#  Ab initio study on magnetism suppression, anharmonicity, rattling mode and
superconductivity in Sc${}_{6}M$Te2 ($M$=Fe, Co, Ni)

Ming-Chun Jiang1,2<EMAIL_ADDRESS>Ryota Masuki3 Guang-Yu Guo1,4
Ryotaro Arita2,5 1Department of Physics and Center for Theoretical Physics,
National Taiwan University, Taipei 10617, Taiwan  2RIKEN Center for Emergent
Matter Science, 2-1 Hirosawa, Wako 351-0198, Japan  3Department of Applied
Physics, University of Tokyo, Tokyo 113-8656, Japan  4Physics Division,
National Center for Theoretical Sciences, Taipei 10617, Taiwan  5Research
Center for Advanced Science and Technology, University of Tokyo, 4-6-1 Meguro-
ku, Tokyo, 153-8904, Japan

###### Abstract

We perform a systematic ab initio study on phonon-mediated superconductivity
in the transition-metal-based superconductors Sc${}_{6}M$Te2 ($M$ = Fe, Co,
Ni). Firstly, our charge analysis reveals significant electron transfer from
Sc to $M$ due to the substantial difference in the electronegativity, filling
the 3$d$ orbitals of $M$ and suppressing magnetic instability. Secondly, we
show that Sc6FeTe2 exhibits strong lattice anharmonicity. Moreover, for $M=$
Fe and Co, we find low-frequency soft phonon bands of $M$ which can be
interpreted as “rattling phonons” in the framework formed by Sc. While not
observed in the case of $M=$ Ni, the rattling phonons give rise to a prominent
peak or plateau in the Eliashberg spectral function and enhance the pairing
instability. By reproducing the experimental trend of superconducting
transition temperatures, our study underscores the potential of designing
phonon-mediated superconductors by strategically combining non-superconducting
and magnetic transition-metal elements.

##  I INTRODUCTION

Identifying common characteristics among known superconductors bears
significance for designing novel superconductors [1]. Despite the abundance of
identified superconductors, platforms remain scarce, where superconductivity
manifests across a series of materials. This is particularly true in compounds
involving $d$ electrons [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Recently, seven
materials from a family of $d$-element-rich compounds, Sc${}_{6}M$Te2, have
been identified as possessing bulk superconductivity [12]. These materials
have the Zr6CoAl2-type structure, with $M$ encompassing 3$d$, 4$d$, and 5$d$
elements. Notably, the compounds with $M=$ Fe, Co, and Ni show a significant
material dependence in their superconducting transition temperatures
($T_{c}$’s) of 4.7 K, 3.6 K, and 2.7 K, respectively. Interestingly, the
pairing mechanism is yet to be fully understood: In particular, it is unclear
why superconductivity occurs in compounds containing magnetic elements and why
$T_{c}$ is highest with $M=$ Fe and lowest with $M=$ Ni.

To address these questions, we perform a systematic ab initio study on the
electronic structures, phonon properties, and the electron-phonon coupling
(EPC) for Sc${}_{6}M$Te2 ($M$ = Fe, Co, Ni). We first perform the charge
analysis based on the electronic structure, which shows an electron transfer
from Sc to $M$. As a result, the $3d$ orbitals of $M$ are almost fully
occupied, which suppresses the local magnetic moments on $M$. By reproducing
the chemical trend of $T_{c}$, we conclude that the target materials are
phonon-mediated superconductors. In addition, we show that the higher $T_{c}$
in Sc6FeTe2 and Sc6CoTe2 can be ascribed to the strong EPC from the low-
frequency phonon modes dominated by the displacements of $M$.

The crystal structure of Sc${}_{6}M$Te2 is characterized by the Sc cages
encapsulating the isolated $M$ atoms, indicating the system to be
host(Sc)-guest($M$) compounds. Due to such structural arrangement and the
emergence of low-frequency phonons with single-atom vibrations, we would
interpret such phonons as “rattling phonons”. This phenomenon is often
observed in materials where large freedom of motion is given to guest atoms
within periodic cages with examples of skutterudites [13, 14], pyrochlores
[15, 16, 17], clathrates [13], or brownmillerites [18].

Non-parabolic behavior of the potential energy surface due to atomic
displacements, namely the lattice anharmonicity, can occur for a rattling
vibrational behavior [19]. While the effect of anharmonicity on
superconductivity has been extensively studied [20, 21, 22, 23, 24], the
combining effect of rattling modes and anharmonicity on superconductivity was
discussed for the $\beta$-pyrochlore oxides $A$Os2O6 [25] and intermetallic
cage compounds $M$V2Al20 [26]. Since rattling modes provide significant atomic
displacement at the flat potential energy surface, the attractive interaction
for the Cooper pair can be large and thus enhance $T_{c}$ [25, 26]. Although
there are some cases where the rattling modes have only a moderate impact on
superconductivity [27], we find that taking advantage of the strong EPC of the
soft rattling phonons is an effective strategy to enhance superconductivity in
Sc${}_{6}M$Te2.

The rest of this paper is organized as follows. Secion II describes the
theoretical methods, the crystalline structure of Sc${}_{6}M$Te2, and the
computational details. Section III presents the calculated electronic
structures, including the orbital projected density of states (DOS), band
structures, and the Fermi surface. Then, we show the phonon band structures,
atom-projected phonon DOS, and the corresponding Eliashberg spectral function
to quantify the EPC strength. Further discussions on the suppression of
magnetism and the emerging superconductivity in Sc${}_{6}M$Te2 are provided in
Sec. IV. Finally, the conclusions drawn from this work are summarized in Sec.
V.

##  II Theory and computational methods

###  II.1 Electron phonon coupling

Based on density functional theory (DFT), we can use density functional
perturbation theory (DFPT) to calculate the EPC matrix element [28, 29, 30].

$\displaystyle g_{mn,\nu}(\textbf{k,q})$
$\displaystyle=\sum_{\kappa\alpha}\sqrt{\frac{\hbar}{2\omega_{\nu\textbf{q}}}}\frac{e_{\kappa\alpha\nu\textbf{q}}}{\sqrt{M_{\kappa}}}$
$\displaystyle\times\left\langle\psi_{m\textbf{k}+\textbf{q}}\left|\sum_{p}\frac{\partial
V}{\partial
u_{\kappa\alpha\textbf{R}_{p}}}\frac{e^{i\textbf{q}\cdot\textbf{R}_{p}}}{\sqrt{N_{p}}}\right|\psi_{n\textbf{k}}\right\rangle,$
(1)

where $\omega_{\nu\textbf{q}}$ is the phonon frequency of mode $\nu$ and
crystal momentum q; $e_{\kappa\alpha\nu\textbf{q}}$ is the polarization vector
of site $\kappa$ in the Cartesian direction $\alpha$; $M_{\kappa}$ is the mass
of the atom at site $\kappa$. $\psi_{n\textbf{k}}$ is the Kohn-Sham orbital
with the band index $n$ and wave vector k. $V$ is the self-consistent
potential, and its first-order derivative with respect to
$u_{\kappa\alpha\textbf{R}_{p}}$ is calculated using DFPT [30, 29, 28]. Note
that $u_{\kappa\alpha\textbf{R}_{p}}$ is the atomic displacement in the
lattice position $\textbf{R}_{p}$ of the $p$-th unit cell. $N_{p}$ is the
number of primitive cells in the Born-von Kármán supercell.

The EPC strength of a specific phonon mode $\nu$ and phonon momentum q can
then be obtained via the Brillouin zone (BZ) integration [28, 29, 31]:

$\displaystyle\lambda_{\nu\textbf{q}}=\frac{1}{N(\epsilon_{\text{F}})\omega_{\nu\textbf{q}}}\sum_{nm}$
$\displaystyle\int_{\rm BZ}\frac{d\textbf{k}}{\Omega_{\rm
BZ}}\left|g_{mn,\nu}(\textbf{k,q})\right|^{2}$
$\displaystyle\times\delta(\epsilon_{n\textbf{k}}-\epsilon_{\text{F}})\delta(\epsilon_{m\textbf{k+q}}-\epsilon_{\text{F}}),$
(2)

where $N(\epsilon_{\text{F}})$ is the electronic DOS at the Fermi level
$\epsilon_{\text{F}}$ and $\epsilon_{n\textbf{k}}$ is the state energy at
momentum k. $\Omega_{\rm BZ}$ is the volume of the BZ. We then introduce the
isotropic Eliashberg spectral function [28, 29, 31]

$\alpha^{2}F(\omega)=\frac{1}{2}\sum_{\nu}\int_{\rm
BZ}\frac{d\textbf{q}}{\Omega_{\rm
BZ}}\omega_{\nu\textbf{q}}\lambda_{\nu\textbf{q}}\delta(\omega-\omega_{\nu\textbf{q}}),$
(3)

from which the EPC constant $\lambda$ is obtained as

$\lambda=2\int\frac{\alpha^{2}F(\omega)}{\omega}d\omega.$ (4)

Through the celebrated McMillan-Allen-Dynes (MAD) formula [32, 33, 34], we can
evaluate the superconducting temperature $T_{c}$ by

$T_{c}=\frac{\omega_{\log}}{1.2}\exp\left[-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\right],$
(5)

in which the prefactor includes the logarithmic average phonon frequency of
the spectrum $\omega_{\log}$ where

$\omega_{\log}=\exp\left[\frac{2}{\lambda}\int\frac{d\omega}{\omega}\alpha^{2}F(\omega)\log{(\omega)}\right]$
(6)

and $\mu^{*}$ is the pseudo-Coulomb parameter introduced to account for the
screened electron-electron interaction.

###  II.2 Self-consistent phonon theory

The self-consistent phonon (SCPH) theory [35, 36, 37] is a mean-field theory
of the lattice anharmonicity, in which we solve a self-consistent equation in
terms of the renormalized effective frequencies $\Omega_{\nu\textbf{q}}$.

$\Omega^{2}_{\nu\textbf{q}}=\omega_{\nu\textbf{q}}^{2}+\frac{1}{2}\sum_{\nu_{1}\textbf{q}_{1}}F_{\textbf{q}\textbf{q}_{1},\nu\nu_{1}}\frac{\hbar[1+2n_{B}(\Omega_{\nu_{1}\textbf{q}_{1}})]}{2\Omega_{\nu_{1}\textbf{q}_{1}}},$
(7)

where $n_{B}(\omega)$ is the Bose-Einstein distribution function.
$F_{\textbf{q}\textbf{q}_{1},\nu\nu_{1}}=\Phi(\textbf{q}\nu;-\textbf{q}\nu;\textbf{q}_{1}\nu_{1};-\textbf{q}_{1}\nu_{1})$
is the fourth-order interatomic force constant (IFC) which is defined in
reciprocal representation as [36]:

$\noindent\Phi(\textbf{q}_{1}\nu_{1};\cdots;\textbf{q}_{4}\nu_{4})=\\\
N_{p}^{-1}\sum_{\kappa\alpha}(M_{\kappa_{1}}\cdots
M_{\kappa_{4}})^{-\frac{1}{2}}e_{\kappa_{1}\alpha_{1}\nu\textbf{q}_{1}}\cdots
e_{\kappa_{4}\alpha_{4}\nu\textbf{q}_{4}}\\\
\times\sum_{p_{2},\cdots,p_{4}}\Phi_{\alpha_{1}\cdots\alpha_{4}}(0\kappa_{1};\cdots;p_{4}\kappa_{4})e^{i(\textbf{q}_{2}\cdot\textbf{R}_{p_{2}}+\cdots+\textbf{q}_{4}\cdot\textbf{R}_{p_{4}})}.$
(8)

where
$\Phi_{\alpha_{1}\cdots\alpha_{4}}(p_{1}\kappa_{1};\cdots;p_{4}\kappa_{4})$ is
the fourth-order derivative, or the quartic term of the Taylor expansion
coefficient, of the Born-Oppenheimer potential energy surface $U$ with respect
to displacement $u_{\kappa\alpha\textbf{R}_{p}}$:

$\Phi_{\alpha_{1}\cdots\alpha_{4}}(p_{1}\kappa_{1};\cdots;p_{4}\kappa_{4})=\left.\frac{\partial^{4}U}{\partial
u_{\kappa_{1}\alpha_{1}\textbf{R}_{1}}\cdots\partial
u_{\kappa_{4}\alpha_{4}\textbf{R}_{4}}}\right|_{u=0}.$ (9)

Equation (7) can be derived using the Dyson equation with a frequency-
independent self-energy [36] or using the variational principle of the free
energy [37]. Note that the polarization vectors are assumed not to change by
the anharmonic effect in this work.

As we discuss later in Section III.2, we obtain imaginary frequency
($\omega_{\nu\textbf{q}}^{2}<0$) phonons in the harmonic phonon calculation
for Sc6FeTe2. This is inconsistent with the experiment because the imaginary
modes indicate the instability of the identified crystal structure. Thus, we
use the SCPH theory to obtain a stable phonon dispersion with real
frequencies. Also, equations (II.1) and (3) indicate that imaginary frequency
phonons would give an ill-defined EPC matrix element and Eliashberg spectral
function, which should be handled carefully for phonon-mediated
superconductors [21, 22, 23, 24, 38, 39]. While one may artificially set the
imaginary phonon frequencies to zero or their absolute values or introduce a
sizeable electronic smearing parameter [40, 41], in this work, we employ a
treatment that uses the SCPH theory. Namely, we obtain the SCPH frequencies
$\Omega_{\nu\textbf{q}}$ and replace the harmonic frequencies
$\omega_{\nu\textbf{q}}$ [42, 43]. Note that using the SCPH theory is not
entirely valid because it considers the finite-temperature renormalization of
the phonon frequencies that originate from the anharmonic effect. However, we
expect that this treatment is helpful because the SCPH frequencies reflect the
overall curvature of the low-energy region of the potential energy surface,
while the harmonic calculation is determined only by the curvature at the
reference structure. Note that we carefully choose the temperature to estimate
$n_{B}(\Omega_{\mu\textbf{q}}$) in the SCPH theory ($T_{\rm SCPH}$) and check
that the $T_{\rm SCPH}$-dependence of the calculated $T_{c}$ is relatively
small, which is further discussed in Appendix B.

###  II.3 CRYSTAL STRUCTURE AND COMPUTATIONAL DETAILS

Figure 1: Hexagonal crystal structure with space group $P\bar{6}2m$ (No. 189)
and Brillouin zone of Sc${}_{6}M$Te2 ($M$= Fe, Co, Ni). (a) Side view and (b)
top view of the crystal structure where one unit cell contains one formula
unit. (c) The hexagonal Brillouin zone.

The crystal structure of Sc${}_{6}M$Te2 ($M$= Fe, Co, Ni) is hexagonal with
the $P\bar{6}2m$ space group (No. 189) which contains one formula unit (f.u.)
per unit cell. The Sc${}_{6}M$Te2 compounds are an ordered variant of the Fe2P
structure, belonging to a large family characterized by the Zr6CoAl2-type
structure [44, 45]. The transition metal elements $M$ are surrounded by the Sc
atoms forming tricapped trigonal prisms [44]. Such a local Sc cage with $M$
bounded inside stack up along the $c$-direction, which forms a linear chain of
$M$. The structures are shown in Fig. 1. In the current study, we perform full
structural relaxation with the results summarized in Table 1.

Table 1: Theoretical lattice constants ($a$, $c$, $c$/$a$) of Sc${}_{6}M$Te2
compared with the experimental data [44]. The number in the parenthesis
denotes the percentage deviation of the theoretical lattice constants from the
experiments.

Sc${}_{6}M$Te2 |  $a$ (Å) |  $c$ (Å) |  $c$/$a$
---|---|---|---
Fe | 7.703 (+0.306%) | 3.788 (-1.272%) | 0.492
| 7.6795a | 3.8368a | 0.4996
Co | 7.732 (+0.446%) | 3.743 (-1.123%) | 0.484
| 7.6977a | 3.7855a | 0.4918
Ni | 7.768 (+0.576%) | 3.729 (-0.972%) | 0.480
| 7.7235a | 3.7656a | 0.4876

aExperimental data taken from Ref. [44]

The structural relaxation and the calculations of electronic and phonon
properties are conducted using DFT, and the corresponding electron-phonon
properties are calculated using DFPT [30], as implemented in the QUANTUM
ESPRESSO (QE) code [46]. All the calculations are performed using the scalar-
relativistic projector-augmented wave (PAW) [47] pseudopotential with the
generalized gradient approximation (GGA) in the form of Perdew-Burke-Ernzerhof
(PBE) [48] from PSLibrary [49]. A plane-wave cutoff value of 952 eV and a
$\Gamma$-centered 12 × 12 × 24 Monkhorst-Pack [50] $k$-mesh was used to
describe the electronic structure. For the DOS calculation, we apply the
tetrahedron method [51]. The valence orbital set is
$3s^{2}3p^{6}4s^{2}3d^{1,7,8,9}$ for Sc, Fe, Co, Ni, and $5s^{2}5p^{4}$ for
Te. The phonon calculations are performed with a $q$-grid of 2 $\times$ 2
$\times$ 2\. Also, the crystal structure is visualized by VESTA [52], and the
Fermi surface is plotted by FermiSurfer [53].

We perform the SCPH calculation for Sc6FeTe2. The IFCs are generated using
Vienna ab initio simulation package (VASP) [54, 55] and ALAMODE package [36,
37, 56], where we use a 2 $\times$ 2 $\times$ 2 supercell containing 72 atoms.
The VASP calculations for the force data are performed using DFT with the PAW
method [47] and the GGA in the form of PBE [48, 57]. A plane wave cut-off
energy of 520 eV is used. For the BZ integration, a $\Gamma$-centered $k$-mesh
of 4 $\times$ 4 $\times$ 4 is used for the supercell. The harmonic IFCs are
calculated by the frozen phonon method with atomic displacements of 0.01 Å.
The anharmonic IFCs are computed using the compressive sensing method, which
enables efficient calculation of IFCs from a small number of supercell
calculations [58]. To generate the displacement-force data from which we
extract the IFCs, we generate configurations with random atomic displacements
using ab initio molecular dynamics (AIMD) simulation. To sample the low-energy
region of the potential surface, we first perform the AIMD in VASP at 300K for
5000 steps with a time step of 1 fs. We discard the first 1000 steps due to
their dependence on the initial setting of the AIMD simulation and extract 80
snapshots from the rest of the trajectory. We further displace the atoms
randomly by 0.04 Å to eliminate the correlation between the snapshots. DFT
calculations of VASP compute the atomic force for each supercell, and up to
the fourth-order IFCs are calculated to fit the displacement-force relation
[37]. Note that the pseudopotentials used in both QE and VASP calculations are
selected based on the best agreement between the results of the phonon
dispersion under the harmonic approximation, and the results are given in Fig.
9 in Appendix A. Based on such a foundation, we can replace the dynamical
matrix data in DFPT, which is used in QE, with the one from SCPH, which is
generated by VASP plus ALAMODE.

Lastly, $T_{c}$ is evaluated using the MAD formula [Eq.(5)], as explained in
the previous section. The pseudo-Coulomb parameter $\mu^{*}$ is set to 0.1 to
account for the screened Coulomb interaction.

##  III RESULTS

###  III.1 Electronic Structures

Figure 2: Total and orbital-decomposed density of states (DOS) of (a)
Sc6FeTe2, (b) Sc6CoTe2, and (c) Sc6NiTe2. The Fermi level is set to zero, as
denoted by the vertical dashed line.  Figure 3: Electronic band structures
around the Fermi level (0 eV) of (a) Sc6FeTe2, (b) Sc6CoTe2, and (c) Sc6NiTe2
color-coded with the $d$-orbital projections. Note that the projected density
of states for the $d$-orbitals not plotted contribute at least three times
less to the Fermi level than those that are plotted. The Fermi surface, color-
coded by band indices, and two sections along the $\Gamma$-M-K and A-L-H plane
for Sc${}_{6}M$Te2 where $M=$ (d)Fe, (e)Co, and (f)Ni.

We examine the electronic structures of Sc${}_{6}M$Te2. Figure 2 shows the
total and orbital-decomposed DOS for Sc${}_{6}M$Te2. We notice that there are
sharp peaks in the DOS around 1.5 eV to 2.2 eV below the Fermi level, which is
dominated by the $M$-3$d$ orbitals. This suggests that the $M$-3$d$ orbitals
are localized in space because the narrow width of the peaks and the weak
hybridization with other orbitals indicates weak hopping to the Sc atoms
around them. We see that these peaks gradually get narrower and deeper below
the Fermi level as we go from Fe toward Ni. This can be explained by the
difference in the electronegativity of Fe(1.83), Co(1.88), and Ni(1.91) [59]
as the electrons will be more localized and more strongly bound in atoms with
larger electronegativity.

Next, we discuss the electronic structures of Sc${}_{6}M$Te2 near the Fermi
level. In Fig. 2, we observe hybridized DOS between Sc1, Sc2, and $M$ within
the energy range from -1 to 1 eV. Notably, there is a peak structure in the
DOS at the Fermi level across all the materials, indicating large DOS at the
Fermi level. Large DOS at the Fermi level is beneficial for phonon-mediated
superconductivity [60].

In Figs. 3(a)-3(c), we show the detailed orbital-projected band structures and
the corresponding Fermi surface in Figs. 3(d)-3(f). In Figs. 3(a)-3(c), all
the band structures possess a hole pocket around the $\Gamma$ point. This hole
pocket at $\Gamma$ primarily comprises hybridized $M$-3$d_{xy,x^{2}-y^{2}}$
and Sc2-3$d_{z^{2}}$ orbitals. From Figs. 3(d)-3(f), we see that the hole
pocket forms a cylindrical Fermi surface around $\Gamma$, and the size of this
cylindrical patch decreases from $M=$ Fe to Ni, as clearly depicted in the
$\Gamma$-M-K section in Figs. 3(d)-3(f). We do not observe any Fermi surface
pockets around the M and K points. On the other hand, the projected band
structures along A-L-H changes through electron filling. For example, from
Figs. 3(a)(b), along A-L-H, we can see that for $M=$ Fe, the
$M$-3$d_{xz,yz}$and Sc2-3$d_{xz,yz}$ bands are not occupied while they are
occupied in the case of $M=$ Co. Moreover, the Fermi surface on the A-L-H
plane is more complicated than the hole pocket surrounding the $\Gamma$ point.
This originates from the less dispersive bands with more crossings across the
Fermi level compared to that of $\Gamma$-M-K plane, as we can see from Figs.
3(a)-(c). The same observation can also be made by looking at the A-L-H
section of the Fermi surface at Figs. 3(d)-3(f).

Table 2: Results of Bader charge analysis for Sc${}_{6}M$Te2 ($M$=Fe, Co, Ni). The closest ideal ionic state, as well as its oxidation state, is also given. The valence orbitals are $4s^{2}3d^{1,7,8,9}$ for Sc, Fe, Co, Ni, and $5s^{2}5p^{4}$ for Te. The Bader charge of each neutral atom is thus 3, 8, 9, 10, 6 for Sc, Fe, Co, Ni, and Te, respectively. Compound | Bader charge | Oxidation state
---|---|---
| (e-) | (e-)
Sc6FeTe2 |  |
Sc1 | 1.95 | 2 (Sc1+)
Sc2 | 1.88 | 2 (Sc1+)
Fe | 10.97 | 11 (Fe3-)
Te | 7.74 | 8 (Te2-)
Sc6CoTe2 |  |
Sc1 | 1.94 | 2 (Sc1+)
Sc2 | 1.91 | 2 (Sc1+)
Co | 11.92 | 12 (Co3-)
Te | 7.76 | 8 (Te2-)
Sc6NiTe2 |  |
Sc1 | 1.96 | 2 (Sc1+)
Sc2 | 1.94 | 2 (Sc1+)
Ni | 12.69 | 12 (Ni2-)
Te | 7.78 | 8 (Te2-)

Last, we discuss the absence of magnetism in Sc${}_{6}M$Te2 ($M=$ Fe, Co, and
Ni). In the experiment, no sign of magnetism has been reported in these
materials [44, 12], which is essential for the emergence of superconductivity
but quite nontrivial because they include magnetic elements $M=$ Fe, Co, and
Ni. In fact, no magnetic states can be stabilized for $M$= Co, Ni in our spin-
polarized self-consistent DFT calculations. For $M=$ Fe, we obtain both a
ferromagnetic and a nonmagnetic solution. Although the energy difference
between them is just $\sim$1 meV, the total energy of the nonmagnetic solution
is lower in our calculations. This result is consistent with the experimental
findings [12]. To investigate the underlying reason, we perform the Bader
charge analysis, which is a useful charge counting analysis done by
segregating atoms along the minimum in the electron density [61]. The Bader
charges, which are the counted valence electrons, and the corresponding
nearest oxidation state are given in Table 2. Interestingly, every Sc atom
donates roughly one electron to the neighboring $M$ and Te, transferring six
electrons. The pair of Te atoms would share three electrons and $M$ would take
the remaining three electrons. Consequently, the 3$d$ electron shell of Co and
Ni is fully occupied. This explains why the potential magnetism brought by Fe,
Co, and Ni does not show up in the Sc${}_{6}M$Te2 systems. Note that this
electron transfer occurs due to the difference in electronegativity where Sc,
Fe, Co, Ni, and Te are of value 1.36, 1.83, 1.88, and 1.91, 2.10, respectively
[59]. Crucially, this electron reallocation prevents the potentially
detrimental effect on superconductivity caused by the partially filled 3$d$
shell. Moreover, since the Fe 3$d$ shell is not fully filled, we expect a more
unstable nonmagnetic state than the rest, resonating with the absence of
stable ferromagnetic results in DFT self-consistent calculation for $M=$ Co
and Ni.

Figure 4: Phonon band structures, total and $M$-projected phonon density of
states (PhDOS) of (a)(b) Sc6FeTe2, (c)(d) Sc6CoTe2, and (e)(f) Sc6NiTe2. The
magenta curves in (a) and (b) are the anharmonicity-renormalized phonon bands.
The displacement pattern of the phonon bands indicated by markers “a”, “b”,
and “c” are given in Fig. 5. Also, the red arrows point to the localized
phonon bands arising from the Fe and Co atoms.

###  III.2 Phonon Properties

Figure 5: Displacement pattern of (a) the imaginary frequency phonon mode at
the M point of Sc6FeTe2 [marker “a” in Figure 4(a)] (b) the doubly-degenerated
phonon bands number 4 and 5 around 75 cm-1 at the $\Gamma$ point of Sc6FeTe2
[marker “b” in Figure 4(a)] and (c) Sc6CoTe2 [marker “c” in Figure 4(c)]. All
displacement modes shown here are in-plane.

Now, we discuss the phonon properties of the Sc${}_{6}M$Te2 systems. The
calculated phonon bands are given in Fig. 4(a)(c)(e). As shown in Fig. 4(a),
under the harmonic approximation, imaginary frequency phonon bands exist along
the $\Gamma$-M-K plane of Sc6FeTe2. We mark the imaginary frequency phonon
bands at the M point as “a” in Fig. 4(a). The displacement amplitude along the
polarization vector of such band is displayed in Fig. 5(a). We see that the
origin of such an imaginary frequency mode is the in-plane vibration of the Fe
atoms. On the other hand, Sc6CoTe2 and Sc6NiTe2 do not have imaginary
frequencies under the harmonic approximation, as shown in Fig. 4(c) and (e).
This suggests that Fe atoms in Sc6FeTe2 will show larger atomic displacements
in the in-plane direction and manifest stronger anharmonicity compared to
$M$=Co and Ni cases. Note that anharmonicity refers to the deviation of the
potential energy surface from the parabolic behavior near the reference
structure.

However, the instability is overestimated in the DFT calculation because no
structural phase transitions have been observed in the experiment [12].
Therefore, we use the SCPH theory to obtain a positive dispersion in a
physically reasonable way, as we discussed in Sec. II.2. The obtained SCPH
dispersion with $T_{\rm SCPH}$ = 20 K is displayed with magenta lines in Fig.
4(a). Comparing the harmonic and SCPH bands, the higher-frequency portion of
the phonon bands remains unchanged, while the imaginary phonon modes are
stabilized, resulting in dispersive phonon bands in the low-frequency region.

Figure 6: Phonon band structure color-coded with the relative size of
electron-phonon coupling strength $\lambda_{\nu\textbf{q}}$ for
Sc${}_{6}M$Te2. Orange lines represent the phonon bands. Note that the
thickness of the line is normalized to $\lambda_{\nu\textbf{q}}$ of each
material.

In Figs. 4(a)(c)(e), we can see common features across $M=$ Fe, Co, and Ni.
All spectral widths are roughly 250 cm-1. Also, the phonon band structures of
Sc${}_{6}M$Te2 are generally dispersed in the low-frequency region and flatter
in higher-frequency regions. For Sc6NiTe2, the phonon bands are more
dispersive than the others. The similarity of the phonon band structures can
also be well observed by the total phonon density of states (PhDOS), as given
in Figs. 4(b)(d)(f). We can see that the peak positions and magnitudes of the
total PhDOS are qualitatively similar for $M=$ Fe, Co, and Ni. Interestingly,
the difference lies in the $M$ atom-projected PhDOS. Compare Figs. 4(b)(d)(f),
as indicated by the red arrows, around 75 cm-1, only the renormalized phonon
bands of Sc6FeTe2 and the harmonic phonon bands of Sc6CoTe2 have a PhDOS peak.
This indicates the emergence of localized $M$ phonon modes for only Sc6FeTe2
and Sc6CoTe2. Thus, we take the phonon bands around 75 cm-1 at the $\Gamma$
point as representatives to investigate their vibration patterns. They are
degenerated phonon bands with index number 4 and 5 for both materials and we
mark them as “b” in Fig. 4(a) and “c” in Fig. 4(c). In Figs. 5(b)(c), we see
that these modes originate from in-plane vibrations of $M$. On the contrary,
the Ni PhDOS of Sc6NiTe2 is fairly uniform throughout the spectrum, as shown
in Fig. 4(f).

Significantly, we realize that such a difference in the distribution of the
$M$-projected PhDOS is related to the EPC strength. Figure 6 displays the
phonon bands of Sc${}_{6}M$Te2 with indication of the relative magnitude of
the EPC strength $\lambda_{\nu\textbf{q}}$ [Eq. (II.1)]. Note that a hot spot
of $\lambda_{\nu\textbf{q}}$ indicates a strong pairing interaction between
electrons at k and k+q mediated by the $\nu$ phonon mode . For $M=$ Fe and Co,
Sc${}_{6}M$Te2 has large $\lambda_{\nu\textbf{q}}$ in phonon modes with
frequencies of $\sim$25-40 cm-1 around M and K. For $M=$ Fe, as is obvious
from Fig. 4(a), the anharmonic effect is significant in this segment of the
phonon bands. The second hot spot lies around 75 cm-1. This segment highlights
the difference in the $M$ atom-projected PhDOS as observed through Figs.
4(b)(d)(f). If we look at Fig. 6 (c), we would see that
$\lambda_{\nu\textbf{q}}$ of Sc6NiTe2 is uniformly distributed with respect to
phonon momentum and frequency. From Fig. 6, we reveal the significance of
these low-frequency phonons in understanding the chemical trend of the EPC and
consequently $T_{c}$ for Sc${}_{6}M$Te2 ($M=$ Fe, Co, Ni). In Sec. IV A, we
will further discuss that we could interpret these low-frequency phonons that
generate strong EPC as “rattling” phonons.

###  III.3 Eliashberg function and Superconductivity

Figure 7: Atom-decomposed phonon density of states (Phonon DOS), as well as
the Eliashberg spectral functions $\alpha^{2}F(\omega)$ and the frequency-
resolved electron-phonon coupling strength $\lambda(\omega)$ of (a)(b)
Sc6FeTe2, (c)(d)Sc6CoTe2, and (e)(f) Sc6NiTe2. Note that
$\lambda(\omega)=2\int^{\omega}_{0}\frac{\alpha^{2}F(\omega^{\prime})}{\omega^{\prime}}d\omega^{\prime}$
and thus the electron-phonon coupling constant $\lambda$ is the value obtained
after integrating over the whole phonon spectrum. Table 3: Calculated
electron-phonon coupling constant $\lambda$, logarithmic average phonon
frequency $\omega_{\rm log}$, total [$N(\epsilon_{\text{F}})$] and
$M$-projected [$N_{M}(\epsilon_{\text{F}})$] density of states at the Fermi
level, and the superconducting transition temperature ($T_{c}$) of
Sc${}_{6}M$Te2 ($M$=Fe, Co, Ni). The last column includes the temperature
($T_{\rm SCPH}$) used in the SCPH theory. The smearing parameter is 0.02 eV,
and the pseudo-Coulomb parameter $\mu^{*}$ is 0.1.

System | $\lambda$ | $\omega_{\rm log}$ | $N(\epsilon_{\text{F}})$ | $N_{M}(\epsilon_{\text{F}})$ |  $T_{c}$ (Exp.a) | $T_{\rm SCPH}$
---|---|---|---|---|---|---
$M$ |  | (K) | (1/eV/f.u.) | (1/eV/$M$) | (K) | (K)
Fe | 1.78 | 72 | 9.04 | 1.16 | 9.4 (4.7) | 20
Fe | 1.67 | 79 | 9.04 | 1.16 | 9.9 (4.7) | 50
Fe | 1.46 | 95 | 9.04 | 1.16 | 10.5 (4.7) | 100
Co | 0.75 | 152 | 16.30 | 2.03 | 6.2 (3.5) | -
Ni | 0.46 | 187 | 12.46 | 0.79 | 1.6 (2.7) | -

aExperimental data taken from Ref. [12]

In Figs. 7(b), 7(d) and 7(f), the calculated Eliashberg spectral function
($\alpha^{2}F(\omega)$) is depicted alongside the PhDOS. $\alpha^{2}F(\omega)$
provides insight into the EPC strength ($\lambda$). Typically, peaks in
$\alpha^{2}F(\omega)$ align with those in PhDOS. However, a significant
$\alpha^{2}F(\omega)$ contribution to $\lambda$ emerges in the 60 to 80 cm-1
range for Sc6FeTe2 and Sc6CoTe2, which is absent in Sc6NiTe2. This distinction
can be attributed to the presence or absence of the $M$-dominated low-
frequency phonons discussed in the previous subsection. For Sc6FeTe2,
$\alpha^{2}F(\omega)$ manifests a wide plateau in 20 to 50 cm-1 frequency
range, yielding significant contributions to $\lambda$. At higher frequency
regions above 80 cm-1, the EPC contribution coincides with the peaks of PhDOS,
initially from the Te phonon bands followed by the Sc phonon bands. Figure 7
illustrates, firstly, the critical role of the $M$ atom in the EPC of
Sc${}_{6}M$Te2. Second, it shows a correlation between low-frequency
vibrations of $M$ and enhanced EPC. Finally, it demonstrates the dominance of
the anharmonicity-driven EPC for $M=$ Fe.

By integrating $\alpha^{2}F(\omega)/\omega$ over frequency, we obtain the EPC
constant $\lambda$ [Eq. (4)]. As shown in Table 3, the resultant $\lambda$
values are 1.78, 0.75, and 0.46 for $M=$ Fe, Co, and Ni, respectively.
Utilizing the calculated $\lambda$, we predict $T_{c}$ values using the MAD
formula [Eq. (5)] [32, 33, 34]. The pseudo-Coulomb interaction ($\mu^{*}$) is
treated as an empirical parameter and is set to 0.1. The predicted $T_{c}$
values for Sc${}_{6}M$Te2 are 9.4 K, 6.2 K, and 1.6 K for $M=$ Fe, Co, and Ni,
respectively. The calculated values of $T_{c}$ align with the observed
chemical trends [12], with $M=$ Fe ranking the highest and $M=$ Ni the lowest.
Also, the predicted values of $T_{c}$ are comparable to the measured values.
Thus, we have revealed that the Sc${}_{6}M$Te2 ($M=$ Fe, Co, Ni) are phonon-
mediated superconductors. Note that the calculated $\lambda$ and $T_{c}$ of
Sc6FeTe2 is little affected by $T_{\rm SCPH}$ as shown in Table 3. While we
use the calculation of $T_{\rm SCPH}=$ 20 K as the representative, the
discussion remains robustly for a wide range of $T_{\rm SCPH}$. See Appendix B
for further discussions on $T_{\rm SCPH}$ of Sc6FeTe2.

Figure 8: Rattling phonon modes of Sc6FeTe2 and Sc6CoTe2. Phonon band
structures in low-frequency regions are color-coded based on their relative
magnitude of atomic displacement for the rattler ($M$ atom) and the cage (Sc
atoms) for $M=$ Fe(a), Co(b), and Ni(c). Insets of each are the phonon band
structures in low-frequency regions color-coded with the relative size of the
electron-phonon coupling strength $\lambda_{\nu\textbf{q}}$. (d) Schematic of
the avoid crossing between the rattling mode and longitudinal acoustic mode as
well as the displacement pattern around the crossing for $M=$ Fe and Co.

Finally, we discuss the possible origins of the deviation between the
calculated and experimental $T_{c}$. One possibility lies in the influence of
spin fluctuation coming from the $d$-electrons in the system. Kawamura et al.
[62] carried out systematic ab initio calculations including both EPC and spin
fluctuation based on density functional theory for superconductors (SCDFT) for
all the elemental metals. Their SCDFT calculations showed that for some
elemental metals, including spin fluctuation could significantly reduce the
phonon-mediated $T_{c}$. In particular, they reported that elemental Sc metal
has an EPC constant $\lambda$ of 0.398 and a $T_{c}$ of 2.7 K when the spin
fluctuation is not considered. However, when the spin fluctuation is taken
into account, the superconductivity disappears, agreeing with the experiments.
We notice that the DOS at the Fermi level in hexagonal Sc is rather large,
being 2.01 states/eV/Sc atom [62], and thus, spin fluctuation would be quite
significant. Figure 2 shows that the projected DOS for Sc at the Fermi level
in Sc${}_{6}M$Te2 is also rather large, being 1.65, 2.06, and 1.38
states/eV/Sc atom, respectively, for $M=$ Fe, Co, and Ni. Since two-thirds of
atoms in Sc${}_{6}M$Te2 ($M$ = Fe, Co, Ni) are Sc, the spin fluctuation in
Sc${}_{6}M$Te2 may affect the predicted phonon-mediated $T_{c}$ in
Sc${}_{6}M$Te2. In particular, this perhaps explains why the calculated
$T_{c}$ values in $M=$ Fe and Co are higher than the experimental values
(Table III).

Also, such deviation of $T_{c}$ could arise from the isotropic gap function
that the MAD formula assumes. Given the complicated Fermi surface of
Sc${}_{6}M$Te2, it is an interesting future problem of how much the predicted
$T_{c}$ is affected if we go beyond the MAD formula.

In conclusion, our findings suggest that superconductivity in Sc${}_{6}M$Te2
($M=$ Fe, Co, Ni) is phonon-mediated, where low-frequency $M$ vibrations and
anharmonic phonon modes play a crucial role. Also, our study suggests by
utilizing the electronegativity difference, we can manipulate the electron
fillings of the $d$-orbitals to make the combination of a non-superconducting
elemental metal, Sc in the present case, and magnetic element become
superconducting. In general, this proposition could serve as a potential
explanation or direction of investigation for Sc-included systems, including
one recent report on the discovery of superconductivity of ScPdGe and ScPdSi
[63].

##  IV Discussions

In this section, we further extend the discussions toward the rattling phonons
and the magnetic suppression of Sc6MnTe2.

###  IV.1 Rattling phonons

From our results (Figs. 4, 6, and 7), we show that low-frequency phonons from
$M$ are the key factor to understand the trend in $T_{c}$ for Sc6MTe2. Also,
note that in Fig. 5, we learn that these phonons are mainly single-atom
vibrations. These results prompt us to interpret this key factor to be
“rattling phonons”, where atoms “rattle” alone in the system, producing low-
frequency vibrational modes and large EPC [64, 25, 26].

Here, we discuss the phonon dispersion of a system with rattling phonons,
which in our case are the $M=$ Fe and Co phonon modes. Christensen et al. [65]
showed by spring model calculations and inelastic neutron scattering
experiments that the hallmark of a flat rattling mode is its avoidance of
crossing with the longitudinal acoustic mode [65]. In Figs. 8(a)-8(c), we
present the low-frequency segment of the phonon band structures. Notably, in
both Sc6FeTe2 and Sc6CoTe2, the flat phonon band around 75 cm-1 predominantly
reflects the $M$ vibration mode. It persists over a wide range along the high
symmetry line, particularly the A-L-H plane. Furthermore, as circled in green
in Figs. 8(a)and(b), we can discern a gap between this flat phonon band and
the longitudinal acoustic mode along the $\Gamma$-M and $\Gamma$-A line,
displaying the hallmark of rattling modes [65]. Fig. 8(d) shows the schematic
of such avoided crossing and the displacement pattern of $M=$ Fe and Co around
the gap. We notice the displacement of the Sc cage to dominate along the
acoustic mode before and after the gap. Moreover, we notice the dominance of
the rattler displacements from Fe and Co to persist on the rattling phonon
band. On the contrary, from Fig. 8(c), we notice the absence of such a feature
in Sc6NiTe2. This is consistent with the PhDOS shown in Figs. 4. The insets of
Fig. 8(a)-(c) are the low-frequency segment of Fig. 6, we clearly see that the
EPC of Sc6FeTe2 and Sc6CoTe2 is highly concentrated in the region of the
rattling phonons. This suggests that Sc${}_{6}M$Te2 is a case to demonstrate
the EPC driven by these localized rattling modes that can be further tuned by
the degree of lattice anharmonicity.

###  IV.2 Magnetic suppression of Sc6MnTe2

Now let us inquire further into the absence of the magnetic moment on $M$ in
Sc${}_{6}M$Te2 [12]. As elaborated in the last part of Sec. III A, our Bader
charge analysis reveals an electron transfer from Sc to $M$ because Sc
displays the lowest electronegativity among the constituent atoms. This
electron transfer provides enough electrons to saturate the 3$d$-orbitals of
Fe, Co, and Ni, thus suppressing the formation of the magnetic moment on $M$.
In the experiments by Shinoda et al. [12], superconductivity is not observed
for $M=$ Mn. Since Mn has a much lower electronegativity of 1.55 compared with
that of Fe, Co, and Ni [59], the large electron transfer from Sc to $M$
mentioned above would be weaker; thus, Sc6MnTe2 may have a magnetic ground
state. This suggests that superconductivity in Sc6MnTe2 is perhaps suppressed
by magnetic pair breaking [66] due to the magnetic moments on the Mn atoms. To
verify this hypothesis, we further perform self-consistent spin-polarized
electronic structure calculations for Sc6MnTe2. Indeed, we find that Sc6MnTe2
has a stable ferromagnetic state with a local magnetic moment of 1.4 $\mu_{B}$
on Mn. The total energy of the ferromagnetic state is 25 meV/f.u. lower than
the nonmagnetic state. Moreover, we perform the Bader charge analysis toward
Sc6MnTe2 with the final results giving us 1.98, 10.01, and 7.71 e- for the Sc,
Mn, and Te atoms, respectively. The Bader charges correspond to the nearest
oxidation state of 2 e- (Sc1+), 10 e- (Mn${}^{3-}=$ Ni valence state
$4s^{2}3d^{8}$), and 8 e- (Te2-).

##  V CONCLUSIONS

In conclusion, we have carried out a systematic ab initio computational study
on the magnetic, electronic, phonon, and superconducting properties of newly
reported transition-metal-based superconductors Sc${}_{6}M$Te2 ($M$ = Fe, Co,
Ni) [12] by using theoretical methods of DFT, DFPT and SCPH. First, our DFT
calculations reveal that Sc${}_{6}M$Te2 ($M$ = Fe, Co, Ni) have nonmagnetic
metallic ground states with multiple Fermi surface pockets. The Bader charge
analysis indicates a significant electron transfer from Sc to $M$ (where $M$ =
Fe, Co, Ni) due to the substantial electronegativity difference. This charge
transfer fills the $M$ 3$d$ orbitals, thus suppressing the magnetism from the
$M$ atoms. Second, combining DFPT and SCPH, we obtain the renormalized phonon
bands for the anharmonic Sc6FeTe2 as we observe flat phonon bands near 75 cm-1
and dispersive low-frequency phonon bands emerging after considering the
anharmonic effect. Interestingly, these two emerging features are rattling
phonon bands holding large EPC at low-frequency regions in Sc${}_{6}M$Te2 when
$M=$ Fe and Co. Quantitatively, they also give rise to prominent peaks in the
calculated Eliashberg spectral function. These rattling mode-enhanced EPCs
lead to a rather high superconducting transition temperature ($T_{c}$) of
$\sim$10 K for $M=$ Fe. Our DFPT calculations also predict Sc${}_{6}M$Te2
($M=$ Co and Ni) to be phonon-mediated superconductors with lower $T_{c}$
values of 6.2 K and 1.6 K, respectively. The chemical trend of the predicted
$T_{c}$ for Sc${}_{6}M$Te2 from $M=$ Fe to Co and Ni agrees well with the
experimental observation [12]. Importantly, we attribute such a chemical trend
to the phonon bands where we observe the progression of increasing rattling
behavior from $M=$ Ni to Co and then the increasing anharmonicity from $M=$ Co
to Fe. In short, our ab initio study not only shows that the superconductivity
observed in 3$d$ transition metal compounds Sc${}_{6}M$Te2 ($M=$ Fe, Co, and
Ni) is phonon-mediated but also suggests a way to design transition-metal-
based superconductors by combining non-superconducting and magnetic metal
elements with different electronegativities.

## APPENDIX A: Comparison of the harmonic phonons using DFPT and the frozen
phonon method

Figure 9: Phonon band structures calculated under the harmonic approximation
using (a) DFPT in the QE package and (b) frozen phonon method in the VASP plus
ALAMODE package.

In Sec. II, we mentioned replacing the harmonic phonon bands with the
renormalized phonon bands from the SCPH theory to evaluate the EPC matrix
element. Eq. (7) shows that the fourth-order IFCs are corrected on the
harmonic phonon frequency. Thus, to merge the information and calculate the
EPC matrix element [Eq. (II.1)], we need to double-check the agreement among
the harmonic phonons. Figure 9 displays the dispersion relations calculated
using DFPT and the frozen phonon method. Clearly, Fig. 9 shows that the two
sets of dispersion relations agree in the frequency range above 80 cm-1. Also,
the imaginary frequency modes on the $\Gamma$-M-K plane appear in both cases.
This is thus an appropriate foundation for further calculating the EPC using
the renormalized phonon frequencies and eigenvectors from the SCPH theory.

## APPENDIX B: Effect of $T_{\rm SCPH}$ on phonon and electron-phonon
properties

In this Appendix, we examine the effect of $T_{\rm SCPH}$ on the phonon
properties and EPC. As mentioned in Sec. II B, a $T_{\rm SCPH}$ of 20 K was
used in the Bose-Einstein distribution of the self-consistent SCPH equation
[Eq. (7)]. We choose the temperature at which the phonon bands are stable but
softer than Sc6CoTe2, reflecting the stronger anharmonicity in Sc6FeTe2. We
consider that the treatment is valid because no structural phase transition
has been experimentally observed for Sc6FeTe2. Furthermore, as Table 3 shows,
$T_{c}$ remains almost the same when $T_{\rm SCPH}$ increases. This indicates
that the choice of $T_{\rm SCPH}=$ 20 K is appropriate.

Figure 10: (a)Phonon band structures and (b) Eliashberg spectral functions
$\alpha^{2}F$ of Sc6FeTe2 calculated using different $T_{\rm SCPH}$ values in
the SCPH treatment of anharmonicity.

Nevertheless, we do notice that $\lambda$ drops and $\omega_{\rm log}$ rises
with increasing $T_{\rm SCPH}$ (Table 3). In Fig. 10, we show the phonon bands
and $\alpha^{2}F(\omega)$ for three different $T_{\rm SCPH}$ values. From Fig.
10(a), we notice that the phonon bands in the frequency region above 100 cm-1
are not affected by the $T_{\rm SCPH}$ value. However, for the flat phonon
band near 75 cm-1 and the dispersive phonon bands below, the frequencies of
the phonon bands, especially near the M point do increase as $T_{\rm SCPH}$
rises. This results in the visible upward shift of the bottom of the
Eliashberg spectral function as $T_{\rm SCPH}$ increases [see Fig. 10(b)].

Finally, we note from Eqs. (3), (II.1) and (6) that
$\lambda\propto\omega^{-1}$ and $\omega_{\rm log}\propto\log(\omega)$. Since
$T_{\rm SCPH}$ only affects the phonon frequency, we can see why $\lambda$
drops as $\omega_{\rm log}$ rises due to the increase of $T_{\rm SCPH}$.
Furthermore, due to the competing scaling relations of $\lambda$ and
$\omega_{\rm log}$ with respect to $\omega$, $T_{c}$ of Sc6FeTe2 evaluated
using Eq. (5) does not significantly depend on $T_{\rm SCPH}$.

## ACKNOWLEDGMENTS

M.-C. J. and G.-Y. G. acknowledge the support from the Ministry of Science and
Technology and the National Center for Theoretical Sciences (NCTS) of the
R.O.C. M.-C. J. was supported by the IPA Program, RIKEN. R. M. was supported
by Grant-in-Aid for JSPS Fellows (No. 22KJ1028). R. A. was supported by JSPS
KAKENHI Grant Number JP24H00190.

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