Spin-polarized Majorana zero modes in proximitized superconducting penta-silicene nanoribbons

Abstract

We theoretically propose penta-silicene nanoribbons (p-SiNRs) with induced p-wave superconductivity as a platform for the emergence of spin-polarized Majorana zero-modes (MZMs). The model explicitly considers the key ingredients of well-known Majorana hybrid nanowire setups: Rashba spin-orbit coupling, magnetic field perpendicular to the nanoribbon plane, and first nearest neighbor hopping with p-wave superconducting pairing. The energy spectrum of the system, as a function of chemical potential, reveals the existence of MZMs with a well-defined spin orientation localized at the opposite ends of both the top and bottom chains of the p-SiNR, associated with well-localized and nonoverlapping wave function profiles. Well-established experimental techniques enable the fabrication of highly ordered p-SiNRs, complemented by a thin lead film on top, responsible for inducing p-wave superconductivity through proximity effect. Moreover, the emergence of MZMs with explicit opposite spin orientations for some set of model parameters opens a new avenue for exploring quantum computing operations, which accounts for both MZMs and spin properties, as well as for new MZMs probe devices based on spin-polarized electronic transport mechanisms.

Full text

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Spin‑polarized Majorana
zero modes in proximitized
superconducting penta‑silicene
nanoribbons
R. C. Bento Ribeiro
1, J. H. Correa
2,3, L. S. Ricco
4, I. A. Shelykh
4,5, Mucio A. Continentino
1,
A. C. Seridonio
6, M. Minissale
7, G. Le Lay
7 & M. S. Figueira
8*
We theoretically propose penta‑silicene nanoribbons (p‑SiNRs) with induced p‑wave superconductivity
as a platform for the emergence of spin‑polarized Majorana zero‑modes (MZMs). The model
explicitly considers the key ingredients of well‑known Majorana hybrid nanowire setups: Rashba
spin‑orbit coupling, magnetic eld perpendicular to the nanoribbon plane, and rst nearest neighbor
hopping with p‑wave superconducting pairing. The energy spectrum of the system, as a function of
chemical potential, reveals the existence of MZMs with a well‑dened spin orientation localized at
the opposite ends of both the top and bottom chains of the p‑SiNR, associated with well‑localized
and nonoverlapping wave function proles. Well‑established experimental techniques enable the
fabrication of highly ordered p‑SiNRs, complemented by a thin lead lm on top, responsible for
inducing p‑wave superconductivity through proximity eect. Moreover, the emergence of MZMs with
explicit opposite spin orientations for some set of model parameters opens a new avenue for exploring
quantum computing operations, which accounts for both MZMs and spin properties, as well as for new
MZMs probe devices based on spin‑polarized electronic transport mechanisms.
Ultra-scaling of nanoelectronic devices, beyond Moore’s law, still using the ubiquitous silicon technology, could
come from silicene1–3, the rst silicon-based graphene-like articial two-dimensional (2D) quantum material,
which further engendered the Xenes family4, and which was used to fabricate an atom-thin channel in a eld
eect transistor5, 6. Moreover, topological silicon nanowires hosting Majorana fermions could be a materials
platform for a quantum computer7. However, like other nanowire candidates, even proximitized ones based
on heavier constituents with larger spin-orbit coupling, until now, no conclusive experimental measurements
guarantee incontrovertibly the existence of topologically protected Majorana zero modes (MZMs) for the pos-
sible realization of qubits8, 9.
Since the appearance of the generic Kitaev model10, several platforms were proposed to realize it, both from
theoretical11–17, and experimental points of view18–24. A helpful review of the experimental state-of-the-art on
this subject can be found in Refs.9, 25, 26. is model considers p-wave superconductor pairing between electrons
in dierent sites of a one-dimensional chain (Kitaev chain) and predicts the existence of unpaired MZMs at
opposite ends of a nite Kitaev chain. However, until now, there are no conclusive experimental measurements
that guarantee without doubt the existence of topologically protected MZMs26–30. e experimental detection
of MZMs remains an elusive problem, and they were not really observed until now. Per se, this situation justies
the search for new platforms.
One possible alternative platform is the one-dimensional honeycomb nanoribbons (HNRs) that have been
receiving growing attention in the literature31–34. Nevertheless, the mono-elemental 2D graphene-like materials
coined Xenes, where X represents elements from group IIIA to group VIA of the periodic table, could constitute
OPEN
1Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud, 150, Urca, Rio de Janeiro, RJ 22290-180,
Brazil. 2Universidad Tecnológica del Perú, Nathalio Sánchez, 125, 15046 Lima, Peru. 3AGH University of Krakow,
Academic Centre for Materials and Nanotechnology, al. A. Mickiewicza 30, 30-059 Kraków, Poland. 4Science
Institute, University of Iceland, Dunhagi-3, 107 Reykjavik, Iceland. 5Russian Quantum Center, Skolkovo IC, Bolshoy
Bulvar 30 bld. 1, Moscow 121205, Russia. 6School of Engineering, Department of Physics and Chemistry, São Paulo
State University (UNESP), Ilha Solteira, SP 15385-000, Brazil. 7Aix-Marseille Université, CNRS, PIIM UMR 7345,
13397 Marseille Cedex, France. 8Instituto de Física, Universidade Federal Fluminense, Av. Litorânea s/N, Niterói,
RJ CEP: 24210-340, Brazil. *email: gueira7255@gmail.com
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possible candidates to build HNRs with the ability to harbor MZMs at their ends35–38. Penta-Silicene (X=Si) is
an up-and-coming candidate in this family for obtaining a p-SiNR geometry that can host MZMs39–41.
A paradigmatic breakthrough would be the experimental implementation of the generic Kitaev toy model
with a silicon platform10. In a previous work34, we addressed the problem of Majorana spin discrimination
employing a double-spin Kitaev zigzag honeycomb nanoribbons (KzHNR), which mimics two parallel Kitaev
chains connected by the hopping t (see gure1 of34). Since such KzHNRs have not been realized in experiments,
we look instead in the present paper at the possibility of obtaining MZMs in p-SiNRs, harboring Dirac fermions,
which have been epitaxially grown on Ag(110) surfaces39, 42–44. Typically, highly perfect, atom thin, massively
aligned single strand p-SiNRs, 0.8nm in width, and with lengths extending to tens of nanometers were obtained
by molecular beam epitaxy upon insitu Si deposition onto Ag(110) surfaces held at room temperature, as shown
in Fig.1a. In scanning tunneling microscopy (STM) and high-resolution nc-AFM images, these p-SiNRs appear
as two shied lines of protrusions along the [110] direction as shown in Fig.1b,c and are separated by twice the
nearest neighbor Ag-Ag distance, i.e., 0.577nm. eir hidden internal atomic structure was initially uncovered
employing thorough density functional theory (DFT) calculations and simulations of the STM images39, point-
ing to an arrangement of pure Si pentagonal building blocks, as displayed in Fig.1d, which denes the missing
pentagonal row (P-MR) model employed in the Supplemental information of reference39 to optimize the angles
and the distance between the silicon atoms in the pentagonal arrangement. is unique atomic geometry was later
directly visualized by high-resolution non-contact atomic force microscopy (Fig.1c from40). We will theoretically
demonstrate that these p-SiNRs could constitute a tantalizing disruptive new Kitaev platform.
eoretically, the most well-established example of a topological superconductor hosting MZMs is a spinless
chain with p-wave superconducting pairing between neighboring sites, as proposed by Kitaev10. is type of
superconductivity seems to be extremely rare in nature45. However, spinless orbital p-wave superconductivity can
be engineered from a conventional spin-singlet s-wave superconductor when associated with a material exhibiting
helical bands46. is was rst achieved experimentally by Mourik etal.19, using a semiconducting nanowire with
strong Rashba spin-orbit interaction in proximity to an s-wave superconductor subject to a magnetic eld aligned
along the axis of the nanowire. Despite signicant advances in sample fabrication, measurement techniques, and
theoretical understanding, a denitive signature of MZMs detected through electronic transport measurements
in such hybrid systems remains challenging47. e main reason stems from inevitable inherent disorder, which
generates trivial zero-energy states, mimicking the MZMs signatures48.
We propose a new experimental platform to circumvent this disorder issue, still within the same conceptual
approach. Compared to previous proposals of Majorana nanowire devices, the paradigmatic dierence is the
replacement of the semiconducting nanowire by a new, highly ordered material, with transport measurements
replaced by scanning tunneling spectroscopy (STS). Specically, our proposal relies on the implementation of
long and atomically precise p-SiNRs, grown insitu under ultra-high-vacuum (UHV) in the cleanest conditions
on the Ag(110) surface, all aligned along the [110] direction39–41.
e localized zero energy states associated with MZMs at the ends of p-SINRs can be detected by low-tem-
perature STS via an STM immersed in a strong parallel magnetic eld, following the methodology of Yazdani
and co-workers49. Since silver is not a superconductor, we will proximitize the p-SiNRs insitu with lead islands.
As mentioned before, lead is a conventional BCS superconductor with a relatively high critical temperature that
can be easily grown at Ag(110) surfaces50, which is known to interact only very weakly with the Si nanoribbons
while preserving its structural and electronic properties51, 52. To this end, a thin lead lm will be evaporated
insitu on top of the p-SiNRs and annealed at temperature
∼200◦C
, as already mentioned in4. Hence, at variance
with the classical fabrication and measurement procedures, all the steps in the experiments and measurements
will be performed in perfectly controlled and highly cleaned conditions in a UHV system, which will comprise
a surface science chamber (with all analytical tools and the two Si and Pb evaporators) directly linked to the
STM/STS chamber.
Moreover, within the present proposal, we characterize the topological phase transitions (TPTs) employing
the spinless version of the model and the inclusion of the p-wave superconducting pairing and the magnetic
eld reveal the emergence of topologically protected MZMs with the spin discriminated at opposite ends of the
p-SiNRs; this result constitutes the main nding of the work. We also calculate the wave function of the MZMs
at the ends of the p-SiNR, showing its topological signature.
Figure1. p-SiNR on Ag(110) surface. (a,b) Experimental STM images (uncorrected dri), (c) High-resolution
nc-AFM image. (d) Top and cross view of the arrangement of the Si pentagonal building blocks. (a,b) Courtesy
Eric Salomon, (c) Reprinted with permission from40. Copyright 2023 American Chemical Society. (d) From
Cerda etal.39.
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The model
Lattice transformations
In Fig.2a, to reduce the geometry complexity of the p-SiNR and facilitate the tight-binding calculations, we rede-
ne its structure using square-shaped pentagons. In the geometry of the pentagons that constitute the p-SiNRs
of Fig.2b, four silicon atoms are located on the missing silver row, and only one exhibits a buckling structure
(pink atoms). We neglect the buckling structure of these atoms and employ a planar conguration composed
of square-shaped pentagons. As the distance between the silicon atoms that constitute the pentagons are close,
we consider them equal to
a0
and identify it as the lattice parameter of the p-SiNR. We also dene the nearest
neighbor hopping as equal to t, which is considered the energy unit in all the calculations.
L≡2Na0
, is the length
of the p-SiNR, and N is the number of sites of the corresponding Kitaev chain (top or bottom), employed in the
calculation, as indicated in Fig.2c, that exhibits the shape of the p-SiNR and the unit cell composed of six atoms
inside the dashed rectangle employed in the calculations. We expect these simplications will not change the
results once we keep the lattice.
Eective Hamiltonian—spinless case
e total Hamiltonian, which describes the spinless p-SiNR of Fig.3 is given by
with
(1)
H=Ht+H,
Figure2. (a) Penta-silicene (p-SiNRs) lattice transformation adopted. (b) Penta-silicene angles. (c) Sketch of
nonequivalent Si atoms placed at the vertices of the “square” pentagonal lattice. e dashed rectangle depicts the
unit cell of the system.
Figure3. Sketch of the p-SiNRs: e penta-silicene system comprises two Kitaev chains, one on the top and
the second on the bottom, hybridized via hopping t. e ellipses represent the superconducting p-wave pairing
between silicon atoms belonging to the same chain, represented by pink (top) and yellow (bottom,) respectively
(in the real material, these atoms correspond to the buckled one). e arrows only indicate the spin polarization
needed to dene a Kitaev chain.
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where
µ
is the chemical potential, the index (−) and (
+
) dierentiate the creation and annihilation operators for
electrons and holes, respectively, and H.c. is the Hermitian conjugate. e system Hamiltonian of Eq.(2) was
built according to the unit cell of nonequivalent Si atoms (a, b, c, d, e, f) shown in Fig.2c.
e p-SiNRs are grown on Ag(110) surfaces in the setup proposed here. However, silver is not a superconduc-
tor, and to generate a p-wave pairing

on the pink and yellow atoms of Fig.3, we evaporate insitu a thin lead
lm over the Ag(110) surface in such a way that the buckled silicon atoms enter in contact with the lead atoms.
Under the presence of a strong Rashba spin orbit coupling (RSOC) arising from the Pb atoms and an applied
magnetic eld, the s-wave Cooper pairs of the Pb lm can enter into the p-SiNR region via proximity eect
(Andreev reections)12, giving rise to a p-wave-induced pairing in the double p-SiNRs structure. By following
the same procedure done in our previous work34 and based on the Kitaev model10, we introduce a spinless p-wave
superconducting pairing

between the “external” pink and yellow atoms of the same type as shown in Fig.3.
e Hamiltonian, which describes such a pairing, reads
Eective Hamiltonian—spinful case
In order to properly account for the spin degree of freedom in the superconducting p-SiNRs, we follow our
previous work34. Considering also the spin degree of freedom on both
Ht
and
H
,
We introduce a Zeeman eect due to the application of an external magnetic eld perpendicular to the p-SiNRs
plane. e Hamiltonian, which accounts for the Zeeman eect, reads:
wherein
Z
is the eective strength of the external Zeeman magnetic eld

B
, and
σ=↑,↓
is the spin index for
each operator.
e extrinsic RSOC induced in the p-SiNRs can be modulated by the action of an external electric eld

E
applied perpendicularly to the nanoribbon plane53–56. Its corresponding general Hamiltonian reads
(2)
H
t=−
N

i=1
µa†
i,+ai,+−a†
i,−ai,−+b†
i,+bi,+−b†
i,−bi,−+c†
i,+ci,+−c†
i,−ci,−
+d†
i,+di,+−d†
i,−di,−+e†
i,+ei,+−e†
i,−ei,−+f†
i,+fi,+−f†
i,−fi,−
−
N

i=1
ta†
i,+bi,+−bi,−a†
i,−+b†
i,+ci,+−ci,−b†
i,−
+c†
i,+di,+−di,−c†
i,−+d†
i,+ei,+−ei,−d†
i,−+e†
i,+fi,+−fi,−e†
i,−
−
N−1

i=1
ta†
i+1,+fi,+−fi,−a†
i+1,−+a†
i+1,+ci,+−ci,−a†
i+1,−+d†
i+1,+fi,+−fi,−d†
i+1,−+
H.c.,
(3)
H
=
N−1

i=1
b†
i,+b†
i+1,−−b†
i+1,−b†
i,++e†
i,+e†
i+1,−−e†
i+1,−e†
i,++
H.c..
(4)
H
t=−
N

i=1,σ
µa†
i,+,σai,+,σ−a†
i,−,σai,−,σ+b†
i,+,σbi,+,σ−b†
i,−,σbi,−,σ+c†
i,+,σci,+,σ−c†
i,−,σci,−,
σ
+d†
i,+,σdi,+,σ−d†
i,−,σdi,−,σ+e†
i,+,σei,+,σ−e†
i,−,σei,−,σ+f†
i,+,σfi,+,σ−f†
i,−,σfi,−,σ
−
N

i=1,σ
ta†
i,+,σbi,+,σ−bi,−,σa†
i,−,σ+b†
i,+,σci,+,σ−ci,−,σb†
i,−,σ+c†
i,+,σdi,+,σ
−di,−,σc†
i,−,σ+d†
i,+,σei,+,σ−ei,−,σd†
i,−,σ+e†
i,+,σfi,+,σ−fi,−,σe†
i,−,σ
−
N−1

i=1,σ
ta†
i+1,+,σfi,+,σ−fi,−,σa†
i+1,−,σ+a†
i+1,+,σci,+,σ−ci,−,σa†
i+1,−,σ
+
d†
i
+
1,
+
,σfi,
+
,σ
−
fi,
−
,σd†
i
+
1,
−
,σ
+
H.c.
(5)
H
�=
N−1

i=1,σ
�b†
i,+,σb†
i+1,−,σ−b†
i+1,−,σb†
i,+,σ+e†
i,+,σe†
i+1,−,σ−e†
i+1,−,σe†
i,+,σ+
H.c..
(6)
H
z=
N

i=1,σ
Z sgn(σ )a†
i,σai,σ+b†
i,σbi,σ+c†
i,σci,σ+d†
i,σdi,σ+e†
i,σei,σ+f†
i,σfi,σ+
H.c.,
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where
�u
i,j
=−
R
a0
ˆ
k
�
δi,
j
, with
R
being the extrinsic RSOC parameter,

δi
,
j
is the vector that connects the adjacent
lattice sites i and j, and
γ
the Pauli matrices. e index
¯σ
indicates the opposite spin direction of
σ
. e Eq.(7)
turns into
where
γ
1
=
−1
2+
i
√3
2
,
γ
2
=
1
2−
i
√3
2
,
γ
3
=
−1
2−
i
√3
2
and
γ
4
=
1
2+
i
√3
2
.
Notice that from Eqs.(6) and(7), we are assuming the external Zeeman magnetic eld

B
perfectly perpen-
dicular to the RSOC, i.e,
B≡B⊥�= 0
and
B�=0
. In Rashba nanowires setups, this condition is responsible for
the vanishing of the induced superconducting gap at zero momentum (inner gap) and the opening of a constant
gap at nite momentum (outer gap), which characterizes the topological phase transition and the concomitant
emergence of MZMs protected by the outer gap12.
However, from the experimental perspective, ensuring that the magnetic eld is applied only in the perpen-
dicular direction of the RSOC eld can be challenging. en, it is natural to consider also the eects of
B= 0
.
In this situation, we have both components of the Zeeman eld, and the critical magnetic eld condition for the
topological phase transition remains the same. However, the behavior of the outer gap is not constant anymore,
which aects the topological protection of the MZMs towards fault-tolerant quantum computing operations.
e eect of
B
in the outer gap is not so detrimental if the RSOC is strong.
It is worth noticing that the opposite cases of
B≡B= 0
and
B⊥=0
can lead to the vanishing of the outer
gap, hence preventing the topological phase and emergence of MZMs. erefore, since our system is qualitatively
described by the similar underlying physics of Rashba nanowires, it is appropriate to experimentally ensure the
dominance of the magnetic eld component perpendicular to the Rashba eld.
We now can dene the total system Hamiltonian as
which can be written in the corresponding Bogolyubov–de Gennes (BdG) form in k-space as
Htotal(k)=�T
H
BdG(k)�
, with
where
Hσ,σ′(±k)
and
H�,σ,σ′(±k)
represent the matrix elements for dierent spin directions and the matrix
elements corresponding to the part of the matrix where superconducting couplings

appear, respectively. e
spinor

was constructed with the fermionic operators in the following order:
e spin alignment for each situation in the next section is computed numerically. We calculate the mean value
of the Pauli matrix in
ˆz
direction
ˆ
Sz
, i.e.,
ˆ
Sz=|ˆ
Sz|
, w here
|�
are the eigenvectors of the total Hamiltonian
given by Eq.(9).
In hybrid semiconducting-superconducting nanowires, sometimes called Majorana nanowires, the following
features strongly suggest the emergence of MZMs at the nanowire ends12:
(a) Closing and subsequent reopening of the superconducting gap in the bulk relation dispersion as the chemi-
cal potential
µ
changes, indicating a TPT;
(b) Emergence of persistent zero-modes for specic system parameter values associated with nonoverlapping
wave functions localized at the opposite ends of the nanowire.
To obtain the TPTs present in the p-SiNRs, we will consider the innite case given by the Hamiltonian of Eq.(1).
We calculate the bulk band structure, discussed in detail in the supplemental material (SM). To investigate the
existence of MZMs in the p-SiNRs, we will analyze the spinless p-SiNRs with nite size
N=100
and calculate
(7)
H
R=
N

i
,
j=
1,σ
ic†
i,σ(�ui,j.�γ)cj,(¯σ) +
H.c.,
(8)
H
R=
N

i=1,σ
γ1(a†
i,σbi+1/2, ¯σ)+γ2(b†
i+1/2,σai,¯σ)
+(a†
i,σci−1, ¯σ)−(c†
i−1,σai,¯σ)+(−i)(a†
i,σfi,¯σ)
+(i)(f†
i,σai,¯σ)+γ3(b†
i+1/2,σci+1, ¯σ)+γ4(c†
i+1,σbi+1/2, ¯σ
)
+(−i)(c†
i+1,σdi+1, ¯σ)+(i)(d†
i+1,σci+1, ¯σ)−(d†
i+1,σfi,¯σ)
+(f†
i,σdi+1, ¯σ)+γ3(d†
i+1,σei+3/2, ¯σ)+γ4(e†
i+3/2,σdi+1, ¯σ
)
+
γ1(e†
i
+
3/2,σfi
+
2,
¯
σ)
+
γ2(f†
i
+
2,σei
+
3/2,
¯
σ)
+
H.c.,
(9)
Htotal =Ht+HZ+HR+H,
(10)
H
BdG(k)=


H
↑,↑
(k)H
↑,↓
(k)H
�,↑,↑
(k)H
�,↑,↓
(k)
H↑,↓(k)H↓,↓(k)H�,↓↑(k)H�,↓,↓(k)
H∗
�,↑,↑(−k)H∗
�,↑,↓(−k)H∗
↑,↑(−k)H∗
↑,↓(−k)
H
∗
�,↓,↑
(
−
k)H
∗
�,↓,↓
(
−
k)H
∗
↓,↑
(
−
k)H
∗
↓,↓
(
−
k)



,
(11)
�T
=(ak,↑,bk,↑,ck,↑,dk,↑,ek,↑,fk,↑,ak,↓,bk,↓,ck,↓,dk,↓,ek,↓,fk,↓,
a†
−k
,
↑
,b†
−k
,
↑
,c†
−k
,
↑
,d†
−k
,
↑
,e†
−k
,
↑
,f†
−k
,
↑
,a†
−k
,
↓
,b†
−k
,
↓
,c†
−k
,
↓
,d†
−k
,
↓
,e†
−k
,
↓
,f†
−k
,
↓
)
.
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the energy spectrum as a function of the chemical potential
µ
and the probability density function
|ψ|2
associ-
ated with the zero-energy states which arise on the real axis of the energy spectrum.
Both the energies
En
and eigenvectors
ψn
per site are obtained by numerically solving the Schrödinger equa-
tion
Hψn=Enψn
for the Hamiltonian of Eq.(1). To evaluate the position dependence of the wave functions
associated with zero energy states, we numerically calculate the eigenvector
ψn
when
En=0
, which allows
obtaining the probability density per lattice site according to
Results and discussion
Finite spinless p‑SiNRs
We employed the following parameter set in all the calculations:
=0.5t
,
Z=0.1t
,
R=0.05t
and
N=100
.
e top panels of Fig.4 show the bulk energy dispersion of the p-SiNRs, in the presence of the superconduct-
ing p-wave pairing, described by Eqs.(1–3), along the
kx
direction, for three representative values of chemical
potential
µ
[vertical lines in panel (d)]. Figure4a depicts the closing of the superconductor (SC) gap at
kx=0
for
µ=0.0t
. As the value of
µ
enhances, the SC gap opens as shown in panels (b) for
µ=0.4t
and closes again
at
kx=0
for
µ=0.7t
as shown in panel (c). is closing and reopening of the SC gap with the tuning of
µ
char-
acterize a topological phase transition. e bulk-boundary correspondence principle57 ensures the topologically
protected MZMs at the ends of the p-SiNRs.
To verify the emergence of MZMs associated with the TPTs depicted in Fig.4a–c, we plot the p-SiNRs
energy spectrum as a function of
µ
in Fig.4d. ere are no zero-energy modes for the values of
µ
where the gap
closes (red and magenta vertical lines). However, for values of
µ
inside the topological gap, for example, when
µ=0.4t
(green vertical line), two zero-energy states appear on the real axis, indicating the presence of MZMs
at the opposite ends of the p-SiNRs, topologically protected by the eective p-wave SC gap (Fig.4b). is nd-
ing is similar to what was obtained in our previous work34, wherein the MZMs emerge at the opposite ends of
a nite double zHNR.
Figure4f shows isolated zero-energy modes for
µ=0.4t
, which are associated with a nonoverlapping wave
function, well-localized at the ends of the p-SiNRs, as depicted in Fig.4j; which together with the topological
phase transition (Fig.4a–c), is a piece of strong evidence that topologically protected MZMs emerge at the
opposite ends of the spinless p-SiNRs. In the Supplemental Material, we developed an extensive analysis of the
topological and trivial phases of the spinless p-wave superconducting p-SiNR, that can be distinguished by the
Zak number topological invariant58. However, we cannot aord to do the same study for the spinful case due to
the extreme mathematical complexity.
Although there are zero-energy modes for other values of
µ
(Fig.4e,g), they are not associated with wave func-
tions well-localized at the ends of the p-SiNR, as can be seen in Fig.4i,k, for
µ=0.0t
and
µ=0.7t
, respect ively.
is implies that a zero mode cannot be exclusively attributed to MZMs. In this context, the genuine nature of
the zero-modes shown in Fig.4e,f, for instance, can be experimentally distinguished by combining STM and
atomic force microscopic (AFM) measurements49, 59. is approach allows associating the zero-bias conductance
(12)
|
ψ
n|2=
ψ
n
ψ
∗
n.
Figure4. Spinless case: (a–c) Bulk energy dispersion for the spinless p-SiNRs as a
kx
function. e colors red,
green, and magenta used in the panels correspond to
µ=0.0t
, 0.4t, and 0.7t, respectively. (d) Energy spectrum
as a function of the chemical potential. (e–g) Zero-energy states spectrum. (i–k) Probability density per lattice
site
|ψ|2
, associated with zero-energy states on the real axis of the Kitaev, top or bottom chains.
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peaks at the p-SINR ends (Fig.4f) with their well-localized wave functions through spatially resolved conduct-
ance maps, cf. Fig.4j.
Furthermore, we highlight that we analyze only one region of all energy spectrum shown in Fig.4d, which
presents other ranges of chemical potentials wherein a zero-energy state, associated with the emergence of MZMs,
arises. A more detailed study of the energy spectra can be found in the SM. We can also observe that, unlike the
system of our previous work34, the energy spectrum of Fig.4d is asymmetric at about
µ=0
.
Finite spinful p‑SiNRs
Now we will analyze how the spinless scenario shown in Fig.4 is aected by the presence of both Zeeman eld
(Eq.6) and extrinsic RSOC (Eq.7) coupling within the spinful description (Eq.9).
Figure5a–e exhibit the energy dispersion of the p-SiNRs given by the eigenenergies of BdG Hamiltonian
(Eq.10) as a function of
kx
, for distinct values of the chemical potential
µ
, indicated by vertical lines in Fig.5f.
e spin polarization is indicated by the vertical color bar, in which the red color represents the spin
↑= 1
, w hile
the blue color stands for spin
↓= −1
, and the light shades of colors mean the spin is neither up nor down. As
µ
is tuned, we can see the opening and closing of the superconducting gap, thus indicating a TPT, as previously
veried in the spinless situation (Fig.4a–c). However, here we can notice that each TPT associated with a specic
value of
µ
has a preferential spin orientation, except Fig.5c, where the system exhibits a conventional band gap.
e spin-polarized TPTs in Fig.5a,b,d,e lead to the appearance of spin-polarized zero-modes in Fig.5f, which
shows the system energy spectrum as a function of
µ
. ese zero-modes indicate the emergence of spin-polarized
MZMs at the ends of the p-SiNRs as
µ
is changed, similar to those found in34.
e panels g–k of Fig.5 depict the corresponding energy levels sorted in ascending order. e dierent values
of
µ
used to calculate the MZMs are indicated by vertical black lines in Fig.5f. For
µ=−2.7t
(Fig.5g), there are
two zero modes on the real axis of spin up (red points), associated with nonoverlapping wave functions shown
in Fig.5l.
For
µ=−2.35t
(Figs.5h and 6h), there are two energy-states in the spin-up direction and the other two
with spin-down, associated with degenerate (blue and red) nonoverlapping wave functions shown in Figs.5m
and6m, respectively. It is worth mentioning that this situation is absent from our previous paper34 and constitutes
a pivotal dierence exclusively related to the p-SiNRs. Here, pairs of MZMs at opposite edges split into top and
bottom chains of the nanoribbon with opposite spins. ese MZMs, acting as an eective two-level electronic
system, would allow the recovery of the spin degree of freedom as a good quantum number for purposes of
quantum computing, as well as to dene the intrinsic spins of the regular fermions built up by these top and
bottom pairs of MZMs edge states, respectively. By considering two p-SiNRs as source and drain reservoirs with
an immersed quantum dot, a type of single electron transistor (SET) could be set15, 60, 61, allowing to detect a
spin-polarized zero-bias conductance provided by one of the spins associated with a given pair of MZMs placed
at one particular nanoribbon chain.
e chemical potential
µ=1.1t
(Fig.5i) represents a non-topological region, there are four spin-down energy
states outside the real axis, there are no MZMs, and the wave functions completely overlap along the ribbon.
e system exhibits a particular “semiconductor” phase with the gap controlled by the
µ
variation and two delta
peaks generated by the zero modes outside the real axis, inside the gap.
Figure5. Spinful case—magnetic eld up: (a–e) Bulk energy dispersion of the superconducting p-SiNRs for
the spinful situation, as a function of
kx
, for
µ=−2.7t
,
−2.35t
, 1.1t, 2.09t and 2.2t, respectively. (f) Energy
spectrum as a function of the chemical potential. Vertical lines indicate the chosen values of chemical potential
shown on top panels. (g)–(k) Zero-energy states spectrum. (l)–(q) Probability density per lattice site
|ψ|2
,
associated with zero-energy states on the real axis of the spin-polarized Kitaev, top or bottom chains.
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For
µ=2.09t
(Fig.5j), there are two zero modes on the real axis of spin up (red points). Finally, for
µ=2.2t
(Fig.5k), there are four MZMs with spin-down energy states on the real axis. is situation happens because,
at
µ=2.09t
, a TPT occurs for spin-up, the gap closes at
k=±π
, and for
µ>2.09t
the gap denes a trivial
band insulator for this spin orientation and MZMs with spin-up are not available anymore. ese well-localized
probability densities describing wave functions centered at the opposite ends of the superconducting p-SiNRs,
associated with zero-energy edge states, indicate the emergence of MZMs in the same way previously found for
the spinless system.
Figure6 represents the same situation as Fig.5 but with the magnetic eld pointing in the opposite direc-
tion. e net eect on the p-SiNRs is to change the MZMs, for all
µ
values, in spin up to down and vice versa.
erefore, it is possible to select the spin polarization of the MZMs by changing the chemical potential
µ
or the
magnetic eld orientation.
In Fig.7, we mainly analyze the dispersion relation, energy spectrum, and nature of the zero-modes at
µ=0
of Fig.5, with the magnetic eld pointing in the up direction. Figure7a depicts E(k) as a function of
kx
, showing
that there is a nite topological superconducting gap only for the spin-down orientation (blue line), while the
spin-up (red line) remains gapless. is behavior suggests a spin-polarized TPT at zero chemical potential, imply-
ing that only the system’s spin-down component is within the topological regime. At the same time, the spin-up
belongs to a metallic phase. Fig.7b,c represent two MZMs of spin-down with its correspondent nonoverlapped
wave function, respectively, and Fig.7d, shows detail at around
µ=0
region.
Additionally, we investigate how the energy spectrum as a function of
µ
is aected by the length of the
p-SiNRs. Fig.8 exhibits the superconducting p-SiNRs’ energy spectrum for increasing nanoribbon length values
N. From the smallest system considered (
N=10
, Fig.8a) to the largest one (
N=100
, Fig.8e), it can be noticed
a decrease of the amplitude of oscillations at around the real axis (
E=0
), and at the same time the denition of
the MZMs on the real axis improves as N increases, and for
N=100
the MZMs are well dened in all the real
axis. It should be mentioned that these oscillations around zero energy are expected for short Majorana nanowires
due to the overlap between Majorana wavefunctions of opposite ends. erefore, such oscillations are expected
to decrease as the system becomes larger. e same behavior was veried in the work34.
Figure6. Spinful case—magnetic eld down: e same situation of Fig.5 but with the magnetic eld pointing
in the opposite direction.
Figure7. Analysis in detail of the
µ=0
case of Fig.5 with the magnetic eld pointing in the up direction.
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Conclusions and perspectives
is paper demonstrates the emergence of topologically protected MZMs at opposite ends of spinless and spinful
p-SiNRs with p-wave superconducting pairing. ese MZMs exhibit spin discrimination, and their polarization
can be controlled by adjusting the nanoribbon chemical potential or the external magnetic eld. To implement
our ndings experimentally, we propose a material engineering of p-SiNRs grown over an Ag(110) surface (cf.
Fig.1a), with a thin Pb lm deposited on top4, 50. In this device, the proximity eect will enable the penetration
of Cooper pairs from the Pb s-wave superconductor into the p-SiNRs12, and in combination with an external
magnetic eld and the extrinsic RSOC modulated by the action of an external electric eld

E
applied perpen-
dicularly to the nanoribbon plane53–56, it will induce p-wave pairing in the buckled atoms of the double p-SiNRs
structure (cf. Fig.1d).
We should highlight the potential applications driven by the spin-polarized MZMs presented in this work,
notably demonstrated in the results of Fig.7, with the down spin component associated with MZMs, while the
up component displays metallic features, resulting in a half-metallic behavior for the system62, 63. is property
could be harnessed to design a single Majorana transistor (SMT) built from a quantum dot (QD) sandwiched by
nite p-SiNR leads6, 64, 65. is setup resembles the conventional single electron transistor (SET)66. e SMT can
be a valuable tool for discerning between MZMs and trivial Andreev bound states15, 17. Particularly, the leakage
of MZMs through the QD67, along with both local and crossed Andreev reections induced by a specic spin
orientation within the p-SiNR-QD-p-SiNR SMT structure, is expected to generate distinct electronic transport
signatures, enabling the identication of MZMs.
In addition to the spin-polarization of MZMs, our proposal also features the emergence of two MZMs located
at opposite ends of the p-SiNR top chain, while another two with opposed spins are at the bottom, as illustrated
in Figs.5h and 6h. ese MZMs acting as an eective two-level electronic system would allow the recovery of the
spin degree of freedom as a good quantum number for purposes of quantum computing implementation, as well
as to dene the intrinsic spins of the regular fermions built-up by these top and bottom couples of edge MZMs,
respectively. It is crucial for implementing quantum computing operations between two qubits, as it requires the
presence of two fermionic sites, i.e., four MZMs68, 69. erefore, our proposal is a promising candidate for realizing
hybrid quantum computing operations70, 71 between conventional qubits and spin-polarized Majorana-based
qubits and paves the way for dening quantum computing operations using Majorana spintronics72.
Data availability
e data that support the ndings of this study are available from the corresponding author upon reasonable
request.
Received: 6 July 2023; Accepted: 11 October 2023
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Acknowledgements
M.S.F., M.A.C., and A.C.S. acknowledge nancial support from the National Council for Scientic and Tech-
nological Development (CNPq) grant numbers 311980/2021-0, 305810/2020-0, 308695/2021-6, respectively.
M.S.F. acknowledges the Foundation for Support of Research in the State of Rio de Janeiro (FAPERJ) processes
number 210 355/2018 and 211.605/2021. M.A.C. acknowledges nancial support to the Foundation for Support
of Research in the State of Rio de Janeiro (FAPERJ) for the fellowship of the Programa Cientistas do Nosso Estado,
E-26/201.223/2021. L.S.R. and I.A.S. acknowledge the Icelandic Research Fund (Rannis), grant No. 163082-051.
Author contributions
All authors participate in the scientic discussion of the work. All authors reviewed the paper. M.S.F., R.C.B.R.,
M.M., G.L.L., L.S.R., A.C.S., and J.H.C. edit the paper. R.C.B.R. performed the numerical calculations. R.C.B.R.,
and L.S.R. performed analytical calculations.
Competing interests
e authors declare no competing interests.
Additional information
Supplementary Information e online version contains supplementary material available at https:// doi. org/
10. 1038/ s41598- 023- 44739-7.
Correspondence and requests for materials should be addressed to M.S.F.
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