# Transmission Line Model for Materials with Spin-Momentum Locking

Shehrin Sayed<EMAIL_ADDRESS>School of Electrical and Computer
Engineering, Purdue University, West Lafayette, IN 47907, USA  Seokmin Hong
Center for Spintronics, Korea Institute of Science and Technology, Seoul
02792, Republic of Korea<EMAIL_ADDRESS>Supriyo Datta<EMAIL_ADDRESS>School of Electrical and Computer Engineering, Purdue University, West
Lafayette, IN 47907, USA

###### Abstract

We provide a transmission line representation for channels exhibiting spin-
momentum locking (SML) which can be used for both time-dependent and steady-
state transport analysis on a wide variety of materials with spin-orbit
coupling such as topological insulators, heavy metals, oxide interfaces, and
narrow bandgap semiconductors. This model is based on a time-dependent four-
component diffusion equation obtained from the Boltzmann transport equation
assuming linear response and elastic scattering in the channel. We classify
all electronic states in the channel into four groups ($U^{+}$, $D^{+}$,
$U^{-}$, and $D^{-}$) depending on the spin index (up ($U$), down ($D$)) and
the sign of the $x$-component of the group velocity ($+,-$) and assign an
average electrochemical potential to each of the four groups to obtain the
four-component diffusion equation. For normal metal channels, the model
decouples into the well-known transmission line model for charge and a time-
dependent version of Valet-Fert equation for spin. We first show that in the
steady-state limit our model leads to simple expressions for charge-spin
interconversion in SML channels in good agreement with existing experimental
data on diverse materials. We then use the full time-dependent model to study
spin-charge separation in the presence of SML ($p_{0}\neq 0$), a subject that
has been controversial in the past. Our model shows that the charge and spin
signals travel with two distinct velocities resulting in well-known spin-
charge separation which is expected to persist even in the presence of SML.
However, our model predicts that the lower velocity signal is purely spin
while the higher velocity signal is largely charge with an additional spin
component proportional to $p_{0}$, which has not been noted before. Finally,
we note that our model can be used within standard circuit simulators like
SPICE to obtain numerical results for complex geometries.

##  I Introduction

Background: Transport properties in materials with spin-orbit coupling (SOC)
are of great interest for potential spintronic applications, especially
because of the unique spin-momentum locking (SML) observed in diverse classes
of materials such as topological insulator (TI) Hasan and Kane (2010); Qi and
Zhang (2011); Fan and Wang (2016), heavy metals Miron  _et al._ (2011); Suzuki
_et al._ (2011); Pai  _et al._ (2012); Liu  _et al._ (2012); Yan  _et al._
(2015), oxide interfaces Lesne  _et al._ (2016); Song  _et al._ (2017), and
narrow bandgap semiconductors Hammar and Johnson (2000); Park  _et al._
(2012); Silsbee (2004). There has been an immense effort to model the
interplay between spin and charge in such materials using time-dependent
classical Tölle  _et al._ (2017) or quantum Boltzmann equation Liu and Sinova
(2012, 2013), nonequilibrium Green’s function Mishchenko  _et al._ (2004);
Burkov  _et al._ (2004); Burkov and Hawthorn (2010); Zainuddin  _et al._
(2011); Hong  _et al._ (2012), phenomenological equations coupled to magnet
dynamics Tserkovnyak and Bender (2014), and time-independent diffusion
equation used to explain bulk spin Hall effect Chen  _et al._ (2013).

Four-Component Diffusion Equation: In this paper, we propose a time-dependent
four-component diffusion equation that can be used for transport analysis on
multi-contact based structures implemented with materials exhibiting SML. The
model is obtained from the Boltzmann transport equation assuming linear
response and elastic scattering processes in the channel. The basic approach
is to assign one electrochemical potential $\mu(\vec{p},s)$ to each of the
eigenstates ($\vec{p},s$) where $\vec{p}$ is the momentum confined in the
$z$-$x$ plane and $s=\pm 1$ is the spin index with $+1$ and $-1$ denoting the
up ($U$) and the down ($D$) spins with respect to the spin quantization axis
$\hat{y}\times\left(\vec{p}-q\vec{A}\right)$ ($\vec{A}$ is the vector magnetic
potential).

We then classify the eigenstates into four groups ($U^{+}$, $D^{+}$, $U^{-}$,
and $D^{-}$) based on the spin index ($U$, $D$) and the sign of the
$x$-component of the group velocity ($+,-$) and define an average
electrochemical potential corresponding to the each of the four groups
resulting in a four-component diffusion equation. This can be viewed as an
extension of the Valet-Fert equation which uses two electrochemical potentials
for $U$ and $D$ states Valet and Fert (1993). The four-component diffusion
equation in steady-state reduces to our prior model Sayed  _et al._ (2016a);
Hong  _et al._ (2016) that we used to predict a unique three resistance state
on SML materials with two FM contacts in a multi-terminal spin valve structure
Sayed  _et al._ (2016a). The prediction has been observed recently on Pt Pham
_et al._ (2016a, b) and InAs Lee  _et al._ (2018) up to room temperature. We
expect the prediction to be observed on any channel exhibiting SML.

In our generalized view, $U^{+}$ (and $U^{-}$) states have same number of
modes $M$ (and $N$) as $D^{-}$ (and $D^{+}$) states due to the time reversal
symmetry. The degree of SML in our model is given by Sayed  _et al._ (2016a);
Hong  _et al._ (2016)

$p_{0}=\dfrac{M-N}{M+N},$ (1)

where $M$ and $N$ are evaluated at Fermi energy for zero temperature and in
general require thermal averaging. For normal metal (NM) channels $p_{0}=0$
i.e. $M=N$. For a perfect topological insulator (TI) $N=0$ leading to
$p_{0}=1$, however, $p_{0}$ gets effectively lowered by the presence of
parallel channels. $p_{0}$ has been quantified for different TIs by a number
of groups Li  _et al._ (2014); Tang  _et al._ (2014); Dankert  _et al._
(2015); Liu  _et al._ (2015); Tian  _et al._ (2015); Lee  _et al._ (2015);
Yang  _et al._ (2016) by measuring the charge current induced spin voltage
using a ferromagnetic (FM) contact, motivated by a theoretical proposal Hong
_et al._ (2012). For a Rashba channel with a coupling coefficient
$\alpha_{R}$, $p_{0}\approx\alpha_{R}k_{F}/(2E_{F})\ll 1$ Silsbee (2004); Hong
_et al._ (2012) which can be quantified with similar spin voltage measurements
Hammar and Johnson (2000); Park  _et al._ (2012). $k_{F}$ and $E_{F}$ are the
Fermi wave vector and Fermi energy respectively. Recently, spin voltage
measurements have been reported on heavy metals like platinum Pham  _et al._
(2016a, b) and gold Li and Appelbaum (2016). These experiments can also be
quantified by $p_{0}$, though the underlying mechanism is subject to active
debate Haney  _et al._ (2013); Sinova  _et al._ (2015); Hoffmann (2013) and
could involve a bulk spin Hall effect Liu  _et al._ (2012); Chen  _et al._
(2013); Wang  _et al._ (2014) or interface Rashba-like channel Yan  _et al._
(2015); Zhang  _et al._ (2014); Hoesch  _et al._ (2004); Tamai  _et al._
(2013).

Transmission Line Model: We translate our four-component ($U^{+}$, $D^{+}$,
$U^{-}$, and $D^{-}$) semiclassical model into a transmission line model with
two component (charge and $z$-component of spin) voltages and currents where
the coupling between charge and spin in a SML channel is characterized by
$p_{0}$ in Eq. (1). The model is compatible within a standard circuit
simulator tool like Simulation Program with Integrated Circuit Emphasis
(SPICE) which will enable straightforward analysis of complex geometries. The
transmission line model is a new addition to our multi-physics spin-circuit
framework Camsari  _et al._ (2013) which has been previously used to explain
experiments and evaluate spin-based device proposals Camsari  _et al._ (2015);
Sayed  _et al._ (2016b).

For NM channels (i.e. $p_{0}=0$), the proposed transmission line model
decouples into the well-known model for charge that has been previously used
to analyze transport in quantum wires Burke (2002, 2003); Salahuddin  _et al._
(2005) and a time-dependent version of Valet-Fert equation Valet and Fert
(1993) for spin. For SML channels (i.e. $p_{0}\neq 0$), our model lead to
several prior results on charge-spin interconversion in the steady-state limit
Hong  _et al._ (2012, 2016) that have been previously used by a number of
experimental groups Li  _et al._ (2014); Tang  _et al._ (2014); Dankert  _et
al._ (2015); Liu  _et al._ (2015); Tian  _et al._ (2015); Lee  _et al._
(2015); Yang  _et al._ (2016); Lee  _et al._ (2018) to quantify their spin
voltage measurements using potentiometric ferromagnetic contacts. We further
derive a simple expression and present SPICE simulation results for a
parameter that has been widely used to quantify the inverse Rashba-Edelstein
effect (IREE) in 2D channels, which are in good agreement with existing
experiments Sánchez  _et al._ (2016); Isasa  _et al._ (2016); Lesne  _et al._
(2016); Wang  _et al._ (2016) on diverse materials.

We then use the full time-dependent transmission line model to study the spin-
charge separation in the presence of SML in materials with SOC, a subject that
has been controversial in the past (see, for example, De Martino  _et al._
(2002); Calzona  _et al._ (2015); Stauber  _et al._ (2013); Moroz  _et al._
(2000)). Our model suggests that depending on the channel cross-section, the
charge signal can travel faster than the spin signal resulting in spin-charge
separation which is well-known for materials without SOC Halperin (2007);
Polini and Vignale (2007); Schroer  _et al._ (2014); Burke (2002) based on the
Luttinger liquid theory. We argue using our model that the spin-charge
separation persists even in the SOC materials exhibiting SML (i.e. $p_{0}\neq
0$). Similar arguments have been made in the past considering the presence of
spin-orbit coupling (SOC) De Martino  _et al._ (2002); Calzona  _et al._
(2015); Stauber  _et al._ (2013) although there exists counter arguments that
the presence of SOC destroys the spin-charge separation Moroz  _et al._
(2000). However, we predict that the high velocity charge signal in SML
channels accompanies an additional spin component proportional to $p_{0}$
having the same velocity as the charge, which has not been discussed before.

Note that the proposed model does not take into account the effects such as
spin precession involving the off-diagonal elements of the density matrix
which we assume to be negligible. An extension of this model to include $x$
and $y$ components of spin could possibly address such issues, as done earlier
for materials without SOC (see Camsari  _et al._ (2015), and references
therein). The assumptions made to derive the model have been discussed in
detail in Section V. Several predictions from our model for steady-state Sayed
_et al._ (2016a) have already received support from experiments Pham  _et al._
(2016b, a); Lee  _et al._ (2018) suggesting that the assumptions are within
the reasonable limits. The assumptions can be revisited as the field evolves
leading to revised model parameters, but the basic model should remain valid.

Outline: The paper is organized as follows. In Section II, we describe the
transmission line model for SML channels and show that special cases lead to
prior well-known models. In Section III, we derive several results on charge-
spin interconversion from our transmission line model in steady-state and
present comparison with SPICE simulations using the full model. We obtain a
simple expression for a parameter that has been widely used to quantify IREE
and show that it is in good agreement with available experiments on diverse
materials. In Section IV, we study the spin-charge separation in terms of spin
and charge signal velocities obtained from our time-dependent transmission
line equations. We show that the separation persists even in SML channels,
however, in SML channels there exists an additional spin component accompanied
by the high velocity charge signal. In Section V, we derive the transmission
line model starting from the Boltzmann transport equation with all the
assumptions clearly stated. We discuss different scattering mechanisms in the
channels and their effects on charge and spin transport. Finally, in Section
VI, we end with a brief summary.

##  II Transmission Line Model

###  II.1 Model Description

We consider the structure and axes in Fig. 1(a) to derive the transmission
line model. The model has two components: charge and $z$-component of spin
with coupling between them characterized by $p_{0}$ in Eq. (1). The charge
model is given by

$\displaystyle\left(\dfrac{1}{C_{E}}+\dfrac{1}{C_{Q}}\right)^{-1}\;\dfrac{\partial}{{\partial
t}}{V_{c}}=-\dfrac{\partial}{{\partial x}}I_{c},$ (2)
$\displaystyle\left(L_{K}+L_{M}\right)\;\dfrac{\partial}{{\partial
t}}I_{c}+R_{c}\,I_{c}=-\dfrac{\partial}{\partial
x}{V_{c}}+{p_{0}}{\eta_{c}}{V_{s}},$

where $I_{c}$ and $V_{s}$ are charge current and voltage along
$\hat{x}$-direction, $C_{E}$ and $C_{Q}$ are the electrostatic and quantum
capacitances per unit length, $L_{M}$ and $L_{K}$ are the magnetic and kinetic
inductances per unit length, and $R_{c}$ is the charge resistance per unit
length.

Figure 1: (a) Structure and corresponding axes of the channel with spin-
momentum locking (SML) under consideration, for which a transmission line
model is derived. The model has two components: (b) charge (corrseponds to Eq.
(2)) and (c) spin (corrseponds to Eq. (3)), with coupling between them
described by degree of SML $p_{0}$ (see Eq. (1)). Coupling between charge and
spin are modeled by dependent voltage and current sources with
$V_{m}=\eta_{c}V_{s}$, $V_{n}=\eta_{s}V_{c}+r_{m}I_{em}$, and
$I_{n}=\gamma_{s}I_{c}-g_{m}V_{em}$.

The spin model is given by

$\displaystyle\dfrac{C_{Q}}{\alpha^{2}}\dfrac{\partial}{{\partial
t}}{V_{s}}+G_{sh}{V_{s}}+p_{0}g_{m}V_{em}=-\dfrac{\partial}{{\partial
x}}{I_{s}}+p_{0}\gamma_{s}I_{c},$ (3)
$\displaystyle\alpha^{2}{L_{K}}\dfrac{\partial}{{\partial
t}}{I_{s}}+{R_{s}}{I_{s}}-p_{0}r_{m}I_{em}=-\dfrac{\partial}{{\partial
x}}{V_{s}}+p_{0}\eta_{s}{V_{c}},$

where $I_{s}$ and $V_{s}$ are spin current and voltage along
$\hat{x}$-direction with spin polarization along the $\hat{z}$-direction. In
this discussion, $\hat{y}$-direction is out-of-plane. Here, $\alpha=2/\pi$ is
an angular averaging factor, $R_{s}$ is the spin resistance per unit length of
the channel and $G_{sh}$ is the shunt conductance per unit length that
captures the spin lost in the channel due to the spin relaxation. Detailed
derivation of Eqs. (2) and (3) from the Boltzmann transport equation will be
discussed in Section V with clearly stated assumptions.

Distributed circuit models for charge and spin are shown in Fig. 1(b) and (c),
which are based on Eqs. (2) and (3) respectively. The dependent sources
proportional to $p_{0}$ represent charge-spin inter-coupling between the two
models. The dependent source parameters in Fig. 1 are given by

$V_{m}=\eta_{c}V_{s},$ (4a) $V_{n}=\eta_{s}V_{c}+r_{m}I_{em},$ (4b)
$\text{and,}\;\,I_{n}=\gamma_{s}I_{c}-g_{m}V_{em}.$ (4c)

The parameters of Eqs. (2) and (3) are given by

$\displaystyle
C_{Q}=\dfrac{2}{R_{B}\left|{\left\langle{{v_{x}}}\right\rangle}\right|},$ (5a)
$\displaystyle
L_{K}=\dfrac{R_{B}}{2\left|{\left\langle{{v_{x}}}\right\rangle}\right|},$ (5b)
$\displaystyle R_{c}=\dfrac{R_{B}}{\lambda},$ (5c) $\displaystyle
R_{s}=\dfrac{\alpha^{2}R_{B}}{\lambda_{0}},$ (5d) $\displaystyle
G_{sh}=\dfrac{4}{\alpha^{2}R_{B}\lambda_{s}},$ (5e) $\displaystyle
g_{m}=\dfrac{2}{\alpha R_{B}},$ (5f) $\displaystyle
r_{m}=\dfrac{\alpha}{\left|{\left\langle{{v_{x}}}\right\rangle}\right|C_{E}},$
(5g) $\displaystyle\eta_{s}=\dfrac{{2\alpha}}{{{\lambda_{0}}}},$ (5h)
$\displaystyle\eta_{c}=\dfrac{2}{\alpha{\lambda_{r}}},$ (5i)
$\displaystyle\gamma_{s}=\dfrac{{2}}{{{\alpha\lambda_{t}}}},$ (5j) and,
$\displaystyle R_{B}=\dfrac{h}{q^{2}}\dfrac{1}{M+N}.$ (5k)

Here, $\lambda$, $\lambda_{0}$, and $\lambda_{s}$ are three distinct mean free
paths that determine $R_{c}$, $R_{s}$, and $G_{sh}$ respectively. $|\langle
v_{x}\rangle|$ is the the magnitude of the thermally averaged electron
velocity $\langle{v}_{x}\rangle$. $\eta_{c}$ represents spin to charge
conversion coefficient and $\eta_{s},\gamma_{s}$ represent charge to spin
conversion coefficients. These coefficients depend on different scattering
mechanisms in the channel, which will be discussed later in Section V. $r_{m}$
and $g_{m}$ are transient charge-spin coupling coefficients which are in the
units of resistance per unit length and conductance per unit length
respectively.

$R_{B}$ is the ballistic resistance of the channel, $h$ is the Planck’s
constant, and $q$ is electron charge. $R_{B}$ is inversely proportional to the
total number of modes ($M + N$) in the channel which represents a material
property and does not imply ballistic transport. The models and related
results discussed in this paper are valid all the way from ballistic to
diffusive regime of operation.

Figure 2: (a) Structure of the spin-momentum locked (SML) channel in the
presence of an external contact. Both (b) charge and (c) spin transmission
line models are modified as compared to Fig. 1, which now correspond to Eqs.
(6) and (7) respectively. The dependent sources are
$V_{m}^{\prime}=\eta_{c}V_{s}+\frac{G_{0}R_{B}}{2}\left(\frac{v_{s}}{\alpha}+p_{f}v_{c}\right)$,
$V_{n}^{\prime}=\eta_{s}V_{c}+r_{m}I_{em}+\frac{G_{0}R_{B}}{2}\left(\alpha
v_{c}+p_{f}v_{s}\right)$, and $I_{n}=\gamma_{s}I_{c}-g_{m}V_{em}$. Note that
the model reduces to that shown in Fig. 1 in the limit $G_{0}\rightarrow 0$.

###  II.2 Presence of an External Contact

In the presence of an external contact on the channel (see Fig. 2(a)), the
charge model in Eq. (2) is modified as

$\displaystyle\left(\dfrac{1}{C_{E}}+\dfrac{1}{C_{Q}}\right)^{-1}\;\dfrac{\partial}{{\partial
t}}{V_{c}}=-\dfrac{\partial}{{\partial x}}I_{c}+i^{c},$ (6)
$\displaystyle\left(L_{K}+L_{M}\right)\;\dfrac{\partial}{{\partial
t}}I_{c}+R_{c}\,I_{c}=-\dfrac{\partial}{\partial
x}{V_{c}}+{p_{0}}{\eta_{c}}{V_{s}}+\Delta v^{c},$

and the spin model in Eq. (3) is modified as

$\displaystyle\dfrac{C_{Q}}{\alpha^{2}}\dfrac{\partial}{{\partial
t}}{V_{s}}+G_{sh}{V_{s}}+p_{0}g_{m}V_{em}=-\dfrac{\partial}{{\partial
x}}{I_{s}}+p_{0}\gamma_{s}I_{c}+i^{s},$ (7)
$\displaystyle\alpha^{2}{L_{K}}\dfrac{\partial}{{\partial
t}}{I_{s}}+{R_{s}}{I_{s}}-p_{0}r_{m}I_{em}=-\dfrac{\partial}{{\partial
x}}{V_{s}}+p_{0}\eta_{s}{V_{c}}+\Delta v^{s},$

where $i^{c}$ and $i^{s}$ represent charge and spin currents entering into the
channel per unit length from the external contact, $\Delta v^{c}$ and $\Delta
v^{s}$ represent the change in channel charge and spin voltages per unit
length in the region under the external contact. They are given as

$\left\\{{\begin{array}[]{*{20}{c}}{{i^{c}}}\\\
{{i^{s}}}\end{array}}\right\\}={G_{0}}\left[{\begin{array}[]{*{20}{c}}1&{\dfrac{p_{f}}{\alpha}}\\\
\\\
\dfrac{p_{f}}{\alpha}&{\dfrac{1}{\alpha^{2}}}\end{array}}\right]\left\\{{\begin{array}[]{*{20}{c}}{{v_{c}}-{V_{c}}}\\\
{{v_{s}}-{V_{s}}}\end{array}}\right\\},\text{ and}$ (8)
$\begin{array}[]{l}\left\\{{\begin{array}[]{*{20}{c}}{\Delta{v^{c}}}\\\
{\Delta{v^{s}}}\end{array}}\right\\}=-\dfrac{{{G_{0}}R_{B}^{2}}}{4}\left[{\begin{array}[]{*{20}{c}}1&{\alpha{p_{f}}}\\\
{\alpha{p_{f}}}&{{\alpha^{2}}}\end{array}}\right]\left\\{{\begin{array}[]{*{20}{c}}{{I_{c}}}\\\
{{I_{s}}}\end{array}}\right\\}\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\dfrac{{{p_{0}}{G_{0}}{R_{B}}}}{2}\left[{\begin{array}[]{*{20}{c}}{{p_{f}}}&{\dfrac{1}{\alpha}}\\\
\alpha&{{p_{f}}}\end{array}}\right]\left\\{{\begin{array}[]{*{20}{c}}{{v_{c}}}\\\
{{v_{s}}}\end{array}}\right\\},\end{array}$ (9)

where $v_{c}=(v_{u}+v_{d})/2$ and $v_{s}=\alpha(v_{u}-v_{d})/2$ are charge and
spin voltages applied at the external contact, $G_{0}$ is the contact
conductance per unit length, and $p_{f}$ is the contact polarization with
$p_{f}=0$ indicating a normal metal contact and $p_{f}\neq 0$ indicating a
ferromagnetic contact. The derivation of the model starting from the Boltzmann
transport equation is shown in Section V with clearly stated assumptions
streamlined with subheadings.

Modified distributed circuit models for charge and spin are shown in Fig. 2(b)
and (c), which are based on Eqs. (6)-(7) respectively. The presence of a
contact with conductance $G_{0}$ adds series resistances
$R_{cont}^{c}=\dfrac{G_{0}R_{B}^{2}}{4}$ and
$R_{cont}^{s}=\dfrac{\alpha^{2}G_{0}R_{B}^{2}}{4}$ in the charge and spin
models as shown in Fig. 2. This effect exists even if the channel is NM. The
presence of the external contact also modulates the dependent sources in Eqs.
(4a) and (4b) as

$V_{m}^{\prime}=\eta_{c}V_{s}+\dfrac{G_{0}R_{B}}{2}\left(\dfrac{v_{s}}{\alpha}+p_{f}v_{c}\right),$
(10a)
$V_{n}^{\prime}=\eta_{s}V_{c}+r_{m}I_{em}+\dfrac{G_{0}R_{B}}{2}\left(\alpha
v_{c}+p_{f}v_{s}\right),$ (10b)

with $V_{a}$ and $V_{b}$ representing the voltage drop across $R_{cont}^{c}$
and $R_{cont}^{s}$ in charge and spin models in Fig. 2 respectively. Note that
the additional terms are proportional to $G_{0}$ and negligible for
potentiometric contacts where $G_{0}$ is very low.

###  II.3 Special Case: Normal Metals ($p_{0}=0$)

We consider a special case of Eqs. (2) and (3) for a normal metal (NM) channel
i.e. $p_{0}=0$. For NM channel, charge and spin model decouples to well-known
models as described below.

####  II.3.1 Charge Model: Transport Model for Quantum Wires

For NM channels ($p_{0}=0$), the charge model in Eq. (2) reduces to the well-
known transmission line model for charge that has been previously used to
analyze transport in quantum wires, given by

$\displaystyle\left(\dfrac{1}{C_{E}}+\dfrac{1}{C_{Q}}\right)^{-1}\;\dfrac{\partial}{{\partial
t}}{V_{c}}=-\dfrac{\partial}{{\partial x}}I_{c},$ (11)
$\displaystyle\left(L_{K}+L_{M}\right)\;\dfrac{\partial}{{\partial
t}}I_{c}+R_{c}\,I_{c}=-\dfrac{\partial}{\partial x}{V_{c}}.$

The model was first derived from Luttinger liquid theory Burke (2002, 2003)
and then from Boltzmann transport equation with one electrochemical potential
Salahuddin  _et al._ (2005). In the quantum wire limit $L_{K}\gg L_{M}$ and
$C_{Q}\ll C_{E}$ while in the classical transmission line limit $L_{K}\ll
L_{M}$ and $C_{Q}\gg C_{E}$ Burke (2002); Salahuddin  _et al._ (2005).

In steady-state ($\partial/\partial t\rightarrow 0$), we get the diffusion
equation for charge from Eq. (11), given by

$\dfrac{d^{2}}{dx^{2}}V_{c}=0.$ (12)

####  II.3.2 Spin Model: Valet-Fert Equation

For NM channels ($p_{0}=0$), the spin model in Eq. (3) becomes a time-
dependent spin-diffusion equation, given by

$\displaystyle\dfrac{C_{Q}}{\alpha^{2}}\dfrac{\partial}{{\partial
t}}{V_{s}}+G_{sh}{V_{s}}=-\dfrac{\partial}{{\partial x}}{I_{s}},$ (13)
$\displaystyle\alpha^{2}{L_{K}}\dfrac{\partial}{{\partial
t}}{I_{s}}+{R_{s}}{I_{s}}=-\dfrac{\partial}{{\partial x}}{V_{s}},$

which in steady-state ($\partial/\partial t\rightarrow 0$), becomes the well-
known Valet-Fert equation Valet and Fert (1993), given by

$\dfrac{d^{2}}{dx^{2}}V_{s}=\dfrac{V_{s}}{\lambda_{sf}^{2}},$ (14)

with the spin diffusion length given by

$\lambda_{sf}=\dfrac{1}{\sqrt{R_{s}G_{sh}}}=\dfrac{\sqrt{\lambda_{0}\lambda_{s}}}{2}.$
(15)

Spin model similar to Eq. (13) has been discussed previously based on
Luttinger liquid theory Burke (2002), however, the model did not take into
account the spin relaxation processes in the channel (the shunt conductance
$G_{sh}$).

##  III Steady-State Transport Results

In this section, we discuss several established steady-state results on
charge-spin interconversion in the potentiometric limit. We start from the
steady-state ($\partial/\partial t\rightarrow 0$) form of the transmission
line model with external contact in Eqs. (6) and (7), given by

$\displaystyle\dfrac{d}{{dx}}I_{c}=i^{c},$ (16a)
$\displaystyle\dfrac{d}{dx}{V_{c}}=-R_{c}\,I_{c}+{p_{0}}{\eta_{c}}{V_{s}}+\Delta
v^{c},$ (16b)
$\displaystyle\dfrac{d}{{dx}}{I_{s}}=-G_{sh}{V_{s}}+p_{0}\gamma_{s}I_{c}+i^{s},$
(16c) and
$\displaystyle\dfrac{d}{{dx}}{V_{s}}=-{R_{s}}{I_{s}}+p_{0}\eta_{s}{V_{c}}+\Delta
v^{s}.$ (16d)

Under the steady-state condition ($\partial/\partial t\rightarrow 0$), the
capacitors and inductors in Fig. 2 become open and short circuits
respectively. The steady-state form in Eq. (16) is equivalent to our prior
time-independent semiclassical equations with four electrochemical potentials
Sayed  _et al._ (2016a) and all our previous results can be reproduced using
Eq. (16).

We first derive a resistance matrix for a three terminal setup (two charge and
one spin) with potentiometric contacts (see Fig. 3) and assuming reflection
with spin-flip to be the dominant scattering mechanism in the channel. We
present dc simulation results on charge-spin interconversion in the SML
channel using the full model (Eqs. (6) and (7)) in SPICE and compare with the
results from the resistance matrix. We then derive a simple expression for a
parameter that has been widely used to quantify inverse Rashba-Edelstein
effect (IREE) in 2D channels. We compare the expression with available
experiments on diverse materials as well as dc SPICE simulation results using
the full model.

###  III.1 Resistance Matrix for Potentiometric Setup

Figure 3: (a) Setup to observe charge current $I_{c}$ induced spin voltage
$v_{s}$ in the SML channel. Charge terminal of contact 3 is kept open and spin
terminals of contacts 1 and 2 are grounded. (b) $v_{s}/I_{c}$ vs. $p_{0}$ for
$i^{s}_{tot}=0$ from SPICE simulation and comparison with Eq. (19). (c) Setup
to observe spin current $i^{s}_{tot}$ induced charge voltage difference
$\Delta V_{c}$ across the SML channel. Charge terminal of contact 3 is kept
open and spin terminals of contacts 1 and 3 are grounded. (d) $\Delta
V_{c}/i^{s}_{tot}$ vs. $p_{0}$ for $I_{c}=0$ from SPICE simulation and
comparison with Eq. (20). The SPICE setup is shown in Fig. 6 of Appendix B.
Parameters: $\alpha=2/\pi$, $G_{0}\approx 0.05G_{B}$, $M+N=100$, and
scattering rate per unit mode $=0.04$ per lattice points. We assumed
reflection with spin-flip to be the dominant scattering process in the
channel.

We consider a structure with three NM contacts ($p_{f}=0$) on top of a SML
channel, as shown in Fig. 3. Contacts 1 and 2 are the charge terminals and
contact 3 is the spin terminal with no charge current flowing through it (i.e.
$i^{c}=0$). We start from Eq. (16) and make two assumptions to drive the
resistance matrix: (i) the contacts are potentiometric i.e. the contact
conductance per unit length $G_{0}$ is very low such that the following
condition is satisfied

$\dfrac{1}{\lambda},\dfrac{1}{\lambda_{0}},\dfrac{1}{\lambda_{r}}\gg\dfrac{G_{0}R_{B}}{4},$
(17)

and (ii) the reflection with spin-flip is the dominant scattering mechanism in
the channel. The details of the derivation are given in Appendix A.

The resistance matrix is given by

$\left\\{{\begin{array}[]{*{20}{c}}\Delta V_{c}\\\
{{v_{s}}}\end{array}}\right\\}=\left[{\begin{array}[]{*{20}{c}}{\dfrac{R_{B}L}{{\lambda}}}+\dfrac{2}{G_{0}^{\prime\prime}}&{-\dfrac{{\alpha{p_{0}}R_{B}}}{{2}}}\\\
{\dfrac{{\alpha{p_{0}}R_{B}}}{{2}}}&{\dfrac{{{\alpha^{2}}}}{{{G_{0}}^{\prime}}}}\end{array}}\right]\left\\{{\begin{array}[]{*{20}{c}}{{I_{c}}}\\\
{i_{tot}^{s}}\end{array}}\right\\},$ (18)

where $\Delta V_{c}$ is the charge voltage difference between contacts 1 and
2, $I_{c}$ is the charge current flowing in the channel with length $L$,
$v_{s}$ is the spin voltage at contact 3, $i^{s}_{tot}=i^{s}L$ is the spin
current at contact 3, contacts 1 and 2 have equal conductance
$G_{0}^{\prime\prime}$, and $G_{0}^{\prime}=G_{0}L$ is the contact conductance
of contact 3. Eq. (18) is the similar to that previously reported in Ref. Hong
_et al._ (2016) with corrections in the diagonal components. The diagonal
components $(1,1)$ and $(2,2)$ in the matrix represent the charge resistance
between contacts 1 and 2 and spin resistance at contact 3 respectively.

Note that Eq. (18) is derived under the assumption that reflection with spin-
flip is the dominant scattering mechanism. In the presence of other scattering
mechanisms, the basic structure of Eq. (18) remains the same, however,
additional factors related to scattering rates multiply each of the components
in the matrix (see Appendix A).

###  III.2 Direct Effects: Charge Current Induced Spin Voltage

For a charge current $I_{c}$ flowing through the SML channel, the open circuit
spin voltage $v_{s}$ at the contact 3 (i.e. $i^{s}_{tot}=0$) can be derived
from Eq. (18) as

$v_{s}=\dfrac{\alpha p_{0}}{2G_{B}}I_{c}.$ (19)

Eq. (19) was originally proposed in Ref. Hong  _et al._ (2012) which was later
confirmed by a number of experiments Li  _et al._ (2014); Tang  _et al._
(2014); Dankert  _et al._ (2015); Liu  _et al._ (2015); Tian  _et al._ (2015);
Lee  _et al._ (2015); Yang  _et al._ (2016); Lee  _et al._ (2018) on different
TI materials using potentiometric ferromagnetic contact. Note that the spin
voltage measured using a contact with higher conductance will be lower than
Eq. (19) due to the current shunting effect in the contact Sayed  _et al._
(2016a). The full model in Eqs. (6) and (7) takes into account such effect
related to contact conductances.

We have simulated the structure in Fig. 3(a) by connecting the full two-
component circuit models given in Figs. 1 and 2 in a distributed manner using
the standard circuit rules. The details of the simulation are discussed in
Appendix B. The simulation results of $v_{s}/I_{c}$ as a function of $p_{0}$
is shown in Fig. 3(b), which is in good agreement with Eq. (19).

###  III.3 Inverse Effects: Spin Current Induced Charge Voltage

The reciprocal effect of Eq. (19) is the spin current $i_{s}$ induced open
circuit charge voltage difference $\Delta V_{c}$ across the SML channel Hong
_et al._ (2016). For $I_{c}=0$ we have from Eq. (18)

$\Delta V_{c}=-\dfrac{\alpha p_{0}}{2G_{B}}i^{s}_{tot}.$ (20)

The reciprocal relation between Eqs. (19) and (20) including the negative sign
has been observed experimentally Liu  _et al._ (2015); Pham  _et al._ (2016a);
Lee  _et al._ (2018). Note that Eq. (20) is exact for a potentiometric
contact. For a contact with higher conductance, $\Delta V_{c}$ will be lower
than Eq. (20) due to current shunting by the same amount as that for the
direct effect (in Eq. (19)) Sayed  _et al._ (2016a), which can be analyzed
using the full model in Eqs. (6) and (7). We have simulated the structure
shown in Fig. 3(c) using the full two-component models in Figs. 1 and 2. The
simulation results of $\Delta V_{c}/i^{s}_{tot}$ as a function of $p_{0}$ is
shown in Fig. 3(d), which show good agreement with Eq. (20). The details of
the simulation are discussed in Appendix B.

Table 1: Estimated Material Parameters.

Material |  $\lambda$ [nm] | $p_{0}$ |  $\lambda_{IREE}$ [nm] |  $\lambda_{IREE}$ [nm]
---|---|---|---|---
|  |  | (Eq. (23)) | (measured)
Ag$|$Bi | 22.6† | 0.05†† | 0.36 | 0.3 Sánchez  _et al._ (2016)
Cu$|$Bi | 0.88† | 0.05†† | 0.014 | 0.009 Isasa  _et al._ (2016)
LAO$|$STO | 180.8† | 0.0616†† | 3.55 | 6.4 Lesne  _et al._ (2016)
Bi2Se3 | 3.2† | 0.025‡ | 0.026 | 0.035 Wang  _et al._ (2016)

†Estimated from the measured sheet resistance of the samples.
††Estimated from the Rashba coupling coefficient and Fermi velocity of the
materials.
‡Estimated from spin-pumping induced voltage and Eq. (20).
The estimations are discussed in detail in Appendix C.

###  III.4 Inverse Rashba-Edelstein Effect (IREE) Length

The phenomena described by Eq. (20) is often known as the inverse Rashba
Edelstein effect (IREE) for 2D channels. IREE is often quantified with a
parameter called IREE length defined as

$\lambda_{IREE}=\dfrac{J_{c}}{J_{s}},$ (21)

where $J_{c}$ is the longitudinal short circuit charge current density in A/m
induced by the injected transverse spin current density $J_{s}$ in A/m2.

We derive a simple expression for IREE length starting from the first row of
Eq. (18) for short circuit condition between contacts 1 and 2 (i.e.
$v_{c1}=v_{c2}$) and assuming large channel resistance compared to contact
resistance (i.e. $L/(G_{B}\lambda)\gg 2/G_{0}^{\prime\prime}$). The expression
is given by

$\lambda_{IREE}=\dfrac{\alpha p_{0}\lambda}{2}.$ (22)

The derivation is given in Appendix A. Note that both $p_{0}$ and $\lambda$
are two completely independent parameters.

Here, $\alpha$ is an angular averaging factor that can vary between 0 and 1
depending on the angular variation of the spin polarization of the eigenstates
and details of the scattering mechanism. We assume that the distribution is
such that the angle between $z$-axis and the spin polarization of the
eigenstates with particular group velocity ($+$ or $-$) vary between $-\pi/2$
to $+\pi/2$, which yields $\alpha=2/\pi$. Thus, from Eq. (22) we have the
following expression

$\lambda_{IREE}=\dfrac{p_{0}\lambda}{\pi},$ (23)

which has been previously reported in Ref. Sayed  _et al._ (2016a). We have
simulated the setup in Fig. 4(a) in order to estimate $\lambda_{IREE}$ of
diverse ranges of $p_{0}$ and $\lambda$. The setup in Fig. 4(a) is same as
that in Fig. 3(c), except we observe the short circuit charge current $I_{c}$
between the charge terminals of contacts 1 and 2 induced by the injected spin
current $i_{s}$ through the spin terminal of contact 3.

Figure 4:  (a) Setup similar to that in Fig. 3(c) but short circuit charge
current between contacts 1 and 2 is being observed. (b) Inverse Rashba
Edelstein effect (IREE) length ($\lambda_{IREE}$) vs. $p_{0}\lambda$ from
SPICE simulation and comparison to Eq. (23) and experiments on different
interfaces with Rashba SOC: Ag$|$Bi Sánchez  _et al._ (2016), Cu$|$Bi Isasa
_et al._ (2016), LaAlO3$|$SrTiO3 (LAO$|$STO) Lesne  _et al._ (2016), and
Bi2Se3 Wang  _et al._ (2016). The back scattering length $\lambda$ is
estimated from measured sheet resistance or resistivity. The degree of spin-
momentum locking $p_{0}$ is estimated from the Rashba coupling coefficient and
the Fermi velocity using Eq. (24). The details of estimations are given in
Appendix B. SPICE simulation parameters: $\alpha=2/\pi$, $G_{0}\approx
0.05G_{B}$, and $M+N=100$. We assumed reflection with spin-flip to be the
dominant scattering process in the channel.

We have compared the simulation results with Eq. (23) as well as available
measurements from spin-pumping and lateral spin valve experiments on different
Rashba interfaces: Ag$|$Bi Sánchez  _et al._ (2016), Cu$|$Bi Isasa  _et al._
(2016), LaAlO3$|$SrTiO3 (LAO$|$STO) Lesne  _et al._ (2016), and Bi2Se3 Wang
_et al._ (2016). The comparison is shown in Fig. 4(b). We have estimated
$\lambda$ from the reported sheet resistance or resistivity of the samples.
$p_{0}$ for Ag$|$Bi, Cu$|$Bi, and LAO$|$STO are estimated using the Rashba
coupling coefficient ($v_{0}$) and the Fermi velocity ($v_{F}$) quoted in the
literature, using the following expression:

$p_{0}=\dfrac{{{v_{0}}}}{{\sqrt{v_{0}^{2}+v_{F}^{2}}}}.$ (24)

For Bi2Se3, we have estimated $p_{0}$ from spin-pumping induced inverse
voltage and Eq. (20). Note that $p_{0}$ estimated for the sample in Ref. Wang
_et al._ (2016) is much lower than previous reports Li  _et al._ (2014); Tang
_et al._ (2014); Dankert  _et al._ (2015); Liu  _et al._ (2015); Tian  _et
al._ (2015); Lee  _et al._ (2015); Yang  _et al._ (2016), which may be due to
the presence of large number of parallel channels. We expect higher
$\lambda_{IREE}$ for Bi2Se3 samples with higher $p_{0}$.

The derivation of Eq. (24) from the Rashba Hamiltonian is shown in Appendix C.
The estimations are summarized in Table 1 and the details are given in
Appendix C. These two independent estimations of $p_{0}$ and $\lambda$ when
applied to Eq. (23), agrees very well with experimentally reported
$\lambda_{IREE}$, as shown in Fig. 4 and Table 1.

##  IV Time-Dependent Transport Results

In this section, we use the full time-dependent model in Eqs. (2) and (3) to
discuss a well-known phenomenon called the spin-charge separation. The spin-
charge separation in the presence of spin-orbit coupling is a subject that has
been controversial in the past. Our model shows that the charge and spin
propagates with two distinct velocities which persist even in the materials
with spin-orbit coupling exhibiting spin-momentum locking ($p_{0}\neq 0)$.
However, we show that the lower velocity signal is purely spin while the
higher velocity signal is largely charge with an additional spin component
proportional to $p_{0}$.

###  IV.1 Velocities of Charge and Spin Signals

The velocities for charge and spin signals can be derived by finding the
eigenvalues of Eqs. (2) and (3) assuming the low loss limit and constant
coefficients. In addition, we find the corresponding eigenvectors as well to
analyze the coupling between spin and charge in the channel due to SML. The
details of the derivation are given in Appendix D.

The lower velocity eigenvalue is given by

$v_{g,s}=\pm\dfrac{1}{\sqrt{L_{K}C_{Q}}}=\pm\langle{v}_{x}\rangle,$ (25)

which is determined by quantum capacitance $C_{Q}$ and kinetic inductance
$L_{K}$, resulting in thermally averaged electron velocity
$\langle{v}_{x}\rangle$. The corresponding eigenvector is given by

$\left\\{{\begin{array}[]{*{20}{c}}{{V_{c}}}\\\
{{I_{c}}}\end{array}}\right\\}=\left\\{{\begin{array}[]{*{20}{c}}0\\\
0\end{array}}\right\\},\text{ and
}\left\\{{\begin{array}[]{*{20}{c}}{{V_{s}}}\\\
{{I_{s}}}\end{array}}\right\\}=\left\\{{\begin{array}[]{*{20}{c}}{\pm\alpha^{2}\sqrt{\dfrac{{{L_{K}}}}{{{C_{Q}}}}}}\\\
1\end{array}}\right\\},$ (26)

which shows that the lower velocity signal is purely spin and no charge
accompanies the signal even in channels with SML i.e. $p_{0}\neq 0$.

The higher velocity eigenvalue is given by

$v_{g,c}=\pm\dfrac{1}{\sqrt{L_{eff}C_{eff}}},$ (27)

where $C_{eff}$ is a series combination of $C_{E}$ and $C_{Q}$ and $L_{eff}$
is a series combination of $L_{M}$ and $L_{K}$. The corresponding eigenvector
is given by

$\begin{array}[]{*{20}{c}}\left\\{{\begin{array}[]{*{20}{c}}{{V_{c}}}\\\
{{I_{c}}}\end{array}}\right\\}=\left\\{{\begin{array}[]{*{20}{c}}{\pm\sqrt{\dfrac{{{L_{eff}}}}{{{C_{eff}}}}}}\\\
1\end{array}}\right\\},\text{ and}\\\
\left\\{{\begin{array}[]{*{20}{c}}{{V_{s}}}\\\
{{I_{s}}}\end{array}}\right\\}=\dfrac{{{p_{0}}}}{{v_{g,c}^{2}-v_{g,s}^{2}}}\left\\{{\begin{array}[]{*{20}{c}}{-\alpha^{2}{g_{m}}\dfrac{{{L_{M}}}}{{{C_{Q}}}}v_{g,c}^{2}+{r_{m}}v_{g,s}^{2}}\\\
{\pm{{{v_{g,c}}{v_{g,s}^{2}}}}{{{C_{Q}}}}\left({-{g_{m}}\dfrac{{{L_{M}}}}{{{C_{Q}}}}+\dfrac{r_{m}}{\alpha^{2}}}\right)}\end{array}}\right\\},\end{array}$
(28)

which shows that the higher velocity signal is largely charge which will be
accompanying an additional spin signal proportional to $p_{0}$, which has not
been discussed before. This additional spin component vanishes in a NM channel
where there is no SML (i.e. $p_{0}=0$) and the signal is purely charge.
Further evaluation of this high velocity spin component we leave as future
work.

The quantum capacitance $C_{Q}$ is proportional to the total number of modes
$(M+N)$ in the channel (see Eq. (5a)) while the kinetic inductance $L_{K}$ is
inversely proportional to $M+N$ (see Eq. (5b)). $M+N$ is proportional to the
channel width (for 2D) or cross-sectional area (for 3D) Datta (2012).

For a conductor with very large cross-section, we may have $C_{E}\ll C_{Q}$
and $L_{M}\gg L_{K}$ which is the standard transmission line limit. In this
limit, the velocity in Eq. (27) becomes

$c=\dfrac{1}{\sqrt{L_{M}C_{E}}},$

which is the velocity predicted by standard transmission line theory and can
be as high as the speed of light.

For a conductor with very small cross-section like quantum wires, we may have
$C_{E}\gg C_{Q}$ and $L_{M}\ll L_{K}$. In this limit, velocity in Eq. (27)
becomes the thermally averaged electron velocity, given by

$\langle{v}_{x}\rangle=\dfrac{1}{\sqrt{L_{K}C_{Q}}},$

which is same as Eq. (25). Note that the two velocity eigenvalues are equal at
this limit.

###  IV.2 Spin-Charge Separation

From Eqs. (25) and (27) we have

$\dfrac{v_{g,s}}{v_{g,c}}=\sqrt{\dfrac{1+\delta\dfrac{C_{Q}}{C_{E}}}{1+\dfrac{C_{Q}}{C_{E}}}},$
(29)

where $\delta=\left(L_{M}C_{E}\right)/\left(L_{K}C_{Q}\right)=\left(\langle
v_{x}\rangle/c\right)^{2}$ (see Eq. 6 of Ref. Salahuddin  _et al._ (2005))
which is usually much less than one, making the spin signal slower than the
charge signal, given by

$v_{g,s}<v_{g,c}.$ (30)

This results in spin-charge separation which is well-established for channels
without SML (i.e. $p_{0}=0$) from Luttinger liquid theory (see for example,
Refs. Halperin (2007); Polini and Vignale (2007); Schroer  _et al._ (2014);
Burke (2002), and references therein). Electrons’ charge excites $C_{E}$ and
$L_{M}$ hence the charge signal velocity is determined by $C_{eff}$ and
$L_{eff}$ given by Eq. (27). However, pure spin signal do not excite $C_{E}$
and $L_{M}$, hence its velocity is determined by $C_{Q}$ and $L_{K}$ only
given by Eq. (25). Similar arguments have been made in the past Burke (2002,
2003) in the context of carbon nanotubes without SOC.

Note that the argument in Eq. (30) is independent of $p_{0}$ (see Appendix D),
which indicates that the spin-charge separation persists even in channels with
SOC exhibiting SML (i.e. $p_{0}\neq 0$). Similar arguments have been discussed
previously by considering SOC De Martino  _et al._ (2002); Calzona  _et al._
(2015); Stauber  _et al._ (2013) although there exists argument that the
presence of SOC may destroy the spin-charge separation Moroz  _et al._ (2000).

In SML channels, an additional spin signal proportional to $p_{0}$ accompanies
the charge signal at the same velocity as the charge ($v_{g,c}$) as seen from
Eq. (28). This additional spin component is induced by the instantaneous
voltage drop across $L_{M}$ and the instantaneous current through $C_{E}$ of
the channel. However, the low velocity signal (see Eq. (25)) remains purely
spin since the spin signal do not excite $L_{M}$ and $C_{E}$ Burke (2002);
Salahuddin  _et al._ (2005) to induce a similar accompanying charge component.

##  V Transmission Line Model from Boltzmann Formalism

In this section, we derive the transmission line model in Eqs. (6) and (7)
starting from the time-dependent Boltzmann transport equation under several
clearly stated assumptions, which allow us to obtain the simple expressions
for the model parameters stated in Eq. (5). Several of our predictions for
steady-state Sayed  _et al._ (2016a) have already received experimental
support Pham  _et al._ (2016b, a); Lee  _et al._ (2018) suggesting that the
assumptions are reasonable, but they could be revisited as the field evolves.

###  V.1 Boltzmann Transport Equation

We assume a structure where the spatial variations and the applied fields are
along $\hat{x}$-direction. The time-dependent Boltzmann transport equation is
given by

$\displaystyle\frac{\partial f}{{\partial t}}+{v}_{x}\frac{\partial
f}{{\partial x}}+{F}_{x}\frac{\partial f}{{\partial
p_{x}}}=\sum_{\vec{p^{\prime}},s^{\prime}}S(\vec{p},s\leftrightarrow\vec{p^{\prime}},s^{\prime})\left(f-f^{\prime}\right),$
(31)

where we have assumed elastic scattering so that the scattering rates are same
in both directions i.e.

$S(\vec{p},s\rightarrow\vec{p^{\prime}},s^{\prime})=S(\vec{p^{\prime}},s^{\prime}\rightarrow\vec{p},s)\equiv
S(\vec{p},s\leftrightarrow\vec{p^{\prime}},s^{\prime}).$

Here, $f\equiv f(x,t,\vec{p},s)$ is the occupation factor of a state for a
particular position $x$, time $t$, momentum $\vec{p}$ and spin index $s=\pm
1$, $f^{\prime}\equiv f(x,t,\vec{p^{\prime}},s^{\prime})$ with momentum
$\vec{p^{\prime}}$ and spin index $s^{\prime}=\pm 1$, ${v}_{x}=\partial
E/\partial p_{x}$ is the $x$-component of the group velocity,
${F}_{x}=-\partial E/\partial x$ is the force on electrons along
$\hat{x}$-direction, and $E$ is the total energy. Note that the spin index
$+1$ and $-1$ correspond to up ($U$) and down ($D$) spin polarized states for
a particular $\vec{p}$.

###  V.2 Occupation Factor

We write the occupation factor $f$ in terms of an electrochemical potential
$\mu\equiv\mu(x,t,\vec{p},{s})$, in the form

$f(x,t,\vec{p},{s})=\dfrac{1}{1+\exp\left(\dfrac{E(x,t,\vec{p},{s})-\mu(x,t,\vec{p},s)}{k_{B}T}\right)},$
(32)

where $k_{B}$ is the Boltzmann constant, $T$ is the temperature. Note that

###  V.3 Linearization

We apply a variable transform $\xi=E-\mu$ on the left hand side of Eq. (31).
On the right hand side of Eq. (31), we expand both $f$ and $f^{\prime}$ into
Taylor series around

$f_{0}=\dfrac{1}{1+\exp\left(\dfrac{(E(x,\vec{p},{s})-\mu_{0})}{k_{B}T}\right)},$
(33)

with constant electrochemical potential $\mu_{0}$ and apply linear response
approximation. Thus Eq. (31) can be written as (see Appendix E for details of
the derivation)

$\begin{array}[]{l}\left(-\dfrac{\partial f_{0}}{\partial
E}\right)\left(\dfrac{\partial\mu}{{\partial t}}-\dfrac{\partial E}{{\partial
t}}+{v}_{x}\dfrac{\partial\mu}{{\partial
x}}+{F}_{x}\dfrac{\partial\mu}{{\partial p_{x}}}\right)\\\
\quad\quad=-\displaystyle\sum_{\vec{p^{\prime}},s^{\prime}}S(\vec{p},s\leftrightarrow\vec{p^{\prime}},s^{\prime})\left(-\dfrac{\partial
f_{0}}{\partial E}\right){\left(\mu-\mu^{\prime}\right)}.\end{array}$ (34)

We assume that there are no internal fields in the present discussion. Hence,
$F_{x}$ comes from the applied voltage and the term
$F_{x}(\partial\mu/\partial p_{x})$ depends on the higher order of the applied
voltage, which can be neglected in the linear response regime Datta (2012).
Thus Eq. (34) is given by

$\begin{array}[]{l}\left(-\dfrac{\partial f_{0}}{\partial
E}\right)\left(\dfrac{\partial\mu}{{\partial t}}-\dfrac{\partial E}{{\partial
t}}+{v}_{x}\dfrac{\partial\mu}{{\partial x}}\right)\\\
\quad\quad=-\displaystyle\sum_{\vec{p^{\prime}},s^{\prime}}S(\vec{p},s\leftrightarrow\vec{p^{\prime}},s^{\prime})\left(-\dfrac{\partial
f_{0}}{\partial E}\right){\left(\mu-\mu^{\prime}\right)}.\end{array}$ (35)

The term $\partial E/\partial t$ can be evaluated from the dispersion relation
of a given Hamiltonian in the semiclassical approximation as discussed below.

###  V.4 Dispersion Relation

We start from the following Rashba Hamiltonian

$\mathcal{H}=\dfrac{|\vec{p}-q\vec{A}|^{2}}{2m}I_{2\times
2}-v_{0}\left(\vec{\sigma}\times(\vec{p}-q\vec{A})\right)\cdot\hat{y}+U_{E}I_{2\times
2}.$ (36)

Here, $I_{2\times 2}$ is a $2\times 2$ identity matrix, $\vec{p}$ and
$\vec{A}$ are the momentum and vector magnetic potential respectively in the
$z$-$x$ plane, $\vec{\sigma}$ is the Pauli’s matrices, $v_{0}$ is the Rashba
coefficient, $U_{E}$ is the electrostatic potential, $m$ is the electron mass,
and $q$ is the electron charge.

Eigenstates of Eq. (36) are given by

$E(\vec{p},{s})=\dfrac{|\vec{p}-q\vec{A}|^{2}}{2m}-s\,v_{0}{|\vec{p}-q\vec{A}|}+U_{E},$
(37)

with $\vec{p}$ is confined to the $z$-$x$ plane.

We assume that $U_{E}$ and $\vec{A}$ varies slowly with $x$ and $t$, so that
in the semiclassical approximation we have

$E(x,t,\vec{p},{s})=\dfrac{|\vec{p}-q\vec{A}(x,t)|^{2}}{2m}-s\,v_{0}{|\vec{p}-q\vec{A}(x,t)|}+U_{E}(x,t).$
(38)

Differentiating Eq. (38) with respect to $t$ yields

$\dfrac{\partial E}{\partial
t}=\vec{v}\cdot\left(-q\dfrac{\partial\vec{A}}{\partial
t}\right)+\dfrac{\partial U_{E}}{\partial t},$ (39)

where $\vec{v}=\nabla_{\vec{p}}E$ (see Appendix F for the derivation).

The electrostatic potential $U_{E}$ and the vector magnetic potential
$\vec{A}$ on the structure of interest can be evaluated from the theory of
electromagnetism.

###  V.5 From Potentials to Charge and Current

The electrostatic potential $U_{E}$ is related to the total charge $Q$ in the
channel by the electrostatic capacitance $C_{E}$ of the structure under
consideration, given by

$\dfrac{U_{E}}{q}=\dfrac{Q}{C_{E}}.$ (40)

We assume that the charge current $I_{c}$ flows along $\hat{x}$-direction
which is uniform in the channel. The vector magnetic potential $\vec{A}$ is
related to $I_{c}$ by the magnetic inductance $L_{M}$ of the channel, given by

$\vec{A}\equiv\hat{x}A_{x}=\hat{x}L_{M}I_{c}.$ (41)

Thus Eq. (39) can be written as

$\dfrac{\partial E}{\partial t}=-qv_{x}L_{M}\dfrac{\partial I_{c}}{\partial
t}+\dfrac{q}{C_{E}}\dfrac{\partial Q}{\partial t}.$ (42)

We combine Eq. (35) with Eq. (42) to get

$\displaystyle\left(\dfrac{\partial f_{0}}{\partial E}\right)$
$\displaystyle\left(\frac{\partial\mu}{{\partial
t}}+{v}_{x}\frac{\partial\mu}{{\partial x}}+qv_{x}L_{M}\dfrac{\partial
I_{c}}{\partial t}-\dfrac{q}{C_{E}}\dfrac{\partial Q}{\partial t}\right)$ (43)
$\displaystyle=-\sum_{\vec{p^{\prime}},s^{\prime}}S(\vec{p},s\leftrightarrow\vec{p^{\prime}},s^{\prime})\left(\dfrac{\partial
f_{0}}{\partial E}\right){\left(\mu-\mu^{\prime}\right)}.$

###  V.6 Classification

We classify all $\vec{p},s$ states into four groups based on the sign of
$v_{x}$ ($+$ or $-$) and the spin index $s=\pm 1$, given by

$\displaystyle\Re:\begin{cases}&U^{+}\in\\{\vec{p},s\,\,\,|\,\,\,v_{x}>0,\,\,\,s=+1\\},\\\
&D^{-}\in\\{\vec{p},s\,\,\,|\,\,\,v_{x}<0,\,\,\,s=-1\\},\\\
&U^{-}\in\\{\vec{p},s\,\,\,|\,\,\,v_{x}<0,\,\,\,s=+1\\},\text{ and}\\\
&D^{+}\in\\{\vec{p},s\,\,\,|\,\,\,v_{x}>0,\,\,\,s=-1\\}.\end{cases}$ (44)

where $s=+1$ and $-1$ denote up ($U$) and down ($D$) spins with respect to the
spin quantization axis defined by
$\hat{y}\times\left(\vec{p}-q\vec{A}\right)$, which is different for each
direction of $\vec{p}$. Such classification can be mapped onto the two Fermi
circles of a Rashba channel (see Fig. 5(a)). The large circle corresponds to
$U^{+}$ and $D^{-}$ groups which share the same number of modes
$n_{m}(U^{+})=n_{m}(D^{-})=M$ satisfying the time-reversal symmetry.
Similarly, the small circle corresponds to $U^{-}$ and $D^{+}$ groups sharing
the same number of modes $n_{m}(U^{-})=n_{m}(D^{+})=N$. Note that the
eigenstates belonging to each of the four half Fermi circles in Fig. 5(a) has
an average spin polarization along $\hat{z}$-direction with an averaging
factor of $\alpha=2/\pi$, which we will use later when writing spin currents
and voltages.

###  V.7 Averaging

We define the thermal average of a variable $\psi\equiv\psi(\vec{p},s)$ within
each of the group $\vec{p},s\in\Re$ as

$\langle\psi\rangle_{\vec{p},s\in\Re}=\dfrac{\displaystyle\sum_{\vec{p},s\in\Re}\left(-\dfrac{\partial
f_{0}}{\partial
E}\right)\psi(\vec{p},s)}{\displaystyle\sum_{\vec{p},s\in\Re}\left(-\dfrac{\partial
f_{0}}{\partial E}\right)}.$ (45)

We sum both sides of Eq. (43) over all $\vec{p}$ states within range
$\vec{p},s\in\Re$ in the $z$-$x$ plane as

$\displaystyle\dfrac{D_{0}(\Re)}{2}\dfrac{\partial\left\langle\mu\right\rangle}{\partial
t}+\dfrac{D_{0}(\Re)}{2}\dfrac{\partial\langle{v_{x}\mu}\rangle}{\partial x}$
(46) $\displaystyle+\dfrac{qD_{0}(\Re)}{2}\langle v_{x}\rangle
L_{M}\dfrac{\partial I_{c}}{\partial
t}-\dfrac{q}{C_{E}}\dfrac{D_{0}(\Re)}{2}\dfrac{\partial Q}{\partial t}$
$\displaystyle=-\sum_{\vec{p},s\in\Re}\,\,\sum_{\vec{p^{\prime}},s^{\prime}}S(\vec{p},s\leftrightarrow\vec{p^{\prime}},s^{\prime})\left(-\dfrac{\partial
f_{0}}{\partial E}\right){\left(\mu-\mu^{\prime}\right)}.$

Here, $D_{0}(\Re)$ is the thermally averaged density of states within
$\vec{p},s\in\Re$ given by

$\dfrac{D_{0}(\Re)}{2}=\displaystyle\sum_{\vec{p},s\in\Re}\left(-\dfrac{\partial
f_{0}}{\partial E}\right),$ (47)

where the factor of 2 appeared since we are summing over all $s$ states. Note
that $D_{0}(U^{+})=D_{0}(D^{-})$ and $D_{0}(U^{-})=D_{0}(D^{+})$ due to time-
reversal symmetry.

We make the following assumption in Eq. (46)

$\langle v_{x}\mu\rangle\approx\langle v_{x}\rangle\langle\mu\rangle,$ (48)

which yields

$\displaystyle\dfrac{D_{0}(\Re)}{2}\dfrac{\partial\left\langle\mu\right\rangle}{\partial
t}+\dfrac{D_{0}(\Re)}{2}\langle
v_{x}\rangle\dfrac{\partial\langle{\mu}\rangle}{\partial x}$ (49)
$\displaystyle+\dfrac{qD_{0}(\Re)}{2}\langle v_{x}\rangle L_{M}\dfrac{\partial
I_{c}}{\partial t}-\dfrac{q}{C_{E}}\dfrac{D_{0}(\Re)}{2}\dfrac{\partial
Q}{\partial t}$
$\displaystyle=-\sum_{\vec{p},s\in\Re}\,\,\sum_{\vec{p^{\prime}},s^{\prime}}S(\vec{p},s\leftrightarrow\vec{p^{\prime}},s^{\prime})\left(-\dfrac{\partial
f_{0}}{\partial E}\right){\left(\mu-\mu^{\prime}\right)}+\dfrac{i_{ext}}{q}.$

Note that the term $i_{ext}/q$ on the right hand side has been added to take
into account the total current entering into $\vec{p},s\in\Re$ states of the
channel an external contact.

Figure 5: (a) Two Fermi circles of a Rashba channel with spin-momentum locking
having opposite spin polarizations at a given energy $E_{F}$. Left (right)
half of each circle represents negative (positive) group velocity. The large
circle corresponds to the large number of modes $M$ in the channel and have a
net spin polarization along $+\hat{z}$ ($-\hat{z}$) on right (left) side. The
small circle corresponds to smaller number of modes $N$ and have net spin
polarization along $-\hat{z}$ ($+\hat{z}$) on right (left) side. We classify
all electronics states into four groups based on the $z$-component of spin
polarization (up ($U$), down ($D$)) and the sign of $x$-component of the group
velocity ($+$, $-$). (b) Assuming high scattering among states in each
individual groups, we assign four electrochemical potentials for these four
groups: $\mu(U^{+})$, $\mu(U^{-})$, $\mu(D^{+})$, and $\mu(D^{-})$. External
contact is modeled as up ($v_{u}$) and down ($v_{d}$) spin voltages applied to
up and down states of the channel through up ($g_{u}$) and down ($g_{d}$) spin
conductances per mode, respectively. Four different currents ($i_{U}^{+}$,
$i_{D}^{+}$, $i_{U}^{-}$, and $i_{D}^{-}$) enter the four different groups in
the channel. The contact can be either normal metal $g_{u}=g_{d}$ or
ferromagnet $g_{u}\neq g_{d}$.

The number of modes within $\vec{p},s\in\Re$ in the channel is given by Datta
(2012)

$n_{m}(\Re)=\dfrac{hD_{0}|\langle v_{x}\rangle|}{2L},$ (50)

where $|\langle v_{x}\rangle|$ is the magnitude of $\langle v_{x}\rangle$ and
$L$ is the channel length. Note that $|\langle v_{x}\rangle|$ is same in all
four groups for the Rashba Hamiltonian considered here (see Appendix F). Thus
Eq. (49) can be written as

$\displaystyle\dfrac{{n_{m}\left(\Re\right)}}{{\left|{\left\langle{{v_{x}}}\right\rangle}\right|}}\dfrac{{\partial\left\langle\mu\right\rangle}}{{\partial
t}}+{\mathop{\rm
sgn}}\left({\left\langle{{v_{x}}}\right\rangle}\right)n_{m}\left({\Re}\right)\dfrac{{\partial\langle\mu\rangle}}{{\partial
x}}-\dfrac{q}{{{C_{E}}}}\frac{{n_{m}\left({\Re}\right)}}{{\left|{\left\langle{{v_{x}}}\right\rangle}\right|}}\frac{{\partial
Q}}{{\partial t}}$ (51) $\displaystyle+{\mathop{\rm
sgn}}\left({\left\langle{{v_{x}}}\right\rangle}\right)q\,{n_{m}\left({\Re}\right)}{L_{M}}\dfrac{{\partial{I_{c}}}}{{\partial
t}}=\tilde{S}(\Re)+\frac{h}{q}\frac{{{i_{ext}}}}{L},$

where

$\tilde{S}(\Re)=-\frac{h}{L}\sum\limits_{\vec{p},s\in\Re}{\mkern 1.0mu}{\mkern
1.0mu}\sum\limits_{\vec{p}^{\prime},s^{\prime}}S(\vec{p},s\leftrightarrow\vec{p}^{\prime},s^{\prime})\left({-\frac{{\partial{f_{0}}}}{{\partial
E}}}\right)\left({\mu-\mu^{\prime}}\right).$ (52)

Eq. (51) applies to each group in Eq. (44) with an average electrochemical
potential given by $\left\langle\mu\right\rangle_{\vec{p},s\in
U^{+}}\equiv\mu\left(U^{+}\right)$,
$\left\langle\mu\right\rangle_{\vec{p},s\in
D^{-}}\equiv\mu\left(D^{-}\right)$,
$\left\langle\mu\right\rangle_{\vec{p},s\in
U^{-}}\equiv\mu\left(U^{-}\right)$, and
$\left\langle\mu\right\rangle_{\vec{p},s\in D^{+}}\equiv\mu\left(D^{+}\right)$
respectively.

###  V.8 Scattering Matrix

We make the following assumption

$\langle S\mu\rangle\approx\langle S\rangle\langle\mu\rangle,$ (53)

which when applied to the term related to the scattering rate $\tilde{S}(\Re)$
in Eq. (52) becomes

$\displaystyle\tilde{S}(\Re)=$
$\displaystyle{\hat{S}_{\Re\leftrightarrow{U^{+}}}}\left({\left\langle{\mu\left({{U^{+}}}\right)}\right\rangle-\left\langle{\mu\left(\Re\right)}\right\rangle}\right)$
(54)
$\displaystyle+{\hat{S}_{\Re\leftrightarrow{D^{-}}}}\left({\left\langle{\mu\left({{D^{-}}}\right)}\right\rangle-\left\langle{\mu\left(\Re\right)}\right\rangle}\right)$
$\displaystyle+{\hat{S}_{\Re\leftrightarrow{U^{-}}}}\left({\left\langle{\mu\left({{U^{-}}}\right)}\right\rangle-\left\langle{\mu\left(\Re\right)}\right\rangle}\right)$
$\displaystyle+{\hat{S}_{\Re\leftrightarrow{D^{+}}}}\left({\left\langle{\mu\left({{D^{+}}}\right)}\right\rangle-\left\langle{\mu\left(\Re\right)}\right\rangle}\right),$

with

${\hat{S}_{\Re\leftrightarrow\Re^{\prime}}}=\frac{h}{L}\sum\limits_{\vec{p},s\in\Re}\sum\limits_{\vec{p}^{\prime},s^{\prime}\in\Re^{\prime}}\left({-\frac{{\partial{f_{0}}}}{{\partial
E}}}\right)S(\vec{p},s\leftrightarrow\vec{p}^{\prime},s^{\prime}).$ (55)

We can evaluate Eq. (54) for each of the group in Eq. (44) i.e. $\Re\equiv
U^{+}$, $D^{-}$, $U^{-}$, and $D^{+}$, which together in the
$\\{\mu(U^{+}),\mu(D^{-}),\mu(U^{-}),\mu(D^{+})\\}^{T}$ basis becomes the
following matrix (see Appendix G for details)

$\displaystyle\left[S\right]=\left[{\begin{array}[]{*{20}{c}}{-{u_{1}}}&{{{\hat{S}}_{{U^{+}}\leftrightarrow{D^{-}}}}}&{{{\hat{S}}_{{U^{+}}\leftrightarrow{U^{-}}}}}&{{{\hat{S}}_{{U^{+}}\leftrightarrow{D^{+}}}}}\\\
{{{\hat{S}}_{{U^{+}}\leftrightarrow{D^{-}}}}}&{-{u_{1}^{\prime}}}&{{{\hat{S}}_{{D^{-}}\leftrightarrow{U^{-}}}}}&{{{\hat{S}}_{{D^{-}}\leftrightarrow{D^{+}}}}}\\\
{{{\hat{S}}_{{U^{+}}\leftrightarrow{U^{-}}}}}&{{{\hat{S}}_{{D^{-}}\leftrightarrow{U^{-}}}}}&{-{u_{2}}}&{{{\hat{S}}_{{U^{-}}\leftrightarrow{D^{+}}}}}\\\
{{{\hat{S}}_{{U^{+}}\leftrightarrow{D^{+}}}}}&{{{\hat{S}}_{{D^{-}}\leftrightarrow{D^{+}}}}}&{{{\hat{S}}_{{U^{-}}\leftrightarrow{D^{+}}}}}&{-{u_{2}^{\prime}}}\end{array}}\right],$
(56)

where

$\displaystyle{u_{1}}={{\hat{S}}_{{U^{+}}\leftrightarrow{D^{-}}}}+{{\hat{S}}_{{U^{+}}\leftrightarrow{U^{-}}}}+{{\hat{S}}_{{U^{+}}\leftrightarrow{D^{+}}}},$
$\displaystyle
u_{1}^{\prime}={{\hat{S}}_{{U^{+}}\leftrightarrow{D^{-}}}}+{{\hat{S}}_{{D^{-}}\leftrightarrow{U^{-}}}}+{{\hat{S}}_{{D^{-}}\leftrightarrow{D^{+}}}},$
$\displaystyle{u_{2}}={{\hat{S}}_{{U^{+}}\leftrightarrow{U^{-}}}}+{{\hat{S}}_{{D^{-}}\leftrightarrow{U^{-}}}}+{{\hat{S}}_{{U^{-}}\leftrightarrow{D^{+}}}},$
and
$\displaystyle{{u_{2}^{\prime}}}={{\hat{S}}_{{U^{+}}\leftrightarrow{D^{+}}}}+{{\hat{S}}_{{D^{-}}\leftrightarrow{D^{+}}}}+{{\hat{S}}_{{U^{-}}\leftrightarrow{D^{+}}}}.$

The scattering matrix is such that the sum of each column is zero satisfying
the charge conservation and the sum of each row is zero satisfying the zero
current requirement under equal potential.

In addition, the time-reversal symmetry requires that

${{\hat{S}}_{{U^{+}}\leftrightarrow{U^{-}}}}={{\hat{S}}_{{D^{-}}\leftrightarrow{D^{+}}}}\text{
and }\hat{S}_{U^{+}\leftrightarrow D^{+}}=\hat{S}_{D^{-}\leftrightarrow
U^{-}}.$ (57)

There are three types of scattering processes considered in the channel: (a)
reflection with spin flip $r_{s1}=\hat{S}_{U^{+}\leftrightarrow D^{-}}$ and
$r_{s2}=\hat{S}_{U^{-}\leftrightarrow D^{+}}$, (b) reflection without spin
flip $r=\hat{S}_{U^{+}\leftrightarrow U^{-}}=\hat{S}_{D^{-}\leftrightarrow
D^{+}}$, and (c) transmission with spin flip
$t_{s}=\hat{S}_{U^{+}\leftrightarrow D^{+}}=\hat{S}_{D^{-}\leftrightarrow
U^{-}}$. They are given by

$r_{s1}=\frac{h}{L}\sum\limits_{\vec{p},s\in
U^{+}}\sum\limits_{\vec{p}^{\prime},s^{\prime}\in
D^{-}}\left({-\frac{{\partial{f_{0}}}}{{\partial
E}}}\right)S(\vec{p},s\leftrightarrow\vec{p}^{\prime},s^{\prime}),$ (58a)
$r_{s1}=\frac{h}{L}\sum\limits_{\vec{p},s\in
U^{-}}\sum\limits_{\vec{p}^{\prime},s^{\prime}\in
D^{+}}\left({-\frac{{\partial{f_{0}}}}{{\partial
E}}}\right)S(\vec{p},s\leftrightarrow\vec{p}^{\prime},s^{\prime}),$ (58b)
$\displaystyle r$ $\displaystyle=\frac{h}{L}\sum\limits_{\vec{p},s\in
U^{+}}\sum\limits_{\vec{p}^{\prime},s^{\prime}\in
U^{-}}\left({-\frac{{\partial{f_{0}}}}{{\partial
E}}}\right)S(\vec{p},s\leftrightarrow\vec{p}^{\prime},s^{\prime}),$ (58c)
$\displaystyle=\frac{h}{L}\sum\limits_{\vec{p},s\in
D^{-}}\sum\limits_{\vec{p}^{\prime},s^{\prime}\in
D^{+}}\left({-\frac{{\partial{f_{0}}}}{{\partial
E}}}\right)S(\vec{p},s\leftrightarrow\vec{p}^{\prime},s^{\prime}),$ and
$\displaystyle t_{s}$ $\displaystyle=\frac{h}{L}\sum\limits_{\vec{p},s\in
U^{+}}\sum\limits_{\vec{p}^{\prime},s^{\prime}\in
D^{+}}\left({-\frac{{\partial{f_{0}}}}{{\partial
E}}}\right)S(\vec{p},s\leftrightarrow\vec{p}^{\prime},s^{\prime}),$ (58d)
$\displaystyle=\frac{h}{L}\sum\limits_{\vec{p},s\in
D^{-}}\sum\limits_{\vec{p}^{\prime},s^{\prime}\in
U^{-}}\left({-\frac{{\partial{f_{0}}}}{{\partial
E}}}\right)S(\vec{p},s\leftrightarrow\vec{p}^{\prime},s^{\prime}).$

Eq. (51) for each group in Eq. (44) is given as

$\displaystyle\frac{1}{\left|{\left\langle{{v_{x}}}\right\rangle}\right|}\frac{\partial}{{\partial
t}}\left\\{{\begin{array}[]{*{20}{c}}{M\tilde{\mu}\left({{U^{+}}}\right)}\\\
{M\tilde{\mu}\left({{D^{-}}}\right)}\\\
{N\tilde{\mu}\left({{U^{-}}}\right)}\\\
{N\tilde{\mu}\left({{D^{+}}}\right)}\end{array}}\right\\}+\frac{\partial}{{\partial
x}}\left\\{{\begin{array}[]{*{20}{c}}{M\tilde{\mu}\left({{U^{+}}}\right)}\\\
{-M\tilde{\mu}\left({{D^{-}}}\right)}\\\
{-N\tilde{\mu}\left({{U^{-}}}\right)}\\\
{N\tilde{\mu}\left({{D^{+}}}\right)}\end{array}}\right\\}$ (59)
$\displaystyle=\left[{\begin{array}[]{*{20}{c}}{-{u_{1}}}&{{r_{s1}}}&r&{{t_{s}}}\\\
{{r_{s1}}}&{-{u_{1}}}&{{t_{s}}}&r\\\ r&{{t_{s}}}&{-{u_{2}}}&{{r_{s2}}}\\\
{{t_{s}}}&r&{{r_{s2}}}&{-{u_{2}}}\end{array}}\right]\left\\{{\begin{array}[]{*{20}{c}}{\tilde{\mu}\left({{U^{+}}}\right)}\\\
{\tilde{\mu}\left({{D^{-}}}\right)}\\\ {\tilde{\mu}\left({{U^{-}}}\right)}\\\
{\tilde{\mu}\left({{D^{+}}}\right)}\end{array}}\right\\}$ $\displaystyle-
qL_{M}{\dfrac{{\partial I_{c}}}{{\partial
t}}}\left\\{{\begin{array}[]{*{20}{c}}{M}\\\ {-M}\\\ {-N}\\\
{N}\end{array}}\right\\}+\dfrac{q}{\left|{\left\langle{{v_{x}}}\right\rangle}\right|C_{E}}\dfrac{{\partial
Q}}{{\partial t}}\left\\{{\begin{array}[]{*{20}{c}}M\\\ M\\\ N\\\
N\end{array}}\right\\}+\frac{h}{q}\left\\{{\begin{array}[]{*{20}{c}}{i_{U}^{+}}\\\
{i_{D}^{-}}\\\ {i_{U}^{-}}\\\ {i_{D}^{+}}\end{array}}\right\\}.$

Note that the electrochemical potentials are referenced with respect to the
constant $\mu_{0}$ i.e. $\tilde{\mu}=\mu-\mu_{0}$. $i_{U}^{+}$, $i_{D}^{-}$,
$i_{U}^{-}$, and $i_{D}^{+}$ are the currents per unit length entering into
the four groups from an external contact (see Fig. 5(b)), which are given as
Sayed  _et al._ (2016a)

$\displaystyle
i_{U}^{+}=\frac{q^{2}}{h}Mg_{u}\left(v_{u}-\frac{\tilde{\mu}(U^{+})}{q}\right),$
(60) $\displaystyle
i_{D}^{+}=\frac{q^{2}}{h}Ng_{d}\left(v_{d}-\frac{\tilde{\mu}(D^{+})}{q}\right),$
$\displaystyle
i_{U}^{-}=\frac{q^{2}}{h}Ng_{u}\left(v_{u}-\frac{\tilde{\mu}(U^{-})}{q}\right),$
and $\displaystyle
i_{D}^{-}=\frac{q^{2}}{h}Mg_{d}\left(v_{d}-\frac{\tilde{\mu}(D^{-})}{q}\right).$

Here, $v_{u}$ and $v_{d}$ are up and down spin voltages at the external
contact respectively. $g_{u}$ and $g_{d}$ are up and down spin conductances
per unit mode per unit length of the contact. The contact can be either NM
($g_{u}=g_{d}$) or FM ($g_{u}\neq g_{d}$). In steady-state, Eq. (59) reduces
to our prior model Sayed  _et al._ (2016a).

###  V.9 Conversion to Charge-Spin Basis

The charge and spin voltages and currents in the channel are defined in terms
of the four average electrochemical potentials as

$\left\\{\begin{array}[]{l}I_{c}{R_{B}}\\\ 2{V_{s}}\\\ {I_{s}}{R_{B}}\\\
2{V_{c}}\end{array}\right\\}\;=\dfrac{q}{h}R_{B}\left[{\begin{array}[]{*{20}{c}}1&-1&-1&1\\\
\alpha&-\alpha&{\alpha}&{-\alpha}\\\
{\dfrac{1}{\alpha}}&{\dfrac{1}{\alpha}}&{-\dfrac{1}{\alpha}}&{-\dfrac{1}{\alpha}}\\\
1&{1}&{1}&1\end{array}}\right]\;\left\\{\begin{array}[]{l}M\tilde{\mu}({U^{+}})\\\
M\tilde{\mu}({D^{-}})\\\ N\tilde{\mu}({U^{-}})\\\
N\tilde{\mu}({D^{+}})\end{array}\right\\},$ (61)

where $R_{B}$ is the ballistic resistance of the channel given in Eq. (5k) and
$\alpha$ is an angular averaging factor. The derivation of Eq. (61) is given
in Appendix H.

We have multiplied the second row of Eq. (61) with a factor $0\leq\alpha\leq
1$ to take into account the angular distribution of the spin polarization of
the eigenstates on the half Fermi circles indicated by $U^{+}$, $U^{-}$,
$D^{+}$, and $D^{-}$ in Fig. 5(a). The net $z$-spin polarization (or $z$-spin
voltage) is expected to be lower by the factor $\alpha$ Hong  _et al._ (2012,
2016); Sayed  _et al._ (2016a) which depends on the distribution of the spin
polarization of the eigenstates and the details of the scattering mechanisms.
The $\alpha$ factor introduced in Eq. (61) appears in Eq. (19) to indicate a
lowering of the charge current induced spin voltage in the channel from the
ideal value due to such angular distribution. Onsager reciprocity requires
that the spin current induced charge voltage in the channel will be lowered by
the same factor $\alpha$ as shown in Eq. (20), which has been taken into
account by multiplying $1/\alpha$ to the third row of Eq. (61). In the
simplest approximation, the angle between the $z$-axis and the spin
polarization of the eigenstates of a particular half Fermi circle in Fig. 5(a)
varies from $-\pi/2$ to $+\pi/2$ which yields $\alpha=2/\pi$.

Combining Eq. (59) with Eq. (61) yields

$\begin{array}[]{l}\dfrac{1}{{\left|{\left\langle{{v_{x}}}\right\rangle}\right|}}\dfrac{\partial}{{\partial
t}}\;\left[{\begin{array}[]{*{20}{c}}0&0&0&1\\\ 0&0&\alpha^{2}&0\\\
0&\dfrac{1}{\alpha^{2}}&0&0\\\
1&0&0&0\end{array}}\right]\left\\{\begin{array}[]{l}I_{c}{R_{B}}\\\
2{V_{s}}\\\ {I_{s}}{R_{B}}\\\
2{V_{c}}\end{array}\right\\}+\dfrac{\partial}{{\partial
x}}\left\\{\begin{array}[]{l}I_{c}{R_{B}}\\\ 2{V_{s}}\\\ {I_{s}}{R_{B}}\\\
2{V_{c}}\end{array}\right\\}\\\ =\;\left[{\begin{array}[]{*{20}{c}}0&0&0&0\\\
0&0&{-\dfrac{2\alpha^{2}}{{{\lambda_{0}}}}}&{\dfrac{2\alpha
p_{0}}{{{\lambda_{0}}}}}\\\
{\dfrac{2}{\alpha\lambda_{s}^{\prime}}}&{-\dfrac{2}{{{\alpha^{2}\lambda_{s}}}}}&0&0\\\
{-\dfrac{2}{\lambda}}&{\dfrac{2}{{\alpha\lambda^{\prime}}}}&0&0\end{array}}\right]\left\\{\begin{array}[]{l}I_{c}{R_{B}}\\\
2{V_{s}}\\\ {I_{s}}{R_{B}}\\\ 2{V_{c}}\end{array}\right\\}\\\
-2{L_{M}}\dfrac{{\partial I_{c}}}{{\partial t}}\left\\{\begin{array}[]{l}0\\\
0\\\ \dfrac{p_{0}}{\alpha}\\\
1\end{array}\right\\}+\dfrac{1}{{\left|{\left\langle{{v_{x}}}\right\rangle}\right|}}\dfrac{2}{{{C_{E}}}}\dfrac{{\partial
Q}}{{\partial t}}\left\\{\begin{array}[]{l}1\\\ \alpha{p_{0}}\\\ 0\\\
0\end{array}\right\\}+\left\\{\begin{array}[]{l}{R_{B}i^{c}}\\\ 2\Delta
v^{s}\\\ R_{B}i^{s}\\\ 2\Delta v^{c}\end{array}\right\\},\end{array}$ (62)

with the external contact terms given by

$\displaystyle i^{c}=i_{U}^{+}+i_{D}^{-}+i_{U}^{-}+i_{D}^{+},$ (63)
$\displaystyle
i^{s}=\dfrac{1}{\alpha}\left(i_{U}^{+}-i_{D}^{-}+i_{U}^{-}-i_{D}^{+}\right)$
$\displaystyle\Delta
v^{c}=\frac{R_{B}}{2}\left(i_{U}^{+}-i_{D}^{-}-i_{U}^{-}+i_{D}^{+}\right),\text{
and}$ $\displaystyle\Delta v^{s}=\frac{\alpha
R_{B}}{2}\left(i_{U}^{+}+i_{D}^{-}-i_{U}^{-}-i_{D}^{+}\right).$

Eq. (63) combined with Eq. (60) yields Eqs. (8) and (9).

###  V.10 Continuity Equation

The term $\partial Q/\partial t$ on the right hand side of Eq. (62) is related
to the charge currents according to the continuity equation given by

$\dfrac{\partial Q}{\partial t}+\dfrac{\partial I_{c}}{\partial x}=i^{c}.$
(64)

The first row of Eq. (62) combined with Eq. (64) becomes

$\frac{\partial}{{\partial
t}}{V_{c}}=\left({\frac{{\left|{\left\langle{{v_{x}}}\right\rangle}\right|{R_{B}}}}{2}+\frac{1}{{{C_{E}}}}}\right)\left({i^{c}}-\frac{\partial}{{\partial
x}}{I_{c}}\right),$ (65)

which is the first equation in Eq. (6).

The second row of Eq. (62) combined with Eq. (64) becomes

$\displaystyle\alpha^{2}\frac{{{R_{B}}}}{{2\left|{\left\langle{{v_{x}}}\right\rangle}\right|}}\frac{\partial}{{\partial
t}}{I_{s}}$ $\displaystyle+\frac{\partial}{{\partial
x}}{V_{s}}=-\frac{{\alpha^{2}{R_{B}}}}{{{\lambda_{0}}}}{I_{s}}+\frac{{2\alpha{p_{0}}}}{{{\lambda_{0}}}}{V_{c}}$
(66)
$\displaystyle+\alpha{p_{0}}\frac{1}{{\left|{\left\langle{{v_{x}}}\right\rangle}\right|}}\frac{1}{{{C_{E}}}}\left({{i^{c}}-\frac{{\partial{I_{c}}}}{{\partial
x}}}\right)+\Delta{v^{s}}.$

We replace the expression for ${{i^{c}}-\dfrac{{\partial}}{{\partial
x}}}{I_{c}}$ from Eq. (65) to get the second equation in Eq. (7). For contact
conductance $G_{0}\rightarrow 0$, Eqs. (6) and (7) reduces to Eqs. (2) and (3)
respectively.

###  V.11 Mean Free Paths

We have three distinct mean free paths in Eq. (62), given by

$\displaystyle\dfrac{1}{\lambda}=\dfrac{1}{2}\left({\dfrac{{{r_{s2}}}}{N}+\dfrac{{{r_{s1}}}}{M}}\right)+\dfrac{r}{2}\left({\dfrac{1}{N}\;+\dfrac{1}{M}}\right),$
(67)
$\displaystyle\dfrac{1}{{{\lambda_{0}}}}=\dfrac{r+{t_{s}}}{2}\left({\dfrac{1}{N}+\dfrac{1}{M}}\right),\;\text{and}$
$\displaystyle\dfrac{1}{{{\lambda_{s}}}}=\dfrac{1}{2}\left({\dfrac{{{r_{s2}}}}{N}+\dfrac{{{r_{s1}}}}{M}}\right)+\dfrac{{t_{s}}}{2}\left({\dfrac{1}{N}+\dfrac{1}{M}}\right),\,$

where $\lambda$, $\lambda_{0}$, and $\lambda_{s}$ determine the series charge
resistance $R_{c}$ (see Eq. (5c)), the series spin resistance $R_{s}$ (see Eq.
(5d)), and the shunt spin conductance $G_{sh}$ (see (5e)) respectively. Note
that $\lambda_{s}$ depends on the spin-flip processes in the channel and
determines the shunt conductance $G_{sh}$ that takes into account the spin
relaxation process.

###  V.12 Charge-Spin Coupling Coefficients

The other terms of Eq. (62) are given by

$\begin{array}[]{l}\dfrac{1}{{\lambda^{\prime}}}=\dfrac{1}{2}\left({\dfrac{{{r_{s2}}}}{N}-\dfrac{{{r_{s1}}}}{M}}\right)+\dfrac{r}{2}\left({\dfrac{1}{N}\;-\dfrac{1}{M}}\right),\;\text{and}\\\
\dfrac{1}{{{{\lambda_{s}^{\prime}}}}}=\dfrac{1}{2}\left({\dfrac{{{r_{s2}}}}{N}-\dfrac{{{r_{s1}}}}{M}}\right)+\dfrac{t_{s}}{2}\left({\dfrac{1}{N}-\dfrac{1}{M}}\right),\end{array}$
(68)

representing coupling coefficients between charge and spin.
$\lambda_{s}^{\prime}$ and $\lambda^{\prime}$ cause a charge induced spin
signal and spin induced charge signal respectively. Note that the first terms
of Eq. (68) indicate a purely scattering induced spin-charge coupling even if
$M=N$ (i.e. $p_{0}=0$) since $r_{s1}$ and $r_{s2}$ are two independent
parameters and there could be situations where $r_{s1}\neq r_{s2}$.

In this paper, we restrict ourselves to SML caused by difference between $M$
and $N$ (i.e. $p_{0}\neq 0$). We can eliminate the first terms in Eq. (68) by
assuming either of the followings:

$r_{s1}=r_{s2}=r_{s},$ (69a)
$\text{or,}\;\;\dfrac{r_{s1}}{M}=\dfrac{r_{s2}}{N}.$ (69b)

The first assumption Eq. (69a) when applied in Eq. (68) yields

$\begin{array}[]{l}\dfrac{1}{{\lambda^{\prime}}}=(r_{s}+r)\left({\dfrac{{{1}}}{N}-\dfrac{{{1}}}{M}}\right)=\dfrac{p_{0}}{\lambda},\;\text{and}\\\
\dfrac{1}{{{{\lambda_{s}^{\prime}}}}}=(r_{s}+t_{s})\left({\dfrac{{{1}}}{N}-\dfrac{{{1}}}{M}}\right)=\dfrac{p_{0}}{\lambda_{s}},\end{array}$
(70)

which in turn gives $\lambda_{r}=\lambda$ and $\lambda_{t}=\lambda_{s}$ in
Eqs. (5i) and (5h) respectively.

The second assumption Eq. (69b) when applied in Eq. (68) yields

$\displaystyle\dfrac{1}{\lambda^{\prime}}=r\left({\dfrac{1}{N}\;-\dfrac{1}{M}}\right)=\dfrac{p_{0}}{\lambda_{r}},\;\text{and}$
(71)
$\displaystyle\dfrac{1}{\lambda_{s}^{\prime}}=t_{s}\left({\dfrac{1}{N}-\dfrac{1}{M}}\right)=\dfrac{p_{0}}{\lambda_{t}}.$

###  V.13 Comments on the Assumptions

The assumptions in Eqs. (48), (53), and (69) result in an effective change in
the transmission line model parameters in Eqs. (5), but does not change the
models in Eqs. (2), (3) (Fig. 1) and Eqs. (6), (7) (Fig. 2) themselves. The
assumptions made to derive the model can be revisited as the field evolves.
However, several predictions from our model for steady-state Sayed  _et al._
(2016a) have already received support from experiments Pham  _et al._ (2016b,
a); Lee  _et al._ (2018) suggesting that the assumptions are within the
reasonable limits.

##  VI Summary

We have proposed a two component (charge and $z$-component of spin)
transmission line model for channels with spin-momentum locking (SML) which is
a new addition to our SPICE compatible multi-physics model library Camsari
_et al._ (2013, 2015); Sayed  _et al._ (2016b). The model enables easy
analysis of complex geometries involving materials with spin-orbit coupling
(SOC) observed in diverse classes of materials e.g. topological insulators,
heavy metals, oxide interfaces, and narrow bandgap semiconductors. The model
is derived from a four-component diffusion equation obtained from the
Boltzmann transport equation assuming linear response and elastic scattering
in the channel. The four-component diffusion equation uses four average
electrochemical potentials based on a classification depending on the sign of
$z$-component of spin (up ($U$) or down ($D$)) and the sign of $x$-component
of group velocity ($+$ or $-$). Such classification can be viewed as an
extension of the Valet-Fert equation Valet and Fert (1993) which uses two
electrochemical potentials for $U$ and $D$ states. For a normal metal channel,
the time-dependent model presented here decouples into (i) the well-known
transmission line model for charge transport in quantum wires Burke (2002,
2003); Salahuddin  _et al._ (2005) and (ii) a time-dependent version of Valet-
Fert equation Valet and Fert (1993) for spin transport. We first derive
several results on charge-spin interconversion starting from our model in
steady-state. The steady-state results show good agreement with existing
experiments on diverse materials. We then study the phenomenon of spin-charge
separation using our full time-dependent model, especially in the materials
with SOC exhibiting SML. Our model shows the expected spin-charge separation
with two distinct velocities for charge and spin, which persist even in
channels exhibiting SML. However, we show that the lower velocity signal is
purely spin while the higher velocity signal is largely charge with an
additional spin component proportional to the degree of SML.

###### Acknowledgements.

This work was in part supported by FAME, one of six centers of STARnet, a
Semiconductor Research Corporation (SRC) program sponsored by MARCO and DARPA
and in part by ASCENT, one of six centers in JUMP, a SRC program sponsored by
DARPA.

##  Appendix A Steady-State Results

This appendix provides the details of the derivation of Eq. (18) and Eq. (23).

###  A.1 Derivation of the Resistance Matrix

Potentiometric NM Contacts: Eqs. (8) and (9) for $p_{f}=0$ becomes

$\displaystyle{i^{c}}={G_{0}}\left({{v_{c}}-{V_{c}}}\right),$ (72a)
$\displaystyle{i^{s}}=\frac{{{G_{0}}}}{{{\alpha^{2}}}}\left({{v_{s}}-{V_{s}}}\right),$
(72b)
$\displaystyle\Delta{v^{c}}=\frac{{{G_{0}}{R_{B}}}}{{2\alpha}}\left({{p_{0}}{v_{s}}-\frac{{\alpha
I_{c}{R_{B}}}}{2}}\right),$ (72c) and
$\displaystyle\Delta{v^{s}}=\frac{{\alpha{G_{0}}{R_{B}}}}{2}\left({{p_{0}}{v_{c}}-\frac{{\alpha{I_{s}}{R_{B}}}}{2}}\right).$
(72d)

We apply Eqs. (72a) and (72b) in Eqs. (72c) and (72d) respectively, which
yields

$\begin{array}[]{l}\Delta{v^{c}}=\dfrac{{\alpha{p_{0}}}}{{2{G_{B}}}}{i^{s}}+{p_{0}}\dfrac{{{G_{0}}{R_{B}}}}{{2\alpha}}{V_{s}}-{R_{cont}^{c}}{I_{c}},\\\
\Delta{v^{s}}=\dfrac{{\alpha{p_{0}}}}{{2{G_{B}}}}{i^{c}}+{p_{0}}\dfrac{{\alpha{G_{0}}{R_{B}}}}{2}{V_{c}}-{R_{cont}^{s}}{I_{s}}.\end{array}$
(73)

We assume that the contact conductance per unit length $G_{0}$ is very low
such that the potentiometric condition in Eq. (17) is satisfied. Thus, we have

$\displaystyle R_{c}\gg R_{cont}^{c},\;\;\;R_{s}\gg R_{cont}^{s},$
$\displaystyle\eta_{c}\gg\frac{G_{0}R_{B}}{2\alpha},\;\;\text{and}\;\;\eta_{s}\gg\frac{\alpha
G_{0}R_{B}}{2}.$

Under this condition, we combine Eq. (16) with Eq. (73) for $i^{c}=0$, which
yields

$\displaystyle\dfrac{d}{{dx}}I_{c}=0,$ (74a)
$\displaystyle\dfrac{d}{dx}{V_{c}}=-R_{c}\,I_{c}+{p_{0}}{\eta_{c}}{V_{s}}+\dfrac{{\alpha{p_{0}}}}{{2{G_{B}}}}{i^{s}},$
(74b)
$\displaystyle\dfrac{d}{{dx}}{I_{s}}=-G_{sh}{V_{s}}+p_{0}\gamma_{s}I_{c}+i^{s},$
(74c) and
$\displaystyle\dfrac{d}{{dx}}{V_{s}}=-{R_{s}}{I_{s}}+p_{0}\eta_{s}{V_{c}}.$
(74d)

Uniform Spin Voltage: We assume that the spin voltage $V_{s}$ is uniform in
the channel region of interest (from $x=0$ to $x=L$ where $L$ is the channel
length)

$\dfrac{d}{dx}V_{s}=0.$ (75)

Thus from Eq. (74d) we can write

${I_{s}}=\frac{{2{p_{0}}{G_{B}}}}{\alpha}V_{c},$ (76)

where we have used the definitions in Eqs. (5d) and (5h).

Differentiating both sides of Eq. (76) with respect to $x$ and combining with
Eqs. (74b) and (74c) yields

$-{G_{sh}}{V_{s}}+{p_{0}}{\gamma_{s}}{I_{c}}+{i^{s}}=\frac{{2{p_{0}}{G_{B}}}}{\alpha}\left({-{R_{c}}{\mkern
1.0mu}{I_{c}}+{p_{0}}{\eta_{c}}{V_{s}}+\frac{{\alpha{p_{0}}}}{{2{G_{B}}}}{i^{s}}}\right),$

which in conjunction with the definitions in Eq. (5) yields

${V_{s}}=\dfrac{{\alpha{p_{0}}}}{{2{G_{B}}}}\dfrac{{\left({\dfrac{1}{{{\lambda_{t}}}}+\dfrac{1}{\lambda}}\right)}}{{\left({\dfrac{1}{{{\lambda_{s}}}}+\dfrac{{p_{0}^{2}}}{{{\lambda_{r}}}}}\right)}}{I_{c}}+\dfrac{{{\alpha^{2}}\left({1-p_{0}^{2}}\right)}}{{4{G_{B}}}}\dfrac{1}{{\left({\dfrac{1}{{{\lambda_{s}}}}+\dfrac{{p_{0}^{2}}}{{{\lambda_{r}}}}}\right)}}{i^{s}}.$
(77)

From Eq. (72b) we have

${V_{s}}={v_{s}}-\frac{{{\alpha^{2}}{i^{s}}}}{{{G_{0}}}},$ (78)

which when combined with Eq. (77), becomes

$\displaystyle{v_{s}}$
$\displaystyle=\dfrac{{\alpha{p_{0}}}}{{2{G_{B}}}}\dfrac{{\left({\dfrac{1}{{{\lambda_{t}}}}+\dfrac{1}{\lambda}}\right)}}{{\left({\dfrac{1}{{{\lambda_{s}}}}+\dfrac{{p_{0}^{2}}}{{{\lambda_{r}}}}}\right)}}{I_{c}}$
(79)
$\displaystyle+\left(\dfrac{{{\alpha^{2}}\left({1-p_{0}^{2}}\right)}}{{4{G_{B}}}}\dfrac{1}{{\left({\dfrac{1}{{{\lambda_{s}}}}+\dfrac{{p_{0}^{2}}}{{{\lambda_{r}}}}}\right)}}+\dfrac{\alpha^{2}}{G_{0}}\right){i^{s}}.$

Charge Voltage in the Channel: Let us assume that the contact 1 is located at
$x=0$ and contact 2 is located at $x=L$ as shown in Fig. 3, where $L$ is the
channel length between the two contacts. The charge voltage in the channel at
$x=0$ and $x=L$ are $V_{c1}$ and $V_{c2}$ respectively. According to (74a),
$I_{c}$ is constant along the channel. Since $V_{s}$ and $i^{s}$ is also
uniform along the channel, we can integrate Eq. (74b) from $x=0$ to $x=L$ as

$V_{c1}-V_{c2}=\dfrac{L}{\lambda}{R_{B}}{I_{c}}-\frac{{2{p_{0}L}}}{{\alpha{\lambda_{r}}}}{V_{s}}-\frac{{\alpha{p_{0}}}}{{2{G_{B}}}}{i^{s}}L.$
(80)

From Eq. (72a), we can write equation for $V_{c1}$ in terms of the terminal
voltage $v_{c1}$ of contact 1 at $x=0$ as

$V_{c1}=v_{c1}-\frac{{{I_{c}}}}{{{G_{0}^{\prime\prime}}}}.$ (81)

noting that $i^{c}=I_{c}$. Here, $G_{0}^{\prime\prime}$ is the contact
conductance of contact 1. Similarly, we can write an equation for contact 2
noting that $i^{c}=-I_{c}$ given by

$V_{c2}=v_{c2}+\frac{{{I_{c}}}}{{{G_{0}^{\prime\prime}}}},$ (82)

where we assume that contact 1 and 2 has same conductance.

We combine Eq. (80) with Eqs. (77), (81), and (82) to have

$\displaystyle{v_{c1}}-{v_{c2}}=\left[{\left\\{{\frac{L}{\lambda}-\frac{{p_{0}^{2}L}}{{{\lambda_{r}}}}\frac{{\left({\frac{1}{{{\lambda_{t}}}}+\frac{1}{\lambda}}\right)}}{{\left({\frac{1}{{{\lambda_{s}}}}+\frac{{p_{0}^{2}}}{{{\lambda_{r}}}}}\right)}}}\right\\}{R_{B}}+\frac{2}{{{G_{0}}^{\prime\prime}}}}\right]{I_{c}}$
(83)
$\displaystyle-\frac{{\alpha{p_{0}}}}{{2{G_{B}}}}\frac{{\left({\dfrac{1}{{{\lambda_{r}}}}+\dfrac{1}{{{\lambda_{s}}}}}\right)}}{{\left({\dfrac{1}{{{\lambda_{s}}}}+\dfrac{{p_{0}^{2}}}{{{\lambda_{r}}}}}\right)}}{i^{s}}L.$

Scattering Condition: We assume that the reflection with spin-flip scattering
mechanism is dominant in the channel

$r_{s1,2}\gg r,\;t_{s}.$ (84)

This condition in Eqs. (67), (70), and (68) yields

$\dfrac{1}{\lambda}\approx\dfrac{1}{\lambda_{s}},\;\;\text{and}\;\;\dfrac{1}{\lambda},\dfrac{1}{\lambda_{s}}\gg\dfrac{1}{\lambda_{0}},\dfrac{1}{\lambda_{r}},\dfrac{1}{\lambda_{t}}.$
(85)

Applying Eq. (85) in Eqs. (83) and (79) we have the first and second rows of
Eq. (18) respectively by setting $\Delta V_{c}=v_{c1}-v_{c2}$,
$G_{0}^{\prime}=G_{0}L$ and $i^{s}_{tot}=i^{s}L$.

###  A.2 Derivation of the IREE Length

We start from the first row of Eq. (18) given by

$\Delta
V_{c}=\left({\frac{L}{{{G_{B}}\lambda}}}+\dfrac{2}{G_{0}^{\prime\prime}}\right)I_{c}-\frac{{\alpha{p_{0}}}}{{2{G_{B}}}}i^{s}_{tot}.$

We assume that the channel resistance is much larger than the contact
resistances, i.e. $L/(G_{B}\lambda)\gg 2/G_{0}^{\prime\prime}$. This yields

$\Delta
V_{c}={\frac{L}{{{G_{B}}\lambda}}}I_{c}-\frac{{\alpha{p_{0}}}}{{2{G_{B}}}}i^{s}_{tot}.$
(86)

We apply the short circuit condition $v_{c1}=v_{c2}$ at contacts 1 and 2 i.e.
$\Delta V_{c}=0$. Thus we get

$I_{c}=\dfrac{\alpha p_{0}\lambda}{2}\dfrac{i^{s}_{tot}}{L}.$ (87)

The IREE length is given by Eq. (21) as

$\lambda_{IREE}=\dfrac{J_{c}}{J_{s}}=\dfrac{I_{c}/w}{I_{s}/(wL)}=\dfrac{\alpha
p_{0}\lambda}{2},$ (88)

which is the expression in Eq. (22). Applying $\alpha=2/\pi$ yields Eq. (23).

##  Appendix B Simulation Setup

This appendix provides the details of the simulation setup in SPICE that was
used to analyze steady-state results of charge-spin interconversion in Section
III.

We have discretized the structure in Figs. 3(a), 3(c), and 4(a) into 100 small
sections and represented each of the small sections with the corresponding
circuit model. For example, Block 1 and block 2 indicated in Fig. 6 are
represented with the models in Figs. 1 and 2 respectively. Note that each of
the nodes in Fig. 6 are two component: charge ($c$) and $z$-component of spin
($s$). We have connected the charge and spin terminals of the models for all
the small sections in a modular fashion using standard circuit rules as shown
in Fig. 6. We perform a dc simulation in SPICE. Note that during dc simulation
in SPICE, capacitors and inductors automatically become open and short circuit
respectively and correspond to the stead-state ($\partial/\partial
t\rightarrow 0$) form in Eq. (16).

The contacts (1, 2, and 3) in this discussion are point contacts. The contact
polarizations $p_{f}=0$ and conductances are in the potentiometric limit with
$G_{0}\approx 0.05G_{B}$. We set the total number of modes $M+N$ in the
channel to be 100. We have assumed that the reflection with spin-flip
scattering mechanism is dominant in the channel i.e. $r_{s1,2}\gg r,t_{s}$.
The scattering rate per unit mode was set to 0.04 per lattice point.

We apply the charge open and spin ground boundary condition at the two
boundaries given by

${\left\\{{\begin{array}[]{*{20}{c}}{{i_{c}}}\\\
{{v_{s}}}\end{array}}\right\\}_{L}}=\left\\{{\begin{array}[]{*{20}{c}}0\\\
0\end{array}}\right\\},\,\,\,\,\text{and}\,\,{\left\\{{\begin{array}[]{*{20}{c}}{{i_{c}}}\\\
{{v_{s}}}\end{array}}\right\\}_{R}}=\left\\{{\begin{array}[]{*{20}{c}}0\\\
0\end{array}}\right\\}.$ (89)

Here, $i_{c}$ and $v_{s}$ indicates boundary charge current and boundary spin
voltage. Indices $L$ and $R$ indicate left and right boundaries respectively.

Setup in Fig. 3(a): For setup in Fig. 3(a), we apply a current $I_{c}$ at the
charge terminals of contacts 1 and 2, given by

${\left\\{{\begin{array}[]{*{20}{c}}{{i_{c}}}\\\
{{v_{s}}}\end{array}}\right\\}_{1}}=\left\\{{\begin{array}[]{*{20}{c}}{{I_{c}}}\\\
0\end{array}}\right\\}\;\;\text{and}\;\;{\left\\{{\begin{array}[]{*{20}{c}}{{i_{c}}}\\\
{{v_{s}}}\end{array}}\right\\}_{2}}=\left\\{{\begin{array}[]{*{20}{c}}{-{I_{c}}}\\\
0\end{array}}\right\\}.$ (90)

where indices $1$ and $2$ represent contacts 1 and 2 respectively. The spin
terminals of contacts 1 and 2 are grounded to take into account the spin
relaxation within the contact. Both charge and spin terminals of contact 3 are
open and we observe open circuit spin voltage at the spin terminal. The
boundary condition at contact 3:

${\left\\{{\begin{array}[]{*{20}{c}}{{i_{c}}}\\\
{{i_{s}}}\end{array}}\right\\}_{3}}=\left\\{{\begin{array}[]{*{20}{c}}0\\\
0\end{array}}\right\\},$ (91)

where index $3$ indicate the contact 3.

Setup in Fig. 3(c): For the setup in Fig. 3(c), the charge terminal of contact
3 is open and we apply a current $i^{s}_{tot}$ at the spin terminal, given by

${\left\\{{\begin{array}[]{*{20}{c}}{{i_{c}}}\\\
{{i_{s}}}\end{array}}\right\\}_{3}}=\left\\{{\begin{array}[]{*{20}{c}}0\\\
{i_{tot}^{s}}\end{array}}\right\\}.$ (92)

The spin terminals of contacts 1 and 2 are grounded to take into account the
spin relaxation within the contact and charge terminals are kept open. Here we
observe the open circuit voltage difference between the charge terminals of
contacts 1 and 2. The boundary conditions are:

${\left\\{{\begin{array}[]{*{20}{c}}{{i_{c}}}\\\
{{v_{s}}}\end{array}}\right\\}_{1}}=\left\\{{\begin{array}[]{*{20}{c}}0\\\
0\end{array}}\right\\}\;\;\text{and}\;\;{\left\\{{\begin{array}[]{*{20}{c}}{{i_{c}}}\\\
{{v_{s}}}\end{array}}\right\\}_{2}}=\left\\{{\begin{array}[]{*{20}{c}}0\\\
0\end{array}}\right\\}.$ (93)

Setup in Fig. 4(a): For the setup in Fig. 4(a), charge terminal of contact 3
is open and we inject a current $i^{s}_{tot}$ through the spin terminal. The
boundary condition at contact 3:

${\left\\{{\begin{array}[]{*{20}{c}}{{i_{c}}}\\\
{{i_{s}}}\end{array}}\right\\}_{3}}=\left\\{{\begin{array}[]{*{20}{c}}0\\\
{i_{tot}^{s}}\end{array}}\right\\}.$ (94)

We short circuit the charge terminals of contacts 1 and 2 and observe the
short circuit charge current $I_{c}$ flowing in the channel induced by
$i^{s}_{tot}$. The spin terminals of contacts 1 and 2 are grounded to take
into account the spin relaxation within the contact. The boundary conditions
at contacts 1 and 2 are:

${\left\\{{\begin{array}[]{*{20}{c}}{{v_{c}}}\\\
{{v_{s}}}\end{array}}\right\\}_{1}}={\left\\{{\begin{array}[]{*{20}{c}}{{v_{c}}}\\\
0\end{array}}\right\\}_{2}}\;\;\text{and}\;\;{\left\\{{\begin{array}[]{*{20}{c}}{{v_{c}}}\\\
{{v_{s}}}\end{array}}\right\\}_{2}}={\left\\{{\begin{array}[]{*{20}{c}}{{v_{c}}}\\\
0\end{array}}\right\\}_{1}}.$ (95)

Figure 6: SPICE setup for dc simulation. The setup connects the transmission
line circuit models in a distributed manner. Block 1 corresponds to model in
Fig. 1 and Block 2 corresponds to model in Fig. 2. All the external contact
have $p_{f}=0$. All terminals are two component: charge ($c$) and spin ($s$).
All indicated spin terminals are grounded except the spin terminal of contact
2.

##  Appendix C Parameter Estimations

This appendix provides the details of the estimations made for the IREE
lengths on diverse materials using Eq. (23).

###  C.1 Estimations

####  C.1.1 Estimation of Fermi Wave Vector ($k_{F}$)

We estimate the Fermi wave vector $k_{F}$ of the channel from the electron
density $n_{3D}$ (units of m-3) or $n_{2D}$ (units of m-2), using the
following expressions

${{k}_{F}}=\sqrt[3]{3{{\pi}^{2}}{{n}_{3D}}},$ (96a) ${{k}_{F}}=\sqrt{2\pi
n_{2D}}.$ (96b)

Estimations are summarized in Table 2.

Table 2: Estimation of $k_{F}$.

Material | Electron Density |  $k_{F}$ [nm-1]
---|---|---
Ag$|$Bi† |  $5.86\times{{10}^{28}}\,{{\text{m}}^{-3}}$ Ashcroft and Mermin (1976) | 12 (Eq. (96a))
Cu$|$Bi† |  $8.49\times{{10}^{28}}\,{{\text{m}}^{-3}}$ Ashcroft and Mermin (1976) | 13.6 (Eq. (96a))
LAO$|$STO |  $2.6\times{{10}^{17}}\,{{\text{m}}^{-2}}$ Lesne  _et al._ (2016) | 1.278 (Eq. (96b))
Bi2Se3 |  $4\times{{10}^{17}}\,{{\text{m}}^{-2}}$ Wang  _et al._ (2016) | 1.59 (Eq. (96b))

†We have used the electron densities of Ag and Cu respectively as they are the
most conductive layer in the corresponding bi-layer.

####  C.1.2 Estimation of Ballistic Conductance ($G_{B}$)

We estimate the total number of modes in the 3D channel using the following
expression

$M+N=\frac{{k_{F}^{2}wt}}{{2\pi}},$ (97)

where $w$ is the width and $t$ is the thickness. For a 2D channel the
expression is

$M+N=\frac{2k_{F}w}{\pi}.$ (98)

We estimate the ballistic conductance of the channel using the following
expression

$G_{B}=\dfrac{q^{2}}{h}(M+N).$ (99)

Estimations are summarized in Table 3.

Table 3: Estimation of $G_{B}$.

Material |  $w$ [$\mu$m] |  $t$ [nm] | $M+N$ | $G_{B}$
---|---|---|---|---
|  |  |  | (Eq. (99))
Ag$|$Bi† | 400 Sánchez  _et al._ (2016) | 5 Sánchez  _et al._ (2016) | $4.6\times 10^{7}$ | 1.77 kS
|  |  | (Eq. (97)) |
Cu$|$Bi† | 0.15 Isasa  _et al._ (2016) | 20 Isasa  _et al._ (2016) | $8.8\times 10^{4}$ | 3.4 S
|  |  | (Eq. (97)) |
LAO$|$STO | 400 Lesne  _et al._ (2016) | - | $3.3\times 10^{5}$ | 12.6 S
|  |  | (Eq. (98)) |
Bi2Se3 | 1000 Wang  _et al._ (2016) | 9†† Wang  _et al._ (2016) | $3.6\times 10^{6}$ | 140 S
|  |  | (Eq. (97)) |

†We have used the thicknesses of the most conductive layer (Ag and Cu
respectively) to estimate $G_{B}$ considering bulk conduction.
††We considered 6 quintuple layer (QL) sample in Ref. Wang  _et al._ (2016). 1
QL $\approx$ 1.5 nm.

####  C.1.3 Estimation of Mean Free Path ($\lambda$)

The mean free path ($\lambda$) is estimated from the measured sheet resistance
$R_{S}$ of the sample using the following expression:

$\frac{{{R_{S}}}}{w}=\frac{1}{{{G_{B}}\lambda}},$ (100)

or from the resistivity $\rho$ of the sample using the following expression:

$\frac{\rho}{{wt}}=\frac{1}{{{G_{B}}\lambda}}.$ (101)

Estimations are summarized in Table 4.

Table 4: Estimation of $\lambda$.

Material |  $R_{S}$ [$\Omega/\square$] |  $\rho$ [$\mu\Omega$-cm] |  $\lambda$ [nm]
---|---|---|---
Ag$|$Bi |  $10$† | - | 22.6 (Eq. (100))
Cu$|$Bi | - |  $100$‡ | 0.88 (Eq. (101))
LAO$|$STO |  $176$†† | - | 180.8 (Eq. (100))
Bi2Se3 | - | 2000‡‡ | 3.2 (Eq. (101))

†Taken from Fig. 3(a) of Ref. Sánchez  _et al._ (2016) for Ag$|$Bi sample with
5 nm Ag.
††Taken from Fig. 1(d) of Ref. Lesne  _et al._ (2016) for LAO$|$STO at 7K.
‡$\rho$ of Bi layer was used which was taken from Ref. Isasa  _et al._ (2016).
‡‡Taken from Fig. 2(b) of Ref. Wang  _et al._ (2016) at $\sim$300K.

####  C.1.4 Estimation of Degree of Spin-Momentum Locking ($p_{0}$)

For Ag$|$Bi, Cu$|$Bi, and LAO$|$STO, the degree of SML $p_{0}$ is estimated
using the Rashba coupling coefficient ($v_{0}=\alpha_{R}/\hbar$) and the Fermi
velocity ($v_{F}$) of the materials, using Eq. (24). For Bi2Se3, $p_{0}$ is
estimated using spin current density reported from spin pumping ($J_{s}$),
measured inverse effect voltage $\Delta V_{c}$, and Eq. (20). The estimations
are summarized in Table 5.

Table 5: Estimation of $p_{0}$.

Material |  $\alpha_{R}$ [eV$\cdot\AA$] |  $v_{F}$ [$\times 10^{6}$ m$\cdot$s-1] | $p_{0}$
---|---|---|---
Ag$|$Bi | 0.56 Sánchez  _et al._ (2016) | 1.39 (Ag) Ashcroft and Mermin (1976) | 0.05
|  | 1.87 (Bi) Ashcroft and Mermin (1976) |
Cu$|$Bi | 0.56 Isasa  _et al._ (2016) | 1.57 (Cu) Ashcroft and Mermin (1976) | 0.05
|  | 1.87 (Bi) Ashcroft and Mermin (1976) |
LAO$|$STO | 0.03 Lesne  _et al._ (2016) | 0.074† | 0.0616
Material |  $\Delta V_{c}$ [$\mu$V] |  $J_{s}$ [A-m-2] | $p_{0}$
Bi2Se3 | 40‡ Wang  _et al._ (2016) | 1.39$\times$105 ‡‡ | 0.025
|  |  | (Eq. (20)).

†We estimate the Fermi velocity using
${{v}_{F}}=\frac{\hbar{{k}_{F}}}{{{m}^{*}}}$. $m^{*}\approx$ 2 $\times$
$9.1\times{{10}^{-31}}$ kg as reported in Ref. Lesne  _et al._ (2016).
$\ddagger$Inverse effect induced voltage due to spin-pumping at 3 GHz.
‡‡Taken from table 1 in supplementary information of Ref. Wang  _et al._
(2016) for 6 QL. The sample dimension is 1 mm $\times$ 5 mm Wang  _et al._
(2016), which yields $i^{s}_{tot}\approx 0.695$ A.

###  C.2 Derivation of Eq. (24)

We start from the eigenstates in Eq. (37) of the Rashba Hamiltonian in Eq.
(36), given by

$E=\dfrac{{p^{\prime}}^{2}}{2m}-s\,v_{0}{p^{\prime}}+U_{E}.$

with ${p^{\prime}}=|\vec{p}-q\vec{A}|$. Solutions for ${p^{\prime}}$ are given
by

$\begin{array}[]{l}{{p^{\prime}}_{1}}(s)=s\,m{v_{0}}+\sqrt{{m^{2}}v_{0}^{2}+2m(E-U_{E})},\\\
{{p^{\prime}}_{2}}(s)=s\,m{v_{0}}-\sqrt{{m^{2}}v_{0}^{2}+2m(E-U_{E})},\end{array}$

noting that $s^{2}=1$. Here, ${{p^{\prime}}_{1}}(s=+1)$ and
${{p^{\prime}}_{1}}(s=-1)$ correspond to $M$ and $N$ respectively. Similarly
${{p^{\prime}}_{2}}(s=-1)$ and ${{p^{\prime}}_{2}}(s=+1)$ correspond to $M$
and $N$ respectively. Thus the degree of SML $p_{0}$ is given by

$\displaystyle p_{0}(E_{F})$
$\displaystyle=\dfrac{{{p^{\prime}}_{1}}(s=+1)-{{p^{\prime}}_{1}}(s=-1)}{{{p^{\prime}}_{1}}(s=+1)+{{p^{\prime}}_{1}}(s=-1)}$
(102)
$\displaystyle=\dfrac{{{v_{0}}}}{{\sqrt{v_{0}^{2}+\dfrac{{2(E_{F}-U_{E})}}{m}}}}.$

Assuming $U_{E}=0$ and applying $E_{F}=\dfrac{1}{2}mv_{F}^{2}$ we have the
expression in Eq. (24).

##  Appendix D Charge and Spin Velocities

This appendix provides the derivation of eigenvalues and eigenvectors of Eqs.
(2) and (3) to find the charge and spin velocities and their coupling while
propagation.

The matrix form of Eqs. (2) and (3) is given by

$\begin{array}[]{l}\left[{\begin{array}[]{*{20}{c}}{{C_{eff}}}&0&0&0\\\
0&{\dfrac{C_{Q}}{\alpha^{2}}}&{p_{0}g_{m}{L_{M}}}&0\\\ 0&0&{{L_{eff}}}&0\\\
{-p_{0}r_{m}{C_{eff}}}&0&0&{\alpha^{2}{L_{K}}}\end{array}}\right]\dfrac{\partial}{{\partial
t}}\left\\{{\begin{array}[]{*{20}{c}}{{V_{c}}}\\\ {{V_{s}}}\\\ {{I_{c}}}\\\
{{I_{s}}}\end{array}}\right\\}\\\ +\left[{\begin{array}[]{*{20}{c}}0&0&0&0\\\
0&{{G_{sh}}}&{-{p_{0}}{\gamma_{s}}}&0\\\ 0&{-{p_{0}}{\eta_{c}}}&{{R_{c}}}&0\\\
{-{p_{0}}{\eta_{s}}}&0&0&{{R_{s}}}\end{array}}\right]\left\\{{\begin{array}[]{*{20}{c}}{{V_{c}}}\\\
{{V_{s}}}\\\ {{I_{c}}}\\\ {{I_{s}}}\end{array}}\right\\}\\\
=-\left[{\begin{array}[]{*{20}{c}}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{array}}\right]\dfrac{\partial}{{\partial
x}}\left\\{{\begin{array}[]{*{20}{c}}{{V_{c}}}\\\ {{V_{s}}}\\\ {{I_{c}}}\\\
{{I_{s}}}\end{array}}\right\\}.\end{array}$ (103)

In the low-loss limit, we assume a solution of the form

$\left\\{\begin{array}[]{*{20}{c}}{{V_{c}}}\\\ {{V_{s}}}\\\ {{I_{c}}}\\\
{{I_{s}}}\end{array}\right\\}\equiv\left\\{\begin{array}[]{*{20}{c}}{{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\\\ {{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\end{array}\right\\}e^{j\left(kx-\omega t\right)},$ (104)

which results in

$\begin{array}[]{l}-j\omega\left[{\begin{array}[]{*{20}{c}}{{C_{eff}}}&0&0&0\\\
0&{\dfrac{C_{Q}}{\alpha^{2}}}&{p_{0}g_{m}{L_{M}}}&0\\\ 0&0&{{L_{eff}}}&0\\\
{-p_{0}r_{m}{C_{eff}}}&0&0&{\alpha^{2}{L_{K}}}\end{array}}\right]\left\\{\begin{array}[]{*{20}{c}}{{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\\\ {{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\end{array}\right\\}\\\
+\left[{\begin{array}[]{*{20}{c}}0&0&0&0\\\
0&{{G_{sh}}}&{-{p_{0}}{\gamma_{s}}}&0\\\ 0&{-{p_{0}}{\eta_{c}}}&{{R_{c}}}&0\\\
{-{p_{0}}{\eta_{s}}}&0&0&{{R_{s}}}\end{array}}\right]\left\\{\begin{array}[]{*{20}{c}}{{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\\\ {{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\end{array}\right\\}\\\
=-jk\left[{\begin{array}[]{*{20}{c}}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{array}}\right]\left\\{\begin{array}[]{*{20}{c}}{{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\\\ {{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\end{array}\right\\}.\end{array}$ (105)

We assume that the co-efficients of the transmission line model (Eqs. (2) and
(3)) are constant with frequency $\omega$ and the propagation vector $k$. We
differentiate both sides of Eq. (105) with respect to $k$ and the following
matrix equation

$v_{g}\left\\{\begin{array}[]{*{20}{c}}{{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\\\ {{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\end{array}\right\\}=\left[{\begin{array}[]{*{20}{c}}0&0&{\dfrac{1}{{{C_{eff}}}}}&0\\\
{-\dfrac{\alpha^{2}p_{0}g_{m}L_{M}}{C_{Q}L_{eff}}}&0&0&{\dfrac{\alpha^{2}}{{{C_{Q}}}}}\\\
{\dfrac{1}{{{L_{eff}}}}}&0&0&0\\\
0&{\dfrac{1}{{\alpha^{2}{L_{K}}}}}&\dfrac{p_{0}r_{m}}{\alpha^{2}L_{K}}&0\end{array}}\right]\left\\{\begin{array}[]{*{20}{c}}{{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\\\ {{\tilde{V}_{c}}}\\\
{{\tilde{V}_{s}}}\end{array}\right\\},$ (106)

where $v_{g}$ is the group velocity given by

$v_{g}=\dfrac{\partial\omega}{\partial k}.$ (107)

Note that the eigenvalues of Eq. (106) give the velocities in Eqs. (25) and
(27). For a particular eigenvalue $v_{g}$, we can write the following equation
from Eq. (106)

$\begin{array}[]{l}\left[{\begin{array}[]{*{20}{c}}{-{v_{g}}}&{\dfrac{1}{{{C_{eff}}}}}\\\
{\dfrac{1}{{{L_{eff}}}}}&{-{v_{g}}}\end{array}}\right]\left\\{{\begin{array}[]{*{20}{c}}{{V_{c}}}\\\
{{I_{c}}}\end{array}}\right\\}=\left\\{{\begin{array}[]{*{20}{c}}0\\\
0\end{array}}\right\\},\\\
\left[{\begin{array}[]{*{20}{c}}{-{v_{g}}}&{\dfrac{1}{{{C_{Q}}}}}\\\
{\dfrac{1}{{{L_{K}}}}}&{-{v_{g}}}\end{array}}\right]\left\\{{\begin{array}[]{*{20}{c}}{{V_{s}}}\\\
{{I_{s}}}\end{array}}\right\\}={p_{0}}\left\\{{\begin{array}[]{*{20}{c}}{\dfrac{{\alpha^{2}{g_{m}}{L_{M}}}}{{{C_{Q}}{L_{eff}}}}{V_{c}}}\\\
{-\dfrac{{{r_{m}}}}{{\alpha^{2}{L_{K}}}}{I_{c}}}\end{array}}\right\\}.\end{array}$
(108)

For the eigenvalue $v_{g}=v_{g,s}={1}/{\sqrt{L_{K}C_{Q}}}$ in Eq. (108) we
have $V_{c}=I_{c}=0$ and assuming $I_{s}=1$ we get Eq. (26). Again, for the
eigenvalue $v_{g}=v_{g,c}={1}/{\sqrt{L_{eff}C_{eff}}}$ we get Eq. (28)
assuming $I_{c}=1$.

##  Appendix E Derivation of semiclassical model

This appendix provides the derivation of Eq. (35), starting from the Boltzmann
transport equation in Eq. (31).

We apply a variable transformation $\xi(x,t,\vec{p},\vec{s})\equiv
E(x,t,\vec{p},\vec{s})-\mu(x,t,\vec{p},\vec{s})$ on the left hand side of Eq.
(31), which yields

$\displaystyle\frac{\partial f}{\partial t}=\frac{\partial
f}{\partial\xi}\left(\frac{\partial E}{\partial t}-\frac{\partial\mu}{\partial
t}\right),$ (109) $\displaystyle v_{x}\frac{\partial f}{\partial
x}=v_{x}\frac{\partial f}{\partial\xi}\left(\frac{\partial E}{\partial
x}-\frac{\partial\mu}{\partial x}\right),\text{ and }$ $\displaystyle
F_{x}\frac{\partial f}{\partial p_{x}}=F_{x}\frac{\partial
f}{\partial\xi}\left(\frac{\partial E}{\partial
p_{x}}-\frac{\partial\mu}{\partial p_{x}}\right).$

We first substitute $F_{x}=-\dfrac{\partial E}{\partial x}$ and
$v_{x}=\dfrac{\partial E}{\partial p_{x}}$. Finally, we set $\dfrac{\partial
f}{\partial\xi}=\dfrac{\partial f_{0}}{\partial E}$ to get the left hand side
of Eq. (35).

On the right hand side of Eq. (31), we expand both $f$ and $f^{\prime}$ into
Taylor series around
$f_{0}=1\big{/}\left(1+\exp\left((E(x,\vec{p},{s})-\mu_{0})/k_{B}T\right)\right)$
with constant electrochemical potential $\mu_{0}$. We set
$\xi^{\prime}(x,t,\vec{p^{\prime}},{s^{\prime}})\equiv
E(x,t,\vec{p^{\prime}},{s^{\prime}})-{\mu}(x,t,\vec{p^{\prime}},{s^{\prime}})$
and $\xi_{0}(x,\vec{p},{s})\equiv E(x,\vec{p},{s})-\mu_{0}$ and assume that
$\xi$ and $\xi^{\prime}$ are close to $\xi_{0}$ which gives

$\begin{array}[]{l}f\approx{f_{0}}+{\left({-\dfrac{{\partial
f}}{{\partial\xi}}}\right)_{\xi={\xi_{0}}}}\left({\xi-{\xi_{0}}}\right),\text{
and}\\\ f^{\prime}\approx{f_{0}}+{\left({-\dfrac{{\partial
f}}{{\partial\xi^{\prime}}}}\right)_{\xi^{\prime}={\xi_{0}}}}\left({\xi^{\prime}-{\xi_{0}}}\right).\end{array}$
(110)

We set ${\left({\dfrac{{\partial
f}}{{\partial\xi}}}\right)_{\xi={\xi_{0}}}}={\left({\dfrac{{\partial
f}}{{\partial\xi^{\prime}}}}\right)_{\xi^{\prime}={\xi_{0}}}}=\dfrac{\partial
f_{0}}{\partial E}$ Thus the right hand side of Eq. (31) becomes

${f-{f^{\prime}}}=\left(-\dfrac{\partial f_{0}}{\partial
E}\right)\left(\xi-{\xi}^{\prime}\right)=\left(\dfrac{\partial f_{0}}{\partial
E}\right){\left(\mu-{\mu^{\prime}}\right)},$ (111)

noting that $E(x,t,\vec{p^{\prime}},s^{\prime})=E(x,t,\vec{p},s)$ in the
elastic scattering limit.

##  Appendix F $E$-$p$ relation

This appendix provides the derivation of Eq. (42), starting from the $E$-$p$
relation in Eq. (38).

Differentiating Eq. (38) with respect to time $t$ yields

$\dfrac{\partial E}{\partial
t}=\left(\dfrac{\vec{p}-q\vec{A}}{m}-s\,v_{0}\dfrac{\vec{p}-q\vec{A}}{|\vec{p}-q\vec{A}|}\right)\cdot\left(-q\dfrac{\partial\vec{A}}{\partial
t}\right)+\dfrac{\partial U_{E}}{\partial t}.$ (112)

The velocity $\vec{v}(E)=\nabla_{\vec{p}}E$ is derived from Eq. (38) as

$\vec{v}(E)=\dfrac{\vec{p}-q\vec{A}}{m}-s\,v_{0}\dfrac{\vec{p}-q\vec{A}}{|\vec{p}-q\vec{A}|}.$
(113)

Combining Eqs. (112) and (113) yields Eq. (39). Note that Eq. (113) can be
written as

$\vec{v}(E)=\left({\frac{{|\vec{p}-q\vec{A}|}}{m}+s\,{v_{0}}}\right)\frac{{\vec{p}-q\vec{A}}}{{|\vec{p}-q\vec{A}|}}.$
(114)

where $\dfrac{{\vec{p}-q\vec{A}}}{{|\vec{p}-q\vec{A}|}}$ is an unit vector
along ${{\vec{p}-q\vec{A}}}$. From Eq. (38) we get

$|\vec{p}-q\vec{A}|=-s\,m\,{v_{0}}\pm
m\sqrt{v_{0}^{2}+\dfrac{2\left({{E}-U_{E}}\right)}{m}},$ (115)

noting that $s^{2}=1$. Thus from Eq. (114) we get

$\vec{v}(E)=\pm\sqrt{v_{0}^{2}+\frac{{2\left({{E}-U_{E}}\right)}}{m}}\frac{{\vec{p}-q\vec{A}}}{{|\vec{p}-q\vec{A}|}}.$
(116)

Note that the magnitude of the velocity for a particular energy of interest is
same for all four groups in Eq. (44) for the Rashba Hamiltonian in Eq. (36)
considered here.

##  Appendix G Scattering Rates

This appendix provides the derivation of the scattering matrix in Eq. (56).

For the group $\Re\equiv U^{+}$, $D^{-}$, $U^{-}$, and $D^{+}$, we have from
Eq. (54)

$\displaystyle\tilde{S}(U^{+})=$
$\displaystyle{\hat{S}_{U^{+}\leftrightarrow{D^{-}}}}\left({\left\langle{\mu\left({{D^{-}}}\right)}\right\rangle-\left\langle{\mu\left(U^{+}\right)}\right\rangle}\right)$
(117)
$\displaystyle+{\hat{S}_{U^{+}\leftrightarrow{U^{-}}}}\left({\left\langle{\mu\left({{U^{-}}}\right)}\right\rangle-\left\langle{\mu\left(U^{+}\right)}\right\rangle}\right)$
$\displaystyle+{\hat{S}_{U^{+}\leftrightarrow{D^{+}}}}\left({\left\langle{\mu\left({{D^{+}}}\right)}\right\rangle-\left\langle{\mu\left(U^{+}\right)}\right\rangle}\right),$

$\displaystyle\tilde{S}(D^{-})=$
$\displaystyle{\hat{S}_{D^{-}\leftrightarrow{U^{+}}}}\left({\left\langle{\mu\left({{U^{+}}}\right)}\right\rangle-\left\langle{\mu\left(D^{-}\right)}\right\rangle}\right)$
(118)
$\displaystyle+{\hat{S}_{D^{-}\leftrightarrow{U^{-}}}}\left({\left\langle{\mu\left({{U^{-}}}\right)}\right\rangle-\left\langle{\mu\left(D^{-}\right)}\right\rangle}\right)$
$\displaystyle+{\hat{S}_{D^{-}\leftrightarrow{D^{+}}}}\left({\left\langle{\mu\left({{D^{+}}}\right)}\right\rangle-\left\langle{\mu\left(D^{-}\right)}\right\rangle}\right),$

$\displaystyle\tilde{S}(U^{-})=$
$\displaystyle{\hat{S}_{U^{-}\leftrightarrow{U^{+}}}}\left({\left\langle{\mu\left({{U^{+}}}\right)}\right\rangle-\left\langle{\mu\left(U^{-}\right)}\right\rangle}\right)$
(119)
$\displaystyle+{\hat{S}_{U^{-}\leftrightarrow{D^{-}}}}\left({\left\langle{\mu\left({{D^{-}}}\right)}\right\rangle-\left\langle{\mu\left(U^{-}\right)}\right\rangle}\right)$
$\displaystyle+{\hat{S}_{U^{-}\leftrightarrow{D^{+}}}}\left({\left\langle{\mu\left({{D^{+}}}\right)}\right\rangle-\left\langle{\mu\left(U^{-}\right)}\right\rangle}\right),$

and

$\displaystyle\tilde{S}(D^{+})=$
$\displaystyle{\hat{S}_{D^{+}\leftrightarrow{U^{+}}}}\left({\left\langle{\mu\left({{U^{+}}}\right)}\right\rangle-\left\langle{\mu\left(D^{+}\right)}\right\rangle}\right)$
(120)
$\displaystyle+{\hat{S}_{D^{+}\leftrightarrow{D^{-}}}}\left({\left\langle{\mu\left({{D^{-}}}\right)}\right\rangle-\left\langle{\mu\left(D^{+}\right)}\right\rangle}\right)$
$\displaystyle+{\hat{S}_{D^{+}\leftrightarrow{U^{-}}}}\left({\left\langle{\mu\left({{U^{-}}}\right)}\right\rangle-\left\langle{\mu\left(D^{+}\right)}\right\rangle}\right).$

Eqs. (117), (118), (119), and (120) together yields the scattering matrix in
Eq. (56) in the $\\{\mu(U^{+}),\mu(D^{-}),\mu(U^{-}),\mu(D^{+})\\}^{T}$ basis.

##  Appendix H Charge and Spin Currents and Voltages

This appendix provides the derivation of the Eq. (61).

The current in any group is given by

$I\left(\Re\right)=\dfrac{q}{L}\sum\limits_{\vec{p},s\in\Re}{{v_{x}}}\,f\left({x,t,\vec{p},s}\right).$
(121)

where $f$ is given by Eq. (32). Under the linear response approximation, we
can write

$f\left({x,t,\vec{p},s}\right)\approx f_{0}+\left(-\frac{\partial
f_{0}}{\partial E}\right)\left(\mu\left({x,t,\vec{p},s}\right)-\mu_{0}\right)$
(122)

where, $f_{0}$ is given by Eq. (33) with constant electrochemical potential
$\mu_{0}$. Thus from Eq. (121), we can write

$I\left(\Re\right)=\frac{q}{L}\,\left({{f_{0}}\sum\limits_{\vec{p},s\in\Re}{{v_{x}}}+\frac{{{D_{0}}\left(\Re\right)}}{2}\left(\left\langle{{v_{x}}\mu}\right\rangle_{\vec{p},s\in\Re}-{\mu_{0}}\left\langle{{v_{x}}}\right\rangle\right)}\right),$
(123)

where $D_{0}(\Re)$ is given by Eq. (47) and the averaging is defined by Eq.
(45).

We assume the following

$\left\langle{{v_{x}}\mu}\right\rangle_{\vec{p},s\in\Re}\approx\left\langle{{v_{x}}}\right\rangle\left\langle{\mu}\right\rangle_{\vec{p},s\in\Re},$

which results in

$\displaystyle I\left(\Re\right)={\mathop{\rm
sgn}}\left({\left\langle{{v_{x}}}\right\rangle}\right){\frac{q}{h}{n_{m}}\left(\Re\right)\left({\left\langle\mu\right\rangle_{\vec{p},s\in\Re}-{\mu_{0}}}\right)}$
(124) $\displaystyle+{\mathop{\rm
sgn}}\left({|{{v_{x}}}|}\right)\dfrac{q}{L}{f_{0}}\sum\limits_{\vec{p},s\in\Re}{{|v_{x}|}},$

where $n_{m}(\Re)$ is given by Eq. (50).

###  H.1 Charge Current

The charge current in the channel is given by

${I_{c}}=I\left({{U^{+}}}\right)+I\left({{D^{+}}}\right)+I\left({{U^{-}}}\right)+I\left({{D^{-}}}\right),$
(125)

which in conjunction to Eq. (124) yields

$\displaystyle{I_{c}}=\dfrac{q}{h}\,\left[{\dfrac{h}{L}{f_{0}}\sum\limits_{\vec{p},s\in{U^{+}}}{{|v_{x}|}}+M\left({\mu\left({{U^{+}}}\right)-{\mu_{0}}}\right)}\right]$
(126)
$\displaystyle+\dfrac{q}{h}\,\left[{\dfrac{h}{L}{f_{0}}\sum\limits_{\vec{p},s\in{D^{+}}}{{|v_{x}|}}+N\left({\mu\left({{D^{+}}}\right)-{\mu_{0}}}\right)}\right]$
$\displaystyle-\dfrac{q}{h}\,\left[{\dfrac{h}{L}{f_{0}}\sum\limits_{\vec{p},s\in{U^{-}}}{{|v_{x}|}}+N\left({\mu\left({{U^{-}}}\right)-{\mu_{0}}}\right)}\right]$
$\displaystyle-\dfrac{q}{h}\,\left[{\dfrac{h}{L}{f_{0}}\sum\limits_{\vec{p},s\in{D^{-}}}{{|v_{x}|}}+M\left({\mu\left({{D^{-}}}\right)-{\mu_{0}}}\right)}\right],$

where $n_{m}(U^{+})=n_{m}(D^{-})=M$ and $n_{m}(U^{-})=n_{m}(D^{+})=N$. Note
that $(U^{+},D^{-})$ and $(U^{-},D^{+})$ are time-reversal symmetric pairs,
hence

$\displaystyle\sum\limits_{\vec{p},s\in{U^{+}}}{{|v_{x}|}}=\sum\limits_{\vec{p},s\in{D^{-}}}{{|v_{x}|}},\text{
and }$ (127)
$\displaystyle\sum\limits_{\vec{p},s\in{U^{-}}}{{|v_{x}|}}=\sum\limits_{\vec{p},s\in{D^{+}}}{{|v_{x}|}},$

Thus the expression for charge current is given by

$I_{c}=\dfrac{q}{h}\left({M\tilde{\mu}\left({U^{+}}\right)-N\tilde{\mu}\left({U^{-}}\right)+N\tilde{\mu}\left({D^{+}}\right)-M\tilde{\mu}\left({D^{-}}\right)}\right),$
(128)

where we defined $\tilde{\mu}=\mu-\mu_{0}$.

###  H.2 Spin Current

The spin current in the channel is given by

${\tilde{I}}_{s}=\dfrac{1}{\alpha}\left(I\left({{U^{+}}}\right)+I\left({{U^{-}}}\right)-I\left({{D^{+}}}\right)-I\left({{D^{-}}}\right)\right),$
(129)

where $\alpha=2/\pi$ is an angular averaging factor to take into account the
average $\hat{z}\cdot\vec{s}$ on a half Fermi circle (see Fig. 5(a)). Eq.
(129) in conjunction to Eq. (124) yields

$\displaystyle{\tilde{I}}_{s}=\dfrac{1}{\alpha}\dfrac{q}{h}\,\left[{\dfrac{h}{L}{f_{0}}\sum\limits_{\vec{p},s\in{U^{+}}}|{{v_{x}}}|+M\left({\mu\left({{U^{+}}}\right)-{\mu_{0}}}\right)}\right]$
(130)
$\displaystyle-\dfrac{1}{\alpha}\dfrac{q}{h}\,\left[{\dfrac{h}{L}{f_{0}}\sum\limits_{\vec{p},s\in{U^{-}}}|{{v_{x}}}|+N\left({\mu\left({{U^{-}}}\right)-{\mu_{0}}}\right)}\right]$
$\displaystyle-\dfrac{1}{\alpha}\dfrac{q}{h}\,\left[{\dfrac{h}{L}{f_{0}}\sum\limits_{\vec{p},s\in{D^{+}}}|{{v_{x}}}|+N\left({\mu\left({{D^{+}}}\right)-{\mu_{0}}}\right)}\right]$
$\displaystyle+\dfrac{1}{\alpha}\dfrac{q}{h}\,\left[{\frac{h}{L}{f_{0}}\sum\limits_{\vec{p},s\in{D^{-}}}|{{v_{x}}}|+M\left({\mu\left({{D^{-}}}\right)-{\mu_{0}}}\right)}\right].$

Applying the condition in Eq. (LABEL:velo_cond), we have the expression for
channel spin current as

$\displaystyle{\tilde{I}}_{s}=\dfrac{1}{\alpha}\dfrac{q}{h}\left({M\tilde{\mu}\left({U^{+}}\right)-N\tilde{\mu}\left({U^{-}}\right)-N\tilde{\mu}\left({D^{+}}\right)+M\tilde{\mu}\left({D^{-}}\right)}\right)$
(131)
$\displaystyle+\dfrac{2}{\alpha}\dfrac{q}{L}f_{0}\left(\sum\limits_{\vec{p},s\in{U^{+}}}|{{v_{x}}}|-\sum\limits_{\vec{p},s\in{U^{-}}}|{{v_{x}}}|\right)$

Note first term is zero at equilibrium condition
$\mu(U^{+})=\mu(U^{-})=\mu(D^{+})=\mu(D^{-})=\mu_{0}$.

However, the second term is non-zero even at equilibrium since

$\sum\limits_{\vec{p},s\in{U^{+}}}|{{v_{x}}}|\neq\sum\limits_{\vec{p},s\in{U^{-}}}|{{v_{x}}}|$

and represent the equilibrium spin current in the channel. We subtract the
equilibrium part from our definition of spin current given by

$\displaystyle{{I}}_{s}=\dfrac{1}{\alpha}\dfrac{q}{h}\left({M\tilde{\mu}\left({U^{+}}\right)-N\tilde{\mu}\left({U^{-}}\right)-N\tilde{\mu}\left({D^{+}}\right)+M\tilde{\mu}\left({D^{-}}\right)}\right).$
(132)

###  H.3 Spin Voltage

The spin voltage in the channel is given by

$q\,V_{s}=\alpha\frac{{M\mu\,(U+)+N\mu\,(U-)-N\mu\,(D+)-M\mu\,(D-)}}{{2\,(M+N)}}.$
(133)

Subtracting each electrochemical potential by $\mu_{0}$ which gives

$q\,V_{s}=\alpha\frac{{M\tilde{\mu}(U+)+N\tilde{\mu}(U-)-N\tilde{\mu}(D+)-M\tilde{\mu}(D-)}}{{2\,(M+N)}}.$
(134)

###  H.4 Charge Voltage

The charge voltage in the channel is given by

$q\,\tilde{V}^{c}=\;\dfrac{{M\mu(U+)+N\mu(U-)+N\mu(D+)+M\mu(D-)}}{{2\,(M+N)}}.$
(135)

Subtracting each electrochemical potential by $\mu_{0}$ according to

$q\,V^{c}=\;\dfrac{{M\tilde{\mu}(U+)+N\tilde{\mu}(U-)+N\tilde{\mu}(D+)+M\tilde{\mu}(D-)}}{{2\,(M+N)}}.$
(136)

where $V_{c}=\tilde{V}_{c}-\dfrac{\mu_{0}}{q}$.

Eqs. (128), (132), (134), and (136) together can be written in a matrix form
given by Eq. (61).

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