#  Spin-polarized voltage probes for helical edge state: a model study

Vivekananda Adak  Department of Physical Sciences, IISER Kolkata, Mohanpur,
West Bengal 741246, India.  Krishanu Roychowdhury  Department of Physics,
Stockholm University, SE-106 91 Stockholm, Sweden.  Sourin Das  Department of
Physical Sciences, IISER Kolkata, Mohanpur, West Bengal 741246, India.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Theoretical models of a spin-polarized voltage probe (SPVP) tunnel-coupled to
the helical edge states (HES) of a quantum spin Hall system (QSHS) are
studied. Our first model of the SPVP comprises $N_{P}$ spin-polarized modes
(subprobes), each of which is locally tunnel-coupled to the HES, while the
SPVP, as a whole, is subjected to a self-consistency condition ensuring zero
average current on the probe. We carry out a numerical analysis which shows
that the optimal situation for reading off spin-resolved voltage from the HES
depends on the interplay of the probe-edge tunnel-coupling and the number of
modes in the probe ($N_{P}$). We further investigate the stability of our
findings by introducing Gaussian fluctuations in (i) the tunnel-coupling
between the subprobes and the HES about a chosen average value and (ii) spin-
polarization of the subprobes about a chosen direction of the net polarization
of SPVP. We also perform a numerical analysis corresponding to the situation
where four such SPVPs are implemented in a self-consistent fashion across a
ferromagnetic barrier on the HES and demonstrate that this model facilitates
the measurements of spin-resolved four-probe voltage drops across the
ferromagnetic barrier. As a second model, we employ the edge state of a
quantum anomalous Hall state (QAHS) as the SPVP which is tunnel-coupled over
an extended region with the HES. A two-dimensional lattice simulation for the
quantum transport of the proposed device setup comprising a junction of QSHS
and QAHS is considered and a feasibility study of using the edge of the QAHS
as an efficient spin-polarized voltage probe is carried out in presence of an
optimal strength of the disorder.

##  I Introduction

Transport of electron spin with minimal loss of polarization and coherence
over networks which are controllable by all electrical means is highly
desirable for spintronics Igor  _et al._ (2004); Hirohata  _et al._ (2020) and
quantum information applications Awschalom and Samarth (2002). In particular,
the surface states of two-dimensional and three-dimensional topological
insulators, which possess spin-momentum locked spectra, can act as a resource
for such applications Pesin and MacDonald (2012); He  _et al._ (2019);
Yongbing Xu (2015). Helical edge states (HES) of quantum spin Hall state
(QSHS) Kane and Mele (2005a, b); Bernevig and Zhang (2006); Bernevig  _et al._
(2006), which are one-dimensional modes lying on the edges carrying electrons
with their spin locked to their momenta (e.g., right movers being spin-up and
left movers being spin-down along a chosen spin-quantization axis), is one
such state that constitutes the main topic of discussion in this manuscript.
The spin-polarization of these states has been addressed experimentally and
found to be polarized mostly in the direction which is perpendicular to the
plane hosting the QSHS Arriola and Broniowski (2016) (we take it to be the
$z$-axis). If we intend to exploit these states for spintronics applications,
it requires to devise a way for carrying our spin-resolved measurements on the
edge. Experimental attempts of probing these states are already in place, for
example, a less invasive detection of these edge states was carried out by
using mesoscale SQUID loop in Ref. Nowack  _et al._ , 2013 whereas a more
invasive one involving the injection of electrical current into the edge was
carried out in Ref. Zhou and Zhang, 2012. In this manuscript, we carry forward
the latter idea and explore the possibility of using a spin-polarized voltage
probe for reading off the local spin-resolved voltages on a helical edge. In
particular, we formulate the problem to address a situation which is analogous
to the six-probe Hall bar setup (two current probes and four voltage probes,
as in Fig. 7) of Ref. Ferry, 2015, involving a quantum point contact (QPC)
which is routinely used in measurements of Hall voltages.

Figure 1: Schematic of the setup involving a SPVP, which is modelled as a
collection of chiral, spin-polarized, one-dimensional modes, tunnel-coupled to
the helical edge state subjected at a finite bias voltage ($V_{R}-V_{L}$). The
red (blue) line represents the spin-up (down) edge. The shaded region in green
represents the QSHS and the two gray patches on the two sides represents the
leads across which the voltage bias is applied.

We consider a situation where a single backscatterer is placed on the HES
which is carrying a net current due to the electrical bias applied across the
edge state. We then place four spin-polarized voltage probes, two on the right
side, and two on the left side of the backscatterer. The polarization
directions of the probes are such chosen that the pair of probes on each side
of the backscatterer has one fully polarized along the $z$-axis (call it an
“up-polarized”) while the other fully polarized along the $-z$-axis (call it a
“down-polarized”). We expect the up-polarized probe would couple primarily to
the right movers (spin-up electrons on the edge) and the down-polarized probe
will couple mostly to the left movers (spin-down electrons on the edge) as we
have assumed spin conserving electron tunneling between the probes and the
helical edge. We will present a detailed analysis which explores the
possibility of such spin-resolved coupling Das and Rao (2011) of the
$z$-polarized voltage probes.

For a pictorial representation of the situation described above, see Fig. 7.
It is straightforward to note the analogy between the situation described in
Fig. 7, (a) and Fig. 7, (c) if one identifies the left (right) movers on the
top and the bottom edge in the Hall setup with the left (right) movers of the
HES. With such a setup, one can measure spin-resolved voltage drops (voltage
drop between a pair of up or down probes) across the impurity which is nothing
but the drop in voltages in the right and the left moving channels of the HES
across the backscatterer. These voltages will be analogous to the longitudinal
voltage drop ($V_{lo}$) in the Hall bar geometry measured across the QPC. We
can also measure the voltage difference between the right and the left mover
on the same side by using the up and the down-polarized probe on the same side
of backscatterer and that will be analogous to the measurement of the Hall
voltage ($V_{H}$) in the Hall bar geometry which should be a robust quantity,
i.e., independent of the strength of the backscatterer provided the voltage
probes are ideal Streda  _et al._ (1987a).

In this manuscript, we will demonstrate via numerical calculations that our
model for the SPVP, when optimized appropriately, can lead to measured values
of $V_{lo}$ and $V_{H}$ which indeed correspond to the spin-resolved voltage
drops across the backscatterer with the latter an analog of Hall voltage on
the HES. This, in turn, implies that our theoretical model for the spin-
polarized probe is capable of measuring the spin-resolved voltages
successfully and hence, could provide useful guidance to future experiments
that are tuned to such objectives.

The rest of the paper is organized as follows: In section II, we lay the
concept of spin-resolved voltage measurements on a HES tunnel-coupled to a
$N_{P}$-subprobe SPVP and discuss the stability of the measurements in
presence of Gaussian disorder in (i) the tunnel-coupling between the subprobes
of the SPVP and the helical edge and (ii) the spin-polarization of the
subprobes about a chosen direction. In section III, we demonstrate the six-
probe setup discussed above and measure Hall response in presence of a
backscatterer including Gaussian disorder in the individual SPVPs. Finally in
section IV, we simulate a device setup using the KWANT package Groth  _et al._
(2014) which comprises a quantum anomalous Hall state (QAHS) acting as the
SPVP for the HES formed at the edge of the QSHS and also present a feasibility
study. We summarize the results and conclude in section V.

##  II Spin-resolved voltage measurement on helical edge states

As already discussed in the introduction, our setup to measure the spin-
resolved voltage on the HES of a QSHS engages an extended SPVP tunnel-coupled
to the HES. The SPVP consists of multiple modes each of which supports spin
injection into the HES as shown in Fig. 1. We also assume that the QSHS is
hosted on the $x$-$y$ plane with the relevant HES of the QSHS laying along the
$x$-axis and described by the Hamiltonian

$\mathcal{H}_{\rm HES}=-\imath\hbar
v_{F}\int_{-\infty}^{\infty}dx~{}(\psi_{R}^{\dagger}\partial_{x}\psi_{R}-\psi_{L}^{\dagger}\partial_{x}\psi_{L}),$
(1)

where $v_{F}$ is the Fermi velocity, the operators $\psi_{R}^{\dagger}$ and
$\psi_{L}^{\dagger}$ create electrons respectively in the right ($R$) and the
left ($L$) propagating edge states with spinors $|n_{R}\rangle=[1~{}0]^{T}$
and $|n_{L}\rangle=[0~{}1]^{T}$ respectively implying the spin-polarization of
the HES being along the $z$-axis Brüne  _et al._ (2012).

We model the SPVP as a collection of one dimensional modes with linear
spectrum each of which is, henceforth, referred to as a subprobe. Each of
these subprobes is regarded as a right moving chiral mode ($R^{\prime}$) with
spin-polarization given by the spinor
$|n_{R^{\prime}}\rangle\equiv[\cos(\theta/2+\pi/4)~{}\sin(\theta/2+\pi/4)e^{i\phi}]^{T}$
($\theta,\phi$ are the polar and the azimuthal angle respectively of
$|n_{R^{\prime}}\rangle$ on the Bloch sphere). As the tunneling between the
subprobes and the HES is taken to be local, hence, the chirality of the
subprobes (being right or left movers) is of no consequence as far as
tunneling current is concerned. The linear spectrum of the subprobes ensures
that the tunneling current, in the weak tunneling limit, does not develop an
energy dependence due to the variation in the density of states of the probe
at the Fermi level as desirable for an ideal probe. Also the modelling of the
voltage probe as a collection of subprobes is motivated from the fact that
this proves a minimal model for the probe accommodating a large number of
electronic degrees of freedom. Hence, the Hamiltonian for a subprobe is given
by

$\mathcal{H}_{\rm subprobe}^{(j)}=-\imath\hbar
v_{F}\int_{-\infty}^{\infty}dx~{}\psi_{R^{\prime}j}^{\dagger}\partial_{x}\psi_{R^{\prime}j},$
(2)

so that
$\mathcal{H}_{\rm{SPVP}}=\sum_{j=1}^{N_{P}}\mathcal{H}_{\rm{subprobe}}^{(j)}$.
Note that an offset of $\pi/4$ is introduced in the expression of
${|n_{R^{\prime}j}\rangle}$ merely to set the range of the polarization angle
of the SPVP to be $\theta\in[-\pi/2,\pi/2]$, symmetric about $\theta=0$. The
tunneling Hamiltonian for electrons between the SPVP with $N_{P}$ number of
subprobes and the HES is taken to be
$\mathcal{H}_{T}=\sum_{j=1}^{N_{P}}\mathcal{H}_{T}^{(j)}$ such that,

$\mathcal{H}_{T}^{(j)}=\sum_{\eta}t_{\eta
R^{\prime}}^{(j)}\psi_{\eta}^{\dagger}(x_{j})\psi_{R^{\prime}j}(0)+{\rm
h.c.},$ (3)

where the tunnel-coupling between the HES and the subprobe is taken to be such
that for each subprobe, the tunneling is happening at $x=0$ of the subprobe
coordinates and at $x=x_{j}$ for the corresponding HES coordinates; $t_{\eta
R^{\prime}}^{(j)}$ is the tunneling strength between the right or the left
moving states [$\eta\in(R,L)$] of the HES and the chiral edge $R^{\prime}$
representing the $j$-th subprobe, further expressed as $t_{\eta
R^{\prime}}^{(j)}=t^{\prime(j)}\gamma_{\eta R^{\prime}}^{(j)}$,
$t^{\prime(j)}$ being real. Note that the form of the tunneling Hamiltonian
preserves the spin rotation symmetry of the electron. The quantity
$\gamma_{\eta R^{\prime}}^{(j)}$ denotes the overlap between the spinors
$|n_{\eta}\rangle$ and $|n_{R^{\prime}j}\rangle$: $\gamma_{\eta
R^{\prime}}^{(j)}=\langle n_{\eta}|n_{R^{\prime}j}\rangle$. For an extended
SPVP with multiple subprobes, we consider the following cases -

  * (i)

All the subprobes have identical polarization set by an angle $\theta$ and
also the tunneling strength $t^{\prime(j)}$ are taken to be uniform across the
junctions (with a magnitude $t^{\prime}$). We refer to this as the uniform
case.

  * (ii)

The tunneling strength $t^{\prime(j)}$ is nonuniform across the junctions and
is characterized by a Gaussian distribution with mean $\mkern
1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu$ and standard
deviation $\sigma_{t^{\prime}}$ while the polarization angle $\theta$ of the
subprobes fluctuates with a mean $\mkern
1.5mu\overline{\mkern-1.5mu\theta\mkern-1.5mu}\mkern 1.5mu$ and standard
deviation $\sigma_{\theta}$. We refer to this as the disordered case.

Electron transport across a given tunneling point between a subprobe and the
HES at $x=x_{j}$ (see Fig. 1) can be quantified in terms of a scattering
matrix ${\cal S}_{j}$ corresponding to the Hamiltonian
$\mathcal{H}=\mathcal{H}_{\rm HES}+\mathcal{H}^{(j)}_{\rm
subprobe}+\mathcal{H}_{T}^{(j)}$. The wavefunctions associated with the
incoming and the outgoing electrons at the tunneling point at $x=x_{j}$
(denoted $\Psi_{\alpha j}^{\rm in}$ and $\Psi_{\alpha j}^{\rm out}$
respectively) are related by the elements of ${\cal S}_{j}$:

$\Psi^{{\rm out}}_{\alpha j}=\sum_{\beta}s^{(j)}_{\alpha\beta}~{}\Psi^{{\rm
in}}_{\beta j},$ (4)

where $\alpha,\beta\in(R,L,R^{\prime})$ and the corresponding currents obey

$I^{\rm out}_{\alpha j}=\sum_{\beta}|s^{(j)}_{\alpha\beta}|^{2}~{}I^{\rm
in}_{\beta j}.$ (5)

For the SPVP to act as an ideal voltage probe, the net current flowing through
it must be zero i.e. $I^{\rm in}_{\rm SPVP}-I^{\rm out}_{\rm SPVP}=0$ is the
voltage probe condition, where $I^{\rm in}_{\rm SPVP}=\sum_{j=1}^{N_{p}}I^{\rm
in}_{R^{\prime}j}$ and $I^{\rm out}_{\rm SPVP}=\sum_{j=1}^{N_{p}}I^{\rm
out}_{R^{\prime}j}$. For an SPVP consisting of one subprobe only ($N_{P}=1$,
tunneling strength $t^{\prime}$, polarization $\theta$), this condition yields
a probe voltage given by

$\displaystyle
V_{R^{\prime}}=\frac{|s_{R^{\prime}R}|^{2}V_{R}+|s_{R^{\prime}L}|^{2}V_{L}}{|s_{R^{\prime}R}|^{2}+|s_{R^{\prime}L}|^{2}},$
(6)

when the HES is connected to a right and a left reservoir maintained at the
voltages $V_{R}$ and $V_{L}$ respectively. In presence of a finite bias
($V_{R}-V_{L}\neq 0$), the HES develops a net magnetization along the
$z$-axis, hence, in the weak tunneling limit ($t^{\prime}\ll\hbar v_{F}$), the
voltage measured by the SPVP is expected to be a sum of the average voltage
$V_{\rm av}=(V_{R}+V_{L})/2$ and an additional contribution which can be
attributed to tunnel magnetoresistance Gerstner (2007). Thus, we get

$\displaystyle
V_{R^{\prime}}|_{{N_{P}}=1}=\frac{1}{2}(V_{R}+V_{L})-\frac{1}{2}(V_{R}-V_{L})\sin\theta.$
(7)

Note that this expression is independent of the azimuthal angle $\phi$ as
expected and the $\sin\theta$ dependence of the magnetoresistance instead of
the standard $\cos\theta$ is due to our phase shifted definition of $\theta$.
This expression of $V_{R^{\prime}}$ is obtained by using the explicit forms of
the scattering matrix elements (see Appendix A for details). In fact, the
above form of $V_{R^{\prime}}$ is valid for all values of $t^{\prime}$ owing
to the fact that the ratios
$|s_{R^{\prime}R}|^{2}/(|s_{R^{\prime}R}|^{2}+|s_{R^{\prime}L}|^{2})=\cos^{2}(\theta/2)$
and
$|s_{R^{\prime}L}|^{2}/(|s_{R^{\prime}R}|^{2}+|s_{R^{\prime}L}|^{2})=\sin^{2}(\theta/2)$
are independent of $t^{\prime}$ though $t^{\prime}$ induces spin flip
scattering in the HES via higher order processes when $\theta\neq\pm\pi/2$.
This leads to the interesting fact that the form of the magnetoresistance
contribution, which is expected to exist in the weak tunneling limit, actually
remains intact even at intermediate and strong tunneling limits. This
magnetoresistance contribution in $V_{R^{\prime}}$ is the key element for
obtaining spin-resolved information of the HES by using the SPVP and it is the
primary focus of this study. Evidently, when $\theta=-\pi/2$, the spinor
$|n_{R^{\prime}}\rangle$ aligns with $|n_{R}\rangle$ and the SPVP measures the
voltage $V_{R^{\prime}}=V_{R}$, while for $\theta=\pi/2$,
$|n_{R^{\prime}}\rangle$ aligns with $|n_{L}\rangle$ yielding a voltage
measurement $V_{R^{\prime}}=V_{L}$. However, these results holds only in the
limit of few subprobes which we will discuss in detail later. For the case of
multiple subprobes ($N_{P}>1$), the voltage probe condition implies

$\displaystyle\sum_{j=1}^{N_{P}}(I^{\rm
out}_{R^{\prime}j}-e^{2}V_{R^{\prime}}/h)\equiv\sum_{j=1}^{N_{P}}I^{(j)}_{n}=0,$
(8)

where $I^{(j)}_{n}$ indicates the net current in the $j$-th subprobe. Also
note that, as we have assumed all subprobes to be in equilibrium with a large
reservoir, which is maintained at a voltage $V_{R^{\prime}}$, hence, the
incoming current in each of the subprobes is given by the Hall relation,
$I^{\rm in}_{R^{\prime}j}=e^{2}V_{R^{\prime}}/h,\forall j$ and the information
regarding the values of $V_{L},V_{R}$ enters Eq. 8 via $I^{\rm
out}_{R^{\prime}j}$ due to Eq. 5. In general, to obtain an analytic expression
for $V_{R^{\prime}}$ by solving Eq. 8 is fairly complicated for a given set of
values for $N_{P}$, $V_{L}$, and $V_{R}$. Hence, we solve Eq. 8 numerically
and also set $e^{2}/h=1$ henceforth, unless stated otherwise.

###  II.1 Transfer matrix approach for numerical analysis

Figure 2: Zoomed-in pictorial representation of a subprobe tunnel-coupled to
the HES where $T$ represents the transfer matrix which connects the
wavefunctions on the right side of the junction to that on its left side.

To obtain $V_{R^{\prime}}$ self-consistently for a given set of values for
$V_{R}$, $V_{L}$, and $t^{\prime}$ numerically, it is efficient to use a
transfer matrix method which connects the left and right going currents on one
side of the tunnel junction between the $j$-th subprobe and the HES at
$x=x_{j}$ to that on the other side as shown in Fig. 2. To proceed further, we
compartmentalize the HES into $N_{P}+1$ segments where the $j$-th segment is
defined as the segment of the HES lying between the tunnel junction at
$x=x_{j-1}$ and $x=x_{j}$ (see Fig. 1). Note that the $1$-st and the
$N_{P}+1$-th segment are connected only to one tunnel junction being the first
and the last one. Also the incoming current in each subprobe (in units of
$e^{2}/h$) is $V_{R^{\prime}}$ irrespective of the position of the subprobe.
Now the relation between currents on different segments can be written as:

$\displaystyle I_{R}^{(j+1)}$
$\displaystyle=T^{(j)}_{RR}I_{R}^{(j)}+T^{(j)}_{RL}I_{L}^{(j)}+T^{(j)}_{RR^{\prime}}V_{R^{\prime}}$
$\displaystyle I_{L}^{(j+1)}$
$\displaystyle=T^{(j)}_{LR}I_{R}^{(j)}+T^{(j)}_{LL}I_{L}^{(j)}+T^{(j)}_{LR^{\prime}}V_{R^{\prime}}$
$\displaystyle I_{R^{\prime}}^{(j)}$
$\displaystyle=T^{(j)}_{R^{\prime}R}I_{R}^{(j)}+T^{(j)}_{R^{\prime}L}I_{L}^{(j)}+T^{(j)}_{R^{\prime}R^{\prime}}V_{R^{\prime}},$
(9)

where $T^{(j)}_{\eta\eta^{\prime}}$ denotes the transfer matrix elements at
the $j$-th junction for the currents while $I_{\eta}^{(j)}$ ($\eta\in R,L$)
corresponds to the right moving or the left moving current on the $j$-th
segment. In this notation, the net current in the $j$-th subprobe is given by
$I_{n}^{(j)}=I_{R^{\prime}}^{(j)}-V_{R^{\prime}}$. The elements
${T}_{\eta\eta^{\prime}}^{(j)}$ can be expressed in terms of the elements of
the scattering matrix, ${\cal S}^{(j)}$, and is given by

$\displaystyle T^{(j)}_{RR}$
$\displaystyle=|s^{(j)}_{RR}|^{2}-|s^{(j)}_{RL}|^{2}|s^{(j)}_{LR}|^{2}/|s^{(j)}_{LL}|^{2}$
$\displaystyle T^{(j)}_{RL}$
$\displaystyle=|s^{(j)}_{RL}|^{2}/|s^{(j)}_{LL}|^{2}$ $\displaystyle
T^{(j)}_{RR^{\prime}}$
$\displaystyle=|s^{(j)}_{RR^{\prime}}|^{2}-|s^{(j)}_{RL}|^{2}|s^{(j)}_{LR^{\prime}}|^{2}/|s^{(j)}_{LL}|^{2}$
$\displaystyle T^{(j)}_{LR}$
$\displaystyle=-|s^{(j)}_{LR}|^{2}/|s^{(j)}_{LL}|^{2}$ $\displaystyle
T^{(j)}_{LL}$ $\displaystyle=1/|s^{(j)}_{LL}|^{2}$ $\displaystyle
T^{(j)}_{LR^{\prime}}$
$\displaystyle=-|s^{(j)}_{LR^{\prime}}|^{2}/|s^{(j)}_{LL}|^{2}$ $\displaystyle
T^{(j)}_{R^{\prime}R}$
$\displaystyle=|s^{(j)}_{R^{\prime}R}|^{2}-|s^{(j)}_{R^{\prime}L}|^{2}|s^{(j)}_{LR}|^{2}/|s^{(j)}_{LL}|^{2}$
$\displaystyle T^{(j)}_{R^{\prime}L}$
$\displaystyle=|s^{(j)}_{R^{\prime}L}|^{2}/|s^{(j)}_{LL}|^{2}$ $\displaystyle
T^{(j)}_{R^{\prime}R^{\prime}}$
$\displaystyle=|s^{(j)}_{R^{\prime}R^{\prime}}|^{2}-|s^{(j)}_{R^{\prime}L}|^{2}|s^{(j)}_{LR^{\prime}}|^{2}/|s^{(j)}_{LL}|^{2},$
(10)

and the expressions of the elements $s^{(j)}_{\eta\eta^{\prime}}$ in terms of
$t^{\prime}$ and $\theta$ are given in Appendix A. Also, we assume that there
is no quantum coherence between the scattering of electrons at successive
junctions between the HES and the subprobes. To obtain $V_{R^{\prime}}$, the
equations in Eq. 9 are solved recursively for $j\in\\{1,\dots,N_{P}\\}$
subject to the voltage probe condition in Eq. 8 and boundary conditions given
by $I_{L}^{N_{P}+1}=V_{L}$ and $I_{R}^{1}=V_{R}$. With this formulation
discussed above, we can now study the cases mentioned previously, i.e., the
uniform case and the disordered case.

Figure 3: Plot of $V_{R^{\prime}}$ against tunneling strength $t^{\prime}$ for
a uniform SPVP with $N_{P}=10$ in (a) and $N_{P}=30$ in (b), where the HES is
maintained at a voltage bias given by $V_{R}=-V_{L}=1$. Different curves
correspond to different values of $\theta$ starting from $\theta=-\pi/2$ (the
top most plot) to $\theta=\pi/2$ (the bottom most plot) in steps of $\pi/40$
(the offset along the $t^{\prime}$-axis of these two figures is
$t^{\prime}=0.04$). (c) Plot of $V_{R^{\prime}}$ as a function of $\theta$ at
different values of $t^{\prime}$ for $N_{P}=25$. (d) Plot of $V_{R^{\prime}}$
as a function of number of subprobes $N_{P}$ in the SPVP at different values
of $t^{\prime}$ for $\theta=-1.37$. The inset shows the scale of this
exponential decay, denoted $N_{0}$, plotted as a function of $t^{\prime}$. The
offset along the $N_{P}$-axis for this plot is $N_{P}=20$.

###  II.2 Uniform case

The results for the uniform case with $N_{P}=10$ and $N_{P}=30$ subprobes are
displayed in Fig. 3 (a) and (b) respectively for $V_{R}=-V_{L}=1$. These
values for $V_{R}$, $V_{L}$ are taken for the numerical calculations as it
corresponds to a zero $V_{\rm av}$ hence leaving behind a neat
magnetoresistance contribution in Eq. 7. From the plots in Fig. 3 we note that
the value of $V_{R^{\prime}}$ decays monotonically towards the average voltage
$V_{\rm av}$ as we increase $t^{\prime}$. This can be understood as follows.
In the weak tunneling limit ($t^{\prime}\ll\hbar v_{F}$), spin-flip scattering
induced by the subprobes in the HES is minimal and hence $V_{R^{\prime}}$
carries spin-resolved information, i.e., it follows Eq. 7. But as we increase
the tunneling strength ($t^{\prime}$), it leads to considerable spin-flip
scattering in the HES at the junctions with the subprobes. The feedback
between various subprobes amplifies the effect, hence, resulting in reduction
of the magnetoresistance response appearing in Eq. 7 and eventually
suppressing it completely in the large $t^{\prime}$ limit. The rate at which
$V_{R^{\prime}}$ drifts towards $V_{\rm av}$ as a function of $t^{\prime}$
depends on two factors: (i) the angle between the spin-polarization axis of
the SPVP and the spin quantization axis of HES, i.e., $\theta$, and (ii) the
number of subprobes in the SPVP. Note that the case corresponding to
$\theta=\pm\pi/2$ are pathological and in these two cases, $V_{R^{\prime}}$
does not decay towards $V_{\rm av}$ as we increase $t^{\prime}$ for the
subprobes do not induce any spin-flip scattering in the HES and, hence,
provides a perfect readout of the spin-resolved voltages.

In Fig. 3 (c) we have plotted $V_{R^{\prime}}$ as a function of $\theta$ for
different values of $t^{\prime}$ and $N_{P}=25$. We note that the value of
$V_{R^{\prime}}$ reduces to $V_{\rm av}=0$ independent of $t^{\prime}$ for
$\theta=0$ owing to fact that the tunnel-coupling strengths of the SPVP with
the spin-up (right mover) and the spin-down (left mover) channel are equal in
this case (see the form of $t_{\eta R^{\prime}}^{(j)}$ below Eq. 3) hence,
leading to a null magnetoresistance. We further note that larger values of
$t^{\prime}$ leads to stronger spin equilibration in the HES and hence, it
forces $V_{R^{\prime}}$ to stay close to $V_{\rm av}=0$ for most values of
$\theta$ which are away from $\theta=\pm\pi/2$. But as we consider the value
of $\theta$ close to $-\pi/2$ or $\pi/2$, the ratio between the tunneling
strength for the spin-up and the spin-down channel with the SPVP becomes very
large or small respectively, hence, leading to a strong suppression of spin
equilibration in the HES. Consequently, $V_{R^{\prime}}$ shows a large
deviation from $V_{\rm av}$ implying that it retains spin-resolved
information. Before we move on to Fig. 3 (d), to elaborate further on the
$\theta$ dependence of $V_{R^{\prime}}$, we will discuss the plot of
$V_{R^{\prime}}$ as a function of $\theta$ presented in Fig. 4 for different
value of $N_{P}$. We note that the plots for the case of $N_{P}=2$ for various
values of $t^{\prime}$ starting from a small value ($t^{\prime}=0.01$) to a
large one ($t^{\prime}=0.76$) closely follows Eq. 3. This can be seen from the
closeness of the plots for $V_{R^{\prime}}$ to that of $\sin\theta$. So it is
clear that the case of two subprobes does not immediately changes the scenario
as far as deviation from Eq. 3 is concerned. Only when we add multiple
subprobes in the SPVP, it leads to a large deviation from Eq. 3 away from
$\theta=\pm\pi/2$ and we observe this very clearly in the cases such as
$N_{P}=20$ and $N_{P}=30$ in Fig. 4.

Figure 4: Variation of $V_{R^{\prime}}$ as a function of $\theta$ for
$N_{P}=2$, $10$, $20$, and $30$ in (a), (b), (c), and (d) respectively.

In Fig. 3 (d), we further demonstrate that for $\theta\neq\pm\pi/2$,
$V_{R^{\prime}}$ falls exponentially to $V_{\rm av}$ with increasing number of
subprobes and, in the large $N_{P}$ limit, is given by

$\displaystyle V_{R^{\prime}}=V_{\rm
av}+[~{}V_{R^{\prime}}|_{{N_{P}}=1}-V_{\rm av}\,]\,e^{-(N_{P}-1)/N_{0}}\,.$
(11)

This exponential decay is visible in the main panel of Fig. 3 (d) for various
values of $t^{\prime}$. This indicates that the contribution to
$V_{R^{\prime}}$ due to the magnetoresistance effect gets exponentially
suppressed with $N_{P}$; $N_{0}$ multiplied by the average spacing between the
subprobes defines a characteristic length scale for the decay. The decay
length, obtained from $N_{0}$, can be though of as a measure of the length
scale over which the spin-up and the spin-down edge equilibrate leading to
loss of spin-polarization of the HES induced by a voltage bias
($V_{L}$-$V_{R}$). The plot in Fig. 3 (d) corresponds to a value of $\theta$
which is shifted from $-\pi/2$ by $0.2$. It is obvious that the length over
which the probe is coupled to the HES needs to be smaller than the
equilibration scale dictated by $N_{0}$ so that the sensitivity of measured
$V_{R}^{\prime}$ to the spin-polarization of the HES survives, which is
equivalent to saying that this will ensure that $V_{R}^{\prime}$ contains a
finite magnetoresistance contribution. Though it is clear that to measure
spin-resolved voltages in a HES, we need a spin-polarized voltage probe, but
to ensure that the probe indeed measures the spin-resolved voltages, we have
to simultaneously optimize the tunneling strength and the physical extent of
the coupling between the probe and the HES so that a complete equilibration of
the spins on the HES induced by the probe does not obstruct the measurement of
a spin-resolved $V_{R}^{\prime}$.

For the above study, we can conclude that the limit, in which an almost
perfect readout of the spin-resolved voltage is possible, is when we are in
the vicinity of $\theta=\pm\pi/2$ and the tunneling strength and the length of
the entire junction (which is proportional to $N_{P}$) or alternatively, the
physical extent of the coupling are chosen appropriately guided by Eq. 11 so
that we are always away form the regime of complete spin equilibration. Also,
the case of $\theta=\pm\pi/2$ with uniform $t^{\prime}$ for the subprobes is
too ideal to be realized in an experimental situation. This naturally compels
us to consider the effects of fluctuations in $\theta$ and $t^{\prime}$ across
the subprobes in the next section where we analyze how such disorders would
influence the readout of the SPVP.

###  II.3 Disordered case

Figure 5: Evolution of the disorder averaged variation of $\mkern
1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu_{R^{\prime}}$ as a
function of the tunneling strength ($\mkern
1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu$) for different
values of $\bar{\theta}$ with $N_{P}=10$. The solid lines in each figure
represent the uniform case: the variation of ${V}_{R^{\prime}}$ as a function
of a uniform value of $t^{\prime}$ for different values of $\theta$. The
scattered datapoints in (a) show the variation of $V_{R^{\prime}}$ for a
single random disorder configuration of $t^{\prime}$ and $\theta$. Figure (b),
(c), and (d) show the same variation after taking a disorder average over $5$,
$10$, and $20$ random configurations of $t^{\prime}$ and $\theta$ about their
respective average values. We have taken $\sigma_{t}$ = $\sigma_{\theta}=0.2$
and the offset along the $x$-axis of all the plots is taken to be $0.01$.
Figure 6:  (a) Plot of $\sigma_{V_{R^{\prime}}}$ vs $\sigma_{\theta}$ at
different values of (uniform) $t^{\prime}$ for a SPVP ($N_{P}=10$) with
Gaussian disorder in $\theta$ across the subprobes with a mean value of
$\theta$ to be $-1.27$. The fluctuations increase in the strong tunneling
limit (averaging performed over $10^{3}$ realizations).(b) Plot of
$\sigma_{V_{R^{\prime}}}$ vs $\sigma_{t^{\prime}}$ at different values of
(uniform) $\theta$ for the SPVP ($N_{P}=10$) with Gaussian disorder in
$t^{\prime}$ across the subprobes with a mean value of $t^{\prime}$ to be
$1.0$. The fluctuations increase for the values of $\theta$ away from
$\pm\pi/2$, but the magnitude is much smaller compared to the disordered
$t^{\prime}$ case shown in (a).

In the disordered case, we first present a comparative study between different
disorder averaging of $V_{R^{\prime}}$ sensed by a SPVP of $N_{P}=10$,
performed over ensembles of various number of disorder configurations of
${t^{\prime}}$ and $\theta$. We denote the resultant voltage as $\mkern
1.5mu\overline{\mkern-1.5muV_{R^{\prime}}\mkern-1.5mu}\mkern 1.5mu$ and plot
it against the average value of the tunneling strength $\mkern
1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu$ for different
values of the average polarization of the SPVP, $\mkern
1.5mu\overline{\mkern-1.5mu\theta\mkern-1.5mu}\mkern 1.5mu$ in Fig. 5 for
ensembles of size 1, 5, 10, and 20. As evident from Fig. 5, we find that the
averaging procedure results in a rapid convergence of the functional
dependence of $\mkern
1.5mu\overline{\mkern-1.5muV_{R^{\prime}}\mkern-1.5mu}\mkern 1.5mu$ to that
for the uniform case as predicted by Eq.11; in fact, an ensemble of size as
few as $20$ realizations is promising enough to obtain a close qualitative
resemblance to the uniform case.

Next we study the stability of the voltage measured by the SPVP as a function
of fluctuations in the tunneling strength $t^{\prime}$ and the polarization
angle $\theta$ of the SPVP. In Fig. 6 (a), we plot the fluctuations in
$V_{R^{\prime}}$ characterized by its standard deviation when the measurement
is performed over many realizations of the subprobe disorders:
$\sigma_{V_{R^{\prime}}}=\sqrt{\mkern
1.5mu\overline{\mkern-1.5muV_{R^{\prime}}^{2}\mkern-1.5mu}\mkern 1.5mu-\mkern
1.5mu\overline{\mkern-1.5muV_{R^{\prime}}\mkern-1.5mu}\mkern 1.5mu^{2}}$, as a
function of $\sigma_{\theta}$ for various values of the tunneling strength
$t^{\prime}$ which is assumed to be uniform across all subprobes (the disorder
averaging is performed over $10^{3}$ realizations). Here $\sigma_{\theta}$
represents the standard deviation in the values of $\theta$ for various
subprobes about the chosen average value of $\theta$. We note that the
fluctuations in $V_{R^{\prime}}$ increase monotonically with the increase in
$\sigma_{\theta}$ as expected but the rate of increase is higher for larger
values of the tunneling strength ($t^{\prime}$), hence, indicating that small
$t^{\prime}$ limit is desirable for implementing such a voltage probe in
presence of fluctuations in $\theta$. Fig. 6 (b) shows the variation of
$\sigma_{V_{R^{\prime}}}$ as a function of $\sigma_{t^{\prime}}$ where
$\sigma_{t^{\prime}}$ represents the standard deviation in the tunneling
strength $t^{\prime}$ across various subprobes about a chosen average value of
$t^{\prime}$. If the value of $\theta$ corresponds to either the parallel or
anti-parallel configuration (i.e., $\pm\pi/2$), we expect
$\sigma_{V_{R^{\prime}}}$ to be strongly suppressed. We emphasize that in the
plot corresponding to $\theta=-1.56$ (a value close to $\theta=-\pi/2$),
increasing $\sigma_{t^{\prime}}$ has a little influence on
$\sigma_{V_{R^{\prime}}}$ and it stays close to zero. But as we move away from
these polarizations, $\sigma_{V_{R^{\prime}}}$ displays a behavior which is
similar to that observed against $\sigma_{\theta}$ discussed above. Also note
that the rate of increase of $\sigma_{V_{R^{\prime}}}$ as a function of
$\sigma_{t^{\prime}}$ is the steepest at $\theta=0$ as expected owing to the
fact that at $\theta=0$, the magnetoresistance response of each subprobe will
be completely suppressed.

Hence, for the above study, we conclude that a strong suppression in
$\sigma_{V_{R^{\prime}}}$ will take place if we take the average value of
$t^{\prime}$ to be small and the average value of $\theta$ to be close to
$\theta=\pm\pi/2$. In what follows in the next section, we carry forward the
discussion and explore the possibility of using a SPVP for reading off the
local spin-resolved voltage drops on a HES using the model discussed above. In
particular, we address a situation which is analogous to the six-probe Hall
bar setup (two current probes and four voltage probes as in Fig.7 Ferry
(2015)) involving a quantum point contact (QPC) which represents an archetypal
setup in the context of quantum Hall experiments.

##  III Measuring Hall-type response in a six-probe setup for HES

Figure 7: (a) Schematic of the six-probe (four voltage probes and two current
probes) setup to measure the Hall conductance between the counterpropagating
states of the QSHS and the longitudinal conductance across an FM barrier
(shaded grey, not to confuse with the FM placed on the bottom edge). The Hall
voltage and the longitudinal voltage are denoted as $V_{H}$ and $V_{lo}$
respectively. The polarization angle for the SPVP are set as
$\theta_{A}=\theta_{C}=-\pi/2=-\theta_{B}=-\theta_{D}$ such that SPVP A and C
sense the voltage on the $\uparrow$-channel (right movers) and SPVP B and D
sense the voltage on the $\downarrow$-channel (left movers) only. A
ferromagnet (FM) is employed to open a gap in the spectrum of the bottom edge
hence, electrically disconnecting the left and the right leads via this edge.
(b) The boxed figure on the right top shows the zoomed-in FM barrier with a
transmission probability of $t=91.39\%$ and the voltage drops across this
barrier depending on $t$ and the reflection probability $r=1-t$ that
characterize the scattering through the barrier. (c) The analogous schematic
of a six-probe Hall bar setup involving a QPC in the middle as discussed in
Ref. Ferry, 2015 where lead 1 and 2 represent the current probes and lead 3,
4, 5, and 6 represent the voltage probes.

Consider the FM barrier, localized at $x=x_{0}$ on the HES, is described by
the Hamiltonian

$\displaystyle{\cal H}_{\rm
FM}=\int_{-\infty}^{\infty}dx~{}\delta(x-x_{0})\big{[}{\cal
B}\psi_{R}^{\dagger}\psi_{L}+{\rm h.c.}\big{]},$ (12)

where $|{\cal B}|$ represents the coupling strength (proportional to the in-
plane magnetic field produced by the FM barrier) and the Hamiltonian for the
HES is already given in Eq. 1. In general, ${\cal B}$ can be a complex number,
however, the transfer matrix that connects the currents across the FM barrier
is independent of the phase of ${\cal B}$, and thus, $\pm|{\cal B}|$ is the
only relevant input that enters the calculations for voltages measured by the
probe. Hence, without loss of generality, we consider ${\cal B}$ to be real.
The currents across the FM barrier are related as

$\displaystyle I_{R}(x_{0}^{+})$
$\displaystyle=T^{\rm(FM)}_{RR}I_{R}(x_{0}^{-})+T^{\rm(FM)}_{RL}I_{L}(x_{0}^{-})$
$\displaystyle I_{L}(x_{0}^{+})$
$\displaystyle=T^{\rm(FM)}_{LR}I_{R}(x_{0}^{-})+T^{\rm(FM)}_{LL}I_{L}(x_{0}^{-}),$
(13)

where the lower index of $R,L$ represents the right and the left movers;
$x_{0}^{\pm}=x_{0}\pm\epsilon$, $\epsilon$ being a vanishingly small positive
number. The corresponding transfer matrix elements for the currents are

$\displaystyle T^{\rm(FM)}_{RR}$
$\displaystyle=|s^{\rm(FM)}_{RR}|^{2}-|s^{\rm(FM)}_{RL}|^{2}|s^{\rm(FM)}_{LR}|^{2}/|s^{\rm(FM)}_{LL}|^{2}$
$\displaystyle T^{\rm(FM)}_{RL}$
$\displaystyle=s^{\rm(FM)}_{RL}|^{2}/|s^{\rm(FM)}_{LL}|^{2}$ $\displaystyle
T^{\rm(FM)}_{LR}$
$\displaystyle=-s^{\rm(FM)}_{LR}|^{2}/|s^{\rm(FM)}_{LL}|^{2}$ $\displaystyle
T^{\rm(FM)}_{LL}$ $\displaystyle=1/|s^{\rm(FM)}_{LL}|^{2},$ (14)

$s^{\rm(FM)}_{ab}$ being the elements of the scattering matrix for the FM
barrier (full expressions given in Appendix B).

Figure 8: Plot of the Hall conductance $G_{H}$ in (a) and the longitudinal
conductance $G_{lo}$ in (b) as a function of $\bar{t^{\prime}}$ and
$\sigma_{\theta}$ (defined in the main text). The variation of $G_{H}$ being
small signifies its robust nature against disorder as illuminated in the main
text.

Let us first consider the situation in which all the subprobes in each of the
SPVPs, denoted by A, B, C, and D in Fig. 7 (a), are coupled to the HES with
uniform tunneling strength ($t^{\prime}$) with equal $N_{P}$ for each of them.
The polarization angle of the SPVPs are further set according to
$\theta_{A}=\theta_{C}=-\theta_{B}=-\theta_{D}=-\pi/2$ such that SPVP A and C
measures the voltage for the $\uparrow$-channel only (note we have considered
a shift in $\theta$ by $\pi/2$ at the starting of our formulation) and SPVP B
and D do so for the $\downarrow$-channel only (see Eq. 7). In Fig. 7 (c) we
have shown a Hall bar geometry where the Hall voltage can be defined as the
voltage difference measured between the voltage probes $2,6$ or $3,5$. In the
setup for the HES in Fig. 7 (a), the analog of Hall voltage $V_{H}$ is
represented by the voltage difference between the $\uparrow$-channel and the
$\downarrow$-channel on the same side of the FM barrier,
$V_{H}=V_{R}-(rV_{R}+tV_{L})=t(V_{R}-V_{L})$ where the FM barrier plays the
role of the QPC in the Hall bar geometry given in Fig. 7 (c) and further
demonstrated in Fig. 7 (b). The net current in the HES is
$I=tV_{R}+rV_{L}-V_{L}=t(V_{R}-V_{L})$ and hence, the Hall resistance can be
defined as $R_{H}=V_{H}/I=1$ (in the units of $e^{2}/h$) which is quantized
Streda  _et al._ (1987b). The longitudinal voltage drop $V_{lo}$ is the
voltage difference along a given spin-polarized edge ($\uparrow$ or
$\downarrow$) across the barrier which is given by
$V_{lo}=V_{R}-(tV_{R}+rV_{L})=r(V_{R}-V_{L})$ and so, the longitudinal
resistance is given by $R_{lo}=V_{lo}/I=r/t$ which is nothing but the four-
probe resistance Imry (2002) for the FM barrier. The sum of these two
resistances satisfies $R_{lo}+R_{H}=1/t$ which is expected from the Landauer
formula Landauer (1957).

Now we will perform a self consistent numerical analysis to implement the
setup described above in terms of our model for the SPVP and check if it
reproduces the expected behaviors of $R_{H}$ and $R_{lo}$. The numerical
calculations are performed considering the coupled equations for the subprobes
given in Eq. 9 for all the four SPVPs. These equations (each for SPVP A, B, C,
and D) are further coupled to each other via transfer matrix for the current
which, in the case between SPVP A and SPVP B, and between SPVP C and SPVP D,
is an identity matrix while between SPVP B and SPVP C, is the transfer matrix
for the FM barrier given in Eq. 14. Finally, the voltage probe condition is
imposed simultaneously on SPVP A, B, C, and D and the full set of equations
are solved numerically. As fluctuations in both $t^{\prime}$ and $\theta$ for
each of the SPVPs are expected to exist in a realistic situation, we perform
our numerical analysis in their presence and check for the stability in the
obtained value of $V_{R^{\prime}}$ for each probe.

As discussed in the previous section, we model such fluctuations with a
Gaussian distribution of $\theta$ among the subprobes in each of the SPVPs
with the mean $\mkern 1.5mu\overline{\mkern-1.5mu\theta_{A}\mkern-1.5mu}\mkern
1.5mu=\mkern 1.5mu\overline{\mkern-1.5mu\theta_{C}\mkern-1.5mu}\mkern
1.5mu=-\mkern 1.5mu\overline{\mkern-1.5mu\theta\mkern-1.5mu}\mkern
1.5mu_{B}=-\mkern 1.5mu\overline{\mkern-1.5mu\theta_{D}\mkern-1.5mu}\mkern
1.5mu=-\pi/2$ and standard deviation $\sigma_{\theta}$ ranging from $10^{-4}$
to $\pi/36$ ($\sim 5.56\%$ of $\pi/2$). We further include such Gaussian
disorder in the tunneling strength $t^{\prime}$ as well with a mean $\mkern
1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu\in[0.01,1]$ and
standard deviation $\sigma_{t^{\prime}}=0.01$ in each of the SPVPs. For the
coupling strength of the barrier, ${\cal B}=0.3$, which corresponds to a
transmission probability of $t=91.39\%$ (using Eq. B in the Appendix), we
study the Hall conductance $G_{H}$ (this involves voltage difference between
SPVP A and B) and the longitudinal conductance $G_{lo}$ (involving voltage
difference between SPVP A and C) as a function of $\mkern
1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu$ and
$\sigma_{\theta}$ following a disorder averaging over $10^{3}$ realizations.
The results are shown in Fig. 8.

For the FM barrier with transmission probability of $91.39\%$, the theoretical
values for $G_{H}=1$ and $G_{lo}=t/r\approx 10.6$. We note from Fig. 8 that
the our numerical analysis reproduces these predicted theoretical values very
well in the limit of $\sigma_{\theta}\rightarrow 0$ independent of the value
of $\mkern 1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu$
where $\sigma_{t^{\prime}}$ is taken to be $0.01$. The plot illuminates the
robust nature of the disorder-averaged value of $G_{H}$ which shows
significantly small variations in its value when plotted as a function of
$\sigma_{\theta}$. An increase in $\mkern
1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu$ tends to
stabilize the value of $G_{H}$ and forces it stay close to the value
corresponding to $\sigma_{\theta}=0$. But we must keep in mind that the
standard deviation in $G_{H}$ should also increase as we increase $\mkern
1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu$ which is known
from the results presented in Fig. 6, i.e., $\sigma_{V_{R^{\prime}}}$
increases with increasing $\sigma_{\theta}$ at a faster rate for larger values
of $\mkern 1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu$ and
hence, $G_{H}$ can develop large fluctuations due to its dependence on
$V_{R^{\prime}}$. Hence, though the average $G_{H}$ stays stable with $\mkern
1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu$, the
fluctuations in its values tend to increase in this limit implying that it
will be safe to operate in the small $\mkern
1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu$ limit for
reproducing the theoretically expected results.

For the longitudinal conductance $G_{lo}$, we observe approximately a constant
value as we increase disorder in the polarization angle ($\sigma_{\theta}$) at
small values of $\bar{t^{\prime}}$, however, the variation increases with
increasing $\bar{t^{\prime}}$ as seen in Fig. 8 (b). This can be understood
better by inspecting the output voltages in the individual SPVPs (measured
with respect to their values for the uniform case, i.e., the disorder free
case) as plotted in Fig. 9 at different values of $\bar{t^{\prime}}$. At very
small values of $\bar{t^{\prime}}$ [e.g. $\bar{t^{\prime}}=0.1$ as in Fig. 9
(a)], the profile of $V_{m}-V_{m}^{(0)}$ ($V_{m}^{(0)}$ denoting the value for
the disorder free case) for $m=A$ and $m=C$ (and likewise, $m=B$ and $m=D$)
almost match resulting in small variation in $G_{lo}$ as observed in Fig. 8
(b) while it can have an adverse effect on $G_{H}$ though small [Fig. 8 (a)].
But with increasing $\bar{t^{\prime}}$ [e.g. $\bar{t^{\prime}}=0.5$ or higher
as in Fig. 9 (b)], the profile of $V_{A}-V_{A}^{(0)}$ and $V_{C}-V_{C}^{(0)}$
(and likewise, $V_{B}-V_{B}^{(0)}$ and $V_{D}-V_{D}^{(0)}$) tend to move away
from each other resulting in $G_{lo}$ to fall off steeply from its uniform
value with increasing $\sigma_{\theta}$.

To conclude, we note that the small $\bar{t^{\prime}}$ limit is very important
for implementing multi-terminal spin-resolved voltage measurements in presence
of disorder (i.e., finite $\sigma_{\theta}$ and $\sigma_{t^{\prime}}$) in the
SPVPs with $N_{P}\gg 1$. Of course in the ideal limit of $N_{P}=1$, the value
of ${t^{\prime}}$ is of no consequence as long as
${\theta_{A}}={\theta_{C}}=-{\theta}_{B}=-{\theta_{D}}=-\pi/2$ but that will
correspond to an unrealistic situation as in any ohmic contact tunnel-coupled
to HES, which could be used as a voltage probe, is expected to host multiple
modes owing to its finite size.

Figure 9: Plot of individual probe voltages with respect to their values for
the uniform case, i.e., $V_{m}-V_{m}^{(0)}$ ($m=A,B,C,D$), as function of
$\sigma_{\theta}$ for $\mkern
1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern 1.5mu=0.1$ in figure
(a) and $\mkern 1.5mu\overline{\mkern-1.5mut^{\prime}\mkern-1.5mu}\mkern
1.5mu=0.5$ in figure (b). A disorder averaging has been performed over 200
realizations in each of the plots.

##  IV A single chiral edge as a SPVP

In the study presented above, we have considered a theoretical model where the
SPVP comprises a number of chiral one-dimensional spin-polarized modes which
are mutually incoherent with each other and are tunnel-coupled to the HES via
local tunneling. In this section, we consider the possibility of using a
simpler model for the SPVP where a single spin-polarized chiral edge is
employed for the purpose and the tunnel-coupling between the spin-polarized
chiral edge and the HES is taken to be spread out over an extended region
rather than being local which makes it complementary to our previous model. We
simulate a lattice model involving a SPVP coupled to the HES of a QSHS using
the software package KWANT Groth  _et al._ (2014). The full lattice model
consists of three parts (schematically shown in Fig. 10): (i) a quantum spin
Hall (QSH) region described by the BHZ model Bernevig  _et al._ (2006) in its
topological phase hosting HES at the edge, (ii) a quantum anomalous Hall (QAH)
region described by the BHZ model with an applied exchange field which drives
it into the QAH phase that supports a single spin-polarized chiral edge mode
whose polarization can be controlled via rotating the direction of the applied
field and, (iii) an insulating barrier between the QAH edge and the HES which
is described by the BHZ model in its trivial phase. The dimensions of the QSH
region are taken to be $L\times W_{S}$ with $L=100a$, $W_{S}=147a$, and $a=3$
nm being the lattice constant. The dimensions of the QAH region is taken as
$L\times W_{A}$ with $W_{A}=50a$, and that of the insulating region is taken
to be of dimensions $3a\times 3a$.

Figure 10: Model of a voltage probe consisting of a single chiral edge acting
as a SPVP (constructed from a QAH system) tunnel-coupled to the HES of a QSHS
as used in the KWANT simulation ($L$, $W_{S}$, and $W_{A}$ specified in the
main text). Lead-1 and Lead-2 represent the leads which act as the electron
reservoirs used to apply a finite voltage bias across the HES while Lead-3 is
used as a voltage probe with probe voltage given by $V_{R^{\prime}}$. The FM
is employed to open a finite gap in the spectrum of the electronic states
lying on the lower HES. An insulating region of dimension $W_{I}$ (specified
in the main text) is used to facilitate a controlled tunneling between the QAH
edge and the HES. The polarization of the HES is along $S_{z}$ while that of
the QAH edge is tilted from $S_{z}$ at an angle $\theta_{P}$ which can be
tuned continuously.

The bulk of the QSH region is given by the BHZ Hamiltonian

$\displaystyle H_{\rm
BHZ}=-Dk^{2}+Ak_{x}\sigma_{z}\tilde{\sigma_{x}}-Ak_{y}\tilde{\sigma_{y}}+(M-Bk^{2})\tilde{\sigma_{z}},$
(15)

where $\sigma$ and $\tilde{\sigma}$ denote the Pauli matrices to describe the
spins ($\uparrow$ or $\downarrow$ along $S_{z}$) and the orbitals ($s$ or $p$
type) respectively and, $A$, $B$, $D$, and $M$ are material-dependent
parameters. For HgTe/CdTe quantum wells, $D=-512.0~{}{\rm nm}^{2}$-meV,
$A=364.5$ nm-meV, $B=-686.0~{}{\rm nm}^{2}$-meV, and $M=-10.0$ meV, however,
in our simulation, we set $D=0$ to place the Dirac point at zero energy. A
discrete form of the Hamiltonian in Eq. 15 can be identified on a square
lattice with a basis of two sites (representing the two orbitals) using
$k^{2}=2a^{-2}[2-\cos(k_{x}a)-\cos(k_{y}a)]$, $k_{x}=a^{-1}\sin(k_{x}a)$, and
$k_{y}=a^{-1}\sin(k_{y}a)$ ($a$ being the lattice constant specified before),
which reads Chen  _et al._ (2016); Adak  _et al._ (2020)

$H_{\rm
tb}=\sum_{i}(c_{i}^{\dagger}H_{i,i+a_{x}}c_{i+a_{x}}+c_{i}^{\dagger}H_{i,i+a_{y}}c_{i+a_{y}}+{\rm
h.c.})+c_{i}^{\dagger}H_{ii}c_{i},$ (16)

where
$c_{i}^{\dagger}\equiv(c^{\dagger}_{i,s,\uparrow},c^{\dagger}_{i,p,\uparrow},c^{\dagger}_{i,s,\downarrow},c^{\dagger}_{i,p,\downarrow})$
denotes the set of creation operators for the electrons in $s$ and $p$ orbital
with $\uparrow$ and $\downarrow$ spins at site $i$ with coordinates
$i=(i_{x},i_{y})$; $a_{x}=a(1,0)$ and $a_{y}=a(0,1)$ are the lattice vectors.
Each of the terms, $H_{ii}$ and $H_{i,i+a_{x}(a_{y})}$, is a $4\times 4$ block
matrices defined by

$\displaystyle H_{ii}$
$\displaystyle=-\frac{4D}{a^{2}}-\frac{4B}{a^{2}}\tilde{\sigma_{z}}+M\tilde{\sigma_{z}}$
$\displaystyle H_{i,i+a_{x}}$
$\displaystyle=\frac{D+B\tilde{\sigma_{z}}}{a^{2}}+\frac{A\sigma_{z}\tilde{\sigma_{x}}}{2ia}$
$\displaystyle H_{i,i+a_{y}}$
$\displaystyle=\frac{D+B\tilde{\sigma_{z}}}{a^{2}}+\frac{iA\tilde{\sigma_{y}}}{2a}.$
(17)

The lattice constant is set to $a=3$ nm to obtain a realistic band structure.
As mentioned earlier, the spin-polarized chiral edge, that models the SPVP in
this case, is obtained from the QSHS by inducing a topological phase
transition via the application of an exchange field of strength $g_{0}$ in the
BHZ Hamiltonian ( $|g_{0}|>|M|$ Liu  _et al._ (2008); Li  _et al._ (2013) ),
where the direction of spin-polarization of the chiral edge can be tuned by
tuning the direction of the applied exchange field and replacing the
coefficient of $A\,k_{x}$ term with an appropriate term as done below. Hence
the model Hamiltonian reads as

$\displaystyle\tilde{H}_{\rm BHZ}=-Dk^{2}+Ak_{x}({\hat{\bf
a}}.{\bf\sigma})\tilde{\sigma_{x}}-Ak_{y}\tilde{\sigma_{y}}$
$\displaystyle+(M-Bk^{2})\tilde{\sigma_{z}}$ $\displaystyle+g_{0}({\hat{\bf
a}}.{\bf\sigma})\tilde{\sigma_{z}},$ (18)

where we have replaced the term $Ak_{x}\sigma_{z}\tilde{\sigma_{x}}$ in Eq. 15
with a new term $Ak_{x}({\hat{\bf a}}.{\bf\sigma})\tilde{\sigma_{x}}$ such
that $S_{\bf a}$ (the component of spin along $\hat{\bf a}$) is conserved and
the unit vector ${\hat{\bf a}}$ deciding the new direction of spin-
polarization on the edge. The corresponding lattice model is given by

$\displaystyle\tilde{H}_{ii}$
$\displaystyle=-\frac{4D}{a^{2}}-\frac{4B}{a^{2}}\tilde{\sigma_{z}}+M\tilde{\sigma_{z}}+g_{0}({\hat{\bf
a}}.{\bf\sigma})\tilde{\sigma_{z}}$ $\displaystyle\tilde{H}_{i,i+a_{x}}$
$\displaystyle=\frac{D+B\tilde{\sigma_{z}}}{a^{2}}+\frac{A({\hat{\bf
a}}.{\bf\sigma})\tilde{\sigma_{x}}}{2ia}$ $\displaystyle\tilde{H}_{i,i+a_{y}}$
$\displaystyle=\frac{D+B\tilde{\sigma_{z}}}{a^{2}}+\frac{iA\tilde{\sigma_{y}}}{2a}.$
(19)

In our case, we consider $g_{0}=-15$ meV in the QAH region which, for
$\theta_{P}=0$, yields a chiral $\uparrow$ edge in the QAH region and for
$\theta_{P}=\pi$, a chiral $\downarrow$ edge. A lead (denoted as Lead-3 in
Fig. 10) consisting of the same Hamiltonian as in Eq. 19 is attached to the
bottom edge (the edge away from the QSH region) of the QAH region through
which the spin-resolved probe voltage $V_{R^{\prime}}$ could be measured as a
function of the polarization angle $\theta_{P}\equiv\cos^{-1}({\hat{\bf
a}}.{\bf\sigma})\in[0,\pi]$ as described below. The insulating region is a
trivial insulator, characterized by the same Hamiltonian as in Eq. 19,
however, $g_{0}=0$ and the sign of the mass term $M$ flipped (taken $M=10.0$
meV) in this region as required for driving a QSHS into a trivial insulating
state.

Figure 11: (a) Plot of the current density on the entire lattice used in the
simulation at two values of the probe polarization angle $\theta_{P}$. The
tick labels in the figure are given in nanometers. The bright regions indicate
the presence of the (chiral) current with its flow guided by the black arrows.
(b) Plots of $T_{\uparrow}$, $T_{\downarrow}$, and $R$ (defined in the main
text) as function of $\theta_{P}$ as obtained from the numerics for an
incident energy of $E=10^{-3}$ meV. The inset shows the ones obtained from an
analytical calculation (details mentioned in the main text).

With this setup, we first study the current density which is proportional to
the transmittance from the chiral (QAH) edge to the helical edge of the QSH
region, denoted $T_{\uparrow}$ and $T_{\downarrow}$ for the $\uparrow$-channel
and $\downarrow$-channel respectively (the reflectance along the chiral
channel is $R=1-T_{\uparrow}-T_{\downarrow}$). We plot the current density in
Fig. 11 (a) for two values of $\theta_{P}$ when electrons are incident at
energy $E=10^{-3}$ meV which clearly indicates the spin-polarization direction
of QAH edge with respect to the two spin-momentum locked modes of the HES. As
mentioned above, at a given value of $\theta_{P}$, the spin-polarization of
the chiral edge has components along that of both the channels of the helical
edge resulting in transmission into the $\uparrow$-channel with magnitude
$T_{\uparrow}$ and into the $\downarrow$ channel with magnitude
$T_{\downarrow}$. An interesting feature to note in this plot is that when
$\theta_{P}$ is changed to $\pi$ from $0$, the positions of the two blocks
characterized by $\sigma_{z}=\pm 1$ (the eigenvalues of $\sigma_{z}$ being
good quantum numbers as $[\sigma_{z},\tilde{H}_{\rm BHZ}]=0$ at
$\theta_{P}=0,\pi$) in the modified BHZ Hamiltonian $\tilde{H}_{\rm BHZ}$ (Eq.
18) are switched, and as a result, when $g_{0}$ is applied with an appropriate
magnitude to counter the mass term $M$, the spin on the (chiral) edge that
survives gets flipped, however, the chirality is retained. This is evident
from the direction of the current density (black arrows) in both the plots at
$\theta_{P}=0$ and $\theta_{P}=\pi$. The spin of the chiral (QAH) edge is
flipped for $\theta_{P}:0\rightarrow\pi$ and consequently the transmittance
changes from $T_{\uparrow}(\theta_{P}=0)\neq 0,T_{\downarrow}(\theta_{P}=0)=0$
to $T_{\uparrow}(\theta_{P}=\pi)=0,T_{\downarrow}(\theta_{P}=\pi)\neq 0$ as
shown in Fig. 11 (b). The zeros in the transmission probabilities demonstrate
a perfect suppression in electron tunneling between the spin orthogonal modes.
The plot in Fig. 11 (b) shows the full behavior of $T_{\uparrow}$,
$T_{\downarrow}$, and $R$ as function of $\theta_{P}$. The inset displays the
behaviors of $T_{\uparrow}$ , $T_{\downarrow}$, and $R$ for the model depicted
in Fig. 2 for a single subprobe with a local tunnel-coupling between the HES
and the subprobe obtained analytically from the scattering matrix at a
tunneling strength of $t^{\prime}=0.5$ (details given in Appendix A). Note
that our two-dimensional transport simulation using KWANT shows a close
qualitative resemblance to the ideal model for the subprobe discussed in the
previous section. The discrepancy in the KWANT results compared to the
analytical ones can be minimized by adjusting the size of the insulating
region and the plots shown above are the closest we could obtain that exhibit
a slight variation of $R$ against $\theta_{P}$ (albeit about a mean value that
matches the analytics). Note that the simple-minded one-dimensional model for
the subprobe assumes a tunnel-coupling which respects spin-rotation symmetry
and this fact leads to the symmetric variation of $T_{\uparrow}$ and
$T_{\downarrow}$ about $\theta_{P}=\pi/2$. But, on the other hand, the
presence of finite spin-nonconserving tunneling events at the junction between
the SPVP and the HES owing to the presence of an applied exchange field at the
microscopic level in the lattice Hamiltonian constituting the junction always
leads to the observed asymmetry in the main figure of Fig. 11 (b).

Now let us consider a situation where $V_{R}>V_{L}$. Then the incoming current
flowing into the probe is $I_{\uparrow}=T_{\uparrow}(V_{R}-V_{R^{\prime}})$
while the outgoing one is
$I_{\downarrow}=T_{\downarrow}(V_{R^{\prime}}-V_{L})$. Assuming that the
transmission probabilities $T_{\uparrow}$ and $T_{\downarrow}$ are independent
of energy in the bias window $V_{R}-V_{L}$, the voltage probe condition,
implying a zero net current in the probe, leads to

$\displaystyle
V_{R^{\prime}}=\frac{T_{\uparrow}V_{R}+T_{\downarrow}V_{L}}{T_{\uparrow}+T_{\downarrow}}=\frac{V_{R}+V_{L}}{2}+\frac{1}{2}\frac{T_{\uparrow}-T_{\downarrow}}{T_{\uparrow}+T_{\downarrow}}(V_{R}-V_{L}),$
(20)

similar to Eq. 6, which we plot against $\theta_{P}$ in Fig. 12 (a). Note that
the expression for $V_{R^{\prime}}$ has two contributions: the first one is
the average voltage of the left and the right moving edge which would have
been the full contribution in absence of electron spin, while the second term,
being proportional to $(V_{R}-V_{L})$, is indeed the magnetoresistance
contribution. In the plot presented in Fig. 12 (a), $V_{R}=-V_{L}$ and hence,
it features only the magnetoresistance response.

Figure 12: (a) Plot of the probe voltage $V_{R^{\prime}}$ vs. $\theta_{P}$ as
calculated from Eq. 20 showing its variation from $V_{R^{\prime}}=V_{R}=1$ at
$\theta_{P}=0$ to $V_{R^{\prime}}=V_{L}=-1$ at $\theta_{P}=\pi$ as expected.
(b) A plot of $T_{\uparrow}$ vs. the incident energy $E$ at different values
of $\theta_{P}$ reveals an energy window $E\in\delta E=[-5,5]$ meV within
which the spin-resolved voltage measurement should be performed.

Finally, it is important to identity an energy window in which the
transmission probabilities have a very week energy dependence for the SPVP to
work and hence, we try to check for the presence of such an energy window in
our lattice simulations. This is expected to work as we are working with edge
states whose spectrum can be approximated to be linear to a large extent. We
provide an estimate of an energy window $\delta E$ for our setup such that
$V_{R/L}\in[-\delta E/2,\delta E/2]$ leads to treating $T_{\uparrow}$ and
$T_{\downarrow}$ as constants under the variation of $E$. This window is shown
in Fig. 12 (b) by the yellow shaded region in the plot of $T_{\uparrow}$
against the incident energy $E$ at various values of $\theta_{P}$. This plot
clearly suggests that, within the band gap $\delta E$, $T_{\uparrow}$ and
$T_{\downarrow}$ can indeed be approximated as constants but beyond this gap,
the transmittances largely deviate from their constant values due to the
contributions of the bulk bands gradually coming into play.

##  V Conclusion

Spin-momentum locked spectrum of surface states of two and three-dimensional
topological insulators is a resource for a variety of spintronics
applications. Though these states have no net polarization in equilibrium,
once a finite bias is applied, they do develop finite polarization at the
Fermi level and this fact can be exploited in a number of device applications.
One way to measure such polarization is to employ spin-polarized voltage
probes which could measure the spin-resolved voltages in these states
subjected to a finite bias. In this work, we have explored two such
possibilities for the surface states of quantum spin Hall system which is a
two-dimensional topological insulator. Within our theoretical model, it is
established that such measurements should be possible if the probes are
designed appropriately Tian  _et al._ (2015).

##  VI Acknowledgments

VA acknowledges support from IISER Kolkata in the form of a subsistence grant.
VA also wants to thank Rafiqul Rahaman for helping him with job submission in
the cluster computing facility. KR thanks the sponsorship, in part, by the
Swedish Research Council. SD would like to acknowledge the MATRICS grant (MTR/
2019/001 043) from the Science and Engineering Research Board (SERB) for
funding. We acknowledge the central computing facility (DIRAC supercomputer)
and the computational facility at the Department of Physics (KEPLER) at IISER
Kolkata.

##  Appendix A Derivation of the scattering matrix elements at a junction of a
chiral edge tunnel-coupled to the HES

The scattering amplitudes for a wavefunction ${\psi}_{\eta}^{(j)}$ with
$\eta\in\\{R,L,R^{\prime}\\}$ across a point contact (PC) located at $x=x_{j}$
can be obtained by studying its equation of motion (e.o.m)

$\imath\hbar\dot{\psi}_{\eta}^{(j)}=[\psi_{\eta}^{(j)},\mathcal{H}^{(j)}],$
(21)

where $\mathcal{H}^{(j)}=\mathcal{H}_{\rm HES}+\mathcal{H}_{\rm
subprobe}^{(j)}+\mathcal{H}_{T}^{(j)}$ is specified in the main text (see Eq.
1-3). Integrating the e.o.m over a region from $-\epsilon$ and $\epsilon$ with
the limit $\epsilon\rightarrow 0$ across $x=x_{j}$, one obtains the required
equations between the incoming and the outgoing amplitudes connected through
the PC Wadhawan  _et al._ (2018). From now on, we drop the superscript $j$
assuming all of the following relations pertain to the $j$-th PC.

The scattering matrix elements $s_{\eta\eta^{\prime}}$, which appear in Eq.
(4), are defined as

$\underbrace{\begin{pmatrix}\psi_{R}(0^{+})\\\ \psi_{L}(0^{-})\\\
\psi_{R^{\prime}}(0^{+})\end{pmatrix}}_{\psi^{\rm
out}}=\underbrace{\begin{pmatrix}s_{RR}&s_{RL}&s_{RR^{\prime}}\\\
s_{LR}&s_{LL}&s_{LR^{\prime}}\\\
s_{R^{\prime}R}&s_{R^{\prime}L}&s_{R^{\prime}R^{\prime}}\end{pmatrix}}_{S-{\rm
matrix}}\underbrace{\begin{pmatrix}\psi_{R}(0^{-})\\\ \psi_{L}(0^{+})\\\
\psi_{R^{\prime}}(0^{-})\end{pmatrix}}_{\psi^{\rm in}},$ (22)

where $s_{RR}=\psi_{R}(0^{+})/\psi_{R}(0^{-})$,
$s_{LR}=\psi_{L}(0^{-})/\psi_{R}(0^{-})$,
$s_{R^{\prime}R}=\psi_{R^{\prime}}(0^{+})/\psi_{R}(0^{-})$, and so on. The
explicit expressions for these elements, after taking $v_{F}=1$, read

$\displaystyle s_{RR}$
$\displaystyle=[8+\iota(\Gamma_{RL}\Gamma_{RR^{{}^{\prime}}}^{\ast}\Gamma_{LR^{{}^{\prime}}}+\Gamma_{RL}^{\ast}\Gamma_{RR^{{}^{\prime}}}\Gamma_{LR^{{}^{\prime}}}^{\ast})$
$\displaystyle~{}~{}~{}~{}~{}~{}-2\\{|\Gamma_{RL}|^{2}+|\Gamma_{RR^{{}^{\prime}}}|^{2}-|\Gamma_{LR^{{}^{\prime}}}|^{2}\\}]/{\cal
D}$ $\displaystyle s_{RL}$
$\displaystyle=\left[-8\iota\Gamma_{RL}-4\Gamma_{RR^{{}^{\prime}}}\Gamma_{LR^{{}^{\prime}}}^{\ast}\right]/{\cal
D}$ $\displaystyle s_{RR^{\prime}}$
$\displaystyle=\left[-8\iota\Gamma_{RR^{{}^{\prime}}}-4\Gamma_{RL}\Gamma_{LR^{{}^{\prime}}}\right]/{\cal
D}$ $\displaystyle s_{LR}$
$\displaystyle=\left[-8\iota\Gamma_{RL}^{\ast}-4\Gamma_{RR^{{}^{\prime}}}^{\ast}\Gamma_{LR^{{}^{\prime}}}\right]/{\cal
D}$ $\displaystyle s_{LL}$
$\displaystyle=[8+\iota(\Gamma_{RL}\Gamma_{RR^{{}^{\prime}}}^{\ast}\Gamma_{LR^{{}^{\prime}}}+\Gamma_{RL}^{\ast}\Gamma_{RR^{{}^{\prime}}}\Gamma_{LR^{{}^{\prime}}}^{\ast})$
$\displaystyle~{}~{}~{}~{}~{}~{}-2\\{|\Gamma_{RL}|^{2}-|\Gamma_{RR^{{}^{\prime}}}|^{2}+|\Gamma_{LR^{{}^{\prime}}}|^{2}\\}]/{\cal
D}$ $\displaystyle s_{LR^{\prime}}$
$\displaystyle=\left[-8\iota\Gamma_{LR^{{}^{\prime}}}-4\Gamma_{RL}^{\ast}\Gamma_{RR^{{}^{\prime}}}\right]/{\cal
D}$ $\displaystyle s_{R^{\prime}R}$
$\displaystyle=\left[-8\iota\Gamma_{RR^{{}^{\prime}}}^{\ast}-4\Gamma_{RL}^{\ast}\Gamma_{LR^{{}^{\prime}}}^{\ast}\right]/{\cal
D}$ $\displaystyle s_{R^{\prime}L}$
$\displaystyle=\left[-8\iota\Gamma_{LR^{{}^{\prime}}}^{\ast}-4\Gamma_{RL}\Gamma_{RR^{{}^{\prime}}}^{\ast}\right]/{\cal
D}$ $\displaystyle s_{R^{{}^{\prime}}R^{{}^{\prime}}}$
$\displaystyle=[8+\iota(\Gamma_{RL}\Gamma_{RR^{{}^{\prime}}}^{\ast}\Gamma_{LR^{{}^{\prime}}}+\Gamma_{RL}^{\ast}\Gamma_{RR^{{}^{\prime}}}\Gamma_{LR^{{}^{\prime}}}^{\ast})$
$\displaystyle~{}~{}~{}~{}~{}~{}-2\\{-|\Gamma_{RL}|^{2}+|\Gamma_{RR^{{}^{\prime}}}|^{2}+|\Gamma_{LR^{{}^{\prime}}}|^{2}\\}]/{\cal
D},$ (23)

where the common denominator, ${\cal D}$ is

$\begin{split}{\cal
D}=8-\iota\left(\Gamma_{RL}\Gamma_{RR^{{}^{\prime}}}^{\ast}\Gamma_{LR^{{}^{\prime}}}+\Gamma_{RL}^{\ast}\Gamma_{RR^{{}^{\prime}}}\Gamma_{LR^{{}^{\prime}}}^{\ast}\right)\\\
+2\\{|\Gamma_{RL}|^{2}+|\Gamma_{RR^{{}^{\prime}}}|^{2}+|\Gamma_{LR^{{}^{\prime}}}|^{2}\\}.\end{split}$
(24)

The functional form of $\Gamma_{\eta\eta^{\prime}}$ in terms of $t^{\prime}$
and $\theta$ are noted in the main text from which all our results readily
follow.

##  Appendix B Derivation of the scattering matrix elements for the FM barrier

The $2\times 2$ scattering matrix defined at the FM barrier of the Fig. 7 can
be easily obtained by following the procedure discussed in appendix A. as

$\underbrace{\begin{pmatrix}\psi_{R}(x_{0}^{+})\\\ \\\
\psi_{L}(x_{0}^{-})\end{pmatrix}}_{\psi^{\rm
out}}=\underbrace{\begin{pmatrix}s_{RR}^{FM}&s_{RL}^{FM}\\\ \\\
s_{LR}^{FM}&s_{LL}^{FM}\end{pmatrix}}_{S-{\rm
matrix}}\underbrace{\begin{pmatrix}\psi_{R}(x_{0}^{-})\\\ \\\
\psi_{L}(x_{0}^{+})\end{pmatrix}}_{\psi^{\rm in}}.$ (25)

The explicit expressions for the $S$-matrix elements are given by

$\displaystyle s_{RR}^{FM}$
$\displaystyle=\left[8-2|\Gamma^{FM}_{RL}|^{2}\right]/{\cal D}^{\prime}$
$\displaystyle s_{RL}^{FM}$
$\displaystyle=\left[-8\iota\Gamma^{FM}_{RL}\right]/{\cal D}^{\prime}$
$\displaystyle s_{LR}^{FM}$
$\displaystyle=\left[-8\iota(\Gamma^{FM}_{RL})^{*}\right]/{\cal D}^{\prime}$
$\displaystyle s_{LL}^{FM}$
$\displaystyle=\left[8-2|\Gamma^{FM}_{RL}|^{2}\right]/{\cal D}^{\prime},$ (26)

where the common denominator, ${\cal D}^{\prime}$ is

${\cal D}^{\prime}=\left[8+2|\Gamma^{FM}_{RL}|^{2}\right].$ (27)

Similar functional forms of $\Gamma^{FM}_{\eta\eta^{\prime}}$ follow with
$t^{\prime}$ being replaced by $\cal B$ and the angular dependence having no
relevance.

## References

  * Igor  _et al._ (2004) Igor, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
  * Hirohata  _et al._ (2020) A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu, B. Diény, P. Pirro, and B. Hillebrands,  _Review on spintronics: Principles and device applications_ (ELSEVIER, 2020).
  * Awschalom and Samarth (2002) D. Awschalom and N. Samarth,  _Semiconductor Spintronics and Quantum Computation_ (Springer, 2002).
  * Pesin and MacDonald (2012) D. Pesin and A. H. MacDonald, Nature Materials 11, 409 (2012).
  * He  _et al._ (2019) M. He, H. Sun, and Q. L. He, Frontiers of Physics 14, 43401 (2019).
  * Yongbing Xu (2015) J. N. Yongbing Xu, David D. Awschalom,  _Spintronics of Topological Insulators_ (Springer, 2015).
  * Kane and Mele (2005a) C. L. Kane and E. J. Mele, Physical review letters 95, 226801 (2005a).
  * Kane and Mele (2005b) C. L. Kane and E. J. Mele, Physical review letters 95, 146802 (2005b).
  * Bernevig and Zhang (2006) B. A. Bernevig and S.-C. Zhang, Physical Review Letters 96, 106802 (2006).
  * Bernevig  _et al._ (2006) B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006).
  * Arriola and Broniowski (2016) E. R. Arriola and W. Broniowski, Few-Body Systems 57, 485 (2016).
  * Nowack  _et al._ (2013) K. C. Nowack, E. M. Spanton, M. Baenninger, M. König, J. R. Kirtley, B. Kalisky, C. Ames, P. Leubner, C. Brüne, H. Buhmann, L. W. Molenkamp, D. Goldhaber-Gordon, and K. A. Moler, Nature Materials 12, 787 (2013).
  * Zhou and Zhang (2012) Y. Zhou and F.-C. Zhang, Nature Physics 8, 448 (2012).
  * Ferry (2015) D. K. Ferry,  _Transport in Semiconductor Mesoscopic Devices_, 2053-2563 (IOP Publishing, 2015).
  * Das and Rao (2011) S. Das and S. Rao, Phys. Rev. Lett. 106, 236403 (2011).
  * Streda  _et al._ (1987a) P. Streda, J. Kucera, and A. H. MacDonald, Phys. Rev. Lett. 59, 1973 (1987a).
  * Groth  _et al._ (2014) C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, New Journal of Physics 16, 063065 (2014).
  * Brüne  _et al._ (2012) C. Brüne, A. Roth, H. Buhmann, E. M. Hankiewicz, L. W. Molenkamp, J. Maciejko, X.-L. Qi, and S.-C. Zhang, Nature Physics 8, 485 (2012).
  * Gerstner (2007) E. Gerstner, Nature Physics 3, 754 (2007).
  * Streda  _et al._ (1987b) P. Streda, J. Kucera, and A. MacDonald, Physical review letters 59, 1973 (1987b).
  * Imry (2002) Y. Imry,  _Introduction to mesoscopic physics_ (Oxford University Press, 2002).
  * Landauer (1957) R. Landauer, IBM Journal of Research and Development 1, 223 (1957).
  * Chen  _et al._ (2016) W. Chen, W.-Y. Deng, J.-M. Hou, D. Shi, L. Sheng, and D. Xing, Physical Review Letters 117, 076802 (2016).
  * Adak  _et al._ (2020) V. Adak, K. Roychowdhury, and S. Das, Physical Review B 102, 035423 (2020).
  * Liu  _et al._ (2008) C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Physical review letters 101, 146802 (2008).
  * Li  _et al._ (2013) H. Li, L. Sheng, R. Shen, L. Shao, B. Wang, D. Sheng, and D. Xing, Physical review letters 110, 266802 (2013).
  * Tian  _et al._ (2015) J. Tian, I. Miotkowski, S. Hong, and Y. P. Chen, Scientific Reports 5, 14293 (2015).
  * Wadhawan  _et al._ (2018) D. Wadhawan, K. Roychowdhury, P. Mehta, and S. Das, Phys. Rev. B 98, 155113 (2018).