# Electronic confinement of surface states in a topological insulator nanowire

Ruchi Saxena  Advanced Technology Institute and Department of Physics,
University of Surrey, Guildford, GU2 7XH, United Kingdom  Eytan Grosfeld
Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 8410501,
Israel  Sebastian E de Graaf  National Physical Laboratory, Hampton Road, TW11
0LW, Teddington, United Kingdom  Tobias Lindstrom  National Physical
Laboratory, Hampton Road, TW11 0LW, Teddington, United Kingdom  Floriana
Lombardi  Quantum Device Physics, Chalmers University of Technology, Sweden
Oindrila Deb  Advanced Technology Institute and Department of Physics,
University of Surrey, Guildford, GU2 7XH, United Kingdom  Eran Ginossar
Advanced Technology Institute and Department of Physics, University of Surrey,
Guildford, GU2 7XH, United Kingdom

(August 28, 2024)

###### Abstract

We analyze the confinement of electronic surface states in a model of a
topological insulator nanowire. Spin-momentum locking in the surface states
reduces unwanted backscattering in the presence of non-magnetic disorder and
is known to counteract localization for certain values of magnetic flux
threading the wire. We show that intentional backscattering can be induced for
a range of conditions in the presence of a nanowire constriction. We propose a
geometry for a nanowire that involves two constrictions and show that these
regions form effective barriers that allow for the formation of a quantum dot.
We analyze the zero-temperature non-interacting electronic transport through
the device using the Landauer-Büttiker approach and show how externally
applied magnetic flux parallel to the nanowire and electrostatic gates can be
used to control the spectrum of the quantum dot and the electronic transport
through the surface states of the model device.

##  I Introduction

Topological insulators (TI) are bulk insulators that exhibit surface states
with unique electronic properties. Three-dimensional (3D) TI nanowires (TINWs)
are promising candidates for studying the electronic surface transport Peng
_et al._ (2010); Jauregui  _et al._ (2016); Matsushita  _et al._ (2019);
Bardarson  _et al._ (2010); they admit two-dimensional (2D) helical surface
states that prohibit backscattering and reduce localisation in the absence of
magnetic disorder while reducing any residual bulk transport due to their
geometry König  _et al._ (2012); Fu and Kane (2012); Ostrovsky  _et al._
(2010); Ryu  _et al._ (2007); Roushan  _et al._ (2009). While this protection
from unintended scattering harbours promise, designing mesoscopic scale
structures that can exploit the topological protection does require control of
the electronic transport, including methods to control transmission and to
generate electronic confinement (e.g., quantum dots, quantum point contacts).
Successful control of charge transport would pave the way for various
applications of TIs ranging from improved spintronic devices, through more
accurate charge pumps for quantum metrology to quantum computing Hasan and
Kane (2010); Zhang  _et al._ (2019); Soumyanarayanan  _et al._ (2016); Schmitt
_et al._ (2022).

There have been various theoretical proposals for inducing a gap in the
surface spectrum of the TIs since this would allow to achieve the electronic
confinement and control needed for practical devices. These include exploiting
the exchange coupling induced by a proximitised magnetic insulator Ferreira
and Loss (2013), by means of surface pairing through a proximitised
superconductor Fu and Kane (2008), or realising a tunnel coupling between the
top and bottom surfaces of an ultra-thin TI Zhang  _et al._ (2010); Linder
_et al._ (2009); Liu  _et al._ (2010); Lu  _et al._ (2010). The latter method
has been experimentally explored in a quantum dot fabricated from a thin film
of a 3D TI with a gate tunable barrier. It was found by varying the gate
voltage that the charge transport ranged from ohmic to tunneling regimes Cho
_et al._ (2012), although signatures of quantised levels localised within the
dot region remained elusive. On the other hand, geometrically constructed
quantum devices for charge confinement have not been fully explored in TINWs.

TINWs are promising with regards to realising full electronic confinement, and
progress in understanding transport was made in recent works which studied
effects of disorder, ripples and magnetic fields Kozlovsky  _et al._ (2020);
Xypakis  _et al._ (2020); d. Juan  _et al._ (2019); Takane and Imura (2013).
Recently, signatures of the sub-bands of the TI nanowire have been detected
experimentally Münning  _et al._ (2021) creating additional motivation to
analyse the potential of such devices for quantum confinement along the
nanowire.

In this work we show, by analysing the conditions for quantum confinement
within a defined section of a long TINW, that a quantum dot can be created
which transport is uniquely based on the nature of the topological surface
states. The emergence of the quantum dot in our work is strictly a quantum
mechanical effect which is based on the quantum interference of the Dirac
surface states. The analysis is based on an effective cylindrical model for a
microstructure with defined radius variations and external electric and
magnetic potentials. A key element is a reduced radius region (constriction)
exhibiting an increased gap in the spectrum of surface states. We show that
this region acts like a potential barrier which can backscatter Dirac
electrons Xypakis  _et al._ (2020). Conditions for charge confinement can be
found when backscattering in the TINW is allowed between non-time reversal
symmetric states as well as for cases where backscattering is facilitated by
the dynamics in the constrictions where time-reversal symmetry (TRS) is
broken. The radius variation is essential when the TINW is flux-biased to the
gapless state of half-flux for which the surface states are topologically
protected from backscattering by scalar potentials. This is considered to be
the point of optimal performance due to the resilience against disorder in the
leads. Based on this, we propose a geometrical construction that can lead to
resonant tunnelling Tsu and Esaki (1973) and form discrete states within a
confined central region, see Fig. 1. We show that the geometry exhibits clear
evidence of quantum dot formation that is manifested in the electronic
transport as sub-gap resonances.

The remainder of this paper is organised as follows. In section II, we present
the geometrical construction of a TINW for a quantum dot and discuss the
scattering between incoming and reflected states that can lead to finite
reflection. In section III, we describe the Hamiltonian and the scattering
matrix for the model device to compute the conductance employing the Landauer-
Büttiker approach. We also investigate the effect of the curved interface on
the motion of the surface states by incorporating the spin connection which is
essential to analyze the motion of Dirac particle on a curved space time
Kozlovsky  _et al._ (2020); Zhang and Vishwanath (2010); Koke  _et al._
(2016); Parente  _et al._ (2011). We discuss the numerical results in section
IV and conclude our findings in section V.

##  II Proposed geometry

We start by presenting the proposed geometry for the TINW quantum dot (Fig.
1a). We consider a cylindrical nanowire of radius $R_{1}$, which is etched at
two regions to a radius $R_{2}$ ($R_{2}<R_{1}$). We label this device NCNCN
where “N” refers to the region with radius $R_{1}$ and “C” denotes the region
with the reduced radius $R_{2}$. Both regions are assumed to have full
rotational-symmetry around the wire axis. The wave functions in each region
satisfy an anti-periodic boundary condition around the nanowire perimeter due
to the curvature-induced $\pi$ Berry phase. As detailed in section III, due to
the radius dependent band gap in the TINW Egger  _et al._ (2010); Kundu  _et
al._ (2011), the band-gap profile experienced by the incoming electronic
states is expected to behave as shown schematically in Fig. 1b. Etching is
expected to have some gradual radius profile, as depicted in Fig. 1c. To
incorporate its effect on the electronic transport we model the interface
using a $z$-dependent radius, as demonstrated in Fig. 1d, with

$\displaystyle R(z)=R_{1}+(R_{2}-R_{1})F(z),$ (1)

where
$F(z)=\frac{1}{2}\left[1+\frac{2}{\pi}\tan^{-1}\left(\frac{z}{a}\right)\right]$
is the smooth Heaviside theta function with the parameter $a$ tuning the
interface from step-like (small $a$) to smooth (large $a$).

Controlled reflection of the electrons from the interfaces is essential to
realising charge confinement. The motion of the particle belonging to a
specific sub-band can be thought of as one-dimensional (only dispersing along
the wire axis) with the gap proportional to the angular momentum $l$. The
incoming and reflected states of same $l$ are not Kramers pairs and hence, are
prone to backscattering due to disorder Egger  _et al._ (2010); Xypakis  _et
al._ (2020); Hong  _et al._ (2014). The unusual finite reflection here is
strikingly different than the zero reflection (unit transmission) situation
where backscattering is prohibited due to the spin-momentum locked gapless and
orthogonal surface states. The scattering between non-orthogonal surface
states in TINW, characterised by $k$ of opposite sign and same $l$, can be
thought to be a result of the finite angular momentum. Similar scattering is
paradigmatic to the suppression of Klein tunnelling in graphene Cheianov and
Falko (2006); Bai  _et al._ (2017) and in TIs Modak  _et al._ (2012); Deb  _et
al._ (2014) when a massless Dirac fermion is incident at an oblique angle to
the potential barrier. A finite transverse component of the momentum arising
from the oblique incidence allows backscattering of the incident Dirac
fermions Gutierrez  _et al._ (2016); Qiao  _et al._ (2017); Liu  _et al._
(2017); Young and Kim (2009); Deb  _et al._ (2014) due to the non-orthogonal
incoming and reflected states.

Figure 1:  Geometrical model of the quantum dot based on a 3D TINW: (a) The
geometry used for the theoretical study of the transport properties. N and C
denote regions of radius $R_{1}$ and the reduced radius regions of radius
$R_{2}$ formed by etching the wire respectively. The interfaces between N and
C regions are drawn flat for simplicity. (b) The bandgap profile (as explained
in the main text) experienced by an incident electron. (c) A smooth interface
connecting the N and C regions of the TINW. (d) The radius dependence of the
TINW as function of the coordinate $z$ along the wire with the parameter $a=8$
corresponding to (c) and $a=0.01$ generating a step-like interface.

##  III Surface Hamiltonian and scattering analysis

The Hamiltonian for the surface states of a non-uniform 3D TINW can be derived
using a field theoretic approach Kozlovsky  _et al._ (2020); Xypakis  _et al._
(2020); d. Juan  _et al._ (2019); Takane and Imura (2013). We provide the
derivation of the Hamiltonian in the Appendix for notational convenience. To
include the coaxial magnetic field we use the symmetric gauge to write the
vector potential $A_{\phi}=-Br/2=-\Phi/2\pi r$, where $\phi$ denotes the
azimuthal direction, r is the radial coordinate, and $\Phi$ is the magnetic
flux along the nanowire axis. With the minimal coupling of the magnetic field
to the transverse motion of the Dirac surface states the Hamiltonian becomes

$\begin{split}H&=\hbar
v_{F}\Bigg{[}\frac{1}{\sqrt{1+{R^{\prime}(z)}^{2}}}\left\\{i\partial_{z}+\frac{iR^{\prime}(z)}{2R(z)}\right\\}\sigma_{y}\\\
&\hskip
85.35826pt+\frac{1}{R(z)}\left(-i\partial_{\phi}-\varphi\right)\sigma_{z}\Bigg{]},\end{split}$
(2)

where $R(z)$ is the radial function which is defined to model the interface
between N and C regions and $\varphi=\Phi/\Phi_{0}$ is the dimensionless
magnetic flux. The term proportional to $R^{\prime}$ is the spin-connection
contribution to the Dirac fermions moving along the curved surface Kozlovsky
_et al._ (2020); Xypakis  _et al._ (2020); d. Juan  _et al._ (2019); Takane
and Imura (2013). This Hamiltonian is particularly useful for finding the
appropriate boundary condition at the curved interface between the normal and
reduced radius regions. We note that we only consider the orbital effect of
the applied magnetic field because at low magnetic field (e.g. $\varphi=0.5$),
the Zeeman energy is negligible compared to the energy spacing of the sub-
bands Cho  _et al._ (2015).

For the cylindrical surfaces in the N and C regions, away from the interface,
$R^{\prime}=0$. For this case, exploiting the rotational symmetry of the
cylinder and translational invariance along the wire axis we can write the
solution of the time-independent Schrödinger equation for this Hamiltonian as

$\Psi_{k,l}(z,\phi)=e^{ikz}\,e^{il\phi}\,\psi_{k,l},$ (3)

here $\psi$ is a two component spinor and $l$ is a half-integer angular
momentum due to the presence of a $\pi$ spin Berry phase acquired by the
$2\pi$ rotation of the spin around the cylindrical surface Zhang  _et al._
(2009); Bardarson  _et al._ (2010); Zhang and Vishwanath (2010). Using the
solution given in Eq.3 the Hamiltonian in N and C regions can be written as

$\begin{split}H_{1,2}&=\hbar
v_{F}\Bigg{[}-k\sigma_{y}+\left(\frac{l-\varphi_{1,2}}{R_{1,2}}\right)\sigma_{z}\Bigg{]}.\end{split}$
(4)

The energy of the surface states can be found by diagonalizing the above
Hamiltonian,

$E_{1,2}=\pm\hbar v_{F}\sqrt{k^{2}_{1,2}+(l-\varphi_{1,2})^{2}/R^{2}_{1,2}},$

where $1,2$ refers to N and C region respectively. Due to the reduced cross
section in the C region, the flux penetrating into this region is given by
$\varphi_{2}=\varphi_{1}R_{2}^{2}/R_{1}^{2}$.

Figure 2:  The allowed and forbidden scattering between orthogonal
($\psi_{k,l-\varphi}$ (pink colour) and $\psi_{-k,-(l-\varphi)}$ (green
colour)) and non-orthogonal states ($\psi_{k,l-\varphi}$ (pink colour) and
$\psi_{-k,l-\varphi}$ (brown colour)) is shown schematically.

We now explain the scattering processes in Fig. 2 where we schematically show
the allowed and forbidden scattering between the right and left moving
spinors. Note that the magnetic flux threaded along the wire axis breaks TRS
and all the surface states become non-degenerate. However, TRS can be restored
at integer and half-integer values of magnetic flux and all bands, except the
$l=\varphi$ band become degenerate Zhang and Vishwanath (2010); Bardarson and
Ilan (2018); d. Juan  _et al._ (2019); Egger  _et al._ (2010). Now, if we fix
the Fermi energy within the bulk gap such that it crosses the lowest
degenerate sub-band (which happens at integer values of magnetic flux), there
always exists two right moving spinors $\psi_{k_{1},\pm(l-\varphi)}$ and two
left moving spinors $\psi_{-k_{1},\pm(l-\varphi)}$ as demonstrated in Fig. 2.
As discussed in section I, although the backscattering is prohibited between
the time-reversal partner $\psi_{k_{1},l-\varphi}$ and
$\psi_{-k_{1},-(l-\varphi)}$, it is allowed between two other available non-
orthogonal states $\psi_{k_{1},l-\varphi}$ and $\psi_{-k_{1},l-\varphi}$. As a
result, the non-magnetic disorder can backscatter the Dirac surface states.

First, we analyse a uniform nanowire for the case of zero magnetic flux. We
consider the scattering of surface states belonging to the $l=1/2$ band in
presence of a double barrier electrostatic potential $V$ induced by gates in a
TINW with uniform radius, see Fig. 3a. Notice that due to a non-zero
potential, there is a uniform upward shift in the energy spectrum as shown in
Fig. 3b which creates a finite energy difference between the conduction bands
belonging to the section with potential V and the section without the
potential. By fixing the energy of the incoming surface state within this
energy window, we compute the transmission through the TINW for zero magnetic
flux and find sub-gap (the gap between the conduction band for V=10 meV and
V=0 in Fig. 3b) resonances in the transmission probability function of the
surface states through the TINW as shown in Fig.3c, which is a typical
signature of bound state formation within the middle region. The formation of
bound states indicate the fact that surface states in a TINW can backscatter
from a scalar disorder (non-magnetic disorder).

Figure 3: (a) The uniform TINW in presence of two electrostatic potentials
$V$. (b) If the energy of the incoming state is fixed between the conduction
band for V=10 meV and V=0 sub-gap transmission resonances are found within the
region between the potential barrier. The inset shows the dispersion of the
$l=1/2$ band in a uniform nanowire of radius 30 nm for zero flux in the
absence (orange colour) and presence (blue colour) of a potential barrier.

For a half-integer value of flux along the nanowire, there is always a
corresponding value of $l$ for which the surface state is non-degenerate and
gapless d. Juan  _et al._ (2019); Egger  _et al._ (2010). In this particular
state, an electrostatic barrier as shown in Fig. 3a, can not backscatter the
Dirac particles. Thus, in order to generate confinement we need an additional
mechanism to ensure a finite gap in the constrictions in the TINW system.
Hence, we propose to have reduced radius regions in the TINW so that the flux
penetrating these regions is not a half-integer as opposed to N region.
Consequentially, the surface states at these constricted regions become gapped
due to the breaking of TRS. To summarize, constricted region ensures a larger
band gap between the surface states as compared to N regions for both with and
without magnetic flux.
The external flux control on the electronic transport through the clean NCNCN
geometry is a unique feature of the TINW quantum dot where each C region has
radius $R_{2}<R_{1}$. The finite reflection due to breaking of TRS in C
regions resembles the backscattering in quantum spin Hall edge in the presence
of local magnetic disorder that breaks the TRS Modak  _et al._ (2012).
Furthermore, the external gate tunability of the band gap in the constricted
region can provide another probe to control the electronic transport through
the model device. In section IV, we discuss both external magnetic flux and
gate dependent transport in detail.
Now, we present the scattering matrix approach to find the scattering
coefficients for the non-interacting electrons. Let us consider the interface
at $z=0$ between an N and a C region in the presence of a finite flux along
the nanowire axis. We compute the transfer matrix that connects the wave
function from the left side of the interface ($\psi(z<0)$) to the right side
of it ($\psi(z>0)$). This can be done by solving the Schrödinger equation
$H\psi=E\psi$, where H is given by Eq.(4). By integrating the Schrödinger
equation across the interface we arrive at the following wave function
matching condition (for the curved interface as shown in Fig.1c)

$\psi(z>0)=T_{z}\,\psi(z<0),$ (5)

where the $2\times 2$ transfer matrix $T_{z}$ is given by

$\begin{split}T_{z}&=\mathcal{P}_{z}\,\text{Exp}\Bigg{[}\int_{z_{0}}^{z_{1}}\Big{\\{}\frac{-R^{\prime}}{2R}-\frac{\sqrt{1+R^{\prime
2}}}{R}\Big{(}l-\varphi\Big{)}\sigma_{x}\\\ &\hskip
99.58464pt-\frac{iE\sqrt{1+R^{\prime 2}}}{\hbar
v}\sigma_{y}\Big{\\}}dz\Bigg{]},\end{split}$ (6)

here $\mathcal{P}_{z}$ is the path ordered product of the exponential factor
along the wire.

Next, considering the elastic scattering within the device we can write the
wave functions in each region of the model device shown in Fig.1a as

$\displaystyle\psi_{1,3,5}$ $\displaystyle=$
$\displaystyle\frac{a_{1,3,5}}{\sqrt{2}}\left({\begin{array}[]{c}\frac{-\chi_{l}+E}{ik_{1}}\\\
1\end{array}}\right)e^{ik_{1}z}+\frac{b_{1,3,5}}{\sqrt{2}}\left({\begin{array}[]{c}\frac{-\chi_{l}+E}{-ik_{1}}\\\
1\end{array}}\right)e^{-ik_{1}z}$ (11) $\displaystyle\psi_{2,4}$
$\displaystyle=$ $\displaystyle
a_{2,4}\left({\begin{array}[]{c}\frac{-\chi_{2}+E}{ik_{2}}\\\
1\end{array}}\right)e^{ik_{2}z}+b_{2,4}\left({\begin{array}[]{c}\frac{-\chi_{2}+E}{-ik_{2}}\\\
1\end{array}}\right)e^{-ik_{2}z},$ (16)

here $\psi_{1,3,5}$ and $\psi_{2,4}$ are the wave functions in the normal N
and reduced radius regions C respectively and
$k_{1,2}=\sqrt{E^{2}-{\chi_{1,2}}^{2}}$,
$\chi_{1,2}=(l-\varphi_{1,2})/R_{1,2}$. $a_{1,2,3,4,5}$ and $b_{1,2,3,4,5}$
are the scattering coefficients. Here 1,3,5 refers to N regions while 2,4
indicates C regions. We use the wave function matching condition given in
Eq.(5) to find the total transfer matrix M that connects the incoming states
in the leftmost N region to the outgoing states in the right most N region and
we get

$\displaystyle
M=\alpha_{1}^{-1}\,T_{z}\,\beta_{1}\,\alpha_{2}^{-1}\,T_{z}^{\dagger}\,\beta_{2}\,\alpha_{3}^{-1}\,T_{z}\,\beta_{3}\,\alpha_{4}^{-1}\,T_{z}^{\dagger}\,\beta_{4},$
(17)

where $\alpha_{1,2,3,4}$ and $\beta_{1,2,3,4}$ are given as

$\displaystyle\alpha_{i}$ $\displaystyle=$
$\displaystyle\left({\begin{array}[]{cc}\frac{-\chi_{i}+E}{ik_{i}}e^{ik_{i}z_{i}}&\frac{-\chi_{i}+E}{-ik_{i}}e^{-ik_{i}z_{i}}\\\
e^{ik_{i}z_{i}}&e^{-ik_{i}z_{1}}\end{array}}\right)$ (20)
$\displaystyle\beta_{i}$ $\displaystyle=$
$\displaystyle\left({\begin{array}[]{cc}\frac{-\chi_{i}+E}{\sqrt{2}ik_{i}}e^{ik_{i}z_{i}}&\frac{-\chi_{i}+E}{-\sqrt{2}ik_{i}}e^{-ik_{i}z_{i}}\\\
e^{ik_{i}z_{i}}&e^{-ik_{i}z_{i}}\end{array}}\right)$ (23)
$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle\alpha_{1}\quad\quad(\text{at}\,z_{1}=z_{3})$
$\displaystyle\alpha_{4}$ $\displaystyle=$
$\displaystyle\alpha_{2}\quad\quad(\text{at}\,z_{2}=z_{4})$
$\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle\beta_{1}\quad\quad(\text{at}\,z_{1}=z_{3})$
$\displaystyle\beta_{4}$ $\displaystyle=$
$\displaystyle\beta_{2}\quad\quad(\text{at}\,z_{2}=z_{4}),$ (24)

Here $i=1,2,3,4,5$ as before. The inverse of the matrix element $M_{11}$ gives
the transmission probability which we use in the next section to discuss the
characteristics of the quantum dot device proposed in this work.

##  IV Numerical results

We now discuss the zero-temperature electronic transport through the nanowire
structure. We consider the energy of the incoming states such that they are
propagating modes in N regions (with radius $R_{1}$) and the radius of C
regions is always smaller than $R_{1}$. The propagating states with total
angular momentum $l=1/2$ coming from the left side of the device can have a
finite reflection at the first interface between N and R. This is due to non-
orthogonality of the incoming state $\psi_{k_{1},(l-\varphi)}$ and the
outgoing states $\psi_{-k_{1},(l-\varphi)}$ which are not the time reversal
partner. Since the C regions have larger gap than in N, these constricted
regions effectively behave like the potential barrier for the Dirac particles
and thus can scatter the incoming particles. The multiple scattering events in
the middle region of NCNCN give rise to resonance effects which manifest in
the total transmission function through the device. We employ the Landauer-
Büttiker approach Datta (1995) to compute the conductance through the NCNCN
device at zero temperature and plot it in Fig.4a (red colour) as a function of
the chemical potential $\mu$ for the flat interface which corresponds to the
boundary condition with $T_{z}=1$. In experiments, $\mu$ is influenced by
substrate, contacts etc. and can be locally tuned using electrostatic gates.
The conductance peaks correspond to the sub-gap resonances in the transmission
probability through the model device and exhibit clear evidence of bound
states formation inside the middle N region of the device. Notice that the N
regions are immune to scalar disorder for $\varphi=l$ due to appearance of
gapless surface states. More generally, when the dimensions of the middle
section is small compared to the mean free path, we do not expect disorder to
significantly affect the resonances.

Figure 4:  (a) Conductance through an etched TI nanowire with flat interfaces
(as sketched in Fig.(1a)) is plotted as a function of chemical potential $\mu$
for a dot length $L_{0}=100\,\text{nm}$. Red colour depicts the result for
flat interface. The position of conductance peaks not only shifts but also
broadens in case of curved interface as shown by the blue (dotted) and green
(solid) curves (here, former and later colour represent the result with and
without incorporating spin connection for the curved manifold). It is due to
the modified boundary conditions (see, Eq.(6)). More detail about the boundary
conditions in all three cases is discussed in the main text. (b) Sub-gap
resonant transmission through the long quantum dot $L_{0}=400$ nm in TI
nanowire NCNCN geometry having flat interfaces. The numerical values of the
system parameters are chosen as ${\text{R}}_{1}=80$ nm, ${\text{R}}_{2}=15$
nm, ${\text{L}}=80$ nm, $\varphi_{1}=0.5$ and
$\varphi_{2}=\varphi_{1}{\text{R}}_{2}^{2}/{\text{R}}_{1}^{2}$. Figure 5:  (a)
The lowest two bound state energies as a function of the magnetic flux
$\varphi_{1}$ for $l=1/2$ and $l=-1/2$. (b) Quantum contribution to the
Coulomb energy ($E_{c}^{q}$) as a function of the magnetic flux. The dotted
line in (b) is drawn to represent the degeneracy point. Other parameters are
the same as in Fig.4b.

The position of the conductance peaks in Fig.4a gives the energy of the bound
states. The width of a conductance peak depends on the length $L$ of the C
regions while the number of these peaks depends on the length $L_{0}$ of the
middle N region (the confined region) where we find the bound states. The
value of the model parameters used in this work are chosen as
${\text{R}}_{1}=80\,\text{nm},{\text{R}}_{2}=15\,\text{nm},\text{L}=80\,\text{nm},{\text{L}_{0}}=100\,\,\text{nm}\,\,\,\text{and}\,\varphi_{1}=0.5$
which are experimentally attainable Peng  _et al._ (2010); Jauregui  _et al._
(2016); Münning  _et al._ (2021). Surface sensitive experiments on
$\text{Bi}_{2}\text{Se}_{3}$ estimate the value of the coherence length to be
500 nm and a typical value of the mean free path to be 100 nm Münning  _et
al._ (2021); Cho  _et al._ (2015); Peng  _et al._ (2010). Therefore, the
electronic transport in the TINR studied in this work is expected to be
ballistic and phase coherent. It is important to note that device parameters
can be found for the case with one non-degenerate sub-gap resonance. In that
case, only one electron can be inside the dot. However, there will be multiple
quantum dot states for a larger quantum dot. We show the transmission through
the device for a larger dot of length $L_{0}=400$ nm and same radius
$R_{2}=15$ nm in Fig. 4b which exhibits many sub-gap resonances in the
transmission function. An useful quantity in the discussion of quantum dot is
the charging energy ${\text{E}}_{\text{c}}={\text{e}}^{2}/\text{C}_{0}$, where
$C_{0}$ is the capacitance that depends on the device dimensions and the
dielectric constant of the material. The value of $C_{0}$ can be inferred from
the experiments based on $\text{Bi}_{2}\text{Se}_{3}$ nanowires. For a
$\text{Bi}_{2}\text{Se}_{3}$ nanowire on a $\text{SiO}_{2}/\text{Si}$
substrate $C_{0}=2\times 10^{-17}$ F for a surface area $8.6\times
10^{-14}\,\text{m}^{2}$ was found Cho  _et al._ (2012), giving an effective
capacitance $C_{\text{eff}}=2.3\times 10^{-4}\,\text{F}\,\text{m}^{-2}$. We
can now estimate the charging energy of the quantum dot (${\text{R}}_{2}=15$
nm, ${\text{L}}_{0}$=400nm) using Kozlovsky  _et al._ (2020),
$\text{E}_{\text{c}}={\text{e}}^{2}/2\pi{\text{R}}_{1}{\text{L}}_{0}{\text{C}}_{\text{eff}}$
and this gives 3.6 meV. I.e. Coulomb blockade like oscillations of the
conductance are expected in transport experiments on a large dot by varying
the gate voltage applied to the central NCNCN region. However, as we pointed
out earlier, the origin of the discrete quantum dot states here uniquely
arises from interference of the surface states, different from the appearance
of quantum dot based on Coulomb blockade. Beenakker (1991); McEuen  _et al._
(1991)

Next, we examine the effect of a curved interface. We utilize the radial
function given in Eq.(1) of the section II to incorporate the curvature
effects on the electronic transport. Here, we use the boundary condition,
$\psi(z>0)={\text{T}}_{z}\,\psi(z<0)$ where ${\text{T}}_{z}$ is the transfer
matrix for $R^{\prime}=0$. It is clear from Fig.4a (blue curve) that not only
the position of the conductance peak gets shifted but also the peak broadens
compared to the situation when we consider the flat interface. Furthermore, we
analyse the motion of the Dirac particle on the curved interface incorporating
the spin connection in the transfer matrix ($R^{\prime}\neq 0$). We plot the
conductance again in Fig. 4a (green curve) which coincides with the plot
computed for $R^{\prime}=0$ (blue colour). The findings suggest that the
electronic transport is unaffected by the inclusion of the spin connection.

Figure 6:  Conductance as a function of chemical potential $\mu$ and the flux
along the wire axis $\varphi_{1}$. Device parameters are fixed at
${\text{R}}_{1}=80\,\text{nm},{\text{R}}_{2}=20\,\text{nm},\text{L}=80\,\text{nm}\,\,\text{and}\,\,{\text{L}}_{0}=100\,\text{nm}$.
The checker board region is non-physical because of the evanescent modes in N
region. Figure 7:  External gate tunability of the conductance is shown as a
function of the electric potential induced on the dot by the electrostatic
gates applied in the C regions and the chemical potential $\mu$. The values of
other parameters are taken as in Fig.6.

Apart from the classical charging energy of the formed quantum dot, which
arises from the long range Coulomb potential, we also estimate the quantum
contribution $\text{E}_{\text{c}}^{\text{q}}$ due to the screened short-range
Coulomb interaction Aleiner  _et al._ (2002) which depends on the quantum dot
states. We study its dependence on the magnetic flux. In Fig.5a, we show how
the energy of the bound states varies as a function of magnetic flux
$\varphi_{1}$. We represent the lowest two bound states corresponding to the
$l=1/2$ and $l=-1/2$ bands as $E_{l=1/2}^{1},\,E_{l=1/2}^{2}$ and
$E_{l=-1/2}^{1},\,E_{l=-1/2}^{2}$ respectively in Fig.5a. We model the short-
range interaction as a contact interaction and compute the Coulomb energy
between electrons which occupy the two lowest energy states. For
$\varphi_{1}<0.3$, the two occupied states correspond to $E_{l=1/2}^{1}$ and
$E_{l=-1/2}^{1}$ while $E_{l=1/2}^{1}$ and $E_{l=1/2}^{2}$ become the lowest
energy states for $\varphi_{1}>0.3$. We plot the Coulomb energy
$\text{E}_{\text{c}}^{\text{q}}$ in Fig.5b as a function of flux
$\varphi_{1}$. We observe a smooth decay in $\text{E}_{\text{c}}^{\text{q}}$
as the magnetic flux increases up until the point of degeneracy at
$\varphi_{1}=0.3$. As flux increases beyond 0.3 we notice slow decrease in
$\text{E}_{\text{c}}^{\text{q}}$. We note that we do not estimate
$\text{E}_{\text{c}}^{\text{q}}$ at one point $\varphi_{1}=0.3$ as the second
lowest quantum dot state is degenerate, see Fig.5a.

The bandgap tunability of the surface state by an external flux enables us to
investigate the electronic transport as a function of the flux threaded along
the wire axis. We plot the conductance through the model device in Fig. 6 for
the flat interfaces between the N and C regions as a function of chemical
potential $\mu$ and the flux $\varphi_{1}$ applied along the wire axis in the
N region. Evidently, if the chemical potential is fixed (consider the dotted
white line in the plot) the transport of the surface states can be externally
controlled by the application of a flux along the wire. The checker board
region in Fig. 6 corresponds to the presence of evanescent modes in the N
region and hence is non-physical. Next, we examine the gate voltage tunability
of the electronic transport in presence of half-integer flux. The two gate
voltages in C regions change the band gap profile by shifting the band
spectrum (with respect to the band spectrum in N regions) upward by the
electrostatic potential V applied by the gate. We plot the conductance as a
function of induced gate potential and the chemical potential in Fig. 7 for
flux $\varphi_{1}=0.5$ and we find that the conductance can be controlled by
varying the gate voltage.

So far, we study the quantum dot considering the flat interfaces between
different sections of the device. However, while etching/pattering nanowire
the interfaces are likely to undergo geometrical deformation. As discussed
earlier in section II, to address the transport in this situation, we model
the interface as shown in Fig.1d and assume a smoothness $a=8$ nm, a
relatively smooth interface that is obtainable using modern nanofabrication
techniques, yet significantly smaller than other dimensions of the NCNCN
structure. We investigate the influence of such interfaces on the electronic
transport as a function of external flux, gate voltages and chemical
potential. First, we show the conductance as we vary flux and chemical
potential in Fig.8. We observe that the resonance line broadens and gets
shifted on the energy axis. The broadening of the conductance peak relates to
the inverse of the life time of the quantum dot state which means that the
particle would quickly escape the dot. Similar features are found when we plot
conductance as a function of chemical potential and the gate voltage in Fig.9.
These results can be attributed to the strong coupling of the dot to the left
and right N regions which determines the shape and position of the conductance
peaks. Thus, from an experimental point of view it is important to engineer as
sharp interfaces as possible.

Figure 8:  The conductance through the NCNCN quantum dot model considering the
smooth interface (taking a=8 nm, as shown in Fig.1d) as a function of chemical
potential and magnetic flux $\varphi_{1}$. The values of all parameters are
the same as in Fig.6. Figure 9:  The conductance through the NCNCN quantum dot
model considering the smooth interface (the same as for Fig.8) as a function
of chemical potential and external gate voltage. The values of all parameters
are the same as in Fig.6.

The external control of quantum transport is a necessary prerequisite in the
fabrication of field-effect transistors and other quantum devices. The
external magnetic flux as well as induced gate voltage tunability makes our
geometrical construction of the TI quantum dot experimentally feasible for
experimental implementation. Moreover, we anticipate that the transmitted
current through the quantum dot states in this device might have different
spinor state for different conditions fixed by chemical potential, magnetic
flux and voltage bias. We leave the thorough study of characterising the
finite temperature current in presence of Coulomb interaction and disorder for
degenerate and non-degenerate dot for future work.

##  V Conclusions

In this work, we exploited the finite reflection processes for designing a
model for a quantum dot in a TINW. We showed that finite reflection is
possible essentially due to the transverse angular momentum modes. We
demonstrated that in the presence of double barrier potential, the uniform
nanowire can host bound states if the Fermi energy lies within the conduction
band minimum for $V\neq 0$ and $V=0$ for a fixed value of angular momentum
$l$. We elucidated the necessity of the constricted region in presence of
half-integer magnetic flux along the wire axis in N region (half-integer
magnetic flux is important to prevent the backscattering due to accident
disorder in N region). In particular, we propose a theoretical model of a
quantum dot in a TINW exploiting the unique characteristics of the surface
states and show that the confining potential due to the larger gap in the
surface states in constricted regions localise the Dirac surface state.
Finally, we explicitly show that the electronic transport through discrete
quantum dot states can be modulated by external magnetic flux and gate
voltage. This will facilitate some of the most desirable quantum devices for
metrology (single-electron devices, charge pumps), spintronics (spin-polarized
current source), quantum optics, and quantum computing.

###### Acknowledgements.

We acknowledge support from the European Union’s Horizon 2020 Research and
Innovation Programme under Grant Agreement No. 766714/HiTIMe. Appendix:
Hamiltonian on the curved surface
The surface of the TINW can be described using the space time metric in 2+1D
($z,\phi,t$)

$g_{\mu,\nu}=(-1,1+{{R}^{\prime}}^{2},R^{2}),$ (25)

where $R^{\prime}$ is the change in radius along the nanowire. This metric is
related to the Minkowski metric for the flat surface by the following relation

$\begin{split}g_{a,b}=g_{\mu,\nu}\,{e^{\mu}}_{a}\,{e^{\nu}}_{b},\end{split}$
(26)

Here Latin letters ($a,b$) denote the local coordinates and Greek letters
($\mu,\nu$) denote general coordinates. The Dirac equation in curved space is

$\begin{split}i\gamma^{\mu}\,D_{\mu}\phi&=0,\end{split}$ (27)

where the covariant derivative,
$D_{\mu}=\partial_{\mu}+\omega_{ab\mu}\,\Sigma^{ab}/4$. It can be shown that
only $\omega_{12\phi}$ and $\omega_{21\phi}$ are non zero d. Juan  _et al._
(2019) and given by

$\omega_{12\phi}=\omega_{21\phi}=\frac{{R}^{\prime}}{\sqrt{1+{{R}^{\prime}}^{2}}},$
(28)

so the covariant derivatives are given by

$\begin{split}D_{t}&=\partial_{t},\\\ D_{z}&=\partial_{z},\\\
D_{\phi}&=\partial_{\phi}+\omega_{12\phi}\,\Sigma^{21}/2.\end{split}$ (29)

Using these expressions we can write the Dirac Eq.(27) as

$\begin{split}i\gamma^{t}\partial_{t}\psi+i\gamma^{z}\partial_{z}\psi+i\gamma^{\phi}\left[\partial_{\phi}+\omega_{12\phi}\Sigma^{21}/2\right]\psi&=0,\end{split}$
(30)

where

$\begin{split}\Sigma^{21}&=\frac{1}{2}\left[\Gamma^{1},\Gamma^{2}\right]\quad\text{and}\quad\gamma^{\mu}\equiv{e^{\mu}}_{a}\,\Gamma^{a}.\end{split}$
(31)

To calculate vierbein or tetrad, $(e^{\mu})_{a}$ we use Eq.(26) and find three
set of equations

$\begin{split}g_{00}&=g_{\mu\nu}\,{e^{\mu}}_{0}\,{e^{\nu}}_{0},\\\
g_{11}&=g_{\mu\nu}\,{e^{\mu}}_{1}\,{e^{\nu}}_{1},\\\
g_{22}&=g_{\mu\nu}\,{e^{\mu}}_{2}\,{e^{\nu}}_{2},\end{split}$ (32)

which can be expended in terms of the components of $g_{\mu\nu}$ as

$\begin{split}-1&=-({e^{t}}_{0})^{2}+(1+{{R}^{\prime}}^{2})({e^{z}}_{0})^{2}+R^{2}({e^{\phi}}_{0})^{2},\\\
-1&=-({e^{t}}_{1})^{2}+(1+{{R}^{\prime}}^{2})({e^{z}}_{1})^{2}+R^{2}({e^{\phi}}_{1})^{2},\,\\\
-1&=-({e^{t}}_{2})^{2}+(1+{{R}^{\prime}}^{2})({e^{z}}_{2})^{2}+R^{2}({e^{\phi}}_{2})^{2},\end{split}$
(33)

now taking only the following tetrad non-zero

$\begin{split}{e^{t}}_{0}&=1,\\\
{e^{z}}_{1}&=\frac{1}{\sqrt{1+{{R}^{\prime}}^{2}}},\\\
{e^{\phi}}_{2}&=\frac{1}{R},\\\ \end{split}$ (34)

we find $\gamma^{\mu}$ using Eq.(31)

$\begin{split}\gamma^{t}&\equiv{e^{t}}_{a}\Gamma^{a}={e^{t}}_{0}\Gamma^{0}+{e^{t}}_{1}\Gamma^{1}+{e^{t}}_{2}\Gamma^{2}=\Gamma^{0},\\\
\gamma^{z}&\equiv{e^{z}}_{a}\Gamma^{a}={e^{z}}_{0}\Gamma^{0}+{e^{z}}_{1}\Gamma^{1}+{e^{z}}_{2}\Gamma^{2}=\frac{1}{\sqrt{1+{{R}^{\prime}}^{2}}}\Gamma^{1},\\\
\gamma^{\phi}&\equiv{e^{\phi}}_{a}\Gamma^{a}={e^{\phi}}_{0}\Gamma^{0}+{e^{\phi}}_{1}\Gamma^{1}+{e^{\phi}}_{2}\Gamma^{2}=\frac{1}{R}\Gamma^{2}.\\\
\end{split}$ (35)

We can put these results in Eq.(30) and get

$\begin{split}i\Gamma^{0}\partial_{t}\psi+i\frac{1}{\sqrt{1+{{R}^{\prime}}^{2}}}\Gamma^{1}\partial_{z}\psi+i\frac{1}{R}\Gamma^{2}\Bigg{[}\partial_{\phi}\\\
+\frac{R^{\prime}}{\sqrt{1+{{R}^{\prime}}^{2}}}\Sigma^{21}/2\Bigg{]}\psi=0.\end{split}$
(36)

Now, we choose the following Dirac matrices

$\begin{split}\Gamma^{0}&=i\sigma_{x},\\\ \Gamma^{1}&=i\sigma_{y},\\\
\Gamma^{2}&=i\sigma_{z},\end{split}$ (37)

and write Eq. (36) as

$\begin{split}-\sigma_{x}\partial_{t}\psi+i\frac{1}{\sqrt{1+{R^{\prime}}^{2}}}\sigma_{z}\partial_{z}\psi+\sigma_{y}\frac{i}{R}\Bigg{[}\partial_{\phi}\\\
+\frac{R^{\prime}}{\sqrt{1+{R^{\prime}}^{2}}}\Sigma^{21}/2\Bigg{]}\psi=0,\end{split}$
(38)

where

$\Sigma^{21}=\frac{1}{2}\left[\Gamma^{2},\Gamma^{1}\right]=i\sigma_{x}.$

Inserting $\hbar$ and $v_{F}$ we arrive at the following Hamiltonian

$\begin{split}-i\hbar\sigma_{x}\partial_{t}\psi=v_{F}\left[\frac{1}{\sqrt{1+{{R}^{\prime}}^{2}}}\sigma_{z}\left\\{ip_{z}+\frac{\hbar
R^{\prime}}{2R}\right\\}+\frac{\hbar}{R}\sigma_{y}\partial_{\phi}\right]\psi.\end{split}$
(39)

We can simplify by multiplying both sides by $-\sigma_{x}$ and get the
following Hamiltonian

$\begin{split}H&=v_{F}\left[\frac{1}{\sqrt{1+{{R}^{\prime}}^{2}}}\left\\{i\hbar\partial_{z}+\frac{i\hbar{R}^{\prime}}{2R}\right\\}\sigma_{y}-\frac{i\hbar}{R}\sigma_{z}\partial_{\phi}\right].\end{split}$
(40)

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