††thanks: These authors contribute equally to this work.††thanks: These
authors contribute equally to this work.

# Bulk superconductivity and Pauli paramagnetism in nearly stoichiometric
CuCo2S4

Yu-Ying Jin  Department of Physics, Zhejiang Province Key Laboratory of
Quantum Technology and Devices, Interdisciplinary Center for Quantum
Information, and State Key Lab of Silicon Materials, Zhejiang University,
Hangzhou 310027, China  Shi-Huai Sun  Department of Physics, Zhejiang Province
Key Laboratory of Quantum Technology and Devices, Interdisciplinary Center for
Quantum Information, and State Key Lab of Silicon Materials, Zhejiang
University, Hangzhou 310027, China  Yan-Wei Cui  Department of Physics,
Zhejiang Province Key Laboratory of Quantum Technology and Devices,
Interdisciplinary Center for Quantum Information, and State Key Lab of Silicon
Materials, Zhejiang University, Hangzhou 310027, China  School of Science,
Westlake Institute for Advanced Study, Westlake University, Hangzhou 310064,
China  Qin-Qing Zhu  Department of Physics, Zhejiang Province Key Laboratory
of Quantum Technology and Devices, Interdisciplinary Center for Quantum
Information, and State Key Lab of Silicon Materials, Zhejiang University,
Hangzhou 310027, China  School of Science, Westlake Institute for Advanced
Study, Westlake University, Hangzhou 310064, China  Liang-Wen Ji  Department
of Physics, Zhejiang Province Key Laboratory of Quantum Technology and
Devices, Interdisciplinary Center for Quantum Information, and State Key Lab
of Silicon Materials, Zhejiang University, Hangzhou 310027, China  Zhi Ren
School of Science, Westlake Institute for Advanced Study, Westlake University,
Hangzhou 310064, China  Guang-Han Cao<EMAIL_ADDRESS>Department of Physics,
Zhejiang Province Key Laboratory of Quantum Technology and Devices,
Interdisciplinary Center for Quantum Information, and State Key Lab of Silicon
Materials, Zhejiang University, Hangzhou 310027, China  Collaborative
Innovation Centre of Advanced Microstructures, Nanjing University, Nanjing
210093, China

###### Abstract

It has long remained elusive whether CuCo2S4 thiospinel shows bulk
superconductivity. Here we clarify the issue by studying on the samples of
sulfur-deficient CuCo2S3.7 and sulfurized CuCo2S4. The sample CuCo2S3.7 has a
smaller lattice constant of $a=9.454$ Å, and it is not superconducting down to
1.8 K. After a full sulfurization, the $a$ axis of the thiospinel phase
increases to 9.475 Å, and the thiospinel becomes nearly stoichiometric
CuCo2S4, although a secondary phase of slightly Cu-doped CoS2 forms. Bulk
superconductivity at 4.2 K and Pauli paramagnetism have been demonstrated for
the sulfurized CuCo2S4 by the measurements of electrical resistivity, magnetic
susceptibility, and specific heat.

###### pacs:

74.70.Xa; 72.80.Ga; 61.66.Fn

##  I Introduction

The discoveries of superconductivity (SC) in the complex copper oxide [1] and
the iron-based pnictide [2] stimulate enthusiasm to search for SC especially
in late 3d-transition-metal (Fe, Co, Ni, and Cu) compounds [3, 4, 5, 6, 7].
Among them, the Co-based superconductors are very limited so far. One example
is the cobalt oxyhydrate NaxCoO${}_{2}\cdot y$H2O ($x\approx 0.35,y\approx
1.3$), which shows SC at $T_{\mathrm{c}}\approx$ 4.5 K [8]. The Co-based
thiospinel CuCo2S4 shows similarities with NaxCoO${}_{2}\cdot y$H2O in the Co
coordination, geometrical frustration, and formal Co valence. However, it is
not clear whether CuCo2S4 superconducts or not. Besides, the magnetism of
CuCo2S4 also remains elusive up to present.

Earlier studies in 1960s suggested Pauli paramagnetism in CuCo2S4 [9, 10], and
no superconducting transition was observed down to 0.05 K [11]. In 1990s,
however, it was reported that CuCo2S4 shows a Curie-Weiss (CW) paramagnetism
with an effective magnetic moment of 0.89 $\mu_{\mathrm{B}}$ per formula unit
(f.u.) [12]. A cusp in the magnetic susceptibility appears at
$T_{\mathrm{N}}=$ 18 K, which was attributed to an antiferromagnetic spin
ordering. In a multiphasic sample with the nominal composition of
Cu1.5Co1.5S4, SC or superconductor-like behavior was observed with an onset
transition temperature of $T_{\mathrm{c}}^{\mathrm{onset}}=$ 4.0 K and a zero-
resistance temperature of $T_{\mathrm{c}}^{\mathrm{zero}}=$ 2.3 K.
Investigations on the 63Cu and 59Co NMR suggested a gapless superconducting
state as well as antiferromagnetic spin correlations, and SC was considered to
be in line with the growth of antiferromagnetic spin correlation [13].
Contrastingly, later NMR study on the Co-rich series samples of
(CuxCo1-x)Co2S4 indicated a full superconducting gap without long-range
magnetic ordering for CuCo2S4 [14]. It was concluded that SC and the
antiferromagnetic spin correlation are associated with the Co-3d and Cu-3d
holes, respectively.

One of the present authors (G.-H.C.) and coworkers [15] attempted to reproduce
SC in CuCo2S4 in 2003. However, no SC was observed above 1.8 K in the single-
phase sample of CuCo2S4, although signature of SC at $T_{\mathrm{c}}=$ 3.5 K
was detected in a multiphasic sample. Aito and Sato [16] reported the
resistivity data of five CuCo2S4 samples from different batches. Two of them
showed a superconducting transition, and the higher $T_{\mathrm{c}}$ value is
3.8 K. Fang et al. [17] synthesized an unusual K-doped sample Cu1.3K0.2Co1.5S4
which showed SC at 4.4 K together with a CW-like susceptibility but without
any antiferromagnetic transition down to 9 K. A very recent work [18] showed
absence of SC with a weak antiferromagnetic transition at about 4 K in
CuCo2S4. In a word, the previous reports on CuCo2S4 appear to be highly
dispersive and even contradictive. To our knowledge, evidence of bulk SC with
specific-heat measurements has not been reported so far in the Cu-Co-S system.

The contradictive results above strongly suggest that the physical properties
are sensitive to the synthesis of samples, and a controlled preparation of
nearly stoichiometric samples of CuCo2S4 is crucial to clarify the intrinsic
properties. Previous studies showed difficulties in obtaining desired samples
of CuCo2S4 [12, 19, 20, 21, 15, 16]. They were commonly synthesized by direct
reacting copper and cobalt powders with sulfur in a sealed evacuated silica
tube at an elevated temperatures. While relatively low reaction temperatures
(500 ∘C - 600 ∘C) were suggested for the preparation of monophasic CuCo2S4
[22], however, a following-up work [19] failed to obtain the single-phase
sample. The synthesized sample tends to form CoS2 impurity, as revealed by the
phase-relation study in the Cu-Co-S system [20, 21]. Indeed, later studies
[12, 14, 15, 16] also showed presence of the CoS2 impurity for the reaction
temperatures from 500 ∘C to 800 ∘C. Note that CoS2 is a ferromagnet with a
Curie temperature of $\sim$120 K [23, 24, 25], which makes it more easily to
be detected by the magnetic measurement [15].

Here we report a novel two-step strategy for the controlled synthesis of
stoichiometric CuCo2S4. First, to minimize the formation of CoS2 impurity, we
prepared sulfur-deficient CuCo2S4-δ with $\delta=$ 0.3. Then, the S-deficient
sample was sulfurized by annealing in the presence of appropriate amount of
sulfur. As a result, the main phase of the annealed sample was found to be
nearly stoichiometric CuCo2S4. Bulk SC at 4.2 K and Pauli paramagnetism in the
normal state were demonstrated in the S-compensated CuCo2S4.

##  II Experimental methods

Polycrystalline sample of S-deficient CuCo2S3.7 was first prepared by high-
temperature reactions of the constituent elements in a sealed evacuated silica
tube. The source materials were powders of copper (99.997%), cobalt (99.998%),
and sulfur (99.999%). The homogenized mixture with the composition of
CuCo2S3.7 was allowed to fire at 750 ∘C for 72 hours. This procedure was
repeated to improve the quality of the sample. In the second step, the
synthesized CuCo2S3.7 was sulfurized in the presence of compensatory sulfur
(0.35 S/f.u.) by annealing at 450 ∘C for 144 hours in a sealed evacuated
silica ampoule. Note that excess of sulfur was necessary to ensure a full
sulfurization. This is because, during the sulfurization, a side reaction that
forms CoS2 always takes place, which additionally consumes sulfur. Besides, in
order to quantize the amount of the CoS2 impurity in the sulfurized sample, we
additionally prepared CoS2 by reacting Co with S in an evacuated silica tube.
The sample is of single phase with the lattice constant of $a=5.535$ Å
(consistent with the previous report [23]), as determined by the powder x-ray
diffractions (XRD).

Powder XRD were carried out using a PANalytical diffractometer (Empyrean
Series 2) with a monochromatic Cu-$K_{\alpha 1}$ radiation. The crystal
structure were refined by a Rietveld analysis using the GSAS+EXPGUI package
[26]. The sulfur content in the crystallites was examined by energy-dispersive
x-ray spectroscopy (EDS, Oxford Instruments X-Max) equipped in a scanning
electron microscope (SEM, Hitachi S-3700N).

The electrical resistivity and specific heat were measured on a Quantum Design
Physical Properties Measurement System, and the magnetic properties were
measured on a Quantum Design Magnetic Property Measurement System. The
resistivity measurement employed a standard four-terminal method. The heat-
capacity measurement utilized a thermal relaxation technique. In the magnetic
measurements, the applied magnetic fields were set to be 20 Oe and 10,000 Oe,
respectively, to detect SC and to study the normal-state magnetism. In the
case of the low-field measurements, both zero-field-cooling and field-cooling
protocols were employed.

##  III Results and discussion

Figure 1: Powder X-ray diffractions with the Rietveld refinement profiles for
samples of sulfur-deficient CuCo2S3.7 (a) and sulfurized CuCo2S4 (b). The
insets (with a logarithmic scale for the intensity) are a close-up of the
marked area, which shows presence of the secondary phases of Cu2S and CoS2,
respectively, in CuCo2S3.7 and sulfurized CuCo2S4.

Figure 1(a) shows the XRD profile for the sulfur-deficient sample of
CuCo2S3.7. Most of the reflections can be well indexed with a face-centered
cubic unit cell of the thiospinel. As is seen in the inset, no reflections
associated with CoS2 are detectable, while tiny amount of Cu2S is possibly
presented. Therefore, with a lack of sulfur we succeeded in avoiding the
appearance of the CoS2 secondary phase. The Rietveld refinement
($R_{\mathrm{wp}}$ = 2.3% and $\chi^{2}$ = 1.31) confirms the normal spinel
structure with $a=9.4544(1)$ Å and $u=0.3865(1)$ for the main phase. Note that
the lattice constant is the smallest among those reported previously for
CuCo2S4 [$9.461(2)$ Å [22], $9.478(5)$ Å [19], and $9.472(1)$ Å [20]]. This
may be attributed the apparent sulfur deficiency and/or the partial
substitution of Cu by Co (hereafter denoted as Co/Cu substitution) [14]. The
latter is implied by the presence of small amount of Cu2S. As a matter of
fact, the Rietveld refinement does not support a significant sulfur vacancy.

Figure 2: Typical scanning electron microscope images of sulfur-deficient
CuCo2S3.7 (a) and sulfurized CuCo2S4 (b). The lower-right insets are the
energy-dispersive x-ray spectrum (EDS) collected with the electron beam
focused on the spots marked. Round-shape grains (indicated by arrows) can be
seen in panel (b), which are identified to be lightly Cu-doped CoS2. The
atomic ratios are given by the EDS analysis.

The XRD pattern of the sulfurized CuCo2S4 is displayed in Fig. 1(b). The main
phase remains to be the cubic thiospinel, although small amount of CoS2-like
phase emerges. With the two-phase Rietveld refinement ($R_{\mathrm{wp}}$ =
2.2% and $\chi^{2}$ = 1.13), the weight percentage of the CoS2-like impurity
was determined to be 14.8(6)%. The lattice constant of the pyrite phase is
refined to be 5.538(1) Å, which is slightly larger than that of CoS2 (5.534 Å
[23]), suggesting that Cu is slightly incorporated. The structural parameters
of the main phase were fitted to be $a=9.4750(2)$ Å and $u=0.3851(1)$. The $a$
axis is remarkably larger than that of the sulfur-deficient CuCo2S3.7,
suggesting a successful sulfurization.

The two samples above were examined by SEM observations in combination with
the EDS measurements. As shown in Fig. 2(a), the crystallites of S-deficient
CuCo2S3.7 are similar in shape. The sulfur content, measured on the basis of
the Co content, is consistent with the nominal composition. However, the Cu
content is substantially lower than the nominal one. The result suggests that
the real composition of the thiospinel phase is something like
(Cu1-x-yCo${}_{x}\Box_{y}$)Co2S4-δ. The SEM image of the sulfurized sample
[Fig. 2(b)] shows additional round-shape crystallites which were identified to
be slightly Cu-doped CoS2 (1-2% Cu) by the EDS analysis. Furthermore, the
sulfur deficiency is fully compensated, and the Cu content is also increased,
as is indicated by the atomic ratio measured. Therefore, we conclude that the
sulfurized sample mainly ($\sim$85% by weight) contains nearly stoichiometric
CuCo2S4.

Figure 3: Temperature dependence of magnetic susceptibility or magnetization
for sulfur-deficient CuCo2S3.7 (a) and sulfurized CuCo2S4 (b). The inset of
(a) is a close-up of the high-temperature data, indicating a positive-
temperature-coefficient behavior (dashed line). In panel (b), the
magnetization of CoS2 (multiplied by a factor of 18.8%) is plotted for
comparison. The inset of (b) compares the magnetic susceptibilities at high
temperatures.

Figure 3(a) shows the temperature dependence of magnetic susceptibility under
a magnetic field of $H=$ 10 kOe for the sulfur-deficient CuCo2S3.7. The
magnetic susceptibility is nearly temperature independent at high
temperatures. No anomaly at $\sim$120 K can be seen, indicating free of the
ferromagnetic impurity of CoS2. There is an upturn tail at low temperatures.
Fitting of the data with the CW formula,
$\chi=\chi_{0}+C/(T-\theta_{\mathrm{CW}})$, yields a temperature-independent
term of $\chi_{0}$ = 0.00047 emu mol-f.u.-1, a Curie constant of $C$ = 0.0043
emu K mol-f.u.-1, and a paramagnetic CW temperature of
$\theta_{\mathrm{CW}}=-1.9$ K. Such a small value of the Curie constant
(corresponding to 0.13 $\mu_{\mathrm{B}}$/Co) is commonly originated from tiny
paramagnetic impurities. Additionally, the positive temperature coefficient of
susceptibility at high temperatures, shown in the inset of Fig. 3(a), also
rules out the possible CW-type paramagnetism in CuCo2S3.7.

Figure 3(b) shows the temperature dependence of magnetization (in the unit of
$\mu_{\mathrm{B}}$/f.u.) of the sulfurized CuCo2S4 under the same magnetic
field of $H=$ 10 kOe. A ferromagnetic transition is seen at about 120 K, which
is attributed to the ferromagnetic impurity of slightly Cu-doped CoS2 that was
identified by the XRD experiment above. To quantify the amount of (Co,Cu)S2
independently, the magnetization data of pure CoS2 are shown for comparison.
One sees that the Curie temperature of (Co,Cu)S2 is slightly lower than that
of CoS2 due to the Cu incorporation. The low-temperature saturation
magnetization is about 19% of that CoS2. At the same time, the high-
temperature magnetic susceptibility basically coincides. Since the Cu content
in (Co,Cu)S2 is only 1-2% according to the EDS measurement, the amount of the
(Co,Cu)S2 impurity should be also around 19%, basically consistent with the
XRD result above.

The high-temperature magnetic susceptibility data are highlighted in the inset
of Fig. 3(b), which shows a CW-type paramagnetism. The CW paramagnetism is
attributed to the (Co,Cu)S2 impurity, because the magnetic susceptibility of
the sulfurized CuCo2S4 shows a similar temperature dependence with that of
18.8% CoS2. The magnetic susceptibility of S-compensated CuCo2S4 phase can be
roughly obtained by a simple substraction. The result indicates a small value
of magnetic susceptibility that is almost temperature independent. Therefore,
CuCo2S4 should be intrinsically Pauli paramagnetic. Nevertheless, the accurate
value of the Pauli-paramagnetic susceptibility cannot be reliably extracted
not only because of the influence of the magnetic impurity, but also due to
the possible Van Vleck paramagnetism involved [10]. According to the
bandstructure calculation of CuCo2S4 which gives the density of states at
Fermi level of 31.88 states/eV/f.u. [27], the calculated Pauli-paramagnetic
susceptibility is derived to be
$\chi_{\mathrm{P}}=\mu_{\mathrm{B}}^{2}N(E_{\mathrm{F}})=1.03\times 10^{-3}$
cm3/mol.

Figure 4: Temperature dependence of magnetic susceptibility for sulfur-
deficient CuCo2S3.7 as well as sulfurized CuCo2S4, measured under a magnetic
field of 20 Oe in both field-cooling (FC) and zero-field-cooling (ZFC) modes.
The inset shows field dependence of magnetization at 2 K for sulfurized
CuCo2S4.

Figure 4 shows the low-temperature susceptibility data for the samples of
CuCo2S3.7 as well as sulfurized CuCo2S4. The S-deficient CuCo2S3.7 exhibits
low values of magnetic susceptibility, and no signal of SC can be detected
down to 1.8 K. By contrast, the sulfurized sample shows a steep decrease in
the magnetic susceptibility at 4.2 K, suggesting a superconducting transition.
Note that the high value of the susceptibility above $T_{\mathrm{c}}$ is due
to the ferromagnetic impurity (Co,Cu)S2. The large magnitude of the ZFC
diamagnetism (exceeding $-100$%) below $T_{\mathrm{c}}$ could also be due to
the extra magnetic field generated by the ferromagnetic (Co,Cu)S2. The inset
shows the field dependence of magnetization at 2 K for the superconducting
sample. An extremely type-II superconductivity with $H_{\mathrm{c2}}\gg
H_{\mathrm{c1}}$ can be concluded. As expected also, the ferromagnetic signal
from (Co,Cu)S2 is superposed on the superconducting loop.

Figure 5(a) shows the temperature dependence of resistivity for the sulfur-
deficient CuCo2S3.7 and the sulfurized CuCo2S4. Both samples show a metallic
behavior, yet the sulfurized CuCo2S4 sample exhibits a lower room-temperature
resistivity with a higher residual resistivity ratio (RRR). The RRR values are
1.4 and 6.4 for CuCo2S3.7 and CuCo2S4, respectively. Although there are about
19% (Co,Cu)S2 impurity in the sulfurized CuCo2S4 sample, no anomaly at
$\sim$120 K associated with the ferromagnetic transition can be detected. At
lower temperatures, while no superconducting transition appears down to 1.8 K
for CuCo2S3.7, a sharp superconducting transition is seen at
$T_{\mathrm{c}}^{\mathrm{onset}}=$ 4.3 K for the S-compensated CuCo2S4. The
observation of SC in relation with a high RRR value was also reported
previously [16]. This could suggest that the nonmagnetic scattering, measured
by the residual resistivity, may destroy SC in the system, resembling the
scenario in Sr2RuO4 [28] and K2Cr3As3 [29]. Besides, the low-temperature
resistivity of CuCo2S4 essentially shows a $T^{2}$ temperature dependence (see
the inset), suggesting dominant electron-electron scattering in the system.

Figure 5: (a) Temperature dependence of electrical resistivity ($\rho$) of the
polycrystalline samples of sulfur-deficient CuCo2S3.7 and sulfurized CuCo2S4.
The inset plots $\rho$ versus $T^{2}$ in the temperature range from 4.5 to 50
K. (b) Resistive superconducting transitions under increased magnetic fields
from which the upper critical fields $H_{\mathrm{c2}}$ were obtained. The
inset plots the resultant $H_{\mathrm{c2}}$ as a function of temperature.

The resistive superconducting transitions are more clearly shown in Fig. 5(b).
One sees that the superconducting transition shifts to lower temperatures with
increasing magnetic fields. Using the criterion of 50% normal-state
resistivity just above $T_{\mathrm{c}}$ for determining $T_{\mathrm{c}}(H)$,
the upper critical magnetic fields $H_{\mathrm{c2}}$ can be extracted. The
resultant $H_{\mathrm{c2}}(T)$ data are shown in the inset of Fig. 5(b), which
shows an essentially linear temperature dependence down to
0.42$T_{\mathrm{c}}$. This result suggests dominant orbital pair-breaking
mechanism over a paramagnetic pair-breaking mechanism. The zero-temperature
upper critical field is estimated to be $H_{\mathrm{c2}}(0)=$ 24.6 kOe from
the linear extrapolation, far below the Pauli-paramagnetic limit
$H_{\mathrm{P}}\approx$ 77 kOe. The coherence length can thus be derived as
$\xi_{0}=$ 11.6 nm using the relation
$H_{\mathrm{c2}}(0)=\Phi_{0}/[2\pi\xi(0)^{2}]$, where $\Phi_{0}(=2.07\times
10^{-15}$ Wb) denotes a magnetic flux quantum.

Figure 6(a) shows the temperature dependence of specific heat for the
sulfurized CuCo2S4 sample. The specific heat tends to approach the value of
$3NR=$ 174.6 J K-1 mol-1 at high temperatures, in accordance with the Dulong-
Petit law. No obvious anomaly is seen at around 120 K where the CoS2 impurity
undergoes a ferromagnetic transition. This observation verifies that the CoS2
impurity is the minor phase. As is seen in the inset of Fig. 6(a), at low
temperatures, a remarkable specific-heat jump is observable at around 4 K,
confirming bulk SC in the sulfurized sample which dominantly contains nearly
stoichiometric CuCo2S4.

Figure 6: Temperature dependence of specific heat for sulfurized CuCo2S4. The
inset of (a) shows a close-up in the low-temperature region. Panel (b) plots
$C/T$ as a function of $T^{2}$, in which a linear fit ($C/T=\gamma+\beta
T^{2}$) is presented for the normal state. Panel (c) shows
$C_{\mathrm{e}}/(\gamma T)$, where $C_{\mathrm{e}}=C-\beta T^{3}$ denotes the
electronic specific heat, as a function of the reduced temperature,
$T/T_{\mathrm{c}}$. The data basically agree with a full-gap BCS $\alpha$
model [$\alpha\equiv\Delta(0)/(k_{\mathrm{B}}T_{\mathrm{c}})$, where
$\Delta(0)$ is the anisotropic superconducting gap at zero temperature] [30]
assuming 77% superconducting phase and a residual electronic specific-heat
coefficient of $\gamma_{0}=0.25\gamma$.

Figure 6(b) shows the plot of $C/T$ versus $T^{2}$, from which the low-
temperature electronic specific heat can be separated out. The linear fit
gives an intercept of $\gamma$ = 32.2 mJ K-2 mol-f.u.-1, corresponding to a
bare density of states of
$N(E_{\mathrm{F}})=3\gamma/(\pi^{2}k_{\mathrm{B}}^{2})=$ 13.6 states/eV/f.u.,
consistent with the electronic structure calculation [27]. Note that the
Sommerfeld constant of CoS2 is 21 mJ K-2 mol-1 [24, 25], somewhat smaller than
the above $\gamma$ value, yet it turns out to be larger on the basis of Co
content. Furthermore, the CoS2 impurity is the minor phase after all.
Therefore, the Sommerfeld coefficient of the CuCo2S4 phase will not change
very much even if corrections due to the existence of CoS2 impurity could be
reliably made. Note that the

Assuming the $\gamma$ value of 32.2 mJ K-2 mol-f.u.-1 and with the electronic
specific heat of $C_{\mathrm{e}}=C-\beta T^{3}$, figure 6(c) was plotted using
$C_{\mathrm{e}}/(\gamma T)$ and $T/T_{\mathrm{c}}$ as the coordinates. Under
the constrain of entropy conservation, i.e.
$\int_{0}^{T_{\mathrm{c}}}[(C_{\mathrm{e}}-\gamma T)/T]\mathrm{d}T=0$, a full-
gap BCS $\alpha$ model [30] can basically fit the data with
$\alpha\equiv\Delta(0)/(k_{\mathrm{B}}T_{\mathrm{c}})=$ 1.5 if a residual
electronic specific-heat coefficient of $\gamma_{0}=0.25\gamma$ due to the
existence of non-superconducting impurity phase of CoS2 is taken into account.
In this circumstance, the superconducting fraction is fitted to be 77(1)%,
which is conversely consistent with $\sim$19% non-superconducting phase.

Here we note that the single-gap BCS model does not account for the data
exclusively. Other models with line energy-gap nodes are also applicable.
However, the present limited data cannot distinguish which model applies.
Interestingly, previous NMR investigations concluded contrasting
superconducting properties in the Cu-Co-S system: one suggested a gapless
superconducting state [13], the other indicated a full superconducting gap
[14]. This discrepancy seems to be due to the big difference in the sample’s
quality. Our present specific-heat result excludes the possibility of gapless
superconductivity in the nearly stoichiometric sample of CuCo2S4. We expect
that future measurements of specific-heat, NMR, and other techniques down to
lower temperatures with using better samples (with less impurity) will be able
to clarify the issue of the superconducting gap.

Above we have clarified that the nearly stoichiometric CuCo2S4 thiospinel is a
superconductor with Pauli paramagnetism. Now let us comment on the previous
dispersive results about “CuCo2S4” [11, 12, 15, 16, 18]. They can be accounted
for in terms of the deviations from the stoichiometry. The actual composition
of the synthesized thiospinel phase should be written as (Cu1-xCox)Co2S4-δ
(because impurity phases such as CoS2 and Cu2S appeared). The S deficiency
obviously decreases the hole concentration in (Cu1-xCox)Co2S4-δ, which
suppresses SC. The Co/Cu substitution (Co2+ partially substitutes Cu+) not
only decreases the hole concentration, but also possibly induces magnetic
impurity of Co2+ at the Cu site, both of which are detrimental to SC. This
could be the main reason for the difficulty to observe SC in the sample with
nominally stoichiometric composition. In the Cu-rich sample of “Cu1.5Co1.5S4”
[13, 14], however, the Co/Cu substitution at the Cu site is greatly reduced
because Co is poor. The possible Cu occupation at the Co site may not destroy
SC because of nonmagnetic Cu+. Thus SC is easily observed in the Cu-rich
samples.

##  IV Concluding remarks

To summarize, with a novel two-step synthesis strategy, we were able to
prepare nearly stoichiometric CuCo2S4 phase which shows bulk SC at 4.2 K with
Pauli paramagnetism in the normal state. We have also revealed that sulfur
deficiency and Co/Cu substitution is detrimental to SC, which may explain the
contradictive results in previous reports. The result calls for further
investigations on the rare Co-based superconductor by optimizing the sample
quality (with the CoS2 impurity as less as possible) and with various
measurements down to lower temperatures.

SC in Co-based compounds is very rare. This work corroborates that CuCo2S4 is
another Co-based superconductor in addition to NaxCoO${}_{2}\cdot y$H2O [8].
Albeit of different crystal structures, interestingly, the two systems show
many similarities including the $T_{\mathrm{c}}$ value, Co coordination,
formal Co valence, and the geometrical frustration. It is of great interest to
clarify whether CuCo2S4 is an unconventional superconductor [31]. On the other
hand, SC is not frequently found in the thiospinel compounds. However, the
Cu$M_{2}$S4 ($M=$ Co, Rh, or Ir) family seem to be the only exception. CuRh2S4
was first discovered to be a superconductor in 1967 with $T_{\mathrm{c}}$ =
4.35-4.8 K [11, 32], which was confirmed in 1990s [33]. CuIr2S4 [34] itself is
not a superconductor, yet it undergoes a metal-insulator transition at 230 K
accompanied with a charge ordering as well as a spin dimerization [35]. SC
with $T_{\mathrm{c}}$ up to 3.4 K can be induced by the suppression of the
metal-insulator transition via Zn/Cu substitution [36, 37]. For the spinel
selenides, SC was reported in CuRh2Se4 ($T_{\mathrm{c}}=$ 3.5 K [11, 32]) and
Cu(Ir0.8Pt0.2)2Se4 ($T_{\mathrm{c}}=$ 1.76 K [38]). Therefore, one may expect
that CuCo2Se4 could be also a superconductor if it can be synthesized with the
stoichiometric composition.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(12050003), National Key Research and Development Program of China
(2017YFA0303002) and the Fundamental Research Funds for the Central
Universities of China.

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