# Full Set of Superconducting Parameters of K3C60

Ren-Shu Wang  School of Science, Harbin Institute of Technology, Shenzhen
518055, China  Center for High Pressure Science and Technology Advanced
Research, Shanghai 201203, China  Di Peng  Center for High Pressure Science
and Technology Advanced Research, Shanghai 201203, China  Li-Na Zong  Center
for High Pressure Science and Technology Advanced Research, Shanghai 201203,
China  Zeng-Wei Zhu  Wuhan National High Magnetic Field Center, Wuhan 430074,
China  Xiao-Jia Chen<EMAIL_ADDRESS>Center for High Pressure Science and
Technology Advanced Research, Shanghai 201203, China  School of Science,
Harbin Institute of Technology, Shenzhen 518055, China

###### Abstract

The superconducting parameters are the key for building or identifying the
theory responsible for the mechanism of superconductivity. Such parameters for
fulleride superconductors have not been well established despite the
tremendous efforts over the past 30 years. Here we provide a full set of
parameters through a systematic study on a well-characterized K3C60 sample.
The obtained high upper critical field of 33.0$\pm$0.5 T from the direct
electrical transport measurements together with the relatively high critical
temperature and large critical current density classifies K3C60 as a promising
three-dimensional superconducting magnet material with the advantage of the
rich carbon abundance on the Earth. This high upper critical field along with
the large reduced superconducting energy gap and strong phonon self-energy
effect supports the strong electron-phonon coupling interactions in this
superconductor. The evaluation of all self-consistently obtained parameters
suggests the unconventional nature of the superconductivity for K3C60 with the
joint contributions from the strong electron-phonon coupling and electron
correlations. These results and findings are important not only for
fundamentally understanding the superconductivity in fullerides but also for
future superconducting magnet developments and applications.

##  I Introduction

The discovery of superconductivity in alkali fullerides with the critical
temperature $T_{c}$ going from 18-19 K for K3C60 Heb  through 28-29 K for
Rb3C60 ross  to 38-40 K for Cs3C60 pals ; taka ; Gan  is an important event in
modern science after the birth of the new form of carbon called
‘buckminsterfullerene’ or in short fullerene krot . Apparently, these three-
dimensional molecular solids are different in structure from the early
discovered high-$T_{c}$ cuprates and recently discovered twisted graphene ycao
. They in reality share the similar superconducting phase diagram emerging
from neighbouring Mott insulating state taka ; Gan ; ycao . This similarity
fuels the hope to solve the long-standing puzzle of the mechanism of
superconductivity in cuprates once knowing the key factors that govern
superconductivity in fullerides due to the relatively easy realization of
superconductivity with high $T_{c}$ in fullerides compared to the need of
precise tuning in graphene, another purely C-based superconductor.

The crucial examination of the theory highly depends on the superconducting
parameters. Take the first fullerene superconductor K3C60 as an example, the
large differences of these parameters from the experiments make the
comparisons with the theory difficult Gun0 ; capo . Conducting electrical
transport measurements to realize the zero resistance state, known as one of
the two essential features of a superconductor, has long been a challenge in
the characterization of its superconductivity. Therefore, the characterization
of fullerene superconductors has mainly been performed based on magnetization
measurements Heb ; Ste ; Hol ; Yoo  due to the challenges in handling the air-
sensitive samples Fle ; Hol1 . Among the several electrical transport studies
on K3C60 Pal ; Xia ; Kle , magnetic characterizations were absent. In fact,
the first experimental evidence for supporting superconductivity in K3C60 from
the zero-resistance state and Meissner effect was obtained from different
samples Heb . The airtight devices were developed to measure the resistivity
for films Heb ; Pal ; Pal1 ; Had1  and single crystals Zha ; Mar  with the
observed zero-resistance state in the superconducting state. Meanwhile, some
residual resistances often appeared at low temperatures in the electrical
transport measurements on single crystals Xia ; Xia1 ; Hou , probably due to
the limited crystallinity in films Heb ; Pal ; Pal1  or uncontrolled alkali
distribution in single crystals Xia ; Xia1 ; Hou . Alternatively, the
resistivity was inferred from some contactless methods Deg ; Kle . However,
these techniques brought about large residual surface resistance Kle . In view
of these facts, the electrical transport measurements are still challenging in
the characterization of superconducting fullerides. The zero-resistance state
and Meissner effect on the same sample have only been realized recently for K-
and Rb-doped C60 through the improvements of the sample quality and
measurement techniques rswang ; zong .

It has been generally accepted that the resistivity measurement at required
high magnetic field and low temperature is a reliable method for accurately
determining the critical magnetic field which is fundamentally important due
to its close relation to the coherent length and the Cooper pairing strength
Tin . At the moment, the upper critical field $H_{c2}$(0) values of K3C60 were
obtained either through the low-magnetic-field measurements of the
magnetization Hol ; Bun  and resistivity Hou ; Hou1 ; Pal  or the high-
magnetic-field non-contact measurements based on the $ac$ magnetization Joh ;
Boe ; Bae  and radiofrequency technique Fon ; Kas . Having accurate
$H_{c2}$(0) of K3C60 through the resistivity measurements at high enough
magnetic fields is still highly desired. Here we choose K-doped C60 with the
considerations of its similarity to other fullerides Fle ; Gun0  and
simplicity due to the absence of the antiferromagnetic state to solve this
long-standing issue with the purpose to provide the constraints on the
identification of the existing theoretical models or the future theory
developments from a full set of superconducting parameters.

Figure 1: Structural and spectroscopic characterizations of K-doped C60. (a)
Experimentally observed (red dots) and calculated (blue solid line) x-ray
diffraction patterns for the doped sample at room temperature. The vertical
purple lines present the Bragg reflection positions of the $fcc$ structure,
and the solid green line at the bottom represents the difference profile.
Inset: Sample picture taken under microscope. (b) Crystal structure of $fcc$
K3C60. The violet and red balls represent C and K atoms, respectively. (c)
Raman scattering spectra of pure and K-doped C60. (d) Schematic diagram of
four representative intramolecular vibrational modes for K3C60. The red arrows
indicate the vibrational directions of the C atoms.

##  II Results

###  II.1 Crystal structure and molecular vibrations of K-doped C60

Wet chemistry method was used to synthesize K-doped C60 detailed in Method.
The quality, structure, and phase of the sample were examined by x-ray
diffraction (XRD) and Raman scattering measurements. The XRD profile can be
well indexed by the $Fm\overline{3}m$ space group [Fig. 1(a)], indicating that
the synthesized K-doped C60 sample is a single-phase compound with the face-
centered cubic ($fcc$) structure [Fig. 1(b)]. The obtained lattice parameter
$a$ is 14.22$\pm$0.01 Å, in good agreement with those reported previously Ste
; Fle .

Figure 1(c) shows Raman spectra of the pristine and K-doped C60 collected at
room temperature. All the Raman-active modes for C60, including two $A_{g}$
modes (at 497 and 1469 cm-1) and eight $H_{g}$ modes: 272, 432, 710, 772,
1100, 1248, 1422, and 1574 cm-1, are observed in the pristine C60, agreeing
well with the literature Zhou . The two low-frequency $H_{g}$ modes and two
$A_{g}$ modes have strong intensities in K-doped C60. The vibrations of these
four modes schlu  are shown schematically in Fig. 1(d). A recognized approach
to determine the doping level of KxC60 is the line-shift of the pinch mode
$A_{g}$(2). The observed 17 cm-1 redshift of this mode yields the doping
concentration of about 3 estimated by the empirical relation of 6 cm-1 per
elementary charge Had . Therefore, the synthesized sample is homogeneous and
has the chemical formula of K3C60 with the $fcc$ structure.

Figure 2: Characterization of superconductivity of K3C60 from the magnetic
susceptibility and electrical transport measurements. (a) DC magnetic
susceptibility ($\chi$) with temperature range of 1.8-100 K at the applied
magnetic field of 10 Oe. Zero-field-cooling (ZFC, red circles) and field-
cooling (FC, blue circles) $\chi$ show an obvious diamagnetism below 19 K. (b)
Real part of the ac magnetic susceptibility ($\chi^{\prime}_{ac}$) in the
temperature range from 1.8 to 50 K. The probe harmonic magnetic field and
frequency are 5 Oe and 234 Hz, respectively. (c) Temperature dependence of the
electrical resistivity ($\rho$). (d) Imaginary component of the ac magnetic
susceptibility ($\chi^{\prime\prime}_{ac}$) as a function of temperature
(1.8-50 K).

###  II.2 Superconductivity of K3C60

The superconductivity of the synthesized K3C60 was identified by the existence
of both the Meissner effect and zero resistance state. Figure 2 shows the
results from the $dc$ and $ac$ susceptibility ($\chi$) and resistivity
measurements. The $dc$ susceptibility directly demonstrates the Meissner
effect of the sample, as shown in Fig. 2(a). Zero-field-cooling (ZFC) and
field-cooling (FC) $\chi$ curves exhibit a clear drop below 18.5 K, which is
defined as $T_{c}$. The shielding fraction (SF) of 80% at 2 K is estimated
from the ZFC $\chi$ curve. This diamagnetic effect is also observed through
the real part ($\chi^{\prime}_{ac}$) of the $ac$ susceptibility [Fig. 2(b)],
similar to the ZFC $\chi$ curve from the $dc$ measurement. As shown in Fig.
2(c), the resistivity drops rapidly below 18.7 K and quickly reaches the
absolute zero value. The zero-resistivity transition of the sample is also
indirectly reflected by the imaginary part ($\chi^{\prime\prime}_{ac}$) of the
$ac$ susceptibility. To response the vortex current signal,
$\chi^{\prime\prime}_{ac}$ can only be induced when the resistivity drops
sharply because of the relatively large resistivity in the normal state and
the flux exclusion in the complete Meissner state Gom . The peak shape of
$\chi^{\prime\prime}_{ac}$ in Fig. 2(d) is the manifestation of the sharp drop
in resistivity. As learned from the literature for the representative
magnetization Heb ; Ste ; Hol ; Hol1 ; Yoo ; Boe ; Bae  and nearly all
electrical transport Heb ; Pal ; Uga ; Zha ; Hes ; Mar ; Xia ; Xia1 ; Hou ;
Pal1  measurements, the evidence for the zero-resistance state and the
Meissner effect taken on the same K3C60 sample is quite rare. The present
study together with our recent efforts rswang ; zong ; rswang2  provides solid
experimental evidence for supporting superconductivity in synthesized
fullerides based on its two essential characters.

###  II.3 Upper critical field determined by resistivity measurements

Figure 3(a) shows the temperature dependence of the resistivity at low
magnetic fields of 0-9 T. With increasing magnetic field, $T_{c}$ shifts
toward lower temperatures [Fig. 3(a)]. The resistivity behaviours at low
temperatures and pulsed magnetic fields up to 50 T are given in Fig. 3(b). The
determination of $H_{c2}$ at a given temperature is illustrated in the inset
of this figure. The $H_{c2}(T)$ data obtained from the high- and low-magnetic
fields can be well described by the Werthamer-Helfand-Hohenberg theory Wer .
Note that the experimentally observed $H_{c2}$’s at low temperatures are
significantly larger than the orbital limit field $H_{c2}^{orb}$(0) of 22 T
estimated from the $H_{c2}(T)$ slope in the low-field regime ($H\leq$1 T) by
using
$H_{c2}^{orb}$(0)=0.69$\times$$T_{c}$$\times\lvert$d$H_{c2}$/d$T$$\rvert$${}_{T=T_{c}}$,
suggesting the strong coupling effect. The experimentally obtained $H_{c2}$(0)
of 33.0$\pm$0.5 T basically agrees with most of the high-field experimental
results for this material Joh ; Boe ; Fon ; Kas .

The reliability of the obtained $H_{c2}$(0) value is guaranteed by the direct
electrical transport measurements at enough high magnetic fields and at low
temperatures. As a key indicator in the applications of superconducting
materials, the $H_{c2}$(0) of K3C60 is little bit higher than that of Nb3Sn
with lower $T_{c}$ of 18 K God , a typical three-dimensional superconductor
well known as superconducting magnets. Interestingly, our $H_{c2}$(0) value
for K3C60 at ambient pressure is even comparable to those of $fcc$ CeH10 with
much higher $T_{c}$=115 K at pressure of 95 GPa Che  and ThH9 with $T_{c}$=151
K at 170 GPa seme . However, the need of high pressures to access the
superconducting state in these superhydrides limits their technological
applications. The larger $H_{c2}$(0) for K3C60 with the higher $T_{c}$
compared to those in Nb3Sn offers the attractive opportunities for the
technological applications due to the rich carbon abundance on the Earth as
well as the low cost. Therefore, the present experimental finding opens a
window for future developments and designs of the three-dimensional
superconducting magnets based on this material.

###  II.4 Lower critical field, penetration depth, and coherence length

The obtained $H_{c2}$(0), combined with other superconducting parameters such
as the lower critical field $H_{c1}$, London penetration depth $\lambda_{L}$,
and coherence length $\xi$, are crucial to the understanding of the physical
properties and the mechanism of superconductivity of a type-II superconductor.
The $H_{c1}$ is determined by the magnetic-field dependence of the $dc$
magnetization $M(H)$ at various temperatures below $T_{c}$ [Fig. 3(c)]. Within
this method, the magnetic field that initially deviates from the linear $M(H)$
behaviour is defined as $H_{c1}(T)$ for a given temperature, as illustrated in
the upper right of Fig. 3(d). The $H_{c1}(T)$ values at different temperatures
plotted in Fig. 3(d) are used to determine $H_{c1}$(0) of 6.9$\pm$0.1 mT
through the empirical relation Tin  $H_{c1}(T)/H_{c1}(0)=1-(T/T_{c})^{2}$.
With these critical fields, $\lambda_{L}$ and the Ginzburg-Landau coherence
length ($\xi_{GL}$) can be determined by using the equations:
$H_{c2}(0)=\Phi_{0}/2\pi\xi_{GL}^{2}$ and
$H_{c1}(0)=(\Phi_{0}/4\pi\lambda_{L}^{2})\ln(\lambda_{L}/\xi_{GL})$ with the
flux quantum $\Phi_{0}=2.0678\times 10^{-15}$ Wb. We thus have
$\lambda_{L}$=3325$\pm$36 Å, $\xi_{GL}$=31.6$\pm$0.3 Å, and the Ginzburg-
Landau parameter $\kappa=\lambda_{L}/\xi_{GL}=105\pm 2$. These results are
compared with those determined by using various techniques such as the
magnetization Hol ; Iro ; Bun ; wong , magnetoresistance Hou , muon spin
relaxation Uem ; Uem1 , nuclear magnetic resonance Tyc ; Sas1 , and optical
reflectivity Deg .

Figure 3: Determination of superconducting parameters of K3C60. (a)
Temperature-dependent resistivities at various magnetic fields up to 9 T. (b)
Temperature dependence of the upper critical field $H_{c2}(T)$. The data
points of the red diamonds and blue pentacles are obtained from the electrical
transport measurements based on the fixed-field (a) and fixed-temperature
scans, respectively. Error bars represent estimated uncertainties in
determining $H_{c2}$. The solid line is the fitting to the Werthamer-Helfand-
Hohenberg theory. Inset: Magnetic field dependent resistivity with scanning
field up to 50 T at temperature of 4.2 K. $H_{c2}$ for a fixed temperature is
determined from the intercept of the linear extrapolations from below and
above the transition. (c) Magnetic-field dependence of the magnetization ($M$)
at various temperatures. (d) Temperature dependence of the lower critical
field $H_{c1}(T)$. The solid curve is the fitting of the measured data points
to the empirical law $H_{c1}(T)/H_{c1}(0)$=1$-$($T/T_{c}$)2. Inset: The
magnetization as a function of magnetic field at 2 K. The derivation from the
initial linear trend is used for the determination of $H_{c1}$. (e) Magnetic
hysteresis (M-H) loops at various temperatures with the applied field up to
$\pm$7 T. (f) Temperature dependence of the critical current density
($J_{c}$). The $J_{c}$ value is determined from the M-H loop (e) based on the
Bean’s critical-state model.

###  II.5 Determination of critical current density

Figure 3(e) displays the $dc$ magnetic hysteresis loops at several fixed
temperatures below $T_{c}$. With increasing temperature, the hysteresis loops
gradually shrink inward until converting to a straight line near $T_{c}$. The
diamond-like loop and temperature-dependent shrinkage are typical features for
a type-II superconductor. The critical current density ($J_{c}$) can be
determined simply from these $dc$ magnetic loops based on Bean’s critical
state model Bea  by using the formula $J_{c}=A\times(M_{+}-M_{-})/r$, where M+
and M- are the magnetizations in the decreasing and increasing circles at a
given field $H$, and $A$ and $r$ are the shape of sample Fie  and the sample
radius, respectively. In our estimation, the radius $r$ is assumed to be 1
$\mu$m according to the XRD results. The $J_{c}$($H$=0) variation with
temperature is shown in Fig. 3(f). The obtained $J_{c}$ value for K3C60 in the
present work differs largely with the early results obtained from the same
magnetization measurements Hol ; Bos ; Zha ; Iro ; Bun ; Lee ; Bun1 ; Bun2 ;
Bae . The difference mainly comes from the uncertainties for the radius $r$,
which has been confirmed by the functional relationship between $J_{c}$ and
the particle size Bos . Meanwhile, the granularity of the sample also affects
the $J_{c}$ value Bun .

It should be emphasized that $J_{c}$ is important for evaluating technological
applications of a superconductor. The material itself possesses a self-field
component $J_{c}$(s) even without applying external field. For a type-II
superconductor, $J_{c}$(s) follows a universal expression Tal
$J_{c}$(s)=$H_{c1}$/$\lambda_{L}$. It is clear that $J_{c}$(s) is independent
of the material geometry. Substituting the obtained $H_{c1}$ and
$\lambda_{L}$, we have $J_{c}$(s) of 1.7$\times$106 A/cm2 for K3C60, slightly
larger than $J_{c}$($H$=0) determined from the magnetization measurements
[Figs. 3(e)-3(f)]. Therefore, the technological parameters of $T_{c}$,
$H_{c2}$, and $J_{c}$ needed for K3C60 to function as a superconducting magnet
material are well established.

Figure 4: Raman spectra in the low-frequency region of K3C60. (a) Raman
spectra at several representative temperatures before and after the
superconducting transition. The scattering peak at around 270 cm-1 is assigned
as the $H_{g}$ (1) mode. (b) Raman spectra normalized by the intensity at 20
K. The intensity shows a linear decrease below 60 cm-1 (as marked by the
arrows) when the temperature is lower than 1/2 $T_{c}$, and it becomes more
obvious at lower temperatures. (c-e) Temperature dependence of the frequency
($\omega$), linewidth ($\Gamma$), and electron-phonon coupling parameter
$\Gamma/\omega^{2}$ of the $H_{g}(1)$ mode, from top to bottom, respectively.
Green shadow areas represent the superconducting state.

###  II.6 Detection of superconducting gap and evaluation of electron-phonon
coupling

The superconducting energy gap ($\Delta$) of K3C60 is determined based on the
Raman spectroscopy measurements. Figure 4(a) shows the representative Raman
spectra in the low-frequency region at a wide temperature range from 5.7 to
100 K. In Fig. 4(b), the Raman spectra at low temperatures below $T_{c}$ were
normalized by dividing the spectrum measured at temperature of 20 K. One can
see clearly a linear decrease in the scattering intensity below 60 cm-1 (as
marked by arrows) when the temperature cools down to 1/2 $T_{c}$. This feature
becomes more obvious at lower temperatures. Such a linear decrease in the
intensity of the electronic scattering can be attributed to the
renormalization of the density of state due to the superconducting transition
Els . The 2$\Delta$ value can be taken from the position where the frequency
starts to deviate the linear behaviour. The reduced energy gap
$2\Delta/k_{B}T_{c}$ of 4.7$\pm$0.3 is thus obtained with $k_{B}$ being the
Boltzmann constant. This value is higher than 3.53 predicted within the BCS
framework. The high $2\Delta/k_{B}T_{c}$ and large $H_{c2}(0)$ together
indicate the strong coupling of the interactions in this superconductor.
Similar conclusions have also been drawn from other techniques including
point-contact tunneling spectroscopy Zhang ; Ren  and nuclear magnetic
resonance Sas . Some other studies based on the optical reflectivity Deg ;
Deg1 , nuclear magnetic resonance Tyc ; Sas , and photoemission Hes  support
weak electron-phonon coupling within the conventional BCS framework.

For type-II spin-singlet superconductors, the orbital and Pauli paramagnetic
effects are two distinct ways for pair-breaking with increasing external
magnetic field. Using $2\Delta/k_{B}T_{c}$=4.7$\pm$0.3, the Pauli-limiting
field $H_{P}$=45.4$\pm$2.2 T is then determined accurately from
$H_{P}=\Delta/(\sqrt{2}\mu_{B})$ Clo  with $\mu_{B}$ being the Bohr magneton.
The obtained orbital component of the upper critical field $H_{c2}^{orb}$ of
22 T is much smaller than $H_{P}$, yielding the Maki parameter Gru
$\gamma=\sqrt{2}H_{c2}^{orb}/H_{P}\sim 0.7$. Since $\gamma$ reflects the
strength of the paramagnetic effect, this indicates that the orbital effect
dominates the $H_{c2}$ behaviour for this superconductor.

Figure 5: Hall effect data of K3C60. Hall resistivity ($\rho_{xy}$) vs. the
applied magnetic field along the two opposite directions up to 6 T in the
temperature range from 20 to 160 K (a) and from 170 to 300 K (b). Numerical
scale is given in the left bottom. The Hall coefficient ($R_{H}$) (c) carrier
concentration ($n_{H}$) (d) and mobility ($\mu_{H}$) (e) as a function of
temperature. The yellow regions in (c-e) denote the boundary for the
occurrence of the orientational transition.

In addition, the electron-phonon contribution to the superconductivity can be
evaluated from the observations of the Raman spectra at temperatures below
$T_{c}$. The low-frequency $H_{g}$(1) intramolecular vibrational mode is
chosen as an example to monitor the superconductivity-induced changes. The
systematic change of this mode can be seen from Fig. 4(a). The temperature
dependence of the frequency ($\omega$), linewidth ($\Gamma$), and electron-
phonon coupling parameter $\Gamma/\omega^{2}$ of this $H_{g}$(1) mode is shown
in Figs. 4(c)-4(e), from top to bottom, respectively. When entering the
superconducting state (the green shaded zone), the $H_{g}$(1) mode exhibits
the downshift (softening) in $\omega$, the increase (widening) in $\Gamma$,
and the enhancement of $\Gamma/\omega^{2}$. These behaviours are due to the
superconductivity-induced phonon self-energy effect. The early proposal prass
regarding the phonon evolution with temperature when this material enters the
superconducting state with high-resolution from an inelastic scattering
technique is finally realized here. The absence of phonon mode near 40 cm-1 in
the Raman spectra at temperatures below $T_{c}$ indicates that it is probably
an acoustic branch and thus inactive to Raman scattering but visible in the
neutron measurements prass . These results provide the strong support for the
important contribution of the electron-phonon coupling to the
superconductivity in K3C60. The observed self-energy effect together with the
large $2\Delta/k_{B}T_{c}$ obtained from the same Raman spectra reveals the
strong electron-phonon coupling in K3C60, which needs to be included in
dealing with the phonon effects varma ; fczhang ; schl ; chen ; han ; Nom ;
dolg .

Table 1: Summary of the superconducting parameters of K3C60. $T_{c}$ (K) |  |  |  |  | 18.5$\pm$0.5
---|---|---|---|---|---
$H_{c1}$(0) (mT) |  |  |  |  | 6.9$\pm$0.1
$H_{c2}$(0) (T) |  |  |  |  | 33.0$\pm 0.5$
$\lambda_{L}$ (Å) |  |  |  |  | 3325$\pm$36
$\xi_{GL}$ (Å) |  |  |  |  | 31.6$\pm$0.3
$J_{c}$ ($\times$104 A/cm2) |  |  |  |  | 100$\pm$2
2$\Delta$/$k_{B}T_{c}$ |  |  |  |  | 4.7$\pm$0.3
$v_{F}$ ($\times$104 m/s) |  |  |  |  | 5.6$\pm$0.4
$E_{F}$ (meV) |  |  |  |  | 55$\pm$13
$T_{F}$ (K) |  |  |  |  | 641$\pm$147

###  II.7 Determination of Hall coefficient

The magnetic-field-dependent Hall resistivities of K3C60 with applied magnetic
fields up to $\pm$6 T at various temperatures from 20 to 300 K are shown in
Figs. 5(a) and 5(b). The Hall resistivity $\rho_{xy}$ versus $H$ curves at all
studied temperatures are essentially linear, ensuring the accurate
determination of the Hall coefficient $R_{H}$ through the linear fitting to
the data points [Fig. 5(c)]. The carrier concentration $n_{H}$ and mobility
$\mu_{H}$ are thus obtained through $n_{H}$=$1/(eR_{H})$ and
$\mu_{H}$=$\sigma_{xx}R_{H}$ with $e$ and $\sigma_{xx}$ being the electron
charge and the electrical conductivity, respectively [Figs. 5(d)-5(e)]. As can
be seen, $R_{H}$ changes sign at the temperature range of 220-250 K. The
change of the carrier character further indicates that the synthesized sample
is in a half-filled state with both electron and hole conduction Pal ; Erw .
Similar sign change has also been observed in K3C60 thin films Pal  but was
absent in the study for K- and Rb-doped C60 single crystals Lu .
Interestingly, around the temperature for the sign change, the orientational
ordering transition has been reported for C60 Hei  and K3C60 rswang ; Bar .
The Hall effect clearly captures the effect of the orientational ordering on
the electronic structure. Most importantly, the obtained evolution of $n_{H}$
with temperature close to $T_{c}$ not only gives the accurate carrier
concentration for K3C60 to superconduct but also determines the electronic
feature at low temperatures rather than the hole character at room
temperature, though the two carrier behaviour was noticed in the band
structure calculations Erw  and the electron doping is generally believed.

Figure 6: Magnetic susceptibility $\chi_{meas}$ of K3C60 in the temperature
range of 20-100 K.

###  II.8 Full set of superconducting parameters

Table I summarizes the obtained superconducting parameters for K3C60. These
parameters obtained from the same sample could settle down the large
contradictions from different groups. For instance, the reported $H_{c1}$(0)
values in the literature are in the range from 1 to 13 mT with the difference
in one order of magnitude Hol ; Iro ; Bun ; Bun3 . Our $H_{c1}$(0) value
locates in the middle of this range. In the case of $2\Delta/k_{B}T_{c}$, the
early results are in the range of 3-3.6 revealed by far-infrared the optical
reflectivity Deg ; Deg1 , nuclear magnetic resonance Tyc ; Sas , and
photoemission Hes , suggestive of the weak BCS coupling. Meanwhile, the strong
coupling effect is supported by the large $2\Delta/k_{B}T_{c}$ values from 4.3
to 6 based on the scanning tunneling microscopy Zhang ; Ren  and nuclear
magnetic resonance Sas  measurements. The obtained value of 4.7$\pm$0.3 for
K3C60 from the present work is in the middle range of these studies, which
supports the strong coupling effect.

Having the knowledge of $\xi_{GL}$ based on the well determined $H_{c2}$(0) as
well as $\Delta$, one can derive the average Fermi velocity $v_{F}$ of
(5.6$\pm$0.4)$\times$104 m/s by using the formula
$v_{F}=\pi\Delta\xi_{GL}/\hbar$, in good agreement with the measurements and
calculations wong ; Erw . The measured magnetic susceptibility $\chi_{meas}$
(Fig. 6) at 20 K gives the Pauli susceptibility
$\chi_{spin}$=(7.5$\pm$0.6)$\times$10-4 emu/mol after the correction of the
core and Landau diamagnetic contributions. Taking the $n_{H}$ value at 20 K
(Fig. 5) for the conduction electron density $n$, we obtain the effective mass
$m^{\ast}$=(6.2$\pm$0.6)$m_{0}$ from the expression
$\chi_{spin}$=$\mu_{B}m^{\ast}(3\pi^{2}n)^{1/3}/\hbar^{2}\pi^{2}$, where
$m_{0}$ and $\hbar$ are the mass of the free electron and the reduced Planck
constant, respectively. The result for $m^{\ast}$ is very close to
(6.4$\pm$1.5)$m_{0}$ in the early study wong . Using the $v_{F}$ and
$m^{\ast}$ value, we have $E_{F}$=55$\pm$13 meV and the Fermi temperature
$T_{F}$=641$\pm$147 K for K3C60 based on the formula
$E_{F}$=$m^{\ast}$$v_{F}^{2}$/2=$k_{B}$$T_{F}$. Note that all these important
parameters are obtained on the same sample in terms of the generally accepted
techniques. Table I is thus updated to include them.

##  III Discussion and conclusions

It was often expected to elucidate the driving force for superconductivity
from the isotope effect. However, the existing isotope exponent $\alpha$
deviates the predicted value of 0.5 from the BCS theory and can be divided two
groups depending on the 13C enrichment, one Che1  with $\alpha$$<$0.5 and the
other Zak ; Aub ; Ric  with $\alpha$$>$1.0. The former suggests the strong
coupling effect, consistent with our Raman data, and the latter favors the
electron correlations chak2 ; ashc . Therefore, the isotope effect suggests
that the superconducting pairing in K3C60 originates from either the strong
electron-phonon interaction or the electronic correlations. However, it is
hard to pin down the exact mechanism due to the scatter of the data.

Figure 7: Relationship between the critical temperature $T_{c}$ and the Fermi
temperature $T_{F}$ of representative superconductors. The data derived from
the current measurements for K3C60 (open square) are compared with those of
various superconductors taken from the previous works ycao ; Uem3 . Bi2223,
YBCO, LSCO, and BKBO represent Bi2Sr2(Ca,Pb)2Cu3O10+δ, YBa2Cu3O7-δ,
La2-xSrxCuO4, and Ba1-xKxBiO3, respectively. TMTSF and BEDT are two organic
charge-transfer salts. TBG denotes twisted bilayer graphene. The broken line
denotes the Bose-Einstein condensation temperature $T_{BEC}$ for an ideal
three-dimensional boson gas.

Now that a full set of parameters for K3C60 (Table I) has been obtained from
the same sample at all the necessary measurement conditions and techniques,
they should provide the reliable constraints on the examination of the
existing theories and the future theory development for the mechanism of
superconductivity. The determined $H_{c2}$(0) not only yields $\xi_{GL}$ and
but also further gives $v_{F}$ with the help of $\Delta$. The magnetization
and Hall effect data give $m^{\ast}$ and thus $T_{F}$ when combining $v_{F}$.
These self-consistently obtained parameters along with $T_{c}$ at least can be
used to see how far away or close the studied K3C60 locates to the well known
superconductors.

A recognized measure for judging the nature of correlated superconductors is
through the change trend for $T_{c}$ and $T_{F}$. All the reported correlated
superconductors discovered so far including cuprates, heavy fermion systems,
organic materials Uem2 ; Uem3 , and even twisted graphene ycao  follow a
linear relationship between $T_{c}$ and $T_{F}$. The common characters for
these superconductors are the existence of some competing order(s) and the
strong electron correlation effect. Thus, the electron correlations are
believed to be the major player for superconductivity in these systems. When
adding the obtained $T_{c}$ and $T_{F}$ values for K3C60 in the map (Fig. 7),
we find that this superconductor nicely follows the same linear trend as the
other correlated superconductors but being very close to Ba1-xKxBiO3. The
trend itself classifies K3C60 as an unconventional correlated superconductor.
The neighbourhood for Ba1-xKxBiO3 and K3C60 implies that their
superconductivity may be governed by the similar factors. For Ba1-xKxBiO3,
till recently it was found Wen  that the electron correlations enhance the
electron-phonon coupling and they together contribute to the high $T_{c}$ in
this system. It is reasonable to believe both the electron correlations and
electron-phonon coupling jointly account for superconductivity in K3C60. This
idea finds the support from the theoretical calculations on alkali-doped
fullerides mainly focusing on Cs3C60 Nom , where the electron-phonon coupling
was found to enhance the electron correlations and they together gave the
correct $T_{c}$ evolution trend with the unit cell volume, though the
calculated $T_{c}$ is generally lower about 10 K than the experimental value.
Since the strong coupling effect in K3C60 has been firmly established from
high $T_{c}$, high $H_{c2}$(0), large $2\Delta/k_{B}T_{c}$, and the Raman
data, the $T_{c}$ and $T_{F}$ relation derived from other parameters further
suggests unconventional correlated superconductivity and the joint
contributions from the electron-phonon coupling and electron correlations Gun0
; capo . Recalling the unconventional isotope effect, the nice collaboration
between the strong electron-phonon coupling and electron correlations is thus
suggested to be responsible for the observed unconventional superconductivity
in K3C60, both in a favorable way. This should be the base and constraints on
the examinations of the existing theories and the future theory development
for the mechanism of superconductivity for fullerides.

##  IV EXPERIMENTAL DETAILS

Sample synthesis. High-quality K-doped C60 sample was synthesized by using a
modified version of a solution-phase reaction process detailed previously Yoo
; rswang ; zong . High-purity potassium (99%, Sinopharm Chemical Reagent) and
fullerene powder (99.9%, Acros Organics) were mixed with the nominal mole
ratio of 3:1 (K:C60). The mixtures, together with certain dose of ultra-dry
tetrahydrofuran (THF) solvent, were loaded into a 15 ml borosilicate vial. The
glass vial was then treated in an ultrasonic device at around 50 ∘C for 5-10
minutes to accelerate the dissolution of alkali metals and the reaction
between potassium and C60 molecules. The colour of the solution turned
reddish-brown after the ultrasound process. It should be noted that the
temperature of the reaction can not exceed 63 ∘C, which is the melting point
of potassium, otherwise potassium would melt into small spheres that are
difficult to react, thus breaking the desired stoichiometric ratio for
superconducting phase. After that, the solution was oscillated in a vortex
mixer (Vortex 3000, Wiggens) with 200 r/min for 10 hours to ensure the
complete reaction and the homogeneity of sample. Upon filtering, we obtained a
black preliminary product. All the above preparation steps were carried out in
an argon-filled glovebox with both O2 and H2O levels less than 0.1 ppm. The
as-prepared K3C60 was then put into quartz tubes and sealed under vacuum about
1$\times$10-4 Pa. The fragile K3C60 sample was obtained after annealing at 250
∘C for 20 hours.

Characterization of crystal structure. The crystal structure was determined
based on the X-ray diffraction spectrometer (Panalytical Emperean) by using Cu
Kα radiation with wavelength of 1.5406 Å. The polyimide film was used to cover
the sample to avoid the oxidation of K3C60. The background signals of the film
were carefully subtracted in the subsequent analysis. The space group and
lattice parameters were determined by using Jana 2020 program jana  based on
the Le Bail method bail  to fit the diffraction patterns.

Raman spectroscopy measurements. The Raman specta were collected in an in-
house system with Charge Coupled Device and Spectrometer from Princeton
Instruments. The laser with the wavelength of 488 nm and power less than 2 mW
was used in the measurements. Pristine and doped C60 were sealed in capillary
tubes when collecting the Raman spectra at room temperature [Fig. 1(c)]. For
cryogenic Raman spectroscopy experiment, the doped sample was loaded into a
sealed cooper holder with an optical quartz window. The sample was then put on
the holder equipped with a heater in a cryogenic vacuum chamber to obtain the
temperature-dependent Raman spectra (Fig. 4).

Magnetization measurements. The magnetization measurements were carried out by
using a Magnetic Properties Measurement System (MPMS3, Quantum Design). The
sample was placed into a nonmagnetic capsule and sealed by GE Varnish to
protect K3C60 from air. The $dc$ magnetic susceptibility $\chi(T)$ curves were
collected with the ZFC and FC runs at the field of 10 Oe at temperature
ranging from 1.8 to 100 K. In the $ac$ magnetic susceptibility measurements,
the used probe harmonic magnetic field and frequency are 5 Oe and 234 Hz,
respectively. The $M(H)$ plots at various fixed temperatures were collected by
two steps. Firstly, the step-size was set to 5 Oe in stable mode from 0 to 400
Oe. Secondly, the $H$ swept from +7 T to -7 T, and then went back to get the
complete hysteresis loop.

Pauli (spin) susceptibility $\chi_{spin}$. The measured magnetic
susceptibility $\chi_{meas}$ is determined by the sum of the paramagnetic
($\chi_{P}$) and diamagnetic ($\chi_{D}$) components. The $\chi_{D}$ component
includes the core and/or conduction electron (Landau) contributions, where the
former can be estimated based on the published Pascal constants, and the
latter usually takes 1/3 for the paramagnetic term. The value of $\chi_{D}$
for K3C60 can be calculated by $\chi_{D}$(K3C60)=60$\chi_{D}$(C
atom)+3$\chi_{D}$(K atom)+$\chi_{Landua}$(conduction electron), in which
$\chi_{D}$(C atom)=-6.0$\times$10-6 emu/mol and $\chi_{D}$(K
atom)=-18.5$\times$10-6 emu/mol. We thus obtain
$\chi_{D}$=-(5.0$\pm$0.3)$\times$10-4 emu/mol. By using
$\chi_{meas}$=(2.5$\pm$0.3$)\times$10-4 emu/mol at 20 K (Fig. 6), we have the
total bulk paramagnetic susceptibility $\chi_{P}$=(7.5$\pm$0.6)$\times$10-4
emu/mol, which can be taken as the Pauli paramagnetic susceptibility
$\chi_{spin}$.

Resistivity and Hall effect measurements. Due to the high sensitivity of the
sample to air, a nonmagnetic Ni-Cr-Al alloy cell equipped with four air-tight
copper leads was developed to measure the electrical and Hall resistivities
when keeping good contact between sample and four electrodes and avoiding
oxidation. The resistivity and Hall coefficient were determined in terms of
the van der Pauw method pauw . The sample thickness for the electrical
transport measurements is 0.2 mm. The low-field resistivity and Hall effect
measurements were performed on Physical Property Measurement System (Quantum
Design, PPMS). The resistivities at paulsed magnetic fields up to 50 T with
the help of a typical four-contact method were measured at the National High
Magnetic Field Center, Wuhan, China.

## Acknowledgements

This work was supported from the Shenzhen Science and Technology Program
(Grant No. KQTD20200820113045081), the Basic Research Program of Shenzhen
(Grant No. JCYJ20200109112810241), and the National Key R$\&$D Program of
China (Grant No. 2018YFA0305900).

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