# Spin transport through a quantum network: Effects of Rashba spin-orbit
interaction and Aharonov-Bohm flux

Moumita Dey  Theoretical Condensed Matter Physics Division, Saha Institute of
Nuclear Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India
Santanu K. Maiti<EMAIL_ADDRESS>Theoretical Condensed Matter
Physics Division, Saha Institute of Nuclear Physics, Sector-I, Block-AF,
Bidhannagar, Kolkata-700 064, India  Department of Physics, Narasinha Dutt
College, 129 Belilious Road, Howrah-711 101, India  S. N. Karmakar
Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear
Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India

###### Abstract

We address spin dependent transport through an array of diamonds in the
presence of Rashba spin-orbit (SO) interaction where each diamond plaquette is
penetrated by an Aharonov-Bohm (AB) flux $\phi$. The diamond chain is attached
symmetrically to two semi-infinite one-dimensional non-magnetic metallic
leads. We adopt a single particle tight-binding Hamiltonian to describe the
system and study spin transport using Green’s function formalism. After
presenting an analytical method for the energy dispersion relation of an
infinite diamond chain in the presence of Rashba SO interaction, we study
numerically the conductance-energy characteristics together with the density
of states of a finite sized diamond network. At the typical flux
$\phi=\phi_{0}/2$, a delocalizing effect is observed in the presence of Rashba
SO interaction, and, depending on the specific choices of SO interaction
strength and AB flux the quantum network can be used as a spin filter. Our
analysis may be inspiring in designing spintronic devices.

###### pacs:

73.23.-b, 72.25.-b

##  I Introduction

In recent times spin transport in low dimensional systems has drawn much
attention both from theoretical as well as experimental point of view, due to
its promising applications in the field of ‘spintronics’ spintronics . It is a
newly developed sub-discipline in condensed matter physics, that deals with
the idea of manipulating spin of the electrons in transport phenomena in
addition to their charge, and holds future promises to integrate memory and
logic into a single device. Since the discovery of Giant Magnetoresistance
(GMR) effect gmr  in Fe-Cr magnetic multilayers revolutionary advancement has
taken place in data processing, device making and quantum computation
techniques. Today generation of pure spin current is a major challenge to us
for further development in quantum computation. A more or less usual way of
realization trend1 ; trend2  of spin filters is by using ferromagnetic leads
or by external magnetic field. But in the first case, spin injection from
ferromagnetic leads is difficult due to large resistivity mismatch and for the
second one the difficulty is to confine a very strong magnetic field into a
small region, like, a quantum dot (QD). Therefore, attention is now being paid
for modeling of spin filters using the intrinsic properties intrinsic1 ;
intrinsic2 ; intrinsic3 ; intrinsic4 ; intrinsic5 ; intrinsic6  of the
mesoscopic systems such as spin-orbit interaction or voltage bias. Studies on
Rashba spin orbit interaction rashba1 ; rashba2 ; rashba3 ; rashba4 , which is
present in asymmetric heterostructures has made a significant impact in
semiconductor spintronics as far as the control of spin dynamics is concerned.
It is generally important in narrow gap semiconductors and its strength can be
tuned by electrostatic means, e.g., applying external gate voltages gate1 ;
gate2 ; gate3 . Rashba SO interaction induces spin flipping through a
mechanism known as D’yakonov-Perel’ Dyakonov  mechanism, which is a slow spin
scattering process in which spin precession takes place around the Rashba
field during transmission.

Over the last few years quantum networks are becoming prospective candidates
for studying transport phenomena because of the manifestation of several
interesting features, like, quantum interference, interplay of AB flux and
network geometry on electron localization, spin-orbit interaction induced
delocalization, effect of disorder, electron-electron interaction, etc. In
2000, Vidal et al. have shown Hubbard interaction can destroy the localization
induced by magnetic field in a diamond network vidal1 . In some other works
vidal2 ; vidal3  they studied the general formalism to obtain conductance of
any quantum networks and the effect of disorder and interaction. Latter in
2002, they considered Josephson-junction chain of diamonds in a magnetic field
to show a local $Z_{2}$ symmetry at half flux-quantum vidal4 . It may be
interesting to study the effects of Rashba SO coupling and AB flux in such
quantum networks. Depending on their topology these geometries exhibit various
striking spectral properties, and the interplay between AB flux and Rashba SO
strength can also be explored. In 2005, Bercioux et al. bercioux  considered
the effect of AB flux and Rashba SO interaction on the energy averaged
conductances of a finite sized diamond chain. They observed that in such a
network spin-orbit interaction or AB flux can induce complete localization,
while the presence of both of them can lead to the effect of weak anti-
localization. The possibility to use such a diamond network as a spin filter
was explored by Aharony et al. in 2008 aharony . In 2009, there was another
work by Chakrabarti et al. sil , where they have shown how such a diamond
network can be implemented as a p-type or n-type semiconductor depending on
the suitable choice of the on-site potentials of the atoms at the vertices of
the network and the strength of magnetic flux penetrating each diamond
plaquette. But the effect of spin-orbit interaction was not considered.
Several other interesting theoretical works have been done considering this
kind of geometry. Gulacsi et al. gul1 ; gul2  in 2007 shown the exact ground
state of diamond Hubbard chain in magnetic field exhibits a wide range of
striking properties, those are tunable by magnetic flux, electron density,
etc. Peeters et al. considered quantum rings in presence of Rashba SO
interaction and magnetic field to obtain various features of
magnetoconductance peeters1 ; peeters2 ; peeters3 . In our present work, we
wish to explore the spectral and transport properties of a diamond network in
the presence of both AB flux and Rashba SO interaction. We calculate spin
conserved and spin flip conductances using single-particle Green’s function
formalism green1  within a tight-binding framework for a finite sized diamond
chain, which is compatible with the analytical dispersion relation obtained by
renormalization group method for an infinite diamond network. Analysis of the
spin-dependent conductances, dispersion relation and the density of states
(DOS) provides an insight about the effect of Rashba SO interaction and AB
flux on the localization behavior of the electrons. Finally, we show that, for
some specific choices of the external parameters this finite sized diamond
network can achieve a high degree of spin polarization.

Our organization of the paper is as follows. Following a brief introduction
(Section I), in Section II, we present the model and the theoretical
formulation. Section III is on our work comprising an analytical form for the
energy dispersion relation for an infinite diamond network, the numerical
calculations of two-terminal conductance, DOS, discussion on delocalizing
effect in presence of SO interaction and demonstration of spin filtering
action for a finite sized diamond array. At the end, the summary of our work
will be available in Section IV.

##  II Model and theoretical formulation

At the beginning of our theoretical formulation we start by describing the
geometry of the quasi one-dimensional nanostructure through which spin
transport properties are being investigated. In Fig. 1 we illustrate
schematically the quantum network, in which the square loops are connected at
the vertices (termed as Diamond Network (DN) or

Figure 1: (Color online). A finite sized diamond network (central region)
connected to two semi-infinite one-dimensional non-magnetic metallic leads,
viz, source and drain. The diamond network is composed of two types of atoms
labeled by filled green and blue circles, where each diamond plaquette is
penetrated by an AB flux $\phi$.

Diamond Chain (DC)). The diamond array is connected symmetrically to two semi-
infinite one-dimensional ($1$D) non-magnetic metallic leads, commonly known as
source and drain which are characterized by the electrochemical potentials
$\mu_{1}$ and $\mu_{2}$ under the non-equilibrium condition when a bias
voltage is applied.

The full Hamiltonian for the complete system, i.e., source-DN-drain can be
written as,

$H=H_{D}+H_{L}+H_{R}+H_{LD}+H_{DR}$ (1)

where, $H_{D}$ represents the Hamiltonian for the diamond network. $H_{L(R)}$
corresponds to the Hamiltonian for the left (right) lead, i.e., source
(drain), and $H_{LD(DR)}$ is the Hamiltonian describing the chain-lead
coupling.

We model the diamond network by the nearest-neighbor tight-binding Hamiltonian
which in Wannier basis can be written as,

$\displaystyle H_{D}$ $\displaystyle=$
$\displaystyle\sum_{l,m}\mbox{\boldmath$c_{l,m}^{\dagger}\epsilon_{0}c_{l,m}$}+\sum_{l,m}\left(\mbox{\boldmath$c_{l,m}^{\dagger}t$}e^{i\alpha}\mbox{\boldmath$c_{l+1,m}$}+h.c.\right)$
(2)
$\displaystyle+\sum_{l,m}\left(\mbox{\boldmath$c_{l,m}^{\dagger}t$}e^{i\alpha}\mbox{\boldmath$c_{l,m+1}$}+h.c.\right)$
$\displaystyle+\sum_{l,m}\left(\mbox{\boldmath$c_{l,m}^{\dagger}(i\sigma_{y})~{}t_{so}$}~{}e^{i\alpha}\mbox{\boldmath$c_{l+1,m}$}+h.c.\right)$
$\displaystyle-\sum_{l,m}\left(\mbox{\boldmath$c_{l,m}^{\dagger}(i\sigma_{x})~{}t_{so}$}~{}e^{i\alpha}\mbox{\boldmath$c_{l,m+1}$}+h.c.\right)$

where,
$c_{l,m}^{\dagger}$=$\left(\begin{array}[]{cc}c_{l,m\uparrow}^{\dagger}&c_{l,m\downarrow}^{\dagger}\end{array}\right);$
$c_{l,m}$=$\left(\begin{array}[]{c}c_{l,m\uparrow}\\\
c_{l,m\downarrow}\end{array}\right);$
$\epsilon_{0}$=$\left(\begin{array}[]{cc}\epsilon_{0}&0\\\
0&\epsilon_{0}\end{array}\right);$ $t$=$t\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right);$ $t_{so}$=$\left(\begin{array}[]{cc}t_{so}&0\\\
0&t_{so}\end{array}\right)$.

Here $\epsilon_{0}$ is the site energy of each atomic site of the diamond
chain. For $A$ type of atoms $\epsilon_{0}=\epsilon_{A}$, while for $B$ type
of atoms we call $\epsilon_{0}$ as $\epsilon_{B}$ (see Fig. 1). The second and
third terms represent the electron hopping along $X$ and $Y$ directions,
respectively, where $t$ is the nearest-neighbor hopping strength and
$\alpha=\frac{2\pi\phi}{4\phi_{0}}$ is the phase factor due to the magnetic
flux $\phi$ threaded by each diamond plaquette. Here we use double indexing to
describe the location of lattice sites in the diamond network, as illustrated
in Fig. 2 for a single plaquette. The fourth and fifth terms are associated
with the spin dependent Rashba interaction, where $t_{so}$ is the isotropic
nearest-neighbor transfer integral that measures the strength of Rashba SO
coupling.

Similarly, the Hamiltonian $H_{L(R)}$ for the two leads can be written as,

$H_{L(R)}=\sum_{i}\mbox{\boldmath$c_{i}^{\dagger}\epsilon_{L(R)}c_{i}$}+\sum_{i}\left(\mbox{\boldmath$c_{i}^{\dagger}t_{L(R)}c_{i+1}$}+h.c.\right).$
(3)

Here also,

$\epsilon_{L(R)}$=$\left(\begin{array}[]{cc}\epsilon_{L(R)}&0\\\
0&\epsilon_{L(R)}\end{array}\right);$
$t_{L(R)}$=$\left(\begin{array}[]{cc}t_{L(R)}&0\\\
0&t_{L(R)}\end{array}\right)$

where, $\epsilon_{L(R)}$ is the site energy and $t_{L(R)}$ is the hopping
strength between the nearest-neighbor sites in the left (right) lead.

The diamond chain-to-lead coupling Hamiltonian is described by,

$H_{LD(DR)}=\left(\mbox{\boldmath$c_{0(NN)}^{\dagger}t_{LD(DR)}c_{11(N+1)}$}+h.c.\right)$
(4)

where, $t_{LD(DR)}$ being the chain-lead coupling strength.

In order to calculate spin dependent transmission probabilities through the
quantum network, we use single particle Green’s

Figure 2: (Color online). Index convention for representing the co-ordinates.
$l$ and $m$ denote the co-ordinates along the $X$ and $Y$ directions,
respectively.

function formalism. Within the regime of coherent transport and in absence of
Coulomb interaction this technique is well applied.

The single particle Green’s function representing the full system for an
electron with energy $E$ is defined as,

$G=(E-H+i\eta)^{-1}$ (5)

where $\eta\rightarrow 0^{+}$.

The matrix representation for the Hamiltonian can be expressed as

$\mbox{\boldmath$H$}=\left(\begin{array}[]{ccc}\mbox{\boldmath$H_{L}$}&\mbox{\boldmath$H_{LD}$}&0\\\
\mbox{\boldmath$H_{LD}^{\dagger}$}&\mbox{\boldmath$H_{D}$}&\mbox{\boldmath$H_{DR}$}\\\
0&\mbox{\boldmath$H_{DR}^{\dagger}$}&\mbox{\boldmath$H_{R}$}\\\
\end{array}\right)$ (6)

where, ${H_{L}}$, ${H_{D}}$ and ${H_{R}}$ are the Hamiltonian matrices for the
left lead, diamond network and right lead, respectively. ${H_{LD}}$ and
${H_{DR}}$ are the coupling matrices between diamond network and the leads.
Since there is no direct coupling between the leads themselves, the corner
elements of $H$ are null matrices. A similar definition holds true for the
Green’s function matrix $G$ as well.

$\mbox{\boldmath$G$}=\left(\begin{array}[]{ccc}\mbox{\boldmath$G_{L}$}&\mbox{\boldmath$G_{LD}$}&0\\\
\mbox{\boldmath$G_{DL}$}&\mbox{\boldmath$G_{D}$}&\mbox{\boldmath$G_{DR}$}\\\
0&\mbox{\boldmath$G_{RD}$}&\mbox{\boldmath$G_{R}$}\\\ \end{array}\right)$ (7)

The problem of finding $G$ in the full Hilbert space of $H$ can be mapped
exactly to a Green’s function $G_{D}^{eff}$ corresponding to an effective
Hamiltonian in the reduced Hilbert space of diamond network and we have

$\mbox{\boldmath${\mathcal{G}}$=$G_{D}^{eff}$}=\left(\mbox{\boldmath$E-H_{D}-\Sigma_{L}-\Sigma_{R}$}\right)^{-1}$
(8)

where, $\Sigma_{L}$ and $\Sigma_{R}$ represent the contact self-energies
introduced to incorporate the effects of semi-infinite leads coupled to the
system. The self-energies are expressed by the relations,

$\Sigma_{L}$ $\displaystyle=$ $H_{LD}^{{\dagger}}G_{L}H_{LD}$ $\Sigma_{R}$
$\displaystyle=$ $H_{DR}^{{\dagger}}G_{R}H_{DR}$. (9)

Thus, the form of self-energies are independent of the nano-structure itself
through which transmission is studied and they completely describe the
influence of the two leads attached to the system. Now, the transmission
probability $(T_{\sigma\sigma^{\prime}})$ of an electron with energy $E$ is
related to the Green’s function as,

$\displaystyle T_{\sigma\sigma^{\prime}}$ $\displaystyle=$
$\displaystyle\Gamma^{1}_{L(\sigma\sigma)}{\mathcal{G}}^{1N}_{r(\sigma\sigma^{\prime})}{\mathcal{G}}^{N1}_{a(\sigma^{\prime}\sigma)}\Gamma^{N}_{R(\sigma^{\prime}\sigma^{\prime})}$
(10) $\displaystyle=$
$\displaystyle\Gamma^{1}_{L(\sigma\sigma)}|{\mathcal{G}}^{1N}_{(\sigma\sigma^{\prime})}|^{2}\Gamma^{N}_{R(\sigma^{\prime}\sigma^{\prime})}$

where, $\Gamma^{1}_{L(\sigma\sigma)}$ = $\langle
11\sigma|{\bf\Gamma_{L}}|11\sigma\rangle$,
$\Gamma^{N}_{R(\sigma^{\prime}\sigma^{\prime})}$ = $\langle
NN\sigma^{\prime}|{\bf\Gamma_{R}}|NN\sigma^{\prime}\rangle$ and
${\mathcal{G}}^{1N}_{\sigma\sigma^{\prime}}=\langle
11\sigma|\mbox{\boldmath${\mathcal{G}}$}|NN\sigma^{\prime}\rangle$. Here,
${\mathcal{G}}_{r}$ and ${\mathcal{G}}_{a}$ are the retarded and advanced
single particle Green’s functions for an electron with energy $E$.
$\bf{\Gamma_{L}}$ and $\bf{{\Gamma_{R}}}$ are the coupling matrices,
representing the coupling of the quantum network to the left and right leads,
respectively, and they are defined by the relation,

$\mbox{\boldmath$\Gamma_{L(R)}$}=i\mbox{\boldmath$\left[\Sigma^{r}_{L(R)}-\Sigma^{a}_{L(R)}\right]$}$
(11)

Here, ${\Sigma^{r}_{L(R)}}$ and ${\Sigma^{a}_{L(R)}}$ are the retarded and
advanced self-energies, respectively, and they are conjugate to each other. It
is shown in literature by Datta et al. green1  that the self-energy can be
expressed as a linear combination of a real and an imaginary part in the form,

$\mbox{\boldmath${\Sigma^{r}_{L(R)}}$}=\mbox{\boldmath$\Lambda_{L(R)}$}-i\mbox{\boldmath$\Delta_{L(R)}$}$
(12)

The real part of self-energy describes the shift of the energy levels and the
imaginary part corresponds to the broadening of the levels. The finite
imaginary part appears due to incorporation of the semi-infinite leads having
continuous energy spectrum. Therefore, the coupling matrices can easily be
obtained from the self-energy expression and is expressed as,

$\mbox{\boldmath$\Gamma_{L(R)}$}=-2~{}{\mbox{Im}}(\mbox{\boldmath$\Sigma_{L(R)}$})$
(13)

Considering linear transport regime, conductance $(g_{\sigma})$ is obtained
using two-terminal Landauer conductance formula,

$g_{\sigma\sigma^{\prime}}=\frac{e^{2}}{h}T_{\sigma\sigma^{\prime}}$ (14)

Throughout our study we choose $c=e=h=1$ for simplicity.

##  III Numerical results and discussion

In this section we study spin dependent transport through a diamond chain in
presence of Rashba spin orbit interaction and magnetic flux and investigate
the interplay between them. An array of diamonds is a bipartite structure

Figure 3: (Color online). Energy dispersion ($E$-$k$) curves for an infinite
diamond chain with $\epsilon_{A}=\epsilon_{B}=0$. The upper, middle and lower
spectra in the $1$st column correspond to $\phi=0$, $0.2$ and $0.4$,
respectively, when $t_{so}=0$. In the $2$nd column three different spectra
from the top represent the results for $t_{so}=0$, $2$ and $4$, respectively,
when $\phi$ is set to $0$. Finally, the three different figures in the last
column refer to the results for the identical values of $t_{so}$ considered in
the $2$nd column when $\phi$ is fixed at $0.4$.

with lattice sites having different co-ordination numbers. Electron
localization plays a significant role even in the absence of disorder in this
kind of geometry due to quantum interference effect. First we obtain
analytically the dispersion relation for an infinite diamond chain in the
presence of magnetic flux and Rashba interaction. Next, we simulate
numerically various features of spin transport using a finite size diamond
chain. Before analyzing the results first we specify the values of the
parameters those are used in the numerical simulations. We consider that the
two non-magnetic side-attached leads are made up of identical materials. The
on-site energies in the two leads ($\epsilon_{L(R)}$) are set to $0$. Hopping
strength between the sites in the leads is chosen as $t_{L(R)}=4$, whereas in
the diamond chain it is set as $t=3$. The Rashba strength ($t_{so}$) is chosen
to be uniform along $X$ and $Y$ directions and throughout the calculation its
magnitude is considered as comparable to $t$. Energy scale is fixed in unit of
$t$. Throughout the analysis we present all the results considering the chain-
to-electrode coupling strength as $t_{LD}=t_{DR}=2.5$.

###  III.1 Energy dispersion relation in presence of Rashba SO interaction and
magnetic flux

The energy dispersion relation for an infinite diamond chain clearly depicts
several significant features of this kind of topology. In order to study the
$E$-$k$ relation theoretically, first we map the quasi one-dimensional diamond
network into a linear chain with modified site energy and hopping strength.

We start with the Schrodinger equation which can be cast in the form of a
difference equation. For an arbitrary site ($n,p$), where $n$ and $p$ denote
the indexing along $X$ and $Y$ directions, respectively, the difference
equation can be expressed as

${(E-\epsilon)\psi_{np}}$ $\displaystyle=$ $\displaystyle e^{\mp
i\alpha}\mbox{\boldmath$t_{x_{+}}\psi_{n+1,p}$}+e^{\mp
i\alpha}\mbox{\boldmath$t_{x_{-}}\psi_{n-1,p}$}+$ (15) $\displaystyle e^{\pm
i\alpha}\mbox{\boldmath$t_{y_{+}}\psi_{n,p+1}$}+e^{\pm
i\alpha}\mbox{\boldmath$t_{y_{-}}\psi_{n,p-1}$}$

where,
$t_{x_{+}}$=$\left(\begin{array}[]{cc}t&t_{so}\\\
-t_{so}&t\end{array}\right);$
$t_{x_{-}}$=$\left(\begin{array}[]{cc}t&-t_{so}\\\
t_{so}&t\end{array}\right);$

$t_{y_{+}}$=$\left(\begin{array}[]{cc}t&it_{so}\\\
it_{so}&t\end{array}\right);$
$t_{y_{-}}$=$\left(\begin{array}[]{cc}t&-it_{so}\\\
-it_{so}&t\end{array}\right);$

$E$=$\left(\begin{array}[]{cc}E&0\\\ 0&E\end{array}\right)$;
$\epsilon$=$\left(\begin{array}[]{cc}\epsilon_{0}&0\\\
0&\epsilon_{0}\end{array}\right)$; and
$\psi_{np}$=$\left(\begin{array}[]{c}\psi_{np,\uparrow}\\\
\psi_{np,\downarrow}\end{array}\right)$

$\phi$ being the AB flux enclosed by each diamond plaquette, $\psi_{np\sigma}$
being the wave function amplitude at the np-th site with spin $\sigma$.
$\phi_{0}=ch/e$, the elementary flux-quantum.

We begin the decimation technique by writing down the difference equations at
the sites containing A and B type atoms (see Fig. 2). The equations are given
below

$(E-\epsilon_{B})\psi_{12}$ $\displaystyle=$ $\displaystyle
e^{i\alpha}\mbox{\boldmath$t_{x_{+}}\psi_{22}$}+e^{-i\alpha}\mbox{\boldmath$t_{y_{-}}\psi_{11}$}$
$(E-\epsilon_{B})\psi_{21}$ $\displaystyle=$ $\displaystyle
e^{i\alpha}\mbox{\boldmath$t_{x_{-}}\psi_{11}$}+e^{-i\alpha}\mbox{\boldmath$t_{y_{+}}\psi_{22}$}$
$(E-\epsilon_{B})\psi_{23}$ $\displaystyle=$ $\displaystyle
e^{i\alpha}\mbox{\boldmath$t_{x_{+}}\psi_{33}$}+e^{-i\alpha}\mbox{\boldmath$t_{y_{-}}\psi_{22}$}$
$(E-\epsilon_{B})\psi_{32}$ $\displaystyle=$ $\displaystyle
e^{i\alpha}\mbox{\boldmath$t_{x_{-}}\psi_{22}$}+e^{-i\alpha}\mbox{\boldmath$t_{y_{+}}\psi_{33}$}$
(16)

and,

$(E-\epsilon_{A})\psi_{22}$ $\displaystyle=$ $\displaystyle
e^{-i\alpha}\mbox{\boldmath$t_{x_{+}}\psi_{32}$}+e^{-i\alpha}\mbox{\boldmath$t_{x_{-}}\psi_{12}$}$
(17)
$\displaystyle+e^{i\alpha}\mbox{\boldmath$t_{y_{+}}\psi_{23}$}+e^{i\alpha}\mbox{\boldmath$t_{y_{-}}\psi_{33}$}$

Substituting $\psi_{32}$, $\psi_{12}$, $\psi_{23}$ and $\psi_{33}$ from Eq.
(16) in Eq. (17) we get,

$\mbox{\boldmath$(E-\epsilon^{\prime})\psi_{22}$}=\mbox{\boldmath$t_{b}\psi_{11}+t_{f}\psi_{33}$}$
(18)

This represents the difference equation for an infinite linear chain with
modified site energy $\epsilon^{\prime}$ and the forward and backward hopping
strengths ${\bf t_{f}}$ and ${\bf t_{b}}$, respectively. These quantities are
expressed as follows.

${\epsilon^{\prime}}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\epsilon_{A}$}+\mbox{\boldmath$t_{x_{+}}.(E-\epsilon_{B})^{-1}.t_{x_{-}}$}$
$\displaystyle+\mbox{\boldmath$t_{x_{-}}.(E-\epsilon_{B})^{-1}.t_{x_{+}}$}+\mbox{\boldmath$t_{y_{+}}.(E-\epsilon_{B})^{-1}.t_{y_{-}}$}$
$\displaystyle+\mbox{\boldmath$t_{y_{-}}.(E-\epsilon_{B})^{-1}.t_{y_{+}}$}$
$t_{b}$ $\displaystyle=$ $\displaystyle
e^{-2i\alpha}\mbox{\boldmath$t_{x_{-}}.(E-\epsilon_{B})^{-1}.t_{y_{-}}$}$
$\displaystyle+e^{2i\alpha}\mbox{\boldmath$t_{y_{-}}.(E-\epsilon_{B})^{-1}.t_{x_{-}}$}$
$t_{f}$ $\displaystyle=$ $\displaystyle
e^{-2i\alpha}\mbox{\boldmath$t_{x_{+}}.(E-\epsilon_{B})^{-1}.t_{y_{+}}$}$
$\displaystyle+e^{2i\alpha}\mbox{\boldmath$t_{y_{+}}.(E-\epsilon_{B})^{-1}.t_{x_{+}}$}$

As the translational invariance is preserved in this decimated infinite linear
chain, the solution will be of Bloch form and can be written as,

$\mbox{\boldmath$\psi_{n}$}=\sum_{k}e^{ikna}\left(\begin{array}[]{c}\psi_{k,\uparrow}\\\
\psi_{k,\downarrow}\end{array}\right)$ (20)

$\psi_{n}$ being a short form of $\psi_{nn}$.

Using this form of $\psi_{n}$, the difference equation for an arbitrary site
$n$ can be expressed as,

$\displaystyle\sum_{k}\mbox{\boldmath$(E-\epsilon^{\prime})$}\left(\begin{array}[]{c}\psi_{k,\uparrow}\\\
\psi_{k,\downarrow}\end{array}\right)e^{ikna}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$t_{f}$}\sum_{k}\left(\begin{array}[]{c}\psi_{k,\uparrow}\\\
\psi_{k,\downarrow}\end{array}\right)e^{ik(n+1)a}$ (25) $\displaystyle+$
$\displaystyle\mbox{\boldmath$t_{b}$}\sum_{k}\left(\begin{array}[]{c}\psi_{k,\uparrow}\\\
\psi_{k,\downarrow}\end{array}\right)e^{ik(n-1)a}$ (28)

For the non-trivial solution of Eq. (LABEL:d7) we have the relation,

${\bf Det[M]}=0$ (30)

where,
$M$=$(\mbox{\boldmath$E$}-\mbox{\boldmath$\epsilon^{\prime}$}-\mbox{\boldmath$t_{f}$}e^{ika}-\mbox{\boldmath$t_{b}$}e^{-ika})$
.

Expanding Eq. (30) we obtain a $4$-th degree polynomial in $E$ and solving it
we get the $E$-$k$ dispersion relation in terms of the parameters $\phi$ and
$t_{so}$. The solutions correspond to energy eigenstates which are linear
combinations of up $(|k\uparrow\rangle)$ and down $(|k\downarrow\rangle)$
states.

Following the above analytical treatment, in Fig. 3 we show the $E$ versus $k$
dispersion curves for some typical parameter values of $\phi$ and $t_{so}$.
The first column corresponds to the results for some specific values of AB
flux $\phi$ in the absence of Rashba SO coupling, i.e., $t_{so}=0$. It is
observed that for zero magnetic flux the spectrum is degenerate and gapless,
whereas a small non-zero flux $(\phi=0.2)$ opens a gap symmetrically around
$E=0$ preserving the degeneracy. Here we choose $\epsilon_{A}=\epsilon_{B}=0$
and the gap appears symmetrically as long as $\phi$ is introduced, but the
point is that for unequal values of $\epsilon_{A}$ and $\epsilon_{B}$ gap
always appears even in the absence of $\phi$ (which is not shown in the
figure). The width of the gap increases symmetrically with the rise in $\phi$.
In the second column of Fig. 3 we present the energy dispersion curves for the
three different values of Rashba SO coupling strength keeping $\phi=0$, where
the upper, middle and lower spectra correspond to $t_{so}=0$, $2$ and $4$,
respectively. The upper spectrum is gapless and degenerate as described
earlier. For non-zero values of $t_{so}$, the energy spectra get splitted
vertically and all the degeneracies are removed except at the points $k=n\pi$,
where $n=0$, $\pm 1$, $\pm 2$, $\dots$. The gap becomes widened with the
increase in Rashba strength as clearly noticed from the middle and lower
spectra. The above features seem to be more interesting when a non-zero
magnetic flux is applied. In this case, each sub-band gets separated
vertically as illustrated in the third column of Fig. 3. With a

Figure 4: (Color online). Variations of (a) $g_{\uparrow\uparrow}$ and (b)
$g_{\uparrow\downarrow}$ with AB flux $\phi$ for a diamond chain considering
$5$ plaquettes at the typical energy $E=5$. The green and red curves
correspond to $t_{so}=0$ and $2$, respectively. Here we set
$\epsilon_{A}=\epsilon_{B}=0$.

sufficiently high magnetic flux ($\phi=0.4$) and Rashba strength ($t_{so}=4$)
two additional gaps occur in the dispersion spectrum along with the previous
one, and these gaps can be controlled externally by tuning the AB flux or the
Rashba strength. We will show that these results are quite significant so far
as the designing of nanoscale spintronic devices are concerned.

###  III.2 Variation of conductances with AB flux

In Fig. 4 we plot conductance-flux characteristics for a diamond network both
in the presence and absence of Rashba SO interaction. The results are computed
at a typical energy $E=5$ considering five diamond plaquettes, where the green
and red curves correspond to $t_{so}=0$ and $2$, respectively. It is observed
that in the absence of Rashba coupling spin flip conductance
$g_{\uparrow\downarrow}$ drops exactly to zero for the entire range of $\phi$
(green curve in Fig. 4(b), coincides with the $\phi$ axis), while spin
conserved conductance $g_{\uparrow\uparrow}$ persists and it provides
$\phi_{0}$ flux-quantum periodicity as a function of $\phi$. Interestingly we
see that $g_{\uparrow\uparrow}$ completely vanishes at $\phi=\phi_{0}/2$ (see
green curve of Fig. 4(a)) due to the complete destructive interference among
the electronic waves passing through different arms of the plaquettes. On the
other hand, in the presence of Rashba SO interaction both
$g_{\uparrow\uparrow}$ and $g_{\uparrow\downarrow}$ have values for wide
ranges of $\phi$ and a significant change in their amplitudes takes place
compared to the case where $t_{so}=0$. In the presence of the SO interaction,
the oscillatory character of the conductances is still preserved providing
traditional $\phi_{0}$ flux-quantum periodicity. The important feature is that
even for non-zero value of $t_{so}$, spin flip conductance disappears at
$\phi=\phi_{0}/2$. Since $g_{\downarrow\downarrow}$ and
$g_{\downarrow\uparrow}$ exhibit exactly identical behavior to those mentioned
for $g_{\uparrow\uparrow}$ and $g_{\uparrow\downarrow}$, respectively, we do
not display these results explicitly.

The above numerical results can be justified from the following mathematical
analysis.

To illustrate the behaviors of AB oscillation both in the presence and absence
of Rashba SO interaction, we consider an ideal $1$D square loop threaded by an
AB flux $\phi$, as shown schematically in Fig. 5. Two semi-infinite one-
dimensional leads are connected at the vertices P and Q of the square loop.

Figure 5: (Color online). A single diamond plaquette threaded by an AB flux
$\phi$. $\psi_{i}$ and $\psi_{o}$ denote the incoming and outgoing waves,
respectively.

Rashba SO interaction is considered to be present only in the loop and not in
the leads. If $\psi_{i}$ and $\psi_{o}$ describe the incoming and outgoing
wave functions at the respective vertices P and Q, then $\psi_{o}$ can be
obtained considering only $1$st order tunneling processes as xie ,

$\mbox{\boldmath$\psi_{0}$}=\frac{1}{2}\left(e^{-i\frac{\gamma}{2}}\mbox{\boldmath$R_{x}$}(\theta)\mbox{\boldmath$R_{y}$}(\theta)+e^{i\frac{\gamma}{2}}\mbox{\boldmath$R_{y}$}(\theta)\mbox{\boldmath$R_{x}$}(\theta)\right)\mbox{\boldmath$\psi_{i}$}$
(31)

where, $\gamma=\frac{2\pi\phi}{\phi_{0}}$. $R_{\hat{r}}$$(\theta)$ is the
rotation operator defined by the relation,

$\mbox{\boldmath$R_{\hat{r}}$}(\theta)=\mbox{\boldmath$I$}\cos\frac{\theta}{2}-i\hat{r}.\vec{\sigma}\sin\frac{\theta}{2}$
(32)

where, $\theta=\frac{2m^{*}\alpha_{R}L}{\hbar^{2}}$ is the spin precession
angle. $\alpha_{R}$ is the strength of Rashba SO interaction and $L$
represents the length of each side of the square loop. The wave functions
$\psi_{i}$ and $\psi_{o}$ used in Eq. (31) are defined as follows.

$\psi_{o}$ $=\left(\begin{array}[]{c}\psi_{o,\uparrow}\\\
\psi_{o,\downarrow}\end{array}\right)$ and $\psi_{i}$
$=\left(\begin{array}[]{c}\psi_{i,\uparrow}\\\
\psi_{i,\downarrow}\end{array}\right)$.

Following Eq. (32) the matrices $R_{x}$$(\theta)$ and $R_{y}$$(\theta)$ can be
written as given below.

$R_{x}$$(\theta)=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}&-i\sin\frac{\theta}{2}\\\
-i\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right)$.

and

$R_{y}$$(\theta)=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}&-\sin\frac{\theta}{2}\\\
\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right)$.

With these matrix forms we can express the wave functions
$|\psi_{o,\uparrow}\rangle$ and $|\psi_{o,\downarrow}\rangle$ as linear
combinations of $|\psi_{i,\uparrow}\rangle$ and $|\psi_{i,\downarrow}\rangle$
by expanding Eq. (31) as,

$\displaystyle|\psi_{o,\uparrow}\rangle$ $\displaystyle=$ $\displaystyle
c_{\uparrow\uparrow}|\psi_{i,\uparrow}\rangle+c_{\downarrow\uparrow}|\psi_{i,\downarrow}\rangle$
$\displaystyle|\psi_{o,\downarrow}\rangle$ $\displaystyle=$ $\displaystyle
c_{\uparrow\downarrow}|\psi_{i,\uparrow}\rangle+c_{\downarrow\downarrow}|\psi_{i,\downarrow}\rangle$
(33)

where, the co-efficients $c_{\sigma\sigma^{\prime}}$ are functions of $\theta$
and $\phi$.

Now the probability of getting an up spin electron at the point Q, for the
incidence of an electron with up spin at the point P, i.e., the spin conserved
transmission probability $T_{\uparrow\uparrow}$ is proportional to
$|\langle\psi_{i,\uparrow}|\psi_{o,\uparrow}\rangle|^{2}$, viz,
$|c_{\uparrow\uparrow}|^{2}$. Similarly, the probability of getting a down
spin electron with up spin incidence, i.e., the spin flip transmission
probability $T_{\uparrow\downarrow}$ is proportional to
$|\langle\psi_{i,\uparrow}|\psi_{o,\downarrow}\rangle|^{2}$, viz,
$|c_{\uparrow\downarrow}|^{2}$. After a few mathematical steps the quantities
$|c_{\uparrow\uparrow}|^{2}$ and $|c_{\uparrow\downarrow}|^{2}$ are expressed
as,

$\displaystyle|c_{\uparrow\uparrow}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{8}e^{-i\frac{2\pi\phi}{\phi_{0}}}\left[1+i\cos\theta+e^{i\frac{2\pi\phi}{\phi_{0}}}(i+\cos\theta)\right]$
(34)
$\displaystyle\times\left[\cos\theta-i+e^{i\frac{2\pi\phi}{\phi_{0}}}(1-i\cos\theta)\right]$

and

$|c_{\uparrow\downarrow}|^{2}=\frac{1}{8}e^{-i\frac{2\pi\phi}{\phi_{0}}}\left(1+e^{i\frac{2\pi\phi}{\phi_{0}}}\right)^{2}\sin^{2}\theta.$
(35)

In the absence of Rashba SO interaction $\theta=0$ and the above two equations
can be simplified as follows.

$|c_{\uparrow\uparrow}|^{2}=\frac{1}{2}\left[1+\cos\left(\frac{2\pi\phi}{\phi_{0}}\right)\right]$
(36)

and

$|c_{\uparrow\downarrow}|^{2}=0$ (37)

With the last four mathematical expressions (Eqs. (34)-(37)) we can clearly
justify the essential features those are presented in Fig. 4. In the absence
of Rashba SO interaction, spin flip conductance vanishes for the entire range
of $\phi$ (coincident green curve of Fig. 4(b) with $\phi$ axis) in accordance
with Eq. (37). On the other hand, a oscillatory character of up spin
conductance with $\phi_{0}$ periodicity in the absence of $t_{so}$ (green
curve of Fig. 4(a)) follows from Eq. (36). The vanishing behavior of
$g_{\uparrow\uparrow}$ at the typical flux $\phi=\phi_{0}/2$ is also justified
from Eq. (36). In the presence of SO interaction, both pure spin transmission
and spin flip transmission get modified satisfying Eqs. (34) and (35),
respectively. For finite value of $t_{so}$, spin flip conductance always
vanishes at $\phi=\phi_{0}/2$ obeying Eq. (35).

###  III.3 Conductance-energy characteristics

Now we focus our attention on the conductance-energy characteristics of a
finite sized diamond network for some specific values of AB flux $\phi$ and
Rashba SO interaction strength $t_{so}$.

In Fig. 6 we plot up spin conductances ($g_{\uparrow\uparrow}$) as a function
of injecting electron energy ($E$) for a diamond network

Figure 6: (Color online). Conductance-energy ($g_{\uparrow\uparrow}$-$E$)
characteristics in the absence of Rashba SO strength for a diamond network
considering $15$ diamond plaquettes with $\epsilon_{A}=\epsilon_{B}=0$. (a),
(b) and (c) correspond to $\phi=0$, $0.2$ and $0.4$, respectively.

considering $15$ identical plaquettes in the absence of Rashba interaction.
The top, middle and bottom spectra correspond to AB flux $\phi=0$, $0.2$ and
$0.4$, respectively. When $\phi=0$, the spectrum is gapless (Fig. 6(a)). The
presence of $\phi$ opens a gap and the width of the gap increases with the
rise in $\phi$ as evident from

Figure 7: (Color online). $g_{\uparrow\uparrow}$ and $g_{\uparrow\downarrow}$
as a function of energy $E$ for a diamond chain with $15$ plaquettes
considering $\epsilon_{A}=\epsilon_{B}=0$ in the absence of AB flux $\phi$.
The $1$st, $2$nd and $3$rd rows represent the results when $t_{so}=0$, $2$ and
$4$, respectively.

Figure 8: (Color online). Spin conserved ($g_{\uparrow\uparrow}$,
$g_{\downarrow\downarrow}$) and spin flip conductances
($g_{\uparrow\downarrow}$, $g_{\downarrow\uparrow}$) as a function of energy
$E$ for a diamond chain with $15$ plaquettes when $\phi$ and $t_{so}$ are
fixed to $0.4$ and $4$, respectively. The parameters $\epsilon_{A}$ and
$\epsilon_{B}$ are set to $0$.

Figs. 6(b) and (c). This gap is symmetric around the energy $E=0$ with the
choice $\epsilon_{A}=\epsilon_{B}=0$. On the other hand, if the site energies
$\epsilon_{A}$ and $\epsilon_{B}$ are unequal then a gap in the conductance
spectrum appears (not shown in the figure) even in the absence of magnetic
flux both for an infinite as well as for a finite sized diamond array. This
gap will be symmetric across $E=0$ provided $\epsilon_{A}$ and $\epsilon_{B}$
are identical, and the width of the gap increases symmetrically about the
center of the gap with the enhancement in $\phi$. In this particular case we
do not consider any Rashba interaction, and therefore, no spin flip
transmission takes place.

In Fig. 7 we show the conductance-energy characteristics of a diamond network
with $15$ plaquettes for different values of $t_{so}$ in the absence of
magnetic flux $\phi$. The $1$st, $2$nd and $3$rd rows correspond to the
results when $t_{so}=0$, $2$ and $4$, respectively. The spin conserved
conductances ($g_{\uparrow\uparrow}$) are plotted in the first column, while
in the second column spin flip conductances ($g_{\uparrow\downarrow}$) are
given. In absence of $t_{so}$, gapless spectrum is observed for spin conserved
conductance, while spin flip conductance vanishes for the entire energy range.
For all other cases, a gap appears in the spectrum and its width can be
regulated by tuning the Rashba coupling strength.

The most interesting features in the conductance-energy characteristics are
observed when we consider the effects of both the AB flux $\phi$ and Rashba SO
coupling $t_{so}$. The results are shown in Fig. 8 for a diamond chain with
$15$ identical diamond plaquettes for $\phi=0.4$ and $t_{so}=4$. For
sufficiently high AB flux and Rashba strength, two additional energy gaps
occur at the flanks on both sides of the conductance spectrum in addition to
the central gap. The energy gaps are positioned identically in all these
spectra.

It is important to note that when anyone of $\phi$ and $t_{so}$ is zero and
other is non-zero, $g_{\uparrow\uparrow}$ becomes exactly identical to
$g_{\downarrow\downarrow}$, and so is $g_{\uparrow\downarrow}$ and
$g_{\downarrow\uparrow}$. On the other hand, when both are non-zero, spin
conserved conductances differ in magnitude, but the spin flip conductances
remain identical. All these conductance-energy characteristics shown in Figs.
6-8 are compatible with the $E$-$k$ diagrams presented in Fig. 3. The gaps of
the conductance spectra of finite sized diamond chain compare well with those
of the dispersion curves obtained earlier for an infinite sized diamond chain.

###  III.4 DOS-energy characteristics

To gain insight into the nature of energy eigenstates of such a quantum
network we address the behavior of average density of states. It is expressed
as,

$\rho_{av}(E)=-\frac{1}{\pi N}{\mbox{Im}}[{\mbox{Tr}}[{\bf G}]]$ (38)

where, $N$ being the total number of atomic sites in the diamond chain.

As illustrative examples, in Fig. 9 we present the variations of average
density of states as a function of energy $E$ for a diamond chain

Figure 9: (Color online). Average density of states as a function of energy
$E$ for a diamond chain considering $15$ plaquettes with different values of
$\phi$ and $t_{so}$ when $\epsilon_{A}=\epsilon_{B}=0$. (a) $\phi=0$,
$t_{so}=0$; (b) $\phi=0.2$, $t_{so}=0$; (c) $\phi=0$, $t_{so}=2$ and (d)
$\phi=0.4$, $t_{so}=4$.

consisting of $15$ plaquettes for different values of AB flux $\phi$ and
Rashba SO coupling strength $t_{so}$. In (a), $\rho$-$E$ spectrum is given
when both $\phi$ and $t_{so}$ are fixed at zero. The spectrum does not exhibit
any gap as expected when $\epsilon_{A}=\epsilon_{B}=0$. A sharp peak is
observed at the band center, i.e., at $E=0$ due to localized states. These
localized states are highly degenerate and in general pinned at the energy
$E=\epsilon_{B}$. The existence of the localized state is a characteristic
feature of diamond network as mentioned in an earlier work sil . In (b) and
(c), energy gaps appear symmetrically around the central peak at $E=0$ by the
AB flux $\phi$ and Rashba coupling strength $t_{so}$, respectively. By tuning
the AB flux $\phi$ or Rashba coupling strength $t_{so}$, the width of the gap
can be controlled. Finally, in (d) we display average DOS when both $\phi$ and
$t_{so}$ are finite. In such situation two extra gaps appear together with the
central ones. The localized states are still situated at the same place as
earlier.

###  III.5 Effect of Rashba spin-orbit interaction on localization

In such a quantum network, AB flux can induce complete localization. At
$\phi=\phi_{0}/2$, conductance drops exactly to zero in the absence of Rashba
SO interaction. This is due to the complete

Figure 10: (Color online). Average density of states as a function of energy
for a diamond network considering $15$ plaquettes with
$\epsilon_{A}=\epsilon_{B}=0$ when AB flux $\phi$ is set at $\phi_{0}/2$. (a)
$t_{so}=0$ and (b) $t_{so}=2$.

destructive interference between the electronic waves passing through
different arms of the network. At $\phi=\phi_{0}/2$, two more sharp peaks
appear in the $\rho$-$E$ characteristics due to localized states (Fig. 10(a))
in addition to the previous one pinned at $E=0$. The positions of these
localized states can be evaluated exactly and they are expressed
mathematically as,

$E=\frac{1}{2}\left[(\epsilon_{A}+\epsilon_{B})\pm\sqrt{(\epsilon_{A}+\epsilon_{B})^{2}-4(\epsilon_{A}\epsilon_{B}-4t^{2})}\right].$
(39)

In the presence of Rashba spin orbit interaction, the interference is not
completely destructive anymore at $\phi=\phi_{0}/2$. The two additional peaks
at the opposite sides of the central one disappear for non-zero Rashba
strength as clearly seen from Fig. 10(b). Rashba spin-orbit coupling affects
the spin dynamics significantly resulting in a non-zero conductance at
$\phi=\phi_{0}/2$.

###  III.6 Rashba induced semi-conducting behavior

Here we address how Rashba SO interaction can induce semi-conducting behavior
in such a quantum network in the absence of $\phi$. A similar type of semi-
conducting nature controlled by AB flux has been established in such a system,
where SO interaction was not considered sil . To establish our idea, in

Figure 11: (Color online). Average density of states as a function of energy
for a diamond chain considering $15$ plaquettes in the absence of AB flux
$\phi$ where $\epsilon_{A}$ and $\epsilon_{B}$ are fixed at $0$ and $2$,
respectively. (a) $t_{so}=0$ and (b) $t_{so}=2$.

Fig. 11 we plot the average density of states as a function of energy
considering $15$ plaquettes where $\epsilon_{A}$ and $\epsilon_{B}$ are set at
$0$ and $2$, respectively. When $\epsilon_{A}$ and $\epsilon_{B}$ are not
same, the diamond network possesses an intrinsic gap in the energy spectrum
even in the absence of $\phi$ and $t_{so}$, as evident from Fig. 11(a). The
sharp peak in the DOS is situated at the edge of the gap, $E=\epsilon_{B}=2$
and it actually corresponds to the localized states. By controlling the
external gate potential, i.e., tuning the Rashba strength to a non-zero value,
the width of the gap can be increased arbitrarily as shown in Fig. 11(b),
keeping the position of the localized states invariant. Now, if the Fermi
level $E_{F}$ is fixed at $E=2$ where we have localized states (see Fig.
11(b)), then for small Rashba coupling strength the gap between the localized
level and the bottom of the right sub-band can be made small enough for the
electrons to bridge. Therefore, the system behaves as a $n$-type
semiconductor. Similarly, if $\epsilon_{B}$ is fixed at $-2$ and the Fermi
level is set at the top of the left sub-band, then the system can be
implemented equivalently as a $p$-type semiconductor. In this case holes are
created in the left sub-band. It is important to mention that when the site
energies ($\epsilon_{A}$ and $\epsilon_{B}$) are fixed at the same value, then
also the system can be used as a semi-conductor depending on the electron
concentration. The detailed analysis is available in Ref. sil .

###  III.7 Spin filtering action

With proper tuning of the external parameters like, magnetic flux $\phi$ and
Rashba strength $t_{so}$, a diamond network can achieve a high degree of spin
polarization as discussed earlier in a theoretical work by Aharony et al.
aharony . Here, we discuss this feature from a different point of view.

When there is no external magnetic field or magnetic flux, time reversal
symmetry is not broken, and the Hamiltonian of the system remains

Figure 12: (Color online). Variation of conductances as a function of energy
for a diamond chain with $3$ plaquettes considering $\phi=0.3$ and $t_{so}=4$
where $\epsilon_{A}=\epsilon_{B}=0$. In (a) $g_{\uparrow\uparrow}$ and
$g_{\downarrow\downarrow}$ and in (b) $g_{\uparrow\downarrow}$ and
$g_{\downarrow\uparrow}$ are superposed to each other.

invariant under time reversal operation. Mathematically it is expressed as
$[H_{SO},T]=0$, where $T$ is the time reversal operator. The Rashba
Hamiltonian ($H_{SO}$) is usually written as,

$\displaystyle H_{SO}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{R}}{\hbar}\left(\vec{\sigma}\times\vec{p}\right)_{Z}$
(40) $\displaystyle=$ $\displaystyle
i\alpha_{R}\left(\sigma_{y}\frac{\partial}{\partial
x}-\sigma_{x}\frac{\partial}{\partial y}\right)$

and $T=i\sigma_{y}\hat{C}$, $\hat{C}$ being the complex conjugation operator.
The second quantized form of Eq. (40) is given by the fourth and fifth terms
of Eq. (2). As electrons are spin-$\frac{1}{2}$ particles, so following
Kramer’s theorem, each eigenstate is at least two-fold degenerate and spin is
no longer a good quantum number in the presence of spin-orbit interaction. In
many cases the degeneracy implied by Kramer’s theorem is merely the degeneracy
between states of spin up and spin down, or something equally obvious. The
theorem is non-trivial for a system with spin-orbit coupling in an
unsymmetrical electric field, so that neither nor angular momentum is
conserved. Kramer’s theorem implies that no such field can split the
degenerate pairs of energy levels ballen .

However, the degeneracy can be removed by applying external magnetic flux or
magnetic field as in this case time reversal symmetry is not conserved anymore
and therefore spin polarization can be achieved. The degree of polarization of
the transmitted electrons is conventionally defined as,

$P(E)=\left|\frac{(g_{\uparrow\uparrow}+g_{\downarrow\uparrow})-(g_{\downarrow\downarrow}+g_{\uparrow\downarrow})}{(g_{\uparrow\uparrow}+g_{\downarrow\uparrow})+(g_{\downarrow\downarrow}+g_{\uparrow\downarrow})}\right|.$
(41)

For our illustrative purpose, in Fig. 12 we plot the variations of
conductances as a function of energy for a diamond network with three
plaquettes considering $\phi=0.3$ and $t_{so}=4$. A significant change is
observed in the magnitudes of spin conserved conductances
($g_{\uparrow\uparrow}$ and $g_{\downarrow\downarrow}$) (Fig. 12(a)), while
the spin flip conductances ($g_{\uparrow\downarrow}$ and
$g_{\downarrow\uparrow}$) are identical as shown in Fig. 12(b). Therefore,
applying a non-zero flux spin polarization is clearly obtained. Following Eq.
(41) we calculate the degree of polarization for an arbitrary energy $E=-5$
(say), and it is about $44\,\%$.

##  IV Closing remarks

To conclude, in the present work we have explored spin dependent transport
through an array of diamonds where Rashba SO interaction is present and each
diamond plaquette is threaded by an AB flux $\phi$. The diamond chain is
directly coupled to two semi-infinite $1$D non-magnetic metallic leads,
namely, source and drain. We have adopted a discrete lattice model within the
tight-binding framework to describe the system and present calculations based
on Green’s function formalism. We have obtained analytical expression for the
$E$-$k$ dispersion relation for an infinite diamond network with Rashba SO
interaction, and, show explicitly the interplay of spin-orbit interaction and
magnetic flux on its band structure. This analysis also gives insight about
the presence of spin dependent localized and extended eigenstates which
crucially controls the spin dependent transport through such device. This
analytical study, in fact, provides us a very good understanding about the
transport behavior of spins across a finite sized array of diamonds. It has
been clearly established that how delocalizing effect sets in due to Rashba SO
interaction when the AB flux $\phi$ is $\phi_{0}/2$. Quite interestingly we
show that depending on the specific choices of SO interaction strength and AB
flux, the quantum network can be utilized as a spin filter.

In the present work we have ignored the effects of temperature, electron-
electron correlation, electron-phonon interaction, disorder, etc. Here, we set
the temperature at $0$K, but the basic features will not change significantly
even at low temperature as long as thermal energy ($k_{B}T$) is less than the
average level spacing of the diamond chain. In this model it is also assumed
that the two side-attached non-magnetic leads have negligible resistance. Our
presented results may be useful in designing spin based nano electronic
devices.

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