2. Length scales and low dimensionality. Electron states and

2. Length scales and low dimensionality.
Electron states and quantum confinement.
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• Contents:
Length scales and low dimensionality.
• Introduction: Nanoscience and Mesoscopic Physics.
• Dimensionality definitions.
• Relevant length scales.
• Examples of low dimensional systems.
• Fabrication and exploring tools.
• New phenomena and new applications.
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• Introduction: Nanoscience and Mesoscopic Physics.
In between an atom and bulk solids. Size below which a solid
• MESOdoes no longer behave bulk-like.
Physics...Physics of small condensed objects (a collection
• Mesoscopic
of atoms)
• Often in the nanometer-size regime ! discipline of “Nanoscience”
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• Introduction: Nanoscience and Mesoscopic Physics.
Nanoscience and Nanotechnology
Why increasing interest for the nanoscale?
1 nm = 0,000000001 m
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• Introduction: Nanoscience and Mesoscopic Physics.
Diameter of human hair
Diameter of red blood cell
Visible light wavelengths
Intel’s newest transistor
ADN
Diameter of DNA, nanotubes
Bohr diameter
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httpp://www.owlnet.rice.edu
• Dimensionality definitions
I: Bond percolation (Microscopic scheme)
(Bottom-up)
•Based on the bonding.
•Strong
covalent bond within regions of structure define the
dimensionality unit and weak (e.g. Van der Waals) between units to
produce the 3D structure overall.
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• Dimensionality definitions
0D: Molecular P4Se3
1D: crystalline SiSe2
2D: crystalline Ge4Se3
3D:
amorphous SiO2
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• Dimensionality definitions
I: Bond percolation (Microscopic scheme)
(Bottom-up)
•Start by considering electrons in single atoms and small molecules.
•Theories to treat electrons in nanostructures:
Huckel theory
“Tight-binding”
Localized orbitals
•Chemistry.
•This point of view will be explored in the last chapter (C nanostructures)
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• Dimensionality definitions
II: Length scales (More macroscopic scheme)
(Top-down)
•Based on size dependence of a physical property, e.g. transport (electrons and
also phonons involved).
•Reduced
dimension if the dimension of the sample is lower than a
characteristic length (e.g. mean free path for transport, Fermi wave-length for
quantization or exciton Bohr radius in semiconductors).
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• Dimensionality definitions
L0 = λ
, characteristic length
L0 > Li ,
i = 1, n
→ (3 − n)D
system
0D: quantum dot
Lx , Ly , Lz < L0
1D: quantum wire
Lx , Ly < L0
2D: quantum well
Lx < L0
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• Dimensionality definitions
II: Length scales (More macroscopic scheme)
(Top-down)
•Start from solid state physics.
•Physics/electrical engineering.
•Shows qualitative features. Not bad for many metals and doped semiconductors.
•Approximations to treat electrons in nanostructures:
• “Free electrons”-no external potential• Independent electron approximation- ignores interactions.
• Many-particle system can be modeled by starting from single particle case
• Starting point: single particle states and energies (next lecture).
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• Relevant length scales
•A few relevant scale lengths:
proccess transistor
(Texas Instruments).
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S. Datta, “Electronic transport in mesoscopic systems”,1995
• Relevant length scales
•Some characteristic lengths:
•De Broglie wave length, Fermi wavelength: λ, λF
Related to kinetic energy of electrons λ = 2π! = 2π
p
k
2π
kF → λF =
Fermi gas: characteristic momentum
kF
!
, ns : sheet density
One single filled band in 2DEG: λ = 2π/ns
√
Boltzamann gas: p = 2mkT
•Mean free path: Lm
Initial momentum of electrons is destroyed
Length between collisions with impurities or phonons
Lm = vτt
typical
velocity
transport
relaxation time
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• Relevant length scales
•Some characteristic lengths:
•Phase-relaxation length : Lϕ
Initial phase of electrons is destroyed
Quantun mechanical: phase of the electron wave function
!
Lϕ = Dτϕ
difusion
constant
typical
time of
elastic
collisions
D = (1/d)vLm
dimensionality of
electron gas
•Thermal dephasing length : LT
Characteristic length of coherent propagation for two electrons
If the energy difference between two electrons is ~kT, they
travel almost coherently during time !/kT
!
LT = !D/kT
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• Relevant length scales
•For example: Transport through a constriction, 3 different regimes:
Lm
Lm << W, L
•Wire dimensions:W, L
•Mean free path: Lm
W < Lm < L
Lm >> W, L
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• Relevant length scales
•For example:conductance quantization in a quantum point contact
By applying voltages to
the planar gate
electrode, the width of
the wire is tuned.
At low T conductance is
quantized in units of
2e2/h.
AFM surface topography of
Ga AS microchip.
A small wire length 140nm,
width 80 nm connects
source and drain. Planar gate
30 nm below its surface.
T. Heinzal, “Mesoscopic electronics in solid state nanostructures”, Wiley
Ballistic regime
• Relevant length scales
•For example:mesoscopic ring used to study Abraronov-Bohm effect
Lϕ ∼ 100µm
(low T)
A significant fraction of
electrons traverse the
ring phase coherently
From a 38 nm film of polycrystalline
gold. Diameter 820 nm. Thickness of
wires 40 nm
S. Washburn and R. A. Webb, Adv. Phys. 35, 375 (1986).
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G. Fraser, “The New Physics for the 21th century”, Y. Imry, ch. 12
• Relevant length scales
•Summary of conditions required for a mesoscopic device
T. Heinzal, “Mesoscopic electronics in solid state nanostructures”, Wiley
• Relevant length scales
•For example:Kondo mirage
Unusual phenomena
due to the wave nature
of electrons and their
correlations around
impurities.
Images of elliptical arrangements of
atoms on a metallic surface, prepared
and visualized with STM microscope.
Placing a magnetic impurity at
a focal point the ellipse created a
shadow in the other focus
(“Kondo mirage”)
D. Eigler et al., IBM Almaden
http://www.almaden.ibm.com/almaden/media/image_mirage.html
(This and more beatiful images)
• Examples of low dimensional systems
•Some quasi-two-dimensional systems:
G. Lehmann, P. Ziesche, “Electronic properties of metals, Esevier, 1990
MCBJ technique to produce
metallic nanowires
E. Sheer et al., Phys. rev. Lett. 78, 3535 (1997)
• Examples of low dimensional systems
•Peroskite-like high temperature superconductors
G. Lehmann, P. Ziesche, “Electronic properties of metals, Esevier, 1990
Superconductivity related to 2D character due lo weakly connected 2D sheets of Cu and O
• Examples of low dimensional systems
•Some quasi-one-dimensional materials:
G. Lehmann, P. Ziesche, “Electronic properties of metals, Esevier, 1990
• Examples of low dimensional systems
Carbon in all dimensions
sp2
Covalent
C-C bonds within
'molecule'
sp2
Variable sp
hybridisation
!+ "
-
pure sp2
+
-
pure sp3
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• Examples of low dimensional systems
•Semiconductor nanostructures starting from GaAs-AlGaAs heterostructures
•Diminishing dimensions...
•2D electron gas
E. Corcoran, TRENDS IN MATERIALS: DIMINISHING DIMENSIONS; November, 1990
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• Examples of low dimensional systems
•Semiconductor nanostructures starting from GaAs-AlGaAs heterostrcutures
•Squeezing 2D electron gas...
QUANTUM
WIRE
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• Examples of low dimensional systems
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• Examples of low dimensional systems
•Also with Si (MOSFET)
Si technology
• Examples of low dimensional systems
•Why GaAs?
C.W.J. Beenakker, H. van Houten, "Quantum Transport in Semiconductor Nanostructures", Solid State Physics 44, 1, 1991.
http://arxiv.org/abs/cond-mat/0412664
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• Examples of low dimensional systems
•OD systems, quantum dots or “artificial atoms”
•Clusters of metallic atoms grown from vapour-phase
•Fullerenes
•Synthetic nanocrystals :CdS, CdSe in glassy matrix, CuCl in
NaCl crystals, Si, Ge...
, Size control (~1nm-> ~200nm)
•Self-assembled QD’s
•QD’s produced from heterostructures and lithographic etching.
applications in nanoelectronics and optoelectronics
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• Examples of low dimensional systems
•OD systems, quantum dots or “artificial atoms”
• Synthetic nanoparticles interesting because of optical properties.
• Reducing the size the gap changes, Higher fusion temperatures,
estructural changes... e.g. the gap of CdSe can be tuned from red
(1.7eV) to green (2.4 eV) when the particle diameter is reduced from
200 nm to 2 nm
•Aplications: lasers, LEDS, biosensors....
increasing size
condensation
• Fabrication and exploring tools
•Nanolithography
•Atomic force microscopy
•Scanning tunneling microscopy
•Molecular beam epitaxy and other
techniques for atomic-scale
layer deposition of material.
•Chemical sysntesis with different methods....
(Described in previouslecture)
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• New phenomena and new applications
•Laboratory for quantum phenomena:
•quantum coherence,
quantum confinement, tunel effect,
electron-electron interactions....
•When we go dowm in dimension properties are not scalable:
•new functional relations among magnitudes,
oscillations of
the physical magnitudes....
•New phenomena
•quantum Hall effect, Coulomb blockade, breakdown of Ohm’s
law, quantum size effects...
•New
operating principles and applications: one electron devices, molecular
electronics, spintronics, nanophotonics. optoelectronic devices, quantum
computing, bio-nano devices for aplications in biomedicine....
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• New phenomena and new applications
•Scalability regimes:
Simulations of breaking of Na
nanowires
Eduardo Ogando, Thesis 2004
• New phenomena and new applications
•General trends or signatures of low dimensionality
•Fermi surface topology for 3D (sphere), 2D (cylinder) and 1D (planes) electron gas
Fermi surface of a
quasi-one -dimensional
electron gas.
Wavy planes due to
weak coupling in real
systems or
(More details in next lecture)
• New phenomena and new applications
•General trends or signatures of low dimensionality
•Density of states
Eduardo Ogando, Thesis 2004
(More details in next lecture)
• New phenomena and new applications
•General trends or signatures of low dimensionality
•Response function, susceptibility
Wave vector dependent response
function for 1”, 2D, and 3D electron gas
at T=0 K
The response function of a 1D free
electron gas at various temperature
(Heeger, 1979)
• New phenomena and new applications
•Response to magnetic fields (quantum Hall effect)
Shubnikov-de Haas oscillations and
the quantum Hall effect.
Measure the longitudinal (Rxx) and Hall
resistance (Rxy) of a 2D electron gas as a
function of the perpendicular magnetic
field.
T=100mK
von Klitzing et al. 1982
G. Fraser, “The New Physics for the 21th century”, Y. Imry, ch. 12
• Summary
Write it yourself and send it to me
(just to fill one slide)
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• Take home exercises
(Be very concise)
•Bibliographic search: Peculiarities or surprises found for other low dimensional
systems. Give paper reference where it is found, describe briefly the system
(composition, size, tempertaure...) and the property studied.
•Find examples of systems behaving as 0D
•Why interest in AsGa? Compare properties of GaAs vs. Si
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