Towards room-temperature superconductivity in low-dimensional C60 nanoarrays: An ab initio study
Dogan Erbahar Dan Liu Savas Berber David Tomanek
















These two authors contributed equally.
Physics and Astronomy Department,
             Michigan State University,
             East Lansing, Michigan 48824, USA
Physics Department,
             Gebze Technical University,
             41400, Kocaeli, Turkey


These two authors contributed equally.
Physics and Astronomy Department,
             Michigan State University,
             East Lansing, Michigan 48824, USA


Physics Department,
             Gebze Technical University,
             41400, Kocaeli, Turkey


[E-mail: ]tomanek@msu.edu

Physics and Astronomy Department,
             Michigan State University,
             East Lansing, Michigan 48824, USA





We propose to raise the critical temperature T_c for
superconductivity in doped C_60 molecular crystals by
increasing the electronic density of states at the Fermi level
N(E_F) and thus the electron-phonon coupling constant in
low-dimensional C_60 nanoarrays. We consider both electron and
hole doping and present numerical results for N(E_F), which
increases with decreasing bandwidth of the partly filled h_u and
t_1u derived frontier bands with decreasing coordination
number of C_60. Whereas a significant increase of N(E_F)
occurs in 2D arrays of doped C_60 intercalated in-between
graphene layers, we propose that the highest T_c values
approaching room temperature may occur in bundles of nanotubes
filled by 1D arrays of externally doped C_60 or La@C_60,
or in diluted 3D crystals, where quasi-1D arrangements of C_60
form percolation paths.










81.05.ub, 
73.22.-f, 
          




74.70.Wz,  
71.15.Mb 
         

 




Towards room-temperature superconductivity
       in low-dimensional C_60 nanoarrays:

       An ab initio study
    David Tománek
    December 30, 2023
================================================================================================================


























The quest for room-temperature superconductivity has lost nothing
of its appeal during the 30-year long intense search following the
observation of superconductivity in cuprate perovskites, with the
critical temperature T_c rising from the 30-K range in the
La-Ba-Cu-O system <cit.> to 77 K in
Y-Ba-Cu-O <cit.>. Current record T_c values of


133 K in the doped HgBa_2Ca_2Cu_3O_8
perovskite <cit.> and 203 K in sulfur
hydride <cit.> have only been observed under high
pressure. Progress in raising T_c significantly further has
lagged behind expectations.

Whereas the microscopic origin of superconductivity is still being
speculated about in high-T_c compounds, the rather high T_c
values observed in doped solid C_60, possibly even exceeding
60 K <cit.>, result from strong electron-phonon
coupling caused by the dynamical Jahn-Teller effect on individual
fullerene molecules <cit.>. In alkali-doped
M_3C_60 molecular solids, T_c could be quantitatively
reproduced <cit.> using the McMillan
equation <cit.>. The key behind a substantial
electron-phonon coupling constant is one of its factors, namely a
high

density of states (DOS) at the Fermi level N(E_F), which depends
on the particular element M used to intercalate bulk C_60.


Here we propose a way to further increase T_c for
superconductivity by increasing N(E_F) and thus the
electron-phonon coupling constant λ by reducing the
C_60 coordination number Z in doped low-dimensional C_60
nanoarrays. We considered intercalation by both electron donors
and acceptors, as well as electron doping in a solid formed of
La@C_60 endohedral complexes. We found that N(E_F) increases
with decreasing bandwidth of the partly filled h_u HOMO and
t_1u LUMO derived frontier bands, which may be achieved by
reducing the coordination number of C_60. Whereas N(E_F)
increases significantly by changing from 3D C_60 crystals to
2D arrays of doped fullerenes intercalated in-between graphene
layers, N(E_F) reaches its maximum in doped quasi-1D arrays of
C_60 molecules inside (10,10) carbon nanotubes (CNTs),
forming C_60@CNT peapods. Whereas partial filling of the
t_1u-derived band of C_60 may be achieved by adsorbing K
atoms on the peapod surface, the desired depopulation of the
h_u-derived band by adsorbed F is not possible. Our results
indicate that the highest T_c value approaching room temperature
may occur in electron-doped C_60 peapod arrays or in diluted
3D crystals, where quasi-1D arrangements of C_60 form
percolation paths.


























We performed density functional theory (DFT) calculations to
obtain insight into the effect of geometrical arrangement of
fullerenes on the electronic structure of C_60 intercalation
compounds. We used the Perdew-Zunger <cit.> form of the
spin-polarized exchange-correlation functional in the local
density approximation to DFT, as implemented in the
SIESTA code <cit.>, which correctly
reproduces the inter-layer spacing and interaction in graphitic
structures. The valence electrons were described by
norm-conserving Troullier-Martins
pseudopotentials <cit.> with partial core corrections
in the Kleinman-Bylander factorized form <cit.>. We
used a double-zeta polarized basis and limited the range of the
localized orbitals in such a way that the energy shift caused by
their spatial confinement was no more than
10 meV <cit.>. The Brillouin zone of a 3D lattice of
C_60 molecules was sampled by 10×10×10
k-points, that of a 2D lattice by 10×10 k-points, and
that of decoupled 1D chains of C_60 molecules inside a
nanotube by 10 k-points. The DOS was convoluted by
0.02 eV^-1. In a periodic arrangement, 1D structures were
separated by 15 Å thick vacuum regions and 2D structures by
13 Å thick vacuum regions. The charge density and the
potentials were determined on a real-space grid with a mesh cutoff
energy of 180 Ry, which was sufficient to achieve a total energy
convergence of better than 2 meV/atom.




















































































In alkali-doped M_3C_60 (M=K, Rb, Cs) fcc crystals,
superconductivity with T_c≲40 K has been
observed <cit.> and explained by electron-phonon
coupling that is modulated by the lattice
constant <cit.>. The same behavior is expected to
occur in the isoelectronic La@C_60 that has been isolated from
raw soot <cit.> and found to be
stable <cit.>. When exohedrally doped M_3C_60 crystals
are exposed to ambient or harsh conditions, atoms from the
environment

may penetrate deep inside the lattice, react with the M atoms and
destroy superconductivity. This is much less likely to occur in
endohedrally doped La@C_60 crystals, since the dopant La atoms
are enclosed inside the protective C_60 cage. As mentioned
above, superconductivity in 3D M_3C_60 crystals is caused by
strong electron-phonon coupling related to a dynamical Jahn-Teller
effect on individual C_60 cages, made possible by retardation.
The dominant role of the intercalated alkali atoms is to partly
fill the t_1u LUMO of C_60 that broadens to a narrow band
in the M_3C_60 molecular solid. Changes in T_c caused by
pressure or changing the element M can be traced back to changes
in the electron-phonon coupling constant λ=VN(E_F) in the
McMillan equation <cit.>. Since the
on-ball Bardeen-Pines interaction V does not change, λ
is proportional to the C_60-projected DOS at the Fermi level
N(E_F), which – for electron doping – is roughly inversely
proportional to the width of the t_1u-derived band. In
hole-doped C_60, E_F is expected to be lowered into the
h_u-derived band with an even higher N(E_F) value, which may
be the cause of the high value T_c60 K that has been
reported earlier <cit.>.





All experimental strategies used so far to raise T_c have been
based on increasing the C_60-C_60 separation d_cc in a
3D fcc crystal, which would lower the width of the LUMO- and
HOMO-derived bands and thus increase N(E_F) in doped crystals.
Our approach is quite different <cit.>: we consider
increasing N(E_F) by reducing the number of C_60 nearest
neighbors. As seen in Fig. <ref>(a), this may be achieved
simply in a 3D crystal by mixing C_60 with clusters of similar
size that do not interact with C_60, such as BN fullerenes. In
this case, the lowered C_60 coordination number would decrease
the width of the t_1u and h_u derived bands and thus
increase N(E_F) in doped crystals. Other C_60 arrangements
with a lower Z include 2D arrays of C_60 that could possibly
be intercalated in graphite <cit.>, as
seen in Fig. <ref>(b), or 1D arrays of C_60 in
C_60@CNT
peapods <cit.>
shown in in Fig. <ref>(c). As seen in Fig. <ref>(d), the
width of the t_1u-derived band decreases both with increasing
the C_60-C_60 separation and with the reduction of
dimensionality that translates to the reduction of Z,

with 1D arrangements appearing optimal. Since superconductivity is
suppressed in truly 1D systems according to the Mermin-Wagner
theorem <cit.>, we consider bundles of weakly interacting
peapods instead of isolated 1D peapods. As we will show in the
following, the main role of the nanotube in these systems is to
provide a suitable enclosure that aligns C_60 molecules and
protects them from the ambient. Due to their weak interaction,
bundles of nanotubes have a very similar DOS as isolated
nanotubes. Since the same applies to peapods, we will consider an
isolated peapod a valid representative of a peapod bundle from the
viewpoint of electronic structure.


Even in pristine systems with no intercalants, we found the
C_60-C_60 separation d_cc to depend on the C_60
orientation and the dimensionality of the system. Different
fullerene orientations, each with a specific optimum d_cc
value, were found to be energetically degenerate within
2 meV/atom and separated by minute activation barriers. At
nonzero temperatures during synthesis and observation, the
fullerenes will explore these orientational degrees of freedom
causing orientational disorder and changing d_cc, as observed
in the 3D lattice <cit.>. The equilibrium value of
d_cc decreases by ≈0.1 Å and its orientational
dependence increases when reducing the dimensionality to 2D and
1D. Also in view of the soft C_60-C_60 interaction, we
always expect a nonzero range of d_cc values in any
experimental sample. In pristine C_60, we expect d_cc
values roughly covering the 9.8-10.1 Å range shown by the
dark band in Fig. <ref>(d).



As seen in Fig. <ref>(e), the geometry is more complex in
alkali intercalated peapods, where intercalant atoms may occupy
sites outside the nanotube, inside the nanotube but outside
C_60, or inside the C_60 molecule such as the La@C_60
metallofullerene <cit.>. Since also these sites are
energetically near-degenerate, the precise geometry may be barely
controllable during synthesis. In the 3D M_3C_60 system,
moreover, d_cc has been found to increase from 9.8 Å to
10.3 Å with increasing atomic number of the alkali element
M <cit.>. We find a similar M-dependent increase in d_cc
also in 2D and 1D systems, where M atoms separate fullerenes. 1D
peapods with the narrowest bandwidth and potentially highest
N(E_F) could be doped by donor or acceptor atoms.



For most of this study, we will focus on donor doping, causing
partial filling of the t_1u-derived band, and will show later
that acceptor doping may be hard to achieve.


The DOS shape of the t_1u LUMO-derived band in quasi-1D, 2D
triangular and 3D fcc lattices of C_60 is depicted in
Fig. <ref>(a) and that of the h_u HOMO-derived band in the
same lattices is shown in Fig. <ref>(d). Clearly, the DOS at
E_F reaches its maximum near half-filling of these bands in
quasi-1D structures. Since the t_1u LUMO-derived band holds up
to 6 electrons and the h_u HOMO-derived band up to 10 electrons,
half-filling of these bands requires either 3 extra electrons

or depletion of 5 electrons from each C_60.

Comparing our results in Figs. <ref>(a) and <ref>(d), we
note that acceptor doping – if achievable – would result in a
significantly higher N(E_F) than donor doping.

The calculated DOS of a C_60@(10,10) peapod doped externally
by 7 K atoms per C_60 is shown in Fig. <ref>(b) and the
DOS of the corresponding peapod doped externally by 8 F atoms per
C_60 is shown in Fig. <ref>(e). Comparing the partial
densities of states in these two cases, we conclude that
C_60-derived states are barely affected by those of the
surrounding nanotube due to a very small hybridization. In the
case of donor doping by K depicted in Fig. <ref>(b), we
clearly observe partial filling of the t_1u-derived band of
C_60 as well as the nearly-free electron bands of the
(10,10) nanotube <cit.>. To get a better feel
for the charge flow in the system, we plotted the charge density
difference defined by
Δρ=ρ_tot(C_60@CNT+7K)-ρ_tot(C_60@CNT)
-∑_atρ_tot(K atom) in Fig. <ref>(c). Obviously,
there is a net electron flow from K atoms to the C_60@(10,10)
peapod, with the excess charge accommodated both by the C_60
and the nanotube. Integration of the C_60-projected DOS in
Fig. <ref>(b) up to E_F indicates a partial population of
the t_1u-derived band by 0.4 electrons. 






The calculated DOS of an acceptor-doped peapod, shown in
Fig. <ref>(e), presents a very different picture. We selected
F as a suitable electron acceptor due to its high
electronegativity. Unlike in previous studies of acceptor-doped
C_60, where covalently bonded halogen atoms disrupted the
π-electron network on the molecules <cit.>, F atoms
were bonded on the outside of the nanotube surrounding C_60
molecules. Thus,

we found the h_u-derived band of C_60 to be essentially
unaffected by the presence of the surrounding nanotube and the 8 F
atoms per C_60 outside the nanotube, but the C_60
molecules remained charge neutral. The h_u-derived narrow band
band remained completely filled, located about 1 eV below E_F.
We found F atoms to bind covalently to the outside of the
nanotube, causing pyramidalization, disrupting its π-electron
network and opening up a gap at the Fermi level, which turned the
system into a semiconductor. This can be clearly seen when
inspecting the charge flow in this system in Fig. <ref>(f).
We found F atoms to strongly hybridize with the C atoms of the
tube, redistributing the charge only within the F/CNT subsystem,
with no effect on the net charge of C_60. Since hole doping of
C_60 appears very difficult, we will focus on electron doping
of the t_1u-derived band of C_60 chains in the following.





As mentioned earlier, the electronic band structure of C_60
arrays should depend to a nontrivial degree on the orientation of
the C_60 molecules that will affect their
interaction <cit.>.

We studied 5 different orientations, identified in
Fig. <ref>(a), which result in a different degree of
inter-ball hybridization.


Due to their energetic near-degeneracy, we expect many C_60
orientations to coexist within a quasi-1D C_60 array inside a
peapod. The DOS for a chain of C_60 molecules at different
orientations and the C_60-C_60 separation
d_cc=9.8 Å is shown in Fig. <ref>(b). We note that
the maximum DOS value changes significantly with orientation.
Therefore, in Fig. <ref>(c), we plotted the range of
achievable maxima of N(E_F) as `error bars' for different
C_60-C_60 orientations.


Depending on the exact position of the intercalant atoms and
fullerene orientation, we found the optimum C_60-C_60
separations to cover the range d_cc≈9.8-10.3 Å,
indicated by the gray strip in Fig. <ref>(c). Higher d_cc
values than in pristine peapods, achieved in case that the
fullerenes are separated by heavy alkalis such as Cs, result in
very high values of N(E_F) for favorable C_60 orientations.






To estimate the critical temperature for superconductivity T_c,
we used McMillan's equation <cit.> 


    T_c = ħω_log/1.2 k_B
    exp[ -1.04 (1+λ)/λ-μ^*-0.62λμ^*]  .


This equation describes the solution of the Eliashberg equations
in superconductors with a strong electron-phonon coupling
(λ2) in a semi-empirical way that is physically
appealing. It also has been found to correctly reproduce the
observed T_c values in M_3C_60 solids as a function of the
C_60-C_60 separation <cit.>, shown by the data
points for 3D systems in Fig. <ref>(d). We used the
parameters of Ref. [DT057], namely
ħω_log/k_B=2800 K, μ^*=0.2 for the effective
mass and V=52 meV for the Bardeen-Pines interaction, which are
not affected by the averaged local arrangement of C_60
molecules. Using λ=VN(E_F) for the electron-phonon
coupling constant, we were able to convert N(E_F) values for 3D,
2D and quasi-1D systems with different C_60-C_60 distances
and C_60 orientations to potentially achievable T_c values
and present our results in Fig. <ref>(d).
Since dynamical orientational disorder and resulting fluctuations
in the C_60-C_60 distance are a natural phenomenon that is
particularly prominent in realistic 1D systems, we can estimate
ranges of d_cc and T_c values at best. 
Our estimates indicate that, in the best imaginable scenario,
T_c near room temperature may be achievable using bundles of
donor-doped peapods.

Clearly, there are limits to the range of C_60-C_60
separations d_cc compatible with superconductivity. Increasing
d_cc decreases the inter-ball hopping integral t, while not
affecting the on-ball Coulomb integral U. At large
C_60-C_60 separations, the U/t ratio should increase
beyond a critical value that would change doped C_60 from a
metal to a Mott-Hubbard
insulator <cit.>.

In summary, we have proposed a viable way to further increase
T_c for superconductivity by increasing the C_60-projected
density of states (DOS) at the Fermi level N(E_F) and thus the
electron-phonon coupling constant in doped low-dimensional
C_60 nanoarrays. We considered intercalation by both electron
donors and acceptors, as well as electron doping in a solid formed
of La@C_60 endohedral complexes. We found that N(E_F)
increases with decreasing bandwidth of the partly filled h_u
HOMO- and t_1u LUMO-derived frontier bands, which may be
achieved by reducing the coordination number of C_60. N(E_F)
increases significantly by changing from 3D C_60 crystals to
2D arrays of doped fullerenes intercalated in-between graphene
layers and reaches its maximum in doped quasi-1D arrays of
C_60 molecules inside C_60@CNT peapods formed of (10,10)
CNTs. Whereas partial filling of the t_1u-derived band may be
achieved by adsorbing alkali atoms outside the 1D peapod, the
desired depopulation of the h_u-derived band could not be
achieved by F atoms adsorbed on the nanotube surrounding the
C_60 molecules. Our results indicate that the highest T_c
values may occur in electron-doped C_60 peapods containing Cs
or in dilute 3D crystals, where quasi-1D arrangements of C_60
form percolation paths. Only experimental evidence will show if
low-dimensional arrays of doped C_60 will become
superconducting with T_c approaching room temperature, or rather
turn to a Mott-Hubbard insulator.




§ ACKNOWLEDGMENTS



D.E. acknowledges the hospitality of MSU, where this research was
performed. D.L. and D.T. acknowledges financial support by the
NSF/AFOSR EFRI 2-DARE grant number EFMA-1433459. Computational
resources have been provided by the 
Michigan State University High Performance Computing Center.























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