# Electron transport in quantum channels with spin-orbit interaction:
Effects of the sign of the Rashba coupling and applications to nanowires

Leonid Gogin  Dipartimento di Scienza Applicata e Tecnologia del Politecnico
di Torino, I-10129 Torino, Italy  Fausto Rossi  Dipartimento di Scienza
Applicata e Tecnologia del Politecnico di Torino, I-10129 Torino, Italy
Fabrizio Dolcini<EMAIL_ADDRESS>Dipartimento di Scienza Applicata
e Tecnologia del Politecnico di Torino, I-10129 Torino, Italy

###### Abstract

We investigate the effects of the sign of the Rashba spin-orbit coupling
(RSOC) on electron transmission through a single-channel Nanowire (NW) in the
quantum coherent regime. We show that, while for a finite length NW with
homogeneous RSOC contacted to two electrodes the sign of its RSOC does not
affect electron transport, the situation can be quite different in the
presence of an inhomogeneous RSOC and a magnetic field applied along the NW
axis. By analyzing transport across an interface between two regions of
different RSOC we find that, if the two regions have equal RSOC signs, the
transmission within the magnetic gap energy range is almost perfect,
regardless of the ratio of the spin-orbit energies to the Zeeman energy. In
contrast, when the two regions have opposite RSOC signs and are Rashba-
dominated, the transmission gets suppressed. Furthermore, we discuss the
implementation on a realistic NW setup where two RSOC regions are realized
with suitably coupled gates separated by a finite distance. We find that the
low-temperature NW conductance exhibits a crossover from a short distance
behavior that strongly depends on the relative RSOC sign of the two regions to
a large distance oscillatory behavior that is independent of such relative
sign. We are thus able to identify the conditions where the NW conductance
mainly depends on the sign of the RSOC and the ones where only the RSOC
magnitude matters.

##  I Introduction

Over the last 10 years most of research on semiconductor Nanowires (NWs) with
Rashba spin-orbit coupling (RSOC) has focussed on the search for Majorana
quasi-particles, which are believed to provide the building block for
topologically protected quantum informationdassarma_2010 ; vonoppen_2010 ;
alicea_review ; aguado_review ; prada-review ; valkov-review . Although
signatures compatible with these exotic quasi-particles have been found in NWs
proximized by superconducting filmskouwenhoven_2012 ; liu_2012 ; heiblum_2012
; xu_2012 ; defranceschi_2014 ; marcus_2016 ; marcus_science_2016 ;
kouwenhoven_2018 , the race to the evidence of Majorana states has somewhat
overshadowed other interesting potentialities of NWs in quantum technologies.

Indeed NWs reaching lengths of various $\mu{\rm m}$ and exhibiting quantum
coherent transport are nowadays fabricated both in the clean
ballistickouwenhoven_2018 ; kallaher_2010 ; kouwenhoven_nanolett_2013 ;
kouwenhoven_nanolett_2016 ; shaepers_nanolett_2016 ; xu_2016 ;
kouwenhoven_2017a ; kouwenhoven_2017b ; DeFranceschi_2018 ; kouwenhoven_2019a
; kouwenhoven_2019b  and in the diffusive regimeschaepers_2020 , and are also
realized in suspended geometriesheiblum_2012 ; giazotto_2019 , in
arrayskouwenhoven_2019b ; bakkers_2012 ; philipose_2019  and in
networkskouwenhoven_2019a . Moreover, they can be used as flexible substrates
for hybrid epitaxial growth on selected facets in order to design
heterostructures with ferromagnets and superconductorskrogstrup_2020 . Also,
NWs represent an extremely versatile and tunable platform for nanoelectronics
since their conduction properties can be controlled both magnetically, e.g. by
applying a magnetic field along the NW axis and thereby opening up a gap in
the spectrum, or electrically by controlling the RSOC through gate voltages.
In particular, the tremendous improvement in gating techniques enables one to
achieve a high control of the RSOCsasaki_2013 ; micolich ; sasaki_2017 ;
das_2019 ; guo_2021 ; sasaki_2021 .

Most of these efforts have been devoted to enhance the magnitude of the RSOC
over a wide range of valuesgao_2012 ; slomski_NJP_2013 ; wimmer_2015 ;
bercioux_review ; nygaard_2016 ; sherman_2016 ; tokatly_PRB_2017 ; loss_2018 ;
goldoni_2018 ; gao-review ; lau_2021 . However, when an electron travels
through a given spin-orbit region, its spin polarization depends not only on
the magnitude, but also on the sign of the local spin-orbit coupling. Are
there any observable effects that can be attributed to the sign? So far, this
question has been addressed only in a few cases. Concerning the Dresselhaus
spin-orbit coupling, its sign reversal has been investigated in InGaAs
ringsnitta-frustaglia , whereas in the case of the RSOC the problem has mainly
been addressed in two-dimensional quantum wells interfaced with other suitable
materialskaindl_2005 ; slomski_2013 ; tsai_2018  or where bulk subbands are
characterized by opposite RSOC signswang-fu_2016 . As far as one-dimensional
single-channel NWs are concerned, it has been shown that a spatially modulated
RSOC with alternating sign can drive the transition from a metallic to an
insulating statejohannesson-japaridze_2011 . Moreover, when a NW is exposed to
a magnetic field along its axis and the RSOC magnitude is large, the
propagating states in the magnetic gap exhibit a locking between propagation
direction and spin, whose helicity is determined by the RSOC signstreda ;
depicciotto_2010 ; loss_PRB_2011 ; lutchyn_2012 ; loss_PRB_2017 .
Nevertheless, when the NW with homogeneous RSOC is contacted to two
electrodes, the electron transport turns out to depend only on the magnitude
of the RSOC, and not on its sign, as we shall see below. Recent studies
suggest that a different scenario may emerge in the presence of inhomogeneous
RSOC profile. A junction between two regions with opposite RSOC signs, for
instance, exhibits interesting spectral and equilibrium properties at the
interface, such as localized bound statesloss_EPJB_2015 ; rossi-dolcini-
rossi_EPJ  and orthogonal spin polarizationrossi-dolcini-rossi_2020 , which
look surprisingly similar to the ones of the topological phase. In
experimental implementations, where the RSOC can be locally controlled by
gates, these inhomogeneous configurations can be realized by coupling various
gates along the NW, providing an additional knob to control the electrical
current flowing through the NW setup.

The question arises whether in such inhomogeneous configurations the sign of
the RSOC can lead to any observable effect on out of equilibrium properties.
In this paper we investigate this problem by analyzing the electron transport
through a single-channel NW with inhomogeneous RSOC in the coherent quantum
limit. The paper is organized as follows. In Sec.II we present the model and
describe the adopted method. Then, in Sec.III we discuss the two necessary
ingredients to observe the effects of the sign of the RSOC, namely the
presence of a uniform magnetic field and the existence of at least two regions
with different RSOC. In Sec.IV we first perform a preliminary analysis of the
transmission in the presence of a single interface separating two regions with
different RSOC, finding a quite different behavior of the electrical
conductance in the cases of equal and opposite RSOC signs. Then, by analyzing
a more realistic configuration, where two differently gated portions of a NW
are separated by a finite distance, we show that such lengthscale can modify
the results obtained in the ideal limit of one single interface case. This
enables us to identify the conditions where the sign of the RSOC affects the
NW transport properties, and the situations where only the RSOC magnitude
matters. Finally, in Sec.VI we discuss our results and draw our conclusions.

##  II Model and method

We consider a one-dimensional electron conduction channel directed along the
$x$ direction, characterized by a spatially varying RSOC profile and exposed
to an external Zeeman magnetic field directed along its axis. The second-
quantized Hamiltonian reads

$\displaystyle\hat{\mathcal{H}}=\int\hat{\Psi}^{\dagger}(x)\left(\frac{p_{x}^{2}}{2m^{*}}\sigma_{0}-\frac{\sigma_{z}}{2\hbar}\\{\alpha(x),p_{x}\\}-\sigma_{x}h_{\perp}\right)\hat{\Psi}(x)$
(1)

where $m^{*}$ denotes the effective electron mass. In order to ensure
Hermiticity, a half of the anticommutator between the momentum operator
$p_{x}$ and the spatially inhomogeneous RSOC $\alpha(x)$ has been introduced,
as customarysherman_PRL_2007 ; brataas_2007 ; sherman_PRB_2013 ;
sherman_PRB_2018 . Here $\sigma_{x},\sigma_{y},\sigma_{z}$ denote spin Pauli
matrices, while $\sigma_{0}$ is the $2\times 2$ identity. Moreover, $z$ is the
direction of the RSOC effective magnetic field, whereas $x$ is the direction
of the actual magnetic field, characterized by an energy coupling
$h_{\perp}={\rm e}g\hbar B/4m_{e}$, where ${\rm e}$ denotes the electron
charge, $g$ the $g$-factor and $m_{e}$ the bare electron mass.

We are interested in describing electron transport along the inhomogeneous
channel. With respect to spin degenerate problems, the charge current operator
exhibits an additional term related to the RSOCrashba_2004 ; dolcini-
rossi_2018

$\displaystyle\hat{J}^{c}(x,t)$ $\displaystyle=$ $\displaystyle-\frac{i{\rm
e}\hbar}{2m^{*}}\left(\hat{\Psi}^{\dagger}(x,t)\partial_{x}\hat{\Psi}(x,t)-\partial_{x}\hat{\Psi}^{\dagger}(x,t)\,\hat{\Psi}(x,t)\right)$
(2) $\displaystyle-\frac{{\rm
e}\alpha(x)}{\hbar}\hat{\Psi}^{\dagger}\sigma_{z}\hat{\Psi}\quad,$

as can be deduced from the very continuity equation
$\partial_{t}\hat{n}+\partial_{x}\hat{J}^{c}=0$ following from the Heisenberg
Equation dictated by the Hamiltonian (1), where the charge density is
$\hat{n}={\rm e}\hat{\Psi}^{\dagger}\hat{\Psi}$.

###  II.1 Relation with a magnetic texture problem

Before discussing any specific RSOC profile, we note a general property of the
Hamiltonian (1). By performing the transformationsanchez_2006 ;
sherman_PRL_2007

$\hat{\Psi}(x)=\displaystyle
e^{i\phi_{SO}(x)\frac{\sigma_{3}}{2}}\,\hat{\Psi}^{\prime}(x)$ (3)

where

$\phi_{SO}(x)=\frac{2m^{*}}{\hbar^{2}}\int_{x_{0}}^{x}\alpha(x^{\prime})\,dx^{\prime}$
(4)

and $x_{0}$ is an arbitrarily fixed reference point, the Hamiltonian (1) is
rewritten in the new fields $\hat{\Psi}^{\prime}(x)$ as

$\displaystyle\hat{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\int{\hat{\Psi}}^{\prime\dagger}(x)\,\Bigg{(}\frac{p^{2}_{x}}{2m^{*}}+U_{SO}(x)-$
$\displaystyle-
h_{\perp}\left(\cos[\phi_{SO}(x)]\sigma_{x}+\sin[\phi_{SO}(x)]\sigma_{y}\right)\Bigg{)}{\hat{\Psi}^{\prime}}(x)\,dx$

where

$U_{SO}(x)=-\frac{{m^{*}}\alpha^{2}(x)}{2\hbar^{2}}\quad.$ (6)

Various aspects are noteworthy. First, in the absence of Zeeman field
($h_{\perp}=0$), while for a homogeneous problem Eq.(6) is just a mere energy
constant and the spin-orbit coupling gets completely gauged out, in an
inhomogeneous problem the spatial profile of the effective scalar potential
$U_{SO}(x)$ does affect electron transmission, although in a spin-independent
way.note-bc  For instance, in the case of a piecewise constant RSOC,
$U_{SO}(x)$ acquires the form of a potential step, a quantum well or a
barrier, leading to an energy dependent transmission. The second aspect is
that, when the Zeeman term $h_{\perp}$ is additionally present, the last term
of Eq.(II.1) acquires the form of a magnetic texture in the spin $x$-$y$ plane
characterized by the rotation angle $\phi_{SO}(x)$ in Eq.(4), which represents
the integrated local spin-orbit wavevector (multiplied by 2). In this case the
role of the spin degree of freedom becomes not trivial because of the
interplay between actual magnetic field and spin-orbit effective magnetic
field. In the following, we shall occasionally refer to the mapping to
Eq.(II.1) in order to interpret some of our results.

###  II.2 Piecewise constant profile

Although for an arbitrary spatial profile the inhomogeneous problem cannot be
solved analytically, any profile $\alpha(x)$ can ultimately be approximated
with a piecewise constant profile, where the solution can be build up by
suitably matching the electron field operator in each region. It is thus worth
recalling briefly the main physical ingredients characterizing each locally
homogeneous region. In a region with RSOC $\alpha_{j}$ one identifies two
energy scales, namely the spin-orbit energy

$E_{SO,j}=\frac{m^{*}\alpha_{j}^{2}}{2\hbar^{2}}$ (7)

and the Zeeman energy

$E_{Z}=|h_{\perp}|$ (8)

which directly impact on the local spectrum. Indeed it consists of two bands
(see Fig.1), separated by a gap $\Delta=2E_{Z}$ at $k=0$, where the lower band
exhibits one minimum or two minima depending on whether the region is in the
Zeeman-dominated regime ($2E_{SO,j}<E_{Z}$) or in the Rashba-dominated regime
($2E_{SO,j}>E_{Z}$).

Figure 1: The local spectrum of the RSOC channel in each piecewise constant
profile region. (a) The case of the Rashba-dominated regime, where the two
local minima of the lower band are at
$E_{min}=-E_{SO}(1+E_{Z}^{2}/4E_{SO}^{2})$; (b) The case of the Zeeman-
dominated regime. In each panel the arrows indicate the direction of
$k$-dependent spin orientation in the $x$-$z$ plane of the sheet, in the case
of $\alpha>0$ (blue arrows) and $\alpha<0$ (red arrows).

The classification of the eigenstates is in principle straightforward,
although it requires a little more care in practice. First, while the spectrum
does not depend on the sign of the RSOC $\alpha_{j}$, the eigenstates do.
Since this will play a role in the results to be presented below, in Fig.1 we
have highlighted the $k$-dependent spin orientation in the $x$-$z$ plane of
the local eigenstates, distinguishing the cases $\alpha>0$ and $\alpha<0$ by
blue and red arrows, respectively. Furthermore, since in the quantum coherent
limit the inhomogeneous solution is characterized by a given energy $E$, in
each region one has to identify all 4 wavevectors corresponding to such
energy, retaining both real wavevectors (propagating modes) and complex
wavevectors (evanescent modes). In particular, the expression of the
evanescent modes, which turn out to play an important role when matching the
wavefunctions in different regions, leads to one further distinction between
the weak Zeeman regime ($2E_{SO,j}<E_{Z}<4E_{SO,j}$) and the strong Zeeman
regime ($E_{Z}>4E_{SO,j}$). After performing such lengthy but straightforward
classification of all eigenstates, whose technical details and results are
reported in Appendix A, the general solution at energy $E$ is built up by
matching the linear superpositions at each interface. Explicitly, at the
$j$-th interface located at position $x_{j}$ and separating two regions with
RSOC values $\alpha_{j}$ (on the left) and $\alpha_{j+1}$ (on the right), the
boundary conditions are

$\left\\{\begin{array}[]{lcl}\hat{\Psi}(x_{j}^{-})&=&\hat{\Psi}(x_{j}^{+})\\\
\\\
\partial_{x}\hat{\Psi}(x_{j}^{-})&=&\partial_{x}\hat{\Psi}(x_{j}^{+})-\frac{im^{*}}{\hbar^{2}}(\alpha_{j+1}-\alpha_{j})\sigma_{z}\hat{\Psi}(x_{j})\end{array}\right.$
(9)

Then, by applying the Scattering Matrix Formalismdatta_book , where the
external regions act like the leads and the inhomogeneous profile determines
the Scattering region, we compute the Scattering Matrix and determine the
Transmission coefficient ${T}(E)$ for various inhomogeneous configurations.
Details and examples are provided in Appendix B. This enables us to determine
how the low temperature linear conductance

$\displaystyle G$ $\displaystyle=\frac{{\rm e}^{2}}{h}{T}(E_{F})$ (10)

depends on the RSOC and the Zeeman field, allowing for an electrical and
magnetic tuning of the transport properties.

##  III Two essential ingredients to observe effects of the RSOC sign

Because our purpose is to analyze the effect of the sign of the RSOC on
transport properties, it is first worth pointing out some general conditions
for this to be observed. We start by noting that the transmission coefficient
is completely independent of a global sign change of the RSOC profile,
$\alpha(x)\rightarrow-\alpha(x)$. Indeed the Hamiltonian (1) with a profile
$-\alpha(x)$ can be mapped back into the one with a profile $+\alpha(x)$
through the transformation $\hat{\Psi}(x)\rightarrow\sigma_{x}\hat{\Psi}(x)$.
This implies, for instance, that for a finite length NW with a homogeneous
RSOC contacted two normal leads with vanishing RSOC, modelled with an
inhomogeneous profile $\alpha(x)=\alpha\,\theta(d/2-|x|)$, where $\theta$
denotes the Heaviside function and $d$ is the NW length, the conductance is
independent of $\mbox{sgn}(\alpha)$. In particular, we note that the effect of
Fano anti-resonances with vanishing transmission occurring in such a case for
suitably chosen parameterssanchez_2006 ; sanchez_2008 ; aguado_2015  is
completely insensitive to the sign of the RSOC in the NW.

Thus, a necessary condition for the effects of the RSOC sign to be observed is
that the sign of the RSOC in a portion of the profile $\alpha(x)$ changes with
respect to the rest. However, it may not be sufficient. Compare for instance
the uniform profile $\alpha(x)\equiv\alpha_{0}$ and the inhomogeneous profile
$\alpha(x)=\alpha_{0}\mbox{sgn}(x)$, where the sign on the left of the origin
is changed with respect to the one on the right. Despite the abrupt sign
change of the RSOC, in the absence of the Zeeman field ($h_{\perp}=0$) the
electron transmission is always perfect and equal to the case of a uniform
profile. This can be straightforwardly deduced from the mapping described in
Sec.II.1, since the potential Eq.(6) reduces to a mere homogeneous constant
$U_{SO}\equiv-{m}^{*}\alpha_{0}^{2}/2\hbar^{2}$. In contrast, if
$h_{\perp}\neq 0$, the term in the second line of Eq.(II.1) is present and the
phase $\phi_{SO}(x)$ in Eq.(4) depends on the sign of $\alpha$.

From these remarks, we conclude that the two essential ingredients to observe
effects of the RSOC sign on electron transport are i) a Zeeman field directed
perpendicularly to the RSOC effective magnetic field and ii) the presence of
at least two regions with different RSOC value. This is what we shall consider
in the following, focussing on the intrinsic transmission properties of the
inhomogeneous RSOC. The presence of spurious resistance due to non-ideal
contacts with external electrodes can in principle be taken into account with
the method outlined in Appendix B. However, it depends on the specific metals
used as electrodes and on the adiabaticity of the contactsrainis-loss_PRB_2014
, and it goes beyond the purpose of this paper.

##  IV The single interface problem

We start by analyzing the transmission across an interface separating two
regions with different RSOC, namely

$\alpha(x)=\left\\{\begin{array}[]{lcl}\alpha_{L}&\mbox{for}&x<0\\\ &&\\\
\alpha_{R}&\mbox{for}&x>0\end{array}\right.\quad.$ (11)

Depending on the strength of the RSOC, each interface side can be in the
Rashba-dominated or in the Zeeman-dominated regime. After solving the
scattering problem, as outlined in the Appendix B, we have determined the
transmission coefficient and the conductance from Eq.(10). Specifically, Fig.2
shows the conductance $G$, in units of the conductance quantum $G_{0}={\rm
e}^{2}/h$, as a function of the Fermi energy $E_{F}$, in units of the Zeeman
energy $E_{Z}$, for the three independent configurations: Zeeman-Zeeman,
Rashba-Rashba, and Rashba-Zeemannota-config . In particular, Fig.2(a)
illustrates the case where the RSOC has the same sign in both sides. Recalling
that the energy range $|E_{F}|<E_{Z}$ corresponds to the magnetic gap (see
Fig.1), we first note that, when both sides are in the Rashba-dominated regime
(red curve), transmission is possible also in the energy range below the
magnetic gap, where it can also exceed $G_{0}$. This is because two
propagating channels exist in the range $E_{min}<E<-E_{Z}$, as one can see
from the mexican-hat shaped lower band in Fig.1(a). At the energies $E_{F}=\pm
E_{Z}$ corresponding to the magnetic gap edges, the conductance of Fig.2(a)
exhibits two sharp anti-resonance suppression cusps, an effect similar to the
one found in Refs.[sanchez_2008, ] and [aguado_2015, ] in a NW coupled to two
leads. We shall comment about this aspect at the end of this section. Apart
from these features, one finds a practically perfect transmission, namely
$G=G_{0}$ when the Fermi energy lies inside the magnetic gap, $|E_{F}|<E_{Z}$,
(one propagating channel) and $G=2G_{0}$ above the gap $E>E_{Z}$ (two
propagating channels), regardless of the specific regime configuration.

Figure 2: The conductance of the single interface problem [see Eq.(11)] is
plotted in units of conductance quantum $G_{0}={\rm e}^{2}/h$ as a function of
the Fermi energy $E_{F}$ (in units of the Zeeman energy $E_{Z}$) for various
spin-orbit configurations. In each panel the black curve refers to the case
where both regions are Zeeman-dominated ($2E_{SO,L}=0.3E_{Z}$;
$2E_{SO,R}=0.5E_{Z}$), the red curve to the case where both regions are
Rashba-dominated ($2E_{SO,L}=2E_{Z}$; $2E_{SO,R}=5E_{Z}$), while the green
curve to the case where the left region is Rashba-dominated and the right
region is Zeeman dominated ($2E_{SO,L}=3E_{Z}$; $2E_{SO,R}=0.5E_{Z}$). While
panel (a) refers to the case where $\alpha_{L},\alpha_{R}>0$, panel (b) refers
to the case of coupling changing sign across the junction, $\alpha_{L}<0$ and
$\alpha_{R}>0$. Insets: the spatial behavior of the spin-orbit angle (4)
describing the rotation of the effective magnetic texture problem (II.1).

A different scenario emerges when the RSOC takes opposite signs across the
interface, as shown in Fig.2(b). In all configurations the conductance in the
magnetic gap range $|E_{F}|<E_{Z}$ remains roughly symmetric with respect to
the midgap energy value $E_{F}=0$. However, we note that, while the
transmission is practically perfect when both sides are in the strong Zeeman
regime (black curve), it reduces when one region enters the Rashba regime
(green curve) and even more when both regions are Rashba-dominated (red
curve). The different behavior of the conductance in the two panels (a) and
(b) can be qualitatively understood through the mapping to the magnetic
texture problem Eq.(II.1). In the present case of the profile Eq.(11) the
rotation angle (4) of the magnetic field acquires the form

$\phi_{SO}(x)=\frac{2m^{*}}{\hbar}\times\left\\{\begin{array}[]{lcl}\alpha_{R}x&\mbox{for}&x>0\\\
&&\\\ \alpha_{L}x&\mbox{for}&x<0\end{array}\right.$ (12)

and its spatial behavior is depicted in the insets of the two panels of Fig.2.
If $\alpha_{L}$ and $\alpha_{R}$ have the same sign [inset of panel (a)], the
incoming wavefunction can easily adapt to the change of slope of
$\phi_{SO}(x)$ from one side to the other by simply stretching or shrinking,
since a coordinate rescaling $x\rightarrow x\,\alpha_{R}/\alpha_{L}$ would
compensate for the slope change. This leads to a very high tranmission. In
contrast, when $\alpha_{L}$ and $\alpha_{R}$ have opposite signs, an actual
cusp appears in $\phi_{SO}$ [inset of panel (b)], which cannot be merely
removed by an affine coordinate transformation. It is therefore more difficult
for the incoming wavefunction spinor to re-adapt to the profile on the other
side of the interface. As long as the values of the RSOC are small (both
regions in the Zeeman dominated regime) this effect is negligible, but when
both regions enter the Rashba-dominated regime the mismatch becomes important
and the transmission is suppressed.

Figure 3:  The single interface problem [see Eq.(11)] with
$E_{SO,R}=0.8E_{SO,L}$. The conductance $G$ in units of the conductance
quantum $G_{0}={\rm e}^{2}/h$ as function of the Fermi energy
$E_{F}/E_{SO,L}$, for different values of the Zeeman energy $E_{Z}$. (a) The
case where the RSOC takes the same sign on both interface sides. Note that,
when both sides are in the Rashba-dominated regime (red curve) a peak of
$2G_{0}$ conductance is present for $E_{min}<E<-E_{Z}$, i.e. for
$-0.878<E/E_{SO,L}<-0.5$, due to the presence of two propagating channels
below the magnetic gap energy (see Fig.1(a)). The inset on the right magnifies
the sharp anti-resonance cusp occurring when the Fermi energy equals the upper
boundary of the magnetic gap $E_{F}=+E_{Z}$. Similar behavior occurs at the
lower boundary $E_{F}=-E_{Z}$. In this case the Zeeman energy $E_{Z}$ only
controls the number of propagating channels (0, 1 or 2) and the conductance
varies discretely in practice. (b) The case where the RSOC has opposite signs
across the interface. In this case the Zeeman energy also enables one to vary
continuously the conductance over the entire energy range.

The role of the RSOC sign can also be highlighted by considering the magnetic
tuning of the conductance $G$ in a Rashba interface where both sides are in
the same regime. As an illustrative example of such situation, Fig.3 describes
the Fermi energy dependence of $G$ for a junction with
$E_{SO,R}/E_{SO,L}=0.8$, for various values of the Zeeman energy $E_{Z}$.
Specifically, panel(a) shows the case where the RSOC takes the same sign
across the interface ($\alpha_{L},\alpha_{R}>0$). When the magnetic field is
absent (green dash-dotted curve) the conductance vanishes for
$E_{F}<-E_{SO,R}$ and then rapidly increases to $2G_{0}$ for
$E_{F}>-E_{SO,R}$. This is exactly the same energy dependence as a spin-
degenerate problem of transmission across a potential step
$(-E_{SO,L}|-E_{SO,R})$, as argued invoking the mapping described in Sec.II.1.
In contrast, when a weak Zeeman field is introduced (red solid curve), the
conductance gets suppressed down to $G_{0}$ for $|E_{F}|<E_{Z}$ due to the
magnetic gap opening that leaves only one propagating channel. Again, very
narrow cusps with vanishing conductance appear at Fermi energies corresponding
to the magnetic gap boundaries $E_{F}=\pm E_{Z}$ (one of them is highlighted
in the inset on the left). For a stronger Zeeman field (black dashed line)
both sides enter the Zeeman-dominated regime and the conductance acquires a
step-like behavior characterizing the number of propagating channels (0, 1 or
2). Thus, in this case of equal RSOC signs, the Zeeman energy in practice
controls only the number of such channels, i.e. the value at which the jump
occurs. Figure 3(b) instead describes the case of opposite RSOC sign across
the interface, i.e. $\alpha_{L}<0$ and $\alpha_{R}>0$. For vanishing Zeeman
field the conductance is insensitive to the sign of the RSOC, so that the
green dash-dotted curve is exactly equal to the one obtained in Fig.3(a).
Again, exploiting the mapping (3) described in Sec.II.1, one can see that in
such situation only the potential (6) is present. In contrast, when the Zeeman
field is introduced, the sign of the RSOC matters and the behavior strongly
differs from panel (a). The conductance varies continuously as a function of
the Fermi energy. In this case the Zeeman energy $E_{Z}$ determines not only
the location of the suppression cusps at the magnetic gap edges, but also the
magnitude of $G$ in the entire energy range. In particular, when the NW is in
the Rashba dominated regime (red curve), the conductance is significantly
suppressed inside and below the magnetic gap.

Anti-resonances in the conductance. We conclude this section by a comment
about the anti-resonance cusps with vanishing conductance displayed in Figs.2
and 3. We emphasize that their presence is not due to the RSOC sign change
across the interface, as they do exist both in the case of equal and opposite
RSOC signs, and turn out to be much sharper in the former case [see inset of
Fig.3(a)]. Although this effect is thus generic and not strictly related to
the main focus of our paper, for the sake of completeness in the presentation
of our results, a brief discussion about their origin is in order. We recall
that the existence of vanishing conductance dips, typically close in energy to
resonance peaks of perfect transmission, is known to occur in the case of a NW
contacted to leadssanchez_2008 ; aguado_2015 , and is attributed to the fact
that the RSOC present in the centrally confined NW region leads to a spin
misalignment of its bound states with respect to the outer leads with
continuum spectrum, causing one bound state to be strongly coupled to the
continuum, while the other one is very weakly coupled. With respect to the
configuration analyzed in Refs.[sanchez_2008, ] and [aguado_2015, ], our case
of a single interface between two RSOC regions exhibits two differences: i)
here, within the energy gap, there are no bound states at the interfacenota-
su-bs ; ii) the anti-resonances with vanishing transmission are always pinned
at the boundaries $E=\pm E_{Z}$ of the magnetic gap. Their origin can be
argued as follows. Within the energy range $|E|<E_{Z}$ of the magnetic gap,
each side of the interface is characterized by propagating modes (see Fig.1),
whose spin orientations lying in the $x$-$z$ spin plane are in general
misaligned, because the RSOC takes different values across the interface.
However, one also has evanescent modes originating from the band spin-
splitting in the magnetic gap [see Fig.8(b) and Fig.9(b) and (c) in App.A],
whose spin orientation lies in the $x$-$y$ spin plane. On each interface side
the wavefunction is thus a superposition of both propagating and evanescent
modes. Although the transmission is carried by the propagating modes only, the
evanescent modes indirectly affect the transmission because they contribute to
realize the wavefunction matching Eq.(9) at the interface. Effectively, one
could consider the two RSOC regions as “leads” with (massless) propagating
modes $\Psi_{pr}$ with misaligned spins, which are coupled to (massive)
evanescent modes $\Psi_{ev}=\xi e^{\pm\kappa_{+}x}$ localized at the
interface, whose spinor $\xi$ effectively performs the spin rotation, thereby
favoring the transmissiongogin_2022 . Of course, in the case of equal RSOC
signs, the misalignment of the propagating modes is less pronounced than in
the case of opposite RSOC signs, and this is why for $|E|<E_{Z}$ the
transmission is typically higher in Fig.2(a) than in 2(b). However, as the
energy approaches (say) the upper edge of the magnetic gap, $E\rightarrow
E_{Z}$, each evanescent mode becomes the state related to the minimum of the
upper band (see Fig.1): Its spin gets locked along $x$ and its decay
lengthscale diverges. This makes it effectively unable to guarantee a finite
transmission, since a spatially uniform mode has a vanishing spatial overlap
with a propagating mode. The transmission thus vanishes with a cusp behavior
because the evanescent mode wavevector $\kappa_{+}$ vanishes with an infinite
slope as a function of energy $E\rightarrow E_{Z}$. To a more quantitative
level, one can see that in such a limit $\Psi_{ev}$ can contribute to the
first interface matching Eq.(9) only with a spin along $x$, and its
contribution to the spatial derivative in the second Eq.(9) vanishes. It turns
out that the only way to realize a wavefunction matching of both Eqs.(9) at
the interface is to have a totally reflected mode, as one can verify by
plotting the wavefunction profile (not shown here). A similar effect occurs at
the lower magnetic gap edge $E\rightarrow-E_{Z}$ when the region is in the
Rashba-dominated regime, since the evanescent mode becomes the state related
to the local maximum of the lower band (see Fig.1).

##  V The doubly gated Nanowire

The analysis of the single interface case carried out in the previous Section
indicates that the conductance heavily depends on the relative sign between
the two RSOC regions. However, a sharp separation between the two RSOC regions
is in fact an idealization. In a realistic NW setup, where the RSOC can be
locally controlled by different gates, a finite distance $d$ separates the
gates, as shown in the configurations sketched in Fig.4. In particular, panel
(a) describes a realization with a NW deposited on a substrate and covered by
two $\Omega$-gateskouwenhoven_2017a .In this kind of setup the NW is usually
separated from the metallic gate by a thin (few nm) dielectric with high
relative dielectric constant, which –at a given gate voltage– enhances the
interface field causing the RSOCsasaki_2017 ; sasaki-NTT-review . In contrast,
Fig.4(b) illustrates a suspended NW coupled to two pairs of side gates with
the advantage of strongly reducing the presence of defectsgiazotto_2019 . In
both cases the gate voltages control the magnitude and the sign of the RSOC.

We now want to take into account the finite size $d$ of the central region
separating the two gated NW regions, which was neglected in the preliminary
analysis of Sec.IV. For definiteness we shall assume that the RSOC in the
central region is negligible, so that it is in the strongly Zeeman-dominated
regime, and we shall adopt the following profile

$\alpha(x)=\left\\{\begin{array}[]{lcl}\alpha_{L}&\mbox{for}&x<-d/2\\\ &&\\\
0&\mbox{for}&|x|<d/2\\\ &&\\\
\alpha_{R}&\mbox{for}&x>d/2\end{array}\right.\quad.$ (13)

Moreover, we shall set the RSOC to be positive in the right region
($\alpha_{R}>0$), while for the left region we will consider both positive and
negative sign of $\alpha_{L}$.

Figure 4: Two possible realizations of a doubly gated NW, where two different
NW regions, characterized by different RSOC, are separated by a distance $d$.
(a) The NW lies on a substrate and is contacted to two $\Omega$-gates (see
e.g. Ref.[kouwenhoven_2017a, ]). (b) The NW is suspended and coupled to two
pairs of side gates (see e.g. Ref.[giazotto_2019, ]).  Figure 5: The
conductance of the double gate configuration [see Eq.(13)], where both gated
regions are in the Rashba-dominated regime ($E_{SO,L}=3E_{Z}$,
$E_{SO,R}=1.5E_{Z}$), is plotted in units of conductance quantum $G_{0}={\rm
e}^{2}/h$ as function of the energy $E_{F}/E_{Z}$. The two panels correspond
to a different value $d$ of the distance between the two gated regions,
expressed in units of the shortest spin-orbit lengthscale $k_{SO,L}^{-1}$,
namely short distance (panel (a)), and large distance (panel (b)). The black
and red curves refer to the case where the two regions have equal and opposite
sign of the RSOC, respectively.

In Fig.5 we analyze the conductance $G$ as a function of the Fermi energy over
the entire spectrum. We focus on the case where the two regions are both in
the Rashba-dominated regime. The two spin-orbit energies Eq.(7) identify two
spin-orbit wavevectors

$k_{SO,j}=\frac{\sqrt{2m^{*}E_{SO,j}}}{\hbar}=\frac{m^{*}|\alpha_{j}|}{\hbar^{2}}\hskip
28.45274ptj=L,R\quad,$ (14)

and the related spin-orbit lengths $k_{SO,j}^{-1}$. For definiteness we take
$E_{SO,L}=2E_{SO,R}$ and express the gate separation $d$ in units of the
shorter spin-orbit length $k_{SO,L}^{-1}$. In particular, Fig.5(a) illustrates
the result for a short gate separation: The conductance is practically perfect
inside the magnetic gap when the two regions have equal RSOC sign (black
curve), whereas it is suppressed when the two regions have opposite RSOC sign
(red curve). Moreover, transmission is almost symmetric in energy and flat
around the middle of the magnetic gap ($E=0$). The result does not deviate
much from the one obtained in Fig.2 for the ideal $d\rightarrow 0$ limit.
However, for large gate separation [see Fig.5(b)], the difference between the
two cases of equal or opposite RSOC signs is strongly reduced, except for a
small difference near the gap edge $E/E_{Z}=+1$. This is due to the fact that,
for larger separation $d$, the electron spin has a spatial room to re-adapt to
the different orientation imposed by the opposite RSOC sign. We also note the
cusps with vanishing conductance discussed in the single interface problem are
modified by the finite distance $d$ between the gates. First, the effect of
vanishing conductance may still be present, but it occurs at values of Fermi
energy that can slightly differ from the magnetic gap boundaries, as is
visible e.g. in the black curve of Fig.5(b), near the upper magnetic gap
boundary $E_{F}=+E_{Z}$. This makes the doubly gated NW configuration similar
to the configuration of a NW contacted to leadssanchez_2008 ; aguado_2015 .
Second, the cusp behavior might occur at a finite conductance value. This can
be the case even for a short distance $d$ [see Fig.5(a)] and is especially
visible near the lower magnetic gap boundary $E_{F}=-E_{Z}$. In this case,
only for very short distance ($k_{SO,L}d<0.01$) one recovers the vanishing
conductance of the single interface problem.

Figure 6: The double gate configuration, where both gated regions are in the
Rashba-dominated regime ($E_{SO,L}=3E_{Z}$, $E_{SO,R}=1.5E_{Z}$). The
conductance, evaluated at three fixed values of Fermi energy inside the gap,
$E_{F}=-0.8E_{Z}$ (green curve) $E_{F}=0$ (black curve) and $E_{F}=+0.8E_{Z}$
(red curve), is plotted as function of the distance $d$ between the gates.
Panels (a) and (b) illustrate the cases of equal and opposite RSOC sign,
respectively

Figure 6 illustrates how the conductance, evaluated at three different Fermi
energy values inside the magnetic gap, behaves as a function of the distance
$d$ between the two gates. At short distance, $k_{SO,L}d\ll 1$, the behavior
of $G$ strongly depends on whether the signs of the two RSOCs are equal [panel
(a)] or opposite [panel (b)]: In the former case it is independent of the
distance $d$ for all energy values ($G\cong G_{0}$), while in the latter case
it linearly grows with the distance and it is small ($G\ll G_{0}$). As the
distance $d$ becomes of the order of the spin-orbit length, $k_{SO,L}d\sim 1$,
in both cases the conductance exhibits a crossover to an oscillatory behavior.
This originates from the fact that at the boundaries of the central region
part of the electron wave is reflected, giving rise to interference conditions
that depend on the distance $d$. Because the electron wavelength depends on
the Fermi energy, such oscillatory pattern depends on $E_{F}$ too, as one can
see from the different period characterizing the various curves. Notably, for
large distance $k_{SO,L}d\gg 1$, such oscillatory behavior becomes independent
of the relative RSOC signs of the two regions, as can be appreciated by
comparing panels (a) and (b).

Implementation with InSb NWs. Let us now consider a specific implementation of
the setup of Fig.4 with a InSb NW, with effective mass $m^{*}=0.015m_{e}$ and
$g$-factor $g\simeq 50$, exposed to a magnetic field corresponding to a Zeeman
energy $E_{Z}=0.1{\rm meV}$. For definiteness, for the RSOC in the right
region we have fixed a positive value $\alpha_{R}\simeq 0.552\,{\rm eV\AA}>0$
corresponding to a spin-orbit energy $E_{SO,R}=0.3\,{\rm meV}=3E_{Z}$, so that
the right region is in the Rashba-dominated regime. Then, by varying
$\alpha_{L}$ over a broad range of positive and negative values, we have
analyzed how the conductance in the middle of the magnetic gap ($E_{F}=0$)
depends on the ratio $\alpha_{L}/\alpha_{R}$. The result is plotted in Fig.7
for four different values of the distance $d$ between the two gates, which can
be compared to the reference lengthscale given by the fixed spin-orbit length
of the right-side is $k^{-1}_{SO,R}\simeq 92\,{\rm nm}$. As one can see, when
$d$ is much shorter than the latter scale (black curve) the conductance $G$
exhibits a strongly asymmetric curve as a function of $\alpha_{L}/\alpha_{R}$.
In this case $G$ depends only weakly on the magnitude of the RSOC, and
strongly on the sign. Indeed a sign switch of $\alpha_{L}$ from positive to
negative changes the transmission from high to low values. In contrast, when
$d$ becomes of the order of the spin-orbit length (red and green curves), the
asymmetry of the curve gradually softens and eventually, for a large
separation $d=1\,{\rm\mu m}$ (blue curve), the conductance exhibits a
completely symmetric Lorentzian-like behavior as a function of
$\alpha_{L}/\alpha_{R}$. In this regime only the magnitude of the RSOC
determines the conductance, while the relative sign of the RSOC plays no role.

Figure 7: The implementation of the double gate configuration with a InSb NW
($m^{*}=0.015m_{e}$) exposed to a magnetic field with an Zeeman energy
$E_{Z}=0.1\,{\rm meV}$. The RSOC in the right region is fixed to be positive,
$\alpha_{R}>0$, and corresponding to a spin-orbit energy $E_{SO,R}=0.3\,{\rm
meV}$ (Rashba dominated regime), corresponding to a spin-orbit length
$k_{SO,R}^{-1}\simeq 92\,{\rm nm}$. The RSOC $\alpha_{L}$ in the left region
is varied in magnitude and sign: The conductance at $E_{F}=0$ (middle of the
magnetic gap) is shown as a function of the ratio $\alpha_{L}/\alpha_{R}$, for
different values of the separation $d$ between the two gates.

The helical limit. We conclude this section by a comment about the helical
limit. As is well known, in the strongly Rashba-dominated regime, $E_{SO}\gg
E_{Z}$, the propagating states in the magnetic gap are helical, i.e. they
exhibit a locking between the propagation directionvonoppen_2010 ;
dassarma_2010 ; streda ; depicciotto_2010 ; loss_PRB_2011 ; lutchyn_2012 ;
loss_PRB_2017 , similarly to the edge states of a 2D Topological Insulator.
Importantly, in a NW the helicity is controlled by the sign of the RSOC.
Explicitly, if $\alpha>0$ right-movers are characterized by spin-$\uparrow$
and left-movers by spin-$\downarrow$, while the opposite occurs for
$\alpha<0$, as is clear from the blue/red arrows in Fig.1(a). This opens up
the possibility to conceive transport configurations that would be quite hard
in 2D Topological Insulators. Let us now consider the inhomogeneous setup
sketched in Fig.4 in the limit where both gated regions are set to the
strongly Rashba-dominated and exhibit opposite RSOC signs. This implements a
peculiar setup, namely a junction of helical states with opposite helicity,
the so called Dirac paradox. As is well known, for a NW with homogeneous RSOC,
the low energy transport within the magnetic gap, i.e. in the energy range
$|E_{F}|\ll E_{Z}$, is well captured by a massless Dirac model describing the
helical states. One may thus wonder whether in such energy window the behavior
of the conductance for the inhomogeneous setup of Fig.4 could be deduced from
the low energy massless Dirac models. This is indeed the case when the RSOC
takes the same sign in both gated regions: The low energy effective theory is
well captured by two massless Dirac models with the same helicity, one on each
region, and one recovers a perfect transmission at low energyperes . However,
in the case of opposite RSOC signs, the customary massless Dirac models are
not sufficient to describe the low-energy transport. Instead it is possible to
showgogin_2022  that additional massive Dirac modes have to be included on
each side to correctly account for the low-energy behavior at $|E_{F}|\ll
E_{Z}$. Although in such energy window these modes are evanescent and carry no
current directly, they do contribute to the current indirectly by favoring the
correct wavefunction matching at the interface. The transmission thus depends
on the parameter $k_{SO}d$ in a non-monotonous waygogin_2022 , in agreement
with the more general results presented here.

##  VI Conclusions

In Conclusion, in this paper we have analyzed the effects of the sign of the
RSOC on electron transport in a one-dimensional electron channel in the
quantum coherent regime. In the case of a NW with homogeneous RSOC and
contacted to two electrodes, despite the spin orientation of the eigenstates
depends on the sign of the RSOC (see Fig.1), we have shown that the transport
properties are independent of $\mbox{sgn}(\alpha)$. In fact, in Sec.III we
have identified two essential ingredients for the effects of the sign to be
observed, namely the presence of i) at least two regions with different RSOC
and ii) an additional Zeeman field that is non collinear with the effective
RSOC magnetic field. In Sec.IV we have first analyzed the electron transport
across an ideal interface separating two regions with different RSOC. We have
shown that, when the sign of the RSOC is the same in both regions, the
transmission is practically perfect, regardless of the regimes (Rashba- or
Zeeman-dominated) of the two regions [see Fig.2(a)]. In this case the magnetic
field essentially determines the number of conducting channels and the
conductance can be tuned to integer values of the conductance quantum [see
Fig.3(a)]. The scenario changes when the RSOC exhibits opposite signs at the
two interface sides. In particular, while transmission remains extremely good
when both regions are in the Zeeman regime, it reduces when one of the two
regions enters the Rashba regime, and it gets suppressed when both regions are
in the Rashba regime [see Fig.2(b)]. In this case, the conductance can be
continuously tuned by varying the Zeeman field [see Fig.3(b)].

Then, in Sec.V we have considered a more realistic NW setup, where two
different RSOC regions are realized by coupling the NW to various gates (see
Fig.4), and we have taken into account the finite distance $d$ between the
gates. The impact of the sign is quite different in the two regimes of small
and large separation $d$ (see Fig.5). Indeed, by analyzing the conductance as
a function of the separation $d$ (see Fig.6), we have found that in the regime
of short separation the conductance strongly depends on the relative RSOC
signs, namely for equal signs one has $G\cong G_{0}$, roughly independently of
$d$, while for opposite signs one has $G\ll G_{0}$ growing linearly with $d$.
In contrast, as the separation becomes comparable with the spin-orbit length,
the conductance exhibits a crossover to an oscillatory behavior, originating
from the quantum interference of the electron backscattering of the ends of
the separation region. In particular, for large separation ($k_{SO,L}d\gg 1$)
such oscillatory behavior becomes independent of the relative RSOC signs.

Furthermore, we have considered a specific implementation of the setup of
Fig.4 with InSb NWs, for four realistic values of separation $d$ between the
gates. Focussing on the midgap conductance $G$ of the setup as a function of
the ratio $\alpha_{L}/\alpha_{R}$ between the two RSOCs (see Fig.7), we have
been able to identify the conditions where $G$ is mainly determined by the
sign or by the magnitude of the RSOC. In particular, for a short separation
$d=30\,{\rm nm}$, the conductance mainly depends on the relative RSOC signs,
and one can tune $G$ from a small to a high fraction of ${\rm e^{2}}/h$ by
varying the ratio $\alpha_{L}/\alpha_{R}$ from negative to positive values
(black curve in Fig.7). In this regime, the RSOC sign thus represents an
actual additional knob to switch on/off the electron current through the NW.
In contrast, for a large separation $d=1\,{\rm\mu m}$, the conductance becomes
insensitive to the sign of $\alpha_{L}/\alpha_{R}$ (see blue curve in Fig.7)
and is mainly dependent on the RSOC magnitude only. Finally, we have discussed
that, for the particular range of small energies $|E_{F}|\ll E_{Z}$ inside the
magnetic gap, our findings are in agreement with the low-energy Dirac theory:
While in the case of two regions with equal RSOC signs the low-energy electron
transport is well described by massless Dirac modes, in the case of opposite
RSOC signs this is not the case, and additional massive Dirac modes have to be
included.

A potentially interesting future development of this work is to extend the
analysis of electron transport through the doubly gated NW to the case of ac
gate voltagessherman_PRB_2013 ; malshukov_2003 ; malshukov_2005 ; dolcini_2012
; loss_2016 , where two time-dependent RSOC are induced along the NW. Work is
in progress along these lines.

###### Acknowledgements.

We are thankful to Lorenzo Rossi for fruitful discussions

##  Appendix A Classification of NW eigenstates

As mentioned in the Main Text, the solution of the inhomogeneous RSOC problem
is built by suitably matching the wavefunctions of each piecewise homogeneous
NW region. To this purpose, for each energy value $E$, one has to classify all
eigenstates (both propagating and evanescent) of a given NW region with RSOC
$\alpha_{j}$. This Appendix is meant to provide this classification. In order
to make the notation lighter, we shall use in this section
$\alpha_{j}\rightarrow\alpha$ and $E_{SO,j}\rightarrow E_{SO}$.

Propagating states. Let us start from the propagating modes
$\psi(x)=\chi\exp[\pm ikx]$, where $\chi$ is a $2\times 1$ spinor. Denoting
$\varepsilon^{0}_{k}=\hbar^{2}k^{2}/2m^{*}$ and inverting the energy spectra
$E_{1,2}(k)=\varepsilon^{0}_{k}\mp\sqrt{E_{Z}^{2}+(\alpha k)^{2}}$ of the two
bands ($\beta=1,2$) in favour of the energy $E$, there are a priori four
wavevectors for each value of $E$, namely two positive ones, $k_{\pm}(E)$, and
two negative ones $-k_{\pm}(E)$, where

$k_{\eta}(E;\alpha)=\frac{\sqrt{2m^{*}}}{\hbar}\sqrt{E+2E_{SO}+\eta\sqrt{4EE_{SO}+4E_{SO}^{2}+E_{Z}^{2}}}\hskip
56.9055pt\eta=\pm\quad.$ (15)

This can explicitly be seen from Fig.8(a) for the Rashba-dominated regime and
in Figs.9(a) for the Zeeman-dominated regime, where $k_{+}$ and $k_{-}$ are
distinguished by thick and thin curves, respectively. However, only real
wavevectors ($k_{\eta}\in\mathbb{R}$) actually describe propagating modes and
must be retained. For these wavevectors, the magnitude
$v=\hbar^{-1}|\partial_{k}E|$ of the group velocity is expressed as a function
of energy as

$v_{\eta}(E;\alpha)=\frac{\hbar
k_{\eta}(E;\alpha)}{m^{*}}\frac{\sqrt{4EE_{SO}+4E_{SO}^{2}+E_{Z}^{2}}}{\left|2E_{SO}+\eta\sqrt{4EE_{SO}+4E_{SO}^{2}+E_{Z}^{2}}\right|}\hskip
56.9055pt\eta=\pm\quad.$ (16)

Figure 8: The Rashba dominated regime ($E_{Z}<2E_{SO}$). (a) Spectrum of the
propagating modes, where thick and thin lines correspond to the the
expressions $\pm k_{+}(E)$ and $\pm k_{-}(E)$ of the wavevector as a function
of energy, respectively [see Eq.(15)]. Here
$E_{min}=-E_{SO}\left(1+E_{Z}^{2}/4E_{SO}^{2}\right)$. (b) Spectrum of the
evanescent modes, where thick and thin lines correspond to $\pm\kappa_{+}(E)$
and to $\pm\kappa_{-}(E)$ [see Eq.(27)], respectively. Note that panel (b) can
also be seen as the spectrum of panel (a) in the magnetic gap and along the
imaginary axis $k=i\kappa$.

For each wavevector related to the energy $E$, the spinor $\chi$ acquires a
different expression depending on the band $\beta$. Explicitly, the two
eigenvectors of the lower band ($\beta=1$) and the upper band ($\beta=2$) are

$\displaystyle\chi_{1}(\pm
k_{\eta}(E;\alpha))=\left(\begin{array}[]{c}\cos\left(\frac{\theta(\pm
k_{\eta}(E;\alpha))}{2}\right)\\\ \\\ \sin\left(\frac{\theta(\pm
k_{\eta}(E;\alpha))}{2}\right)\end{array}\right)$ (20)
$\displaystyle\chi_{2}(\pm
k_{\eta}(E;\alpha))=\left(\begin{array}[]{c}-\sin\left(\frac{\theta(\pm
k_{\eta}(E;\alpha))}{2}\right)\\\ \\\ \cos\left(\frac{\theta(\pm
k_{\eta}(E;\alpha))}{2}\right)\end{array}\right)$ (24)

respectively, where

$\displaystyle\cos\left(\frac{\theta(\pm k_{\eta}(E;\alpha))}{2}\right)$
$\displaystyle=\sqrt{\frac{1}{2}\left(1\pm
sgn(\alpha)\frac{\sqrt{4E_{SO}(E+2E_{SO})+\eta
4E_{SO}\sqrt{4EE_{SO}+4E_{SO}^{2}+E_{Z}^{2}}}}{\left|\sqrt{4EE_{SO}+4E_{SO}^{2}+E_{Z}^{2}}+\eta
2E_{SO}\right|}\right)}$ (25) $\displaystyle\sin\left(\frac{\theta(\pm
k_{\eta}(E;\alpha))}{2}\right)$
$\displaystyle=sgn(h_{\perp})\sqrt{\frac{1}{2}\left(1\mp
sgn(\alpha)\frac{\sqrt{4E_{SO}(E+2E_{SO})+\eta
4E_{SO}\sqrt{4EE_{SO}+4E_{SO}^{2}+E_{Z}^{2}}}}{\left|\sqrt{4EE_{SO}+4E_{SO}^{2}+E_{Z}^{2}}+\eta
2E_{SO}\right|}\right)}$ (26)

The eigenstates (20)-(24) identify a spin lying in the $x$-$z$ plane and
forming an angle $\theta$ with the $z$-axis. This enables one to obtain the
pattern of spin-orientation in the two bands in Fig.1.

Evanescent states. One can proceed in a similar way for the evanescent modes
$\psi(x)=\xi\exp[\pm\kappa x]$, where $\xi$ is a spinor. Their spectrum
$E_{1,2}(\kappa)=-\varepsilon^{0}_{\kappa}\mp\sqrt{E_{Z}^{2}-(\alpha\kappa)^{2}}$
consists of two bands ($\beta=1,2$) touching at the energy
$E_{min}=-E_{Z}^{2}/4E_{SO}$ for $|\kappa|=E_{Z}/|\alpha|$, as depicted in
Fig.8(b) for the Rashba-dominated regime and in Figs.9(b) and (c) for the weak
and strong Zeeman regime, respectively. When inverting the spectrum in favour
of energy, one obtains

$\kappa_{\eta}(E;\alpha)=\frac{\sqrt{2m^{*}}}{\hbar}\sqrt{-E-2E_{SO}+\eta\sqrt{4EE_{SO}+4E_{SO}^{2}+E_{Z}^{2}}}\hskip
56.9055pt\eta=\pm$ (27)

Figure 9:  The Zeeman dominated regime ($E_{Z}>2E_{SO}$). (a) The spectrum of
the propagating modes. Thick and thin lines correspond to $\pm k_{+}(E)$ and
$\pm k_{-}(E)$, respectively [see Eq.(15)]. (b) The spectrum of the evanescent
modes in the weak Zeeman regime $2E_{SO}<E_{Z}<4E_{SO}$; (c) The spectrum of
the evanescent modes in the strong Zeeman regime ($4E_{SO}<E_{Z}$). Thick and
thin lines correspond to $\pm\kappa_{+}(E)$ and to $\pm\kappa_{-}(E)$ [see
Eq.(27)]. In all plots $E_{min}=-E_{SO}(1+E_{Z}^{2}/4E_{SO}^{2})$.

and the spinors acquire the form

$\displaystyle\xi_{1}(\pm\kappa_{\eta}(E;\alpha))$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}e^{\mp
i\,\arctan(\sinh(\theta(\kappa_{\eta}(E;\alpha))))}\\\ 1\end{array}\right)$
(30) $\displaystyle\xi_{2}(\pm\kappa_{\eta}(E;\alpha))$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}-e^{\pm
i\,\arctan(\sinh(\theta(\kappa_{\eta}(E;\alpha))))}\\\ 1\end{array}\right)$
(33)

where

$\displaystyle\sinh(\theta(\kappa_{\eta}(E;\alpha)))=sgn(\alpha)\,\frac{\sqrt{-4E_{SO}(E+2E_{SO})+\eta\,4E_{SO}\sqrt{4EE_{SO}+4E_{SO}^{2}+E_{Z}^{2}}}}{\left|\sqrt{4EE_{SO}+4E_{SO}^{2}+E_{Z}^{2}}-\eta\,2E_{SO}\right|}$
(34)

Note that, differently from the spinors of the propagating modes, the
eigenstates (30)-(33) identify a spin lying in the $x$-$y$ plane.

Rashba dominated regime ($2E_{SO}>E_{Z}$)
($E_{min}=-E_{SO}\left(1+E_{Z}^{2}/4E_{SO}^{2}\right)$)
energy range propagating modes velocity evanescent modes $E_{min}<E<-E_{Z}$ 4
propagating states $\chi_{1}(\pm k_{-}(E))e^{\pm ik_{-}(E)x}$ $\chi_{1}(\pm
k_{+}(E))e^{\pm ik_{+}(E)x}$ [$k_{\pm}(E)\in E_{1}$ band] $\mp v_{-}(E)$ $\pm
v_{+}(E)$ No evanescent mode $-E_{Z}<E<-\frac{E_{Z}^{2}}{4E_{SO}}$ 2
propagating states $\chi_{1}(\pm k_{+}(E))e^{\pm ik_{+}(E)x}$ [$k_{+}(E)\in
E_{1}$ band] $\pm v_{+}(E)$ $\xi_{1}(\pm\kappa_{+}(E))e^{\pm\kappa_{+}(E)}$
[$\kappa_{+}(E)\in E_{1}$ band] $-\frac{E_{Z}^{2}}{4E_{SO}}<E<E_{Z}$ 2
propagating states $\chi_{1}(\pm k_{+}(E))e^{\pm ik_{+}(E)x}$ [$k_{+}(E)\in
E_{1}$ band] $\pm v_{+}(E)$ $\xi_{2}(\pm\kappa_{+}(E))e^{\pm\kappa_{+}(E)}$
[$\kappa_{+}(E)\in E_{2}$ band] $E_{Z}<E$ 4 propagating modes $\chi_{2}(\pm
k_{-}(E))e^{\pm ik_{-}(E)x}$ $\chi_{1}(\pm k_{+}(E))e^{\pm ik_{+}(E)x}$
[$k_{+}(E)\in E_{1}$ band] [$k_{-}(E)\in E_{2}$ band] $\pm v_{-}(E)$ $\pm
v_{+}(E)$ No evanescent mode

Table 1:  Eigenstates of the NW in the Rashba dominated regime
($2E_{SO}>E_{Z}$). For the propagating modes the wavevectors $k_{\pm}$ are
given in Eqs.(15), the velocities in Eq.(16) and the spinors $\chi_{1,2}$ in
Eqs.(20)-(24), while for the evanescent modes the wavevectors $\kappa_{\pm}$
are given in Eq.(27) and the spinors $\xi_{1,2}$ are given in Eqs.(30, 33).

Weak Zeeman regime ($2E_{SO}<E_{Z}<4E_{SO}$)
energy range propagating modes velocity evanescent modes
$-E_{SO}\left(1+\frac{E_{Z}^{2}}{4E_{SO}^{2}}\right)<E<-E_{Z}$ no propagating
states $\xi_{1}(\pm\kappa_{+}(E))e^{\pm\kappa_{+}(E)x}$
$\xi_{1}(\pm\kappa_{-}(E))e^{\pm\kappa_{-}(E)x}$ [$\kappa_{\pm}(E)\in E_{1}$
band] $-E_{Z}<E<-\frac{E_{Z}^{2}}{4E_{SO}}$ 2 propagating states $\chi_{1}(\pm
k_{+}(E))e^{\pm ik_{+}(E)x}$ [$k_{+}(E)\in E_{1}$ band] $\pm v_{+}(E)$
$\xi_{1}(\pm\kappa_{+}(E))e^{\pm\kappa_{+}(E)}$ [$\kappa_{+}(E)\in E_{1}$
band] $-\frac{E_{Z}^{2}}{4E_{SO}}<E<E_{Z}$ 2 propagating states $\chi_{1}(\pm
k_{+}(E))e^{\pm ik_{+}(E)x}$ [$k_{+}(E)\in E_{1}$ band] $\pm v_{+}(E)$
$\xi_{2}(\pm\kappa_{+}(E))e^{\pm\kappa_{+}(E)}$ [$\kappa_{+}(E)\in E_{2}$
band] $E>E_{Z}$ 4 propagating states $\chi_{2}(\pm k_{-}(E))e^{\pm
ik_{-}(E)x}$ $\chi_{1}(\pm k_{+}(E))e^{\pm ik_{+}(E)x}$ [$k_{-}(E)\in E_{2}$
band] [$k_{+}(E)\in E_{1}$ band] $\pm v_{-}(E)$ $\pm v_{+}(E)$ no evanescent
states

Table 2:  Eigenstates of the NW in the weak Zeeman regime
($2E_{SO}<E_{Z}<4E_{SO}$). The meaning of symbols is the same as in Table 1.

Strong Zeeman regime ($E_{Z}>4E_{SO}$)
energy range propagating modes velocity evanescent modes
$-E_{SO}\left(1+\frac{E_{Z}^{2}}{4E_{SO}^{2}}\right)<E<-\frac{E_{Z}^{2}}{4E_{SO}}$
no propagating mode $\xi_{1}(\pm\kappa_{-}(E))e^{\pm\kappa_{-}(E)x}$
$\xi_{1}(\pm\kappa_{+}(E))e^{\pm\kappa_{+}(E)x}$ [$\kappa_{\pm}(E)\in E_{1}$
band] $-\frac{E_{Z}^{2}}{4E_{SO}}<E<-E_{Z}$ no propagating states
$\xi_{1}(\pm\kappa_{-}(E))e^{\pm\kappa_{-}(E)x}$
$\xi_{2}(\pm\kappa_{+}(E))e^{\pm\kappa_{+}(E)x}$ [$\kappa_{-}(E)\in E_{1}$
band] [$\kappa_{+}(E)\in E_{2}$ band] $-E_{Z}<E<E_{Z}$ 2 propagating states
$\chi_{1}(\pm k_{+}(E))e^{\pm ik_{+}(E)x}$ [$k_{+}(E)\in E_{1}$ band] $\pm
v_{+}(E)$ $\xi_{2}(\pm\kappa_{+}(E))e^{\pm\kappa_{+}(E)}$ [$\kappa_{+}(E)\in
E_{2}$ band] $E>E_{Z}$ 4 propagating states $\chi_{2}(\pm k_{-}(E))e^{\pm
ik_{-}(E)x}$ $\chi_{1}(\pm k_{+}(E))e^{\pm ik_{+}(E)x}$ [$k_{-}(E)\in E_{2}$
band] [$k_{+}(E)\in E_{1}$ band] $\pm v_{-}(E)$ $\pm v_{+}(E)$ no evanescent
states

Table 3:  Eigenstates of the NW in the strong Zeeman regime ($E_{Z}>4E_{SO}$).
The meaning of symbols is the same as in Table 1.

Three possible regimes. Depending on the local value of $E_{SO}$ and $E_{Z}$,
the NW region can be in three possible regimes, namely Rashba-dominated
($2E_{SO}>E_{Z}$), weak Zeeman regime ($2E_{SO}<E_{Z}<4E_{SO}$) and strong
Zeeman regime ($E_{Z}>4E_{SO}$). For each of these regimes, there are various
energy ranges where the eigenstates acquire different expressions. In Table 1
we have summarized the result for the Rashba-dominated regime. In particular,
propagating states exist for $E>E_{min}$, as can be see from Fig.8(a). Their
wavevectors are given by Eq.(15), the spinors $\chi_{1}$ and $\chi_{2}$,
related to the lower and upper band, respectively, are given in Eqs.(20)-(24),
while the velocities are given in Eq.(16). In contrast, evanescent modes exist
for $|E|<E_{Z}$, as shown in Fig.9(a), their wavevectors are given in
Eqs.(27), and the related spinors, related to the upper and lower band, are
given in Eqs.(30)-(33).
As far as the Zeeman-dominated regime is concerned ($E_{Z}>2E_{SO}$),
propagating modes exist for $E>-E_{Z}$ (see Fig.8(b)), while evanescent modes
exist for $E_{min}<E<E_{Z}$. However, the explicit expression and type
(upper/lower band) of the NW states also depends whether the NW is in the weak
Zeeman subregime ($2E_{SO}<E_{Z}<4E_{SO}$) or in the strong Zeeman subregime
($4E_{SO}<E_{Z}$), as shown in Figs.9(b) and (c). The expressions of
eigenstates in the weak and strong Zeeman regimes are summarized in Tables 2
and 3, respectively.

##  Appendix B Details about the evaluation of the Scattering Matrix and the
Transmission coefficient

Here we provide the technical details concerning the determination of the
transmission coefficient for the system with inhomogeneous RSOC setup, whence
the conductance is straightforwardly determined from Eq.(10). To this purpose,
one first has compute the Transfer Matrix. We exploit the fact that, since the
system is mesoscopic, energy is conserved, and we can write the general
solution as a superposition over the energy $E$ of stationary solutions
$\hat{\Psi}_{E}(x,t)=\hat{\Psi}_{E}(x)\,e^{-iEt/\hbar}$. In turn, in the
$j$-th region ($j=L,R=0,1$) the field $\hat{\Psi}_{E}(x)$ is expressed as a
superposition of all possible modes (propagating and evanescent)
characterizing the region at a fixed energy $E$. These modes can be selected
from Tables 1, 2 and 3, depending on the parameter regime (Rashba-dominated,
weakly or strongly Zeeman dominated) and the energy sub-range.

The single interface case. As an example, let us consider the case of one
single interface at $x=0$, and assume that the left side is in the Rashba-
dominated regime, while the right side is in the strongly Zeeman-dominated
regime. Moreover, let us consider an energy value inside the magnetic gap,
$|E|<E_{Z}$. As can be seen from Tables 1 and 3, in this energy range there
are two propagating modes and two evanescent modes on each side of the
interface. Then, a scattering state has the form

$\hat{\Psi}_{E}(x)=\frac{1}{\sqrt{2\pi\hbar}}\left\\{\begin{array}[]{lcl}\begin{array}[]{l}\frac{1}{\sqrt{v_{+}(E;\alpha_{L})}}\bigg{(}\hat{a}_{1E}^{(L)}\chi_{1}(k_{+}(E;\alpha_{L}))\,e^{ik_{+}(E;\alpha_{L})x}+\hat{b}_{1E}^{(L)}\chi_{1}(-k_{+}(E;\alpha_{L}))\,e^{-ik_{+}(E;\alpha_{L})x}\bigg{)}+\\\
\hskip 14.22636pt+\hat{f}_{\beta_{E}E}^{(L)}\
\xi_{\beta_{E}}(\kappa_{+}(E;\alpha_{L}))e^{\kappa_{+}(E;\alpha_{L})\,x}\end{array}&&x<0\\\
&&\\\
\begin{array}[]{l}\frac{1}{\sqrt{v_{+}(E;\alpha_{R})}}\,\bigg{(}\hat{b}_{1E}^{(R)}\chi_{1}(k_{+}(E;\alpha_{R}))\,e^{ik_{+}(E;\alpha_{R})x}+\hat{a}_{1E}^{(R)}\chi_{1}(-k_{+}(E;\alpha_{R}))\,e^{-ik_{+}(E;\alpha_{R})x}\bigg{)}+\\\
\hskip 14.22636pt+\hat{g}_{2E}^{(R)}\
\xi_{2}(-\kappa_{+}(E;\alpha_{R}))e^{-\kappa_{+}(E;\alpha_{R})\,x}\end{array}&&x>0\end{array}\right.$
(35)

where $\hat{a}_{1E}^{(L/R)}$ denotes the fermionic operator related to the
propagating mode incoming from the left/right side (originating from the lower
band $\beta=1$), $\hat{b}_{1E}^{(L/R)}$ describes the one related to the
propagating mode outgoing to the left/right (also originating from the lower
band $\beta=1$), with $\chi_{1}$ being the related spinor given in Eq.(20).
Similarly $\hat{f}_{\beta_{E}E}^{(L)}$ and $\hat{g}_{2E}^{(R)}$ are the
evanescent mode operators and $\xi_{1},\xi_{2}$ denote the related spinors
given in Eqs.(30)-(33). For the Rashba-dominated side on the left, from Table
1 one sees that the evanescent mode originates from the lower complex band
($\beta_{E}=1$) for $E<-E_{Z}^{2}/4E_{SO}$ and from the upper complex band
($\beta_{E}=2$) for $E>-E_{Z}^{2}/4E_{SO}$, whereas for the strongly Zeeman-
dominated side on the right we deduce from Table 3 that the evanescent mode
originates from the upper complex band $(\beta=2$) on the Zeeman-side.
Inserting Eq.(35) into the boundary conditions (9) at the interface $x=0$, the
latter can be rewritten in a matrix form as

$M_{E}^{(L)}(0)\begin{pmatrix}\hat{a}_{1E}^{(L)}\\\ \hat{b}_{1E}^{(L)}\\\
\hat{f}_{\beta_{E}E}^{(L)}\\\
0\end{pmatrix}=M_{E}^{(R)}(0)\begin{pmatrix}\hat{b}_{1E}^{(R)}\\\
\hat{a}_{1E}^{(R)}\\\ 0\\\ \hat{g}_{2E}^{(R)}\end{pmatrix}$ (36)

where $M^{(L/R)}_{E}(0)\in\mathbb{C}^{4\times 4}$ are two $4\times 4$ complex
boundary matrices at energy $E$ given by

$M_{E}^{(L)}(0)=\begin{pmatrix}\frac{\chi_{1}(k_{+})}{\sqrt{v_{+}}}&\frac{\chi_{1}(-k_{+})}{\sqrt{v_{+}}}&\xi_{2}(\kappa_{+})&\xi_{2}(-\kappa_{+})\\\
&&&\\\ i\frac{(k_{+}-\sigma_{L}k_{SO,L})\
\chi_{1}(k_{+})}{\sqrt{v_{+}}}&-i\frac{(k_{+}+\sigma_{L}k_{SO,L})\
\chi_{1}(-k_{+})}{\sqrt{v_{+}}}&(\kappa_{+}-i\sigma_{L}k_{SO,L})\xi_{2}(\kappa_{+})&-(\kappa_{+}+i\sigma_{L}k_{SO,L})\xi_{2}(-\kappa_{+})\end{pmatrix}$
(37)

and

$M_{E}^{(R)}(0)=\begin{pmatrix}\frac{\chi_{1}(k_{+})}{\sqrt{v_{+}}}&\frac{\chi_{1}(-k_{+})}{\sqrt{v_{+}}}&\xi_{\beta_{E}}(\kappa_{+})&\xi_{\beta_{E}}(-\kappa_{+})\\\
&&&\\\ i\frac{(k_{+}-\sigma_{R}k_{SO,R})\
\chi_{1}(k_{+})}{\sqrt{v_{+}}}&-i\frac{(k_{+}+\sigma_{R}k_{SO,R})\
\chi_{1}(-k_{+})}{\sqrt{v_{+}}}&(\kappa_{+}-i\sigma_{R}k_{SO,R})\xi_{\beta_{E}}(\kappa_{+})&-(\kappa_{+}+i\sigma_{R}k_{SO,R})\xi_{\beta_{E}}(-\kappa_{+})\end{pmatrix}$
(38)

where $\chi_{1}$, $\xi_{1}$ and $\xi_{2}$ are the $2\times 1$ spinors in
Eq.(20), (30)-(33), respectively, and the dependence of $\kappa_{+}$ and
$v_{+}$ on $\alpha_{L}$ and $E$ [in Eq.(37)] and on $\alpha_{R}$ and $E$ [in
Eq.(38)] has been omitted for simplicity. Moreover, we have denoted
$\sigma_{R,L}=\mbox{sgn}(\alpha_{R,L})$ and we have used the fact that the
second Eq.(9) at the interface $x=0$ can be rewritten as

$\partial_{x}\hat{\Psi}(0^{-})=\partial_{x}\hat{\Psi}(0^{+})-i\left(\sigma_{R}k_{SO,R}-\sigma_{L}k_{SO,L}\right)\sigma_{z}\hat{\Psi}(0)$
(39)

in terms of the spin-orbit wavevectors (14). From Eq.(36), the transfer matrix
relating the operators on the right side of interface to the operators on the
left side,

$\begin{pmatrix}\hat{b}_{1E}^{(R)}\\\ \hat{a}_{1E}^{(R)}\\\ 0\\\
\hat{g}_{2E}^{(R)}\end{pmatrix}=W_{E}\begin{pmatrix}\hat{a}_{1E}^{(L)}\\\
\hat{b}_{1E}^{(L)}\\\ \hat{f}_{\beta_{E}E}^{(L)}\\\ 0\end{pmatrix}\quad,\hskip
56.9055pt|E|<E_{Z}$ (40)

is straightforwardly found to be $W_{E}=[M_{E}^{(R)}(0)]^{-1}M_{E}^{(L)}(0)$.
Since Eqs.(40) represent four constraints for the six unknown operators, one
can express four of them in terms of the incoming operators
$\hat{a}_{1E}^{(L)}$ and $\hat{a}_{1E}^{(R)}$. In particular, one can thus
compute the Scattering Matrix $S_{E}$, which relates the outgoing propagating
modes to the incoming ones

$\left(\begin{array}[]{c}\hat{b}_{1E}^{(L)}\vspace{3pt}\\\
\hat{b}_{1E}^{(R)}\end{array}\right)=\underbrace{\begin{pmatrix}r_{E}&t^{\prime}_{E}\\\
&\\\
t_{E}&r^{\prime}_{E}\end{pmatrix}}_{S_{E}}\left(\begin{array}[]{c}\hat{a}_{1E}^{(L)}\vspace{3pt}\\\
\hat{a}_{1E}^{(R)}\end{array}\right)$ (41)

The transmission coefficient is computed as
$T(E)=|t_{E}|^{2}=|t_{E}^{\prime}|^{2}$ and the linear conductance in the
considered range $|E_{F}|<E_{Z}$ is obtained from Eq.(10).

In a similar way, one can find the transmission in the energy range $E>E_{Z}$
above the gap. In such a case, on each side of the interface there are two
propagating modes and no evanescent mode on each side, so that the Scattering
state solution now acquires the form

$\hat{\Psi}_{E}(x)=\frac{1}{\sqrt{2\pi\hbar}}\left\\{\begin{array}[]{lcl}\begin{array}[]{l}\frac{1}{\sqrt{v_{+}(E;\alpha_{L})}}\bigg{(}\hat{a}_{1E}^{(L)}\chi_{1}(k_{+}(E;\alpha_{L}))\,e^{ik_{+}(E;\alpha_{L})x}+\hat{b}_{1E}^{(L)}\chi_{1}(-k_{+}(E;\alpha_{L}))\,e^{-ik_{+}(E;\alpha_{L})x}\bigg{)}+\\\
\hskip
14.22636pt+\frac{1}{\sqrt{v_{-}(E;\alpha_{L})}}\bigg{(}\hat{a}_{2E}^{(L)}\chi_{2}(k_{-}(E;\alpha_{L}))\,e^{ik_{-}(E;\alpha_{L})x}+\hat{b}_{2E}^{(L)}\chi_{2}(-k_{-}(E;\alpha_{L}))\,e^{-ik_{-}(E;\alpha_{L})x}\bigg{)}\end{array}&&x<0\\\
&&\\\
\begin{array}[]{l}\frac{1}{\sqrt{v_{+}(E;\alpha_{R})}}\,\bigg{(}\hat{b}_{1E}^{(R)}\chi_{1}(k_{+}(E;\alpha_{R}))\,e^{ik_{+}(E;\alpha_{R})x}+\hat{a}_{1E}^{(R)}\chi_{1}(-k_{+}(E;\alpha_{R}))\,e^{-ik_{+}(E;\alpha_{R})x}\bigg{)}+\\\
\hskip
14.22636pt+\frac{1}{\sqrt{v_{-}(E;\alpha_{R})}}\bigg{(}\hat{b}_{2E}^{(R)}\chi_{2}(k_{-}(E;\alpha_{R}))\,e^{ik_{-}(E;\alpha_{R})x}+\hat{a}_{2E}^{(R)}\chi_{2}(-k_{-}(E;\alpha_{R}))\,e^{-ik_{-}(E;\alpha_{R})x}\bigg{)}\end{array}&&x>0\end{array}\right.$
(42)

Inserting Eq.(42) into the boundary conditions (9) and proceeding as outlined
above, one can find the Transfer Matrix

$\left(\begin{array}[]{c}\hat{a}_{1E}^{(R)}\vspace{3pt}\\\
\hat{b}_{1E}^{(R)}\vspace{3pt}\\\ \hat{a}_{2E}^{(R)}\vspace{3pt}\\\
\hat{b}_{2E}^{(R)}\end{array}\right)=W_{E}\left(\begin{array}[]{c}\hat{a}_{1E}^{(L)}\vspace{3pt}\\\
\hat{b}_{1E}^{(L)}\vspace{3pt}\\\ \hat{a}_{2E}^{(L)}\vspace{3pt}\\\
\hat{b}_{2E}^{(L)}\end{array}\right)$ (43)

whence the Scattering matrix $S_{E}$, which is now a $4\times 4$ matrix, can
straightforwardly be obtained

$\begin{pmatrix}\hat{b}_{1E}^{(L)}\vspace{3pt}\\\
\hat{b}_{2E}^{(L)}\vspace{3pt}\\\ \hat{b}_{1E}^{(R)}\vspace{3pt}\\\
\hat{b}_{2E}^{(R)}\end{pmatrix}=\underbrace{\begin{pmatrix}\mathsf{r}_{E}&\mathsf{t}^{\prime}_{E}\\\
&\\\
\mathsf{t}_{E}&\mathsf{r}^{\prime}_{E}\end{pmatrix}}_{S_{E}}\begin{pmatrix}\hat{a}_{1E}^{(L)}\vspace{3pt}\\\
\hat{a}_{2E}^{(L)}\vspace{3pt}\\\ \hat{a}_{1E}^{(R)}\vspace{3pt}\\\
\hat{a}_{2E}^{(R)}\end{pmatrix}$ (44)

with $\mathsf{t},\mathsf{t}^{\prime}$ and $\mathsf{r},\mathsf{r}^{\prime}$
denoting the $2\times 2$ transmission and reflection blocks. The transmission
coefficient is now obtained as
$T(E)={\rm}{tr}[\mathsf{t}_{E}^{\dagger}\mathsf{t}_{E}]={\rm}{tr}[\mathsf{t}^{\prime\,\dagger}_{E}\mathsf{t}^{\prime}_{E}]$.

Generalization to multiple interfaces. It is straightforward to generalise the
same approach to the case of $N$ interfaces. Let us consider an interface
located at $x_{j}$ separating a region with RSOC $\alpha_{j}$ (on the left
side) from a region with RSOC $\alpha_{j+1}$ on the right, as depicted in
Fig.10. Similarly to the single interface problem [see Eqs.(35) and (42)], on
each side of the interface the electron field operator $\hat{\Psi}_{E}(x)$ is
written as an expansion of (at most) four modes, which can be propagating or
evanescent, depending on the specific regime of that region and on the energy
range, according to Tables 1, 2, 3. Denoting by $\hat{c}_{E}^{(j)}$,
$\hat{d}_{E}^{(j)}$, $\hat{f}_{E}^{(j)}$, and $\hat{g}_{E}^{(j)}$ the
operators related to the above four modes on the $j$-th region and the
boundary conditions (9) can be rewritten in a matrix form

$M_{E}^{(j)}(x_{j})\left(\begin{array}[]{c}\hat{c}_{E}^{(j)}\vspace{3pt}\\\
\hat{d}_{E}^{(j)}\vspace{3pt}\\\ \hat{f}_{E}^{(j)}\vspace{3pt}\\\
\hat{g}_{E}^{(j)}\end{array}\right)=M_{E}^{(j+1)}(x_{j})\left(\begin{array}[]{c}\hat{c}_{E}^{(j+1)}\vspace{3pt}\\\
\hat{d}_{E}^{(j+1)}\vspace{3pt}\\\ \hat{f}_{E}^{(j+1)}\vspace{3pt}\\\
\hat{g}_{E}^{(j+1)}\end{array}\right)$ (45)

Figure 10:  Generalization of the transfer matrix to multiple interfaces. The
total transfer matrix is computed as the product of the transfer matrices
across each interface.

The transfer matrix across the interface at $x_{j}$ is obtained as
$W_{E}(x_{j})=[M_{E}^{(j+1)}(x_{j})]^{-1}M_{E}^{(j)}(x_{j})$. Then, by
observing that the operators on right of the interface $x_{j}$ are the very
same operators as the ones on the left of the interface at $x_{j+1}$ (see
Fig.10), the total transfer matrix $W_{E}$ across the entire inhomogeneous
scattering region

$\left(\begin{array}[]{c}\hat{c}_{E}^{(N)}\vspace{3pt}\\\
\hat{d}_{E}^{(N)}\vspace{3pt}\\\ \hat{f}_{E}^{(N)}\vspace{3pt}\\\
\hat{g}_{E}^{(N+1)}\end{array}\right)=W_{E}\left(\begin{array}[]{c}\hat{c}_{E}^{(0)}\vspace{3pt}\\\
\hat{d}_{E}^{(0)}\vspace{3pt}\\\ \hat{f}_{E}^{(0)}\vspace{3pt}\\\
\hat{g}_{E}^{(0)}\end{array}\right)$ (46)

can be easily found as the product $W_{E}=\prod_{j=0}^{N-1}W_{E}(x_{N-j-1})$.
For instance, for the doubly gated NW problem of Sec.V, the total transfer
matrix is the product $W_{E}=W_{E}(x_{1})W_{E}(x_{0})$ of the two single-
interface transfer matrices and contains the distance $d=x_{1}-x_{0}$ between
the two interfaces at $x_{0}$ and $x_{1}$. Finally, one observes that in the
outer region $j=0$ (left lead) the modes propagating rightwards (leftwards)
are incoming to (outgoing from) the scattering region and can thus be
relabelled as $\hat{a}^{(L)}_{\beta E}$ ($\hat{b}^{(L)}_{\beta E}$). Moreover,
in order to ensure the normalizability of the state, only the evanescent mode
decaying to the left is admitted, like in Eq.(35). Effectively, this means
that in Eq.(45), the operator related to the mode diverging on the left has to
be set to zero, like in Eq.(40). Similarly, in the outer region $j=N$ (right
lead) the modes propagating leftwards (rightwards) are incoming (outgoing) and
are relabelled as $\hat{a}^{(R)}_{\beta E}$ ($\hat{b}^{(R)}_{\beta E}$), while
only the evanescent mode decaying on the right, if any, is admitted.
Implementing these aspects in Eq.(45), and re-expressing the outgoing mode
operators $\hat{b}^{(L)}_{\beta E},\hat{b}^{(R)}_{\beta E}$ in terms of the
incoming ones $\hat{a}^{(L)}_{\beta E},\hat{a}^{(R)}_{\beta E}$, one finds the
Scattering Matrix similarly to what was done in Eqs.(41) and Eqs.(44) for the
single interface case.

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