# Spin-helical detection in a semiconductor quantum device with ferromagnetic
contacts

Zedong Yang  Paul A. Crowell  Vlad S. Pribiag<EMAIL_ADDRESS>School of
Physics and Astronomy, University of Minnesota

(August 29, 2024)

###### Abstract

Spin-helical states, which arise in quasi-one-dimensional (1D) channels with
spin-orbital (SO) coupling, underpin efforts to realize topologically-
protected quantum bits based on Majorana modes in semiconductor nanowires.
Detecting helical states is challenging due to non-idealities present in real
devices. Here we show by means of tight-binding calculations that by using
ferromagnetic contacts it is possible to detect helical modes with high
sensitivity even in the presence of realistic device effects, such as quantum
interference. This is possible because of the spin-selective transmission
properties of helical modes. In addition, we show that spin-polarized contacts
provide a unique path to investigate the spin texture and spin-momentum
locking properties of helical states. Our results are of interest not only for
the ongoing development of Majorana qubits, but also as for realizing possible
spin-based quantum devices, such as quantum spin modulators and interconnects
based on spin-helical channels.

Semiconductor nanowire-superconductor hybrid systems are a highly promising
materials platform for investigating Majorana zero modes (MZMs)[1, 2, 3, 4, 5,
6, 7, 8], as well as spin transport in quantum channels [9, 10]. Because MZMs
arise in odd-parity topological superconductors, realizing them in hybrid
semiconductor nanowire-superconductor devices requires removing the spin
degeneracy to achieve an effective odd-parity state. This can be achieved by
opening a spin-helical gap in the semiconductor channel, thereby introducing a
correlation between spin and momentum in certain energy ranges of the nanowire
subbands [1, 2, 11, 12, 13]. A large helical gap requires strong spin-orbit
(SO) materials with long mean free paths, typically InSb or InAs nanowires, in
conjunction with a Zeeman interaction, usually achieved by using an external
magnetic field. Despite the rapid pace of research on MZMs in nanowire-
superconductor devices in recent years, there remain important open questions
about the individual components. For example, a clear detection of the spin-
helical state, the necessary precursor to Majorana modes in nanowire systems,
is currently lacking. Conventional normal metal leads can in principle be used
to investigate quantized conductance patterns that should serve as signatures
of a helical gap, however imperfect quantization and quantum interference
effects present in real devices lead to complex conductance patterns that
cannot be unambiguously ascribed to a helical gap [14, 15, 16, 17]. Normal
metal contacts are also not suitable for studying the rich spin texture that
makes the helical state relevant for realizing MZMs and for investigating
other quantum spin-based effects. Recently, there has been growing interest in
integrating ferromagnetic (FM) elements with high SO coupling materials to
realize topological states [18, 19, 20] and to demonstrate quantum transport
effects with spin-polarized currents, including spin-filtering in the regime
of few quantum modes and possible spin-dependent signatures of the helical
state [9, 10]. However, an approach for reliably detecting spin-helical states
and exploring their spin texture is currently lacking. In this work, we model
quasi-1D quantum devices semiconductor devices with strong SO and FM contacts
using numerical tight-binding calculations. We find that using FM contacts can
improve substantially the detectability of the helical state and circumvent
limitations that arise naturally from quantum interference effects in real
devices with normal contacts. Moreover, this approach can also reveal
information about spin states in quantum wires, which is inaccessible with
normal contacts. These results can help guide the development and
interpretation of experiments on MZMs and quantum spintronics devices based on
semiconductors with strong SO coupling, and could help enable future on-chip
quantum interconnects for modular quantum computation [21, 22, 23, 24].

Figure 1: (a) Device structure showing the semiconductor (blue lattice) and
magnetic leads (red lattice). Substrate and metallic leads are represented by
the green and gray blocks respectively. (b-e) The lowest subbands of the 1D
strip with (b) no SO coupling or Zeeman splitting ($\alpha=0$ and $B_{\rm
x}=0$), (c) with Zeeman splitting from a finite $B_{x}$ ($g\mu_{B}B_{\rm
x}=0.02t$), (d) with SO coupling (effective $\vec{B}_{\rm SO}$ along -z
direction), and (e) with both finite Zeeman splitting and SO coupling, showing
helical gap. (f-g) Conductance ($G$) vs. Fermi energy ($E$) for the three
subband structures from (b), (c) and (e) respectively. (f) $G$ vs. $E$
considering normal leads with spin-mixing. (g) $G$ vs. $E$ in the absence of
spin-mixing in the leads. Traces in (f) and (g) are offset vertically by
$2e^{2}/h$ for clarity.

To investigate spin-dependent transport in the helical regime, we set up a
quasi-1D nanostrip with magnetic contacts on a square lattice (Fig. 1(a)) and
compute the transmission matrix using KWANT [25]. The Hamiltonian is described
by:

$H=\frac{p^{2}}{2m^{\ast}}-i\alpha_{\rm R}\sigma_{\rm
z}\frac{\partial}{\partial x}-\frac{g\mu_{\rm
B}}{2}{\vec{\sigma}}\cdot{\vec{B}},$ (1)

where the second term describes the SO coupling, with $\alpha_{\rm R}$ the
Rashba spin-orbit coefficient and ${\sigma_{\rm z}}$ the $z$-component of the
Pauli matrix, and the third term describes the Zeeman coupling to an applied
magnetic field $\vec{B}$. The Rashba spin-orbit effective field $\vec{B}_{\rm
SO}=\alpha_{\rm R}\vec{k}\times\vec{E}$ points along the $z$-direction, since
$\vec{k}$ is along $+x$, the electron propagation direction, and the electric
field $\vec{E}$ is along $-y$, normal to the substrate. In our simulation, we
implement the scattering region [blue lattice in Fig. 1(a)] as a $50\times 10$
lattice mesh with lattice constant $a=10$ nm. It is noteworthy that although
the scattering region consists of a two-dimensional lattice, the system is
still effectively 1D because the electronic states are quantized into discrete
levels along the strip width ($z$-direction). The resulting subbands are
parabolic and behave as in a 1D conductor (see also Supplementary Section I
[26]). With this model, the hopping energy
$t=\frac{\hbar^{2}}{2m_{e}^{\ast}a^{2}}=27.2$meV, where
$m_{e}^{\ast}=0.014m_{e}$ is the effective mass of electrons in InSb, a
typical semiconductor candidate for hybrid Majorana devices. This corresponds
to a nanostrip diameter of approximately 100 nm and a lead separation of
approximately 500 nm, typical dimensions for experiments on InSb and InAs
nanowires. To open a helical gap in the scattering region, we consider a
magnetic field $B_{x}$ applied along the $x$-direction, with resulting Zeeman
splitting $\Delta E_{Z}=g\mu B_{x}=0.02t$, and a Rashba coefficient
$\alpha_{R}=0.4ta$. This corresponds to an applied field $B\approx 200$ mT and
a spin-orbit energy $E_{so}\approx 1$ meV, consistent with experimental values
[3, 27]. Fig.1(b)-(e) show the nanostrip subbands computed for different
combinations of magnetic fields and Rashba SO coupling. We focus here on the
first subbands for clarity, but additional subbands are plotted in
Supplemental Section I [26]. The key effects remain the same as more subbands
are considered.

With the Hamiltonian set up in the lattice, we can calculate the transmission
matrix of the system. We compute the transmission rate from the left lead to
the right lead, which yields the conductance, $G$ after multiplying with a
factor of $e^{2}/h$. In the absence of both external field and Rashba
coupling, the subbands are parabolic (Fig. 1(b)) and the conductance increase
step-wisely in $2e^{2}/h$ increments (blue curve in Fig. 1(f)). When a small
finite field is applied, the subband degeneracy is lifted by the Zeeman
interaction leading to a splitting of size $E_{\rm Z}$ (Fig. 1(c)) and an
intermediate $e^{2}/h$ conductance plateau develops as a result (green curve
in Fig. 1(f)). In contrast, including Rashba SO coupling introduces the
effective magnetic field $\vec{B}_{\rm SO}$ along the -$z$ direction, which
shifts the subbands horizontally leading to a $k=0$ crossing at the spin-
orbital energy, $E_{\rm SO}$ (Fig. 1(d)). In the presence of both finite
$\vec{B}_{\rm SO}$ and $B_{\rm x}$ a helical gap of size $E_{\rm H}$ is
opened, with a pre-helical gap bump feature of size $E_{\rm B}$ (Fig. 1(e)).
This results in the re-entrant conductance pattern $0\rightarrow
2e^{2}/h\rightarrow e^{2}/h\rightarrow 2e^{2}/h$ (red curve in Fig. 1(f)),
providing a theoretical fingerprint of the helical state in measurements with
normal contacts.

The conductance traces shown in Fig. 1(f) were obtained by considering the
same Hamiltonian in the leads as in the scattering region. However, this
typical treatment is not suitable for the case of FM leads. We model each FM
contact as two independent leads, each of which can only emit or accept one
spins species, either up or down [28]. The spin polarization, $P$, of the
source and drain is assumed to take on the bulk value for the ferromagnet.
Within the KWANT framework, FMs can be implemented by attaching one pair of
leads at each end of the scattering region, which we label L$\uparrow$,
L$\downarrow$ , R$\uparrow$, R$\downarrow$, for the up-spin ($\uparrow$) and
down-spin ($\downarrow$) left (L) (source) and right (R) (drain) leads,
respectively. This allows computation of the conductances $G_{ss^{\prime}}$
between spin species $s^{\prime}$ and $s$ across the device, where
$s^{\prime}$ and $s$ can be either $\uparrow$ or $\downarrow$. For the case of
spin-unpolarized FM leads, which corresponds to both spin components $s$ and
$s’$ having equal weights, the conductance traces are shown in Fig. 1(g).
Interestingly, a softening of the conductance plateaus occurs due to
conductance oscillations superimposed on the plateaus. These oscillations
arise from quantum interference of partially-reflected electrons as a result
of the energy mismatch between the semiconducting scattering region, in which
spin-mixing is allowed, and the FM leads, in which spin-mixing is not allowed
under the independent-spins FM contacts model [28]. Importantly, these soft
plateaus are reproduced within the typical treatment of normal leads if the
spin-mixing Zeeman term vanishes in the leads, which serves as a useful test
case validating our model for FM leads (more details of the implementation of
the FM leads are available in Supplemental Section II [26]). We emphasize that
considering different spin-mixing properties for leads and semiconductor is
not only relevant for FM leads, but also enables a more realistic description
of experiments with normal leads. For example, the leads of a typical MZM
device are made of a metal whose Landé $g$-factor (of order 2) is
substantially smaller than that of the typical semiconductors (InAs or InSb,
with $g$-factors of order 10-50) and thus the Zeeman splitting for a given
applied field is substantially smaller in the leads. We thus expect that the
red trace in Fig. 1(g) provides a more accurate depiction of the helical gap
fingerprint that can be expected in the best devices with normal contacts than
the commonly-considered idealized case of Fig. 1(f).

Figure 2: (a, b) Conductance of the spin-conserving channels as a function of
bias window $-eV_{\rm sd}$ and Fermi energy $E$. (a) Spin-down to spin-down
conductance, $G_{\rm\downarrow\downarrow}$, (b) spin-up to spin-up
conductance, $G_{\rm\uparrow\uparrow}$. White dashed lines indicate the
boundaries between different regions as defined in the main text. Green dashed
lines denote the linecuts displayed in (c)-(e). The corresponding energy
ranges of the pre-helical bump ($E_{\rm B}$) and the helical gap ($E_{\rm H}$)
are labelled. (c)-(e) Vertical linecuts corresponding to $V_{\rm sd}$ values
of 0, $-0.07t$, and $-0.14t$, showing (c) minor-spin conductance suppression
effects (small bias), (d) spin splitting effects (intermediate bias) and (e)
spin-conductance overlap (large bias) respectively. Insets in (c)-(e): subband
diagrams showing the relative position of the electrochemical potentials in
the leads (solid lines) and $E$ in the semiconductor scattering region (dotted
lines).

Recent experimental work [9, 10] has demonstrated that spin-filtering can be
achieved in strong SO nanowire devices with magnetic contacts, however a
comprehensive understanding and description of the spin transport picture in
such devices is still missing. Below, we show that FM contacts can provide
access to key information about the helical state and its spin texture, which
cannot be obtained in devices with normal contacts. We thus expect that these
insights will be of high importance for guiding future experiments. For
simplicity, we begin by considering FM leads with 100% spin polarization
($P=1$) and subsequently discuss the case of intermediate values of $P$. We
begin by examining the role of the source-drain bias voltage ($V_{\rm sd}$) on
transport across the device. $V_{\rm sd}$ effects on conductance are important
because spin is coupled to the net electron propagation direction in the
helical gap, i.e. the group velocity. With helical spin-momentum coupling,
changes in bias magnitude and sign are expected to influence spin-filtering
effects, similar to the case of topological insulator devices [29, 30, 31]. In
the presence of an applied bias, the voltages on the two leads are given by:
$\absolutevalue{e}V_{\rm L}=E+\absolutevalue{e}V_{\rm sd}/2$ and
$\absolutevalue{e}V_{\rm R}=E-\absolutevalue{e}V_{\rm sd}/2$, where $\mu_{\rm
L}=\absolutevalue{e}V_{\rm L}$ and $\mu_{\rm R}=\absolutevalue{e}V_{\rm R}$
are the electrochemical potentials of the left and right leads respectively.
We first investigate the spin-filtering effects on the spin-conserving
conductance channels $G_{\rm\uparrow\uparrow}$, and
$G_{\rm\downarrow\downarrow}$, which can be experimentally measured by
aligning the magnetizations in the two leads parallel to each other. Fig. 2(a)
and Fig. 2(b) display the corresponding conductance maps as a function of
$V_{\rm sd}$ and $E$. Individual conductance traces for different values of
$V_{\rm sd}$ are plotted in Fig. 2(c-e). At zero bias (Fig. 2(c)) the
conductance in the $G_{\rm\downarrow\downarrow}$ channel displays a stepwise
increase by $e^{2}/h$ as $E$ reaches the bottom of the subbands, then remains
relatively constant with increasing energy. In contrast, the
$G_{\rm\uparrow\uparrow}$ channel displays a strong dip to $\sim 0.3e^{2}/h$.

As a result of the helical gap, the nanowire acts as a spin filter, owing to
the spin-imbalance between right-going modes (group velocity $v_{\rm
g}=dE/dk>0$), which are spin-$\downarrow$, and the left-going modes ($v_{\rm
g}=dE/dk<0$), which are spin-$\uparrow$ (Fig. 1(e)). Two energy scales are
important to clearly illustrate the spin-filtering transport: $E_{\rm B}$, the
pre-helical bump size, and $E_{\rm H}$, the helical gap size. This gives rise
to three distinct regions based on the value of $V_{\rm sd}$: $|eV_{\rm
sd}|/2<E_{\rm B}$ (Region I), $E_{\rm B}<|eV_{\rm sd}|/2<E_{\rm H}+E_{\rm B}$
(Region II), and $|eV_{\rm sd}|/2>E_{\rm H}+E_{\rm B}$ (Region III). The spin-
filtering conductance is readily apparent when comparing
$G_{\rm\uparrow\uparrow}$ with $G_{\rm\downarrow\downarrow}$ (Fig. 2). In
Region I, the bias window is small enough to resolve both the pre-helical bump
feature and the helical gap as $E$ is varied. In this region
$G_{\rm\uparrow\uparrow}$ displays a non-monotonic behavior because the right-
going spin-$\uparrow$ mode is not transmitted when $E$ lies in the helical gap
[green arrow in Fig. 2(c)]. When $V_{\rm sd}$ is increased further and Region
II becomes accessible, the pre-helical bump feature in
$G_{\rm\uparrow\uparrow}$ [orange arrow in Fig. 2(c)] drops significantly
below $e^{2}/h$. This occurs because once $\mu_{\rm L}$ reaches the subband
edge from below, $E$ lies in the helical gap, where the right-going
spin-$\uparrow$ modes cannot propagate. However, instead of the
$e^{2}/h\rightarrow 0.3e^{2}/h\rightarrow e^{2}/h$ conductance evolution seen
in $G_{\rm\uparrow\uparrow}$ in Region I, Region II shows a prominent
difference between $G_{\rm\uparrow\uparrow}$ and $G_{\rm\downarrow\downarrow}$
as the conductance rises above 0, indicative of spin-filtering. When the bias
window is increased further (Region III), for similar reasons, once $\mu_{\rm
L}$ reaches the bottom of the subbands (Fig. 2(e) inset), $E$ crosses above
the helical gap and consequently the $G_{\rm\uparrow\uparrow}$ trace has a
stepwise $e^{2}/h$ increment. It is useful to compare the above results for FM
leads to the conventional normal-contact conductance measurement technique.
Whereas with normal contacts it is necessary to consecutively resolve both the
pre-helical-bump ($2e^{2}/h$ conductance) and the helical gap (close to
$e^{2}/h$ conductance) with a small bias window in Region I, the use of FM
contacts leverages the spin-filtering properties of the helical gap to enable
its clear detection over a large range of $V_{\rm sd}$ extending over both
Region I and Region II.

Figure 3: (a, b) Conductance of the spin-flip channels as a function of bias
window $-eV_{\rm sd}$ and energy. (a) Spin-up to spin-down
($G_{\rm\uparrow\downarrow}$). (b) Spin-down to spin-up
($G_{\rm\downarrow\uparrow}$). White dashed lines mark the boundaries of the
regions defined as in Figure 2. Green dash lines denote the axis along which
(c-g) are extracted. The corresponding energy ranges of the pre-helical bump
($E_{\rm B}$) and the helical gap ($E_{\rm H}$) are labelled. (c-g) Vertical
linecuts for $-eV_{\rm sd}$ values of $-0.14t$, $-0.07t$, 0, $0.07t$, and
$0.14t$, highlighting the evolution of the bias-asymmetric transport feature
in the spin-flip transport channels.

When $E$ is located in the the helical gap, the spin is coupled to the
electron propagation direction, which suggests that the spin-filtering
properties of the system should depend on the sign of $V_{\rm sd}$. Fig.
2(a,b) show that the spin-conserving conductance ($G_{\rm\uparrow\uparrow}$,
$G_{\rm\downarrow\downarrow}$) is independent of bias polarity. To observe a
dependence on bias polarity it is necessary to consider the two spin-flip
channels, $G_{\rm\uparrow\downarrow}$ and $G_{\rm\downarrow\uparrow}$, which
describe the conductance from $L\uparrow$ to $R\downarrow$ and $L\downarrow$
to $R\uparrow$ respectively. For fully spin-polarized leads, this corresponds
to experimental conductance measurements with antiparallel magnetizations in
the two FM leads. Surprisingly, although the bias-asymmetric effects are
missing in the spin-conserving transport channel, a substantial correlation
between bias polarity and spin orientation exists in the spin-flip channel.
Fig. 3(a,b) show the corresponding conductance maps for the lowest subband as
a function of $E$ and $V_{\rm sd}$, revealing a clear bias-polarity
dependence. For example, Fig. 3(a) shows the evolution of
$G_{\rm\uparrow\downarrow}$ ($L\uparrow$ to $R\downarrow$), revealing an
enhancement of the spin-flip conductance along a ridge at the edge of Region
II, where the lower electrochemical potential is aligned to the subband bottom
and and $E$ is in the helical gap. An intuitive understanding of the bias
polarity dependence is that the bias window applied on the two leads drives
electrons moving from the higher electrochemical potential to the lower one,
resulting in an effective $\vec{k}$ along this direction. This consequently
influences the $\vec{B}_{\rm SO}$ direction: for example, in Fig. 3(a), the
positive bias window leads to an effective $\vec{k}$ along $+x$ ($\vec{B}_{\rm
SO}$ along $-z$), thus increasing the spin-up to spin-down transmission.
Similar effects occur for $G_{\rm\downarrow\uparrow}$ under reversed bias
polarity. The exact mechanisms leading to the observed ridge resonances and
their bias asymmetry require further study, however this numerical observation
suggests a novel experimental signature of the helical state that arises only
in devices with FM contacts.

Figure 4: Dependence of the conductance difference ration (CDR) on spin
polarization, for the spin-conserving channel (blue squares) and the spin-flip
channel (orange disks).

In the preceding discussion, we have considered fully spin-polarized leads. We
now evaluate the impact of finite $P$ on the spin conductance contrast. To
model spin-polarized transport we introduce the constants $t_{\rm maj}$
($t_{\rm min}$), with $t_{\rm maj}>t_{\rm min}$ and $t_{\rm maj}+t_{\rm
min}=1$, which describe the coupling of the majority (minority) spins from the
leads to the semiconductor region. Majority (minority) spins are aligned
parallel (anti-parallel) to the lead’s magnetization. The polarization of the
leads is defined as $P=\frac{t_{\rm maj}-t_{\rm min}}{t_{\rm maj}+t_{\rm
min}}$ and describes the imbalance between the two spin channels in transport.
A normal (non-FM) lead has $P=0$ and $t_{\rm maj}=t_{\rm min}=0.5$; a fully
spin-polarized lead has $P=1$, $t_{\rm maj}=1$ and $t_{\rm min}=0$. Within
this framework we calculate the spin-dependent conductance by using the
coupling constants as weights for each of the two spin-conserving and each of
the two spin-flip channels (calculation details and conductance color plots
are shown in the supplementary section III [26]). For simplicity, we consider
the typical case where the left and right leads have the same spin
polarization, but this assumption can be relaxed in a straightforward manner
within our model.

In order to quantitatively evaluate the impact of finite spin polarization on
the spin-resolved conductance for parallel and antiparallel lead
magnetizations, we compute the conductance difference ratio (CDR), defined as
the maximum of $\frac{\absolutevalue{G_{1}-G_{2}}}{G_{1}+G_{2}}$ extracted
from the conductance maps as shown in Fig. 2(a,b) and Fig. 3(a,b) for $P=1$,
where $G_{1}$ and $G_{2}$ correspond to $G_{\rm\uparrow\uparrow}$ and
$G_{\rm\downarrow\downarrow}$ for the spin-conserving channel (measured with
parallel magnetization), and to $G_{\rm\downarrow\uparrow}$ and
$G_{\rm\uparrow\downarrow}$ for the spin-flip channel (measured with
antiparallel magnetization). Fig. 4 displays the dependence of the CDR on $P$
for the spin-conserving and spin-flip channels. For the spin-conserving
channel, the CDR has a maximum value of ${\rm CDR}=0.57$, at $P=1$, and
decreases gradually with decreasing $P$ to reach $80\%$ of its maximum value
at $P=0.5$, and 0 for completely unpolarized (normal) leads. In contrast, for
the spin-flip channel, the CDR drops rapidly from a value of 1, at $P=1$,
reaching less than $20\%$ of its maximum value at $P=0.5$. Thus, the bias-
asymmetric signatures of the helical state, which as discussed earlier arise
only in the spin-flip channel, depend crucially on the presence of a
substantial spin polarization, although a finite contrast remains present for
any non-zero value of $P$ provided FM leads are used.

Figure 5: (a) Total conductance map for normal metallic leads. (b) Conductance
difference between the major spin transport channel ($\downarrow$ to
$\downarrow$) and the minor channel ($\uparrow$ to $\uparrow$) for FM leads.
(c) Vertical linecuts extracted from (a) for different interfacial barrier
heights. (d) Vertical linecuts extracted from (b) for different interfacial
barrier heights. Linecuts are indicated by green dash lines in the color
plots. The traces in (c) and (d) are offeset vertically for clarity.

In 1D quantum systems with partially-reflecting ends the interference of
propagating and reflected particle waves leads to quantum interference,
analogous to that of photons in a laser cavity. Such Fabry-Pérot intereference
is common for electron waves in mesoscopic systems, such as nanotube- and
nanowire-based quantum devices [32, 33, 34, 35, 9], where the electrostatic
potential can be spatially dependent in the neighborhood of
metal/semiconductor interfaces, leading to partial reflection of the electron
wavefunctions, e.g. near the contacts. Fabry-Pérot interference can severely
limit or even prevent the detection of helical gaps using normal (non-FM)
leads by limiting the visibility of conductance plateaus or the ability to
distinguish conductance fingerprints of helical states from interference-
related conductance modulation [14, 15, 16, 17, 9]. Below, we demonstrate that
these challenges can be greatly mitigated by using FM contacts, which can
substantially improve the contrast between helical features and quantum
interference. To investigate Fabry-Pérot intereference, we now introduce a
finite energy barrier at the interface between leads and nanostrip by
increasing the onsite potential energy for the interfacial lattice sites (one-
layer of lattice along the $z$-direction in the scattering region next to the
leads) by a value $E_{\rm I}$, which we tune between $0$ to $t$. This induces
partial reflection at the lead-semiconductor interface, which leads to
characteristic Fabry-Pérot interference oscillations in the transmission as a
function of energy, as shown in Fig. 5. The effects of the interference are
very different, depending on whether the leads are normal or FM. Since helical
gap detection using normal leads is based on using the sequence of conductance
steps associated with the gap as the distinguishing feature (Fig. 1(f)), this
method requires a clear observation of: i) the subband pre-helical bump
(conductance at 2$e^{2}/h$), ii) the $e^{2}/h$ conductance suppression in the
helical gap and iii) the subsequent increase back to 2$e^{2}/h$. However, as
shown in Fig. 5(a), where the total conductance is computed, for normal
contacts this conductance fingerprint is engulfed by quantum interference
oscillations even for interfacial barriers as low as 0.2$t$. For even stronger
potential distortions near the contacts, the interference-induced oscillations
make it all but impossible to resolve the conductance fingerprint of the
helical gap. In contrast, by using two FM contacts in a spin-valve
configuration, the spin filtering characteristics of the helical gap can be
leveraged, allowing the helical gap to be detected across a substantially
wider range of interfacial barrier strengths (Fig. 5(b)). This is because of
the contrast between the conductances $G_{\rm\uparrow\uparrow}$ and
$G_{\rm\downarrow\downarrow}$, which can be obtained experimentally by
comparing the conductances obtained before and after flipping the
magnetization directions of both FM contacts. Thus, using FM contacts not only
enables detection of the spin properties of the helical gap, but it can also
substantially alleviate the difficulties in detecting the gap in the first
place. These difficulties are largely unavoidable in the conventional, normal-
contact devices under realistic conditions because quantum interference is
nearly always present. The calculations presented here were performed for
fully polarized leads, $P=1$. For $P<1$, the signal shown in Fig. 5(b,d),
which is obtained from the conductance difference in the spin-conserving
channels, will decrease slowly following the tendency shown in Fig. 4, but an
advantage over normal leads is still expected for $P>0$.

In this work we modeled quantum transport in the helical gap regime of
semiconductor nanowires with strong Rashba spin-orbital interaction and spin-
selective contacts by using a one-dimensional tight-binding model. Our
quantitative numerical results show that FM contacts can be used to
successfully detect the helical gap in the presence of realistic device
effects that would prohibit detection with conventional normal contacts.
Moreover, we show that magnetic contacts provide access to new physics
associated with the spin structure of the helical gap. In particular, we found
that spin-conserving and spin-flip transport are affected in qualitatively
distinct ways by applied bias, bias polarity and interface barriers. This work
reveals the importance of experimentally integrating FM elements, especially
materials with high spin polarization, such as certain Heusler alloys, within
nanowire-based topological quantum devices for creating a new generation of
robust experiments on Majorana zero-modes and for developing 1D quantum
spintronic devices.

We acknowledge B. Heischmidt and J. Etheridge for helpful discussions. This
work was supported by the Department of Energy under award no. DE-SC-0019274.

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