# Weak antilocalization and Shubnikov$-$de Haas oscillations in CaCuSb single
crystal

Souvik Sasmal<EMAIL_ADDRESS>Department of Condensed Matter Physics
and Materials Science, Tata Institute of Fundamental Research, Homi Bhabha
Road, Colaba, Mumbai 400 005, India.  Vikas Saini  Department of Condensed
Matter Physics and Materials Science, Tata Institute of Fundamental Research,
Homi Bhabha Road, Colaba, Mumbai 400 005, India.  Nicolas Bruyant  Laboratoire
National des Champs Magnétiques Intenses, CNRS-UGA-UPS-INSA, 31400 Toulouse,
France  Rajib Mondal  UGC-DAE Consortium for Scientific Research, Kolkata
Centre, Bidhannagar, Kolkata 700 106, India.  Ruta Kulkarni  Department of
Condensed Matter Physics and Materials Science, Tata Institute of Fundamental
Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India.  Bahadur Singh
<EMAIL_ADDRESS>Department of Condensed Matter Physics and Materials
Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba,
Mumbai 400 005, India.  Vikram Tripathi  Department of Theoretical Physics,
Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400
005, India.  A. Thamizhavel<EMAIL_ADDRESS>Department of Condensed
Matter Physics and Materials Science, Tata Institute of Fundamental Research,
Homi Bhabha Road, Colaba, Mumbai 400 005, India.

###### Abstract

Quantum oscillations in both linear and Hall resistivities and weak
antilocalization (WAL) are barely observed in bulk single crystals. Here we
report the transport properties of a CaCuSb single crystal that crystallizes
in the hexagonal crystal structure. The magnetotransport studies reveal WAL
and Shubnikov$-$de Haas (SdH) quantum oscillations with a unique frequency at
$314$ T. A cusp-like behavior in the low field regime of magnetotransport for
$J~{}\parallel~{}(ab)$-plane and $B~{}\parallel$ [0001] confirms the WAL in
CaCuSb. Angular-dependent normalized magnetoconductance and SdH oscillations
studies reveal that the observed phenomena originate from the $2D$ transport
channels. The high magnetic field (up to $45$ T) experiments demonstrate
plateau-like features in the Hall measurements. The first-principles
calculations unfold that CaCuSb is a non-topological semimetal with dominant
hole carries at the Fermi level. Our study reveals that CaCuSb is a promising
candidate to explore the quasi$-2D$ quantum transport phenomenon in the
transition metal pnictide materials.

##  I INTRODUCTION

Since the discovery of topological insulators (TIs) with nontrivial metallic
surface states Hasan and Kane (2010); Moore (2010), substantial research
efforts are devoted in searching new topological semimetals owing to their
novel electronic structure and interesting physical properties Hu  _et al._
(2019); Burkov (2016); Wang  _et al._ (2018); Zhang  _et al._ (2017); Berger
_et al._ (2018); Wu  _et al._ (2019); Hao  _et al._ (2018); Cai  _et al._
(2018). Following the foot-print of graphene Meyer  _et al._ (2007), $2D$
materials are widely investigated due to their exotic $2D$ transport
properties which seed from carrier confinement effects Zhang  _et al._ (2019);
Pi  _et al._ (2019). It has been found that in addition to $2D$ materials, the
topological materials with layered crystal structure also show the $2D$
behavior of carriers. In this regard, the transition metal-pnictides layered
ternary compounds of $ABC$-type offer a prominent setup for realizing exotic
topological and transport propertiesSingh  _et al._ (2018); Mardanya  _et al._
(2019); Sasmal  _et al._ (2020). For example, CaAgBi Sasmal  _et al._ (2020)
exhibits Dirac-Weyl semimetal with weak antilocalization (WAL) effects,
whereas LuPdBi Pavlosiuk  _et al._ (2015) and YPtBi Butch  _et al._ (2011)
exhibit WAL as well as superconductivity at low temperature. When the phase
coherence length ($l_{\Phi}$) is larger than the mean free path ($l$), quantum
interference results in either the weak localization (WL) or WAL effects
depending on whether the interference is constructive or destructive,
respectively Lu and Shen (2014); Datta (1995); Zhao  _et al._ (2019a). The
strong spin-orbit coupling (SOC) effects lead to spin-rotation in materials
whenever an electron gets scattered due to impurity, and thereby causing the
phase change of the electron wave function, realizing a destructive
interference. The destructive interference enhances the conductivity and a
cusp-like negative conductivity is observed in the single crystals of various
ternary materials Shrestha  _et al._ (2017); Sasmal  _et al._ (2020); Laha
_et al._ (2019). At a sufficiently high magnetic field, if the system is in a
quantum diffusive regime ($\hbar\omega>k_{B}T$), the Fermi energy crosses the
Landau levels with increasing field, leading to Shubnikov$-$de Haas (SdH)
oscillations in the magnetoresistance. There are a few reports that show both
the WAL and quantum oscillations in longitudinal resistivity, although
oscillations in Hall data are usually absent in those materials Bao  _et al._
(2012); Bhardwaj  _et al._ (2018).

In this study, we report magnetotransport properties and demonstrate the WAL
and SdH quantum oscillations in both the linear and Hall resistivities in a
single crystal of ${\rm CaCuSb}$. CaCuSb belongs to a large family of
$ABC$-type hexagonal materials with the space group $P6_{3}/mmc$ (No. 194). A
cusp-like behavior is observed in the low field region of magnetoresistance
(MR), confirming the WAL effect in CaCuSb. At sufficiently high magnetic
fields, SdH oscillations are observed which enable us to reveal its conduction
and Fermi surface properties. Interestingly, the Hall effect measurements show
the signatures of Hall plateaus at definite intervals. Our analyses suggest
that the observed transport phenomena in $\rm{CaCuSb}$ are due to the
quasi-$2D$ transport channels in the layered crystal structure, consisting of
Cu-Sb layers that are separated by Ca layers.

##  II Experimental Methods

Good quality single crystals of CaCuSb were synthesized by flux method using
Cu:Sb eutectic composition as a flux. High purity starting elements of Ca
(99.98% ingot from Alfa Aesar), Cu (99.999% shot from Alfa Aesar) and Sb
(99.999% shot from Alfa Aesar) were taken in 1:4.5:7.5 molar ratio in an
alumina crucible. The alumina crucible was sealed in a quartz ampoule under a
partial pressure of Ar gas. The sealed ampoule was placed in a box type
furnace, and heated to 1050∘C at the rate of 30∘C/h and held at this
temperature for 24 h, for proper homogenization. Then the furnace was allowed
to slow cool at the rate of 2∘C/h to 630∘C at which point the flux was removed
by centrifuging. Electrical transport studies have been performed on a cut and
aligned bar-shaped sample with current along the principle crystallographic
directions $viz.,$ in the basal plane and along the $c$-axis. Electrical
transport measurements were carried out on a sample with dimension $1.5\times
1\times 0.12$ mm3, using Quantum Design Physical Property Measurement System
(PPMS) instrument with magnetic field up to 14 T, in the temperature range
$2-300$ K. High magnetic field Hall effect measurement was performed at the
European Magnetic Field Laboratory (EMFL) at the Laboratoire National des
Champs Magnétiques Intenses (LNCMI) at Toulouse, France, up to a mangetic
field of $45$ T.

##  III Results and Discussions

###  III.1 X-ray diffraction

The room temperature powder x-ray diffraction (XRD) and Rietveld refinement of
the XRD data are shown in Fig. 1(a). From the refinement we estimate the
lattice parameters as $a~{}=~{}4.452$ Å and $c~{}=~{}8.133$ Å which are in
good agreement with the previously published data Eisenmann  _et al._ (1974).
The Rietveld analysis also confirmed the space group as $P6_{3}/mmc$ (No. 194)
and the Wyckoff positions of Ca, Cu, and Sb as $2a$, $2d$ and $2c$
respectively. The crystal structure of CaCuSb is shown in the inset of Fig.
1(a). It is evident that the crystal structure is layered along the hexagonal
$c$-axis where Cu$-$Sb layers are sandwiched between Ca-layers and each
Cu$-$Sb layer is rotated by $60^{\circ}$ relative to the neighbouring layer.
These Cu$-$Sb layers may act as a unique charge transport layers which are
confined in the $(ab)$-plane. As grown single crystal with the flat plane
corresponding to (0001) is shown in Fig 1(b). The Laue diffraction pattern of
the flat plane is shown in Fig. 1(c). Well defined Laue diffraction pattern
together with the $6$-fold symmetry confirms the good quality of the single
crystal.

Figure 1: (a) Powder x-ray diffraction pattern along with the Rietveld
refinement of CaCuSb. Inset: The hexagonal crystal structure of CaCuSb. Each
Cu$-$Sb layer is rotated by 60∘ relative to the neighbouring Cu$-$Sb layer.
(b) As grown crystal image (c) Laue diffraction pattern of CaCuSb depicting
the (0001)-plane. Figure 2: (a) Temperature dependence of the electrical
resistivities $\rho_{xx}$ and $\rho_{zz}$ of CaCuSb in zero magnetic field.
The inset shows the anisotropy ratio ($\rho_{zz}$/$\rho_{xx}$) as function of
temperature. (b) The magnetoresistance at $T=2$ K, for
$J~{}\parallel~{}$[12̄10] and $J~{}\parallel~{}[0001]$. A cusp like behavior
in low field region indicates the WAL effect. (c) Hall resistivity $\rho_{xy}$
as a function of magnetic field ($B$) at $1.8$ K. The black solid line show
the linear fit to the data to extract the carrier concentration. The linear
resistivity $\rho_{xx}$ up to $14$ T is also shown. (d) Magnetoconductivity
($\Delta\sigma_{xx}$) at $2$ K (red markers) and HLN fit (dash black line)
(see text for details). (e) Phase coherence length ($L_{\Phi}$) and
coefficient $\alpha$ as a function of temperature. Black solid line indicates
the phase coherence length fitting using Eq.( 2). (f) Normalized
magnetoconductivity ($\sigma_{\rm N}$(Normalized)) at different tilt angles
($\theta$) between current ($J$ $\parallel$ [12̄10]) and $B$.

###  III.2 Electrical transport and Hall resistivity

The temperature dependence of electrical resistivity is shown in Fig 2(a) for
both in-plane ($\rho_{xx}$) and out-of-plane ($\rho_{zz}$) current directions
which shows a metallic nature. The electrical resistivity is anisotropic
depending on the direction of the current flow and a large anisotropy is
observed for in-plane and out-of-plane current directions, thus it refers to a
quasi-$2D$ nature of the electrical transport. Figure 2(b) shows the
magnetoresistance (MR) at $T=2$ K, for $J~{}\parallel~{}$[12̄10] ($ab$-plane)
and $B~{}\parallel~{}$[0001], showing a cusp-like behavior that increases
rapidly at low fields and linearly above $3$ T, and the MR reaches 41% at 14
T. This cusp-like behavior in MR is reminiscent of the WAL effect seen in
similar equiatomic systems He  _et al._ (2011); Liu  _et al._ (2012); Sasmal
_et al._ (2020); Pavlosiuk  _et al._ (2015); Butch  _et al._ (2011).
Generally, the WAL phenomenon is observed due to $2D$ charge diffusion in the
system. However, the presence of such WAL for $J~{}\parallel$ [12̄10] here
suggests the possibility of destructive interference in the $ab$\- plane. For
$J~{}\parallel$ [0001] and $B~{}\parallel$ [101̄0], MR is 3.5% at 14 T (12
times smaller than $ab$-plane) and almost linear in the entire field range
without any sign of WAL in this direction, indicating the absence of such
interference effects. These results support that the observed WAL has a $2D$
nature. The WAL behavior vanishes for $T>50~{}K$, as shown in the Supplemental
Material Figure S1 sup . Furthermore, in the range of $10-14$ T, SdH
oscillation has been observed for $J~{}\parallel$ [12̄10] and $B~{}\parallel$
[0001], but it is absent for $J~{}\parallel$ [0001] and $B~{}\parallel$
[101̄0]. Here, we have symmetrized the linear resistivity
$\rho_{xx}=[\rho_{xx}^{raw}(+B)+\rho_{xx}^{raw}(-B)]/2$ and antisymmetrized
the Hall resistivity $\rho_{xy}=[\rho_{xy}^{raw}(+B)-\rho_{xy}^{raw}(-B)]/2$.
The Hall resistivity is measured in a pulsed magnetic field of $45$ T at
$T=1.8$ K is shown in Fig. 2(c), which remains positive, suggesting that the
hole carriers are dominant in the system. From the slope of Hall resistivity
we estimate the carrier density and the mobility using a single band model as
$n_{\rm h}\sim 5.81~{}\times~{}10^{26}$ m-3 and $\mu_{\rm h}\sim 0.043$
m2V-1s-1, respectively. The two-band model gives contribution from both the
electron and hole (See Supplemental Material Figure S2 sup ).

###  III.3 Weak antilocalization

The calculated magnetoconductivity (MC),
$\Delta\sigma_{xx}(B)=\sigma_{xx}(B)-\sigma_{xx}(0)$ with
$\sigma_{xx}(B)=\frac{\rho_{xx}(B)}{\rho_{xx}^{2}(B)+\rho_{xy}^{2}(B)}$, for
$J~{}\parallel~{}$[12̄10] and $B~{}\parallel~{}[0001]$ is shown in Fig. 2(d).
It is obvious that the calculated MC also exhibits a cusp-like feature
suggesting WAL in this quasi-$2D$ CaCuSb. Usually, this cusp-like feature can
be seen at very low field for low dimensional systems. In the present case at
$B=4$ T, the value of $\omega\tau=\frac{eB}{m^{*}}\tau=0.26$, which is less
than 1, where $m^{*}=0.4m_{e}$ and $\tau=1.5\times 10^{-13}$ s, and $e$ is the
electronic charge (the quantum parameters $m^{*}$ and $\tau$ have been
obtained from the SdH oscillations, as discussed later). This suggests the
charge transport is in quantum diffusive regime, so that WAL has been observed
even at $B=3$ T. Notably, the observed WAL in MC at low fields can be
explained based on the Hikami-Larkin-Nagaoka (HLN) model Hikami  _et al._
(1980). In the strong SOC limit, the HLN model is given as

$\Delta\sigma_{xx}(B)=-\frac{\alpha
e^{2}}{2\pi^{2}\hbar}\left[\psi\left(\frac{\hbar}{4eL_{\phi}^{2}B}+\frac{1}{2}\right)\\\
-ln\left(\frac{\hbar}{4eL_{\phi}^{2}B}\right)\right],$ (1)

where $e$, $\psi$, and $L_{\phi}$ are the electronic charge, digamma function,
and phase coherence length, respectively. From this HLN fit, the phase
coherence length is estimated as $L_{\phi}=53.6$ nm and
${\alpha}=3.41~{}\times~{}10^{10}$ m-1 (shown in in Fig 2(d)). Here, $\alpha$
indicates the number of conduction channels per unit length. Considering the
sample thickness of 120 $\mu m$, the obtained total conduction channels in the
system is $\sim 4\times 10^{6}$. Such a large number of conduction channels
has been reported in several semimetallic systems like Bi2Te3 Shrestha  _et
al._ (2017), YbCdGe Laha  _et al._ (2019), LuPdBi Pavlosiuk  _et al._ (2015),
YPtBi Pavlosiuk  _et al._ (2016). Temperature dependence of phase coherence
length ($L_{\Phi}$) is shown in Fig 2(e). Because of phononic contribution the
$L_{\Phi}$ decreases with increasing temperature and it can be described by
the relation

$\frac{1}{L_{\Phi}^{2}(T)}=\frac{1}{L_{\Phi}^{2}(0)}+A_{ee}T+A_{ep}T^{2},$ (2)

where $L_{\Phi}(0)$ the zero temperature phase coherence length, and $A_{ee}T$
and $A_{ep}T^{2}$ are the electron-electron and electron-phonon contribution,
respectively. The obtained fitting parameters are $L_{\Phi}(0)=53.7$ nm,
$A_{ee}=3.35\times 10^{-7}$ nm${}^{-2}K^{-1}$ and $A_{ep}=2.32\times 10^{-7}$
nm${}^{-2}K^{-2}$. The coefficient $\alpha$ is almost constant through the
temperature regime $2-20$ K, this suggests the number of conduction channels
are almost unaltered with temperature. Furthermore, we have also performed the
angular dependence of MC at $T=2$ K to understand the origin of WAL (Fig.
2(f)). The normalized MC $\sigma_{N}$(Normalized)$=\frac{R(0)}{R(B,\theta)}$
is plotted as a function $B\cdot{\rm sin}(\theta)$ at different tilt angles
$\theta$ ($\theta$ is the angle between the direction of current and magnetic
field). The $\sigma_{\rm N}$ vs $B\cdot{\rm sin}{(\theta)}$ plot shows that
the low field MC falls onto a single curve for all the measured field angles.
This suggests that WAL effect is originated from $2D$ parallel conduction
channels in the bulk Chen  _et al._ (2011); Shrestha  _et al._ (2017); Xu  _et
al._ (2014). Similar behavior of MC curve is seen in other measured samples
(Supplemental Material Figure S3 sup ). The $2D$ parallel conduction channels
are presumably due to the Cu$-$Sb layers which is further confirmed from the
band structure studies (to be discussed later) where the density-of-states
(DOS) of the Cu and Sb atoms is much higher than Ca atoms at the Fermi level.

###  III.4 Shubnikov$-$de Haas (SdH) quantum oscillations

It is interesting to note that the longitudinal resistance of CaCuSb at
sufficiently large magnetic fields ($B~{}\parallel~{}[0001]$) oscillates
periodically in $1/B$, depicting SdH quantum oscillations. The raw SdH
oscillation observed in the field range $10-14$ T at $T=2$ K is shown in Fig.
3(a). We further obtain SdH frequency by subtracting the background signal
using a polynomial function and plot $\Delta R_{xx}$ vs $B$ in the top inset
of Fig. 3(a). A simple fast Fourier transform (FFT) of $\Delta R_{xx}$ as a
function of $1/B$ reveals a unique frequency at $314$ T. A similar quantum
oscillation in linear resistivity has been observed on a different sample B2,
see Figure S4 in Supplemental Material sup . From the Onsager relation
$F=(\Phi_{0}/2\pi^{2})A_{ext}$ we have estimated extremal area $A_{ext}$ of
the Fermi surface, where $\Phi_{0}$ is quantum of flux. $\Delta R_{xx}$ and
FFT amplitude at different temperatures are shown in Fig 3(b)-(c). The
oscillating part of the $\Delta R_{xx}$ has been analyzed using the standard
Lifshitz-Kosevich (LK) equation which is given by Shoenberg (2009)

$\Delta R_{xx}(T,B)\propto~{}R_{\rm B}~{}R_{T}~{}{\rm
cos}\left[2\pi\left(\frac{F}{B}+\frac{1}{2}+\beta\right)\right].$ (3)

Figure 3: (a) SdH quantum oscillation at $2$ K. The solid black line shows the estimated background using a polynomial function. The top inset shows the background-subtracted SdH oscillations and the lower inset illustrates observed single quantum oscillation frequency at $314$ T. (b) SdH oscillation at different temperature (c) oscillation frequency at different temperature. The corresponding mass plot is shown in the inset. (d) Dingle plot ($T_{\rm D}=8$ K) for estimating the Dingle temperature in CaCuSb. (e) $\Delta R_{xx}$ versus $1/B$ at various field angles. (f) Angular-dependent SdH oscillation frequencies of experimental data (black solid square), calculated electron (red) and hole (blue) pockets.  Figure 4: (a) $\Delta R_{xx}$ and $\Delta R_{xy}$ as a function of $1/B$. $\Delta R_{xx}$ is 90∘ out-of-phase with $\Delta R_{xy}$. (b) $1/\rho_{xy}$ vs $1/B$ plot (measured up to a field of 45 T). The horizontal lines mark the Hall plateaus with LL index $n$. Inset shows the plateau position of $1/\rho_{xy}$ as a function of $1/B$. The linear behavior establishes the quantization nature of Hall data.  Table 1: List of parameters estimated from the SdH quantum oscillations measurements. See the main text for the details. $F({\rm T})$ | $A_{ext}({\rm m}^{-2})$ | $\mu_{SdH}({\rm m}^{2}{\rm V}^{-1}{\rm s}^{-1})$ | $m^{*}(m_{e})$ | $T_{D}({\rm K})$ | $\tau~{}({\rm s})$
---|---|---|---|---|---
$314$ | $3\times 10^{18}$ | 0.0652 | 0.4 | 8 | 1.5$\times 10^{-13}$

Here, $R_{\rm B}={\rm exp}(-\lambda T_{\rm D}m^{*}/B)$ is the field induced
damping factor where $\lambda=(2\pi^{2}k_{\rm B}/(e\hbar))$ is a constant, $F$
is the frequency of oscillation, and $T_{\rm D}$ is the Dingle temperature.
$R_{T}$ is the thermal damping factor given by the LK formula $R_{T}=(X/{\rm
sinh(X)})$ where $X=(\lambda Tm^{*}/B)$ and $\beta$ is the phase shift that
includes Berry phase information. The temperature variation of the FFT
amplitude is fitted to thermal damping factor $R_{T}$ which resulted in an
effective carrier mass $m^{*}\sim 0.4m_{e}$. From the slope of $ln~{}[D]$ vs
$1/B$ plot, where $D=\Delta R_{xx}B$ sinh(X), we obtain the Dingle temperature
($T_{\rm D}$) as $8$ K (Fig. 3 (d)). The carrier life time
$\tau=\frac{\hbar}{2\pi k_{B}T_{\rm D}}$ and mobility
$\mu_{SdH}=\frac{e\tau}{m^{*}}$ are also estimated. All the parameters
obtained from SdH oscillations are listed in Table 1. To confirm the
quasi-$2D$ nature of CaCuSb, we have performed the angular dependence of SdH
oscillation, as shown in Fig. 3(e), where the angle $\theta$ represents the
angle between magnetic field and current. However, as given in Ref. Xiang  _et
al._ (2015); Zhao  _et al._ (2019b, a), there is another approach of rotation
of sample which can also give the Fermi surface information. The angular
dependence of the FFT frequency obtained from the experimentally observed
quantum oscillations is shown as black filled square in Fig. 3(f). For angles
below $60^{\circ}$, the quantum oscillations are not discernible. This
suggests a possibility of an elongated ellipsoidal Fermi surface at the center
of the Brillouin zone. A much larger magnetic field and lower temperature are
necessary to observe the SdH oscillations at low angles and oscillations from
other Fermi pockets as well. The calculated quantum oscillations using first-
principles Fermi surface are shown in Fig. 3(f) (see Sec. III.5 for discussion
on Fermi surface). From the detailed band structure calculations, we find that
the Brillouin zone contains two hole pockets centered at $\Gamma$ point and
small electron pockets located at the $M$ point (see Fig. 5). The calculated
quantum oscillations frequency for $B~{}\parallel~{}[0001]$ is 305 T (40 T)
for outer (inner) hole pocket and 160 T for the electron pockets. The extremal
area for the outer (inner) hole pocket is $2.9\times 10^{18}$ m-2 ($3.8\times
10^{17}$ m-2) and for the electron pocket is $1.5\times 10^{18}$ m-2. The
experimentally observed frequency of 314 T corresponds to an area of $3\times
10^{18}$ m-2. These results indicate the central large hole pocket at $\Gamma$
point is compatible with the experimental SdH oscillation for
$J~{}\parallel~{}$[12̄10] and $B~{}\parallel~{}[0001]$. Furthermore, a
detailed comparison of our experimental results shows that the observed Fermi
pocket may have an elongated ellipsoidal shape. The anisotropy in the Fermi
surface is very well reflected in the large anisotropy in the electrical
resistivity (Fig. 2(a)) where $\rho_{zz}$ is larger than $\rho_{xx}$,
suggesting a quasi-$2D$ character.

Background subtracted $\Delta R_{xy}$ (right axis) and $\Delta R_{xx}$ (left
axis) are shown in Fig 4(a). It is evident from the figure that quantum
oscillations are observed even for the Hall resistance $R_{xy}$, however there
is a phase shift of about $90^{\circ}$ between $\Delta R_{xx}$ and $\Delta
R_{xy}$. Interestingly, at sufficiently high magnetic fields ($10$ T
$\leq~{}B\leq 45$ T), the Hall resistivity shows step-like features (plateaus)
in the $1/\rho_{xy}$ vs $1/B$ and the plateau size corresponding to two
consecutive steps is $\Delta(1/\rho_{xy})/(e^{2}/h)$ = $7.09\times 10^{9}$ m-1
(Fig 4(b)). In Fig 4(b), $n$ refers to the LL index which is obtained from
$F/B$ where $F$ is the frequency of the quantum oscillation and $B$ is the
field at which $\Delta R_{xx}$ shows minimum in the oscillations. The
oscillations observed in Hall data up to a magnetic field of 45 T also have
the FFT frequency of 314 T. For multilayer quantum Hall effect, the step size
of inverse Hall resistivity can be written as $\Delta(1/\rho_{xy})=Ze^{2}/h$,
where $Z$ is the average $2D$ conduction channels per unit thickness
contributing to this multilayer transport. If we assume these $2D$ parallel
conduction channels result in the observed plateau-like feature in the Hall
measurement, then the step size in the scale of $e^{2}/h$ is
$\Delta(1/\rho_{xy})/(e^{2}/h)=Z\sim 7.09\times 10^{9}$ m-1. From the $Z$
value, we can calculate the bulk carrier density $((eF)/h)Z=5.43\times
10^{26}~{}m^{-3}$ which matches well with the bulk carrier density estimated
from the slope of the Hall resistivity data ($5.81\times 10^{26}~{}m^{-3}$).
From the HLN model, the obtained average number of conduction channels per
unit thickness is $3\times 10^{10}$ m-1, which is nearly similar to the
average number of conduction channels ($Z$) obtained from the Hall plateau.
Even at such high magnetic fields, no additional frequency or any kind of peak
splitting is observed (see Supplementary Materials Fig. S5 sup ). The step-
like features observed in Fig. 4(b) are something similar to quantized Hall
plateaus with high filling factor Xu  _et al._ (2017); Sasaki  _et al._
(1999). Additionally, the $90^{\circ}$ phase difference between $\Delta
R_{xx}$ and $\Delta R_{xy}$, and $\omega\tau=\frac{eB}{m^{*}}\tau~{}\sim 11$
for $B=40~{}\rm{T}$, indicating the quantum Hall regime. All these suggest
that the observed steps in the Hall data may be due to quantum Hall effect.
The linearity in the ($1/\rho_{xy}$) $vs.$ $1/B$ plot (Fig. 4(b)) gives
additional testimony for the quantization nature of the Hall effect. Notably,
the quantized Hall steps in $3D$ systems are usually attributed to the $2D$
parallel conduction channels and have been observed in various layered bulk
systems such as Bi2Se3 Cao  _et al._ (2012); Busch  _et al._ (2018), HfTe5
Galeski  _et al._ (2020), and EuMnBi2 Masuda  _et al._ (2016). CaCuSb
possesses a layered crystal structure where the Cu-Sb layers are separated by
Ca layers. Our experimental results and band structure calculations also
indicate that the Cu-Sb layers act as $2D$ conduction channels, which leads to
the observation of Hall plateaus in CaCuSb.

Figure 5: (a) First-principles bulk band structure of CaCuSb with SOC. Sizes
of gray, red, green, and blue markers represent contributions from Ca $d$, Cu
$s$, Sb $p_{xy}$, and Sb $p_{z}$ states to various bands. (b) The total and
partial DOS calculated with SOC. Black, red, blue, and green lines mark total,
Ca, Cu, and Sb DOS, respectively. (c) Fermi surface with the Fermi velocity of
associated electronic states. The blue and red regions highlight Fermi surface
area with low and high Fermi velocity, respectively.

###  III.5 Band structure of CaCuSb

First-principles calculations were performed based on the density-functional
theory Hohenberg and Kohn (1964) with the projector augmented wave (PAW)
method Blöchl (1994), as implemented in the Vienna $abinitio$ simulation
package (VASP) Kresse and Furthmüller (1996); Kresse and Joubert (1999). The
generalized gradient approximation (GGA) was used for the exchange-correlation
effects Perdew  _et al._ (1996). The SOC was considered self-consistently. A
plane wave cutoff energy of 420 eV was used and a $10\times 10\times 8$
$\Gamma$-centered $k$ mesh was employed to sample bulk Brillouin zone. The
FermiSurfer program was used to visualize the Fermi surface Kawamura (2019).
We present the calculated band structure CaCuSb in Fig. 5(a). It is seen to be
semimetallic in which small electron pockets located at $M$ overlap in energy
with the large hole bands centered at the $\Gamma$ point. The orbital resolved
band structure shows that hole bands are dominated by Sb $p$ states whereas
the electron bands near the Fermi level are mainly composed of Cu $s$ states.
Notably, the bands do not constitute a band inversion, showing that CaCuSb is
a nontopological semimetal. The density of states (DOS) in Fig. 5(b) further
reveal a small DOS at the Fermi level which is a reminiscent feature of the
semimetallic state. Nevertheless, we find that Ca has vanishing DOS whereas
the Cu-Sb has notable DOS at the Fermi level. The Fermi surface is shown in
Fig. 5(c) where the color scale marks the Fermi velocity on each band. It
consists of a big ellipsoidal shaped hole pocket with a major axis along the
hexagonal $c$ axis and three small electron pockets at the $M$ points.
Importantly, the Fermi velocity is higher for the hole bands whereas it is
minimum for the electron pockets. This is in accord with the dominant hole-
carriers observed in our experimental samples. Furthermore, the finite Cu-Sb
states constituted hole bands with relatively higher DOS at the Fermi level.
It should be noted that both the Cu and Sb atoms have large SOC which splits
the Sb $p_{xy}$ states, leading to two-hole pockets at the $\Gamma$-point.
Thus, while the system is topologically trivial, due to inherent large SOC, it
can show WAL as seen in our experiments as well as reported widely in other
materials Hu  _et al._ (2014); Bergmann (1984); Lin and Bird (2002).

##  IV Summary

In summary, the flux-grown CaCuSb single crystal exhibits a huge anisotropy in
electrical transport. This anisotropy can be related to the layered crystal
structure where the two Cu$-$Sb layers are separated by Ca-layers of the
hexagonal unit cell. WAL and quantum oscillations have been observed in the
linear and Hall resistivities. The WAL behavior is confirmed from the
magnetotransport and low field magnetoconductance analysis. Interestingly, the
SdH oscillation is observed in the field range $10-14$ T with a frequency at
$314$ T, which is primarily due to the large hole-pocket centered at
$\Gamma$\- point. Low field normalized magnetoconductance falls onto a single
curve for different angles and SdH oscillation vanishes below 60∘ suggest the
quasi-$2D$ nature of the electrical transport. The high magnetic field Hall
resistivity data exhibits step-like features at regular intervals. The
angular-dependent magnetoconductance, anisotropy in magnetoresistance, and
band structure indicate the presence of multiple $2D$ conduction channels.
First-principles calculations have revealed that CaCuSb is a normal semimetal
with dominant hole carriers and higher Cu$-$Sb DOS at the Fermi level. A
comparison of calculated quantum oscillations with the observed oscillations
suggests that the Fermi pockets may have an elongated ellipsoidal shape. It is
clear from these results that although CaCuSb possesses a $3D$ layered
structure, it can be considered as a pseudo-$2D$ system. The present study
would pave way for identifying layered materials as a new platform to
investigate quasi-$2D$ transport properties.

##  V ACKNOWLEDGEMENTS

We thank Department of Atomic Energy (DAE) of Government of India for
financial support. We also acknowledge the support of the LNCMI-CNRS,member of
the European Magnetic Field Laboratory (EMFL) for high magnetic field
measurements.

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# Robust superconductivity and fragile magnetism induced by
the strong Cu impurity scattering in the high-pressure phase of FeSe

Z. Zajicek<EMAIL_ADDRESS>Clarendon Laboratory, Department
of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK  S. J. Singh
Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road,
Oxford OX1 3PU, UK  Current address: Institute of High Pressure Physics
(IHPP), Polish Academy of Sciences, Sokolowska 29/37, 01-142, Warsaw, Poland
A. I. Coldea<EMAIL_ADDRESS>Clarendon Laboratory, Department
of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK

###### Abstract

Superconductivity in FeSe is strongly enhanced under applied pressure and it
is proposed to emerge from anomalously coupled structural and magnetic phases.
Small impurities inside the Fe plane can strongly disrupt the pair formation
in FeSe at ambient pressure and can also reveal the interplay between normal
and superconducting phases. Here, we investigate how an impurity inside the Fe
plane induced by the Cu substitution can alter the balance between competing
electronic phases of FeSe at high pressures. In the absence of an applied
magnetic field, at low pressures the nematic and superconducting phases are
suppressed by a similar factor. On the other hand, at high pressures, above 10
kbar, the superconductivity remains unaltered despite the lack of any
signature in transport associated to a magnetic phase in zero-magnetic field.
However, by applying a magnetic field, the resistivity displays an anomaly
preceding the activated behaviour in temperature, assigned to a magnetic
anomaly. We find that the high-pressure superconducting phase of FeSe is
robust and remains enhanced in the presence of Cu impurity, whereas the
magnetic phase is not. This could suggest that high-$T_{\rm c}$
superconductivity has a sign-preserving order parameter in a presence of a
rather glassy magnetic phase.

## Introduction.

Hydrostatic pressure is an invaluable tool to stabilize novel electronic
phases as well as to enhance superconductivity towards room temperature [1].
Among unconventional superconductors, FeSe displays the signature of a nematic
electronic phase before becoming superconducting at low temperatures below 9 K
[2]. However, with applied pressure the nematic phase is suppressed and the
superconducting transition temperature is enhanced towards 37 K close to 6.3
GPa [3, 4, 5, 6, 7, 8, 9]. This enhanced superconductivity at high pressures
occurs in a region in which a new electronic phase, believed to be of magnetic
origin (spin-density wave (SDW) phase), is present [10, 11]. The volume and
the local magnetic field of this magnetic phase is strongly dependent on the
applied pressure [10] and it coincides with a first-order structural
transition at high pressures, suggesting a potential magnetoelastic coupling
and phase coexistence [11]. Interestingly, this magnetic phase follows the
superconducting phase very closely at high pressures, and it raises the
question whether bulk superconductivity coexists or competes on short-length
scales with the magnetic order [10]. Furthermore, the enhancement of
superconductivity is also present in FeSe1-xSx under pressure, even though the
signatures associated with any magnetic order are strongly reduced with
increased isoelectronic substitution [12].

Another tuning parameter of superconducting and nematic phases of FeSe is the
chemical substitution either inside or outside the conducting Fe plane. The Cu
substitution is highly disruptive, due to the larger size of the Cu relative
to the Fe ions inside the conducting planes, which introduces significant
impurity scattering. The nematic and superconducting phases are suppressed
linearly up to 3% Cu substitution, whereas the carrier mobilities are quickly
reduced even with a small amount of Cu substitution [13, 14, 15]. By
increasing the Cu substitution the system undergoes a metal-to-insulator
transition which could lead to the stabilization of local magnetic moments at
Fe sites [15, 16, 17]. However, under applied hydrostatic pressure the
superconductivity of Fe1-xCuxSe could be restored by suppressing the
insulating behaviour in powder samples [18] or by quenching from high
pressures in single crystals [19]. These reports highlight that the high-
pressure superconductivity stabilized under pressure is robust while
signatures of any magnetic order are not detected. Theoretical studies suggest
that non-magnetic disorder can have the potential to enhance superconductivity
in a multi-band system [20], and thus the Cu substitution can be used to
assess this proposal and to understand its unusual manifestation.

In this paper, we report a transport study of the effect of a small amount of
Cu substitution ($x\sim$ 0.0025) in FeSe under applied hydrostatic pressure up
to 20 kbar to understand the response of the superconducting, nematic and
magnetic phases. Despite the suppression of the nematic and superconducting
phases at low pressures compared to FeSe, the high pressure superconducting
phase is robust against the Cu substitution. In the absence of an applied
magnetic field, we observe no anomaly in resistivity, often associated to a
SDW phase in FeSe, but a broad superconducting transition invoking that the
electronic conduction at high pressures above 10 kbar is still affected by the
Cu substitution. However, in strong magnetic fields we detect an upturn in
resistivity which is strongly hysteretic suggestive of glassy behaviour.
Furthermore, we detect a reduction of the charge carrier mobilities and
increased scattering rate with increasing pressure, in this region remains
with robust superconductivity.

## Methods.

Single crystals of Fe1-xCuxSe, with the nominal composition of $x$ = 0.0025(4)
were grown using the KCl/AlCl3 chemical vapour transport method [21, 22], as
reported previously in [14]. Magnetotransport studies under pressure were
carried out in a 16 T Quantum Design PPMS. The in-plane resistivity
$\rho_{xx}$ and Hall $\rho_{xy}$ components were measured using a low-
frequency five-probe technique and were separated by (anti)symmetrizing data
measured in positive and negative magnetic fields with the magnetic field
applied along the $c$-axis ($I||$($ab$)). Good electrical contacts were
achieved by In soldering and currents up to 1 mA (peak-to-peak) were used to
avoid heating. The pressure studies were performed using a piston-cylinder
cell and Daphne Oil 7373 up to 21 kbar. The pressure at low temperatures was
determined via the superconducting transition temperature of Sn after
cancelling the remnant magnetic field [23]. Errors in the exact contact
positions and size result in errors of up to 13% of absolute values of
resistivity. The superconducting transitions, $T_{\rm c}$ and $H_{\rm c2}$,
are defined as the offset temperature or field, respectively, unless stated
otherwise. The nematic structural transition, $T_{\rm s}$, and the high
pressure magnetic phase transition, $T_{\rm m}$, are defined by the minimum in
the derivative of the resistivity as a function of temperature, as in previous
studies [24].

## Transport studies under applied pressure.

Fig. 1(a) shows the temperature dependence of the resistivity in the absence
of a magnetic field for pressures up to 20 kbar (see shifted curves in Fig. S1
in the Supplemental Material (SM) [25]). Due to the presence of Cu, the
nematic transition $T_{\rm s}$, at 80 K, and superconducting transition,
$T_{c}$, at 6.2 K are suppressed, as compared with FeSe [14]. Under applied
pressure, the nematic transition is linearly suppressed with applied pressure
up to 15 kbar, similar to FeSe [26], whereas the superconductivity is
continuously enhanced with applied pressure up to 20 kbar, without showing any
dome like-feature visible for FeSe [26]. The superconducting transition width,
$\Delta T_{\rm c}$, is around 1.3(2) K, before increasing by a factor of 2 at
higher pressures above 10 kbar, as shown in Fig. 1(b). In FeSe under pressure
the superconducting transition width is sharp ($<$1 K) at low pressures before
increasing significantly towards 7 K for pressures where the magnetic phase is
present [24]. This suggests that either the high pressure superconducting
phase becomes inhomogeneous in the presence of another competing phase, or
that strong superconducting and/or magnetic fluctuations are present in this
high pressure regime [26].

Figure 1: Zero-field transport properties of Fe1-xCuxSe under pressure. (a)
Temperature dependence of the resistivity as a function of applied pressure.
(b) The evolution of the superconducting transition width, $\Delta
T_{c}=T_{\rm on}-T_{\rm off}$ (black circles), and the resistivity at 300 K
(blue triangles) with pressure. Solid lines are guides to the eye. (c)
Temperature-pressure phase diagram in zero magnetic field tuned by pressure.
The nematic phase occurs at $T_{\rm s}$ (blue triangles) and superconductivity
has the offset temperature at $T_{\rm c}$ (red squares). The phase diagram can
be divided into three different regions: the low pressure region up to
$p_{1}=10$ kbar, inside the nematic phase, the high-pressure region above
$p_{2}=15$ kbar, once the nematic phase is fully suppressed, and the
intermediate pressure region between $p_{1}$ and $p_{2}$. The shaded areas
indicate different phase boundaries.

The resistivity behaviour provides additional information about the normal
electronic phases as a function of pressure. At room temperature, the
resistivity initially decreases with increasing pressure, indicating that the
in-plane transfer integrals are larger under pressure which increases the
bandwidth and the system becomes a better metal, as shown in Fig. 1(b).
However, at higher pressures above 15 kbar this trend reverses with
resistivity increasing again, also observed at low temperatures (see Fig.
S1(b) in the SM [25]). The substitution of Cu in FeSe increases the impurity
scattering and the residual resistivity, leading to the suppression of
superconductivity and inducing a linear dependence down to 0.4 K [14]. Under
pressure, the linear dependence is found both inside the nematic phase at low
pressures as well as in the high pressure phase, as also illustrated by the
temperature dependence of the resistivity exponent, $n$ in Figs. S1(c) and (d)
in the SM [25]. At high temperatures in the tetragonal phase, the resistivity
can be described by a power law of $T^{1.5}$ which is rather similar to that
observed for both FeSe1-xSx, tuned either by the isoelectronic substitution or
applied pressure [27, 28]. Interestingly, in the high pressure regime above 10
kbar the resistivity of Cu-substituted FeSe shows no additional anomaly
associated to other phase transitions (see Figs.1(a) and S1 in the SM [25]).
This is in contrast to FeSe where a clear upturn in the resistivity anomalies
was associated with the presence of a magnetic transition at a temperature
above $T_{\rm c}$ [9, 29, 26, 24].

Figure 2: The transport behaviour in magnetic field of the Fe1-xCuxSe under
pressure. (a) The temperature dependence of the longitudinal resistivity in
different magnetic fields at $p$ = 17 kbar. (b) The resistivity upturn at
$T_{\rm m}$ in 15 T at 17 kbar using various temperature sweep rates. The
solid lines correspond to the warming sweeps whereas the dashed lines
correspond to the cooling sweeps. (c) The temperature dependence of
resistivity in a magnetic field of 15 T ($H||c$) for different applied
pressures and (d) their corresponding derivatives. The value of $T_{\rm m}$ is
defined as the minimum in the derivative and indicated by arrows, similar to
FeSe [12]. Curves are offset for clarity. Field dependence of (e) the
longitudinal resistivity, $\rho_{\rm xx}$, and (f) the Hall resistivity,
$\rho_{\rm xy}$ at a pressure of $p$ = 13.3 kbar. (g) Mobility and (h) carrier
density of each charge carrier (hole, $h$ and electron $e_{1}$) considering a
two-band compensated model ($n_{h}=n_{e_{1}}$) at different pressures. The
two-band model at ambient pressure uses data below $<7$ T and a three-band
could be used to described the full field window, as reported in Ref. [14] and
shown in Fig. S7 in the SM. Solid and dashed lines are guides to the eye.

Based on these transport studies, the zero-field temperature-pressure phase
diagram of the Fe1-xCuxSe up to 20 kbar is constructed, as shown in Fig. 1(c).
At high pressures, the superconducting transition temperature, $T_{\rm c}$,
increases once the nematicity is suppressed even in the presence of the Cu
impurity. This behaviour is in agreement with previous studies in powder
samples of Cu substituted FeSe, where an insulating phase is transformed into
a superconductor with applied pressure, although not to a zero resistance
state [18]. Furthermore, the phase diagram in zero-magnetic field in the
presence of the Cu impurity inside the conducting plane is remarkably similar
to that of FeSe1-xSx, in which the isoelectronic substitution takes place
outside the conducting plane [28, 12, 30, 31].

## The high-pressure electronic phase in high magnetic fields.

Next, we use high magnetic fields to suppress the superconductivity to reveal
any hidden electronic phases and to explore the normal electronic behaviour at
lower temperatures, once the superconductivity is suppressed. Fig. 2(a) shows
the temperature dependence of resistivity in different magnetic fields at 17
kbar, in which an upturn occurs at $T_{\rm m}$, as the temperature is reduced
in large enough magnetic field. This anomaly shows hysteretic behaviour
between the cooling and warming curves, independent of the temperature sweep
rate, indicative of a first-order phase transition, as shown in Fig. 2(b). In
FeSe, the high pressure magnetic phase displays evidence of hysteresis in heat
capacity and NMR studies, implying the existence of a concomitant structural
transition via a spin-lattice coupling [26, 32]. The temperature dependence of
the resistivity in 15 T shows smooth changes in the upturn anomaly as a
function of the applied pressure, as shown in Fig. 2(c) and the corresponding
derivatives in Fig. 2(d). As the pressure increases, this anomaly shifts to
higher temperatures and it tracks closely the zero field superconducting
transition temperature. In FeSe, this anomaly was associated to a magnetic
phase and it occurs already in zero-field at a higher transition temperature
than the superconducting transition, $T_{\rm c}$, but its appearance seems to
vary strongly with pressure, either as an increase, decrease or change in
slope [9].

## Magnetotransport behaviour.

The magnetotransport studies can provide valuable insight into the changes in
the electronic structure and scattering with applied pressure. Figs. 2(e) and
(f) show the field dependence of the longitudinal and Hall resistivity,
$\rho_{xx}$ and $\rho_{xy}$, at 13.3 kbar. The presence of the linear field-
dependence of the Hall component indicates a two-band behaviour which describe
the behaviour inside the high pressure and the tetragonal phase (see Figs. S3,
S4, and S5 in the SM [25]). At high pressures, the Hall coefficient extracted
from the slope of the Hall resistivity in low fields, $\rho_{xy}$, is positive
and generally increases with decreasing temperature, as shown in Fig. S7(f) in
the SM [25]. This is in stark contrast to FeSe which displays negative Hall
coefficient at low temperatures inside the nematic phase both at ambient and
at low pressures [33, 34, 35].

Magnetotransport measurements enable the extraction of the charge carrier
density and mobilities by simultaneously fitting the $\rho_{xx}$ and
$\rho_{xy}$ components to a two-band compensated model. We use the initial
input parameters from the mobility spectrum, whose peaks indicate that the
mobilities are slightly reduced with increasing pressure (see Fig. S6 in the
SM [25]). In order to compare the parameters between different pressure
regions, we use a two-band compensated model in low-field regime ($<7$ T) for
ambient pressure rather the three-band compensated model which can account for
the non-linear Hall resistivity, similar to FeSe [14, 36, 34]. Figs. 2(g) and
(h) compare the temperature dependence of the charge carrier mobilities and
carrier densities at various pressures (see also Fig. S7 in the SM [25]). Upon
entering the nematic phase from high temperatures there is a significant
decrease in the apparent carrier density, $n$, due to the development of
anisotropic scattering, as found for FeSe [36, 34]. In the high pressure
phase, the carrier density is significantly larger indicative of an increase
of the Fermi surface size similar to findings for FeSe1-xSx [28]. Furthermore,
the mobilities of both holes and electrons are suppressed with applied
pressure, with stronger suppression for the negative charge carriers, similar
to findings in thin flakes of FeSe [34]. This could indicate an increased
scattering rate of spin fluctuations at high pressures and/or an increase in
effective mass, which is consistent with the enhanced resistivity found in the
high pressure phase of Cu-substituted FeSe shown in Fig. 1(b). This behaviour
is rather similar to that of FeSe under pressure, where a three-band model is
required to explain its low temperature behaviour for all pressures, but the
carrier densities are enhanced and the mobilities decreased with increasing
pressure [37].

## The pressure-temperature $p-T$ phase diagrams.

To summarize we compare the temperature-pressure phase diagram of Cu-FeSe with
that of FeSe [24] in 0 T and 15 T, as shown in Figs. 3(a) and (b). The phase
diagram can be split into three distinct regions: firstly, in the low pressure
region, $p<p_{1}\sim 10$ kbar, the superconducting phase emerges from the
nematic phase, secondly at intermediate pressures, $p_{1}<p<p_{2}\sim 15$
kbar, signatures of the magnetic phase are detected in FeSe in zero-field and
finally, at higher pressures, $p>p_{2}$ the magnetic and superconducting
phases coexist and the nematic phase is suppressed at $p_{2}$. Interestingly,
as the pressure increases beyond $p_{1}$, $T_{\rm c}$ displays a broadly
similar pressure dependence for the two systems, indicating a robustness of
superconductivity to impurity scattering. The increase in $T_{\rm c}$ occurs
in a regime where the carrier density $n$ increases at high pressure, as shown
in Fig. 2(h). On the other hand, the magnetic phase above $p_{2}$ is highly
sensitive to impurity scattering, being washed out by the small amount of the
Cu substitution and its signatures are only visible in magnetic field, as
shown in Fig. 3(b). The upper critical field across these phases shows a
remarkable scaling to $T_{\rm c}$ in the presence of the Cu impurity across
the phase diagram (Fig. S7 in the SM).

In order to account for the relative changes in superconductivity due to the
Cu substitution, the temperature-pressure diagrams are scaled to their ambient
value of $T_{\rm s}$, as shown in Fig. 3(c). We notice a remarkable scaling of
the nematic and superconducting temperatures between FeSe and Cu-FeSe, up to
$p_{1}$, suggesting that the impurity scattering affects the two phases in a
similar manner and have a similar origin, as found also for higher Cu
substitution [14, 13]. As the superconductivity at low pressure is strongly
suppressed by the Cu substitution, this is consistent with sign-changing
$s_{\pm}$-type of order parameter in the low pressure regime [14].

Figure 3:  The pressure-temperature phase diagrams of Cu substituted FeSe. The
comparison between the different electronic phases of FeSe (open symbols) from
Ref. 24 and Cu-FeSe (solid symbols) measured in (a) 0 T and (b) in 15 T for
Cu-FeSe and 16 T for FeSe (after Ref. [24]). The shaded areas in all panels
reflect the phase boundaries for Cu-FeSe, as described in Fig. 1(c). The solid
lines in (b) are guides to the eye for FeSe. Nematic shaded area in (a) and
(b) are for Cu-FeSe, assuming a field-independent $T_{\rm s}$. (c) The scaling
of the temperature-pressure phase diagram from (a) by using the reduced
temperature $t=T/T_{\rm s}$, where $T_{\rm s}$ is the nematic transition at
ambient pressure for the two systems. The two shaded regions reflect
schematically the two proposed superconducting regions with the boundaries
given by the maximum value of $T_{\rm c}$.

Interestingly, at high pressure above $p_{2}$ the magnetic and superconducting
phases of FeSe enhance their transition temperatures with increasing pressure
[26]. However, the very small Cu substitution is sufficient to break down this
trend and only superconductivity is present, whereas the signatures of the
magnetic phase in resistivity have disappeared. This suggest the fragility of
the magnetic phase which is only detected and stabilized by the presence of a
magnetic field. In high magnetic fields ($\sim 15$ T), the magnetic transition
is strongly suppressed in the Cu-FeSe as compared with FeSe, whereas the
superconducting domes are rather similar, as shown in Fig. 3(b). This suggest
that the superconducting phase at high pressure is robust in the presence of
the Cu impurity.

Significant disruption of the high pressure magnetic phase was also detected
under pressure in FeSe crystals with higher disorder [38, 39], as well as in
thin flakes of FeSe under pressure, where the magnetic phase is suppressed as
the thickness decreases [40]. Interestingly, in the presence of sulphur
substitution outside the Fe planes, the magnetic phase is also suppressed at
lower pressures in FeSe1-xSx, although it was suggested to be stabilized at
higher pressures around 50 kbar [12]. The robust superconductivity at high
pressure occur up to 31 K also in the presence of large amount of the Cu
substitution towards 4% but there are no signatures of a magnetic phase [38,
39, 18]. This implies that the superconductivity in the high pressure phase of
FeSe and Cu-FeSe is rather robust to impurity scattering, which would be
consistent with an $s_{++}$ pairing symmetry or that the high-pressure
superconductivity is strongly inhomogeneous. In the latter case, the
superconducting order could still have a $s_{\pm}$ pairing symmetry since
extended emergent disorder regions are unable to provide large-momentum
transfer interband scattering connecting opposite signs of the superconducting
gap function on electron and hole pockets, and therefore strongly limiting any
substantial pair-breaking processes from the inhomogeneous disorder landscape.
However, for superconductors with a sign-reversal order parameter the $T_{\rm
c}$ is strongly suppressed by impurity scattering, as found for the Cu-
substituted FeSe inside the nematic phase [14]. Theoretically, it is suggested
that disorder in multi-band superconductors disorder can cause a transition
between $s_{+-}$ and $s_{++}$ pairing symmetry if one of two gaps changes sign
and $T_{\rm c}$ is predicted to remain finite and almost independent of the
impurity scattering rate [41]. Additionally, superconductivity could be
enhanced for a sign-preserving gap, even in the presence of large enough
disorder, as disorder can induce spatial modulations where the coherence
length is a few nanometers [20, 42]. Fig. 3(c) indicates the separation into
two superconducting domes of potentially different symmetry and response to
impurity scattering of the $p-T$ phase diagram of Cu-FeSe. Interestingly, two
superconducting domes are also present for FeSe0.89S0.11, in which the
isoelectronic substitution occurs outside the conducting plane, and no
signature of a magnetic phase is detected at the high-pressure phase up to 22
kbar [28].

The enhanced resistivity at high pressures, combined with the reduction of
mobilities while the apparent carrier density increases, imply that either the
high-pressure phase is highly inhomogeneous or that there are very strong spin
fluctuations that do not permit the stabilization of the long-range magnetic
order. These effects are also evident in the evolution of the superconducting
transition width which becomes extremely broad above $p_{1}$ both in FeSe and
Cu-FeSe, as shown in Fig. 1b. The large hysteresis effects observed in the
resistivity curves for different cooling rates in the high pressure phase
suggest that the local short-range magnetic order is affected by the presence
of Cu ions, by freezing of dynamical spin fluctuations of the Fe magnetic
moments in the presence of non-magnetic disorder [43]. Evidence of short-range
local magnetism was found in FeSe under pressure using $\mu$SR as well as
Mossbauer spectroscopy [10, 11], but neutron diffraction studies have not been
able to resolve any long-range magnetic order [44].

Theoretical calculations of non-magnetic impurities in FeSe show that point-
like disorder sites locally induce short-range magnetism [45, 46]. The
presence of the Cu impurity inside the SDW phase can lead to phase
fragmentation or glassy behaviour, similar to a SDW glass proposed to arise in
systems with weak non-magnetic disorder [47]. Microscopic calculations inside
the SDW phase of iron-based superconductors have suggested that a strong non-
magnetic potentials, like Cu, can generate extended magnetic nematogens, which
are regions of a a competing magnetic phase centered at the impurity site
(typical lengths of $\sim 10$ lattice constants) [46]. Such entities could be
short-circuited in transport studies at high pressure leading to the vanishing
of the SDW signatures, despite the tiny amount of Cu in the samples. An
external field, however, could potentially re-align and couple the induced
magnetic puddles and thereby introduce additional spin-dependent scattering
which can enhance resistivity and mimic the behaviour in the absence of the Cu
impurities. Spatially-resolved probes would be very valuable for obtaining a
detailed understanding of the fascinating interplay between disorder,
magnetism and superconductivity in the high-pressure region of FeSe.
Theoretical work addressing the nature and the robustness of the high-pressure
superconducting phase, specific to FeSe and its related systems, would be very
valuable to guide experimental findings and to help identify the essential
ingredients to design a high-temperature superconductor.

## Conclusions

Our study provides new insight into the nature of unconventional
superconductivity of FeSe in different regimes and its sensitivity to impurity
scattering in the presence of a small amount of Cu substitution. The magnetic
phase at high pressures is profoundly affected by the impurity scattering that
can lead to a phase separation between the superconducting and magnetic phases
with a hysteretic first-order transition. At low pressures, nematic and
superconducting phases are suppressed by the Cu impurity, consistent with the
presence of a sign-changing pairing. On the other hand, the robustness of the
high-pressure superconducting phase, despite the fragility of the magnetic
phase, implies either a non-sign changing superconducting phase, or a highly
inhomogeneous phase with mainly extended scattering regions with suppressed
pair-breaking effects for $s_{\pm}$ superconductivity. Further theoretical
work is required to understand how to stabilize such a robust
superconductivity, in the absence of applied pressure.

## Acknowledgements

We thank Brian Andersen, Peter Hirschfeld, Maria Gastiasoro, and Pascal Reiss
for very useful discussions and comments on the manuscript. This work was
mainly supported by EPSRC (EP/I004475/1) and Oxford Centre for Applied
Superconductivity. We also acknowledge financial support of the John Fell Fund
of the Oxford University. Z.Z. acknowledges financial support from the EPSRC
studentship (EP/N509711/1). AIC acknowledges an EPSRC Career Acceleration
Fellowship (EP/I004475/1).

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Supplemental Material

Figure S1: The zero-field resistivity and the local resistivity exponent. (a)
Temperature dependence of the resistivity as a function of applied pressure.
The curves are shifted for clarity (by 75$\mu\Omega cm$ each). The pressure of
each run is given by the corresponding colour. The non-shifted curves are
shown in Fig. 1(a). The resistivity can be expressed using the form
$\rho=\rho_{0}+\alpha T^{n}$, where $n$ is the local resistivity temperature
exponent. At ambient pressure the zero-temperature resistivity, $\rho_{0}$, is
obtained by using high magnetic fields to suppress the superconductivity at
temperatures below $T_{\rm c}$ and extrapolating the high-field dependence at
zero-field for each temperature [14]. The mean free path, $\ell=\frac{\pi
c\hbar}{Ne^{2}k_{\rm F}\rho_{\rm 0}}$, is calculated using the ambient
pressure values for $c$ and $k_{\rm F}$ [14]. (b) The pressure dependence of
the zero-temperature resistivity, $\rho_{0}$, extracted from a linear
extrapolation of the zero-field resistivity above the onset of
superconductivity similar to the ambient pressure dependence [14]. The red
triangles are the mean free path, using the $\rho_{0}$ values. (c) The
temperature dependence of the local resistivity exponent, $n$, for different
pressures in zero magnetic field and (d) its interpolated colour map for
different pressures.  Figure S2: Magnetotransport behaviour in the normal
phase. The field dependence of (a) $\rho_{xx}$ and (b) $\rho_{xy}$ at a fixed
temperature close to the onset of superconductivity for different pressures
and their corresponding derivatives in relation to the magnetic field in (c)
and (d), respectively. The derivatives in magnetic field of the $\rho_{xy}$
component shows a constant behaviour at high-pressures above 10 kbar
indicative of a two-band behaviour.

Figure S3: Magnetotransport behaviour at different pressures. The field-
dependence of (a) $\rho_{xx}$ and (b) $\rho_{xy}$ measured a different
constant temperatures. Each panel corresponds to different applied pressure.

Figure S4: The derivative of magnetotransport curves in relation to magnetic
field at different pressures. Derivatives with respect to magnetic field for
(a) $\rho_{xx}$ and (b) $\rho_{xy}$ as a function of magnetic field measured
at different constant temperatures. Each panel corresponds to different
applied pressures. Examples of how the critical field was extracted are shown
in (h). The peak of the derivative is referred to as $H^{\rm p}_{\rm c2}$, and
the zero extrapolation as $H^{\rm 0}_{\rm c2}$.  Figure S5: Mobility spectrum
at different pressures. (a) The mobility spectrum extracted from
magnetotransport data [48] for different pressures at 30 K. Arrows indicated
the positions of mobilities for the positive and negative charge carriers.
Mobility spectrum extracted for constant temperatures at different pressures
of (b) 0 kbar, (c) 13.3 kbar and (d) 18 kbar, respectively. Figure S6:
Mobility spectrum at different pressures. The mobility temperature dependence
for (a) positive charge carriers (holes) and (b) the negative charge carriers
(electrons). The ambient pressure data is modelled using a there-band model,
described in the Ref. 14, whilst the two-band model is taken from low-field
limit ($<7$ T) only. The mobility of the second electron carrier, e2, in (b)
is scaled by a factor 1/3. The inverse mobility against a $T^{2}$ dependence
for (c) holes and (d) electrons. (e) cot($\theta_{\rm H}$), cotangent of the
Hall angle defined, temperature dependence. Solid lines are fit to linear
regions at higher temperatures. The values are taken from field-dependent
measurements and chosen at a fixed field value for all temperatures. (f) The
Hall coefficient, $R_{\rm H}$, temperature dependence for different pressures
of Cu-FeSe. The dashed lines are guide to the eye. The solid line for 13.3
kbar is from a temperature sweep measured in constant field of 1 T. $T^{*}$
marks the minimum in $R_{\rm H}$ for 0 kbar.