Spin polarization induced by decoherence in a tunneling one-dimensional Rashba
model

S. Varela1, M. Peralta1, V. Mujica2, B. Berche3, E. Medina4*

1 Institute of Materials Science and Max Bergmann Center of Biomaterials,
Dresden University of Technology, 01062 Dresden, Germany

2 School of Molecular Sciences, Arizona State University, Tempe, AZ 85281,
United States

3 Laboratoire de Physique et Chimie Théoriques, Université de Lorraine, Nancy,
France

4 Departamento de Física, Colegio de Ciencias e Ingeniería, Universidad San
Francisco de Quito, Diego de Robles y Vía Interoceánica, Quito, 170901,
Ecuador

<EMAIL_ADDRESS>
August 27, 2024

## Abstract

Basic questions on the nature of spin polarization in two terminal systems and
the way in which decoherence breaks Time-Reversal Symmetry (TRS) are analyzed.
We exactly solve several one-dimensional models of tunneling electrons and
show the interplay of spin precession and decay of the wavefunction in either
a $U(1)$ magnetic field or an effective Spin-Orbit (SO) magnetic field. Spin
polarization is clearly identified as the emergence of a spin component
parallel to either magnetic field. We show that Onsager’s reciprocity is
fulfilled when time reversal symmetry is present and no spin polarization
arises, no matter the barrier parameters or the SO strength. Introducing a
Büttiker’s decoherence probe, that preserves unitarity of time evolution, we
show that breaking of TRS results in a strong spin polarization for realistic
SO, and barrier strengths. We discuss the significance of these results as a
very general scenario for the onset of the Chiral-Induced Spin Selectivity
effect (CISS).

###### Contents

  1. 1 Introduction
  2. 2 Barrier model with a magnetic field
    1. 2.1 Spectrum, eigenfunctions, and wavevectors
    2. 2.2 Spin precession under the barrier with magnetic field
  3. 3 Barrier model with a Rashba term
    1. 3.1  Spectrum, eigenfunctions, and wavevectors
    2. 3.2 Spin precession under the barrier for Rashba
  4. 4 Other helicity Hamiltonians
  5. 5 Decoherence with Büttiker’s probe
  6. 6 Summary and Discussion
  7. A Coupling Büttiker’s Probe

##  1 Introduction

The Spin-Orbit (SO) coupling is many times neglected in electron transport
because of the energy scale of the coupling, meV for C, N, O in chiral
molecules and e.g. Si, Ga, and Ge semiconductors in the bulk. Although the
source of SO coupling in many technologically relevant materials is atomic,
how this coupling translates to transport depends on the geometry of the
connection between spin active atoms. This way in flat graphene, while atomic
SO coupling is meV, the effective transport SO coupling is $\mu$eV. In a
tight-binding model the SO coupling cancels from the nearest neighbour
contribution due to interference effects and it is only when the second
neighbours coupling is introduced one gets a meV interaction. On the other
hand, bending the graphene sheet and producing e.g. nanotubes, increases the
SO coupling by three orders of magnitude, as it becomes a first neighbours
interaction[1]. The same enhancement is seen in silicene which has a
corrugated surface structure that breaks the orthogonality of $\pi$ orbitals
and the $\sigma$ structure in graphene[2].

In the case of electron transport in molecules, the SO coupling has been
largely disregarded but has come into light due to large spin activity
reported as Chiral-Induced Spin Selectivity (CISS) effect[3]. CISS effect is
observed for both point chiral[4] in amino acids, helical chirality such as
DNA[5], and helicene[6, 7]. Biological molecules generally combine both as in
e.g. oligopeptides[8, 9].

There exists an enormous gap in understanding between the size of the SO
coupling in molecules, in the meV range, and the magnitude of the spin
polarization effect in CISS effect experiments, with spin-polarized
transmissions above 40%[5]. This percentage exceeds the polarization strength
produced by transmission through ferromagnets. The SO coupling is almost
universally regarded as the spin active ingredient in CISS effect and
theoretical estimates have yielded the correct qualitative behavior i.e.
helicity states with the propagation axis as the quantization axis of the
electron spin. On the other hand, the prediction for the magnitude of the spin
polarization is at least ten times smaller, when correct atomic SO coupling
strengths are contemplated [10, 11, 12, 13].

An important issue on the symmetries involved in electron transmission with
two terminals with SO coupling was pointed out by Yang, van der Wal and van
Wees [14, 15]. As was clearly argued, Onsager’s reciprocity precludes the
possibility of spin polarization in the two terminal setting in the linear
regime, in contrast with the results of many works in the literature, both
experimentally and numerically. It is then important to observe how symmetry
arguments play out in specific calculations as reference results[14, 16].
Symmetry arguments alone cannot say how sensitive these results will be in the
face of weak symmetry-breaking perturbations, in this case, of Time-Reversal
Symmetry (TRS).

In this work we will discuss the simplest transmission model through a SO
active barrier, as tunneling is a very common electron transfer mechanism in
large molecules [17]. We will explore the possibilities of spin-polarized
electron transmission in a one-dimensional two-probe setting for an exactly
solved model. The action of a $U(1)$ field on tunneling electrons[18] will be
contrasted with the effective momentum-dependent magnetic field arising from
the SO coupling. A very important issue in the latter case is the velocity
operator’s non-diagonal nature that secures the flux’s continuity through
boundaries[19]. The strong result is that while a $U(1)$ magnetic field
polarizes spin along its direction, under the action of the barrier, no spin
polarization results under the action of a spin-orbit magnetic field.
Following, we solve the model for the spin-orbit magnetic field under the
effects of weak TRS breaking decoherence effects, enacted through Büttiker’s
probe. Large spin polarization highlights a high sensitivity to TRS breaking
with realistic SO couplings. These findings address the core issue of CISS
effect.

To clarify the nature of the contribution in this work: The first three
sections are qualitative, as the parameters are not chosen to fit the actual
physical ones for chiral molecules. In section 3, independently of the
parameters chosen, the null results for polarization are exact results
verifying the symmetry requirements of the Onsager relations. A few previous
numerical computations resulted in transport spin filtering associated with
non-hermiticities that crept into the calculations. Although we refer to
previous papers with the correct conclusion for the polarization of the Rashba
case, these contributions have shortcomings that are overcome here realizing
the correct boundary conditions.

The meaning of polarization in terms of an asymmetric treatment of opposite
spin orientations is also clarified in this paper, along with an exact
solution to the problem (with time-reversal symmetry in place), in light of
Buttiker’s magnetic field case that breaks time reversal. Finally, in section
5, on Buttiker’s probe, the parameters are fitted to a polaron model of
transport. Nevertheless, the coupling to the reservoir is not parameterized
quantitatively and we only show qualitatively the sensitivity of the
polarization to decoherence. We also note that in more realistic situation,
the Rashba coupling can be a varying function of the coordinate[20, 21], since
it can be modulated by local geometrical configurations[22], but we believe
that our findings where this is neglected retain the essential of the physics
at work..

##  2 Barrier model with a magnetic field

###  2.1 Spectrum, eigenfunctions, and wavevectors

We will first fully solve analytically for the emblematic problem of a
magnetic field under a barrier for spinful particles[18]. The correct solution
to this problem allowed for properly addressing the tunneling time problem.
Büttiker realised that spin precession in the field is modulated by the spin-
dependent decay of the wavefunctions under the barrier, generating
polarization in the direction of the magnetic field. The Hamiltonian in this
case is given by

$\cal{H}=\begin{cases}(\frac{p_{x}^{2}}{2m}+V_{0})\mathds{1}_{\sigma}-\Gamma\sigma_{z},&\text{if\leavevmode\nobreak\
$0<x<a$}\\\
(\frac{p_{x}^{2}}{2m})\mathds{1}_{\sigma},&\text{otherwise},\end{cases}$ (1)

where $\mathds{1}_{\sigma}$ is the unit matrix in spin space and $\sigma_{i}$
are the Pauli spin matrices, with $i=x,y,z$. $\Gamma=\hbar\omega_{L}/2$ where
$\omega_{L}$ is the Larmor frequency, and $V_{0}$ is the barrier height.
$\cal{H}$ acts on the spinors $\psi=\left(\psi_{+}(x)\leavevmode\nobreak\
\psi_{-}(x)\right)$ where $|\psi_{\pm}|^{2}dx$ is the probability of find a
particle between $x$ and $x+dx$ with spin $\pm\hbar/2$. The Hamiltonian inside
the barrier has the dispersion relation depicted in Fig. 1. The choice of
coordinates is slightly different from that of Büttiker so we can discuss all
one-dimensional models with the same notation.

The incoming wavefunction we choose to be

$\psi=\frac{1}{\sqrt{1+|s|^{2}}}\begin{pmatrix}1\\\ {is}\end{pmatrix}.$ (2)

Figure 1: The dispersion relation for the barrier with a magnetic field. The
figure depicts the degenerate $\kappa_{\sigma}^{\lambda}$ vectors that occur
in the barrier range.

The values of $s=\pm 1$ correspond to the two eigenfunctions of the
$\sigma_{y}$ matrix, and $s=\pm i$ correspond to the two eigenfunctions of the
$\sigma_{x}$ matrix, appropriately normalized. The eigenvalues of the
Hamiltonian are

$E=\frac{p_{x}^{2}}{2m}-\sigma\Gamma+V_{0},$ (3)

where $\sigma=\pm 1$ is the spin degree of freedom. Using
$E=\hbar^{2}k^{2}/2m$, we define the wavevector outside the barrier $k$. This
Hamiltonian is not time-reversal invariant since inverting time flips $p_{x}$
and $\sigma$, and these flips change the energy. From the eigenvalue equation,
one can then distinguish between the different wavevectors under the barrier

$\kappa_{\sigma}^{\lambda}=\lambda\left(k^{2}-k_{0}^{2}+\sigma
k_{B}^{2}\right)^{1/2},$ (4)

where $k_{B}^{2}=2m\Gamma/\hbar^{2}$ and $k_{0}^{2}=2mV_{0}/\hbar^{2}$. As can
be seen, $\kappa_{\sigma}^{\lambda}$ can only be real or imaginary. Thus we
have either exponentially decaying solutions for $k^{2}<k_{0}^{2}-\sigma
k_{B}^{2}$ or plane waves otherwise. As the Hamiltonian commutes with
$\sigma_{z}$ in the barrier region we can superpose eigenfunctions of
$\sigma_{z}$ as

$\displaystyle\psi_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+|s|^{2}}}\begin{pmatrix}1\\\
is\end{pmatrix}e^{ikx}+\begin{pmatrix}A_{+}\\\ A_{-}\end{pmatrix}e^{-ikx},$
(5) $\displaystyle\psi_{2}$ $\displaystyle=$
$\displaystyle\epsilon\begin{pmatrix}1\\\
0\end{pmatrix}e^{i\kappa_{+}^{+}x}+\zeta\begin{pmatrix}0\\\
1\end{pmatrix}e^{i\kappa_{-}^{+}x}+\eta\begin{pmatrix}1\\\
0\end{pmatrix}e^{i\kappa_{+}^{-}x}+\theta\begin{pmatrix}0\\\
1\end{pmatrix}e^{i\kappa_{-}^{-}x},$ (6) $\displaystyle\psi_{3}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}D_{+}\\\
D_{-}\end{pmatrix}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(7)

The boundary conditions are

$\displaystyle\psi_{i}(x_{b})$ $\displaystyle=$
$\displaystyle\psi_{i+1}(x_{b}),$ (8)
$\displaystyle{\hat{v}}_{x}\psi_{i}(x_{b})$ $\displaystyle=$
$\displaystyle{\hat{v}}_{x}\psi_{i+1}(x_{b}),$ (9)

where ${\hat{v}}_{x}=(\partial{\cal H}/\partial p_{x})$ is the velocity
operator and $x_{b}$ the boundary between the different space regions. We
match the wavefunction and the amplitude flux. This latter boundary condition
is very important to realize and has often been confused in the literature.
Any dependence of the mass on position (effective mass) should be carefully
considered to yield the appropriate hermitian velocity operator[23]. If the
mass is a constant, then the boundary conditions amount to matching the
wavefunctions and the first derivatives thereof. Although this problem can be
separated into two spinless tunneling problems with different barrier
heights[18], we have decided to phrase somewhat more cumbersomely to make a
few important points when considering spin-orbit coupling.

The system of equations above can be solved to yield

$\displaystyle
D_{+}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{t_{+}}{\sqrt{1+|s|^{2}}}}$
$\displaystyle=$
$\displaystyle\frac{2k({\kappa_{+}^{-}}-{\kappa_{+}^{+}})e^{-ia(-{\kappa_{+}^{-}}-{\kappa_{+}^{+}}+k)}}{\sqrt{|s|^{2}+1}\left(e^{ia{\kappa_{+}^{-}}}({\kappa_{+}^{-}}-k)({\kappa_{+}^{+}}+k)-e^{ia{\kappa_{+}^{+}}}({\kappa_{+}^{-}}+k)({\kappa_{+}^{+}}-k)\right)},$
$\displaystyle
D_{-}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{ist_{-}}{\sqrt{1+|s|^{2}}}}$
$\displaystyle=$
$\displaystyle\frac{2iks({\kappa_{-}^{-}}-{\kappa_{-}^{+}})e^{-ia(-{\kappa_{-}^{-}}-{\kappa_{-}^{+}}+k)}}{\sqrt{|s|^{2}+1}\left(e^{ia{\kappa_{-}^{-}}}({\kappa_{-}^{-}}-k)({\kappa_{-}^{+}}+k)-e^{ia{\kappa_{-}^{+}}}({\kappa_{-}^{-}}+k)({\kappa_{-}^{+}}-k)\right)},$
$\displaystyle
A_{+}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{r_{+}}{\sqrt{1+|s|^{2}}}}$
$\displaystyle=$
$\displaystyle\frac{({\kappa_{+}^{-}}-k)(k-{\kappa_{+}^{+}})\left(e^{ia{\kappa_{+}^{-}}}-e^{ia{\kappa_{+}^{+}}}\right)}{\sqrt{|s|^{2}+1}\left(e^{ia{\kappa_{+}^{-}}}({\kappa_{+}^{-}}-k)({\kappa_{+}^{+}}+k)-e^{ia{\kappa_{+}^{+}}}({\kappa_{+}^{-}}+k)({\kappa_{+}^{+}}-k)\right)},$
$\displaystyle
A_{-}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{isr_{-}}{\sqrt{1+|s|^{2}}}}$
$\displaystyle=$
$\displaystyle\frac{is({\kappa_{-}^{-}}-k)(k-{\kappa_{-}^{+}})\left(e^{ia{\kappa_{-}^{-}}}-e^{ia{\kappa_{-}^{+}}}\right)}{\sqrt{|s|^{2}+1}\left(e^{ia{\kappa_{-}^{-}}}({\kappa_{-}^{-}}-k)({\kappa_{-}^{+}}+k)-e^{ia{\kappa_{-}^{+}}}({\kappa_{-}^{-}}+k)({\kappa_{-}^{+}}-k)\right)}.$
(10)

where the $t_{\pm}$ and $r_{\pm}$ denote the transmission and reflection
amplitudes. We recall
$\kappa_{\sigma}^{\lambda}=\lambda\sqrt{\left(k^{2}-k_{0}^{2}+\sigma
k_{B}^{2}\right)}$. In the next section, we obtain the behavior of the spin as
a function of the magnetic field strength consistent with tunneling and we see
both regular Larmor precession with $V_{0}=0$, and precession combined with
spin alignment in the field direction when tunneling occurs.

###  2.2 Spin precession under the barrier with magnetic field

One readily verifies the differential decay of the transmission with the
length of the barrier as $T_{+}\sim e^{-2\kappa_{+}^{+}a}$, and $T_{-}\sim
e^{-2\kappa_{-}^{+}a}$ as long as $E<V_{0}\mp\hbar\omega_{L}/2$. The following
relations quantify the polarization of the electron, the transmitted (T) wave
is

$\psi_{T}=\frac{1}{\sqrt{|D_{+}|^{2}+|D_{-}|^{2}}}\begin{pmatrix}D_{+}\\\
D_{-}\end{pmatrix}e^{ikx},$ (11)

and the spin averages are defined by

$\displaystyle\langle s_{z}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\langle\psi_{T}|\sigma_{z}|\psi_{T}\rangle=\frac{\hbar}{2}\frac{|D_{+}|^{2}-|D_{-}|^{2}}{|D_{+}|^{2}+|D_{-}|^{2}},$
$\displaystyle\langle s_{y}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\langle\psi_{T}|\sigma_{y}|\psi_{T}\rangle=i\frac{\hbar}{2}\frac{D_{+}D_{-}^{*}-D_{+}^{*}D_{-}}{|D_{+}|^{2}+|D_{-}|^{2}},$
$\displaystyle\langle s_{x}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\langle\psi_{T}|\sigma_{x}|\psi_{T}\rangle=\frac{\hbar}{2}\frac{D_{+}D_{-}^{*}+D_{+}^{*}D_{-}}{|D_{+}|^{2}+|D_{-}|^{2}}.$
(12)

Analogous relations can be written for the reflected wave. A spin oriented in
the $y$ direction (corresponding to $s=-1$, see Eq. (2) will Larmor precess
around the magnetic field (in $z$ direction) when $V_{0}=0$ as shown in Fig.
2.

Figure 2: Precession of the spin as it goes through an increasing barrier
length $a$ starting from the $1/\sqrt{2}\left(1\leavevmode\nobreak\
\leavevmode\nobreak\ -i\right)$ state, for $V_{0}=0$. This is the simple
Larmor precession initially surmised for tunneling times[18]. Figure 3:
Relaxation of spin toward magnetic field direction due to tunneling when
$k^{2}<k_{0}^{2}\mp k_{B}^{2}$[18]. For long enough barriers the spin becomes
completely polarized in the direction of the field. The reference values taken
for the plot are $k=2/a$, $k_{0}=3/a$, and $k_{B}=1/a$, where $a$ is the
barrier length.

On the other hand, for $V_{0}>E\pm\hbar\omega_{L}/2$, spin precession around
the magnetic field is only part of the average spin motion, since each spin
component decays at a different rate under the barrier. This gives rise to a
$z$-component that aligns with the direction of the field[18]. Figure 3
depicts the qualitative motion for the latter case. For
$V_{0}<E\pm\hbar\omega_{L}/2$ only Larmor precession follows.

Finally, Fig. 4 shows the transmitted probability in $z$ quantization axis.
The input spin orientation is along the negative $y$ axis and the transmitted
wave selects the up spin orientation due to the slower decay of the lower
energy state under the barrier. This produces spin alignment with the magnetic
field.

Figure 4: Transmission contrast between spin components for a magnetic field
under the barrier. One can see the preferred spin polarization due to the
slower decay of the lower energy spin configuration under the barrier, leading
to an alignment of the entering spin to the magnetic field. The parameters
chosen are the same as in Fig. 3.

A very important relationship to check is that of the conservation of angular
momentum. As proven in reference [18], the conservation can be stated exactly
as

$(R_{+}+R_{-})\langle s_{z}\rangle_{R}=-\langle s_{z}\rangle(T_{+}+T_{-}),$
(13)

where $T_{+}=|t_{+}|^{2}$ and $R_{+}=|r_{+}|^{2}$ and $\langle
s_{z}\rangle_{R}$ is the reflected (R) spin component in the $z$ direction.

Such a relationship is verified in Fig. 5 where the changed angular momentum
transmitted is compensated by the opposite angular momentum reflected. We have
thus verified Büttiker’s scenario for tunneling with a magnetic field under
the barrier. Before addressing the case of the SO coupling in one dimension,
some useful gauge concepts will be introduced.

Figure 5: Angular momentum conservation balancing transmitted spin up
($z$-component) and reflected spin down. As there is no incident spin-up
current the two previous components (Eq. (13)) must balance.

##  3 Barrier model with a Rashba term

###  3.1  Spectrum, eigenfunctions, and wavevectors

We solve the scattering problem for the following model

$\cal{H}=\begin{cases}(\frac{p_{x}^{2}}{2m}+V_{0})\mathds{1}_{\sigma}+\Lambda
p_{x}\sigma_{y},&\text{if\leavevmode\nobreak\ $0<x<a$}\\\
(\frac{p_{x}^{2}}{2m})\mathds{1}_{\sigma},&\text{otherwise},\end{cases}$ (14)

where $\mathds{1}_{\sigma}$ is the unit matrix in spin space and $\sigma_{i}$
are the Pauli spin matrices. This Hamiltonian can be obtained from a helical
model of a molecule with $p$ wave overlaps and SO active Carbon/Nitrogen
atoms[24, 25, 26]. The magnitude of the SO coupling considered is in fact
derived from the overlaps that are only feasible for the chiral structure
considered in those models.

We take the incident beam to have an amplitude

$\psi=\frac{\sqrt{2}}{\sqrt{1+s^{2}}}\begin{pmatrix}\frac{1+s}{2}\\\
\frac{1-s}{2}\end{pmatrix},$ (15)

where $s=1$ corresponds to the up-spin normalized eigenstate of the
$\sigma_{z}$ matrix and $s=-1$ to the down-spin state. The normalization also
allows access to all spin states in the $x-z$ plane of the Bloch sphere. Here
we will illustrate how the $SU(2)$ gauge vector for the one-dimensional Rashba
Hamiltonian becomes crucial in the barrier boundary conditions[19] which has
been missed in previous treatments.

We can faithfully rewrite the Hamiltonian in the following form

${\cal
H}=\frac{1}{2m}\left({\hat{p}}_{x}\mathds{1}_{\sigma}+m\Lambda(x)\sigma_{y}\right)^{2}+V_{0}-\frac{m\Lambda^{2}}{2},$
(16)

where we can identify the $SU(2)$ gauge field ${\cal
A}_{x}=A_{x}^{y}\sigma_{y}=m\Lambda(x)\sigma_{y}$. The velocity operator
defined by $v_{x}=\partial{\cal H}/\partial
p_{x}=((p_{x}/m)\mathds{1}_{\sigma}+\Lambda\sigma_{y})$, where no effective
mass differences are considered[23] for the different scattering regions.
Solving for the eigenvalues of this Hamiltonian, we arrive at

$E=\frac{1}{2m}\left(p_{x}+m\sigma\Lambda\right)^{2}-\frac{m\Lambda^{2}}{2}+V_{0},$
(17)

where $\sigma=\pm 1$ is the spin quantum number (eigenvalue label of $SU(2)$
Hamiltonian). Equating $E=\hbar^{2}k^{2}/2m$ we define the wavevector outside
the barrier region as $k$. Starting from the eigenvalue, we can solve for the
possible values of $p_{x}=\hbar q$. A new quantum number arises that
distinguishes right and left propagating waves. The resulting possible values
of the wavevector under the barrier are

$q_{\sigma}^{\lambda}=\lambda\sqrt{k^{2}+k^{2}_{\rm so}-k^{2}_{0}}-\sigma
k_{\rm so},$ (18)

where $k_{\rm so}=m\Lambda/\hbar$ and $k_{0}^{2}=2mV_{0}/\hbar^{2}$. The
meaning of the quantum numbers is depicted in Fig. 6, where the degeneracy of
two Kramer’s pairs is evident. Note that for each direction of propagation,
there are two distinct wavevectors with opposite spin labels and that the
previous wave vector can be real or complex depending on the values of the
incoming wavevector (with energy $E=\hbar^{2}k^{2}/2m$) and the height of the
potential barrier.

Figure 6: Dispersion for Hamiltonian in Eq. (14) The labels correspond to the
wavevectors in the barrier region with $q_{\sigma}^{\lambda}$

As is easily derived from Eq. (14), the Hamiltonian commutes with
$\sigma_{y}$, and $\hat{p}_{x}$ so it has common eigenstates with $\sigma_{y}$
and the $\hat{p}_{x}$ eigenstates. In the $\sigma_{z}$ basis the wavefunctions
in the different regions are parameterized as follows

$\displaystyle\psi_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\frac{1+s}{2}\\\
\frac{1-s}{2}\end{pmatrix}e^{ikx}+\begin{pmatrix}A_{+}\\\
A_{-}\end{pmatrix}e^{-ikx},$ $\displaystyle\psi_{2}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\sqrt{2}}\begin{pmatrix}1\\\
i\end{pmatrix}e^{iq_{+}^{+}x}+\frac{\beta}{\sqrt{2}}\begin{pmatrix}1\\\
-i\end{pmatrix}e^{iq_{-}^{+}x}+\frac{\gamma}{\sqrt{2}}\begin{pmatrix}1\\\
i\end{pmatrix}e^{iq_{+}^{-}x}+\frac{\delta}{\sqrt{2}}\begin{pmatrix}1\\\
-i\end{pmatrix}e^{iq_{-}^{-}x},$ $\displaystyle\psi_{3}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}D_{+}\\\
D_{-}\end{pmatrix}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}e^{ikx}},$
(19)

where the coupling between the direction of propagation and spin orientation
has been implemented by the appropriate $q^{\lambda}_{\sigma}$ wavevectors.
The boundary conditions are the same as in Eqs.9[19] where
${\hat{v}}_{x}=\left({\hat{p}}_{x}+m\Lambda\sigma_{y}\right)/m$. The linear
system of eight unknowns can be explicitly solved for the transmission and
reflection amplitudes

$\displaystyle t_{+}$ $\displaystyle=$ $\displaystyle\frac{(1+i)\Delta
ke^{-iak}\left((1-is)e^{ik_{\rm so}a}+(s-1)e^{-ik_{\rm
so}a}\right)}{\left(e^{-ia\Delta}(\Delta+k)^{2}-e^{ia\Delta}(k-\Delta)^{2}\right)},$
$\displaystyle t_{-}$ $\displaystyle=$ $\displaystyle-\frac{(1+i)\Delta
ke^{-iak}\left((s+i)e^{ik_{\rm so}a}-(1+is)e^{-ik_{\rm
so}a}\right)}{\left(e^{-ia\Delta}(\Delta+k)^{2}-e^{ia\Delta}(k-\Delta)^{2}\right)},$
$\displaystyle r_{+}$ $\displaystyle=$
$\displaystyle\frac{(k-\Delta)(\Delta+k)(s+1)\left((k-\Delta)^{2}e^{2ia\Delta}+(\Delta+k)^{2}e^{-2ia\Delta}-2(k^{2}+\Delta^{2})\right)}{2\left(e^{-ia\Delta}(\Delta+k)^{2}-e^{ia\Delta}(k-\Delta)^{2}\right)^{2}},$
$\displaystyle r_{-}$ $\displaystyle=$
$\displaystyle-\frac{(k-\Delta)(\Delta+k)(s-1)\left((k-\Delta)^{2}e^{2ia\Delta}+(\Delta+k)^{2}e^{-2ia\Delta}-2\left(k^{2}+\Delta^{2}\right)\right)}{2\left(e^{-ia\Delta}(\Delta+k)^{2}-e^{ia\Delta}(k-\Delta)^{2}\right)^{2}},$
(20)

where $\Delta=\sqrt{k^{2}+k_{\rm so}^{2}-k_{0}^{2}}$, and as before $t_{\pm}$
and $r_{\pm}$ are the spin dependent transmission and reflection amplitudes.
Such amplitudes will be very important to understand how decoherence effects
generate spin polarization. The barrier region amplitudes are

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{(-1)^{1/4}e^{iaq_{+}^{-}}k(s-i)(k+\Delta)}{-e^{iaq_{+}^{+}}(k-\Delta)^{2}+e^{iaq_{+}^{-}}(k+\Delta)^{2}},$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{(-1)^{1/4}k(1-is)e^{iaq_{-}^{-}}(\Delta+k)}{e^{iaq_{-}^{-}}(\Delta+k)^{2}-e^{iaq_{-}^{+}}(k-\Delta)^{2}},$
(21)

and

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{(-1)^{1/4}k(s-i)e^{iaq_{+}^{+}}(k-\Delta)}{e^{iaq_{+}^{+}}(k-\Delta)^{2}-e^{iaq_{+}^{-}}(\Delta+k)^{2}},$
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{(-1)^{1/4}k(1-is)e^{iaq_{-}^{+}}(k-\Delta)}{e^{iaq_{-}^{+}}(k-\Delta)^{2}-e^{iaq_{-}^{-}}(\Delta+k)^{2}}.$
(22)

We recall that $q_{\sigma}^{\lambda}=\lambda\sqrt{k^{2}+k^{2}_{\rm
so}-k^{2}_{0}}-\sigma k_{\rm so}$.

###  3.2 Spin precession under the barrier for Rashba

The transmission of up-spin as a function of the entry spin polarization is

$\displaystyle|t_{+}|^{2}$ $\displaystyle=$
$\displaystyle\frac{8|\Delta|^{2}k^{2}[1+s\cos(2k_{\rm
so}a)]}{\left|e^{-ia\Delta}(\Delta+k)^{2}-e^{ia\Delta}(k-\Delta)^{2}\right|^{2}},$
$\displaystyle|t_{-}|^{2}$ $\displaystyle=$
$\displaystyle\frac{8|\Delta|^{2}k^{2}[1-s\cos(2k_{\rm
so}a)]}{\left|e^{-ia\Delta}(\Delta+k)^{2}-e^{ia\Delta}(k-\Delta)^{2}\right|^{2}}$
(23)

from where we can see that $T=|t_{+}|^{2}+|t_{-}|^{2}$ so the total
conductance is

$\displaystyle G_{{\rm total}}$ $\displaystyle=$ $\displaystyle G_{+}+G_{-}$
$\displaystyle=$
$\displaystyle\frac{e^{2}}{h}T=\frac{16e^{2}|\Delta|^{2}k^{2}}{h\left|e^{-ia\Delta}(\Delta+k)^{2}-e^{ia\Delta}(k-\Delta)^{2}\right|^{2}},$

where $G_{\pm}=(e^{2}/h)|t_{\pm}|^{2}$, which is spin independent[19, 27] even
in the presence of a barrier and an open system at either end of the barrier.

Following the definitions for average spin components in Eq. (.12), we can now
see the behavior of the injected spin into the spin-orbit active barrier. Fig.
7 shows how the average spin traverses the spin-active barrier region with SO
coupling. In analogy with Büttiker’s $U(1)$ magnetic field, we can define a
new momentum-dependent magnetic field ${\bf B}_{\rm SO}$ given the mapping
$\lambda p_{x}\sigma_{y}=-\gamma{\bf B}_{\rm SO}\cdot{\bm{\sigma}}$ that
results in ${\bf B}_{\rm SO}=-(\Lambda/\gamma)p_{x}{\bf u}_{y}$. ${\bf B}_{\rm
SO}$ lies in the negative $y$ direction for the model Hamiltonian. Precession
follows correctly the torque equation $d\langle{\bf
s\rangle}/dt=\gamma\langle{\bf s}\rangle\times{\bf B}_{\rm SO}$. Note that, as
even under the barrier, the wavevector is complex, unlike the magnetic field
case, precession proceeds with no generation of a spin component along the
${\bf B}_{\rm SO}$ direction. Also, both spin components suffer the same decay
within the barrier (although dependent on SO) independent of their spin
orientation (see Eq. (18)).

Figure 7: Precession of spin around the spin-orbit magnetic field. Note that
as the $k$ vector inside the barrier has always a real part, we have pure
precession with no tilting toward the magnetic field as with the magnetic
field in the previous section. This happens below and above the barrier.

We can see from comparing the two cases (magnetic field and SO) that one
superposes different $k$ vectors corresponding with the same energy when
traversing the spin active region. The two wavevectors, having different real
parts, cause precession due to a torque around the direction of ${\bf B}_{\rm
SO}$. In the case of a real magnetic field, under the barrier, the $k$ vector
is purely imaginary, and only a spin-dependent decay ensues (see Eq. (4)),
producing an alignment of the spin in the magnetic field direction. In the
case of the SO, there is always a real part to the $k$ vector (even under the
barrier) so that precession occurs for energies above and below the barrier.
No spin polarization along ${\bf B}_{\rm SO}$ follows from this scenario in
tune with time-reversal symmetry.

##  4 Other helicity Hamiltonians

Varying the Hamiltonian under the barrier to the case where the eigenstates
are projected along the direction of propagation (helicity states), can be
interpreted directly from the previous results. The Hamiltonian in this case
is

$\cal{H}=\begin{cases}(\frac{p_{x}^{2}}{2m}+V_{0})\mathds{1}_{\sigma}+\Lambda
p_{x}\sigma_{x},&\text{if\leavevmode\nobreak\ $0<x<a$}\\\
(\frac{p_{x}^{2}}{2m})\mathds{1}_{\sigma},&\text{otherwise}.\end{cases}$ (25)

The SO magnetic field is now in the $x$ direction as ${\bf B}_{\rm
SO}=-\Lambda p_{x}/\gamma\leavevmode\nobreak\ {\bf u}_{x}$. Working out the
eigenstates in the SO active region, the eigenstates will be those of
$\sigma_{x}$ matrix, and the possible $k$ vectors will be
$\kappa_{\sigma}^{\lambda}=\lambda\sqrt{k^{2}+k_{\rm so}^{2}-k_{0}^{2}}-\sigma
k_{\rm so}$ as before. The $k$-vector under the barrier always has a real part
that results in a spin precession. If we start from a spin orientation in the
$z$-axis, then the spin will precess around the $x$ axis without generating a
spin component in the $x$ direction. So no changes from the conclusion in the
previous section follow in this case.

##  5 Decoherence with Büttiker’s probe

The spin-orbit coupling does not contrast between spin species, so it cannot,
alone, account for polarised spin polarization as expected in CISS effect.
Nevertheless, the perfect conditions under which these results are valid i.e.,
no coupling to a TRS breaking probe beyond the two terminals, are not met,
especially at room temperature conditions. A thermalization of electron
transport to the environment through the electron-phonon or electron-electron
interactions is inevitable. This environment can be modeled as a lumped probe
that disrupts the delicate coherences that yield Bardarson’s theorem that
translates into transport as the Onsager reciprocity relations in the linear
regime. This turns our attention to a tunneling molecular system to a three-
probe scenario.

Figure 8: Büttiker’s probe under the spin-orbit active barrier with electrons
impinging on the left and leaving on the rigth with wavevector $k$. The
function $\Gamma(\alpha,\beta,\gamma,\delta)$ represents the wavefunction
combination under the barrier as in Eq. 27. The probe absorbs each eigenstate
spin species under the barrier with the same scattering matrix, so no spurious
spin selection is induced. Flux conditions are imposed on building the S
matrix for a wideband Büttiker probe. Figure 9: Precession of input spin
orientation when the decoherence probe couples to a particular point $x_{0}$
under the barrier. A noticeable disruption of spin precession is observed,
generating a spin polarization toward $B_{SO}$ analogous to an actual magnetic
field (see Fig. 3, realizing broken time-reversal symmetry). As here
$x_{0}=0.8$, we begin the plot at $a=1$ so that the probe is always under the
barrier.

The Büttiker’s voltage probe[28] is an ingenious way to introduce decoherence
processes through the scattering matrix for an exactly solved model. Here we
introduce a generalization of the probe used previously in the context of
persistent currents[29, 30, 31] (see Fig. 8). The probe is spin insensitive,
so we do not introduce extraneous sources of spin selection. This is achieved
by introducing two probes, one for each spin species at the same point
connected to a third reservoir thermalized to a Fermi distribution at
temperature $T$. The probe is wide-band, supporting the wavevectors injected
by the barrier channels. The probe scattering matrix returns an amplitude
consistent with a simple electron reservoir unrelated to the input amplitude
(while preserving unitarity) so that a disruption to the interferences occurs
according to the local fluxes of each spin orientation. Such a probe
introduces TRS breaking that generates the Büttiker tilting of the spin in the
${\bf B}_{\rm SO}$ direction producing net spin polarization.

The behavior of the Büttiker probe follows the combination of an ideal lead
with $v=\hbar\kappa/m$ that supports a current $dI=ev(dN/dE)f(E)dE$ in the
energy interval $dE$, where $f(E)$ is the Fermi distribution,
$dN/dE=1/2\pi\hbar v$ is the density of states. This model can then induce
level broadening[29, 30], ([31] for Hamiltonian version) and level shifts
under the barrier, and also depends on where the decoherence event occurs.
Besides the coupling of the probe to the barrier, we can also control the
temperature through the Fermi distribution of the attached reservoir. The
Büttiker probe is also discussed in [32], where they consider inelastic events
i.e., energy changes but conservation of particles. Here $S^{\dagger}S=1$ so
that only decoherence is contemplated.

Figure 8 shows the four regions that must be matched for continuity and flux.
Under the barrier, the matching occurs at position $(x_{0},y_{0})=(x_{0},0)$
where $y$ describes the coordinate of the third probe. The Scattering ($S$)
matrix can then emulate a generic dephasing process[29]. Matching flux
conditions at $x_{0}$ yields the following $S$ matrix between input and output
amplitudes for each spin species i.e., spin eigenstates under the barrier.
Here we only show the spin down matrix equation (see Appendix A for details)

$\Psi_{out}=\begin{pmatrix}\sqrt{\cal N}\zeta_{-}\\\ \beta^{\prime}\\\
\delta\end{pmatrix}=S_{-}\Psi_{in}=\begin{pmatrix}-(\mathcal{A}+\mathcal{B})&-\sqrt{\varepsilon}e^{iq_{-}^{-}x_{0}}&\sqrt{\varepsilon}e^{iq_{-}^{+}x_{0}}\\\
\sqrt{\varepsilon}e^{-iq_{-}^{+}x_{0}}&-\mathcal{A}e^{-2i\Delta
x_{0}}&\mathcal{B}\\\
-\sqrt{\varepsilon}e^{-iq_{-}^{-}x_{0}}&\mathcal{B}&-\mathcal{A}e^{2i\Delta
x_{0}}\end{pmatrix}\begin{pmatrix}\sqrt{\cal N}\\\ \delta^{\prime}\\\
\beta\end{pmatrix},$ (26)

where $\Psi_{in,out}$ represent the input/output amplitudes to the junction
and $S_{-}$ is the scattering matrix for the spin-down label. The labels
follow the usage previously introduced where $q_{\sigma}^{\lambda}$, with
$\sigma$ the spin label and $\lambda$ the sense of propagation label.
$\mathcal{A}=(\sqrt{1-2\varepsilon}-1)/2$ and
$\mathcal{B}=(\sqrt{1-2\varepsilon}+1)/2$, while ${{\cal N}=ef(E)dE/2\pi\hbar
v}$ with $f(E)$ the Fermi distribution, $e$ the electron charge, $E$ the
energy and $v$ the velocity of the carriers in the lead[28].
$0<\varepsilon<0.5$ describes the coupling of the probe to the barrier from
uncoupled to fully coupled.

The results regarding the influence of decoherence, barrier length, and SO
coupling is depicted in Figs. 9 thru 11. In Fig. 9, where only qualitative
parameters are used so that precession can be appreciated, one can see that
the SO coupled to the third probe produces a smooth disruption of the spin
precession which is no longer in the $(x,z)$ plane but yields a $\langle
s\rangle_{y}$ component. The tunneling electrons now achieve a large
polarization in the direction of the SO magnetic field (see inset). Thus, the
${\bf B}_{\rm SO}$ acts as a symmetry-breaking interaction such as a real
magnetic field in Fig. 3.

Figure 10: Transmission contrast of spin orientations along the $y$
quantization axis (${\bf B}_{\rm SO}$ field orientation). The input spin
wavefunction in the $(1\leavevmode\nobreak\ 0)$ orientation, which has equal
amplitudes in the latter basis, acquires a preferred orientation aligned with
the ${\bf B}_{\rm SO}$ field analogously to the $U(1)$ magnetic field case.
Figure 11: Spin polarization for $k=0.4\leavevmode\nobreak\ {\rm nm}^{-1}$,
$k_{0}=1\leavevmode\nobreak\ {\rm nm}^{-1}$, $a=5\leavevmode\nobreak\ {\rm
nm}$, $x_{0}=1.5\leavevmode\nobreak\ {\rm nm}$ and ${\cal N}=0.1$, generated
by decoherence analogous to that caused by a real external magnetic field. The
contour plot shows the effect of a spin-orbit and decoherence coupling,
consistent with estimates of ref.[24] for tunneling. The appearance of
alignment of the spin to the ${\bf B}_{\rm SO}$ is very sensitive to the
coupling to the Büttiker probe, producing up to 16% polarization with
realistic values of SO coupling.

Figure 10 depicts the transmission contrast between spin components along the
$y$ axis in which the ${\bf B}_{\rm SO}$ field is oriented. As can be seen,
the spin acquires a preferred direction along the $-y$ direction aligning with
the SO magnetic field. This is analogous to the case of the TRS breaking
$U(1)$ magnetic field considered at the outset. When the SO strength is zero,
the two components merge, showing no spin activity, and only express the
transmission through the barrier.

Figure 11 shows the dependence of the new polarization as a function $k_{\rm
so}$, and the coupling to the reservoir. In contrast to previous figures
depicting qualitative precession features, here we have set the parameters to
realistic ranges for spin-orbit strength, barrier height, and barrier-probe
coupling[24]. It is evident that even for weak coupling ($\varepsilon$) and
$E_{\rm SO}\sim 10$ meV, one can achieve polarizations of $40\%$. The
polarization effects can yield positive and negative polarizations depending
on the length of the barrier $a$ and exhibit a non-trivial temperature
dependence. No spin polarization is produced without the SO interaction, no
matter the coupling to the third probe.

Figure 12: Spin polarized component for $k=0.6\leavevmode\nobreak\ {\rm
nm}^{-1}$, $k_{0}=2\leavevmode\nobreak\ {\rm nm}^{-1}$,
$a=4\leavevmode\nobreak\ {\rm nm}$, $x_{0}=1.5\leavevmode\nobreak\ {\rm nm}$
and ${\cal N}=0.1$. Manifest interference effects in the spin polarization
through the coupling to the third probe. Small couplings to the probe can
produce large polarizations while larger couplings degrade the polarization.
Note the dependence on the probe position $x_{0}$ and the high sensitivity to
the probe coupling that cannot be surmised solely based on symmetry arguments.

Figure 12 shows the non-monotone/interference effects of coupling to the third
probe and the sensitivity to the coupling to the third probe. Sufficiently
large values of SO wavevector $k_{\rm so}$, can produce a large polarization
at low coupling, while large couplings degrade the polarization.

Figure 13: Temperature dependence of spin polarization as a function of the
barrier length and occupation of the Büttiker probe for $k=0.4$ ${\rm
nm}^{-1}$, $k_{\rm so}=0.1$ ${\rm nm}^{-1}$, $k_{0}=1$ ${\rm nm}^{-1}$,
$x_{0}=1.5$ ${\rm nm}$ and $\varepsilon=0.1$ (same as those of Fig. 11). The
polarization increases with temperature for the range of parameters chosen.

Figure 13 depicts the temperature dependence of the polarization as a function
of the barrier length. The temperature dependence is expressed through the
parameter ${\cal N}=ef(E)dE/2\pi\hbar v$ proportional to the thermal
occupation of the probe. As temperature rises $\cal N$ is reduced so that as
the temperature is increased, the polarization increases for fixed barrier
length. Non-monotone effects with the probe coupling can change this
temperature dependence. They will be determined by the specific nature of how
the electron spin current thermalizes to the environment e.g., electron-
phonon, electron-electron interactions.

A parameter that may be the least determined physically is the probe’s
position $x_{0}$, and how many of these probes should be placed in the
tunneling/spin active region. Although it is beyond the scope of this work,
one may estimate how many electron-phonon events may occur as a function of
temperature and the particular electron-phonon coupling within the tunneling
region and statistically place as many probes as the estimate dictates. As to
the specific location of the probe, the most appropriate procedure is to
average its effect over different positions under the barrier. We have not
performed these procedures but show the smooth behavior of the polarization as
a function of the probe position so that the previous procedures can be
readily implemented. Figure 14 shows the smooth probe position dependence of
the decoherence-generated spin polarization for the parameters of Fig. 11. The
figure also displays the sensitivity of the generated polarization as a
function of probe-barrier coupling $\varepsilon$. Such smooth dependence lends
itself to the averaging procedures proposed and will not change qualitatively
the one probe results in an essential way.

Figure 14:  Dependence of the polarization on the probe position $x_{0}$ under
the barrier, for the parameters of Fig.11, $k=0.4$ ${\rm nm}^{-1}$, $k_{\rm
so}=0.1$ ${\rm nm}^{-1}$, $k_{0}=1$ ${\rm nm}^{-1}$ for different probe
barrier couplings $\varepsilon$. We see the sensitivity of the polarization to
the coupling of the probe and the smooth behavior of the polarization with the
probe position.

##  6 Summary and Discussion

We have discussed a one-dimensional system with SO interaction which can be
derived from a three-dimensional model ignoring the orbital degree of
freedom[24, 25]. The spin-orbit coupling arises from the geometrical
arrangement of the $p$ orbitals of the helical model’s chiral structure.
Without it, the SO coupling would be orders of magnitude smaller[25], as
happens comparing SO coupling of planar graphene and carbon nanotubes[1]. Due
to the atomic origin of the SO coupling in this model, the helix’s spin and
orbital degrees of freedom are uncoupled at the lowest order and in the half-
filling model[25, 26]. Thus the orbital degree of freedom only modulates the
kinetic energy and adds orbital angular momentum, which can also enhance spin-
orbit effects, as shown in ref.[33]. So chirality has a role in the present
CISS model in generating its $\sim$10 meV strength.

In the succession of models presented for transmission through a spin-active
barrier, we have expressed spin polarization as the manifestation of a lack of
TRS that selects a spin orientation. We first discussed Büttiker’s model of a
$U(1)$ magnetic field under a barrier. The differential decay of the
amplitudes for spin-up and spin-down produces a reorientation of the spin
along the magnetic field. It serves as the trademark of TRS breaking in the
spin polarization. In the following model we assessed the SO coupling where an
effective magnetic field can also be identified ${\bf B}_{\rm SO}$, mapping
the SO coupling to a Zeeman-like term. Nevertheless, this effective field
depends on the propagation direction and does not break TRS. The exact
solution of the tunneling problem, which has not been satisfactorily solved
before in the literature in this form, yields nevertheless the expected
result, implied by Onsager’s reciprocity for two terminal devices, i.e. no
spin polarization independent of the magnitude of the SO coupling. Spin
precession around ${\bf B}_{\rm SO}$ with no differential decay of different
spin components is shown. Thus, in the two terminal setup, at $T=0$, the
chiral structure and SO will then not be enough for spin selectivity since, as
we have shown above, the spin-orbit coupling only makes for spin precession
due to the spin torque of the SO magnetic field ${\bf B}_{\rm SO}$ with no
asymmetric treatment of both spin species.

One of the most emblematic features of the CISS effect is that it is measured
at room temperature (although see[34, 35]) in molecular systems that are
strongly coupled to the thermal environment. Strictly coherent quantum models
can only be part of the story. The final model addresses minimal coupling to
the environment through a third probe, making for decoherent/dephasing albeit
unitary processes. Symmetry arguments cannot assess beforehand, how sensitive
the system will be to weak-TRS-breaking events. These events are incorporated
in the last model as a time reversal symmetric interaction (SO coupling) under
a barrier coupled to a Büttiker probe. Our exact results show a high
sensitivity of spin polarization, where the combination of SO coupling and
decoherence acts analogously to a $U(1)$ magnetic field. The input spin
reorients to the effective magnetic field producing a net spin polarization.
This is a very different mechanism from the first model considered, where
differential decay of each spin orientation gives polarization. Here delicate
interferences that guarantee time-reversal symmetry are disrupted, producing a
large effect of even 40% for small couplings to the Büttiker probe.

The proposed scenario, in the context of an exactly solved model, addresses
many issues that have related to the CISS effect: i) The size of the SO
coupling is set to realistic values in agreement with theoretical estimates in
the meV range[25]; ii) The models reproduce the Onsager relations for two
terminal devices for TRS interactions; iii) Coupling weakly to the environment
through a third probe (other than terminals) produces highly sensitive effects
of the polarization capacity of chiral molecules, that can match the order of
magnitude found in experiments. Additionally, we note that other interactions
such as electron- and spin-phonon have been modeled[36] to investigate their
effect on electron transport and spin polarization through chiral molecules.
These works show that non-zero coupling to a thermal reservoir is necessary to
have spin selectivity[37, 38, 39].

Of course, this is the bare-bones model for the spintronics of chiral
molecules. The nature of the environment coupling to electron transport should
be developed in much more detail to assess its quantitative correctness.
Incorporating the interplay between orbital and spin degrees of freedom, not
yet addressed to our knowledge, should enrich further the possibilities of the
theory to describe, more completely, the CISS effect.

In summary, one of our main conclusions in that decoherence effects, even
those associated with small coupling constants, can translate into significant
changes in spin polarization. This important result allows for a Büttiker-
probe representation of several mechanisms, including electron-electron and
electron-phonon interactions, that can be related to the CISS effect. There is
an important connection between our treatment and a Liouville equation
description of electron transfer in molecules [40], that we will explore
extensively in a forthcoming article.

## Acknowledgements

#### Funding information

E.M. acknowledges support from Project 17617 Spin Active Molecular electronics
by the Universidad San Francisco de Quito. M.P. acknowledge the support given
by the Dresden Junior Fellowship Programme by the Chair of Materials Science
and Nanotechnology at the Technische Universität Dresden. S.V. acknowledges
the support given by the Eleonore-Trefftz-Programm at the Technische
Universität Dresden. V.M acknowledges the support of Ikerbasque, the Basque
Foundation for Science, and the W.M. Keck Foundation through the grant
“Chirality, spin coherence and entanglement in quantum biology”.

##  Appendix A Coupling Büttiker’s Probe

To implement Büttiker’s probe, we divide the scattering problem into four
regions, two inside the barrier to accommodate for a probe at site $x_{0}$.

$\displaystyle\psi_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\frac{1+s}{2}\\\
\frac{1-s}{2}\end{pmatrix}e^{ikx}+\begin{pmatrix}A_{+}\\\
A_{-}\end{pmatrix}e^{-ikx},$ $\displaystyle\psi_{2}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\sqrt{2}}\begin{pmatrix}1\\\
i\end{pmatrix}e^{iq_{+}^{+}x}+\frac{\beta}{\sqrt{2}}\begin{pmatrix}1\\\
-i\end{pmatrix}e^{iq_{-}^{+}x}+\frac{\gamma}{\sqrt{2}}\begin{pmatrix}1\\\
i\end{pmatrix}e^{iq_{+}^{-}x}+\frac{\delta}{\sqrt{2}}\begin{pmatrix}1\\\
-i\end{pmatrix}e^{iq_{-}^{-}x},$ $\displaystyle\psi_{\pm}^{probe}$
$\displaystyle=$ $\displaystyle\sqrt{\cal{N}}(\zeta_{\pm}e^{i\kappa
y}+e^{-i\kappa y})\begin{pmatrix}1\\\ {\pm}i\end{pmatrix},$
$\displaystyle\psi_{3}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}}{\sqrt{2}}\begin{pmatrix}1\\\
i\end{pmatrix}e^{iq_{+}^{+}x}+\frac{\beta^{\prime}}{\sqrt{2}}\begin{pmatrix}1\\\
-i\end{pmatrix}e^{iq_{-}^{+}x}+\frac{\gamma^{\prime}}{\sqrt{2}}\begin{pmatrix}1\\\
i\end{pmatrix}e^{iq_{+}^{-}x}+\frac{\delta^{\prime}}{\sqrt{2}}\begin{pmatrix}1\\\
-i\end{pmatrix}e^{iq_{-}^{-}x},$ $\displaystyle\psi_{4}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}D_{+}\\\
D_{-}\end{pmatrix}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}e^{ikx}},$
(27)

where now $\alpha,\beta,\gamma,\delta$, and
$\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\delta^{\prime}$ correspond to
amplitudes at either sides of $x_{0}$. We assume the wideband limit for the
probe leads so that the reservoir lead can carry any wavevector under the
barrier without wavevector-dependent scattering[41]. ${\cal A}$, and ${\cal
B}$ were defined below Eq.26 Such amplitude are now related by the scattering
matrix

$\begin{pmatrix}\sqrt{\cal N}\zeta_{+}\\\ \alpha^{\prime}\\\
\gamma\end{pmatrix}=\begin{pmatrix}-(\mathcal{A}+\mathcal{B})&-\sqrt{\varepsilon}e^{iq_{+}^{-}x_{0}}&\sqrt{\varepsilon}e^{iq_{+}^{+}x_{0}}\\\
\sqrt{\varepsilon}e^{-iq_{+}^{+}x_{0}}&-\mathcal{A}e^{-2i\Delta
x_{0}}&\mathcal{B}\\\
-\sqrt{\varepsilon}e^{-iq_{+}^{-}x_{0}}&\mathcal{B}&-\mathcal{A}e^{2i\Delta
x_{0}}\end{pmatrix}\begin{pmatrix}\sqrt{\cal N}\\\ \gamma^{\prime}\\\
\alpha\end{pmatrix},$ (28)

and

$\begin{pmatrix}\sqrt{\cal N}\zeta_{-}\\\ \beta^{\prime}\\\
\delta\end{pmatrix}=\begin{pmatrix}-(\mathcal{A}+\mathcal{B})&-\sqrt{\varepsilon}e^{iq_{-}^{-}x_{0}}&\sqrt{\varepsilon}e^{iq_{-}^{+}x_{0}}\\\
\sqrt{\varepsilon}e^{-iq_{-}^{+}x_{0}}&-\mathcal{A}e^{-2i\Delta
x_{0}}&\mathcal{B}\\\
-\sqrt{\varepsilon}e^{-iq_{-}^{-}x_{0}}&\mathcal{B}&-\mathcal{A}e^{2i\Delta
x_{0}}\end{pmatrix}\begin{pmatrix}\sqrt{\cal N}\\\ \delta^{\prime}\\\
\beta\end{pmatrix}.$ (29)

The parameters were all defined below Eq.26. Each matrix equation involves and
single spin orientation in a separate lead. ${\cal N}$ involves the reservoir
parameters and is common to the two reservoir leads. The system has fourteen
unknowns and fourteen conditions from which all variables can be found
explicitly.

Here we give the expressions for the polarized transmission amplitudes:

$\displaystyle(De^{iak}/\Delta)t_{+}=$ $\displaystyle{\cal
B}^{3}(k-\Delta)^{2}\Bigl{\\{}$ $\displaystyle
k(1+i)(1-is)e^{i(a+x_{0})(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}q_{+}^{-}}$
(30)
$\displaystyle+k(1+i)(s-i)e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}q_{-}^{-}}$
$\displaystyle+\sqrt{2{\cal
N}\epsilon}(k-\Delta)[e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{+})+i(aq_{+}^{-}+x_{0}q_{-}^{-})}+e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{+})+i(aq_{-}^{-}+x_{0}q_{+}^{-})}]\Bigr{\\}}$
$\displaystyle+{\cal B}(k+\Delta)^{2}\Bigl{\\{}$ $\displaystyle
k(1+i)(i-s)e^{i(a+x_{0})(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}q_{-}^{+}}$
$\displaystyle-k(1-i)(i+s)e^{i(a+x_{0})(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}q_{+}^{+}}$
$\displaystyle-\sqrt{2{\cal
N}\epsilon}[(k-\Delta)(e^{i(a+x_{0})(q_{-}^{-}+q_{+}^{+})+i(aq_{+}^{-}+x_{0}q_{-}^{+})}+e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{-})+i(aq_{-}^{-}+x_{0}q_{+}^{+})})]\Bigr{\\}}$
$\displaystyle+{\cal A}^{2}(k-\Delta)^{2}\Bigl{\\{}$ $\displaystyle{\cal
B}k(1+i)(i-s)e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}q_{-}^{-}}$
$\displaystyle-{\cal
B}k(1-i)(i+s)e^{i(a+x_{0})(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}q_{+}^{-}}$
$\displaystyle-\sqrt{2{\cal
N}\epsilon}(k-\Delta)e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{+})}(e^{i(aq_{+}^{-}+x_{0}q_{-}^{-})}+e^{i(aq_{-}^{-}+x_{0}q_{+}^{-})})$
$\displaystyle+\sqrt{2{\cal
N}\epsilon}(k+\Delta)[e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})}+e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{+}+2q_{+}^{-})}$
$\displaystyle
e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{-}+q_{+}^{+})}+e^{ia(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{+}+q_{+}^{+})}$
$\displaystyle
e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})}+e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}(q_{-}^{+}+2q_{+}^{+})}]\Bigr{\\}}$
$\displaystyle+{\cal A}(k-\Delta)\Bigl{\\{}$ $\displaystyle{\cal
B}k(k+\Delta)(1+i)(1-is)e^{i(a+x_{0})(q_{-}^{-}+q_{-}^{+})+i(aq_{+}^{+}+2x_{0}q_{+}^{-})}$
$\displaystyle+{\cal
B}k(k+\Delta)(1+i)(1-is)e^{i(a+x_{0})(q_{-}^{-}+q_{-}^{+})+i(aq_{+}^{-}+2x_{0}q_{+}^{+})}$
$\displaystyle+k(1+i)(s-i)e^{i(a+x_{0})(q_{+}^{-}+q_{+}^{+})}(e^{i(aq_{-}^{+}+2x_{0}q_{-}^{-})}+e^{i(aq_{-}^{-}+2x_{0}q_{-}^{+})})$
$\displaystyle+\sqrt{2{\cal
N}\epsilon}(k-\Delta)[e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{+}+2q_{+}^{-})}+e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{-}+q_{+}^{+})}$
$\displaystyle+e^{ia(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{+}+q_{+}^{+})}+e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}(q_{-}^{+}+2q_{+}^{+})}]$
$\displaystyle-{\cal B}^{2}\sqrt{2{\cal
N}\epsilon}(k-\Delta)^{2}e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{+})}(e^{i(aq_{+}^{-}+x_{0}q_{-}^{-})}+e^{i(aq_{-}^{-}+x_{0}q_{+}^{-})})$
$\displaystyle+\sqrt{2{\cal
N}\epsilon}(k+\Delta)^{2}[e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{-}+q_{+}^{-})}+e^{ia(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{+}+q_{+}^{-})}$
$\displaystyle
e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{-}+2q_{+}^{-})}+e^{ia(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})}$
$\displaystyle
e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})}+e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}(q_{-}^{-}+2q_{+}^{+})}]$
$\displaystyle+{\cal A}^{3}\sqrt{2{\cal
N}\epsilon}(k-\Delta)^{3}e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{+})}(e^{i(aq_{+}^{-}+x_{0}q_{-}^{-})}+e^{i(aq_{-}^{-}+x_{0}q_{+}^{-})})$
$\displaystyle+{\cal B}^{2}\sqrt{2{\cal
N}\epsilon}(k-\Delta)^{2}(k+\Delta)[e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{-})+i(aq_{+}^{+}+x_{0}q_{-}^{-})}$
$\displaystyle+e^{i(a+x_{0})(q_{-}^{-}+q_{+}^{+})+i(aq_{-}^{+}+x_{0}q_{+}^{-})}]$
$\displaystyle-\sqrt{2{\cal
N}\epsilon}(k+\Delta)^{3}e^{i(a+x_{0})(q_{-}^{-}+q_{+}^{-})}[e^{i(aq_{+}^{+}+x_{0}q_{-}^{+})}+e^{i(aq_{-}^{+}+x_{0}q_{+}^{+})}]\Bigr{\\}}$

$\displaystyle(De^{iak}/k)t_{-}=$ $\displaystyle{\cal
B}(k+\Delta)^{2}\Bigl{\\{}$
$\displaystyle\Delta(1-i)(s-i)e^{i(a+x_{0})(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}q_{-}^{+}}$
(31)
$\displaystyle+\Delta(1+i)(i+s)e^{i(a+x_{0})(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}q_{+}^{+}}$
$\displaystyle-i\sqrt{2{\cal
N}\epsilon}(k-\Delta)[e^{i(a+x_{0})(q_{-}^{-}+q_{+}^{+})+i(aq_{+}^{-}+x_{0}q_{-}^{+})}+e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{-})+i(aq_{-}^{-}+x_{0}q_{+}^{+})}]\Bigr{\\}}$
$\displaystyle+{\cal B}^{3}(k-\Delta)^{2}\Bigl{\\{}$
$\displaystyle\Delta(1+i)(1+is)e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}q_{-}^{-}}$
$\displaystyle-\Delta(1+i)(i+s)e^{i(a+x_{0})(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}q_{+}^{-}}$
$\displaystyle+i\sqrt{2{\cal
N}\epsilon}(k-\Delta)[e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{+})+i(aq_{+}^{-}+x_{0}q_{-}^{-})}-e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{+})+i(aq_{-}^{-}+x_{0}q_{+}^{-})}]\Bigr{\\}}$
$\displaystyle+{\cal A}^{2}(k-\Delta)^{2}\Bigl{\\{}$ $\displaystyle{\cal
B}\Delta(1-i)(s-i)e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}q_{-}^{-}}$
$\displaystyle+{\cal
B}\Delta(1+i)(i+s)e^{i(a+x_{0})(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}q_{+}^{-}}$
$\displaystyle-i{\cal B}\sqrt{2{\cal
N}\epsilon}(k-\Delta)e^{i(a+x_{0})(q_{-}^{+}+q_{+}^{+})}(e^{i(aq_{+}^{-}+x_{0}q_{-}^{-})}-e^{i(aq_{-}^{-}+x_{0}q_{+}^{-})})$
$\displaystyle-i\sqrt{2{\cal
N}\epsilon}(k+\Delta)[e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})}-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{+}+2q_{+}^{-})}$
$\displaystyle+e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{-}+q_{+}^{+})}+e^{ia(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{+}+q_{+}^{+})}$
$\displaystyle-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})}-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}(q_{-}^{+}+2q_{+}^{+})}]\Bigr{\\}}$
$\displaystyle+{\cal A}(k-\Delta)\Bigl{\\{}$ $\displaystyle{\cal
B}\Delta(k+\Delta)(1+i)(1+is)e^{i(a+x_{0})(q_{+}^{-}+q_{+}^{+})}[e^{i(aq_{-}^{+}+2x_{0}q_{-}^{-})}+e^{i(aq_{-}^{-}+2x_{0}q_{-}^{+})}]$
$\displaystyle-{\cal
B}\Delta(k+\Delta)(1+i)(i+s)e^{i(a+x_{0})(q_{-}^{-}+q_{-}^{+})}[e^{i(aq_{+}^{+}+2x_{0}q_{+}^{-})}+e^{i(aq_{+}^{-}+2x_{0}q_{+}^{+})}]$
$\displaystyle+i{\cal B}(k^{2}-\Delta^{2})\sqrt{2{\cal
N}\epsilon}[-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{+}+2q_{+}^{-})}+e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{-}+q_{+}^{+})}$
$\displaystyle+e^{ia(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{+}+q_{+}^{+})}-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}(q_{-}^{+}+2q_{+}^{+})}]$
$\displaystyle-i{\cal B}^{2}(k-\Delta)^{2}\sqrt{2{\cal
N}\epsilon}[e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})}-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})}]$
$\displaystyle+i(k+\Delta)^{2}\sqrt{2{\cal
N}\epsilon}[e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{-}+q_{+}^{-})}+e^{ia(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}(2q_{-}^{+}+q_{+}^{-})}$
$\displaystyle-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{-}+2q_{+}^{-})}+e^{ia(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})}$
$\displaystyle-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})}-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}(q_{-}^{-}+2q_{+}^{+})}]$
$\displaystyle+i{\cal A}^{3}(k-\Delta)^{3}\sqrt{2{\cal
N}\epsilon}[e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})}-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})}]$
$\displaystyle+i{\cal B}^{2}(k-\Delta)^{2}(k+\Delta)\sqrt{2{\cal
N}\epsilon}[e^{ia(q_{-}^{+}+q_{+}^{-}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})}-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})}]$
$\displaystyle-i(k+\Delta)^{3}\sqrt{2{\cal
N}\epsilon}[e^{ia(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})+ix_{0}(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})}-e^{ia(q_{-}^{-}+q_{-}^{+}+q_{+}^{-})+ix_{0}(q_{-}^{-}+q_{+}^{-}+q_{+}^{+})}]\Bigr{\\}}$
$\displaystyle D=$ $\displaystyle\Bigl{[}-{\cal A}^{2}(k-\Delta)^{2}$
$\displaystyle e^{i(aq_{-}^{+}+x_{0}(q_{-}^{-}+q_{-}^{+}))}+{\cal
A}(k^{2}-\Delta^{2})(e^{i(aq_{-}^{+}+2x_{0}q_{-}^{-})}+e^{i(aq_{-}^{-}+2x_{0}q_{-}^{+})})$
(32) $\displaystyle+e^{ix_{0}(q_{-}^{-}+q_{-}^{+})}({\cal
B}^{2}(k-\Delta)^{2}e^{iaq_{-}^{+}}-(k+\Delta)^{2}e^{iaq_{-}^{-}})\Bigr{]}$
$\displaystyle\times\Bigl{[}{\cal A}^{2}(k-\Delta)^{2}$ $\displaystyle
e^{i(aq_{+}^{+}+x_{0}(q_{+}^{-}+q_{+}^{+}))}-{\cal
A}(k^{2}-\Delta^{2})(e^{i(aq_{+}^{+}+2x_{0}q_{+}^{-})}+e^{i(aq_{+}^{-}+2x_{0}q_{+}^{+})})$
$\displaystyle+e^{ix_{0}(q_{+}^{-}+q_{+}^{+})}(-{\cal
B}^{2}(k-\Delta)^{2}e^{iaq_{+}^{+}}+(k+\Delta)^{2}e^{iaq_{+}^{-}})\Bigr{]}$

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