# Antiferromagnetic helix as an efficient spin polarizer: Interplay between
electric field and higher ordered hopping

Debjani Das Gupta  Physics and Applied Mathematics Unit, Indian Statistical
Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India  Santanu K.
Maiti<EMAIL_ADDRESS>Physics and Applied Mathematics Unit, Indian
Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India

###### Abstract

We report spin filtration operation considering an antiferromagnetic helix
system, possessing zero net magnetization. Common wisdom suggests that for
such a system, a spin-polarized current is no longer available from a beam of
unpolarized electrons. But, once we apply an electric field perpendicular to
the helix axis, a large separation between up and down spin energy channels
takes place which yields a high degree of spin polarization. Such a
prescription has not been reported so far to the best of our concern.
Employing a tight-binding framework to illustrate the antiferromagnetic helix,
we compute spin filtration efficiency by determining spin selective currents
using Landauer-Büttiker formalism. Geometrical conformation plays an important
role in spin channel separation, and here we critically investigate the
effects of short-range and long-range hoppings of electrons in presence of the
electric field. We find that the filtration performance gets improved with
increasing the range of hopping of electrons. Moreover, the phase of spin
polarization can be altered selectively by changing the strength and direction
of the electric field, and also by regulating the physical parameters that
describe the antiferromagnetic helix. Finally, we explore the specific role of
dephasing, to make the system more realistic and to make the present
communication a self-contained one. Our analysis may provide a new route of
getting conformation-dependent spin polarization possessing longer range
hopping of electrons, and can be generalized further to different kinds of
other fascinating antiferromagnetic systems.

##  I Introduction

After the discovery of the giant magnetoresistance (GMR) effect gmr1 ; gmr2 ;
prinz , spintronics becomes a new field of research in the discipline of
condensed matter physics where the spin degree of freedom of an electron is
explored along with its charge wolf ; spin1 ; spin2 ; spin3 . In conventional
electronic devices, Joule heating is an inevitable effect due to the flow of
electrons which causes a sufficient power loss. But, if we utilize electron
spin instead of the charge, then power will be consumed and at the same time
operation will be much faster spin1 ; spin2 . Two pivotal features for the
consideration of spin-based electronic devices rather than conventional
charged-based ones are: (i) saving more power and information can be
transferred at a much faster rate which undoubtedly reduces cost price by a
significant amount and (ii) size of the devices becomes too small so that a
large number of functional elements can be integrated into a small dimension
wolf ; spin1 . For instance, the hard disk drive made in 1957 was able to
store data only up to $3.75$ megabytes, and it occupied a volume of $68$ cubic
feet. Whereas, a recent hard disk drive possessing a volume of the order of
$2.1$ cubic inches can even store data up to several terabytes hd . Using the
GMR phenomenon, not only in storing devices but significant development has
been made in different other technologies involving electron spin spin1 ;
spin2 ; spin3 .

One of the most fundamental issues in spintronics is to find an efficient
route for the separation of two spins, or more precisely we can say, the
generation of polarized spin current from a completely unpolarized electron
beam. Several propositions have already been made along this line curr1 ;
curr2 ; fm1 . The most common practice is to use ferromagnetic materials,
though there are several unavoidable limitations fm2 . For instance, a large
resistivity mismatch occurs across a junction formed by ferromagnetic and non-
magnetic materials, which hinders the proper injection of electrons into the
system fm2 ; fm3 . The other crucial limitation arises when we think about the
tuning spin selective junction currents. Usually, this is done by means of an
external magnetic field, but for a quantum regime, it is very hard to confine
a strong magnetic field, and the problem still persists even today. Over the
last few years, the use of ferromagnetic materials significantly gets
suppressed when spin-orbit (SO) coupled systems came into the picture. Two
different kinds of SO interactions are taken into account in solid-state
materials, one is known as Rashba rashba and the other one is referred to as
Dresselhaus SO interaction dres ; soi . The latter type appears due to the
breaking of bulk inversion symmetry of a system, whereas the previous one
arises due to the breaking of the symmetry in confining potential. Among these
two, the Rashba strength can be tuned externally by suitable setups gate1 ;
gate2 , and therefore, the Rashba SO coupled systems draw significant
attention than the Dresselhaus ones in the field of spintronics. Different
kinds of systems starting from tailor-made geometries, organic and inorganic
molecules have been considered as functional elements in two-terminal as well
as multi-terminal setups, and many interesting features have been explored
multi1 ; multi2 ; multi3 . But, in most of these cases, especially in
molecular systems, the major concern is that the SO coupling strength is too
weak compared to the electronic hopping strength, almost an order of magnitude
smaller strength . Moreover, the variation of the SO coupling strength is also
quite limited by external means. Because of these facts, a high degree of spin
polarization and its possible tuning in a wide range are quite difficult to
achieve in SO coupled systems, though there are of course many other
advantages that make these systems promising functional elements in
spintronics.

To avoid all these issues, modern machinery has concentrated on
antiferromagnetic (AF) materials that possess an alternate type of magnetic
ordering and have zero net magnetization afm1 ; afm2 ; afm3 . Several key
prospects of using an AF system as a spin-polarized functional element are
there. For instance, these materials are insensitive to external magnetic
fields, and they are much faster and can be operated up to a high-frequency
range ($\sim$THz) than the traditional ferromagnetic systems thz . Moreover,
due to the absence of any stray fields, a large number of closely packed
functional elements can be accommodated in a small region which leads to
several important advantages in designing efficient electronic devices based
on spin-based transport phenomena. Nowadays, antiferromagnetic spintronics
evolves as a cutting-edge research field, and, may lead to new prospects in
the magnetic community afm4 ; afm5 ; afm6 ; afm7 .

In the present work, we propose a new prescription for efficient spin
filtration considering an antiferromagnetic system, which we refer to as an
“antiferromagnetic helix” (AFH). The role of chirality on spin filtration
first came into realization based on the experimental work of Göhler et al.
dna where they have shown that almost $60\%$ spin polarization can be achieved
through a self-assembled monolayers of double-stranded DNA molecules deposited
on a gold substrate. They have described this effect as chiral induced spin
selectivity (CISS). After this realization, several experimental and
theoretical research groups have paid significant attention to this CISS
effect, considering different kinds of molecular as well as artificially
designed systems, possessing helical geometry prl ; protein ; bacteria ; dna1
; helix ; aprotein ; edge ; sarkar ; magfield . But, to the best of our
knowledge, no effort has been made so far to address the phenomenon of spin
filtration considering a ‘magnetic helix structure with vanishing net
magnetization’, and this is precisely the fundamental motivation behind the
present work. For our AFH, simulated by a tight-binding framework, magnetic
moments in alternate lattice sites are arranged in opposite directions,
resulting in a zero net magnetization. In such a system, common wisdom
suggests that spin filtration is no longer possible. But, interestingly we
find that, once we apply an electric field perpendicular to the helix axis, a
large separation between two spin channels takes place, which results in a
high degree of spin polarization. The central mechanism relies on the helicity
and the applied electric field. In the absence of any of these two, helicity
and electric field, no such phenomenon is observed.

Determining spin-dependent transmission probabilities using the well-known
Green’s function formalism gf1 ; gff1 ; gf2 , we compute spin selective
currents through the AFH following the Landauer-Büttiker prescription gf3 ;
gf4 . From the currents, we evaluate the spin polarization coefficient. The
geometrical conformation plays a significant role in spin filtration, and we
investigate it by considering both short- and long-range hopping cases. From
our analysis, we find that the spin filtration efficiency gets enhanced with
increasing electron hopping among more lattice sites. The specific roles of
all other physical quantities are thoroughly discussed which lead to several
interesting features. Finally, to make the quantum system more realistic and
for the sake of completeness of our study, we include the effects of electron
dephasing dephase1 ; dephase2 ; dephase3 ; dephase4 ; dephase5 on spin
polarization. Our analysis may provide some key inputs towards designing
efficient spintronic devices considering different kinds of antiferromagnetic
helical systems, possessing longer range hopping of electrons.

The rest part of the work is arranged as follows.

Figure 1: (Color online). (a) A right-handed antiferromagnetic helix is
clamped between two one-dimensional non-magnetic electrodes, namely, source
and drain. $R$ being the radius of the helix, and $\Delta\phi$ and $\Delta h$
correspond to the twisting angle and stacking distance between the magnetic
sites (colored balls), respectively. Successive magnetic moments, shown by the
blue arrows, associated with the magnetic sites are arranged along $+Z$ and
$-Z$ directions, resulting in a vanishing net magnetization. An electric field
of strength $E_{g}$ is applied perpendicular to the helix axis, which plays a
pivotal role in spin filtration. (b) Orientation of the net spin
$\langle\boldsymbol{\vec{S}_{n}}\rangle$ in spherical polar coordinate system.
(c) Projection of the right-handed helix on the $X$-$Y$ plane.

Section II includes the spin-polarized setup, tight-binding (TB) Hamiltonian
of the junction, and the required theoretical prescriptions for the
calculations. All the results are presented and thoroughly discussed in Sec.
III. Finally, the essential findings are summarized in Sec. IV.

##  II Quantum system, TB Hamiltonian and theoretical formulation

###  II.1 Junction setup and the TB Hamiltonian

Let us begin with the spin-polarized setup shown in Fig. 1(a), where an
antiferromagnetic (AF) system is coupled to the source and drain electrodes.
The magnetic sites (filled colored balls) in the AF system are arranged in a
helical pattern. Each of these sites contains a finite magnetic moment,
denoted by the blue arrow, associated with a net spin
$\langle\vec{\boldsymbol{S}_{n}}\rangle$. The successive magnetic moments are
aligned in opposite directions ($\pm Z$), and therefore, the net magnetization
of the helix becomes zero. We refer to this system as an “antiferromagnetic
helix” (AFH), and here in our work, we will show how such an AFH acts for spin
filtration. The general orientation of any local spin
$\langle\vec{\boldsymbol{S}_{n}}\rangle$, and hence the magnetic moment, can
be illustrated by the usual co-ordinate system as presented in Fig. 1(b),
where $\theta_{n}$ and $\varphi_{n}$ are the polar and azimuthal angles,
respectively.

Two essential physical parameters characterize the helical geometry prl ;
protein those are $\Delta\phi$ and $\Delta h$ (see Fig. 1), where the first
one represents the twisting angle and the other parameter denotes the stacking
distance. Depending on $\Delta h$, we can have a helical system where lower or
higher-order hopping of electrons becomes significant. When $\Delta h$ is very
small i.e., atoms in the helix are densely packed, electrons can hop at
multiple sites, which is referred to as ‘long-range hopping antiferromagnetic
helix’ (LRH AFH). On the other hand, when the atoms are less densely packed
viz, $\Delta h$ is quite large, electrons can hop between a few neighboring
magnetic sites. Such a system is called a ‘short-range hopping
antiferromagnetic helix’ (SRH AFH). In the present work, we consider these two
different kinds of AF helices and investigate the results.

The antiferromagnetic helix is subjected to an electric field, having strength
$E_{g}$, perpendicular to the helix axis. It plays a central role in spin
filtration, and it can be understood from our forthcoming discussion. With a
suitable setup, one can tune the strength of this field as well as its
direction.

In order to investigate spin dependent transport phenomena and to exhibit the
spin filtration operation, the helical system is clamped between source and
drain electrodes. We describe this nanojuntion using the tight-binding
framework sk1 ; sk2 ; sk3 ; skm1 . The TB Hamiltonian of the full system is
written as a sum

$\displaystyle\boldsymbol{\mathit{H}}=\boldsymbol{\mathit{H}}_{AFH}+\boldsymbol{\mathit{H}}_{S}+\boldsymbol{\mathit{H}}_{D}+\boldsymbol{\mathit{H}}_{tun}$
(1)

where different sub-Hamiltonians in the right side of Eq. 1 are associated
with different parts of the nanojunction and they are described as follows.

The term $\boldsymbol{\mathit{H}}_{AFH}$ corresponds to the Hamiltonian of the
antiferromagnetic helix. For an AFH, be it a short-range or a long-range one,
the TB Hamiltonian is expressed as

$\displaystyle\boldsymbol{\mathit{H}}_{AFH}=\sum_{n}\boldsymbol{c}_{n}^{\dagger}\left(\boldsymbol{\epsilon}_{n}-\boldsymbol{\vec{\mathpzc{h}}}_{n}\cdot\boldsymbol{\vec{\sigma}}\right)\boldsymbol{c}_{n}$
$\displaystyle+\sum_{n=1}^{N-1}\sum_{m=1}^{N-n}\left(\boldsymbol{c}_{n}^{\dagger}\boldsymbol{t}_{m}\boldsymbol{c}_{n+m}+h.c.\right).$
(2)

where $\boldsymbol{c}_{n}^{\dagger}=\left(c_{n\uparrow}^{\dagger}\quad
c_{n\downarrow}^{\dagger}\right)$. $c_{n\sigma}^{\dagger}$ $(c_{n\sigma})$ is
the creation (annihilation) operator of an electron at site $n$ with spin
$\sigma\left(=\uparrow,\downarrow\right)$.
$\boldsymbol{\epsilon}_{n}-\vec{\boldsymbol{\mathpzc{h}}}_{n}\cdot\vec{\boldsymbol{\sigma}}$
is the effective site energy matrix which looks like

$\boldsymbol{\epsilon}_{n}-\vec{\boldsymbol{\mathpzc{h}}}_{n}\cdot\vec{\boldsymbol{\sigma}}=\begin{pmatrix}\epsilon_{n}-\mathpzc{h}\cos\theta_{n}&-\mathpzc{h}\sin\theta_{n}e^{-i\varphi_{n}}\\\
-\mathpzc{h}\sin\theta_{n}e^{i\varphi_{n}}&\epsilon_{n}+\mathpzc{h}\cos\theta_{n}\end{pmatrix}$
(3)

where $\epsilon_{n}$ is the on-site energy in the absence of any kind of
magnetic scattering. $\boldsymbol{\vec{\mathpzc
h}}_{n}=J\langle\boldsymbol{\vec{S}}_{n}\rangle$, called as the spin-flip
scattering parameter where $J$ is the coupling strength strength between the
coupling of an itinerant electron with local magnetic moment, associated with
the average spin $\langle\boldsymbol{\vec{S}}_{n}\rangle$.
$\vec{\boldsymbol{\sigma}}$ is the Pauli spin vector. Here we assume that
$\boldsymbol{\sigma}_{z}$ is diagonal. The term
$\vec{\boldsymbol{\mathpzc{h}}}_{n}\cdot\vec{\boldsymbol{\sigma}}$ represents
the spin-dependent scattering and it is widely used in literature sk1 ; sk2 ;
sk3 ; skm1 . The key point is that the strength ‘$J$’ is reasonably large than
the other spin-dependent scattering parameters viz, SO coupling, Zeeman
splitting in presence of magnetic field, etc strength . Thus, there is a
possibility of getting a high degree of spin filtration under suitable input
condition(s), in the presence of a spin-moment scattering mechanism.

The second term of Eq. 2, involving $\boldsymbol{t}_{m}$, is quite tricky, not
like usual nearest-neighbor hopping case, and the summations over $n$ and $m$
need to take carefully. $\boldsymbol{t}_{m}$ is a $(2\times 2)$ hopping
matrix, and it becomes

$\displaystyle\boldsymbol{t}_{m}=\begin{pmatrix}t_{m}&0\\\
0&t_{m}\end{pmatrix}$ (4)

where $t_{m}$ represents the hopping between the sites $n$ and $(n+m)$. The
hopping strength $t_{m}$ is written as prl ; protein

$t_{m}=t_{1}\textrm{e}^{-(l_{m}-l_{1})/l_{c}}$ (5)

where $t_{1}$ is the nearest-neighbor hopping integral, $l_{m}$ is the
distance of separation between the sites $n$ and $(n+m)$, $l_{1}$ is the
distance among the nearest-neighbor sites and $l_{c}$ is the decay constant.
In terms of the radius $R$ (see Fig. 1(c) where the projection of the helix in
the $X$-$Y$ plane is shown), twisting angle $\Delta\phi$ and the stacking
distance $\Delta h$, $l_{m}$ gets the form prl ; protein ; edge

$l_{m}=\sqrt{{[2R\sin(m\Delta\phi/2)}]^{2}+(m\Delta h)^{2}}.$ (6)

When the AFH is subjected to a transverse electric field, its site energies
get modified. The effective site energy for any site $n$ becomes aprotein ;
edge

$\epsilon_{n}^{\rm eff}=\epsilon_{n}+eV_{g}\cos(n\Delta\phi-\beta)$ (7)

where $e$ is the electronic charge, and, $V_{g}$ ($=2E_{g}R$) is the gate
voltage, responsible for the generation of the electric field. $\beta$
represents the angle between the incident electric field and the positive
$X$-axis prl ; protein .

The TB Hamiltonians of the side attached source (S) and drain (D) electrodes,
$\boldsymbol{\mathit{H}}_{S}$ and $\boldsymbol{\mathit{H}}_{D}$ and their
coupling with the AFH ($\boldsymbol{\mathit{H}}_{tun}$) look quite simple than
what is described above for the antiferromagnetic helix. The electrodes are
assumed to be perfect, one-dimensional, and non-magnetic in nature. They are
expressed as,

$\displaystyle\boldsymbol{\mathit{H}}_{S}=\boldsymbol{\mathit{H}}_{D}=\sum_{n}\boldsymbol{a}_{n}^{\dagger}\boldsymbol{\epsilon}_{0}\boldsymbol{a}_{n}+\sum_{n}\left(\boldsymbol{a}_{n+1}^{\dagger}\boldsymbol{t}_{0}\boldsymbol{a}_{n}+h.c.\right)$
(8)

where $\boldsymbol{\epsilon}_{0}={\rm diag}(\epsilon_{0},\epsilon_{0})$ and
$\boldsymbol{t}_{0}={\rm diag}(t_{0},t_{0})$. $\epsilon_{0}$ and $t_{0}$ are
the on-site energy and nearest-neighbor hopping integral, respectively.
$\boldsymbol{a}_{n}^{\dagger}=\left(a_{n\uparrow}^{\dagger}\quad
a_{n\downarrow}^{\dagger}\right)$, $a_{n\sigma}$’s are the usual fermionic
operators in the electrodes.

Finally, the tunneling Hamiltonian is expressed as

$\displaystyle\boldsymbol{\mathit{H}}_{tun}=t_{S}\left(\boldsymbol{c}_{1}^{\dagger}\boldsymbol{a}_{-1}+h.c.\right)+t_{D}\left(\boldsymbol{c}_{N}^{\dagger}\boldsymbol{a}_{N+1}+h.c.\right)$
(9)

where $t_{S}$ and $t_{D}$ are the coupling strengths of the AFH with S and D,
respectively. We refer to the lattice site of the source which is coupled to
the helix as $-1$ and the site of the drain which is attached to the helix as
$N+1$. The sites $1,2\dots N$ ($N$ being the total number of magnetic sites)
are used for the AFH.

###  II.2 Theoretical formulation

In order to inspect spin-dependent transport phenomena and spin polarization
coefficient, the first and foremost thing that we need to calculate is the
two-terminal transmission probability. We compute it using the well known non-
equilibrium Green’s function formalism gf1 ; gff1 ; gf2 ; gf3 ; gf4 . In terms
of retarded and advanced Green’s functions, $\boldsymbol{\mathit{G}}^{r}$ and
$\boldsymbol{\mathit{G}}^{a}$, the spin-dependent transmission coefficient is
obtained from the expression gf1 ; gff1 ; gf4

$T_{\sigma\sigma^{\prime}}=\textrm{Tr}\left[\boldsymbol{\Gamma}_{S}\boldsymbol{\mathit{G}}^{r}\boldsymbol{\Gamma}_{D}\boldsymbol{\mathit{G}}^{a}\right]$
(10)

where,

$\boldsymbol{\mathit{G}}^{r}=\left(\boldsymbol{\mathit{G}}^{a}\right)^{\dagger}=\left[\boldsymbol{E}-\boldsymbol{\mathit{H}}_{AFH}-\boldsymbol{\Sigma}_{S}-\boldsymbol{\Sigma}_{D}\right]^{-1}.$
(11)

$\boldsymbol{\Sigma}_{S}$ and $\boldsymbol{\Sigma}_{D}$ are the self-energy
matrices gf1 ; gff1 ; gf2 which capture all the essential information of the
electrodes and their coupling with the helix. $\boldsymbol{\Gamma}_{S}$ and
$\boldsymbol{\Gamma}_{D}$ are the coupling matrices.

For an incoming electron with spin $\uparrow$, two things may occur. We can
have a finite possibility of getting up spin electron as a up spin or it can
be flipped. Similar options are also available for an injected down spin
electron. Thus, considering pure $(T_{\sigma\sigma})$ and spin flip
$(T_{\sigma\sigma^{\prime}})$ transmissions, we can write the net up and down
spin transmission probabilities ($T_{\uparrow}$ and $T_{\downarrow}$) as

$\displaystyle T_{\uparrow}$
$\displaystyle=T_{\uparrow,\uparrow}+T_{\downarrow,\uparrow}$ (12a)
$\displaystyle T_{\downarrow}$
$\displaystyle=T_{\downarrow,\downarrow}+T_{\uparrow,\downarrow}.$ (12b)

From the transmission coefficients $T_{\uparrow}$ and $T_{\downarrow}$, we
evaluate up and down spin junction currents, using the Landauer Büttiker
prescription gf1 ; gff1 ; gf2 ; gf3 ; gf4 . The spin-dependent current, when a
finite bias $V$ is applied across the AFH, is expressed as

$I_{\sigma}=\frac{e}{h}\int T_{\sigma}\left(f_{S}-f_{D}\right)dE$ (13)

where, $f_{S}$ and $f_{D}$ are the Fermi functions, associated with S and D,
respectively, and they are

$f_{S(D)}=\frac{1}{1+\textrm{e}^{\left(E-\mu_{S(D)}\right)/k_{B}\mathsf{T}}}.$
(14)

Here $\mu_{S}$ and $\mu_{D}$ are the electro-chemical potentials of S and D,
respectively, and $k_{B}\mathsf{T}$ is the thermal energy.

Determining $I_{\uparrow}$ and $I_{\downarrow}$, we evaluate spin filtration
efficiency following the relation dephase3

$P=\frac{I_{\uparrow}-I_{\downarrow}}{I_{\uparrow}+I_{\uparrow}}\times 100\%.$
(15)

When only up spin electrons propagate we get $P=100\%$, while for the
situation where only down spin electrons get transferred through the AFH, we
get $P=-100\%$. For the situation where both up and down spin electrons
propagate equally, no spin filtration occurs. We want to reach the limiting
value where $P=100\%$ or $-100\%$, which is usually very hard to achieve.

Inclusion of dephasing: Dephasing is an important factor and in many cases, it
cannot be avoided especially when we think about the experimental realization
of a theoretical proposal. There are different possible routes through which a
system is disturbed by dephasing, and it is thus required to incorporate its
effect in our analysis. Several methodologies dephase1 ; dephase2 ; dephase3 ;
dephase4 ; dephase5 are available for the inclusion of dephasing and most of
them are very complex. Büttiker on the other hand predicted phenomenologically
a very simple but elegant way to incorporate the dephasing effect into the
system dephase1 ; dephase2 , and here we use the same procedure. In this
prescription it is assumed that each lattice site of the AFH is attached to a
dephasing electrode, commonly referred to as Büttiker probe. The key concept
is that the dephasing electrodes will not drag or inject any finite number of
electrons into the system i.e., the net current passing through such
electrodes becomes exactly zero dephase1 ; dephase2 . Electrons from the AFH
enter into the dephasing electrodes, and after losing their phase memories,
they eventually come back to the parent system.

In order to achieve the zero current condition in different dephasing
electrodes, we need to choose the voltages $V_{n}$ ($V_{n}$ being the voltage
at $n$th dephasing electrode) in the appropriate way aprotein ; gff1 . The
$V_{n}$’s are determined following the Landauer-Büttiker current expression
gf1 ; gff1 associated with each dephasing electrode, and evaluating the bias
drop at different lattice sites of the helix. It is crucial to point out that,
the evaluation of this bias drop is quite complicated as it is a non-linear
problem. The prescription can be simplified to some extent by considering a
linear profile along the helix, which is most commonly used in literature, and
here in our present work we also follow it. Suppose a finite voltage $V_{0}$
is applied between the real electrodes $S$ and $D$, and (say) $V_{S}=V_{0}$
and $V_{D}=0$, without loss of any generality. Then, the voltages $V_{n}$ at
different lattice sites can be calculated without much difficulty, as we
assume the linear drop, and adjusting these voltages across the dephasing
electrodes, the zero-current condition is established (a more detailed
discussion about it is available in aprotein ; gff1 ).

In presence of the Bütiker probes, the transmission probability of getting
electrons at the drain electrode (D) is modified aprotein ; dephase5 , and it
becomes

$T_{\sigma\sigma^{\prime}}^{eff}(V)=\sum_{\alpha=S,n}T_{\sigma\sigma^{\prime}}^{\alpha
D}(V)\frac{V_{\alpha}}{V_{0}}.$ (16)

The transmission probabilities are now voltage dependent, and thus, special
care has to be taken to calculate these quantities vt1 ; vt2 . Here
$T_{\sigma\sigma^{\prime}}^{SD}(V)$ and $T_{\sigma\sigma^{\prime}}^{nD}(V)$
denote the transmission probabilities from the source electrode (S) and from
the $n$-th dephasing electrode to the drain end, respectively. The dephasing
electrodes are connected at all the lattice sites of the antiferromagnetic
helix, apart from the sites $1$ and $N$ where the real electrodes (S, D) are
attached. In our formulation, the coupling strength between the AFH and the
dephasing electrode is mentioned by the parameter $\eta$, and it describes the
dephasing strength.

To compute $T_{\sigma\sigma^{\prime}}^{eff}(V)$, an important step must be
performed which is as follows. For a biased system, since the scattering
states become the eigenstates of the biased Hamiltonian, the site energy
$\epsilon_{0}$ needs to be shifted by $V_{0}$ in the source electrode, and by
$V_{n}$ in the $n$-th Büttiker probe vt1 ; vt2 . Using Eq. 16, we get the
effective up and down spin transmission probabilities, at different voltages,
from the relations

$\displaystyle T_{\uparrow}^{eff}(V)$
$\displaystyle=T_{\uparrow,\uparrow}^{eff}(V)+T_{\downarrow,\uparrow}^{eff}(V)$
(17a) $\displaystyle T_{\downarrow}^{eff}(V)$
$\displaystyle=T_{\downarrow,\downarrow}^{eff}(V)+T_{\uparrow,\downarrow}^{eff}(V).$
(17b)

The effective spin-dependent current in presence of dephasing can thus be
obtained through the expression

$I_{\sigma}(V)=\frac{e}{h}\int T_{\sigma}^{eff}(V)\left(f_{S}-f_{D}\right)dE.$
(18)

With the effective spin-dependent currents, the same definition is followed as
mentioned in Eq. 15, to compute spin polarization coefficient $P$ in the
presence of dephasing.

In the extreme low biased condition, the above current equation (Eq. 18) for
any $q$-th electrode, be it real or virtual, boils down to aprotein

$I_{\sigma}^{q}(V)=\frac{e^{2}}{h}\sum_{\alpha}T_{\sigma}^{\alpha
q}(V_{\alpha}-V_{q})$ (19)

where the voltages $V_{\alpha}$ and $V_{q}$ can be derived from the
prescription given above. In this limiting condition, the current is
‘linearly’ proportional to the voltage. On the other hand, in the limit of
high bias, Eq. 18 cannot be simplified in the linear form like what is given
in Eq. 19, and we get the non-linear behavior. For the sake of completeness of
our analysis, we discuss the accuracy of the above prescription in the
appropriate sub-section.

##  III Numerical Results and Discussion

Now we present our results and investigate the specific role of the external
electric field on spin filtration, under different input conditions. Both the
short-range and long-range antiferromagnetic helices are taken into account.
For these two types of AFHs, we choose the geometrical parameters $R$, $\Delta
h$, and $\Delta\phi$ as given in Table 1. These parameter values are analogous
to the real helical systems like single-stranded DNA and protein molecules,
and they are the most suitable examples where respectively the short-range and
long-range hopping models are taken into account prl ; protein . A large
amount of investigation has already been done in the literature considering
this particular set of parameter values in different contemporary works, and
accordingly, here also we select these typical values. Other sets of parameter
values represent the SRH and LRH of electrons can also be considered, and all
the physical pictures studied here will remain unaltered.

The other physical parameters those are common throughout the analysis are as
follows. In the absence of electric field, the on-site

Table 1: Geometrical parameters describing the SRH and LRH AFHs. System |  $R$ (nm) |  $\Delta h$ (nm) |  $\Delta\phi$ (rad) |  $l_{c}$ (nm)
---|---|---|---|---
SRH AFH | $0.7$ | $0.34$ | $\pi/5$ | $0.09$
LRH AFH | $0.25$ | $0.10$ | $5\pi/9$ | $0.09$

energies ($\epsilon_{n}$) in the AFH are set to zero, and we fix the NNH
strength $t_{1}=1\,$eV. The spin dependent scattering parameters $\mathpzc{h}$
is set at $1\,$eV. As already mentioned, the successive magnetic moments in
the helix system are aligned in opposite directions ($\pm Z$) (see the
schematic diagram given in Fig. 1(a)). We set $\theta_{n}=0$ for all odd $n$
sites, and $\theta_{n}=\pi$ for all the even $n$ sites. The azimuthal angle
$\varphi_{n}$ is fixed to zero for all $n$. In the side attached electrodes,
we choose $\epsilon_{0}=0$, $t_{0}=2\,$eV. The coupling parameters $t_{S}$ and
$t_{D}$ are set at $1\,$eV. All the other energies are also measured in units
of electron-volt (eV). Unless specified, the results are worked out
considering a right-handed antiferromagnetic helix with $\beta=0$ and in the
absence of dephasing. We set the system temperature at $100\,$K, throughout
the discussion.

###  III.1 Spin dependent transmission probabilities and spin polarization
coefficient

Let us begin with spin-dependent transmission probabilities, shown in the
first column of Fig. 2, where Fig. 2(a) and Fig. 2(c) are associated with the
SRH and LRH AFHs respectively. The corresponding spin polarization
coefficients are presented in the right column. All these results are
evaluated for the electric field-free condition i.e. when $V_{g}=0$. For both
the helix systems we find that up and down spin transmission probabilities,
shown by the black and cyan colors respectively, exactly overlap with each
other, resulting in a vanishing spin polarization. This is expected, as for
such helices the symmetry between the up and down spin sub-Hamiltonians
($\boldsymbol{H}_{\uparrow}$ and $\boldsymbol{H}_{\downarrow}$) is preserved.
Accordingly, we get an identical set of energy eigenvalues for the two
different spin electrons, and thus the transmission probabilities, as the
transmission peaks are directly related to the energy eigenvalues of the
bridging system. The appearance of identical energy eigenvalues can easily be
checked by writing the Hamiltonian of the AFH
($\boldsymbol{\mathit{H}}_{AFH}$) as a sum of the two sub-Hamiltonians (viz,
$\boldsymbol{\mathit{H}}_{AFH}=\boldsymbol{\mathit{H}}_{\uparrow}+\boldsymbol{\mathit{H}}_{\downarrow}$),
one is associated with up spin electrons and the other is involved with the
down spin ones. In the absence of the electric field, these two sub-
Hamiltonians are effectively identical to each other, and hence, the same set
of energy channels is obtained. For a particular AFH, the sharpness of
different transmission peaks depends on the coupling ($t_{S}$ and $t_{D}$) of
the helix to the side attached electrodes. With increasing the coupling, the
transmission peaks get more broadened, and the broadening due to this coupling
is always higher than

Figure 2: (Color online). Zero field case: Spin dependent transmission
probabilities as a function of energy for (a) SRH and (c) LRH AFHs, where the
black and cyan colors are associated with the up and down spin electrons,
respectively. The up and down spin transmission spectra get exactly overlapped
with each other for each AFH. The corresponding spin polarization coefficients
with bias voltage are shown in (b) and (d), respectively. Spin polarization
drops to zero for the entire voltage window. Here we choose $E_{F}=-0.9\,$eV
and set $N=30$. The dephasing strength $\eta=0$.

that caused by the thermal effect. In our numerical calculation since $t_{S}$
and $t_{D}$ are comparable to $t_{1}$ (strong coupling limit), any significant
change with increasing temperature is not expected and therefore we restrict
our calculation at a moderate temperature.

Depending on the specific range of electron hopping, we get a contrasting
nature in the transmission peaks and their arrangements over the energy
window. For the chosen set of parameter values, electrons can able to hop in a
few neighboring lattice sites in the SRH system, and for this case, the
transmission peaks are quite uniformly spaced and peaked as well. More regular
behavior is obtained as we move towards the NNH model. A finite and large gap
is obtained across $E=0$, following the energy gap in the SRH system. This is
quite analogous to the binary alloy system where alternate sites possess two
different energies and repeat it throughout the system, due to the
antiferromagnetic ordering. But, once the higher-order hopping of electrons is
taken into account, like what is considered for the LRH system, the sharp gap
around $E=0$ disappears. At the same time, the uniformity is lost
significantly. Along one edge of the energy window, the transmission peaks are
closely packed whereas large gaps between the peaks are obtained along the
other edge of the window protein . All these characteristics are the generic
features of a long-range hopping system. This asymmetric distribution, on the
other hand, plays an important role to achieve a favorable response in spin
polarization, which can be visualized in the forthcoming discussion.

The situation drastically changes, once we apply an electric field. In Fig. 3
we show the results, for the same set of systems as taken in Fig. 2,
considering the gate voltage $V_{g}=0.6$ V. Several notable features are
obtained those are illustrated one by one as follows. At a first glance, we
find that up and down spin transmission probabilities get separated both for
the SRH and LRH helices, as clearly reflected from the spectra given in Figs.
3(a) and (c), where two different colors are associated with two

Figure 3: (Color online). Results in the presence of electric field with
$V_{g}=0.6\,$V. In (a) and (c), spin selective transmission probabilities as a
function of energy are shown for (a) SRH and (c) LRH AFHs, where the black and
cyan colors are associated with up and down spin electrons, respectively. The
corresponding $P$-$V$ curves are shown in (b) and (d), respectively. All the
other physical parameters are the same as taken in Fig. 2.

different spin electrons. In the presence of electric field, site energies get
modified in a cosine form, following Eq. 7, which makes the system a
disordered (correlated) one aah1 ; aah2 ; aah3 . Due to this disorder, the
symmetry between up and down spin sub-Hamiltonians gets broken, which provides
different sets of spin-specific energy channels. Under this situation, a
mismatch occurs between the spin-dependent transmission spectra. Apparently,
it seems that the separation between the up and down spin transmission peaks
is not that much large, what we generally expect from ferromagnetic systems,
but selectively placing the Fermi energy, we can have the possibility of
getting a reasonably high degree of spin filtration. This is precisely shown
in Figs. 3(b) and (d). Both for the SRH and LRH AFHs, large spin polarization
is obtained over a particular voltage range, but the response becomes more
favorable for the case of LRH AFH. This is entirely due to the non-uniform
distribution of the transmission peaks around the center of the spectrum. So,
naturally, starting from an NNH AFH, we can expect a better response whenever
we include an additional hopping of electrons, and we confirm it through our
detailed numerical calculations. Moreover, it is pertinent to note that, the
cosine modulation in site energies due to the electric field makes the system
a non-trivial one, as it generates a fragmented energy spectrum (which is the
generic feature of the well-known Aubry-André-Harper (AAH) model aah1 ; aah2 ;
aah3 ). This behavior helps us to find a high degree of spin polarization even
at multiple Fermi energies.

The key conclusion that is drawn from the above analysis is that the breaking
of the symmetry between up and down spin sub-Hamiltonians in the AFH entirely
depends on the external electric field which makes the system a disordered
(correlated) one. In the absence of helicity, the site energies become
uniform. Under this situation, the symmetry between the sub-Hamiltonians
associated with up and down spin electrons gets preserved, and hence, no such
spin filtration phenomenon is obtained.

###  III.2 Possible tuning of spin polarization

This sub-section deals with the possible tuning of spin filtration efficiency,
by adjusting different parameter values associated with the junction setup.

Figure 4: (Color online). Simultaneous variations of $P$ with bias voltage $V$
and gate voltage $V_{g}$ for a $50$-site LRH AFH when $E_{F}$ is fixed at
$1\,$eV. The dephasing parameter $\eta=0$.

From the above analysis since it is already established that LRH AFH is
superior to the SRH AFH, in the rest part of our discussion we concentrate
only on the LRH AFH systems unless stated otherwise.

Figure 4 shows simultaneous variations of spin polarization coefficient $P$
with the bias voltage $V$ and the gate voltage $V_{g}$. Both these two factors
play a significant role in spin filtration efficiency. When $V_{g}$ is
absolutely zero (i.e., in the absence of electric field) there is no spin
polarization as up and down spin sub-Hamiltonians are symmetric. With
increasing $V_{g}$, we are expecting a favorable spin polarization, but
attention has to be given to the localizing behavior of energy eigenstates.
The inclusion of an electric field transforms the AFH from a completely
perfect to a correlated disordered system, and thus, for large $V_{g}$ the
eigenstates will be almost localized. In that limit, we cannot get spin
filtration operation. Hence, we need to restrict $V_{g}$ in such a way that
the energy channels are conducting in behavior (see Ref. sarkar for a
comprehensive analysis of electronic localization in SRH and LRH helices in
presence of transverse electric field). For a fixed $V_{g}$, when a finite
voltage drop is introduced across the junction, we get a non-zero spin
polarization depending on the dominating energy channels among up and down
spin electrons. With the increase of the bias window, more and more number of
both up and down spin channels are available that contribute to the current,
and hence, the possibility of mutual cancellations becomes higher which can
reduce the degree of spin polarization. Thus, both $V_{g}$ and the bias
voltage need to be considered selectively, to have a favorable response.

Like bias voltage $V$, the choice of $E_{F}$ is also very crucial. Our aim is
to find a suitable $E_{F}$ where any of the two spin channels dominates over
the other, as maximum as it is possible. This, on the other hand, is directly
linked with the gate voltage $V_{g}$ as it

Figure 5: (Color online). Dependence of $P$ with $E_{F}$ and $V_{g}$ when the
bias voltage is fixed at $0.2\,$V. The other parameters are: $N=50$ and
$\eta=0$.

influences the site energies of the AFH. To have an idea about the selection
of $V_{g}$ and $E_{F}$, in Fig. 5 we show the dependence of $P$ on these
quantities. The results are computed, setting the voltage drop $V=0.2\,$V.
This typical bias voltage is considered due to the fact that here we can get a
favorable response as reflected from Fig. 4. We vary $E_{F}$ almost over the
entire allowed energy window, and it is seen that for a wide range of $E_{F}$
($\sim-2\leq E_{F}\leq\sim 2$), a reasonably large $P$ is obtained. Thus, fine
tuning of $E_{F}$ is no longer required, which is of course quite important in
the context of experimental realization of our proposed setup.

In the same footing it is indeed required to check the filtration efficiency
when we simultaneously vary $E_{F}$ and the bias voltage $V$, keeping the gate
voltage constant. The results are shown in Fig. 6, where we fix
$V_{g}=0.8\,$V. This typical value of $V_{g}$ is chosen observing the
favorable response from Figs. 4 and 5. Here also we find that a high degree of
spin polarization is achieved at different bias drops across the junction for
several distinct choices of Fermi energy $E_{F}$. All these favorable
responses are associated with the modifications of the up and down spin energy
channels.

Along with the favorable spin polarization, a complete phase reversal (change
of sign) of $P$ is also obtained from all these figures (Figs. 4-6) which is
due to the swapping of dominating spin channels with the change of the
physical parameters.

Now we concentrate on the effect of field direction, which is changed by the
parameter $\beta$. The dependence of spin polarization with $\beta$ is
presented in Fig. 7, by varying $\beta$ from $0$ to $2\pi$. Almost a regular
oscillation is shown providing high peaks ($P\sim 100\%$) and dips
($P\sim-100\%$). The magnitude and sign reversal of $P$ are due to the
modifications of spin specific energy

Figure 6: (Color online). Dependence of $P$ with $E_{F}$ and $V$ when the gate
voltage is fixed at $0.8\,$V. The other parameters are: $N=50$ and $\eta=0$.

Figure 7: (Color online). Variation of spin polarization ($P$) with the field
direction ($\beta$). The other physical parameters are: $E_{F}=0.5\,$eV,
$V=0.2\,$V, $V_{g}=0.8\,$V, $N=50$ and $\eta=0$.

channels of the helix with $\beta$ as it is directly related to the site
energy (see Eq. 7). It suggests that the spin filtration efficiency can be
monitored selectively by changing the field direction, keeping all the other
factors unchanged.

Following the above analysis it is found that the helicity and electric field
are strongly correlated with each other, and in the absence of any of these
two, spin filtration is no longer possible. At this stage, it is indeed
required to check the effect of helicity on a more deeper level. In Fig. 8 we
show the conformational effect of the helical geometry on spin polarization.
We vary $\Delta h$ and $\Delta\phi$ in a reasonable range around the chosen
values of these parameters for the LRH AFH, as mentioned in Table 1. The
radius $R$ is kept constant which is $0.25\,$nm. The twisting angle
$\Delta\phi$ has an important role as it modulates the site energy as well as
the hopping integrals. But the more pronounced effect is observed by changing
the stacking distance $\Delta h$. With the enhancement of

Figure 8: (Color online). Simultaneous variations of $P$ with $\Delta\phi$ and
$\Delta h$. Here we choose $E_{F}=0.5\,$eV, $V=0.2\,$V, $V_{g}=0.8\,$V,
$N=50$, $\beta=0$ and $\eta=0$.

$\Delta h$, for a fixed $\Delta\phi$, the hopping of electrons in larger sites
gradually decreases, and the system approaches the NNH model, providing
reduced spin polarization. In the limiting region when electrons can hop only
between the nearest-neighbor sites, the spin polarization becomes vanishingly
small. Carefully inspecting the density plot given in Fig. 8, it is inferred
that the best performance is obtained for low $\Delta h$ when the twisting
angle is confined within the range $\sim\pi/2\leq\phi\leq\sim 2\pi/3$. Thus,
the helicity and higher-order hopping of electrons are the key aspects to
having a high degree of spin filtration.

###  III.3 Effect of dephasing

To have a more realistic situation, especially considering the experimental
facts, in this sub-section we explore the specific role of dephasing dephase1
; dephase2 ; dephase3 ; dephase4 ; dephase5 that may enter into the system in
many ways on spin polarization. Different kinds of interactions of the
physical system with external factors can phenomenologically be incorporated
through the dephasing effect.

$\blacksquare$ Accuracy checking of theoretical prescription in presence of
dephasing: Before presenting the results in the presence of dephasing, it is
indeed required to check the accuracy of the theoretical formulation
illustrated above in Sec. II. To do that, we consider a spin-less LRH helix
with $15$ lattice sites, as a typical example, and compute the currents in all
the Büttiker probes (such probes are connected in all the sites of the helix
from the site number $2$ to $14$), along with the drain current (red curve).
The results are shown in Fig. 9 for the two distinct dephasing strengths that
are presented in (a) and (b), setting the Fermi energy $E_{F}=0.3\,$eV. It is
clearly seen that the drain current is reasonably large than the currents in
all the Büttiker probes, and most interestingly we find that the currents in
the Büttiker probes are almost zero,

Figure 9: (Color online). Current in the drain electrode (red curve) along
with the currents in all the Büttiker probes (other colored curves, not
clearly seen as they almost overlap to each other) as a function of bias
voltage for a spin-less LRH helix, where (a) $\eta=0.25$ and (b) $\eta=0.5$.
The other physical parameters are: $N=15$, $E_{F}=0.3\,$eV and $V_{g}=0$.

even for too high voltages. This is exactly what we are expecting i.e., the
vanishing current condition in each dephasing electrode. Thus, we can argue
that the theoretical prescription given here can safely be used to study the
effect of dephasing.

Now come to the results of AFH in presence of dephasing. Like Fig. 6, in Fig.
10 we show the simultaneous variations of $P$ on the bias voltage $V$ and the
Fermi energy $E_{F}$, fixing the dephasing strength $\eta=0.1$. All the other
physical parameters are kept unchanged as taken in Fig. 6. The effect of
dephasing is quite appreciable. What we see is that the Fermi energy and the
bias windows for which a high degree of spin polarization is available in the
absence of dephasing (see Fig. 6), get reduced when the dephasing effect is
taken into account. The reduction of spin polarization in presence of $\eta$
can be explained as follows. In the Büttiker probe prescription, the effect of
dephasing is incorporated by connecting each and every lattice site of the AFH
with virtual electrodes. Due to the coupling of the AFH to these virtual
electrodes, transmission peaks get broadened, and thus more overlap takes
place between the up and down spin channels. Therefore, these two spin-
dependent channels contribute to the current, and hence, the spin polarization
decreases. A similar kind of dephasing effect (viz, reduction of $P$ with
$\eta$) is also obtained when we observe simultaneous variations of $P$ with
$V_{g}$ and $V$ keeping $E_{F}$ constant, and, $V_{g}$ and $E_{F}$ considering
a fixed bias voltage. Accordingly, here we do not present the density plots of
$P$, like what are shown in Fig. 4 and Fig. 5, in the presence of $\eta$ as
the role $\eta$ can be guessed in these cases.

Here it is relevant to check the dependence of spin current
($I_{S}=I_{\uparrow}-I_{\downarrow}$) and spin polarization for other values
of the dephasing strength $\eta$ as well. The results are given in Figs. 11
(a) and (b), respectively, by varying $\eta$ in a

Figure 10: (Color online). A similar kind of density plot, as given in Fig. 6,
in the presence of dephasing, with the dephasing strength $\eta=0.1$.

Figure 11: (Color online). (a) Spin current and (b) spin polarization
coefficient as a function of dephasing strength $\eta$, at three typical Fermi
energies. Here we set $N=50$, $V=0.2\,$V and $V_{g}=0.8\,$V.

reasonable range. Three different Fermi energies are taken into account, those
are represented by three different colored curves. It is clearly observed that
the spin current decreases with increasing $\eta$, following the above
argument of the broadening of both up and down spin transmission peaks and
their overlap (Fig. 11(a)). This reduction of spin current can also be
implemented by means of phase randomization of the electrons in the virtual
electrodes, as originally put forward by Büttiker dephase1 ; dephase2 ;
dephase5 . Electrons enter into these electrodes and they come back to the
helix system after randomizing their phases, and thus more mixing of the two
opposite spin electrons occurs, resulting in a reduced spin current. In the
same analogy, the degree of spin polarization gets decreased with increasing
the dephasing strength (Fig. 11(b)). From $P$-$\eta$ characteristics it is
found that though the spin filtration efficiency gets reduced with $\eta$,
still a sufficiently large value of $P$ is obtained even when $\eta$ is
reasonably large. Thus, we can safely claim that our proposed quantum system
can be utilized as an efficient functional element for spin filtration
operation under strong environmental interactions as well as the limit of high
temperatures.

###  III.4 Possible experimental routes of designing AFH

Finally, we refer to some experimental works where different kinds of anti-
ferromagnetic systems have been used, aiming to establish confidence that our
proposed magnetic helix system can also be realized experimentally. In
presence of an external magnetic field, Johnston has reported afhelix1
magnetic properties and other related phenomena considering an AFH system.
There are several experimental works performed by Sangeetha and co-workers,
where they have found helical antiferromagnet in different compounds. For
instance in Ref. afhelix2 , Sangeetha et al. have established a transition
from antiferromagnetic to paramagnetic phase using the compound ${\rm
EuCo}_{2-y}{\rm As}_{2}$ with spin $S=7/2$ which possesses a helical shape
afhelix2 . In this work, they have measured different physical quantities like
magnetic susceptibility, heat capacities, etc. A high nuclear magneto
resistance (NMR) has also been found in that sample nmr . In another work,
Sangeetha and co-workers have established helical antiferromagnetic ordering
considering ${\rm EuNi}_{1.95}{\rm As}_{2}$ single crystal afhelix3 . Goetsch
et al. have reported the same in polycrystalline sample ${\rm Lu}_{1-x}{\rm
Sc}_{x}{\rm MnSi}$ at higher Neél’s temperature afhelix4 . The
antiferromagnetic helical pattern has also been noticed in an organic
molecule. Lin et al. have reported the canted antiferromagnetic behavior in
${\rm[M(mtpo)_{2}(H_{2}O)_{n}]}$, ${\rm M=Co^{2+}}$ or ${\rm Ni^{+}}$ with a
helical topology afhelix5 . Pylypovskyi et al. have suggested an idea about
the tailoring of the geometry of curvilinear antiferromagnet afhelix6 . This
prescription allows substantiating a chiral helimagnet in presence of
Dzyloshinskii-Moriya interaction. Considering the helical antiferromagnet
sample ${\rm SrFeO}_{3-\delta}$, Zhao et al. have discussed a metal-insulator
transition afhelix7 . There exist several other antiferromagnetic helices as
well.

Considering all such examples of antiferromagnetic helical systems, we believe
that our proposed spin-polarized AFH system can be designed with modern
technology and with a suitable laboratory setup.

Here it is relevant to note that all the above-mentioned experimental
references contain heavy magnetic elements, and thus one may think whether the
tight-binding Hamiltonian mentioned in Eq. 2 can be used to describe our helix
systems or not. But the theoretical work studied by Takahashi and Igarashi
taka gives us confidence that Eq. 2 can safely be considered, as in that work
they have also taken a similar kind of tight-binding Hamiltonian to describe
${\rm La_{2}CuO_{4}}$ and ${\rm Sr_{2}CuO_{2}Cl_{2}}$. There exist several
other references as well pic ; cao where tight-binding Hamiltonians have been
taken into account for such types of heavy magnetic elements.

##  IV Summary and outlook

For the first time, we report spin filtration operation considering an
antiferromagnetic helical system in presence of an external electric field.
Both the short-range and long-range hopping cases are taken into account,
associated with the geometrical conformation. Simulating the spin-polarized
nanojunction (source-AFH-drain) within a tight-binding framework, we compute
spin-dependent transmission probabilities following the well-known Green’s
function formalism and the spin-dependent junction currents through the
Landauer-Büttiker prescription. From the currents, we evaluate the spin
polarization coefficient. To make the proposed quantum system more realistic,
we also include the effects of dephasing, and to get the confidence, the
accuracy of the theoretical prescription in presence of dephasing is
critically checked. Finally, we discuss the possible routes of realizing such
an antiferromagnetic helical geometry in a laboratory. Different aspects of
spin-dependent transmission probabilities and spin polarization coefficient
under different input conditions are critically investigated. The essential
findings are listed as follows.
$\bullet$ In the absence of the external electric field, up and down spin sub-
Hamiltonians become symmetric and thus no spin polarization is obtained. Once
the symmetry is broken by applying the electric field, finite spin
polarization is found.
$\bullet$ Comparing the results between the SRH and LRH AFHs, it is noticed
that LRH AFH is much superior to the other. This is essentially due to the
irregular distribution of the resonant peaks in the transmission-energy
spectrum. The irregularity gradually decreases with lowering the electron
hopping among the lattice sites.
$\bullet$ The degree of spin polarization and its phase can be tuned
selectively by means of the input parameters, and the notable thing is that a
high degree of spin polarization persists over a reasonable range of physical
parameters. It clearly suggests that fine tuning of the parameter values is no
longer required, and hence we hope that the studied results can be examined in
a laboratory.
$\bullet$ The geometrical conformation plays an important role. For the
situation when $\Delta\phi$ becomes zero i.e., in the absence of twisting,
spin filtration is no longer available.
$\bullet$ Though the degree of spin filtration efficiency gets reduced with
increasing the dephasing strength $\eta$, still, reasonably large spin
polarization is available, even for moderate $\eta$. It suggests that the
proposed functional element can safely be used for spin polarization under
strong environmental interactions as well as in the limit of high
temperatures.

Our present proposition may help to design efficient spintronic devices using
antiferromagnetic helices with longer-range hopping of electrons, and can be
generalized to other correlated antiferromagnetic systems as well.

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[a, b]Ekaterina Kriukova

#  INR-TH-2024-004 Dark photon emission in elastic proton bremsstrahlung

###### Abstract

We study the production of dark photons with masses in range 0.4-1.5 GeV in
elastic proton bremsstrahlung. In contrast to the known approaches to
calculating the bremsstrahlung cross section, we consider the non-zero
momentum transfer between protons. The numerical estimates show that the
refined result agrees well with the generalized Weizsacker-Williams
approximation, although it significantly differs from the result obtained by
Blumlein and Brunner [1] which is widely used in proton beam-dump
phenomenology.

##  1 Dark photons

One of the most popular ways to describe the structure of any extension of the
Standard Model (SM) of particle physics is to consider it as the coexistence
of the SM particles, the so-called portal and the hidden sector (HS). The
portal is the way to write down the interaction between the SM and the HS. It
is realized by new physics particles called mediators that can interact both
with the particles from the SM sector and the hidden one.

There are only three renormalizable portals — scalar, vector and fermion ones
[2]. Scalar portal is produced by the cubic and quartic interaction of dark
scalars with the SM Higgs doublet. The fermion portal includes heavy neutral
leptons as mediators that couple to SM leptons and the Higgs doublet via
Yukawa-like interaction. Finally, the vector portal, that we consider in this
paper, consists of the dark photons which are the massive vector particles
that kinetically mix with the SM photons.

In this work we consider the so-called minimal dark photon model (BC1 in
Beyond Colliders at CERN classification [3]), where dark photons interact only
with the SM particles and any interaction of dark photons with the dark matter
is neglected. Thus one can write part of the Lagrangian relevant for our study
as [4]

$\mathcal{L}=\mathcal{L}_{\text{SM}}-\frac{1}{4}F^{\prime}_{\mu\nu}F^{\prime\mu\nu}+\frac{\epsilon}{2}F^{\prime}_{\mu\nu}B^{\mu\nu}+\frac{m_{\gamma^{\prime}}^{2}}{2}A^{\prime}_{\mu}A^{\prime\mu},$
(1)

where $\mathcal{L}_{\text{SM}}$ is the SM Lagrangian, $F^{\prime}_{\mu\nu}$ is
the dark photon strength tensor, $B_{\mu\nu}$ is the SM $U(1)_{Y}$ strength
tensor. Then the model is characterized by two parameters — the kinetic mixing
parameter $\epsilon$ and the dark photon mass $m_{\gamma^{\prime}}$.

Among the experiments that have performed searches for minimal dark photons
(see [5] for review) there are NA64 [6], PS191 [7], BaBar [8], LHCb [9], CMS
[10], etc. It is important to note that despite being intensively studied, the
minimal dark photon parameter plane $(\epsilon,m_{\gamma^{\prime}})$ is still
weakly constrained for $m_{\gamma^{\prime}}>0.8\text{ GeV}$. It will be
possible to study the phenomenology of $\mathcal{O}(1)$ GeV dark photons
produced in $pp$-collisions in the near detectors of neutrino experiments
being prepared such as DUNE [11] and T2K [12]. The proposed SHiP experiment at
CERN could also search for visible decays of dark photon [13].

Figure 1

Figure 1: The dominant mechanisms of dark photon production in $pp$-collisions
depending on the dark photon mass $m_{\gamma^{\prime}}$ from left to right:
meson decays, proton bremsstrahlung and Drell-Yan process.

depicts three possible mechanisms of dark photon production in
$pp$-collisions. Which mechanism prevails is determined by the dark photon
mass $m_{\gamma^{\prime}}$ [14]. Dark photons lighter than 0.4 GeV are mostly
produced via $\pi^{0}$, $\eta$ meson decays as shown in green in figure 1. The
analogue of QCD Drell-Yan process shown in blue is dominant for heavier dark
photons with masses above about 1.8 GeV. Red boxes illustrate the process of
proton bremsstrahlung that is crucial for dark photon masses in range 0.4-1.8
GeV and was studied in this work.

The paper is organized as follows. Section 2 reviews the previously known
estimates of bremsstrahlung cross section. In section 3 we present the result
of our calculation of elastic proton bremsstrahlung taking into account the
non-zero momentum transfer between protons. The numerical results and
comparison with other methods are summarized in section 4. Finally, section 5
contains conclusions.

##  2 Previously suggested approximations for proton bremsstrahlung

Figure 2(a)

(a)

(b)

Figure 2: Feynman diagrams schematically illustrating the known
approximations: (a) the Weizsacker-Williams approximation for electron
bremsstrahlung [15], (b) the Blumlein-Brunner approximation for proton
bremsstrahlung [1]. The subdiagram relevant to splitting function
$w_{p\gamma^{\prime}}$ is shown in red.

shows the idea of generalized Wiezsacker-Williams (WW) approximation for the
process of electron bremsstrahlung
$e(a)p(P_{i})\rightarrow\gamma^{\prime}(b)e(c)p(P_{f})$ with one-photon
exchange [15]. Since the impact of photon propagator to the cross section is
proportional to ${1}/{t^{2}}$, where $t\equiv-q^{2}$ is the minus square of
photon momentum $q$ transferred between protons, the largest contribution is
given at the minimal square of momentum transfer $t_{\text{min}}$,

$\sqrt{t_{\text{min}}}=\frac{(a\cdot
b)-m^{2}_{\gamma^{\prime}}/2}{a_{0}-b_{0}}.$ (2)

This allows one to connect the answer with the cross section of the
$2\rightarrow 2$ subprocess $e\gamma\rightarrow\gamma^{\prime}e$ that is
highlighted with green ellipse in figure 2(a),

$\left[\frac{\differential^{2}\sigma(ep\rightarrow\gamma^{\prime}ep)}{\differential(a\cdot
b)\differential(b\cdot
P_{i})}\right]_{WW}=\left[\frac{\differential\sigma(e\gamma\rightarrow\gamma^{\prime}e)}{\differential(a\cdot
b)}\right]_{t=t_{min}}\frac{\alpha}{\pi}\frac{\chi}{(c\cdot P_{i})},$ (3)

where due to the smallness of $t_{\text{min}}$ the photon is usually taken on-
shell, $\alpha$ is the electromagnetic coupling constant and $\chi$ is the
flux of photons emitted by the target proton

$\chi=\int_{t_{\text{min}}}^{t_{\text{max}}}\frac{t-t_{\text{min}}}{t^{2}}G_{2}(t)\differential
t.$ (4)

Here $G_{2}(t)$ is the proton electric form factor (for details see [15]). The
generalized WW approximation has been used earlier to calculate the dark
photon production in electron [16] and elastic proton [17] bremsstrahlung.

Another widely used approximation schematically shown in figure 2(b) was
formulated by Blumlein and Brunner for inelastic bremsstrahlung [1], but can
be applied to the elastic process
$p(p)p(P_{i})\rightarrow\gamma^{\prime}(k)p(p^{\prime})p(P_{f})$ along the
same lines. In the BB approach the $pp$-interaction is considered as the
exchange of hypothetical massless vector particles $b$. On the first step in
the framework of WW approximation the matrix element of proton bremsstrahlung
is connected to the amplitude of the $2\rightarrow 2$ subprocess
$pb\rightarrow\gamma^{\prime}p$. Then one puts the 4-momentum of $b$-boson
exactly to 0 and associates the probability of dark photon emission in the
subprocess $p\rightarrow\gamma^{\prime}p$ with the splitting function
$w_{p\gamma^{\prime}}(z,k_{\perp}^{2})$ in the style of DGLAP QED equation.
The first argument of the splitting function $z\equiv k_{z}/p_{z}$ is the
ratio of the longitudinal component of dark photon’s 3-momentum to the same
component of incident proton’s 3-momentum in the lab frame. The second
argument $k_{\perp}^{2}$ is the square of the transverse component of dark
photon’s 3-momentum. The final result for cross section is

$\left[\frac{\differential^{2}\sigma(pp\rightarrow\gamma^{\prime}pp)}{\differential
z\differential
k^{2}_{\perp}}\right]_{BB}=w_{\gamma^{\prime}p}(z,k_{\perp}^{2})\sigma_{pp}(\overline{s}),$
(5)

where the splitting function

$w_{\gamma^{\prime}p}(z,k_{\perp}^{2})=\frac{\epsilon^{2}\alpha}{2\pi
H}\left[\frac{1+(1-z)^{2}}{z}-2z(1-z)\left(\frac{2M^{2}+m^{2}_{\gamma^{\prime}}}{H}-z^{2}\frac{2M^{4}}{H^{2}}\right)+\right.\\\
\left.+2z(1-z)(1+(1-z)^{2})\frac{M^{2}m_{\gamma^{\prime}}^{2}}{H^{2}}+2z(1-z)^{2}\frac{m^{4}_{\gamma^{\prime}}}{H^{2}}\right],$
(6)

the frequently used kinematic combination $H\equiv
k_{\perp}^{2}+(1-z)m_{\gamma^{\prime}}^{2}+z^{2}M^{2}$ includes the proton
mass $M$ and $\sigma_{pp}(\overline{s})$ is the elastic $pp$-scattering cross
section taken at the square of protons center-of-mass energy. It was shown in
[17] that the BB approach does not agree with another available estimate for
inelastic bremsstrahlung. The result obtained in this approach for elastic
bremsstrahlung was also criticized in [18]. Despite this, it is actively used
for estimating experimental sensitivities of future and proposed fixed-target
experiments [13, 19, 20].

The main thing that WW and BB approximations have in common is that the minus
square of momentum transferred between protons $t$ or the momentum $q$ itself
is set to 0. One can think about the validity of such approach for the
composite particle such as proton. As it is shown in figure 1 in [18], the
maximum of the differential flux of hypothetical bosons that mediate elastic
$pp$-interaction lies at $\sqrt{t}\simeq 0.35\text{ GeV}$. In contrast to
electron bremsstrahlung, where the maximum of analogous virtual photon flux
lies at $\sqrt{t_{\text{min}}}\ll\Lambda_{\text{QCD}}$ [15], this scale cannot
be treated as negligible for the proton bremsstrahlung. Thus in [18] we
considered the elastic $2\rightarrow 3$ bremsstrahlung taking into account
non-zero momentum transfer $q$. We briefly summarize our main analytical
results in the next section.

##  3 Bremsstrahlung with non-zero momentum transfer

The full calculation of the elastic bremsstrahlung with non-zero momentum
transfer is presented in [18]. There we considered the sum of two Feynman
diagrams with the dark photon emission in the initial state and in the final
state (for the last one see figure 2(b)). In the lab frame the momenta marked
in figure 2(b) are as follows

$\displaystyle P_{i}^{\mu}$ $\displaystyle=\\{M,0,0,0\\},$ (7) $\displaystyle
p^{\mu}$ $\displaystyle=\\{P+\frac{M^{2}}{2P},0,0,P\\},$ (8) $\displaystyle
k^{\mu}$
$\displaystyle=\\{zP+\frac{m^{2}_{\gamma^{\prime}}+k^{2}_{\perp}}{2zP},k_{x},k_{y},zP\\},$
(9) $\displaystyle p^{\prime\mu}$
$\displaystyle=\\{p^{\prime}_{0},-k_{x}-q_{x},-k_{y}-q_{y},P(1-z)-q_{z}\\},$
(10) $\displaystyle q^{\mu}$ $\displaystyle=\\{q_{0},q_{x},q_{y},q_{z}\\},$
(11)

where

$p^{\prime}_{0}=P(1-z)+\frac{M^{2}+k^{2}_{\perp}}{2P\left(1-z\right)}+\frac{k_{x}q_{x}+k_{y}q_{y}}{P\left(1-z\right)}-q_{z}+\frac{q^{2}_{\perp}}{2P\left(1-z\right)},$
(12)

$q^{\mu}$ is the hypothetical boson momentum going down from incident proton
$p$ to the target proton $P_{i}$ in terms of figure 2(b) and we assume that
longitudinal components of momenta $\vec{p}$, $\vec{k}$ and $\vec{p^{\prime}}$
are much greater than their transverse components and the masses of these
particles. We consider the incident proton as a Dirac fermion as a whole and
model the $pp$-interaction using photon-like propagator and the data on
elastic $pp$-scattering.

The final result for differential cross section of proton bremsstrahlung takes
the form

$\frac{\differential[2]{\sigma}}{\differential{k_{\perp}^{2}}\differential{z}}=\frac{\epsilon^{2}\alpha}{32\left(2\pi\right)^{2}zP\tilde{S}^{2}\sqrt{P^{2}(1-z)^{2}+k_{\perp}^{2}}}\int\displaylimits_{t_{\text{min}}}^{t_{\text{max}}}\differential
t\absolutevalue{T_{+}+T_{+}^{c}}^{2}\times\left[-b_{0}-\right.\\\
\left.-\frac{b_{1}t}{2M}+b_{4}t+\left(b_{2}+\frac{b_{5}t}{2M}\right)\frac{k^{2}_{\perp}|\vec{q}|\cos\hat{\theta}_{q}}{\sqrt{P^{2}(1-z)^{2}+k_{\perp}^{2}}}-\frac{b_{3}k_{\perp}^{2}}{P^{2}(1-z)^{2}+k_{\perp}^{2}}\times\right.\\\
\left.\times\left(\frac{t}{2}\left(\frac{t}{4M^{2}}+1\right)P^{2}(1-z)^{2}+|\vec{q}|^{2}\cos^{2}\hat{\theta}_{q}\left(k_{\perp}^{2}-\frac{P^{2}}{2}(1-z)^{2}\right)\right)\right],$
(13)

where
$\tilde{S}^{2}\equiv\left(\bar{s}-2M^{2}\right)^{2}+q^{2}\left(\bar{s}-M^{2}\right)$,
the limits of integration are
$\sqrt{t_{\text{min}}}={H}/\left(2z(1-z)P\right)$ and
$\sqrt{t_{\text{max}}}\sim m_{\gamma^{\prime}}$,
$T_{+}\left(\bar{s},q^{2}\right)$ and $T_{+}^{c}\left(\bar{s},q^{2}\right)$
are the fit functions for the differential cross section of elastic proton
scattering defined as in [21], the coefficients $b_{0},\dots,b_{5}$ are

$\displaystyle b_{0}$ $\displaystyle\equiv\frac{1+\left(1-z\right)^{2}}{1-z},$
(14) $\displaystyle b_{1}$
$\displaystyle\equiv\frac{4zP}{H}\left(1+\left(1-z\right)^{2}\right),$ (15)
$\displaystyle b_{2}$
$\displaystyle\equiv-\frac{8\left(1-z\right)z^{2}}{H^{3}}\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)\left(m^{2}_{\gamma^{\prime}}+k^{2}_{\perp}-z^{2}M^{2}\right)-$
(16)
$\displaystyle\quad\,-\frac{2z\left(-1+\left(1-z\right)^{2}\right)}{H\left(1-z\right)},$
(17) $\displaystyle b_{3}$
$\displaystyle\equiv\frac{16\left(1-z\right)^{2}z^{4}}{H^{4}}\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)P^{2}+\frac{z^{2}\left(-12+4\left(1-z\right)^{2}\right)}{H^{2}\left(1-z\right)}+$
(18)
$\displaystyle\quad\,+\frac{32}{H^{4}}\left(1-z\right)z^{3}\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)\left(m^{2}_{\gamma^{\prime}}+k^{2}_{\perp}-z^{2}M^{2}\right),$
(19) $\displaystyle b_{4}$
$\displaystyle\equiv\frac{4\left(1-z\right)z^{2}P^{2}}{H^{2}}\left(1+\left(1-z\right)^{2}\right),$
(20) $\displaystyle b_{5}$
$\displaystyle\equiv-\frac{8z^{2}P}{H^{2}}\left(\left(1-z\right)^{2}+\frac{z^{2}}{H}\left(1-z\right)\left(2M^{2}+m^{2}_{\gamma^{\prime}}\right)\right)$
(21)

and the cosine of the angle between $\vec{q}$ and $\vec{p}-\vec{k}$ is fixed
by the $\delta$-function

$\cos\hat{\theta}_{q}=\frac{1}{2|\vec{p}-\vec{k}||\vec{q}|}\left(\frac{H}{z}+\frac{t}{M}\left(P(1-z)+M+\frac{M^{2}}{2P}-\frac{m^{2}_{\gamma^{\prime}}+k^{2}_{\perp}}{2zP}\right)\right).$
(22)

##  4 Numerical results

In this section we present the numerical results obtained for the incident
proton beam with $P=120\text{ GeV}$ corresponding to the DUNE experiment.
Figures 3(a), 3(b)

(a)

(b)

Figure 3: Differential cross section of elastic proton bremsstrahlung (13)
depending on (a) the ratio of $z$-component of dark photon momentum to the
momentum of incident proton, (b) the square of transverse momentum of a dark
photon, both in the lab frame for different values of dark photon mass listed
in the legend and the incident proton momentum $P=120\text{ GeV}$.

show the differential cross section (13) integrated over the square of dark
photon transverse momentum $k_{\perp}^{2}$ in the region 0.01-1 GeV2
($\differential\sigma/\differential z$ in figure 3(a)) or over the momentum
ratio $z$ in the region where $zP$ and $(1-z)P$ are much greater than
transverse dark photon momentum and particle masses $m_{\gamma^{\prime}}$, $M$
($\differential\sigma/\differential k^{2}_{\perp}$ in figure 3(b)). One can
see that depending on the dark photon mass (we varied it in range 0.4-1.5 GeV)
the differential cross section $\differential\sigma/\differential z$ can
behave in qualitatively different ways: either reaching the maximum at the
intermediate values of $z$ or at $z\rightarrow 1$. Another finding is that the
differential cross section $\differential\sigma/\differential k^{2}_{\perp}$
quickly decreases with the growth of transverse momentum squared
$k_{\perp}^{2}$ which is important for understanding which part of produced
dark photons will reach a detector.

In figures 4(a), 4(b)

(a)

(b)

Figure 4: The comparison of differential cross section with non-zero momentum
transfer (13) [18] (dashed line) with the generalized WW approximation (light
blue line) [15] and the result of BB (green line) [1] depending on (a) the
ratio of $z$-component of dark photon momentum to the momentum of incident
proton, (b) the square of transverse momentum of a dark photon, both in the
lab frame for the incident proton momentum $P=120\text{ GeV}$ and the dark
photon mass $m_{\gamma^{\prime}}=1\text{ GeV}$.

we compare the differential cross section of elastic proton bremsstrahlung
obtained in our work [18] with the result in the generalized Weizsacker-
Williams approximation [15] and the one of Blumlein and Brunner [1]. In the
latter case we multiply the splitting function
$w_{\gamma^{\prime}p}(z,k_{\perp}^{2})$ (6) by the elastic $pp$-cross section.
The growth of BB cross section with $z\rightarrow 0$ is tightly connected with
the structure of the splitting function (6) containing the terms proportional
to $1/z$. One can observe the good agreement (within 3-9 %) between our result
taking into account $q\neq 0$ and the WW approximation. On the contrary, the
result of BB, as was also earlier noticed in [17] for inelastic
bremsstrahlung, significantly differs from other available estimates.

In order to justify the observed agreement, one can study the dependence of
the differential cross section $\differential^{3}\sigma/\differential
z\differential k^{2}_{\perp}\differential t$ on $t$ for the generalized WW
approximation and for our refined answer. The former is analogous to the
differential flux of hypothetical bosons in the elastic $pp$-scattering
$\differential\chi_{b}/\differential t$ depicted in figure 1 in [18]. They
have the same shape, and the maxima of these distributions for proton elastic
bremsstrahlung lie at close $\sqrt{t}\sim\Lambda_{\text{QCD}}$. The
generalized WW approximation can be obtained from our answer by neglecting the
terms with coefficients $b_{1}$, $b_{2}$, $b_{5}$ and partly $b_{3}$ in (13)
and using that $\sqrt{t_{\text{min}}}\ll M$, $k^{2}_{\perp}\ll P^{2}(1-z)^{2}$
and $t\ll 4M^{2}$ for minus square of transferred momentum $t$ with
$\differential\chi_{b}/\differential t$ that is significantly different from
0. It is important to note that the last condition is stronger than the naive
one $\sqrt{t}\lesssim M$, coming from the comparison of typical scales in this
problem mentioned in the end of section 2.

The generalized WW approximation, originally formulated for one-photon
exchange processes with electrons, explicitly utilized the fact that the
maximal contribution to the differential cross section is at
$\sqrt{t}\sim\sqrt{t_{\text{min}}}$ due to the singular behaviour of the
matrix element at small $t$. Opposite to this, we see that for proton
bremsstrahlung the differential cross section maximum is reached at the
momentum transfer $\sqrt{t}\sim\Lambda_{\text{QCD}}$ that is small compared to
the momentum $P$, but is still non-negligible and that there is no singularity
at $\sqrt{t}/M\rightarrow 0$. Generally speaking, since small momentum
transfers can be associated with interaction at large distances, this is the
illustration of the fact that the electromagnetic interaction is long-range
(photon exchange in electron bremsstrahlung), while the strong interaction
playing the key role in proton bremsstrahlung has a finite range. Despite
this, the coincidence of distribution shapes and the conditions mentioned
earlier like $t\ll 4M^{2}$ allow one to successfully use the WW approximation.

##  5 Conclusions

To sum up, we have obtained the elastic bremsstrahlung cross section
considering the non-zero momentum transfer between incident and target
protons. Our result agrees well with the generalized WW approximation [15].
Thus the generalized WW approximation is applicable to protons and can give
reliable results.

## Acknowledgements

We thank D. Gorbunov for helpful advice, continuous support and patience. We
are also grateful to O. Teryaev and I. Timiryasov for valuable comments and
discussion. The work is partially supported by the Russian Science Foundation
RSF grant 21-12-00379 (analytical calculation) and the grant of Foundation for
the Advancement of Theoretical Physics and Mathematics “BASIS” no.
21-2-10-37-1 (numerical results).

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