# Rashba spin-orbit coupling enhanced magnetoresistance in junctions with one
ferromagnet

Chenghao Shen  Department of Physics, University at Buffalo, State University
of New York, Buffalo, NY 14260, USA  Ranran Cai  International Center for
Quantum Materials, School of Physics, Peking University, Beijing 100871, P. R.
China  Alex Matos-Abiague  Department of Physics and Astronomy, Wayne State
University, Detroit, MI 48201, USA  Wei Han  International Center for Quantum
Materials, School of Physics, Peking University, Beijing 100871, P. R. China
Jong E. Han  Department of Physics, University at Buffalo, State University of
New York, Buffalo, NY 14260, USA  Igor Žutić  Department of Physics,
University at Buffalo, State University of New York, Buffalo, NY 14260, USA

###### Abstract

We explain how Rashba spin-orbit coupling (SOC) in a two-dimensional electron
gas (2DEG), or in a conventional $s$-wave superconductor, can lead to a large
magnetoresistance even with one ferromagnet. However, such enhanced
magnetoresistance is not generic and can be nonmonotonic and change its sign
with Rashba SOC. For an in-plane rotation of magnetization, it is typically
negligibly small for a 2DEG and depends on the perfect transmission which
emerges from a spin-parity-time symmetry of the scattering states, while this
symmetry is generally absent from the Hamiltonian of the system. The key
difference from considering the normal-state magnetoresistance is the presence
of the spin-dependent Andreev reflection at superconducting interfaces. In the
fabricated junctions of quasi-2D van der Waals ferromagnets with conventional
$s$-wave superconductors (Fe0.29TaS2/NbN) we find another example of enhanced
magnetoresistance where the presence of Rashba SOC reduces the effective
interfacial strength and is responsible for an equal-spin Andreev reflection.
The observed nonmonotonic trend in the out-of-plane magnetoresistance with the
interfacial barrier is an evidence for the proximity-induced equal-spin-
triplet superconductivity.

##  I I. Introduction

The magnetoresistance (MR) is a key figure of merit in spintronics and its
enhancement is associated with the major advances in magnetically sensing and
storing information Maekawa and Gafvert (1982); Moodera  _et al._ (1995);
Žutić  _et al._ (2004); Parkin  _et al._ (2004); Tsymbal and Žutić (2019).
Typically, a large MR is sought in structures with multiple ferromagnetic
regions, where the resulting spin-valve effect implies that the resistance of
the systems depends of the relative magnetization, M orientation of those
ferromagnets.

However, as first discovered in 1857 by Lord Kelvin Thomson (1857),
anisotropic MR (AMR) shows that with spin-orbit coupling (SOC) there is a
change of the electrical resistivity with the relative direction of the charge
current with respect to M of a single bulk ferromagnet (F), such as Ni or Fe.
Another example of MR with a single F region is the tunneling AMR (TAMR) Gould
_et al._ (2004); Moser  _et al._ (2007); Chantis  _et al._ (2007); Fabian  _et
al._ (2007), also a manifestation of the interplay between SOC and M.
Unfortunately, while a single F region simplify scaled-down devices both AMR
and TAMR are limited by their small magnitudes (typically $<1\,\%$ for in-
plane M rotation) Shen  _et al._ (2020); Fabian  _et al._ (2007).

Figure 1: (a) Lateral geometry, the current flows in the 2DEG. (b) Action of
the time reversal, $\mathcal{T}$, and the space inversion operator,
$\mathcal{P}$, on the incident wave. By applying the $\mathcal{P}\mathcal{T}$
operator, the incident wave on the left is transformed into a transmitted wave
on the right. For a spinless barrier system, the $\mathcal{P}\mathcal{T}$
symmetry gives the perfect transmission. (c) Action of the
$\mathcal{P}\sigma_{z}\mathcal{T}$ operator and the
$\mathcal{P}\sigma_{z}\mathcal{T}$ symmetry generalizes the condition from (b)
for an incident wave with an in-plane spin and a barrier with the in-plane
magnetization ${\mathbf{M}}$, defined by the polar angle $\phi$.

In this work we explore a possibility for a much larger MR with a single F
region and Bychkov-Rashba, also known as Rashba SOC Bychkov and Rashba (1984),
in both normal and superconducting state. The resulting enhanced MR is not
only significant for the potential spintronic applications, but could also
distinguish between the trivial and topological states Shen  _et al._ (2020),
or provide a signature of equal-spin-triplet superconductivity, sought to
realize coexistence of ferromagnetism and superconductivity, dissipationless
spin currents, and Majorana bound states for fault-tolerant topological
quantum computing Cai  _et al._ (2023); Amudsen  _et al._ (2022); Eschrig
(2015); Linder and Robinson (2015); Buzdin (2005); Bergeret  _et al._ (2005);
Kitaev (2001); Aasen  _et al._ (2016); Güngördü and Kovalev (2022); Laubscher
and Klinovaja (2021).

To realize an enhanced normal-state MR, we consider a lateral geometry from
Fig. 1(a). The resulting TAMR is not determined by highly spin-polarized
carriers or a large exchange energy in the F region, but by the high
interfacial transmission from the interplay of the Rashba SOC, the barrier
strength at the F/2DEG interface, and the proximity-induced exchange field,
which can reach tens of meV and extend over tens of nm Žutić  _et al._ (2019);
Akimov  _et al._ (2017); Korenev  _et al._ (2019); Takiguchi  _et al._ (2019);
Betthausen  _et al._ (2012).

For a simple spinless case of a square barrier with thickness $d$ and height
$V_{0}$, a standard expression for transmission, $T$, with energy, $E>V_{0}$,
is Griffiths (2005); Sakurai and Napolitano (2011)

$T=\left(1+V_{0}^{2}\sin^{2}(kd)/[4E(E-V_{0})]\right)^{-1},$ (1)

where $k=\sqrt{2m(E-V_{0})}/\hbar$ is the wave vector. The perfect
transmission, $T=1$, requires (i) $V_{0}=0$ or (ii) $kd=n\pi$, where
$n=1,2,3,...$, gives the resonance condition. This well-known behavior
Griffiths (2005); Sakurai and Napolitano (2011) is usually not connected to
the parity-time symmetry $\mathcal{P}\mathcal{T}$, depicted in Fig. 1(b),
which satisfies both cases (i) and (ii). Such a symmetry, where $\mathcal{P}$
is the parity operator, $\mathcal{T}=\mathcal{K}$ is the time-reversal
operator, and $\mathcal{K}$ is the complex conjugation operator, which ensures
that the incoming and transmitted spinless wave are identical, up to a phase.

However, generalization of the perfect transmission for spin-1/2 carriers,
where $\mathcal{T}=-i{\sigma_{y}}\mathcal{K}$, and $\sigma_{y}$ is the Pauli
matrix, is much less explored with SOC and magnetic barriers. It was recently
found that the perfect transmission emerges when the eigenstates of the F/2DEG
Hamiltonian, $H_{\text{F/2DEG}}$, satisfy $\mathcal{P}\sigma_{z}\mathcal{T}$,
the spin-parity-time symmetry, where [$\mathcal{P}{\sigma_{z}}\mathcal{T}$,
$H_{\text{F/2DEG}}$]$\neq 0$ Shen  _et al._ (2020). Intuitively, the emerging
perfect transmission for the eigenstates of $H_{\text{F/2DEG}}$ which satisfy
$\mathcal{P}\sigma_{z}\mathcal{T}$ can be understood from Fig. 1(c).
$\mathcal{T}$ reverses the spin and motion of the incident wave, while
$\mathcal{P}\sigma_{z}$ inverts both the spin (through $\sigma_{z}$) and
position (through $\mathcal{P}$) of the wave. By applying the
$\mathcal{P}\sigma_{z}\mathcal{T}$ operator the incident wave on the left is
transformed to itself, but as a transmitted wave on the right. Therefore,
scattering states which are eigenfunctions of
$\mathcal{P}\sigma_{z}\mathcal{T}$ experience perfect transmission. The
resulting in-plane TAMR amplitude

${\rm TAMR_{\|}}=\frac{G(\phi=0)-G(\phi=\pi/2)}{G(\phi=\pi/2)},$ (2)

where the $\phi$-dependent conductance $G$ is the inverse of the resistance,
$G=1/R$, with $\phi$ defined in Fig. 1(c), can be enhanced by one or two
orders of magnitude.

To study an enhanced out-of-plane MR in the superconducting state, we focus on
the recent transport experiments in junctions of quasi-2D van der Waals F with
conventional $s$-wave superconductors (S) (Fe0.29TaS2/NbN) Cai  _et al._
(2021). Compared to the normal state MR, the key difference is the presence of
Andreev reflection at the F/S interface with Rashba SOC. In this process an
incoming electron is reflected as a hole and the Cooper pair enters the S
region. With SOC, in addition to the conventional Andreev reflection, where
the incoming and reflected particle have an opposite spin, a spin-flip or
equal-spin Andreev reflection is also possible in which the incoming and
reflected particles have the same spin Žutić and Das Sarma (1999); Högl  _et
al._ (2015). While the first process is responsible for the spin-singlet
proximity-induced superconductivity, the second yields the spin-triplet
counterpart.

Figure 2: Schematic of the geometry including a quasi-2D vdW F/S junction
Fe0.29TaS2/NbN, with an Al2O3 insulating barrier. MR measurements are
performed for an out-of-plane ${\mathbf{M}}$ rotation, defined by angle
$\theta$ with respect to the applied magnetic field, B. The interfacial
resistance is obtained as $V/I$.

The corresponding anisotropic behavior, magnetoanisotropic Andreev reflection
(MAAR) can be viewed as the generalization of TAMR Högl  _et al._ (2015);
Vezin  _et al._ (2020); Lv  _et al._ (2018), which is recovered in the normal
state of the F/S junction for a large bias, $V$, applied magnetic field, $B$,
or a high temperature. From a full MAAR calculation for the F/S junction, such
a TAMR value can be obtained by taking the superconducting gap to vanish.
Similar to Eq. (2), the out-of-plane MAAR amplitude in F/S junctions is

${\rm MAAR_{\bot}}=\frac{G(\theta=0)-G(\theta=\pi/2)}{G(\theta=\pi/2)},$ (3)

where $\theta$ is the angle defined in Fig. 2. As we later show, F/S
measurements for the MAAR can be orders of magnitude larger than TAMR in the
normal state. This enhancement is dominated by the spin-flip Andreev
reflection with Rashba SOC at the F/S interface.

Despite the different geometries for the considered F/2DEG and F/S structures,
which also differ by the in-plane vs out-of-plane rotation of M, we find
important common features and surprising nonmonotonic trends in TAMR in both
systems. Some of these identified trends arise from the role of Rashba SOC
which modifies the effective barrier strength and determines the condition for
perfect transmission or an enhanced contribution of the spin-flip Andreev
reflection.

Following this Introduction, we explore the structures from Fig. 1(a). In Sec.
II we describe the F/2DEG Hamiltonian, consider the influence of SOC on the
scattering states and analyze the conductance which reveals different
resonant-like behavior, peaked at different parameters. The calculation of in-
plane TAMR in Sec. III reveals different trends as a function of the barrier
and SOC strengths as well as the proximity-induced exchange splitting. In Sec.
IV we examine the superconducting structures from Fig. 2 and describe the
corresponding F/S Hamiltonian, followed by the measured and calculated
conductance using a simple F/S Hamiltonian. In Sec. V we describe how the
predicted influence of the Rashba SOC on the effective barrier strength
provides a good description of the measured out-of-plane MR and confirms the
evidence for the equal-spin-triplet superconductivity in the considered
heterostructures. In Sec. VI we provide additional discussion for the
relevance of enhanced MR in the normal and superconducting state and note some
open questions for future work.

##  II II. F/2DEG HAMILTONIAN AND CONDUCTANCE ANALYSIS

To explore the interplay of the proximity-modified 2DEG and SOC, together with
the magnetic anisotropy of the transport properties, we consider a model
Hamiltonian of the system represented in Fig. 1(a) given by

$H_{\text{F/2DEG}}=\frac{p^{2}}{2m^{\ast}}+\frac{\alpha}{\hbar}\left(\bm{\sigma}\times\mathbf{p}\right)\cdot\mathbf{\hat{z}}+[V_{0}-\Delta_{\text{xc}}(\mathbf{m}\cdot\bm{\sigma})]h(x),$
(4)

where $m^{\ast}$ is the effective mass, $\alpha$ is the Rashba SOC strength,
$\mathbf{\hat{z}}$ is the unit vector along the $z$-axis,
$\mathbf{p}=(p_{x},p_{y})$ is the 2D momentum operator, $\bm{\sigma}$ is the
vector of Pauli matrices, $V_{0}$ describes the potential barrier,
$\Delta_{\text{xc}}$ and $\mathbf{m}$ are the magnitude and direction of the
proximity-induced ferromagnetic exchange field. The function
$h(x)=\Theta(d/2+x)\Theta(d/2-x)$ describes a square barrier of thickness $d$.
We focus on electrons, not holes Winkler (2003); Rozhansky  _et al._ (2016);
Liu  _et al._ (2018); Arovas and Lyanda-Geller (1998), with the effective
barrier region and typical band structure shown in Fig. 3. In Fig. 3(a) the
resulting spin-dependent barrier is shown in a weak SOC limit, $\alpha
k_{F}\ll\Delta_{\text{xc}}$, where $k_{F}=\sqrt{2m^{\ast}E_{F}/\hbar}$ is the
Fermi wave vector averaged over the inner and outer contours in the leads,
while $E_{F}$ is the Fermi energy. The blue (yellow) contours in Fig. 3(c)
denote lower (upper) bands. Inside the barrier the upper band is irrelevant
since its bottom is above $E_{F}$. The spin orientations are marked by arrows.

Due to Rashba SOC, the wave functions can be classified by the helicity index,
where $\lambda=1~{}(-1)$ refers to the inner (outer) Fermi contour as depicted
in Fig. 3(c). The scattering states for the finite square barrier model can be
written as
${\psi_{\lambda}}\left({x,y}\right)=(1/\sqrt{2A}){e^{i{k_{y}}y}}{\phi_{\lambda}}\left(x\right)$,
with sample area $A$ and the conserved parallel component of the wave vector
$k_{y}$ in the ballistic transport. Right from the barrier,
${\phi_{\lambda}}\left(x\right)$ is a combination of the two plane waves with
transmission coefficient, $t_{\lambda\lambda}$ and $t_{\lambda\bar{\lambda}}$,
where $\bar{\lambda}=-\lambda$, to describe intraband and interband scattering
processes.

By matching ${\phi_{\lambda}}\left(x\right)$ and
$d{\phi_{\lambda}}\left(x\right)/dx$ in the regions, $x<-d/2$, $-d/2<x<-d/2$,
and $x>d/2$, we obtain $t_{\lambda\lambda}$, $t_{\lambda\bar{\lambda}}$, and
the particle current density of the $\lambda$ channel

${j_{\lambda}}{\text{ =
}}\operatorname{Re}\left[v/A{\left({{{\left|{{t_{\lambda\lambda}}}\right|}^{2}}\cos{\varphi_{\lambda}}+{{\left|{{t_{\lambda\bar{\lambda}}}}\right|}^{2}}\cos{\varphi_{\bar{\lambda}}}}\right)}\right],$
(5)

here, the group velocity of the scattered particle,
$v=\sqrt{({\alpha/\hbar)^{2}+2E/m^{\ast}}}$, has the same magnitude for the
two bands. This current contains contributions from the intra- and inter-
channel transmission, where $\varphi_{\lambda}$ is the incident and
$\varphi_{\bar{\lambda}}$ the propagation angle of the cross-channel wave with
the conservation of the $k_{y}$ component and $\varphi_{\bar{\lambda}}$ is
related to $\varphi_{\lambda}$ by
$\cos{\varphi_{\bar{\lambda}}}={\sqrt{k_{\bar{\lambda}}^{2}-k_{\lambda}^{2}{{\sin}^{2}}\varphi_{\lambda}}}/{k_{\bar{\lambda}}}$,
where

$k_{\lambda}=\sqrt{{{\left({\frac{{\alpha
m^{\ast}}}{{{\hbar^{2}}}}}\right)}^{2}}+\frac{{2m^{\ast}E}}{{{\hbar^{2}}}}}-\frac{{\lambda\alpha
m^{\ast}}}{{{\hbar^{2}}}}.$ (6)

Figure 3: (a) Schematic of the spin-dependent barrier. Its effective strength
is modified by the proximity-induced exchange field $\Delta_{\text{xc}}$,
shown in a weak SOC limit. (b) 2DEG band structure with the Fermi energy
$E_{F}$ and the effective barrier region (middle) of height $V_{0}$ and
$\Delta_{\text{xc}}$. (c) Corresponding Fermi contours, arrows denote the spin
orientation. Dashed lines: the range of a conserved wave vector $k_{y}$ in the
scattering states. For incident angles exceeding $\varphi_{c}$ backscattering
is suppressed.

In the low-bias limit, i.e. $\left|{eV}\right|\ll{E_{F}}$, we get the
expression for the conductance $G_{\lambda}$ in the $\lambda$ channel Girvin
and Yang (2019)

${G_{\lambda}}=\frac{e^{2}}{h}\frac{D}{2\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{d{\varphi_{\lambda}}k_{F}^{\lambda}{T_{\lambda}}\cos{\varphi_{\lambda}}},$
(7)

where $D$ is the sample width, $k_{F}^{\lambda}$ the $\lambda$-channel Fermi
wave vector [$E=E_{F}$ in Eq. (6)] and the transmission is

${T_{\lambda}}=\operatorname{Re}\left[{\left|{t_{\lambda\lambda}}\right|^{2}}+{\left|{t_{\lambda\bar{\lambda}}}\right|^{2}}\left({\cos{\varphi_{\bar{\lambda}}}/\cos{\varphi_{\lambda}}}\right)\right].$
(8)

The total conductance is the sum of the two channels,

${G}=\sum\limits_{\lambda=\pm 1}{G_{\lambda}}.$ (9)

To explore the evolution of the conductance as a function of the proximity-
induced field and its direction, we consider its normalized value in Eq. (9)
expressed in terms of the Sharvin conductance for the 2DEG system

$G_{0}^{\text{2D}}=\frac{e^{2}}{h}\frac{2D}{\pi}\sqrt{{{\left({\frac{\alpha\,m^{\ast}}{\hbar^{2}}}\right)}^{2}}+\frac{2m^{\ast}\,E_{F}}{\hbar^{2}}.}$
(10)

As shown in Fig. 1(c), the $\mathcal{P}\sigma_{z}\mathcal{T}$ symmetry leads
to perfect transmission and influences the conductance for the 2DEG system.
Specifically, scattering states which satisfy
$\mathcal{P}\sigma_{z}\mathcal{T}\psi(x,y)=\xi\psi(x,y)$, with eigenvalues of
the form $\xi=e^{i\eta}$, include the perfect transmission either due to
effectively vanishing barrier or the resonances expected from the standing-
wave condition. However, $\mathcal{P}\sigma_{z}\mathcal{T}$ symmetry is not an
intrinsic symmetry of the system and $\mathcal{P}\sigma_{z}\mathcal{T}$ does
not commute with $H_{\text{F/2DEG}}$ in Eq. (4). Instead, the
$\mathcal{P}\sigma_{z}\mathcal{T}$ symmetry emerges only for certain specific
system parameters and scattering states satisfying,

$[H_{\text{F/2DEG}},\mathcal{P}\sigma_{z}\mathcal{T}]\psi_{R}(x,y)=0,$ (11)

where the index $R$ emphasizes that the relation is restricted only to the
cases of the perfect transmission (for vanishing of the effective barrier or
at resonances). This symmetry generalizes a simple case of the perfect
transmission in a potential barrier (or a spinless) system Griffiths (2005).

Figure 4:  Normalized conductance, $G$, as a function of $\Delta_{\text{xc}}$,
for $E_{F}=5\,$meV, $V_{0}=10\,$meV, $\alpha=0.093\,$eVÅ, and $d=26\,$nm.
Fermi contours at different $\Delta_{\text{xc}}$, ${\bf m}||{\bf x}$: Blue and
red circles are in the lead and the dashed contours in the barrier (only one
band exists at the given $E$). Each of the three peaks in $G$ satisfies the
$\mathcal{P}\sigma_{z}\mathcal{T}$ symmetry for the perfect transmission.

To understand the relation between conductance, barrier parameters, and the
perfect transmission, we perform a Fermi contour analysis illustrated in Fig.
4. In the spinless case, with the better matching between Fermi contours in
the lead and barrier a higher transmission is expected. For a simple 1D
geometry (normal incidence or a fixed $k_{y}$) and $\delta$-barrier, the
transmission is Blonder  _et al._ (1982)

$T{\text{ = 1/}}\left({1+{Z^{2}}}\right),$ (12)

where $Z$ is the effective barrier strength which combines the influence of a
native barrier and the Fermi wave vectors mismatch in the two regions Žutić
and Das Sarma (1999); Vezin  _et al._ (2020); Blonder and Tinkham (1983);
Žutić and Valls (2000). Since the difference between the lead and barrier
Fermi contours is associated with the effective barrier potential, a larger
mismatch between Fermi contours corresponds to a larger effective $Z$ and thus
to a lower transmission.

We now generalize this situation to a spin-dependent case in the presence of
$\Delta_{\text{xc}}$ and $\alpha$, as depicted in Figs. 3(b) and (c). The
effective barrier at the F/2DEG interface becomes spin-dependent since the
energy bands in the barrier are split by the exchange field and spinless $Z$
becomes $Z_{\pm}^{\text{eff}}$ Vezin  _et al._ (2020). For the situation from
Fig. 3, we can decompose the incident spinor in the basis constructed by the
corresponding barrier eigenspinor ($k_{x,y}$ is the same as in the incident
state) for the lower band $\chi_{\uparrow}$ and its antiparallel partner from
the upper band $\chi_{\downarrow}$, i.e.,
$\left|\chi_{\text{in}}\right\rangle=\left\langle\chi_{\uparrow}{\left|\right.}\chi_{\text{in}}\right\rangle\left|\chi_{\uparrow}\right\rangle+\left\langle\chi_{\downarrow}{\left|\right.}\chi_{\text{in}}\right\rangle\left|\chi_{\downarrow}\right\rangle$.
The first term undergoes a weak effective barrier $Z_{-}^{\text{eff}}$ while
the second term experiences a strong barrrier $Z_{+}^{\text{eff}}$.

The effective interfacial barrier is inequivalent for two helicities (for
outer/inner Fermi contours) leading to an important influence of spin mismatch
on transmission, not just the mismatch of the Fermi contours Vezin  _et al._
(2020). Considering that the direction of the proximity-induced exchange
field, ${\bf m}$ [see Eq. (4)] is determined by $\phi$, for an in-plane
${\mathbf{M}}$ as depicted in Fig. 1(c), the effective barrier for
$\left|\chi_{\text{in}}\right\rangle$ from the outer band of the lead is

$Z_{\pm}^{\text{eff}}\propto{V_{0}}\pm\sqrt{{{\left({\alpha{k_{x}}+{\Delta_{\text{xc}}}\sin\phi}\right)}^{2}}+{{\left({\alpha{k_{y}}-{\Delta_{\text{xc}}}\cos\phi}\right)}^{2}}}-\alpha
k.$ (13)

Equivalently, for $\left|\chi_{\text{in}}\right\rangle$ from the inner band of
the lead

$Z_{\pm}^{\text{eff}}\propto{V_{0}}\pm\sqrt{{{\left({\alpha{k_{x}}+{\Delta_{\text{xc}}}\sin\phi}\right)}^{2}}+{{\left({\alpha{k_{y}}-{\Delta_{\text{xc}}}\cos\phi}\right)}^{2}}}+\alpha
k.$ (14)

In Eqs. (13) and (14) the impact of the reflected states is neglected, and
this treatment is most accurate for small effective $Z$, i.e., for a good
matching of Fermi contours. We note that the effective barrier becomes energy-
dependent when $\Delta_{xc}\neq 0$ and $\alpha\neq 0$, opening the possibility
for resonant tunneling to occur when the energy of the incident carrier is
such that $Z_{-}^{\text{eff}}\rightarrow 0$ (or $Z_{+}^{\text{eff}}\rightarrow
0$).

Due to the spin mismatch, we need to include a correction of
${\left|{\left\langle{{\chi_{\uparrow}}|{\chi_{{\text{in}}}}}\right\rangle}\right|^{2}}$
in the transmission, i.e.,
$T\approx{\left|{\left\langle{{\chi_{\uparrow}}|{\chi_{{\text{in}}}}}\right\rangle}\right|^{2}}T_{\chi_{\uparrow}}$,
where $T_{\chi_{\uparrow}}$ is the transmission without the spin mismatch. We
can see from Fig. 1(c) that the spin mismatch is much smaller for the incident
state from the outer lead band $(\lambda=-1)$, i.e.,
$\left|{\left\langle{{\chi_{\uparrow}}}\mathrel{\left|{\vphantom{{\chi_{\uparrow}}{\chi_{\text{in}},\lambda=-1}}}\right.\kern-1.2pt}{{\chi_{\text{in}}},\lambda=-1}\right\rangle}\right|\gg\left|{\left\langle{{{\chi_{\uparrow}}}}\mathrel{\left|{\vphantom{{\chi_{\uparrow}}{{\chi_{\text{in}}},\lambda=1}}}\right.\kern-1.2pt}{{\chi_{\text{in}}},\lambda=1}\right\rangle}\right|$,
which indicates that the most of $G$ is contributed by the incident states
from the outer lead band. When the state inside the barrier is the same (up to
a phase) as that in the lead, perfect transmission is achieved.

With vanishing Rashba SOC, $\alpha\rightarrow 0$, outer and inner lead Fermi
contours coincide and the spin-dependent barrier,
$V_{\pm}=V_{0}\pm\Delta_{\text{xc}}$, from Fig. 3(a) is recovered. We focus on
$V_{0}>0$, $V_{-}=0$ leads to the perfect transmission,
$T_{\chi_{\uparrow}}=1$, from our discussion above. From Eqs. (13) and (14),
$Z_{\pm}^{\text{eff}}\propto V_{0}\pm\Delta_{\text{xc}}$, so one can still get
$Z_{-}^{\text{eff}}\rightarrow 0$ when $V_{0}=\Delta_{\text{xc}}$. We will
further address the importance of vanishing $V_{-}$ in F/S junctions. At a
fixed $k_{y}$, the eigenstate for the perfect transmission $T(k_{y})=1$,
precisely satisfies the $\mathcal{P}\sigma_{z}\mathcal{T}$ symmetry:
$\mathcal{P}\sigma_{z}\mathcal{T}\psi(x,y)=\xi\psi(x,y)$, with eigenvalue
$\xi=e^{i\eta}$ and an arbitrary phase $\eta$.

This analysis and the relevance of the $\mathcal{P}\sigma_{z}\mathcal{T}$
symmetry can be applied to Fig. 4 with parameters of a typical InGaAs/InAlAs
2DEG, $m^{\ast}=0.05m_{0}$, where $m_{0}$ is the free-electron mass, and
$\alpha=0.093\,$eVÅ Nikolić  _et al._ (2005). It is crucial to note that
$\mathcal{P}\sigma_{z}\mathcal{T}$ symmetry arguments for a perfect
transmission pertain to a single $k_{y}$ (single channel), while the geometry
from Figs. 1(a) and 3 corresponds to a 2D system where the total $G$ reflects
the contribution from all allowed $k_{y}$. In the spinless case, maximizing
$G$ means an overall contour matching, while even with adding spin our Fermi
contour analysis provide a valuable tool to visualize and understand various
trends in $G$.

With conserved $k_{y}$, from Fig. 3(c) we can recognize many similarities with
the Snell’s law Žutić and Valls (2000). We may get propagating or decaying
(evanescent) states in different regions, depending on the incident angle,
$\varphi$ and the particular contour (outer, inner) considered. For example,
for carriers from the $\lambda=-1$ band (outer contour) and incident angles
$\varphi$ exceeding the critical angle,

$\varphi_{c}=\pm\arcsin(k_{1}/k_{-1}),$ (15)

where $k_{\lambda}=\pm 1$ is given by Eq. (6), the transmission and reflection
to the $\lambda=1$ band are not allowed because there are no such propagating
states. In this regime, back scattering is suppressed while $T_{\lambda=-1}$
is enhanced.

The overall maximum in $G(\Delta_{\text{xc}})$ is related to the perfect
transmission due to the Fermi contour matching depicted by the example (2) in
Fig. 4. As shown for examples (1) and (3) a small change in
$\Delta_{\text{xc}}$ leads to worse Fermi contour matching and reduced $G$.
The origin of the overall peak in $G$ near $\Delta_{\text{xc}}=V_{0}$ can be
further understood in the small SOC limit. The barrier contour reduces to a
shifted circle with radius
$k_{F}\sqrt{\left({{E_{F}}+\Delta-{V_{0}}}\right)/{E_{F}}}$, where we recall
that $k_{F}$ is average of the outer and inner lead circles. The barrier
contour can then be approximated by

$\begin{gathered}{\left[{k_{x}-(m^{\ast}\alpha/\hbar^{2})\sin\phi}\right]^{2}}+{\left[{k_{y}+(m^{\ast}\alpha/\hbar^{2})\cos\phi}\right]^{2}}\hfill\\\
\hskip 90.0pt\approx(2m^{\ast}/\hbar^{2})\left({E_{F}+\Delta-
V_{0}}\right).\hfill\\\ \end{gathered}$ (16)

In the region near the $\Delta_{\text{xc}}=V_{0}$, both $G(\phi=0)$ and
$G(\phi=\pi/2)$, which correspond to ${\bf m}\parallel{\bf x}$ and ${\bf
m}\parallel{\bf y}$, reach their maxima because of the best matching of Fermi
contours between lead and barrier. The shift of barrier circle is always
$\perp$ to $\mathbf{m}$ and is of the first order in $\alpha$, while the
deformation is a higher-order correction.

Remarkably, the other local maxima in $G$, for larger $\Delta_{\text{xc}}$ in
Fig. 4, also satisfy the $\mathcal{P}\sigma_{z}\mathcal{T}$ symmetry. However,
instead of the contour matching, they are due to standing de Broglie waves in
the barrier and the constructive interference. With outer/inner lead contours,
there are four eigenstates in the leads with $\pm k_{x\lambda},\pm
k_{x\tilde{\lambda}}$, we can distinguish cases: (i) 4 (4), (ii) 4 (2), (iii)
2 (4), and (iv) 2 (2) propagating states in the leads (barrier) Shen  _et al._
(2020).

In cases (i) and (ii) [arising for an incident state from the inner band of
for the outer band and $\varphi<\varphi_{c}$], the perfect transmission means
that no reflected wave should exist in the left lead, while the state in the
right lead should match, up to a phase, the corresponding one in the left
lead. For case (i), under $\mathcal{P}{\sigma_{z}}\mathcal{T}$ symmetry
operation, up to a phase, all the propagating states inside the barrier, with
$x$ component of the wave vector, $\tilde{k_{j}}$, j=1,…,4, remain the same.
By matching ${\phi_{\lambda}}\left(x\right)$ and
$d{\phi_{\lambda}}\left(x\right)/dx$, as noted in Sec. II, the resonance
condition is satisfied when

$\left({{{\widetilde{k}}_{j}}-{{\widetilde{k}}_{1}}}\right)d=2\pi{n_{j}},\quad
j=2,3,4,$ (17)

where $n_{j}=1,2,...$ However, for the system we consider,
${{\widetilde{k}}_{j}}d\sim 10$ and Eq. (17) only holds in few special cases.

In case (ii) the resonance condition similar to Eq. (17) is not possible,
unless for the normal incidence when $\mathbf{m}\parallel\mathbf{y}$. In that
case, the coefficients of the decaying barrier states vanish and the spins of
all scattering states become parallel to each other which makes the system
“spinless” and perfect transmission becomes possible.

In cases (iii) and (iv), the presence of the decaying states in the leads can
be understood for the incident particle from the $\lambda=-1$ band at an angle
$\varphi>\varphi_{c}$. Since such decaying states in the leads do not carry
any current Žutić and Valls (2000), as can be understood from the Snell’s law,
the perfect transmission and the resonance conditions are still possible. In
case (iv), under the symmetry $\mathcal{P}{\sigma_{z}}\mathcal{T}$ operation,
up to a phase, the two propagating states inside the barrier remain the same,
while the two decaying states in the barrier become, up to a phase, the two
decaying states in the leads. The resonance condition is

$\left({\tilde{k}}_{2}-{\tilde{k}}_{1}\right)d=2n\pi-\delta,\quad
n=1,2,3\ldots,$ (18)

where for the $x$ component of the propagating wave vectors in the barrier we
assume ${\tilde{k}}_{2}>{\tilde{k}}_{1}$, while the correction
$\delta\propto(\alpha/\hbar)\sqrt{m^{\ast}/\Delta_{\text{xc}}}\ll 1$. Since
the resonances occur at ${\Delta_{\text{xc}}}>E_{F}$, the magnitude of the
propagating barrier wave vector is much larger than $k_{y}$. Therefore, the
$x$ component of the propagating barrier wave vector is almost the same for
all the incoming states with different incident angles, which means the
transmission resonances occur for all the incoming states almost
simultaneously when the resonance condition is satisfied and thus the maximum
$G$ is reached.

The two local maxima in $G$ near $\Delta_{\text{xc}}=20\,$meV and 55 meV,
shown in Fig. 4, both correspond to the case (iv) and a large
$\Delta_{\text{xc}}$ regime, where for $\mathbf{m}\parallel\mathbf{x}$ the
propagating wave vectors are
${{\tilde{k}}_{1,2}}\approx\mp\sqrt{2m^{\ast}\Delta}/\hbar$. Applying the
resonance condition from Eq. (18) the first (second) of these maxima
corresponds to $n=1$ ($n=2$). However, with large $\Delta_{\text{xc}}$, the
correction $\delta$ in Eq. (18) is very small such that the resonance
condition is accurately described by the spinless case where $\delta=0$.

Figure 5: (a) Dependence of the TAMR amplitude on the Rashba SOC $\alpha$ and
the barrier strength $V_{0}$ for a 2DEG system with a $d=13\,$nm thick
barrier, $E_{F}=10\,$meV, and proximity-induced exchange splitting
$\Delta_{\text{xc}}=15\,$meV. (b) Dependence of the TAMR amplitude on
$\Delta_{\text{xc}}$ and $V_{0}$, for the same $d$ and $E_{F}$ as in (a), with
$\alpha=0.093\,$eVÅ for a typical InGaAs/InAlAs 2DEG. (c) The same as (b), but
$\alpha=0.93\,$eVÅ. The blue and red dashed lines in (a) are for the same
parameters as the ones in (b) and (c).

##  III III. F/2DEG In-Plane TAMR

A large value of TAMR, as defined in Eq. (2), has the origin in the difference
for the maximum $G(\phi=0)$ and $G(\phi=\pi/2)$, that is for the
$\mathbf{m}\parallel\mathbf{x}$ and $\mathbf{m}\parallel\mathbf{y}$
orientations. In Fig. (4), for $G(\phi=0)$ and shown examples (1)-(3), we see
the expected barrier Fermi contour (broken line) shifted downward
($\perp\mathbf{x}$). This broken $k_{y}$ symmetry leads to an unusual
tunneling planar Hall effect, where the transverse voltage is maximum for the
$\mathbf{M}\parallel\mathbf{x}$, while other Hall effects would vanish Scharf
_et al._ (2016). Due to the same asymmetry, the perfect match for both upper
and lower half of the circle cannot be achieved simultaneously. We see in
examples (1)-(3) that the upper (lower) half of the barrier contour tends to
match the inner (outer) lead contour since the spin mismatch is smaller for
these states. The best possible simultaneous match of Fermi contours yields
the large transmission at $\Delta_{\text{xc}}=V_{0}$, resulting in the first
peak in $G$, even for large $\alpha$ when the barrier contour is strongly
deformed from a circle, as in Fig. 3(c).

As we see from Fig. 4 with multiple peaks in $G$, there also exist several
resonances which could influence the TAMR. In the limit of $\alpha\to 0$, the
condition for such resonances can be derived from Eq. (18) with $\delta=0$

$\Delta_{\text{xc}}={V_{0}}-{E_{F}}+\frac{\pi^{2}\hbar^{2}n^{2}}{2m^{\ast}d^{2}},$
(19)

for both $G(\phi=0)$ and $G(\phi=\pi/2)$ which correspond to ${\bf
m}\parallel{\bf x}$ and ${\bf m}\parallel{\bf y}$, where $n=1,2,...$ describes
the order of the resonance that pertains to the case (iv) in Sec. II. In this
situation, of vanishing Rashba SOC, the conductance no longer depends on the
in-plane ${\bf m}$ and we expect a vanishing TAMR, as inferred from Eq. (2).

However, already for a small and nonvanishing $\alpha$ the maximum
$G(\phi=\pi/2)$ is achieved for a different condition when the barrier circle
shares the same size as the outer circle of the lead Shen  _et al._ (2020). Up
to the first order in $\alpha$ the condition leading to the maximum
$G(\phi=\pi/2)$ is given by $\Delta_{\text{xc}}=V_{0}+\alpha k_{F}$. When such
a condition is satisfied, the barrier Fermi contour matches the outer lead
contour instead of the inner one, because the main $G$ contribution is from
the incident particles on the outer Fermi contour. Correspondingly, near
$\Delta_{\text{xc}}={V_{0}}$, due to contour matching and the perfect
transmission, we expect enhanced TAMR, as can be seen from Fig. 5.

Furthermore, when $\alpha\neq 0$, ${{\tilde{k}}_{2}-{\tilde{k}}_{1}}$, which
determine the resonance condition in Eq. (18), varies as ${\mathbf{M}}$ is
rotated. This means the maxima for $G(\phi=0)$ and $G(\phi=\pi/2)$ will be
achieved at different $\Delta_{\text{xc}}$. Therefore, similar to the
situation near $\Delta_{\text{xc}}=V_{0}$, an enhanced TAMR will arise from
the small difference in the peak conditions. With fixed $E_{F}$ and $V_{0}$,
up to the lowest-order correction of $\alpha$, we can derive the difference in
$\Delta_{\text{xc}}$ at the same order maxima ($n$) of $G(\phi=0)$ and
$G(\phi=\pi/2)$

$\Delta^{\text{diff}}_{\text{xc}}\propto\alpha^{2}m^{\ast}/\hbar^{2}.$ (20)

Unlike the perfect transmission condition due to contour matching, the
difference in now quadratic in $\alpha$.

With this conductance analysis and nonmonotonic $G(\Delta_{\text{xc}})$ in
Fig. 4, a consequence of the collective contributions of multiple $T\approx 1$
states corresponding to different propagation directions of the tunneling
carriers, we expect various nonmonotonic TAMR trends and a strong influence of
Rashba SOC. We can see from Fig. 5(a) that increasing $\alpha$ can enhance
both TAMR and the range of $V_{0}$ for such enhancement. However, if $\alpha$
gets too large ($>1.5\,$eVÅ), the absolute value of TAMR decreases, for a
large range of $V_{0}$.

The reason for this unexpected trend in TAMR($\alpha$) can be understood from
calculated results in Figs. (5)(b) and (c), which are shown for two different
values of $\alpha$, as indicated by blue and red vertical lines in Fig.
(5)(a). From the previous discussion of a small $\alpha$ limit, we know that
$G$ peaks near $\Delta_{\text{xc}}=V_{0}$ and such a peak will be shifted by
$\alpha k_{F}$ when ${\bf M}$ is rotated. The sensitive dependence of $G$ on
the ${\bf M}$ orientation leads to a large TAMR near the diagonal line in the
parameter space, shown in Figs. (5)(b) and (c). Since such a shift is
proportional to $\alpha$, by comparing Figs. (5)(b) and (c) there is a wider
range of the enhanced TAMR for a larger $\alpha$.

However, if $\alpha$ is too large (compared to $V_{0}/k_{F}$ and
$\Delta_{\text{xc}}/k_{F}$), the impact of the ${\bf M}$ orientation becomes
negligible, and TAMR starts to decrease. This can be seen at the lower-left
corner in Fig. (5(c), where the resonant TAMR is much smaller when both
$V_{0}/k_{F}$, $\Delta_{\text{xc}}/k_{F}<\alpha$. Even for a fixed $V_{0}$,
for example at 5 meV, we see that with $\alpha$ the absolute value of TAMR is
nonmonotonic and TAMR also changes sign.

The calculated in-plane TAMR reveals various other peculiar trends. For a
fixed $\alpha$ [Fig. (5(a)] or $\Delta_{\text{xc}}/k_{F}$ [Fig. (5(b) and
(c)], we can see that TAMR can be nonmonotonic in $V_{0}$ and even change its
sign. Furthermore, there is as a clear nonmonotonic amplitude with
$\Delta_{\text{xc}}$ at a fixed $V_{0}$ in Fig. (5(b). Different barrier
thickness in Fig. 5 (26 nm) from that in Fig. 4 (13 nm) modify the respective
$\Delta_{\text{xc}}$ values for the perfect transmission. These different TAMR
trends in Fig. 5 were primarily related to the best contour matching, near
$\Delta_{\text{xc}}=V_{0}$ [as the first peak in $G$, seen in Fig. 4)], while
the role of $n=1$ resonance can be seen in the lower-right corner in Fig.
5(c).

From the angular dependence of TAMR, it is possible to obtain valuable
information about the interfacial crystallographic symmetry Žutić  _et al._
(2019). For a 2DEG system, in the limit and $\Delta_{\text{xc}}/V_{0}\ll 1$,
the leading contribution to the in-plane angular dependence of $G$ from the
two incoming channels with helicity $\lambda=\pm 1$, is $\pm\sin\phi$.
However, with their opposite signs, these leading contributions cancel in the
total $G$, which becomes significantly smaller, quadratic in the small
parameter, and has a different angular dependence, resulting in

$\text{TAMR}_{\text{F/2DEG}}\left(\phi\right)\sim(\Delta_{\text{xc}}/V_{0})^{2}\sin^{2}\phi.$
(21)

While this angular dependence also coincides with the results from the surface
states of 3D topological insulators, their junctions with ferromagnets do not
have a resonant TAMR behavior of a 2DEG and thus lead to different trends in
$\Delta_{\text{xc}}$ and $V_{0}$ Shen  _et al._ (2020).

Another striking signature of the underlying resonant behavior of the
calculated TAMR is its magnitude. In commonly considered vertical tunneling
devices, the in-plane TAMR is rather small (typically $<1$ %) even for large
carrier spin polarization Gould  _et al._ (2004) and exchange energies $>$ eV
Moser  _et al._ (2007); Park  _et al._ (2008); Uemura  _et al._ (2009). In
contrast, with much smaller proximity-induced exchange fields,
$\Delta_{\text{xc}}\sim 10\,$meV, we find a much larger TAMR in our lateral
(planar) structures, $>10\,$% even for SOC for a typical InAs-based 2DEG.

As we next focus on the F/S junctions and their superconducting out-of-plane
TAMR analog [recall Eq. (3)], we will see that some of the calculated
nonomonotonic trends, similar to those we discussed for F/2DEG systems, are
experimentally verified, while the analysis of effective barriers allows us to
give simple estimates of the enhanced MR which can even reach 100 % in the
measured samples.

##  IV IV. F/S HAMILTONIAN AND CONDUCTANCE

Motivated by the recent experiments demonstrating a large MR enhancement in
F/S junctions Cai  _et al._ (2021), depicted in Fig. 2, we consider a simple
model for F/S Hamiltonian, $H_{\text{F/S}}$. It shares several similarities
with $H_{\text{F/2DEG}}$ in Eq. (4), used to analyze in-plane TAMR. The key
difference in the superconducting state is the need to simultaneously include
the presence of electrons and holes and the pair potential $\Delta$, which
couples electronlike and holelike components of the underlying wavefunction.

Additionally, while Rashba SOC was inherent for whole 2DEG region, in F/S
junction it is only an interfacial contribution, along with the potential
barrier, between F and S region. Now we study vertical transport and out-of-
plane MR [recall Eq. (3)], which is dominated by the process of Andreev
reflection, introduced in Sec. I and it is the microscopic mechanism for the
superconducting proximity effects Žutić  _et al._ (2019).

Before specifying $H_{\text{F/S}}$ and the resulting equation for
quasiparticle states, it is instructive to note a similarity between the two-
component transport in normal metal (N)/S junctions (for electronlike and
holelike quasiparticles) and F/N junctions (for spin $\uparrow$,
$\downarrow$), which both lead to current conversion, accompanied by
additional boundary resistance Žutić  _et al._ (2004). In the N/S junction
Andreev reflection is responsible for the conversion between the normal and
the supercurrent, characterized by the superconducting coherence length, while
in the F/N case a conversion between spin-polarized and unpolarized current is
characterized by the spin diffusion length. In the N/S charge transport,
$\Delta$, which couples electronlike and holelike states, plays a similar role
of the spin-flip potential which couples $\uparrow$, $\downarrow$ states in
the F/N spin transport.

We consider ballistic F/S junction, depicted in Fig. 2, having a flat
interface at $z=0$ with potential and Rashba SOC scattering. Similar to the
approach in Sec. II of matching the wavefunctions for the scattering states in
different regions, we generalize the Griffin-Demers-Blonder-Tinkham-Klapwijk
formalismGriffin and Demers (1971); Blonder  _et al._ (1982); Kashiwaya and
Tanaka (2000); Wu and Samokhin (2009) to solve Bogoliubov-de Gennes equation
for quasiparticle states
$H_{\text{F/S}}\Psi\left(\bm{r}\right)=E\Psi\left(\bm{r}\right)$, with energy
$E$ Högl  _et al._ (2015), where

$H_{\text{F/S}}=\left({\begin{array}[]{*{20}{c}}{{H_{e}}}&{\Delta\Theta(z){I_{2\times
2}}}\\\ {{\Delta^{*}}\Theta(z){I_{2\times 2}}}&{{H_{h}}}\end{array}}\right),$
(22)

with $\Delta$ the $s$-wave pair potential which, for a homogeneous S region,
is also the value of the superconducting gap, and the single-particle
Hamiltonian for electrons is

$\displaystyle
H_{e}=-\frac{\hbar^{2}}{2}\bm{\nabla}{\frac{1}{m(z)}}\bm{\nabla}-\mu(z)-\Delta_{\text{xc}}(\bm{m}\cdot{\bm{\sigma}})\Theta(-z)+H_{B},$
(23)

while for holes $H_{h}=-\sigma_{y}H_{e}^{*}\sigma_{y}$. In Eq. (22) $m(z)$ is
the effective mass and $\mu(z)$ is the chemical potential. The interfacial
scattering is modeled by delta-like potential barrier, with the effective
barrier height $V_{0}$ and width $d$ and the interfacial Rashba SOC of
strength $\alpha$ has different units than Rashba SOC in F/2DEG Moser  _et
al._ (2007)

$\displaystyle
H_{B}=[V_{0}d+\alpha(k_{y}{\hat{\sigma}}_{x}-k_{x}{\bf\sigma}_{y})]\delta(z).$
(24)

As in Sec. II, $\Delta_{xc}$ is the exchange spin splitting and we denote the
orientation of the magnetization, $\bf{M}$, by $\bf{m}$, but we now also
consider its out-of-plane rotation
$\bf{m}=\left({\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta}\right)$.

Since the in-plane wave vector $\bm{k}_{\parallel}$ is conserved, the
scattering states for incident spin $\sigma$ electron are given by
$\Psi_{\sigma}\left(\bm{r}\right)=e^{i\bm{k}_{\parallel}\cdot\bm{r}_{\parallel}}\psi_{\sigma}\left(z\right)$
in a four-component basis, they include Andreev and specular reflection, each
without and with spin flip, given by the amplitudes $a_{\sigma}$,
$b_{\sigma}$, $\bar{a}_{\sigma}$, and $\bar{b}_{\sigma}$ Žutić and Das Sarma
(1999).

In the F region, the eigenspinors for electrons and holes are
$\chi_{\sigma}^{e}={\left({{\chi_{\sigma}},0}\right)^{T}}$ and
$\chi_{\sigma}^{h}={\left({0,{\chi_{-\sigma}}}\right)^{T}}$ with

${\chi_{\sigma}}=(1/\sqrt{2}){\left({\sigma\sqrt{1+\sigma\cos\theta}{e^{-i\phi}},\sqrt{1-\sigma\cos\theta}}\right)}^{T},$
(25)

where $\sigma=1(-1)$ or $\uparrow$ ($\downarrow$) refer to spin parallel
(antiparallel) to $\bf{M}$ and the $z$ components of the wave vector are

$k^{e\,(h)}_{\sigma}=\sqrt{k_{F}^{2}+\frac{2m_{F}}{\hbar^{2}}\left[(-)E+\sigma\Delta_{\text{xc}}\right]-k_{\parallel}^{2}},$
(26)

with a spin-averaged Fermi wave vector, $k_{F}$ Žutić and Valls (2000). At an
interface, with conserved ${\bm{k}}_{\|}$, the $H_{B}$ barrier eigenspinors in
the helicity basis are given by Fabian  _et al._ (2007); Matos-Abiague and
Fabian (2009),

${\chi_{\pm}}=\frac{1}{{\sqrt{2}}}\left({\begin{array}[]{*{20}{c}}1\\\ {\mp
i{e^{i\gamma}}}\end{array}}\right),$ (27)

where $\gamma$ is the angle between $\bm{k_{\|}}$ and $\hat{\bm{k}}_{x}$.

In the S region, coherence factors, $u$, $v$, satisfy
${u^{2}}=1-{v^{2}}=\left({1+\sqrt{{E^{2}}-{\Delta^{2}}}/E}\right)/2$, while
the $z$-components of the wave vector are

$q^{e\,(h)}=\sqrt{q_{F}^{2}+(-)\frac{2m_{S}}{\hbar^{2}}\sqrt{E^{2}-\Delta^{2}}-k_{\parallel}^{2}},$
(28)

with $q_{F}$ the Fermi wave vector. Similar to Snell’s law Žutić and Valls
(2000), for a large $k_{\parallel}$ these $z$-components can become imaginary
representing evanescent states which carry no net current. The vanishing of
evanescent states at infinity requires
$\operatorname{Im}[{k_{\sigma}^{h}}]<0$, so the sign of the $z$-component of
the wave vectors needs to be chosen correctly.

From the charge current conservation, the zero-temperature differential
conductance at applied bias, $V$, can be expressed as Högl  _et al._ (2015)

$G(V)=\sum\limits_{\sigma}\int\frac{d\bm{k}_{\parallel}}{2\pi
k_{F}^{2}}\left[1+R_{\sigma}^{h}(-eV)-R_{\sigma}^{e}(eV)\right].$ (29)

Here $G(V)$ is normalized by the Sharvin conductance Žutić  _et al._ (2004)

$G_{0}^{\text{3D}}=\frac{e^{2}k_{F}^{2}A}{2\pi h},$ (30)

where $A$ is the interfacial area, while the form is different than the
previous expression in 2D [recall Eq. (10)], where SOC is present in the whole
2D region, not just at an interface. Only the probability amplitudes from the
F region are needed, for Andreev
$R_{\sigma}^{h}=\operatorname{Re}[(k_{-\sigma}^{h}/k_{\sigma}^{e}){\left|a_{\sigma}\right|^{2}}+(k_{\sigma}^{h}/k_{\sigma}^{e}){\left|{\bar{a}_{\sigma}}\right|^{2}}]$
and specular reflection
$R_{\sigma}^{e}=\operatorname{Re}[{\left|b_{\sigma}\right|^{2}+(k_{-\sigma}^{e}/k_{\sigma}^{e})\left|\bar{b}_{\sigma}\right|^{2}}]$.
A finite-temperature correction is straightforward by adding Fermi functions
in Eq. (29), which smear the calculated $G(V)$ at $T=0\,$K. The integration
kernel in Eq. (29), $[1+R_{\sigma}^{h}-R_{\sigma}^{e}]$ can be viewed as the
effective transmission.

Figure 6: (a) Measured and (b) calculated finite-temperature and $B=0$
differential conductance for Fe0.29TaS2/NbN junction, normalized to its value
at $V=5\,$mV, corresponding to the normal-state result. For $T=2\,$K
comparison, in our calculations we choose $T/T_{c}=0.16$, $P=0.42$, and
$\Delta=2.7\,$meV.

We now turn to the measured bias-dependence for differential conductance,
$dI/dV$, for a quasi-2D vdW F/S junction [recall its geometry in Fig. 2],
shown in Fig. 6(a) for several $T$. Some trends are expected, $dI/dV$ peaks
reflect the peaks in the density of states, near $|eV|=\Delta$ Blonder  _et
al._ (1982), moving inward with $T$ as $\Delta(T)$ is also suppressed with
$T$. While the perfect transmission in 1D N/S geometry near $V=0$ implies the
doubling of the normal-state conductance Blonder  _et al._ (1982), since an
incident electron through Andreev reflection contributes to the transfer of an
electron pair across the N/S interface, the F/S transmission should be
diminished, both by the interfacial barrier, parametrized by $Z$ [recall Eq.
(12)] as well by the spin polarization, $P=\Delta_{\text{xc}}/\mu_{F}\approx
0.4-0.5$, expected for Fe0.29TaS2 and measured in a similar Fe0.26TaS2
ferromagnet Yu  _et al._ (2019). Only a fraction of the incident majority spin
electrons can find the opposite spin partner for Andreev reflection Soulen Jr.
_et al._ (1998); Žutić and Valls (2000).

With the NbN critical temperature, $T_{c}=12.5\,$K, we focus on $dI/dV$ at the
lowest $T=2\,$K and theoretically explore the role of interfacial properties
using dimensional parameters for barrier strength and Rashba SOC

$\displaystyle Z$ $\displaystyle=$ $\displaystyle
V_{0}d\sqrt{m_{F}m_{S}}/(\hbar^{2}\sqrt{k_{F}q_{F}}),$ (31)
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle
2\alpha\sqrt{m_{F}m_{S}}/\hbar^{2},$ (32)

since we are interested in identifying trends with these interfacial
parameters, rather than obtaining the best fit for $dI(V)/dV$, we further
simplify this approach by considering the case for $m_{F}=m_{S}=m$ and
$k_{F}=q_{F}$.

Our calculated results are given in Fig. 6(b) and reflect these
simplifications. For example, assuming that, $k_{F}=q_{F}$ and that $m$ is the
free electron mass, the chosen dimensionless barrier strenght $Z=0.8$
corresponds to $k_{F}=1.4\times 10^{10}\,$m-1, consistent with NbN values
Chockalingam  _et al._ (2008), while $V_{0}=0.5\,$eV, and $d=1.7\,$nm is an
average thickness among our studied F/S samples. The value of Rashba SOC,
$\alpha=5.7\,$eVÅ2, which follows from the choice of $\lambda=1.5$, is also
consistent with separate fits of the angular dependence of MR measured near
zero bias in Ref. Cai  _et al._ (2021).

By comparing $\lambda=0$ and $\lambda=1.5$ results at a fixed $Z=0.8$ above
the gap the changes are very small. However, the inclusion of Rashba SOC can
strongly enhance the low-bias conductance. To understand the origin of this
SOC effect, and similar trends from Sec. II, we note that in the normal state
for the barrier region the dispersion is SOC split and shifted by the barrier
potential (assuming $V_{0}>0$). As in Sec. II, a spinor of an incident
electron with $\bm{k}_{\parallel}$ can be decomposed into barrier
eigenspinors,
$\left|\chi_{\sigma}\right\rangle=\left\langle\chi_{+}{\left|\right.}\chi_{\sigma}\right\rangle\left|\chi_{+}\right\rangle+\left\langle\chi_{-}{\left|\right.}\chi_{\sigma}\right\rangle\left|\chi_{-}\right\rangle$,
where $\chi_{\pm}$ from Eq. (27) has helicity $\pm 1$. The two helicities have
inequivalent effective barriers [recall Eqs. (13) and (14)]

$Z_{\pm}=Z\pm\lambda k_{\parallel}/2k_{F}.$ (33)

Since $Z,\lambda k_{\parallel}/(2k_{F})\geq 0$, for positive helicity the
barrier is enhanced, $Z^{+}_{\rm{eff}}\geq Z$. However, for negative helicity,
at $Z=\lambda k_{\parallel}/(2k_{F})$, $Z_{-}$ becomes effectively completely
transparent, such $k_{\|}$ can be viewed as “open channels” and are
responsible for a strongly increased $dI/dV$.

Some peculiar conductance trends can already be understood at $V=0$ and
$T=0\,$K, where the charge transport in F/S junctions is determined by
conventional and spin-flip Andreev reflection with opposite and equal spin
projection of the incident and reflected particle, respectively, corresponding
to the spin-singlet and spin-triplet interfacial superconducting correlations.
For $G(V=0)$ plotted in the plane defined by $Z>0$ and $\lambda>0$, a striking
behavior was found for $|P|<1$ in the triangular region approximately
delimited with lines T1 and T2 Vezin  _et al._ (2020)

$\text{T1:}\>\>\lambda=2Z/\sqrt{1-P},\quad\quad\text{T2:}\>\>\lambda=2Z,$ (34)

where $G$ is dominated by the spin-flip Andreev reflection and proximity-
induced spin-triplet superconductivity.

Before showing next that a region in $G$ delimited by the lines T1 and T2 also
pertains to the enhanced calculated MAAR, we comment on relaxing the
assumption of equal masses and Fermi velocities in the F and S regions. In a
simple 1D N/S case, without SOC, different Fermi velocities,
$F_{v}=(v_{S}/v_{N})\neq 1$, are known to increase the effective barrier
strength, $Z_{\text{eff}}^{2}\rightarrow Z^{2}+(1-F_{v})^{2}/4F_{v}>Z^{2}$
Blonder and Tinkham (1983). While this argument is often used to also ignore a
Fermi velocity mismatch in the F/S case Soulen Jr.  _et al._ (1998), as it is
accounted for by simply enhancing $Z$, some subtleties exist Žutić and Valls
(2000); Žutić and Das Sarma (1999). The mismatch of effective masses,
$F_{m}=m_{S}/m_{F}$, together with the mismatch of the Fermi wave vectors,
$F_{k}=q_{F}/k_{F}$, can then be viewed as determining $F_{v}$ and enhancing
the effective $Z$. In our studies with SOC, $F_{k}\neq 1$ implies that Eq.
(33) should be generalized by replacing $k_{F}$ with $\sqrt{k_{F}q_{F}}$,
including in the expression for open channels and a vanishing $Z_{-}$. Even
with such a wave vector mismatch, the spin-triplet contribution remains
enhanced within a triangular region, as long as $F_{k}>\sqrt{1-P}$ i.e.,
$q_{F}>k_{\downarrow}$. The delimiting lines T1 and T2, for both $F_{k}<1$ and
$F_{k}>1$, have slopes modified by $F_{k}$ as compared to those given by Eq.
(34).

##  V V. F/S Out-Of-Plane MAAR

Just as in the F/2DEG structures, the interplay between exchange field and
Rashba SOC will lead to various nonmonotonic trends Högl  _et al._ (2015);
Vezin  _et al._ (2020); Cai  _et al._ (2021); Mizuno  _et al._ (2009). Such
trends can be inferred from Eqs. (33) and (34), as well as in the MR, which in
F/S junctions corresponds to the superconducting counterpart of TAMR, and its
term, MAAR (recall Sec. I), identified the key role of the Andreev reflection.

Similar to the significance of the condition for perfect transmission and
contour matching in understanding the origin of an enhanced TAMR, for MAAR it
is important to identify the influence of open channels Vezin  _et al._
(2020). From Eq. (33), we see that $Z,\lambda>0$ yields vanishing $Z_{-}$ with
open channels located at the Fermi contour of radius

$k_{\|}^{\text{open}}=(2Z/\lambda)k_{F}.$ (35)

Open channels give the dominant contribution to $G$ and are also expected to
determine the amplitude of MAAR.

Figure 7: Amplitude of out-of-plane magnetoanisotropic Andreev reflection
(MAAR), given by Eq. (3) calculated at zero bias and $T=0\,$K, shown as a
function of interfacial parameters $Z$ and $\lambda$ for $P=0.47$, consistent
with Fe0.29TaS2. Lines T1 and T2 are defined in Eq. (34).

Since evanescent state for large $k_{\|}$ do not contribute to $G$ they should
be excluded in open channels. From the Snell’s law Žutić and Valls (2000), for
the incident majority spin electron this implies that to have propagating
scattering states, $k_{\parallel}\leq k_{\downarrow}$ is required for
conventional Andreev reflections and $k_{\parallel}\leq q_{F}$ for dominant
spin-flip Andreev reflections Žutić and Das Sarma (1999). The maximum of the
total $G$ is achieved when the amount of the open channels $\propto
k_{\parallel}$, is maximized. For $k_{F}=q_{F}$, in the two limiting case
$k_{\|}^{\text{open}}=k_{\downarrow}$ and $k_{\|}^{\text{open}}=k_{F}$ we
recover the conditions for lines T1 and T2 from Eq. (34), such that the
maximum spin-flip Andreev reflection is located near
$k_{\|}^{\text{open}}=k_{F}$ i.e., $\lambda=2Z$.

From the calculated MAAR in Fig. 7 we see that the triangular region delimited
by lines T1, T2 is also identifying the region of an enhanced MAAR and
confirming the influence of open channels. Similar to our contour matching in
TAMR, as noted in Sec. II., for a full picture the role of spin mismatch also
needs to be included. Two characteristic features are easy to see from Fig. 7:
(i) a large MAAR enhancement, which can reach 100 % and (ii) MAAR is
nonomontonic with both $Z$ and $\lambda$, when the other parameter is kept
constant.

To understand (i), we recall that in the triangular region, the dominant
contribution to the total $G$ is from the open channels that is associated
with spin-flip Andreev reflection and spin-triplet pairing
$\left|{\uparrow\uparrow}\right\rangle$. Therefore, the total conductance can
be approximately written as

$\displaystyle G(V=0)$ $\displaystyle\approx$
$\displaystyle\int{{k_{\parallel}}d{k_{\parallel}}\int_{0}^{2\pi}{\frac{{d\gamma}}{{\pi
k_{F}^{2}}}R_{\sigma=1}^{h}\left({{k_{\parallel}},\gamma}\right)}}$
$\displaystyle\approx$ $\displaystyle
k_{\parallel}^{\text{open}}\Delta{k_{\parallel}}\int_{0}^{2\pi}{\frac{{d\gamma}}{{\pi
k_{F}^{2}}}R_{\sigma=1}^{h}\left({k_{\parallel}^{\text{open}},\gamma}\right)}$

where $\Delta{k_{\parallel}}$ is the width of the open channels in
$\bm{k_{\parallel}}$-plane, $\gamma$ is the angle between $\bm{k_{\|}}$ and
$\hat{\bm{k}}_{x}$, recall Eq. (27). For out-of-plane M,
$R_{\sigma=1}^{h}\left({k_{\parallel}^{\text{open}},\gamma}\right)\equiv
R_{\sigma=1}^{h,\text{open}}$ does not depend on $\gamma$ due to the
rotational symmetry and we can write

$G\left({\theta=0}\right)\approx\frac{2k_{\parallel}^{\text{open}}\Delta{k_{\parallel}}}{k_{F}^{2}}R_{\sigma=1}^{h,\text{open}}.$
(37)

Figure 8: (a) and (b) Atomic force microscope images of 2 nm Al2O3 on
Fe0.29TaS2 thin flake with RMS roughness of $\sim 0.270\,$nm indicate that
Al2O3 is relatively flat and without obvious pin holes or discontinuities.
(c), (e), and (g) schematic of the incident spin-polarized electrons tunneling
across different effective interfacial barriers with Rashba SOC into the
proximity-induced spin-triplet superconductivity. (d), (f), and (h) the
corresponding magnetoresistance (MR) [recall $\theta$ in Fig. 2], for
different junction resistance-area products $R_{J}S$, a proxy for
dimensionless barrier strength $Z$. Red lines: theoretically predicted two-
fold symmetry. The measurements were at $T=2\,$K, $B=9\,$T, and $V=1\,$mV.

For in-plane M, considering the interplay between the incident electron spin
and the barrier eigenspinor, the spin-flip Andreev reflection of an incident
$\uparrow$ electron is given by
$R_{\sigma=1}^{h}\left({k_{\parallel}^{\text{open}},\gamma}\right)\approx
R_{\sigma=1}^{h,\text{open}}{\cos^{2}}\left({\gamma-\phi}\right)$ Vezin  _et
al._ (2020). As a result, for in-plane M, we obtain

$G\left(\theta=\frac{\pi}{2}\right)\approx\frac{k_{\parallel}^{\text{open}}\Delta{k_{\parallel}}}{k_{F}^{2}}R_{\sigma=1}^{h,\text{open}}=\frac{G\left({\theta=0}\right)}{2}.$
(38)

From Eq. (3), this means that MAAR in the triangular region is 100%, which
explains that the previous prediction Högl  _et al._ (2015) can be attributed
to spin-triplet superconductivity and is also consistent with the largest
values in the experimental results for Fe0.29TaS2/NbN junctions with a
perpendicular magnetic anisotropy Cai  _et al._ (2021).

Despite our simple theoretical model to include the interplay of exchange
field and SOC on the proximity-induced superconductivity in F/S junctions,
such a description provides MAAR which is in a good agreement with the
measured angular dependence of MR in Fig. 8. Remarkably, just as the variation
of $Z$ shows a nonmonotonic MAAR in Fig. 7 and a huge increase of MAAR at
intermediate $Z$ values, which with Rashba SOC can lead to a vanishing
$Z_{-}$, we see that this trend in Figs. 8(d), (e), and (h) is also retained
in the measured results obtained by varying the Al2O3 barrier thickness ($\sim
1-2.5\,$nm).

While we cannot rule out a non-MAAR contribution in the measured low-bias MR
for Fe0.29TaS2/NbN junctions, we can still examine its magnitude from several
reported control measurements Cai  _et al._ (2021). For example, by increasing
$T$ above $T_{c}$, MR is drastically reduced from its maximum at $\sim 100\,$%
in Fig. 8(f) and we can attribute the main MR part to a superconducting
response. By considering MR in Al/Al2O3/NbN samples, we no longer have the
interplay between the ferromagnetism and SOC, responsible for MAAR, but the
resulting small MR of just few $\%$ at $T=2\,$K is encouraging that the
presence of vortices at $B=9\,$T has only a modest MR effect. Taken together,
these findings confirm that MAAR is indeed the dominant source of the measured
MR in Fig. 8, which further indicates that F0.29TaS2/NbN junctions, provide an
important platform to realize elusive equal-spin-triplet superconductivity.

##  VI VI. Discussion and Conclusions

We have shown that Rashba SOC can lead to strongly enhanced magnetoresistance
in junctions with one ferromagnet. Both in the normal-state and the
superconducting response the calculated magnetoresistance is characterized by
various nonmonotonic trends. While some of these trends and an enhanced
magnetoresistance have been measured in superconducting junctions Martinez
_et al._ (2020); Cai  _et al._ (2021), experiments in the normal state are
largely unexplored.

To realize magnetic proximity effects for the in-plane transport, magnetic
insulators are desirable Žutić  _et al._ (2019); Jiang  _et al._ (2015); Wei
_et al._ (2013); Lee  _et al._ (2014). This precludes current flow in the more
resistive F region [Fig. 1(a)] and minimizes hybridization with the 2DEG to
enable a gate-tunable proximity-induced exchange splitting in their respective
states. However, as shown in graphene Lazić  _et al._ (2016); Zollner  _et
al._ (2016); Xu  _et al._ (2018) for tunable magnetic proximity effects one
could also employ ferromagnetic metals, separated by an insulating layer from
the 2DEG .

For a suitable materials platform, which would support large magnetoresistive
effects, we could extend our focus on simple Rashba SOC to a growing class of
van der Waals (vdW) materials. Their heterostructure offer both transport in
materials with strong SOC as well 2D ferromagnets Gong  _et al._ (2017); Huang
_et al._ (2017); Gong and Zhang (2019) with atomically-sharp interfaces and
not limited to lattice-matching constraints. For example, transition-metal
dichalcogenides in addition to their inherent SOC also provide spin-orbit
proximity Avsar  _et al._ (2014); Yan  _et al._ (2016); Han  _et al._ (2014);
Dankert and Dash (2017); Wang  _et al._ (2015); Antonio Benitez  _et al._
(2019); Douli  _et al._ (2020) and thereby alter spin textures and expected
TAMR, while 2D ferromagnets support a versatile gate control Burch  _et al._
(2018); Deng  _et al._ (2018); Xing  _et al._ (2017).

While we have focused on a longitudinal transport in a very simple system, the
behavior emerging from a spin parity-time symmetry of the scattering states
with perfect transmission is important not just in explaining a surprisingly
large TAMR, but also as a sensitive probe to distinguish between trivial and
topological states Shen  _et al._ (2020). It would be interesting to know if
our predicted nonmonotonic trends with interfacial parameters for TAMR could
be also relevant for other transport phenomena in junctions with a single
ferromagnet, such as spin-orbit torque and spin-Hall magnetoresistance Tsymbal
and Žutić (2019); Belashchenko  _et al._ (2020).

In the superconducting state, the observed large magnetoresistance has
important implications as it provides an alternative probe for spin-triplet or
mixed singlet-triplet superconductivity Gor’kov and Rashba (2001) and such a
large signal is possible to realize even systems with only a modest SOC and
negligible normal-state magnetoresistance Martinez  _et al._ (2020). Since
this work shows that spin-triplet superconductivity, desirable both for
superconducting spintronics and Majorana bound states Amudsen  _et al._
(2022), is feasible in simple structures with a single F and S region, it
would also be important to extend its analysis to Josephson junctions, where
enhanced magnetoresistance was predicted Costa  _et al._ (2017), but not
connected to the spin-triplet superconductivity, which was extensively studied
in other Josephson junctions Khaire  _et al._ (2010); Robinson  _et al._
(2010); Banerjee  _et al._ (2014); Valls (2022); Cai  _et al._ (2023); Amudsen
_et al._ (2022).

In our work, the nonmonotonic trends with the interfacial barrier strength
were observed by comparing samples with different thickness of the (Al2O3). It
would be desirable, to realize systems in which such changes, as well as the
tunability of the Rashba SOC strength could be controlled in the same sample.
A desirable progress is realized in Josephson junctions where currently there
are no ferromagnetic regions, but the Zeeman splitting is due to an applied
magnetic field Amudsen  _et al._ (2022); Fornieri  _et al._ (2019); Ren  _et
al._ (2019); Dartiailh  _et al._ (2021); Zhou  _et al._ (2022); Banerjee  _et
al._ (2022). Related experimental support in junctions with Al as a
superconductor and InAs-based 2DEG already suggests an observation of
topological spin-triplet superconductivity Dartiailh  _et al._ (2021). In the
same sample that both a reentrant superconductivity with an applied magnetic
field and the jump in the superconducting phase are measured, but an enhanced
magnetoresistance as another signature of spin-triplet superconductivity has
not yet been studied.

##  VII Acknowledgments

We thank Thomas Vezin for useful discussions. We had a privilege of first
meeting Professor Emmanuel Rashba at the Spintronics 2001: International
Conference on Novel Aspects of Spin-Polarized Transport and Spin Dynamics in
Washington, D.C. (chaired by one of us Žutić (2002)). He gave an insightful
talk sharing his personal experience about relevant historic developments as
well as a vision for the future opportunities in spintronics, conveyed further
in his article Rashba (2002). Since then, through his valuable comments, we
were able to better understand our own work and its broader context. His
advice and guidance was not limited to semiconductor systems, and we again
benefited from his comments which came soon after posting online our recent
manuscript on spin-triplet superconductivity Cai  _et al._ (2021). This work
is supported by U.S. Department of Energy, Office of Science, Basic Energy
Sciences under Award No. DE-SC0004890, the UB Center for Computational
Research, National Basic Research Programs of China (No. 2019YFA0308401), and
National Natural Science Foundation of China (No. 11974025).

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