# Theory of Chirality Induced Spin Selectivity: Progress and Challenges

Ferdinand Evers<EMAIL_ADDRESS>Institute of
Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany
Amnon Aharony  School of Physics and Astronomy, Tel Aviv University, Tel Aviv
6997801, Israel  Nir Bar-Gill  Department of Applied Physics, Racah Institute
of Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel  Ora
Entin-Wohlman  Raymond and Beverly Sackler School of Physics and Astronomy,
Tel Aviv University, Tel Aviv 6997801, Israel  Per Hedegård  Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark  Oded Hod
Department of Physical Chemistry, School of Chemistry, The Raymond and Beverly
Sackler Faculty of Exact Sciences and The Sackler Center for Computational
Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel
Pavel Jelinek  Nanosurf Lab, Institute of Physics of the Czech Academy of
Sciences, Prague 6, CZ 162 00, Czech Republic  Grzegorz Kamieniarz  Department
of Physics, Adam Mickiewicz University, Poznań 61-614, Poland  Mikhail
Lemeshko  IST Austria (Institute of Science and Technology Austria), Am Campus
1, 3400 Klosterneuburg, Austria  Karen Michaeli  Department of Condensed
Matter Physics, Weizmann Institute of Science, Rehovoth 7610001, Israel
Vladimiro Mujica  School of Molecular Sciences, Arizona State University,
Tempe, AZ 85287-1604, USA  Ron Naaman  Department of Chemical and Biological
Physics, Weizmann Institute of Science, Rehovoth 76100, Israel  Yossi Paltiel
Department of Applied Physics, Racah Institute of Physics, The Hebrew
University of Jerusalem, Jerusalem 9190401, Israel  Sivan Refaely-Abramson
Department of Molecular Chemistry and Materials Science, Weizmann Institute of
Science, Rehovoth 76100, Israel  Oren Tal  Department of Chemical and
Biological Physics, Weizmann Institute of Science, Rehovoth 76100, Israel  Jos
Thijssen  3 Kavli Institute of Nanoscience Delft, Delft University of
Technology, Lorentzweg 1, Delft, 2628 CJ The Netherlands  Michael Thoss
Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104
Freiburg, Germany  Jan M. van Ruitenbeek  Huygens-Kamerlingh Onnes Laboratory,
Leiden University, Niels Bohrweg 2, 2333 CA Leiden, Netherlands  Latha
Venkataraman  Department of Applied Physics and Department of Chemistry,
Columbia University, New York, New York 10027, USA  David H. Waldeck
Department of Chemistry, University of Pittsburgh, Pittsburgh, PA 15260, USA
Binghai Yan  Department of Condensed Matter Physics, Weizmann Institute of
Science, Rehovoth 7610001, Israel  Leeor Kronik<EMAIL_ADDRESS>Department of Molecular Chemistry and Materials Science, Weizmann Institute of
Science, Rehovoth 76100, Israel

###### Abstract

We provide a critical overview of the theory of the chirality-induced spin
selectivity (CISS) effect, i.e., phenomena in which the chirality of molecular
species imparts significant spin selectivity to various electron processes.
Based on discussions in a recently held workshop, and further work published
since, we review the status of CISS effects - in electron transmission,
electron transport, and chemical reactions. For each, we provide a detailed
discussion of the state-of-the-art in theoretical understanding and identify
remaining challenges and research opportunities.

††preprint: APS/123-QED

Chirality-induced spin selectivity (CISS), first discovered some two decades
ago in the context of photoemission [1], is now an umbrella term used to
interpret a wide range of experimental phenomena in which the chirality of
molecular species imparts significant spin selectivity to various electron
processes [2, 3, 4, 5, 6, 7, 8, 9]. The interplay between molecular handedness
and electron spin suggests spin-orbit coupling (SOC) as a natural mechanism
for CISS, given that it is easy to show that “current through a coil”
arguments yield an effect that is lower by many orders of magnitude. Early
theoretical efforts have indeed confirmed that SOC may provide a qualitative
explanation for some aspects of the experimental findings. Quantitatively,
however, such calculations have consistently predicted effects that were
smaller by up to several orders of magnitude than those observed
experimentally. While additional theoretical research efforts, described in
more detail below, have shed more light on CISS, a complete quantitative
theory of the effect remains elusive and its microscopic origins are
insufficiently understood.

The authors of this article have addressed this puzzle in a workshop hosted by
the Weizmann Institute of Science in early 2020. In an attempt to form a new
community with a joint research agenda, the workshop brought together
theoretical and computational physicists and chemists of diverse backgrounds
in different model Hamiltonian and first principles approaches, and with an
interest in various relevant phenomena, including magnetic materials,
strongly-correlated systems, topological materials, quantum transport, and
molecular electronics and spintronics. These theorists were joined by several
experimentalists with an interest in CISS. This article provides an overview
of CISS theory, based on discussions in the workshop and further work
published since. It provides a brief overview of the status of CISS
experiments, followed by a detailed critical discussion of the state-of-the-
art in theoretical understanding of CISS. Finally, it identifies remaining
challenges and research opportunities.

##  I Survey of Experimental CISS Studies

We start our considerations by presenting a brief survey of the status of CISS
experimental work. This survey is not meant to be comprehensive. Rather, it
aims to provide sufficient context for a meaningful analysis of the advantages
and disadvantages of various theoretical approaches to CISS. For additional
aspects of CISS studies, the reader is referred to past review articles, and
references therein [2, 3, 4, 5, 6, 7, 8, 9].

Relations between chirality and magnetic phenomena have a long history.
Magnetically induced optical activity in crystals, as well as natural optical
activity in chiral crystals, have been known since the nineteenth century,
leading Pasteur himself to search, unsuccessfully, for a link between the two
[10]. It was not until 1997, however, that this link was finally found in the
form of magneto-chiral dichroism, i.e., a difference in the magnetic optical
activity of the two enantiomers of a chiral medium [11]. CISS, discovered only
two years after that [1], takes a significant step further by establishing a
direct link between chirality and spin, even in the absence of an external
magnetic field and/or circularly polarized illumination. Generally speaking,
CISS effects that have been observed since the original discovery can be
divided into three broad categories: The first - also historically - involves
transmission of unbound electrons (typically photoexcited from an underlying
substrate) through a chiral medium to vacuum. The second - where most work to
date has been done - involves transport of bound electrons, between leads,
through a chiral medium. The third - and relatively new - category concerns
relations between electron spin and chemical reactions.

###  I.1 CISS in electron transmission

CISS in electron transmission was first observed experimentally by Ray et al.
[1], who considered how electrons, emitted from an Au substrate upon
excitation with circularly polarized light, are transmitted through an
organized adsorbed monolayer comprising chiral molecules. They found that
electrons excited using clockwise (cw) light exhibited a significant asymmetry
in the transmission probability as compared to those excited with counter-
clockwise (ccw) light, with electrons excited using linearly polarized light
exhibiting an intermediate transmission probability [1]. This is demonstrated
here using the later work of Carmeli et al. [12], shown in Fig. 1a. In these
studies, the illumination (cw or ccw) for which transmission through a
polyalanine layer is preferred was found to depend on the handedness of the
peptide (L or D, which exhibit left- and right-handed chirality,
respectively). Based on the well-known connection between the direction (cw or
ccw) of circularly polarized light and the spin polarization (up or down) of
the excited electrons, it was inferred that the molecular chirality induces
spin selectivity in the transmision, i.e., CISS is observed. This conjecture,
while reasonable, took another 12 years to verify. This was finally achieved
by Göhler et al. [13], who used a Mott polarimeter to measure directly the
spin of photoelectrons transmitted from an Au substrate through a monolayer of
double-stranded DNA. As shown in Fig. 1b, significant spin-polarization was
found even when the photoelectrons were generated with linearly polarized
light. Further direct confirmation for the CISS effect in transmission came
from the work of Niño et al. [14], who found significant differences in spin
polarization between two enantiomers of the same molecule
(1,2-diphenyl-1,2-ethanediol).

Figure 1: (a) Energy distribution for photoelectrons transmitted through (i)
L, C-terminus connected, (ii) D, N-terminus connected, and (iii) D, C-terminus
connected helical polyalanine films and excited using a cw (negative spin
polarization; red, solid), ccw circularly (positive spin polarization; blue,
dashed), or linearly (no spin polarization; black, dotted) polarized light.
Taken from Ref. 12, used with permission. (b) Photoelectron polarization,
measured for electrons ejected from a Au-coated substrate with a monolayer of
78-base pair double-stranded DNA, for cw circularly polarized light [(–54.5
$\pm$ 7.0%); top, green], linearly polarized light [(-57.2 $\pm$ 5.9)%;
middle, blue] and ccw circularly polarized light [(-60.8 $\pm$ 5.8%); bottom,
red]. Taken from Ref. 13, used with permission.

Importantly, the essential effect does not depend on the specifics of the
molecule and has been observed, e.g., in DNA of varying lengths, different
oligopeptides, and helicenes (see Ref. 15 and references therein).
Interestingly, heptahelicene, composed of only carbon and hydrogen atoms and
exhibiting only a single helical turn, was found to already show a
longitudinal spin polarization of about 6% to 8% in the transmission of
initially spin-balanced electrons [16]. Note that, as also shown in Fig. 1a
for polyalanine, the direction of spin polarization can be inferred to depend
not only on the handedness but also on the molecular dipole direction, as it
changes sign depending on whether the molecule is bound to the substrate
through the N-terminus (amine group side) or C-terminus (carboxyl group side)
[12].

###  I.2 CISS in electron transport

The above studies suggest that spin selectivity could be observed also for
bound electrons traveling through a chiral medium, i.e., also for electron
transport and not just for electron transmission. Experimental verification of
this conjecture, however, requires proper contacting of the chiral medium -
often an organic molecular layer - to metallic leads, which is not trivial.
This was first accomplished by using a ferromagnetic substrate to inject spin
(in this case to a DNA layer), with the tip of a conductive-probe atomic force
microscope (CP-AFM) serving as the top contact [17]. As chiral media are often
grown on non-magnetic substrates, more generally a magnetic CP-AFM (mCP-AFM)
tip can be used, as shown in Fig. 2a [18]. Many other techniques have been
used to detect spin imbalance due to CISS in electron transport. Two notable
ones are magnetoresistance in a film where at least one electrode is
ferromagnetic [19, 20] and electrochemical measurements with a ferromagnetic
electrode [21].

Several recent observations of CISS in electron transport are directly
relevant for the theoretical discussion that follows. First, a sizable signal
interpreted as being due to CISS effect has been reported in many articles
[2]. The effect has been observed in electron transport through many media,
including different types of chiral molecular layers (including biological
ones) [17, 22, 23, 24], carbon nanotubes [25, 26], and chiral materials [27,
18, 28, 29, 30, 8]. It has even been reported in single-molecule experiments
using a break-junction [31]. Second, it has been repeatedly demonstrated that
the effect generally increases with medium length [32] and that it can be very
large. For example, a spin-selectivity of up to 80% was reported for electrons
traversing ca. 2-6 $\mu$m-long self-assembled superhelical conducting
polyaniline micro-fiber channels at room temperature [24]. Spin polarization
exceeding 85% was achieved using $\pi$-conjugated molecular materials based on
coronene bisimide and tetra-amidated porphyrin cores appended with
alkoxyphenyl groups [27]. Even higher numbers were reported in recent studies
of spin-dependent charge transport through 2D chiral hybrid organic-inorganic
perovskite materials. Using ((R/S-)methylbenzylammonium (MBA) lead-iodide (see
Fig. 2), a highest spin-polarization transport of up to 86% was obtained [18]
and with (R-/S)MBA-tin-iodide the efficiency was as high as 94% [28]. Third,
as in CISS in transmission, also here the sign of the preferred spin depends
on the direction of the molecular dipole [33], in addition to the usual
dependence on the handedness. Fourth, from a mechanistic point of view, two
observations are important: CISS is repeatedly found to correlate with optical
activity [27, 32, 34] and CISS is prominent and clearly detected in the non-
linear current-voltage regime (as also seen in Fig. 2) [35, 7, 36].

We also note that the spin-selectivity is increasingly explored practically
for demonstrating spintronic effects and devices that can be used for logic
and memory [37, 5]. Two recent examples are a spin filter [38] and magnetless
Hall voltage measurements [33, 23].

Finally, it is important to note that a constructive and at the same time
critical debate scrutinizing all measurements is still ongoing. In particular,
CISS is not always observed in transport through chiral media and at present a
broad consensus as to the precise experimental conditions under which a
specific manifestation of CISS is expected does not exist.

Figure 2: (A) Setup for mCP-AFM room-temperature measurements of the chirality
dependence in out-of-plane charge transport through chiral $\sim$50 nm two-
dimensional hybrid perovskite (MBA)2PbI4 thin films deposited on fluorine-
doped tin oxide (FTO) substrates. (B-D) Current-voltage curves for S (left-
handed chiral), achiral, and R (right-handed chiral) films, with the tip
magnetized north (blue), south (red), or not magnetized (black). The curves
for each film were averaged over 100 scans and the shaded region around the
lines marks the 95% confidence limits for the average results. Taken from Ref.
18, used with permission.

###  I.3 CISS in chemical reactions

Beyond electron transmission or transport through a chemically stable medium,
a third category of CISS uses the electron spin as an enantio-selective
chemical reagent [39, 40]. The origins of this idea can be traced back to the
work of Rosenberg et al. [41]. They showed that use of low-energy spin-
polarized secondary electrons, produced by irradiation of a magnetic
substrate, results in different bond cleavage rates for R and S enantiomers of
a chiral molecule adsorbed on the substrate. Based on CISS in transmission,
the magnetic substrate was later replaced by a non-magnetic substrate with a
chiral DNA overlayer acting as a spin filter, with similar consequences for
enantio-selective chemistry [42]. In the same manner, one can use CISS in
transport, typically in an electrochemical setting, to employ spin in order to
promote chemical reactions. For example, it was shown that when
electrochemical water splitting occurs with an anode that accepts
preferentially one spin owing to CISS, the process is enhanced and the
formation of hydrogen peroxide is diminished [43, 44, 45, 46]. More recently,
using Hall voltage measurements it was observed that charge displacement in
chiral molecules (in this case L- and D-oligopeptides) creates transient spin
polarization [47], which in turns imparts an enantio-selective inter-molecular
interaction even without electron injection. This transient spin-polarization
is then also expected to affect properties at a ferromagnet/chiral molecule
interface via spin-exchange interactions. The most dramatic demonstration of
this principle so far was the separation of enantiomers by their interaction
with a magnetic substrate, shown in Fig. 3 [48]. The same idea, with
appropriate experimental modifications, was then used for enantioselective
crystallization of amino acids [49]. We also note that an inverse phenomenon,
namely magnetization reversal in a thin-film ferromagnet solely by
chemisorption of a chiral molecular monolayer, has also been reported [50],
with the effect possibly persisting for extended periods of time (hours) [51].

Figure 3: Adsorption of a polyalanine oligopeptide [shown in inset of (v)] on
ferromagnetic samples (silicon with a 1.8-nm Co film and a 5-nm Au film),
magnetized with the magnetic dipole pointing up (H+) or down (H–) relative to
the substrate surface. SiO2 nanoparticles were attached to the adsorbed
oligopeptides. Panels (i,ii) and (iii,iv) exhibit L-polyalanine and
D-polyalanine, respectively, adsorbed for 2 s on a substrate magnetized up
(i,iii) or down (ii,iv). Panel (v) summarizes the nanoparticle adsorption
densities shown in (i) to (iv), compared with the adsorption density on Au
with an applied external magnetic field (red bars). Double-headed arrows
represent error bars. The errors are the standard deviation among 10
measurements conducted on each of the 10 samples, hence a total of 100
measurements. Taken from Ref. 48, used with permission.

To summarize, this short survey of experimental work shows that CISS is an
important fundamental effect, with many manifestations and various practical
consequences in a number of areas, from spintronic devices to chemical
reactions. There is therefore great merit in theoretical understanding of CISS
origins.

##  II Status of Theoretical CISS studies

Inspired by the early experiments on CISS in transmission, theoretical studies
have focused initially on scattering theory of photoelectrons off helical
potentials. This was followed by a theoretical and computational focus on CISS
in electron transport through helical wires, which constitutes the bulk of
theory reported so far. Theory related to the more recent CISS in chemical
reactions has been much more limited and is only now starting to emerge. We
now provide a critical discussion of these efforts.

###  II.1 Theory of CISS in transmission

Historically, scattering off asymmetric potentials is a theoretical topic
almost as old as quantum mechanics itself. Of special importance has been the
scattering of light, where Ref. 52 is an early example. In particular,
motivated by the need to provide a theoretical underpinning for the detection
of chemical helicity with polarized light, scattering of light by helical
potential shapes has been a longstanding topic, especially in theoretical
chemistry and biophysics [53, 54], but also in the electromagnetic theory of
chiral materials [55]. The scattering of electrons off helical - or chiral -
obstacles appears to have received much less attention but more recently has
been motivated by CISS experiments.

An analytical scattering theory, which includes the effect of SOC, has been
developed for electrons traversing a helical molecule [56] or a self-assembled
monolayer (SAM) thereof [57, 58, 59], or even a chiral molecule that is not
necessarily helical [60]. Qualitatively, SOC is indeed found to induce spin-
polarization. We note that dissipation can also polarize angular momenta. This
can be illustrated within a classical scattering model, where the role of spin
is played by classical angular momentum and SOC is replaced by friction [61]),
leading us to speculate that a similar mechanism could also contribute to
CISS. Unfortunately, for realistic model parameters, in particular for the
SOC, the resulting polarization is too small [62], much smaller in magnitude
than that found in some of the above-discussed experiments.

Figure 4: (a) Sketch of a helical scattering potential (black), indicating the
pitch (p) and the polar scattering angle, $\theta$. (b) Dependence of the
(normalized) scattering cross-section on the angular momentum of the incident
particle, for wires of increasing length $L$. Depending on clockwise or
counter-clockwise entry ($n\pm 1$), the scattering cross-section differs by
two orders of magnitude. Potential parameters have been adjusted to the case
of DNA; impinging energy is 0.5 eV. Taken from Ref. 62, used with permission.

In light of this discrepancy, Gersten et al. have extended the scattering
approach [62] to include the SOC of the substrate that supports the SAM. This
introduces a concept of “induced” spin filtering, which expands the notion of
“current transfer”, developed earlier by Skourtis et al. [63]. The main idea
is that an electron migrating through an obstacle retains a “memory” of its
initial momentum. In the context of spin-filtering, this idea is used for
angular momentum, implying that the helical molecule is “filtering” angular
momentum. This means that spin-filtering occurs if the angular momentum of the
impinging electron was at least partly spin-selected to begin with, as is
indeed the case with a substrate possessing strong spin-orbit coupling - see
Fig. 4. While the authors expressed a hope that their theory could at least
roughly account for the magnitude of the experimental observations [62], there
appears to be no consensus concerning this claim [64]. Experimentally, CISS
has been observed using substrates with negligible SOC.[21, 65, 16] Therefore
substrate SOC may indeed be an important contribution where it exists, but
cannot explain the whole effect.

###  II.2 Theory of CISS in transport

As discussed above, many CISS-related transport phenomena have been reported
experimentally, often through the measurement of spin-resolved current-voltage
(I-V) characteristics of electrons flowing through a chiral medium, as for
example in Fig. 2 above. It is therefore only natural that much of the
theoretical effort has been focused in the same direction.

Before considering any specifics of the these efforts, we consider what basic
aspects distinguish scattering and transport experiments on a conceptual
level. Taking a rough perspective, the passage of electrons from a source to a
drain appears similar for bound and unbound particles: in both cases electrons
migrate through a region with obstacles and therefore conventional scattering
terminology applies in either situation, emphasizing a notion of similarity.
However, significant conceptual differences enter upon considering that: (i)
The passage of bound particles through a thin wire is quasi-one-dimensional,
while scattering of an unbound particle is intrinsically three dimensional;
(ii) The bound particle experiences the properties of the underlying material
much more strongly than the unbound particle. For example, the electron
dispersion relation will, in general, no longer be parabolic and interactions
with other degrees of freedom (e.g., inelastic ”multi-scattering processes”)
tend to be much stronger.

We begin by recalling that general conditions under which a two-terminal
device can be expected to exhibit spin filtering, even in principle, have been
worked out in the field of spintronics. This analysis includes CISS as a
special case thereof and therefore provides an important framework for our
discussion.

We emphasize two basic facts arising from it: (i) It is well known that in
single channel wires SOC can be removed by a gauge transformation [66].
Mathematically, this is because SOC can be written as an SU(2) gauge field.
Intuitively, this means that due to the absence of loops (and magnetic fields)
the spin rotates in one-to-one correspondence with a position-space shift.
Because the spin degree of freedom can be gauged out, a spin filtering
functionality based solely on SOC is not expected [66]. A particular
consequence of this is that spin-filtering owing to SOC is not expected to
arise in tight-binding models of single-stranded DNA that afford only a single
orbital per site. Therefore, Refs. 67, 68, 69 have emphasized the importance
of two channels for the observation of CISS.

(ii) Even in two-terminal molecular junctions supporting several channels,
spin-selectivity is still suppressed in the linear regime because of time-
reversal symmetry, as emphasized in Ref. 70 (and see Refs. 71, 72 for
additional discussion). The proof of this claim uses an Onsager-type argument:
Consider a two-terminal device with a non-magnetic source. Let the drain
exhibit magnetism, ${\mathbf{M}}$, so that it can act as a spin-analyzer, but
there is no magnetic field otherwise. The argument proceeds via reductio ad
absurdum: Assume that the device could act as a spin-filter. Then the
conductance would be sensitive to the direction of ${\mathbf{M}}$ in the
analyzer; in particular $G({\mathbf{M}})\neq G(-{\mathbf{M}})$. However, an
Onsager relation protected by time-reversal invariance implies
$G({\mathbf{M}})=G(-{\mathbf{M}})$, leading to a contradiction and thereby
proving the claim. We also note that a special case of both of the above-
statements, for non-interacting single-channel wires in the linear regime, is
known as the ”single-channel no-go-theorem” [73, 74].

Below we survey many studies that have reported chirality-selective spin
transport in models of single-channel wires. In many (but not all [67]) cases,
violation of these restrictions is reported. A detailed analysis of each
individual theoretical/computational model is beyond the scope of this
overview. However, it is of utmost importance to assess model predictions
against the above general restrictions [70, 75]. In some cases, violations may
be rationalized in terms of the computed quantities or the fundamental model
assumptions (some examples are given below), while in others this may reveal
mistakes in the analysis.

A considerable number of studies based their analysis on transmission
calculations for tight-binding models. [67, 76, 77, 68, 78, 79, 80, 81, 82] In
an early study, Gutierrez et al. [77] considered numerically a single-channel
tight-binding model with nearest neighbor hopping and SOC. They reported a
very large spin polarization near the band edges, reaching up to 100%. The
degree of spin polarization obviously depends on the tight-binding model
parameters. The authors motivated their choice based on DNA, with the hopping
parameter reported to be in the range of 20–40 meV and with chirality entering
the model indirectly via its feedback into the SOC. To determine the latter, a
heuristic argument was exploited, which yields for light atoms (C, B, N, O)
typical (about 2 meV) values for coupling strength. Indeed, scales of meV can
be reached with light elements, e.g., when promoting a carbon atom in graphene
from sp2\- to sp3-hybridization [83]. However, as compared to this promotion,
the chirality-induced symmetry breaking should be weaker by a geometric factor
that incorporates the helical parameters of pitch and diameter.

Clearly, further insight into a quantitative estimate of the effect can come
from first principles studies that do not utilize model parameters. However,
these are challenging because full helicity usually implies large molecules
that need to be treated at a level of theory that includes proper relativistic
corrections to the electronic structure.

Facing these difficulties, Maslyuk et al. [84] have attempted a first
principles transport calculation for a DNA molecule. In their calculations,
the SOC has been implemented employing pseudopotentials and the zero order
regular approximation (ZORA) [85]. Spin-filtering of an $\alpha-$helix was
compared to that of a $\beta-$strand, the latter corresponding to an enforced
linear geometry. The authors found that spin-filtering was stronger in the
helical conformation, as compared to the linear one, which is expected due to
the symmetry reduction. Quantitatively, however, the observed polarization was
an order of magnitude below the experimental reports. This observation has
been shared by Rebergen and Thijssen [86]: At a qualitative level, the
existence of CISS is confirmed, but quantitatively the native SOC in the
molecules considered appears to be too small, possibly by an order of
magnitude, in order to account for the experimental results. Thus, an
additional and important challenge is the same quantitative issue faced by the
theory of CISS in transmission. In light of this difficulty, and inspired by
the early work of Gersten et al. [62], Liu et al. [87] proposed an orbital-
polarization model with SOC from the electrode to interpret CISS in transport.
In their model, by going through the chiral molecule electrons become orbital-
polarized (an effect also obtained in model calculations [88]) and the orbital
polarization is converted to spin polarization by the SOC in the electrodes.
This leads, in the non-linear regime, to unidirectional magnetoresistance,
rationalizing CISS to some extent.

Quantitative issues notwithstanding, the vexing problem of the microscopic
origins of CISS-type phenomena in transport has motivated many authors to
investigate situations that definitively avoid the Onsager-based no-go
theorem. In a recent example, Utsumi et al. calculated time-reversal symmetric
charge and spin transport through a molecule comprising two-orbital channels
and connected to two leads. They demonstrated that spin-resolved currents are
generated when spin-flip processes are accompanied by a flip of the orbital
channels.[89]

Guo et al. [67, 78] (and in a different context independently also 87) and 90
have suggested a “symmetry workaround”, further explored in Refs. 69, 64. This
involves a third bath that the electrons traversing the chiral molecule may
couple to. The bath gives rise, in general, to non-unitary effects such as
‘dephasing’ or ‘leakage’, such that time-reversal symmetry is effectively
broken and Onsager’s theorem no longer applies and spin polarization ensues.
An example is shown in Fig. 5. Technically, this effect has been modeled in
these studies by introducing an anti-hermitian self-energy
${\mathfrak{i}}\Gamma_{\text{d}}$. This was found to bring about spin-
selective transport in the presence of SOC. An intuitive understanding of this
finding can proceed from the observation that, in the presence of SOC and
leakage, evanescent waves associated with opposite spins have different decay
lengths [69].

Figure 5: (a) Tight binding model of a single helical molecule with radius
$R$, pitch $h$, and twist angle $\varphi$. Electrons can hop between adjacent
sites along the helix with hopping amplitude $J$ or vertically to the
$N^{\mathrm{th}}$ neighbor with hopping amplitude $\tilde{J}$. Spin-orbit
interaction is assumed to act only between nearest neighbor sites. (i)
Schematic view of the helical molecule. (ii) Mapping of the model onto a one-
dimensional chain of $M$ unit cells, each containing $N$ sites. (b) Spin
polarization (solid blue) in a chain of $M$=6 unit cells (other parameters are
$N$=2, $J$=1.5$J_{0}$, $\tilde{J}$=0.6$J_{0}$, with a lateral component
$J_{x}$=0.2$J_{0}$, using a complex self-energy. (i) as a function of energy
(in units of $J_{0}$), with $\theta$, an indicator of SOC strength, given by
0.4$\pi$. (ii) as a function of $\theta$, with $E$=0 (center of the band). The
spin polarization vanishes (dashed magenta) if either $J_{x}$=0 (unitary
chain), $\tilde{J}$=0 (nearest neighbor chain), or $\theta$= 0 (no SOC).
Inset: Reflection (black) and transmission (green) coefficients for spin up
(solid) and spin down (dashed). The red line is the sum of these coefficients.
Taken from Ref. 69, used with permission.

Quantitatively, this approach typically uses [67, 76, 78, 79, 91] model
parameters similar to those of Ref. 77, so that a quantitative uncertainty
carries over. Furthermore, the overall magnitude of the filtering effect thus
brought about is very sensitive to the leakage rate $\Gamma_{\text{d}}$. While
the very existence of this rate is physically well motivated, its magnitude is
still difficult to establish in realistic terms. The rates employed in the
simulations to achieve spin-polarization in the experimentally reported regime
are usually not small, typically a few percent of the hopping integral. Since
the resulting leakage is very significant, the overall effect thus achieved
still appears to be too small to explain the main experimental features, and
one would further expect significant resistance effects.

A different approach to the idea of a bath has been proposed recently by
Volosniev et al., who proposed that the CISS-effect is a many-body phenomenon
that arises from a bath that manifests in the carrier dynamics as friction
[92]. To illustrate their idea, they adopted a qualitative single-channel
model, in which friction enters as an effective electric field that is
proportional to the product of friction constant and momentum expectation
value. The latter is non-vanishing in the presence of spin-orbit interaction
and points in opposite directions for different spin orientations. By
construction, the model produces a quasi-stationary state with spins pointing
at opposite directions for a wire of finite length. While the main idea is
transparent, the relation to the CISS-effect remains uncertain for two
reasons: (a) The microscopic source of the friction, and therefore the
relevance of the qualitative model, remains an open issue. (b) Spin-separation
is brought about by a friction-controlled dynamical process, which requires
the supply of a sufficient amount of energy, the source of which remains
unspecified.

A natural candidate for a physical bath that participates in the carrier
dynamics are the atomic nuclei [93, 94, 95]. As recently pointed out [96, 97],
the effect of the nuclei on electronic spin separation can be enhanced due to
conical intersections, so that the effect could indeed contribute to the CISS-
phenomenon.

Exploring a different approach, Dalum and Hedegård [98] considered situations
in which the source feeds a current into the device, i.e. the helical
molecule, with an occupation of incoming scattering states that is out of
equilibrium. In fact, such a situation arises very naturally when photo-
electrons traverse a helical SAM; it is, however, more difficult to motivate
in the context of conventional conductance experiments. Dalum and Hedegard
point out, in addition, that even after a workaround has been implemented, one
still faces the problem that SOC is small as compared to all other native
energy scales, so a sizable polarization is not expected. To overcome this
difficulty, the authors invoke degeneracies, the consequences of which they
study by adopting helical polyacetylene as a paradigm system.

An altogether different approach towards understanding CISS-type effects has
been undertaken by Yang et al. [70]. They take the existence of CISS-effects
for granted and construct, based on this assumption, phenomenological models
that describe typical CISS measurements. The model is formulated in terms of
different versions of transfer matrices that represent different elements of
the measurement circuit, such as magnetic and non-magnetic barriers, the
helical molecule, etc. Due to the model simplicity, analytical calculations
are feasible. By construction, Onsager’s reciprocity theorems are satisfied by
the model. Therefore, an explicit calculation of the two-terminal conductance
yields the expected negative result, i.e., no spin-filtering in a linear-
response two-terminal calculation. However, a positive result is found in
multi-terminal calculations.

A puzzling aspect of Ref. 70 is that on the one hand the existence of CISS is
deduced from experiments, which have been performed in two-point geometries,
while on the other hand it is found that two-point conductance measurements
will not show the CISS effect. Routes escaping this dilemma are proposed in
Ref. 36. Specifically, two natural routes are discussed : (i) The reciprocity
theorem makes no statement about non-linear effects; therefore, traces of CISS
can exist, and have been identified in Ref. 36, also in two-terminal
measurements if they are operated in the non-linear regime of bias voltages.
(ii) Also, the reciprocity theorem does apply if time reversal invariance is
broken. Hence, one might expect that aligning two helical molecules in series
with a small resistor in between will yield different results, depending on
whether the molecules have the same or opposite helicities. Along this idea
another set of experiments has been proposed in Ref. 36, which could show CISS
in a two-terminal setup, here even in the linear regime.

Finally, we mention a conceptually interesting field-theoretical approach
towards CISS that has recently been put forward [99]. It considers the Dirac
equation in a one-dimensional curved space-time. In this framework, the usual
Foldy-Wouthousen transformation can be used in order to project into the non-
relativistic (low kinetic energy) sector. The authors show that as a result of
curvature, the kinetic energy acquires an SU(2)-gauge field, which plays a
role analogous to the SOC in single-channel wires. Embarking on this
observation, the authors apply their theory to transport in helical molecules,
where the (quasi-)one-dimensional molecule is interpreted as a physical
realization of a curved space-time for the traversing electron. The basic idea
of this application is conceptually appealing, but the space-time considered
in Ref. 99 carries only a single parameter, the curvature $\kappa$ of the
helical path. As a consequence, the theory does not describe the three-
dimensional nature of experimental observations, which manifests as an
emergence of molecular properties when gradually adding atom by atom.
Furthermore, the results depend on the order of limits (first dimensional
reduction, then non-relativistic limit), a point also emphasized by Geyer et
al. [100].

###  II.3 Theory of CISS in chemical reactions

As explained above, CISS in chemical reactions is a very new field, even
experimentally. Accordingly, theory is still limited. First principles
calculations have repeatedly shown that if one assumes that chirality indeed
begets spin polarization, then the latter can explain chemical enantio-
selectivity [47, 48]. Recent first principles calculations for a chiral
monolayer on a magnetic substrate provided first indications of an emergent
electronic structure and emphasized the role of exchange interactions [101].
However, the degree to which the emergent structure is affected by the choice
of density functional [33] was not investigated. In addition, chiral symmetry
breaking was enforced “manually” via introduction of an external Ni atom, so
that the effect of the intrinsic chirality remains to be clarified. To the
best of our knowledge, a more complete theoretical framework that describes
how CISS emerges in such scenarios has yet to be provided.

##  III Conclusions and Outlook

In this overview, we have surveyed the three main types of CISS - in
transmission, transport, and chemical reactions. For each, we critically
overviewed existing theoretical approaches, while emphasizing advantages and
disadvantages with respect to qualitative and quantitative agreement with
known experimental results.

At present, a unifying scheme that would allow one to interpret all
experiments in terms of only a single microscopic effect – the “CISS effect” -
has not yet been identified. While such a framework cannot be ruled out,
chirality-induced spin selectivity may perhaps be thought of as a set of
phenomena that have a unifying scheme only in the sense that they all derive
from the interplay of spin-orbit interaction and chirality. For example, it
has been suggested theoretically that spin-orbit interaction leads to non-
conservation of spin currents in a two-terminal junction and consequently to a
mechanical torque [102], which is a different experimental observable than the
ones surveyed above. Such a CISS-induced torque has indeed been suggested as
an explanation for a recent experiment demonstrating the use of a chiral
molecule as a molecular motor [103].

Correspondingly, a large gap remains between the experimental observations and
the quantitative estimates from theory. Further experiments are required for
guiding the theory and for limiting the possible interpretations for this
potentially very important phenomenon. Further exploration of recent
suggestions of finite temperature effects, which go beyond pure electronic
ones, is also of interest.

Finally, theoretical studies have focused on steady-state transport, but very
little theory addresses the growing number of experiments that report
transient CISS phenomena (e.g. [47]). Likewise, the hypothesized role that
such transient phenomena may play in CISS in chemical reactions [48] has not
been sufficiently explored yet.

We believe that consolidating the field, in the sense of bringing theory and
experiment much closer, with the goal of achieving a detailed microscopic
understanding of CISS, is an ongoing challenge that provides many research
opportunities. To this end, one can perhaps concentrate on model systems,
where CISS can be studied in great detail in both experiment and theory.

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