Mechanism for Electrostatically Generated Magnetoresistance in Chiral Systems without Spin-Dependent Transport

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 the theoretical interpretations, most notably the apparent failure of the Onsager reciprocity relations in the linear response regime. We provide an alternative mechanism for the two terminal MR in chiral systems coupled to a ferromagnet. For this, we point out that it was observed experimentally that the electrostatic contact potential of chiral materials on a ferromagnet depends on the magnetization direction and chirality. The mechanism that we provide 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 and 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.


INTRODUCTION
−4 It is a collective term for the interpretation of a diverse range of experiments in which chirality is assumed to give rise to various spin-dependent effects.−7 Although a wide range of theories have been proposed, 8−16 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 fundamental physics and for possible applications of chiral systems in the field of spintronics.
Currently, the large MR is assumed to originate from spindependent transport in or into the chiral system.The chiral system is presumed to be a spin polarizer/filter which, together with the ferromagnet as a 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 their incompatibility in terms of spin transport.Then, we provide an alternative mechanism for the origin of the MR of chiral systems coupled to a ferromagnet.−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 is 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 discusses a modification of the charge transport barrier between the chiral system and the ferromagnet. 4,16,22However, it is argued that this is due to bias current-induced charge trapping that can only result in an MR in the nonlinear bias regime. 9,23It therefore cannot explain the experimental observation of MR in the linear response.

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 5−7,24−33 and scanning tunneling microscopy 34,35 ) 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 magnetizations 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%, 27,32 are reported.Importantly, in many reports the measured MR is (almost) bias-independent 7,27,30,32,36,37 for a bias up to several volts.Alternatively, junction geometries are used 5,22,25,26,36−41 where the chiral layer is sandwiched between two microfabricated electrodes, of which one is ferromagnetic with an oxide barrier in between it and the chiral system.An external magnetic field orients the magnetization.In these two terminal experiments, the VI characteristic is measured, and the bias dependence of the MR is calculated; from this the MR is obtained as a function of the out-of-plane magnetization component.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. 32,37ompared 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 a chiral solid state crystal system was studied by four terminal measurements; 42−44 see Appendix B. For recent theoretical overviews of spin selectivity of a chiral solid state system, see papers by Sławinśka 45 and Yang et al. 46

PROBLEMS WITH THE SPIN TRANSPORT INTERPRETATION OF TWO TERMINAL MAGNETORESISTANCE
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 9,47,48 generated by the entire circuit (chiral system + ferromagnet) from mechanisms that generate the spindependent electron transport in the chiral systems themselves. 49,50In 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.

Unknown Mechanism for Spin Injection and Detection by the Ferromagnet.
In past decades, significant effort was dedicated to electrical injection/detection of spinpolarized electrons 51,52 into metals and semiconductors, which involves two different transport mechanisms.The first is spindependent conductivity determined by the polarization of the ferromagnet at the Fermi energy, which is the mechanism of giant magnetoresistance (GMR). 17The second is spindependent tunneling proportional to the density of states at the Fermi energy commonly used in tunnel magnetoresistance (TMR). 18However, for most chiral systems, it is unclear which of these mechanisms could be responsible for the assumed spin injection/detection from the ferromagnet.
3.2.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, known as the conductivity mismatch. 51This 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, how the conductivity mismatch between the ferromagnetic metal and the chiral system is overcome is especially relevant since most of the chiral systems studied are molecules with comparatively low conductivity, making the mismatch problem more significant.

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. 27,32A conventional TMR spin valve consist of two ferromagnetic electrodes and operates based on spin-dependent tunneling for which the Juliere model connects the spin polarization of the density of states of the ferromagnets, P FM , to the MR. 48Using this model, it can be shown that, where I ↑ ↓ is the measured current with magnetization direction indicated by the subscript and P Chiral is 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 48 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 Juliere model.Note also that due to the comparatively low conductivity of the typical chiral (molecular) systems, the spin relaxation length is expected to be short, thus suppressing spin-dependent transport processes.

Bias Voltage
Dependence of the Magnetoresistance.Theoretical effort is dedicated to explaining 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.−13,15,23,49,50,53−56 This is in stark contrast with the experimental results, which show, in many reports, an (almost) bias-independent MR, 7,27,30,32,36,37 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 the expected symmetry, which should be "odd" in bias, meaning a change in sign of the MR when the bias is reversed; 9,23 see Appendix C.
3.5.Failure of the Onsager Reciprocity Relations.In the 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, 57 because these are different concepts.Figure 1(a) and (b) shows the spin-dependent transmission model developed by Yang et al. 8−10 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 Figure 4 in Appendix B. Only in the nonlinear regime can an MR be detected, which requires a combination of energydependent transport and energy relaxation (e.g., possible electron−electron and electron−phonon interactions 12,55,56,58 ); see Appendix C.
However, several experimental results for a wide range of chiral systems 7,31,32,36,37,59 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 mechanism for the origin of the MR in terms of a chirality-dependent modification of the contact potential by magnetization, not involving spin transport.

MAGNETOCHIRAL CONTACT POTENTIAL MODIFICATION
−21 It was found that the contact potential of chiral molecules on a ferromagnet is modified either by reversal of 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 mV when the direction of the magnetization is reversed, which is similar in magnitude to the values previously reported by KPFM measurements 20 and photoemission experiments. 21We note that no electron transport is involved in these measurements of the contact potential shifts.The large experimental time scales, in our opinion, exclude transient effects, justifying that this is an equilibrium phenomenon.The magnitude of the contact potential shift of ∼50 mV is large compared to k B T, Figure 1.(a) Elementary spin-dependent transmission and reflection model for a chiral system.The "favored" electron spin is transmitted, and the "unfavored" 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) 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 Δ ψ ⇆ that is built up over a distance d.This then modifies the electrostatic potential profile of the chiral molecules.
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.An electrostatic contact potential modification by reversal of the magnetization direction or by changing the chirality of the system is not expected based on time reversal symmetry and parity inversion considerations. 60−21 Another recent experiment by Volpi et al. 61 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 mechanism 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 in the electron affinity of the molecules at the interface with the ferromagnet.This strongly influences the electrostatic potential landscape of the junction between a ferromagnet and 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 the metallic ferromagnet and the chiral molecular system, a Schottky-type barrier, where the HOMO/LUMO bands bend, is expected.In Figure 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 Δψ ⇆ , over a distance given by the screening length λ.
To check the feasibility of this mechanism, we calculate the areal electron density, N, that is required to build up a potential difference of Δψ ⇆ = 50 mV 19 over a typical length scale of a molecule d = 1 nm, using a parallel plate capacitor approximation, where ϵ s = ϵ r ϵ 0 is the permittivity and e is the electron charge.With ϵ r = 5, we find that N = 10 −2 nm −2 , which corresponds to 2.5 × 10 −3 electron charge per molecule for a molecular packing density of 4 nm −2 .This indicates that only a small fraction of the charge shift per single molecule is required to build up a potential of 50 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 λ, which depends on the shift of contact potential.We estimate the length scale λ with the Poisson− Boltzmann equation, where ψ(z) is the potential as a function of the distance z and ρ(z) is the local electric charge density.With the boundary conditions ψ(0) = Δψ ⇆ and ψ(λ) = 0, the solution to eq 3 yields Here, N D is the doping concentration in the chiral system and λ is the screening length, given by In Figure 2(a) we plot ψ(x) for the case where Δ ψ ⇆ > k B T with a contact potential modification of 50 mV at N D = 10 15 cm −3 to illustrate the length scales on which the electrostatic potential is modified.The length λ is dependent on N D , as seen in the inset graph in Figure 2(a).From the graph of ψ(z), it can be seen that the length scale of the potential modification is ∼150 nm, making it comparable to if not greater than the molecular thin film thickness, which is typically less than 20 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 a linear response.
When the ferromagnet-chiral molecule junction is under bias, the contact Fermi energy is shifted by eV.Three different mechanisms can carry the transport; see Figure 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 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 nonmagnetic, as is illustrated in Figure 1(c).In the other configuration, illustrated in Figure 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 nonferromagnetic.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 mV for a 10 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 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 the chiral film, and therefore this electrostatic effect can dominate the electronic (but not the spin) transport.
With a tunnel barrier between the chiral molecular and the ferromagnet, the modification of the contact potential adjusts the potential landscape as is illustrated in Figure 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 Figure 3(a)) corresponds to a decrease in the average tunnel barrier height (dashed line in Figure 3(b)).Some results from local probe and tunnel junction experiments give an opposite sign of the MR for the same chiral system and seem to support this. 26,33However, 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.

CONCLUSIONS AND OUTLOOK
We have provided an alternative mechanism 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 the magnetization direction.We argue that as a result, the transport barrier between the chiral molecules and the ferromagnet is modified by the magnetization direction 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 in which 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) 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.
already 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 a (large) MR in the linear response regime.
With this simplified model, we propose an alternative mechanism for the MR of a 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.Also we have not made any statements regarding possible effects related to spin transport or transfer in chiral systems that could be relevant for out of equilibrium phenomena in the nonlinear regime such as, for example, circularly polarized electroluminescence (CP-EL) in chiral light-emitting diodes. 62 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 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 from the ferromagnet.
The second remaining question concerns the biasindependent nature of the MR at biases even up to several volts.For a larger bias, both the Schottky barrier type the tunnel barrier potential profiles will be adjusted by the biasinduced 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. 63A 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 64,65 (with different magnetization directions) as well as in field-effect transistor devices with a chiral channel 61 or other experiments 66 involving electrostatic gating of a chiral system.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 conclude, 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 by 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 Figure 3(c).In this geometry, any possible modulation of spin transport cannot generate an 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 behavior of chiral systems in nonlocal device geometries where the charge and spin transport can be fully decoupled, for instance, in nonlocal 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 Figure 1(c) and Figure 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 an MR with opposite signs.

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. 14A conservative energy range for the linear response regime is eV ≈ k B T, where k B is the Boltzmann constant and T is the temperature. 67However, 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. 68As 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 the following: where the subscripts ij correspond to the current contacts and nm correspond to the voltage contacts of the sample.In a two terminal connection, this reduces to 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 associated 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 eqs A1 and A2.
The Onsager reciprocity relations are very well established and have been tested in many experiments, 69 including systems with strong spin−orbit interaction, 70 organic single-molecule junctions 71 and when a Buẗtiker (dephasing) probe is included. 23,72e are not aware of any experimental results nor theory that deviates from eqs A1 and A2 except the MR measured in chiral systems coupled to a ferromagnet (see main text).

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 spindependent transmission model developed by Yang et al. 8−10 This model is based on spin-dependent transmission and reflection coefficients as illustrated in Figure 1(a) and (b).The "favored" spin is transmitted, and the "unfavored" 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 an 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 favored spin; see Figure 1(b).
Figure 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 favored 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 Figure 4(a) assumes an ideal spin-dependent transport for both the ferromagnet and the chiral system, the same conclusion holds for 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 Figure 4(b).When the magnetizations of both ferromagnets are reversed, the total resistance is unaffected, and the two terminal Onsager relation of eq A2 is satisfied.
We note that electron transport in chiral crystals has been studied in devices where the current flows along a crystallographic axis, 42−44 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 measurements where voltage and current contacts are interchanged were observed to satisfy eq A1, 42−44 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. 7,31,32,36,37,59Furthermore, they show a difference in the linear response resistance when the magnetization is reversed, giving a nonzero MR, contradicting the Onsager reciprocity relations.We note that, in a number of these results, the bias dependence of the MR shows anomalous behavior around zero bias voltage. 7,27,30,32,33,37We 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 2, 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.−21 We note that the macroscopic time scales that are used in these experiments exclude transient effects and justify that this is a persistent equilibrium phenomenon.Furthermore, there is no charge transport in these experiment, which excludes a current-induced (nonlinear) effect.The reversal of the magnetization does not affect the total charge current, and hence eq A2 does hold.Note that 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. 9(b) This is fundamentally different from two connected ferromagnets and the magnetization of one of them being reversed.

APPENDIX C: LINEAR RESPONSE COMPARED TO NONLINEAR 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 behavior.This changes when the applied bias modifies the properties of the system under study and we enter the nonlinear 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, an MR can be generalized in the nonlinear regime when the magnetization is reversed.Furthermore, the MR is strongly bias-dependent, has to be strictly zero at zero bias and has to 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) 60,73,74 where the resistance of a chiral system has an additional chirality-dependent (D/L) term ΔR that is proportional to the applied current I and the magnetic field B, 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 an MR, which is, in many reports, independent of the applied bias, down to zero bias, making it an even function of the applied bias.

Figure 2 .
Figure 2. (a) Calculated potential ψ(z) for a contact potential change of 50 mV.We use N D = 10 15 cm −3 and ϵ r = 5.The inserted graph shows the screening length λ as a function of the doping concentration on a semilogarithmic 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, and (3) transport within the screening length.

Figure 4 .
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 A2 does hold.Note that 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. 9(b) This is fundamentally different from two connected ferromagnets and the magnetization of one of them being reversed.