# Realizing attractive interacting topological surface fermions: A resonating
TI- thin film hybrid platform

Saran Vijayan and Fei Zhou  Department of Physics and Astronomy, University of
British Columbia, 6224 Agricultural Road, Vancouver, BC, V6T 1Z1, Canada

###### Abstract

In this article, we propose a practical way to realize topological surface
Dirac fermions with tunable attractive interaction between them. The approach
involves coating the surface of a topological insulator with a thin film metal
and utilizing the strong-electron phonon coupling in the metal to induce
interaction between the surface fermions. We found that for a given TI and
thin film, the attractive interaction between the surface fermions can be
maximally enhanced when the Dirac point of the TI surface resonates with one
of the quasi-2D quantum-well bands of the thin film. This effect can be
considered to be an example of ’quantum-well resonance’. We also demonstrate
that the superconductivity of the resonating surface fermions can be further
enhanced by choosing a strongly interacting thin film metal or by tuning the
spin-orbit coupling of the TI. This TI-thin film hybrid configuration holds
promise for applications in Majorana-based quantum computations and for the
study of quantum critical physics of strongly attractively interacting surface
topological matter with emergent supersymmetry.

††preprint: APS/123-QED

##  I Introduction

Topological Insulators(TI)[1, 2, 3, 4, 5, 6, 7, 8] belong to the class of
symmetry-protected topological phases, where the gapless boundary states are
protected by Time-Reversal Symmetry(TRS). One interesting feature of these
surface states is that their low-energy excitations can resemble a single
flavor of 2-component massless Dirac fermions($N_{f}=1/2$). This is unique
because it is impossible to realize an odd number of flavors of 2-component
Dirac fermions in a bulk lattice because of the fermion doubling problem
intrinsic to lattice models[9, 10]. Therefore topological surface provides an
interesting platform to study various interactions involving a single flavor
of 2-component Dirac fermions, provided the interactions do not break the
time-reversal symmetry.

Of particular interest is when there is an effective attractive interaction
between the surface fermions. For a finite chemical potential (i.e. when the
Fermi level is above or below the Dirac point), the U(1) symmetry breaking
leading to the superconducting phase can happen for arbitrarily weak
attractive interaction due to the Cooper instability at the surface. On the
other hand at the zero chemical potential (Fermi level aligned with the Dirac
point), the interaction strength must be greater than a critical value for the
phase transition to occur. In both these cases, the resulting superconducting
phase can be of non-trivial topological character[11, 12]. Specifically, it
implies that the vortex core of the superconductor can host Majorana zero
modes[13, 14, 15, 16, 17]. They are considered to be an ideal candidate for
fault-tolerant quantum computing because of their non-Abelian statistics.

Another interesting feature of the attractively interacting surface fermions
is that surface dynamics have an emergent Lorentz symmetry when the chemical
potential is zero. It has been demonstrated that the effective field theory of
the surface states further exhibits emergent supersymmetry (SUSY) when the
coupling constant of the attractive interactions is tuned to be quantum
critical [18, 19, 20, 21, 22, 23]. Supersymmetry is the symmetry between
bosons and fermions and had been speculated to exist as a fundamental symmetry
in elementary particle physics. Emergent supersymmetry in lattice models is
difficult to realize, at least in $d>1$ spatial dimensions, because fermions
typically have more degrees of freedom than bosons in lattices, a consequence
of the fermion doubling problem. But at a quantum critical point of
topological surfaces, an emergent SUSY exists between the charge 2e bosons
that naturally emerge as quasi-particles and the 2-component Dirac fermions in
the semi-metallic phase, both of which can be strongly self-interacting and
mutually interacting. Therefore, the topological surface provides an ideal
platform to study the dynamics of supersymmetric quantum matter.

However, realizing a topological surface with net attractive interactions
between them is not straightforward and can be challenging. One reason is the
unscreened nature of repulsive Coulomb interactions in an insulator. In
addition, many topological insulator materials do not have strong electron-
phonon interactions. In this article, we propose coating the 3D TI surface
with a metallic thin film as a practical way to realize a ground state of
interacting surface fermions with net attractive interaction between them. A
thin film is characterized by the quasi-2-dimensional quantum-well bands due
to the quantum confinement of the electronic states in the third dimension.
Due to the screened nature of Coulomb repulsion, the phonon-mediated
attractive interaction can be the dominant form of interaction between
electrons at zero temperature. On depositing the thin film to the TI surface,
the 2D surface Dirac fermions and the quasi-2D quantum-well fermions start
hybridizing. These hybrid fermions are a quantum superposition of the quantum-
well states and the TI surface states. Hence the hybrid fermions not only
acquire a helical spin-texture from the surface side but will also experience
a net attractive interaction due to coupling with the phonons in the thin
film. In a way, hybridization causes the surface Dirac cone to be exported to
the thin film which results in the helical Dirac fermions experiencing a
phonon-mediated attractive interaction between them. Alternatively, one can
show that the hybridization leads to variable or tunable attractive
interactions among topological surface Dirac fermions.

We have observed that this attractive interaction between the helical fermions
is maximally enhanced when the Dirac point of the TI surface resonates with
one of the quantum-well states of the thin film. While at resonance, there is
no clear distinction between the TI surface and the thin film states as the
electronic states are strongly hybridized, we do show that in the wide range
of parameter space, the low energy physics effectively becomes that of
strongly interacting surface Dirac fermions. We study the superconductivity of
these resonating hybrid states at different thickness regimes. Consider the
ultra-thin limit of the film, when only a single quantum-well (QW) band
crosses the Fermi level (we shall call this the $N=1$ limit, where $N$ is the
number of QW bands crossing the Fermi level). Then we effectively have a four-
band model of the interacting helical hybrid states. Following the bulk-
boundary relations(BBR) of interactions obtained before[24], we find that
effective phonon-mediated interaction scales as $1/D$, $D$ being the thickness
of the film and hence the interaction strength is at its strongest in the
$N=1$ ultrathin limit. We show that for a wide range of chemical potentials,
it is possible to construct an effective field theory of attractively
interacting 2-component Dirac fermions. We then studied possible ways of
enhancing the superconducting gap by tuning the bulk coupling constant of the
thin film and the Dirac velocity of the surface fermions.

When the thin film thickness is increased further, in addition to the Fermi
surfaces formed by the resonating hybrid bands, there also exists Fermi
surfaces formed by the QW bands that were off-resonance. Therefore the
superconducting gap in this limit is formed not just due to attractive
interaction between the surface fermions but also because of the scattering of
the singlet pair of electrons from these background off-resonance QW Fermi
surfaces. In the very thick limit (large-$N$ limit), we explicitly show that
the superconductivity on these resonating hybrid bands is dominated by the
scattering of the singlet pair of electrons from the off-resonance Fermi
surfaces. However, the interaction between the surface fermions can further
enhance the surface superconductivity. And when the interactions are
sufficiently strong, the enhancement can be very substantial.

It should be noted here that if one’s prime focus is mainly to realize a
topological superconducting phase, then it is not necessary to have attractive
interaction between the surface fermions[12]. Superconductivity can be induced
on the surface by the proximity effect, implemented by depositing a bulk
s-wave superconductor on the TI surface. The interface between the TI and the
s-wave superconductor had been shown to be in the topological superconducting
phase even though the surface electrons are non-interacting. As mentioned
before, the main objective of our work is to realize a platform of strongly
interacting surface fermions. The attractive interactions between surface
fermions can lead to emergent SUSY at its QCP, a phenomenon that can
potentially have a lot of impacts on the fundamental understanding of the
building blocks of nature.

However, the TI-thin film hybrid has richer physics over the conventional
proximity structures even if our objective is only to realize a topological
superconducting phase. Due to the strong single-particle hybridization, there
is a finite probability of finding the surface fermions on the thin film side,
sometimes called the ’topological proximity effect’ [25, 26, 27]. Thus in this
structure, the topological superconducting phase can proliferate across the
interface, and can even be observed on the thin film side and not just at the
interface, making it easy to detect in the experiments [28].

We like to note here that the effect of tunneling of the TI surface fermions
on the superconductivity in the thin film has been extensively studied in
refs.[29, 30, 31]. Ref.[29] studied the superconductivity in the monolayer
thin film-TI hybrid as a function of tunneling strength and the chemical
potential. They found a suppression of the superconducting gap in the thin
film when the Fermi momentum of the thin film and the TI surface matched and
non-hybridized surface fermions were integrated out. Ref.[31] found an
enhancement in the superconducting order when the Fermi level crosses the
bottom of the double-well hybrid bands. This result is encouraging in the
context of thin film superconductivity. However, the Lifshitz transition
leading to the enhancement results in two additional fermion surfaces and does
not affect the topological aspect of superconducting pairing.

The main focus of this article on the other hand is to understand Dirac
fermions in the topological surface and their interactions and pairing
dynamics mediated by coated thin films. When the superconductivity of non-
interacting surface fermions (before the tunneling is turned on) is concerned,
we find in this work that the superconductivity on the surface fermions can be
induced and greatly enhanced if surface fermions are in resonance with
electrons in thin films. Although in the limit of resonance physically it is
not possible to entirely isolate the surface fermions from the thin film
electrons, the effective field-theory description in the most interesting
limit is simply of the form of interacting Dirac fermions but with various
substantially renormalized parameters. These renormalization effects
especially the fermion-field renormalization are one of the main focuses of
our studies and discussions below as they directly set the strength of
interactions mediated by the thin films. These renormalization effects can
either lead to surface superconductivity that otherwise won’t exist because of
the absence of direct pairing dynamics or further enhance the well-known Fu-
Kane proximity effects of non-interacting surface fermions[12]. The induced
surface fermion interactions are also shown to follow explicitly the generic
scaling law indicated in the general bulk-boundary interaction relation
obtained in a previous article[24].

The article is organized as follows: In the section II, we discuss the single
particle tunneling physics at the interface. We write down the single-particle
Hamiltonian for the hybrid fermions on the helicity basis. In section III,
starting from the fundamental electron-phonon coupling Hamiltonian of the thin
film, we derive a general short-ranged pairing Hamiltonian that explains the
interactions of the hybrid fermions with one another and with the thin film
electrons belonging to the off-resonance bands. Here we find that the hybrid
fermions acquire an effective attractive interaction between them and the
interaction strength is renormalized by a $Z$-factor. The $Z$ factor is
essentially a measure of the probability amplitude of the hybrid fermions to
be in the thin film side of the interface. In section IV, we study the
evolution of this $Z$-factor as a function of the dimensionless detuning
parameter $\tilde{\delta}$ and find that the attractive interaction between
the surface fermions is enhanced at the quantum-well resonance
($\tilde{\delta}=0)$. Section V is dedicated to the mean-field approximation.
Here we derive the superconducting gap equation under the assumption that the
Debye frequency $\omega_{D}\ll\mu$, where $\mu$ is the chemical potential.
Essentially, we assume that only electronic states near the Fermi level are
interacting. In section VI, we consider the limit when the surface states
hybridize with the $N=1$ QW band of the thin film. We construct an effective
theory followed by exploring various ways to enhance the superconducting gap
in this limit. Section VII discusses the large-$N$ limit of the theory. Here
we make connections to Fu-Kane’s model in the perturbative limit of tunneling.
In Section VIII, we studied the evolution of the superconductivity on the
resonating hybrid states as a function of thickness (parametrized by the band
index $N$).

Figure 1:  a)TI-thin film hybrid considered in this paper. b)Left: Schematic
picture of the surface states of the topological insulator(Red lines). The
gapped bulk bands are indicated by the green-shaded area. Here, $m$ stands for
the bulk energy gap. Right: Schematic picture of the effective 2D quantum-well
bands of the thin film considered in this article. The chemical potential
$\mu$(dashed line) is set within the bulk energy gap of the TI. We only
consider the case when the $N$th band that is energetically closest to the
Dirac point of the TI surface is separated from the two adjacent QW bands by
an energy separation much greater than the $m$, the bulk gap of the TI (See
Eqn.7). In this limit, the tunneling effects on the $N\pm 1$ QW bands from the
surface Dirac cone are negligible and can be conveniently ignored.

##  II Non-interacting theory

####  II.0.1 Model Hamiltonian

We start by defining a minimal theoretical model to understand the essential
tunneling physics at the thin film-topological insulator interface. Let the
thin film - TI interface be at $z=0$. The topological insulator(TI) occupies
the bottom half-plane defined by $z<0$. Consider a thin film of thickness $d$
deposited over the TI surface, so that it occupies the space $0<z<d$.

Let us first write down a simple model for thin film electrons. In the XY
plane, we apply periodic boundary conditions. The electron confinement in the
z-direction is usually characterized by an infinite well potential with its
boundaries at $z=0$ and $z=d$. However this model cannot permit tunneling of
thin film electrons to the TI side since the amplitude of the electron
wavefunction is zero at the interface. To allow for tunneling, a simple way is
to impose open boundary conditions at the interface so that the amplitude is
maximum at the interface. As a result, the momentum in the z-direction gets
quantized as $k_{z}=(n-1/2)\pi/d$ where $n=1,2,..$, and the z-dependence of
the electron wavefunction becomes $\psi_{n}(z)=\sqrt{2/d}\cos\left((n-1/2)\pi
z/d\right)$. Thus, the Hamiltonian governing the dynamics of thin film
electrons deposited over the TI has the form,

$\displaystyle\mathcal{H}^{\text{tf}}=$
$\displaystyle\sum_{n,s}\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}c^{\dagger}_{\textbf{k},n}h^{\text{tf}}_{\textbf{k},n}c_{\textbf{k},n}$
(1)

where

$\displaystyle h^{\text{tf}}_{\textbf{k},n}$ $\displaystyle=$
$\displaystyle\epsilon^{\text{tf}}_{\textbf{k},n}\hat{I}$ $\displaystyle=$
$\displaystyle\biggl{[}\frac{\hbar^{2}k^{2}}{2m^{*}}+(n-1/2)^{2}\frac{\pi^{2}\hbar^{2}}{2m^{*}d^{2}}\biggr{]}\hat{I}$

where
$c_{\textbf{k},n}=\begin{bmatrix}c_{\textbf{k},n,\uparrow}&c_{\textbf{k},n,\downarrow}\end{bmatrix}$
is the creation operator for an electron at the $n$th quantum well state with
in-plane momentum $\textbf{k}=(k_{x},k_{y})$ in the thin film. $\hat{I}$ is
just an identity matrix to emphasize that $h^{\text{tf}}$ is a $2\times 2$
matrix in the spin-1/2 space.

The effective Hamiltonian that describes the surface states of a topological
insulator is,

$\displaystyle\mathcal{H}^{\text{surf}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\chi^{\dagger}_{\textbf{k}}h^{\text{surf}}_{\textbf{k}}\chi_{\textbf{k}}$
(3)

where

$\displaystyle
h^{\text{surf}}_{\textbf{k}}=A_{0}\left(s_{x}k_{y}-s_{y}k_{x}\right)+E_{0}$

where
$\chi^{\dagger}=\begin{bmatrix}\chi^{\dagger}_{\uparrow}&\chi^{\dagger}_{\downarrow}\end{bmatrix}$
is the creation operator of the surface electron. $s_{x},s_{y}$ are Pauli
matrices in the spin-1/2 space. $A_{0}$ describes the strength of spin-orbit
coupling. $E_{0}$ is the energy at the Dirac point. Due to the presence of
spin-orbit coupling, the Hamiltonian does not have spin-rotation symmetry.
Rather it is diagonal in the helicity basis. The energy eigenstates in the
helicity basis are given by,

$\displaystyle\epsilon^{\text{surf}}_{\textbf{k},\pm}=\pm
A_{0}|\textbf{k}|+E_{0}$ (4)

Assuming that the tunneling process is spin-independent, the simplest model
that can describe the hybridization of the surface states of the TI with the
quantum well states of the thin film is given by,

$\displaystyle\mathcal{H}^{t}=t\int
d^{2}\textbf{r}\,\left(\chi^{\dagger}(\textbf{r})\Psi(\textbf{r},z=0)+h.c\right)$
(5)

here $\chi^{\dagger}(\textbf{r})$ is the spinor field operator that creates a
topological surface electron at in-plane position $\textbf{r}=(x,y)$.
$\Psi(\textbf{r})$ is the spinor field operator thin film electrons with open
boundary conditions. In the k-space, the Hamiltonian is of the form,

$\displaystyle\mathcal{H}^{t}$ $\displaystyle=$ $\displaystyle
t_{d}\,\sum_{n}\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\chi^{\dagger}_{\textbf{k}}c_{\textbf{k},n}+h.c$
(6) $\displaystyle t_{d}$ $\displaystyle=$ $\displaystyle\frac{t}{\sqrt{d}}$

We find that the effective tunneling strength scales as a function of the thin
film thickness $d$ as a result of quantum confinement in the $z$-direction.
The surface area in the $xy$-plane given by $L_{x}L_{y}$ is set to unity
throughout this paper.

In this article, we ignore the possibility of multi-band tunneling. This is a
good approximation provided we work in the limit where the energy difference
between successive thin film QW bands is greater than the bulk energy gap of
the topological insulator. In this limit, there is effectively only one QW
band on which the tunneling effects due to TI surface electrons are
significant. Since the chemical potential is aligned within the bulk energy
gap of the TI, this QW band will be the topmost conduction band of the thin
film. In other words, this band will be the one closest to the Dirac point of
the TI surface in terms of energy. Tunneling effects on other QW bands are
perturbative which is not the focus of our study in this section.
Quantitatively, the effective model that we introduce in this article works
well only when the following condition is satisfied,

$\displaystyle|\epsilon^{\text{tf}}_{\textbf{k}=0,N}-\epsilon^{\text{tf}}_{\textbf{k}=0,N\pm
1}|\geq m$ (7)

where $m$ is the mass gap of the topological insulator and $N$ is the band
index of the thin film QW band that is energetically closest to the Dirac
point of the TI surface. Once this condition is satisfied, we can conveniently
ignore the tunneling effects on all other $n\neq N$ bands. This setup is
illustrated schematically in Fig.1(b). Then the simplified effective
Hamiltonian of the electronic states involved in tunneling becomes,

$\displaystyle\mathcal{H}^{\text{hbd}}=$
$\displaystyle\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\biggl{[}c^{\dagger}_{\textbf{k},N}h^{\text{tf}}_{\textbf{k},N}c_{\textbf{k},N}+\chi^{\dagger}_{\textbf{k}}h^{\text{surf}}_{\textbf{k}}\chi_{\textbf{k}}$
$\displaystyle+$ $\displaystyle
t_{d}\,c^{\dagger}_{\textbf{k},N}\chi_{\textbf{k}}+\text{h.c}\biggr{]}$ (8)

####  II.0.2 Hybridization at the interface

Turning on $t$ results in thin film electrons tunneling to the TI surface side
and vice versa. Tunneling effects will be significant when
$\frac{t}{|\triangle E|}>1$, where $\triangle E$ is the difference in energy
between the initial and the final state. In this case, a perturbative
treatment won’t be sufficient. Here we shall understand the effects of
tunneling in a non-perturbative manner. The full Hamiltonian is diagonalized
exactly and the properties of the resulting hybrid electrons are studied. To
diagonalize the Hamiltonian, we shall define a $SU(2)$ space
$\sigma_{i}(i=x,y,z)$ to model the spatial profile of the electrons. In this
space, the single-particle Hamiltonian in the momentum space becomes,

$\displaystyle h^{\text{hbd}}_{\textbf{k}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}h^{\text{tf}}_{\textbf{k},N}&t_{d}\\\
t_{d}&h^{\text{surf}}_{\textbf{k}}\end{array}\right)$ (11) $\displaystyle=$
$\displaystyle I\otimes
M_{\textbf{k},N}+\sigma_{z}\otimes\delta_{\textbf{k},N}+\sigma_{x}\otimes
It_{d}$ (12)

in the basis
$\Gamma_{\textbf{k},N}^{\dagger}=\left(\begin{array}[]{cccc}c^{\dagger}_{\textbf{k},N,\uparrow}&c^{\dagger}_{\textbf{k},N,\downarrow}&\chi^{\dagger}_{\textbf{k},\uparrow}&\chi^{\dagger}_{\textbf{k},\downarrow}\end{array}\right)$.
Here $\delta_{\textbf{k},N}$ and $M_{\textbf{k},N}$ are $2\times 2$ matrices
in the spin-1/2 space with the respective definitions:

$\displaystyle\delta_{\textbf{k},N}$ $\displaystyle=$
$\displaystyle\left(h^{\text{tf}}_{\textbf{k},N}-h^{\text{surf}}_{\textbf{k}}\right)/2$
$\displaystyle M_{\textbf{k},N}$ $\displaystyle=$
$\displaystyle\left(h^{\text{tf}}_{\textbf{k},N}+h^{\text{surf}}_{\textbf{k}}\right)/2$
(13)

Since we discuss the hybridization effect only on the $N$th band in the thin
film, the index $N$ will be dropped from now on. But do note that the unitary
matrix elements do depend on the value of $N$ which in turn is connected to
the thickness of the thin film. The Hamiltonian can be diagonalized in the
$\sigma$ space by performing a unitary transformation with the following
unitary matrix,

$\displaystyle
U_{\textbf{k}}=\left(\begin{array}[]{ccc}\cos{\frac{\theta_{\textbf{k}}}{2}}&\sin{\frac{\theta_{\textbf{k}}}{2}}\\\
-\sin{\frac{\theta_{\textbf{k}}}{2}}&\cos{\frac{\theta_{\textbf{k}}}{2}}\end{array}\right),\,\,\cos\theta_{\textbf{k}}=\frac{\delta_{\textbf{k}}}{\sqrt{\delta^{2}_{\textbf{k}}+\frac{t^{2}}{d}}}$
(16)

The Hamiltonian after rotation attains the following diagonal form,

$\displaystyle\mathcal{H}^{\text{hbd}}=\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\left[d^{\dagger}_{\textbf{k},t}h_{\textbf{k},t}d_{\textbf{k},t}+d^{\dagger}_{\textbf{k},b}h_{\textbf{k},b}d_{\textbf{k},b}\right]$
(17)

where
$d^{\dagger}_{\textbf{k},t(b)}=\begin{bmatrix}d^{\dagger}_{\textbf{k},t(b),\uparrow}&d^{\dagger}_{\textbf{k},t(b),\downarrow}\end{bmatrix}$
are two-component spinors in the spin basis. $h_{\textbf{k},t(b)}$ have the
following definitions,

$\displaystyle
h_{\textbf{k},t}=M_{\textbf{k}}+\sqrt{\delta^{2}_{\textbf{k}}+t^{2}_{d}}$
(18a) $\displaystyle
h_{\textbf{k},b}=M_{\textbf{k}}-\sqrt{\delta^{2}_{\textbf{k}}+t^{2}_{d}}$
(18b)

Figure 2: Energy spectrum of the $N$th band of the thin film and the surface
Dirac cone before and after the tunneling is turned on. Here $N$ is set to
unity. The definitions of the creation operators shown in the picture are
contained in the main text.(See Eqns.1, 3 and 23).

here the index $t$ and $b$ represent the ’top’ and ’bottom’ bands
respectively. This splitting is a result of the tunneling of single-particle
states between the two sides of the hybrid. In addition, we find that
$h_{t(b)}$ are $2\times 2$ matrices in the spin-1/2 space.
$h_{\textbf{k},t(b)}$ has terms proportional to $s_{x}k_{y}-s_{y}k_{x}$
implying that the hybrid states acquired an emergent spin-orbit coupling.
Tunneling essentially resulted in hybridizing the thin film QW state and the
TI surface state. Due to this induced helical spin structure of the hybrid
states, it is better to write the full Hamiltonian on a helicity basis. We
define the following set of creation operators,

$\displaystyle
d^{\dagger}_{\textbf{k},t(b)}=a^{\dagger}_{\textbf{k},t(b)}\Pi^{\dagger}_{\textbf{k}},$
$\displaystyle\Pi_{\textbf{k}}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
e^{i\phi_{\textbf{k}}}&-e^{i\phi_{\textbf{k}}}\end{array}\right)$ (21)
$\displaystyle e^{i\phi_{k}}$ $\displaystyle=$
$\displaystyle\frac{k_{y}-ik_{x}}{|\textbf{k}|}$ (22)

where
$a^{\dagger}_{\textbf{k},t(b)}=\begin{bmatrix}a^{\dagger}_{\textbf{k},t(b),+}&a^{\dagger}_{\textbf{k},t(b),-}\end{bmatrix}$.
Here $(+)$ and $(-)$ represent states with positive and negative helicity
respectively. In this helicity basis, the single-particle Hamiltonian has the
following diagonal representation,

$\displaystyle\mathcal{H}^{\text{hbd}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\biggl{[}a^{\dagger}_{\textbf{k},t,+}\epsilon^{\text{hbd}}_{\textbf{k},t,+}a_{\textbf{k},t,+}$
(23a) $\displaystyle+$ $\displaystyle
a^{\dagger}_{\textbf{k},t,-}\epsilon^{\text{hbd}}_{\textbf{k},t,-}a_{\textbf{k},t,-}+a^{\dagger}_{\textbf{k},b,+}\epsilon^{\text{hbd}}_{\textbf{k},+}a_{\textbf{k},b,+}$
$\displaystyle+$ $\displaystyle
a^{\dagger}_{\textbf{k},b,-}\epsilon^{\text{hbd}}_{\textbf{k},b,-}a_{\textbf{k},b,-}\biggr{]}$
$\displaystyle\epsilon^{\text{hbd}}_{\textbf{k},t,\pm}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{\text{tf}}_{\textbf{k},N}+\epsilon^{\text{surf}}_{\textbf{k},\pm}}{2}+\sqrt{\left(\frac{\epsilon^{\text{tf}}_{\textbf{k},N}-\epsilon^{\text{surf}}_{\textbf{k},\pm}}{2}\right)^{2}+t^{2}_{d}}$
$\displaystyle\epsilon^{\text{hbd}}_{\textbf{k},b,\pm}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{\text{tf}}_{\textbf{k},N}+\epsilon^{\text{surf}}_{\textbf{k},\pm}}{2}-\sqrt{\left(\frac{\epsilon^{\text{tf}}_{\textbf{k},N}-\epsilon^{\text{surf}}_{\textbf{k},\pm}}{2}\right)^{2}+t^{2}_{d}}$

Fig.2 shows an example of the energy spectrum before and after the tunneling.
Given that the condition in Eqn.7 is satisfied, the tunneling effect on the
thin QW bands of index $n\neq N$ is perturbative and hence they are ignored.
Therefore, the single-particle Hamiltonian of all these $n\neq N$ QW bands is
unaffected by the tunneling and retains the form given in Eqn.1. The
electronic states in these bands will play huge role in the pairing physics
especially in the large-$N$ limit, as we shall see later in this article.

##  III Effective Pairing Hamiltonian

We examined the physics of single-particle tunneling at the thin film-TI
hybrid in the preceding section. We discovered that non-perturbative tunneling
results in the hybridization of the surface bands with the thin film’s
resonant quantum-well band. We now have a four-band model with single particle
states that are a linear superposition of the thin film state and the surface
state. As a result, it is possible that the hybrid states couple with the
phonons in the thin film. The effective short-ranged pairing Hamiltonian that
explains the interactions of the hybrid electrons with one another and with
the thin film electrons belonging to the inner bands is derived in this
section starting with the fundamental electron-phonon coupling Hamiltonian.

###  III.1 Phonon-mediated interaction potential between thin film electrons

####  III.1.1 2D electron-phonon coupling Hamiltonian

Similar to electrons, phonons in the thin film are also spatially confined
within the range $z=0$ and $z=d$. As a result, the phonon spectrum also gets
quantized resulting in the formation of 2D QW bands indexed by the integer
$l$. We implement open boundary conditions at the thin film-TI interface. The
phonon spectrum becomes, $E_{\text{ph}}(\textbf{q},l)=\hbar
c\sqrt{q^{2}+\left(\frac{(l-1/2)\pi}{d}\right)^{2}}$, where $l$ is an integer
identifying the confined slab phonon mode. The electron-phonon coupling
Hamiltonian in 3D has the form,

$\displaystyle\mathcal{H}_{\text{e-ph}}=G_{\text{fp}}\int
d^{2}\textbf{r}dz\,\Psi^{\dagger}(\textbf{r},z)\vec{\nabla}.\vec{\Phi}(\textbf{r},z)\Psi(\textbf{r},z)$
(24)

where $\Psi(\textbf{r})$ is the 2-component electron field operator and
$\Phi_{i}(\textbf{r})(i=x,y,z)$ is the phonon field operator in the thin film
with the following definitions,

$\displaystyle\Psi(\textbf{r},z)$ $\displaystyle=$
$\displaystyle\sum_{n}\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\,\psi_{n}(z)e^{i\textbf{k}.\textbf{r}}c_{\textbf{k},n}$
$\displaystyle\Phi_{i}(\textbf{r},z)$ $\displaystyle=$
$\displaystyle\sum_{l}\int\frac{d^{2}\textbf{q}}{(2\pi)^{2}}\phi_{l}(z)\frac{e^{i\textbf{q}.\textbf{r}}}{2\sqrt{E_{ph}(\textbf{q},l)}}\left[b_{\textbf{q},l,i}+b^{\dagger}_{-\textbf{q},l,i}\right]$

where $\psi_{n}(z)=\sqrt{\frac{2}{d}}\cos\left(\frac{(n-1/2)\pi z}{d}\right)$
and $\phi_{l}(z)=\sqrt{\frac{2}{d}}\cos\left(\frac{(l-1/2)\pi z}{d}\right)$.
Integrating out the z-degrees of freedom, we obtain the following effective 2D
Hamiltonian,

$\mathcal{H}_{\text{e-ph}}=\sum_{n,n^{\prime},l}g^{l}_{n,n^{{}^{\prime}}}(d)\int
d^{2}\textbf{r}\,\Psi^{\dagger}_{n^{\prime}}(\textbf{r})\vec{\nabla}.\vec{\Phi}_{l}(\textbf{r})\Psi_{n}(\textbf{r})$
(26)

Here $\Psi_{n}$ is the effective 2D electron field operator for an electron
with band index $n$. Similar definition holds for $\vec{\Phi}_{l}$. The
scattering matrix $g^{l}_{n,n^{{}^{\prime}}}(d)$ is given by

$\displaystyle g^{l}_{n,n^{{}^{\prime}}}(d)$ $\displaystyle=$
$\displaystyle(-1)^{n+n^{\prime}-l}\frac{G_{\text{fp}}}{\pi}\sqrt{\frac{2}{d}}\biggl{[}\frac{l-\frac{1}{2}}{(l-\frac{1}{2})^{2}-(n-n^{\prime})^{2}}$
(27) $\displaystyle-$
$\displaystyle\frac{l-\frac{1}{2}}{(l-\frac{1}{2})^{2}-(n+n^{\prime}-1)^{2}}\biggr{]}$

We find here that coupling with phonons can lead to interband scattering of
electrons in the thin film.

####  III.1.2 Pairing potential matrix

It is well known that coupling with phonons leads to an effective electron-
electron interaction that could be attractive under certain conditions. The
minimal BCS pairing Hamiltonian that emerges out of the coupling term in
Eqn.26, has the following form,

$\displaystyle\mathcal{H}_{I}$ $\displaystyle=$
$\displaystyle\sum_{n,n^{\prime}}\mathcal{H}_{I}(n,n^{\prime})$ (28)
$\displaystyle=$
$\displaystyle-\sum_{n,n^{\prime}}\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\frac{d^{2}\textbf{p}}{(2\pi)^{2}}$
$\displaystyle
V^{n,n^{\prime}}_{\textbf{k},\textbf{p}}c^{\dagger}_{\textbf{p},n}s_{y}c^{\dagger
T}_{-\textbf{p},n}c^{T}_{-\textbf{k},n^{\prime}}s_{y}c_{\textbf{k},n^{\prime}}$

where the pairing potential $V_{n,n^{\prime}}$ has the form,

$\displaystyle
V^{n,n^{\prime}}_{\textbf{k},\textbf{p}}=\begin{cases}\sum\limits_{l=1}^{l_{max}}|g^{l}_{n,n^{\prime}}(d)|^{2},&-\omega_{D}<\xi_{\textbf{k},n},\xi_{\textbf{p},n^{\prime}}<\omega_{D}\\\
0,&\text{else}\end{cases}$ (29)

Here $\omega_{D}$ is the Debye frequency of the thin film.
$\xi_{\textbf{k},n}$ is the single-particle energy of the thin film electrons
measured from the chemical potential. The electron-phonon coupling matrix
$g_{n,n^{\prime}}$ is summed over all the slab phonon modes up to $l_{max}$.
It is the maximum value that a phonon mode could have in the thin film at a
given thickness ${}^{\prime}d^{\prime}$. To find its value, recall that Debye
frequency sets the UV cut-off for the energy of lattice vibrations. Hence
$l_{max}$ can be calculated by taking the integer part of the expression
$d\left(k_{D}/\pi\right)$, where $k_{D}$ is the Debye momentum. A
comprehensive study of the thin film superconductivity with attractive
interaction mediated by confined phonons was conducted in ref.[32].

An important consequence of the dimensional reduction applied in the context
of interactions[24] is that the effective 2D interaction potential acquires a
scaling dependence on the thin film thickness as,

$V^{n,n^{\prime}}_{\textbf{k},\textbf{p}}\propto\frac{1}{d}$ (30)

Thus the attractive interaction increases with reducing thickness. This
implies that the attractive interaction is maximum in the ultrathin($N=1$)
limit of the thin film. We shall use this scaling relation in the later part
of this paper in order to enhance the attractive interaction between surface
fermions.

###  III.2 The general interaction Hamiltonian of the thin film-TI hybrid

When the tunneling is turned on, the thin film band which is close to the
Dirac point of the TI surface is hybridized. Let $N$ be the index of the band
that is hybridized. As mentioned before, we consider only the limit when the
$N\pm 1$ bands are separated from the $N$th band by a magnitude of at least
the order of bulk energy gap of the TI (See Eqn.7). So, the effects of
hybridization on all these $n\neq N$ bands are ignored. Now coming back to the
$N$th band, hybridization with the surface Dirac cone implies that the
electronic states in that QW band are no longer diagonal in the thin film
basis. The hybrid states are in a linear superposition of the thin film and
the TI surface states. The emergent excitations of this hybrid system are the
states $d^{\dagger}_{\textbf{k},t(b)}\ket{0}$ in the spin basis. It is even
easier to study the interaction if we could rotate the states to the helicity
basis since the hybrid states are diagonal in the helicity basis. So we
project the interaction Hamiltonian $\mathcal{H}_{I}$ of the resonant band
indexed by $N$ into the basis spanned by $a_{\textbf{k},t(b),\pm}$ states
(defined in Eqn.22).

After the projection, the full Hamiltonian $\mathcal{H}_{I}$ can be divided
into essentially three terms. The first term is the Hamiltonian describing the
attractive interaction between the helical hybrid fermions. Secondly, we have
the term describing attractive interaction between the hybridized fermions and
the trivial fermions of all the $n\neq N$ thin film transverse bands. Lastly,
we have the interaction Hamiltonian for the fermions in the thin film
unaffected by hybridization. In doing this projection, terms that describe
interband pairing between the helical fermions have been ignored. This is a
good approximation in the BCS limit. We shall write down the three terms in
the Hamiltonian explicitly below,

$\displaystyle\mathcal{H}_{I}=\mathcal{H}^{\text{hbd-
hbd}}_{I}+\mathcal{H}^{\text{hbd-tf}}_{I}+\mathcal{H}^{\text{tf-tf}}_{I}$ (31)

Now we shall derive these three terms in the Hamiltonian starting from the
fundamental s-wave pairing Hamiltonian in the thin film. The details of the
derivation are given in the appendix A.

####  III.2.1 Hamiltonian for Interaction between hybrid fermions
($\mathcal{H}^{\text{hbd}-\text{hbd}}$)

Here we shall derive the pairing Hamiltonian that describes the attractive
interaction between the helical hybrid fermions. Before the tunneling was
switched on, the interaction between electrons in the $N$th band of the thin
film is described by the following Hamiltonian,

$\displaystyle\mathcal{H}_{I}(N,N)$ $\displaystyle=$
$\displaystyle-\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\frac{d^{2}\textbf{p}}{(2\pi)^{2}}$
(32)
$\displaystyle\,V^{N,N}_{\textbf{k},\textbf{p}}c^{\dagger}_{\textbf{p},N}s_{y}c^{\dagger
T}_{-\textbf{p},N}c^{T}_{-\textbf{k},N}s_{y}c_{\textbf{k},N}$

Once the tunneling is switched on, the electronic states in the $N$th band are
hybridized and we have a 4-band model with a helical spin texture. So, it is
better that the interaction Hamiltonian be written down in the helicity basis.
Before we write down the Hamiltonian, we shall define the notations used to
identify all four hybrid bands. Let $m$, $m^{\prime}$ run over the band
indices $t$(top) and $b$(bottom). Similarly, $\lambda$ and $\lambda^{\prime}$
run over the $+$ and $-$ helicity branches. Using this set of indices, we can
write down the following interaction Hamiltonian that describes all possible
pairing interactions(except the inter-band pairing) between the four hybrid
bands:

$\mathcal{H}^{\text{hbd-
hbd}}_{I}=-\sum_{\alpha,\beta}\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\frac{d^{2}\textbf{p}}{(2\pi)^{2}}e^{i\left(\phi_{\textbf{p}}-\phi_{\textbf{k}}\right)}\\\
\lambda\lambda^{\prime}J^{\alpha,\beta}_{\textbf{k},\textbf{p}}a^{\dagger}_{\textbf{k},\alpha}a^{\dagger}_{-\textbf{k},\alpha}a_{-\textbf{p},\beta}a_{\textbf{p},\beta}$
(33a) $\displaystyle
J^{\alpha,\beta}_{\textbf{k},\textbf{p}}=V^{N,N}_{\textbf{k},\textbf{p}}Z^{\alpha}_{\textbf{k}}Z^{\beta}_{\textbf{p}}$
(33b)

Here $\alpha=(m,\lambda)\,\,\beta=(m^{\prime},\lambda^{\prime})$ is used as a
shorthand notation to denote the band indices. Note that
$\lambda\lambda^{\prime}=-1$ if the scattering is between bands of opposite
helicity.

$Z^{\alpha}_{\textbf{k}}$ can be identified as the wavefunction
renormalization of a hybridized electronic state as a result of tunneling with
respect to a thin film state without tunneling. This implies that
$Z^{\alpha}_{\textbf{k}}=1$ for a thin film state and
$Z^{\alpha}_{\textbf{k}}=0$ for TI surface state before the tunneling was
turned on. They have the following structure,

$\displaystyle Z^{(t,\pm)}_{\textbf{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1+\frac{\delta_{\textbf{k},\pm}}{\sqrt{\delta^{2}_{\textbf{k},\pm}+\frac{t^{2}}{d}}}\right)$
$\displaystyle Z^{(b,\pm)}_{\textbf{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1-\frac{\delta_{\textbf{k},\pm}}{\sqrt{\delta^{2}_{\textbf{k},\pm}+\frac{t^{2}}{d}}}\right)$
$\displaystyle\delta_{\textbf{k},\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\epsilon^{\text{tf}}_{\textbf{k},N}-\epsilon^{\text{surf}}_{\textbf{k},\pm}\right)$

So we find here that, as a result of tunneling, a pairing potential exists
between the helical hybrid fermions and it is proportional to the square of
the renormalization factors of the bands corresponding to the initial and
final states of the Kramers pair of electrons involved in pairing. This makes
physical sense because the $Z$-factor determines the probability that an
electron is in the thin film side of the interface. Only the electrons in the
thin film side of the interface will experience an attractive interaction
mediated by phonons. If $Z^{\alpha}_{\textbf{k}}=1$ for an electronic state of
momentum k and in a hybrid band indexed by $\alpha$, the electronic state is
completely in the thin film side of the interface and experience the full
attractive interaction. But in this case, the electronic state will not have
the helical spin texture induced by the TI surface. On the other hand, if
$Z^{\alpha}_{\textbf{k}}=0$ for an electronic state in the hybrid band, then
the electron is entirely on the TI side of the interface and does not
experience an attractive interaction. So we have to fine-tune the material
parameters such that both the effects, the helical spin texture, and the
attractive interaction are substantial. We shall show in this article
quantitatively that this can be achieved by fine-tuning the thickness to
’quantum well resonance’ at the Dirac point. A detailed discussion of this
phenomenon will be presented in the next section.

####  III.2.2 Interaction between hybrid fermions and the thin film fermions
in the $n\neq N$ band $\mathcal{H}^{\text{hbd-tf}}$

In the limit that we are working, hybridization effects are substantial only
for the thin film QW band at $n=N$. All the other $n\neq N$ bands are much
above or much below the Dirac point of the TI surface so that the tunneling
effects due to surface fermions are negligible. But it is possible that the
hybrid fermions can still experience attractive interaction with the thin film
electrons lying in all of the $n\neq N$ bands. This effect is captured by the
interband scattering terms of the thin film interaction Hamiltonian given in
Eqn.28. Before tunneling is introduced, it is possible that a singlet Cooper
pair of electrons in the $N$th band can scatter to any of the $n\neq N$ bands.
The Hamiltonian describing such a process can be read out from the full
interaction Hamiltonian given in Eqn.28 by fixing $n^{\prime}$ to $N$ and
letting $n$ run over all $n\neq N$.

$\displaystyle\sum_{n\neq N}\mathcal{H}_{I}(N,n)=-\sum_{n\neq
N}\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\frac{d^{2}\textbf{p}}{(2\pi)^{2}}$
$\displaystyle\,V^{N,N}_{\textbf{k},\textbf{p}}c^{\dagger}_{\textbf{p},n}s_{y}c^{\dagger
T}_{-\textbf{p},n}c^{T}_{-\textbf{k},N}s_{y}c_{\textbf{k},N}$ (35)

Once the tunneling is switched on, the Cooper pair
$c^{T}_{-\textbf{k},N}s_{y}c_{\textbf{k},N}$ is projected to the helicity
basis of the $t$ and $b$ hybrid bands. In doing this, we arrive at an
interaction Hamiltonian that describes the attractive interaction between the
hybrid fermions and the off-resonance thin film fermions. Let us call the
Hamiltonian by the name $\mathcal{H}^{\text{hbd-tf}}_{I}$ and has the
following definition,

$\displaystyle\mathcal{H}^{\text{hbd-tf}}_{I}=-\sum_{n\neq
N}\sum_{\alpha}\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\frac{d^{2}\textbf{p}}{(2\pi)^{2}}e^{i\phi_{\textbf{p}}}$
$\displaystyle\,\lambda
K^{n,\alpha}_{\textbf{k},\textbf{p}}c^{\dagger}_{\textbf{k},n}(-is_{y})c^{\dagger
T}_{-\textbf{k},n}a_{-\textbf{p},\alpha}a_{\textbf{p},\alpha}$ (36a)
$\displaystyle
K^{n,\alpha}_{\textbf{k},\textbf{p}}=V^{n,N}_{\textbf{k},\textbf{p}}Z^{n}_{\textbf{k}}Z^{\alpha}_{\textbf{p}}$
(36b)

Note that $Z^{n}_{\textbf{k}}=1$ for all $k$ and $n\neq N$ since it
corresponds to the renormalization factor of the thin film electrons which did
not participate in tunneling. It has been included in the expression only for
the purpose of generality. So here we find that even though the thin film
electrons in the $n\neq N$ bands do not participate in tunneling, they do
contribute to the superconducting phase of the hybrid fermions.

####  III.2.3 Interaction between all the $n\neq N$ band thin film fermions
($\mathcal{H}^{\text{tf-tf}}_{I}$)

It is also important to consider the attractive interaction between the
electrons in the $n\neq N$ bands that were not part of the tunneling. It is
just the trivial BCS singlet pairing Hamiltonian. It is found by summing over
$\mathcal{H}_{I}(n,n^{\prime})$ defined in Eqn.28 for all $n,n^{\prime}\neq
N$. Let us call this Hamiltonian as $\mathcal{H}^{\text{tf-tf}}_{I}$. It has
the form,

$\displaystyle\mathcal{H}^{\text{tf-tf}}_{I}$ $\displaystyle=$
$\displaystyle-\sum_{n,n^{\prime}\neq
N}\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\frac{d^{2}\textbf{p}}{(2\pi)^{2}}$
(37) $\displaystyle
V^{n,n^{\prime}}_{\textbf{k},\textbf{p}}c^{\dagger}_{\textbf{p},n}s_{y}c^{\dagger
T}_{-\textbf{p},n}c^{T}_{-\textbf{k},n^{\prime}}s_{y}c_{\textbf{k},n^{\prime}}$

The full interaction Hamiltonian of the TI-thin film hybrid is now the sum of
all three terms as given in Eqn.31.

##  IV Z-factor and the quantum-well resonance

In section II, we studied the single-particle tunneling of electronic states
in the topological surface to the QW thin film band lying closest to it. The
tunneling effectively results in the hybridization of the electronic states
and leads to the formation of four spin-split hybrid bands, with an emergent
helical spin texture for each of them.

In section III, we found that these helical hybrid electrons can couple with
the confined phonons of the thin film and could result in an effective
attractive interaction between them. The effect of tunneling is taken into
account in the interaction strength by the renormalization factor
$Z^{\alpha}_{\textbf{k}}$ defined in Eqns.LABEL:Zfactor. For instance, one can
show that the type of pairing between two electrons with renormalization
factors equal to unity will be trivial s-wave-like. This is because these
electrons lie entirely in the thin film side and the tunneling effect on them
is negligible. The other extreme is when the renormalization factor of the
electrons is zero. This corresponds to the non-interacting surface electrons.

From these intuitive arguments, one can anticipate that the ideal choice for
the renormalization factor of an electronic state will be $1/2$. It is at this
limit the tunneling effect is maximum. This implies that the surface states
that are initially non-interacting will acquire maximum attractive interaction
in this limit. This is because it is the tunneling that actually induces an
effective attractive interaction between surface fermions. In order to realize
this maximum tunneling effect, the corresponding electronic states on both
sides of the interface must be degenerate. In other words, the electronic
states should be in quantum-well resonance. In this section, we will show this
explicitly by studying the behavior of the renormalization factor as a
function of the detuning parameter defined at the Dirac point.

The renormalization factors were defined in Eqn.LABEL:Zfactor as a function of
the band indices and momentum k. Since there are four hybrid bands, we have
four renormalization factors for a fixed momentum k. One can show that they
follow a general relationship,

$\displaystyle Z^{(t,+)}_{\textbf{k}}+Z^{(b,+)}_{\textbf{k}}$ $\displaystyle=$
$\displaystyle 1$ $\displaystyle
Z^{(t,-)}_{\textbf{k}}+Z^{(b,-)}_{\textbf{k}}$ $\displaystyle=$ $\displaystyle
1$ (38)

for any momentum state k. This implies that for a fixed helicity if one hybrid
band is on the thin film side, the other band lies on the TI surface side. We
are mostly interested in the interacting dynamics of the electronic states
near the Dirac point. Therefore we set the momentum $k=0$ in the above
equations and study the evolution of the $Z$-factors as a function of the
detuning parameter also defined at zero momentum. At $k=0$, there is a further
simplification. We find that due to the crossing of the two helicity branches
at the Dirac point, the respective $Z$ factors turn out to be equal. That is,

Figure 3:  a) Plot showing the evolution of the wavefunction renormalization
Z-factors defined at the Dirac point, $Z^{t}$ and $Z^{b}$(defined in
Eqns.LABEL:Zfactor,40) corresponding to the top and the bottom hybrid bands
respectively as a function of the detuning parameter $\tilde{\delta}$. The
black dashed lines trace the Z-factor of the surface fermions. The Z-factor
and hence the surface attractive interaction that is proportional to $Z^{2}$
(Eqns.33b) is maximized at resonance($\tilde{\delta}=0$). Also shown are the
energy spectra of the thin film QW band and the surface Dirac cone at three
different values of the detuning parameter before (b) and after (c) the
tunneling is turned on. The arrowheads show the respective Dirac points at
which we calculated $Z^{t}$ and $Z^{b}$ of the top and bottom bands. The
detuning parameter is tuned by fine-tuning the thin film thickness.

$\displaystyle
Z^{(t,+)}_{\textbf{k}=0}=Z^{(t,-)}_{\textbf{k}=0},\,\,\text{and}\,\,Z^{(b,+)}_{\textbf{k}=0}=Z^{(b,-)}_{\textbf{k}=0}$
(39)

So at $k=0$, we essentially have ended up with just two $Z$ factors subject to
the constraint that their sum must be equal to unity. We shall make the
following redefinitions,

$\displaystyle Z^{t}=Z^{(t,\pm)}_{\textbf{k}=0}\,\,$
$\displaystyle\text{and}\,\,Z^{b}=Z^{(b,\pm)}_{\textbf{k}=0}$ (40)
$\displaystyle\text{so we have}\,\,Z^{t}+Z^{b}$ $\displaystyle=$
$\displaystyle 1$

Now we shall define the detuning parameter at $k=0$. It has the form,

$\displaystyle\tilde{\delta}(d)=\frac{\delta_{\textbf{k}=0,N}(d)}{t_{d}}$ (41)

here $\delta_{\textbf{k}}$ is defined in Eqn.13 in the section II as a
$2\times 2$ matrix in the spin space. But at $\textbf{k}=0$, it turns out to
be an identity matrix that can be treated as a number. $\tilde{\delta}$
essentially gives the energy difference between the electronic state in the
thin film band closest(indexed by $n=N$) to the Dirac point and the Dirac
point of the TI surface. When the energy difference is zero, the electrons at
$k=0$ are in quantum-well resonance and the tunneling effect will be maximum.
Moving away from $\tilde{\delta}=0$ is equivalent to detuning away from
resonance. We defined the detuning parameter at $\textbf{k}=0$ because we are
mostly interested in studying the interacting dynamics of the electrons near
the Dirac point. In general, one can define a detuning parameter for any
general k. Here we use thin film thickness $d$ to tune the detuning parameter.

From Eqns.LABEL:Zfactor and 41, we could deduce the following simple
relationship between renormalization factors and the dimensionless detuning
parameter $\tilde{\delta}$ at zero momentum,

$\displaystyle
Z^{t}(\tilde{\delta})=\frac{1}{2}\left(1+\frac{\tilde{\delta}}{\sqrt{1+\tilde{\delta}^{2}}}\right)\,\,Z^{b}(\tilde{\delta})=\frac{1}{2}\left(1-\frac{\tilde{\delta}}{\sqrt{1+\tilde{\delta}^{2}}}\right)$
(42)

Fig.3 shows the results. In b) we plotted $Z^{t}$ and $Z^{b}$ as a function of
the detuning parameter. a) part shows the band spectrum of the thin film and
the TI surface at the three different limits of detuning. When
$\tilde{\delta}\ll 0$, $Z^{b}\approx 1$ and $Z^{t}\approx 0$. This implies
that the bottom hybrid band is the thin film transverse band while the top
band is the surface Dirac cone. On the other hand, when $\tilde{\delta}\gg 0$,
the bottom band is the surface Dirac cone and the top band is the thin film
transverse band. This is clearly understood once we look at the band
dispersion shown in Fig.3(a). In these two limits, the tunneling effects are
perturbative. One can notice that the renormalization factor $Z^{b}$, which
follows the surface band when $\tilde{\delta}\ll 0$. is nearly zero in this
limit. Similar is the case with $Z^{t}$ when $\tilde{\delta}\gg 0$. This
implies that the surface electrons do not experience a substantial attractive
interaction when $|\tilde{\delta}|\gg 0$.

But as $\tilde{\delta}\rightarrow 0$ from either side, things begin to change.
We find that both the renormalization factors approach $1/2$ from either side.
This clearly implies that the tunneling gets stronger and is non-perturbative.
One can trace the surface Dirac cone by $Z^{b}$ when $\tilde{\delta}<0$ and
$Z^{t}$ when $\tilde{\delta}>0$. We see that both the quantities rise up as
$\tilde{\delta}$ approaches zero and reach a maximum equal to $1/2$ at
$\tilde{\delta}=0$. Recall that the interaction strength between the helical
fermions is proportional to $Z^{2}$. Thus this spike at $\tilde{\delta}=0$ is
clear evidence of the surface fermions experiencing a maximum effective
attractive interaction at $\tilde{\delta}=0$.

On the other hand, the electrons that used to be in the thin film side when
tunneling was zero now experience comparatively weaker attractive interaction.
This is evident if we observe the evolution of $Z^{b}$ when $\tilde{\delta}<0$
and $Z^{t}$ when $\tilde{\delta}>0$. The two renormalization factors reach a
minimum at $\tilde{\delta}=0$ implying that the effective attractive
interaction got weaker.

In conclusion, by studying the evolution of the renormalization factors as a
function of the detuning parameter, we showed that the effective attractive
interaction acquired by the surface fermions near the Dirac point is the
strongest when the thin film QW band is in quantum-well resonance with the
surface Dirac cone. The fact that the $Z$-factors approach 1/2 at resonance
suggests that there is no clear difference between the thin film fermions and
the surface fermions at quantum-well resonance. This is clear evidence of our
earlier proposition that the electronic states at the quantum-well resonance
are hybridized. The eigenstates are a quantum superposition of the thin film
and the surface states. They acquire a helical spin structure from the surface
side and an effective attractive interaction between them from the thin film
side. We shall be studying the superconductivity of these helical hybridized
fermions within the BCS mean-field theory in the coming sections.

##  V Effective mean-field Hamiltonian and the gap equation

###  V.1 Mean-field approximation

Here we shall use the mean-field theory to decouple the four-fermion
interaction Hamiltonian. Let $\triangle^{\text{hbd}}_{\alpha}(\textbf{k})$ be
the order parameter on the helical hybrid band of index $m$($t$ or $b$) and
helicity $\lambda$(=$+$ or $-$). Note that $\alpha=(m,\lambda)$. Similarly,
define $\triangle^{\text{tf}}_{n}$ be the order parameter on the thin film
band of index $n\neq N$. Now we apply mean-field approximation to the
4-fermion interaction Hamiltonian in Eqn.37,

$\displaystyle\triangle^{\text{hbd}}_{\textbf{k},\alpha}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\left[\sum_{\beta=\\{m^{\prime},\lambda^{\prime}\\}}\lambda^{\prime}e^{i\phi_{\textbf{p}}}J^{\alpha,\beta}_{\textbf{k},\textbf{p}}\left<a_{\textbf{p},\beta}a_{-\textbf{p},\beta}\right>+\sum_{n\neq
N}K^{n,\alpha}_{\textbf{k},\textbf{p}}\left<c^{T}_{\textbf{p},n}(is_{y})c_{-\textbf{p},n}\right>\right]$
(43a) $\displaystyle\triangle^{\text{tf}}_{\textbf{k},n}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\left[\sum_{\alpha=\\{m,\lambda\\}}\lambda
e^{i\phi_{\textbf{p}}}K^{n,\alpha}_{\textbf{k},\textbf{p}}\left<a_{\textbf{p},\alpha}a_{-\textbf{p},\alpha}\right>+\sum_{n^{\prime}\neq
N}V^{n,n^{\prime}}_{\textbf{k},\textbf{p}}\left<c^{T}_{\textbf{p},n^{\prime}}(is_{y})c_{-\textbf{p},n^{\prime}}\right>\right]$
(43b) $\displaystyle\mathcal{H}_{\textbf{MF}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\left[\sum_{\alpha=\\{m,\lambda\\}}\lambda\triangle^{\text{hbd}}_{\textbf{k},\alpha}e^{-i\phi_{\textbf{k}}}a^{\dagger}_{\textbf{k},\alpha}a^{\dagger}_{-\textbf{k},\alpha}+\sum_{n\neq
N}\triangle^{\text{tf}}_{\textbf{k},n}c^{\dagger}_{\textbf{k},n}(-is_{y})c^{\dagger
T}_{-\textbf{k},n}\right]$ (43c)

Interestingly, the order parameters on the helical bands are of odd parity. So
we find that the helical fermions have an ’effective’ p-wave pairing even
though we started with a purely s-wave interaction. This is because the spin
rotation symmetry(SRS) is broken by the induced spin-orbit coupling, while the
time-reversal symmetry is preserved[33, 34, 35]. On the other hand, the
pairing amplitude on the $n\neq N$ thin film transverse bands are of even
parity.

###  V.2 The superconducting gap equation

Using the mean-field theory, we derived the most general expression for the
superconducting order parameter on the four helical hybrid bands and the
remaining spin-degenerate thin film transverse bands. Note here that in our
case, the fundamental origin of the attractive interaction is the electrons
coupling to phonons. Since Debye frequency sets the UV cut-off for phonon
modes, only electrons whose energy lies within the range
$\left[\mu-\omega_{D},\mu+\omega_{D}\right]$ can experience the attractive
interaction. Here $\mu$ is the chemical potential. Here we focus on the limit
$\omega_{D}\ll\mu$. This puts a strict constraint on the number of bands and
the number of electrons participating in the pairing interaction. Only those
bands that cross the Fermi level needed to be considered for pairing
interaction. All those bands that lie above the Fermi level can be ignored.
Before hybridization, the number of bands that cross the Fermi level can be
calculated by taking the integer value of the expression,
$d/\pi\sqrt{2m\mu/\hbar^{2}}+1/2$. This integer will turn out to be the same
as $N$, the index of the band that is hybridized with the surface Dirac cone.
Hence before hybridization, we essentially have $2N$ Fermi surfaces because
the thin film bands are spin-degenerate.

Figure 4: Schematic picture of the different Fermi surfaces in the thin film -
TI hybrid before (a) and after (b) the tunneling is introduced. Also shown are
the possible coupling matrix elements between the Fermi surfaces. In a), the
blue circles represent the Fermi surfaces formed by the thin film QW bands,
while the red circle corresponds to the Fermi surface formed by the surface
Dirac cone. Each Fermi surface is twofold degenerate. $V^{n,n^{\prime}}$ gives
the pairing matrix elements between the thin film Fermi surfaces. The
electrons in the surface Dirac cone FS are non-interacting. After the
tunneling is introduced in (b), $N$th band is hybridized with surface
fermions. The tunneling effect on all the $n\neq N$ QW bands is ignored in our
effective model given the condition in Eqn.7 is satisfied. So the Fermi
surfaces formed by all the $n<N$ QW bands remain unaffected by tunneling,
hence spin-degenerate. $\mathcal{J}^{i,j}$ represents the coupling matrix
element between the $i$th and $j$th hybrid FS. $\mathcal{K}^{i,n}$ represent
the coupling matrix element between the $i$th hybrid FS and $n$th thin film
Fermi surface. The definitions of the coupling matrix elements are given in
Eqns.45,46,47.

Now the chemical potential should be set within the bulk energy gap of the
topological insulator. Once the thin film is deposited over the TI surface,
the $N$th band is hybridized and we effectively have a 4-band model within the
bulk gap. By fine-tuning the chemical potential further, it is possible that
one can have the system with either three hybrid Fermi surfaces or just one
Fermi surface (see fig.5). In the latter case, both the positive and negative
helicity branches of the top band lie above the Fermi level and therefore do
not participate in pairing. We shall derive the superconducting gap equation
for these two cases separately here.

####  V.2.1 3 hybrid Fermi surfaces + 2N - 2 thin film Fermi surfaces

Now consider the case when the Fermi level is adjusted such that the hybrid
has three Fermi surfaces within the thickness regime that we like to explore.
We shall write down a gap equation for this specific case. The innermost Fermi
surface(FS) was formed by the positive helicity branch of the $t$ (top) band
while the next FS was formed by the negative helicity branch of the $t$ band.
The outermost FS is formed by the positive branch of the $b$ (bottom) band. At
this point, it is more convenient to express the superconducting gap and the
coupling strength as functions of Fermi surface indices rather than the band
indices. In the weak-pairing limit ($\omega_{D}\ll\mu$), only electronic
states very close to the Fermi surface take part in pairing. Thus, the
electron renormalization factor that enters the pairing potential matrix can
be re-expressed in terms of the Fermi momenta of the respective Fermi surfaces
rather than the band indices. To support this, let us define three quantities
$Z_{1}$, $Z_{2}$ and $Z_{3}$ for the three Fermi surfaces such that,

$\displaystyle Z_{1}$ $\displaystyle=$ $\displaystyle
Z^{(t,+)}_{\textbf{k}_{F1}},\,\,\,\,\,\,Z_{2}=Z^{(t,-)}_{\textbf{k}_{F2}}$
$\displaystyle Z_{3}$ $\displaystyle=$ $\displaystyle
Z^{(b,+)}_{\textbf{k}_{F3}}$ (44)

where $1,2$, and $3$ are the hybrid Fermi surface indices from smallest to
largest in terms of size. Thus, $\textbf{k}_{F1},\textbf{k}_{F2}$ and
$\textbf{k}_{F3}$ are the Fermi momenta on these three hybrid Fermi surfaces.
Since the renormalization factor depends only on the magnitude of momentum,
$Z_{i}$ is the same for all electrons in the Fermi surface indexed by $i$. The
approximation we will do here is that we assume $Z_{i}$ factor is the same for
all the electronic states lying within the energy window
$[-\omega_{D},\omega_{D}]$ measured from the chemical potential, given that
the electronic states lie near the $i$th hybrid Fermi surface. This
approximation allows us to re-express the interaction potential matrix in
terms of the Fermi surface indices rather than the band indices. Let us
define,

$\displaystyle\mathcal{J}^{i,j}_{\textbf{k},\textbf{p}}=V^{N,N}_{\textbf{k},\textbf{p}}Z_{i}Z_{j}$
(45)

$\mathcal{J}^{ij}_{\textbf{k},\textbf{p}}$ is the interaction matrix element
that gives the scattering strength of Cooper pair from the $i$th hybrid Fermi
surface to the $j$th hybrid Fermi surface. One can also redefine
$K^{n\alpha}_{\textbf{k},\textbf{p}}$ in terms of the Fermi surface indices.
From Eqn.44, we have,

$\mathcal{K}^{n,i}_{\textbf{k},\textbf{p}}=V^{n,N}_{\textbf{k},\textbf{p}}Z_{n}Z_{i}$
(46)

where $\mathcal{K}^{n,i}_{\textbf{k},\textbf{p}}$ determines the scattering of
Kramer’s doublets from the $i$th hybrid Fermi surface to the $2n$th or
$(2n-1)$th ($n<N$) thin film Fermi surface. Here $2n$th and $(2n-1)$th Fermi
surfaces are formed by the helicity subbands of the $n$th spin-degenerate
band. Due to this spin-degeneracy, the two helical Fermi surfaces overlap and
hence the interaction parameters are the same for both.

From the definition of $V^{N,N}_{\textbf{k},\textbf{p}}$ in Eqn.29, we find
that the matrix elements $\mathcal{J}^{i,j}_{\textbf{k},\textbf{p}}$ and
$\mathcal{K}^{n,i}_{\textbf{k},\textbf{p}}$ are independent of momenta for
electronic states lying within the Debye frequency measured from the Fermi
level and zero otherwise. That is, we can write down the effective interaction
potential in the following simple way,

$\displaystyle V^{n,n^{\prime}}_{\textbf{k},\textbf{p}}$ $\displaystyle=$
$\displaystyle
V^{n,n^{\prime}}\theta\left(\omega_{D}-\xi^{\text{tf}}_{\textbf{k},n}\right)\theta\left(\omega_{D}-\xi^{\text{tf}}_{\textbf{p},n^{\prime}}\right)$
$\displaystyle\mathcal{J}^{i,j}_{\textbf{k},\textbf{p}}$ $\displaystyle=$
$\displaystyle\mathcal{J}^{i,j}\theta\left(\omega_{D}-\xi^{\text{hbd}}_{\textbf{k},i}\right)\theta\left(\omega_{D}-\xi^{\text{hbd}}_{\textbf{p},j}\right)$
$\displaystyle\mathcal{K}^{n,i}_{\textbf{k},\textbf{p}}$ $\displaystyle=$
$\displaystyle\mathcal{K}^{n,i}\theta\left(\omega_{D}-\xi^{\text{tf}}_{\textbf{k},n}\right)\theta\left(\omega_{D}-\xi^{\text{hbd}}_{\textbf{p},i}\right)$
(47)

where $\theta(x)$ is the Heavyside step function and the coupling matrix
elements $V^{n,n^{\prime}}$, $\mathcal{J}^{i,j}$ and $\mathcal{K}^{n,i}$ are
independent of momenta. Also
$\xi^{\text{tf(hbd)}}_{\textbf{k},n}=\epsilon^{\text{tf(hbd)}}_{\textbf{k},n}-\mu$
is just the energy of the thin film(hybrid) fermions measured from the
chemical potential, involved in the interaction.

With these definitions, it is straightforward to derive the superconducting
gap equation. We shall also redefine the superconducting order parameters of
the hybrid fermions also in terms of the Fermi surface indices as follows:

$\displaystyle\triangle^{\text{hbd}}_{\textbf{k},1}$ $\displaystyle\approx$
$\displaystyle\triangle^{\text{hbd}}_{\textbf{k},t,+},\,\,\,\,\triangle^{\text{hbd}}_{\textbf{k},2}\approx\triangle^{\text{hbd}}_{\textbf{k},t,-}$
$\displaystyle\triangle^{\text{hbd}}_{\textbf{k},3}$ $\displaystyle\approx$
$\displaystyle\triangle^{\text{hbd}}_{\textbf{k},b,+}$ (48)

It has the form,

$\displaystyle\triangle^{\text{hbd}}_{\textbf{k},i}$ $\displaystyle-$
$\displaystyle\sum^{3}_{j=1}\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\frac{\mathcal{J}^{i,j}_{\textbf{k},\textbf{p}}\triangle^{\text{hbd}}_{\textbf{p},j}}{2\sqrt{\left(\xi^{\text{hbd}}_{\textbf{p},j}\right)^{2}+(\triangle^{\text{hbd}}_{\textbf{p},j})^{2}}}$
(49a) $\displaystyle=$
$\displaystyle\sum^{N-1}_{n=1}\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\frac{\mathcal{K}^{n,i}_{\textbf{k},\textbf{p}}\triangle^{\text{tf}}_{\textbf{p},n}}{2\sqrt{(\xi^{\text{tf}}_{\textbf{p},n})^{2}+(\triangle^{\text{tf}}_{\textbf{p},n})^{2}}}$
$\displaystyle\triangle^{\text{tf}}_{\textbf{k},n}$ $\displaystyle-$
$\displaystyle\sum^{N-1}_{n^{\prime}=1}\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\frac{V^{n,n^{\prime}}_{\textbf{k},\textbf{p}}\triangle^{\text{tf}}_{\textbf{p},n^{\prime}}}{2\sqrt{(\xi^{\text{tf}}_{\textbf{p},n^{\prime}})^{2}+(\triangle^{\text{tf}}_{\textbf{p},n^{\prime}})^{2}}}$
(49b) $\displaystyle=$
$\displaystyle\sum^{3}_{i=1}\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\frac{\mathcal{K}^{n,i}_{\textbf{k},\textbf{p}}\triangle^{\text{hbd}}_{\textbf{p},j}}{2\sqrt{\left(\xi^{\text{hbd}}_{\textbf{p},i}\right)^{2}+(\triangle^{\text{hbd}}_{\textbf{p},i})^{2}}}$
$\displaystyle\xi^{\text{hbd}}_{\textbf{k},1}$ $\displaystyle=$
$\displaystyle\epsilon^{\text{hbd}}_{\textbf{k},t,+}-\mu,\,\,\,\,\xi^{\text{hbd}}_{\textbf{k},2}=\epsilon^{\text{hbd}}_{\textbf{k},t,-}-\mu$
$\displaystyle\xi^{\text{hbd}}_{\textbf{k},3}$ $\displaystyle=$
$\displaystyle\epsilon^{\text{hbd}}_{\textbf{k},b,+}-\mu,\,\,\,\,\xi^{\text{tf}}_{\textbf{k},n}=\epsilon^{\text{tf}}_{\textbf{k},n}-\mu$

With the weak-pairing approximation discussed above, the magnitude of the
superconducting order parameters at all the Fermi surfaces turns out to be
momentum-independent. The only possible momentum dependence on the gap
magnitude could come from the restriction set by the Debye frequency. With
this in mind, we shall define the parameters $\triangle^{\text{hbd}}_{i}$ and
$\triangle^{\text{tf}}_{n}$ such that,

$\displaystyle\triangle^{\text{hbd}}_{\textbf{k},i}$ $\displaystyle=$
$\displaystyle\triangle^{\text{hbd}}_{i}\,\theta\left(\omega_{D}-\xi^{\text{hbd}}_{\textbf{k},i}\right)$
$\displaystyle\triangle^{\text{tf}}_{\textbf{k},n}$ $\displaystyle=$
$\displaystyle\triangle^{\text{tf}}_{n}\,\theta\left(\omega_{D}-\xi^{\text{tf}}_{\textbf{k},n}\right)$
(50)

where $\theta(x)$ is the Heavyside step function. In all the future
computations, we shall be representing the order parameters in dimensionless
form as
$\tilde{\triangle}^{\text{hbd}}_{i}=\triangle^{\text{hbd}}_{i}/\omega_{D}$ and
$\tilde{\triangle}^{\text{tf}}_{n}=\triangle^{\text{tf}}_{n}/\omega_{D}$ where
$\omega_{D}$ is the Debye frequency of the thin film metal.

A schematic picture of the coupling of Cooper pairs of electrons between
different Fermi surfaces within the weak-coupling approximation before and
after the tunneling is introduced is shown in fig.4.

####  V.2.2 1 hybrid Fermi surface + 2N - 2 thin film Fermi surfaces

Suppose that the Fermi level is fine-tuned to one hybrid Fermi surface within
the bulk gap. That is, both the helicity branches of the top band are above
the Fermi level (see fig.5). Hence the top band does not contribute to the
pairing at all. It is only the positive ( or the negative) helicity branch of
the bottom band that crosses the Fermi level. One can observe that in the
$N=1$ limit when there are no QW bands crossing the Fermi level, we
effectively have a single band of helical fermions subject to attractive
interaction. We shall study this limit more carefully in the next section.

Since there is just one hybrid band crossing the Fermi level, the
superconducting gap equation becomes far easier in this limit. Consider that
it is the positive helicity branch of the $b$ band that crosses the Fermi
level. In this case, only the coupling constant
$\mathcal{J}^{33}_{\textbf{k},\textbf{p}}$ survives. All the other elements
vanish in this limit. For the interaction with thin film fermions, only
$\mathcal{K}^{n,3}$ is needed to be taken into account. Hence in this limit,
the superconducting gap equation becomes,

$\displaystyle\triangle^{\text{hbd}}_{\textbf{k}}$ $\displaystyle-$
$\displaystyle\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\frac{\mathcal{J}^{3,3}_{\textbf{k},\textbf{p}}\triangle^{\text{hbd}}_{\textbf{p}}}{2\sqrt{\xi^{2}_{\textbf{p},3}+(\triangle^{\text{hbd}}_{\textbf{p}})^{2}}}$
(51a) $\displaystyle=$
$\displaystyle\sum^{N-1}_{n=1}\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\frac{\mathcal{K}^{n,3}_{\textbf{k},\textbf{p}}\triangle^{\text{tf}}_{\textbf{p},n}}{2\sqrt{(\xi^{\text{tf}}_{\textbf{p},n})^{2}+(\triangle^{\text{tf}}_{\textbf{p},n})^{2}}}$
$\displaystyle\triangle^{\text{tf}}_{\textbf{k},n}$ $\displaystyle-$
$\displaystyle\sum^{N-1}_{n^{\prime}=1}\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\frac{V^{n,n^{\prime}}_{\textbf{k},\textbf{p}}\triangle^{\text{tf}}_{\textbf{p},n^{\prime}}}{2\sqrt{(\xi^{\text{tf}}_{\textbf{p},n^{\prime}})^{2}+(\triangle^{\text{tf}}_{\textbf{p},n^{\prime}})^{2}}}$
(51b) $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\frac{\mathcal{K}^{n,3}_{\textbf{k},\textbf{p}}\triangle^{\text{hbd}}_{\textbf{p}}}{2\sqrt{\xi^{2}_{\textbf{p},3}+(\triangle^{\text{hbd}}_{\textbf{p}})^{2}}}$

where
$\triangle^{\text{hbd}}_{\textbf{k}}\approx\triangle^{\text{hbd}}_{\textbf{k},b,+}$.
Here too we define $\triangle^{\text{hbd}}$ such that,

$\triangle^{\text{hbd}}_{\textbf{k}}=\triangle^{\text{hbd}}\theta(\omega_{D}-\xi^{\text{hbd}}_{\textbf{k}})$
(52)

Figure 5:  Here we schematically show that by fine-tuning the chemical
potential, it is possible to obtain cases with either 1) one Fermi surface or
2) three Fermi surfaces. The electrons on all these Fermi surfaces are
helical, that is the spin-states are locked to their momentum direction. The
arrowheads show the spin orientation of the Fermi electrons.
$\mathcal{J}^{i,j}$ (where $i,j=1,2,3$) represents the coupling matrix element
between the Fermi surfaces indexed by $i$ and $j$. Its exact definition is
given in Eqns.45. For the single Fermi surface case, the Fermi level crosses
the positive helicity branch of the ’bottom’ band and the coupling matrix
element is $\mathcal{J}^{3,3}$. Also shown here is the definition of
hybridized band indices. Here $t(b)$ means the ’top’(’bottom’) band and
’$\pm$’ identifies the corresponding helicity.

##  VI The N = 1 four band model

Here we shall present our work’s simple yet most interesting result. Consider
the case when the thin film transverse band of quantum number $N=1$ is in
resonance with the Dirac point of the topological insulator. Quantitatively
from Eqns.1 and 4, we find that the following condition should be satisfied:
$\epsilon^{\text{tf}}_{0,n=1}=\epsilon^{\text{surf}}_{0,\pm}$. In other words,
the detuning parameter $\tilde{\delta}(d)=0$. If the material parameters of
the topological insulator are fixed, then a practical way to achieve this
condition is to tune the thin film thickness. So once the thickness is set and
the thin film is deposited over the TI surface, the tunneling results in the
hybridization of the electronic states near $\textbf{k}=0$ resulting in the
formation of four hybrid bands. Since we are in the $N=1$ limit, there are no
trivial (or off-resonance) QW bands of index $n\neq N$ crossing the Fermi
level. That is, only the hybridized fermions are present near the Fermi level.
We know that the thin film favors an effective attractive interaction between
electrons at zero temperature mediated by phonons. Therefore, we essentially
have an effective model with helical hybridized fermions interacting via an
effective attractive interaction between them. The full BCS interaction
Hamiltonian in this $N=1$ limit attains the form,

$\mathcal{H}_{I}=\mathcal{H}^{\text{hbd-hbd}}_{I}\\\
=-\sum_{\alpha,\beta}\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\frac{d^{2}\textbf{p}}{(2\pi)^{2}}e^{i\left(\phi_{\textbf{p}}-\phi_{\textbf{k}}\right)}\\\
\lambda\lambda^{\prime}J^{\alpha,\beta}_{\textbf{k},\textbf{p}}a^{\dagger}_{\textbf{k},\alpha}a^{\dagger}_{-\textbf{k},\alpha}a_{-\textbf{p},\beta}a_{\textbf{p},\beta}$
(53)

We have seen in the previous section that by fine-tuning the Fermi level, we
essentially have phases with either three hybrid Fermi surfaces or just one
hybrid Fermi surface as shown in Fig.5. In this $N=1$ limit, these are the
only Fermi surfaces present in the system. In the first part, we shall put
forward the theoretical model in the two cases separately. In the last part,
we shall tune various material parameters and look for possible enhancement of
the superconducting gap.

###  VI.1 Theoretical models

####  VI.1.1 Single Fermi surface model

Here we consider the case when the Fermi level is tuned to one Fermi surface.
This Fermi surface can be formed by either the positive or negative helicity
branch of the bottom band. Since the interaction is mediated by the phonons,
only the electronic states that lie within the energy window $\omega_{D}$
measured from the Fermi level experiences an attractive interaction. In this
context, if the magnitude of the energy difference between the chemical
potential and the emergent Dirac point of the bottom band is greater than the
Debye frequency, then only the positive(negative) helicity states of the $b$
band experience attractive interaction. The negative(positive) branch is
essentially non-interacting. Therefore, the projected Hamiltonian in the
helicity basis resembles a single-band BCS problem for ’spinless fermions’. If
the Fermi level crosses the positive helicity branch as shown in Fig.6 the
Hamiltonian attains the following simple form,

$\mathcal{H}=\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\biggl{[}a^{\dagger}_{\textbf{k},\alpha}\left[\epsilon^{\text{hbd}}_{\textbf{k},\alpha}-\mu\right]a_{\textbf{k},\alpha}\\\
-\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\mathcal{J}^{3,3}_{\textbf{k},\textbf{p}}e^{i\left(\phi_{\textbf{p}}-\phi_{\textbf{k}}\right)}a^{\dagger}_{\textbf{k},\alpha}a^{\dagger}_{-\textbf{k},\alpha}a_{-\textbf{p},\alpha}a_{\textbf{p},\alpha}\biggr{]}$
(54)

where $\mathcal{J}^{3,3}_{\textbf{k},\textbf{p}}$ is defined in Eqn.45.
$\alpha=\\{b,+\\}$ is the band index. Following the procedure explained in
Section V, mean-field Hamiltonian becomes,

$\displaystyle\mathcal{H}_{MF}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\left[\triangle^{\text{hbd}}_{\textbf{k}}e^{-i\phi_{\textbf{k}}}a^{\dagger}_{\textbf{k},\alpha}a^{\dagger}_{-\textbf{k},\alpha}+h.c\right]$
$\displaystyle\triangle^{\text{hbd}}_{\textbf{k}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\mathcal{J}^{3,3}_{\textbf{k},\textbf{p}}e^{i\phi_{\textbf{p}}}\left<a_{\textbf{p},\alpha}a_{-\textbf{p},\alpha}\right>$
$\displaystyle\mathcal{J}^{3,3}_{\textbf{k},\textbf{p}}$ $\displaystyle=$
$\displaystyle V^{N,N}_{\textbf{k},\textbf{p}}Z_{3}Z_{3}$ (55)

Here $N=1$ and $\mathcal{J}^{3,3}$ is the renormalized interaction potential
between the helical fermions. Recall that $V^{N,N}$ is the thin film
interaction potential matrix element between electrons in the $N$th band.
$Z_{3}$ is the renormalization factor of the electrons in the positive
helicity branch of the bottom hybrid band at the Fermi momentum $k_{F}$.
$Z_{3}$ essentially calculates the probability amplitude of a Kramer’s pair of
fermions to be in the thin film side of the interface. Since the hybrid
fermions are a linear superposition of the thin film and the TI surface
states, they acquire a helical spin texture from the TI surface side while
also experiencing an effective attractive interaction mediated by the thin
film phonons. The superconducting order is of odd parity as expected.

Figure 6:  Energy spectrum of the hybrid bands when the Dirac point of the TI
surface is resonating with the $N=1$ thin film QW band. Here we show the case
when the Fermi level(Black dashed lines) is tuned to a single Fermi surface
and it crosses the positive helicity branch of the $b$ band. $\mu_{b}$ is the
Fermi energy measured from the emergent Dirac point of the bottom band (band
index - $(b,+)$). Its definition is given in Eqn.57.

Here we shall present certain limits where simple analytical results for the
superconducting gap can be derived. We will also show a limit where the
effective pairing essentially goes back to singlet order. To identify these
limits, let us define a parameter called $\mu_{b}$ with the following
definition,

$\mu_{b}=\mu-\epsilon^{\text{hbd}}_{0,b,+}$ (56)

It is the difference in energy between the Fermi level and the emergent Dirac
point of the bottom band. $\mu_{b}=0$ implies the Fermi level is aligned with
the Dirac point and the Fermi surface reduces to just a Fermi point. So one
can call this term an ’effective’ chemical potential of the bottom band. Let
us represent $\mu_{b}$ in dimensionless form by dividing it with the tunneling
strength $t_{d}$ defined in Eqn.6. That is,

$\displaystyle\tilde{\mu}_{b}=\frac{\mu_{b}}{t_{d}}$ (57)

Here the thickness $d$ is fixed. When $\tilde{\mu}_{b}\ll 1$, we find that the
energy dispersion of the states that cross the Fermi level is essentially a
linear function of k. That is, the energy of Fermi electrons can be
approximated as,

$\displaystyle\epsilon^{\text{hbd}}_{\textbf{k},b,+}-\mu\approx+A_{b}|\textbf{k}|-\mu_{b}$
(58)

$A_{b}=A_{0}/2$ is the effective spin-orbit coupling on the helical fermions
in the $b$ band near the Dirac point. When the Debye frequency
$\omega_{D}<\mu_{b}$, only the positive helicity branch is interacting. In
this limit, one can solve Eqn.55 analytically to arrive at a simple expression
for the magnitude of the p-wave pairing gap,

$\displaystyle\triangle^{\text{hbd}}=2\omega_{D}\text{Exp}\left[-\frac{4\pi
A^{2}_{b}}{\mu_{b}\mathcal{J}^{3,3}}\right]$ (59)

Note here that if $\mu_{b}<\omega_{D}$, then both the negative and the
positive helicity branches of the $b$ band fall within the energy window
$\left[\mu_{b}-\omega_{D},\mu_{b}+\omega_{D}\right]$. This implies that the
electronic states of both helicities that fall within this window will be
interacting. The effective theory described in Eqn.55 does not explain the
full physics in this limit.

A rather interesting limit is when the chemical potential $\mu_{b}=0$. In this
limit, hybrid electronic states of both the helicity branches experience
attractive interaction on an equal footing. Therefore, the triplet component
of the order parameter cancels out. That is, we essentially have a purely
singlet-pairing superconducting phase of helical Dirac fermions. In the limit
when $\omega_{D}\ll t_{d}$, the effective low-energy interacting Hamiltonian
in this limit has the form:

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\,\biggl{[}A_{b}\,d^{\dagger}_{\textbf{k},b}\left[\textbf{s}\times\textbf{k}.\hat{z}\right]d_{\textbf{k},b}$
(60) $\displaystyle-$
$\displaystyle\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\,\mathcal{V}_{\textbf{k},\textbf{p}}\,d^{\dagger}_{\textbf{p},b}s_{y}d^{\dagger
T}_{-\textbf{p},b}d^{T}_{-\textbf{k},b}s_{y}d_{\textbf{k},b}\biggr{]}$
$\displaystyle\mathcal{V}_{\textbf{k},\textbf{p}}$ $\displaystyle\approx$
$\displaystyle\frac{V^{1,1}_{\textbf{k},\textbf{p}}}{4}$

where $V^{1,1}_{\textbf{k},\textbf{p}}$ is the thin film phonon-mediated
interaction potential between the electronic states in the transverse bands
indexed by $N=1$. Its definition is given in Eqn.29. The factor of $4$ is
because in the limit when $\omega_{D}\ll t_{d}$, the renormalization factor is
diagonal in the spin basis with both the diagonal elements equal to $1/2$. In
other words, the electrons involved in the interaction are in quantum-well
resonance.
$d_{\textbf{k},b}=\begin{bmatrix}d_{\textbf{k},b,\uparrow}&d_{\textbf{k},b,\downarrow}\end{bmatrix}$
is the 2-component spinor representing the annihilation operator for emergent
Dirac fermions of the $b$ band in the spin basis. This effective theory has
extra emergent symmetries in contrast to the finite chemical potential case.
One can see that it has both the particle-hole symmetry and the Lorentz
symmetry. Since there are no Fermi electrons in this limit to induce Cooper
instability, the coupling constant must be greater than a critical value for
the superconducting phase transition to happen[35]. The critical value of the
interaction strength is given by,

$\mathcal{V}_{c}=\frac{4\pi A^{2}_{b}}{\omega_{D}}$ (61)

If the interaction strength is tuned to the quantum critical point, the
effective theory possesses emergent surface supersymmetry(SUSY). So what we
have here is essentially a very practical platform to study the dynamics of
the emergent supersymmetric quantum matter.

####  VI.1.2 Three Fermi surface model

Figure 7:  Similar setup as in Fig.6 but here we tuned the Fermi level to
three Fermi surfaces. $t_{d}$ is the tunneling strength (defined in Eqn.57).
The numbers here represent the Fermi surface indices.

Now consider the case when the Fermi level is adjusted in such a way that we
effectively have three Fermi surfaces. A schematic picture of such a
possibility is shown in Fig.5. To realize a three Fermi surface model, the
effective chemical potential of the $b$ band defined as $\mu_{b}$ in Eqn.56
has to be greater than $2t_{d}$. In this limit, the Fermi surface closest to
the Dirac point is formed by either positive or negative helicity branches of
the $t$ band depending on the fine-tuning of the chemical potential. This is
indexed by $1$. The second and third Fermi surfaces are formed by the negative
helicity branch of the $t$ band (band index - $(t,-)$) and the positive
helicity branch of the $b$ band (band index - $(b,+)$) respectively. They are
indexed as $2$ and $3$ respectively. Since the attractive interaction is
mediated by phonons, only the electronic states lying within the energy window
$\pm\omega_{D}$ measured from the chemical potential actually experience an
attractive interaction. Since we are working in the limit where
$\omega_{D}\ll\mu$, the absolute value of the chemical potential, essentially
only the electrons in and around the Fermi level take part in the interaction.
Also note that, since we are in the $N=1$ limit, only the helical hybrid
fermions are present in the system. The mean field Hamiltonian then takes the
form,

$\displaystyle\mathcal{H}_{\text{MF}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\biggl{[}\triangle^{\text{hbd}}_{\textbf{k},1}e^{-i\phi_{\textbf{k}}}a^{\dagger}_{\textbf{k},t,+}a^{\dagger}_{-\textbf{k},t,+}$
$\displaystyle-$
$\displaystyle\triangle^{\text{hbd}}_{\textbf{k},2}e^{-i\phi_{\textbf{k}}}a^{\dagger}_{\textbf{k},t,-}a^{\dagger}_{-\textbf{k},t,-}$
$\displaystyle+$
$\displaystyle\triangle^{\text{hbd}}_{\textbf{k},3}e^{-i\phi_{\textbf{k}}}a^{\dagger}_{\textbf{k},b,+}a^{\dagger}_{-\textbf{k},b,+}+h.c\biggr{]}$

Here we assumed that the Fermi level crosses the positive helicity branch of
the top band to form the Fermi surface that is closest to the Dirac point as
shown in Fig.12(a). Also, the energy difference between the Dirac point of the
$t$ band and the Fermi level must be higher than the Debye frequency for the
above Hamiltonian to effectively describe the pairing physics. Otherwise, the
electrons in the negative helicity branch of the $t$ band near $k=0$ will also
be interacting. This is not taken into account in the effective Hamiltonian
defined here.

As long as the three hybrid Fermi surfaces do not overlap in the momentum
space, the superconducting order on each of them is of p-wave symmetry. Notice
that since the Fermi surface indexed by $2$ is formed by the negative helicity
branch of the $t$ band, the sign of the order parameter is negative. That is,
it differs from the order parameter on the positive helicity branch by a phase
of $\pi$. If this Fermi surface happens to overlap with a positive helicity
branch of the $b$ band, which could happen in case the tunneling is zero or
negligible, then one can find that the triplet component of the order
parameter cancels out. In that case, we are left with an even-parity spin-
singlet pairing phase.

The superconducting gap equation satisfied by $\triangle^{\text{hbd}}_{i}$’s
is similar to what is given in Eqn.49a. But since there are no thin film FSs,
the RHS of Eqn.49a vanishes. So we finally obtain a simple form for the gap
equation which we shall write down below for clarity,

$\displaystyle\triangle^{\text{hbd}}_{\textbf{k},i}$ $\displaystyle-$
$\displaystyle\sum^{3}_{j=1}\int\frac{d^{2}\textbf{p}}{(2\pi)^{2}}\frac{\mathcal{J}^{ij}_{\textbf{k},\textbf{p}}\triangle^{\text{hbd}}_{\textbf{p},j}}{2\sqrt{\xi^{2}_{\textbf{p},j}+(\triangle^{\text{hbd}}_{\textbf{p},j})^{2}}}$
(63) $\displaystyle=$ $\displaystyle 0$

where $i=1,2,3$. The matrix elements of $\hat{J}$ are given in Eqn.45. It
describes the scattering strength of Kramer’s doublets from the Fermi surface
indexed by $i$ to $j$.

So we find here that we have to effectively solve a set of 3 non-linear
coupled integral equations to find the superconducting order parameters in
each Fermi surface. A simple analytical solution as was done in the single
Fermi surface case is difficult to realize here.

###  VI.2 Numerical results: Solving the gap equation

The objective of this part of the section is to study the evolution of the
superconducting order in the $N=1$ limit as a function of various tuning
parameters. Basically, our goal is to look for various ways to enhance the
superconductivity. The role of the thin film in this hybrid system is to
induce an effective attractive interaction between the helical surface
fermions. Therefore, a straightforward way to enhance the pairing interaction
between the helical hybrid fermions will be to tune the electron-phonon
coupling strength of the thin film metal. In the case of a topological
insulator, it is the spin-orbit interaction that decides the Fermi velocity of
the surface Dirac fermions. So understanding the evolution of the
superconducting order as a function of the spin-orbit coupling strength is
important.

Here we begin by emphasizing again the role played by quantum-well resonance
in realizing a ground state with attractively interacting helical fermions and
in enhancing the superconducting order. This is a continuation of the physics
discussed in section IV. There we discussed how the effective attractive
interaction attained by the surface fermions through tunneling reaches its
maximum when the two systems are in quantum-well resonance. We used the
evolution of the $Z$-factors of the two hybrid bands as a function of the
detuning parameter to prove this point. Having derived the pairing gap
equation, we can finally study how the pairing gap on the Fermi surfaces
evolves as a function of the detuning parameter. This will give a rather
concrete idea of why we must tune the thin film thickness to quantum-well
resonance for a given $N$ to study the interacting physics of surface
fermions.

In short, we essentially write down the p-wave superconducting gap on the
hybrid bands as a function of the three tuning parameters,

$\displaystyle\triangle^{\text{hbd}}_{i}\equiv\triangle^{\text{hbd}}_{i}(\tilde{\delta}(d),\tilde{\lambda}^{\text{bulk}},\tilde{v})$
(64)

Here $\tilde{\lambda}^{\text{bulk}}$ is the dimensionless form of the phonon-
mediated interaction strength of the 3D bulk counterpart of the metal thin
film. In terms of the electron-phonon coupling strength $G_{fp}$ defined in
Eqn.26,

$\displaystyle\tilde{\lambda}^{\text{bulk}}=\frac{mk^{\text{bulk}}_{F}}{2\pi^{2}\hbar^{2}}G^{2}_{\text{fp}}$
(65)

$k^{\text{bulk}}_{F}$ is the bulk Fermi momentum of the metal for a given
chemical potential. In the calculations here, we shall only tune the electron-
phonon coupling strength of the metal while keeping all other parameters
constant. The dimensionless detuning parameter is defined in Eqn.41.
$\tilde{v}$ here is the dimensionless form to represent the Fermi velocity of
the surface fermions. For the class of topological insulators that we
consider, it is proportional to the SOC strength of the TI. It has the
following definition,

$\tilde{v}=\frac{A_{0}}{\hbar c}$ (66)

where $A_{0}$ is the SOC strength of the topological insulator and $c$ is the
speed of light. Tuning down $\tilde{v}$ is essentially equivalent to moving
towards the flat band limit of the TI surface. Now we shall study the
evolution of the superconducting order as a function of these dimensionless
tuning parameters. For numerical purposes, we shall be using material
parameters corresponding to Pb(lead) for the thin film except in the section
where we tune the interaction strength.

Figure 8: Resonance effect: Plots showing the emergence of topological
superconductivity on the thin film-TI hybrid due to effective attractively
interacting surface fermions when the detuning parameter $\tilde{\delta}(d)$
approaches zero. (a) shows the energy spectra of the TI surface Dirac cone and
the thin film transverse band at three regimes of the detuning parameter
before the tunneling is turned on. (c) shows the energy spectra of the hybrid
bands at the same three values of the detuning parameter after the tunneling
is turned on. The grey horizontal line denotes the Fermi level. The numbers
denote the Fermi surface indices as defined in the main text. (b) shows the
evolution of the superconducting order parameters
$\triangle^{\text{hbd}}_{i}$’s on the three Fermi surfaces indexed by
$i=1,2,3$ as a function of the detuning parameter before tunneling is turned
on. (d) shows the evolution of the same set of order parameters after
tunneling is turned on.

####  VI.2.1 Resonance effect

Here we shall study the evolution of the p-wave pairing gaps as a function of
the dimensionless detuning parameter at $\textbf{k}=0$ defined in Eqn.41. The
detuning parameter is varied by tuning the thin film thickness. We shall solve
the gap equation both before and after the tunneling is turned on. The Fermi
level is set at $0.05\text{eV}$ above the Dirac point of the topological
insulator. Essentially, we set the Fermi level close to the Dirac point
because we are tuning the detuning parameter defined at $k=0$. If the Fermi
level is much above or below the Dirac point, then the detuning parameter
should be defined at the Fermi momentum instead of at $k=0$.

Fig.41 shows the results. Here the detuning parameter is varied from $-2$ to
$2$. We have studied the evolution of the pairing gaps
$\triangle^{\text{hbd}}_{1}$(Red), $\triangle^{\text{hbd}}_{2}$(Green) and
$\triangle^{\text{hbd}}_{3}$(Blue) on the three Fermi surfaces(if present)
before and after the tunneling is switched on. Before the tunneling is turned
on, the innermost Fermi surface is formed by the surface Dirac cone. The
second and third Fermi surfaces are formed entirely by the two helicity
branches of the thin film band and hence they overlap. Essentially in this
limit, the TI surface is non-interacting, which means we are studying just the
thin film superconductivity. The purpose is just to set a benchmark for the
study of the superconducting order once the tunneling is turned on. Therefore,
$\triangle^{\text{hbd}}_{1}$ is always zero. And we have
$\triangle^{\text{hbd}}_{2}=\triangle^{\text{hbd}}_{3}$. The triplet component
of the order parameter cancels out and we have the trivial s-wave
superconducting order as expected. When the detuning parameter is increased,
the thin film band starts moving up. This is because, in our convention,
increasing the detuning parameter is equivalent to reducing the thin film
thickness. At a particular thickness, the bottom of the band crosses the Fermi
level. Beyond this point, there are no interacting Fermi electrons. Hence
superconductivity vanishes as the detuning parameter is increased further.

Now when the tunneling is turned on, the surface band and the thin film band
get hybridized. From the figure, we understand that the pairing physics is not
very different from the zero-tunneling result when $|\tilde{\delta}|\gg 0$.
But as we fine-tune $\tilde{\delta}$ to zero, we start seeing the effects of
electronic hybridization. The electrons in the innermost Fermi surface, which
essentially is the surface Dirac cone start interacting and a superconducting
gap opens up. The magnitude of the gap increases as we fine-tune to
$\tilde{\delta}=0$ from the left side. One can identify that
$\triangle^{\text{hbd}}_{1}$(the red points in the plot) is the effective
pairing gap on the Dirac cone. Note that the contribution to the pairing gap
also comes from the scattering of Cooper pairs to the other two Fermi surfaces
as well.

When the detuning parameter is increased further, the bottom of the $t$ hybrid
band crosses the Fermi level. This means, there is essentially a crossover
from the three Fermi surface to the single Fermi surface limit. Both the 1st
and the 2nd Fermi surfaces vanish beyond this limit. When the tunneling was
zero, there was no superconductivity in this limit because the surface was
essentially non-interacting. But here we see that a superconducting gap exists
on the Fermi surface formed by the Dirac cone (the blue-colored points on the
plot). This is clear evidence of the effective attractive interaction between
the surface Dirac fermions. Also, we see that the magnitude of the gap
decreases as the detuning parameter is tuned away from zero. This clearly
proves that the quantum-well resonance is the ideal point to study the
attractive interacting physics of surface Dirac fermions.

Figure 9:  Single hybrid Fermi surface limit (see fig.6 for the energy
spectrum): Plot showing the evolution of the superconducting order parameter
of the resonating hybrid electrons as a function of the bulk coupling constant
of the thin film $\tilde{\lambda}^{\text{bulk}}$(defined in Eqn.65). The
evolution is studied at three different values of the effective chemical
potential expressed in a dimensionless form as $\tilde{\mu}_{b}$(defined in
Eqn.57). The SOC strength is fixed at $A_{0}=1.5\text{eV}$Åand the tunneling
strength $t_{d}=0.2\text{eV}$.

####  VI.2.2 Dependence on the interaction strength

In part 1, we understood the importance of quantum-well resonance to realize a
phase with attractively interacting helical surface fermions. So from here
onwards, we fine-tune the thickness to quantum-well resonance at the Dirac
point. In this limit, the electronic states close to the Dirac point on both
sides of the interface are strongly hybridized. There is no clear difference
between the thin film and the TI surface fermions. These resonating hybrid
fermions acquire the emergent spin-orbit coupling from the thin film side and
an effective attractive interaction from the thin film side. We effectively
have helical fermions with an effective attractive interaction between them.

Here we tune the electron-phonon coupling strength $G_{\text{fp}}$ of the thin
film metal and study the evolution of the pairing gap on the hybrid Fermi
surfaces. To represent the tuning parameter in a dimensionless form, we
defined the bulk coupling constant of the metal
$\tilde{\lambda}^{\text{bulk}}$ in Eqn.65. We keep all other material
parameters including Debye frequency, effective electron mass, etc. constant.
Here we used the material parameters of the Pb metal for numerical
calculations. The cases of single and three Fermi surfaces were considered
separately. The effective chemical potential was fine-tuned further for each
of the two cases to understand its significance.

##### Single Fermi surface

Fig.9 shows the results in the case when the chemical potential is tuned to a
single Fermi surface. Here we plotted the magnitude of the p-wave
superconducting gap represented in a dimensionless form(with respect to the
Debye frequency) at three different chemical potential values,
$\tilde{\mu}_{b}=0.25,0.50,0.75$. Here chemical potential is expressed in a
dimensionless form as $\tilde{\mu}_{b}=\mu_{b}/t_{d}$ where the tunneling
strength $t_{d}$ is fixed at $t_{d}=0.2\text{eV}$. The chemical potential is
set very close to the Dirac point because the Fermi electrons then will be at
quantum well resonance. In addition, the electron band will be linear,
resembling a surface Dirac cone. The corresponding energy spectrum is shown in
fig.6 We set the spin-orbit coupling strength at $A_{0}=1.5\text{eV}$Å. To
arrive at this result, we numerically solved Eqn.55 self-consistently at
different values of the coupling strength.

Figure 10: Three hybrid Fermi surfaces (see fig.7 for the energy specctrum):
Similar setting as in fig.9 but here the effective chemical potential
$\tilde{\mu}_{b}>2$ for all the three cases. Thus we have three hybrid Fermi
surfaces. The evolution of the SC order parameters on these three Fermi
surfaces has been studied as a function of the bulk interaction strength of
the thin film.

As expected, we find an exponential enhancement of the superconducting gap as
the coupling constant $\tilde{\lambda}^{\text{bulk}}$ is increased. Increasing
chemical potential also enhances the superconducting gap. The results can be
explained in the following way: Since the chemical potential is set close to
the Dirac point($\tilde{\mu}_{b}<1$), the band is nearly linear when it
crosses the Fermi level. Hence the approximate analytical expression for the
pairing gap magnitude derived in Eqn.59 works well in these cases. There we
found that $\triangle^{\text{hbd}}\propto e^{-1/\mu_{b}\mathcal{J}^{3,3}}$.
Here $\mathcal{J}^{3,3}$ is proportional to the electron-phonon coupling
constant. Thus both the chemical potential and the interaction strength have a
similar enhancement effect on the superconducting gap magnitude. This is in
contrast to a 2D quadratic electronic dispersion. There the density of states
is independent of the chemical potential. Note here that, if $\mu_{b}\geq
t_{d}$, then the band is no longer linear. In this case, the analytical result
derived in Eqn.59 is no longer a good approximation. In addition, since the
Fermi electrons lie away from the quantum-well resonance, the tunneling
effects will be perturbative.

##### Three Fermi surfaces

Fig.10 shows the results when the chemical potential is tuned to three Fermi
surfaces. As discussed before, the effective chemical potential, $\mu_{b}$
must be of the order of $2t_{d}$ or greater than that to realize a three Fermi
surface model. The corresponding energy spectrum is given in Eqn.7 Here we
studied the evolution of the p-wave pairing gaps on the three Fermi surfaces
as a function of the electron-phonon coupling strength of the thin film metal
at three different values of the chemical potential. Here
$\triangle^{\text{hbd}}_{i}$(i=1,2,3) is the SC gap magnitude on the $i$th
Fermi surface. Here $1$ is the closest and $3$ is the farthest from the Dirac
point. They are represented in a dimensionless form by dividing them with the
Debye frequency of the thin film metal. We used the dimensionless parameter
$\tilde{\mu}_{b}$ to represent the chemical potential. The tunneling strength
and the spin-orbit coupling strength of the TI surface are all fixed with the
same numerical values as in the single Fermi surface case. We numerically
solved the coupled set of superconducting gap equations given in Eqn.63 to
arrive at these results.

We see a much-anticipated enhancement in the superconducting gap magnitude as
the interaction strength is increased. We also notice that the magnitude of
the superconducting gap is substantially larger compared to the single Fermi
surface case for a given strength of interaction. This is because there is a
larger number of Fermi electrons involved in the interaction for the three
Fermi surface cases, leading to an enhancement in the superconducting order.

####  VI.2.3 Dependence on the spin-orbit coupling strength

Here we shall study the evolution of the superconducting order on the helical
hybrid bands as a function of the spin-orbit coupling strength of the TI
surface. As we did in the previous part, the Dirac point of the TI surface is
fixed at quantum-well resonance with the $N=1$ transverse band of the thin
film. The bulk interaction strength is fixed at
$\tilde{\lambda}^{\text{bulk}}=0.39$. As before, the tunneling strength is
fixed at $t_{d}=0.2\text{eV}$. The spin-orbit coupling strength is expressed
in a dimensionless form given by $\tilde{v}=A_{0}/\hbar c$. The logic here is
that for a given SOC strength $A_{0}$, the Dirac velocity of the surface
fermions is given by $v=A_{0}/\hbar$. So tuning the SOC strength is equivalent
to tuning the Dirac velocity of the surface fermions.

We study the SOC dependence for the two cases separately: when the Fermi level
is set to a single Fermi surface and when the Fermi level is set to three
Fermi surfaces. Even though we expect a monotonic increase in the
superconducting gap as the SOC strength is decreased due to the obvious
increase in the density of states, we shall find here that it is not the case.
The change in the hybrid band structure has huge consequences on the
renormalization factors $Z_{i}$ which substantially affects the pairing
interaction.

Figure 11: Single Fermi surface model (See fig.6): Plot (a) shows the
evolution of the superconducting order parameter as a function of $\tilde{v}$.
Here $\tilde{v}=\frac{A_{0}}{\hbar c}$ is used to represent the spin-orbit
coupling strength of the TI in a dimensionless way. It also indicates the
Dirac velocity of the surface fermions. We study the pairing at three
different values of the effective chemical potential $\tilde{\mu}$(Definition
given in Eqn.57). (b) shows the evolution of the density of states at the
Fermi level as $\tilde{v}$ is tuned. (c) shows the evolution of the $Z$-factor
of the hybrid Fermi electronic states involved in pairing. It represents the
probability amplitude of the Fermi electrons to be in the thin film side of
the interface. The expression is given in Eqn.44. The tunneling strength is
fixed at $t_{d}=0.2\text{eV}$.

##### Single Fermi surface

Here we shall study the evolution of the pairing gap as a function of the
spin-orbit coupling parametrized by $\tilde{v}$ at different values of
$\tilde{\mu}_{b}$. Since the magnitude of the superconducting gap in our case
is mostly decided by the density of states at the Fermi level and the
renormalization factor $Z_{3}$, we have plotted both of them as a function of
$\tilde{v}$. This helps us better understand the behavior of
$\tilde{\triangle}^{\text{hbd}}$ as $\tilde{v}$ is tuned. The density of
states at the Fermi level has the following definition,

$\displaystyle\mathcal{N}^{\text{hbd}}=\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\delta(\epsilon^{\text{hbd}}_{\textbf{k},b,+}-\mu)$
(67)

By tuning $\tilde{v}$, we shall expect the density of states at the Fermi
level to increase thus enhancing the superconductivity. But here we shall find
that it is not always the case as evident from Fig.11. Here we plotted
$\triangle^{\text{hbd}}$ as a function of $\tilde{v}$ at two different values
of the dimensionless chemical potential $\tilde{\mu}_{b}$. We find that the
pairing gap increases when $\tilde{v}$ is reduced, reaches a peak, and then
decreases to zero in the flat band limit when chemical potential
$\tilde{\mu}_{b}=0.50$ and $\tilde{\mu}_{b}=0.75$. But when chemical potential
is very low($\tilde{\mu}_{b}=0.25$), the peak is reached only when
$\tilde{v}\approx 0$.

This rather surprising result has to do with the renormalization factor in the
interaction constant. It essentially gives the probability amplitude of the
given electronic state to be in the thin film side of the interface. Its
definition is given in Eqn.LABEL:Zfactor. Here $Z_{3}$ is defined as the
renormalization factor of the electrons on the Fermi surface. Since the Dirac
point is in resonance with the thin film transverse band, $Z_{3}$ is exactly
$1/2$ at $k=0$. But if the Fermi momentum is much greater than zero, then the
renormalization factor changes from $1/2$. This is equivalent to detuning away
from resonance. If the hybrid band is adiabatically connected to the thin film
band at large $k$, then $Z_{3}\rightarrow 1$ at large Fermi momentum. On the
other hand, if the hybrid band is connected to the surface Dirac cone, then
$Z_{3}\rightarrow 1$ at large Fermi momentum. This change in the
renormalization factor can substantially affect the magnitude of the SC gap.

So what we observe here essentially is an interplay between the density of
states at the Fermi level and the renormalization factor of the electronic
states on the thin film side. The density of states increases with decreasing
$\tilde{v}$ in a monotonic fashion for any value of $\tilde{\mu}_{b}$. This is
evident from the density of states plot in Fig.11(b). The density of states
increases in a power law fashion in both cases of chemical potential as the
$\tilde{v}$ is lowered.

On the other hand, the renormalization factor $Z_{3}$ decreases as $\tilde{v}$
is lowered(see Fig.11(c)). This can be explained in the following way: Here
the Fermi level crosses the positive helicity branch of the bottom band(band
index - $(b,+)$). Consider the large $\tilde{v}$ limit, which is defined as
the limit when $Z_{3}(\tilde{v})>1/2$. In this limit, the electrons in this
band are adiabatically connected to the thin film band at a large $k$ limit,
where they are out-of-resonance. So if the Fermi level crosses this band at
large $k$, then $Z_{3}\approx 1$. Also, notice that the range of momentum
states around the Dirac point which experience strong hybridization decreases
as $\tilde{v}$ is increased. As a result of these two factors, one can see why
$Z_{3}$ increases when $\tilde{v}$ is increased. On the other hand, in the
limit of $\tilde{v}$ when $Z_{3}(\tilde{v})<1/2$, the hybrid band under
consideration(band index - $(b,+)$) is adiabatically connected to the non-
interacting surface Dirac cone. This is the reason why $Z_{3}\rightarrow 0$ as
$\tilde{v}\rightarrow 0$. At $Z_{3}(\tilde{v})=1/2$, the electrons in the
Fermi surface are in quantum-well resonance.

The variation in $Z_{3}$ will be more substantial for cases with higher
chemical potential than those with lower ones. Due to the higher chemical
potential, the Fermi electrons are detuned away from resonance and hence the
$Z_{3}$ factor will be different from $1/2$. This is the reason why we see a
peak in the pairing gap for $\tilde{\mu}_{b}=0.50,0.75$(fig.11)(a). On the
other hand, $Z_{3}\sim 1/2$ for $\tilde{\mu}_{b}=0.25$ at all values of
$\tilde{v}$, implying that the electrons lying in the Fermi surface are in
quantum-well resonance throughout. As a result, the monotonic behavior of the
density of states $\mathcal{N}^{\text{hbd}}$ is also reflected in the
evolution of the pairing gap.

Figure 12: Three Fermi surface model(see fig.7): Similar setup as in Fig.11
but here the Fermi level is set to three Fermi surfaces. The effective
chemical potential is fixed at $\tilde{\mu}_{b}=2.25$. In (b), we plotted the
density of states at the Fermi level of the three Fermi surfaces as a function
of $\tilde{v}$. The exact definition of $N_{i}$(i=1,2,3) is given in Eqn.68.
In (c), we plotted the $Z$-factors of the Fermi electrons on the three Fermi
surfaces.

##### Three Fermi surfaces

Here we study the evolution of the p-wave pairing gaps on the three FSs as a
function of $\tilde{v}$. Just like in the previous case of a single FS, the
tunneling strength and the thin film’s material parameters are kept fixed.
Note that since the tunneling results in a 2-level splitting of the top and
bottom bands at $k=0$ by a factor of $2t_{d}$, the effective chemical
potential $\mu_{b}$(defined in Eqn.56) should be of the order of $2t_{d}$ or
greater than that to realize a three Fermi surface model. In other words, the
dimensionless parameter $\tilde{\mu}_{b}\geq 2$. See the energy spectrum in
fig.7 for details.

In our calculations, we fix the effective chemical potential at
$\tilde{\mu}_{b}=2.25$. We fix the tunneling strength at $t_{d}=0.2\text{eV}$.
The results are shown in Fig.12(a). Here we plotted the p-wave superconducting
gaps on the three FSs as a function of the SOC strength of the TI surface,
represented in a dimensionless form as $\tilde{v}$ (defined in Eqn.66). As
before, we numerically solved the self-consistent superconducting gap
equations defined in Eqn.63 to calculate the pairing amplitudes on the three
Fermi surfaces. The pairing gaps have been represented in a dimensionless form
by dividing it with the Debye frequency of the thin film metal. In Fig.12(b),
we plotted the density of states at the Fermi level for each Fermi surface as
a function of $\tilde{v}$. The definitions are given by,

$\displaystyle\mathcal{N}^{\text{hbd}}_{1}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\delta(\epsilon^{\text{hbd}}_{\textbf{k},t,+}-\mu)$
$\displaystyle\mathcal{N}^{\text{hbd}}_{2}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\delta(\epsilon^{\text{hbd}}_{\textbf{k},t,-}-\mu)$
$\displaystyle\mathcal{N}^{\text{hbd}}_{3}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\textbf{k}}{(2\pi)^{2}}\delta(\epsilon^{\text{hbd}}_{\textbf{k},b,+}-\mu)$
(68)

where $\mathcal{N}^{\text{hbd}}_{i}$(i=1,2,3) implies the density of states at
the Fermi surface indexed by $i$ with $i=1$ being the closest to the Dirac
point. In Fig.12(c), we plotted the renormalization factor $Z_{i}$(i=1,2,3) of
the three Fermi surfaces. We studied the variation of the renormalization
factors of the Fermi electrons on each Fermi surface as a function of
$\tilde{v}$.

Similar to what we saw in the single Fermi surface case, the magnitude of the
SC gaps on the three Fermi surfaces is determined by the interplay of the
electron density of states at the Fermi level and the renormalization factors
$Z_{i}$. One can notice here by observing the Figs.12(a) and 12(c) that it is
the $Z$-factors in three FSs that play the dominant role here. To realize a
three-Fermi surface model, we require $\tilde{\mu}_{b}\geq 2$. Thus the Fermi
momentum of the 2nd and 3rd FSs are already much greater than zero. Thus the
tunneling effect on these Fermi electrons becomes lesser and lesser
significant as the spin-orbit coupling strength is tuned up, no matter what
the absolute value of the tunneling strength is. In addition, we also notice
that the two Fermi surfaces get closer with increasing $\tilde{v}$. This is
also reflected in the magnitude of the pairing gap. We find here that
$|\tilde{\triangle}^{\text{hbd}}_{2}-\tilde{\triangle}^{\text{hbd}}_{3}|\rightarrow
0$ as $\tilde{v}\rightarrow 1$. One can notice here that the triplet component
of the pairing amplitude, which is proportional to the difference in the
pairing amplitude on the positive and negative helicity branches for a given
k, vanishes as a result. Thus as $\tilde{v}\rightarrow 1$, the tunneling
effect on the two Fermi surfaces is negligible, effectively leading to a
trivial singlet pairing order on the two Fermi surfaces which essentially
overlaps. On the other hand, the electrons on the 1st Fermi surface have their
Z-factor nearly equal to $1/2$, implying the electronic states are near
resonance even if we increase $\tilde{v}$. This is because the Fermi momentum
is very close to zero. But notice here that the density of states
$\mathcal{N}_{1}$ is nearly zero as $\tilde{v}$ is increased. This implies
that the superconducting gap is dominated by the scattering of Cooper pairs
from the other two Fermi surfaces, rather than the intra-band scattering.

When $\tilde{v}$ is decreased, we are effectively moving toward the flat band
limit of the TI surface. The density of states at each hybrid Fermi surface
shows a monotonic increase as expected. However, this is not reflected in the
SC gap magnitude. We find here that the pairing amplitude on the third Fermi
surface vanishes in the limit $\tilde{v}\rightarrow 0$. On the other hand, the
pairing amplitudes on the first and the second Fermi surfaces converge. That
is, we observe that
$|\tilde{\triangle}^{\text{hbd}}_{1}-\tilde{\triangle}^{\text{hbd}}_{2}|\rightarrow
0$ as $\tilde{v}\rightarrow 0$. This implies that the two Fermi surfaces
overlap to form the trivial thin QW band and the superconductivity on them
will turn out to be of the trivial s-wave order. Since the superconductivity
on the third Fermi surface vanishes as $\tilde{v}\rightarrow 0$, the
topological superconductivity is absent in the flat band limit.

So in conclusion, we explored the evolution of the pairing gaps as a function
of the SOC strength on the three Fermi surfaces at a fixed chemical potential
and tunneling strength. We found that in the limit of large
$\tilde{v}$($\tilde{v}\rightarrow 1$), the second and the third Fermi surfaces
overlap and the pairing on them is of spin-singlet order. The SC pairing on
the innermost Fermi surface still maintains the p-wave character. Thus the
topological character is still maintained. In the limit when
$\tilde{v}\rightarrow 0$, we found that the electrons in the 3rd Fermi surface
lie entirely on the TI surface side. Hence they are effectively non-
interacting. The first and the second Fermi surfaces overlap and we
effectively have singlet pairing superconductivity on them. Hence in the flat
band limit, the hybrid is no longer topological.

##  VII The large $N$ limit

Here we consider the situation when the thin film band which is in quantum-
well resonance with the surface Dirac point has its band index $N$ very much
greater than one. Physically, this limit can be realized by increasing the
thickness of the thin film. This is because, the energy difference between the
successive quantum well bands,
$|\epsilon_{\textbf{k},n}-\epsilon_{\textbf{k},n-1}|\propto 1/d^{2}$. In this
situation, given that the Fermi level is adjusted close to the Dirac point,
there will be $N-1$ off-resonance degenerate thin film bands crossing the
Fermi level. Hence after hybridization, we shall have $2N-2$ off-resonance
Fermi surfaces plus one or three hybrid Fermi surfaces. When $N\gg 1$, we
anticipate that the dominant contribution to the superconducting gap on the
hybrid bands is coming from the scattering of the singlet pair of electrons
from the trivial thin film Fermi surfaces. The pairing between the helical
fermions of the hybrid bands will only have a negligible effect on the pairing
gap on off-resonance thin film bands in this limit. Effectively, one can
describe this limit as equivalent to an external s-wave pairing field acting
on the hybrid bands. So this is similar to the well-known superconducting
proximity effect but in the momentum space.

In the first part, we shall derive an analytical expression for the pairing
gap on the hybrid Fermi surface(s) by employing the large $N$ approximation.
Using this, we essentially study how far the interaction between the hybrid
fermions can enhance the superconducting gap on the hybrid Fermi surface.

In the last part of this section, we show that the momentum space proximity
effect smoothly transforms into the real space proximity effect in the
perturbative limit of tunneling. The surface interaction only gives a higher-
order correction to the proximity-induced superconducting gap.

###  VII.1 Momentum space proximity effect

Consider the case when the Fermi level is adjusted such that it crosses just a
single hybrid Fermi surface. So we have $2N-2$ off-resonance Fermi surfaces
and one hybrid Fermi surface. The exact gap equation in the limit when
$\omega_{D}\ll\mu$ is given in Eqns.51b, 51a. In the large $N$ limit, we could
make substantial simplifications to arrive at an analytical expression. Recall
that in all our calculations, we considered the attractive interaction in the
thin film to be mediated by confined phonons as explained in section IIIA. But
as $N\rightarrow\infty$ which is attained by increasing the film thickness, it
is a good approximation to replace the confined phonons with the bulk phonons.
This essentially makes the interaction potential
$V^{n,n^{\prime}}_{\textbf{k},\textbf{p}}$ isotropic. In the limit when the
thickness $d\rightarrow\infty$, the interaction potential is defined in Eqn.29
attains the following isotropic form,

$V^{n,n^{\prime}}_{\textbf{k},\textbf{p}}\approx\frac{G^{2}_{\text{fp}}}{d}\left(1+\frac{\delta_{n,n^{\prime}}}{2}\right)\theta(\omega_{D}-\xi^{\text{tf}}_{\textbf{k}})\theta(\omega_{D}-\xi^{\text{tf}}_{\textbf{p}})$

where $\delta_{n.n^{\prime}}$ here is the Dirac-delta function. Since the
interaction potential is isotropic, the superconducting gap will also turn out
to be the same on all the thin film QW bands. Now we shall plug this back into
Eqn. 51b. Also in the large $N$ limit, scattering of Cooper pairs from the
hybrid Fermi surface will have only a negligible effect on the s-wave thin
film superconducting gap. This means the second term in the LHS of Eqn.51b is
neglected. With all these approximations, we obtain the following simple
analytical form for the thin film s-wave superconducting gap,

$\displaystyle\triangle^{\text{tf}}$ $\displaystyle\approx$ $\displaystyle
2\omega_{D}\text{Exp}\left[-\frac{d}{G^{2}_{\text{fp}}\mathcal{N}^{\text{tf}}(N-1/2)}\right]$
(69) $\displaystyle\text{where}\,\,\triangle^{\text{tf}}_{n}$ $\displaystyle=$
$\displaystyle\triangle^{\text{tf}}_{n^{\prime}}=\triangle^{\text{tf}},\forall
n,n^{\prime}\leq N$

Here we used $\triangle^{\text{tf}}$ for the s-wave superconducting gap on the
thin film bands. $\mathcal{N}^{\text{tf}}=\frac{m}{2\pi\hbar^{2}}$ is the
density of states at the Fermi level of a thin film transverse band, given the
electronic dispersion is quadratic. Now let us plug this back into the gap
equation for the magnitude of the effective p-wave superconducting order
parameter on the hybrid Fermi surface. After doing some algebra, we get,

$\displaystyle\triangle^{\text{hbd}}$ $\displaystyle=$
$\displaystyle\frac{Z_{3}\triangle^{\text{tf}}}{1-\tilde{\lambda}^{\text{hbd}}\text{ln}\frac{2\omega_{D}}{\triangle^{\text{hbd}}}}$
(70) $\displaystyle\text{where}\,\,\tilde{\lambda}^{\text{hbd}}$
$\displaystyle=$ $\displaystyle\mathcal{J}^{3,3}\mathcal{N}^{\text{hbd}}$

Here $\tilde{\lambda}^{\text{hbd}}$ is the dimensionless coupling strength of
interaction between the helical hybrid fermions. $\mathcal{N}^{\text{hbd}}$ is
the density of states at the hybrid Fermi surface. $\mathcal{J}^{3,3}$ defined
in Eqn.45 is the renormalized interaction potential between the hybrid
fermions. $Z_{3}$ in the numerator is the renormalization factor of the hybrid
Fermi electrons(defined in Eqns.LABEL:Zfactor, 44). This factor comes from the
scattering matrix element $\mathcal{K}^{n,3}$ that determines the scattering
of singlet pair of electrons from the off-resonance thin film Fermi surface to
the hybrid Fermi surface.

Figure 13:  An illustration of the enhancement effect on the superconducting
gap on the hybrid Fermi surface due to the surface interaction. Here
$x=\triangle^{\text{hbd}}/\omega_{D}$, the SC gap on the hybrid Fermi surface
is taken as a variable. $y_{\text{prxmt}}(x)$(blue) takes into account the
contribution to the pairing gap due to the scattering of Cooper pair from the
off-resonance thin film Fermi surfaces. $y_{\text{hbd}}(x)$(red) is the
contribution to the SC gap due to interaction between the hybrid fermions. The
exact solution is at the point where the two curves cross each other. Figure
14: Here we solved the superconducting gap equation in the large-$N$
limit(Eqn.13) and plotted the SC order parameter $\triangle^{\text{hbd}}$(blue
curve) as a function of the coupling strength between the hybrid fermions
$\tilde{\lambda}^{\text{hbd}}$. The red dashed lines show the contribution to
the SC gap due to the momentum space proximity effect. The black dashed line
shows the solution solely due to the Cooper instability on the hybrid Fermi
surface.

Let us analyze the large $N$ result given in Eqn.70 more carefully. The
numerator and the denominator come from different sources. The numerator is
essentially the contribution to the superconducting gap due to the scattering
of singlet-pair electrons from the off-resonance thin film Fermi surfaces. The
denominator is due to the attractive interaction between the helical hybrid
fermions. Hence it is this term that actually results in the Cooper
instability on the hybrid Fermi surface. The numerator could open up a gap but
does not lead to actual Cooper instability.

The numerator in the above expression is analogous to the proximity-induced
superconductivity observed in several TI-SC heterostructures[12]. In the
proximity effect, the superconducting gap opens up on the Dirac cone due to
the tunneling of Cooper pairs across the junction. The difference here is that
the coefficient $Z$ here is nearly equal to $1/2$. In fact, as we shall
demonstrate soon, the numerator turns out to be the proximity-induced
superconducting gap in the perturbative limit of tunneling. The difference we
notice in the resonance regime is that we observe an enhancement in the
superconducting gap due to the attractive interaction between the helical
hybrid fermions. The amount of enhancement is determined by the coupling
strength $\tilde{\lambda}^{\text{hbd}}$.

The fig.13 below illustrates this enhancement effect on the p-wave pairing gap
due to the interaction between the helical fermions. To do this, we defined
the following functions,

$\displaystyle y_{\text{hbd}}(x)$ $\displaystyle=$ $\displaystyle
1-\tilde{\lambda}^{\text{hbd}}\text{ln}\frac{2}{x}$ (71a) $\displaystyle
y_{\text{prxmt}}(x)$ $\displaystyle=$
$\displaystyle\frac{Z_{3}\tilde{\triangle}^{\text{tf}}}{x}$ (71b)

Here we replaced $\triangle^{\text{hbd}}/\omega_{D}$ in Eqn.70 by a variable
$x$. So $\triangle^{\text{tf}}$ is also represented in a dimensionless form as
$\tilde{\triangle}^{\text{tf}}=\triangle^{\text{tf}}/\omega_{D}$.
$y_{\text{hbd}}(x)$ is the contribution to the pairing gap due to the
interaction between hybrid fermions. $y_{\text{prxmt}}(x)$ is the contribution
due to the momentum-space proximity effect. The actual value for $x$ is found
by solving the equation $y_{\text{hbd}}(x)=y_{\text{prxmt}}(x)$. We shall call
the actual solution by $x_{0}$. One can call the solution to the equation
$y_{\text{prxmt}}(x)=1$ as the proximity limit of the superconductivity. This
would have been the actual solution if the coupling constant
$\tilde{\lambda}^{\text{hbd}}=0$. Then we plotted the function
$y_{\text{hbd}}(x)$ at different values of the coupling constant
$\tilde{\lambda}^{\text{hbd}}$ in fig.13. Here we find that as the coupling
constant is increased, the crossing point moves farther away from the
proximity limit. This shows strong evidence of enhancement in the
superconducting order due to interaction between hybridized fermions

To further emphasize this enhancement effect due to interaction between the
hybrid fermions, we solved the equation
$y_{\text{hbd}}(x)=y_{\text{prxmt}}(x)$ and plotted the resulting
superconducting order parameter magnitude $\tilde{\triangle}^{\text{hbd}}$ as
a function of the hybrid coupling constant $\tilde{\lambda}^{\text{hbd}}$. The
results are shown in fig.14. Here the red dashed lines are the proximity limit
of the superconductivity obtained by solving $y_{\text{prxmt}}=1$, while the
black dashed lines give the BCS limit of the hybrid FS given by
$y_{\text{hbd}}(x)=0$. The enhancement due to the surface interaction exists
even in the weakly interacting limit. As $\tilde{\lambda}^{\text{hbd}}$
approaches unity, we find that the order parameter attains an exponential
form.

But there are practical limitations in enhancing
$\tilde{\lambda}^{\text{hbd}}$ to strongly interacting limit. The interaction
potential $\mathcal{J}^{3,3}$ is predetermined by the bulk coupling constant
of the thin film. At resonance, it is of the form
$\mathcal{J}^{3,3}=Z^{2}_{3}V^{N,N}\approx V^{N,N}/4$, which means it is
always less than $\mathcal{K}^{n,3}$ for any $n$. So the only tunable
parameter is the density of states at the Fermi level given by
$\mathcal{N}^{\text{hbd}}$. If the energy dispersion of the hybrid band is
linear when it crosses the Fermi level, then
$\mathcal{N}^{\text{hbd}}=\frac{\mu_{b}}{2\pi A^{2}_{b}}$(Refer to Eqn.59).
Ideally, one could tune down the SOC strength of the TI surface to enhance the
surface interaction. But as we discussed in section VIB, reducing the SOC
strength will detune the Fermi electrons away from resonance for a fixed
chemical potential, driving the Fermi surface back to the perturbative limit
of tunneling. In short, what we like to convey here is that there are
practical limitations in increasing the coupling strength
$\tilde{\lambda}^{\text{hbd}}$. So in the large $N$ limit, the dominant
contribution to the superconducting gap on the hybrid Fermi surface comes from
the momentum space proximity effect due to the off-resonance thin film bands.
There is an enhancement due to the Cooper instability on the hybrid Fermi
surface, but that is not very substantial compared to the proximity effect.

###  VII.2 Perturbative limit of tunneling: Connection to the Fu-Kane model

Here we shall consider the perturbative limit of tunneling by detuning away
from the quantum-well resonance of the TI-thin film hybrid. Our objective here
is to show that the momentum space proximity effect discussed in the previous
section transforms into the real-space superconducting proximity effect in the
perturbative limit of tunneling.

The perturbative regime is characterized by the limit $\tilde{\delta}\gg 0$.
Here $\tilde{\delta}$ is the dimensionless detuning parameter at $k=0$ defined
in Eqn.41. So for convenience, we shall define a new parameter to study the
perturbative limit given by,

$\displaystyle\tilde{t}=\frac{1}{\tilde{\delta}}$ (72)

where we can call $\tilde{t}$ as the dimensionless tunneling strength. This
quantity essentially gives the probability amplitude of an electronic state in
the thin film side to tunnel to the TI surface and vice versa.

In the perturbative regime, the single-particle hybridization effects are
negligible. This implies that we should treat the surface fermions and the
thin film fermions separately. This is evident from the discussions we had in
section IV regarding the Z-effect. There we saw that on tuning
$\tilde{\delta}\rightarrow-\infty$, the top hybrid band transforms to the
surface Dirac cone and the bottom hybrid band transforms to the thin film
band. Correspondingly $Z^{b}$ approaches unity while $Z^{t}$ approaches zero.
It happens the other way when $\tilde{\delta}\rightarrow+\infty$.

Now we shall see how the expression for $\triangle^{\text{hbd}}$ derived in
the large $N$ limit at quantum-well resonance(see Eqn.70) changes when detuned
to the perturbative limit. We shall be studying the perturbative limit for the
case when $\tilde{\delta}\geq 0$. But the qualitative conclusions do not
change when $\tilde{\delta}\leq 0$ also. If the Fermi momentum of the surface
Dirac cone is very small, then $Z_{3}$ is essentially equal to $Z^{b}$ defined
in Eqn.42. For clarity, let us rewrite the expression again here. When Fermi
momentum of the surface Dirac cone $k_{F}\approx 0$,

$\displaystyle
Z_{3}=Z^{b}(\tilde{\delta})=\frac{1}{2}\left(1-\frac{\tilde{\delta}}{\sqrt{1+\tilde{\delta}^{2}}}\right)$
(73)

Now expanding $Z_{3}$ in powers of $\tilde{t}$, we arrive at,

$\displaystyle Z_{3}=\tilde{t}^{2}+\mathcal{O}(\tilde{t}^{4})$ (74)

Thus $Z_{3}$ scales as $\tilde{t}^{2}$ in the perturbative limit of tunneling.
Recall that the coupling strength $\tilde{\lambda}^{\text{hbd}}$ determines
the interaction between the surface fermions. Since there is no hybridization
in this limit, let us call $\tilde{\lambda}^{\text{hbd}}$ as
$\tilde{\lambda}^{\text{surf}}$. This is to emphasize that the coupling
constant determines the attractive interaction strength between the surface
fermions. Since the interaction potential is proportional to the square of the
$Z$ factor, we see that in the perturbative limit,

$\displaystyle\tilde{\lambda}^{\text{surf}}$ $\displaystyle=$
$\displaystyle\alpha\tilde{t}^{4}$ (75)

where $\alpha=V^{N,N}\mathcal{N}^{\text{surf}}$. Here $N$ is the index of the
thin film band that is closest to the TI surface. $\mathcal{N}^{\text{surf}}$
is the density of states at the Fermi level of the surface Dirac cone.
Plugging this back to the Eqn.70, the expression for the superconducting gap
at the surface Dirac cone when expanded in powers of $\tilde{t}$ has the form,