Spin-dependent magnetotransport through a mesoscopic ring
in the presence of spin-orbit interaction
X. F. Wang* and P. Vasilopoulos
†
Department of Physics, Concordia University, 1455 de Maisonneuve Ouest, Montréal, Québec, Canada, H3G 1M8
Received 2 June 2005; revised manuscript received 28 July 2005; published 26 October 2005
The Schrödinger equation for an electron in a mesoscopic ring, in the presence of the Rashba and linear
Dresselhaus terms of the spin-orbit interaction SOI and of a magnetic field B, is solved exactly. The effective
electric fields of these terms as well as B have perpendicular and radial components. The interplay between
them and B and their influence on the spectrum is studied. The transmission through such a ring, with two leads
connected to it, is evaluated as a function of the SOI strengths and of the orientations of these fields. The
Rashba and Dresselhaus terms affect the transmission in different ways. The transmission through a series of
rings with different radii and with SOI in both arms of the rings or only in one of them is also evaluated. For
weak magnetic fields B 1 T the influence of the Zeeman term on the transmission, assessed by perturbation
theory, is negligible.
DOI: 10.1103/PhysRevB.72.165336 PACS numbers: 72.25.-b, 71.70.Ej, 03.65.Vf, 85.35.-p
I. INTRODUCTION
The concept of the geometric phase
1–4
has attracted con-
siderable interest since it is established in a general way. In
mesoscopic rings, the geometric phase of electrons in mag-
netic and electric fields can be obtained by solving the time-
independent Schrödinger equation.
4
From the point of view
of the stationary Schrödinger equation, the energy dispersion
relation of electrons changes as the configuration of the sys-
tem varies. As a result, electrons of the same energy have
different wave vectors in different systems and may accumu-
late different phases after passing even the same path in real
space. In the presence of external magnetic and electric
fields, the one-particle Hamiltonian can be expressed as
5
H = p - eA -
B
E/2c
2
2
/2m . 1
The contribution from the vector potential A corresponds to
the Aharonov-Bohm AB phase
6
and the contribution from
the spin-orbit interaction SOI to the Aharonov-Casher AC
phase.
7
In mesoscopic systems of semiconductor heterostruc-
tures, however, the macroscopic SOI results from the asym-
metry of the microscopic crystal field and may appear in
different forms depending on the materials and structures
involved. In materials with asymmetric crystal structure, the
cubic Dresselhaus SOI term exists in bulk materials while an
extra linear Dresselhaus SOI DSOI term appears in con-
fined, low-dimensional systems due to the change of the
crystal structure along the direction of the confinement. In
systems with asymmetric confinement, the Rashba SOI
RSOI term results from a nonvanishing confining electric
field as well as from various other SOI mechanisms such as
the one related to differing band discontinuities at the hetero-
structure interfaces considered in k · p models. The aggregate
strength of all these SOI mechanisms is denoted by .
8
By inserting a mescoscopic ring into a circuit, we can
study the quantum transport through the ring when the in-
elastic diffusion length is larger than the size of the ring. In
the two-terminal configuration, the transmission depends on
the interference between electrons propagating through the
ring’s two arms and the transmission properties through the
two junctions connecting the ring and the leads. In general, a
symmetric junction can be described by a 3 3 scattering
matrix with the transmission through each arm as a
parameter.
9–11
For a ballistic, one-dimensional 1D ring con-
nected to two leads, the scattering matrix can be determined
by imposing the continuity of the wave function and of the
spin flux at each junction.
12,13
In the presence of SOI the
transmission through the ring as well as the geometric phases
are spin dependent and the ring can be used as a spin-
interference device.
14
Similar considerations apply to a
square loop or arrays of such loops.
15
Theoretically this spin
interference was further studied in 1D and 2D rings
16
but
only in the presence of the RSOI.
In this paper we study ballistic transport through one or
more 1D rings, symmetrically connected to two leads, in the
presence of a magnetic field, with components along the ra-
dial and perpendicular direction, and of both terms of the
SOI, RSOI, and DSOI. The corresponding effective electric
fields have perpendicular and radial components. For weak
magnetic fields, B 1 T, the influence of the Zeeman term is
validly assessed by perturbation theory. In Sec. II we present
the one-electron energy spectrum and formulate the corre-
sponding transfer-matrix transmission problem. In Sec. III
we present numerical results for the transmission, through
one or more rings, and in Sec. IV concluding remarks.
II. ONE-ELECTRON PROBLEM
A. Hamiltonian
We consider a one-dimensional ring, of radius a, in the
x - y or r - plane and in a magnetic field with compo-
nents B
z
= B cos
3
and B
r
= B sin
3
.
For the vector potential we choose the gauge A
= A
r
, A
, A
z
= 0, B
z
r /2- B
r
z ,0 with z =0 in the plane of the
ring. The one-electron Hamiltonian is
17
H =
2
/2m
*
a
2
- i / + /
0
2
+ g
B
· B/2, 2
where =
x
,
y
,
z
r
,
,
z
are the Pauli matrices,
= B
z
a
2
is the magnetic flux passing through the ring,
0
PHYSICAL REVIEW B 72, 165336 2005
1098-0121/2005/7216/1653368/$23.00 ©2005 The American Physical Society 165336-1