# Magnetotransport Signatures of the Radial Rashba Spin-Orbit Coupling in
Proximitized Graphene

Wun-Hao Kang (康文豪)  Department of Physics, National Cheng Kung University,
Tainan 70101, Taiwan  Center of Quantum Frontiers of Research and Technology
(QFort), National Cheng Kung University, Tainan 70101, Taiwan  Michael Barth
Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg,
Germany  Aitor Garcia-Ruiz  Department of Physics, National Cheng Kung
University, Tainan 70101, Taiwan  National Graphene Institute, University of
Manchester, M13 9PL Manchester, U.K.  Alina Mreńca-Kolasińska  AGH University,
Faculty of Physics and Applied Computer Science, 30-059 Kraków, Poland  Ming-
Hao Liu (劉明豪<EMAIL_ADDRESS>Department of Physics, National
Cheng Kung University, Tainan 70101, Taiwan  Center of Quantum Frontiers of
Research and Technology (QFort), National Cheng Kung University, Tainan 70101,
Taiwan  Denis Kochan<EMAIL_ADDRESS>Institute of Physics, Slovak
Academy of Sciences, 84511 Bratislava, Slovakia  Center of Quantum Frontiers
of Research and Technology (QFort), National Cheng Kung University, Tainan
70101, Taiwan

###### Abstract

Graphene-based van der Waals heterostructures take advantage of tailoring
spin-orbit coupling (SOC) in the graphene layer by proximity effect. At long-
wavelength—saddled by the electronic states near the Dirac points—the
proximitized features can be effectively modeled by the Hamiltonian involving
novel SOC terms and allow for an admixture of the tangential and radial spin
textures—by the so-called _Rashba angle_ $\theta_{\text{R}}$. Taking such
effective models we perform realistic large-scale magneto-transport
calculations—transverse magnetic focusing and Dyakonov-Perel spin
relaxation—and show that there are unique qualitative and quantitative
features allowing for an unbiased experimental disentanglement of the
conventional Rashba SOC from its novel radial counterpart, called here the
_radial Rashba SOC_. Along with that, we propose a scheme for a direct
estimation of the Rashba angle by exploring the magneto-response symmetries
when swapping an in-plane magnetic field. To complete the story, we analyze
the magneto-transport signatures in the presence of an emergent _Dresselhaus
SOC_ and also provide some generic ramifications about possible scenarios of
the _radial superconducting diode effect_.

Graphene-based van der Waals heterostructures provide an ideal platform for
exploring various physical phenomena Han  _et al._ (2014); Roche  _et al._
(2015); Žutić  _et al._ (2019); Avsar  _et al._ (2020); Sierra  _et al._
(2021) that are intertwining single-particle and many-body properties,
including magnetic, ferro-electric, topological and even superconducting
states of matter Cao  _et al._ (2018); Kennes  _et al._ (2021); Lin  _et al._
(2022); Szałowski  _et al._ (2023); Tschirhart  _et al._ (2023); Garcia-Ruiz
_et al._ (2023); Tao  _et al._ (2024). A key ingredient for mediating many of
these features is spin-orbit coupling (SOC). Although the latter is small in
pristine graphene Gmitra  _et al._ (2009); Konschuh  _et al._ (2012), it can
be substantially enhanced by proximity effects of either adatoms or other
layers Weeks  _et al._ (2011); Gmitra  _et al._ (2013); Kaloni  _et al._
(2014); Wang  _et al._ (2015); Irmer  _et al._ (2015); Calleja  _et al._
(2015); Gmitra  _et al._ (2016); Alsharari  _et al._ (2016); Wang  _et al._
(2016); Zollner  _et al._ (2016); Gmitra  _et al._ (2018); Frank  _et al._
(2017); Kochan  _et al._ (2017); Völkl  _et al._ (2017); Zollner  _et al._
(2018); Zihlmann  _et al._ (2018); Völkl  _et al._ (2019); Ghiasi  _et al._
(2019); Safeer  _et al._ (2020); Benítez  _et al._ (2020); Ingla-Aynés  _et
al._ (2021); Tiwari  _et al._ (2022); Ontoso  _et al._ (2023). The strong SOC
can be achieved by proximitizing graphene with transition metal
dichalcogenides, a further interplay of twisting, stacking and of long
supercell-periodicities allows one to engineer and tune the strengths of
valley-Zeeman (VZ) and Rashba SOC Gmitra  _et al._ (2016, 2018); Cummings  _et
al._ (2017); Offidani  _et al._ (2017); Li and Koshino (2019); David  _et al._
(2019); Naimer  _et al._ (2021); Veneri  _et al._ (2022); Lee  _et al._
(2022); Péterfalvi  _et al._ (2022).

The new kid on the block is a possibility to engender the Rashba SOC with the
radial component. As shown in Frank  _et al._ (2024), there are very specific
twist angles allowing for pure radial Rashba (RR) SOC. However, in general,
the resulting spin texture admixes both Szałowski  _et al._ (2023); Zollner
_et al._ (2023); Yang  _et al._ (2023), the tangential and radial components,
quantified by a Rashba angle $\theta_{\text{R}}$, and as such it gives rise to
a tilted spin-momentum locking triggering, for example, unconventional spin-
to-charge conversion Lee  _et al._ (2022); Veneri  _et al._ (2022); Zollner
_et al._ (2023). These novel spin textures are going beyond the ordinary SOC-
classification based on a reduction of just the graphene-point group
symmetries Kochan  _et al._ (2017). Indeed, the long-wavelength electronic
states with momenta near the valley centers experience much more complex
interaction landscape and the inherent supercell symmetries differ from an
ordinary graphene allowing for more complex SOC Hamiltonians Li and Koshino
(2019); David  _et al._ (2019).

Figure 1: Fermi contours around the $K$ point at energy $E=-100$ meV in the
absence (upper panels) and presence (lower panels) of an in-plane magnetic
field $\mathbf{B}_{\parallel}=(B_{x},0,0)$ with $B_{x}=3$ T. The directions
how Fermi contours shift in the $k$-space corroborate with an increase or
decrease of the spin-Zeeman energy, $\pm g\mu_{\text{B}}\langle s_{x}\rangle
B_{x}$, as inherently imprinted by the underlying spin-textures: (a) CR, (b)
RR and (c) Dresselhaus SOC, considering
$\lambda_{\text{CR}}=\lambda_{\text{RR}}=\lambda_{\text{D}}=6\,$meV. The
arrows on the contours display in-plane spin expectation values. The black
dashed circles in the lower panels serve as guides to the eyes to contrast the
original and shifted Fermi circles. Figure 2: Calculated energy band
structures and spin textures near the Dirac point of the proximitized graphene
described by $\mathcal{H}_{0}+\mathcal{H}_{\text{SOC}}$, considering (a)
$\mathcal{H}_{\text{SOC}}=\mathcal{H}_{\text{VZ}}+\mathcal{H}_{\text{CR}}$,
(b)
$\mathcal{H}_{\text{SOC}}=\mathcal{H}_{\text{VZ}}+\mathcal{H}_{\text{RR}}$,
and (c)
$\mathcal{H}_{\text{SOC}}=\mathcal{H}_{\text{VZ}}+\mathcal{H}_{\text{D}}$, in
all three cases with $\lambda_{\text{VZ}}=3.5\,\text{meV}$ for
$\mathcal{H}_{\text{VZ}}$, and separately with
$\lambda_{\text{CR}}=\lambda_{\text{RR}}=\lambda_{\text{D}}=6\,\text{meV}$.
The side panels at the right of (a)–(c) show spin-textures—in-plane black
arrows and red-blue color shading are displaying, respectively, $\langle
s_{x},s_{y}\rangle$ and $\langle s_{z}\rangle$, with the latter calibrated by
the common color bar shown at the right of (c). Panels (d) and (e) display
modeled devices: (d) three terminal transverse magnetic focusing with
schematics of spin-split skipping orbits, as well as, with the corresponding
real-space spin-textures in the case of RR SOC—rotated by $90^{\circ}$
compared to $k$-space textures shown at (b), (e) Dyakonov-Perel spin
relaxation setup with schematics of Gaussian-type disorder.

The main goal of this _Letter_ is to identify robust transport features
allowing for experimentally unique discrimination between the conventional
Rashba (CR) and RR SOC, and, moreover, also for the determination of the
Rashba angle $\theta_{\text{R}}$. We show that SOC-imprinted spin
textures—tangential and radial—typical for the CR and RR SOC—respond
differently when exposed to an in-plane magnetic field
$\mathbf{B}_{\parallel}=(B_{x},B_{y},0)$ rendering an additional spin-Zeeman
interaction. Correspondingly, the initial spin-split Fermi contours shift—for
CR SOC in perpendicular, and for RR SOC in parallel direction with respect to
$\mathbf{B}_{\parallel}$; see Fig. 1 exemplified with
$\mathbf{B}_{\parallel}=(B_{x},0,0)$, to be further explained later.

Therefore, an in-plane magnetic field $\mathbf{B}_{\parallel}$ can resolve the
contribution to a transport quantity $Q$ (probed along a direction given by
unit vector $\hat{\mathbf{n}}$) coming from these two different spin-orbit
textures. In particular, $Q$ shows perfect swap-symmetry,
$Q(\hat{\mathbf{n}},\mathbf{B}_{\parallel})=Q(\hat{\mathbf{n}},-\mathbf{B}_{\parallel})$
in two complementary cases: CR SOC when
$\hat{\mathbf{n}}\parallel\mathbf{B}_{\parallel}$ and RR SOC when
$\hat{\mathbf{n}}\perp\mathbf{B}_{\parallel}$, which offer their unbiased
experimental discrimination. Moreover, in the case when both CR and RR SOC
admix via $\theta_{\text{R}}$ by changing an angle between $\hat{\mathbf{n}}$
and $\mathbf{B}_{\parallel}$ one finds a sweet-spot angle
$\varphi=\sphericalangle(\hat{\mathbf{n}},\mathbf{B}_{\parallel})$ at which
$Q(\hat{\mathbf{n}},\mathbf{B}_{\parallel})=Q(\hat{\mathbf{n}},-\mathbf{B}_{\parallel})$.
This sweet-spot angle is related to the Rashba angle
$\theta_{\text{R}}=-\varphi$, up to modulus of adding $\pi$. The above
symmetries persist even in the case the SOC Hamiltonian admixes a non-zero VZ
SOC that offers unique experimental signatures for unveiling CR, RR and their
admix angle $\theta_{\text{R}}$ while swapping the orientation and magnitude
of the in-plane field $\mathbf{B}_{\parallel}$.

Although our discussion and conclusions are general, for providing
quantitative results we particularly focus on large-scale simulations Liu  _et
al._ (2015); Rickhaus  _et al._ (2015); Mreńca-Kolasińska  _et al._ (2023);
Rao  _et al._ (2023) of _transverse magnetic focusing_ (TMF) and the
_Dyakonov-Perel spin relaxation_ (DPSR) D’yakonov and Perel’ (1971, 1972);
Meier and Zakharchenya (1984); Fabian  _et al._ (2007); Boross  _et al._
(2013). For the former, $Q$ stands for the TMF-conductance $G$ (or more
precisely its second peak, $G_{\text{2nd}}$), and a uniform out-of-plane
magnetic field $\mathbf{B}_{\perp}=(0,0,B_{z})$ is required. For the latter,
no $\mathbf{B}_{\perp}$ is required, and $Q$ stands for the DPSR-rate,
$\tau_{s}^{-1}$. For completeness, we also consider, in addition to CR and RR,
the Dresselhaus SOC Dresselhaus (1955), and reveal magnetotransport signatures
of them. Even though there are not yet reported experimental, neither DFT
evidences of such proximity-engineered term in graphene, the existing rush in
the field of van der Waals materials can not preclude its future exposure.

## Model Hamiltonian.

The effective tight-binding description of these spin features can be modeled
by
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{\text{Z}}+\mathcal{H}_{\text{SOC}}$.
Here, $\mathcal{H}_{0}$ represents the graphene Hamiltonian composed of
conventional hopping $-t$ (taken as $-3$ eV in the following) between all
pairs of nearest-neighbor sites and
$\mathcal{H}_{\text{Z}}=g\mu_{\text{B}}\mathbf{s}\cdot\mathbf{B}$ stands for
the spin-Zeeman coupling, where $\mu_{\text{B}}$ is the Bohr magneton,
$\mathbf{s}=(s_{x},s_{y},s_{z})$ is a vector of the $2\times 2$ Pauli
matrices,
$\mathbf{B}=\mathbf{B}_{\parallel}+\mathbf{B}_{\perp}=(B_{x},B_{y},B_{z})$ is
the net external magnetic field, and $g$ is an isotropic g-factor, we use
$g=42.5$ as the elevated values of $g$ are typical for the confined graphene-
based heterostructures with proximity engineered band splittings Frank and
Fabian (2020); Ge  _et al._ (2023); Xiang  _et al._ (2023). The SOC
Hamiltonian, $\mathcal{H}_{\text{SOC}}$, comprises VZ, CR, RR and Dresselhaus
SOC terms, which can be respectively written as

$\displaystyle\mathcal{H}_{\text{VZ}}$
$\displaystyle=\frac{i}{3\sqrt{3}}\sum_{\langle\\!\langle
m,n\rangle\\!\rangle}\sum_{\sigma}\lambda_{o(m)}c^{\dagger}_{n\sigma}\,\nu_{n\leftarrow
m}\left[s_{z}\right]_{\sigma\sigma}\,c^{\phantom{\dagger}}_{m\sigma}\ ,$ (1a)
$\displaystyle\mathcal{H}_{\text{CR}}$
$\displaystyle=\frac{2i}{3}\lambda_{\text{CR}}\sum_{\langle
m,n\rangle}\sum_{\sigma,\sigma^{\prime}}c^{\dagger}_{n\sigma}\,\left[\left(\mathbf{s}\times\hat{\mathbf{e}}_{n\leftarrow
m}\right)\cdot\hat{\mathbf{e}}_{z}\right]_{\sigma\sigma^{\prime}}\,c_{m\sigma^{\prime}}\
,$ (1b) $\displaystyle\mathcal{H}_{\text{RR}}$
$\displaystyle=-\frac{2i}{3}\lambda_{\text{RR}}\sum_{\langle
m,n\rangle}\sum_{\sigma,\sigma^{\prime}}c^{\dagger}_{n\sigma}\,\left[\mathbf{s}\cdot\hat{\mathbf{e}}_{n\leftarrow
m}\right]_{\sigma\sigma^{\prime}}\,c_{m\sigma^{\prime}}\ ,$ (1c)
$\displaystyle\mathcal{H}_{\text{D}}$
$\displaystyle=\frac{2i}{3}\lambda_{\text{D}}\sum_{\langle
m,n\rangle}\sum_{\sigma,\sigma^{\prime}}c^{\dagger}_{n\sigma}\,\left[\mathbf{s}^{*}\cdot\hat{\mathbf{e}}_{n\leftarrow
m}\right]_{\sigma\sigma^{\prime}}\,c_{m\sigma^{\prime}}\ ,$ (1d)

where $\sum_{\langle m,n\rangle}$ ($\sum_{\langle\\!\langle
m,n\rangle\\!\rangle}$) denotes a summation over all pairs of lattice sites
$m,n$ that are nearest (second-nearest) to each other, $c_{m\sigma}$
($c^{\dagger}_{m\sigma}$) annihilates (creates) an electron with spin $\sigma$
on lattice site $m$, $\hat{\mathbf{e}}_{n\leftarrow m}$ appearing in (1b)–(1d)
is the unit vector pointing from site $m$ to site $n$, and
$\left[\cdots\right]_{\sigma\sigma^{\prime}}$ is the matrix element of
$\left[\cdots\right]$ of the $\sigma$th row and $\sigma^{\prime}$th column.

Figure 3:  Simulated conductance, as a function of $B_{z}$ for a hole-doped
graphene at energy $E=-100$ meV, between the injector and detector (spaced by
$1\,\mathrm{\mu m}$) of a three-terminal TMF device schematically shown in
Fig. 2(d), considering the CR SOC [(a) and (b)], RR SOC [(c) and (d)], and
Dresselhaus SOC [(e) and (f)], all subtended by a varying in-plane magnetic
field of $\mathbf{B}_{x}$ for the upper panels and $\mathbf{B}_{y}$ for the
lower panels. Color codes for different strengths of the in-plane magnetic
field, $B_{\parallel}$, are shown at the right. The insets show the dependence
of the second TMF conductance peaks, $G_{\text{2nd}}$, i.e., $G$ at
$B_{z}\approx-0.4$ T, on the strength of the in-plane magnetic fields ($B_{x}$
for the upper panels and $B_{y}$ for the lower panels).

In the VZ term, Eq. (1a), $o(m)=A,B$ is the sublattice index of site $m$, and
the sign factor $\nu_{n\leftarrow m}$ is $-1$ ($+1$) whenever the second-
nearest hopping from site $m$ to site $n$ via their common nearest-neighbor
forms a clockwise (counterclockwise) path. Note that the VZ Hamiltonian, Eq.
(1a), involves sublattice-resolved couplings, $\lambda_{A}$ and
$\lambda_{B}=-\lambda_{A}$, such that
$\lambda_{\text{VZ}}\equiv(\lambda_{B}-\lambda_{A})/2$. The coupling constant
$\lambda_{\text{CR}}$ in (1b), where $\hat{\mathbf{e}}_{z}$ is the unit vector
along the $z$ axis (and similarly $\hat{\mathbf{e}}_{x},\hat{\mathbf{e}}_{y}$
to appear later), and $\lambda_{\text{RR}}$ in (1c) are parameterized by
$\lambda_{\text{CR}}=\lambda_{\text{R}}\cos\theta_{\text{R}}$ and
$\lambda_{\text{RR}}=\lambda_{\text{R}}\sin\theta_{\text{R}}$ via the Rashba
coupling $\lambda_{\text{R}}>0$ and the Rashba angle $\theta_{\text{R}}$. In
Eq. (1d), $\lambda_{\text{D}}$ is the Dresselhaus SOC coupling strength, and
$\mathbf{s}^{*}=(s_{x},s_{y},s_{z})^{*}=(s_{x},-s_{y},s_{z})$. Throughout the
main text, we use
$\lambda_{\text{CR}}=\lambda_{\text{RR}}=\lambda_{\text{D}}=6$ meV, while
$\lambda_{\text{VZ}}$ is set to be either $0$ or $3.5$ meV. The low-energy
band structure (centered at the $K$ point of the hexagonal Brillouin zone) of
$\mathcal{H}_{0}+\mathcal{H}_{\text{SOC}}$ considering individually
$\mathcal{H}_{\text{SOC}}=\mathcal{H}_{\text{VZ}}+\mathcal{H}_{\text{CR}}$,
$\mathcal{H}_{\text{SOC}}=\mathcal{H}_{\text{VZ}}+\mathcal{H}_{\text{RR}}$,
and $\mathcal{H}_{\text{SOC}}=\mathcal{H}_{\text{VZ}}+\mathcal{H}_{\text{D}}$
is shown in Fig. 2(a), 2(b), and 2(c), respectively, along with the
corresponding spin expectation values, $\langle\mathbf{s}\rangle=(\langle
s_{x}\rangle,\langle s_{y}\rangle,\langle s_{z}\rangle)$, of the four bands,
showing tangential, radial, and mixed in-plane spin textures.

To confirm the expected symmetry
$Q(\hat{\mathbf{n}},\mathbf{B}_{\parallel})=Q(\hat{\mathbf{n}},-\mathbf{B}_{\parallel})$
that corroborates the given type of Rashba/Dresselhaus SOC, we performed a
series of bench marking large-scale magneto-transport simulations for TMF and
DPSR schematically shown in Fig. 2(d) and 2(e), respectively, incorporating
$\mathcal{H}_{0}+\mathcal{H}_{\text{Z}}+\mathcal{H}_{\text{SOC}}$ varying
Fermi energy, SOC strengths, disorder, and
$\mathbf{B}=\mathbf{B}_{\parallel}+\mathbf{B}_{\perp}$. Moreover, when
employing the out-of-plane magnetic field $\mathbf{B}_{\perp}$ we also
minimally couple the latter by means of the the Peierl’s phase, as compared to
the case with only $\mathbf{B}_{\parallel}$ that couples solely by means of
the spin-Zeeman term $\mathcal{H}_{\text{Z}}$.

## Rationale.

Before showing the transport simulations, let us come back to Fig. 1 (where
$\lambda_{\text{VZ}}=0$) to further explain the shifts of the Fermi contours
associated with an interplay of $\mathcal{H}_{\text{SOC}}$ and
$\mathcal{H}_{\text{Z}}$ and its consequences for the swap symmetries. The
upper row of Fig. 1 shows Fermi contours at $E=-100$ meV around the $K$ valley
without $\mathbf{B}_{\parallel}$, along with the in-plane spin textures that
are consistent with the lowest two bands of Figs. 2(a)–(c). With
$\mathbf{B}_{\parallel}=(B_{x},0,0)\equiv\mathbf{B}_{x}$ applied
($B_{x}=3\,$T), the lower panels of Fig. 1 show distinct responses arising
from the applied $\mathbf{B}_{\parallel}$ as expected from the characteristic
SOC-imprinted spin-textures: Fermi surfaces in Fig. 1(a) for the CR SOC with
the tangential textures move along $\pm\hat{\mathbf{e}}_{y}$, i.e.
perpendicular to $\mathbf{B}_{x}$, while these in Fig. 1(b) for the RR SOC
with the radial textures, as well as in Fig. 1(c) for the Dresselhaus SOC,
move along $\pm\hat{\mathbf{e}}_{x}$, i.e. parallel to $\mathbf{B}_{x}$.
Changing the polarity from $\mathbf{B}_{x}$ to $-\mathbf{B}_{x}$ the contours
will shift into the opposite sides (not shown). Importantly, for both
$\pm\mathbf{B}_{x}$, the figures in Fig. 1(a) stay mirror-symmetric with
respect to $\hat{\mathbf{e}}_{y}$, while those in Figs. 1(b) and 1(c) are
mirror-symmetric with respect to $\hat{\mathbf{e}}_{x}$. Therefore, probing a
transport quantity $Q$ along the $x$-axis for the in-plane field
$\mathbf{B}_{x}$ will furnish fully symmetric magneto-responses,
$Q(\hat{\mathbf{e}}_{x},\mathbf{B}_{x})=Q(-\hat{\mathbf{e}}_{x},\mathbf{B}_{x})$
and
$Q(\hat{\mathbf{e}}_{x},\mathbf{B}_{x})=Q(\hat{\mathbf{e}}_{x},-\mathbf{B}_{x})$,
for CR SOC, while a priori asymmetric ones for RR and Dresselhaus SOC.
Complementarily, probing $Q$ along the $y$-axis will give
$Q(\hat{\mathbf{e}}_{y},\mathbf{B}_{x})=Q(-\hat{\mathbf{e}}_{y},\mathbf{B}_{x})$
and
$Q(\hat{\mathbf{e}}_{y},\mathbf{B}_{x})=Q(\hat{\mathbf{e}}_{y},-\mathbf{B}_{x})$,
for RR and Dresselhaus SOC, but not for CR SOC. The latter can be also recast
in the alternative way, i.e. as
$Q(\hat{\mathbf{e}}_{x},\mathbf{B}_{y})=Q(-\hat{\mathbf{e}}_{x},\mathbf{B}_{y})$
and
$Q(\hat{\mathbf{e}}_{x},\mathbf{B}_{y})=Q(\hat{\mathbf{e}}_{x},-\mathbf{B}_{y})$
where $\mathbf{B}_{y}\equiv(0,B_{y},0)$ if one interchanges roles of the in-
plane field and transport direction. This is a very robust and experimentally-
sizable effect allowing to disentangle CR from RR SOC and even to determine
$\theta_{\text{R}}$.

Figure 4:  Simulated DPSR rate as a function of Fermi energy $E$ for a two-
terminal disordered graphene schematically shown in Fig. 2(e), considering the
CR [(a) and (b)], RR [(c) and (d)], and Dresselhaus SOC [(e) and (f)], all
subtended by a varying in-plane magnetic field of $\mathbf{B}_{x}$ for the
upper panels and $\mathbf{B}_{y}$ for the lower panels. Color codes for
different strengths of the in-plane magnetic field, $B_{\parallel}$, are shown
at the right. The scattering region is a zigzag ribbon with the width $w=131a$
and length $L=300a$, where $a=2.46\,\AA$ is graphene lattice constant. The
insets show the variation of $\tau_{s}^{-1}$ with the in-plane magnetic fields
($B_{x}$ for the upper panels and $B_{y}$ for the lower panels) at $E=100$
meV.

In our transport simulations, all based on the open-source Python package
Kwant Groth  _et al._ (2014), we explicitly verify these swap-symmetries
[panels (a), (d) and (f) in Figs. 3 and 4] by taking $Q$ as the second
conductance peak $G_{\text{2nd}}$ in the TMF simulation and the spin
relaxation rate $\tau_{s}^{-1}$ in the DPSR simulation, considering separately
CR, RR and Dresselhaus SOC with
$\lambda_{\text{CR}}=\lambda_{\text{RR}}=\lambda_{\text{D}}=6\,\text{meV}$ and
varying in-plane field components $B_{x},B_{y}\in[-3,+3]$ T. Moreover, we
confirm that in the case when CR and RR SOC terms admix with an angle
$\theta_{\text{R}}$, one gets symmetry of
$G_{\text{2nd}}(\hat{\mathbf{e}}_{x},\mathbf{B}_{\parallel}(\hat{\mathbf{u}}))=G_{\text{2nd}}(\hat{\mathbf{e}}_{x},-\mathbf{B}_{\parallel}(\hat{\mathbf{u}}))$
and
$\tau_{s}^{-1}(\hat{\mathbf{e}}_{x},\mathbf{B}_{\parallel}(\hat{\mathbf{u}}))=\tau_{s}^{-1}(\hat{\mathbf{e}}_{x},-\mathbf{B}_{\parallel}(\hat{\mathbf{u}}))$
by probing a transport along $\hat{\mathbf{e}}_{x}$ while swapping the in-
plane field $\mathbf{B}_{\parallel}(\hat{\mathbf{u}})$ along
$\hat{\mathbf{u}}=\pm\left(\cos\theta_{\text{R}},-\sin\theta_{\text{R}},0\right)$.
This is not hard to see when plotting the underlying Fermi contours and their
spin textures using the SOC Hamiltonian consisting of both CR and RR terms.
Generally, for a transport axis along $\hat{\mathbf{n}}$ one recovers symmetry
$Q(\hat{\mathbf{n}},\mathbf{B}_{\parallel})=Q(\hat{\mathbf{n}},-\mathbf{B}_{\parallel})$
if
$\varphi=\sphericalangle(\hat{\mathbf{n}},\mathbf{B}_{\parallel})=-\theta_{\text{R}}+(\pi)$.
As $\mathcal{H}_{\text{RR}}$ and $\mathcal{H}_{\text{D}}$, are unitary
equivalent, $s_{x}\mathcal{H}_{\text{RR}}s_{x}=\mathcal{H}_{\text{D}}$
assuming $\lambda_{\text{RR}}=-\lambda_{\text{D}}$ (or
$\mathcal{H}_{\text{RR}}^{*}=\mathcal{H}^{\phantom{*}}_{\text{D}}$ assuming
$\lambda_{\text{RR}}=\lambda_{\text{D}}$), TMF and DPSR cannot uniquely
disentangle RR from the Dresselhaus SOC as both display the same swap-symmetry
with $\mathbf{B}_{\parallel}$.

## TMF.

Figure 3 shows the conductance $G$ as a function of $B_{z}\in[-0.5,0]$ T for a
symmetric three-terminal TMF setup, schematically shown in Fig. 2(d), with
proximity-induced CR [panels (a) and (b)], RR [panels (c) and (d)] and
Dresselhaus [panels (e) and (f)] SOC. Results are displayed for the hole-doped
graphene at energy $E=-100$ meV; for corresponding figures with
$\lambda_{\text{VZ}}\neq 0$ and different values of $E$,
$\lambda_{\text{CR}}$, $\lambda_{\text{RR}}$, and $\lambda_{\text{D}}$, see
the Supplemental Material 111See Supplemental Material for …. The insets in
each panel show variations of the second conductance peak,
$G_{\text{2nd}}(|\mathbf{B}_{\parallel}|)$, with either
$\mathbf{B}_{\parallel}=\mathbf{B}_{x}$ or
$\mathbf{B}_{\parallel}=\mathbf{B}_{y}$ 222We focus on the second conductance
TMF peak, because there two skipping orbits of the spin-split Fermi contour
coalesce, see schematics in Fig. 2(d), what contrasts the first TMF peak that
is becoming spin-split at elevated values of $\lambda_{\text{CR/RR/D}}$’s or
at larger in-plane magnetic/Zeeman fields—see Fig. 3 and also SM. Similarly,
the higher TMF peaks are suffering from a more pronounced smearing—a finite
width of Injector allows for an injection of electrons with varying incidence
angles, while this variation grows with a propagation time, and with a
bouncing of the skipping orbits off the edge the final signal gets more
smeared. It shall be emphasized that our results about the non-spin-split
second TMF peak are valid assuming the Rashba and Dresselhaus SOC terms
dominate over the in-plane Zeeman coupling, in the opposite case, the spin-
textures of the skipping orbits and the Fermi contours subordinate to
$\mathcal{H}_{\text{Z}}$. displaying the expected swap-symmetry at panels (a),
(d) and (f), correlating the transport direction with the direction of
$\mathbf{B}_{\parallel}$ and the given SOC. In weaker in-plane fields
($|\mathbf{B}_{\parallel}|\lesssim 1\,\text{T}$), the $G_{\text{2nd}}$ shows
no pronounced variations with $|\mathbf{B}_{\parallel}|$, but at higher fields
the variations can approach up to $30\%$. For more details, see the
Supplemental Material Note (1).

## DPSR.

Figure 4 shows the simulated DPSR-rates, $\tau_{s}^{-1}$, as functions of
energy $E\in[-0.3,+0.3]$ eV for a symmetric two-terminal setup, schematically
shown in Fig. 2(e), considering CR [panels (a) and (b)], RR [panels (c) and
(d)] and Dresselhaus SOC [panels (e) and (f)], in the presence of non-magnetic
on-site disorder. We follow and extend a methodology developed in Refs.
Bundesmann  _et al._ (2015); Kochan  _et al._ (2020); Barth  _et al._ (2022)
by incorporating an effect of the spin-Zeeman coupling,
$\mathcal{H}_{\text{Z}}$, due to $\mathbf{B}_{\parallel}$. Electronic states
in the scattering region are given in terms of the full model Hamiltonian,
$\mathcal{H}_{0}+\mathcal{H}_{\text{Z}}+\mathcal{H}_{\text{SOC}}$, with
twisted-boundary conditions across the zigzag edges Bundesmann  _et al._
(2015); Kochan  _et al._ (2020); Barth  _et al._ (2022) to account for 2D
transport features along $\hat{\mathbf{e}}_{x}$, while DPSR is due to a non-
magnetic spatially-correlated Gaussian disorder $U(x,y)$ generated by means of
the Thorsos-Monte-Carlo method Mack (2013); Note (1) 333Disorder is defined by
the on-site potential—on-site potential correlation function
$C(|\mathbf{r}-\mathbf{r}^{\prime}|)^{\mathrm{2D}}=\langle
U(\mathbf{r}),U(\mathbf{r}^{\prime})\rangle=K_{0}\frac{(\hbar
v_{\mathrm{F}})^{2}}{2\pi\xi^{2}}\exp\left(-\frac{|\mathbf{r}-\mathbf{r}^{\prime}|^{2}}{2\xi^{2}}\right)$,
as already used in other Dirac systems Dufouleur  _et al._ (2018). The
parameter $\xi$ is the spatial correlation length, $\xi=10a$, such that
effective impurity concentration $\eta=\frac{\pi\xi^{2}}{wL}=0.799\%$.
Furthermore, $K_{0}$ is a dimensionless scaling parameter which is set to 0.1,
and we define Fermi velocity $v_{\mathrm{F}}$ such that
$\hbar\tfrac{v_{\mathrm{F}}}{a}=\tfrac{\sqrt{3}}{2}t$ where $t=3$ eV. Finally,
we introduce periodic boundary conditions in a direction perpendicular to the
transport axis, where we include a phase factor $\exp(i\phi)$ analogously to
Refs. Bundesmann  _et al._ (2015); Kochan  _et al._ (2020); Barth  _et al._
(2022). The spin-flip relaxation rates are then obtained by averaging over 20
phases $\phi\in[0,2\pi]$ and 30 different disorder configurations. with an
effective impurity concentration $\eta=0.799\%$. Insets show
$\tau_{s}^{-1}(\mathbf{B}_{\parallel})$ at $E=100$ meV, with expected swap-
symmetry at panels (a), (d) and (f), correlating transport direction, with the
direction of the in-plane field and with the type of SOC 444As
$\mathcal{H}_{\text{RR}}^{*}=\mathcal{H}^{\phantom{*}}_{\text{D}}$ Dresselhaus
plots follow from RR ones by swapping the sign of $E$..

## Radial supercurrent diode effect (SDE).

Finally, let us offer a short perspective how engendering exotic RR SOC in a
superconducting two-dimensional electron gas (2DEG) would manifest in SDE.
Considering quadratic 2DEG dispersion set by an effective mass $m^{\ast}$ and
Fermi energy $E$, in the presence of RR SOC,
$\alpha_{\text{RR}}(k_{x}s_{x}+k_{y}s_{y})$, and an additional Zeeman-
coupling, $\mathcal{H}_{\text{Z}}$, due to the in-plane field
$\mathbf{B}_{\parallel}$, the Fermi contours acquire longitudinal shifts along
$\mathbf{B}_{\parallel}$; see Fig. 1 with 2DEG Fermi contours around the
$\Gamma$ point. Therefore, turning on the superconducting correlations will
lead to an emergence of the helical phase,
$\psi_{\text{RR}}(\mathbf{r})=e^{i\mathbf{P}_{\text{RR}}\cdot\mathbf{r}/\hbar}|\psi|$,
with a finite center of mass momentum, $\mathbf{P}_{\text{RR}}\propto
g\mu_{\text{B}}\alpha_{\text{RR}}\tfrac{m^{\ast}}{\hbar
E}\mathbf{B}_{\parallel}$, aligned along $\mathbf{B}_{\parallel}$. This
opposes the CR case Edelshtein (1989); Yuan and Fu (2022); Kochan  _et al._
(2023), $\alpha_{\text{CR}}(k_{x}s_{y}-k_{y}s_{x})$, where the center of mass
momentum, $\mathbf{P}_{\text{CR}}$, develops in the direction perpendicular to
$\mathbf{B}_{\parallel}$, i.e.
$\psi_{\text{CR}}(\mathbf{r})=e^{i\mathbf{P}_{\text{CR}}\cdot\mathbf{r}/\hbar}|\psi|$,
with $\mathbf{P}_{\text{CR}}\propto
g\mu_{\text{B}}\alpha_{\text{CR}}\tfrac{m^{\ast}}{\hbar
E}(\mathbf{B_{\parallel}}\times\hat{\mathbf{e}}_{z})$. The immediate
ramifications lead to different magneto-chiral responses of the SDE critical
current, $I^{c}_{\text{SDE}}$, and the Josephson inductance, $L_{\text{J}}$,
on a probing current $\mathbf{j}$ and the in-plane field
$\mathbf{B}_{\parallel}$. So both,
$I^{c}_{\text{SDE}}=I^{c}_{0}+\gamma_{I}f(\mathbf{j},\mathbf{B}_{\parallel})$
and
$L_{\text{J}}=L_{\text{J}0}+\gamma_{L}f(\mathbf{j},\mathbf{B}_{\parallel})$,
will depend on magneto-chiral factor $f(\mathbf{j},\mathbf{B}_{\parallel})$
that, correspondingly, for RR or CR SOC equals to
$\mathbf{j}\cdot\mathbf{B}_{\parallel}$ and
$\mathbf{j}\cdot(\mathbf{B}_{\parallel}\times\hat{\mathbf{e}}_{z})$, the
magneto-chiral coefficients $\gamma_{I/L}$ encapsulate all material
dependencies. This unique magneto-chiral feature offered by RR SOC, if
possible to realize in 2DEG, would allow to unveil the roots of SDE, namely,
its SOC or screening-current origins.

## Conclusions.

Although spectrally CR, RR and Dresselhaus SOC manifest similarly, the
underlying wave functions and spin-textures discern when exposed to a Zeeman
coupling triggered by an in-plane magnetic field. We showed that such SOC-
imprinted features allow to disentangle CR and RR SOC components, quantified
by the Rashba angle $\theta_{\text{R}}$, when exploring the magneto-transport
symmetries of TMF and DPSR signals by swapping the orientation of in-plane
field. While the simulated TMF provides different quantitative responses for
all three SOC cases allowing for their mutual discrimination, the DPSR lacks
this ability in the case of RR and Dresselhaus SOC, however it allows to
discriminate the latter from the CR SOC. Even though we focused on graphene
with the proximity engendered SOC our conclusions are valid in general and
offer novel magneto-chiral response of the SDE.

###### Acknowledgements.

We thank Jaroslav Fabian and Paulo E. Faria Junior for useful discussions.
W.-H.K., A.G.-R., and M.-H.L. gratefully acknowledge National Science and
Technology Council of Taiwan (grant no. NSTC 112-2112-M-006-019-MY3) for
financial support and National Center for High-performance Computing (NCHC)
for providing computational and storage resources. M.B. acknowledges support
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
within Project-ID 314695032–SFB 1277 (project A07). A.M.-K. acknowledges
partial support by program “Excellence initiative - research university” for
the AGH University of Krakow, and by PL-Grid Infrastructure. D.K. acknowledges
partial support from the project IM-2021-26 (SUPERSPIN) funded by the Slovak
Academy of Sciences via the programme IMPULZ 2021, and Grant DESCOM VEGA
2/0183/21. A.G.-R., A.M.-K., M.-H.L. and D.K. acknowledge the hospitality of
Ming-Yu Liu for organizing a trip to Fo Guang Shan Monastery issuing in many
fruitful discussions.

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