Semicircular Rashba arc spin polarizerZhuo Bin Siu, Mansoor B. A. Jalil, and Seng Ghee Tan

Citation: Journal of Applied Physics 115, 17C513 (2014); doi: 10.1063/1.4866388 View online: http://dx.doi.org/10.1063/1.4866388 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin polarization in one dimensional ring with Rashba spin-orbit interaction J. Appl. Phys. 115, 044313 (2014); 10.1063/1.4863466 Strain enhanced spin polarization in graphene with Rashba spin-orbit coupling and exchange effects J. Appl. Phys. 111, 033705 (2012); 10.1063/1.3679568 Spin-polarized transport in zigzag graphene nanoribbons with Rashba spin–orbit interaction J. Appl. Phys. 110, 103702 (2011); 10.1063/1.3660704 Spin-polarized current separator based on a fork-shaped Rashba nanostructure J. Appl. Phys. 108, 093717 (2010); 10.1063/1.3504246 Spin-polarized transport through an Aharonov-Bohm interferometer with Rashba spin-orbit interaction J. Appl. Phys. 100, 113703 (2006); 10.1063/1.2365379

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

142.104.240.194 On: Wed, 04 Jun 2014 16:19:42

Semicircular Rashba arc spin polarizer

Zhuo Bin Siu,1,2,a) Mansoor B. A. Jalil,1 and Seng Ghee Tan2

1NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore,Singapore 1174562Advanced Concepts and Nanotechnology, Data Storage Institute, DSI Building, 5 Engineering Drive 1(Off Kent Ridge Crescent, NUS), Singapore 117608

(Presented 5 November 2013; received 22 September 2013; accepted 15 November 2013; published

online 24 February 2014)

In this work, we study the generation of spin polarized currents using curved arcs of finite widths,

in which the Rashba spin orbit interaction (RSOI) is present. Compared to the 1-dimensional RSOI

arcs with zero widths studied previously, the finite width presents charge carriers with another

degree of freedom along the transverse width of the arc, in addition to the longitudinal degree of

freedom along the circumference of the arc. The asymmetry in the transverse direction due to the

difference in the inner and outer radii of the arc breaks the antisymmetry of the longitudinal spin zcurrent in a straight RSOI segment. This property can be exploited to generate spin z polarized

current output from the RSOI arc by a spin unpolarized current input. The sign of the spin current

can be manipulated by varying the arc dimensions. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4866388]

I. INTRODUCTION

The generation of spin polarized current is an important

prerequisite for the viability of spintronics, where the spin

degree of freedom of charge carriers is used to store, trans-

mit, and process information.1 The use of the Rashba spin-

orbit interaction (RSOI), present in 2-dimensional electron

gases (2DEG) in heterostructures lacking bulk inversion

symmetry,2 is a common approach to generate spin polarized

currents without the use of ferromagnetic materials or exter-

nal magnetic fields. The Hamiltonian of a 2DEG with RSOI

can be written as

H ¼ p2

2m�þ að~p � zÞ �~r; (1)

where a represents the strength of the RSOI. The form of the

RSOI interaction að~p � zÞ �~r invites the interpretation of the

RSOI as a momentum-dependent effective magnetic field.

The effective magnetic field leads to spin precession around

an axis defined by (py, –px, 0). The coupling of the momentum

and spin degrees of freedom enables spin to be controlled by

manipulating the momentum of the electrons. One way of

doing this is to force electrons to assume momenta in certain

directions by modifying the geometric shape of the nanostruc-

tures through which the charge carriers flow, for example, by

introducing curvature in an otherwise straight structure.

In this work, we investigate the spin and charge trans-

port in a simple curved structure with RSOI, the semi-

circular arc illustrated in Fig. 1. While transport in

1-dimensional RSOI arcs of zero widths have been previ-

ously studied,3–5 transport in RSOI arcs of finite widths has

attracted far less attention. The study of arcs with finite

widths is pertinent because experimentally fabricable struc-

tures have finite widths.

In contrast to the 1-dimensional arcs studied previously,

the finite widths of the RSOI arcs studied here allow electron

motion and spin precession to occur along both the transverse

width as well as along the longitudinal circumference of the

arc. The electrons along the inner and outer radii of an arc

travel different path lengths and thus undergo differential spin

evolution for a given angular displacement along the arc. We

find that the asymmetry between the inner and outer radii

leads to the generation of a finite spin z polarization at the nor-

mal metallic (NM) output drain lead from spin unpolarized

current flowing into the normal metallic input source lead.

II. NUMERICAL DETAILS

We use the standard real space finite-difference Non-

Equilibrium Green’s function (NEGF) formalism to calculate.

the spin and charge densities and currents in our system.6 In

our numerical calculations, we take the InAs values of

a¼ 0.3 eV nm and m*¼ 0.03 me.7 A first step in the applica-

tion of NEGF is to construct the finite difference

FIG. 1. A schematic representation of the system studied in this work. The

system consists of spin unpolarized current flowing from a NM lead without

RSOI into the semicircular arc where RSOI is present. Spin polarized cur-

rent flows out of the NM drain lead.

a)Author to whom correspondence should be addressed. Electronic mail:

a0018876@nus.edu.sg.

0021-8979/2014/115(17)/17C513/3/$30.00 VC 2014 AIP Publishing LLC115, 17C513-1

JOURNAL OF APPLIED PHYSICS 115, 17C513 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

142.104.240.194 On: Wed, 04 Jun 2014 16:19:42

approximation to the Hamiltonian. The NM leads are mod-

eled as a 2DEG using the Hamiltonian Eq. (1) with the same

effective mass m* as the RSOI segment and the RSOI

strength a set to 0. The spatial dimensions of the systems con-

sidered are on the order of a few hundred nanometres in line

with the Rashba rings fabricated in experimental studies of

the Aharonov-Casher effect.8,9

The arc was described mathematically using curvilinear

coordinates. In order to have a unified set of coordinates

which have the physical units of length for both the straight

leads as well as the curved arc, we introduce the coordinates

ql which increases along the longitudinal direction of current

flow (i.e., along the circumference of the arc), and qt which

increases along the transverse direction of current flow (i.e.,

along the radial direction in the arc). At the upper source

lead, ql (qt) corresponds to the x (y) coordinate. Within the

arc, ql and qt are related to the (x, y) Cartesian coordinates

via x ¼ ðRi þ qtÞcosðql=RiÞ and y ¼ ðRi þ qtÞsinðql=RiÞ,where Ri is the inner radius of the arc.

The Laplacian operator occurring in the p2/2 m* kinetic

energy term of Eq. (1) on a curved manifold is generically

given by

r2w ¼ @igij þ 1

2ð@i ln gÞgij

� �@jwþ gij@i@jw; (2)

where gij is the metric tensor, and g is the determinant of its

inverse. A na€ıve discretization of Eq. (2) is non-Hermitian.

This results in the unphysical breaking of current conserva-

tion. It is, however, possible to construct a Hermitian

approximation for the Laplacian operator by following the

scheme suggested by Meredith and Koonin10 (details to be

reported elsewhere). Denoting our coordinates as q1 and q2,ffiffiffigp

evaluated over the finite difference lattice site with coor-

dinates q1, q2 as a½q1;q2�, and the lattice point spacing as dq1

and dq2, the Hermitian finite difference approximation for

the Laplacian operator on a curved 2-dimensional manifold

reads

r2wð~rðq1; q2ÞÞ� ððg11½q1�dq1=2;q2�a½q1�dq1=2;q2� þ g11

½q1þdq1=2;q2�a½q1þdq1=2;q2�Þ=a½q1;q2�w½q1;q2�

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2½q1þdq1=2;q2�

a½q1;q2�a½q1þdq1;q2�

sg11½q1þdq1=2;q2�w½q1þdq1;q2� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2½q1�dq1=2;q2�

a½q1;q2�a½q1�dq1;q2�

sg11½q1�dq1=2;q2�w½q1�dq1;q2�Þ=ðdq1Þ2

þðanalogous term for contributions due to g22Þþ ðcontributions due to g12 and g21Þ;

where the contributions due to g12 are�� g12

½q1;q2þdq2�a½q1;q2þdq2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a½q1;q2�a½q1þdq1;q2þdq2�p þ g12

½q1þdq1;q2�a½q1þdq1;q2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a½q1;q2�a½q1þdq1;q2þdq2�p

� �w½q1þdq1;q2þdq2�

þ ðanalogous term for w½q1�dq1;q2�dq2� Þ þ�

g12½q1þdq1;q2�

a½q1þdq1;q2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia½q1;q2�a½q1þdq1;q2�dq2�p þg12

½q1;q2�dq2�a½q1;q2�dq2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a½q1;q2�a½q1þdq1;q2�dq2�p

�w½q1þdq1;q2�dq2�

þ analogous term for w½q1�dq1;q2þdq2� Þ�=ð4dq1dq2Þ;

and an analogous expression term holds for the contributions

due to g21.

III. RESULTS AND DISCUSSION

Fig. 2 shows an exemplary calculation of the spin z cur-

rent flowing in the longitudinal direction within a curved

RSOI segment, and that in a straight RSOI segment of the

same width and length equal to the distance along the cir-

cumferential middle line of the RSOI arc.

A straight RSOI segment with its length parallel to the xaxis and lying between �W=2 < y < W=2 is symmetrical

upon reflection about the x axis. This symmetry is reflected

in the antisymmetry of the spin z current.11 The net spin zcurrent across a transverse cross section of the straight seg-

ment hence cancels out to 0. We showed in an earlier work

that breaking the reflection symmetry about the central longi-

tudinal axis, in that case by introducing an off-center defect,

FIG. 2. The spin z current densities flowing along the longitudinal direction

of (a) a semicircular arc and (b) a straight RSOI segment of the same width

and length equal to middle circumference of the RSOI arc, (c) shows the

longitudinal spin z current density flowing across the middle of the straight

segment with the central qt¼W/2 and the jzl ¼ 0 lines drawn to emphasize

the antisymmetry of the spin current. For both systems E¼ 0.05 eV.

17C513-2 Siu, Jalil, and Tan J. Appl. Phys. 115, 17C513 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

142.104.240.194 On: Wed, 04 Jun 2014 16:19:42

results in finite spin z current polarization.12 The reflection

symmetry about the central longitudinal axis parallel to the

circumference of the arc and located at the transverse center

of the arc is also absent in the semicircular arc. This breaks

the antisymmetry of the spin z current, and results in a finite

spin z current flowing across the transverse cross section.

The asymmetry of the spin z current can be understood in

terms of the different distances that a spin travels across, and

hence precesses, along the inner and outer edges of the arc.

We next present illustrative examples of how the spin

polarization changes with the inner radius and width of the

arc. Fig. 3 shows the difference between the spin up and spin

down transmissions plotted as functions of the energy E for

two arcs of different widths and the same inner radius. There

is a threshold energy range below which the spin polarization

is zero. The absence of spin polarization when there is only a

single spin-degenerate pair of propagating modes in the

source leads is a general feature of SOI systems, which stems

from the symmetries of the SOI system and charge conserva-

tion.13,14 The threshold energy hence corresponds to the

energy at which the second spin-degenerate pair of propagat-

ing source modes emerges. It decreases with increasing trans-

verse width in accordance with the relation En ¼ 12m�

npW

� �2for

the nth eigenmode of the infinite potential square well.

The smaller energy spacings between successive trans-

verse modes for a wider segment lead to the presence of

more propagating modes within the RSOI segment at a given

value of energy. The propagating modes interfere in a pair-

wise manner. This results in the precession of the spin with a

characteristic length scale corresponding to the wavevector

difference between the two members in each pair of interfer-

ing propagating modes. The resultant evolution of the spin as

it propagates down the length of the segment is then due to

the combined effect of the precessions due to interference

between all pairwise combinations of propagating modes.

The spin polarization hence exhibits a greater oscillatory

behavior with energy for wider leads.

In general, the sign and magnitude of the spin polarization

vary with energy. It is however possible to tune device param-

eters to minimize the variation of the sign of the spin polariza-

tion over certain energy ranges. For example, the spin

polarization is largely positive for the 120 nm arc of Fig. 3 at

energies above 0.03 eV. A constant sign of the spin polariza-

tion ensures that spin polarization does not cancel out when

the transmission is integrated over the energy falling in

between the electrochemical potentials of the source and drain

leads when calculating the total current flowing between them.

It is also possible to tune the device parameters in order

to switch the sign of the spin polarization. This is shown in

Fig. 4, where arcs of the same width but different inner radii

exhibit spin polarizations of opposite signs. (The total cur-

rent for a given bias voltage in the graphs is calculated by

integrating the transmission from E¼ 0 to E¼Vb.) The mag-

nitude of the spin polarization for the larger inner radius is

significantly smaller partly due to the greater averaging

effect after summing across more oscillations in the spin cur-

rent as the energy is varied.

IV. CONCLUSION

In this work, we presented a Hermitian finite difference

approximation to the Laplacian operator on a curved mani-

fold. Using this approximation, we performed NEGF calcu-

lations for the spin transport properties of a curved RSOI arc

of finite width. We find that the asymmetry between the

inner and outer radii of the arc leads to a finite spin z polar-

ized current. The magnitude and sign of the spin polarization

may be tuned by adjusting device parameters such as the

inner radius and width of the arc.

1S. A. Wolf et al., Science 16, 1488 (2001).2Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).3M. P. Trushin and A. L. Chudnovskiy, JETP Lett. 83, 318 (2006).4E. Zhang et al., Phys. Rev. B 75, 085308 (2007).5M.-H. Liu et al., Phys. Rev. B 84, 085307 (2011).6S. Datta, Quantum Transport: Atom to Transistor (Cambridge University

Press, 2005).7T. Matsuyama et al., Phys. Rev. B 61, 15588 (2000).8M. K€onig et al., Phys. Rev. Lett. 96, 076804 (2006).9J. Nitta et al. J. Phys: Conf. Ser. 302, 012002 (2011).

10S. E. Koonin and D. C. Meredith, Computational Physics (Addisson-

Wesley, Reading, 1990), Chap. 6.11Y. Jiang and L. Hu, Phys. Rev. B 75, 195343 (2007).12Z. B. Siu, M.Sc. dissertation, National University of Singapore, Singapore,

2008.13A. A. Kiselev and K. W. Kim, Phys. Rev. B 71, 153315 (2005).14F. Zhai and H. Q. Xu, Phys. Rev. Lett. 94, 246601 (2005).

FIG. 3. The difference between the spin up transmission T" and the spin

down transmission for two RSOI arcs of inner radii 45 nm, and transverse

widths as indicated in the graph legends. FIG. 4. The spin z current Iz as a function of the applied bias Vb calculated

for RSOI segments of width 120 nm and inner radii as indicated in the plot.

17C513-3 Siu, Jalil, and Tan J. Appl. Phys. 115, 17C513 (2014)

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

142.104.240.194 On: Wed, 04 Jun 2014 16:19:42