††thanks: The authors contribute equally.††thanks: The authors contribute
equally.

# Tomasch Oscillations as Above-Gap Signature of Topological Superconductivity

Antonio Štrkalj  TCM Group, Cavendish Laboratory, J. J. Thomson Avenue,
Cambridge CB3 0HE, United Kingdom  Xi-Rong Chen  National Laboratory of Solid
State Microstructures, School of Physics, and Collaborative Innovation Center
of Advanced Microstructures, Nanjing University, Nanjing 210093, China  School
of Microelectronics and Physics, Hunan University of Technology and Business,
Changsha, Hunan Province, 410205, China  Wei Chen  Corresponding author:
<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures, School
of Physics, and Collaborative Innovation Center of Advanced Microstructures,
Nanjing University, Nanjing 210093, China  D. Y. Xing  National Laboratory of
Solid State Microstructures, School of Physics, and Collaborative Innovation
Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
Oded Zilberberg  Department of Physics, University of Konstanz, 78464
Konstanz, Germany

###### Abstract

The identification of topological superconductors usually involves searching
for in-gap modes that are protected by topology. However, in current
experimental settings, the smoking-gun evidence of these in-gap modes is still
lacking. In this work, we propose to support the distinction between two-
dimensional conventional $s$-wave and topological $p$-wave superconductors by
above-gap transport signatures. Our method utilizes the emergence of Tomasch
oscillations of quasiparticles in a junction consisting of a superconductor
sandwiched between two metallic leads. We demonstrate that the behavior of the
oscillations in conductance as a function of the interface barriers provides a
distinctive signature for $s$-wave and $p$-wave superconductors. Specifically,
the oscillations become weaker as the barrier strength increases in $s$-wave
superconductors, while they become more pronounced in $p$-wave
superconductors, which we prove to be a direct manifestation of the pairing
symmetries. Our method can serve as a complimentary probe for identifying some
classes of topological superconductors through the above-gap transport.

Introduction.– At the heart of superconductivity is the pairing of conduction
electrons into Cooper pairs that form a bosonic condensate [1]. These Cooper
pairs can be in either a spin-singlet state, with a total spin 0, or a spin-
triplet state, with a spin 1. The spin-singlet state is characterized by a
wavefunction with even angular momentum, such as $s$-wave or $d$-wave, while
the spin-triplet state supports a wavefunction with odd angular momentum, such
as $p$-wave or $f$-wave. In conventional $s$-wave superconductivity the
pairing function $\Delta(\mathbf{k})=\Delta_{s}$ is constant irrespective of
the direction of the momentum vector $\mathbf{k}$. The Cooper pair, in this
case, consists of two electrons with opposite spins. On the other hand, in
unconventional $p$-wave superconductors, electrons with the same spin form
Cooper pairs and the pairing $\Delta(\mathbf{k})$ is no longer constant with
$\mathbf{k}$ [2, 3, 4, 5, 6, 7].

Over the past several decades, there has been significant interest in
unconventional $p$-wave superconductors [8, 2, 3, 4, 5, 7], mostly due to
their unique topological properties [9, 10, 11]. The topology implies, for
example, the presence of exotic quasiparticles, such as Majorana zero-energy
modes in one-dimensional systems [12, 13], and in-gap Majorana states in two-
dimensional systems [14, 15, 16, 17, 18, 19, 20]. These quasiparticles have
potential applications in topological quantum computing [21, 22, 23], and are
used as a key signature for discerning between topologically nontrivial
$p$-wave and trivial $s$-wave superconductors. As a result, the search for
topological superconductors encompass two main approaches. The first approach
involves searching for topological superconductivity in specific materials,
such as Sr2RuO4 [24, 25, 3, 26], UTe2 [27], Pb3Bi [28], and hybrid systems
such as Pb/Co/Si(111) [29, 20]. The second approach involves using engineered
metamaterials that share some of the properties of topological superconductors
[30, 31, 32, 33, 34, 35, 9, 36, 37, 38, 39, 40, 41, 19].

The behavior of superconductors can be largely understood through the
Bogoliubov-de-Gennes (BdG) formalism [42], which describes the mean-field
behavior of quasiparticles in the superconductor that form through
hybridization between electrons and holes. The resulting band structure
resembles a ”sombrero” shape, with an energy gap that depends on the details
of the pairing function $\Delta(\mathbf{k})$. Notably, the band structure of
quasiparticles in superconductors resembles that of band-inverted
semiconductors [43, 44, 45, 46], with the band gap of the latter corresponding
to the superconducting gap in the former. Recently, Fabry-Pérot oscillations
were observed in a two-dimensional junction made out of an inverted InAs/GaSb
double quantum well [46]. The mechanism leading to such Fabry-Pérot
oscillations in a two-dimensional junction stems from the sombrero-shaped band
structure. Specifically, the interference is dominated by the scattering
between the electron-like and hole-like states at energies close to the band
gap [46, 47]. Such interference is quite ubiquitous and applies to a variety
of condensed matter systems with inverted-band dispersions [46, 48].
Interestingly, similar Fabry-Pérot oscillations are also studied in
superconducting junctions and go under the name of Tomasch oscillations [49,
50, 51, 52, 53, 54, 55].

In this work, we demonstrate that Tomasch oscillations in the conductance
across the two-dimensional NSN junctions (two normal metals sandwiching a
superconductor) provide a signature that can act as a complementary probe to
discern between conventional and topological superconductors. As such, we
focus on transport with energies above the superconducting gap and investigate
the effects of the interface barriers on the Tomasch oscillations for
superconductors with different pairing symmetries. In the weak barrier limit,
we find that the inverted-band mechanism responsible for the oscillations is
the same for both $s$-wave and $p$-wave superconductors. Interestingly, the
oscillations are crucially different in the strong barrier limit, i.e., in the
tunneling limit. This distinction arises from differing pairings in the BdG
Hamiltonians of $s$\- and $p$-wave superconductors, affecting the visibility
of the oscillations. Our result offers an alternative experimental probe using
the above-gap transport signatures for distinguishing between conventional and
topological superconductors, in contrast to the commonly studied in-gap
signatures [56].

Figure 1:  (a) Sketch of the NSN junction consisting of a superconductor (S)
coupled to two normal metal leads (N) with the interface barrier $Z$. (top
panels) The quasiparticle dispersion for $k_{y}=0$ in all three regions. We
consider an incident particle from the left lead with energy $E$ (dashed
line). (b) Differential conductance as a function of energy for both the $s$\-
and $p$-wave superconductors in the absence of barriers ($Z=0$) and with the
magnitude of the pair potential set to be equal, i.e.,
$|\Delta(\mathbf{k})|=\Delta_{s}=\Delta_{p}k_{\rm F}$. The period of the
conductance oscillations, $\delta(E)$, increases with energy. (c) Same as (b)
in the presence of finite barriers with equal strengths $Z=4$. Small grey
arrows denote the minimum and the maximum of a single oscillation that we use
in Fig. 3 when calculating the averaged visibility. For (b) and (c), we used
the following parameters: $L=10\xi_{0}$, $k_{\rm F}=2000\xi_{0}^{-1}$,
$\mu_{S}=\mu_{\rm N}=1000|\Delta(\mathbf{k})|$, where $|\Delta(\mathbf{k})|$
is the magnitude of the pair potential and $\xi_{0}$ is the superconducting
coherence length.

Setup.– We study a two-dimensional NSN junction made of two normal metals (N),
and a superconductor (S) sandwiched between them, see Fig. 1(a). We
concentrate on a ballistic case, where the mean free path of particles is the
largest scale in the system. The BdG Hamiltonian around $\Gamma$ point in the
continuum limit of the whole system is

$\displaystyle H=H_{\rm N}+H_{\rm S}+U(x)\sigma_{z}\,,$ (1)

where the Pauli matrix $\sigma_{z}$ (and later $\sigma_{x}$) operates in the
Nambu space and $H_{\rm N}$ describes the metallic leads

$\displaystyle H_{\rm N}(\mathbf{k})$
$\displaystyle=\begin{pmatrix}\frac{\hbar^{2}}{2m_{\rm
N}}\mathbf{k}^{2}-\mu_{\rm N}&0\\\ 0&-\frac{\hbar^{2}}{2m_{\rm
N}}\mathbf{k}^{2}+\mu_{\rm N}\end{pmatrix}\,,$ (2)

with $\mathbf{k}=(k_{x},k_{y})$ the wave vector, $m_{\rm N}$ the effective
mass of electrons in the metallic leads, and $\mu_{\rm N}$ their chemical
potential. The Hamiltonian of the superconductor is given by [57]

$\displaystyle H_{\rm S}(\mathbf{k})$
$\displaystyle=\begin{pmatrix}\frac{\hbar^{2}}{2m_{\rm
S}}\mathbf{k}^{2}-\mu_{\rm S}&\Delta(\mathbf{k})\\\
\Delta^{*}(\mathbf{k})&-\frac{\hbar^{2}}{2m_{\rm S}}\mathbf{k}^{2}+\mu_{\rm
S}\end{pmatrix}\,,$ (3)

with $m_{\rm S}$ and $\mu_{\rm S}$ the corresponding effective mass and
chemical potential, respectively. We introduce a pairing potential
$\Delta(\mathbf{k})$ corresponding to two types of superconductors, namely a
time-reversal symmetric $s$-wave superconductor with a constant pairing
$\Delta(\mathbf{k})=\Delta_{s}$, and a time-reversal broken $p$-wave
superconductor with $\Delta(\mathbf{k})=i\Delta_{p}(k_{x}+ik_{y})$. Note that
for a vanishing pairing potential, the band structure takes a parabolic shape.
Furthermore, at the N-S interfaces, we introduce sharp barriers
$U(x)=U(\delta(x)+\delta(x-L))$ to account for the materials’ mismatch or
imperfections [58, 59]. Alternatively, the barriers can be introduced and
adjusted by local strip gates. For simplicity, we assume that the barriers are
perfectly flat in the $y$-direction, such that they do not break the
translational invariance in the $y$-direction. In the following, we employ the
often-used dimensionless barrier strength $Z\equiv mU/(\hbar^{2}k_{\rm F})$
and, without loss of generality consider $m\equiv m_{\rm N}=m_{\rm S}$ and
$\mu\equiv\mu_{\rm N}=\mu_{\rm S}=\hbar^{2}k_{\rm F}^{2}/(2m)$.

We calculate the differential conductance across the junction using the
Blonder-Tinkham-Klapwijk formula [58]

$G(E)=G_{0}\int^{K(E)}_{-K(E)}\frac{\differential
k_{y}}{2K(E)}\left[1+|a_{L}(k_{y},E)|^{2}-|b_{L}(k_{y},E)|^{2}\right]\,,$ (4)

where $a_{L}$ and $b_{L}$ denote the amplitudes of Andreev [60] and normal
reflections, respectively. The factor $K(E)=\sqrt{2m_{\rm N}(E+\mu)}$ is the
maximal value of $k_{y}$ for a given incident energy $E$, and
$G_{0}=2e^{2}KW/(\pi h)$ is the conductance of the metallic lead with width
$W$ in the $y$-direction. The microscopic analysis of the scattering
amplitudes appears below, cf. Eq. (5). Using formula (4), we calculate the
conductance for $s$\- and $p$-wave superconductors for vanishing barrier
($Z=0$) and strong barrier ($Z=4$), see Figs. 1(b) and (c), respectively. The
case of a vanishing/weak barrier shows the same conductance behavior for both
the $s$\- and $p$-wave case, namely strong oscillations with an energy-
dependent period $\delta(E)$. The oscillations come from constructive
interference of multiple scattering paths inside the superconducting cavity,
where electronic modes in the outer branch of the band structure scatter to
hole modes in the inner part of the band structure. Therefore, it is possible
to analytically obtain the position of each peak in the conductance, as well
as their period, by solving the following interference condition
$k_{x}^{e/h}(E)-k_{x}^{h/e}(E)=2\pi n/L$, where $n$ is an integer. This limit
of vanishing barriers is studied in detail in Refs. [46, 61]. On the other
hand, in the opposite limit of strong barriers [cf. the $Z=4$ case in Fig.
1(c)], the conductance oscillations drastically differ between the $s$\- and
$p$-wave cases. Specifically, they are suppressed in the former and
significantly enhanced in the latter. This signature in bulk transport above
the gap is the main result of our work and it can be used to distinguish
between different types of superconductors.

Figure 2:  (a) and (b) Transmission probability with $Z=0$ for $s$\- and
$p$-wave superconductor, respectively. Two main features are visible: ① inside
the gap, perfect transmission occurs with $T=2$, and ② both electron- and
hole-like quasiparticles are present, hybridize with one another, and produce
oscillations that only weakly depend on $k_{y}$. (c) and (d) Transmission
probability with $Z=2$ for $s$\- and $p$-wave superconductor, respectively. In
(d) the topological in-gap mode, marked with ③, is visible, while in (c) there
are no in-gap modes present. (e) and (f) The enlarged regions of (c) and (d).
A new differentiating feature, marked with ④, appears in a case of strong
barriers: in (e), the transmission maxima of different electron- and hole-like
modes cross, while in (f) avoided crossings appear, leading to secondary gaps
and flat transmission bands appear. For all plots, we used the same parameters
as in Fig. 1.

Momentum resolved transmission.– To understand the microscopic origin of the
results shown in Figs. 1(b) and (c), we turn to the analysis of the momentum-
resolved transmission probability
$T(k_{y},E)=1+|a_{L}(k_{y},E)|^{2}-|b_{L}(k_{y},E)|^{2}$, i.e., the integrand
of Eq. (4), which describes the transmission of charge in the left lead. We
start by formulating the junction’s scattering equations by considering an
electron incident from the left lead with energy $E$ and transverse momentum
$k_{y}$. We describe the states in the left/right metallic leads (L/R) and the
superconducting region (S)

$\begin{split}\Psi_{L}=&\left[\vec{\Phi}_{N}^{e}e^{iq_{x}^{e}x}+a_{L}\vec{\Phi}_{N}^{h}e^{iq_{x}^{h}x}+b_{L}\vec{\Phi}_{N}^{e}e^{-iq_{x}^{e}x}\right]e^{ik_{y}y},\\\
\Psi_{R}=&\left[a_{R}\vec{\Phi}_{N}^{h}e^{-iq_{x}^{h}x}+b_{R}\vec{\Phi}_{N}^{e}e^{iq_{x}^{e}x}\right]e^{ik_{y}y},\\\
\Psi_{S}=&\sum_{\eta=\pm}\left[s^{e}_{\eta}\vec{\Phi}_{S}^{e}e^{i\eta
k^{e}_{x}x}+s^{h}_{\eta}\vec{\Phi}_{S}^{h}e^{-i\eta
k_{x}^{h}x}\right]e^{ik_{y}y},\end{split}$ (5)

where $q_{x}^{e/h}$ and $k_{x}^{e/h}$ are the $x$-components of the
quasiparticle’s momentum in the metallic leads and the superconductor,
respectively. In the metallic leads, the spinors
$\vec{\Phi}_{N}^{e}=(1,0)^{T},\vec{\Phi}_{N}^{h}=(0,1)^{T}$ describe an
electron in the outer dispersion branch and a hole in the inner dispersion
branch, respectively.
$\vec{\Phi}_{S}^{e}=[u(k^{e}_{x},k_{y}),v(k^{e}_{x},k_{y})]^{T}$ and
$\vec{\Phi}_{S}^{h}=[u(k^{h}_{x},k_{y}),v(k^{h}_{x},k_{y})]^{T}$ are spinors
of electron- and hole-like quasiparticles in the superconductor and $u,v$ are
electron and hole wave components. The coefficients $a_{L},a_{R},b_{L},b_{R}$
denote the amplitudes of Andreev reflection [60], cross Andreev reflection,
normal reflection, and elastic cotunneling, respectively. The coefficients
$s^{e,h}_{\pm}$ are scattering amplitudes inside the superconductor.

To find the scattering amplitudes above, we impose the following boundary
conditions on the two N-S interfaces: $\Psi_{L/R}=\Psi_{S}$ and
$\partial_{x}\Psi_{S}-\partial_{x}\Psi_{L/R}=\pm 2Zk_{F}\Psi_{S}$ for the
$s$-wave superconductor, and $\Psi_{L/R}=\Psi_{S}$ and
$\partial_{x}\Psi_{S}-\partial_{x}\Psi_{L/R}=\pm
2Zk_{F}\Psi_{S}+(m\Delta/\hbar)\sigma_{x}\Psi_{S}$ for the $p$-wave
superconductor, where “$\pm$” correspond to the left and right interfaces at
$x=0,L$, respectively. Note that due to the perfectly flat barriers in
$y$-direction, the momentum $k_{y}$ is preserved for all scattering processes.

We solve the scattering equations (5), with the aforementioned boundary
conditions, for $a_{L}$ and $b_{L}$, and show the result for $T(k_{y},E)$ in
Fig. 2 for both the $s$\- and $p$-wave superconductors and in the limits of
weak [Figs. 2(a) and (b)] and strong [Figs. 2(c-f)] barriers. In the former,
$T(k_{y},E)$ is identical for $s$\- and $p$-wave superconductors and we
identify two main features (marked ① and ② in the figure). In region ①, the
energy resides in the superconducting gap, i.e., the main gap, see Fig. 1(a)
and (b), and the transmission through the NSN junction is constant and equal
to 2 without the effect of the interface barriers [60, 58]. In region ②, both
electron and hole-like modes coexist; due to the predominant electron-to-hole
scattering, the transmission maxima exhibit a relatively weak dependence on
$k_{y}$. Consequently, strong oscillations manifest in the conductance, see
Fig. 1(b). Note that $T(k_{y},E)=T(-k_{y},E)$, and therefore in Fig. 2, we
show only positive $k_{y}$ plane.

In the opposite limit of strong barriers – or equivalently, weak coupling to
the leads – the transmission throughout region ① is now strongly suppressed
for the $s$-wave superconductor, with a power-law scaling with $Z$ [58]. At
the same time, for the $p$-wave superconductor on top of the suppressed
transmission, a clear sign of a topological edge mode, marked with ③, can be
seen [11]. On the other hand, the maxima of $T(k_{y},E)$ in region ② become
sharper for both $s$-wave and $p$-wave superconductor and they also acquire an
additional structure that was smeared out by the strong coupling with the
leads, see Figs. 2(c-f). Moreover, gaps between maxima – dubbed secondary gaps
– close in the case of the $s$-wave superconductor, while in the $p$-wave case
they remain open even for very large barrier strengths $Z$, as marked by ④ in
Figs. 2(e) and (f).

Opening of secondary gaps.– To better understand the mechanisms responsible
for the different behavior of the secondary gaps in transmission – and with
that the difference in conductance oscillations – between the $s$\- and
$p$-wave superconductors, we employ a perturbative approach. We first
concentrate on the limit of strong barriers. In this limit, the gap structure
in the transmission plots [cf. Figs. 2(c-f)] is determined by the eigenmodes
of the superconducting cavity, which are only perturbatively affected by the
coupling to the leads. Therefore, it is sufficient to analyze the isolated
superconductor, which is finite in the $x$-direction with length $L$ and
infinite in the $y$-direction. Doing so, we discover that the momentum
dependence of the pairing potential in Eq. (3) is responsible for the
selective hybridization of particle- and hole-like modes of the cavity, see
Supplemental Material [62] for more details. Crucially, in the $p$-wave
superconductor, secondary gaps are opened even without the presence of the
leads due to the $k_{x}$ dependence of the pairing term in $H_{\rm S}$. The
Hamiltonian of the $s$-wave superconductor, on the other hand, has constant
off-diagonal elements and the particle- and hole-like modes do not hybridize.
As a result, the secondary gaps in $T(k_{y},E)$ are closed in that case.

In the limit of weak barriers, i.e. the strong hybridization with the leads,
secondary gaps are opened for both $s$-wave and $p$-wave superconductors, see
Figs. 2(a) and (b). To understand this, we include the leads in our analytical
analysis via the weak tunnel coupling to the superconductor. Such treatment,
which relies on a calculation of the self-energy, gives rise to the finite
coupling between the electron- and hole-like modes of the superconductor,
which is second-order in the tunneling. As a result, secondary gaps are opened
between all degenerate modes of the closed cavity, see Supplemental Material
[62] for details. This conclusion is also valid when the tunnel coupling is
strong, i.e., when there are no barriers at all, cf. Figs. 2(a) and (b).

To quantify the impact of the barrier in both $s$\- and $p$-wave cases, we
study the visibility of conductance oscillations defined as

$\nu=\frac{1}{N}\sum_{i=1}^{N}\frac{G_{i}^{\text{max}}-G_{i}^{\text{min}}}{G_{i}^{\text{max}}+G_{i}^{\text{min}}}\,,$
(6)

where $G_{i}^{\text{max}}$ and $G_{i}^{\text{min}}$ are the neighboring local
maximum and minimum values of the conductance, see grey arrows in Figs. 1(b)
and (c). We numerically calculate the visibility of the first five periods of
oscillation ($N=5$) as a function of the barrier strength $Z$, see Fig. 3.
While in the absence of the barriers, $Z=0$, both types of superconductors
have the same values of visibility, at large $Z$ the visibility increases with
$Z$ for the $p$-wave superconductor and saturates to a small constant for the
$s$-wave superconductor. Such behavior reflects the analytical discussion
above on secondary gaps.

Figure 3:  (a) Visibility of the conductance oscillations defined in Eq. (6)
as a function of the barrier strength for the $s$-wave (red dots) and the
$p$-wave (blue squares) cases. (b) Visibility of the first oscillation, i.e.,
the one closest to the gap for both $s$-and $p$-wave cases. (c) The height of
the first oscillation, $\Delta G\equiv G_{1}^{\max}-G_{1}^{\min}$, for both
cases. We used the same parameters as in Fig. 1.

Note that when $Z>2$ in the $s$-wave case [dashed lines inFig. 3(a)], the
height of the conductance oscillations is so small that it becomes comparable
to fluctuations caused by the finite-element numerical integration over Eq.
(4). On the other hand, the height of the first oscillation is distinguishable
for much larger $Z$ in both the $s$\- and $p$-wave cases. We plot the
visibility of that first oscillation peak in Fig. 3(b), from which the same
trend can be extracted as in Fig. 3(a). Lastly, in Fig. 3(c), we plot the
height of the aforementioned first oscillation – defined as $\Delta G\equiv
G_{1}^{\max}-G_{1}^{\min}$ – for both cases. The height decays for large $Z$,
but with a slower rate in the $p$-wave case.

In conclusion, we have demonstrated that Tomasch oscillations of Bogoliubov
quasiparticles provide a promising method to distinguish between topological
$p$-wave superconductivity and conventional $s$-wave superconductivity.
Specifically, the resulting conductance oscillations display contrasting
behavior for the two types of superconductors as the interface barriers
increase, which is a direct manifestation of the pairing symmetries. Our study
introduces bulk probes for identifying topological superconductivity, which is
usually overlooked. Our proposed above-gap transport signature can serve as an
essential supplement to in-gap measurements [62]. Interestingly, Tomasch
oscillations were first reported approximately 60 years ago in both Pb and In
films with thicknesses ranging from 3 to 30 $\mu$m [49, 50], making their
observation in junctions with topological superconductors highly promising
using state-of-the-art techniques. Furthermore, some recent experiments on
normal-metal/insulator/superconductor junctions [63] reported high tunability
of the barrier strength $Z$. By changing the thickness of the barrier, $Z$ is
easily tuned to $Z\gg 10$, which lies deeply in the regime that we discuss in
our work. Last, our mechanism is established for continuum single-band models
without the presence of spin-orbit coupling, where the band inversion and
pairing mechanism dictate the appearance of topology; it would be interesting
to extend the discussion to more complicated multi-band systems as well as
lattice models with anisotropy. There, anisotropy can cause a significant
morphing of the sombrero-shaped band structure, which can even lead to
topological phase transitions by gap closing at momenta away from the $\Gamma$
point [64]. Whether our method can be applied to distinguish between different
phases in that case will be the focus of future work.

## Acknowledgements

We are grateful to W. Belzig for insightful discussions. W.C. acknowledges
financial support from the National Natural Science Foundation of China under
Grant No. 12074172 and No. 12222406, and the National Key Projects for
Research and Development of China under Grant No. 2022YFA120470. X.R.C.
acknowledges financial support from the National Natural Science Foundation of
China under Grant No. 12304064. D.Y.X. acknowledges financial support from the
State Key Program for Basic Researches of China under Grants No.
2021YFA1400403. A.Š. acknowledges financial support from the Swiss National
Science Foundation (Grant No. 199969). O.Z. acknowledges financial support
from the Deutsche Forschungsgemeinschaft (DFG) - project number 449653034. All
data that support the plots within this paper are available from the
corresponding author upon request.

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## Supplemental Material for “Tomasch Oscillations as Above-Gap Signature of
Topological Superconductivity”

The Supplemental Material is structured as follows: in Sec. .1, we first
elucidate the mechanism of opening of secondary gaps for the $p$-wave case in
an isolated superconducting cavity (SC) and the lack of it in the $s$-wave
case. This explains the differences in transmission patterns (Fig. 2 in the
main text) and conductance oscillations [Fig. 1(c) in the main text] in the
strong barrier limit ($Z\gg 1$). In Sec. .2, we show how coupling to the
metallic leads provides the mechanism for opening secondary gaps also in the
$s$-wave case. In Sec. .3, we show the dependence of the oscillations in
conductance on the length of the SC. Lastly, in Sec. .4, we briefly discuss
some clear advantages and potential drawbacks of our method.

###  .1 Opening of secondary gaps in an isolated semi-infinite superconducting
cavity

Figure S.1:  Band structure for an isolated superconducting cavity that is
infinite in $y$-direction, and finite in $x$-direction with width $L$. The
upper plot shows the case of a $p$-wave superconductor where the secondary
gaps open between electronic and hole modes with different parities. Full
black circles mark secondary gaps that open, while dashed circles mark the
ones that remain closed even for finite $\Delta_{p}$. The lower plot shows the
case of an $s$-wave superconductor where the secondary gaps remain closed for
any value of $\Delta_{s}$. To emphasize all the features of the band
structure, i.e. shape of the sombrero hat, the main gap, regimes
inside/outside the sombrero hat where only particles/both particles and holes
exist and secondary gaps, we changed the parameters – compared to the main
text – and used the following: $L=2\xi_{0}$, $k_{\rm F}=20\xi_{0}^{-1}$,
$\mu_{S}=10\Delta_{s}$ ($\mu_{S}=10\Delta_{p}k_{F}$ for the $p$-wave SC),
where $\xi_{0}$ is the superconducting coherence length. Note that due to the
relatively small $L$, the two in-gap modes in the upper plot are hybridised
around $k_{y}\xi_{0}=0$, which results in a small gap opening and vanishing of
the dispersion at zero energy. This effect is not observed for much larger
cavities discussed in the main text, see Fig. 2(d).

In this section, we study the Hamiltonian of an isolated superconducting
cavity (SC), restricted to a strip geometry with a finite length $L$ in the
$x$-direction and infinite width in the $y$-direction. The spectrum of the SC
drastically changes depending on the type of superconductivity involved,
namely $s$\- or $p$-wave pairing, which introduces coupling between electron
and hole [cf. Eq. (3) of the main text]. Neglecting the pairing potential
altogether, independent standing modes are formed in the $x$-direction, while
in the $y$-direction the wavefunctions are simply free waves. Once $s$-wave
coupling is introduced, we show that only the standing modes with the same
orbital quantum number hybridize, and open the main gap $\Delta_{s}$ in the
energy spectrum. On the other hand, if the coupling is of $p$-wave type, each
standing mode in the $x$-direction is coupled to all other modes that have the
opposite parity (including itself). Therefore, besides the main gap that opens
in the spectrum, additional secondary gaps appear at higher energies, see Fig.
S.1. This finding explains the behavior of the transmission in Fig. 2 of the
main text in the limit of a high barrier ($Z\gg 1$), and consequentially the
behavior of the visibility of the oscillations in Fig. 3 of the main text.

Let us start by rewriting the BdG Hamiltonian of the isolated SC in real space
in terms of fermionic field operators as

$\displaystyle\mathcal{H}_{\rm SC}$
$\displaystyle=\int\differential\mathbf{r}\,\Psi^{{\dagger}}(\mathbf{r})\,H_{\rm
SC}(\hat{\mathbf{k}})\,\Psi(\mathbf{r}),$ (S.1) $\displaystyle H_{\rm
SC}(\hat{\mathbf{k}})$ $\displaystyle=\begin{pmatrix}\frac{-\hbar^{2}}{2m_{\rm
S}}\mathbf{\nabla}^{2}-\mu_{\rm S}&\Delta(-i\nabla)\\\
\Delta^{\dagger}(-i\nabla)&\frac{\hbar^{2}}{2m_{\rm
S}}\mathbf{\nabla}^{2}+\mu_{\rm S}\end{pmatrix}\,,$ (S.2)

where $\Psi(\mathbf{r})=(\psi(\mathbf{r}),\psi^{\dagger}(\mathbf{r}))^{T}$ is
the Nambu spinor, and the wave vector is replaced by the operator
$\hat{\mathbf{k}}\equiv{-i\mathbf{\nabla}}$. Here, we have dropped the spin
index for brevity which is irrelevant to our analysis. Physically, $s$-wave
paired electrons are in the spin-singlet states with opposite spins while the
$p$-wave pairing occurs either between spinless fermions or between spinful
fermions in the spin-triplet states. Here, we only concentrate on the orbital
part of the wave function. The field operators for the infinite strip geometry
can be expanded as

$\displaystyle\psi(\mathbf{r})$
$\displaystyle=\sqrt{\frac{2}{LW}}\sum_{\alpha,k_{y}}e^{ik_{y}y}\sin(\frac{\alpha\pi}{L}x)d_{\alpha,k_{y}},$
(S.3) $\displaystyle\psi^{{\dagger}}(\mathbf{r})$
$\displaystyle=\sqrt{\frac{2}{LW}}\sum_{\beta,k_{y}^{\prime}}e^{-ik_{y}^{\prime}y}\sin(\frac{\beta\pi}{L}x)d^{{\dagger}}_{\beta,k_{y}^{\prime}}\,,$
(S.4)

with $d,d^{{\dagger}}$ the fermionic annihilation and creation operators, and
$\alpha,\beta\in\mathbb{N}$ are the quantum numbers associated with the
standing modes in the $x$-direction. We can now make use of the orthogonal
normalization conditions

$\displaystyle\frac{2}{L}\int_{0}^{L}\differential
x\sin(\frac{\alpha\pi}{L}x)\sin(\frac{\beta\pi}{L}x)$
$\displaystyle=\delta_{\alpha,\beta},$
$\displaystyle\frac{1}{W}\int_{-\infty}^{\infty}\differential
y\,e^{i(k_{y}-k_{y}^{\prime})y}$
$\displaystyle=\delta_{k_{y},k_{y}^{\prime}},$ (S.5)

to obtain the Hamiltonians in the second quantization

$\displaystyle\mathcal{H}_{\rm SC}^{s}=$
$\displaystyle\sum_{\alpha,k_{y}}D^{\dagger}_{\alpha,k_{y}}\begin{pmatrix}\varepsilon_{\alpha}(k_{y})&\Delta_{s}\\\
\Delta_{s}^{*}&-\varepsilon_{\alpha}(k_{y})\end{pmatrix}D_{\alpha,k_{y}}\,,$
(S.6) $\displaystyle\mathcal{H}_{\rm SC}^{p}=$
$\displaystyle\sum_{\alpha,\beta,k_{y}}D^{\dagger}_{\alpha,k_{y}}\begin{pmatrix}\varepsilon_{\alpha}(k_{y})\delta_{\alpha,\beta}&-\Delta_{p}[k_{y}\delta_{\alpha,\beta}+q(\alpha,\beta)]\\\
-\Delta_{p}^{*}[k_{y}\delta_{\alpha,\beta}-q(\alpha,\beta)]&-\varepsilon_{\alpha}(k_{y})\delta_{\alpha,\beta}\end{pmatrix}D_{\beta,k_{y}}\,,$
(S.7)

where the fermionic operator is defined as
$D_{\alpha/\beta,k_{y}}=(d_{\alpha/\beta,k_{y}},d_{\alpha/\beta,-k_{y}}^{\dagger})^{T}$,
the superscript $s,p$ denote the $s$\- and $p$-wave pairing, respectively, and

$\displaystyle\varepsilon_{\alpha}(k_{y})$
$\displaystyle=\frac{\hbar^{2}k_{y}^{2}+\hbar^{2}\alpha^{2}\pi^{2}/L^{2}}{2m_{\rm
S}}-\mu_{\rm S}\,,$ (S.8) $\displaystyle q(\alpha,\beta)$
$\displaystyle=\frac{2\alpha\beta}{L(\alpha^{2}-\beta^{2})}[\cos(\pi\alpha)\cos(\pi\beta)-1].$
(S.9)

From Eq. (S.6) it follows that in the case of an $s$-wave superconductor,
standing modes with different quantum numbers $\alpha$ remain uncoupled and
the band structure is simply
$E_{\alpha}(k_{y})=\sqrt{\varepsilon_{\alpha}^{2}(k_{y})+\Delta_{s}^{2}}$, see
the lower panel of Fig. S.1. The main gap is opened due to the coupling
between electron and hole modes with the same $\alpha$. On the other hand, the
situation is more complicated in the $p$-wave superconductor. Besides the
opening of the main gap, caused by the off-diagonal term $-\Delta_{p}k_{y}$ in
Eq. (S.7), there are secondary gaps opening at crossings of modes with
different $\alpha$ quantum numbers due to the additional off-diagonal term
$-\Delta_{p}q(\alpha,\beta)$, see the upper panel of Fig. S.1. Furthermore,
the factor $\cos(\alpha\pi)\cos(\beta\pi)-1$ in Eq. (S.9) vanishes if $\alpha$
and $\beta$ are of the same parity and it is equal to -2 if the parities are
opposite. This explains the double helix shape that bands exhibit at low
$k_{y}$ in the upper panel of Fig. S.1. From the prefactor
$\alpha\beta/(\alpha^{2}-\beta^{2})$ in Eq. (S.9) it follows that the coupling
between different standing modes reduces with increasing difference between
the quantum numbers, which results in the reduction of the size of secondary
gaps when energy is increased away from the main gap. As a consequence, the
width of the bands resembling a double helix reduces for bands closer to the
main gap, and they become flatter. The same effect occurs in the transmission
plot for high barriers, see Fig. 2(d) and (f) of the main text. Note that the
secondary gaps, being opened by the pairing potential, are robust to the
change of coupling with the external metallic leads, which explains the high
values for visibility for $Z\gg 1$ in Fig. 3 of the main text.

###  .2 Introducing the coupling to leads as a small perturbation

In Sec. .1, we showed how the momentum-dependence of the $p$-wave
superconducting pairing opens the secondary gaps in the band structure of an
SC that is finite in the $x$-direction and infinite in the $y$-direction. On
the other hand, the $s$-wave coupling does not hybridize different standing
modes of the SC, and the secondary gaps remain closed for any value of
$\Delta_{s}$. In this section, we introduce weak tunnel-coupling $t$ between
the SC and the metallic leads, i.e., high, but finite barrier $Z>1$. Thus, we
show that hybridization with the metals can open secondary gaps, which
explains the structure of transmission in Fig.2(a) – and with that the
existence of oscillations in conductance – for the $s$-wave superconductor.

Here, we concentrate on the $s$-wave SC when it is coupled to two-dimensional
metallic leads that are infinite in the $y$-direction and semi-infinite in the
$x$-direction. In the following, we treat the tunneling Hamiltonian as a small
perturbation and obtain an effective Hamiltonian for the SC. The BdG
Hamiltonian of a whole system is

$\mathcal{H}=\mathcal{H}^{s}_{\rm SC}+\mathcal{H}_{\rm N}+\mathcal{H}_{\rm
T}\,,$ (S.10)

where $\mathcal{H}^{s}_{\rm SC}$ is given in Eq. (S.6), the metallic leads are
described by

$\displaystyle\mathcal{H}_{\rm N}$
$\displaystyle=\sum_{\mathbf{k}}C_{\mathbf{k}}^{\dagger}H_{\rm
N}(\mathbf{k})C_{\textbf{k}}\,,$ (S.11)

with $H_{\rm N}(\mathbf{k})$ given by Eq. (2) of the main text and
$C_{\textbf{k}}=(c_{\textbf{k}},c_{-\textbf{k}}^{\dagger})$ is the fermionic
operator in the leads, and the tunnel coupling term reads

$\displaystyle\mathcal{H}_{\rm T}$
$\displaystyle=\sum_{\textbf{k},\alpha}C^{\dagger}_{\textbf{k}}H_{\rm
T}D_{\alpha,k_{y}}+h.c.\,,$ (S.12) $\displaystyle H_{\rm T}$
$\displaystyle=\begin{pmatrix}t&0\\\ 0&-t^{*}\end{pmatrix},$ (S.13)

with the coupling strength being $t$ which, for the moment, we assume to be
constant.

Applying the unitary transformation,

$\displaystyle d_{\alpha,k_{y}}$
$\displaystyle=u_{\alpha}\gamma_{\alpha,k_{y}}+v_{\alpha}\gamma_{\alpha,-k_{y}}^{\dagger}\,,$
(S.14)

with

$\displaystyle
u_{\alpha}^{2}=1-v_{\alpha}^{2}=\frac{1}{2}\left(1+\frac{\varepsilon_{\alpha}(k_{y})}{E_{\alpha}(k_{y})}\right)\,,$
(S.15)

and

$\displaystyle
E_{\alpha}(k_{y})=\sqrt{\varepsilon_{\alpha}(k_{y})^{2}+|\Delta_{s}|^{2}}\,,$
(S.16)

we can diagonalize $\mathcal{H}^{s}_{\rm SC}$ [cf. Eq. (S.6)] as

$\displaystyle\mathcal{H}^{s}_{\rm SC}=$
$\displaystyle\sum_{\alpha,k_{y}}\Gamma_{\alpha,k_{y}}^{\dagger}\begin{pmatrix}E_{\alpha}(k_{y})&0\\\
0&-E_{\alpha}(k_{y})\end{pmatrix}\Gamma_{\alpha,k_{y}}\,,$ (S.17)

where
$\Gamma_{\alpha,k_{y}}=(\gamma_{\alpha,k_{y}},\gamma^{\dagger}_{\alpha,-k_{y}})^{T}$.
Note that $\mathcal{H}^{s}_{\rm SC}$ is diagonal in the basis of the standing
modes inside the superconducting region. At the same time, the tunneling
Hamiltonian is transformed into

$\displaystyle\mathcal{H}_{\rm T}=$
$\displaystyle\sum_{\textbf{k},\alpha}C_{\textbf{k}}^{\dagger}\tilde{H}^{\alpha}_{\rm
T}\Gamma_{\alpha,k_{y}}+h.c.,$ (S.18) $\displaystyle\tilde{H}^{\alpha}_{\rm
T}=$ $\displaystyle\begin{pmatrix}u_{\alpha}(k_{y})t&v_{\alpha}(k_{y})t\\\
-v_{\alpha}(k_{y})t^{*}&-u_{\alpha}(k_{y})t^{*}\end{pmatrix}.$ (S.19)

Figure S.2:  Band structure of an $s$-wave SC [infinite in the $y$-direction,
and finite in the $x$-direction with length $L$]. (left plot) No coupling to
the leads. (right plot) Coupling to the leads lifts the degeneracy of all
crossings and opens secondary gaps in the spectrum also away from the main
gap. To emphasize all the features of the band structure, i.e. shape of the
sombrero hat, the main gap, regimes inside/outside the sombrero hat where only
particles/both particles and holes exist and secondary gaps opening when
coupling to the leads is included, we changed the parameters – compared to the
main text – and used the following: $L=2\xi_{0}$, $k_{\rm F}=20\xi_{0}^{-1}$,
$\mu_{S}=10\Delta_{s}$, where $\xi_{0}$ is the superconducting coherence
length.

Let us now calculate the self-energy correction to the SC Hamiltonian that
arises because of the coupling with the leads. We choose
$\left(\Gamma_{\\{\alpha=1,\cdots,n\\},k_{y}},C_{\\{k_{x}\in(-\infty,\infty)\\},k_{y}}\right)$
as the basis and the matrix elements of the self-energy of the SC due to the
tunnel coupling is calculated by

$\displaystyle\Sigma_{\alpha\beta}(\omega,k_{y})$
$\displaystyle=\sum_{k_{x}}\tilde{H}_{\rm T}^{\alpha{\dagger}}(k_{y})G_{\rm
N}(\textbf{k},\omega)\tilde{H}_{\rm T}^{\beta}(k_{y}),$ (S.20) $\displaystyle
G_{\rm N}(\textbf{k},\omega)$
$\displaystyle=\frac{1}{\omega-\epsilon(\textbf{k})\sigma_{z}+i0^{+}},$ (S.21)

where $G_{\rm N}(\textbf{k},\omega)$ is the retarded Green’s function for the
leads with $\epsilon(\textbf{k})=\hbar^{2}(k_{x}^{2}+k_{y}^{2})/(2m_{\rm
N})-\mu_{\rm N}$ the energy of electron therein.

We are interested in the opening of the secondary gaps between different
standing modes in the SC. The positive and negative eigenvalues $\\{\pm
E_{\alpha}(k_{y})\\}$ have already been separated by the main gap $\Delta_{s}$
so that it is sufficient to consider the matrix elements of the self-energy
that couple different modes with positive energy $\\{+E_{\alpha}\\}$, which
reads

$\displaystyle\Sigma_{\alpha\beta}^{++}(k_{y})$
$\displaystyle=T_{\alpha\beta}(k_{y})=T_{0}(u_{\alpha}u_{\beta}+v_{\alpha}v_{\beta}),$
$\displaystyle T_{0}$
$\displaystyle=\sum_{k_{x}}\frac{|t|^{2}}{\omega-\epsilon(\textbf{k})+i0^{+}}.$
(S.22)

Moreover, given that the high interface barrier has pushed the electronic
states in the leads (in the vicinity of the interface) far away above the gap
$\Delta_{s}$, $T_{0}$ can be assumed to be real and constant in the interested
energy interval $\omega\sim\Delta_{s}$. We can now write down the effective
Hamiltonian which describes the bands of the SC with positive energy as

$\displaystyle\mathcal{H}^{s+}_{\rm SC}=$
$\displaystyle\sum_{\alpha,\beta,k_{y}}\gamma_{\alpha,k_{y}}^{\dagger}\left[E_{\alpha}(k_{y})\delta_{\alpha,\beta}+T_{\alpha\beta}(k_{y})\right]\gamma_{\beta,k_{y}},$
(S.23)

or in its matrix form as

$\displaystyle H^{s+}_{\rm
SC}=\left(\begin{array}[]{cccc}E_{1}+T_{11}&T_{12}&\cdots&T_{1n}\\\
T_{21}&E_{2}+T_{22}&\cdots&\vdots\\\ \vdots&\vdots&\ddots&T_{n-1,n}\\\
T_{n1}&\cdots&T_{n,n-1}&E_{n}+T_{nn}\end{array}\right)_{n\times n}\,,$ (S.28)

where the truncation is made to the lowest $n$ standing modes with the energy
much higher than $\Delta_{s}$. One can see that the self-energy introduces the
off-diagonal elements that couple different standing modes and thus open
secondary gaps between neighboring standing modes inside the cavity. We plot
the energy modes of Eq. (S.28) numerically in Fig. S.2.

###  .3 Conductance as a function of the length of the superconducting cavity

Figure S.3:  Differential conductance as a function of energy through the NSN
junction containing a $p$-wave superconductor for different lengths $L$ and
for (a) $Z=0$ and (b) $Z=4$.

In Fig. S.3, we show differential conductance obtained from Eq. (4) for the
case of a $p$-wave superconducting cavity for different cavity lengths. In
Fig. S.3(a), a transparent limit with $Z=0$ is presented, while in Fig. S.3(b)
we show the tunnelling limit with $Z=4$. For both weak and strong barriers,
the energy dependent period of oscillations doubles when the length $L$ is
halved, as it is expected from the interference condition
$k_{x}^{e/h}(E)-k_{x}^{h/e}(E)=2\pi n/L$, where $n$ is an integer. More
details about this limit can be found in Refs. [46, 61].

Furthermore, in both cases the height of oscillations compared to the
background, and therefore their visibility, reduces with increasing $L$. For
the transparent limit shown in Fig. S.3(a) this effect can be contributed to
the broadening of the resonances in the superconducting cavity caused by the
hybridization with the external leads, similarly to the case of 1D Fabry-Pérot
oscillations. Bringing the resonances closer together by increasing $L$
results in increase of the background conductance in the energy interval
between the resonances.

On the other hand, in the tunnelling limit, shown in the Fig. S.3(b), besides
the small broadening of the resonances caused by a weak coupling to the leads,
the height of oscillations additionally reduces because of the fact that the
off-diagonal elements in Eq. (S.7), which are responsible for opening the
secondary gaps, scale as $1/L$, see Eq. (S.9).

###  .4 A short survey of the advantages and drawbacks of contemporary methods
proposed for the identification of topological superconductors

Heretofore, several methods have been proposed with the goal of unequivocally
detecting topological superconductivity. These detection schemes are based on
unique signatures that these systems should have, such as surface states,
zero-bias peak in vortex cores where the Majorana mode is expected to exist,
and the quantisation of the aforementioned states. All of these in-gap
signatures, unfortunately, can emerge from other trivial reasons, e.g., from
disorder that can introduce trivial surface states and result in a zero-bias
peak, Caroli-de Gennes-Matricon states in vortices [65, 66], and Yu-Shiba-
Rusinov states [67, 68, 69], that can generate similar experimental signatures
as those of Majorana modes. Our proposal is immune to the aforementioned in-
gap disorder-induced effects, since it concentrates on interference effects
above the gap. The same disorder, that would produce false-positive results in
other measurements, will close the secondary gaps above the bottom of the
band, and reduce the oscillations in conductance – therefore giving a correct,
negative result in this case.

Other phase-sensitive probes that provide some information about the gap
symmetry make use of cleverly designed SQUID interferometers, e.g., corner
[70] or GLB SQUID interferometers [71] attached to 3D superconducting
crystals. However, such experiments are notoriously difficult to perform and
include many effects induced by contact resistance between two different types
of superconductors. Moreover, they can discern the parity of the gap phase,
but cannot resolve the $p$\- versus $f$-wave issue [3]. Furthermore, the
aforementioned experiments are relevant for 3D superconducting samples, but
their generalisation to 2D systems is still lacking.

Lastly, let us mention some clear advantages and disadvantages of our method.
In contrast to the common in-gap measurements that can give false-positive
results caused by trivial effects of disorder, the Tomasch oscillations, being
a coherent phenomenon, would be washed out in the presence of disorder and,
therefore, correctly identify the trivial superconductor. However, in the case
of a chiral $d$-wave superconductivity – where the pairing is an even function
of momenta – the secondary gaps will remain closed, and therefore, we do not
expect to observe any oscillations in conductance. This would clearly be a
false negative result of our method.