# A mechanism for electrostatically generated magnetoresistance in chiral
systems without spin-dependent transport

Sytze H. Tirion<EMAIL_ADDRESS>Zernike Institute for Advanced Materials,
University of Groningen, NL-9747AG Groningen, The Netherlands  Bart J. van
Wees  Zernike Institute for Advanced Materials, University of Groningen,
NL-9747AG Groningen, The Netherlands

###### Abstract

Significant attention has been drawn to electronic transport in chiral
materials coupled to ferromagnets in the chirality induced spin selectivity
(CISS) effect. A large magnetoresistance (MR) is usually observed which is
widely interpreted to originate from spin (dependent) transport. However,
there are severe discrepancies between the experimental results and
theoretical interpretations, most notably the apparent failure of the Onsager
reciprocity relation in the linear response regime. We provide an alternative
explanation for the mechanism of the two terminal MR in chiral systems coupled
to a ferromagnet. For this we point out that it was observed that the
electrostatic contact potential of chiral materials on a ferromagnet depends
on the magnetization direction and chirality. In our explanation this causes
the transport barrier to be modified by the magnetization direction, already
in equilibrium, in the absence of a bias current. This strongly alters the
charge transport through/over the barrier, not requiring spin transport. This
provides a mechanism that allows the linear response resistance to be
sensitive to the magnetization direction and also explains the failure of the
Onsager reciprocity relations. We propose experimental configurations to
confirm our alternative mechanism for MR.

††preprint: APS/123-QED

##  I Introduction

In recent years the chirality of materials has been connected to the electron
spin in an effect titled the chirality-induced spin selectivity (CISS) effect
[1, 2, 3, 4]. It is a collective term for the interpretation of a diverse
range of experiments where chirality is assumed to give rise to various spin-
dependent effects. A major direction in the field of CISS is formed by
experiments on electronic transport through chiral systems coupled to a
ferromagnet, which usually show a large magnetoresistance (MR) [5, 6, 7].
Although a wide range of theories have been proposed, [8, 9, 10, 11, 12, 13,
14, 15, 16], the discrepancies between the experimental results and the
theories remain large [1]. It is crucial to understand the origin of the MR
that is experimentally observed for chiral systems connected to a ferromagnet,
both for understanding of fundamental physics, as well as possible
applications of chiral systems in the field of spintronics.

Currently, the large MR is assumed to originate from spin-dependent transport
in, or into the chiral system. The chiral system is presumed to be a spin
polarizer/filter which, together with the ferromagnet as spin
polarizer/analyzer, will give a modification of the spin transport when either
the chirality of the system is changed or the magnetization direction is
reversed. This then causes an MR similar to giant magnetoresistance (GMR) [17]
or tunnel magnetoresistance (TMR) [18] in purely ferromagnetic systems.

Here, we first review the experimental results of the MR measured for chiral
systems connected to a ferromagnet and identify the incompatibility of them in
terms of spin transport. Then, we provide an alternative explanation for the
origin of the MR of chiral systems coupled to a ferromagnet. We argue that
experiments measured the electrostatic contact potential of the system in
equilibrium that is altered by either reversal of the magnetization direction
or chirality [19, 20, 21]. We investigate the implications of this for the
modification of the potential profiles of a Schottky barrier-like junction
where the chiral system is in direct contact with a ferromagnetic metal, as
well as the case where they are separated by a tunnel barrier. We find that
the barrier height and the energy bands within the screening length are
adjusted and thus can modify the charge transport. This gives an MR which does
not dependent on spin dependent transport.

Although we do not understand its origin, the fact that the electrostatic
equilibrium properties of a chiral system coupled to a ferromagnet are
modified by the magnetization direction also causes the Onsager reciprocity
relations to fail and gives an MR in the linear response.

A recent theoretical discussion by Xiao  _et al._ [16] of the large MR also
discuss a modification of the charge transport barrier between the chiral
system and the ferromagnet [22, 16, 4]. However, it is argued that this is due
to bias current induced charge trapping that can only result in an MR in the
non-linear bias regime [9, 23]. It therefore cannot explain the experimental
observation of MR in linear response.

##  II CISS magnetoresistance, overview of experimental techniques and results

The electronic transport in chiral systems coupled to ferromagnets is studied
by two main experimental techniques.

Local scanning probe techniques (conductive atomic force microscopy [24, 25,
5, 26, 6, 27, 28, 29, 30, 7, 31, 32] and scanning tunneling microscopy [33,
34]) where a thin film of chiral material (in most cases molecules) is
absorbed on a conducting substrate and a local probe is used to measure the
two terminal voltage-current (VI) characteristics of the chiral system with
typical currents of nanoamperes, for applied biases up to several Volts.
Either the substrate or the probe is ferromagnetic and the VI characteristics
of the chiral system are measured with the magnetization pointing either up or
down. The (averaged) VI curves of the up and down magnetization are compared,
and the MR is calculated as a function of the applied bias. Comparatively
large MR values, typically above 50% and in some cases close to 100% [26, 31],
are reported. Importantly, in many reports the measured MR is (almost) bias
independent [35, 26, 36, 31, 29, 7] for a bias up to several Volts.

Alternatively, junction geometries are used [24, 37, 5, 25, 35, 38, 36, 39,
22, 40] where the chiral layer is sandwiched between two microfabricated
electrodes, one ferromagnetic with an oxide barrier in between it and the
chiral system. An external magnetic field orients the magnetization. In these
two terminal experiments, either the VI characteristic is measured, and the
bias dependence of the MR is calculated. In most cases the reported MR from
the junctions is an order of magnitude smaller compared to the local scanning
probe techniques, but recently some junctions also show high MR values [31,
36].

Compared to the two terminal geometry, we note that it is difficult to perform
four terminal measurements on the chiral systems that have been studied so
far. However, the electronic transport of chiral solid state crystal system
was studied by four terminal measurements [41, 42, 43], see Appendix B. For
recent theoretical overviews of spin selectivity of chiral solid state system
see Ref [44]& [45].

##  III Problems with the spin transport interpretation of two terminal
magnetoresistance

Figure 1:  (a) Elementary spin-dependent transmission and reflection model for
a chiral system. The “favoured” electron spin is transmitted and the
“unfavoured” electron spin is reflected but has to spin-flip [8, 9] because
directional spin transmission always requires a spin-flip reflection process
to avoid spin currents in equilibrium. (b) A ferromagnet gives rise to spin-
dependent transport but this is fundamentally different because the preferred
spin is set by the magnetization direction, not the propagation direction. (c)
An energy level diagram of a chiral system contacted by a ferromagnet tip (FM
Tip) and a metal substrate (NM) to illustrate the alternative origin for the
MR. At the interface between the tip and the chiral system a Schottky like
barrier is formed. The magnetization direction affects the contact potential
$\Delta\psi_{\leftrightarrows}$ that is build up over a distance $d$. This
then modifies the electrostatic potential profile of the chiral molecules.

The interpretation of the experimental results assumes that electron transport
in chiral systems is spin-dependent, where the chirality sets the preferred
spin direction which is coupled to a given propagation direction. However, the
measured MR is often directly interpreted in terms of the spin polarization of
the transmitted electrons. It is crucial to distinguish the MR [9, 46, 47]
generated by the entire circuit (chiral system + ferromagnet) from mechanisms
that generate the spin-dependent electron transport in the chiral systems
themselves [48, 49]. In this work we assume the presence of directional spin-
dependent transmission and reflection in the chiral systems itself. However,
despite this assumption, below we identify five main problems with the common
interpretation of the measured MR in terms of spin-dependent transport.

###  III.1 Unknown mechanism for spin injection and detection by the
ferromagnet

In past decades, significant effort was dedicated to electrical
injection/detection of spin-polarized electrons [50, 51] into metals and
semiconductors, which involves two different transport mechanisms. The first
is spin-dependent conductivity determined by the polarization of the
ferromagnet at the Fermi energy, which is the mechanism of giant
magnetoresistance (GMR) [17]. The second is spin-dependent tunneling
proportional to the density of states at the Fermi energy commonly used in
tunnel magnetoresistance (TMR) [18]. However, for most chiral systems it is
unclear which of these mechanisms could be responsible for the assumed spin
injection/detection from the ferromagnet.

###  III.2 Conductivity mismatch

The (low bias) injection of spin-polarized electrons by a ferromagnetic metal
into semiconductors or other low-conductivity materials is strongly hindered
by the large difference in conductivity, know as the “conductivity mismatch”
[50]. This problem can be overcome by impedance matching the ferromagnet and
the semiconductor e.g. by a tunnel junction. This will make spin-dependent
tunneling a possible mechanism for spin injection and detection from a
ferromagnet into a chiral system. However, it is unaddressed how the
conductivity mismatch between the ferromagnetic metal and the chiral system is
overcome. This is especially relevant since most of the chiral systems studied
are molecules with comparatively low conductivity, making the mismatch problem
only more significant.

###  III.3 Large magnetoresistance

The reported MR for chiral systems coupled to a ferromagnet is typically above
50% and approaches 100% in a number of recent experimental results [26, 31]. A
conventional TMR spin valve consist of two ferromagnetic electrodes and
operates based on spin-dependent tunneling for which the Julière model
connects the spin polarization of the density of states of the ferromagnets,
$P_{FM}$, to the MR [47]. Using this model, it can be shown that,

$MR=\frac{I_{\uparrow}-I_{\downarrow}}{I_{\uparrow}+I_{\downarrow}}=P_{FM}P_{Chiral},$
(1)

where $I_{\uparrow\downarrow}$ is the measured current with magnetization
direction indicated by the subscript and $P_{Chiral}$ the“spin polarization”
of the chiral system. This clearly shows that the total MR can never exceed
the polarization of the ferromagnet, even when we assume a perfect spin
polarization of the chiral system ($P_{Chiral}=1$), as was recently also
pointed out by Liu and Weiss [47] and others. Note that here the MR definition
commonly used for CISS is applied, which is different from the conventional
definition of the TMR ratio applied in the Julière model.

###  III.4 Bias voltage dependence of the magnetoresistance

Theoretical effort is dedicated to explain the high MR in terms of the spin
polarization of transmitted electrons from the chiral system, arguing why
$P_{Chiral}$ could be close to 1. However, as far as we know the theoretically
predicted spin polarization of both the chiral systems as well as for
ferromagnets, is strongly energy-dependent and hence bias dependent [48, 52,
11, 49, 12, 13, 15, 53, 23, 54, 55]. This is in stark contrast with the
experimental results which shows in many reports, an (almost) bias independent
MR [35, 26, 36, 31, 29, 7], for a bias range of several Volts. This also makes
the experimentally observed MR an ’even’ function of the bias, meaning it does
not change sign when the bias is reversed. This does not agree with expected
symmetry which should be ’odd’ in bias, meaning a change in sign of the MR
when bias is reversed [9, 23], see Appendix C.

###  III.5 Failure of the Onsager reciprocity relations

In linear response the charge and spin transport parameters can be fully
determined by the equilibrium properties of the sample, making it adhere to
the Onsager reciprocity relations (see Appendix A).

It is important to disentangle the MR generated by chiral systems coupled to a
ferromagnet from the possible spin-dependent transport in the chiral system
itself [56], because these are different concepts. FIG. 1(a)&(b) shows the
spin-dependent transmission model developed by Yang  _et al._ [8, 9, 10], that
includes both the spin-dependent transmission and the reflection processes of
chiral and ferromagnetic systems. It was found that the linear response
resistance is unaffected by reversal of the magnetization, see FIG. 4 in
Appendix B. Only in the nonlinear regime a MR can be detected which requires a
combination of energy dependent transport and energy relaxation (e.g. possible
electron-electron and electron-phonon interactions [57, 12, 54, 55]), Appendix
C.

However, several experimental results for a wide range of chiral systems [35,
58, 36, 31, 7, 30] now show a clear difference in the linear response
resistance when the magnetization is reversed, which is at odds with Onsager
reciprocity. This makes the origin of the MR reported for chiral systems
coupled to a ferromagnet one of the main enigmas in the field of CISS.

Based on the discussion above we argue that the origin of the MR reported in
(two terminal) CISS transport experiments is not due to a (modification of
the) spin injection/transport. Instead, we provide an alternative explanation
for the origin of the MR in terms of a chirality dependent modification of the
contact potential by the magnetization, not involving spin transport.

##  IV Magnetochiral contact potential modification

Figure 2:  (a) The calculated potential $\psi(z)$ for a contact potential
change of $50\text{\,}\mathrm{mV}$. We use $N_{D}=$
${10}^{15}\text{\,}{\mathrm{cm}}^{-3}$ and $\epsilon_{r}=5$. The inserted
graph shows the screening length $\lambda$ as function of the doping
concentration on a semi-logarithmic x-axis. (b) Energy level diagram of a
chiral system in contact with a ferromagnet. When a bias is applied the
(modified) transport can occur via three mechanisms: 1. Thermionic emission 2.
Tunneling through the barrier 3. Transport within the screening length.

Recent experimental results report a modification of the (electrostatic)
equilibrium properties of a chiral system coupled to a ferromagnet in the
absence of a bias current [21, 20, 19]. It was found that the contact
potential of chiral molecules on a ferromagnet is modified by reversal of
either the magnetization direction or by interchanging the chirality of the
system. Theiler  _et al._ [19] recently reported a shift of the contact
potential measured by Kelvin probe force microscopy (KPFM) of
$50\text{\,}\mathrm{mV}$ when the direction of the magnetization is reversed,
which is similar in magnitude previously reported by KPFM measurements [20]
and photo-emission experiments [21]. We point out that no electron transport
is involved in these measurements of the contact potential shifts. The large
experimental timescales, in our opinion, exclude transient effects, justifying
that this is an equilibrium phenomenon. The magnitude of the contact potential
shift of $\sim$ $50\text{\,}\mathrm{mV}$, is large compared to $k_{b}T$, where
$k_{b}$ is the Boltzmann constant and $T$ is the (room) temperature, which
makes the change in contact potential relevant for the (low bias) electron
transport and the MR.

Another recent experiment by Volpi  _et al._ [59] has investigated field
effect transistors with a chiral semiconductor channel and ferromagnetic
source and drain electrodes. It was found that the threshold gate voltage is
strongly modified by the magnetization direction, highlighting the
significance of electrostatic effects for chiral systems coupled to a
ferromagnet.

Taking the modification of the contact potential by the magnetization
direction as a starting point, we give an alternative explanation for the
origin of the MR reported for chiral systems coupled to a ferromagnet, for
which we use a semiconductor analogy. We assume a chirality dependent
modification of the contact potential when the magnetization direction is
reversed, which can be seen as a change of the electron affinity of the
molecules at the interface with the ferromagnet. This is strongly influences
the electrostatic potential landscape of the junction between a ferromagnet a
the chiral molecular system, for which we use a semiconductor band picture
with the HOMO and LUMO bands. Therefore, the electronic charge transport (but
not the spin transport) through the ferromagnet/molecular junction is affected
by the magnetization direction.

For a direct contact between metallic ferromagnet and the chiral molecular
system a Schottky-type barrier, where the HOMO/LUMO bands bend, is expected.
In FIG. 1(c), the effect of the contact potential modification on the barrier
between the ferromagnet and a molecular system is depicted. In equilibrium,
with a constant Fermi energy, the magnetization direction changes the contact
potential and, therefore modifies the electrostatic potential and the barrier
height by $\Delta\psi_{\leftrightarrows}$, over a distance given by the
screening length $\lambda$.

To check the feasibility of this explanation, we calculate the areal electron
density, $N$, that is required to build up a potential difference of
$\Delta\psi_{\leftrightarrows}=$ $50\text{\,}\mathrm{mV}$ [19] over a typical
length scale of a molecule $d=$ $1\text{\,}\mathrm{nm}$, using a parallel
plate capacitor approximation,

$N=\frac{\epsilon_{s}\Delta\psi_{\leftrightarrows}}{ed},$ (2)

where $\epsilon_{s}=\epsilon_{r}\epsilon_{0}$ is the permittivity and $e$ the
electron charge. With $\epsilon_{r}=$ $5\text{\,}$, we find that $N=$
${10}^{-2}\text{\,}{\mathrm{nm}}^{-2}$, which corresponds to
$2.5\text{\times}{10}^{-3}\text{\,}$ electron charge per molecule for a
molecular packing density of $4\text{\,}{\mathrm{nm}}^{-2}$. This indicates
that only a small fraction of charge shift per single molecule is required to
build up a potential of $50\text{\,}\mathrm{mV}$.

In a Schottky like junction the HOMO and LUMO bands bend when they return to
their bulk position. This happens over the length scale, $\lambda$, which
depends on the shift of contact potential. We estimate the length scale
$\lambda$ with the the Poisson-Boltzmann equation,

Figure 3:  Effect of the potential profile modification with a spacer between
the ferromagnetic substrate and the HOMO/LUMO band of the chiral system. (a)
In most experiments that use a ferromagnet substrate, there is a gold (Au)
layer between the ferromagnet and the chiral system. However, this is not
significantly different from the case where there is direct contact between
the ferromagnet and the chiral system. (b) The potential profile of an oxide
(Ox) tunnel barrier between the chiral system and the ferromagnet is also
modified by the magnetization direction. Here, we ignore possible band bending
of the HOMO/LUMO bands of the chiral system and show the potential profile
only close to the ferromagnetic substrate, ignoring the normal metallic
tip/substrate. (c) A proposed sample configuration with the chiral layer
contacted by a gold coated ferromagnet on one side and by a tunnel barrier on
the other side. In this geometry spin-dependent transport cannot generate an
MR but the contact potential modification can.

$\nabla^{2}\psi(z)=\frac{\partial^{2}\psi(z)}{\partial
z^{2}}=-\frac{\rho(z)}{\epsilon_{s}},$ (3)

where $\psi(z)$ is the potential as function of the distance $z$ and $\rho(z)$
is the local electric charge density. With the boundary conditions
$\psi(0)=\Delta\psi_{\leftrightarrows}$ and $\psi(\lambda)=0$, the solution to
Eq. (3) yields,

$\psi(z)=\frac{eN_{d}}{2\epsilon_{s}}\left[\lambda^{2}-(\lambda-z)^{2}\right],$
(4)

here $N_{D}$ is the doping concentration in the chiral system and $\lambda$ is
the screening length, given by,

$\displaystyle\lambda$
$\displaystyle=\sqrt{\frac{2\epsilon_{s}\Delta\psi_{\leftrightarrows}}{eN_{D}}}\;,\qquad\Delta\psi_{\leftrightarrows}>k_{B}T,$
(5) $\displaystyle\lambda$
$\displaystyle=\sqrt{\frac{2\epsilon_{s}k_{B}T}{e^{2}N_{D}}}\;,\qquad\Delta\psi_{\leftrightarrows}<k_{B}T.$
(6)

In FIG. 2(a) we plot $\psi(x)$ for the case where
$\Delta\psi_{\leftrightarrows}>k_{B}T$ with a contact potential modification
of $50\text{\,}\mathrm{mV}$ at $N_{D}=$ ${10}^{15}\text{\,}{\mathrm{cm}}^{-3}$
to illustrate the length scales on which the electrostatic potential is
modified. The length $\lambda$, is dependent on $N_{D}$, as seen in the
inserted graph in FIG. 2(a). From the graph of $\psi(z)$, it can be seen that
the length scale of the potential modification is $\approx$
$150\text{\,}\mathrm{nm}$, making it comparable to if not greater than the
molecular thin film thickness, which is typically less than
$20\text{\,}\mathrm{nm}$. This illustrates that the potential landscape of the
entire molecular thin film can be modified by the reversal of the
magnetization direction, which will strongly alter the charge transport, also
in linear response.

When the ferromagnet-chiral molecule junction is under bias, the contact Fermi
energy gets shifted by $eV$. Three different mechanisms can carry the
transport; see FIG. 2(b). In the first mechanism, (1) thermionic emission, the
carriers need to have a comparatively high energy to overcome the barrier. The
second transport mechanism is (2) quantum mechanical tunneling through the
barrier, which is extremely sensitive to the potential landscape of the
barrier. The third mechanism is (3) transport in the region within the
screening length due to modification of the occupation of the states in the
bands.

Since the magnetization direction modifies the electrostatic potential
landscape as well as the barrier height, all three transport mechanisms will
be strongly influenced by the magnetization direction. Furthermore, we
emphasize that both the transport via the HOMO and LUMO bands is modified by
the magnetization direction. When the barrier height is increased, transport
via the LUMO is reduced, but the opposite is true for transport via the HOMO.
Therefore, the sign of the MR is expected to change when the doping or the
carrier type (electrons or holes) is changed in the chiral system.

Two different experimental configurations are used for the local probe
measurements. In one, the tip is ferromagnetic, and the substrate is non-
magnetic, as is illustrated in FIG. 1(c). In the other configuration,
illustrated in FIG. 3(a), the substrate is ferromagnetic, and in almost all
experiments covered by a thin (2 to 10 nm) gold (Au) layer, and the tip is
non-ferromagnetic. Relevant here is experimental work by Ghosh  _et al._ [20],
where the effect of an Au spacer layer between the ferromagnet and the chiral
molecules was investigated. It was found that the shift in contact potential
induced by reversing the magnetization direction is around
$40\text{\,}\mathrm{mV}$ for a $10\text{\,}\mathrm{nm}$ thick Au layer, a
comparatively large distance. Furthermore, it was found that reduction of the
Au thickness increases the contact potential shift to more than
$100\text{\,}\mathrm{mV}$. Therefore these two seemingly different
configurations yield similar results (but note our final remarks in the
conclusions and outlook section). We estimate that the potential landscape is
modified on a length scale comparable to that of the thickness of chiral film
and therefore this electrostatic effect can dominate the electronic (but not
the spin) transport.

With a tunnel barrier between the chiral molecular system and the ferromagnet,
the modification of the contact potential adjusts the potential landscape as
is illustrated in FIG. 3(b), where we neglect possible bending of the
HOMO/LUMO bands. The modified contact potential also alters the potential
profile and height of the tunnel barrier. Interestingly, the magnetization
direction that corresponds to an increased barrier height for a Schottky type
junction (dashed lines in FIG. 3(a)) corresponds to a decrease in the average
tunnel barrier height (dashed line in FIG. 3(b)). Some experimental results,
where local probe and tunnel junction experiments are performed, give an
opposite sign of the MR for the same chiral system and seem to support this
[25, 32]. However, it is of vital importance to make the comparison between
Schottky barrier type junctions and tunnel junctions with the same definition
of up and down magnetization, which is not always clear in the literature.

##  V Conclusions and Outlook

We have provided an alternative explanation for the origin of the linear
response MR of chiral systems coupled to a ferromagnet. Experimental results
show a shift in the contact potential of chiral molecules in proximity to a
ferromagnet when reversing magnetization direction. We argue that the
transport barrier between the chiral molecules and the ferromagnet is modified
by the magnetization direction in equilibrium without any bias current. The
length scale on which the potential is modified is of similar magnitude, if
not greater than the thickness of the measured chiral thin film. Therefore,
the electronic transport is strongly modified by the magnetization direction,
resulting in an (large) MR in the linear response regime.

With this simplified model we propose an alternative explanation for the MR of
chiral system coupled to a ferromagnet that does not require spin-dependent
transport and explains the failure of the Onsager reciprocity relations.
However, we emphasize that in principle nonlinear/strongly out of equilibrium
directional spin-dependent transport in chiral systems can lead to an MR [9]
(see Appendix C) purely based on spin transport.

A number of questions remain to be addressed. The first is the origin of the
contact potential modification by the magnetization direction, as of yet it is
not understood what physical mechanism causes this, in particular in
equilibrium. A (yet unknown) magnetic exchange mechanism could be responsible
for the chirality dependent contact potential modification in the case of
direct contact between the chiral system and the ferromagnet. It could also be
applied to the gold coated ferromagnets. However the effect was shown to
extend over $10\text{\,}\mathrm{nm}$ of gold, a comparatively large distance
for which the mechanism is unclear. It is also unclear how an exchange
mechanism could be applied to the tunnel barrier that separates the chiral
system and the ferromagnet.

The second remaining question concerns the bias independent nature of the MR
at biases even up to several Volts. For a larger bias both the Schottky
barrier type and the tunnel barrier potential profiles will be adjusted by the
bias induced electric field. How the MR stays relatively independent of the
applied bias is a question that remains to be answered.

The third question is if and how our explanation can be applied to other types
of experiments (beyond two terminal configurations) where chiral systems are
coupled to a ferromagnet. A recent experiment has investigated the MR measured
from the conduction in the plane of a ferromagnetic/chiral molecule bilayer
[60]. A chirality-dependent asymmetry in the MR of the ferromagnet is found
that is independent of the applied current, showing that coupling between
(molecular) chirality and ferromagnetism can induce an effect on the measured
MR that is not current dependent. This could also be relevant for measurements
of adhesion forces between chiral systems and ferromagnets (with different
magnetization directions) [61, 62] or in field-effect transistor devices with
a chiral channel [59]. This shows that the electrostatic interactions between
chiral systems and ferromagnets can be a widely applicable phenomena with
important fundamental physics to be explored as well as possible application
in nanotechnology.

To finish, we identify five main approaches to further confirm the proposed
mechanism for the MR. A systematic study should be made where the shift in
contact potential is compared to the Schottky barrier height extracted
temperature dependent electronic transport measurements. A second approach is
to change the carrier type in the chiral system by means of doping or a gate
electrode because we expect that the MR changes sign when the transport in
HOMO changes to the LUMO. As a third approach we propose a yet to be explored
geometry where the chiral material is directly contacted by a ferromagnet on
one side and by an oxide barrier and nonmagnetic electrode on the other, as
can be seen in FIG. 3(c). In this geometry any possible modulation of spin
transport cannot generate a MR but the electrostatic potential modification
can generate an MR which allows the two origins to be separated. The fourth
approach is to investigate the transport behaviour of chiral systems in non
local device geometries where the charge and spin transport can be fully
decoupled. For instance in non local devices [8] where ferromagnetic
electrodes inject spin and electrodes made of a chiral system to detect the
spin signal. Last but not least, a comparison of FIG. 1(c) and FIG. 3(a) shows
that the electrostatic modification is, for the same chirality and
magnetization direction of the system, opposite for a magnetic electrode on
top or below the chiral system. This implies that an experiment with a
magnetic tip or a magnetic substrate would show MR with opposite signs.

##  VI acknowledgments

This work is supported by the Zernike Institute for Advanced Materials (ZIAM),
and the Spinoza prize awarded to Professor B.J. van Wees by the Nederlandse
Organisatie voor Wetenschappelijk Onderzoek (NWO). The authors like to
acknowledge useful discussions during the Gordon Research Conference ’Electron
Spin Interactions with Chiral Molecules and Materials’ in August 2023. In
particular with P.M. Theiler. We also thank M.H. Diniz Guimaraes and C.H. van
der Wal for useful discussion and comments on the manuscript.

##  Appendix A Linear response regime

In linear response, an applied bias $eV$ only causes a gradient/difference in
the electrochemical potential that drives electron transport. Therefore the
charge and spin transport parameters of the sample such as the conductivity
can be fully determined by the equilibrium properties of the sample. This
implies that the associated current induced electric field does not matter for
the transport and that other current induced effects, such as the spin
transfer torque, are not relevant in linear response [14]. A conservative
energy range for the linear response regime is $eV\approx k_{b}T$, where
$k_{b}$ is the Boltzmann constant and $T$ is the temperature [63]. However,
$k_{b}T$ is a lower bound estimate; the linear response regime can extend to
energies far above $k_{b}T$ e.g. if inelastic relaxation processes take place
in the sample.

Owing to microscopic reversibility, the linear response follows several
fundamental symmetry properties, most notably the Onsager reciprocity
relations [64]. As a consequence of the Onsager reciprocity relations, a
system with a magnetization ($M$) in an external magnetic field ($B$) requires
the linear response resistance to satisfy:

$R_{ij,nm}(B,M)=R_{nm,ij}(-B,-M),$ (7)

where the subscripts $ij$ correspond to the current contacts and $nm$ to the
voltage contacts of the sample. In a two terminal connection this reduces to,

$R_{2T}(B,M)=R_{2T}(-B,-M),$ (8)

such that the resistance is unchanged when the magnetization and external
magnetic field are reversed, meaning no MR in linear response. Note that this
is under the assumption that only the magnetization is reversed and not
possible associate electric fields. A disadvantage of a two-terminal
measurement is that in addition to the sample resistance also the resistance
of the contacts will be measured, significantly increasing the risk of
spurious effects being detected. However, in the linear response regime, the
resistance of the contacts also satisfies Eq. (7) & (8).

The Onsager reciprocity relations are very well established and have been
tested in many experiments [65], including systems with strong spin-orbit
interaction [66], organic single-molecule junctions [67] and when a Büttiker
(dephasing) probe is included [23, 68].

We are not aware of any experimental results nor theory that deviates from Eq.
(7) & Eq. (8) except the MR measured in chiral systems coupled to a
ferromagnet (see main text).

Figure 4:  (a) Spin-dependent electron transmission model for electrons
propagating from left-to-right though a chiral system coupled to a
ferromagnet. The reversal of the magnetization does not affect the total
charge current and hence Eq. (8) does hold. Note there are spin currents on
both sides of the system which depend on the chirality and the magnetization
direction but that the charge transport is unaffected by the magnetization
direction, see Ref [9]. (b) This is fundamentally different from two connected
ferromagnets and the magnetization of one of them being reversed.

##  Appendix B Spin-dependent transmission for linear response
magnetoresistance

It is key to distinguish the MR generated by the entire circuit of chiral
system plus ferromagnet from the possible mechanisms which cause the spin-
dependent electron transport in chiral systems, because they are fundamentally
different questions. Unfortunately, the two concepts are sometimes mixed up.

The absence of MR in the linear response for a chiral system coupled to a
ferromagnet was illustrated with a general spin-dependent transmission model
developed by Yang  _et al._ [8, 9, 10]. This model is based on spin-dependent
transmission and reflection coefficients as illustrated in FIG. 1(a)&(b). The
“favoured” spin is transmitted and the “unfavoured” spin gets reflected with
spin-flip, [8] which is required by the absence of spin currents in
equilibrium. Owing to the spin-dependent transport, spin currents are
generated in the chiral system when it is biased, due to the spin-dependent
transport in the chiral system. However, this does not result in a MR when the
chiral system is connected to a ferromagnet. Transport through a ferromagnet
is also spin-dependent. However, the crucial difference is that the
magnetization direction sets the direction of the favoured spin, see FIG.
1(b).

FIG. 4(a) shows spin-dependent transport in a chiral system coupled to a
ferromagnet, including spin transmission, reflection and spin-flip, for both
magnetization directions. When the ferromagnet transmits electrons that are
favoured by the chiral system, they are transmitted by the chiral system. With
the magnetization reversed, the ferromagnet transmits electrons unfavored by
the chiral system. However, owing to the spin-flip reflection process of the
chiral system and the spin-conserving reflection of the ferromagnet, the total
charge current is unchanged by the magnetization reversal, giving a strict
zero MR in the linear response regime. Although the illustrated electron
transmission and reflection in FIG. 4(a) assumes an ideal spin-dependent
transport for both the ferromagnet and the chiral system, the same conclusion
holds for a any ferromagnet and chiral system.

Fundamentally different is the spin valve consisting of two ferromagnets,
where the total number of transmitted electrons is affected when the
magnetization of only one of the ferromagnets is reversed, as can be seen in
FIG. 4(b). When the magnetizations of both ferromagnets are reversed the total
resistance is unaffected, and the two terminal Onsager relation of Eq. (8) is
satisfied.

We note that electron transport in chiral crystals has been studied in devices
where the current flows along a crystallographic axis [41, 42, 43] and
perpendicular to the current direction a contact with strong spin-orbit
interaction measures a voltage proportional to the spin signal from the
inverse spin Hall effect in the contact. For these devices, that do not have a
magnetic electrode, the reciprocal measurement where voltage and current
contacts are interchanged were observed to satisfy Eq. (7) [41, 42, 43], the
four terminal Onsager relations.

However, despite the low conductivity of many chiral systems hindering the
studies of the transport at low bias, there are now several experiments that
do show a clear linear response regime [35, 58, 36, 31, 7, 30]. Furthermore,
they show a difference in the linear response resistance when the
magnetization is reversed, giving a non-zero MR, contradicting the Onsager
reciprocity relations. We note that, in a number of these results, the bias
dependence of the MR shows anomalous behaviour around zero bias voltage [26,
31, 36, 29, 7, 32]. We attribute this to possible offset effects in the
applied bias voltage and/or the measured current which can lead to incorrect
experimental determination of the MR around zero bias.

Based on the discussion above and in section II, we conclude that the
experimental results of charge transport in chiral systems connected to a
ferromagnet are inconsistent with the spin transport interpretation that is
commonly used. This emphasises that an alternative explanation for the MR in
the linear response is required, as we give in this paper.

##  Appendix C Linear response compared to non-linear response regime

In the linear response regime, the bias current does not significantly modify
the properties of the sample compared to equilibrium of the system, and
Onsager reciprocity relations are strict. It is important to note that the
gradient in electrochemical potential drives the bias current and that the
electric field is not relevant for the transport behaviour. This changes when
the applied bias modifies the properties of the system under study and we
enter the non-linear regime where the Onsager reciprocity relations are not
strict. For a chiral system coupled to a ferromagnet, it was shown [9, 23]
that, due to a combination of energy-dependent transport and energy
relaxation, a MR can be generalized in the non-linear regime when the
magnetization is reversed. Furthermore, the MR is strongly bias dependent, has
to be strictly zero at zero bias and change sign when bias is reversed,
meaning that the MR is an odd function in the bias.

This has similarity to the well-established electrical magnetochiral
anisotropy (EMCA) [69, 70] where the resistance of a chiral system has an
additional chirality-dependent ($D/L$) term $\Delta R$ that is proportional to
the applied current $I$ and the magnetic field $B$,

$\Delta R=R_{o}^{D/L}I\cdot B$ (9)

, making the MR an odd function of $I$ and $B$.

However, this is entirely inconsistent with the experimental observations of
transport in chiral systems coupled to a ferromagnet because they show a MR,
which is, in many reports, independent of the applied bias, down to zero bias,
making it an even function of the applied bias.

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