# Influence of structural deformations on the reentrant conductance feature in
semiconducting nanowires

Iann Cunha  Leonardo Villegas-Lelovsky  Leonardo Kleber Castelano
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de São
Carlos, 13565-905, São Carlos, SP, Brazil  Departamento de Física, IGCE,
Universidade Estadual Paulista, 13506-900 Rio Claro SP, Brazil

###### Abstract

Helical states can be measured through the observation of the reentrant
behaviour, which is a dip in the conductance probed in semiconducting
nanowires (NWs) with strong spin-orbit coupling (SOC) under the presence of an
external perpendicular magnetic field. We investigate the effects of
deformations in the electronic transport in NWs considering the coupling
between different transverse modes. Within this approach, we show that the dip
in the conductance of a NW is affected by the presence of a local
constriction. Moreover, we find that the reentrant feature in the conductance
can appear in NWs with a local expansion of its radius, even in the absence of
SOC and magnetic field. Furthermore, we develop a numerical approach to
calculate transport properties, which is able to include the deformation and
the coupling among several scattering channels.

##  I Introduction

Majorana zero modes can be employed to perform topologically protected quantum
computation Nayak et al. (2008). The observation of Majorana states was
predicted to occur in a system composed of a semiconducting nanowire (NW) with
strong spin-orbit coupling (SOC) in proximity to an s-wave superconductor and
under the presence of an external magnetic field Alicea et al. (2011); Oreg et
al. (2010); Lutchyn et al. (2010). The experimental realization and
characterization of semiconductor NWs based on InAs and InSb have been
explored to check the existence of helical states Heedt et al. (2017);
Kammhuber et al. (2017); van Weperen et al. (2013); Sun et al. (2018), which
are related to Majorana zero modes. The verification of helical states can be
probed by the so-called reentrant feature, which appears as a measurable dip
in the conductance. The reentrant feature was predicted to be observed when
the electron energy reaches the pseudo gap occurring in NWs with strong Rashba
SOC and subjected to an external magnetic field Streda and Seba (2003). In our
approach Cunha et al. (2020), we reinterpret the dip in the conductance as a
resonant reflection, which occurs if the NW contains a localized attractive
potential and the coupling between different scattering channels is different
from zero GURVITZ and LEVINSON (1993). Furthermore, Sanchez et al Sanchez and
Serra (2006) showed that the Rashba Hamiltonian limited to a finite region is
equivalent to an attractive potential and promotes the coupling of different
transport channels, when a gauge transformation is taken into account.
Therefore, the dip in the conductance can exist even in the absence of the
magnetic field Heedt et al. (2017); Cunha et al. (2020). In this paper, we
show that these fundamental ingredients can also be found in NWs having only a
local structural deformation. The study of conductance considering structural
deformations has a long history and several interesting phenomena have been
found over the past decadesTorres et al. (1994); Scherbakov et al. (1996); Kim
et al. (1999); Nikolić and Allen (1999); Avotina et al. (2005); Dacal et al.
(2005); Racec et al. (2009); Wołoszyn et al. (2014); Shchamkhalova and
Sablikov (2005). Perhaps, the most important one, was the discovery of quantum
point contactsvan Wees et al. (1988); Wharam et al. (1988) which are narrow
constrictions in a two dimensional electron gas that exhibit quantized
conductance and can be used as sensitive charge detectors. As already
mentioned, here we show that the appearance of the dip in the conductance can
be observed even when the Rashba SOC is lacking, but the NW must have a
limited region of space where its radius suffers an expansion. On the other
hand, a local radius compression does not trigger the dips in the conductance
because the fundamental ingredient of having an attractive potential is
lacking in this case. We also probe the NW with structural deformations and
the Rashba SOC occurring in the same region of space. Moreover, we consider a
perpendicular magnetic field in the whole NW together with Rashba SOC and
structural deformations. Results, in this case, show that structural
deformations can affect the observation of the reentrant feature and even
mitigate it. Lastly, we provide a scheme to numerically solve the transport of
charges considering the coupling among several scattering channels and the
spatial dependence of the localized potential, which made the realization of
the study of structural deformations in NW possible. This method is very
useful and can be employed to investigate other systems, e.g. to study the
transport of holes with different effective masses coupled between each other
in a heterostructure using the Luttinger Hamiltonian Winkler (2003).

##  II Theoretical Model

Our model to study the electronic transport in an NW with a structural
deformation is described by the following Hamiltonian.

$\left[\frac{-\hbar^{2}}{2m^{*}}\nabla^{2}+V_{c}(x,y)+V(x,y,z)\right]\Psi(x,y,z)=E\Psi(x,y,z),$
(1)

where $m^{*}$ is the effective mass, $V_{c}(x,y)$ is the lateral confining
potential, and $V(x,y,z)$ is the potential describing the structural
deformation. To solve (1), we use the transverse modes $\Phi_{n,m}(x,y)$,
which are solutions of the following equation

$\left[\frac{-\hbar^{2}\nabla^{2}_{\perp}}{2m^{*}}+V_{c}(x,y)-E_{n,m}\right]\Phi_{n,m}(x,y)=0,$
(2)

where $\nabla^{2}_{\perp}=\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)$. The wave-function can be
expanded in the complete basis formed by the transverse modes (channels) such
as $\Psi(x,y,z)=\sum_{n,m}\chi_{n,m}(z)\Phi_{n,m}(x,y)$, where $\chi_{n,m}(z)$
are the scattering wave-functions, which are solutions of the coupled-channels
equations

$\left[\frac{\partial^{2}}{\partial
z^{2}}+k^{2}_{n,m}\right]\chi_{n,m}(z)-\frac{2m^{*}}{\hbar^{2}}\sum_{n^{\prime},m^{\prime}}V^{n,m}_{n^{\prime},m^{\prime}}(z)\chi_{n^{\prime},m^{\prime}}(z)=0,$
(3)

where $k^{2}_{n,m}=\frac{2m^{*}}{\hbar^{2}}(E-E_{n,m})$ and

$V^{n,m}_{n^{\prime},m^{\prime}}(z)=\int dx\int
dy\;\Phi^{*}_{n,m}(x,y)V(x,y,z)\Phi_{n^{\prime},m^{\prime}}(x,y).$ (4)

To be able to solve (3), we assume that $V(x,y,z)$ is different from zero only
in the region $|z|\leq L/2$. Therefore, there are three distinct regions where
we must calculate the wave-functions. Usually, wave-functions are written in a
vectorial form, but here we must use the matrix form to be able to implement
the numerical scheme. This matrix form also has the advantage of considering
the injection of electrons in different channels and solving these different
channels at once. In the region I, the matrix elements of the wave-function
$\chi_{\text{I}}$ are given by

$\displaystyle\chi^{\text{I}}_{i,j}=\begin{cases}e^{izk_{i}}+r_{i,i}e^{-izk_{i}},&\text{if
}i=j\\\ r_{i,j}e^{-izk_{i}},&\text{if }i\neq j.\end{cases}$ (5)

In regions II and III, the matrix form for the wave-function respectively are

$\displaystyle\chi^{\text{II}}=\left(\begin{array}[]{ccccc}\phi_{1,1}(z)&\phi_{1,2}(z)&\ldots&\phi_{1,N}(z)\\\
\phi_{2,1}(z)&\phi_{2,2}(z)&\ldots&\phi_{2,N}(z)\\\
\phi_{3,1}(z)&\phi_{3,2}(z)&\ldots&\phi_{3,N}(z)\\\
\vdots&\vdots&\ddots&\vdots\\\
\phi_{N,1}(z)&\phi_{N,2}(z)&\ldots&\phi_{N,N}(z)\\\ \end{array}\right),$ (11)

$\displaystyle\chi^{\text{III}}=\left(\begin{array}[]{ccccc}t_{1,1}e^{izk_{1}}&t_{1,2}e^{izk_{1}}&\ldots&t_{1,N}e^{izk_{1}}\\\
t_{2,1}e^{izk_{2}}&t_{2,2}e^{izk_{2}}&\ldots&t_{2,N}e^{izk_{2}}\\\
t_{3,1}e^{izk_{3}}&t_{3,2}e^{izk_{3}}&\ldots&t_{3,N}e^{izk_{3}}\\\
\vdots&\vdots&\ddots&\vdots\\\
t_{N,1}e^{izk_{N}}&t_{N,2}e^{izk_{N}}&\ldots&t_{N,N}e^{izk_{N}}\end{array}\right).$
(17)

The wave-function $\chi^{R}$ is the solution of (3) in the region $R$, where
$R=$I corresponds to $z\leq-L$/2, $R=$II corresponds to $|z|\leq L/2$, and
$R=$III corresponds to $z\geq L/2$. We also assume that electrons are injected
from the left ($z=-\infty$) in the channel $i$. The channel labeled by $i$
corresponds to the quantum numbers $(n,m,s)$ in (3) in ascending order of the
energy of transverse modes $E_{n,m}$, where $s=\pm$ designates the spin degree
of freedom. Particularly, the Hamiltonian (1) does not depend on spin and
transport equations can be solved to each spin separately. Although there are
an infinite number of transverse modes, we truncate the problem to a certain
size $N$. The coefficients $r_{i,j}$ and $t_{i,j}$ represent the reflection
and transmission coefficients of an electron injected in $j$-th channel and
scattered in $i$-th channel. The $\phi_{i,j}(z)$ are the wave-functions in the
scattering region and depend on the potential form, which can be rewritten in
the matrix form as

$\displaystyle\mathbf{V}=\left(\begin{array}[]{ccccc}V_{1,1}(z)&V_{1,2}(z)&\ldots&V_{1,N}(z)\\\
V_{2,1}(z)&V_{2,2}(z)&\ldots&V_{2,N}(z)\\\
V_{3,1}(z)&V_{3,2}(z)&\ldots&V_{3,N}(z)\\\ \vdots&\vdots&\ddots&\vdots\\\
V_{N,1}(z)&V_{N,2}(z)&\ldots&V_{N,N}(z)\\\ \end{array}\right).$ (23)

To numerically solve the transport problem, we must be able to find $r_{i,j}$,
$t_{i,j}$, and $\phi_{i,j}(z)$. By means of the numerical integration of the
coupled second-order ordinary differential equations, one needs to know the
wave-functions and its derivatives in a certain point of space to be able to
propagate the solution in the whole space. The wave-functions described in
equations(5-17) cannot be univocally determined in a certain point of space
because they depend on the $r_{i,j}$, $t_{i,j}$, and $\phi_{i,j}(z)$. Thus, we
need to rewrite (17) in the following way

$\displaystyle\chi^{\text{III}}=\left(\begin{array}[]{cccc}e^{izk_{1}}&0&\ldots&0\\\
0&e^{izk_{2}}&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&e^{izk_{N}}\end{array}\right)\mathbf{t},$ (28)

where

$\displaystyle\mathbf{t}=\left(\begin{array}[]{ccccc}t_{1,1}&t_{1,2}&\ldots&t_{1,N}\\\
t_{2,1}&t_{2,2}&\ldots&t_{2,N}\\\ \vdots&\vdots&\ddots&\vdots\\\
t_{N,1}&t_{N,2}&\ldots&t_{N,N}\end{array}\right),$ (33)

and redefine the wave-functions such as: $\xi^{k}=\chi^{k}\mathbf{t}^{-1}$,
where $k=$(I,II, or III) and $\mathbf{t}^{-1}$ is the inverse of the
transmission matrix given in (33). With this redefinition, the wave-function
in region III does not depend on the transmission coefficients and it is
univocally determined; thereby, enabling us to match the wave-function and its
derivative at $z=L/2$, which give the following matrix elements

$\xi^{\text{II}}_{i,j}(L/2)=\begin{cases}e^{ik_{i}L/2},&\text{if }i=j\\\
0,&\text{if }i\neq j,\end{cases}$
$D[\xi^{\text{II}}_{i,j}(z),L/2]=\begin{cases}ik_{i}e^{ik_{i}L/2},&\text{if
}i=j\\\ 0,&\text{if }i\neq j.\end{cases}$

Hereafter, to represent the derivative of a function $f(z)$ evaluated at the
point $a$, we use the following definition
$D[f(z),a]=\left.\frac{df(z)}{dz}\right|_{a}$ .

Through the matching conditions above, we are able to numerically propagate
the solution from $z=L/2$ to $z=-L/2$. Such a propagation can be performed,
for example, by means of the 4th order Runge-Kutta method applied to (3). By
using the numerically evaluated $\xi^{\text{II}}(-L/2)$ and
$D[\xi^{\text{II}}(z),-L/2]$, we can find
$\chi_{\text{II}}(-L/2)=\xi_{\text{II}}(-L/2)\mathbf{t}$ and
$D[\chi^{\text{II}}(z),-L/2]=D[\xi^{\text{II}}(z),-L/2]\mathbf{t}$. At
$z=-L/2$, we also have to impose the boundary conditions
$\chi_{\text{II}}(-L/2)=\chi_{\text{I}}(-L/2)$ and
$D[\chi^{\text{II}},-L/2]=D[\chi^{\text{I}},-L/2]$. These two last equations
form a system of linear equations with $N\times N$ equations and variables,
which can be solved numerically. If the wave-functions were written in a
vectorial form from the beginning, the matching conditions at $z=-L/2$ would
lead to a system of linear equations with different number of equations and
variables, which is inconsistent. The reflection $r_{i,j}$ and transmission
$t_{i,j}$ coefficients are extracted from the solutions to the system of
linear equations. The conductance can be evaluated through the Landauer-
Buttiker formula $G=G_{0}\mathrm{Tr}[tt^{\dagger}]$, where $G_{0}=e^{2}/\hbar$
and $t$ is the transmission matrix, whose matrix elements are $t_{i,j}$.

To demonstrate that the numerical scheme works, we have to check if the
current density probability is conserved, which can be verified through

$\sum_{i,j=(prop.modes)}\frac{v_{i}}{v_{j}}|r_{i,j}|^{2}+\frac{v_{i}}{v_{j}}|t_{i,j}|^{2}=1,$
(34)

where $v_{i}=\hbar k_{i}/m^{*}$ and the sum above should be evaluated over the
propagating modes, i.e., only if $k_{i}$ is real. The evanescent modes,
characterized by the imaginary $k_{i}$, have an important role in the
transport properties but do not contribute to the conservation of the current
density probability.

Figure 1: Schematic picture showing the longitudinal cut of the NW containing
an expansion of the radius in the region II with length $L$. Panel (a) shows
the cone-shape deformation $\Delta r_{1}(z)$, while panel (b) shows the
square-shape deformation $\Delta r_{2}(z)$.

By employing this numerical scheme, we can investigate the NW containing a
variable radius within the region II. We model the z-dependence of the radius
by the function $r(z)=r_{0}+\Delta r(z)$, where $r_{0}$ is the NW radius and
$\Delta r(z)$ is the variation of the radius within region II. We choose two
different forms for the radius variation, $\Delta r_{1}(z)=\Delta
r_{0}(1-2|z|/L)$ for $|z|\leq L/2$ and

$\displaystyle\Delta r_{2}(z)=\Delta
r_{0}\begin{cases}\sin{\left(\frac{\pi(z+L/2)}{2L_{0}}\right)},\;-\frac{L}{2}\leq
z<-\frac{L}{2}+L_{0}\\\ 1,\qquad\qquad\quad-\frac{L}{2}+L_{0}\leq
z\leq\frac{L}{2}-L_{0}\\\
\sin\left(\frac{\pi(L/2-z)}{2L_{0}}\right),\;\frac{L}{2}-L_{0}<z\leq\frac{L}{2}\end{cases}$

where $\Delta r_{0}$ is the maximum variation of the radius at $z=0$ and
$L_{0}$ defines the the width of the ramp between zero and $\Delta r_{0}$. The
first form $\Delta r_{1}(z)$ describes a NW with a cone-shape deformation
(panel (a) of figure 1), while the second form $\Delta r_{2}(z)$ produces a
square-shape deformation (panel (b) of figure 1). We choose these two forms to
probe the effects of different shapes of deformations in the electronic
transport properties. The main difference between both shapes is that the
square-shape deformation has a bigger area than the cone-shape deformation.

To find analytical solutions for the transverse modes, we employ the harmonic
oscillator potential to describe the lateral confining potential
$V_{c}(x,y)=m^{*}\omega_{0}^{2}(x^{2}+y^{2})/2$, where
$\omega_{0}=\hbar/(m^{*}r_{0}^{2})$. These transverse modes are solutions of
the following equation:

$\left[\frac{-\hbar^{2}\nabla^{2}_{\perp}}{2m^{*}}+\frac{m^{*}\omega_{0}^{2}}{2}(x^{2}+y^{2})-E_{n,m}\right]\Phi_{n,m}(x,y)=0.$
(35)

By using creation and annihilation operators:

$a_{q}=\sqrt{\frac{m^{*}\omega_{0}}{2\hbar}}q+\frac{i}{\sqrt{2m^{*}\hbar\omega_{0}}}p_{q},$
(36)
$a^{\dagger}_{q}=\sqrt{\frac{m^{*}\omega_{0}}{2\hbar}}q-\frac{i}{\sqrt{2m^{*}\hbar\omega_{0}}}p_{q},$
(37)

where $q=x,y$ and $p_{q}=-i\hbar\frac{\partial}{\partial q}$, we can rewrite
(35) as

$\hbar\omega_{0}\left[a_{x}^{\dagger}a_{x}+a_{y}^{\dagger}a_{y}+1\right]|\Phi_{n,m}\rangle=E_{n,m}|\Phi_{n,m}\rangle,$
(38)

whose eigenvalues are $E_{n,m}=\hbar\omega_{0}(n+m+1)$, where
$n=m=0,1,2,\ldots$, where $n$ ($m$) are associated to the number operator
$N_{x}=a_{x}^{\dagger}a_{x}$ ($N_{y}=a_{y}^{\dagger}a_{y}$). In region II, we
have a change in the radius that causes a change in the confinement, which can
be described by $\omega(z)=\frac{\hbar}{m^{*}r(z)^{2}}$. Therefore, the
scattering potential can be modelled as
$V(x,y,z)=m^{*}[\omega(z)^{2}-\omega_{0}^{2}](x^{2}+y^{2})/2$ , whose matrix
elements (23) are given by

$\displaystyle
V_{n^{\prime},m^{\prime}}^{n,m}(z)=v(z)\langle\Phi_{n,m}|(a_{x}+a_{x}^{\dagger})^{2}+(a_{y}+a_{y}^{\dagger})^{2}|\Phi_{n^{\prime},m^{\prime}}\rangle$
$\displaystyle=v(z)\left[d^{n}_{m}\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}+c^{n}_{n^{\prime}}\delta_{m,m^{\prime}}+c^{m}_{m^{\prime}}\delta_{n,n^{\prime}}\right],$
(39)

where $v(z)=\frac{\hbar[\omega(z)^{2}-\omega_{0}^{2}]}{4\omega_{0}}$,
$d^{n}_{m}=(2n+2m+1)$, and
$c^{n}_{n^{\prime}}=\left(\sqrt{n^{\prime}(n^{\prime}-1)}\delta_{n,n^{\prime}-2}+\sqrt{(n^{\prime}+1)(n^{\prime}+2)}\delta_{n,n^{\prime}+2}\right)$.
In (39), there are the diagonal terms proportional to
$V^{d}_{n,m}(z)=d^{n}_{m}v(z)$ and the off-diagonal terms that couples
channels that obey the selection rules $n=n^{\prime}\pm 2$ with $m=m^{\prime}$
or $m=m^{\prime}\pm 2$ with $n=n^{\prime}$. The diagonal terms can be
rewritten as a function of the radius variation,
$V^{d}_{n,m}(z)=d^{n}_{m}\hbar\omega_{0}\frac{[r_{0}^{4}-r(z)^{4}]}{4r(z)^{4}}$;
thereby, showing that it is an attractive potential if the radius within the
region II increases ($r(z)\geq r_{0}$). For a narrowed NW in the region II, we
have that $r(z)\leq r_{0}$ and the diagonal term becomes a repulsive
potential.

##  III Numerical Results

###  III.1 Deformation effects

We use the following channels in all numerical calculations using a variation
in the quantum number $m$: $\\{(0,0,\pm),(0,1,\pm),(0,2,\pm),(0,3,\pm)\\}$,
which is a good approximation for Fermi energy $E\leq 4\varepsilon_{0}$, where
$\varepsilon_{0}=\hbar\omega_{0}$. The channels with variation in the quantum
number $n$, such as $\\{(0,0,\pm),(1,0,\pm),(2,0,\pm),(3,0,\pm)\\}$, are
decoupled from the channels where the variation occurs only in $m$; therefore,
the conductance considering the change of both quantum numbers $(n,m)$ is
simply obtained by multiplying the results from the variation in the quantum
number $m$ by two. We start our analysis by exploring the effect of a narrowed
NW ($\Delta r_{0}<0$). In figure (2), we plot the normalized conductance
$G/G_{0}$ as a function of the normalized Fermi energy $E/\varepsilon_{0}$ for
different values of the $\Delta r_{0}$ and for a fixed size of the region II
($L=8r_{0}$), considering both shapes of deformation. By increasing the value
of $|\Delta r_{0}|$, one can notice that (i) the Fermi energy where the
conductance becomes different from zero shifts towards higher energies as the
compression is increased and (ii) the conductance does not exhibit dips and it
is smoothed for the cone-shape deformation. Both results are expected because
the appearance of the repulsive potential in the diagonal terms of (39) for a
narrowed NW, which becomes stronger for higher values of $|\Delta r_{0}|$. The
conductance for square-shape deformation (dashed lines in figure (2))
resembles the results for a square barrier potential, as would be expected.

Figure 2: Normalized conductance as a function of the normalized Fermi energy
for different radius compression $\Delta r_{0}/r_{0}=-0.1,-0.2,-0.3,-0.4,$ and
-0.5, considering the length $L/r_{0}=8.0$. Solid (dashed) lines describe the
results for the square (cone)-shape deformation model. The curves are offset
for clarity according to the indicated values. Figure 3: Normalized
conductance as a function of the normalized Fermi energy for different radius
expansion $\Delta r_{0}/r_{0}=0.1,0.2,0.3,0.4,$ and 0.5, considering the
length $L/r_{0}=8.0$. Solid (dashed) lines describe the results for the square
(cone)-shape deformation model. The curves are offset for clarity according to
the indicated values.

On the other hand, dips can be observed when an expansion occurs in the radius
of the NW in region II. In figure (3), the normalized conductance is plotted
as a function of the normalized Fermi energy considering $\Delta r_{0}>0$ for
$L=8r_{0}$. The conductance is almost constant within the first plateau
($\varepsilon_{0}<E<2\varepsilon_{0}$) and the dip is only observed for
$\Delta r_{0}\geq 0.3r_{0}$ for the cone-shape deformation (dashed curves in
figure (3)). However, many dips are present for an expanded NW for
$E>2\varepsilon_{0}$, which indicates strong reflection of electrons,
manifesting the reentrance behaviour within the second and third plateaus. As
the value of $\Delta r_{0}$ increases the position of the reentrance changes
in the normalized energy axis as can be observed in figure (3). The square-
shape deformation presents more dips in the conductance than the cone-shape
deformation because the dips can be related to quasi-bound states localized in
region II GURVITZ and LEVINSON (1993) that interfere to the scattering wave-
functions; thus, the wider the potential, the higher the number of quasi-bound
states. To understand the role of the size of the region II, we fix $\Delta
r_{0}=0.3r_{0}$ and we vary the length $L=4.4,5.6,6.8,8.0,$ and$,9.2r_{0}$.
Such results are shown in figure (4). For $L=$ 4.4 and 5.6$r_{0}$, the dip in
the first plateau is absent while two dips are present within the second
plateau, for the cone-shape deformation (dashed curves in figure (4)). On the
other hand, the dip in the first plateau appears for $L=$4.4 and 5.6$r_{0}$
when the square-shape deformation is considered (solid curves in figure (4)).
For $L=$6.8, 8.0, and 9.2$r_{0}$, a dip in the first plateau is present for
$\Delta r_{1}(z)$ (dashed curves in figure(4)). On the contrary, the dip in
the first plateau becomes very narrow for $L=$6.8, 8.0, and 9.2$r_{0}$ when
$\Delta r_{2}(z)$ is employed (solid curves in figure (4)). It can be noted
that the number of dips within second and third plateaus are really influenced
by the length $L$ due to interference phenomena occurring between quasi-bound
and scattering sates GURVITZ and LEVINSON (1993). The results of this section
demonstrate that the reentrance feature can indeed be observed even in the
absence of the SOC and the role of deformations in the transport properties in
NWs.

Figure 4: Normalized conductance as a function of the normalized Fermi energy
for different length $L/r_{0}=4.4,5.6,6.8,8.0,$ and 9.2, and fixed radius
deformation $\Delta r_{0}/r_{0}=0.3$. Solid (dashed) lines describe the
results for the square (cone)-shape deformation model. The curves are offset
for clarity according to the indicated values.

###  III.2 Rashba SOC

In this section, we want to explore the role of the structural deformation in
the NW when the Rashba SOC is also taken into account. In this case, we must
add the Rashba term to (1), which is given by the following equation

$H_{SO}=\alpha(z)(k_{y}\sigma_{z}-k_{z}\sigma_{y}),$ (40)

where $\sigma_{y}$ ($\sigma_{z}$) and $k_{y}$ ($k_{z}$) are respectively the
Pauli spin matrix and the wave vector in the y-direction (z-direction). We
assume that the Rashba interaction only acts in the region II of length L
along the z-direction; thus,
$\alpha(z)=\alpha\left[\Theta(z+L)-\Theta(z-L)\right]$ and $\Theta(z)$ is the
Heaviside function and $\alpha$ is the Rashba coupling strength. The reason
for considering a localized Rashba SOC is related to experimental setups where
an electric field is applied between top and bottom gates, which induces the
emergence of a structural inversion asymmetry SOC Moroz and Barnes (1999). We
must explicitly add the spin degree of freedom when the Rashba SOC is
considered; therefore, the matrix elements evaluated in (39) must be
multiplied by $\delta_{s,s^{\prime}}$, where $s,s^{\prime}\in\\{-,+\\}$. Using
the transverse modes, which are solutions of (2), we can calculate the matrix
elements for the Rashba SOC, as follows

$H_{j,j^{\prime}}^{SO}=-is\left[b^{m}_{m^{\prime}}\delta_{n,n^{\prime}}\delta_{s,s^{\prime}}+\\{\alpha(z),k_{z}\\}\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}\delta_{s,-s^{\prime}}\right],$
(41)

with
$b^{m}_{m^{\prime}}=\frac{\sqrt{2}\alpha(z)}{r_{0}}\left[\sqrt{m^{\prime}}\delta_{m,m^{\prime}-1}-\sqrt{m^{\prime}+1}\delta_{m,m^{\prime}+1}\right]$,
$\\{\alpha(z),k_{z}\\}=(\alpha(z)k_{z}+k_{z}\alpha(z))/2$, and $j=(n,m,s)$.

Figure 5: Normalized conductance as a function of the normalized Fermi energy
for different radius compression $\Delta
r_{0}/r_{0}=0,-0.02,-0.04,-0.06,-0.08,$ and -0.1, considering the length
$L/r_{0}=8.0$ and the Rashba constant $\alpha/\alpha_{0}=0.4$. Solid (dashed)
lines describe the results for the square (cone)-shape deformation model. The
curves are offset for clarity according to the indicated values. Figure 6:
Normalized conductance as a function of the normalized Fermi energy for
different radius expansion $\Delta r_{0}/r_{0}=0,0.02,0.04,0.06,0.08$ and 0.1,
considering the length $L/r_{0}=8.0$ and the Rashba constant
$\alpha/\alpha_{0}=0.4$. Solid (dashed) lines describe the results for the
square (cone)-shape deformation model. The curves are offset for clarity
according to the indicated values. Figure 7: Normalized conductance as a
function of the normalized Fermi energy for a fixed value of the radius
expansion $\Delta r_{0}/r_{0}=0.04$ and length $L/r_{0}=8.0$, considering the
variation of the Rashba constant $\alpha/\alpha_{0}=0.0,0.2,0.4,0.6,0.8$, and
1.0. The curves are offset for clarity according to the indicated values.
Solid (dashed) lines describe the results for the square (cone)-shape
deformation model.

Due to the Rashba SOC, the boundary conditions for the derivative of wave-
functions must be reformulated as Xiao et al. (2014):
$D[\chi^{\text{II}},-L/2]=D[\chi^{\text{I}},-L/2]+\frac{i\mathbf{M}}{2}\chi_{II}(L/2)$
and
$D[\chi^{\text{II}},L/2]=D[\chi^{\text{I}},L/2]-\frac{i\mathbf{M}}{2}\chi_{II}(L/2)$,
where $\mathbf{M}$ is a matrix, whose elements are given by
$\mathbf{M}_{i,j}=-2\alpha\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}\delta_{s,s^{\prime}}$.
Results including the SOC are shown in Figs. (5-7). In figure (5), we plot the
normalized conductance as a function of the Fermi energy considering
$\alpha=0.4\alpha_{0}$ ($\alpha_{0}=\varepsilon_{0}r_{0}$) and for different
values for the radius compression, $\Delta
r_{0}/r_{0}=0.0,-0.02,-0.04,\ldots,-0.1$. When $\Delta r_{0}=0.0$, there are
two dips in the first plateau at $E\approx 1.8\varepsilon_{0}$ and $E\approx
2.0\varepsilon_{0}$ and one dip in the second plateau at $E\approx
3.0\varepsilon_{0}$ only due to the Rashba SOC. For $\Delta r_{0}=-0.02r_{0}$,
all dips in the conductance are reduced. By further compressing the radius of
the NW, one can notice in figure (5) that the dips completely disappear for
$\Delta r_{0}\geq-0.04r_{0}$ when the square-shape model is considered (solid
curves in figure 5). These results are expected because the local radius
compression works as a repulsive potential, which can cancel the attractive
potential given by the SOC Sanchez and Serra (2006). When the radius of the NW
suffers a local expansion in the same region where the SOC takes place, we
have the appearance of more dips and a shift towards smaller energies as the
expansion increases, as can be seen in figure (6). In figure (7), we plot the
normalized conductance as a function of the Fermi energy considering $\Delta
r_{0}=0.04r_{0}$ and different values for the Rashba constant
($\alpha/\alpha_{0}=0.0-1.0$ with step 0.2). For these parameters, there is no
dip in the first plateau for $\alpha=0.0$, but dips appear for $\alpha\geq
0.2\alpha_{0}$. One can also notice that both deformation models present
different results, even considering a small deformation $\Delta
r_{0}=0.04r_{0}$. All results presented in Figs. (5-7) demonstrate that there
is a competition between Rashba SOC and structural deformations, which can
cause a misinterpretation of experimental data when related to the reentrant
feature.

###  III.3 Magnetic Field

Figure 8: Normalized conductance as a function of the normalized Fermi energy
for different radius compression $\Delta
r_{0}/r_{0}=0.0,-0.02,-0.04,-0.06,-0.08,$ and -0.1, considering the length
$L/r_{0}=8.0$, the Rashba constant $\alpha/\alpha_{0}=0.4$, and Zeeman energy
$E_{Z}/\varepsilon_{0}=0.2$. The curves are offset for clarity according to
the indicated values. Solid (dashed) lines describe the results for the square
(cone)-shape deformation model. Figure 9: Normalized conductance as a function
of the normalized Fermi energy for different radius expansion $\Delta
r_{0}/r_{0}=0.0,0.02,0.04,0.06,0.08,$ and 0.1, considering the length
$L/r_{0}=8.0$, the Rashba constant $\alpha/\alpha_{0}=0.4$, and Zeeman energy
$E_{Z}/\varepsilon_{0}=0.2$. The curves are offset for clarity according to
the indicated values. Solid (dashed) lines describe the results for the square
(cone)-shape deformation model.  Figure 10: Normalized conductance as a
function of the normalized Fermi energy for different length $L/r_{0}=0.2$ to
$9.4$ with step $\Delta L/r_{0}=0.2$, considering $\Delta r_{1}(z)$ for the
radius variation function with $\Delta r_{0}/r_{0}=0.04$, the Rashba constant
$\alpha/\alpha_{0}=0.4$, and Zeeman energy $E_{Z}/\varepsilon_{0}=0.2$. The
curves are offset for clarity. Figure 11: Normalized conductance as a function
of the normalized Fermi energy for different Rashba constant
$\alpha/\alpha_{0}=0.0$ to $0.2$ with step $\Delta\alpha/\alpha_{0}=0.02$,
considering $\Delta r_{1}(z)$ for the radius variation function with $\Delta
r_{0}/r_{0}=0.3$, length $L/r_{0}=8.0$, and Zeeman energy
$E_{Z}/\varepsilon_{0}=0.2$. The curves are offset for clarity. Figure 12:
Normalized conductance as a function of the normalized Fermi energy for
different Zeeman energy $E_{Z}/\varepsilon_{0}=0.0$ to $0.2$ with step $\Delta
E_{Z}/\varepsilon_{0}=0.02$, considering $\Delta r_{2}(z)$ for the radius
variation function with $\Delta r_{0}/r_{0}=0.3$, the Rashba constant
$\alpha/\alpha_{0}=0.2$, and length $L/r_{0}=8.0$. The curves are offset for
clarity. Figure 13: Normalized conductance as a function of the normalized
Fermi energy for different Zeeman energy $E_{Z}/\varepsilon_{0}=0.0$ to $0.2$
with step $\Delta E_{Z}/\varepsilon_{0}=0.02$, considering $\Delta r_{2}(z)$
for the radius variation function with $\Delta r_{0}/r_{0}=0.3$, the Rashba
constant $\alpha/\alpha_{0}=0.8$, and length $L/r_{0}=8.0$. The curves are
offset for clarity.

Ideal helical states occur when an external magnetic field is applied to the
NW. In this section, we add a uniform magnetic field applied in the
x-direction to the previous NW configuration, in accordance to the experiment
of Heedt et al., 2017. In this case, we must add the Zeeman Hamiltonian
$H_{Z}=E_{Z}\sigma_{x}$ to (1), where $E_{Z}$ is the Zeeman energy. The matrix
elements for this Hamiltonian are given by
$H^{Z}_{j,j^{\prime}}=E_{Z}\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}\delta_{s,-s^{\prime}}$.
When the magnetic field is included, the subband energy split as
$E^{\pm}_{n,m}=E_{n,m}\pm E_{Z}$. This would move the energy position where
the first step in the conductance takes place. Yet, the electron source will
be affected by the uniform magnetic field in the same way, thereby canceling
out this energy shift. Thus, we have compensated for this difference and
considered that the total energy where the first channel opens does not change
with the variation of the magnetic field. In figure 8, we plot the normalized
conductance for a fixed Zeeman energy, $E_{Z}=0.2\varepsilon_{0}$, as a
function of the total energy considering $\alpha=0.4\alpha_{0}$, $L=8r_{0}$
and $\Delta r_{0}/r_{0}=$0.0,-0.02,-0.04,-0.06,-0.08,and -0.1. In this case,
many dips appear when $\Delta r_{0}=$0.0, but they fade out even for a small
compression $\Delta r_{0}=$-0.04 $r_{0}$. By further compressing the NW, we
clearly observe the disappearance of the dips for $\Delta r_{0}=$-0.08 $r_{0}$
using the square-model (solid curves in figure 8). On the other hand, more
dips appear for a expanded NW due to interference phenomenon between
scattering and quasi-bound states GURVITZ and LEVINSON (1993), as can be
observed in figure 9. Such results demonstrate how a small structural
deformation can play an important role in the observation of the reentrant
feature in NWs when a magnetic field and Rashba SOC coexist. To further
explore our method, we consider some extra cases to understand the dependence
of the dips in the conductance as a function of different parameters. First,
we vary the length of the region II within the range $L\in[0.2,9.4]r_{0}$ with
step $\Delta L=0.2r_{0}$, assuming $\alpha/\alpha_{0}=0.4$,
$E_{Z}=0.2\varepsilon_{0}$, and the cone-shape deformation with $\Delta
r_{0}/r_{0}=0.04$, whose results for the conductance are shown in figure 10.
As already discussed, the Rashba SOC works as an attractive potential Sanchez
and Serra (2006), specifically as a quantum well with depth
$V_{0}=-m\alpha^{2}/\hbar^{2}$ and length $L$. When $L\leq 0.6r_{0}$ and
$\alpha=0.4\alpha_{0}$ there is no dip in the conductance because there is no
quasibound state within this quantum well. According to the approximate two-
channel model described in Ref. GURVITZ and LEVINSON, 1993, the dip in the
conductance occurs when an attractive potential induces the formation of a
quasi-bound state. In figure 10 there is only one dip in the plateaus for
$0.6r_{0}<L\lessapprox 4.8r_{0}$, which shifts towards smaller energies in a
quadratic way as a function of the length $L$, which is qualitatively in
accordance to the behavior of eigenenergies in a quantum well that represent
the position of the quasibound states. For $L>4.8r_{0}$, we start to observe
two dips within the same plateau, which indicates the presence of two
quasibound states within the same range of energy where the plateau takes
place. To further explore effects of the Rashba SOC, we vary
$\alpha=0.0\rightarrow 0.2\alpha_{0}$ with step $\Delta\alpha=0.02\alpha_{0}$
assuming $L>8r_{0}$, $E_{Z}=0.2\varepsilon_{0}$, and the cone-shape
deformation with $\Delta r_{0}/r_{0}=0.3$. Such results are plotted in figure
11, which shows that there is no dip in the first plateau for $\alpha\leq
0.02\alpha_{0}$. Yet, dips appear for $\alpha>0.02\alpha_{0}$ resulting only
from the Rashba SOC. Within others plateaus, dips coming from the radius
expansion already appear for $\alpha=0$ and these dips interfere with dips
descendant from Rashba SOC. Finally, we probe the conductance for different
magnetic fields, considering $L>8r_{0}$, $E_{Z}=0\rightarrow
0.2\varepsilon_{0}$ with step $\Delta E_{z}=0.02\varepsilon_{0}$, the square-
shape deformation with $\Delta r_{0}/r_{0}=0.3$, and for two different values
of Rashba constant $\alpha=0.2\alpha_{0}$ (figure 12) and
$\alpha=0.8\alpha_{0}$ (figure 13). When the Rashba SOC energy
$E_{R}=m^{*}\alpha^{2}/2\hbar^{2}$ is smaller than the Zeeman energy, the
conductance is more affected by the magnetic field and more dips appear when
$E_{Z}$ is increased, as shown in figure 12. On the other hand, figure 13
shows that dips are not very affected by the magnetic field because in this
case the Rashba SOC dominates over the external magnetic field.

##  IV Conclusions

In this paper, we provided a method to numerically calculate the conductance
considering an arbitrary 3D-spatial shaped potential and including the
coupling between scattering channels. By employing this method, we were able
to calculate the conductance for an NW containing structural deformations. We
also showed the appearance of reentrant behavior without the Rashba SOC and
magnetic field. Furthermore, we investigated the effects of structural
deformations taking into account an external magnetic field. We found that
these deformations affect the observation of the reentrant feature.
Particularly, the compression of the NW radius can destroy the dip in the
conductance. Here, we focused on structural deformation of the NW, but similar
effects would occur in NWs under the presence of a localized potential. We
believe the deformations can be experimentally achieved by employing
nonuniform top-bottom gates in experiments such as those reported in Refs.
[Heedt et al., 2017; Kammhuber et al., 2017]. Also, III–V NWs grown in the
[111] direction by the vapour-liquid-solid mechanism that exhibit planar
stacking faults Caroff; Algra are a good platform to probe our findings.

###### Acknowledgements.

The authors are grateful for financial support by the Brazilian Agencies
FAPESP, CNPq and CAPES. LKC thanks to the Brazilian Agencies FAPESP and CNPq
(grant No 2019/09624-3 and No 311450/2019-9) for supporting this research.

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##  Appendix A Fit spectra of the sample

The spectral analysis of the 14 objects of our sample is shown in Figures
11-14. The spectral fit for the Ciii] spectral range of Mrk 110 is absent
because there are no Ciii] data for this object.

Figure 11: Spectral fits of (from top to bottom) Mrk 335, Mrk 110, Mrk 509 and
Ark 564 in the wavelength ranges considered, with specific fluxes measured in
units of $10^{-15}$ erg s-1 cm-2 ${\rm\AA}^{-1}$ and the wavelength measured
in Å. A lower box describing the residuals is associated with each spectral
fit, with wavelength expressed in units of radial velocities (km s-1). The
original spectrum is plotted with a black line and the modeled spectrum is
plotted in magenta. Each additional line component is plotted with a different
colour, according to the legend. Dashed vertical lines represent the rest-
frame wavelengths of the main lines of the selected spectral range. Left:
$\lambda$1320-1450 Å range. Middle-left: $\lambda$1450-1700 Å range. Middle-
right: $\lambda$1800-2100 Å range. Right: $\lambda$4430-5300 Å range.

Figure 12: Same as in Figure 11 but for Mrk 478, Ton 28, LB 2522 and LEDA
51016, respectively.

Figure 13: Same as in Figure 11 but for I Zw 1, PHL 1092, Fairall 9 and Ark
120, respectively.

Figure 14: Same as in Figure 11 but for NGC 3783 and TXS 0042+101,
respectively.

##  Appendix B Distribution of $U$, $n_{H}$ and $Z$ for each object

The distribution of the physical parameters of $U$, $n_{H}$ and $Z$ for the 14
objects of our sample is shown in Figures 15-19.

Figure 15: Two-dimensional parameter spaces illustrating the physical
conditions of the gas in the BLR of the objects. Each plot represents the BC
with a green dot indicating the minimum $\chi^{2}$ computed between measured
diagnostic ratios and CLOUDY simulations. The orange region corresponds to
1$\sigma$ accuracy. Blue and red dots and shades represent the minimum
$\chi^{2}$ and associated regions for the BLUE and VBC, respectively. Left:
Logarithmic plane of $n_{H}$ and $Z$. Middle: Logarithmic plane of $n_{H}$ and
$U$. Right: Logarithmic plane of $U$ and $Z$.

Figure 16: Continued.

Figure 17: Continued.

Figure 18: Continued.

Figure 19: Continued.


undergoes frequent L2 separatrix-crossings to become perturber-phylic. Note
how the regular, long-period oscillations in energy corresponding to its Pac-
Man phase are interspersed with rapid oscillations corresponding to its
perturber-phylic phase.

Figure 12: Examples of Chimera orbits. From top to bottom the panels depict
(i) a Chimera orbit initially classified as a horse-shoe, which occasionally
undergoes separatrix crossing to transform into a Pac-Man, (ii) an initial
horse-shoe that transforms into a tadpole, (iii) an initial Pac-Man that
transforms into perturber-phylic and COM-phylic orbits, and (iv) an initial
Pac-Man that occasionally transforms into a perturber-phylic orbit. See text
for details.


Electron transport through a quantum wire coupled with a mesoscopic ring

Santanu K. Maiti1,2,∗

1Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear
Physics,
1/AF, Bidhannagar, Kolkata-700 064, India
2Department of Physics, Narasinha Dutt College, 129, Belilious Road,
Howrah-711 101, India

Abstract

Electronic transport through a quantum wire sandwiched between two metallic
electrodes and coupled to a quantum ring, threaded by a magnetic flux $\phi$,
is studied. An analytic approach for the electron transport through the bridge
system is presented based on the tight-binding model. The transport properties
are discussed in three aspects: (a) presence of an external magnetic filed,
(b) strength of the wire to electrode coupling, and (c) presence of in-plane
electric field.

PACS No.: 73.23.-b; 73.63.-b; 73.21.Hb

Keywords: Green’s function; Conductance; $I$-$V$ characteristic; Electric
field.

∗Corresponding Author: Santanu K. Maiti

Electronic mail<EMAIL_ADDRESS>
##  1 Introduction

With the advancement in nanoscience and nanotechnology, the fabrication of
sub-micron devices has become possible and has allowed one to study the
electron transport through quantum systems in a very controllable way. These
quantum systems have attracted much more attention since they constitute
promising building blocks for future generation of electronic devices and
directed attention on the study of discrete structures, such as a single
molecule, arrays of molecules, quantum dots, quantum wires and mesoscopic
rings. The electron transport through a bridge system was first studied
theoretically in $1974$ [1]. Later, several numerous experiments [2, 3, 4, 5,
6] have been performed through quantum systems placed between two metallic
electrodes with few nanometer separation. The operation of such two-terminal
devices is due to an applied bias. Current passing across the junction is
strongly nonlinear function of applied bias voltage and its detailed
description is a very complex problem. Though lot of theoretical as well as
experimental papers have been available in the literature, yet the complete
knowledge of the conduction mechanism in this scale is not well understood
even today. The transport properties of these systems are associated with some
quantum effects like, quantization of energy levels, quantum interference of
electron waves, etc. A quantitative understanding of the physical mechanisms
underlying the operation of nanoscale devices remains a major challenge in the
present nanoelectronics research.

The aim of the present article is to reproduce an analytic approach based on
the tight-binding model to investigate the electronic transport properties
through a quantum wire coupled to a mesoscopic ring. There exist some ab
initio methods for the calculation of conductance [7, 8, 9, 10, 11, 12], yet
it is needed the simple parametric approaches [13, 14, 17, 15, 16, 18, 19, 20,
21, 22, 23] for this calculation, especially for the case of larger molecular
bridge systems. The parametric study is motivated by the fact that the ab
initio theories are computationally too expensive and here we focus our
attention on the qualitative effects rather than the quantitative ones. This
is why we restrict our calculations on the simple analytical formulation of
the transport problem.

We organize the paper as follow. Following the introduction (Section $1$), in
Section $2$, we present the model system under consideration and give a very
brief description for the calculation of conductance and current-voltage
characteristics through the bridge system. Section $3$ presents the results of
the system taken into account. Finally, we summarize our results in Section
$4$.

##  2 The model and a brief description onto the theoretical formulation

We begin by referring to Fig. 1. The system considered here is a quantum wire
coupled to a mesoscopic ring with $N$ atomic sites and the wire is attached to
two semi-infinite one-dimensional ($1$D) metallic electrodes, namely, source
and drain.

Figure 1: Schematic view of a quantum wire coupled to a mescopic ring,
threaded by a magnetic flux $\phi$, and the wire is attached to two $1$D
metallic electrodes.

The full system (quantum wire with ring) is described by a single-band tight-
binding Hamiltonian within a non-interacting electron picture, and it can be
written in the form,

$H_{C}=H_{W}+H_{R}+H_{WR}$ (1)

where, $H_{W}$, $H_{R}$ and $H_{WR}$ correspond to the Hamiltonians for the
wire, ring and wire-to-ring coupling, respectively, and they can be expressed
as,

$H_{W}=\sum_{i}\epsilon_{i}d_{i}^{\dagger}d_{i}+\sum_{<ij>}t_{w}\left(d_{i}^{\dagger}d_{j}+d_{j}^{\dagger}d_{i}\right)$
(2)
$H_{R}=\sum_{k}\epsilon_{k}c_{k}^{\dagger}c_{k}+\sum_{<kl>}t_{r}\left(c_{k}^{\dagger}c_{l}e^{i\theta}+c_{l}^{\dagger}c_{k}e^{-i\theta}\right)$
(3) $H_{WR}=t_{0}\left(c_{1}^{\dagger}d_{0}+d_{0}^{\dagger}c_{1}\right)$ (4)

Here, $\epsilon_{i}$’s ($\epsilon_{k}$’s) are the on-site energies of the ring
(wire), $d_{i}^{\dagger}$ and $c_{k}^{\dagger}$ are the creation operators of
an electron at site $i$ and $k$ in the wire and ring. $\theta=2\pi\phi/N$ is
the phase factor due to the flux $\phi$ threaded by the ring. $t_{w}$
($t_{r}$) is the hopping integral between two nearest-neighbor sites in the
ring (wire) and $t_{0}$ is the wire-to-ring tunneling coupling.

At much low temperatures and bias voltage, the linear conductance of the wire-
ring system can be calculated by using one-channel Landauer conductance
formula,

$g=\frac{2e^{2}}{h}T$ (5)

where $T$ is the transmission probability of an electron from the source to
drain through the wire including the ring, and it is defined as [24],

$T(E,V)={\mbox{Tr}}\left[\left(\Sigma_{S}^{r}-\Sigma_{S}^{a}\right)G^{r}\left(\Sigma_{D}^{a}-\Sigma_{D}^{r}\right)G^{a}\right]$
(6)

Now the Green’s function $G$ of the full system (wire with ring) is given by
the relation,

$G=\left[E-H_{C}-\Sigma_{S}-\Sigma_{D}\right]^{-1}$ (7)

where $E$ is the energy of injecting electrons from the source and $H$ is the
Hamiltonian of the full system described above (Eq. 1). In Eq. 7,
$\Sigma_{S}=h_{SC}^{\dagger}g_{S}h_{SC}$ and
$\Sigma_{D}=h_{DC}g_{D}h_{DC}^{\dagger}$, are the self-energy terms due to the
two electrodes. $g_{S}$ and $g_{D}$ correspond to the Green’s functions for
the source and drain, respectively. $h_{SC}$ and $h_{DC}$ are the coupling
matrices and they are non-zero only for the adjacent points of the quantum
wire and the electrodes. The coupling terms $\Gamma_{S}$ and $\Gamma_{D}$ for
the full system can be calculated through the expression [24],

$\Gamma_{\\{S,D\\}}=i\left[\Sigma_{\\{S,D\\}}^{r}-\Sigma_{\\{S,D\\}}^{a}\right]$
(8)

where $\Sigma_{\\{S,D\\}}^{r}$ and $\Sigma_{\\{S,D\\}}^{a}$ are the retarded
and advanced self-energies respectively and they are conjugate to each other.
Datta et al. [25] have shown that the self-energies can be expressed like,

$\Sigma_{\\{S,D\\}}^{r}=\Lambda_{\\{S,D\\}}-i\Delta_{\\{S,D\\}}$ (9)

where $\Lambda_{\\{S,D\\}}$ are the real parts of the self-energies which
correspond to the shift of the energy eigenvalues of the full system (quantum
wire with ring) and the imaginary parts $\Delta_{\\{S,D\\}}$ of the self-
energies represent the broadening of the energy levels. Since this broadening
is much larger than the thermal broadening, we restrict our all calculations
only at absolute zero temperature. By doing some simple calculations, these
real and imaginary parts of the self-energies can be determined in terms of
the coupling strength ($\tau_{\\{S,D\\}}$) between the wire and two
electrodes, injecting electron energy ($E$) and hopping strength ($v$) between
nearest-neighbor sites in the electrodes. Using Eq. 9, the coupling terms
$\Gamma_{S}$ and $\Gamma_{D}$ can be written in terms of the retarded self-
energy as,

$\Gamma_{\\{S,D\\}}=-2{\mbox{Im}}\left[\Sigma_{\\{S,D\\}}^{r}\right]$ (10)

All the information regarding the wire to electrode coupling are included into
the two self energies stated above and is analyzed through the use of Newns-
Anderson chemisorption theory [13, 14]. The detailed description of this
theory is obtained in these two references.

Thus, by calculating the self-energies, the coupling terms $\Gamma_{S}$ and
$\Gamma_{D}$ can be easily obtained and then the transmission probability $T$
will be calculated from the expression given in Eq. 6.

The current passing through the bridge is depicted as a single-electron
scattering process between the two reservoirs of charge carriers. The current-
voltage relation is evaluated from the following expression [24],

$I(V)=\frac{e}{\pi\hbar}\int\limits_{E_{F}-eV/2}^{E_{F}+eV/2}T(E,V)~{}dE$ (11)

where $E_{F}$ is the equilibrium Fermi energy. For the sake of simplicity,
here we assume that the entire voltage is dropped across the wire-electrode
interfaces and this assumption does not greatly affect the qualitative aspects
of the $I$-$V$ characteristics. Throughout the article we set $E_{F}$ to $0$
and use the units $c=e=h=1$.

##  3 Results and discussion

Here we describe conductance-energy and current-voltage characteristics
through the quantum wire coupled to a mesoscopic ring at absolute zero
temperature. Electron transport properties through the system are strongly
affected by the magnetic flux $\phi$, wire-to-electrode coupling strength and
the in-plane electric field. In the presence of in-plane electric filed and
assuming it along the perpendicular direction of the wire, the dependence of
the site energies on the electric field $\mathcal{E}$ is written within the
tight-binding approximation as [15],

$\displaystyle\epsilon_{i}$ $\displaystyle=$
$\displaystyle\left(e\mathcal{E}aN/2\pi\right)\cos\left[2\pi(i-1)/N\right]$
(12) $\displaystyle=$
$\displaystyle\left(et_{r}\right)\left(\mathcal{E}^{\star}N/2\pi\right)\cos\left[2\pi(i-1)/N\right]$

where, $a$ is lattice spacing in the mesoscopic ring and $\mathcal{E}^{\star}$
is the dimensionless electric field strength defined by $\mathcal{E}a/t_{r}$.
For simplicity, here we assume $t_{w}$, $t_{r}$ and $t_{0}$ are identical to
each other in magnitude and specify them by the symbol $t$. We investigate all
the essential features of electron transport for the two limiting cases. One
is the weak-coupling limit, defined as $\tau_{\\{S,D\\}}<<t$ and the other one
is the

Figure 2: Conductance $g$ as a function of energy $E$ in the weak-coupling
limit for the system with ring size $N=10$, where (a) in the absence of any
electric filed with $\phi=0$ (solid line) and $0.4$ (dotted line) and (b) in
the presence of $\phi=0.4$ with $\mathcal{E}=2$ (solid line) and $4$ (dotted
line).

strong-coupling limit and defined it as $\tau_{\\{S,D\\}}\sim t$. The
parameters $\tau_{S}$ and $\tau_{D}$ correspond to the couplings of the wire
to the source and drain, respectively. The common set of values of these
parameters in the two limiting cases are as follow: $\tau_{S}=\tau_{D}=0.5$,
$t=3$ (weak-coupling) and $\tau_{S}=\tau_{D}=2$, $t=3$ (strong-coupling).

In Fig. 2, we plot the conductance ($g$) as a function of the injecting
electron energy ($E$) for the bridge system in the limit of weak-coupling.
Figure 2(a) corresponds to the spectrum in the absence of any electric filed
where, the solid and dotted curves are respectively for $\phi=0$ and $0.4$. In
Fig. 2(b), the spectrum is shown for the non-zero value of the electric field
with $\phi=0.4$ where, the solid and dotted curves represent the results for
the electric filed strengths $\mathcal{E}=2$ and $4$, respectively.
Conductance vanishes almost for all energies except at resonances where it
approaches to $2$. At these resonances, the transmission probability $T$
becomes unity, since $g=2T$ (from the Landauer formula with $e=h=1$). The
resonant peaks in the conductance spectrum coincide with eigenenergies of the
system (wire including the ring), and thus the spectrum manifests itself the
energy levels of the system. For zero electric field strength and in the
absence of magnetic flux $\phi$, the conductance exhibits a single resonant
peak across $E=0$ (see solid curve of Fig. 2(a)), while, in the presence of
$\phi$ more resonant peaks appear in the spectrum (see dotted curve of Fig.
2(a)). It reveals that for non-zero value of $\phi$ more resonating states
appear in the system. This is due to the removal of all the degeneracies in
the energy eigenstates for any non-zero value of $\phi$. In the presence of
in-plane electric field, these resonant peaks are shifted and the conductance
spectrum becomes asymmetric with respect to the energy $E$ (see Fig. 2(b)).

For the strong wire-to-electrode coupling, resonant peaks get substantial
widths as presented in Fig. 3 where, the solid and dotted curves

Figure 3: Conductance $g$ as a function of energy $E$ in the strong-coupling
limit for the system with ring size $N=10$, where (a) in the absence of any
electric filed with $\phi=0$ (solid line) and $0.4$ (dotted line) and (b) in
the presence of $\phi=0.4$ with $\mathcal{E}=2$ (solid line) and $4$ (dotted
line).

correspond to the identical meaning as earlier. The increment of the resonant
widths is due to the broadening of the energy levels of the wire including the
ring, where the contribution comes from the imaginary parts of the two self-
energies [24].

The scenario of electron transfer through the bridge becomes much more clearly
visible by studying the current $I$ as a function of the applied bias voltage
$V$.

Figure 4: Current $I$ as a function of bias voltage $V$ in the limit of weak
wire-to-electrode coupling for the system with ring size $N=10$ and
$\phi=0.4$. The solid and dotted lines correspond to the currents for
$\mathcal{E}=0$ and $3$, respectively.

The Current is computed from the integration procedure of the transmission
function $T$ which shows the same variation, differ only in magnitude by the
factor $2$,

Figure 5: Current $I$ as a function of bias voltage $V$ in the limit of strong
wire-to-electrode coupling for the system with ring size $N=10$ and
$\phi=0.4$. The solid and dotted curves correspond to the currents for
$\mathcal{E}=0$ and $3$, respectively.

like as the conductance spectra (Figs. 2 and 3). The current-voltage
characteristic in the weak-coupling limit for the bridge system is shown in
Fig. 4 where, the solid curve corresponds to the current in the absence of any
electric field and the dotted curve denotes the same for $\mathcal{E}=3$. Here
we take $\phi=0.4$. The current shows staircase-like behavior with sharp
steps, which is associated with the discrete nature of the resonant spectrum
(Fig. 2). The shape and width of the current steps depend on the width of the
resonant spectrum since the hight of a step in $I$-$V$ curve is directly
proportional to the area of the corresponding peak in the conductance
spectrum. On the other hand, the current varies continuously with the applied
bias voltage and achieves much bigger values in the strong-coupling limit, as
shown in Fig. 5 where, the solid and dotted curves correspond to the same
meaning as earlier. From both Figs. 4 and 5 it is clearly observed that the
in-plane electric field suppresses the current amplitude (see the dotted
curves). This feature may be utilized to control externally the amplitude of
the current through the bridge system.

##  4 Concluding remarks

To summarize, we have introduced parametric approach based on the tight-
binding model to investigate the electron transport properties at absolute
zero temperature through a quantum wire coupled to a mesoscopic ring threaded
by a magnetic flux $\phi$. A simple parametric approach is given to study
electron transport properties through the system, and it can be used to study
the transport behavior in any complicated molecular bridge system. Electronic
conduction through the quantum wire is strongly influenced by the flux $\phi$
threaded by the ring and the wire-to-electrode coupling strength. The effects
of in-plane electric field have also been studied in this context and it has
been predicted that the current amplitude can be controlled externally through
the bridge system by means of this electric field.

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