Suppression of the AharonovBohm effect in semiconductor nanorings

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Suppression of the Aharonov-Bohm effect in
semiconductor nano-rings O.Voskoboynikov Depart
ment of Electronics Engineering and Institute of
Electronics National Chiao Tung
University Hsinchu, Taiwan
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In collaboration with Prof. C.P. Lee NCTU,
Taiwan Prof. S. M. Sze NDL, Taiwan Prof. O.
Tretyak KU, Ukraine Dr.Y. M. Li NCTU, Taiwan C.
F. Shih NCTU, Taiwan H.M. Lu NTHU, Taiwan
Outline

• Motivation.
• What is the semiconductor nano-rings?
• Theory of semiconductor nano-rings in magnetic
  fields.
• Energy states and wave functions of electrons.
• Magnetization of nano-rings.
• Conclusion.

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• Motivation.
• What is the semiconductor nano-rings?

Advances in the fabrication of semiconductor
nano-structures have allowed us to construct
nano-scale systems with a wide range of
geometries. Recent experimental results on InGaAs
torus shaped quantum nano-rings demonstrated such
capabilities. The typical lateral sizes and
height are about 50 nm and 2 nm J.M. Garsia at
al., Appl. Phys. Lett. 71, 2014 (1997) A. Lorke
at al., Physica B 256-258, 424 (1998) Phys. Rev.
Lett. 84, 2223 (2000) The realization of such
semiconductor nano-rings bridges the gap between
quantum dots and meso-scale ring structures.
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• Possible scenario for the self-organized ring
  formation
• InAs dot are partly covered by a thin layer of
  GaAs.
• during the annealing time, In diffuses away from
  its
• original location and forms a volcano-like
  structure on the surface (c).
• (A. Lorke at al., Jpn. J. Appl. Phys. 40, 1857
  (2001))

Capacitance-voltage traces for samples with
growth interruption after different coverages of
1, 3, and 5 nm, respectively.
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Nano-rings are viewed as quasi-0D systems and
show very different physical behavior from their
quasi-1D meso-scopic analogs (meso-rings). Nano-r
ings also posses different from quantum dots
electrical, and optical properties. Nano-rings
posses the unique property of trapping magnetic
flux and persistent current which is not affected
by the presence of random scatterers.
Absorption per density of absorbers is plotted
here against photon energy for nano dots and
rings at 4.2 K (H. Petterson at al., Physica E 6,
510 (2000))
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Manifestation of the Aharonov-Bohm effect at
meso-scale rings
(a) Energy levels of an ideal one-dimensional
meso-scale ring as a function of the magnetic
flux in the ring area.
Is the flux quanta
(b) Calculated energy levels in a parabolic
wire bent into a circular ring (see inset). The
data points (right-hand scale) give
the gate-voltage shift of the lowest capacitance
maximum. from A. Lorke at al., Phys. Rev. Lett.
84, 2223 (2000).
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• Theory of semiconductor nano-rings
• in magnetic fields.

• Most of theoretical studies either used the
  traditional one-dimensional model or assume that
  electrons move in two-dimensional parabolic
  potentials. Those models neglect at least two
  factors finite width and finite hard wall
  confinement potential.
• Only recently few three-dimensional simulations
  where performed for torus shaped rings. Those
  calculations provide some explanations for FIR
  spectra of the rings.
• In this work, we go beyond one-dimensional and
  two-dimensional pictures to study magnetic
  properties of nano-rings.

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The energy states and magnetization of
three-dimensional InAs/GaAs nano-rings are
calculated by solving the Shrödinger equation
with the effective one-band Hamiltonian
Where
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B
The Ben Daniel-Duke boundary conditions
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Because the cylindrical symmetry of the
problem, the wave function can be represented as
The Shrödinger equation
And the boundary conditions
The energy states and wave functions are found by
the finite difference and inverse iteration
methods.
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• Energy states and wave functions of electrons.

InAs/GaAs quantum ring h2nm ?R10nm
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InAs/GaAs ellipsoidal quantum ring 3D surface
plot and 2D contour plot of the ground state wave
function
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The energy state classification
R018nm
R08nm
R038nm
Electron states for the torus shaped InAs
nano-rings with various inner radii (h2.4nm,
?R24 nm) (O. Voskoboynikov at al. , Phys. Rev. B
66, 155306 (2002))
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Electron states for the cut- torus (volcano)
shaped InAs nano-rings with (h10 nm, R010 nm,
?R50nm)
Experimental
Theoretical
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Magnetic field dependence of the energy gap for
InAs/GaAs nano-ring with ellipsoidal
cross-section ( h 2.4 nm, inner radius R1 8 nm,
width ?R 24 nm)
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• Magnetization of nano-rings.

Magnetization (magnetic moment) of the system
where
Magnetic susceptibility
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Aharonov-Bohn periodic oscillation of
traditional one-dimensional one-electron
meso-scopic quantum ring when
The non-periodic oscillations of the one-electron
volcano nano-ring h2.4 nm, R010 nm, ?R20
nm.
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The crossing points between states (jumps in
magnetization) occur when
The k parameter for the cut- torus ring with
h2.4 nm, R010 nm, ?R20 nm.
The magnetization jump (?M/?B)
Crossing point
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Magnetization of multi-electron rings
N2
N3
N4
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The fist paramagnetic peak in magnetic
susceptibility of the one-electron
nano-ring h2.4 nm, R010 nm, ?R20 nm.
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• Conclusion.

• Semiconductor nano-ring is a new and interesting
  nano-system
• for quantum topological effects
• Three-dimensional realistic modeling is required
  for the ring description
• The rings should have unusual and interesting
  magnetic properties
• the magnetization oscillates aperiodically
  with suppression
• of the Aharonov-Bohn effect

QTMP Jan. 9-11, 2003