.

###  II.2 Non-Fermi Liquid Behaviour

Despite the clear changes in the resistive and superconducting properties as
well as their mutual correlation as a function of pressure, our multiple
studies do not reveal signatures of any additional phase transition. The
smooth temperature dependence of the isobar resistivity $\rho(T)$ and the TDO
resonant frequency $f(T)$ (see Fig. 1), are in marked contrast to previous
studies of FeSe1-xSx under high pressures. In the case of FeSe, the
resistivity displays kink-like signatures at the onset of a SDW phase from
pressure above $\approx 0.8$ GPa [8, 35, 29, 22], whereas FeSe0.88S0.12 shows
weaker anomalies in resistivity in the range $\approx 5-7$ GPa [15]. We can
rule out that any thermodynamic transition involving changes in the magnetic
susceptibility occurs even within the superconducting phase as the TDO
technique would be capable of detecting such transitions [36]. Furthermore,
superconductivity survives in the high-pressure phase of FeSe and FeSe1-xSx
for similar compositions when studied using a cubic anvil cell and no
insulating behavior occurs up to $8$ GPa [8, 15] which clearly differs from
our observations. In contrast, the observed loss of superconductivity under
pressure observed here is consistent with a previous $ac$ susceptibility
study, using a similar pressure technique [18]. These seemingly contradictory
results may point at a strong sensitivity of FeSe-based materials to uniaxial
pressure arising from the thermal contraction of opposing anvil cells as
employed here [25, 26] and in Ref. 18, combined with the additional effect
induced by the sample thickness [22, 21].

Yet, despite the absence of a signature of any long-range phase transition in
the temperature dependence of the resistivity and TDO resonant frequency, we
observe strong changes in the normal state properties at low temperatures
which indicate that our samples are in the proximity to a pressure-induced
electronic instability. Figure 4 summarizes the presence of a clear fan of
non-Fermi liquid behavior in the resistivity, which is often regarded as a
hall-mark of putative quantum critical behavior. The resistivity in the
vicinity of $p_{s}$ displays a power-law form $\rho(T)=\rho_{0}+aT^{k}$ with
$k\approx 0.75$ for multiple samples, ranging from the onset of
superconductivity up to $\approx 130$ K. To trace the pressure and temperature
regimes where the exponent $k=0.75$ provides a valid description of the
resistivity, we extract $k$ using a second derivative method as
$k=1+\partial/\partial(\log T)(\partial\rho/\partial T)$. A key advantage of
this method is that it is independent of the knowledge of $\rho_{0}$ which is
often unreliable to extract from high-temperature data. Figure 4(b) shows the
pressure and temperature dependence of the exponent $k$ which reveals a fan-
like structure above the superconducting dome for sample B, before the fan is
truncated by non-metallic behavior from $\approx 5$ GPa onwards. Linearly
extrapolating the low- and high-pressure edges of the fan to zero temperature,
we find the fan to collapse to a point at $p_{Q}=4.3(2)$ GPa, close to the
pressure scale $p_{s}$ identified above which is also the pressure close to
the maximum of $T_{c}^{0}$. The presence of a fan-like dependence of the
resistivity exponent has been reported in many systems and it is typically
regarded as a strong indicator for the presence of a putative quantum critical
point (QCP) buried underneath the superconducting dome. A potential candidate
for such behaviour could be an antiferromagnetic QCP, suggested to exist in
the FeSe1-xSx under pressure, measured using a cubic anvil cell [8, 15].

Figure 4(c) shows that the analysis for the resistivity exponent $k$ gives
consistent result for all samples measured using DACs, but differs in the
pressure dependence from sample P measured in the piston cylinder cell. Here,
$k$ was extracted over a fixed temperature interval between $40$ K and $100$ K
outside the quantum critical fan from low pressures $\lesssim 2$ GPa and
across the quantum critical fan at $p_{Q}$. At the lowest pressures, outside
the nematic phase, we find $k\approx 1.5$ for the DAC, in excellent agreement
with sample P measured inside the piston cylinder cell, consistent with
previous reports [37, 16]. For all samples measured in the DACs the exponent
$k$ has a sample-independent pressure dependence and it is continuously
reduced upon entering the quantum critical fan to reach a minimum of
$k=0.75\pm 0.1$ for $p_{Q}=4.3(2)$ GPa. Evidently, we detect a pressure offset
in the evolution of $k$ in sample P (measured in a piston cylinder cell) by
$\Delta p\approx 1$ GPa around $p\approx 2$ GPa (Figure 4(c)), which is
consistent with the offset in $T_{c}$ and the absolute value of $\rho$ in this
pressure range, shown in Figs. 2(b) and 3(b).

Figure 5: Critical current study in the high-pressure phase of FeSe0.89S0.11
for sample N. (a) The temperature dependence of zero-field resistivity during
cooling as a function of applied current $I$ under a pressure of $p\approx
5.0$ GPa. (b-c) $I$-$V$-curves as a function of temperature for $p=5.0$ GPa
(panel b) and $p=6.1$ GPa (panel c). The solid lines are guide-to-the-eyes.
The dashed black line is a linear dependence $V\sim I$ indicating an ohmic
resistance.

Figure 4(d) provides an additional test of the location of the quantum
critical fan close to $p_{Q}\approx 4.3$ GPa. The inset shows the extracted
values for $\rho_{0}$ as obtained from the fixed-exponent fits presented in
Fig. 4(a). Evidently, for small pressures $p<p_{Q}\approx 4.3$ GPa, $\rho_{0}$
appears negative which is unphysical, in contrast to larger pressure
$p>p_{Q}\approx 4.3$ GPa where a positive $\rho_{0}$ is obtained. This finding
implies that for pressure lower than the putative quantum critical point, the
resistivity must return to a larger exponent at low temperatures, e.g. return
to Fermi liquid behavior with $k=2$. We can make a rough estimate for such a
Fermi liquid cross-over temperature $T_{\text{FL}}$ by matching Fermi liquid
and non-Fermi liquid behavior and requiring $\rho_{0}=0$. The resulting
$T_{\text{FL}}\approx 10-20$ K is shown in the main panel of Fig. 4(d). These
values for $T_{\text{FL}}$ are similar to our previous estimate close to the
nematic quantum critical point where $T_{\text{FL}}=11$ K was extracted for
sample P [16]. Moreover, we find that $T_{\text{FL}}$ is always lower than
$T_{c}^{\text{on}}$ and thus cannot be resolved in Figure 4(b). This analysis
also shows that a collapsing $T_{\text{FL}}$ in the vicinity of $p_{Q}$ is
consistent within the experimental resolution.

Finally, we note that the extracted value of the resistivity exponent
$k\approx 0.75=3/4$ is unusual and differs from typical values for
ferromagnetic or antiferromagnetic quantum criticality, in either 2D or 3D,
and for clean or dirty systems [38, 39]. Hence, we cannot infer the nature of
the QCP from this value. However, theoretical predictions for quantum critical
exponents in transport measurements actually relate to the temperature
dependence of the scattering rate $\tau^{-1}$, and not to the resistivity
$\rho$. Since the resistivity $\rho\sim m^{\star}\tau^{-1}/n$ also depends on
the charge carrier concentration $n$ and the effective mass $m^{\star}$, an
explicit temperature dependence of these quantities will alter the observed
resistivity exponent $k$. Indeed, FeSe and related systems display a strong
temperature dependence of the chemical potential [40] which manifests as a
variation of the charge carrier density as a function of temperature.
Additionally, inside the nematic phase, anisotropic scattering due to spin
fluctuations may also become important [41, 42]. Further magnetotransport
studies will be required to assess changes in scattering in the high-pressure
phase of FeSe1-xSx.

##  III $I$-$V$ Characteristic

We now turn to the nature of the high-pressure, low-temperature phase which
displays a finite but incomplete superconducting phase. In order to separate
normal metallic from partially superconducting resistances, we perform a
$I$-$V$-study to search for non-ohmic conduction and to determine the
superconducting critical current density. Assuming an inhomogeneous
superconducting phase with only filamentary superconducting paths, we would
expect a strongly reduced critical current density $j$, when measured across
the entire sample, which should be the smallest close to $T_{c}$. Figure 5(a)
shows cooling curves on sample N at a pressure of $p\approx 5.0$ GPa under
different applied currents across the broad superconducting transition. At
this pressure, the sample is tuned beyond $p_{s}\approx p_{Q}$, but it shows a
distinct superconducting onset, and eventually a zero-resistance for all
currents studied, see also Fig. 2. However, in contrast to expectations, we
find a strongly current-dependent sample resistivity only for temperatures
below $T\approx 20$ K, whereas close to the onset $T_{c}^{\text{on}}$ and
sharpest-drop $T_{c}^{\text{p}}$, the resistivity is essentially current-
independent, as shown in Fig. 5(b). By increasing the pressure towards $6.2$
GPa, shown in Fig. 5(c), we observe ohmic resistance at any temperature,
despite a continuously well defined superconducting onset and sharp drop in
the zero-field resistivity, cf. Fig. 1(a). This demonstrates that the high-
pressure, low-temperature phase beyond $p_{s}\approx p_{Q}$ cannot simply be
understood as filamentary superconductivity, but a more complex model is
required, as discussed below.

##  IV Summary and Discussion

Figure 6: Proposed phase diagram covering the nematic, superconducting,
tetragonal, quantum critical and coexistence regions. Real space sketches
indicate how the superconducting volume fraction is lost.

Our high-pressure study of the superconducting and normal state properties of
FeSe0.89S0.11 reveals a complex pressure-temperature phase diagram, as
summarized in Fig. 6. At very low pressures, the system displays a well-
established nematic order below $T_{s}$ and $p_{c}$ [16, 17], before it
becomes superconducting below $10$ K. Outside the nematic phase, $T_{c}$ first
reaches a minimum of $6$ K at a pressure of $1$ GPa, before it surges towards
$\approx 40$ K with increasing pressure. At high pressures, we observe a
number of signatures in our transport data which suggest concomitant
electronic and structural instabilities across $p_{Q}=4.3$ GPa and
$p_{s}(T)\approx 4-4.5$ GPa, respectively. Firstly, reaching up to room
temperature, the resistivity shows a marked up-turn at $p_{s}$, and the system
develops a non-metallic temperature dependence. Secondly, we uncover clear
signatures of non-Fermi liquid behavior at $p_{Q}$ which suggests the presence
of an electronic quantum critical point located at this pressure. Thirdly, the
unusual $I$-$V$ dependence clearly suggests a loss of the superconducting
volume fraction, consistent with a previous $ac$ susceptibility study [18].

The signatures of the high pressure phase could be explained considering a
phase separation scenario, as sketched in Fig. 6. At low pressures, $p<p_{s}$,
the superconducting phase has sharp superconducting transitions and reaches
the zero-resistance state, as shown in Fig. 1(a). Moreover, there is a large
change in the TDO resonant frequency, as evident from Fig. 1(c), and a sizable
response in a previous $ac$ susceptibility study [18]. Therefore, this low-
pressure superconducting phase reflects the bulk nature of the superconducting
state of the single crystals. In contrast, in the high-pressure phase, a
finite resistance remains at lowest temperatures which displays clear non-
ohmic behaviour only close to $p_{s}\approx p_{Q}$, whereas the onset of the
superconducting transition remains sharp and the finite resistance displays
ohmic behaviour. This suggests that for temperatures just below
$T_{c}^{\text{on}}$, only isolated superconducting islands develop within an
emerging non-metallic matrix but they fail to connect to provide a fully
superconducting current path. Thus, a drop in the resistance is observed when
the islands become superconducting, and any finite resistance measured in this
temperature regime arises solely from the normal-state non-metallic matrix. In
this case, the resistance should have ohmic behaviour and $T_{c}^{\text{on}}$
should essentially be current independent, as observed in Fig. 5. Upon further
lowering the temperature, the remaining normal-state matrix gradually develops
filamentary connections between the superconducting islands, which induce a
strongly current dependent phase and non-ohmic resistance, as observed in Fig.
5. With increasing pressure, the superconducting islands shrink and gradually
disappear which makes a filamentary connection unlikely. This development is
fully consistent with the remaining but shrinking sharp drop of the sample
resistance at $T_{c}^{\text{on}}$ and $T_{c}^{\text{p}}$, even under a
pressure of $p=6.2$ GPa, the fully ohmic resistance observed at any
temperatures, and the lack of a zero-resistance phase. By increasing the
pressure even further, the islands are lost eventually, and the drop in
resistance disappears, as shown in Figs. 1(a).

The occurrence of a phase separation across $p_{s}\approx p_{Q}$ is consistent
with a structural first-order transition. Yet, the observation of signatures
associated with quantum criticality at $p_{Q}$ implies a second-order
transition. The coexistence of such distinct transitions is highly unusual.
Previous studies have reported pressure-induced structural transitions in
FeSe1-xSx crystals at room temperature, however, they occur at much higher
pressures than the low-temperature electronic instabilities. For example, FeSe
displays signatures of a low-temperature SDW phase in the pressure range
$\approx 2-6$ GPa which occurs together with a weak orthorhombic lattice
distortion associated with the onset of the magnetic order. Yet, the system
remains metallic in this pressure regime [8, 43, 23]. In contrast, a room-
temperature structural instability from tetragonal towards orthorhombic and
hexagonal phases in FeSe could be induced only at much higher pressures above
$10$ GPa where non-metallic behavior was observed [8, 34, 23]. Similarly for
FeSe0.88S0.12, it was suggested that a SDW phase is stabilized in the pressure
range $\approx 4-7$ GPa from transport data anomalies, but no signatures of a
non-metallic phase emerges at high pressures when using a cubic anvil cell
[15]. A pressure-induced structural instability was detected for a similar
compound at room temperatures beyond $9$ GPa only [44].

One interesting aspect of the pressure studies of FeSe1-xSx is that any
discrepancies could be related to the different pressure media and pressure
cell designs used in the experiments as well as the sample thickness. In the
case of structural studies, often Helium is used as pressure medium which
ensures a much more hydrostatic environment, as compared with Glycerol and 4:1
Methanol:Ethanol used here [23, 44]. Cubic anvil cells are also less prone to
uniaxial thermal contractions than the opposing anvil cell used here [25, 26],
and the resistivities display a zero-resistivity state beyond $4$ GPa [8, 15].
In contrast, studied performed with opposing anvil cell designs revealed a
gradual loss of superconductivity around $\approx 4$ GPa and show good
agreement with our study [18, 19]. This suggests that the concomitant
structural and electronic instabilities could arise from finite uniaxial
pressure components unavoidable in an opposing anvil cell setup. This
assumption is supported by studies which demonstrated that the sample
thickness and the choice of pressure medium can have profound influence on the
stability of the SDW phase in FeSe [45, 21]. Thus, it is conceivable that
finite uniaxial pressures along the crystallographic $c$ axis might trigger
the structural transition around $p_{s}\approx 4.2$ GPa. Indeed, many iron-
based superconductors are very sensitive to uniaxial pressure components [46]
and the electronic structure changes significantly with varying chalcogen
height above the conducting Fe planes $h=z\cdot c$ [47]. Interestingly,
$p_{s}\approx p_{Q}$ is remarkably similar to the hydrostatic limit of
Glycerol which solidifies around $4$ GPa at room temperatures [28]. However,
our TDO study demonstrates that superconductivity survives only to marginally
higher pressures of $\approx 5$ GPa using a 4:1 mixture of Methanol:Ethanol as
pressure medium, which has a solidification around $10$ GPa [27, 28]. This
suggests that the structural and electronic instabilities are not directly
triggered by the solidification of Glycerol.

A consequence of the concomitant first-order structural transition at $p_{s}$
and the second order electronic transition of possibly magnetic origin around
$p_{Q}$ is the inherent domain formation that would naturally lead to
additional scattering of the charger-carriers. Interestingly, the resistivity
trends observed under pressures can be compared with the evolution of the
resistivity in the presence of strong impurity scattering in CuxFe1-xSe [48,
14]. The primary effect of the Cu-substitution is to disturb significantly the
charge carrier transport in the Fe-planes, and thus to lead to a significant
enhancement of scattering. As a result, the resistivity shows similar
enhancements both in temperature dependence and in absolute values, as
compared with those in the high-pressure phase, and additionally there is a
strong reduction of charge carrier mobilities [14, 20]. On the other hand, the
loss of superconductivity appears different in CuxFe1-xSe as the
superconducting transition is strongly reduced with increasing Cu content.
However, these distinct observations may be reconciled assuming the
aforementioned phase separation between metallic/superconducting and non-
metallic domains, respectively.

In summary, we have explored the high-pressure phase of the iron-based
superconductor FeSe0.89S0.11. Our transport and TDO measurements revealed a
strongly enhanced superconducting phase at high pressures, which gradually
disappears beyond $\approx 4.2$ GPa as the system becomes non-metallic. This
trend suggests the development of a phase separation triggered by an uniaxial
pressure component induced first-order structural transition. At low
temperatures, we revealed a fan-like structure of non-Fermi liquid behaviour
towering above the superconducting phase which points towards a putative
quantum critical point buried below. Our results reveal the occurrence of
accidentally concomitant structural (first-order) and electronic (second-
order) phase transitions which could explain the strongly reduced critical
current density of the superconducting phase. These effects reported here
emphasize the sensitivity of high-$T_{c}$ superconductivity of iron-
chalcogenides to the structural and electronic degrees of freedom induced by
uniaxial and hydrostatic pressures.

##  V Acknowledgments

We thank Sven Friedemann and Patricia Alireza for their valuable technical
support setting up the high-pressure techniques in Oxford. We further
acknowledge Matthew Watson for his previous measurement of sample E. This work
was mainly supported by the EPSRC (EP/I004475/1, EP/I017836/1). P.R. and
A.A.H. acknowledge financial support of the Oxford Quantum Materials Platform
Grant (EP/M020517/1). Part of this work was supported by HFML-RU and LNCMI-
CNRS, members of the European Magnetic Field Laboratory (EMFL) and by EPSRC
(UK) via its membership to the EMFL (grant no. EP/N01085X/1). We acknowledge
financial support of Oxford University John Fell Fund. Z.Z. acknowledges
financial support from the EPSRC studentship (EP/N509711/1 and EP/R513295/1).
A.I.C. acknowledges an EPSRC Career Acceleration Fellowship (EP/I004475/1) and
Oxford Centre for Applied Superconductivity.

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